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3f4593c3aea71bbff600bff2e5ad24a178fa3e893e2874456a65978118fbe4ef | """Geometrical Points.
Contains
========
Point
Point2D
Point3D
When methods of Point require 1 or more points as arguments, they
can be passed as a sequence of coordinates or Points:
>>> from sympy.geometry.point import Point
>>> Point(1, 1).is_collinear((2, 2), (3, 4))
False
>>> Point(1, 1).is_collinear(Point(2, 2), Point(3, 4))
False
"""
from __future__ import division, print_function
import warnings
from sympy.core import S, sympify, Expr
from sympy.core.compatibility import is_sequence
from sympy.core.containers import Tuple
from sympy.simplify import nsimplify, simplify
from sympy.geometry.exceptions import GeometryError
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.complexes import im
from sympy.matrices import Matrix
from sympy.core.numbers import Float
from sympy.core.evaluate import global_evaluate
from sympy.core.add import Add
from sympy.utilities.iterables import uniq
from sympy.utilities.misc import filldedent, func_name, Undecidable
from .entity import GeometryEntity
class Point(GeometryEntity):
"""A point in a n-dimensional Euclidean space.
Parameters
==========
coords : sequence of n-coordinate values. In the special
case where n=2 or 3, a Point2D or Point3D will be created
as appropriate.
evaluate : if `True` (default), all floats are turn into
exact types.
dim : number of coordinates the point should have. If coordinates
are unspecified, they are padded with zeros.
on_morph : indicates what should happen when the number of
coordinates of a point need to be changed by adding or
removing zeros. Possible values are `'warn'`, `'error'`, or
`ignore` (default). No warning or error is given when `*args`
is empty and `dim` is given. An error is always raised when
trying to remove nonzero coordinates.
Attributes
==========
length
origin: A `Point` representing the origin of the
appropriately-dimensioned space.
Raises
======
TypeError : When instantiating with anything but a Point or sequence
ValueError : when instantiating with a sequence with length < 2 or
when trying to reduce dimensions if keyword `on_morph='error'` is
set.
See Also
========
sympy.geometry.line.Segment : Connects two Points
Examples
========
>>> from sympy.geometry import Point
>>> from sympy.abc import x
>>> Point(1, 2, 3)
Point3D(1, 2, 3)
>>> Point([1, 2])
Point2D(1, 2)
>>> Point(0, x)
Point2D(0, x)
>>> Point(dim=4)
Point(0, 0, 0, 0)
Floats are automatically converted to Rational unless the
evaluate flag is False:
>>> Point(0.5, 0.25)
Point2D(1/2, 1/4)
>>> Point(0.5, 0.25, evaluate=False)
Point2D(0.5, 0.25)
"""
is_Point = True
def __new__(cls, *args, **kwargs):
evaluate = kwargs.get('evaluate', global_evaluate[0])
on_morph = kwargs.get('on_morph', 'ignore')
# unpack into coords
coords = args[0] if len(args) == 1 else args
# check args and handle quickly handle Point instances
if isinstance(coords, Point):
# even if we're mutating the dimension of a point, we
# don't reevaluate its coordinates
evaluate = False
if len(coords) == kwargs.get('dim', len(coords)):
return coords
if not is_sequence(coords):
raise TypeError(filldedent('''
Expecting sequence of coordinates, not `{}`'''
.format(func_name(coords))))
# A point where only `dim` is specified is initialized
# to zeros.
if len(coords) == 0 and kwargs.get('dim', None):
coords = (S.Zero,)*kwargs.get('dim')
coords = Tuple(*coords)
dim = kwargs.get('dim', len(coords))
if len(coords) < 2:
raise ValueError(filldedent('''
Point requires 2 or more coordinates or
keyword `dim` > 1.'''))
if len(coords) != dim:
message = ("Dimension of {} needs to be changed"
"from {} to {}.").format(coords, len(coords), dim)
if on_morph == 'ignore':
pass
elif on_morph == "error":
raise ValueError(message)
elif on_morph == 'warn':
warnings.warn(message)
else:
raise ValueError(filldedent('''
on_morph value should be 'error',
'warn' or 'ignore'.'''))
if any(coords[dim:]):
raise ValueError('Nonzero coordinates cannot be removed.')
if any(a.is_number and im(a) for a in coords):
raise ValueError('Imaginary coordinates are not permitted.')
if not all(isinstance(a, Expr) for a in coords):
raise TypeError('Coordinates must be valid SymPy expressions.')
# pad with zeros appropriately
coords = coords[:dim] + (S.Zero,)*(dim - len(coords))
# Turn any Floats into rationals and simplify
# any expressions before we instantiate
if evaluate:
coords = coords.xreplace(dict(
[(f, simplify(nsimplify(f, rational=True)))
for f in coords.atoms(Float)]))
# return 2D or 3D instances
if len(coords) == 2:
kwargs['_nocheck'] = True
return Point2D(*coords, **kwargs)
elif len(coords) == 3:
kwargs['_nocheck'] = True
return Point3D(*coords, **kwargs)
# the general Point
return GeometryEntity.__new__(cls, *coords)
def __abs__(self):
"""Returns the distance between this point and the origin."""
origin = Point([0]*len(self))
return Point.distance(origin, self)
def __add__(self, other):
"""Add other to self by incrementing self's coordinates by
those of other.
Notes
=====
>>> from sympy.geometry.point import Point
When sequences of coordinates are passed to Point methods, they
are converted to a Point internally. This __add__ method does
not do that so if floating point values are used, a floating
point result (in terms of SymPy Floats) will be returned.
>>> Point(1, 2) + (.1, .2)
Point2D(1.1, 2.2)
If this is not desired, the `translate` method can be used or
another Point can be added:
>>> Point(1, 2).translate(.1, .2)
Point2D(11/10, 11/5)
>>> Point(1, 2) + Point(.1, .2)
Point2D(11/10, 11/5)
See Also
========
sympy.geometry.point.Point.translate
"""
try:
s, o = Point._normalize_dimension(self, Point(other, evaluate=False))
except TypeError:
raise GeometryError("Don't know how to add {} and a Point object".format(other))
coords = [simplify(a + b) for a, b in zip(s, o)]
return Point(coords, evaluate=False)
def __contains__(self, item):
return item in self.args
def __div__(self, divisor):
"""Divide point's coordinates by a factor."""
divisor = sympify(divisor)
coords = [simplify(x/divisor) for x in self.args]
return Point(coords, evaluate=False)
def __eq__(self, other):
if not isinstance(other, Point) or len(self.args) != len(other.args):
return False
return self.args == other.args
def __getitem__(self, key):
return self.args[key]
def __hash__(self):
return hash(self.args)
def __iter__(self):
return self.args.__iter__()
def __len__(self):
return len(self.args)
def __mul__(self, factor):
"""Multiply point's coordinates by a factor.
Notes
=====
>>> from sympy.geometry.point import Point
When multiplying a Point by a floating point number,
the coordinates of the Point will be changed to Floats:
>>> Point(1, 2)*0.1
Point2D(0.1, 0.2)
If this is not desired, the `scale` method can be used or
else only multiply or divide by integers:
>>> Point(1, 2).scale(1.1, 1.1)
Point2D(11/10, 11/5)
>>> Point(1, 2)*11/10
Point2D(11/10, 11/5)
See Also
========
sympy.geometry.point.Point.scale
"""
factor = sympify(factor)
coords = [simplify(x*factor) for x in self.args]
return Point(coords, evaluate=False)
def __neg__(self):
"""Negate the point."""
coords = [-x for x in self.args]
return Point(coords, evaluate=False)
def __sub__(self, other):
"""Subtract two points, or subtract a factor from this point's
coordinates."""
return self + [-x for x in other]
@classmethod
def _normalize_dimension(cls, *points, **kwargs):
"""Ensure that points have the same dimension.
By default `on_morph='warn'` is passed to the
`Point` constructor."""
# if we have a built-in ambient dimension, use it
dim = getattr(cls, '_ambient_dimension', None)
# override if we specified it
dim = kwargs.get('dim', dim)
# if no dim was given, use the highest dimensional point
if dim is None:
dim = max(i.ambient_dimension for i in points)
if all(i.ambient_dimension == dim for i in points):
return list(points)
kwargs['dim'] = dim
kwargs['on_morph'] = kwargs.get('on_morph', 'warn')
return [Point(i, **kwargs) for i in points]
@staticmethod
def affine_rank(*args):
"""The affine rank of a set of points is the dimension
of the smallest affine space containing all the points.
For example, if the points lie on a line (and are not all
the same) their affine rank is 1. If the points lie on a plane
but not a line, their affine rank is 2. By convention, the empty
set has affine rank -1."""
if len(args) == 0:
return -1
# make sure we're genuinely points
# and translate every point to the origin
points = Point._normalize_dimension(*[Point(i) for i in args])
origin = points[0]
points = [i - origin for i in points[1:]]
m = Matrix([i.args for i in points])
return m.rank()
@property
def ambient_dimension(self):
"""Number of components this point has."""
return getattr(self, '_ambient_dimension', len(self))
@classmethod
def are_coplanar(cls, *points):
"""Return True if there exists a plane in which all the points
lie. A trivial True value is returned if `len(points) < 3` or
all Points are 2-dimensional.
Parameters
==========
A set of points
Raises
======
ValueError : if less than 3 unique points are given
Returns
=======
boolean
Examples
========
>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 2)
>>> p2 = Point3D(2, 7, 2)
>>> p3 = Point3D(0, 0, 2)
>>> p4 = Point3D(1, 1, 2)
>>> Point3D.are_coplanar(p1, p2, p3, p4)
True
>>> p5 = Point3D(0, 1, 3)
>>> Point3D.are_coplanar(p1, p2, p3, p5)
False
"""
if len(points) <= 1:
return True
points = cls._normalize_dimension(*[Point(i) for i in points])
# quick exit if we are in 2D
if points[0].ambient_dimension == 2:
return True
points = list(uniq(points))
return Point.affine_rank(*points) <= 2
def distance(self, other):
"""The Euclidean distance between self and another GeometricEntity.
Returns
=======
distance : number or symbolic expression.
Raises
======
TypeError : if other is not recognized as a GeometricEntity or is a
GeometricEntity for which distance is not defined.
See Also
========
sympy.geometry.line.Segment.length
sympy.geometry.point.Point.taxicab_distance
Examples
========
>>> from sympy.geometry import Point, Line
>>> p1, p2 = Point(1, 1), Point(4, 5)
>>> l = Line((3, 1), (2, 2))
>>> p1.distance(p2)
5
>>> p1.distance(l)
sqrt(2)
The computed distance may be symbolic, too:
>>> from sympy.abc import x, y
>>> p3 = Point(x, y)
>>> p3.distance((0, 0))
sqrt(x**2 + y**2)
"""
if not isinstance(other, GeometryEntity):
try:
other = Point(other, dim=self.ambient_dimension)
except TypeError:
raise TypeError("not recognized as a GeometricEntity: %s" % type(other))
if isinstance(other, Point):
s, p = Point._normalize_dimension(self, Point(other))
return sqrt(Add(*((a - b)**2 for a, b in zip(s, p))))
distance = getattr(other, 'distance', None)
if distance is None:
raise TypeError("distance between Point and %s is not defined" % type(other))
return distance(self)
def dot(self, p):
"""Return dot product of self with another Point."""
if not is_sequence(p):
p = Point(p) # raise the error via Point
return Add(*(a*b for a, b in zip(self, p)))
def equals(self, other):
"""Returns whether the coordinates of self and other agree."""
# a point is equal to another point if all its components are equal
if not isinstance(other, Point) or len(self) != len(other):
return False
return all(a.equals(b) for a, b in zip(self, other))
def evalf(self, prec=None, **options):
"""Evaluate the coordinates of the point.
This method will, where possible, create and return a new Point
where the coordinates are evaluated as floating point numbers to
the precision indicated (default=15).
Parameters
==========
prec : int
Returns
=======
point : Point
Examples
========
>>> from sympy import Point, Rational
>>> p1 = Point(Rational(1, 2), Rational(3, 2))
>>> p1
Point2D(1/2, 3/2)
>>> p1.evalf()
Point2D(0.5, 1.5)
"""
coords = [x.evalf(prec, **options) for x in self.args]
return Point(*coords, evaluate=False)
def intersection(self, other):
"""The intersection between this point and another GeometryEntity.
Parameters
==========
other : Point
Returns
=======
intersection : list of Points
Notes
=====
The return value will either be an empty list if there is no
intersection, otherwise it will contain this point.
Examples
========
>>> from sympy import Point
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0)
>>> p1.intersection(p2)
[]
>>> p1.intersection(p3)
[Point2D(0, 0)]
"""
if not isinstance(other, GeometryEntity):
other = Point(other)
if isinstance(other, Point):
if self == other:
return [self]
p1, p2 = Point._normalize_dimension(self, other)
if p1 == self and p1 == p2:
return [self]
return []
return other.intersection(self)
def is_collinear(self, *args):
"""Returns `True` if there exists a line
that contains `self` and `points`. Returns `False` otherwise.
A trivially True value is returned if no points are given.
Parameters
==========
args : sequence of Points
Returns
=======
is_collinear : boolean
See Also
========
sympy.geometry.line.Line
Examples
========
>>> from sympy import Point
>>> from sympy.abc import x
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> p3, p4, p5 = Point(2, 2), Point(x, x), Point(1, 2)
>>> Point.is_collinear(p1, p2, p3, p4)
True
>>> Point.is_collinear(p1, p2, p3, p5)
False
"""
points = (self,) + args
points = Point._normalize_dimension(*[Point(i) for i in points])
points = list(uniq(points))
return Point.affine_rank(*points) <= 1
def is_concyclic(self, *args):
"""Do `self` and the given sequence of points lie in a circle?
Returns True if the set of points are concyclic and
False otherwise. A trivial value of True is returned
if there are fewer than 2 other points.
Parameters
==========
args : sequence of Points
Returns
=======
is_concyclic : boolean
Examples
========
>>> from sympy import Point
Define 4 points that are on the unit circle:
>>> p1, p2, p3, p4 = Point(1, 0), (0, 1), (-1, 0), (0, -1)
>>> p1.is_concyclic() == p1.is_concyclic(p2, p3, p4) == True
True
Define a point not on that circle:
>>> p = Point(1, 1)
>>> p.is_concyclic(p1, p2, p3)
False
"""
points = (self,) + args
points = Point._normalize_dimension(*[Point(i) for i in points])
points = list(uniq(points))
if not Point.affine_rank(*points) <= 2:
return False
origin = points[0]
points = [p - origin for p in points]
# points are concyclic if they are coplanar and
# there is a point c so that ||p_i-c|| == ||p_j-c|| for all
# i and j. Rearranging this equation gives us the following
# condition: the matrix `mat` must not a pivot in the last
# column.
mat = Matrix([list(i) + [i.dot(i)] for i in points])
rref, pivots = mat.rref()
if len(origin) not in pivots:
return True
return False
@property
def is_nonzero(self):
"""True if any coordinate is nonzero, False if every coordinate is zero,
and None if it cannot be determined."""
is_zero = self.is_zero
if is_zero is None:
return None
return not is_zero
def is_scalar_multiple(self, p):
"""Returns whether each coordinate of `self` is a scalar
multiple of the corresponding coordinate in point p.
"""
s, o = Point._normalize_dimension(self, Point(p))
# 2d points happen a lot, so optimize this function call
if s.ambient_dimension == 2:
(x1, y1), (x2, y2) = s.args, o.args
rv = (x1*y2 - x2*y1).equals(0)
if rv is None:
raise Undecidable(filldedent(
'''can't determine if %s is a scalar multiple of
%s''' % (s, o)))
# if the vectors p1 and p2 are linearly dependent, then they must
# be scalar multiples of each other
m = Matrix([s.args, o.args])
return m.rank() < 2
@property
def is_zero(self):
"""True if every coordinate is zero, False if any coordinate is not zero,
and None if it cannot be determined."""
nonzero = [x.is_nonzero for x in self.args]
if any(nonzero):
return False
if any(x is None for x in nonzero):
return None
return True
@property
def length(self):
"""
Treating a Point as a Line, this returns 0 for the length of a Point.
Examples
========
>>> from sympy import Point
>>> p = Point(0, 1)
>>> p.length
0
"""
return S.Zero
def midpoint(self, p):
"""The midpoint between self and point p.
Parameters
==========
p : Point
Returns
=======
midpoint : Point
See Also
========
sympy.geometry.line.Segment.midpoint
Examples
========
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(13, 5)
>>> p1.midpoint(p2)
Point2D(7, 3)
"""
s, p = Point._normalize_dimension(self, Point(p))
return Point([simplify((a + b)*S.Half) for a, b in zip(s, p)])
@property
def origin(self):
"""A point of all zeros of the same ambient dimension
as the current point"""
return Point([0]*len(self), evaluate=False)
@property
def orthogonal_direction(self):
"""Returns a non-zero point that is orthogonal to the
line containing `self` and the origin.
Examples
========
>>> from sympy.geometry import Line, Point
>>> a = Point(1, 2, 3)
>>> a.orthogonal_direction
Point3D(-2, 1, 0)
>>> b = _
>>> Line(b, b.origin).is_perpendicular(Line(a, a.origin))
True
"""
dim = self.ambient_dimension
# if a coordinate is zero, we can put a 1 there and zeros elsewhere
if self[0] == S.Zero:
return Point([1] + (dim - 1)*[0])
if self[1] == S.Zero:
return Point([0,1] + (dim - 2)*[0])
# if the first two coordinates aren't zero, we can create a non-zero
# orthogonal vector by swapping them, negating one, and padding with zeros
return Point([-self[1], self[0]] + (dim - 2)*[0])
@staticmethod
def project(a, b):
"""Project the point `a` onto the line between the origin
and point `b` along the normal direction.
Parameters
==========
a : Point
b : Point
Returns
=======
p : Point
See Also
========
sympy.geometry.line.LinearEntity.projection
Examples
========
>>> from sympy.geometry import Line, Point
>>> a = Point(1, 2)
>>> b = Point(2, 5)
>>> z = a.origin
>>> p = Point.project(a, b)
>>> Line(p, a).is_perpendicular(Line(p, b))
True
>>> Point.is_collinear(z, p, b)
True
"""
a, b = Point._normalize_dimension(Point(a), Point(b))
if b.is_zero:
raise ValueError("Cannot project to the zero vector.")
return b*(a.dot(b) / b.dot(b))
def taxicab_distance(self, p):
"""The Taxicab Distance from self to point p.
Returns the sum of the horizontal and vertical distances to point p.
Parameters
==========
p : Point
Returns
=======
taxicab_distance : The sum of the horizontal
and vertical distances to point p.
See Also
========
sympy.geometry.point.Point.distance
Examples
========
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(4, 5)
>>> p1.taxicab_distance(p2)
7
"""
s, p = Point._normalize_dimension(self, Point(p))
return Add(*(abs(a - b) for a, b in zip(s, p)))
def canberra_distance(self, p):
"""The Canberra Distance from self to point p.
Returns the weighted sum of horizontal and vertical distances to
point p.
Parameters
==========
p : Point
Returns
=======
canberra_distance : The weighted sum of horizontal and vertical
distances to point p. The weight used is the sum of absolute values
of the coordinates.
Examples
========
>>> from sympy.geometry import Point
>>> p1, p2 = Point(1, 1), Point(3, 3)
>>> p1.canberra_distance(p2)
1
>>> p1, p2 = Point(0, 0), Point(3, 3)
>>> p1.canberra_distance(p2)
2
Raises
======
ValueError when both vectors are zero.
See Also
========
sympy.geometry.point.Point.distance
"""
s, p = Point._normalize_dimension(self, Point(p))
if self.is_zero and p.is_zero:
raise ValueError("Cannot project to the zero vector.")
return Add(*((abs(a - b)/(abs(a) + abs(b))) for a, b in zip(s, p)))
@property
def unit(self):
"""Return the Point that is in the same direction as `self`
and a distance of 1 from the origin"""
return self / abs(self)
n = evalf
__truediv__ = __div__
class Point2D(Point):
"""A point in a 2-dimensional Euclidean space.
Parameters
==========
coords : sequence of 2 coordinate values.
Attributes
==========
x
y
length
Raises
======
TypeError
When trying to add or subtract points with different dimensions.
When trying to create a point with more than two dimensions.
When `intersection` is called with object other than a Point.
See Also
========
sympy.geometry.line.Segment : Connects two Points
Examples
========
>>> from sympy.geometry import Point2D
>>> from sympy.abc import x
>>> Point2D(1, 2)
Point2D(1, 2)
>>> Point2D([1, 2])
Point2D(1, 2)
>>> Point2D(0, x)
Point2D(0, x)
Floats are automatically converted to Rational unless the
evaluate flag is False:
>>> Point2D(0.5, 0.25)
Point2D(1/2, 1/4)
>>> Point2D(0.5, 0.25, evaluate=False)
Point2D(0.5, 0.25)
"""
_ambient_dimension = 2
def __new__(cls, *args, **kwargs):
if not kwargs.pop('_nocheck', False):
kwargs['dim'] = 2
args = Point(*args, **kwargs)
return GeometryEntity.__new__(cls, *args)
def __contains__(self, item):
return item == self
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
return (self.x, self.y, self.x, self.y)
def rotate(self, angle, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
See Also
========
rotate, scale
Examples
========
>>> from sympy import Point2D, pi
>>> t = Point2D(1, 0)
>>> t.rotate(pi/2)
Point2D(0, 1)
>>> t.rotate(pi/2, (2, 0))
Point2D(2, -1)
"""
from sympy import cos, sin, Point
c = cos(angle)
s = sin(angle)
rv = self
if pt is not None:
pt = Point(pt, dim=2)
rv -= pt
x, y = rv.args
rv = Point(c*x - s*y, s*x + c*y)
if pt is not None:
rv += pt
return rv
def scale(self, x=1, y=1, pt=None):
"""Scale the coordinates of the Point by multiplying by
``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) --
and then adding ``pt`` back again (i.e. ``pt`` is the point of
reference for the scaling).
See Also
========
rotate, translate
Examples
========
>>> from sympy import Point2D
>>> t = Point2D(1, 1)
>>> t.scale(2)
Point2D(2, 1)
>>> t.scale(2, 2)
Point2D(2, 2)
"""
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
return Point(self.x*x, self.y*y)
def transform(self, matrix):
"""Return the point after applying the transformation described
by the 3x3 Matrix, ``matrix``.
See Also
========
geometry.entity.rotate
geometry.entity.scale
geometry.entity.translate
"""
if not (matrix.is_Matrix and matrix.shape == (3, 3)):
raise ValueError("matrix must be a 3x3 matrix")
col, row = matrix.shape
valid_matrix = matrix.is_square and col == 3
x, y = self.args
return Point(*(Matrix(1, 3, [x, y, 1])*matrix).tolist()[0][:2])
def translate(self, x=0, y=0):
"""Shift the Point by adding x and y to the coordinates of the Point.
See Also
========
rotate, scale
Examples
========
>>> from sympy import Point2D
>>> t = Point2D(0, 1)
>>> t.translate(2)
Point2D(2, 1)
>>> t.translate(2, 2)
Point2D(2, 3)
>>> t + Point2D(2, 2)
Point2D(2, 3)
"""
return Point(self.x + x, self.y + y)
@property
def x(self):
"""
Returns the X coordinate of the Point.
Examples
========
>>> from sympy import Point2D
>>> p = Point2D(0, 1)
>>> p.x
0
"""
return self.args[0]
@property
def y(self):
"""
Returns the Y coordinate of the Point.
Examples
========
>>> from sympy import Point2D
>>> p = Point2D(0, 1)
>>> p.y
1
"""
return self.args[1]
class Point3D(Point):
"""A point in a 3-dimensional Euclidean space.
Parameters
==========
coords : sequence of 3 coordinate values.
Attributes
==========
x
y
z
length
Raises
======
TypeError
When trying to add or subtract points with different dimensions.
When `intersection` is called with object other than a Point.
Examples
========
>>> from sympy import Point3D
>>> from sympy.abc import x
>>> Point3D(1, 2, 3)
Point3D(1, 2, 3)
>>> Point3D([1, 2, 3])
Point3D(1, 2, 3)
>>> Point3D(0, x, 3)
Point3D(0, x, 3)
Floats are automatically converted to Rational unless the
evaluate flag is False:
>>> Point3D(0.5, 0.25, 2)
Point3D(1/2, 1/4, 2)
>>> Point3D(0.5, 0.25, 3, evaluate=False)
Point3D(0.5, 0.25, 3)
"""
_ambient_dimension = 3
def __new__(cls, *args, **kwargs):
if not kwargs.pop('_nocheck', False):
kwargs['dim'] = 3
args = Point(*args, **kwargs)
return GeometryEntity.__new__(cls, *args)
def __contains__(self, item):
return item == self
@staticmethod
def are_collinear(*points):
"""Is a sequence of points collinear?
Test whether or not a set of points are collinear. Returns True if
the set of points are collinear, or False otherwise.
Parameters
==========
points : sequence of Point
Returns
=======
are_collinear : boolean
See Also
========
sympy.geometry.line.Line3D
Examples
========
>>> from sympy import Point3D, Matrix
>>> from sympy.abc import x
>>> p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1)
>>> p3, p4, p5 = Point3D(2, 2, 2), Point3D(x, x, x), Point3D(1, 2, 6)
>>> Point3D.are_collinear(p1, p2, p3, p4)
True
>>> Point3D.are_collinear(p1, p2, p3, p5)
False
"""
return Point.is_collinear(*points)
def direction_cosine(self, point):
"""
Gives the direction cosine between 2 points
Parameters
==========
p : Point3D
Returns
=======
list
Examples
========
>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 3)
>>> p1.direction_cosine(Point3D(2, 3, 5))
[sqrt(6)/6, sqrt(6)/6, sqrt(6)/3]
"""
a = self.direction_ratio(point)
b = sqrt(Add(*(i**2 for i in a)))
return [(point.x - self.x) / b,(point.y - self.y) / b,
(point.z - self.z) / b]
def direction_ratio(self, point):
"""
Gives the direction ratio between 2 points
Parameters
==========
p : Point3D
Returns
=======
list
Examples
========
>>> from sympy import Point3D
>>> p1 = Point3D(1, 2, 3)
>>> p1.direction_ratio(Point3D(2, 3, 5))
[1, 1, 2]
"""
return [(point.x - self.x),(point.y - self.y),(point.z - self.z)]
def intersection(self, other):
"""The intersection between this point and another point.
Parameters
==========
other : Point
Returns
=======
intersection : list of Points
Notes
=====
The return value will either be an empty list if there is no
intersection, otherwise it will contain this point.
Examples
========
>>> from sympy import Point3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 0, 0)
>>> p1.intersection(p2)
[]
>>> p1.intersection(p3)
[Point3D(0, 0, 0)]
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=3)
if isinstance(other, Point3D):
if self == other:
return [self]
return []
return other.intersection(self)
def scale(self, x=1, y=1, z=1, pt=None):
"""Scale the coordinates of the Point by multiplying by
``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) --
and then adding ``pt`` back again (i.e. ``pt`` is the point of
reference for the scaling).
See Also
========
translate
Examples
========
>>> from sympy import Point3D
>>> t = Point3D(1, 1, 1)
>>> t.scale(2)
Point3D(2, 1, 1)
>>> t.scale(2, 2)
Point3D(2, 2, 1)
"""
if pt:
pt = Point3D(pt)
return self.translate(*(-pt).args).scale(x, y, z).translate(*pt.args)
return Point3D(self.x*x, self.y*y, self.z*z)
def transform(self, matrix):
"""Return the point after applying the transformation described
by the 4x4 Matrix, ``matrix``.
See Also
========
geometry.entity.rotate
geometry.entity.scale
geometry.entity.translate
"""
if not (matrix.is_Matrix and matrix.shape == (4, 4)):
raise ValueError("matrix must be a 4x4 matrix")
col, row = matrix.shape
valid_matrix = matrix.is_square and col == 4
from sympy.matrices.expressions import Transpose
x, y, z = self.args
m = Transpose(matrix)
return Point3D(*(Matrix(1, 4, [x, y, z, 1])*m).tolist()[0][:3])
def translate(self, x=0, y=0, z=0):
"""Shift the Point by adding x and y to the coordinates of the Point.
See Also
========
rotate, scale
Examples
========
>>> from sympy import Point3D
>>> t = Point3D(0, 1, 1)
>>> t.translate(2)
Point3D(2, 1, 1)
>>> t.translate(2, 2)
Point3D(2, 3, 1)
>>> t + Point3D(2, 2, 2)
Point3D(2, 3, 3)
"""
return Point3D(self.x + x, self.y + y, self.z + z)
@property
def x(self):
"""
Returns the X coordinate of the Point.
Examples
========
>>> from sympy import Point3D
>>> p = Point3D(0, 1, 3)
>>> p.x
0
"""
return self.args[0]
@property
def y(self):
"""
Returns the Y coordinate of the Point.
Examples
========
>>> from sympy import Point3D
>>> p = Point3D(0, 1, 2)
>>> p.y
1
"""
return self.args[1]
@property
def z(self):
"""
Returns the Z coordinate of the Point.
Examples
========
>>> from sympy import Point3D
>>> p = Point3D(0, 1, 1)
>>> p.z
1
"""
return self.args[2]
|
2fdd675ab43c016a046dc32040eba93eb975ead136e6db62b4b2ee74d8fcf7d5 | """Elliptical geometrical entities.
Contains
* Ellipse
* Circle
"""
from __future__ import division, print_function
from sympy import Expr, Eq
from sympy.core import S, pi, sympify
from sympy.core.evaluate import global_evaluate
from sympy.core.logic import fuzzy_bool
from sympy.core.numbers import Rational, oo
from sympy.core.compatibility import ordered
from sympy.core.symbol import Dummy, _uniquely_named_symbol, _symbol
from sympy.simplify import simplify, trigsimp
from sympy.functions.elementary.miscellaneous import sqrt, Max
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.functions.special.elliptic_integrals import elliptic_e
from sympy.geometry.exceptions import GeometryError
from sympy.geometry.line import Ray2D, Segment2D, Line2D, LinearEntity3D
from sympy.polys import DomainError, Poly, PolynomialError
from sympy.polys.polyutils import _not_a_coeff, _nsort
from sympy.solvers import solve
from sympy.solvers.solveset import linear_coeffs
from sympy.utilities.misc import filldedent, func_name
from .entity import GeometryEntity, GeometrySet
from .point import Point, Point2D, Point3D
from .line import Line, Segment
from .util import idiff
import random
class Ellipse(GeometrySet):
"""An elliptical GeometryEntity.
Parameters
==========
center : Point, optional
Default value is Point(0, 0)
hradius : number or SymPy expression, optional
vradius : number or SymPy expression, optional
eccentricity : number or SymPy expression, optional
Two of `hradius`, `vradius` and `eccentricity` must be supplied to
create an Ellipse. The third is derived from the two supplied.
Attributes
==========
center
hradius
vradius
area
circumference
eccentricity
periapsis
apoapsis
focus_distance
foci
Raises
======
GeometryError
When `hradius`, `vradius` and `eccentricity` are incorrectly supplied
as parameters.
TypeError
When `center` is not a Point.
See Also
========
Circle
Notes
-----
Constructed from a center and two radii, the first being the horizontal
radius (along the x-axis) and the second being the vertical radius (along
the y-axis).
When symbolic value for hradius and vradius are used, any calculation that
refers to the foci or the major or minor axis will assume that the ellipse
has its major radius on the x-axis. If this is not true then a manual
rotation is necessary.
Examples
========
>>> from sympy import Ellipse, Point, Rational
>>> e1 = Ellipse(Point(0, 0), 5, 1)
>>> e1.hradius, e1.vradius
(5, 1)
>>> e2 = Ellipse(Point(3, 1), hradius=3, eccentricity=Rational(4, 5))
>>> e2
Ellipse(Point2D(3, 1), 3, 9/5)
"""
def __contains__(self, o):
if isinstance(o, Point):
x = Dummy('x', real=True)
y = Dummy('y', real=True)
res = self.equation(x, y).subs({x: o.x, y: o.y})
return trigsimp(simplify(res)) is S.Zero
elif isinstance(o, Ellipse):
return self == o
return False
def __eq__(self, o):
"""Is the other GeometryEntity the same as this ellipse?"""
return isinstance(o, Ellipse) and (self.center == o.center and
self.hradius == o.hradius and
self.vradius == o.vradius)
def __hash__(self):
return super(Ellipse, self).__hash__()
def __new__(
cls, center=None, hradius=None, vradius=None, eccentricity=None, **kwargs):
hradius = sympify(hradius)
vradius = sympify(vradius)
eccentricity = sympify(eccentricity)
if center is None:
center = Point(0, 0)
else:
center = Point(center, dim=2)
if len(center) != 2:
raise ValueError('The center of "{0}" must be a two dimensional point'.format(cls))
if len(list(filter(lambda x: x is not None, (hradius, vradius, eccentricity)))) != 2:
raise ValueError(filldedent('''
Exactly two arguments of "hradius", "vradius", and
"eccentricity" must not be None.'''))
if eccentricity is not None:
if hradius is None:
hradius = vradius / sqrt(1 - eccentricity**2)
elif vradius is None:
vradius = hradius * sqrt(1 - eccentricity**2)
if hradius == vradius:
return Circle(center, hradius, **kwargs)
if hradius == 0 or vradius == 0:
return Segment(Point(center[0] - hradius, center[1] - vradius), Point(center[0] + hradius, center[1] + vradius))
return GeometryEntity.__new__(cls, center, hradius, vradius, **kwargs)
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG ellipse element for the Ellipse.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
c = N(self.center)
h, v = N(self.hradius), N(self.vradius)
return (
'<ellipse fill="{1}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" cx="{2}" cy="{3}" rx="{4}" ry="{5}"/>'
).format(2. * scale_factor, fill_color, c.x, c.y, h, v)
@property
def ambient_dimension(self):
return 2
@property
def apoapsis(self):
"""The apoapsis of the ellipse.
The greatest distance between the focus and the contour.
Returns
=======
apoapsis : number
See Also
========
periapsis : Returns shortest distance between foci and contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.apoapsis
2*sqrt(2) + 3
"""
return self.major * (1 + self.eccentricity)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the functions.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point2D(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(filldedent('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name))
return Point(self.center.x + self.hradius*cos(t),
self.center.y + self.vradius*sin(t))
@property
def area(self):
"""The area of the ellipse.
Returns
=======
area : number
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.area
3*pi
"""
return simplify(S.Pi * self.hradius * self.vradius)
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
h, v = self.hradius, self.vradius
return (self.center.x - h, self.center.y - v, self.center.x + h, self.center.y + v)
@property
def center(self):
"""The center of the ellipse.
Returns
=======
center : number
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.center
Point2D(0, 0)
"""
return self.args[0]
@property
def circumference(self):
"""The circumference of the ellipse.
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.circumference
12*elliptic_e(8/9)
"""
if self.eccentricity == 1:
# degenerate
return 4*self.major
elif self.eccentricity == 0:
# circle
return 2*pi*self.hradius
else:
return 4*self.major*elliptic_e(self.eccentricity**2)
@property
def eccentricity(self):
"""The eccentricity of the ellipse.
Returns
=======
eccentricity : number
Examples
========
>>> from sympy import Point, Ellipse, sqrt
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, sqrt(2))
>>> e1.eccentricity
sqrt(7)/3
"""
return self.focus_distance / self.major
def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.
Notes
-----
Being on the border of self is considered False.
Parameters
==========
p : Point
Returns
=======
encloses_point : True, False or None
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Ellipse, S
>>> from sympy.abc import t
>>> e = Ellipse((0, 0), 3, 2)
>>> e.encloses_point((0, 0))
True
>>> e.encloses_point(e.arbitrary_point(t).subs(t, S.Half))
False
>>> e.encloses_point((4, 0))
False
"""
p = Point(p, dim=2)
if p in self:
return False
if len(self.foci) == 2:
# if the combined distance from the foci to p (h1 + h2) is less
# than the combined distance from the foci to the minor axis
# (which is the same as the major axis length) then p is inside
# the ellipse
h1, h2 = [f.distance(p) for f in self.foci]
test = 2*self.major - (h1 + h2)
else:
test = self.radius - self.center.distance(p)
return fuzzy_bool(test.is_positive)
def equation(self, x='x', y='y', _slope=None):
"""
Returns the equation of an ellipse aligned with the x and y axes;
when slope is given, the equation returned corresponds to an ellipse
with a major axis having that slope.
Parameters
==========
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
_slope : Expr, optional
The slope of the major axis. Ignored when 'None'.
Returns
=======
equation : sympy expression
See Also
========
arbitrary_point : Returns parameterized point on ellipse
Examples
========
>>> from sympy import Point, Ellipse, pi
>>> from sympy.abc import x, y
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> eq1 = e1.equation(x, y); eq1
y**2/4 + (x/3 - 1/3)**2 - 1
>>> eq2 = e1.equation(x, y, _slope=1); eq2
(-x + y + 1)**2/8 + (x + y - 1)**2/18 - 1
A point on e1 satisfies eq1. Let's use one on the x-axis:
>>> p1 = e1.center + Point(e1.major, 0)
>>> assert eq1.subs(x, p1.x).subs(y, p1.y) == 0
When rotated the same as the rotated ellipse, about the center
point of the ellipse, it will satisfy the rotated ellipse's
equation, too:
>>> r1 = p1.rotate(pi/4, e1.center)
>>> assert eq2.subs(x, r1.x).subs(y, r1.y) == 0
References
==========
.. [1] https://math.stackexchange.com/questions/108270/what-is-the-equation-of-an-ellipse-that-is-not-aligned-with-the-axis
.. [2] https://en.wikipedia.org/wiki/Ellipse#Equation_of_a_shifted_ellipse
"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
dx = x - self.center.x
dy = y - self.center.y
if _slope is not None:
L = (dy - _slope*dx)**2
l = (_slope*dy + dx)**2
h = 1 + _slope**2
b = h*self.major**2
a = h*self.minor**2
return l/b + L/a - 1
else:
t1 = (dx/self.hradius)**2
t2 = (dy/self.vradius)**2
return t1 + t2 - 1
def evolute(self, x='x', y='y'):
"""The equation of evolute of the ellipse.
Parameters
==========
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
Returns
=======
equation : sympy expression
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> e1.evolute()
2**(2/3)*y**(2/3) + (3*x - 3)**(2/3) - 5**(2/3)
"""
if len(self.args) != 3:
raise NotImplementedError('Evolute of arbitrary Ellipse is not supported.')
x = _symbol(x, real=True)
y = _symbol(y, real=True)
t1 = (self.hradius*(x - self.center.x))**Rational(2, 3)
t2 = (self.vradius*(y - self.center.y))**Rational(2, 3)
return t1 + t2 - (self.hradius**2 - self.vradius**2)**Rational(2, 3)
@property
def foci(self):
"""The foci of the ellipse.
Notes
-----
The foci can only be calculated if the major/minor axes are known.
Raises
======
ValueError
When the major and minor axis cannot be determined.
See Also
========
sympy.geometry.point.Point
focus_distance : Returns the distance between focus and center
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.foci
(Point2D(-2*sqrt(2), 0), Point2D(2*sqrt(2), 0))
"""
c = self.center
hr, vr = self.hradius, self.vradius
if hr == vr:
return (c, c)
# calculate focus distance manually, since focus_distance calls this
# routine
fd = sqrt(self.major**2 - self.minor**2)
if hr == self.minor:
# foci on the y-axis
return (c + Point(0, -fd), c + Point(0, fd))
elif hr == self.major:
# foci on the x-axis
return (c + Point(-fd, 0), c + Point(fd, 0))
@property
def focus_distance(self):
"""The focal distance of the ellipse.
The distance between the center and one focus.
Returns
=======
focus_distance : number
See Also
========
foci
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.focus_distance
2*sqrt(2)
"""
return Point.distance(self.center, self.foci[0])
@property
def hradius(self):
"""The horizontal radius of the ellipse.
Returns
=======
hradius : number
See Also
========
vradius, major, minor
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.hradius
3
"""
return self.args[1]
def intersection(self, o):
"""The intersection of this ellipse and another geometrical entity
`o`.
Parameters
==========
o : GeometryEntity
Returns
=======
intersection : list of GeometryEntity objects
Notes
-----
Currently supports intersections with Point, Line, Segment, Ray,
Circle and Ellipse types.
See Also
========
sympy.geometry.entity.GeometryEntity
Examples
========
>>> from sympy import Ellipse, Point, Line, sqrt
>>> e = Ellipse(Point(0, 0), 5, 7)
>>> e.intersection(Point(0, 0))
[]
>>> e.intersection(Point(5, 0))
[Point2D(5, 0)]
>>> e.intersection(Line(Point(0,0), Point(0, 1)))
[Point2D(0, -7), Point2D(0, 7)]
>>> e.intersection(Line(Point(5,0), Point(5, 1)))
[Point2D(5, 0)]
>>> e.intersection(Line(Point(6,0), Point(6, 1)))
[]
>>> e = Ellipse(Point(-1, 0), 4, 3)
>>> e.intersection(Ellipse(Point(1, 0), 4, 3))
[Point2D(0, -3*sqrt(15)/4), Point2D(0, 3*sqrt(15)/4)]
>>> e.intersection(Ellipse(Point(5, 0), 4, 3))
[Point2D(2, -3*sqrt(7)/4), Point2D(2, 3*sqrt(7)/4)]
>>> e.intersection(Ellipse(Point(100500, 0), 4, 3))
[]
>>> e.intersection(Ellipse(Point(0, 0), 3, 4))
[Point2D(3, 0), Point2D(-363/175, -48*sqrt(111)/175), Point2D(-363/175, 48*sqrt(111)/175)]
>>> e.intersection(Ellipse(Point(-1, 0), 3, 4))
[Point2D(-17/5, -12/5), Point2D(-17/5, 12/5), Point2D(7/5, -12/5), Point2D(7/5, 12/5)]
"""
# TODO: Replace solve with nonlinsolve, when nonlinsolve will be able to solve in real domain
x = Dummy('x', real=True)
y = Dummy('y', real=True)
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
elif isinstance(o, (Segment2D, Ray2D)):
ellipse_equation = self.equation(x, y)
result = solve([ellipse_equation, Line(o.points[0], o.points[1]).equation(x, y)], [x, y])
return list(ordered([Point(i) for i in result if i in o]))
elif isinstance(o, Polygon):
return o.intersection(self)
elif isinstance(o, (Ellipse, Line2D)):
if o == self:
return self
else:
ellipse_equation = self.equation(x, y)
return list(ordered([Point(i) for i in solve([ellipse_equation, o.equation(x, y)], [x, y])]))
elif isinstance(o, LinearEntity3D):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Intersection not handled for %s' % func_name(o))
def is_tangent(self, o):
"""Is `o` tangent to the ellipse?
Parameters
==========
o : GeometryEntity
An Ellipse, LinearEntity or Polygon
Raises
======
NotImplementedError
When the wrong type of argument is supplied.
Returns
=======
is_tangent: boolean
True if o is tangent to the ellipse, False otherwise.
See Also
========
tangent_lines
Examples
========
>>> from sympy import Point, Ellipse, Line
>>> p0, p1, p2 = Point(0, 0), Point(3, 0), Point(3, 3)
>>> e1 = Ellipse(p0, 3, 2)
>>> l1 = Line(p1, p2)
>>> e1.is_tangent(l1)
True
"""
if isinstance(o, Point2D):
return False
elif isinstance(o, Ellipse):
intersect = self.intersection(o)
if isinstance(intersect, Ellipse):
return True
elif intersect:
return all((self.tangent_lines(i)[0]).equals((o.tangent_lines(i)[0])) for i in intersect)
else:
return False
elif isinstance(o, Line2D):
return len(self.intersection(o)) == 1
elif isinstance(o, Ray2D):
intersect = self.intersection(o)
if len(intersect) == 1:
return intersect[0] != o.source and not self.encloses_point(o.source)
else:
return False
elif isinstance(o, (Segment2D, Polygon)):
all_tangents = False
segments = o.sides if isinstance(o, Polygon) else [o]
for segment in segments:
intersect = self.intersection(segment)
if len(intersect) == 1:
if not any(intersect[0] in i for i in segment.points) \
and all(not self.encloses_point(i) for i in segment.points):
all_tangents = True
continue
else:
return False
else:
return all_tangents
return all_tangents
elif isinstance(o, (LinearEntity3D, Point3D)):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Is_tangent not handled for %s' % func_name(o))
@property
def major(self):
"""Longer axis of the ellipse (if it can be determined) else hradius.
Returns
=======
major : number or expression
See Also
========
hradius, vradius, minor
Examples
========
>>> from sympy import Point, Ellipse, Symbol
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.major
3
>>> a = Symbol('a')
>>> b = Symbol('b')
>>> Ellipse(p1, a, b).major
a
>>> Ellipse(p1, b, a).major
b
>>> m = Symbol('m')
>>> M = m + 1
>>> Ellipse(p1, m, M).major
m + 1
"""
ab = self.args[1:3]
if len(ab) == 1:
return ab[0]
a, b = ab
o = b - a < 0
if o == True:
return a
elif o == False:
return b
return self.hradius
@property
def minor(self):
"""Shorter axis of the ellipse (if it can be determined) else vradius.
Returns
=======
minor : number or expression
See Also
========
hradius, vradius, major
Examples
========
>>> from sympy import Point, Ellipse, Symbol
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.minor
1
>>> a = Symbol('a')
>>> b = Symbol('b')
>>> Ellipse(p1, a, b).minor
b
>>> Ellipse(p1, b, a).minor
a
>>> m = Symbol('m')
>>> M = m + 1
>>> Ellipse(p1, m, M).minor
m
"""
ab = self.args[1:3]
if len(ab) == 1:
return ab[0]
a, b = ab
o = a - b < 0
if o == True:
return a
elif o == False:
return b
return self.vradius
def normal_lines(self, p, prec=None):
"""Normal lines between `p` and the ellipse.
Parameters
==========
p : Point
Returns
=======
normal_lines : list with 1, 2 or 4 Lines
Examples
========
>>> from sympy import Line, Point, Ellipse
>>> e = Ellipse((0, 0), 2, 3)
>>> c = e.center
>>> e.normal_lines(c + Point(1, 0))
[Line2D(Point2D(0, 0), Point2D(1, 0))]
>>> e.normal_lines(c)
[Line2D(Point2D(0, 0), Point2D(0, 1)), Line2D(Point2D(0, 0), Point2D(1, 0))]
Off-axis points require the solution of a quartic equation. This
often leads to very large expressions that may be of little practical
use. An approximate solution of `prec` digits can be obtained by
passing in the desired value:
>>> e.normal_lines((3, 3), prec=2)
[Line2D(Point2D(-0.81, -2.7), Point2D(0.19, -1.2)),
Line2D(Point2D(1.5, -2.0), Point2D(2.5, -2.7))]
Whereas the above solution has an operation count of 12, the exact
solution has an operation count of 2020.
"""
p = Point(p, dim=2)
# XXX change True to something like self.angle == 0 if the arbitrarily
# rotated ellipse is introduced.
# https://github.com/sympy/sympy/issues/2815)
if True:
rv = []
if p.x == self.center.x:
rv.append(Line(self.center, slope=oo))
if p.y == self.center.y:
rv.append(Line(self.center, slope=0))
if rv:
# at these special orientations of p either 1 or 2 normals
# exist and we are done
return rv
# find the 4 normal points and construct lines through them with
# the corresponding slope
x, y = Dummy('x', real=True), Dummy('y', real=True)
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
norm = -1/dydx
slope = Line(p, (x, y)).slope
seq = slope - norm
# TODO: Replace solve with solveset, when this line is tested
yis = solve(seq, y)[0]
xeq = eq.subs(y, yis).as_numer_denom()[0].expand()
if len(xeq.free_symbols) == 1:
try:
# this is so much faster, it's worth a try
xsol = Poly(xeq, x).real_roots()
except (DomainError, PolynomialError, NotImplementedError):
# TODO: Replace solve with solveset, when these lines are tested
xsol = _nsort(solve(xeq, x), separated=True)[0]
points = [Point(i, solve(eq.subs(x, i), y)[0]) for i in xsol]
else:
raise NotImplementedError(
'intersections for the general ellipse are not supported')
slopes = [norm.subs(zip((x, y), pt.args)) for pt in points]
if prec is not None:
points = [pt.n(prec) for pt in points]
slopes = [i if _not_a_coeff(i) else i.n(prec) for i in slopes]
return [Line(pt, slope=s) for pt, s in zip(points, slopes)]
@property
def periapsis(self):
"""The periapsis of the ellipse.
The shortest distance between the focus and the contour.
Returns
=======
periapsis : number
See Also
========
apoapsis : Returns greatest distance between focus and contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.periapsis
3 - 2*sqrt(2)
"""
return self.major * (1 - self.eccentricity)
@property
def semilatus_rectum(self):
"""
Calculates the semi-latus rectum of the Ellipse.
Semi-latus rectum is defined as one half of the the chord through a
focus parallel to the conic section directrix of a conic section.
Returns
=======
semilatus_rectum : number
See Also
========
apoapsis : Returns greatest distance between focus and contour
periapsis : The shortest distance between the focus and the contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.semilatus_rectum
1/3
References
==========
[1] http://mathworld.wolfram.com/SemilatusRectum.html
[2] https://en.wikipedia.org/wiki/Ellipse#Semi-latus_rectum
"""
return self.major * (1 - self.eccentricity ** 2)
def auxiliary_circle(self):
"""Returns a Circle whose diameter is the major axis of the ellipse.
Examples
========
>>> from sympy import Circle, Ellipse, Point, symbols
>>> c = Point(1, 2)
>>> Ellipse(c, 8, 7).auxiliary_circle()
Circle(Point2D(1, 2), 8)
>>> a, b = symbols('a b')
>>> Ellipse(c, a, b).auxiliary_circle()
Circle(Point2D(1, 2), Max(a, b))
"""
return Circle(self.center, Max(self.hradius, self.vradius))
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.plot_interval()
[t, -pi, pi]
"""
t = _symbol(parameter, real=True)
return [t, -S.Pi, S.Pi]
def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
Examples
========
>>> from sympy import Point, Ellipse, Segment
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point2D(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point2D(2.1, 1.4)
Notes
=====
When creating a random point, one may simply replace the
parameter with a random number. When doing so, however, the
random number should be made a Rational or else the point
may not test as being in the ellipse:
>>> from sympy.abc import t
>>> from sympy import Rational
>>> arb = e1.arbitrary_point(t); arb
Point2D(3*cos(t), 2*sin(t))
>>> arb.subs(t, .1) in e1
False
>>> arb.subs(t, Rational(.1)) in e1
True
>>> arb.subs(t, Rational('.1')) in e1
True
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
"""
from sympy import sin, cos, Rational
t = _symbol('t', real=True)
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
# simplify this now or else the Float will turn s into a Float
r = Rational(rng.random())
c = 2*r - 1
s = sqrt(1 - c**2)
return Point(x.subs(cos(t), c), y.subs(sin(t), s))
def reflect(self, line):
"""Override GeometryEntity.reflect since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle, Line
>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
>>> from sympy import Ellipse, Line, Point
>>> Ellipse(Point(3, 4), 1, 3).reflect(Line(Point(0, -4), Point(5, 0)))
Traceback (most recent call last):
...
NotImplementedError:
General Ellipse is not supported but the equation of the reflected
Ellipse is given by the zeros of: f(x, y) = (9*x/41 + 40*y/41 +
37/41)**2 + (40*x/123 - 3*y/41 - 364/123)**2 - 1
Notes
=====
Until the general ellipse (with no axis parallel to the x-axis) is
supported a NotImplemented error is raised and the equation whose
zeros define the rotated ellipse is given.
"""
if line.slope in (0, oo):
c = self.center
c = c.reflect(line)
return self.func(c, -self.hradius, self.vradius)
else:
x, y = [_uniquely_named_symbol(
name, (self, line), real=True) for name in 'xy']
expr = self.equation(x, y)
p = Point(x, y).reflect(line)
result = expr.subs(zip((x, y), p.args
), simultaneous=True)
raise NotImplementedError(filldedent(
'General Ellipse is not supported but the equation '
'of the reflected Ellipse is given by the zeros of: ' +
"f(%s, %s) = %s" % (str(x), str(y), str(result))))
def rotate(self, angle=0, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
Note: since the general ellipse is not supported, only rotations that
are integer multiples of pi/2 are allowed.
Examples
========
>>> from sympy import Ellipse, pi
>>> Ellipse((1, 0), 2, 1).rotate(pi/2)
Ellipse(Point2D(0, 1), 1, 2)
>>> Ellipse((1, 0), 2, 1).rotate(pi)
Ellipse(Point2D(-1, 0), 2, 1)
"""
if self.hradius == self.vradius:
return self.func(self.center.rotate(angle, pt), self.hradius)
if (angle/S.Pi).is_integer:
return super(Ellipse, self).rotate(angle, pt)
if (2*angle/S.Pi).is_integer:
return self.func(self.center.rotate(angle, pt), self.vradius, self.hradius)
# XXX see https://github.com/sympy/sympy/issues/2815 for general ellipes
raise NotImplementedError('Only rotations of pi/2 are currently supported for Ellipse.')
def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since it is the major and minor
axes which must be scaled and they are not GeometryEntities.
Examples
========
>>> from sympy import Ellipse
>>> Ellipse((0, 0), 2, 1).scale(2, 4)
Circle(Point2D(0, 0), 4)
>>> Ellipse((0, 0), 2, 1).scale(2)
Ellipse(Point2D(0, 0), 4, 1)
"""
c = self.center
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
h = self.hradius
v = self.vradius
return self.func(c.scale(x, y), hradius=h*x, vradius=v*y)
def tangent_lines(self, p):
"""Tangent lines between `p` and the ellipse.
If `p` is on the ellipse, returns the tangent line through point `p`.
Otherwise, returns the tangent line(s) from `p` to the ellipse, or
None if no tangent line is possible (e.g., `p` inside ellipse).
Parameters
==========
p : Point
Returns
=======
tangent_lines : list with 1 or 2 Lines
Raises
======
NotImplementedError
Can only find tangent lines for a point, `p`, on the ellipse.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Line
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.tangent_lines(Point(3, 0))
[Line2D(Point2D(3, 0), Point2D(3, -12))]
"""
p = Point(p, dim=2)
if self.encloses_point(p):
return []
if p in self:
delta = self.center - p
rise = (self.vradius**2)*delta.x
run = -(self.hradius**2)*delta.y
p2 = Point(simplify(p.x + run),
simplify(p.y + rise))
return [Line(p, p2)]
else:
if len(self.foci) == 2:
f1, f2 = self.foci
maj = self.hradius
test = (2*maj -
Point.distance(f1, p) -
Point.distance(f2, p))
else:
test = self.radius - Point.distance(self.center, p)
if test.is_number and test.is_positive:
return []
# else p is outside the ellipse or we can't tell. In case of the
# latter, the solutions returned will only be valid if
# the point is not inside the ellipse; if it is, nan will result.
x, y = Dummy('x'), Dummy('y')
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
slope = Line(p, Point(x, y)).slope
# TODO: Replace solve with solveset, when this line is tested
tangent_points = solve([slope - dydx, eq], [x, y])
# handle horizontal and vertical tangent lines
if len(tangent_points) == 1:
assert tangent_points[0][
0] == p.x or tangent_points[0][1] == p.y
return [Line(p, p + Point(1, 0)), Line(p, p + Point(0, 1))]
# others
return [Line(p, tangent_points[0]), Line(p, tangent_points[1])]
@property
def vradius(self):
"""The vertical radius of the ellipse.
Returns
=======
vradius : number
See Also
========
hradius, major, minor
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.vradius
1
"""
return self.args[2]
def second_moment_of_area(self, point=None):
"""Returns the second moment and product moment area of an ellipse.
Parameters
==========
point : Point, two-tuple of sympifiable objects, or None(default=None)
point is the point about which second moment of area is to be found.
If "point=None" it will be calculated about the axis passing through the
centroid of the ellipse.
Returns
=======
I_xx, I_yy, I_xy : number or sympy expression
I_xx, I_yy are second moment of area of an ellise.
I_xy is product moment of area of an ellipse.
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.second_moment_of_area()
(3*pi/4, 27*pi/4, 0)
References
==========
https://en.wikipedia.org/wiki/List_of_second_moments_of_area
"""
I_xx = (S.Pi*(self.hradius)*(self.vradius**3))/4
I_yy = (S.Pi*(self.hradius**3)*(self.vradius))/4
I_xy = 0
if point is None:
return I_xx, I_yy, I_xy
# parallel axis theorem
I_xx = I_xx + self.area*((point[1] - self.center.y)**2)
I_yy = I_yy + self.area*((point[0] - self.center.x)**2)
I_xy = I_xy + self.area*(point[0] - self.center.x)*(point[1] - self.center.y)
return I_xx, I_yy, I_xy
class Circle(Ellipse):
"""A circle in space.
Constructed simply from a center and a radius, from three
non-collinear points, or the equation of a circle.
Parameters
==========
center : Point
radius : number or sympy expression
points : sequence of three Points
equation : equation of a circle
Attributes
==========
radius (synonymous with hradius, vradius, major and minor)
circumference
equation
Raises
======
GeometryError
When the given equation is not that of a circle.
When trying to construct circle from incorrect parameters.
See Also
========
Ellipse, sympy.geometry.point.Point
Examples
========
>>> from sympy import Eq
>>> from sympy.geometry import Point, Circle
>>> from sympy.abc import x, y, a, b
A circle constructed from a center and radius:
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.hradius, c1.vradius, c1.radius
(5, 5, 5)
A circle constructed from three points:
>>> c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
>>> c2.hradius, c2.vradius, c2.radius, c2.center
(sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point2D(1/2, 1/2))
A circle can be constructed from an equation in the form
`a*x**2 + by**2 + gx + hy + c = 0`, too:
>>> Circle(x**2 + y**2 - 25)
Circle(Point2D(0, 0), 5)
If the variables corresponding to x and y are named something
else, their name or symbol can be supplied:
>>> Circle(Eq(a**2 + b**2, 25), x='a', y=b)
Circle(Point2D(0, 0), 5)
"""
def __new__(cls, *args, **kwargs):
from sympy.geometry.util import find
from .polygon import Triangle
evaluate = kwargs.get('evaluate', global_evaluate[0])
if len(args) == 1 and isinstance(args[0], Expr):
x = kwargs.get('x', 'x')
y = kwargs.get('y', 'y')
equation = args[0]
if isinstance(equation, Eq):
equation = equation.lhs - equation.rhs
x = find(x, equation)
y = find(y, equation)
try:
a, b, c, d, e = linear_coeffs(equation, x**2, y**2, x, y)
except ValueError:
raise GeometryError("The given equation is not that of a circle.")
if a == 0 or b == 0 or a != b:
raise GeometryError("The given equation is not that of a circle.")
center_x = -c/a/2
center_y = -d/b/2
r2 = (center_x**2) + (center_y**2) - e
return Circle((center_x, center_y), sqrt(r2), evaluate=evaluate)
else:
c, r = None, None
if len(args) == 3:
args = [Point(a, dim=2, evaluate=evaluate) for a in args]
t = Triangle(*args)
if not isinstance(t, Triangle):
return t
c = t.circumcenter
r = t.circumradius
elif len(args) == 2:
# Assume (center, radius) pair
c = Point(args[0], dim=2, evaluate=evaluate)
r = args[1]
# this will prohibit imaginary radius
try:
r = Point(r, 0, evaluate=evaluate).x
except:
raise GeometryError("Circle with imaginary radius is not permitted")
if not (c is None or r is None):
if r == 0:
return c
return GeometryEntity.__new__(cls, c, r, **kwargs)
raise GeometryError("Circle.__new__ received unknown arguments")
@property
def circumference(self):
"""The circumference of the circle.
Returns
=======
circumference : number or SymPy expression
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.circumference
12*pi
"""
return 2 * S.Pi * self.radius
def equation(self, x='x', y='y'):
"""The equation of the circle.
Parameters
==========
x : str or Symbol, optional
Default value is 'x'.
y : str or Symbol, optional
Default value is 'y'.
Returns
=======
equation : SymPy expression
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.equation()
x**2 + y**2 - 25
"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
t1 = (x - self.center.x)**2
t2 = (y - self.center.y)**2
return t1 + t2 - self.major**2
def intersection(self, o):
"""The intersection of this circle with another geometrical entity.
Parameters
==========
o : GeometryEntity
Returns
=======
intersection : list of GeometryEntities
Examples
========
>>> from sympy import Point, Circle, Line, Ray
>>> p1, p2, p3 = Point(0, 0), Point(5, 5), Point(6, 0)
>>> p4 = Point(5, 0)
>>> c1 = Circle(p1, 5)
>>> c1.intersection(p2)
[]
>>> c1.intersection(p4)
[Point2D(5, 0)]
>>> c1.intersection(Ray(p1, p2))
[Point2D(5*sqrt(2)/2, 5*sqrt(2)/2)]
>>> c1.intersection(Line(p2, p3))
[]
"""
return Ellipse.intersection(self, o)
@property
def radius(self):
"""The radius of the circle.
Returns
=======
radius : number or sympy expression
See Also
========
Ellipse.major, Ellipse.minor, Ellipse.hradius, Ellipse.vradius
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.radius
6
"""
return self.args[1]
def reflect(self, line):
"""Override GeometryEntity.reflect since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle, Line
>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
"""
c = self.center
c = c.reflect(line)
return self.func(c, -self.radius)
def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle
>>> Circle((0, 0), 1).scale(2, 2)
Circle(Point2D(0, 0), 2)
>>> Circle((0, 0), 1).scale(2, 4)
Ellipse(Point2D(0, 0), 2, 4)
"""
c = self.center
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
c = c.scale(x, y)
x, y = [abs(i) for i in (x, y)]
if x == y:
return self.func(c, x*self.radius)
h = v = self.radius
return Ellipse(c, hradius=h*x, vradius=v*y)
@property
def vradius(self):
"""
This Ellipse property is an alias for the Circle's radius.
Whereas hradius, major and minor can use Ellipse's conventions,
the vradius does not exist for a circle. It is always a positive
value in order that the Circle, like Polygons, will have an
area that can be positive or negative as determined by the sign
of the hradius.
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.vradius
6
"""
return abs(self.radius)
from .polygon import Polygon
|
84a9ab00ec8e1c41842537cfde3346330e886a66d4288a2ed47477c6686359d3 | """The definition of the base geometrical entity with attributes common to
all derived geometrical entities.
Contains
========
GeometryEntity
GeometricSet
Notes
=====
A GeometryEntity is any object that has special geometric properties.
A GeometrySet is a superclass of any GeometryEntity that can also
be viewed as a sympy.sets.Set. In particular, points are the only
GeometryEntity not considered a Set.
Rn is a GeometrySet representing n-dimensional Euclidean space. R2 and
R3 are currently the only ambient spaces implemented.
"""
from __future__ import division, print_function
from sympy.core.basic import Basic
from sympy.core.compatibility import is_sequence
from sympy.core.containers import Tuple
from sympy.core.sympify import sympify
from sympy.functions import cos, sin
from sympy.matrices import eye
from sympy.multipledispatch import dispatch
from sympy.sets import Set
from sympy.sets.handlers.intersection import intersection_sets
from sympy.sets.handlers.union import union_sets
from sympy.utilities.misc import func_name
# How entities are ordered; used by __cmp__ in GeometryEntity
ordering_of_classes = [
"Point2D",
"Point3D",
"Point",
"Segment2D",
"Ray2D",
"Line2D",
"Segment3D",
"Line3D",
"Ray3D",
"Segment",
"Ray",
"Line",
"Plane",
"Triangle",
"RegularPolygon",
"Polygon",
"Circle",
"Ellipse",
"Curve",
"Parabola"
]
class GeometryEntity(Basic):
"""The base class for all geometrical entities.
This class doesn't represent any particular geometric entity, it only
provides the implementation of some methods common to all subclasses.
"""
def __cmp__(self, other):
"""Comparison of two GeometryEntities."""
n1 = self.__class__.__name__
n2 = other.__class__.__name__
c = (n1 > n2) - (n1 < n2)
if not c:
return 0
i1 = -1
for cls in self.__class__.__mro__:
try:
i1 = ordering_of_classes.index(cls.__name__)
break
except ValueError:
i1 = -1
if i1 == -1:
return c
i2 = -1
for cls in other.__class__.__mro__:
try:
i2 = ordering_of_classes.index(cls.__name__)
break
except ValueError:
i2 = -1
if i2 == -1:
return c
return (i1 > i2) - (i1 < i2)
def __contains__(self, other):
"""Subclasses should implement this method for anything more complex than equality."""
if type(self) == type(other):
return self == other
raise NotImplementedError()
def __getnewargs__(self):
"""Returns a tuple that will be passed to __new__ on unpickling."""
return tuple(self.args)
def __ne__(self, o):
"""Test inequality of two geometrical entities."""
return not self == o
def __new__(cls, *args, **kwargs):
# Points are sequences, but they should not
# be converted to Tuples, so use this detection function instead.
def is_seq_and_not_point(a):
# we cannot use isinstance(a, Point) since we cannot import Point
if hasattr(a, 'is_Point') and a.is_Point:
return False
return is_sequence(a)
args = [Tuple(*a) if is_seq_and_not_point(a) else sympify(a) for a in args]
return Basic.__new__(cls, *args)
def __radd__(self, a):
"""Implementation of reverse add method."""
return a.__add__(self)
def __rdiv__(self, a):
"""Implementation of reverse division method."""
return a.__div__(self)
def __repr__(self):
"""String representation of a GeometryEntity that can be evaluated
by sympy."""
return type(self).__name__ + repr(self.args)
def __rmul__(self, a):
"""Implementation of reverse multiplication method."""
return a.__mul__(self)
def __rsub__(self, a):
"""Implementation of reverse substraction method."""
return a.__sub__(self)
def __str__(self):
"""String representation of a GeometryEntity."""
from sympy.printing import sstr
return type(self).__name__ + sstr(self.args)
def _eval_subs(self, old, new):
from sympy.geometry.point import Point, Point3D
if is_sequence(old) or is_sequence(new):
if isinstance(self, Point3D):
old = Point3D(old)
new = Point3D(new)
else:
old = Point(old)
new = Point(new)
return self._subs(old, new)
def _repr_svg_(self):
"""SVG representation of a GeometryEntity suitable for IPython"""
from sympy.core.evalf import N
try:
bounds = self.bounds
except (NotImplementedError, TypeError):
# if we have no SVG representation, return None so IPython
# will fall back to the next representation
return None
svg_top = '''<svg xmlns="http://www.w3.org/2000/svg"
xmlns:xlink="http://www.w3.org/1999/xlink"
width="{1}" height="{2}" viewBox="{0}"
preserveAspectRatio="xMinYMin meet">
<defs>
<marker id="markerCircle" markerWidth="8" markerHeight="8"
refx="5" refy="5" markerUnits="strokeWidth">
<circle cx="5" cy="5" r="1.5" style="stroke: none; fill:#000000;"/>
</marker>
<marker id="markerArrow" markerWidth="13" markerHeight="13" refx="2" refy="4"
orient="auto" markerUnits="strokeWidth">
<path d="M2,2 L2,6 L6,4" style="fill: #000000;" />
</marker>
<marker id="markerReverseArrow" markerWidth="13" markerHeight="13" refx="6" refy="4"
orient="auto" markerUnits="strokeWidth">
<path d="M6,2 L6,6 L2,4" style="fill: #000000;" />
</marker>
</defs>'''
# Establish SVG canvas that will fit all the data + small space
xmin, ymin, xmax, ymax = map(N, bounds)
if xmin == xmax and ymin == ymax:
# This is a point; buffer using an arbitrary size
xmin, ymin, xmax, ymax = xmin - .5, ymin -.5, xmax + .5, ymax + .5
else:
# Expand bounds by a fraction of the data ranges
expand = 0.1 # or 10%; this keeps arrowheads in view (R plots use 4%)
widest_part = max([xmax - xmin, ymax - ymin])
expand_amount = widest_part * expand
xmin -= expand_amount
ymin -= expand_amount
xmax += expand_amount
ymax += expand_amount
dx = xmax - xmin
dy = ymax - ymin
width = min([max([100., dx]), 300])
height = min([max([100., dy]), 300])
scale_factor = 1. if max(width, height) == 0 else max(dx, dy) / max(width, height)
try:
svg = self._svg(scale_factor)
except (NotImplementedError, TypeError):
# if we have no SVG representation, return None so IPython
# will fall back to the next representation
return None
view_box = "{0} {1} {2} {3}".format(xmin, ymin, dx, dy)
transform = "matrix(1,0,0,-1,0,{0})".format(ymax + ymin)
svg_top = svg_top.format(view_box, width, height)
return svg_top + (
'<g transform="{0}">{1}</g></svg>'
).format(transform, svg)
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the GeometryEntity.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
raise NotImplementedError()
def _sympy_(self):
return self
@property
def ambient_dimension(self):
"""What is the dimension of the space that the object is contained in?"""
raise NotImplementedError()
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
raise NotImplementedError()
def encloses(self, o):
"""
Return True if o is inside (not on or outside) the boundaries of self.
The object will be decomposed into Points and individual Entities need
only define an encloses_point method for their class.
See Also
========
sympy.geometry.ellipse.Ellipse.encloses_point
sympy.geometry.polygon.Polygon.encloses_point
Examples
========
>>> from sympy import RegularPolygon, Point, Polygon
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
>>> t2 = Polygon(*RegularPolygon(Point(0, 0), 2, 3).vertices)
>>> t2.encloses(t)
True
>>> t.encloses(t2)
False
"""
from sympy.geometry.point import Point
from sympy.geometry.line import Segment, Ray, Line
from sympy.geometry.ellipse import Ellipse
from sympy.geometry.polygon import Polygon, RegularPolygon
if isinstance(o, Point):
return self.encloses_point(o)
elif isinstance(o, Segment):
return all(self.encloses_point(x) for x in o.points)
elif isinstance(o, Ray) or isinstance(o, Line):
return False
elif isinstance(o, Ellipse):
return self.encloses_point(o.center) and \
self.encloses_point(
Point(o.center.x + o.hradius, o.center.y)) and \
not self.intersection(o)
elif isinstance(o, Polygon):
if isinstance(o, RegularPolygon):
if not self.encloses_point(o.center):
return False
return all(self.encloses_point(v) for v in o.vertices)
raise NotImplementedError()
def equals(self, o):
return self == o
def intersection(self, o):
"""
Returns a list of all of the intersections of self with o.
Notes
=====
An entity is not required to implement this method.
If two different types of entities can intersect, the item with
higher index in ordering_of_classes should implement
intersections with anything having a lower index.
See Also
========
sympy.geometry.util.intersection
"""
raise NotImplementedError()
def is_similar(self, other):
"""Is this geometrical entity similar to another geometrical entity?
Two entities are similar if a uniform scaling (enlarging or
shrinking) of one of the entities will allow one to obtain the other.
Notes
=====
This method is not intended to be used directly but rather
through the `are_similar` function found in util.py.
An entity is not required to implement this method.
If two different types of entities can be similar, it is only
required that one of them be able to determine this.
See Also
========
scale
"""
raise NotImplementedError()
def reflect(self, line):
"""
Reflects an object across a line.
Parameters
==========
line: Line
Examples
========
>>> from sympy import pi, sqrt, Line, RegularPolygon
>>> l = Line((0, pi), slope=sqrt(2))
>>> pent = RegularPolygon((1, 2), 1, 5)
>>> rpent = pent.reflect(l)
>>> rpent
RegularPolygon(Point2D(-2*sqrt(2)*pi/3 - 1/3 + 4*sqrt(2)/3, 2/3 + 2*sqrt(2)/3 + 2*pi/3), -1, 5, -atan(2*sqrt(2)) + 3*pi/5)
>>> from sympy import pi, Line, Circle, Point
>>> l = Line((0, pi), slope=1)
>>> circ = Circle(Point(0, 0), 5)
>>> rcirc = circ.reflect(l)
>>> rcirc
Circle(Point2D(-pi, pi), -5)
"""
from sympy import atan, Point, Dummy, oo
g = self
l = line
o = Point(0, 0)
if l.slope == 0:
y = l.args[0].y
if not y: # x-axis
return g.scale(y=-1)
reps = [(p, p.translate(y=2*(y - p.y))) for p in g.atoms(Point)]
elif l.slope == oo:
x = l.args[0].x
if not x: # y-axis
return g.scale(x=-1)
reps = [(p, p.translate(x=2*(x - p.x))) for p in g.atoms(Point)]
else:
if not hasattr(g, 'reflect') and not all(
isinstance(arg, Point) for arg in g.args):
raise NotImplementedError(
'reflect undefined or non-Point args in %s' % g)
a = atan(l.slope)
c = l.coefficients
d = -c[-1]/c[1] # y-intercept
# apply the transform to a single point
x, y = Dummy(), Dummy()
xf = Point(x, y)
xf = xf.translate(y=-d).rotate(-a, o).scale(y=-1
).rotate(a, o).translate(y=d)
# replace every point using that transform
reps = [(p, xf.xreplace({x: p.x, y: p.y})) for p in g.atoms(Point)]
return g.xreplace(dict(reps))
def rotate(self, angle, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
The default pt is the origin, Point(0, 0)
See Also
========
scale, translate
Examples
========
>>> from sympy import Point, RegularPolygon, Polygon, pi
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
>>> t # vertex on x axis
Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
>>> t.rotate(pi/2) # vertex on y axis now
Triangle(Point2D(0, 1), Point2D(-sqrt(3)/2, -1/2), Point2D(sqrt(3)/2, -1/2))
"""
newargs = []
for a in self.args:
if isinstance(a, GeometryEntity):
newargs.append(a.rotate(angle, pt))
else:
newargs.append(a)
return type(self)(*newargs)
def scale(self, x=1, y=1, pt=None):
"""Scale the object by multiplying the x,y-coordinates by x and y.
If pt is given, the scaling is done relative to that point; the
object is shifted by -pt, scaled, and shifted by pt.
See Also
========
rotate, translate
Examples
========
>>> from sympy import RegularPolygon, Point, Polygon
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
>>> t
Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
>>> t.scale(2)
Triangle(Point2D(2, 0), Point2D(-1, sqrt(3)/2), Point2D(-1, -sqrt(3)/2))
>>> t.scale(2, 2)
Triangle(Point2D(2, 0), Point2D(-1, sqrt(3)), Point2D(-1, -sqrt(3)))
"""
from sympy.geometry.point import Point
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
return type(self)(*[a.scale(x, y) for a in self.args]) # if this fails, override this class
def translate(self, x=0, y=0):
"""Shift the object by adding to the x,y-coordinates the values x and y.
See Also
========
rotate, scale
Examples
========
>>> from sympy import RegularPolygon, Point, Polygon
>>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
>>> t
Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
>>> t.translate(2)
Triangle(Point2D(3, 0), Point2D(3/2, sqrt(3)/2), Point2D(3/2, -sqrt(3)/2))
>>> t.translate(2, 2)
Triangle(Point2D(3, 2), Point2D(3/2, sqrt(3)/2 + 2), Point2D(3/2, 2 - sqrt(3)/2))
"""
newargs = []
for a in self.args:
if isinstance(a, GeometryEntity):
newargs.append(a.translate(x, y))
else:
newargs.append(a)
return self.func(*newargs)
def parameter_value(self, other, t):
"""Return the parameter corresponding to the given point.
Evaluating an arbitrary point of the entity at this parameter
value will return the given point.
Examples
========
>>> from sympy import Line, Point
>>> from sympy.abc import t
>>> a = Point(0, 0)
>>> b = Point(2, 2)
>>> Line(a, b).parameter_value((1, 1), t)
{t: 1/2}
>>> Line(a, b).arbitrary_point(t).subs(_)
Point2D(1, 1)
"""
from sympy.geometry.point import Point
from sympy.core.symbol import Dummy
from sympy.solvers.solvers import solve
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if not isinstance(other, Point):
raise ValueError("other must be a point")
T = Dummy('t', real=True)
sol = solve(self.arbitrary_point(T) - other, T, dict=True)
if not sol:
raise ValueError("Given point is not on %s" % func_name(self))
return {t: sol[0][T]}
class GeometrySet(GeometryEntity, Set):
"""Parent class of all GeometryEntity that are also Sets
(compatible with sympy.sets)
"""
def _contains(self, other):
"""sympy.sets uses the _contains method, so include it for compatibility."""
if isinstance(other, Set) and other.is_FiniteSet:
return all(self.__contains__(i) for i in other)
return self.__contains__(other)
@dispatch(GeometrySet, Set)
def union_sets(self, o):
""" Returns the union of self and o
for use with sympy.sets.Set, if possible. """
from sympy.sets import Union, FiniteSet
# if its a FiniteSet, merge any points
# we contain and return a union with the rest
if o.is_FiniteSet:
other_points = [p for p in o if not self._contains(p)]
if len(other_points) == len(o):
return None
return Union(self, FiniteSet(*other_points))
if self._contains(o):
return self
return None
@dispatch(GeometrySet, Set)
def intersection_sets(self, o):
""" Returns a sympy.sets.Set of intersection objects,
if possible. """
from sympy.sets import Set, FiniteSet, Union
from sympy.geometry import Point
try:
# if o is a FiniteSet, find the intersection directly
# to avoid infinite recursion
if o.is_FiniteSet:
inter = FiniteSet(*(p for p in o if self.contains(p)))
else:
inter = self.intersection(o)
except NotImplementedError:
# sympy.sets.Set.reduce expects None if an object
# doesn't know how to simplify
return None
# put the points in a FiniteSet
points = FiniteSet(*[p for p in inter if isinstance(p, Point)])
non_points = [p for p in inter if not isinstance(p, Point)]
return Union(*(non_points + [points]))
def translate(x, y):
"""Return the matrix to translate a 2-D point by x and y."""
rv = eye(3)
rv[2, 0] = x
rv[2, 1] = y
return rv
def scale(x, y, pt=None):
"""Return the matrix to multiply a 2-D point's coordinates by x and y.
If pt is given, the scaling is done relative to that point."""
rv = eye(3)
rv[0, 0] = x
rv[1, 1] = y
if pt:
from sympy.geometry.point import Point
pt = Point(pt, dim=2)
tr1 = translate(*(-pt).args)
tr2 = translate(*pt.args)
return tr1*rv*tr2
return rv
def rotate(th):
"""Return the matrix to rotate a 2-D point about the origin by ``angle``.
The angle is measured in radians. To Point a point about a point other
then the origin, translate the Point, do the rotation, and
translate it back:
>>> from sympy.geometry.entity import rotate, translate
>>> from sympy import Point, pi
>>> rot_about_11 = translate(-1, -1)*rotate(pi/2)*translate(1, 1)
>>> Point(1, 1).transform(rot_about_11)
Point2D(1, 1)
>>> Point(0, 0).transform(rot_about_11)
Point2D(2, 0)
"""
s = sin(th)
rv = eye(3)*cos(th)
rv[0, 1] = s
rv[1, 0] = -s
rv[2, 2] = 1
return rv
|
e8888bd1e020362be4f55c9f767414c778dcd3945faf6110c7043dbfc3dc1376 | """Line-like geometrical entities.
Contains
========
LinearEntity
Line
Ray
Segment
LinearEntity2D
Line2D
Ray2D
Segment2D
LinearEntity3D
Line3D
Ray3D
Segment3D
"""
from __future__ import division, print_function
from sympy import Expr
from sympy.core import S, sympify
from sympy.core.compatibility import ordered
from sympy.core.numbers import Rational, oo
from sympy.core.relational import Eq
from sympy.core.symbol import _symbol, Dummy
from sympy.functions.elementary.trigonometric import (_pi_coeff as pi_coeff, acos, tan, atan2)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.logic.boolalg import And
from sympy.simplify.simplify import simplify
from sympy.geometry.exceptions import GeometryError
from sympy.core.containers import Tuple
from sympy.core.decorators import deprecated
from sympy.sets import Intersection
from sympy.matrices import Matrix
from sympy.solvers.solveset import linear_coeffs
from .entity import GeometryEntity, GeometrySet
from .point import Point, Point3D
from sympy.utilities.misc import Undecidable, filldedent
from sympy.utilities.exceptions import SymPyDeprecationWarning
class LinearEntity(GeometrySet):
"""A base class for all linear entities (Line, Ray and Segment)
in n-dimensional Euclidean space.
Attributes
==========
ambient_dimension
direction
length
p1
p2
points
Notes
=====
This is an abstract class and is not meant to be instantiated.
See Also
========
sympy.geometry.entity.GeometryEntity
"""
def __new__(cls, p1, p2=None, **kwargs):
p1, p2 = Point._normalize_dimension(p1, p2)
if p1 == p2:
# sometimes we return a single point if we are not given two unique
# points. This is done in the specific subclass
raise ValueError(
"%s.__new__ requires two unique Points." % cls.__name__)
if len(p1) != len(p2):
raise ValueError(
"%s.__new__ requires two Points of equal dimension." % cls.__name__)
return GeometryEntity.__new__(cls, p1, p2, **kwargs)
def __contains__(self, other):
"""Return a definitive answer or else raise an error if it cannot
be determined that other is on the boundaries of self."""
result = self.contains(other)
if result is not None:
return result
else:
raise Undecidable(
"can't decide whether '%s' contains '%s'" % (self, other))
def _span_test(self, other):
"""Test whether the point `other` lies in the positive span of `self`.
A point x is 'in front' of a point y if x.dot(y) >= 0. Return
-1 if `other` is behind `self.p1`, 0 if `other` is `self.p1` and
and 1 if `other` is in front of `self.p1`."""
if self.p1 == other:
return 0
rel_pos = other - self.p1
d = self.direction
if d.dot(rel_pos) > 0:
return 1
return -1
@property
def ambient_dimension(self):
"""A property method that returns the dimension of LinearEntity
object.
Parameters
==========
p1 : LinearEntity
Returns
=======
dimension : integer
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> l1 = Line(p1, p2)
>>> l1.ambient_dimension
2
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0, 0), Point(1, 1, 1)
>>> l1 = Line(p1, p2)
>>> l1.ambient_dimension
3
"""
return len(self.p1)
def angle_between(l1, l2):
"""Return the non-reflex angle formed by rays emanating from
the origin with directions the same as the direction vectors
of the linear entities.
Parameters
==========
l1 : LinearEntity
l2 : LinearEntity
Returns
=======
angle : angle in radians
Notes
=====
From the dot product of vectors v1 and v2 it is known that:
``dot(v1, v2) = |v1|*|v2|*cos(A)``
where A is the angle formed between the two vectors. We can
get the directional vectors of the two lines and readily
find the angle between the two using the above formula.
See Also
========
is_perpendicular, Ray2D.closing_angle
Examples
========
>>> from sympy import Point, Line, pi
>>> e = Line((0, 0), (1, 0))
>>> ne = Line((0, 0), (1, 1))
>>> sw = Line((1, 1), (0, 0))
>>> ne.angle_between(e)
pi/4
>>> sw.angle_between(e)
3*pi/4
To obtain the non-obtuse angle at the intersection of lines, use
the ``smallest_angle_between`` method:
>>> sw.smallest_angle_between(e)
pi/4
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.angle_between(l2)
acos(-sqrt(2)/3)
>>> l1.smallest_angle_between(l2)
acos(sqrt(2)/3)
"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')
v1, v2 = l1.direction, l2.direction
return acos(v1.dot(v2)/(abs(v1)*abs(v2)))
def smallest_angle_between(l1, l2):
"""Return the smallest angle formed at the intersection of the
lines containing the linear entities.
Parameters
==========
l1 : LinearEntity
l2 : LinearEntity
Returns
=======
angle : angle in radians
See Also
========
angle_between, is_perpendicular, Ray2D.closing_angle
Examples
========
>>> from sympy import Point, Line, pi
>>> p1, p2, p3 = Point(0, 0), Point(0, 4), Point(2, -2)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.smallest_angle_between(l2)
pi/4
See Also
========
angle_between, Ray2D.closing_angle
"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')
v1, v2 = l1.direction, l2.direction
return acos(abs(v1.dot(v2))/(abs(v1)*abs(v2)))
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the Line.
Parameters
==========
parameter : str, optional
The name of the parameter which will be used for the parametric
point. The default value is 't'. When this parameter is 0, the
first point used to define the line will be returned, and when
it is 1 the second point will be returned.
Returns
=======
point : Point
Raises
======
ValueError
When ``parameter`` already appears in the Line's definition.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.arbitrary_point()
Point2D(4*t + 1, 3*t)
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 1)
>>> l1 = Line3D(p1, p2)
>>> l1.arbitrary_point()
Point3D(4*t + 1, 3*t, t)
"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(filldedent('''
Symbol %s already appears in object
and cannot be used as a parameter.
''' % t.name))
# multiply on the right so the variable gets
# combined with the coordinates of the point
return self.p1 + (self.p2 - self.p1)*t
@staticmethod
def are_concurrent(*lines):
"""Is a sequence of linear entities concurrent?
Two or more linear entities are concurrent if they all
intersect at a single point.
Parameters
==========
lines : a sequence of linear entities.
Returns
=======
True : if the set of linear entities intersect in one point
False : otherwise.
See Also
========
sympy.geometry.util.intersection
Examples
========
>>> from sympy import Point, Line, Line3D
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> p3, p4 = Point(-2, -2), Point(0, 2)
>>> l1, l2, l3 = Line(p1, p2), Line(p1, p3), Line(p1, p4)
>>> Line.are_concurrent(l1, l2, l3)
True
>>> l4 = Line(p2, p3)
>>> Line.are_concurrent(l2, l3, l4)
False
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 5, 2)
>>> p3, p4 = Point3D(-2, -2, -2), Point3D(0, 2, 1)
>>> l1, l2, l3 = Line3D(p1, p2), Line3D(p1, p3), Line3D(p1, p4)
>>> Line3D.are_concurrent(l1, l2, l3)
True
>>> l4 = Line3D(p2, p3)
>>> Line3D.are_concurrent(l2, l3, l4)
False
"""
common_points = Intersection(*lines)
if common_points.is_FiniteSet and len(common_points) == 1:
return True
return False
def contains(self, other):
"""Subclasses should implement this method and should return
True if other is on the boundaries of self;
False if not on the boundaries of self;
None if a determination cannot be made."""
raise NotImplementedError()
@property
def direction(self):
"""The direction vector of the LinearEntity.
Returns
=======
p : a Point; the ray from the origin to this point is the
direction of `self`
Examples
========
>>> from sympy.geometry import Line
>>> a, b = (1, 1), (1, 3)
>>> Line(a, b).direction
Point2D(0, 2)
>>> Line(b, a).direction
Point2D(0, -2)
This can be reported so the distance from the origin is 1:
>>> Line(b, a).direction.unit
Point2D(0, -1)
See Also
========
sympy.geometry.point.Point.unit
"""
return self.p2 - self.p1
def intersection(self, other):
"""The intersection with another geometrical entity.
Parameters
==========
o : Point or LinearEntity
Returns
=======
intersection : list of geometrical entities
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(7, 7)
>>> l1 = Line(p1, p2)
>>> l1.intersection(p3)
[Point2D(7, 7)]
>>> p4, p5 = Point(5, 0), Point(0, 3)
>>> l2 = Line(p4, p5)
>>> l1.intersection(l2)
[Point2D(15/8, 15/8)]
>>> p6, p7 = Point(0, 5), Point(2, 6)
>>> s1 = Segment(p6, p7)
>>> l1.intersection(s1)
[]
>>> from sympy import Point3D, Line3D, Segment3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(7, 7, 7)
>>> l1 = Line3D(p1, p2)
>>> l1.intersection(p3)
[Point3D(7, 7, 7)]
>>> l1 = Line3D(Point3D(4,19,12), Point3D(5,25,17))
>>> l2 = Line3D(Point3D(-3, -15, -19), direction_ratio=[2,8,8])
>>> l1.intersection(l2)
[Point3D(1, 1, -3)]
>>> p6, p7 = Point3D(0, 5, 2), Point3D(2, 6, 3)
>>> s1 = Segment3D(p6, p7)
>>> l1.intersection(s1)
[]
"""
def intersect_parallel_rays(ray1, ray2):
if ray1.direction.dot(ray2.direction) > 0:
# rays point in the same direction
# so return the one that is "in front"
return [ray2] if ray1._span_test(ray2.p1) >= 0 else [ray1]
else:
# rays point in opposite directions
st = ray1._span_test(ray2.p1)
if st < 0:
return []
elif st == 0:
return [ray2.p1]
return [Segment(ray1.p1, ray2.p1)]
def intersect_parallel_ray_and_segment(ray, seg):
st1, st2 = ray._span_test(seg.p1), ray._span_test(seg.p2)
if st1 < 0 and st2 < 0:
return []
elif st1 >= 0 and st2 >= 0:
return [seg]
elif st1 >= 0: # st2 < 0:
return [Segment(ray.p1, seg.p1)]
elif st2 > 0: # st1 <= 0
return [Segment(ray.p1, seg.p2)]
def intersect_parallel_segments(seg1, seg2):
if seg1.contains(seg2):
return [seg2]
if seg2.contains(seg1):
return [seg1]
# direct the segments so they're oriented the same way
if seg1.direction.dot(seg2.direction) < 0:
seg2 = Segment(seg2.p2, seg2.p1)
# order the segments so seg1 is "behind" seg2
if seg1._span_test(seg2.p1) < 0:
seg1, seg2 = seg2, seg1
if seg2._span_test(seg1.p2) < 0:
return []
return [Segment(seg2.p1, seg1.p2)]
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if other.is_Point:
if self.contains(other):
return [other]
else:
return []
elif isinstance(other, LinearEntity):
# break into cases based on whether
# the lines are parallel, non-parallel intersecting, or skew
pts = Point._normalize_dimension(self.p1, self.p2, other.p1, other.p2)
rank = Point.affine_rank(*pts)
if rank == 1:
# we're collinear
if isinstance(self, Line):
return [other]
if isinstance(other, Line):
return [self]
if isinstance(self, Ray) and isinstance(other, Ray):
return intersect_parallel_rays(self, other)
if isinstance(self, Ray) and isinstance(other, Segment):
return intersect_parallel_ray_and_segment(self, other)
if isinstance(self, Segment) and isinstance(other, Ray):
return intersect_parallel_ray_and_segment(other, self)
if isinstance(self, Segment) and isinstance(other, Segment):
return intersect_parallel_segments(self, other)
elif rank == 2:
# we're in the same plane
l1 = Line(*pts[:2])
l2 = Line(*pts[2:])
# check to see if we're parallel. If we are, we can't
# be intersecting, since the collinear case was already
# handled
if l1.direction.is_scalar_multiple(l2.direction):
return []
# find the intersection as if everything were lines
# by solving the equation t*d + p1 == s*d' + p1'
m = Matrix([l1.direction, -l2.direction]).transpose()
v = Matrix([l2.p1 - l1.p1]).transpose()
# we cannot use m.solve(v) because that only works for square matrices
m_rref, pivots = m.col_insert(2, v).rref(simplify=True)
# rank == 2 ensures we have 2 pivots, but let's check anyway
if len(pivots) != 2:
raise GeometryError("Failed when solving Mx=b when M={} and b={}".format(m, v))
coeff = m_rref[0, 2]
line_intersection = l1.direction*coeff + self.p1
# if we're both lines, we can skip a containment check
if isinstance(self, Line) and isinstance(other, Line):
return [line_intersection]
if self.contains(line_intersection) and other.contains(line_intersection):
return [line_intersection]
return []
else:
# we're skew
return []
return other.intersection(self)
def is_parallel(l1, l2):
"""Are two linear entities parallel?
Parameters
==========
l1 : LinearEntity
l2 : LinearEntity
Returns
=======
True : if l1 and l2 are parallel,
False : otherwise.
See Also
========
coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> p3, p4 = Point(3, 4), Point(6, 7)
>>> l1, l2 = Line(p1, p2), Line(p3, p4)
>>> Line.is_parallel(l1, l2)
True
>>> p5 = Point(6, 6)
>>> l3 = Line(p3, p5)
>>> Line.is_parallel(l1, l3)
False
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(3, 4, 5)
>>> p3, p4 = Point3D(2, 1, 1), Point3D(8, 9, 11)
>>> l1, l2 = Line3D(p1, p2), Line3D(p3, p4)
>>> Line3D.is_parallel(l1, l2)
True
>>> p5 = Point3D(6, 6, 6)
>>> l3 = Line3D(p3, p5)
>>> Line3D.is_parallel(l1, l3)
False
"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')
return l1.direction.is_scalar_multiple(l2.direction)
def is_perpendicular(l1, l2):
"""Are two linear entities perpendicular?
Parameters
==========
l1 : LinearEntity
l2 : LinearEntity
Returns
=======
True : if l1 and l2 are perpendicular,
False : otherwise.
See Also
========
coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 1)
>>> l1, l2 = Line(p1, p2), Line(p1, p3)
>>> l1.is_perpendicular(l2)
True
>>> p4 = Point(5, 3)
>>> l3 = Line(p1, p4)
>>> l1.is_perpendicular(l3)
False
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(-1, 2, 0)
>>> l1, l2 = Line3D(p1, p2), Line3D(p2, p3)
>>> l1.is_perpendicular(l2)
False
>>> p4 = Point3D(5, 3, 7)
>>> l3 = Line3D(p1, p4)
>>> l1.is_perpendicular(l3)
False
"""
if not isinstance(l1, LinearEntity) and not isinstance(l2, LinearEntity):
raise TypeError('Must pass only LinearEntity objects')
return S.Zero.equals(l1.direction.dot(l2.direction))
def is_similar(self, other):
"""
Return True if self and other are contained in the same line.
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 1), Point(3, 4), Point(2, 3)
>>> l1 = Line(p1, p2)
>>> l2 = Line(p1, p3)
>>> l1.is_similar(l2)
True
"""
l = Line(self.p1, self.p2)
return l.contains(other)
@property
def length(self):
"""
The length of the line.
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> l1 = Line(p1, p2)
>>> l1.length
oo
"""
return S.Infinity
@property
def p1(self):
"""The first defining point of a linear entity.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p1
Point2D(0, 0)
"""
return self.args[0]
@property
def p2(self):
"""The second defining point of a linear entity.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.p2
Point2D(5, 3)
"""
return self.args[1]
def parallel_line(self, p):
"""Create a new Line parallel to this linear entity which passes
through the point `p`.
Parameters
==========
p : Point
Returns
=======
line : Line
See Also
========
is_parallel
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.parallel_line(p3)
>>> p3 in l2
True
>>> l1.is_parallel(l2)
True
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> l2 = l1.parallel_line(p3)
>>> p3 in l2
True
>>> l1.is_parallel(l2)
True
"""
p = Point(p, dim=self.ambient_dimension)
return Line(p, p + self.direction)
def perpendicular_line(self, p):
"""Create a new Line perpendicular to this linear entity which passes
through the point `p`.
Parameters
==========
p : Point
Returns
=======
line : Line
See Also
========
is_perpendicular, perpendicular_segment
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(2, 3, 4), Point3D(-2, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
"""
p = Point(p, dim=self.ambient_dimension)
if p in self:
p = p + self.direction.orthogonal_direction
return Line(p, self.projection(p))
def perpendicular_segment(self, p):
"""Create a perpendicular line segment from `p` to this line.
The enpoints of the segment are ``p`` and the closest point in
the line containing self. (If self is not a line, the point might
not be in self.)
Parameters
==========
p : Point
Returns
=======
segment : Segment
Notes
=====
Returns `p` itself if `p` is on this linear entity.
See Also
========
perpendicular_line
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 2)
>>> l1 = Line(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point(4, 0))
Segment2D(Point2D(4, 0), Point2D(2, 2))
>>> from sympy import Point3D, Line3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, 2, 0)
>>> l1 = Line3D(p1, p2)
>>> s1 = l1.perpendicular_segment(p3)
>>> l1.is_perpendicular(s1)
True
>>> p3 in s1
True
>>> l1.perpendicular_segment(Point3D(4, 0, 0))
Segment3D(Point3D(4, 0, 0), Point3D(4/3, 4/3, 4/3))
"""
p = Point(p, dim=self.ambient_dimension)
if p in self:
return p
l = self.perpendicular_line(p)
# The intersection should be unique, so unpack the singleton
p2, = Intersection(Line(self.p1, self.p2), l)
return Segment(p, p2)
@property
def points(self):
"""The two points used to define this linear entity.
Returns
=======
points : tuple of Points
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 11)
>>> l1 = Line(p1, p2)
>>> l1.points
(Point2D(0, 0), Point2D(5, 11))
"""
return (self.p1, self.p2)
def projection(self, other):
"""Project a point, line, ray, or segment onto this linear entity.
Parameters
==========
other : Point or LinearEntity (Line, Ray, Segment)
Returns
=======
projection : Point or LinearEntity (Line, Ray, Segment)
The return type matches the type of the parameter ``other``.
Raises
======
GeometryError
When method is unable to perform projection.
Notes
=====
A projection involves taking the two points that define
the linear entity and projecting those points onto a
Line and then reforming the linear entity using these
projections.
A point P is projected onto a line L by finding the point
on L that is closest to P. This point is the intersection
of L and the line perpendicular to L that passes through P.
See Also
========
sympy.geometry.point.Point, perpendicular_line
Examples
========
>>> from sympy import Point, Line, Segment, Rational
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(Rational(1, 2), 0)
>>> l1 = Line(p1, p2)
>>> l1.projection(p3)
Point2D(1/4, 1/4)
>>> p4, p5 = Point(10, 0), Point(12, 1)
>>> s1 = Segment(p4, p5)
>>> l1.projection(s1)
Segment2D(Point2D(5, 5), Point2D(13/2, 13/2))
>>> p1, p2, p3 = Point(0, 0, 1), Point(1, 1, 2), Point(2, 0, 1)
>>> l1 = Line(p1, p2)
>>> l1.projection(p3)
Point3D(2/3, 2/3, 5/3)
>>> p4, p5 = Point(10, 0, 1), Point(12, 1, 3)
>>> s1 = Segment(p4, p5)
>>> l1.projection(s1)
Segment3D(Point3D(10/3, 10/3, 13/3), Point3D(5, 5, 6))
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
def proj_point(p):
return Point.project(p - self.p1, self.direction) + self.p1
if isinstance(other, Point):
return proj_point(other)
elif isinstance(other, LinearEntity):
p1, p2 = proj_point(other.p1), proj_point(other.p2)
# test to see if we're degenerate
if p1 == p2:
return p1
projected = other.__class__(p1, p2)
projected = Intersection(self, projected)
# if we happen to have intersected in only a point, return that
if projected.is_FiniteSet and len(projected) == 1:
# projected is a set of size 1, so unpack it in `a`
a, = projected
return a
# order args so projection is in the same direction as self
if self.direction.dot(projected.direction) < 0:
p1, p2 = projected.args
projected = projected.func(p2, p1)
return projected
raise GeometryError(
"Do not know how to project %s onto %s" % (other, self))
def random_point(self, seed=None):
"""A random point on a LinearEntity.
Returns
=======
point : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Line, Ray, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> line = Line(p1, p2)
>>> r = line.random_point(seed=42) # seed value is optional
>>> r.n(3)
Point2D(-0.72, -0.432)
>>> r in line
True
>>> Ray(p1, p2).random_point(seed=42).n(3)
Point2D(0.72, 0.432)
>>> Segment(p1, p2).random_point(seed=42).n(3)
Point2D(3.2, 1.92)
"""
import random
if seed is not None:
rng = random.Random(seed)
else:
rng = random
t = Dummy()
pt = self.arbitrary_point(t)
if isinstance(self, Ray):
v = abs(rng.gauss(0, 1))
elif isinstance(self, Segment):
v = rng.random()
elif isinstance(self, Line):
v = rng.gauss(0, 1)
else:
raise NotImplementedError('unhandled line type')
return pt.subs(t, Rational(v))
class Line(LinearEntity):
"""An infinite line in space.
A 2D line is declared with two distinct points, point and slope, or
an equation. A 3D line may be defined with a point and a direction ratio.
Parameters
==========
p1 : Point
p2 : Point
slope : sympy expression
direction_ratio : list
equation : equation of a line
Notes
=====
`Line` will automatically subclass to `Line2D` or `Line3D` based
on the dimension of `p1`. The `slope` argument is only relevant
for `Line2D` and the `direction_ratio` argument is only relevant
for `Line3D`.
See Also
========
sympy.geometry.point.Point
sympy.geometry.line.Line2D
sympy.geometry.line.Line3D
Examples
========
>>> from sympy import Point, Eq
>>> from sympy.geometry import Line, Segment
>>> from sympy.abc import x, y, a, b
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line2D(Point2D(2, 3), Point2D(3, 5))
>>> L.points
(Point2D(2, 3), Point2D(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)
Instantiate with keyword ``slope``:
>>> Line(Point(0, 0), slope=0)
Line2D(Point2D(0, 0), Point2D(1, 0))
Instantiate with another linear object
>>> s = Segment((0, 0), (0, 1))
>>> Line(s).equation()
x
The line corresponding to an equation in the for `ax + by + c = 0`,
can be entered:
>>> Line(3*x + y + 18)
Line2D(Point2D(0, -18), Point2D(1, -21))
If `x` or `y` has a different name, then they can be specified, too,
as a string (to match the name) or symbol:
>>> Line(Eq(3*a + b, -18), x='a', y=b)
Line2D(Point2D(0, -18), Point2D(1, -21))
"""
def __new__(cls, *args, **kwargs):
from sympy.geometry.util import find
if len(args) == 1 and isinstance(args[0], Expr):
x = kwargs.get('x', 'x')
y = kwargs.get('y', 'y')
equation = args[0]
if isinstance(equation, Eq):
equation = equation.lhs - equation.rhs
xin, yin = x, y
x = find(x, equation) or Dummy()
y = find(y, equation) or Dummy()
a, b, c = linear_coeffs(equation, x, y)
if b:
return Line((0, -c/b), slope=-a/b)
if a:
return Line((-c/a, 0), slope=oo)
raise ValueError('neither %s nor %s were found in the equation' % (xin, yin))
else:
if len(args) > 0:
p1 = args[0]
if len(args) > 1:
p2 = args[1]
else:
p2=None
if isinstance(p1, LinearEntity):
if p2:
raise ValueError('If p1 is a LinearEntity, p2 must be None.')
dim = len(p1.p1)
else:
p1 = Point(p1)
dim = len(p1)
if p2 is not None or isinstance(p2, Point) and p2.ambient_dimension != dim:
p2 = Point(p2)
if dim == 2:
return Line2D(p1, p2, **kwargs)
elif dim == 3:
return Line3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, p2, **kwargs)
def contains(self, other):
"""
Return True if `other` is on this Line, or False otherwise.
Examples
========
>>> from sympy import Line,Point
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> l = Line(p1, p2)
>>> l.contains(p1)
True
>>> l.contains((0, 1))
True
>>> l.contains((0, 0))
False
>>> a = (0, 0, 0)
>>> b = (1, 1, 1)
>>> c = (2, 2, 2)
>>> l1 = Line(a, b)
>>> l2 = Line(b, a)
>>> l1 == l2
False
>>> l1 in l2
True
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
return Point.is_collinear(other, self.p1, self.p2)
if isinstance(other, LinearEntity):
return Point.is_collinear(self.p1, self.p2, other.p1, other.p2)
return False
def distance(self, other):
"""
Finds the shortest distance between a line and a point.
Raises
======
NotImplementedError is raised if `other` is not a Point
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> s = Line(p1, p2)
>>> s.distance(Point(-1, 1))
sqrt(2)
>>> s.distance((-1, 2))
3*sqrt(2)/2
>>> p1, p2 = Point(0, 0, 0), Point(1, 1, 1)
>>> s = Line(p1, p2)
>>> s.distance(Point(-1, 1, 1))
2*sqrt(6)/3
>>> s.distance((-1, 1, 1))
2*sqrt(6)/3
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if self.contains(other):
return S.Zero
return self.perpendicular_segment(other).length
@deprecated(useinstead="equals", issue=12860, deprecated_since_version="1.0")
def equal(self, other):
return self.equals(other)
def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
if not isinstance(other, Line):
return False
return Point.is_collinear(self.p1, other.p1, self.p2, other.p2)
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of line. Gives
values that will produce a line that is +/- 5 units long (where a
unit is the distance between the two points that define the line).
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.plot_interval()
[t, -5, 5]
"""
t = _symbol(parameter, real=True)
return [t, -5, 5]
class Ray(LinearEntity):
"""A Ray is a semi-line in the space with a source point and a direction.
Parameters
==========
p1 : Point
The source of the Ray
p2 : Point or radian value
This point determines the direction in which the Ray propagates.
If given as an angle it is interpreted in radians with the positive
direction being ccw.
Attributes
==========
source
See Also
========
sympy.geometry.line.Ray2D
sympy.geometry.line.Ray3D
sympy.geometry.point.Point
sympy.geometry.line.Line
Notes
=====
`Ray` will automatically subclass to `Ray2D` or `Ray3D` based on the
dimension of `p1`.
Examples
========
>>> from sympy import Point, pi
>>> from sympy.geometry import Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray2D(Point2D(2, 3), Point2D(3, 5))
>>> r.points
(Point2D(2, 3), Point2D(3, 5))
>>> r.source
Point2D(2, 3)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.slope
2
>>> Ray(Point(0, 0), angle=pi/4).slope
1
"""
def __new__(cls, p1, p2=None, **kwargs):
p1 = Point(p1)
if p2 is not None:
p1, p2 = Point._normalize_dimension(p1, Point(p2))
dim = len(p1)
if dim == 2:
return Ray2D(p1, p2, **kwargs)
elif dim == 3:
return Ray3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, *pts, **kwargs)
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" '
'marker-start="url(#markerCircle)" marker-end="url(#markerArrow)"/>'
).format(2. * scale_factor, path, fill_color)
def contains(self, other):
"""
Is other GeometryEntity contained in this Ray?
Examples
========
>>> from sympy import Ray,Point,Segment
>>> p1, p2 = Point(0, 0), Point(4, 4)
>>> r = Ray(p1, p2)
>>> r.contains(p1)
True
>>> r.contains((1, 1))
True
>>> r.contains((1, 3))
False
>>> s = Segment((1, 1), (2, 2))
>>> r.contains(s)
True
>>> s = Segment((1, 2), (2, 5))
>>> r.contains(s)
False
>>> r1 = Ray((2, 2), (3, 3))
>>> r.contains(r1)
True
>>> r1 = Ray((2, 2), (3, 5))
>>> r.contains(r1)
False
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
if Point.is_collinear(self.p1, self.p2, other):
# if we're in the direction of the ray, our
# direction vector dot the ray's direction vector
# should be non-negative
return bool((self.p2 - self.p1).dot(other - self.p1) >= S.Zero)
return False
elif isinstance(other, Ray):
if Point.is_collinear(self.p1, self.p2, other.p1, other.p2):
return bool((self.p2 - self.p1).dot(other.p2 - other.p1) > S.Zero)
return False
elif isinstance(other, Segment):
return other.p1 in self and other.p2 in self
# No other known entity can be contained in a Ray
return False
def distance(self, other):
"""
Finds the shortest distance between the ray and a point.
Raises
======
NotImplementedError is raised if `other` is not a Point
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(1, 1)
>>> s = Ray(p1, p2)
>>> s.distance(Point(-1, -1))
sqrt(2)
>>> s.distance((-1, 2))
3*sqrt(2)/2
>>> p1, p2 = Point(0, 0, 0), Point(1, 1, 2)
>>> s = Ray(p1, p2)
>>> s
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 2))
>>> s.distance(Point(-1, -1, 2))
4*sqrt(3)/3
>>> s.distance((-1, -1, 2))
4*sqrt(3)/3
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if self.contains(other):
return S.Zero
proj = Line(self.p1, self.p2).projection(other)
if self.contains(proj):
return abs(other - proj)
else:
return abs(other - self.source)
def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
if not isinstance(other, Ray):
return False
return self.source == other.source and other.p2 in self
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ray. Gives
values that will produce a ray that is 10 units long (where a unit is
the distance between the two points that define the ray).
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Ray, pi
>>> r = Ray((0, 0), angle=pi/4)
>>> r.plot_interval()
[t, 0, 10]
"""
t = _symbol(parameter, real=True)
return [t, 0, 10]
@property
def source(self):
"""The point from which the ray emanates.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2 = Point(0, 0), Point(4, 1)
>>> r1 = Ray(p1, p2)
>>> r1.source
Point2D(0, 0)
>>> p1, p2 = Point(0, 0, 0), Point(4, 1, 5)
>>> r1 = Ray(p2, p1)
>>> r1.source
Point3D(4, 1, 5)
"""
return self.p1
class Segment(LinearEntity):
"""A line segment in space.
Parameters
==========
p1 : Point
p2 : Point
Attributes
==========
length : number or sympy expression
midpoint : Point
See Also
========
sympy.geometry.line.Segment2D
sympy.geometry.line.Segment3D
sympy.geometry.point.Point
sympy.geometry.line.Line
Notes
=====
If 2D or 3D points are used to define `Segment`, it will
be automatically subclassed to `Segment2D` or `Segment3D`.
Examples
========
>>> from sympy import Point
>>> from sympy.geometry import Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment2D(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1))
>>> s.points
(Point2D(4, 3), Point2D(1, 1))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)
>>> Segment((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment(Point(4, 3, 9), Point(1, 1, 7)); s
Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.points
(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)
"""
def __new__(cls, p1, p2, **kwargs):
p1, p2 = Point._normalize_dimension(Point(p1), Point(p2))
dim = len(p1)
if dim == 2:
return Segment2D(p1, p2, **kwargs)
elif dim == 3:
return Segment3D(p1, p2, **kwargs)
return LinearEntity.__new__(cls, p1, p2, **kwargs)
def contains(self, other):
"""
Is the other GeometryEntity contained within this Segment?
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> s = Segment(p1, p2)
>>> s2 = Segment(p2, p1)
>>> s.contains(s2)
True
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 1, 1), Point3D(3, 4, 5)
>>> s = Segment3D(p1, p2)
>>> s2 = Segment3D(p2, p1)
>>> s.contains(s2)
True
>>> s.contains((p1 + p2) / 2)
True
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
if Point.is_collinear(other, self.p1, self.p2):
d1, d2 = other - self.p1, other - self.p2
d = self.p2 - self.p1
# without the call to simplify, sympy cannot tell that an expression
# like (a+b)*(a/2+b/2) is always non-negative. If it cannot be
# determined, raise an Undecidable error
try:
# the triangle inequality says that |d1|+|d2| >= |d| and is strict
# only if other lies in the line segment
return bool(simplify(Eq(abs(d1) + abs(d2) - abs(d), 0)))
except TypeError:
raise Undecidable("Cannot determine if {} is in {}".format(other, self))
if isinstance(other, Segment):
return other.p1 in self and other.p2 in self
return False
def equals(self, other):
"""Returns True if self and other are the same mathematical entities"""
return isinstance(other, self.func) and list(
ordered(self.args)) == list(ordered(other.args))
def distance(self, other):
"""
Finds the shortest distance between a line segment and a point.
Raises
======
NotImplementedError is raised if `other` is not a Point
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 1), Point(3, 4)
>>> s = Segment(p1, p2)
>>> s.distance(Point(10, 15))
sqrt(170)
>>> s.distance((0, 12))
sqrt(73)
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 3), Point3D(1, 1, 4)
>>> s = Segment3D(p1, p2)
>>> s.distance(Point3D(10, 15, 12))
sqrt(341)
>>> s.distance((10, 15, 12))
sqrt(341)
"""
if not isinstance(other, GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if isinstance(other, Point):
vp1 = other - self.p1
vp2 = other - self.p2
dot_prod_sign_1 = self.direction.dot(vp1) >= 0
dot_prod_sign_2 = self.direction.dot(vp2) <= 0
if dot_prod_sign_1 and dot_prod_sign_2:
return Line(self.p1, self.p2).distance(other)
if dot_prod_sign_1 and not dot_prod_sign_2:
return abs(vp2)
if not dot_prod_sign_1 and dot_prod_sign_2:
return abs(vp1)
raise NotImplementedError()
@property
def length(self):
"""The length of the line segment.
See Also
========
sympy.geometry.point.Point.distance
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.length
5
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.length
sqrt(34)
"""
return Point.distance(self.p1, self.p2)
@property
def midpoint(self):
"""The midpoint of the line segment.
See Also
========
sympy.geometry.point.Point.midpoint
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(4, 3)
>>> s1 = Segment(p1, p2)
>>> s1.midpoint
Point2D(2, 3/2)
>>> from sympy import Point3D, Segment3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(4, 3, 3)
>>> s1 = Segment3D(p1, p2)
>>> s1.midpoint
Point3D(2, 3/2, 3/2)
"""
return Point.midpoint(self.p1, self.p2)
def perpendicular_bisector(self, p=None):
"""The perpendicular bisector of this segment.
If no point is specified or the point specified is not on the
bisector then the bisector is returned as a Line. Otherwise a
Segment is returned that joins the point specified and the
intersection of the bisector and the segment.
Parameters
==========
p : Point
Returns
=======
bisector : Line or Segment
See Also
========
LinearEntity.perpendicular_segment
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2, p3 = Point(0, 0), Point(6, 6), Point(5, 1)
>>> s1 = Segment(p1, p2)
>>> s1.perpendicular_bisector()
Line2D(Point2D(3, 3), Point2D(-3, 9))
>>> s1.perpendicular_bisector(p3)
Segment2D(Point2D(5, 1), Point2D(3, 3))
"""
l = self.perpendicular_line(self.midpoint)
if p is not None:
p2 = Point(p, dim=self.ambient_dimension)
if p2 in l:
return Segment(p2, self.midpoint)
return l
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Segment gives
values that will produce the full segment in a plot.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Segment
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> s1 = Segment(p1, p2)
>>> s1.plot_interval()
[t, 0, 1]
"""
t = _symbol(parameter, real=True)
return [t, 0, 1]
class LinearEntity2D(LinearEntity):
"""A base class for all linear entities (line, ray and segment)
in a 2-dimensional Euclidean space.
Attributes
==========
p1
p2
coefficients
slope
points
Notes
=====
This is an abstract class and is not meant to be instantiated.
See Also
========
sympy.geometry.entity.GeometryEntity
"""
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
verts = self.points
xs = [p.x for p in verts]
ys = [p.y for p in verts]
return (min(xs), min(ys), max(xs), max(ys))
def perpendicular_line(self, p):
"""Create a new Line perpendicular to this linear entity which passes
through the point `p`.
Parameters
==========
p : Point
Returns
=======
line : Line
See Also
========
is_perpendicular, perpendicular_segment
Examples
========
>>> from sympy import Point, Line
>>> p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
>>> l1 = Line(p1, p2)
>>> l2 = l1.perpendicular_line(p3)
>>> p3 in l2
True
>>> l1.is_perpendicular(l2)
True
"""
p = Point(p, dim=self.ambient_dimension)
# any two lines in R^2 intersect, so blindly making
# a line through p in an orthogonal direction will work
return Line(p, p + self.direction.orthogonal_direction)
@property
def slope(self):
"""The slope of this linear entity, or infinity if vertical.
Returns
=======
slope : number or sympy expression
See Also
========
coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(0, 0), Point(3, 5)
>>> l1 = Line(p1, p2)
>>> l1.slope
5/3
>>> p3 = Point(0, 4)
>>> l2 = Line(p1, p3)
>>> l2.slope
oo
"""
d1, d2 = (self.p1 - self.p2).args
if d1 == 0:
return S.Infinity
return simplify(d2/d1)
class Line2D(LinearEntity2D, Line):
"""An infinite line in space 2D.
A line is declared with two distinct points or a point and slope
as defined using keyword `slope`.
Parameters
==========
p1 : Point
pt : Point
slope : sympy expression
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point
>>> from sympy.abc import L
>>> from sympy.geometry import Line, Segment
>>> L = Line(Point(2,3), Point(3,5))
>>> L
Line2D(Point2D(2, 3), Point2D(3, 5))
>>> L.points
(Point2D(2, 3), Point2D(3, 5))
>>> L.equation()
-2*x + y + 1
>>> L.coefficients
(-2, 1, 1)
Instantiate with keyword ``slope``:
>>> Line(Point(0, 0), slope=0)
Line2D(Point2D(0, 0), Point2D(1, 0))
Instantiate with another linear object
>>> s = Segment((0, 0), (0, 1))
>>> Line(s).equation()
x
"""
def __new__(cls, p1, pt=None, slope=None, **kwargs):
if isinstance(p1, LinearEntity):
if pt is not None:
raise ValueError('When p1 is a LinearEntity, pt should be None')
p1, pt = Point._normalize_dimension(*p1.args, dim=2)
else:
p1 = Point(p1, dim=2)
if pt is not None and slope is None:
try:
p2 = Point(pt, dim=2)
except (NotImplementedError, TypeError, ValueError):
raise ValueError(filldedent('''
The 2nd argument was not a valid Point.
If it was a slope, enter it with keyword "slope".
'''))
elif slope is not None and pt is None:
slope = sympify(slope)
if slope.is_finite is False:
# when infinite slope, don't change x
dx = 0
dy = 1
else:
# go over 1 up slope
dx = 1
dy = slope
# XXX avoiding simplification by adding to coords directly
p2 = Point(p1.x + dx, p1.y + dy, evaluate=False)
else:
raise ValueError('A 2nd Point or keyword "slope" must be used.')
return LinearEntity2D.__new__(cls, p1, p2, **kwargs)
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" '
'marker-start="url(#markerReverseArrow)" marker-end="url(#markerArrow)"/>'
).format(2. * scale_factor, path, fill_color)
@property
def coefficients(self):
"""The coefficients (`a`, `b`, `c`) for `ax + by + c = 0`.
See Also
========
sympy.geometry.line.Line.equation
Examples
========
>>> from sympy import Point, Line
>>> from sympy.abc import x, y
>>> p1, p2 = Point(0, 0), Point(5, 3)
>>> l = Line(p1, p2)
>>> l.coefficients
(-3, 5, 0)
>>> p3 = Point(x, y)
>>> l2 = Line(p1, p3)
>>> l2.coefficients
(-y, x, 0)
"""
p1, p2 = self.points
if p1.x == p2.x:
return (S.One, S.Zero, -p1.x)
elif p1.y == p2.y:
return (S.Zero, S.One, -p1.y)
return tuple([simplify(i) for i in
(self.p1.y - self.p2.y,
self.p2.x - self.p1.x,
self.p1.x*self.p2.y - self.p1.y*self.p2.x)])
def equation(self, x='x', y='y'):
"""The equation of the line: ax + by + c.
Parameters
==========
x : str, optional
The name to use for the x-axis, default value is 'x'.
y : str, optional
The name to use for the y-axis, default value is 'y'.
Returns
=======
equation : sympy expression
See Also
========
LinearEntity.coefficients
Examples
========
>>> from sympy import Point, Line
>>> p1, p2 = Point(1, 0), Point(5, 3)
>>> l1 = Line(p1, p2)
>>> l1.equation()
-3*x + 4*y + 3
"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
p1, p2 = self.points
if p1.x == p2.x:
return x - p1.x
elif p1.y == p2.y:
return y - p1.y
a, b, c = self.coefficients
return a*x + b*y + c
class Ray2D(LinearEntity2D, Ray):
"""
A Ray is a semi-line in the space with a source point and a direction.
Parameters
==========
p1 : Point
The source of the Ray
p2 : Point or radian value
This point determines the direction in which the Ray propagates.
If given as an angle it is interpreted in radians with the positive
direction being ccw.
Attributes
==========
source
xdirection
ydirection
See Also
========
sympy.geometry.point.Point, Line
Examples
========
>>> from sympy import Point, pi
>>> from sympy.geometry import Ray
>>> r = Ray(Point(2, 3), Point(3, 5))
>>> r
Ray2D(Point2D(2, 3), Point2D(3, 5))
>>> r.points
(Point2D(2, 3), Point2D(3, 5))
>>> r.source
Point2D(2, 3)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.slope
2
>>> Ray(Point(0, 0), angle=pi/4).slope
1
"""
def __new__(cls, p1, pt=None, angle=None, **kwargs):
p1 = Point(p1, dim=2)
if pt is not None and angle is None:
try:
p2 = Point(pt, dim=2)
except (NotImplementedError, TypeError, ValueError):
from sympy.utilities.misc import filldedent
raise ValueError(filldedent('''
The 2nd argument was not a valid Point; if
it was meant to be an angle it should be
given with keyword "angle".'''))
if p1 == p2:
raise ValueError('A Ray requires two distinct points.')
elif angle is not None and pt is None:
# we need to know if the angle is an odd multiple of pi/2
c = pi_coeff(sympify(angle))
p2 = None
if c is not None:
if c.is_Rational:
if c.q == 2:
if c.p == 1:
p2 = p1 + Point(0, 1)
elif c.p == 3:
p2 = p1 + Point(0, -1)
elif c.q == 1:
if c.p == 0:
p2 = p1 + Point(1, 0)
elif c.p == 1:
p2 = p1 + Point(-1, 0)
if p2 is None:
c *= S.Pi
else:
c = angle % (2*S.Pi)
if not p2:
m = 2*c/S.Pi
left = And(1 < m, m < 3) # is it in quadrant 2 or 3?
x = Piecewise((-1, left), (Piecewise((0, Eq(m % 1, 0)), (1, True)), True))
y = Piecewise((-tan(c), left), (Piecewise((1, Eq(m, 1)), (-1, Eq(m, 3)), (tan(c), True)), True))
p2 = p1 + Point(x, y)
else:
raise ValueError('A 2nd point or keyword "angle" must be used.')
return LinearEntity2D.__new__(cls, p1, p2, **kwargs)
@property
def xdirection(self):
"""The x direction of the ray.
Positive infinity if the ray points in the positive x direction,
negative infinity if the ray points in the negative x direction,
or 0 if the ray is vertical.
See Also
========
ydirection
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, -1)
>>> r1, r2 = Ray(p1, p2), Ray(p1, p3)
>>> r1.xdirection
oo
>>> r2.xdirection
0
"""
if self.p1.x < self.p2.x:
return S.Infinity
elif self.p1.x == self.p2.x:
return S.Zero
else:
return S.NegativeInfinity
@property
def ydirection(self):
"""The y direction of the ray.
Positive infinity if the ray points in the positive y direction,
negative infinity if the ray points in the negative y direction,
or 0 if the ray is horizontal.
See Also
========
xdirection
Examples
========
>>> from sympy import Point, Ray
>>> p1, p2, p3 = Point(0, 0), Point(-1, -1), Point(-1, 0)
>>> r1, r2 = Ray(p1, p2), Ray(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0
"""
if self.p1.y < self.p2.y:
return S.Infinity
elif self.p1.y == self.p2.y:
return S.Zero
else:
return S.NegativeInfinity
def closing_angle(r1, r2):
"""Return the angle by which r2 must be rotated so it faces the same
direction as r1.
Parameters
==========
r1 : Ray2D
r2 : Ray2D
Returns
=======
angle : angle in radians (ccw angle is positive)
See Also
========
LinearEntity.angle_between
Examples
========
>>> from sympy import Ray, pi
>>> r1 = Ray((0, 0), (1, 0))
>>> r2 = r1.rotate(-pi/2)
>>> angle = r1.closing_angle(r2); angle
pi/2
>>> r2.rotate(angle).direction.unit == r1.direction.unit
True
>>> r2.closing_angle(r1)
-pi/2
"""
if not all(isinstance(r, Ray2D) for r in (r1, r2)):
# although the direction property is defined for
# all linear entities, only the Ray is truly a
# directed object
raise TypeError('Both arguments must be Ray2D objects.')
a1 = atan2(*list(reversed(r1.direction.args)))
a2 = atan2(*list(reversed(r2.direction.args)))
if a1*a2 < 0:
a1 = 2*S.Pi + a1 if a1 < 0 else a1
a2 = 2*S.Pi + a2 if a2 < 0 else a2
return a1 - a2
class Segment2D(LinearEntity2D, Segment):
"""A line segment in 2D space.
Parameters
==========
p1 : Point
p2 : Point
Attributes
==========
length : number or sympy expression
midpoint : Point
See Also
========
sympy.geometry.point.Point, Line
Examples
========
>>> from sympy import Point
>>> from sympy.geometry import Segment
>>> Segment((1, 0), (1, 1)) # tuples are interpreted as pts
Segment2D(Point2D(1, 0), Point2D(1, 1))
>>> s = Segment(Point(4, 3), Point(1, 1)); s
Segment2D(Point2D(4, 3), Point2D(1, 1))
>>> s.points
(Point2D(4, 3), Point2D(1, 1))
>>> s.slope
2/3
>>> s.length
sqrt(13)
>>> s.midpoint
Point2D(5/2, 2)
"""
def __new__(cls, p1, p2, **kwargs):
p1 = Point(p1, dim=2)
p2 = Point(p2, dim=2)
if p1 == p2:
return p1
return LinearEntity2D.__new__(cls, p1, p2, **kwargs)
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the LinearEntity.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
verts = (N(self.p1), N(self.p2))
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1}".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" />'
).format(2. * scale_factor, path, fill_color)
class LinearEntity3D(LinearEntity):
"""An base class for all linear entities (line, ray and segment)
in a 3-dimensional Euclidean space.
Attributes
==========
p1
p2
direction_ratio
direction_cosine
points
Notes
=====
This is a base class and is not meant to be instantiated.
"""
def __new__(cls, p1, p2, **kwargs):
p1 = Point3D(p1, dim=3)
p2 = Point3D(p2, dim=3)
if p1 == p2:
# if it makes sense to return a Point, handle in subclass
raise ValueError(
"%s.__new__ requires two unique Points." % cls.__name__)
return GeometryEntity.__new__(cls, p1, p2, **kwargs)
ambient_dimension = 3
@property
def direction_ratio(self):
"""The direction ratio of a given line in 3D.
See Also
========
sympy.geometry.line.Line.equation
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.direction_ratio
[5, 3, 1]
"""
p1, p2 = self.points
return p1.direction_ratio(p2)
@property
def direction_cosine(self):
"""The normalized direction ratio of a given line in 3D.
See Also
========
sympy.geometry.line.Line.equation
Examples
========
>>> from sympy import Point3D, Line3D
>>> p1, p2 = Point3D(0, 0, 0), Point3D(5, 3, 1)
>>> l = Line3D(p1, p2)
>>> l.direction_cosine
[sqrt(35)/7, 3*sqrt(35)/35, sqrt(35)/35]
>>> sum(i**2 for i in _)
1
"""
p1, p2 = self.points
return p1.direction_cosine(p2)
class Line3D(LinearEntity3D, Line):
"""An infinite 3D line in space.
A line is declared with two distinct points or a point and direction_ratio
as defined using keyword `direction_ratio`.
Parameters
==========
p1 : Point3D
pt : Point3D
direction_ratio : list
See Also
========
sympy.geometry.point.Point3D
sympy.geometry.line.Line
sympy.geometry.line.Line2D
Examples
========
>>> from sympy import Point3D
>>> from sympy.geometry import Line3D, Segment3D
>>> L = Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L
Line3D(Point3D(2, 3, 4), Point3D(3, 5, 1))
>>> L.points
(Point3D(2, 3, 4), Point3D(3, 5, 1))
"""
def __new__(cls, p1, pt=None, direction_ratio=[], **kwargs):
if isinstance(p1, LinearEntity3D):
if pt is not None:
raise ValueError('if p1 is a LinearEntity, pt must be None.')
p1, pt = p1.args
else:
p1 = Point(p1, dim=3)
if pt is not None and len(direction_ratio) == 0:
pt = Point(pt, dim=3)
elif len(direction_ratio) == 3 and pt is None:
pt = Point3D(p1.x + direction_ratio[0], p1.y + direction_ratio[1],
p1.z + direction_ratio[2])
else:
raise ValueError('A 2nd Point or keyword "direction_ratio" must '
'be used.')
return LinearEntity3D.__new__(cls, p1, pt, **kwargs)
def equation(self, x='x', y='y', z='z', k=None):
"""Return the equations that define the line in 3D.
Parameters
==========
x : str, optional
The name to use for the x-axis, default value is 'x'.
y : str, optional
The name to use for the y-axis, default value is 'y'.
z : str, optional
The name to use for the z-axis, default value is 'z'.
Returns
=======
equation : Tuple of simultaneous equations
Examples
========
>>> from sympy import Point3D, Line3D, solve
>>> from sympy.abc import x, y, z
>>> p1, p2 = Point3D(1, 0, 0), Point3D(5, 3, 0)
>>> l1 = Line3D(p1, p2)
>>> eq = l1.equation(x, y, z); eq
(-3*x + 4*y + 3, z)
>>> solve(eq.subs(z, 0), (x, y, z))
{x: 4*y/3 + 1}
"""
if k is not None:
SymPyDeprecationWarning(
feature="equation() no longer needs 'k'",
issue=13742,
deprecated_since_version="1.2").warn()
from sympy import solve
x, y, z, k = [_symbol(i, real=True) for i in (x, y, z, 'k')]
p1, p2 = self.points
d1, d2, d3 = p1.direction_ratio(p2)
x1, y1, z1 = p1
v = (x, y, z)
eqs = [-d1*k + x - x1, -d2*k + y - y1, -d3*k + z - z1]
# eliminate k from equations by solving first eq with k for k
for i, e in enumerate(eqs):
if e.has(k):
kk = solve(eqs[i], k)[0]
eqs.pop(i)
break
return Tuple(*[i.subs(k, kk).as_numer_denom()[0] for i in eqs])
class Ray3D(LinearEntity3D, Ray):
"""
A Ray is a semi-line in the space with a source point and a direction.
Parameters
==========
p1 : Point3D
The source of the Ray
p2 : Point or a direction vector
direction_ratio: Determines the direction in which the Ray propagates.
Attributes
==========
source
xdirection
ydirection
zdirection
See Also
========
sympy.geometry.point.Point3D, Line3D
Examples
========
>>> from sympy import Point3D
>>> from sympy.geometry import Ray3D
>>> r = Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r
Ray3D(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r.points
(Point3D(2, 3, 4), Point3D(3, 5, 0))
>>> r.source
Point3D(2, 3, 4)
>>> r.xdirection
oo
>>> r.ydirection
oo
>>> r.direction_ratio
[1, 2, -4]
"""
def __new__(cls, p1, pt=None, direction_ratio=[], **kwargs):
from sympy.utilities.misc import filldedent
if isinstance(p1, LinearEntity3D):
if pt is not None:
raise ValueError('If p1 is a LinearEntity, pt must be None')
p1, pt = p1.args
else:
p1 = Point(p1, dim=3)
if pt is not None and len(direction_ratio) == 0:
pt = Point(pt, dim=3)
elif len(direction_ratio) == 3 and pt is None:
pt = Point3D(p1.x + direction_ratio[0], p1.y + direction_ratio[1],
p1.z + direction_ratio[2])
else:
raise ValueError(filldedent('''
A 2nd Point or keyword "direction_ratio" must be used.
'''))
return LinearEntity3D.__new__(cls, p1, pt, **kwargs)
@property
def xdirection(self):
"""The x direction of the ray.
Positive infinity if the ray points in the positive x direction,
negative infinity if the ray points in the negative x direction,
or 0 if the ray is vertical.
See Also
========
ydirection
Examples
========
>>> from sympy import Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(1, 1, 1), Point3D(0, -1, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.xdirection
oo
>>> r2.xdirection
0
"""
if self.p1.x < self.p2.x:
return S.Infinity
elif self.p1.x == self.p2.x:
return S.Zero
else:
return S.NegativeInfinity
@property
def ydirection(self):
"""The y direction of the ray.
Positive infinity if the ray points in the positive y direction,
negative infinity if the ray points in the negative y direction,
or 0 if the ray is horizontal.
See Also
========
xdirection
Examples
========
>>> from sympy import Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0
"""
if self.p1.y < self.p2.y:
return S.Infinity
elif self.p1.y == self.p2.y:
return S.Zero
else:
return S.NegativeInfinity
@property
def zdirection(self):
"""The z direction of the ray.
Positive infinity if the ray points in the positive z direction,
negative infinity if the ray points in the negative z direction,
or 0 if the ray is horizontal.
See Also
========
xdirection
Examples
========
>>> from sympy import Point3D, Ray3D
>>> p1, p2, p3 = Point3D(0, 0, 0), Point3D(-1, -1, -1), Point3D(-1, 0, 0)
>>> r1, r2 = Ray3D(p1, p2), Ray3D(p1, p3)
>>> r1.ydirection
-oo
>>> r2.ydirection
0
>>> r2.zdirection
0
"""
if self.p1.z < self.p2.z:
return S.Infinity
elif self.p1.z == self.p2.z:
return S.Zero
else:
return S.NegativeInfinity
class Segment3D(LinearEntity3D, Segment):
"""A line segment in a 3D space.
Parameters
==========
p1 : Point3D
p2 : Point3D
Attributes
==========
length : number or sympy expression
midpoint : Point3D
See Also
========
sympy.geometry.point.Point3D, Line3D
Examples
========
>>> from sympy import Point3D
>>> from sympy.geometry import Segment3D
>>> Segment3D((1, 0, 0), (1, 1, 1)) # tuples are interpreted as pts
Segment3D(Point3D(1, 0, 0), Point3D(1, 1, 1))
>>> s = Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7)); s
Segment3D(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.points
(Point3D(4, 3, 9), Point3D(1, 1, 7))
>>> s.length
sqrt(17)
>>> s.midpoint
Point3D(5/2, 2, 8)
"""
def __new__(cls, p1, p2, **kwargs):
p1 = Point(p1, dim=3)
p2 = Point(p2, dim=3)
if p1 == p2:
return p1
return LinearEntity3D.__new__(cls, p1, p2, **kwargs)
|
981f100245118b1e78e42e2945ff56f145a85847fa73fc3ca3073f3db14b4c8b | from __future__ import division, print_function
from sympy.core import Expr, S, Symbol, oo, pi, sympify
from sympy.core.compatibility import as_int, range, ordered
from sympy.core.symbol import _symbol, Dummy
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import cos, sin, tan
from sympy.geometry.exceptions import GeometryError
from sympy.logic import And
from sympy.matrices import Matrix
from sympy.simplify import simplify
from sympy.utilities import default_sort_key
from sympy.utilities.iterables import has_dups, has_variety, uniq, rotate_left, least_rotation
from sympy.utilities.misc import func_name
from .entity import GeometryEntity, GeometrySet
from .point import Point
from .ellipse import Circle
from .line import Line, Segment, Ray
from sympy import sqrt
import warnings
class Polygon(GeometrySet):
"""A two-dimensional polygon.
A simple polygon in space. Can be constructed from a sequence of points
or from a center, radius, number of sides and rotation angle.
Parameters
==========
vertices : sequence of Points
Attributes
==========
area
angles
perimeter
vertices
centroid
sides
Raises
======
GeometryError
If all parameters are not Points.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment, Triangle
Notes
=====
Polygons are treated as closed paths rather than 2D areas so
some calculations can be be negative or positive (e.g., area)
based on the orientation of the points.
Any consecutive identical points are reduced to a single point
and any points collinear and between two points will be removed
unless they are needed to define an explicit intersection (see examples).
A Triangle, Segment or Point will be returned when there are 3 or
fewer points provided.
Examples
========
>>> from sympy import Point, Polygon, pi
>>> p1, p2, p3, p4, p5 = [(0, 0), (1, 0), (5, 1), (0, 1), (3, 0)]
>>> Polygon(p1, p2, p3, p4)
Polygon(Point2D(0, 0), Point2D(1, 0), Point2D(5, 1), Point2D(0, 1))
>>> Polygon(p1, p2)
Segment2D(Point2D(0, 0), Point2D(1, 0))
>>> Polygon(p1, p2, p5)
Segment2D(Point2D(0, 0), Point2D(3, 0))
The area of a polygon is calculated as positive when vertices are
traversed in a ccw direction. When the sides of a polygon cross the
area will have positive and negative contributions. The following
defines a Z shape where the bottom right connects back to the top
left.
>>> Polygon((0, 2), (2, 2), (0, 0), (2, 0)).area
0
When the the keyword `n` is used to define the number of sides of the
Polygon then a RegularPolygon is created and the other arguments are
interpreted as center, radius and rotation. The unrotated RegularPolygon
will always have a vertex at Point(r, 0) where `r` is the radius of the
circle that circumscribes the RegularPolygon. Its method `spin` can be
used to increment that angle.
>>> p = Polygon((0,0), 1, n=3)
>>> p
RegularPolygon(Point2D(0, 0), 1, 3, 0)
>>> p.vertices[0]
Point2D(1, 0)
>>> p.args[0]
Point2D(0, 0)
>>> p.spin(pi/2)
>>> p.vertices[0]
Point2D(0, 1)
"""
def __new__(cls, *args, **kwargs):
if kwargs.get('n', 0):
n = kwargs.pop('n')
args = list(args)
# return a virtual polygon with n sides
if len(args) == 2: # center, radius
args.append(n)
elif len(args) == 3: # center, radius, rotation
args.insert(2, n)
return RegularPolygon(*args, **kwargs)
vertices = [Point(a, dim=2, **kwargs) for a in args]
# remove consecutive duplicates
nodup = []
for p in vertices:
if nodup and p == nodup[-1]:
continue
nodup.append(p)
if len(nodup) > 1 and nodup[-1] == nodup[0]:
nodup.pop() # last point was same as first
# remove collinear points
i = -3
while i < len(nodup) - 3 and len(nodup) > 2:
a, b, c = nodup[i], nodup[i + 1], nodup[i + 2]
if Point.is_collinear(a, b, c):
nodup.pop(i + 1)
if a == c:
nodup.pop(i)
else:
i += 1
vertices = list(nodup)
if len(vertices) > 3:
return GeometryEntity.__new__(cls, *vertices, **kwargs)
elif len(vertices) == 3:
return Triangle(*vertices, **kwargs)
elif len(vertices) == 2:
return Segment(*vertices, **kwargs)
else:
return Point(*vertices, **kwargs)
@property
def area(self):
"""
The area of the polygon.
Notes
=====
The area calculation can be positive or negative based on the
orientation of the points. If any side of the polygon crosses
any other side, there will be areas having opposite signs.
See Also
========
sympy.geometry.ellipse.Ellipse.area
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.area
3
In the Z shaped polygon (with the lower right connecting back
to the upper left) the areas cancel out:
>>> Z = Polygon((0, 1), (1, 1), (0, 0), (1, 0))
>>> Z.area
0
In the M shaped polygon, areas do not cancel because no side
crosses any other (though there is a point of contact).
>>> M = Polygon((0, 0), (0, 1), (2, 0), (3, 1), (3, 0))
>>> M.area
-3/2
"""
area = 0
args = self.args
for i in range(len(args)):
x1, y1 = args[i - 1].args
x2, y2 = args[i].args
area += x1*y2 - x2*y1
return simplify(area) / 2
@staticmethod
def _isright(a, b, c):
"""Return True/False for cw/ccw orientation.
Examples
========
>>> from sympy import Point, Polygon
>>> a, b, c = [Point(i) for i in [(0, 0), (1, 1), (1, 0)]]
>>> Polygon._isright(a, b, c)
True
>>> Polygon._isright(a, c, b)
False
"""
ba = b - a
ca = c - a
t_area = simplify(ba.x*ca.y - ca.x*ba.y)
res = t_area.is_nonpositive
if res is None:
raise ValueError("Can't determine orientation")
return res
@property
def angles(self):
"""The internal angle at each vertex.
Returns
=======
angles : dict
A dictionary where each key is a vertex and each value is the
internal angle at that vertex. The vertices are represented as
Points.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.LinearEntity.angle_between
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.angles[p1]
pi/2
>>> poly.angles[p2]
acos(-4*sqrt(17)/17)
"""
# Determine orientation of points
args = self.vertices
cw = self._isright(args[-1], args[0], args[1])
ret = {}
for i in range(len(args)):
a, b, c = args[i - 2], args[i - 1], args[i]
ang = Ray(b, a).angle_between(Ray(b, c))
if cw ^ self._isright(a, b, c):
ret[b] = 2*S.Pi - ang
else:
ret[b] = ang
return ret
@property
def ambient_dimension(self):
return self.vertices[0].ambient_dimension
@property
def perimeter(self):
"""The perimeter of the polygon.
Returns
=======
perimeter : number or Basic instance
See Also
========
sympy.geometry.line.Segment.length
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.perimeter
sqrt(17) + 7
"""
p = 0
args = self.vertices
for i in range(len(args)):
p += args[i - 1].distance(args[i])
return simplify(p)
@property
def vertices(self):
"""The vertices of the polygon.
Returns
=======
vertices : list of Points
Notes
=====
When iterating over the vertices, it is more efficient to index self
rather than to request the vertices and index them. Only use the
vertices when you want to process all of them at once. This is even
more important with RegularPolygons that calculate each vertex.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.vertices
[Point2D(0, 0), Point2D(1, 0), Point2D(5, 1), Point2D(0, 1)]
>>> poly.vertices[0]
Point2D(0, 0)
"""
return list(self.args)
@property
def centroid(self):
"""The centroid of the polygon.
Returns
=======
centroid : Point
See Also
========
sympy.geometry.point.Point, sympy.geometry.util.centroid
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.centroid
Point2D(31/18, 11/18)
"""
A = 1/(6*self.area)
cx, cy = 0, 0
args = self.args
for i in range(len(args)):
x1, y1 = args[i - 1].args
x2, y2 = args[i].args
v = x1*y2 - x2*y1
cx += v*(x1 + x2)
cy += v*(y1 + y2)
return Point(simplify(A*cx), simplify(A*cy))
def second_moment_of_area(self, point=None):
"""Returns the second moment and product moment of area of a two dimensional polygon.
Parameters
==========
point : Point, two-tuple of sympifiable objects, or None(default=None)
point is the point about which second moment of area is to be found.
If "point=None" it will be calculated about the axis passing through the
centroid of the polygon.
Returns
=======
I_xx, I_yy, I_xy : number or sympy expression
I_xx, I_yy are second moment of area of a two dimensional polygon.
I_xy is product moment of area of a two dimensional polygon.
Examples
========
>>> from sympy import Point, Polygon, symbols
>>> a, b = symbols('a, b')
>>> p1, p2, p3, p4, p5 = [(0, 0), (a, 0), (a, b), (0, b), (a/3, b/3)]
>>> rectangle = Polygon(p1, p2, p3, p4)
>>> rectangle.second_moment_of_area()
(a*b**3/12, a**3*b/12, 0)
>>> rectangle.second_moment_of_area(p5)
(a*b**3/9, a**3*b/9, a**2*b**2/36)
References
==========
https://en.wikipedia.org/wiki/Second_moment_of_area
"""
I_xx, I_yy, I_xy = 0, 0, 0
args = self.args
for i in range(len(args)):
x1, y1 = args[i-1].args
x2, y2 = args[i].args
v = x1*y2 - x2*y1
I_xx += (y1**2 + y1*y2 + y2**2)*v
I_yy += (x1**2 + x1*x2 + x2**2)*v
I_xy += (x1*y2 + 2*x1*y1 + 2*x2*y2 + x2*y1)*v
A = self.area
c_x = self.centroid[0]
c_y = self.centroid[1]
# parallel axis theorem
I_xx_c = (I_xx/12) - (A*(c_y**2))
I_yy_c = (I_yy/12) - (A*(c_x**2))
I_xy_c = (I_xy/24) - (A*(c_x*c_y))
if point is None:
return I_xx_c, I_yy_c, I_xy_c
I_xx = (I_xx_c + A*((point[1]-c_y)**2))
I_yy = (I_yy_c + A*((point[0]-c_x)**2))
I_xy = (I_xy_c + A*((point[0]-c_x)*(point[1]-c_y)))
return I_xx, I_yy, I_xy
@property
def sides(self):
"""The directed line segments that form the sides of the polygon.
Returns
=======
sides : list of sides
Each side is a directed Segment.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.sides
[Segment2D(Point2D(0, 0), Point2D(1, 0)),
Segment2D(Point2D(1, 0), Point2D(5, 1)),
Segment2D(Point2D(5, 1), Point2D(0, 1)), Segment2D(Point2D(0, 1), Point2D(0, 0))]
"""
res = []
args = self.vertices
for i in range(-len(args), 0):
res.append(Segment(args[i], args[i + 1]))
return res
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
verts = self.vertices
xs = [p.x for p in verts]
ys = [p.y for p in verts]
return (min(xs), min(ys), max(xs), max(ys))
def is_convex(self):
"""Is the polygon convex?
A polygon is convex if all its interior angles are less than 180
degrees and there are no intersections between sides.
Returns
=======
is_convex : boolean
True if this polygon is convex, False otherwise.
See Also
========
sympy.geometry.util.convex_hull
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.is_convex()
True
"""
# Determine orientation of points
args = self.vertices
cw = self._isright(args[-2], args[-1], args[0])
for i in range(1, len(args)):
if cw ^ self._isright(args[i - 2], args[i - 1], args[i]):
return False
# check for intersecting sides
sides = self.sides
for i, si in enumerate(sides):
pts = si.args
# exclude the sides connected to si
for j in range(1 if i == len(sides) - 1 else 0, i - 1):
sj = sides[j]
if sj.p1 not in pts and sj.p2 not in pts:
hit = si.intersection(sj)
if hit:
return False
return True
def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.
Notes
=====
Being on the border of self is considered False.
Parameters
==========
p : Point
Returns
=======
encloses_point : True, False or None
See Also
========
sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.encloses_point
Examples
========
>>> from sympy import Polygon, Point
>>> from sympy.abc import t
>>> p = Polygon((0, 0), (4, 0), (4, 4))
>>> p.encloses_point(Point(2, 1))
True
>>> p.encloses_point(Point(2, 2))
False
>>> p.encloses_point(Point(5, 5))
False
References
==========
[1] http://paulbourke.net/geometry/polygonmesh/#insidepoly
"""
p = Point(p, dim=2)
if p in self.vertices or any(p in s for s in self.sides):
return False
# move to p, checking that the result is numeric
lit = []
for v in self.vertices:
lit.append(v - p) # the difference is simplified
if lit[-1].free_symbols:
return None
poly = Polygon(*lit)
# polygon closure is assumed in the following test but Polygon removes duplicate pts so
# the last point has to be added so all sides are computed. Using Polygon.sides is
# not good since Segments are unordered.
args = poly.args
indices = list(range(-len(args), 1))
if poly.is_convex():
orientation = None
for i in indices:
a = args[i]
b = args[i + 1]
test = ((-a.y)*(b.x - a.x) - (-a.x)*(b.y - a.y)).is_negative
if orientation is None:
orientation = test
elif test is not orientation:
return False
return True
hit_odd = False
p1x, p1y = args[0].args
for i in indices[1:]:
p2x, p2y = args[i].args
if 0 > min(p1y, p2y):
if 0 <= max(p1y, p2y):
if 0 <= max(p1x, p2x):
if p1y != p2y:
xinters = (-p1y)*(p2x - p1x)/(p2y - p1y) + p1x
if p1x == p2x or 0 <= xinters:
hit_odd = not hit_odd
p1x, p1y = p2x, p2y
return hit_odd
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the polygon.
The parameter, varying from 0 to 1, assigns points to the position on
the perimeter that is that fraction of the total perimeter. So the
point evaluated at t=1/2 would return the point from the first vertex
that is 1/2 way around the polygon.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the Polygon's definition.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Polygon, S, Symbol
>>> t = Symbol('t', real=True)
>>> tri = Polygon((0, 0), (1, 0), (1, 1))
>>> p = tri.arbitrary_point('t')
>>> perimeter = tri.perimeter
>>> s1, s2 = [s.length for s in tri.sides[:2]]
>>> p.subs(t, (s1 + s2/2)/perimeter)
Point2D(1, 1/2)
"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name)
sides = []
perimeter = self.perimeter
perim_fraction_start = 0
for s in self.sides:
side_perim_fraction = s.length/perimeter
perim_fraction_end = perim_fraction_start + side_perim_fraction
pt = s.arbitrary_point(parameter).subs(
t, (t - perim_fraction_start)/side_perim_fraction)
sides.append(
(pt, (And(perim_fraction_start <= t, t < perim_fraction_end))))
perim_fraction_start = perim_fraction_end
return Piecewise(*sides)
def parameter_value(self, other, t):
from sympy.solvers.solvers import solve
if not isinstance(other,GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if not isinstance(other,Point):
raise ValueError("other must be a point")
if other.free_symbols:
raise NotImplementedError('non-numeric coordinates')
unknown = False
T = Dummy('t', real=True)
p = self.arbitrary_point(T)
for pt, cond in p.args:
sol = solve(pt - other, T, dict=True)
if not sol:
continue
value = sol[0][T]
if simplify(cond.subs(T, value)) == True:
return {t: value}
unknown = True
if unknown:
raise ValueError("Given point may not be on %s" % func_name(self))
raise ValueError("Given point is not on %s" % func_name(self))
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the polygon.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Polygon
>>> p = Polygon((0, 0), (1, 0), (1, 1))
>>> p.plot_interval()
[t, 0, 1]
"""
t = Symbol(parameter, real=True)
return [t, 0, 1]
def intersection(self, o):
"""The intersection of polygon and geometry entity.
The intersection may be empty and can contain individual Points and
complete Line Segments.
Parameters
==========
other: GeometryEntity
Returns
=======
intersection : list
The list of Segments and Points
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment
Examples
========
>>> from sympy import Point, Polygon, Line
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly1 = Polygon(p1, p2, p3, p4)
>>> p5, p6, p7 = map(Point, [(3, 2), (1, -1), (0, 2)])
>>> poly2 = Polygon(p5, p6, p7)
>>> poly1.intersection(poly2)
[Point2D(1/3, 1), Point2D(2/3, 0), Point2D(9/5, 1/5), Point2D(7/3, 1)]
>>> poly1.intersection(Line(p1, p2))
[Segment2D(Point2D(0, 0), Point2D(1, 0))]
>>> poly1.intersection(p1)
[Point2D(0, 0)]
"""
intersection_result = []
k = o.sides if isinstance(o, Polygon) else [o]
for side in self.sides:
for side1 in k:
intersection_result.extend(side.intersection(side1))
intersection_result = list(uniq(intersection_result))
points = [entity for entity in intersection_result if isinstance(entity, Point)]
segments = [entity for entity in intersection_result if isinstance(entity, Segment)]
if points and segments:
points_in_segments = list(uniq([point for point in points for segment in segments if point in segment]))
if points_in_segments:
for i in points_in_segments:
points.remove(i)
return list(ordered(segments + points))
else:
return list(ordered(intersection_result))
def distance(self, o):
"""
Returns the shortest distance between self and o.
If o is a point, then self does not need to be convex.
If o is another polygon self and o must be convex.
Examples
========
>>> from sympy import Point, Polygon, RegularPolygon
>>> p1, p2 = map(Point, [(0, 0), (7, 5)])
>>> poly = Polygon(*RegularPolygon(p1, 1, 3).vertices)
>>> poly.distance(p2)
sqrt(61)
"""
if isinstance(o, Point):
dist = oo
for side in self.sides:
current = side.distance(o)
if current == 0:
return S.Zero
elif current < dist:
dist = current
return dist
elif isinstance(o, Polygon) and self.is_convex() and o.is_convex():
return self._do_poly_distance(o)
raise NotImplementedError()
def _do_poly_distance(self, e2):
"""
Calculates the least distance between the exteriors of two
convex polygons e1 and e2. Does not check for the convexity
of the polygons as this is checked by Polygon.distance.
Notes
=====
- Prints a warning if the two polygons possibly intersect as the return
value will not be valid in such a case. For a more through test of
intersection use intersection().
See Also
========
sympy.geometry.point.Point.distance
Examples
========
>>> from sympy.geometry import Point, Polygon
>>> square = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
>>> triangle = Polygon(Point(1, 2), Point(2, 2), Point(2, 1))
>>> square._do_poly_distance(triangle)
sqrt(2)/2
Description of method used
==========================
Method:
[1] http://cgm.cs.mcgill.ca/~orm/mind2p.html
Uses rotating calipers:
[2] https://en.wikipedia.org/wiki/Rotating_calipers
and antipodal points:
[3] https://en.wikipedia.org/wiki/Antipodal_point
"""
e1 = self
'''Tests for a possible intersection between the polygons and outputs a warning'''
e1_center = e1.centroid
e2_center = e2.centroid
e1_max_radius = S.Zero
e2_max_radius = S.Zero
for vertex in e1.vertices:
r = Point.distance(e1_center, vertex)
if e1_max_radius < r:
e1_max_radius = r
for vertex in e2.vertices:
r = Point.distance(e2_center, vertex)
if e2_max_radius < r:
e2_max_radius = r
center_dist = Point.distance(e1_center, e2_center)
if center_dist <= e1_max_radius + e2_max_radius:
warnings.warn("Polygons may intersect producing erroneous output")
'''
Find the upper rightmost vertex of e1 and the lowest leftmost vertex of e2
'''
e1_ymax = Point(0, -oo)
e2_ymin = Point(0, oo)
for vertex in e1.vertices:
if vertex.y > e1_ymax.y or (vertex.y == e1_ymax.y and vertex.x > e1_ymax.x):
e1_ymax = vertex
for vertex in e2.vertices:
if vertex.y < e2_ymin.y or (vertex.y == e2_ymin.y and vertex.x < e2_ymin.x):
e2_ymin = vertex
min_dist = Point.distance(e1_ymax, e2_ymin)
'''
Produce a dictionary with vertices of e1 as the keys and, for each vertex, the points
to which the vertex is connected as its value. The same is then done for e2.
'''
e1_connections = {}
e2_connections = {}
for side in e1.sides:
if side.p1 in e1_connections:
e1_connections[side.p1].append(side.p2)
else:
e1_connections[side.p1] = [side.p2]
if side.p2 in e1_connections:
e1_connections[side.p2].append(side.p1)
else:
e1_connections[side.p2] = [side.p1]
for side in e2.sides:
if side.p1 in e2_connections:
e2_connections[side.p1].append(side.p2)
else:
e2_connections[side.p1] = [side.p2]
if side.p2 in e2_connections:
e2_connections[side.p2].append(side.p1)
else:
e2_connections[side.p2] = [side.p1]
e1_current = e1_ymax
e2_current = e2_ymin
support_line = Line(Point(S.Zero, S.Zero), Point(S.One, S.Zero))
'''
Determine which point in e1 and e2 will be selected after e2_ymin and e1_ymax,
this information combined with the above produced dictionaries determines the
path that will be taken around the polygons
'''
point1 = e1_connections[e1_ymax][0]
point2 = e1_connections[e1_ymax][1]
angle1 = support_line.angle_between(Line(e1_ymax, point1))
angle2 = support_line.angle_between(Line(e1_ymax, point2))
if angle1 < angle2:
e1_next = point1
elif angle2 < angle1:
e1_next = point2
elif Point.distance(e1_ymax, point1) > Point.distance(e1_ymax, point2):
e1_next = point2
else:
e1_next = point1
point1 = e2_connections[e2_ymin][0]
point2 = e2_connections[e2_ymin][1]
angle1 = support_line.angle_between(Line(e2_ymin, point1))
angle2 = support_line.angle_between(Line(e2_ymin, point2))
if angle1 > angle2:
e2_next = point1
elif angle2 > angle1:
e2_next = point2
elif Point.distance(e2_ymin, point1) > Point.distance(e2_ymin, point2):
e2_next = point2
else:
e2_next = point1
'''
Loop which determines the distance between anti-podal pairs and updates the
minimum distance accordingly. It repeats until it reaches the starting position.
'''
while True:
e1_angle = support_line.angle_between(Line(e1_current, e1_next))
e2_angle = pi - support_line.angle_between(Line(
e2_current, e2_next))
if (e1_angle < e2_angle) is True:
support_line = Line(e1_current, e1_next)
e1_segment = Segment(e1_current, e1_next)
min_dist_current = e1_segment.distance(e2_current)
if min_dist_current.evalf() < min_dist.evalf():
min_dist = min_dist_current
if e1_connections[e1_next][0] != e1_current:
e1_current = e1_next
e1_next = e1_connections[e1_next][0]
else:
e1_current = e1_next
e1_next = e1_connections[e1_next][1]
elif (e1_angle > e2_angle) is True:
support_line = Line(e2_next, e2_current)
e2_segment = Segment(e2_current, e2_next)
min_dist_current = e2_segment.distance(e1_current)
if min_dist_current.evalf() < min_dist.evalf():
min_dist = min_dist_current
if e2_connections[e2_next][0] != e2_current:
e2_current = e2_next
e2_next = e2_connections[e2_next][0]
else:
e2_current = e2_next
e2_next = e2_connections[e2_next][1]
else:
support_line = Line(e1_current, e1_next)
e1_segment = Segment(e1_current, e1_next)
e2_segment = Segment(e2_current, e2_next)
min1 = e1_segment.distance(e2_next)
min2 = e2_segment.distance(e1_next)
min_dist_current = min(min1, min2)
if min_dist_current.evalf() < min_dist.evalf():
min_dist = min_dist_current
if e1_connections[e1_next][0] != e1_current:
e1_current = e1_next
e1_next = e1_connections[e1_next][0]
else:
e1_current = e1_next
e1_next = e1_connections[e1_next][1]
if e2_connections[e2_next][0] != e2_current:
e2_current = e2_next
e2_next = e2_connections[e2_next][0]
else:
e2_current = e2_next
e2_next = e2_connections[e2_next][1]
if e1_current == e1_ymax and e2_current == e2_ymin:
break
return min_dist
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the Polygon.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
from sympy.core.evalf import N
verts = map(N, self.vertices)
coords = ["{0},{1}".format(p.x, p.y) for p in verts]
path = "M {0} L {1} z".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" />'
).format(2. * scale_factor, path, fill_color)
def _hashable_content(self):
D = {}
def ref_list(point_list):
kee = {}
for i, p in enumerate(ordered(set(point_list))):
kee[p] = i
D[i] = p
return [kee[p] for p in point_list]
S1 = ref_list(self.args)
r_nor = rotate_left(S1, least_rotation(S1))
S2 = ref_list(list(reversed(self.args)))
r_rev = rotate_left(S2, least_rotation(S2))
if r_nor < r_rev:
r = r_nor
else:
r = r_rev
canonical_args = [ D[order] for order in r ]
return tuple(canonical_args)
def __contains__(self, o):
"""
Return True if o is contained within the boundary lines of self.altitudes
Parameters
==========
other : GeometryEntity
Returns
=======
contained in : bool
The points (and sides, if applicable) are contained in self.
See Also
========
sympy.geometry.entity.GeometryEntity.encloses
Examples
========
>>> from sympy import Line, Segment, Point
>>> p = Point(0, 0)
>>> q = Point(1, 1)
>>> s = Segment(p, q*2)
>>> l = Line(p, q)
>>> p in q
False
>>> p in s
True
>>> q*3 in s
False
>>> s in l
True
"""
if isinstance(o, Polygon):
return self == o
elif isinstance(o, Segment):
return any(o in s for s in self.sides)
elif isinstance(o, Point):
if o in self.vertices:
return True
for side in self.sides:
if o in side:
return True
return False
class RegularPolygon(Polygon):
"""
A regular polygon.
Such a polygon has all internal angles equal and all sides the same length.
Parameters
==========
center : Point
radius : number or Basic instance
The distance from the center to a vertex
n : int
The number of sides
Attributes
==========
vertices
center
radius
rotation
apothem
interior_angle
exterior_angle
circumcircle
incircle
angles
Raises
======
GeometryError
If the `center` is not a Point, or the `radius` is not a number or Basic
instance, or the number of sides, `n`, is less than three.
Notes
=====
A RegularPolygon can be instantiated with Polygon with the kwarg n.
Regular polygons are instantiated with a center, radius, number of sides
and a rotation angle. Whereas the arguments of a Polygon are vertices, the
vertices of the RegularPolygon must be obtained with the vertices method.
See Also
========
sympy.geometry.point.Point, Polygon
Examples
========
>>> from sympy.geometry import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r
RegularPolygon(Point2D(0, 0), 5, 3, 0)
>>> r.vertices[0]
Point2D(5, 0)
"""
__slots__ = ['_n', '_center', '_radius', '_rot']
def __new__(self, c, r, n, rot=0, **kwargs):
r, n, rot = map(sympify, (r, n, rot))
c = Point(c, dim=2, **kwargs)
if not isinstance(r, Expr):
raise GeometryError("r must be an Expr object, not %s" % r)
if n.is_Number:
as_int(n) # let an error raise if necessary
if n < 3:
raise GeometryError("n must be a >= 3, not %s" % n)
obj = GeometryEntity.__new__(self, c, r, n, **kwargs)
obj._n = n
obj._center = c
obj._radius = r
obj._rot = rot % (2*S.Pi/n) if rot.is_number else rot
return obj
@property
def args(self):
"""
Returns the center point, the radius,
the number of sides, and the orientation angle.
Examples
========
>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.args
(Point2D(0, 0), 5, 3, 0)
"""
return self._center, self._radius, self._n, self._rot
def __str__(self):
return 'RegularPolygon(%s, %s, %s, %s)' % tuple(self.args)
def __repr__(self):
return 'RegularPolygon(%s, %s, %s, %s)' % tuple(self.args)
@property
def area(self):
"""Returns the area.
Examples
========
>>> from sympy.geometry import RegularPolygon
>>> square = RegularPolygon((0, 0), 1, 4)
>>> square.area
2
>>> _ == square.length**2
True
"""
c, r, n, rot = self.args
return sign(r)*n*self.length**2/(4*tan(pi/n))
@property
def length(self):
"""Returns the length of the sides.
The half-length of the side and the apothem form two legs
of a right triangle whose hypotenuse is the radius of the
regular polygon.
Examples
========
>>> from sympy.geometry import RegularPolygon
>>> from sympy import sqrt
>>> s = square_in_unit_circle = RegularPolygon((0, 0), 1, 4)
>>> s.length
sqrt(2)
>>> sqrt((_/2)**2 + s.apothem**2) == s.radius
True
"""
return self.radius*2*sin(pi/self._n)
@property
def center(self):
"""The center of the RegularPolygon
This is also the center of the circumscribing circle.
Returns
=======
center : Point
See Also
========
sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.center
Examples
========
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.center
Point2D(0, 0)
"""
return self._center
centroid = center
@property
def circumcenter(self):
"""
Alias for center.
Examples
========
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.circumcenter
Point2D(0, 0)
"""
return self.center
@property
def radius(self):
"""Radius of the RegularPolygon
This is also the radius of the circumscribing circle.
Returns
=======
radius : number or instance of Basic
See Also
========
sympy.geometry.line.Segment.length, sympy.geometry.ellipse.Circle.radius
Examples
========
>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.radius
r
"""
return self._radius
@property
def circumradius(self):
"""
Alias for radius.
Examples
========
>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.circumradius
r
"""
return self.radius
@property
def rotation(self):
"""CCW angle by which the RegularPolygon is rotated
Returns
=======
rotation : number or instance of Basic
Examples
========
>>> from sympy import pi
>>> from sympy.abc import a
>>> from sympy.geometry import RegularPolygon, Point
>>> RegularPolygon(Point(0, 0), 3, 4, pi/4).rotation
pi/4
Numerical rotation angles are made canonical:
>>> RegularPolygon(Point(0, 0), 3, 4, a).rotation
a
>>> RegularPolygon(Point(0, 0), 3, 4, pi).rotation
0
"""
return self._rot
@property
def apothem(self):
"""The inradius of the RegularPolygon.
The apothem/inradius is the radius of the inscribed circle.
Returns
=======
apothem : number or instance of Basic
See Also
========
sympy.geometry.line.Segment.length, sympy.geometry.ellipse.Circle.radius
Examples
========
>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.apothem
sqrt(2)*r/2
"""
return self.radius * cos(S.Pi/self._n)
@property
def inradius(self):
"""
Alias for apothem.
Examples
========
>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.inradius
sqrt(2)*r/2
"""
return self.apothem
@property
def interior_angle(self):
"""Measure of the interior angles.
Returns
=======
interior_angle : number
See Also
========
sympy.geometry.line.LinearEntity.angle_between
Examples
========
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.interior_angle
3*pi/4
"""
return (self._n - 2)*S.Pi/self._n
@property
def exterior_angle(self):
"""Measure of the exterior angles.
Returns
=======
exterior_angle : number
See Also
========
sympy.geometry.line.LinearEntity.angle_between
Examples
========
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.exterior_angle
pi/4
"""
return 2*S.Pi/self._n
@property
def circumcircle(self):
"""The circumcircle of the RegularPolygon.
Returns
=======
circumcircle : Circle
See Also
========
circumcenter, sympy.geometry.ellipse.Circle
Examples
========
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.circumcircle
Circle(Point2D(0, 0), 4)
"""
return Circle(self.center, self.radius)
@property
def incircle(self):
"""The incircle of the RegularPolygon.
Returns
=======
incircle : Circle
See Also
========
inradius, sympy.geometry.ellipse.Circle
Examples
========
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 7)
>>> rp.incircle
Circle(Point2D(0, 0), 4*cos(pi/7))
"""
return Circle(self.center, self.apothem)
@property
def angles(self):
"""
Returns a dictionary with keys, the vertices of the Polygon,
and values, the interior angle at each vertex.
Examples
========
>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.angles
{Point2D(-5/2, -5*sqrt(3)/2): pi/3,
Point2D(-5/2, 5*sqrt(3)/2): pi/3,
Point2D(5, 0): pi/3}
"""
ret = {}
ang = self.interior_angle
for v in self.vertices:
ret[v] = ang
return ret
def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.
Notes
=====
Being on the border of self is considered False.
The general Polygon.encloses_point method is called only if
a point is not within or beyond the incircle or circumcircle,
respectively.
Parameters
==========
p : Point
Returns
=======
encloses_point : True, False or None
See Also
========
sympy.geometry.ellipse.Ellipse.encloses_point
Examples
========
>>> from sympy import RegularPolygon, S, Point, Symbol
>>> p = RegularPolygon((0, 0), 3, 4)
>>> p.encloses_point(Point(0, 0))
True
>>> r, R = p.inradius, p.circumradius
>>> p.encloses_point(Point((r + R)/2, 0))
True
>>> p.encloses_point(Point(R/2, R/2 + (R - r)/10))
False
>>> t = Symbol('t', real=True)
>>> p.encloses_point(p.arbitrary_point().subs(t, S.Half))
False
>>> p.encloses_point(Point(5, 5))
False
"""
c = self.center
d = Segment(c, p).length
if d >= self.radius:
return False
elif d < self.inradius:
return True
else:
# now enumerate the RegularPolygon like a general polygon.
return Polygon.encloses_point(self, p)
def spin(self, angle):
"""Increment *in place* the virtual Polygon's rotation by ccw angle.
See also: rotate method which moves the center.
>>> from sympy import Polygon, Point, pi
>>> r = Polygon(Point(0,0), 1, n=3)
>>> r.vertices[0]
Point2D(1, 0)
>>> r.spin(pi/6)
>>> r.vertices[0]
Point2D(sqrt(3)/2, 1/2)
See Also
========
rotation
rotate : Creates a copy of the RegularPolygon rotated about a Point
"""
self._rot += angle
def rotate(self, angle, pt=None):
"""Override GeometryEntity.rotate to first rotate the RegularPolygon
about its center.
>>> from sympy import Point, RegularPolygon, Polygon, pi
>>> t = RegularPolygon(Point(1, 0), 1, 3)
>>> t.vertices[0] # vertex on x-axis
Point2D(2, 0)
>>> t.rotate(pi/2).vertices[0] # vertex on y axis now
Point2D(0, 2)
See Also
========
rotation
spin : Rotates a RegularPolygon in place
"""
r = type(self)(*self.args) # need a copy or else changes are in-place
r._rot += angle
return GeometryEntity.rotate(r, angle, pt)
def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since it is the radius that must be
scaled (if x == y) or else a new Polygon must be returned.
>>> from sympy import RegularPolygon
Symmetric scaling returns a RegularPolygon:
>>> RegularPolygon((0, 0), 1, 4).scale(2, 2)
RegularPolygon(Point2D(0, 0), 2, 4, 0)
Asymmetric scaling returns a kite as a Polygon:
>>> RegularPolygon((0, 0), 1, 4).scale(2, 1)
Polygon(Point2D(2, 0), Point2D(0, 1), Point2D(-2, 0), Point2D(0, -1))
"""
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
if x != y:
return Polygon(*self.vertices).scale(x, y)
c, r, n, rot = self.args
r *= x
return self.func(c, r, n, rot)
def reflect(self, line):
"""Override GeometryEntity.reflect since this is not made of only
points.
Examples
========
>>> from sympy import RegularPolygon, Line
>>> RegularPolygon((0, 0), 1, 4).reflect(Line((0, 1), slope=-2))
RegularPolygon(Point2D(4/5, 2/5), -1, 4, atan(4/3))
"""
c, r, n, rot = self.args
v = self.vertices[0]
d = v - c
cc = c.reflect(line)
vv = v.reflect(line)
dd = vv - cc
# calculate rotation about the new center
# which will align the vertices
l1 = Ray((0, 0), dd)
l2 = Ray((0, 0), d)
ang = l1.closing_angle(l2)
rot += ang
# change sign of radius as point traversal is reversed
return self.func(cc, -r, n, rot)
@property
def vertices(self):
"""The vertices of the RegularPolygon.
Returns
=======
vertices : list
Each vertex is a Point.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.vertices
[Point2D(5, 0), Point2D(0, 5), Point2D(-5, 0), Point2D(0, -5)]
"""
c = self._center
r = abs(self._radius)
rot = self._rot
v = 2*S.Pi/self._n
return [Point(c.x + r*cos(k*v + rot), c.y + r*sin(k*v + rot))
for k in range(self._n)]
def __eq__(self, o):
if not isinstance(o, Polygon):
return False
elif not isinstance(o, RegularPolygon):
return Polygon.__eq__(o, self)
return self.args == o.args
def __hash__(self):
return super(RegularPolygon, self).__hash__()
class Triangle(Polygon):
"""
A polygon with three vertices and three sides.
Parameters
==========
points : sequence of Points
keyword: asa, sas, or sss to specify sides/angles of the triangle
Attributes
==========
vertices
altitudes
orthocenter
circumcenter
circumradius
circumcircle
inradius
incircle
exradii
medians
medial
nine_point_circle
Raises
======
GeometryError
If the number of vertices is not equal to three, or one of the vertices
is not a Point, or a valid keyword is not given.
See Also
========
sympy.geometry.point.Point, Polygon
Examples
========
>>> from sympy.geometry import Triangle, Point
>>> Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
Triangle(Point2D(0, 0), Point2D(4, 0), Point2D(4, 3))
Keywords sss, sas, or asa can be used to give the desired
side lengths (in order) and interior angles (in degrees) that
define the triangle:
>>> Triangle(sss=(3, 4, 5))
Triangle(Point2D(0, 0), Point2D(3, 0), Point2D(3, 4))
>>> Triangle(asa=(30, 1, 30))
Triangle(Point2D(0, 0), Point2D(1, 0), Point2D(1/2, sqrt(3)/6))
>>> Triangle(sas=(1, 45, 2))
Triangle(Point2D(0, 0), Point2D(2, 0), Point2D(sqrt(2)/2, sqrt(2)/2))
"""
def __new__(cls, *args, **kwargs):
if len(args) != 3:
if 'sss' in kwargs:
return _sss(*[simplify(a) for a in kwargs['sss']])
if 'asa' in kwargs:
return _asa(*[simplify(a) for a in kwargs['asa']])
if 'sas' in kwargs:
return _sas(*[simplify(a) for a in kwargs['sas']])
msg = "Triangle instantiates with three points or a valid keyword."
raise GeometryError(msg)
vertices = [Point(a, dim=2, **kwargs) for a in args]
# remove consecutive duplicates
nodup = []
for p in vertices:
if nodup and p == nodup[-1]:
continue
nodup.append(p)
if len(nodup) > 1 and nodup[-1] == nodup[0]:
nodup.pop() # last point was same as first
# remove collinear points
i = -3
while i < len(nodup) - 3 and len(nodup) > 2:
a, b, c = sorted(
[nodup[i], nodup[i + 1], nodup[i + 2]], key=default_sort_key)
if Point.is_collinear(a, b, c):
nodup[i] = a
nodup[i + 1] = None
nodup.pop(i + 1)
i += 1
vertices = list(filter(lambda x: x is not None, nodup))
if len(vertices) == 3:
return GeometryEntity.__new__(cls, *vertices, **kwargs)
elif len(vertices) == 2:
return Segment(*vertices, **kwargs)
else:
return Point(*vertices, **kwargs)
@property
def vertices(self):
"""The triangle's vertices
Returns
=======
vertices : tuple
Each element in the tuple is a Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy.geometry import Triangle, Point
>>> t = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t.vertices
(Point2D(0, 0), Point2D(4, 0), Point2D(4, 3))
"""
return self.args
def is_similar(t1, t2):
"""Is another triangle similar to this one.
Two triangles are similar if one can be uniformly scaled to the other.
Parameters
==========
other: Triangle
Returns
=======
is_similar : boolean
See Also
========
sympy.geometry.entity.GeometryEntity.is_similar
Examples
========
>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -3))
>>> t1.is_similar(t2)
True
>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -4))
>>> t1.is_similar(t2)
False
"""
if not isinstance(t2, Polygon):
return False
s1_1, s1_2, s1_3 = [side.length for side in t1.sides]
s2 = [side.length for side in t2.sides]
def _are_similar(u1, u2, u3, v1, v2, v3):
e1 = simplify(u1/v1)
e2 = simplify(u2/v2)
e3 = simplify(u3/v3)
return bool(e1 == e2) and bool(e2 == e3)
# There's only 6 permutations, so write them out
return _are_similar(s1_1, s1_2, s1_3, *s2) or \
_are_similar(s1_1, s1_3, s1_2, *s2) or \
_are_similar(s1_2, s1_1, s1_3, *s2) or \
_are_similar(s1_2, s1_3, s1_1, *s2) or \
_are_similar(s1_3, s1_1, s1_2, *s2) or \
_are_similar(s1_3, s1_2, s1_1, *s2)
def is_equilateral(self):
"""Are all the sides the same length?
Returns
=======
is_equilateral : boolean
See Also
========
sympy.geometry.entity.GeometryEntity.is_similar, RegularPolygon
is_isosceles, is_right, is_scalene
Examples
========
>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_equilateral()
False
>>> from sympy import sqrt
>>> t2 = Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3)))
>>> t2.is_equilateral()
True
"""
return not has_variety(s.length for s in self.sides)
def is_isosceles(self):
"""Are two or more of the sides the same length?
Returns
=======
is_isosceles : boolean
See Also
========
is_equilateral, is_right, is_scalene
Examples
========
>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4))
>>> t1.is_isosceles()
True
"""
return has_dups(s.length for s in self.sides)
def is_scalene(self):
"""Are all the sides of the triangle of different lengths?
Returns
=======
is_scalene : boolean
See Also
========
is_equilateral, is_isosceles, is_right
Examples
========
>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4))
>>> t1.is_scalene()
True
"""
return not has_dups(s.length for s in self.sides)
def is_right(self):
"""Is the triangle right-angled.
Returns
=======
is_right : boolean
See Also
========
sympy.geometry.line.LinearEntity.is_perpendicular
is_equilateral, is_isosceles, is_scalene
Examples
========
>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_right()
True
"""
s = self.sides
return Segment.is_perpendicular(s[0], s[1]) or \
Segment.is_perpendicular(s[1], s[2]) or \
Segment.is_perpendicular(s[0], s[2])
@property
def altitudes(self):
"""The altitudes of the triangle.
An altitude of a triangle is a segment through a vertex,
perpendicular to the opposite side, with length being the
height of the vertex measured from the line containing the side.
Returns
=======
altitudes : dict
The dictionary consists of keys which are vertices and values
which are Segments.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment.length
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.altitudes[p1]
Segment2D(Point2D(0, 0), Point2D(1/2, 1/2))
"""
s = self.sides
v = self.vertices
return {v[0]: s[1].perpendicular_segment(v[0]),
v[1]: s[2].perpendicular_segment(v[1]),
v[2]: s[0].perpendicular_segment(v[2])}
@property
def orthocenter(self):
"""The orthocenter of the triangle.
The orthocenter is the intersection of the altitudes of a triangle.
It may lie inside, outside or on the triangle.
Returns
=======
orthocenter : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.orthocenter
Point2D(0, 0)
"""
a = self.altitudes
v = self.vertices
return Line(a[v[0]]).intersection(Line(a[v[1]]))[0]
@property
def circumcenter(self):
"""The circumcenter of the triangle
The circumcenter is the center of the circumcircle.
Returns
=======
circumcenter : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcenter
Point2D(1/2, 1/2)
"""
a, b, c = [x.perpendicular_bisector() for x in self.sides]
if not a.intersection(b):
print(a,b,a.intersection(b))
return a.intersection(b)[0]
@property
def circumradius(self):
"""The radius of the circumcircle of the triangle.
Returns
=======
circumradius : number of Basic instance
See Also
========
sympy.geometry.ellipse.Circle.radius
Examples
========
>>> from sympy import Symbol
>>> from sympy.geometry import Point, Triangle
>>> a = Symbol('a')
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, a)
>>> t = Triangle(p1, p2, p3)
>>> t.circumradius
sqrt(a**2/4 + 1/4)
"""
return Point.distance(self.circumcenter, self.vertices[0])
@property
def circumcircle(self):
"""The circle which passes through the three vertices of the triangle.
Returns
=======
circumcircle : Circle
See Also
========
sympy.geometry.ellipse.Circle
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcircle
Circle(Point2D(1/2, 1/2), sqrt(2)/2)
"""
return Circle(self.circumcenter, self.circumradius)
def bisectors(self):
"""The angle bisectors of the triangle.
An angle bisector of a triangle is a straight line through a vertex
which cuts the corresponding angle in half.
Returns
=======
bisectors : dict
Each key is a vertex (Point) and each value is the corresponding
bisector (Segment).
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment
Examples
========
>>> from sympy.geometry import Point, Triangle, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> from sympy import sqrt
>>> t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1))
True
"""
# use lines containing sides so containment check during
# intersection calculation can be avoided
# This would reduce the processing time for calculating the bisectors
s = [Line(l) for l in self.sides]
v = self.vertices
c = self.incenter
l1 = Segment(v[0], Line(v[0], c).intersection(s[1])[0])
l2 = Segment(v[1], Line(v[1], c).intersection(s[2])[0])
l3 = Segment(v[2], Line(v[2], c).intersection(s[0])[0])
return {v[0]: l1, v[1]: l2, v[2]: l3}
@property
def incenter(self):
"""The center of the incircle.
The incircle is the circle which lies inside the triangle and touches
all three sides.
Returns
=======
incenter : Point
See Also
========
incircle, sympy.geometry.point.Point
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.incenter
Point2D(1 - sqrt(2)/2, 1 - sqrt(2)/2)
"""
s = self.sides
l = Matrix([s[i].length for i in [1, 2, 0]])
p = sum(l)
v = self.vertices
x = simplify(l.dot(Matrix([vi.x for vi in v]))/p)
y = simplify(l.dot(Matrix([vi.y for vi in v]))/p)
return Point(x, y)
@property
def inradius(self):
"""The radius of the incircle.
Returns
=======
inradius : number of Basic instance
See Also
========
incircle, sympy.geometry.ellipse.Circle.radius
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(4, 0), Point(0, 3)
>>> t = Triangle(p1, p2, p3)
>>> t.inradius
1
"""
return simplify(2 * self.area / self.perimeter)
@property
def incircle(self):
"""The incircle of the triangle.
The incircle is the circle which lies inside the triangle and touches
all three sides.
Returns
=======
incircle : Circle
See Also
========
sympy.geometry.ellipse.Circle
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(2, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.incircle
Circle(Point2D(2 - sqrt(2), 2 - sqrt(2)), 2 - sqrt(2))
"""
return Circle(self.incenter, self.inradius)
@property
def exradii(self):
"""The radius of excircles of a triangle.
An excircle of the triangle is a circle lying outside the triangle,
tangent to one of its sides and tangent to the extensions of the
other two.
Returns
=======
exradii : dict
See Also
========
sympy.geometry.polygon.Triangle.inradius
Examples
========
The exradius touches the side of the triangle to which it is keyed, e.g.
the exradius touching side 2 is:
>>> from sympy.geometry import Point, Triangle, Segment2D, Point2D
>>> p1, p2, p3 = Point(0, 0), Point(6, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.exradii[t.sides[2]]
-2 + sqrt(10)
References
==========
[1] http://mathworld.wolfram.com/Exradius.html
[2] http://mathworld.wolfram.com/Excircles.html
"""
side = self.sides
a = side[0].length
b = side[1].length
c = side[2].length
s = (a+b+c)/2
area = self.area
exradii = {self.sides[0]: simplify(area/(s-a)),
self.sides[1]: simplify(area/(s-b)),
self.sides[2]: simplify(area/(s-c))}
return exradii
@property
def medians(self):
"""The medians of the triangle.
A median of a triangle is a straight line through a vertex and the
midpoint of the opposite side, and divides the triangle into two
equal areas.
Returns
=======
medians : dict
Each key is a vertex (Point) and each value is the median (Segment)
at that point.
See Also
========
sympy.geometry.point.Point.midpoint, sympy.geometry.line.Segment.midpoint
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medians[p1]
Segment2D(Point2D(0, 0), Point2D(1/2, 1/2))
"""
s = self.sides
v = self.vertices
return {v[0]: Segment(v[0], s[1].midpoint),
v[1]: Segment(v[1], s[2].midpoint),
v[2]: Segment(v[2], s[0].midpoint)}
@property
def medial(self):
"""The medial triangle of the triangle.
The triangle which is formed from the midpoints of the three sides.
Returns
=======
medial : Triangle
See Also
========
sympy.geometry.line.Segment.midpoint
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medial
Triangle(Point2D(1/2, 0), Point2D(1/2, 1/2), Point2D(0, 1/2))
"""
s = self.sides
return Triangle(s[0].midpoint, s[1].midpoint, s[2].midpoint)
@property
def nine_point_circle(self):
"""The nine-point circle of the triangle.
Nine-point circle is the circumcircle of the medial triangle, which
passes through the feet of altitudes and the middle points of segments
connecting the vertices and the orthocenter.
Returns
=======
nine_point_circle : Circle
See also
========
sympy.geometry.line.Segment.midpoint
sympy.geometry.polygon.Triangle.medial
sympy.geometry.polygon.Triangle.orthocenter
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.nine_point_circle
Circle(Point2D(1/4, 1/4), sqrt(2)/4)
"""
return Circle(*self.medial.vertices)
@property
def eulerline(self):
"""The Euler line of the triangle.
The line which passes through circumcenter, centroid and orthocenter.
Returns
=======
eulerline : Line (or Point for equilateral triangles in which case all
centers coincide)
Examples
========
>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.eulerline
Line2D(Point2D(0, 0), Point2D(1/2, 1/2))
"""
if self.is_equilateral():
return self.orthocenter
return Line(self.orthocenter, self.circumcenter)
def rad(d):
"""Return the radian value for the given degrees (pi = 180 degrees)."""
return d*pi/180
def deg(r):
"""Return the degree value for the given radians (pi = 180 degrees)."""
return r/pi*180
def _slope(d):
rv = tan(rad(d))
return rv
def _asa(d1, l, d2):
"""Return triangle having side with length l on the x-axis."""
xy = Line((0, 0), slope=_slope(d1)).intersection(
Line((l, 0), slope=_slope(180 - d2)))[0]
return Triangle((0, 0), (l, 0), xy)
def _sss(l1, l2, l3):
"""Return triangle having side of length l1 on the x-axis."""
c1 = Circle((0, 0), l3)
c2 = Circle((l1, 0), l2)
inter = [a for a in c1.intersection(c2) if a.y.is_nonnegative]
if not inter:
return None
pt = inter[0]
return Triangle((0, 0), (l1, 0), pt)
def _sas(l1, d, l2):
"""Return triangle having side with length l2 on the x-axis."""
p1 = Point(0, 0)
p2 = Point(l2, 0)
p3 = Point(cos(rad(d))*l1, sin(rad(d))*l1)
return Triangle(p1, p2, p3)
|
709c266f504980b21d58df83b6c935185cb2cf26c1dfae0d1d3e4e826aeed871 | """
This module defines tensors with abstract index notation.
The abstract index notation has been first formalized by Penrose.
Tensor indices are formal objects, with a tensor type; there is no
notion of index range, it is only possible to assign the dimension,
used to trace the Kronecker delta; the dimension can be a Symbol.
The Einstein summation convention is used.
The covariant indices are indicated with a minus sign in front of the index.
For instance the tensor ``t = p(a)*A(b,c)*q(-c)`` has the index ``c``
contracted.
A tensor expression ``t`` can be called; called with its
indices in sorted order it is equal to itself:
in the above example ``t(a, b) == t``;
one can call ``t`` with different indices; ``t(c, d) == p(c)*A(d,a)*q(-a)``.
The contracted indices are dummy indices, internally they have no name,
the indices being represented by a graph-like structure.
Tensors are put in canonical form using ``canon_bp``, which uses
the Butler-Portugal algorithm for canonicalization using the monoterm
symmetries of the tensors.
If there is a (anti)symmetric metric, the indices can be raised and
lowered when the tensor is put in canonical form.
"""
from __future__ import print_function, division
from collections import defaultdict
import operator
import itertools
from sympy import Rational, prod, Integer
from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, \
bsgs_direct_product, canonicalize, riemann_bsgs
from sympy.core import Basic, Expr, sympify, Add, Mul, S
from sympy.core.compatibility import string_types, reduce, range, SYMPY_INTS
from sympy.core.containers import Tuple, Dict
from sympy.core.decorators import deprecated
from sympy.core.symbol import Symbol, symbols
from sympy.core.sympify import CantSympify, _sympify
from sympy.core.operations import AssocOp
from sympy.matrices import eye
from sympy.utilities.exceptions import SymPyDeprecationWarning
import warnings
@deprecated(useinstead=".replace_with_arrays", issue=15276, deprecated_since_version="1.4")
def deprecate_data():
pass
class _IndexStructure(CantSympify):
"""
This class handles the indices (free and dummy ones). It contains the
algorithms to manage the dummy indices replacements and contractions of
free indices under multiplications of tensor expressions, as well as stuff
related to canonicalization sorting, getting the permutation of the
expression and so on. It also includes tools to get the ``TensorIndex``
objects corresponding to the given index structure.
"""
def __init__(self, free, dum, index_types, indices, canon_bp=False):
self.free = free
self.dum = dum
self.index_types = index_types
self.indices = indices
self._ext_rank = len(self.free) + 2*len(self.dum)
self.dum.sort(key=lambda x: x[0])
@staticmethod
def from_indices(*indices):
"""
Create a new ``_IndexStructure`` object from a list of ``indices``
``indices`` ``TensorIndex`` objects, the indices. Contractions are
detected upon construction.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, _IndexStructure
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> _IndexStructure.from_indices(m0, m1, -m1, m3)
_IndexStructure([(m0, 0), (m3, 3)], [(1, 2)], [Lorentz, Lorentz, Lorentz, Lorentz])
In case of many components the same indices have slightly different
indexes:
>>> _IndexStructure.from_indices(m0, m1, -m1, m3)
_IndexStructure([(m0, 0), (m3, 3)], [(1, 2)], [Lorentz, Lorentz, Lorentz, Lorentz])
"""
free, dum = _IndexStructure._free_dum_from_indices(*indices)
index_types = [i.tensor_index_type for i in indices]
indices = _IndexStructure._replace_dummy_names(indices, free, dum)
return _IndexStructure(free, dum, index_types, indices)
@staticmethod
def from_components_free_dum(components, free, dum):
index_types = []
for component in components:
index_types.extend(component.index_types)
indices = _IndexStructure.generate_indices_from_free_dum_index_types(free, dum, index_types)
return _IndexStructure(free, dum, index_types, indices)
@staticmethod
def _free_dum_from_indices(*indices):
"""
Convert ``indices`` into ``free``, ``dum`` for single component tensor
``free`` list of tuples ``(index, pos, 0)``,
where ``pos`` is the position of index in
the list of indices formed by the component tensors
``dum`` list of tuples ``(pos_contr, pos_cov, 0, 0)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, \
_IndexStructure
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2, m3 = tensor_indices('m0,m1,m2,m3', Lorentz)
>>> _IndexStructure._free_dum_from_indices(m0, m1, -m1, m3)
([(m0, 0), (m3, 3)], [(1, 2)])
"""
n = len(indices)
if n == 1:
return [(indices[0], 0)], []
# find the positions of the free indices and of the dummy indices
free = [True]*len(indices)
index_dict = {}
dum = []
for i, index in enumerate(indices):
name = index._name
typ = index.tensor_index_type
contr = index._is_up
if (name, typ) in index_dict:
# found a pair of dummy indices
is_contr, pos = index_dict[(name, typ)]
# check consistency and update free
if is_contr:
if contr:
raise ValueError('two equal contravariant indices in slots %d and %d' %(pos, i))
else:
free[pos] = False
free[i] = False
else:
if contr:
free[pos] = False
free[i] = False
else:
raise ValueError('two equal covariant indices in slots %d and %d' %(pos, i))
if contr:
dum.append((i, pos))
else:
dum.append((pos, i))
else:
index_dict[(name, typ)] = index._is_up, i
free = [(index, i) for i, index in enumerate(indices) if free[i]]
free.sort()
return free, dum
def get_indices(self):
"""
Get a list of indices, creating new tensor indices to complete dummy indices.
"""
return self.indices[:]
@staticmethod
def generate_indices_from_free_dum_index_types(free, dum, index_types):
indices = [None]*(len(free)+2*len(dum))
for idx, pos in free:
indices[pos] = idx
generate_dummy_name = _IndexStructure._get_generator_for_dummy_indices(free)
for pos1, pos2 in dum:
typ1 = index_types[pos1]
indname = generate_dummy_name(typ1)
indices[pos1] = TensorIndex(indname, typ1, True)
indices[pos2] = TensorIndex(indname, typ1, False)
return _IndexStructure._replace_dummy_names(indices, free, dum)
@staticmethod
def _get_generator_for_dummy_indices(free):
cdt = defaultdict(int)
# if the free indices have names with dummy_fmt, start with an
# index higher than those for the dummy indices
# to avoid name collisions
for indx, ipos in free:
if indx._name.split('_')[0] == indx.tensor_index_type.dummy_fmt[:-3]:
cdt[indx.tensor_index_type] = max(cdt[indx.tensor_index_type], int(indx._name.split('_')[1]) + 1)
def dummy_fmt_gen(tensor_index_type):
fmt = tensor_index_type.dummy_fmt
nd = cdt[tensor_index_type]
cdt[tensor_index_type] += 1
return fmt % nd
return dummy_fmt_gen
@staticmethod
def _replace_dummy_names(indices, free, dum):
dum.sort(key=lambda x: x[0])
new_indices = [ind for ind in indices]
assert len(indices) == len(free) + 2*len(dum)
generate_dummy_name = _IndexStructure._get_generator_for_dummy_indices(free)
for ipos1, ipos2 in dum:
typ1 = new_indices[ipos1].tensor_index_type
indname = generate_dummy_name(typ1)
new_indices[ipos1] = TensorIndex(indname, typ1, True)
new_indices[ipos2] = TensorIndex(indname, typ1, False)
return new_indices
def get_free_indices(self):
"""
Get a list of free indices.
"""
# get sorted indices according to their position:
free = sorted(self.free, key=lambda x: x[1])
return [i[0] for i in free]
def __str__(self):
return "_IndexStructure({0}, {1}, {2})".format(self.free, self.dum, self.index_types)
def __repr__(self):
return self.__str__()
def _get_sorted_free_indices_for_canon(self):
sorted_free = self.free[:]
sorted_free.sort(key=lambda x: x[0])
return sorted_free
def _get_sorted_dum_indices_for_canon(self):
return sorted(self.dum, key=lambda x: x[0])
def _get_lexicographically_sorted_index_types(self):
permutation = self.indices_canon_args()[0]
index_types = [None]*self._ext_rank
for i, it in enumerate(self.index_types):
index_types[permutation(i)] = it
return index_types
def _get_lexicographically_sorted_indices(self):
permutation = self.indices_canon_args()[0]
indices = [None]*self._ext_rank
for i, it in enumerate(self.indices):
indices[permutation(i)] = it
return indices
def perm2tensor(self, g, is_canon_bp=False):
"""
Returns a ``_IndexStructure`` instance corresponding to the permutation ``g``
``g`` permutation corresponding to the tensor in the representation
used in canonicalization
``is_canon_bp`` if True, then ``g`` is the permutation
corresponding to the canonical form of the tensor
"""
sorted_free = [i[0] for i in self._get_sorted_free_indices_for_canon()]
lex_index_types = self._get_lexicographically_sorted_index_types()
lex_indices = self._get_lexicographically_sorted_indices()
nfree = len(sorted_free)
rank = self._ext_rank
dum = [[None]*2 for i in range((rank - nfree)//2)]
free = []
index_types = [None]*rank
indices = [None]*rank
for i in range(rank):
gi = g[i]
index_types[i] = lex_index_types[gi]
indices[i] = lex_indices[gi]
if gi < nfree:
ind = sorted_free[gi]
assert index_types[i] == sorted_free[gi].tensor_index_type
free.append((ind, i))
else:
j = gi - nfree
idum, cov = divmod(j, 2)
if cov:
dum[idum][1] = i
else:
dum[idum][0] = i
dum = [tuple(x) for x in dum]
return _IndexStructure(free, dum, index_types, indices)
def indices_canon_args(self):
"""
Returns ``(g, dummies, msym, v)``, the entries of ``canonicalize``
see ``canonicalize`` in ``tensor_can.py``
"""
# to be called after sorted_components
from sympy.combinatorics.permutations import _af_new
n = self._ext_rank
g = [None]*n + [n, n+1]
# ordered indices: first the free indices, ordered by types
# then the dummy indices, ordered by types and contravariant before
# covariant
# g[position in tensor] = position in ordered indices
for i, (indx, ipos) in enumerate(self._get_sorted_free_indices_for_canon()):
g[ipos] = i
pos = len(self.free)
j = len(self.free)
dummies = []
prev = None
a = []
msym = []
for ipos1, ipos2 in self._get_sorted_dum_indices_for_canon():
g[ipos1] = j
g[ipos2] = j + 1
j += 2
typ = self.index_types[ipos1]
if typ != prev:
if a:
dummies.append(a)
a = [pos, pos + 1]
prev = typ
msym.append(typ.metric_antisym)
else:
a.extend([pos, pos + 1])
pos += 2
if a:
dummies.append(a)
return _af_new(g), dummies, msym
def components_canon_args(components):
numtyp = []
prev = None
for t in components:
if t == prev:
numtyp[-1][1] += 1
else:
prev = t
numtyp.append([prev, 1])
v = []
for h, n in numtyp:
if h._comm == 0 or h._comm == 1:
comm = h._comm
else:
comm = TensorManager.get_comm(h._comm, h._comm)
v.append((h._symmetry.base, h._symmetry.generators, n, comm))
return v
class _TensorDataLazyEvaluator(CantSympify):
"""
EXPERIMENTAL: do not rely on this class, it may change without deprecation
warnings in future versions of SymPy.
This object contains the logic to associate components data to a tensor
expression. Components data are set via the ``.data`` property of tensor
expressions, is stored inside this class as a mapping between the tensor
expression and the ``ndarray``.
Computations are executed lazily: whereas the tensor expressions can have
contractions, tensor products, and additions, components data are not
computed until they are accessed by reading the ``.data`` property
associated to the tensor expression.
"""
_substitutions_dict = dict()
_substitutions_dict_tensmul = dict()
def __getitem__(self, key):
dat = self._get(key)
if dat is None:
return None
from .array import NDimArray
if not isinstance(dat, NDimArray):
return dat
if dat.rank() == 0:
return dat[()]
elif dat.rank() == 1 and len(dat) == 1:
return dat[0]
return dat
def _get(self, key):
"""
Retrieve ``data`` associated with ``key``.
This algorithm looks into ``self._substitutions_dict`` for all
``TensorHead`` in the ``TensExpr`` (or just ``TensorHead`` if key is a
TensorHead instance). It reconstructs the components data that the
tensor expression should have by performing on components data the
operations that correspond to the abstract tensor operations applied.
Metric tensor is handled in a different manner: it is pre-computed in
``self._substitutions_dict_tensmul``.
"""
if key in self._substitutions_dict:
return self._substitutions_dict[key]
if isinstance(key, TensorHead):
return None
if isinstance(key, Tensor):
# special case to handle metrics. Metric tensors cannot be
# constructed through contraction by the metric, their
# components show if they are a matrix or its inverse.
signature = tuple([i.is_up for i in key.get_indices()])
srch = (key.component,) + signature
if srch in self._substitutions_dict_tensmul:
return self._substitutions_dict_tensmul[srch]
array_list = [self.data_from_tensor(key)]
return self.data_contract_dum(array_list, key.dum, key.ext_rank)
if isinstance(key, TensMul):
tensmul_args = key.args
if len(tensmul_args) == 1 and len(tensmul_args[0].components) == 1:
# special case to handle metrics. Metric tensors cannot be
# constructed through contraction by the metric, their
# components show if they are a matrix or its inverse.
signature = tuple([i.is_up for i in tensmul_args[0].get_indices()])
srch = (tensmul_args[0].components[0],) + signature
if srch in self._substitutions_dict_tensmul:
return self._substitutions_dict_tensmul[srch]
#data_list = [self.data_from_tensor(i) for i in tensmul_args if isinstance(i, TensExpr)]
data_list = [self.data_from_tensor(i) if isinstance(i, Tensor) else i.data for i in tensmul_args if isinstance(i, TensExpr)]
coeff = prod([i for i in tensmul_args if not isinstance(i, TensExpr)])
if all([i is None for i in data_list]):
return None
if any([i is None for i in data_list]):
raise ValueError("Mixing tensors with associated components "\
"data with tensors without components data")
data_result = self.data_contract_dum(data_list, key.dum, key.ext_rank)
return coeff*data_result
if isinstance(key, TensAdd):
data_list = []
free_args_list = []
for arg in key.args:
if isinstance(arg, TensExpr):
data_list.append(arg.data)
free_args_list.append([x[0] for x in arg.free])
else:
data_list.append(arg)
free_args_list.append([])
if all([i is None for i in data_list]):
return None
if any([i is None for i in data_list]):
raise ValueError("Mixing tensors with associated components "\
"data with tensors without components data")
sum_list = []
from .array import permutedims
for data, free_args in zip(data_list, free_args_list):
if len(free_args) < 2:
sum_list.append(data)
else:
free_args_pos = {y: x for x, y in enumerate(free_args)}
axes = [free_args_pos[arg] for arg in key.free_args]
sum_list.append(permutedims(data, axes))
return reduce(lambda x, y: x+y, sum_list)
return None
@staticmethod
def data_contract_dum(ndarray_list, dum, ext_rank):
from .array import tensorproduct, tensorcontraction, MutableDenseNDimArray
arrays = list(map(MutableDenseNDimArray, ndarray_list))
prodarr = tensorproduct(*arrays)
return tensorcontraction(prodarr, *dum)
def data_tensorhead_from_tensmul(self, data, tensmul, tensorhead):
"""
This method is used when assigning components data to a ``TensMul``
object, it converts components data to a fully contravariant ndarray,
which is then stored according to the ``TensorHead`` key.
"""
if data is None:
return None
return self._correct_signature_from_indices(
data,
tensmul.get_indices(),
tensmul.free,
tensmul.dum,
True)
def data_from_tensor(self, tensor):
"""
This method corrects the components data to the right signature
(covariant/contravariant) using the metric associated with each
``TensorIndexType``.
"""
tensorhead = tensor.component
if tensorhead.data is None:
return None
return self._correct_signature_from_indices(
tensorhead.data,
tensor.get_indices(),
tensor.free,
tensor.dum)
def _assign_data_to_tensor_expr(self, key, data):
if isinstance(key, TensAdd):
raise ValueError('cannot assign data to TensAdd')
# here it is assumed that `key` is a `TensMul` instance.
if len(key.components) != 1:
raise ValueError('cannot assign data to TensMul with multiple components')
tensorhead = key.components[0]
newdata = self.data_tensorhead_from_tensmul(data, key, tensorhead)
return tensorhead, newdata
def _check_permutations_on_data(self, tens, data):
from .array import permutedims
if isinstance(tens, TensorHead):
rank = tens.rank
generators = tens.symmetry.generators
elif isinstance(tens, Tensor):
rank = tens.rank
generators = tens.components[0].symmetry.generators
elif isinstance(tens, TensorIndexType):
rank = tens.metric.rank
generators = tens.metric.symmetry.generators
# Every generator is a permutation, check that by permuting the array
# by that permutation, the array will be the same, except for a
# possible sign change if the permutation admits it.
for gener in generators:
sign_change = +1 if (gener(rank) == rank) else -1
data_swapped = data
last_data = data
permute_axes = list(map(gener, list(range(rank))))
# the order of a permutation is the number of times to get the
# identity by applying that permutation.
for i in range(gener.order()-1):
data_swapped = permutedims(data_swapped, permute_axes)
# if any value in the difference array is non-zero, raise an error:
if any(last_data - sign_change*data_swapped):
raise ValueError("Component data symmetry structure error")
last_data = data_swapped
def __setitem__(self, key, value):
"""
Set the components data of a tensor object/expression.
Components data are transformed to the all-contravariant form and stored
with the corresponding ``TensorHead`` object. If a ``TensorHead`` object
cannot be uniquely identified, it will raise an error.
"""
data = _TensorDataLazyEvaluator.parse_data(value)
self._check_permutations_on_data(key, data)
# TensorHead and TensorIndexType can be assigned data directly, while
# TensMul must first convert data to a fully contravariant form, and
# assign it to its corresponding TensorHead single component.
if not isinstance(key, (TensorHead, TensorIndexType)):
key, data = self._assign_data_to_tensor_expr(key, data)
if isinstance(key, TensorHead):
for dim, indextype in zip(data.shape, key.index_types):
if indextype.data is None:
raise ValueError("index type {} has no components data"\
" associated (needed to raise/lower index)".format(indextype))
if indextype.dim is None:
continue
if dim != indextype.dim:
raise ValueError("wrong dimension of ndarray")
self._substitutions_dict[key] = data
def __delitem__(self, key):
del self._substitutions_dict[key]
def __contains__(self, key):
return key in self._substitutions_dict
def add_metric_data(self, metric, data):
"""
Assign data to the ``metric`` tensor. The metric tensor behaves in an
anomalous way when raising and lowering indices.
A fully covariant metric is the inverse transpose of the fully
contravariant metric (it is meant matrix inverse). If the metric is
symmetric, the transpose is not necessary and mixed
covariant/contravariant metrics are Kronecker deltas.
"""
# hard assignment, data should not be added to `TensorHead` for metric:
# the problem with `TensorHead` is that the metric is anomalous, i.e.
# raising and lowering the index means considering the metric or its
# inverse, this is not the case for other tensors.
self._substitutions_dict_tensmul[metric, True, True] = data
inverse_transpose = self.inverse_transpose_matrix(data)
# in symmetric spaces, the traspose is the same as the original matrix,
# the full covariant metric tensor is the inverse transpose, so this
# code will be able to handle non-symmetric metrics.
self._substitutions_dict_tensmul[metric, False, False] = inverse_transpose
# now mixed cases, these are identical to the unit matrix if the metric
# is symmetric.
m = data.tomatrix()
invt = inverse_transpose.tomatrix()
self._substitutions_dict_tensmul[metric, True, False] = m * invt
self._substitutions_dict_tensmul[metric, False, True] = invt * m
@staticmethod
def _flip_index_by_metric(data, metric, pos):
from .array import tensorproduct, tensorcontraction
mdim = metric.rank()
ddim = data.rank()
if pos == 0:
data = tensorcontraction(
tensorproduct(
metric,
data
),
(1, mdim+pos)
)
else:
data = tensorcontraction(
tensorproduct(
data,
metric
),
(pos, ddim)
)
return data
@staticmethod
def inverse_matrix(ndarray):
m = ndarray.tomatrix().inv()
return _TensorDataLazyEvaluator.parse_data(m)
@staticmethod
def inverse_transpose_matrix(ndarray):
m = ndarray.tomatrix().inv().T
return _TensorDataLazyEvaluator.parse_data(m)
@staticmethod
def _correct_signature_from_indices(data, indices, free, dum, inverse=False):
"""
Utility function to correct the values inside the components data
ndarray according to whether indices are covariant or contravariant.
It uses the metric matrix to lower values of covariant indices.
"""
# change the ndarray values according covariantness/contravariantness of the indices
# use the metric
for i, indx in enumerate(indices):
if not indx.is_up and not inverse:
data = _TensorDataLazyEvaluator._flip_index_by_metric(data, indx.tensor_index_type.data, i)
elif not indx.is_up and inverse:
data = _TensorDataLazyEvaluator._flip_index_by_metric(
data,
_TensorDataLazyEvaluator.inverse_matrix(indx.tensor_index_type.data),
i
)
return data
@staticmethod
def _sort_data_axes(old, new):
from .array import permutedims
new_data = old.data.copy()
old_free = [i[0] for i in old.free]
new_free = [i[0] for i in new.free]
for i in range(len(new_free)):
for j in range(i, len(old_free)):
if old_free[j] == new_free[i]:
old_free[i], old_free[j] = old_free[j], old_free[i]
new_data = permutedims(new_data, (i, j))
break
return new_data
@staticmethod
def add_rearrange_tensmul_parts(new_tensmul, old_tensmul):
def sorted_compo():
return _TensorDataLazyEvaluator._sort_data_axes(old_tensmul, new_tensmul)
_TensorDataLazyEvaluator._substitutions_dict[new_tensmul] = sorted_compo()
@staticmethod
def parse_data(data):
"""
Transform ``data`` to array. The parameter ``data`` may
contain data in various formats, e.g. nested lists, sympy ``Matrix``,
and so on.
Examples
========
>>> from sympy.tensor.tensor import _TensorDataLazyEvaluator
>>> _TensorDataLazyEvaluator.parse_data([1, 3, -6, 12])
[1, 3, -6, 12]
>>> _TensorDataLazyEvaluator.parse_data([[1, 2], [4, 7]])
[[1, 2], [4, 7]]
"""
from .array import MutableDenseNDimArray
if not isinstance(data, MutableDenseNDimArray):
if len(data) == 2 and hasattr(data[0], '__call__'):
data = MutableDenseNDimArray(data[0], data[1])
else:
data = MutableDenseNDimArray(data)
return data
_tensor_data_substitution_dict = _TensorDataLazyEvaluator()
class _TensorManager(object):
"""
Class to manage tensor properties.
Notes
=====
Tensors belong to tensor commutation groups; each group has a label
``comm``; there are predefined labels:
``0`` tensors commuting with any other tensor
``1`` tensors anticommuting among themselves
``2`` tensors not commuting, apart with those with ``comm=0``
Other groups can be defined using ``set_comm``; tensors in those
groups commute with those with ``comm=0``; by default they
do not commute with any other group.
"""
def __init__(self):
self._comm_init()
def _comm_init(self):
self._comm = [{} for i in range(3)]
for i in range(3):
self._comm[0][i] = 0
self._comm[i][0] = 0
self._comm[1][1] = 1
self._comm[2][1] = None
self._comm[1][2] = None
self._comm_symbols2i = {0:0, 1:1, 2:2}
self._comm_i2symbol = {0:0, 1:1, 2:2}
@property
def comm(self):
return self._comm
def comm_symbols2i(self, i):
"""
get the commutation group number corresponding to ``i``
``i`` can be a symbol or a number or a string
If ``i`` is not already defined its commutation group number
is set.
"""
if i not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[n][0] = 0
self._comm[0][n] = 0
self._comm_symbols2i[i] = n
self._comm_i2symbol[n] = i
return n
return self._comm_symbols2i[i]
def comm_i2symbol(self, i):
"""
Returns the symbol corresponding to the commutation group number.
"""
return self._comm_i2symbol[i]
def set_comm(self, i, j, c):
"""
set the commutation parameter ``c`` for commutation groups ``i, j``
Parameters
==========
i, j : symbols representing commutation groups
c : group commutation number
Notes
=====
``i, j`` can be symbols, strings or numbers,
apart from ``0, 1`` and ``2`` which are reserved respectively
for commuting, anticommuting tensors and tensors not commuting
with any other group apart with the commuting tensors.
For the remaining cases, use this method to set the commutation rules;
by default ``c=None``.
The group commutation number ``c`` is assigned in correspondence
to the group commutation symbols; it can be
0 commuting
1 anticommuting
None no commutation property
Examples
========
``G`` and ``GH`` do not commute with themselves and commute with
each other; A is commuting.
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, TensorManager
>>> Lorentz = TensorIndexType('Lorentz')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> A = tensorhead('A', [Lorentz], [[1]])
>>> G = tensorhead('G', [Lorentz], [[1]], 'Gcomm')
>>> GH = tensorhead('GH', [Lorentz], [[1]], 'GHcomm')
>>> TensorManager.set_comm('Gcomm', 'GHcomm', 0)
>>> (GH(i1)*G(i0)).canon_bp()
G(i0)*GH(i1)
>>> (G(i1)*G(i0)).canon_bp()
G(i1)*G(i0)
>>> (G(i1)*A(i0)).canon_bp()
A(i0)*G(i1)
"""
if c not in (0, 1, None):
raise ValueError('`c` can assume only the values 0, 1 or None')
if i not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[n][0] = 0
self._comm[0][n] = 0
self._comm_symbols2i[i] = n
self._comm_i2symbol[n] = i
if j not in self._comm_symbols2i:
n = len(self._comm)
self._comm.append({})
self._comm[0][n] = 0
self._comm[n][0] = 0
self._comm_symbols2i[j] = n
self._comm_i2symbol[n] = j
ni = self._comm_symbols2i[i]
nj = self._comm_symbols2i[j]
self._comm[ni][nj] = c
self._comm[nj][ni] = c
def set_comms(self, *args):
"""
set the commutation group numbers ``c`` for symbols ``i, j``
Parameters
==========
args : sequence of ``(i, j, c)``
"""
for i, j, c in args:
self.set_comm(i, j, c)
def get_comm(self, i, j):
"""
Return the commutation parameter for commutation group numbers ``i, j``
see ``_TensorManager.set_comm``
"""
return self._comm[i].get(j, 0 if i == 0 or j == 0 else None)
def clear(self):
"""
Clear the TensorManager.
"""
self._comm_init()
TensorManager = _TensorManager()
class TensorIndexType(Basic):
"""
A TensorIndexType is characterized by its name and its metric.
Parameters
==========
name : name of the tensor type
metric : metric symmetry or metric object or ``None``
dim : dimension, it can be a symbol or an integer or ``None``
eps_dim : dimension of the epsilon tensor
dummy_fmt : name of the head of dummy indices
Attributes
==========
``name``
``metric_name`` : it is 'metric' or metric.name
``metric_antisym``
``metric`` : the metric tensor
``delta`` : ``Kronecker delta``
``epsilon`` : the ``Levi-Civita epsilon`` tensor
``dim``
``eps_dim``
``dummy_fmt``
``data`` : a property to add ``ndarray`` values, to work in a specified basis.
Notes
=====
The ``metric`` parameter can be:
``metric = False`` symmetric metric (in Riemannian geometry)
``metric = True`` antisymmetric metric (for spinor calculus)
``metric = None`` there is no metric
``metric`` can be an object having ``name`` and ``antisym`` attributes.
If there is a metric the metric is used to raise and lower indices.
In the case of antisymmetric metric, the following raising and
lowering conventions will be adopted:
``psi(a) = g(a, b)*psi(-b); chi(-a) = chi(b)*g(-b, -a)``
``g(-a, b) = delta(-a, b); g(b, -a) = -delta(a, -b)``
where ``delta(-a, b) = delta(b, -a)`` is the ``Kronecker delta``
(see ``TensorIndex`` for the conventions on indices).
If there is no metric it is not possible to raise or lower indices;
e.g. the index of the defining representation of ``SU(N)``
is 'covariant' and the conjugate representation is
'contravariant'; for ``N > 2`` they are linearly independent.
``eps_dim`` is by default equal to ``dim``, if the latter is an integer;
else it can be assigned (for use in naive dimensional regularization);
if ``eps_dim`` is not an integer ``epsilon`` is ``None``.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> Lorentz.metric
metric(Lorentz,Lorentz)
"""
def __new__(cls, name, metric=False, dim=None, eps_dim=None,
dummy_fmt=None):
if isinstance(name, string_types):
name = Symbol(name)
obj = Basic.__new__(cls, name, S.One if metric else S.Zero)
obj._name = str(name)
if not dummy_fmt:
obj._dummy_fmt = '%s_%%d' % obj.name
else:
obj._dummy_fmt = '%s_%%d' % dummy_fmt
if metric is None:
obj.metric_antisym = None
obj.metric = None
else:
if metric in (True, False, 0, 1):
metric_name = 'metric'
obj.metric_antisym = metric
else:
metric_name = metric.name
obj.metric_antisym = metric.antisym
sym2 = TensorSymmetry(get_symmetric_group_sgs(2, obj.metric_antisym))
S2 = TensorType([obj]*2, sym2)
obj.metric = S2(metric_name)
obj._dim = dim
obj._delta = obj.get_kronecker_delta()
obj._eps_dim = eps_dim if eps_dim else dim
obj._epsilon = obj.get_epsilon()
obj._autogenerated = []
return obj
@property
@deprecated(useinstead="TensorIndex", issue=12857, deprecated_since_version="1.1")
def auto_right(self):
if not hasattr(self, '_auto_right'):
self._auto_right = TensorIndex("auto_right", self)
return self._auto_right
@property
@deprecated(useinstead="TensorIndex", issue=12857, deprecated_since_version="1.1")
def auto_left(self):
if not hasattr(self, '_auto_left'):
self._auto_left = TensorIndex("auto_left", self)
return self._auto_left
@property
@deprecated(useinstead="TensorIndex", issue=12857, deprecated_since_version="1.1")
def auto_index(self):
if not hasattr(self, '_auto_index'):
self._auto_index = TensorIndex("auto_index", self)
return self._auto_index
@property
def data(self):
deprecate_data()
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
deprecate_data()
# This assignment is a bit controversial, should metric components be assigned
# to the metric only or also to the TensorIndexType object? The advantage here
# is the ability to assign a 1D array and transform it to a 2D diagonal array.
from .array import MutableDenseNDimArray
data = _TensorDataLazyEvaluator.parse_data(data)
if data.rank() > 2:
raise ValueError("data have to be of rank 1 (diagonal metric) or 2.")
if data.rank() == 1:
if self.dim is not None:
nda_dim = data.shape[0]
if nda_dim != self.dim:
raise ValueError("Dimension mismatch")
dim = data.shape[0]
newndarray = MutableDenseNDimArray.zeros(dim, dim)
for i, val in enumerate(data):
newndarray[i, i] = val
data = newndarray
dim1, dim2 = data.shape
if dim1 != dim2:
raise ValueError("Non-square matrix tensor.")
if self.dim is not None:
if self.dim != dim1:
raise ValueError("Dimension mismatch")
_tensor_data_substitution_dict[self] = data
_tensor_data_substitution_dict.add_metric_data(self.metric, data)
delta = self.get_kronecker_delta()
i1 = TensorIndex('i1', self)
i2 = TensorIndex('i2', self)
delta(i1, -i2).data = _TensorDataLazyEvaluator.parse_data(eye(dim1))
@data.deleter
def data(self):
deprecate_data()
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
if self.metric in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self.metric]
def _get_matrix_fmt(self, number):
return ("m" + self.dummy_fmt) % (number)
@property
def name(self):
return self._name
@property
def dim(self):
return self._dim
@property
def delta(self):
return self._delta
@property
def eps_dim(self):
return self._eps_dim
@property
def epsilon(self):
return self._epsilon
@property
def dummy_fmt(self):
return self._dummy_fmt
def get_kronecker_delta(self):
sym2 = TensorSymmetry(get_symmetric_group_sgs(2))
S2 = TensorType([self]*2, sym2)
delta = S2('KD')
return delta
def get_epsilon(self):
if not isinstance(self._eps_dim, (SYMPY_INTS, Integer)):
return None
sym = TensorSymmetry(get_symmetric_group_sgs(self._eps_dim, 1))
Sdim = TensorType([self]*self._eps_dim, sym)
epsilon = Sdim('Eps')
return epsilon
def __lt__(self, other):
return self.name < other.name
def __str__(self):
return self.name
__repr__ = __str__
def _components_data_full_destroy(self):
"""
EXPERIMENTAL: do not rely on this API method.
This destroys components data associated to the ``TensorIndexType``, if
any, specifically:
* metric tensor data
* Kronecker tensor data
"""
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def delete_tensmul_data(key):
if key in _tensor_data_substitution_dict._substitutions_dict_tensmul:
del _tensor_data_substitution_dict._substitutions_dict_tensmul[key]
# delete metric data:
delete_tensmul_data((self.metric, True, True))
delete_tensmul_data((self.metric, True, False))
delete_tensmul_data((self.metric, False, True))
delete_tensmul_data((self.metric, False, False))
# delete delta tensor data:
delta = self.get_kronecker_delta()
if delta in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[delta]
class TensorIndex(Basic):
"""
Represents an abstract tensor index.
Parameters
==========
name : name of the index, or ``True`` if you want it to be automatically assigned
tensortype : ``TensorIndexType`` of the index
is_up : flag for contravariant index
Attributes
==========
``name``
``tensortype``
``is_up``
Notes
=====
Tensor indices are contracted with the Einstein summation convention.
An index can be in contravariant or in covariant form; in the latter
case it is represented prepending a ``-`` to the index name.
Dummy indices have a name with head given by ``tensortype._dummy_fmt``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, TensorIndex, TensorSymmetry, TensorType, get_symmetric_group_sgs
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i = TensorIndex('i', Lorentz); i
i
>>> sym1 = TensorSymmetry(*get_symmetric_group_sgs(1))
>>> S1 = TensorType([Lorentz], sym1)
>>> A, B = S1('A,B')
>>> A(i)*B(-i)
A(L_0)*B(-L_0)
If you want the index name to be automatically assigned, just put ``True``
in the ``name`` field, it will be generated using the reserved character
``_`` in front of its name, in order to avoid conflicts with possible
existing indices:
>>> i0 = TensorIndex(True, Lorentz)
>>> i0
_i0
>>> i1 = TensorIndex(True, Lorentz)
>>> i1
_i1
>>> A(i0)*B(-i1)
A(_i0)*B(-_i1)
>>> A(i0)*B(-i0)
A(L_0)*B(-L_0)
"""
def __new__(cls, name, tensortype, is_up=True):
if isinstance(name, string_types):
name_symbol = Symbol(name)
elif isinstance(name, Symbol):
name_symbol = name
elif name is True:
name = "_i{0}".format(len(tensortype._autogenerated))
name_symbol = Symbol(name)
tensortype._autogenerated.append(name_symbol)
else:
raise ValueError("invalid name")
is_up = sympify(is_up)
obj = Basic.__new__(cls, name_symbol, tensortype, is_up)
obj._name = str(name)
obj._tensor_index_type = tensortype
obj._is_up = is_up
return obj
@property
def name(self):
return self._name
@property
@deprecated(useinstead="tensor_index_type", issue=12857, deprecated_since_version="1.1")
def tensortype(self):
return self.tensor_index_type
@property
def tensor_index_type(self):
return self._tensor_index_type
@property
def is_up(self):
return self._is_up
def _print(self):
s = self._name
if not self._is_up:
s = '-%s' % s
return s
def __lt__(self, other):
return (self.tensor_index_type, self._name) < (other.tensor_index_type, other._name)
def __neg__(self):
t1 = TensorIndex(self.name, self.tensor_index_type,
(not self.is_up))
return t1
def tensor_indices(s, typ):
"""
Returns list of tensor indices given their names and their types
Parameters
==========
s : string of comma separated names of indices
typ : ``TensorIndexType`` of the indices
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
"""
if isinstance(s, string_types):
a = [x.name for x in symbols(s, seq=True)]
else:
raise ValueError('expecting a string')
tilist = [TensorIndex(i, typ) for i in a]
if len(tilist) == 1:
return tilist[0]
return tilist
class TensorSymmetry(Basic):
"""
Monoterm symmetry of a tensor
Parameters
==========
bsgs : tuple ``(base, sgs)`` BSGS of the symmetry of the tensor
Attributes
==========
``base`` : base of the BSGS
``generators`` : generators of the BSGS
``rank`` : rank of the tensor
Notes
=====
A tensor can have an arbitrary monoterm symmetry provided by its BSGS.
Multiterm symmetries, like the cyclic symmetry of the Riemann tensor,
are not covered.
See Also
========
sympy.combinatorics.tensor_can.get_symmetric_group_sgs
Examples
========
Define a symmetric tensor
>>> from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = TensorSymmetry(get_symmetric_group_sgs(2))
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
def __new__(cls, *args, **kw_args):
if len(args) == 1:
base, generators = args[0]
elif len(args) == 2:
base, generators = args
else:
raise TypeError("bsgs required, either two separate parameters or one tuple")
if not isinstance(base, Tuple):
base = Tuple(*base)
if not isinstance(generators, Tuple):
generators = Tuple(*generators)
obj = Basic.__new__(cls, base, generators, **kw_args)
return obj
@property
def base(self):
return self.args[0]
@property
def generators(self):
return self.args[1]
@property
def rank(self):
return self.args[1][0].size - 2
def tensorsymmetry(*args):
"""
Return a ``TensorSymmetry`` object.
One can represent a tensor with any monoterm slot symmetry group
using a BSGS.
``args`` can be a BSGS
``args[0]`` base
``args[1]`` sgs
Usually tensors are in (direct products of) representations
of the symmetric group;
``args`` can be a list of lists representing the shapes of Young tableaux
Notes
=====
For instance:
``[[1]]`` vector
``[[1]*n]`` symmetric tensor of rank ``n``
``[[n]]`` antisymmetric tensor of rank ``n``
``[[2, 2]]`` monoterm slot symmetry of the Riemann tensor
``[[1],[1]]`` vector*vector
``[[2],[1],[1]`` (antisymmetric tensor)*vector*vector
Notice that with the shape ``[2, 2]`` we associate only the monoterm
symmetries of the Riemann tensor; this is an abuse of notation,
since the shape ``[2, 2]`` corresponds usually to the irreducible
representation characterized by the monoterm symmetries and by the
cyclic symmetry.
Examples
========
Symmetric tensor using a Young tableau
>>> from sympy.tensor.tensor import TensorIndexType, TensorType, tensorsymmetry
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1, 1])
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
Symmetric tensor using a ``BSGS`` (base, strong generator set)
>>> from sympy.tensor.tensor import get_symmetric_group_sgs
>>> sym2 = tensorsymmetry(*get_symmetric_group_sgs(2))
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
from sympy.combinatorics import Permutation
def tableau2bsgs(a):
if len(a) == 1:
# antisymmetric vector
n = a[0]
bsgs = get_symmetric_group_sgs(n, 1)
else:
if all(x == 1 for x in a):
# symmetric vector
n = len(a)
bsgs = get_symmetric_group_sgs(n)
elif a == [2, 2]:
bsgs = riemann_bsgs
else:
raise NotImplementedError
return bsgs
if not args:
return TensorSymmetry(Tuple(), Tuple(Permutation(1)))
if len(args) == 2 and isinstance(args[1][0], Permutation):
return TensorSymmetry(args)
base, sgs = tableau2bsgs(args[0])
for a in args[1:]:
basex, sgsx = tableau2bsgs(a)
base, sgs = bsgs_direct_product(base, sgs, basex, sgsx)
return TensorSymmetry(Tuple(base, sgs))
class TensorType(Basic):
"""
Class of tensor types.
Parameters
==========
index_types : list of ``TensorIndexType`` of the tensor indices
symmetry : ``TensorSymmetry`` of the tensor
Attributes
==========
``index_types``
``symmetry``
``types`` : list of ``TensorIndexType`` without repetitions
Examples
========
Define a symmetric tensor
>>> from sympy.tensor.tensor import TensorIndexType, tensorsymmetry, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> sym2 = tensorsymmetry([1, 1])
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
"""
is_commutative = False
def __new__(cls, index_types, symmetry, **kw_args):
assert symmetry.rank == len(index_types)
obj = Basic.__new__(cls, Tuple(*index_types), symmetry, **kw_args)
return obj
@property
def index_types(self):
return self.args[0]
@property
def symmetry(self):
return self.args[1]
@property
def types(self):
return sorted(set(self.index_types), key=lambda x: x.name)
def __str__(self):
return 'TensorType(%s)' % ([str(x) for x in self.index_types])
def __call__(self, s, comm=0):
"""
Return a TensorHead object or a list of TensorHead objects.
``s`` name or string of names
``comm``: commutation group number
see ``_TensorManager.set_comm``
Examples
========
Define symmetric tensors ``V``, ``W`` and ``G``, respectively
commuting, anticommuting and with no commutation symmetry
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorsymmetry, TensorType, canon_bp
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> sym2 = tensorsymmetry([1]*2)
>>> S2 = TensorType([Lorentz]*2, sym2)
>>> V = S2('V')
>>> W = S2('W', 1)
>>> G = S2('G', 2)
>>> canon_bp(V(a, b)*V(-b, -a))
V(L_0, L_1)*V(-L_0, -L_1)
>>> canon_bp(W(a, b)*W(-b, -a))
0
"""
if isinstance(s, string_types):
names = [x.name for x in symbols(s, seq=True)]
else:
raise ValueError('expecting a string')
if len(names) == 1:
return TensorHead(names[0], self, comm)
else:
return [TensorHead(name, self, comm) for name in names]
def tensorhead(name, typ, sym=None, comm=0):
"""
Function generating tensorhead(s).
Parameters
==========
name : name or sequence of names (as in ``symbol``)
typ : index types
sym : same as ``*args`` in ``tensorsymmetry``
comm : commutation group number
see ``_TensorManager.set_comm``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[1]*2])
>>> A(a, -b)
A(a, -b)
If no symmetry parameter is provided, assume there are not index
symmetries:
>>> B = tensorhead('B', [Lorentz, Lorentz])
>>> B(a, -b)
B(a, -b)
"""
if sym is None:
sym = [[1] for i in range(len(typ))]
sym = tensorsymmetry(*sym)
S = TensorType(typ, sym)
th = S(name, comm)
return th
class TensorHead(Basic):
r"""
Tensor head of the tensor
Parameters
==========
name : name of the tensor
typ : list of TensorIndexType
comm : commutation group number
Attributes
==========
``name``
``index_types``
``rank``
``types`` : equal to ``typ.types``
``symmetry`` : equal to ``typ.symmetry``
``comm`` : commutation group
Notes
=====
A ``TensorHead`` belongs to a commutation group, defined by a
symbol on number ``comm`` (see ``_TensorManager.set_comm``);
tensors in a commutation group have the same commutation properties;
by default ``comm`` is ``0``, the group of the commuting tensors.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorhead, TensorType
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> A = tensorhead('A', [Lorentz, Lorentz], [[1],[1]])
Examples with ndarray values, the components data assigned to the
``TensorHead`` object are assumed to be in a fully-contravariant
representation. In case it is necessary to assign components data which
represents the values of a non-fully covariant tensor, see the other
examples.
>>> from sympy.tensor.tensor import tensor_indices, tensorhead
>>> from sympy import diag
>>> i0, i1 = tensor_indices('i0:2', Lorentz)
Specify a replacement dictionary to keep track of the arrays to use for
replacements in the tensorial expression. The ``TensorIndexType`` is
associated to the metric used for contractions (in fully covariant form):
>>> repl = {Lorentz: diag(1, -1, -1, -1)}
Let's see some examples of working with components with the electromagnetic
tensor:
>>> from sympy import symbols
>>> Ex, Ey, Ez, Bx, By, Bz = symbols('E_x E_y E_z B_x B_y B_z')
>>> c = symbols('c', positive=True)
Let's define `F`, an antisymmetric tensor, we have to assign an
antisymmetric matrix to it, because `[[2]]` stands for the Young tableau
representation of an antisymmetric set of two elements:
>>> F = tensorhead('F', [Lorentz, Lorentz], [[2]])
Let's update the dictionary to contain the matrix to use in the
replacements:
>>> repl.update({F(-i0, -i1): [
... [0, Ex/c, Ey/c, Ez/c],
... [-Ex/c, 0, -Bz, By],
... [-Ey/c, Bz, 0, -Bx],
... [-Ez/c, -By, Bx, 0]]})
Now it is possible to retrieve the contravariant form of the Electromagnetic
tensor:
>>> F(i0, i1).replace_with_arrays(repl, [i0, i1])
[[0, -E_x/c, -E_y/c, -E_z/c], [E_x/c, 0, -B_z, B_y], [E_y/c, B_z, 0, -B_x], [E_z/c, -B_y, B_x, 0]]
and the mixed contravariant-covariant form:
>>> F(i0, -i1).replace_with_arrays(repl, [i0, -i1])
[[0, E_x/c, E_y/c, E_z/c], [E_x/c, 0, B_z, -B_y], [E_y/c, -B_z, 0, B_x], [E_z/c, B_y, -B_x, 0]]
Energy-momentum of a particle may be represented as:
>>> from sympy import symbols
>>> P = tensorhead('P', [Lorentz], [[1]])
>>> E, px, py, pz = symbols('E p_x p_y p_z', positive=True)
>>> repl.update({P(i0): [E, px, py, pz]})
The contravariant and covariant components are, respectively:
>>> P(i0).replace_with_arrays(repl, [i0])
[E, p_x, p_y, p_z]
>>> P(-i0).replace_with_arrays(repl, [-i0])
[E, -p_x, -p_y, -p_z]
The contraction of a 1-index tensor by itself:
>>> expr = P(i0)*P(-i0)
>>> expr.replace_with_arrays(repl, [])
E**2 - p_x**2 - p_y**2 - p_z**2
"""
is_commutative = False
def __new__(cls, name, typ, comm=0, **kw_args):
if isinstance(name, string_types):
name_symbol = Symbol(name)
elif isinstance(name, Symbol):
name_symbol = name
else:
raise ValueError("invalid name")
comm2i = TensorManager.comm_symbols2i(comm)
obj = Basic.__new__(cls, name_symbol, typ, **kw_args)
obj._name = obj.args[0].name
obj._rank = len(obj.index_types)
obj._symmetry = typ.symmetry
obj._comm = comm2i
return obj
@property
def name(self):
return self._name
@property
def rank(self):
return self._rank
@property
def symmetry(self):
return self._symmetry
@property
def typ(self):
return self.args[1]
@property
def comm(self):
return self._comm
@property
def types(self):
return self.args[1].types[:]
@property
def index_types(self):
return self.args[1].index_types[:]
def __lt__(self, other):
return (self.name, self.index_types) < (other.name, other.index_types)
def commutes_with(self, other):
"""
Returns ``0`` if ``self`` and ``other`` commute, ``1`` if they anticommute.
Returns ``None`` if ``self`` and ``other`` neither commute nor anticommute.
"""
r = TensorManager.get_comm(self._comm, other._comm)
return r
def _print(self):
return '%s(%s)' %(self.name, ','.join([str(x) for x in self.index_types]))
def __call__(self, *indices, **kw_args):
"""
Returns a tensor with indices.
There is a special behavior in case of indices denoted by ``True``,
they are considered auto-matrix indices, their slots are automatically
filled, and confer to the tensor the behavior of a matrix or vector
upon multiplication with another tensor containing auto-matrix indices
of the same ``TensorIndexType``. This means indices get summed over the
same way as in matrix multiplication. For matrix behavior, define two
auto-matrix indices, for vector behavior define just one.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[1]*2])
>>> t = A(a, -b)
>>> t
A(a, -b)
"""
tensor = Tensor(self, indices, **kw_args)
return tensor.doit()
def __pow__(self, other):
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
if self.data is None:
raise ValueError("No power on abstract tensors.")
deprecate_data()
from .array import tensorproduct, tensorcontraction
metrics = [_.data for _ in self.args[1].args[0]]
marray = self.data
marraydim = marray.rank()
for metric in metrics:
marray = tensorproduct(marray, metric, marray)
marray = tensorcontraction(marray, (0, marraydim), (marraydim+1, marraydim+2))
return marray ** (Rational(1, 2) * other)
@property
def data(self):
deprecate_data()
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
deprecate_data()
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
deprecate_data()
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def __iter__(self):
deprecate_data()
return self.data.__iter__()
def _components_data_full_destroy(self):
"""
EXPERIMENTAL: do not rely on this API method.
Destroy components data associated to the ``TensorHead`` object, this
checks for attached components data, and destroys components data too.
"""
# do not garbage collect Kronecker tensor (it should be done by
# ``TensorIndexType`` garbage collection)
if self.name == "KD":
return
# the data attached to a tensor must be deleted only by the TensorHead
# destructor. If the TensorHead is deleted, it means that there are no
# more instances of that tensor anywhere.
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def _get_argtree_pos(expr, pos):
for p in pos:
expr = expr.args[p]
return expr
class TensExpr(Expr):
"""
Abstract base class for tensor expressions
Notes
=====
A tensor expression is an expression formed by tensors;
currently the sums of tensors are distributed.
A ``TensExpr`` can be a ``TensAdd`` or a ``TensMul``.
``TensAdd`` objects are put in canonical form using the Butler-Portugal
algorithm for canonicalization under monoterm symmetries.
``TensMul`` objects are formed by products of component tensors,
and include a coefficient, which is a SymPy expression.
In the internal representation contracted indices are represented
by ``(ipos1, ipos2, icomp1, icomp2)``, where ``icomp1`` is the position
of the component tensor with contravariant index, ``ipos1`` is the
slot which the index occupies in that component tensor.
Contracted indices are therefore nameless in the internal representation.
"""
_op_priority = 12.0
is_commutative = False
def __neg__(self):
return self*S.NegativeOne
def __abs__(self):
raise NotImplementedError
def __add__(self, other):
return TensAdd(self, other).doit()
def __radd__(self, other):
return TensAdd(other, self).doit()
def __sub__(self, other):
return TensAdd(self, -other).doit()
def __rsub__(self, other):
return TensAdd(other, -self).doit()
def __mul__(self, other):
"""
Multiply two tensors using Einstein summation convention.
If the two tensors have an index in common, one contravariant
and the other covariant, in their product the indices are summed
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t1 = p(m0)
>>> t2 = q(-m0)
>>> t1*t2
p(L_0)*q(-L_0)
"""
return TensMul(self, other).doit()
def __rmul__(self, other):
return TensMul(other, self).doit()
def __div__(self, other):
other = _sympify(other)
if isinstance(other, TensExpr):
raise ValueError('cannot divide by a tensor')
return TensMul(self, S.One/other).doit()
def __rdiv__(self, other):
raise ValueError('cannot divide by a tensor')
def __pow__(self, other):
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
if self.data is None:
raise ValueError("No power without ndarray data.")
deprecate_data()
from .array import tensorproduct, tensorcontraction
free = self.free
marray = self.data
mdim = marray.rank()
for metric in free:
marray = tensorcontraction(
tensorproduct(
marray,
metric[0].tensor_index_type.data,
marray),
(0, mdim), (mdim+1, mdim+2)
)
return marray ** (Rational(1, 2) * other)
def __rpow__(self, other):
raise NotImplementedError
__truediv__ = __div__
__rtruediv__ = __rdiv__
def fun_eval(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
``index_types`` list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.fun_eval((i, k),(-j, l))
A(k, L_0)*B(-L_0, l)
"""
expr = self.xreplace(dict(index_tuples))
expr = expr.replace(lambda x: isinstance(x, Tensor), lambda x: x.args[0](*x.args[1]))
# For some reason, `TensMul` gets replaced by `Mul`, correct it:
expr = expr.replace(lambda x: isinstance(x, (Mul, TensMul)), lambda x: TensMul(*x.args).doit())
return expr
def get_matrix(self):
"""
DEPRECATED: do not use.
Returns ndarray components data as a matrix, if components data are
available and ndarray dimension does not exceed 2.
"""
from sympy import Matrix
deprecate_data()
if 0 < self.rank <= 2:
rows = self.data.shape[0]
columns = self.data.shape[1] if self.rank == 2 else 1
if self.rank == 2:
mat_list = [] * rows
for i in range(rows):
mat_list.append([])
for j in range(columns):
mat_list[i].append(self[i, j])
else:
mat_list = [None] * rows
for i in range(rows):
mat_list[i] = self[i]
return Matrix(mat_list)
else:
raise NotImplementedError(
"missing multidimensional reduction to matrix.")
@staticmethod
def _get_indices_permutation(indices1, indices2):
return [indices1.index(i) for i in indices2]
def expand(self, **hints):
return _expand(self, **hints).doit()
def _expand(self, **kwargs):
return self
def _get_free_indices_set(self):
indset = set([])
for arg in self.args:
if isinstance(arg, TensExpr):
indset.update(arg._get_free_indices_set())
return indset
def _get_dummy_indices_set(self):
indset = set([])
for arg in self.args:
if isinstance(arg, TensExpr):
indset.update(arg._get_dummy_indices_set())
return indset
def _get_indices_set(self):
indset = set([])
for arg in self.args:
if isinstance(arg, TensExpr):
indset.update(arg._get_indices_set())
return indset
@property
def _iterate_dummy_indices(self):
dummy_set = self._get_dummy_indices_set()
def recursor(expr, pos):
if isinstance(expr, TensorIndex):
if expr in dummy_set:
yield (expr, pos)
elif isinstance(expr, (Tuple, TensExpr)):
for p, arg in enumerate(expr.args):
for i in recursor(arg, pos+(p,)):
yield i
return recursor(self, ())
@property
def _iterate_free_indices(self):
free_set = self._get_free_indices_set()
def recursor(expr, pos):
if isinstance(expr, TensorIndex):
if expr in free_set:
yield (expr, pos)
elif isinstance(expr, (Tuple, TensExpr)):
for p, arg in enumerate(expr.args):
for i in recursor(arg, pos+(p,)):
yield i
return recursor(self, ())
@property
def _iterate_indices(self):
def recursor(expr, pos):
if isinstance(expr, TensorIndex):
yield (expr, pos)
elif isinstance(expr, (Tuple, TensExpr)):
for p, arg in enumerate(expr.args):
for i in recursor(arg, pos+(p,)):
yield i
return recursor(self, ())
@staticmethod
def _match_indices_with_other_tensor(array, free_ind1, free_ind2, replacement_dict):
from .array import tensorcontraction, tensorproduct, permutedims
index_types1 = [i.tensor_index_type for i in free_ind1]
# Check if variance of indices needs to be fixed:
pos2up = []
pos2down = []
free2remaining = free_ind2[:]
for pos1, index1 in enumerate(free_ind1):
if index1 in free2remaining:
pos2 = free2remaining.index(index1)
free2remaining[pos2] = None
continue
if -index1 in free2remaining:
pos2 = free2remaining.index(-index1)
free2remaining[pos2] = None
free_ind2[pos2] = index1
if index1.is_up:
pos2up.append(pos2)
else:
pos2down.append(pos2)
else:
index2 = free2remaining[pos1]
if index2 is None:
raise ValueError("incompatible indices: %s and %s" % (free_ind1, free_ind2))
free2remaining[pos1] = None
free_ind2[pos1] = index1
if index1.is_up ^ index2.is_up:
if index1.is_up:
pos2up.append(pos1)
else:
pos2down.append(pos1)
if len(set(free_ind1) & set(free_ind2)) < len(free_ind1):
raise ValueError("incompatible indices: %s and %s" % (free_ind1, free_ind2))
# TODO: add possibility of metric after (spinors)
def contract_and_permute(metric, array, pos):
array = tensorcontraction(tensorproduct(metric, array), (1, 2+pos))
permu = list(range(len(free_ind1)))
permu[0], permu[pos] = permu[pos], permu[0]
return permutedims(array, permu)
# Raise indices:
for pos in pos2up:
metric = replacement_dict[index_types1[pos]]
metric_inverse = _TensorDataLazyEvaluator.inverse_matrix(metric)
array = contract_and_permute(metric_inverse, array, pos)
# Lower indices:
for pos in pos2down:
metric = replacement_dict[index_types1[pos]]
array = contract_and_permute(metric, array, pos)
if free_ind1:
permutation = TensExpr._get_indices_permutation(free_ind2, free_ind1)
array = permutedims(array, permutation)
if hasattr(array, "rank") and array.rank() == 0:
array = array[()]
return free_ind2, array
def replace_with_arrays(self, replacement_dict, indices=None):
"""
Replace the tensorial expressions with arrays. The final array will
correspond to the N-dimensional array with indices arranged according
to ``indices``.
Parameters
==========
replacement_dict
dictionary containing the replacement rules for tensors.
indices
the index order with respect to which the array is read. The
original index order will be used if no value is passed.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices
>>> from sympy.tensor.tensor import tensorhead
>>> from sympy import symbols, diag
>>> L = TensorIndexType("L")
>>> i, j = tensor_indices("i j", L)
>>> A = tensorhead("A", [L], [[1]])
>>> A(i).replace_with_arrays({A(i): [1, 2]}, [i])
[1, 2]
Since 'indices' is optional, we can also call replace_with_arrays by
this way if no specific index order is needed:
>>> A(i).replace_with_arrays({A(i): [1, 2]})
[1, 2]
>>> expr = A(i)*A(j)
>>> expr.replace_with_arrays({A(i): [1, 2]})
[[1, 2], [2, 4]]
For contractions, specify the metric of the ``TensorIndexType``, which
in this case is ``L``, in its covariant form:
>>> expr = A(i)*A(-i)
>>> expr.replace_with_arrays({A(i): [1, 2], L: diag(1, -1)})
-3
Symmetrization of an array:
>>> H = tensorhead("H", [L, L], [[1], [1]])
>>> a, b, c, d = symbols("a b c d")
>>> expr = H(i, j)/2 + H(j, i)/2
>>> expr.replace_with_arrays({H(i, j): [[a, b], [c, d]]})
[[a, b/2 + c/2], [b/2 + c/2, d]]
Anti-symmetrization of an array:
>>> expr = H(i, j)/2 - H(j, i)/2
>>> repl = {H(i, j): [[a, b], [c, d]]}
>>> expr.replace_with_arrays(repl)
[[0, b/2 - c/2], [-b/2 + c/2, 0]]
The same expression can be read as the transpose by inverting ``i`` and
``j``:
>>> expr.replace_with_arrays(repl, [j, i])
[[0, -b/2 + c/2], [b/2 - c/2, 0]]
"""
from .array import Array
indices = indices or []
replacement_dict = {tensor: Array(array) for tensor, array in replacement_dict.items()}
# Check dimensions of replaced arrays:
for tensor, array in replacement_dict.items():
if isinstance(tensor, TensorIndexType):
expected_shape = [tensor.dim for i in range(2)]
else:
expected_shape = [index_type.dim for index_type in tensor.index_types]
if len(expected_shape) != array.rank() or (not all([dim1 == dim2 if
dim1 is not None else True for dim1, dim2 in zip(expected_shape,
array.shape)])):
raise ValueError("shapes for tensor %s expected to be %s, "\
"replacement array shape is %s" % (tensor, expected_shape,
array.shape))
ret_indices, array = self._extract_data(replacement_dict)
last_indices, array = self._match_indices_with_other_tensor(array, indices, ret_indices, replacement_dict)
#permutation = self._get_indices_permutation(indices, ret_indices)
#if not hasattr(array, "rank"):
#return array
#if array.rank() == 0:
#array = array[()]
#return array
#array = permutedims(array, permutation)
return array
def _check_add_Sum(self, expr, index_symbols):
from sympy import Sum
indices = self.get_indices()
dum = self.dum
sum_indices = [ (index_symbols[i], 0,
indices[i].tensor_index_type.dim-1) for i, j in dum]
if sum_indices:
expr = Sum(expr, *sum_indices)
return expr
class TensAdd(TensExpr, AssocOp):
"""
Sum of tensors
Parameters
==========
free_args : list of the free indices
Attributes
==========
``args`` : tuple of addends
``rank`` : rank of the tensor
``free_args`` : list of the free indices in sorted order
Notes
=====
Sum of more than one tensor are put automatically in canonical form.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorhead, tensor_indices
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b = tensor_indices('a,b', Lorentz)
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(a) + q(a); t
p(a) + q(a)
>>> t(b)
p(b) + q(b)
Examples with components data added to the tensor expression:
>>> from sympy import symbols, diag
>>> x, y, z, t = symbols("x y z t")
>>> repl = {}
>>> repl[Lorentz] = diag(1, -1, -1, -1)
>>> repl[p(a)] = [1, 2, 3, 4]
>>> repl[q(a)] = [x, y, z, t]
The following are: 2**2 - 3**2 - 2**2 - 7**2 ==> -58
>>> expr = p(a) + q(a)
>>> expr.replace_with_arrays(repl, [a])
[x + 1, y + 2, z + 3, t + 4]
"""
def __new__(cls, *args, **kw_args):
args = [_sympify(x) for x in args if x]
args = TensAdd._tensAdd_flatten(args)
obj = Basic.__new__(cls, *args, **kw_args)
return obj
def doit(self, **kwargs):
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
if not args:
return S.Zero
if len(args) == 1 and not isinstance(args[0], TensExpr):
return args[0]
# now check that all addends have the same indices:
TensAdd._tensAdd_check(args)
# if TensAdd has only 1 element in its `args`:
if len(args) == 1: # and isinstance(args[0], TensMul):
return args[0]
# Remove zeros:
args = [x for x in args if x]
# if there are no more args (i.e. have cancelled out),
# just return zero:
if not args:
return S.Zero
if len(args) == 1:
return args[0]
# Collect terms appearing more than once, differing by their coefficients:
args = TensAdd._tensAdd_collect_terms(args)
# collect canonicalized terms
def sort_key(t):
x = get_index_structure(t)
if not isinstance(t, TensExpr):
return ([], [], [])
return (t.components, x.free, x.dum)
args.sort(key=sort_key)
if not args:
return S.Zero
# it there is only a component tensor return it
if len(args) == 1:
return args[0]
obj = self.func(*args)
return obj
@staticmethod
def _tensAdd_flatten(args):
# flatten TensAdd, coerce terms which are not tensors to tensors
a = []
for x in args:
if isinstance(x, (Add, TensAdd)):
a.extend(list(x.args))
else:
a.append(x)
args = [x for x in a if x.coeff]
return args
@staticmethod
def _tensAdd_check(args):
# check that all addends have the same free indices
indices0 = set([x[0] for x in get_index_structure(args[0]).free])
list_indices = [set([y[0] for y in get_index_structure(x).free]) for x in args[1:]]
if not all(x == indices0 for x in list_indices):
raise ValueError('all tensors must have the same indices')
@staticmethod
def _tensAdd_collect_terms(args):
# collect TensMul terms differing at most by their coefficient
terms_dict = defaultdict(list)
scalars = S.Zero
if isinstance(args[0], TensExpr):
free_indices = set(args[0].get_free_indices())
else:
free_indices = set([])
for arg in args:
if not isinstance(arg, TensExpr):
if free_indices != set([]):
raise ValueError("wrong valence")
scalars += arg
continue
if free_indices != set(arg.get_free_indices()):
raise ValueError("wrong valence")
# TODO: what is the part which is not a coeff?
# needs an implementation similar to .as_coeff_Mul()
terms_dict[arg.nocoeff].append(arg.coeff)
new_args = [TensMul(Add(*coeff), t).doit() for t, coeff in terms_dict.items() if Add(*coeff) != 0]
if isinstance(scalars, Add):
new_args = list(scalars.args) + new_args
elif scalars != 0:
new_args = [scalars] + new_args
return new_args
def get_indices(self):
indices = []
for arg in self.args:
indices.extend([i for i in get_indices(arg) if i not in indices])
return indices
@property
def rank(self):
return self.args[0].rank
@property
def free_args(self):
return self.args[0].free_args
def _expand(self, **hints):
return TensAdd(*[_expand(i, **hints) for i in self.args])
def __call__(self, *indices):
"""Returns tensor with ordered free indices replaced by ``indices``
Parameters
==========
indices
Examples
========
>>> from sympy import Symbol
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> D = Symbol('D')
>>> Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> g = Lorentz.metric
>>> t = p(i0)*p(i1) + g(i0,i1)*q(i2)*q(-i2)
>>> t(i0,i2)
metric(i0, i2)*q(L_0)*q(-L_0) + p(i0)*p(i2)
>>> from sympy.tensor.tensor import canon_bp
>>> canon_bp(t(i0,i1) - t(i1,i0))
0
"""
free_args = self.free_args
indices = list(indices)
if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
index_tuples = list(zip(free_args, indices))
a = [x.func(*x.fun_eval(*index_tuples).args) for x in self.args]
res = TensAdd(*a).doit()
return res
def canon_bp(self):
"""
canonicalize using the Butler-Portugal algorithm for canonicalization
under monoterm symmetries.
"""
expr = self.expand()
args = [canon_bp(x) for x in expr.args]
res = TensAdd(*args).doit()
return res
def equals(self, other):
other = _sympify(other)
if isinstance(other, TensMul) and other._coeff == 0:
return all(x._coeff == 0 for x in self.args)
if isinstance(other, TensExpr):
if self.rank != other.rank:
return False
if isinstance(other, TensAdd):
if set(self.args) != set(other.args):
return False
else:
return True
t = self - other
if not isinstance(t, TensExpr):
return t == 0
else:
if isinstance(t, TensMul):
return t._coeff == 0
else:
return all(x._coeff == 0 for x in t.args)
def __getitem__(self, item):
deprecate_data()
return self.data[item]
def contract_delta(self, delta):
args = [x.contract_delta(delta) for x in self.args]
t = TensAdd(*args).doit()
return canon_bp(t)
def contract_metric(self, g):
"""
Raise or lower indices with the metric ``g``
Parameters
==========
g : metric
contract_all : if True, eliminate all ``g`` which are contracted
Notes
=====
see the ``TensorIndexType`` docstring for the contraction conventions
"""
args = [contract_metric(x, g) for x in self.args]
t = TensAdd(*args).doit()
return canon_bp(t)
def fun_eval(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
Parameters
==========
index_types : list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j) + A(i, -j)
>>> t.fun_eval((i, k),(-j, l))
A(k, L_0)*B(-L_0, l) + A(k, l)
"""
args = self.args
args1 = []
for x in args:
y = x.fun_eval(*index_tuples)
args1.append(y)
return TensAdd(*args1).doit()
def substitute_indices(self, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
Parameters
==========
index_types : list of tuples ``(old_index, new_index)``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.substitute_indices((i,j), (j, k))
A(j, L_0)*B(-L_0, -k)
"""
args = self.args
args1 = []
for x in args:
y = x.substitute_indices(*index_tuples)
args1.append(y)
return TensAdd(*args1).doit()
def _print(self):
a = []
args = self.args
for x in args:
a.append(str(x))
a.sort()
s = ' + '.join(a)
s = s.replace('+ -', '- ')
return s
def _extract_data(self, replacement_dict):
from sympy.tensor.array import Array, permutedims
args_indices, arrays = zip(*[
arg._extract_data(replacement_dict) if
isinstance(arg, TensExpr) else ([], arg) for arg in self.args
])
arrays = [Array(i) for i in arrays]
ref_indices = args_indices[0]
for i in range(1, len(args_indices)):
indices = args_indices[i]
array = arrays[i]
permutation = TensMul._get_indices_permutation(indices, ref_indices)
arrays[i] = permutedims(array, permutation)
return ref_indices, sum(arrays, Array.zeros(*array.shape))
@property
def data(self):
deprecate_data()
return _tensor_data_substitution_dict[self.expand()]
@data.setter
def data(self, data):
deprecate_data()
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
deprecate_data()
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
def __iter__(self):
deprecate_data()
if not self.data:
raise ValueError("No iteration on abstract tensors")
return self.data.flatten().__iter__()
def _eval_rewrite_as_Indexed(self, *args):
return Add.fromiter(args)
class Tensor(TensExpr):
"""
Base tensor class, i.e. this represents a tensor, the single unit to be
put into an expression.
This object is usually created from a ``TensorHead``, by attaching indices
to it. Indices preceded by a minus sign are considered contravariant,
otherwise covariant.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType("Lorentz", dummy_fmt="L")
>>> mu, nu = tensor_indices('mu nu', Lorentz)
>>> A = tensorhead("A", [Lorentz, Lorentz], [[1], [1]])
>>> A(mu, -nu)
A(mu, -nu)
>>> A(mu, -mu)
A(L_0, -L_0)
"""
is_commutative = False
def __new__(cls, tensor_head, indices, **kw_args):
is_canon_bp = kw_args.pop('is_canon_bp', False)
indices = cls._parse_indices(tensor_head, indices)
obj = Basic.__new__(cls, tensor_head, Tuple(*indices), **kw_args)
obj._index_structure = _IndexStructure.from_indices(*indices)
obj._free_indices_set = set(obj._index_structure.get_free_indices())
if tensor_head.rank != len(indices):
raise ValueError("wrong number of indices")
obj._indices = indices
obj._is_canon_bp = is_canon_bp
obj._index_map = Tensor._build_index_map(indices, obj._index_structure)
return obj
@staticmethod
def _build_index_map(indices, index_structure):
index_map = {}
for idx in indices:
index_map[idx] = (indices.index(idx),)
return index_map
def doit(self, **kwargs):
args, indices, free, dum = TensMul._tensMul_contract_indices([self])
return args[0]
@staticmethod
def _parse_indices(tensor_head, indices):
if not isinstance(indices, (tuple, list, Tuple)):
raise TypeError("indices should be an array, got %s" % type(indices))
indices = list(indices)
for i, index in enumerate(indices):
if isinstance(index, Symbol):
indices[i] = TensorIndex(index, tensor_head.index_types[i], True)
elif isinstance(index, Mul):
c, e = index.as_coeff_Mul()
if c == -1 and isinstance(e, Symbol):
indices[i] = TensorIndex(e, tensor_head.index_types[i], False)
else:
raise ValueError("index not understood: %s" % index)
elif not isinstance(index, TensorIndex):
raise TypeError("wrong type for index: %s is %s" % (index, type(index)))
return indices
def _set_new_index_structure(self, im, is_canon_bp=False):
indices = im.get_indices()
return self._set_indices(*indices, is_canon_bp=is_canon_bp)
def _set_indices(self, *indices, **kw_args):
if len(indices) != self.ext_rank:
raise ValueError("indices length mismatch")
return self.func(self.args[0], indices, is_canon_bp=kw_args.pop('is_canon_bp', False)).doit()
def _get_free_indices_set(self):
return set([i[0] for i in self._index_structure.free])
def _get_dummy_indices_set(self):
dummy_pos = set(itertools.chain(*self._index_structure.dum))
return set(idx for i, idx in enumerate(self.args[1]) if i in dummy_pos)
def _get_indices_set(self):
return set(self.args[1].args)
@property
def is_canon_bp(self):
return self._is_canon_bp
@property
def indices(self):
return self._indices
@property
def free(self):
return self._index_structure.free[:]
@property
def free_in_args(self):
return [(ind, pos, 0) for ind, pos in self.free]
@property
def dum(self):
return self._index_structure.dum[:]
@property
def dum_in_args(self):
return [(p1, p2, 0, 0) for p1, p2 in self.dum]
@property
def rank(self):
return len(self.free)
@property
def ext_rank(self):
return self._index_structure._ext_rank
@property
def free_args(self):
return sorted([x[0] for x in self.free])
def commutes_with(self, other):
"""
:param other:
:return:
0 commute
1 anticommute
None neither commute nor anticommute
"""
if not isinstance(other, TensExpr):
return 0
elif isinstance(other, Tensor):
return self.component.commutes_with(other.component)
return NotImplementedError
def perm2tensor(self, g, is_canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
return perm2tensor(self, g, is_canon_bp)
def canon_bp(self):
if self._is_canon_bp:
return self
expr = self.expand()
g, dummies, msym = expr._index_structure.indices_canon_args()
v = components_canon_args([expr.component])
can = canonicalize(g, dummies, msym, *v)
if can == 0:
return S.Zero
tensor = self.perm2tensor(can, True)
return tensor
@property
def index_types(self):
return list(self.component.index_types)
@property
def coeff(self):
return S.One
@property
def nocoeff(self):
return self
@property
def component(self):
return self.args[0]
@property
def components(self):
return [self.args[0]]
def split(self):
return [self]
def _expand(self, **kwargs):
return self
def sorted_components(self):
return self
def get_indices(self):
"""
Get a list of indices, corresponding to those of the tensor.
"""
return list(self.args[1])
def get_free_indices(self):
"""
Get a list of free indices, corresponding to those of the tensor.
"""
return self._index_structure.get_free_indices()
def as_base_exp(self):
return self, S.One
def substitute_indices(self, *index_tuples):
return substitute_indices(self, *index_tuples)
def __call__(self, *indices):
"""Returns tensor with ordered free indices replaced by ``indices``
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> A = tensorhead('A', [Lorentz]*5, [[1]*5])
>>> t = A(i2, i1, -i2, -i3, i4)
>>> t
A(L_0, i1, -L_0, -i3, i4)
>>> t(i1, i2, i3)
A(L_0, i1, -L_0, i2, i3)
"""
free_args = self.free_args
indices = list(indices)
if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
t = self.fun_eval(*list(zip(free_args, indices)))
# object is rebuilt in order to make sure that all contracted indices
# get recognized as dummies, but only if there are contracted indices.
if len(set(i if i.is_up else -i for i in indices)) != len(indices):
return t.func(*t.args)
return t
# TODO: put this into TensExpr?
def __iter__(self):
deprecate_data()
return self.data.__iter__()
# TODO: put this into TensExpr?
def __getitem__(self, item):
deprecate_data()
return self.data[item]
def _extract_data(self, replacement_dict):
from .array import Array
for k, v in replacement_dict.items():
if isinstance(k, Tensor) and k.args[0] == self.args[0]:
other = k
array = v
break
else:
raise ValueError("%s not found in %s" % (self, replacement_dict))
# TODO: inefficient, this should be done at root level only:
replacement_dict = {k: Array(v) for k, v in replacement_dict.items()}
array = Array(array)
dum1 = self.dum
dum2 = other.dum
if len(dum2) > 0:
for pair in dum2:
# allow `dum2` if the contained values are also in `dum1`.
if pair not in dum1:
raise NotImplementedError("%s with contractions is not implemented" % other)
# Remove elements in `dum2` from `dum1`:
dum1 = [pair for pair in dum1 if pair not in dum2]
if len(dum1) > 0:
indices2 = other.get_indices()
repl = {}
for p1, p2 in dum1:
repl[indices2[p2]] = -indices2[p1]
other = other.xreplace(repl).doit()
array = _TensorDataLazyEvaluator.data_contract_dum([array], dum1, len(indices2))
free_ind1 = self.get_free_indices()
free_ind2 = other.get_free_indices()
return self._match_indices_with_other_tensor(array, free_ind1, free_ind2, replacement_dict)
@property
def data(self):
deprecate_data()
return _tensor_data_substitution_dict[self]
@data.setter
def data(self, data):
deprecate_data()
# TODO: check data compatibility with properties of tensor.
_tensor_data_substitution_dict[self] = data
@data.deleter
def data(self):
deprecate_data()
if self in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self]
if self.metric in _tensor_data_substitution_dict:
del _tensor_data_substitution_dict[self.metric]
def _print(self):
indices = [str(ind) for ind in self.indices]
component = self.component
if component.rank > 0:
return ('%s(%s)' % (component.name, ', '.join(indices)))
else:
return ('%s' % component.name)
def equals(self, other):
if other == 0:
return self.coeff == 0
other = _sympify(other)
if not isinstance(other, TensExpr):
assert not self.components
return S.One == other
def _get_compar_comp(self):
t = self.canon_bp()
r = (t.coeff, tuple(t.components), \
tuple(sorted(t.free)), tuple(sorted(t.dum)))
return r
return _get_compar_comp(self) == _get_compar_comp(other)
def contract_metric(self, g):
# if metric is not the same, ignore this step:
if self.component != g:
return self
# in case there are free components, do not perform anything:
if len(self.free) != 0:
return self
antisym = g.index_types[0].metric_antisym
sign = S.One
typ = g.index_types[0]
if not antisym:
# g(i, -i)
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
else:
# g(i, -i)
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
dp0, dp1 = self.dum[0]
if dp0 < dp1:
# g(i, -i) = -D with antisymmetric metric
sign = -sign
return sign
def contract_delta(self, metric):
return self.contract_metric(metric)
def _eval_rewrite_as_Indexed(self, tens, indices):
from sympy import Indexed
# TODO: replace .args[0] with .name:
index_symbols = [i.args[0] for i in self.get_indices()]
expr = Indexed(tens.args[0], *index_symbols)
return self._check_add_Sum(expr, index_symbols)
class TensMul(TensExpr, AssocOp):
"""
Product of tensors
Parameters
==========
coeff : SymPy coefficient of the tensor
args
Attributes
==========
``components`` : list of ``TensorHead`` of the component tensors
``types`` : list of nonrepeated ``TensorIndexType``
``free`` : list of ``(ind, ipos, icomp)``, see Notes
``dum`` : list of ``(ipos1, ipos2, icomp1, icomp2)``, see Notes
``ext_rank`` : rank of the tensor counting the dummy indices
``rank`` : rank of the tensor
``coeff`` : SymPy coefficient of the tensor
``free_args`` : list of the free indices in sorted order
``is_canon_bp`` : ``True`` if the tensor in in canonical form
Notes
=====
``args[0]`` list of ``TensorHead`` of the component tensors.
``args[1]`` list of ``(ind, ipos, icomp)``
where ``ind`` is a free index, ``ipos`` is the slot position
of ``ind`` in the ``icomp``-th component tensor.
``args[2]`` list of tuples representing dummy indices.
``(ipos1, ipos2, icomp1, icomp2)`` indicates that the contravariant
dummy index is the ``ipos1``-th slot position in the ``icomp1``-th
component tensor; the corresponding covariant index is
in the ``ipos2`` slot position in the ``icomp2``-th component tensor.
"""
identity = S.One
def __new__(cls, *args, **kw_args):
is_canon_bp = kw_args.get('is_canon_bp', False)
args = list(map(_sympify, args))
# Flatten:
args = [i for arg in args for i in (arg.args if isinstance(arg, (TensMul, Mul)) else [arg])]
args, indices, free, dum = TensMul._tensMul_contract_indices(args, replace_indices=False)
# Data for indices:
index_types = [i.tensor_index_type for i in indices]
index_structure = _IndexStructure(free, dum, index_types, indices, canon_bp=is_canon_bp)
obj = TensExpr.__new__(cls, *args)
obj._indices = indices
obj._index_types = index_types
obj._index_structure = index_structure
obj._ext_rank = len(obj._index_structure.free) + 2*len(obj._index_structure.dum)
obj._coeff = S.One
obj._is_canon_bp = is_canon_bp
return obj
@staticmethod
def _indices_to_free_dum(args_indices):
free2pos1 = {}
free2pos2 = {}
dummy_data = []
indices = []
# Notation for positions (to better understand the code):
# `pos1`: position in the `args`.
# `pos2`: position in the indices.
# Example:
# A(i, j)*B(k, m, n)*C(p)
# `pos1` of `n` is 1 because it's in `B` (second `args` of TensMul).
# `pos2` of `n` is 4 because it's the fifth overall index.
# Counter for the index position wrt the whole expression:
pos2 = 0
for pos1, arg_indices in enumerate(args_indices):
for index_pos, index in enumerate(arg_indices):
if not isinstance(index, TensorIndex):
raise TypeError("expected TensorIndex")
if -index in free2pos1:
# Dummy index detected:
other_pos1 = free2pos1.pop(-index)
other_pos2 = free2pos2.pop(-index)
if index.is_up:
dummy_data.append((index, pos1, other_pos1, pos2, other_pos2))
else:
dummy_data.append((-index, other_pos1, pos1, other_pos2, pos2))
indices.append(index)
elif index in free2pos1:
raise ValueError("Repeated index: %s" % index)
else:
free2pos1[index] = pos1
free2pos2[index] = pos2
indices.append(index)
pos2 += 1
free = [(i, p) for (i, p) in free2pos2.items()]
free_names = [i.name for i in free2pos2.keys()]
dummy_data.sort(key=lambda x: x[3])
return indices, free, free_names, dummy_data
@staticmethod
def _dummy_data_to_dum(dummy_data):
return [(p2a, p2b) for (i, p1a, p1b, p2a, p2b) in dummy_data]
@staticmethod
def _tensMul_contract_indices(args, replace_indices=True):
replacements = [{} for _ in args]
#_index_order = all([_has_index_order(arg) for arg in args])
args_indices = [get_indices(arg) for arg in args]
indices, free, free_names, dummy_data = TensMul._indices_to_free_dum(args_indices)
cdt = defaultdict(int)
def dummy_fmt_gen(tensor_index_type):
fmt = tensor_index_type.dummy_fmt
nd = cdt[tensor_index_type]
cdt[tensor_index_type] += 1
return fmt % nd
if replace_indices:
for old_index, pos1cov, pos1contra, pos2cov, pos2contra in dummy_data:
index_type = old_index.tensor_index_type
while True:
dummy_name = dummy_fmt_gen(index_type)
if dummy_name not in free_names:
break
dummy = TensorIndex(dummy_name, index_type, True)
replacements[pos1cov][old_index] = dummy
replacements[pos1contra][-old_index] = -dummy
indices[pos2cov] = dummy
indices[pos2contra] = -dummy
args = [arg.xreplace(repl) for arg, repl in zip(args, replacements)]
dum = TensMul._dummy_data_to_dum(dummy_data)
return args, indices, free, dum
@staticmethod
def _get_components_from_args(args):
"""
Get a list of ``Tensor`` objects having the same ``TIDS`` if multiplied
by one another.
"""
components = []
for arg in args:
if not isinstance(arg, TensExpr):
continue
if isinstance(arg, TensAdd):
continue
components.extend(arg.components)
return components
@staticmethod
def _rebuild_tensors_list(args, index_structure):
indices = index_structure.get_indices()
#tensors = [None for i in components] # pre-allocate list
ind_pos = 0
for i, arg in enumerate(args):
if not isinstance(arg, TensExpr):
continue
prev_pos = ind_pos
ind_pos += arg.ext_rank
args[i] = Tensor(arg.component, indices[prev_pos:ind_pos])
def doit(self, **kwargs):
is_canon_bp = self._is_canon_bp
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
args = [arg for arg in args if arg != self.identity]
# Extract non-tensor coefficients:
coeff = reduce(lambda a, b: a*b, [arg for arg in args if not isinstance(arg, TensExpr)], S.One)
args = [arg for arg in args if isinstance(arg, TensExpr)]
if len(args) == 0:
return coeff
if coeff != self.identity:
args = [coeff] + args
if coeff == 0:
return S.Zero
if len(args) == 1:
return args[0]
args, indices, free, dum = TensMul._tensMul_contract_indices(args)
# Data for indices:
index_types = [i.tensor_index_type for i in indices]
index_structure = _IndexStructure(free, dum, index_types, indices, canon_bp=is_canon_bp)
obj = self.func(*args)
obj._index_types = index_types
obj._index_structure = index_structure
obj._ext_rank = len(obj._index_structure.free) + 2*len(obj._index_structure.dum)
obj._coeff = coeff
obj._is_canon_bp = is_canon_bp
return obj
# TODO: this method should be private
# TODO: should this method be renamed _from_components_free_dum ?
@staticmethod
def from_data(coeff, components, free, dum, **kw_args):
return TensMul(coeff, *TensMul._get_tensors_from_components_free_dum(components, free, dum), **kw_args).doit()
@staticmethod
def _get_tensors_from_components_free_dum(components, free, dum):
"""
Get a list of ``Tensor`` objects by distributing ``free`` and ``dum`` indices on the ``components``.
"""
index_structure = _IndexStructure.from_components_free_dum(components, free, dum)
indices = index_structure.get_indices()
tensors = [None for i in components] # pre-allocate list
# distribute indices on components to build a list of tensors:
ind_pos = 0
for i, component in enumerate(components):
prev_pos = ind_pos
ind_pos += component.rank
tensors[i] = Tensor(component, indices[prev_pos:ind_pos])
return tensors
def _get_free_indices_set(self):
return set([i[0] for i in self.free])
def _get_dummy_indices_set(self):
dummy_pos = set(itertools.chain(*self.dum))
return set(idx for i, idx in enumerate(self._index_structure.get_indices()) if i in dummy_pos)
def _get_position_offset_for_indices(self):
arg_offset = [None for i in range(self.ext_rank)]
counter = 0
for i, arg in enumerate(self.args):
if not isinstance(arg, TensExpr):
continue
for j in range(arg.ext_rank):
arg_offset[j + counter] = counter
counter += arg.ext_rank
return arg_offset
@property
def free_args(self):
return sorted([x[0] for x in self.free])
@property
def components(self):
return self._get_components_from_args(self.args)
@property
def free(self):
return self._index_structure.free[:]
@property
def free_in_args(self):
arg_offset = self._get_position_offset_for_indices()
argpos = self._get_indices_to_args_pos()
return [(ind, pos-arg_offset[pos], argpos[pos]) for (ind, pos) in self.free]
@property
def coeff(self):
return self._coeff
@property
def nocoeff(self):
return self.func(*[t for t in self.args if isinstance(t, TensExpr)]).doit()
@property
def dum(self):
return self._index_structure.dum[:]
@property
def dum_in_args(self):
arg_offset = self._get_position_offset_for_indices()
argpos = self._get_indices_to_args_pos()
return [(p1-arg_offset[p1], p2-arg_offset[p2], argpos[p1], argpos[p2]) for p1, p2 in self.dum]
@property
def rank(self):
return len(self.free)
@property
def ext_rank(self):
return self._ext_rank
@property
def index_types(self):
return self._index_types[:]
def equals(self, other):
if other == 0:
return self.coeff == 0
other = _sympify(other)
if not isinstance(other, TensExpr):
assert not self.components
return self._coeff == other
return self.canon_bp() == other.canon_bp()
def get_indices(self):
"""
Returns the list of indices of the tensor
The indices are listed in the order in which they appear in the
component tensors.
The dummy indices are given a name which does not collide with
the names of the free indices.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m1)*g(m0,m2)
>>> t.get_indices()
[m1, m0, m2]
>>> t2 = p(m1)*g(-m1, m2)
>>> t2.get_indices()
[L_0, -L_0, m2]
"""
return self._indices
def get_free_indices(self):
"""
Returns the list of free indices of the tensor
The indices are listed in the order in which they appear in the
component tensors.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m1)*g(m0,m2)
>>> t.get_free_indices()
[m1, m0, m2]
>>> t2 = p(m1)*g(-m1, m2)
>>> t2.get_free_indices()
[m2]
"""
return self._index_structure.get_free_indices()
def split(self):
"""
Returns a list of tensors, whose product is ``self``
Dummy indices contracted among different tensor components
become free indices with the same name as the one used to
represent the dummy indices.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(a,b)*B(-b,c)
>>> t
A(a, L_0)*B(-L_0, c)
>>> t.split()
[A(a, L_0), B(-L_0, c)]
"""
if self.args == ():
return [self]
splitp = []
res = 1
for arg in self.args:
if isinstance(arg, Tensor):
splitp.append(res*arg)
res = 1
else:
res *= arg
return splitp
def _expand(self, **hints):
# TODO: temporary solution, in the future this should be linked to
# `Expr.expand`.
args = [_expand(arg, **hints) for arg in self.args]
args1 = [arg.args if isinstance(arg, (Add, TensAdd)) else (arg,) for arg in args]
return TensAdd(*[
TensMul(*i) for i in itertools.product(*args1)]
)
def __neg__(self):
return TensMul(S.NegativeOne, self, is_canon_bp=self._is_canon_bp).doit()
def __getitem__(self, item):
deprecate_data()
return self.data[item]
def _get_args_for_traditional_printer(self):
args = list(self.args)
if (self.coeff < 0) == True:
# expressions like "-A(a)"
sign = "-"
if self.coeff == S.NegativeOne:
args = args[1:]
else:
args[0] = -args[0]
else:
sign = ""
return sign, args
def _sort_args_for_sorted_components(self):
"""
Returns the ``args`` sorted according to the components commutation
properties.
The sorting is done taking into account the commutation group
of the component tensors.
"""
cv = [arg for arg in self.args if isinstance(arg, TensExpr)]
sign = 1
n = len(cv) - 1
for i in range(n):
for j in range(n, i, -1):
c = cv[j-1].commutes_with(cv[j])
# if `c` is `None`, it does neither commute nor anticommute, skip:
if c not in [0, 1]:
continue
if (cv[j-1].component.types, cv[j-1].component.name) > \
(cv[j].component.types, cv[j].component.name):
cv[j-1], cv[j] = cv[j], cv[j-1]
# if `c` is 1, the anticommute, so change sign:
if c:
sign = -sign
coeff = sign * self.coeff
if coeff != 1:
return [coeff] + cv
return cv
def sorted_components(self):
"""
Returns a tensor product with sorted components.
"""
return TensMul(*self._sort_args_for_sorted_components()).doit()
def perm2tensor(self, g, is_canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
return perm2tensor(self, g, is_canon_bp=is_canon_bp)
def canon_bp(self):
"""
Canonicalize using the Butler-Portugal algorithm for canonicalization
under monoterm symmetries.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> A = tensorhead('A', [Lorentz]*2, [[2]])
>>> t = A(m0,-m1)*A(m1,-m0)
>>> t.canon_bp()
-A(L_0, L_1)*A(-L_0, -L_1)
>>> t = A(m0,-m1)*A(m1,-m2)*A(m2,-m0)
>>> t.canon_bp()
0
"""
if self._is_canon_bp:
return self
expr = self.expand()
if isinstance(expr, TensAdd):
return expr.canon_bp()
if not expr.components:
return expr
t = expr.sorted_components()
g, dummies, msym = t._index_structure.indices_canon_args()
v = components_canon_args(t.components)
can = canonicalize(g, dummies, msym, *v)
if can == 0:
return S.Zero
tmul = t.perm2tensor(can, True)
return tmul
def contract_delta(self, delta):
t = self.contract_metric(delta)
return t
def _get_indices_to_args_pos(self):
"""
Get a dict mapping the index position to TensMul's argument number.
"""
pos_map = dict()
pos_counter = 0
for arg_i, arg in enumerate(self.args):
if not isinstance(arg, TensExpr):
continue
assert isinstance(arg, Tensor)
for i in range(arg.ext_rank):
pos_map[pos_counter] = arg_i
pos_counter += 1
return pos_map
def contract_metric(self, g):
"""
Raise or lower indices with the metric ``g``
Parameters
==========
g : metric
Notes
=====
see the ``TensorIndexType`` docstring for the contraction conventions
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> m0, m1, m2 = tensor_indices('m0,m1,m2', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(m0)*q(m1)*g(-m0, -m1)
>>> t.canon_bp()
metric(L_0, L_1)*p(-L_0)*q(-L_1)
>>> t.contract_metric(g).canon_bp()
p(L_0)*q(-L_0)
"""
expr = self.expand()
if self != expr:
expr = expr.canon_bp()
return expr.contract_metric(g)
pos_map = self._get_indices_to_args_pos()
args = list(self.args)
antisym = g.index_types[0].metric_antisym
# list of positions of the metric ``g`` inside ``args``
gpos = [i for i, x in enumerate(self.args) if isinstance(x, Tensor) and x.component == g]
if not gpos:
return self
# Sign is either 1 or -1, to correct the sign after metric contraction
# (for spinor indices).
sign = 1
dum = self.dum[:]
free = self.free[:]
elim = set()
for gposx in gpos:
if gposx in elim:
continue
free1 = [x for x in free if pos_map[x[1]] == gposx]
dum1 = [x for x in dum if pos_map[x[0]] == gposx or pos_map[x[1]] == gposx]
if not dum1:
continue
elim.add(gposx)
# subs with the multiplication neutral element, that is, remove it:
args[gposx] = 1
if len(dum1) == 2:
if not antisym:
dum10, dum11 = dum1
if pos_map[dum10[1]] == gposx:
# the index with pos p0 contravariant
p0 = dum10[0]
else:
# the index with pos p0 is covariant
p0 = dum10[1]
if pos_map[dum11[1]] == gposx:
# the index with pos p1 is contravariant
p1 = dum11[0]
else:
# the index with pos p1 is covariant
p1 = dum11[1]
dum.append((p0, p1))
else:
dum10, dum11 = dum1
# change the sign to bring the indices of the metric to contravariant
# form; change the sign if dum10 has the metric index in position 0
if pos_map[dum10[1]] == gposx:
# the index with pos p0 is contravariant
p0 = dum10[0]
if dum10[1] == 1:
sign = -sign
else:
# the index with pos p0 is covariant
p0 = dum10[1]
if dum10[0] == 0:
sign = -sign
if pos_map[dum11[1]] == gposx:
# the index with pos p1 is contravariant
p1 = dum11[0]
sign = -sign
else:
# the index with pos p1 is covariant
p1 = dum11[1]
dum.append((p0, p1))
elif len(dum1) == 1:
if not antisym:
dp0, dp1 = dum1[0]
if pos_map[dp0] == pos_map[dp1]:
# g(i, -i)
typ = g.index_types[0]
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
else:
# g(i0, i1)*p(-i1)
if pos_map[dp0] == gposx:
p1 = dp1
else:
p1 = dp0
ind, p = free1[0]
free.append((ind, p1))
else:
dp0, dp1 = dum1[0]
if pos_map[dp0] == pos_map[dp1]:
# g(i, -i)
typ = g.index_types[0]
if typ._dim is None:
raise ValueError('dimension not assigned')
sign = sign*typ._dim
if dp0 < dp1:
# g(i, -i) = -D with antisymmetric metric
sign = -sign
else:
# g(i0, i1)*p(-i1)
if pos_map[dp0] == gposx:
p1 = dp1
if dp0 == 0:
sign = -sign
else:
p1 = dp0
ind, p = free1[0]
free.append((ind, p1))
dum = [x for x in dum if x not in dum1]
free = [x for x in free if x not in free1]
# shift positions:
shift = 0
shifts = [0]*len(args)
for i in range(len(args)):
if i in elim:
shift += 2
continue
shifts[i] = shift
free = [(ind, p - shifts[pos_map[p]]) for (ind, p) in free if pos_map[p] not in elim]
dum = [(p0 - shifts[pos_map[p0]], p1 - shifts[pos_map[p1]]) for i, (p0, p1) in enumerate(dum) if pos_map[p0] not in elim and pos_map[p1] not in elim]
res = sign*TensMul(*args).doit()
if not isinstance(res, TensExpr):
return res
im = _IndexStructure.from_components_free_dum(res.components, free, dum)
return res._set_new_index_structure(im)
def _set_new_index_structure(self, im, is_canon_bp=False):
indices = im.get_indices()
return self._set_indices(*indices, is_canon_bp=is_canon_bp)
def _set_indices(self, *indices, **kw_args):
if len(indices) != self.ext_rank:
raise ValueError("indices length mismatch")
args = list(self.args)[:]
pos = 0
is_canon_bp = kw_args.pop('is_canon_bp', False)
for i, arg in enumerate(args):
if not isinstance(arg, TensExpr):
continue
assert isinstance(arg, Tensor)
ext_rank = arg.ext_rank
args[i] = arg._set_indices(*indices[pos:pos+ext_rank])
pos += ext_rank
return TensMul(*args, is_canon_bp=is_canon_bp).doit()
@staticmethod
def _index_replacement_for_contract_metric(args, free, dum):
for arg in args:
if not isinstance(arg, TensExpr):
continue
assert isinstance(arg, Tensor)
def substitute_indices(self, *index_tuples):
return substitute_indices(self, *index_tuples)
def __call__(self, *indices):
"""Returns tensor product with ordered free indices replaced by ``indices``
Examples
========
>>> from sympy import Symbol
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> D = Symbol('D')
>>> Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
>>> i0,i1,i2,i3,i4 = tensor_indices('i0:5', Lorentz)
>>> g = Lorentz.metric
>>> p, q = tensorhead('p,q', [Lorentz], [[1]])
>>> t = p(i0)*q(i1)*q(-i1)
>>> t(i1)
p(i1)*q(L_0)*q(-L_0)
"""
free_args = self.free_args
indices = list(indices)
if [x.tensor_index_type for x in indices] != [x.tensor_index_type for x in free_args]:
raise ValueError('incompatible types')
if indices == free_args:
return self
t = self.fun_eval(*list(zip(free_args, indices)))
# object is rebuilt in order to make sure that all contracted indices
# get recognized as dummies, but only if there are contracted indices.
if len(set(i if i.is_up else -i for i in indices)) != len(indices):
return t.func(*t.args)
return t
def _extract_data(self, replacement_dict):
args_indices, arrays = zip(*[arg._extract_data(replacement_dict) for arg in self.args if isinstance(arg, TensExpr)])
coeff = reduce(operator.mul, [a for a in self.args if not isinstance(a, TensExpr)], S.One)
indices, free, free_names, dummy_data = TensMul._indices_to_free_dum(args_indices)
dum = TensMul._dummy_data_to_dum(dummy_data)
ext_rank = self.ext_rank
free.sort(key=lambda x: x[1])
free_indices = [i[0] for i in free]
return free_indices, coeff*_TensorDataLazyEvaluator.data_contract_dum(arrays, dum, ext_rank)
@property
def data(self):
deprecate_data()
dat = _tensor_data_substitution_dict[self.expand()]
return dat
@data.setter
def data(self, data):
deprecate_data()
raise ValueError("Not possible to set component data to a tensor expression")
@data.deleter
def data(self):
deprecate_data()
raise ValueError("Not possible to delete component data to a tensor expression")
def __iter__(self):
deprecate_data()
if self.data is None:
raise ValueError("No iteration on abstract tensors")
return self.data.__iter__()
def _eval_rewrite_as_Indexed(self, *args):
from sympy import Sum
index_symbols = [i.args[0] for i in self.get_indices()]
args = [arg.args[0] if isinstance(arg, Sum) else arg for arg in args]
expr = Mul.fromiter(args)
return self._check_add_Sum(expr, index_symbols)
class TensorElement(TensExpr):
"""
Tensor with evaluated components.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensorhead
>>> from sympy import symbols
>>> L = TensorIndexType("L")
>>> i, j, k = symbols("i j k")
>>> A = tensorhead("A", [L, L], [[1], [1]])
>>> A(i, j).get_free_indices()
[i, j]
If we want to set component ``i`` to a specific value, use the
``TensorElement`` class:
>>> from sympy.tensor.tensor import TensorElement
>>> te = TensorElement(A(i, j), {i: 2})
As index ``i`` has been accessed (``{i: 2}`` is the evaluation of its 3rd
element), the free indices will only contain ``j``:
>>> te.get_free_indices()
[j]
"""
def __new__(cls, expr, index_map):
if not isinstance(expr, Tensor):
# remap
if not isinstance(expr, TensExpr):
raise TypeError("%s is not a tensor expression" % expr)
return expr.func(*[TensorElement(arg, index_map) for arg in expr.args])
expr_free_indices = expr.get_free_indices()
name_translation = {i.args[0]: i for i in expr_free_indices}
index_map = {name_translation.get(index, index): value for index, value in index_map.items()}
index_map = {index: value for index, value in index_map.items() if index in expr_free_indices}
if len(index_map) == 0:
return expr
free_indices = [i for i in expr_free_indices if i not in index_map.keys()]
index_map = Dict(index_map)
obj = TensExpr.__new__(cls, expr, index_map)
obj._free_indices = free_indices
return obj
@property
def free(self):
return [(index, i) for i, index in enumerate(self.get_free_indices())]
@property
def dum(self):
# TODO: inherit dummies from expr
return []
@property
def expr(self):
return self._args[0]
@property
def index_map(self):
return self._args[1]
def get_free_indices(self):
return self._free_indices
def get_indices(self):
return self.get_free_indices()
def _extract_data(self, replacement_dict):
ret_indices, array = self.expr._extract_data(replacement_dict)
index_map = self.index_map
slice_tuple = tuple(index_map.get(i, slice(None)) for i in ret_indices)
ret_indices = [i for i in ret_indices if i not in index_map]
array = array.__getitem__(slice_tuple)
return ret_indices, array
def canon_bp(p):
"""
Butler-Portugal canonicalization
"""
if isinstance(p, TensExpr):
return p.canon_bp()
return p
def tensor_mul(*a):
"""
product of tensors
"""
if not a:
return TensMul.from_data(S.One, [], [], [])
t = a[0]
for tx in a[1:]:
t = t*tx
return t
def riemann_cyclic_replace(t_r):
"""
replace Riemann tensor with an equivalent expression
``R(m,n,p,q) -> 2/3*R(m,n,p,q) - 1/3*R(m,q,n,p) + 1/3*R(m,p,n,q)``
"""
free = sorted(t_r.free, key=lambda x: x[1])
m, n, p, q = [x[0] for x in free]
t0 = S(2)/3*t_r
t1 = - S(1)/3*t_r.substitute_indices((m,m),(n,q),(p,n),(q,p))
t2 = S(1)/3*t_r.substitute_indices((m,m),(n,p),(p,n),(q,q))
t3 = t0 + t1 + t2
return t3
def riemann_cyclic(t2):
"""
replace each Riemann tensor with an equivalent expression
satisfying the cyclic identity.
This trick is discussed in the reference guide to Cadabra.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead, riemann_cyclic
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> R = tensorhead('R', [Lorentz]*4, [[2, 2]])
>>> t = R(i,j,k,l)*(R(-i,-j,-k,-l) - 2*R(-i,-k,-j,-l))
>>> riemann_cyclic(t)
0
"""
t2 = t2.expand()
if isinstance(t2, (TensMul, Tensor)):
args = [t2]
else:
args = t2.args
a1 = [x.split() for x in args]
a2 = [[riemann_cyclic_replace(tx) for tx in y] for y in a1]
a3 = [tensor_mul(*v) for v in a2]
t3 = TensAdd(*a3).doit()
if not t3:
return t3
else:
return canon_bp(t3)
def get_lines(ex, index_type):
"""
returns ``(lines, traces, rest)`` for an index type,
where ``lines`` is the list of list of positions of a matrix line,
``traces`` is the list of list of traced matrix lines,
``rest`` is the rest of the elements ot the tensor.
"""
def _join_lines(a):
i = 0
while i < len(a):
x = a[i]
xend = x[-1]
xstart = x[0]
hit = True
while hit:
hit = False
for j in range(i + 1, len(a)):
if j >= len(a):
break
if a[j][0] == xend:
hit = True
x.extend(a[j][1:])
xend = x[-1]
a.pop(j)
continue
if a[j][0] == xstart:
hit = True
a[i] = reversed(a[j][1:]) + x
x = a[i]
xstart = a[i][0]
a.pop(j)
continue
if a[j][-1] == xend:
hit = True
x.extend(reversed(a[j][:-1]))
xend = x[-1]
a.pop(j)
continue
if a[j][-1] == xstart:
hit = True
a[i] = a[j][:-1] + x
x = a[i]
xstart = x[0]
a.pop(j)
continue
i += 1
return a
arguments = ex.args
dt = {}
for c in ex.args:
if not isinstance(c, TensExpr):
continue
if c in dt:
continue
index_types = c.index_types
a = []
for i in range(len(index_types)):
if index_types[i] is index_type:
a.append(i)
if len(a) > 2:
raise ValueError('at most two indices of type %s allowed' % index_type)
if len(a) == 2:
dt[c] = a
#dum = ex.dum
lines = []
traces = []
traces1 = []
#indices_to_args_pos = ex._get_indices_to_args_pos()
# TODO: add a dum_to_components_map ?
for p0, p1, c0, c1 in ex.dum_in_args:
if arguments[c0] not in dt:
continue
if c0 == c1:
traces.append([c0])
continue
ta0 = dt[arguments[c0]]
ta1 = dt[arguments[c1]]
if p0 not in ta0:
continue
if ta0.index(p0) == ta1.index(p1):
# case gamma(i,s0,-s1) in c0, gamma(j,-s0,s2) in c1;
# to deal with this case one could add to the position
# a flag for transposition;
# one could write [(c0, False), (c1, True)]
raise NotImplementedError
# if p0 == ta0[1] then G in pos c0 is mult on the right by G in c1
# if p0 == ta0[0] then G in pos c1 is mult on the right by G in c0
ta0 = dt[arguments[c0]]
b0, b1 = (c0, c1) if p0 == ta0[1] else (c1, c0)
lines1 = lines[:]
for line in lines:
if line[-1] == b0:
if line[0] == b1:
n = line.index(min(line))
traces1.append(line)
traces.append(line[n:] + line[:n])
else:
line.append(b1)
break
elif line[0] == b1:
line.insert(0, b0)
break
else:
lines1.append([b0, b1])
lines = [x for x in lines1 if x not in traces1]
lines = _join_lines(lines)
rest = []
for line in lines:
for y in line:
rest.append(y)
for line in traces:
for y in line:
rest.append(y)
rest = [x for x in range(len(arguments)) if x not in rest]
return lines, traces, rest
def get_free_indices(t):
if not isinstance(t, TensExpr):
return ()
return t.get_free_indices()
def get_indices(t):
if not isinstance(t, TensExpr):
return ()
return t.get_indices()
def get_index_structure(t):
if isinstance(t, TensExpr):
return t._index_structure
return _IndexStructure([], [], [], [])
def get_coeff(t):
if isinstance(t, Tensor):
return S.One
if isinstance(t, TensMul):
return t.coeff
if isinstance(t, TensExpr):
raise ValueError("no coefficient associated to this tensor expression")
return t
def contract_metric(t, g):
if isinstance(t, TensExpr):
return t.contract_metric(g)
return t
def perm2tensor(t, g, is_canon_bp=False):
"""
Returns the tensor corresponding to the permutation ``g``
For further details, see the method in ``TIDS`` with the same name.
"""
if not isinstance(t, TensExpr):
return t
elif isinstance(t, (Tensor, TensMul)):
nim = get_index_structure(t).perm2tensor(g, is_canon_bp=is_canon_bp)
res = t._set_new_index_structure(nim, is_canon_bp=is_canon_bp)
if g[-1] != len(g) - 1:
return -res
return res
raise NotImplementedError()
def substitute_indices(t, *index_tuples):
"""
Return a tensor with free indices substituted according to ``index_tuples``
``index_types`` list of tuples ``(old_index, new_index)``
Note: this method will neither raise or lower the indices, it will just replace their symbol.
Examples
========
>>> from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
>>> Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
>>> i, j, k, l = tensor_indices('i,j,k,l', Lorentz)
>>> A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
>>> t = A(i, k)*B(-k, -j); t
A(i, L_0)*B(-L_0, -j)
>>> t.substitute_indices((i,j), (j, k))
A(j, L_0)*B(-L_0, -k)
"""
if not isinstance(t, TensExpr):
return t
free = t.free
free1 = []
for j, ipos in free:
for i, v in index_tuples:
if i._name == j._name and i.tensor_index_type == j.tensor_index_type:
if i._is_up == j._is_up:
free1.append((v, ipos))
else:
free1.append((-v, ipos))
break
else:
free1.append((j, ipos))
t = TensMul.from_data(t.coeff, t.components, free1, t.dum)
return t
def _expand(expr, **kwargs):
if isinstance(expr, TensExpr):
return expr._expand(**kwargs)
else:
return expr.expand(**kwargs)
|
8cfc02d563afbe6e8b133cab12d91b92936ba243d9d9589eb4a45fc06dbe629d | """A module that handles matrices.
Includes functions for fast creating matrices like zero, one/eye, random
matrix, etc.
"""
from .common import ShapeError, NonSquareMatrixError
from .dense import (
GramSchmidt, casoratian, diag, eye, hessian, jordan_cell,
list2numpy, matrix2numpy, matrix_multiply_elementwise, ones,
randMatrix, rot_axis1, rot_axis2, rot_axis3, symarray, wronskian,
zeros)
from .dense import MutableDenseMatrix
from .matrices import DeferredVector, MatrixBase
Matrix = MutableMatrix = MutableDenseMatrix
from .sparse import MutableSparseMatrix
from .sparsetools import banded
from .immutable import ImmutableDenseMatrix, ImmutableSparseMatrix
ImmutableMatrix = ImmutableDenseMatrix
SparseMatrix = MutableSparseMatrix
from .expressions import (
MatrixSlice, BlockDiagMatrix, BlockMatrix, FunctionMatrix, Identity,
Inverse, MatAdd, MatMul, MatPow, MatrixExpr, MatrixSymbol, Trace,
Transpose, ZeroMatrix, blockcut, block_collapse, matrix_symbols, Adjoint,
hadamard_product, HadamardProduct, HadamardPower, Determinant, det,
diagonalize_vector, DiagonalizeVector, DiagonalMatrix, DiagonalOf, trace,
DotProduct, kronecker_product, KroneckerProduct)
|
9d6c6c53c62f94b736a6f5c250ba8dd124fd4013a18ebc793b3b5ff0e1c0468b | """
Basic methods common to all matrices to be used
when creating more advanced matrices (e.g., matrices over rings,
etc.).
"""
from __future__ import division, print_function
from collections import defaultdict
from inspect import isfunction
from sympy.assumptions.refine import refine
from sympy.core.basic import Atom
from sympy.core.compatibility import (
Iterable, as_int, is_sequence, range, reduce)
from sympy.core.decorators import call_highest_priority
from sympy.core.expr import Expr
from sympy.core.function import count_ops
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.functions import Abs
from sympy.simplify import simplify as _simplify
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import flatten
from sympy.utilities.misc import filldedent
class MatrixError(Exception):
pass
class ShapeError(ValueError, MatrixError):
"""Wrong matrix shape"""
pass
class NonSquareMatrixError(ShapeError):
pass
class MatrixRequired(object):
"""All subclasses of matrix objects must implement the
required matrix properties listed here."""
rows = None
cols = None
shape = None
_simplify = None
@classmethod
def _new(cls, *args, **kwargs):
"""`_new` must, at minimum, be callable as
`_new(rows, cols, mat) where mat is a flat list of the
elements of the matrix."""
raise NotImplementedError("Subclasses must implement this.")
def __eq__(self, other):
raise NotImplementedError("Subclasses must implement this.")
def __getitem__(self, key):
"""Implementations of __getitem__ should accept ints, in which
case the matrix is indexed as a flat list, tuples (i,j) in which
case the (i,j) entry is returned, slices, or mixed tuples (a,b)
where a and b are any combintion of slices and integers."""
raise NotImplementedError("Subclasses must implement this.")
def __len__(self):
"""The total number of entries in the matrix."""
raise NotImplementedError("Subclasses must implement this.")
class MatrixShaping(MatrixRequired):
"""Provides basic matrix shaping and extracting of submatrices"""
def _eval_col_del(self, col):
def entry(i, j):
return self[i, j] if j < col else self[i, j + 1]
return self._new(self.rows, self.cols - 1, entry)
def _eval_col_insert(self, pos, other):
cols = self.cols
def entry(i, j):
if j < pos:
return self[i, j]
elif pos <= j < pos + other.cols:
return other[i, j - pos]
return self[i, j - other.cols]
return self._new(self.rows, self.cols + other.cols,
lambda i, j: entry(i, j))
def _eval_col_join(self, other):
rows = self.rows
def entry(i, j):
if i < rows:
return self[i, j]
return other[i - rows, j]
return classof(self, other)._new(self.rows + other.rows, self.cols,
lambda i, j: entry(i, j))
def _eval_extract(self, rowsList, colsList):
mat = list(self)
cols = self.cols
indices = (i * cols + j for i in rowsList for j in colsList)
return self._new(len(rowsList), len(colsList),
list(mat[i] for i in indices))
def _eval_get_diag_blocks(self):
sub_blocks = []
def recurse_sub_blocks(M):
i = 1
while i <= M.shape[0]:
if i == 1:
to_the_right = M[0, i:]
to_the_bottom = M[i:, 0]
else:
to_the_right = M[:i, i:]
to_the_bottom = M[i:, :i]
if any(to_the_right) or any(to_the_bottom):
i += 1
continue
else:
sub_blocks.append(M[:i, :i])
if M.shape == M[:i, :i].shape:
return
else:
recurse_sub_blocks(M[i:, i:])
return
recurse_sub_blocks(self)
return sub_blocks
def _eval_row_del(self, row):
def entry(i, j):
return self[i, j] if i < row else self[i + 1, j]
return self._new(self.rows - 1, self.cols, entry)
def _eval_row_insert(self, pos, other):
entries = list(self)
insert_pos = pos * self.cols
entries[insert_pos:insert_pos] = list(other)
return self._new(self.rows + other.rows, self.cols, entries)
def _eval_row_join(self, other):
cols = self.cols
def entry(i, j):
if j < cols:
return self[i, j]
return other[i, j - cols]
return classof(self, other)._new(self.rows, self.cols + other.cols,
lambda i, j: entry(i, j))
def _eval_tolist(self):
return [list(self[i,:]) for i in range(self.rows)]
def _eval_vec(self):
rows = self.rows
def entry(n, _):
# we want to read off the columns first
j = n // rows
i = n - j * rows
return self[i, j]
return self._new(len(self), 1, entry)
def col_del(self, col):
"""Delete the specified column."""
if col < 0:
col += self.cols
if not 0 <= col < self.cols:
raise ValueError("Column {} out of range.".format(col))
return self._eval_col_del(col)
def col_insert(self, pos, other):
"""Insert one or more columns at the given column position.
Examples
========
>>> from sympy import zeros, ones
>>> M = zeros(3)
>>> V = ones(3, 1)
>>> M.col_insert(1, V)
Matrix([
[0, 1, 0, 0],
[0, 1, 0, 0],
[0, 1, 0, 0]])
See Also
========
col
row_insert
"""
# Allows you to build a matrix even if it is null matrix
if not self:
return type(self)(other)
pos = as_int(pos)
if pos < 0:
pos = self.cols + pos
if pos < 0:
pos = 0
elif pos > self.cols:
pos = self.cols
if self.rows != other.rows:
raise ShapeError(
"`self` and `other` must have the same number of rows.")
return self._eval_col_insert(pos, other)
def col_join(self, other):
"""Concatenates two matrices along self's last and other's first row.
Examples
========
>>> from sympy import zeros, ones
>>> M = zeros(3)
>>> V = ones(1, 3)
>>> M.col_join(V)
Matrix([
[0, 0, 0],
[0, 0, 0],
[0, 0, 0],
[1, 1, 1]])
See Also
========
col
row_join
"""
# A null matrix can always be stacked (see #10770)
if self.rows == 0 and self.cols != other.cols:
return self._new(0, other.cols, []).col_join(other)
if self.cols != other.cols:
raise ShapeError(
"`self` and `other` must have the same number of columns.")
return self._eval_col_join(other)
def col(self, j):
"""Elementary column selector.
Examples
========
>>> from sympy import eye
>>> eye(2).col(0)
Matrix([
[1],
[0]])
See Also
========
row
col_op
col_swap
col_del
col_join
col_insert
"""
return self[:, j]
def extract(self, rowsList, colsList):
"""Return a submatrix by specifying a list of rows and columns.
Negative indices can be given. All indices must be in the range
-n <= i < n where n is the number of rows or columns.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(4, 3, range(12))
>>> m
Matrix([
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11]])
>>> m.extract([0, 1, 3], [0, 1])
Matrix([
[0, 1],
[3, 4],
[9, 10]])
Rows or columns can be repeated:
>>> m.extract([0, 0, 1], [-1])
Matrix([
[2],
[2],
[5]])
Every other row can be taken by using range to provide the indices:
>>> m.extract(range(0, m.rows, 2), [-1])
Matrix([
[2],
[8]])
RowsList or colsList can also be a list of booleans, in which case
the rows or columns corresponding to the True values will be selected:
>>> m.extract([0, 1, 2, 3], [True, False, True])
Matrix([
[0, 2],
[3, 5],
[6, 8],
[9, 11]])
"""
if not is_sequence(rowsList) or not is_sequence(colsList):
raise TypeError("rowsList and colsList must be iterable")
# ensure rowsList and colsList are lists of integers
if rowsList and all(isinstance(i, bool) for i in rowsList):
rowsList = [index for index, item in enumerate(rowsList) if item]
if colsList and all(isinstance(i, bool) for i in colsList):
colsList = [index for index, item in enumerate(colsList) if item]
# ensure everything is in range
rowsList = [a2idx(k, self.rows) for k in rowsList]
colsList = [a2idx(k, self.cols) for k in colsList]
return self._eval_extract(rowsList, colsList)
def get_diag_blocks(self):
"""Obtains the square sub-matrices on the main diagonal of a square matrix.
Useful for inverting symbolic matrices or solving systems of
linear equations which may be decoupled by having a block diagonal
structure.
Examples
========
>>> from sympy import Matrix
>>> from sympy.abc import x, y, z
>>> A = Matrix([[1, 3, 0, 0], [y, z*z, 0, 0], [0, 0, x, 0], [0, 0, 0, 0]])
>>> a1, a2, a3 = A.get_diag_blocks()
>>> a1
Matrix([
[1, 3],
[y, z**2]])
>>> a2
Matrix([[x]])
>>> a3
Matrix([[0]])
"""
return self._eval_get_diag_blocks()
@classmethod
def hstack(cls, *args):
"""Return a matrix formed by joining args horizontally (i.e.
by repeated application of row_join).
Examples
========
>>> from sympy.matrices import Matrix, eye
>>> Matrix.hstack(eye(2), 2*eye(2))
Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2]])
"""
if len(args) == 0:
return cls._new()
kls = type(args[0])
return reduce(kls.row_join, args)
def reshape(self, rows, cols):
"""Reshape the matrix. Total number of elements must remain the same.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(2, 3, lambda i, j: 1)
>>> m
Matrix([
[1, 1, 1],
[1, 1, 1]])
>>> m.reshape(1, 6)
Matrix([[1, 1, 1, 1, 1, 1]])
>>> m.reshape(3, 2)
Matrix([
[1, 1],
[1, 1],
[1, 1]])
"""
if self.rows * self.cols != rows * cols:
raise ValueError("Invalid reshape parameters %d %d" % (rows, cols))
return self._new(rows, cols, lambda i, j: self[i * cols + j])
def row_del(self, row):
"""Delete the specified row."""
if row < 0:
row += self.rows
if not 0 <= row < self.rows:
raise ValueError("Row {} out of range.".format(row))
return self._eval_row_del(row)
def row_insert(self, pos, other):
"""Insert one or more rows at the given row position.
Examples
========
>>> from sympy import zeros, ones
>>> M = zeros(3)
>>> V = ones(1, 3)
>>> M.row_insert(1, V)
Matrix([
[0, 0, 0],
[1, 1, 1],
[0, 0, 0],
[0, 0, 0]])
See Also
========
row
col_insert
"""
# Allows you to build a matrix even if it is null matrix
if not self:
return self._new(other)
pos = as_int(pos)
if pos < 0:
pos = self.rows + pos
if pos < 0:
pos = 0
elif pos > self.rows:
pos = self.rows
if self.cols != other.cols:
raise ShapeError(
"`self` and `other` must have the same number of columns.")
return self._eval_row_insert(pos, other)
def row_join(self, other):
"""Concatenates two matrices along self's last and rhs's first column
Examples
========
>>> from sympy import zeros, ones
>>> M = zeros(3)
>>> V = ones(3, 1)
>>> M.row_join(V)
Matrix([
[0, 0, 0, 1],
[0, 0, 0, 1],
[0, 0, 0, 1]])
See Also
========
row
col_join
"""
# A null matrix can always be stacked (see #10770)
if self.cols == 0 and self.rows != other.rows:
return self._new(other.rows, 0, []).row_join(other)
if self.rows != other.rows:
raise ShapeError(
"`self` and `rhs` must have the same number of rows.")
return self._eval_row_join(other)
def diagonal(self, k=0):
"""Returns the kth diagonal of self. The main diagonal
corresponds to `k=0`; diagonals above and below correspond to
`k > 0` and `k < 0`, respectively. The values of `self[i, j]`
for which `j - i = k`, are returned in order of increasing
`i + j`, starting with `i + j = |k|`.
Examples
========
>>> from sympy import Matrix, SparseMatrix
>>> m = Matrix(3, 3, lambda i, j: j - i); m
Matrix([
[ 0, 1, 2],
[-1, 0, 1],
[-2, -1, 0]])
>>> _.diagonal()
Matrix([[0, 0, 0]])
>>> m.diagonal(1)
Matrix([[1, 1]])
>>> m.diagonal(-2)
Matrix([[-2]])
Even though the diagonal is returned as a Matrix, the element
retrieval can be done with a single index:
>>> Matrix.diag(1, 2, 3).diagonal()[1] # instead of [0, 1]
2
See Also
========
diag - to create a diagonal matrix
"""
rv = []
k = as_int(k)
r = 0 if k > 0 else -k
c = 0 if r else k
for i in range(min(self.shape) - max(r, c)):
rv.append(self[r + i, c + i])
if not rv:
raise ValueError(filldedent('''
The %s diagonal is out of range [%s, %s]''' % (
k, 1 - self.rows, self.cols - 1)))
return self._new(1, len(rv), rv)
def row(self, i):
"""Elementary row selector.
Examples
========
>>> from sympy import eye
>>> eye(2).row(0)
Matrix([[1, 0]])
See Also
========
col
row_op
row_swap
row_del
row_join
row_insert
"""
return self[i, :]
@property
def shape(self):
"""The shape (dimensions) of the matrix as the 2-tuple (rows, cols).
Examples
========
>>> from sympy.matrices import zeros
>>> M = zeros(2, 3)
>>> M.shape
(2, 3)
>>> M.rows
2
>>> M.cols
3
"""
return (self.rows, self.cols)
def tolist(self):
"""Return the Matrix as a nested Python list.
Examples
========
>>> from sympy import Matrix, ones
>>> m = Matrix(3, 3, range(9))
>>> m
Matrix([
[0, 1, 2],
[3, 4, 5],
[6, 7, 8]])
>>> m.tolist()
[[0, 1, 2], [3, 4, 5], [6, 7, 8]]
>>> ones(3, 0).tolist()
[[], [], []]
When there are no rows then it will not be possible to tell how
many columns were in the original matrix:
>>> ones(0, 3).tolist()
[]
"""
if not self.rows:
return []
if not self.cols:
return [[] for i in range(self.rows)]
return self._eval_tolist()
def vec(self):
"""Return the Matrix converted into a one column matrix by stacking columns
Examples
========
>>> from sympy import Matrix
>>> m=Matrix([[1, 3], [2, 4]])
>>> m
Matrix([
[1, 3],
[2, 4]])
>>> m.vec()
Matrix([
[1],
[2],
[3],
[4]])
See Also
========
vech
"""
return self._eval_vec()
@classmethod
def vstack(cls, *args):
"""Return a matrix formed by joining args vertically (i.e.
by repeated application of col_join).
Examples
========
>>> from sympy.matrices import Matrix, eye
>>> Matrix.vstack(eye(2), 2*eye(2))
Matrix([
[1, 0],
[0, 1],
[2, 0],
[0, 2]])
"""
if len(args) == 0:
return cls._new()
kls = type(args[0])
return reduce(kls.col_join, args)
class MatrixSpecial(MatrixRequired):
"""Construction of special matrices"""
@classmethod
def _eval_diag(cls, rows, cols, diag_dict):
"""diag_dict is a defaultdict containing
all the entries of the diagonal matrix."""
def entry(i, j):
return diag_dict[(i, j)]
return cls._new(rows, cols, entry)
@classmethod
def _eval_eye(cls, rows, cols):
def entry(i, j):
return S.One if i == j else S.Zero
return cls._new(rows, cols, entry)
@classmethod
def _eval_jordan_block(cls, rows, cols, eigenvalue, band='upper'):
if band == 'lower':
def entry(i, j):
if i == j:
return eigenvalue
elif j + 1 == i:
return S.One
return S.Zero
else:
def entry(i, j):
if i == j:
return eigenvalue
elif i + 1 == j:
return S.One
return S.Zero
return cls._new(rows, cols, entry)
@classmethod
def _eval_ones(cls, rows, cols):
def entry(i, j):
return S.One
return cls._new(rows, cols, entry)
@classmethod
def _eval_zeros(cls, rows, cols):
def entry(i, j):
return S.Zero
return cls._new(rows, cols, entry)
@classmethod
def diag(kls, *args, **kwargs):
"""Returns a matrix with the specified diagonal.
If matrices are passed, a block-diagonal matrix
is created (i.e. the "direct sum" of the matrices).
kwargs
======
rows : rows of the resulting matrix; computed if
not given.
cols : columns of the resulting matrix; computed if
not given.
cls : class for the resulting matrix
unpack : bool which, when True (default), unpacks a single
sequence rather than interpreting it as a Matrix.
strict : bool which, when False (default), allows Matrices to
have variable-length rows.
Examples
========
>>> from sympy.matrices import Matrix
>>> Matrix.diag(1, 2, 3)
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
The current default is to unpack a single sequence. If this is
not desired, set `unpack=False` and it will be interpreted as
a matrix.
>>> Matrix.diag([1, 2, 3]) == Matrix.diag(1, 2, 3)
True
When more than one element is passed, each is interpreted as
something to put on the diagonal. Lists are converted to
matricecs. Filling of the diagonal always continues from
the bottom right hand corner of the previous item: this
will create a block-diagonal matrix whether the matrices
are square or not.
>>> col = [1, 2, 3]
>>> row = [[4, 5]]
>>> Matrix.diag(col, row)
Matrix([
[1, 0, 0],
[2, 0, 0],
[3, 0, 0],
[0, 4, 5]])
When `unpack` is False, elements within a list need not all be
of the same length. Setting `strict` to True would raise a
ValueError for the following:
>>> Matrix.diag([[1, 2, 3], [4, 5], [6]], unpack=False)
Matrix([
[1, 2, 3],
[4, 5, 0],
[6, 0, 0]])
The type of the returned matrix can be set with the ``cls``
keyword.
>>> from sympy.matrices import ImmutableMatrix
>>> from sympy.utilities.misc import func_name
>>> func_name(Matrix.diag(1, cls=ImmutableMatrix))
'ImmutableDenseMatrix'
A zero dimension matrix can be used to position the start of
the filling at the start of an arbitrary row or column:
>>> from sympy import ones
>>> r2 = ones(0, 2)
>>> Matrix.diag(r2, 1, 2)
Matrix([
[0, 0, 1, 0],
[0, 0, 0, 2]])
See Also
========
eye
diagonal - to extract a diagonal
.dense.diag
.expressions.blockmatrix.BlockMatrix
"""
from sympy.matrices.matrices import MatrixBase
from sympy.matrices.dense import Matrix
from sympy.matrices.sparse import SparseMatrix
klass = kwargs.get('cls', kls)
strict = kwargs.get('strict', False) # lists -> Matrices
unpack = kwargs.get('unpack', True) # unpack single sequence
if unpack and len(args) == 1 and is_sequence(args[0]) and \
not isinstance(args[0], MatrixBase):
args = args[0]
# fill a default dict with the diagonal entries
diag_entries = defaultdict(int)
rmax = cmax = 0 # keep track of the biggest index seen
for m in args:
if isinstance(m, list):
if strict:
# if malformed, Matrix will raise an error
_ = Matrix(m)
r, c = _.shape
m = _.tolist()
else:
m = SparseMatrix(m)
for (i, j), _ in m._smat.items():
diag_entries[(i + rmax, j + cmax)] = _
r, c = m.shape
m = [] # to skip process below
elif hasattr(m, 'shape'): # a Matrix
# convert to list of lists
r, c = m.shape
m = m.tolist()
else: # in this case, we're a single value
diag_entries[(rmax, cmax)] = m
rmax += 1
cmax += 1
continue
# process list of lists
for i in range(len(m)):
for j, _ in enumerate(m[i]):
diag_entries[(i + rmax, j + cmax)] = _
rmax += r
cmax += c
rows = kwargs.get('rows', None)
cols = kwargs.get('cols', None)
if rows is None:
rows, cols = cols, rows
if rows is None:
rows, cols = rmax, cmax
else:
cols = rows if cols is None else cols
if rows < rmax or cols < cmax:
raise ValueError(filldedent('''
The constructed matrix is {} x {} but a size of {} x {}
was specified.'''.format(rmax, cmax, rows, cols)))
return klass._eval_diag(rows, cols, diag_entries)
@classmethod
def eye(kls, rows, cols=None, **kwargs):
"""Returns an identity matrix.
Args
====
rows : rows of the matrix
cols : cols of the matrix (if None, cols=rows)
kwargs
======
cls : class of the returned matrix
"""
if cols is None:
cols = rows
klass = kwargs.get('cls', kls)
rows, cols = as_int(rows), as_int(cols)
return klass._eval_eye(rows, cols)
@classmethod
def jordan_block(kls, size=None, eigenvalue=None, **kwargs):
"""Returns a Jordan block
Parameters
==========
size : Integer, optional
Specifies the shape of the Jordan block matrix.
eigenvalue : Number or Symbol
Specifies the value for the main diagonal of the matrix.
.. note::
The keyword ``eigenval`` is also specified as an alias
of this keyword, but it is not recommended to use.
We may deprecate the alias in later release.
band : 'upper' or 'lower', optional
Specifies the position of the off-diagonal to put `1` s on.
cls : Matrix, optional
Specifies the matrix class of the output form.
If it is not specified, the class type where the method is
being executed on will be returned.
rows, cols : Integer, optional
Specifies the shape of the Jordan block matrix. See Notes
section for the details of how these key works.
.. note::
This feature will be deprecated in the future.
Returns
=======
Matrix
A Jordan block matrix.
Raises
======
ValueError
If insufficient arguments are given for matrix size
specification, or no eigenvalue is given.
Examples
========
Creating a default Jordan block:
>>> from sympy import Matrix
>>> from sympy.abc import x
>>> Matrix.jordan_block(4, x)
Matrix([
[x, 1, 0, 0],
[0, x, 1, 0],
[0, 0, x, 1],
[0, 0, 0, x]])
Creating an alternative Jordan block matrix where `1` is on
lower off-diagonal:
>>> Matrix.jordan_block(4, x, band='lower')
Matrix([
[x, 0, 0, 0],
[1, x, 0, 0],
[0, 1, x, 0],
[0, 0, 1, x]])
Creating a Jordan block with keyword arguments
>>> Matrix.jordan_block(size=4, eigenvalue=x)
Matrix([
[x, 1, 0, 0],
[0, x, 1, 0],
[0, 0, x, 1],
[0, 0, 0, x]])
Notes
=====
.. note::
This feature will be deprecated in the future.
The keyword arguments ``size``, ``rows``, ``cols`` relates to
the Jordan block size specifications.
If you want to create a square Jordan block, specify either
one of the three arguments.
If you want to create a rectangular Jordan block, specify
``rows`` and ``cols`` individually.
+--------------------------------+---------------------+
| Arguments Given | Matrix Shape |
+----------+----------+----------+----------+----------+
| size | rows | cols | rows | cols |
+==========+==========+==========+==========+==========+
| size | Any | size | size |
+----------+----------+----------+----------+----------+
| | None | ValueError |
| +----------+----------+----------+----------+
| None | rows | None | rows | rows |
| +----------+----------+----------+----------+
| | None | cols | cols | cols |
+ +----------+----------+----------+----------+
| | rows | cols | rows | cols |
+----------+----------+----------+----------+----------+
References
==========
.. [1] https://en.wikipedia.org/wiki/Jordan_matrix
"""
if 'rows' in kwargs or 'cols' in kwargs:
SymPyDeprecationWarning(
feature="Keyword arguments 'rows' or 'cols'",
issue=16102,
useinstead="a more generic banded matrix constructor",
deprecated_since_version="1.4"
).warn()
klass = kwargs.pop('cls', kls)
band = kwargs.pop('band', 'upper')
rows = kwargs.pop('rows', None)
cols = kwargs.pop('cols', None)
eigenval = kwargs.get('eigenval', None)
if eigenvalue is None and eigenval is None:
raise ValueError("Must supply an eigenvalue")
elif eigenvalue != eigenval and None not in (eigenval, eigenvalue):
raise ValueError(
"Inconsistent values are given: 'eigenval'={}, "
"'eigenvalue'={}".format(eigenval, eigenvalue))
else:
if eigenval is not None:
eigenvalue = eigenval
if (size, rows, cols) == (None, None, None):
raise ValueError("Must supply a matrix size")
if size is not None:
rows, cols = size, size
elif rows is not None and cols is None:
cols = rows
elif cols is not None and rows is None:
rows = cols
rows, cols = as_int(rows), as_int(cols)
return klass._eval_jordan_block(rows, cols, eigenvalue, band)
@classmethod
def ones(kls, rows, cols=None, **kwargs):
"""Returns a matrix of ones.
Args
====
rows : rows of the matrix
cols : cols of the matrix (if None, cols=rows)
kwargs
======
cls : class of the returned matrix
"""
if cols is None:
cols = rows
klass = kwargs.get('cls', kls)
rows, cols = as_int(rows), as_int(cols)
return klass._eval_ones(rows, cols)
@classmethod
def zeros(kls, rows, cols=None, **kwargs):
"""Returns a matrix of zeros.
Args
====
rows : rows of the matrix
cols : cols of the matrix (if None, cols=rows)
kwargs
======
cls : class of the returned matrix
"""
if cols is None:
cols = rows
klass = kwargs.get('cls', kls)
rows, cols = as_int(rows), as_int(cols)
return klass._eval_zeros(rows, cols)
class MatrixProperties(MatrixRequired):
"""Provides basic properties of a matrix."""
def _eval_atoms(self, *types):
result = set()
for i in self:
result.update(i.atoms(*types))
return result
def _eval_free_symbols(self):
return set().union(*(i.free_symbols for i in self))
def _eval_has(self, *patterns):
return any(a.has(*patterns) for a in self)
def _eval_is_anti_symmetric(self, simpfunc):
if not all(simpfunc(self[i, j] + self[j, i]).is_zero for i in range(self.rows) for j in range(self.cols)):
return False
return True
def _eval_is_diagonal(self):
for i in range(self.rows):
for j in range(self.cols):
if i != j and self[i, j]:
return False
return True
# _eval_is_hermitian is called by some general sympy
# routines and has a different *args signature. Make
# sure the names don't clash by adding `_matrix_` in name.
def _eval_is_matrix_hermitian(self, simpfunc):
mat = self._new(self.rows, self.cols, lambda i, j: simpfunc(self[i, j] - self[j, i].conjugate()))
return mat.is_zero
def _eval_is_Identity(self):
def dirac(i, j):
if i == j:
return 1
return 0
return all(self[i, j] == dirac(i, j) for i in range(self.rows) for j in
range(self.cols))
def _eval_is_lower_hessenberg(self):
return all(self[i, j].is_zero
for i in range(self.rows)
for j in range(i + 2, self.cols))
def _eval_is_lower(self):
return all(self[i, j].is_zero
for i in range(self.rows)
for j in range(i + 1, self.cols))
def _eval_is_symbolic(self):
return self.has(Symbol)
def _eval_is_symmetric(self, simpfunc):
mat = self._new(self.rows, self.cols, lambda i, j: simpfunc(self[i, j] - self[j, i]))
return mat.is_zero
def _eval_is_zero(self):
if any(i.is_zero == False for i in self):
return False
if any(i.is_zero is None for i in self):
return None
return True
def _eval_is_upper_hessenberg(self):
return all(self[i, j].is_zero
for i in range(2, self.rows)
for j in range(min(self.cols, (i - 1))))
def _eval_values(self):
return [i for i in self if not i.is_zero]
def atoms(self, *types):
"""Returns the atoms that form the current object.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.matrices import Matrix
>>> Matrix([[x]])
Matrix([[x]])
>>> _.atoms()
{x}
"""
types = tuple(t if isinstance(t, type) else type(t) for t in types)
if not types:
types = (Atom,)
return self._eval_atoms(*types)
@property
def free_symbols(self):
"""Returns the free symbols within the matrix.
Examples
========
>>> from sympy.abc import x
>>> from sympy.matrices import Matrix
>>> Matrix([[x], [1]]).free_symbols
{x}
"""
return self._eval_free_symbols()
def has(self, *patterns):
"""Test whether any subexpression matches any of the patterns.
Examples
========
>>> from sympy import Matrix, SparseMatrix, Float
>>> from sympy.abc import x, y
>>> A = Matrix(((1, x), (0.2, 3)))
>>> B = SparseMatrix(((1, x), (0.2, 3)))
>>> A.has(x)
True
>>> A.has(y)
False
>>> A.has(Float)
True
>>> B.has(x)
True
>>> B.has(y)
False
>>> B.has(Float)
True
"""
return self._eval_has(*patterns)
def is_anti_symmetric(self, simplify=True):
"""Check if matrix M is an antisymmetric matrix,
that is, M is a square matrix with all M[i, j] == -M[j, i].
When ``simplify=True`` (default), the sum M[i, j] + M[j, i] is
simplified before testing to see if it is zero. By default,
the SymPy simplify function is used. To use a custom function
set simplify to a function that accepts a single argument which
returns a simplified expression. To skip simplification, set
simplify to False but note that although this will be faster,
it may induce false negatives.
Examples
========
>>> from sympy import Matrix, symbols
>>> m = Matrix(2, 2, [0, 1, -1, 0])
>>> m
Matrix([
[ 0, 1],
[-1, 0]])
>>> m.is_anti_symmetric()
True
>>> x, y = symbols('x y')
>>> m = Matrix(2, 3, [0, 0, x, -y, 0, 0])
>>> m
Matrix([
[ 0, 0, x],
[-y, 0, 0]])
>>> m.is_anti_symmetric()
False
>>> from sympy.abc import x, y
>>> m = Matrix(3, 3, [0, x**2 + 2*x + 1, y,
... -(x + 1)**2 , 0, x*y,
... -y, -x*y, 0])
Simplification of matrix elements is done by default so even
though two elements which should be equal and opposite wouldn't
pass an equality test, the matrix is still reported as
anti-symmetric:
>>> m[0, 1] == -m[1, 0]
False
>>> m.is_anti_symmetric()
True
If 'simplify=False' is used for the case when a Matrix is already
simplified, this will speed things up. Here, we see that without
simplification the matrix does not appear anti-symmetric:
>>> m.is_anti_symmetric(simplify=False)
False
But if the matrix were already expanded, then it would appear
anti-symmetric and simplification in the is_anti_symmetric routine
is not needed:
>>> m = m.expand()
>>> m.is_anti_symmetric(simplify=False)
True
"""
# accept custom simplification
simpfunc = simplify
if not isfunction(simplify):
simpfunc = _simplify if simplify else lambda x: x
if not self.is_square:
return False
return self._eval_is_anti_symmetric(simpfunc)
def is_diagonal(self):
"""Check if matrix is diagonal,
that is matrix in which the entries outside the main diagonal are all zero.
Examples
========
>>> from sympy import Matrix, diag
>>> m = Matrix(2, 2, [1, 0, 0, 2])
>>> m
Matrix([
[1, 0],
[0, 2]])
>>> m.is_diagonal()
True
>>> m = Matrix(2, 2, [1, 1, 0, 2])
>>> m
Matrix([
[1, 1],
[0, 2]])
>>> m.is_diagonal()
False
>>> m = diag(1, 2, 3)
>>> m
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
>>> m.is_diagonal()
True
See Also
========
is_lower
is_upper
is_diagonalizable
diagonalize
"""
return self._eval_is_diagonal()
@property
def is_hermitian(self, simplify=True):
"""Checks if the matrix is Hermitian.
In a Hermitian matrix element i,j is the complex conjugate of
element j,i.
Examples
========
>>> from sympy.matrices import Matrix
>>> from sympy import I
>>> from sympy.abc import x
>>> a = Matrix([[1, I], [-I, 1]])
>>> a
Matrix([
[ 1, I],
[-I, 1]])
>>> a.is_hermitian
True
>>> a[0, 0] = 2*I
>>> a.is_hermitian
False
>>> a[0, 0] = x
>>> a.is_hermitian
>>> a[0, 1] = a[1, 0]*I
>>> a.is_hermitian
False
"""
if not self.is_square:
return False
simpfunc = simplify
if not isfunction(simplify):
simpfunc = _simplify if simplify else lambda x: x
return self._eval_is_matrix_hermitian(simpfunc)
@property
def is_Identity(self):
if not self.is_square:
return False
return self._eval_is_Identity()
@property
def is_lower_hessenberg(self):
r"""Checks if the matrix is in the lower-Hessenberg form.
The lower hessenberg matrix has zero entries
above the first superdiagonal.
Examples
========
>>> from sympy.matrices import Matrix
>>> a = Matrix([[1, 2, 0, 0], [5, 2, 3, 0], [3, 4, 3, 7], [5, 6, 1, 1]])
>>> a
Matrix([
[1, 2, 0, 0],
[5, 2, 3, 0],
[3, 4, 3, 7],
[5, 6, 1, 1]])
>>> a.is_lower_hessenberg
True
See Also
========
is_upper_hessenberg
is_lower
"""
return self._eval_is_lower_hessenberg()
@property
def is_lower(self):
"""Check if matrix is a lower triangular matrix. True can be returned
even if the matrix is not square.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(2, 2, [1, 0, 0, 1])
>>> m
Matrix([
[1, 0],
[0, 1]])
>>> m.is_lower
True
>>> m = Matrix(4, 3, [0, 0, 0, 2, 0, 0, 1, 4 , 0, 6, 6, 5])
>>> m
Matrix([
[0, 0, 0],
[2, 0, 0],
[1, 4, 0],
[6, 6, 5]])
>>> m.is_lower
True
>>> from sympy.abc import x, y
>>> m = Matrix(2, 2, [x**2 + y, y**2 + x, 0, x + y])
>>> m
Matrix([
[x**2 + y, x + y**2],
[ 0, x + y]])
>>> m.is_lower
False
See Also
========
is_upper
is_diagonal
is_lower_hessenberg
"""
return self._eval_is_lower()
@property
def is_square(self):
"""Checks if a matrix is square.
A matrix is square if the number of rows equals the number of columns.
The empty matrix is square by definition, since the number of rows and
the number of columns are both zero.
Examples
========
>>> from sympy import Matrix
>>> a = Matrix([[1, 2, 3], [4, 5, 6]])
>>> b = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> c = Matrix([])
>>> a.is_square
False
>>> b.is_square
True
>>> c.is_square
True
"""
return self.rows == self.cols
def is_symbolic(self):
"""Checks if any elements contain Symbols.
Examples
========
>>> from sympy.matrices import Matrix
>>> from sympy.abc import x, y
>>> M = Matrix([[x, y], [1, 0]])
>>> M.is_symbolic()
True
"""
return self._eval_is_symbolic()
def is_symmetric(self, simplify=True):
"""Check if matrix is symmetric matrix,
that is square matrix and is equal to its transpose.
By default, simplifications occur before testing symmetry.
They can be skipped using 'simplify=False'; while speeding things a bit,
this may however induce false negatives.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(2, 2, [0, 1, 1, 2])
>>> m
Matrix([
[0, 1],
[1, 2]])
>>> m.is_symmetric()
True
>>> m = Matrix(2, 2, [0, 1, 2, 0])
>>> m
Matrix([
[0, 1],
[2, 0]])
>>> m.is_symmetric()
False
>>> m = Matrix(2, 3, [0, 0, 0, 0, 0, 0])
>>> m
Matrix([
[0, 0, 0],
[0, 0, 0]])
>>> m.is_symmetric()
False
>>> from sympy.abc import x, y
>>> m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3])
>>> m
Matrix([
[ 1, x**2 + 2*x + 1, y],
[(x + 1)**2, 2, 0],
[ y, 0, 3]])
>>> m.is_symmetric()
True
If the matrix is already simplified, you may speed-up is_symmetric()
test by using 'simplify=False'.
>>> bool(m.is_symmetric(simplify=False))
False
>>> m1 = m.expand()
>>> m1.is_symmetric(simplify=False)
True
"""
simpfunc = simplify
if not isfunction(simplify):
simpfunc = _simplify if simplify else lambda x: x
if not self.is_square:
return False
return self._eval_is_symmetric(simpfunc)
@property
def is_upper_hessenberg(self):
"""Checks if the matrix is the upper-Hessenberg form.
The upper hessenberg matrix has zero entries
below the first subdiagonal.
Examples
========
>>> from sympy.matrices import Matrix
>>> a = Matrix([[1, 4, 2, 3], [3, 4, 1, 7], [0, 2, 3, 4], [0, 0, 1, 3]])
>>> a
Matrix([
[1, 4, 2, 3],
[3, 4, 1, 7],
[0, 2, 3, 4],
[0, 0, 1, 3]])
>>> a.is_upper_hessenberg
True
See Also
========
is_lower_hessenberg
is_upper
"""
return self._eval_is_upper_hessenberg()
@property
def is_upper(self):
"""Check if matrix is an upper triangular matrix. True can be returned
even if the matrix is not square.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(2, 2, [1, 0, 0, 1])
>>> m
Matrix([
[1, 0],
[0, 1]])
>>> m.is_upper
True
>>> m = Matrix(4, 3, [5, 1, 9, 0, 4 , 6, 0, 0, 5, 0, 0, 0])
>>> m
Matrix([
[5, 1, 9],
[0, 4, 6],
[0, 0, 5],
[0, 0, 0]])
>>> m.is_upper
True
>>> m = Matrix(2, 3, [4, 2, 5, 6, 1, 1])
>>> m
Matrix([
[4, 2, 5],
[6, 1, 1]])
>>> m.is_upper
False
See Also
========
is_lower
is_diagonal
is_upper_hessenberg
"""
return all(self[i, j].is_zero
for i in range(1, self.rows)
for j in range(min(i, self.cols)))
@property
def is_zero(self):
"""Checks if a matrix is a zero matrix.
A matrix is zero if every element is zero. A matrix need not be square
to be considered zero. The empty matrix is zero by the principle of
vacuous truth. For a matrix that may or may not be zero (e.g.
contains a symbol), this will be None
Examples
========
>>> from sympy import Matrix, zeros
>>> from sympy.abc import x
>>> a = Matrix([[0, 0], [0, 0]])
>>> b = zeros(3, 4)
>>> c = Matrix([[0, 1], [0, 0]])
>>> d = Matrix([])
>>> e = Matrix([[x, 0], [0, 0]])
>>> a.is_zero
True
>>> b.is_zero
True
>>> c.is_zero
False
>>> d.is_zero
True
>>> e.is_zero
"""
return self._eval_is_zero()
def values(self):
"""Return non-zero values of self."""
return self._eval_values()
class MatrixOperations(MatrixRequired):
"""Provides basic matrix shape and elementwise
operations. Should not be instantiated directly."""
def _eval_adjoint(self):
return self.transpose().conjugate()
def _eval_applyfunc(self, f):
out = self._new(self.rows, self.cols, [f(x) for x in self])
return out
def _eval_as_real_imag(self):
from sympy.functions.elementary.complexes import re, im
return (self.applyfunc(re), self.applyfunc(im))
def _eval_conjugate(self):
return self.applyfunc(lambda x: x.conjugate())
def _eval_permute_cols(self, perm):
# apply the permutation to a list
mapping = list(perm)
def entry(i, j):
return self[i, mapping[j]]
return self._new(self.rows, self.cols, entry)
def _eval_permute_rows(self, perm):
# apply the permutation to a list
mapping = list(perm)
def entry(i, j):
return self[mapping[i], j]
return self._new(self.rows, self.cols, entry)
def _eval_trace(self):
return sum(self[i, i] for i in range(self.rows))
def _eval_transpose(self):
return self._new(self.cols, self.rows, lambda i, j: self[j, i])
def adjoint(self):
"""Conjugate transpose or Hermitian conjugation."""
return self._eval_adjoint()
def applyfunc(self, f):
"""Apply a function to each element of the matrix.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(2, 2, lambda i, j: i*2+j)
>>> m
Matrix([
[0, 1],
[2, 3]])
>>> m.applyfunc(lambda i: 2*i)
Matrix([
[0, 2],
[4, 6]])
"""
if not callable(f):
raise TypeError("`f` must be callable.")
return self._eval_applyfunc(f)
def as_real_imag(self):
"""Returns a tuple containing the (real, imaginary) part of matrix."""
return self._eval_as_real_imag()
def conjugate(self):
"""Return the by-element conjugation.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> from sympy import I
>>> a = SparseMatrix(((1, 2 + I), (3, 4), (I, -I)))
>>> a
Matrix([
[1, 2 + I],
[3, 4],
[I, -I]])
>>> a.C
Matrix([
[ 1, 2 - I],
[ 3, 4],
[-I, I]])
See Also
========
transpose: Matrix transposition
H: Hermite conjugation
D: Dirac conjugation
"""
return self._eval_conjugate()
def doit(self, **kwargs):
return self.applyfunc(lambda x: x.doit())
def evalf(self, prec=None, **options):
"""Apply evalf() to each element of self."""
return self.applyfunc(lambda i: i.evalf(prec, **options))
def expand(self, deep=True, modulus=None, power_base=True, power_exp=True,
mul=True, log=True, multinomial=True, basic=True, **hints):
"""Apply core.function.expand to each entry of the matrix.
Examples
========
>>> from sympy.abc import x
>>> from sympy.matrices import Matrix
>>> Matrix(1, 1, [x*(x+1)])
Matrix([[x*(x + 1)]])
>>> _.expand()
Matrix([[x**2 + x]])
"""
return self.applyfunc(lambda x: x.expand(
deep, modulus, power_base, power_exp, mul, log, multinomial, basic,
**hints))
@property
def H(self):
"""Return Hermite conjugate.
Examples
========
>>> from sympy import Matrix, I
>>> m = Matrix((0, 1 + I, 2, 3))
>>> m
Matrix([
[ 0],
[1 + I],
[ 2],
[ 3]])
>>> m.H
Matrix([[0, 1 - I, 2, 3]])
See Also
========
conjugate: By-element conjugation
D: Dirac conjugation
"""
return self.T.C
def permute(self, perm, orientation='rows', direction='forward'):
"""Permute the rows or columns of a matrix by the given list of swaps.
Parameters
==========
perm : a permutation. This may be a list swaps (e.g., `[[1, 2], [0, 3]]`),
or any valid input to the `Permutation` constructor, including a `Permutation()`
itself. If `perm` is given explicitly as a list of indices or a `Permutation`,
`direction` has no effect.
orientation : ('rows' or 'cols') whether to permute the rows or the columns
direction : ('forward', 'backward') whether to apply the permutations from
the start of the list first, or from the back of the list first
Examples
========
>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.permute([[0, 1], [0, 2]], orientation='rows', direction='forward')
Matrix([
[0, 0, 1],
[1, 0, 0],
[0, 1, 0]])
>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.permute([[0, 1], [0, 2]], orientation='rows', direction='backward')
Matrix([
[0, 1, 0],
[0, 0, 1],
[1, 0, 0]])
"""
# allow british variants and `columns`
if direction == 'forwards':
direction = 'forward'
if direction == 'backwards':
direction = 'backward'
if orientation == 'columns':
orientation = 'cols'
if direction not in ('forward', 'backward'):
raise TypeError("direction='{}' is an invalid kwarg. "
"Try 'forward' or 'backward'".format(direction))
if orientation not in ('rows', 'cols'):
raise TypeError("orientation='{}' is an invalid kwarg. "
"Try 'rows' or 'cols'".format(orientation))
# ensure all swaps are in range
max_index = self.rows if orientation == 'rows' else self.cols
if not all(0 <= t <= max_index for t in flatten(list(perm))):
raise IndexError("`swap` indices out of range.")
# see if we are a list of pairs
try:
assert len(perm[0]) == 2
# we are a list of swaps, so `direction` matters
if direction == 'backward':
perm = reversed(perm)
# since Permutation doesn't let us have non-disjoint cycles,
# we'll construct the explicit mapping ourselves XXX Bug #12479
mapping = list(range(max_index))
for (i, j) in perm:
mapping[i], mapping[j] = mapping[j], mapping[i]
perm = mapping
except (TypeError, AssertionError, IndexError):
pass
from sympy.combinatorics import Permutation
perm = Permutation(perm, size=max_index)
if orientation == 'rows':
return self._eval_permute_rows(perm)
if orientation == 'cols':
return self._eval_permute_cols(perm)
def permute_cols(self, swaps, direction='forward'):
"""Alias for `self.permute(swaps, orientation='cols', direction=direction)`
See Also
========
permute
"""
return self.permute(swaps, orientation='cols', direction=direction)
def permute_rows(self, swaps, direction='forward'):
"""Alias for `self.permute(swaps, orientation='rows', direction=direction)`
See Also
========
permute
"""
return self.permute(swaps, orientation='rows', direction=direction)
def refine(self, assumptions=True):
"""Apply refine to each element of the matrix.
Examples
========
>>> from sympy import Symbol, Matrix, Abs, sqrt, Q
>>> x = Symbol('x')
>>> Matrix([[Abs(x)**2, sqrt(x**2)],[sqrt(x**2), Abs(x)**2]])
Matrix([
[ Abs(x)**2, sqrt(x**2)],
[sqrt(x**2), Abs(x)**2]])
>>> _.refine(Q.real(x))
Matrix([
[ x**2, Abs(x)],
[Abs(x), x**2]])
"""
return self.applyfunc(lambda x: refine(x, assumptions))
def replace(self, F, G, map=False):
"""Replaces Function F in Matrix entries with Function G.
Examples
========
>>> from sympy import symbols, Function, Matrix
>>> F, G = symbols('F, G', cls=Function)
>>> M = Matrix(2, 2, lambda i, j: F(i+j)) ; M
Matrix([
[F(0), F(1)],
[F(1), F(2)]])
>>> N = M.replace(F,G)
>>> N
Matrix([
[G(0), G(1)],
[G(1), G(2)]])
"""
return self.applyfunc(lambda x: x.replace(F, G, map))
def simplify(self, ratio=1.7, measure=count_ops, rational=False, inverse=False):
"""Apply simplify to each element of the matrix.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy import sin, cos
>>> from sympy.matrices import SparseMatrix
>>> SparseMatrix(1, 1, [x*sin(y)**2 + x*cos(y)**2])
Matrix([[x*sin(y)**2 + x*cos(y)**2]])
>>> _.simplify()
Matrix([[x]])
"""
return self.applyfunc(lambda x: x.simplify(ratio=ratio, measure=measure,
rational=rational, inverse=inverse))
def subs(self, *args, **kwargs): # should mirror core.basic.subs
"""Return a new matrix with subs applied to each entry.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.matrices import SparseMatrix, Matrix
>>> SparseMatrix(1, 1, [x])
Matrix([[x]])
>>> _.subs(x, y)
Matrix([[y]])
>>> Matrix(_).subs(y, x)
Matrix([[x]])
"""
return self.applyfunc(lambda x: x.subs(*args, **kwargs))
def trace(self):
"""
Returns the trace of a square matrix i.e. the sum of the
diagonal elements.
Examples
========
>>> from sympy import Matrix
>>> A = Matrix(2, 2, [1, 2, 3, 4])
>>> A.trace()
5
"""
if self.rows != self.cols:
raise NonSquareMatrixError()
return self._eval_trace()
def transpose(self):
"""
Returns the transpose of the matrix.
Examples
========
>>> from sympy import Matrix
>>> A = Matrix(2, 2, [1, 2, 3, 4])
>>> A.transpose()
Matrix([
[1, 3],
[2, 4]])
>>> from sympy import Matrix, I
>>> m=Matrix(((1, 2+I), (3, 4)))
>>> m
Matrix([
[1, 2 + I],
[3, 4]])
>>> m.transpose()
Matrix([
[ 1, 3],
[2 + I, 4]])
>>> m.T == m.transpose()
True
See Also
========
conjugate: By-element conjugation
"""
return self._eval_transpose()
T = property(transpose, None, None, "Matrix transposition.")
C = property(conjugate, None, None, "By-element conjugation.")
n = evalf
def xreplace(self, rule): # should mirror core.basic.xreplace
"""Return a new matrix with xreplace applied to each entry.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.matrices import SparseMatrix, Matrix
>>> SparseMatrix(1, 1, [x])
Matrix([[x]])
>>> _.xreplace({x: y})
Matrix([[y]])
>>> Matrix(_).xreplace({y: x})
Matrix([[x]])
"""
return self.applyfunc(lambda x: x.xreplace(rule))
_eval_simplify = simplify
def _eval_trigsimp(self, **opts):
from sympy.simplify import trigsimp
return self.applyfunc(lambda x: trigsimp(x, **opts))
class MatrixArithmetic(MatrixRequired):
"""Provides basic matrix arithmetic operations.
Should not be instantiated directly."""
_op_priority = 10.01
def _eval_Abs(self):
return self._new(self.rows, self.cols, lambda i, j: Abs(self[i, j]))
def _eval_add(self, other):
return self._new(self.rows, self.cols,
lambda i, j: self[i, j] + other[i, j])
def _eval_matrix_mul(self, other):
def entry(i, j):
try:
return sum(self[i,k]*other[k,j] for k in range(self.cols))
except TypeError:
# Block matrices don't work with `sum` or `Add` (ISSUE #11599)
# They don't work with `sum` because `sum` tries to add `0`
# initially, and for a matrix, that is a mix of a scalar and
# a matrix, which raises a TypeError. Fall back to a
# block-matrix-safe way to multiply if the `sum` fails.
ret = self[i, 0]*other[0, j]
for k in range(1, self.cols):
ret += self[i, k]*other[k, j]
return ret
return self._new(self.rows, other.cols, entry)
def _eval_matrix_mul_elementwise(self, other):
return self._new(self.rows, self.cols, lambda i, j: self[i,j]*other[i,j])
def _eval_matrix_rmul(self, other):
def entry(i, j):
return sum(other[i,k]*self[k,j] for k in range(other.cols))
return self._new(other.rows, self.cols, entry)
def _eval_pow_by_recursion(self, num):
if num == 1:
return self
if num % 2 == 1:
return self * self._eval_pow_by_recursion(num - 1)
ret = self._eval_pow_by_recursion(num // 2)
return ret * ret
def _eval_scalar_mul(self, other):
return self._new(self.rows, self.cols, lambda i, j: self[i,j]*other)
def _eval_scalar_rmul(self, other):
return self._new(self.rows, self.cols, lambda i, j: other*self[i,j])
def _eval_Mod(self, other):
from sympy import Mod
return self._new(self.rows, self.cols, lambda i, j: Mod(self[i, j], other))
# python arithmetic functions
def __abs__(self):
"""Returns a new matrix with entry-wise absolute values."""
return self._eval_Abs()
@call_highest_priority('__radd__')
def __add__(self, other):
"""Return self + other, raising ShapeError if shapes don't match."""
other = _matrixify(other)
# matrix-like objects can have shapes. This is
# our first sanity check.
if hasattr(other, 'shape'):
if self.shape != other.shape:
raise ShapeError("Matrix size mismatch: %s + %s" % (
self.shape, other.shape))
# honest sympy matrices defer to their class's routine
if getattr(other, 'is_Matrix', False):
# call the highest-priority class's _eval_add
a, b = self, other
if a.__class__ != classof(a, b):
b, a = a, b
return a._eval_add(b)
# Matrix-like objects can be passed to CommonMatrix routines directly.
if getattr(other, 'is_MatrixLike', False):
return MatrixArithmetic._eval_add(self, other)
raise TypeError('cannot add %s and %s' % (type(self), type(other)))
@call_highest_priority('__rdiv__')
def __div__(self, other):
return self * (S.One / other)
@call_highest_priority('__rmatmul__')
def __matmul__(self, other):
other = _matrixify(other)
if not getattr(other, 'is_Matrix', False) and not getattr(other, 'is_MatrixLike', False):
return NotImplemented
return self.__mul__(other)
def __mod__(self, other):
return self.applyfunc(lambda x: x % other)
@call_highest_priority('__rmul__')
def __mul__(self, other):
"""Return self*other where other is either a scalar or a matrix
of compatible dimensions.
Examples
========
>>> from sympy.matrices import Matrix
>>> A = Matrix([[1, 2, 3], [4, 5, 6]])
>>> 2*A == A*2 == Matrix([[2, 4, 6], [8, 10, 12]])
True
>>> B = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> A*B
Matrix([
[30, 36, 42],
[66, 81, 96]])
>>> B*A
Traceback (most recent call last):
...
ShapeError: Matrices size mismatch.
>>>
See Also
========
matrix_multiply_elementwise
"""
other = _matrixify(other)
# matrix-like objects can have shapes. This is
# our first sanity check.
if hasattr(other, 'shape') and len(other.shape) == 2:
if self.shape[1] != other.shape[0]:
raise ShapeError("Matrix size mismatch: %s * %s." % (
self.shape, other.shape))
# honest sympy matrices defer to their class's routine
if getattr(other, 'is_Matrix', False):
return self._eval_matrix_mul(other)
# Matrix-like objects can be passed to CommonMatrix routines directly.
if getattr(other, 'is_MatrixLike', False):
return MatrixArithmetic._eval_matrix_mul(self, other)
# if 'other' is not iterable then scalar multiplication.
if not isinstance(other, Iterable):
try:
return self._eval_scalar_mul(other)
except TypeError:
pass
return NotImplemented
def __neg__(self):
return self._eval_scalar_mul(-1)
@call_highest_priority('__rpow__')
def __pow__(self, num):
if self.rows != self.cols:
raise NonSquareMatrixError()
a = self
jordan_pow = getattr(a, '_matrix_pow_by_jordan_blocks', None)
num = sympify(num)
if num.is_Number and num % 1 == 0:
if a.rows == 1:
return a._new([[a[0]**num]])
if num == 0:
return self._new(self.rows, self.cols, lambda i, j: int(i == j))
if num < 0:
num = -num
a = a.inv()
# When certain conditions are met,
# Jordan block algorithm is faster than
# computation by recursion.
elif a.rows == 2 and num > 100000 and jordan_pow is not None:
try:
return jordan_pow(num)
except MatrixError:
pass
return a._eval_pow_by_recursion(num)
elif not num.is_Number and num.is_negative is None and a.det() == 0:
from sympy.matrices.expressions import MatPow
return MatPow(a, num)
elif isinstance(num, (Expr, float)):
return jordan_pow(num)
else:
raise TypeError(
"Only SymPy expressions or integers are supported as exponent for matrices")
@call_highest_priority('__add__')
def __radd__(self, other):
return self + other
@call_highest_priority('__matmul__')
def __rmatmul__(self, other):
other = _matrixify(other)
if not getattr(other, 'is_Matrix', False) and not getattr(other, 'is_MatrixLike', False):
return NotImplemented
return self.__rmul__(other)
@call_highest_priority('__mul__')
def __rmul__(self, other):
other = _matrixify(other)
# matrix-like objects can have shapes. This is
# our first sanity check.
if hasattr(other, 'shape') and len(other.shape) == 2:
if self.shape[0] != other.shape[1]:
raise ShapeError("Matrix size mismatch.")
# honest sympy matrices defer to their class's routine
if getattr(other, 'is_Matrix', False):
return other._new(other.as_mutable() * self)
# Matrix-like objects can be passed to CommonMatrix routines directly.
if getattr(other, 'is_MatrixLike', False):
return MatrixArithmetic._eval_matrix_rmul(self, other)
# if 'other' is not iterable then scalar multiplication.
if not isinstance(other, Iterable):
try:
return self._eval_scalar_rmul(other)
except TypeError:
pass
return NotImplemented
@call_highest_priority('__sub__')
def __rsub__(self, a):
return (-self) + a
@call_highest_priority('__rsub__')
def __sub__(self, a):
return self + (-a)
@call_highest_priority('__rtruediv__')
def __truediv__(self, other):
return self.__div__(other)
def multiply_elementwise(self, other):
"""Return the Hadamard product (elementwise product) of A and B
Examples
========
>>> from sympy.matrices import Matrix
>>> A = Matrix([[0, 1, 2], [3, 4, 5]])
>>> B = Matrix([[1, 10, 100], [100, 10, 1]])
>>> A.multiply_elementwise(B)
Matrix([
[ 0, 10, 200],
[300, 40, 5]])
See Also
========
cross
dot
multiply
"""
if self.shape != other.shape:
raise ShapeError("Matrix shapes must agree {} != {}".format(self.shape, other.shape))
return self._eval_matrix_mul_elementwise(other)
class MatrixCommon(MatrixArithmetic, MatrixOperations, MatrixProperties,
MatrixSpecial, MatrixShaping):
"""All common matrix operations including basic arithmetic, shaping,
and special matrices like `zeros`, and `eye`."""
_diff_wrt = True
class _MinimalMatrix(object):
"""Class providing the minimum functionality
for a matrix-like object and implementing every method
required for a `MatrixRequired`. This class does not have everything
needed to become a full-fledged SymPy object, but it will satisfy the
requirements of anything inheriting from `MatrixRequired`. If you wish
to make a specialized matrix type, make sure to implement these
methods and properties with the exception of `__init__` and `__repr__`
which are included for convenience."""
is_MatrixLike = True
_sympify = staticmethod(sympify)
_class_priority = 3
is_Matrix = True
is_MatrixExpr = False
@classmethod
def _new(cls, *args, **kwargs):
return cls(*args, **kwargs)
def __init__(self, rows, cols=None, mat=None):
if isfunction(mat):
# if we passed in a function, use that to populate the indices
mat = list(mat(i, j) for i in range(rows) for j in range(cols))
if cols is None and mat is None:
mat = rows
rows, cols = getattr(mat, 'shape', (rows, cols))
try:
# if we passed in a list of lists, flatten it and set the size
if cols is None and mat is None:
mat = rows
cols = len(mat[0])
rows = len(mat)
mat = [x for l in mat for x in l]
except (IndexError, TypeError):
pass
self.mat = tuple(self._sympify(x) for x in mat)
self.rows, self.cols = rows, cols
if self.rows is None or self.cols is None:
raise NotImplementedError("Cannot initialize matrix with given parameters")
def __getitem__(self, key):
def _normalize_slices(row_slice, col_slice):
"""Ensure that row_slice and col_slice don't have
`None` in their arguments. Any integers are converted
to slices of length 1"""
if not isinstance(row_slice, slice):
row_slice = slice(row_slice, row_slice + 1, None)
row_slice = slice(*row_slice.indices(self.rows))
if not isinstance(col_slice, slice):
col_slice = slice(col_slice, col_slice + 1, None)
col_slice = slice(*col_slice.indices(self.cols))
return (row_slice, col_slice)
def _coord_to_index(i, j):
"""Return the index in _mat corresponding
to the (i,j) position in the matrix. """
return i * self.cols + j
if isinstance(key, tuple):
i, j = key
if isinstance(i, slice) or isinstance(j, slice):
# if the coordinates are not slices, make them so
# and expand the slices so they don't contain `None`
i, j = _normalize_slices(i, j)
rowsList, colsList = list(range(self.rows))[i], \
list(range(self.cols))[j]
indices = (i * self.cols + j for i in rowsList for j in
colsList)
return self._new(len(rowsList), len(colsList),
list(self.mat[i] for i in indices))
# if the key is a tuple of ints, change
# it to an array index
key = _coord_to_index(i, j)
return self.mat[key]
def __eq__(self, other):
try:
classof(self, other)
except TypeError:
return False
return (
self.shape == other.shape and list(self) == list(other))
def __len__(self):
return self.rows*self.cols
def __repr__(self):
return "_MinimalMatrix({}, {}, {})".format(self.rows, self.cols,
self.mat)
@property
def shape(self):
return (self.rows, self.cols)
class _MatrixWrapper(object):
"""Wrapper class providing the minimum functionality
for a matrix-like object: .rows, .cols, .shape, indexability,
and iterability. CommonMatrix math operations should work
on matrix-like objects. For example, wrapping a numpy
matrix in a MatrixWrapper allows it to be passed to CommonMatrix.
"""
is_MatrixLike = True
def __init__(self, mat, shape=None):
self.mat = mat
self.rows, self.cols = mat.shape if shape is None else shape
def __getattr__(self, attr):
"""Most attribute access is passed straight through
to the stored matrix"""
return getattr(self.mat, attr)
def __getitem__(self, key):
return self.mat.__getitem__(key)
def _matrixify(mat):
"""If `mat` is a Matrix or is matrix-like,
return a Matrix or MatrixWrapper object. Otherwise
`mat` is passed through without modification."""
if getattr(mat, 'is_Matrix', False):
return mat
if hasattr(mat, 'shape'):
if len(mat.shape) == 2:
return _MatrixWrapper(mat)
return mat
def a2idx(j, n=None):
"""Return integer after making positive and validating against n."""
if type(j) is not int:
jindex = getattr(j, '__index__', None)
if jindex is not None:
j = jindex()
else:
raise IndexError("Invalid index a[%r]" % (j,))
if n is not None:
if j < 0:
j += n
if not (j >= 0 and j < n):
raise IndexError("Index out of range: a[%s]" % (j,))
return int(j)
def classof(A, B):
"""
Get the type of the result when combining matrices of different types.
Currently the strategy is that immutability is contagious.
Examples
========
>>> from sympy import Matrix, ImmutableMatrix
>>> from sympy.matrices.common import classof
>>> M = Matrix([[1, 2], [3, 4]]) # a Mutable Matrix
>>> IM = ImmutableMatrix([[1, 2], [3, 4]])
>>> classof(M, IM)
<class 'sympy.matrices.immutable.ImmutableDenseMatrix'>
"""
priority_A = getattr(A, '_class_priority', None)
priority_B = getattr(B, '_class_priority', None)
if None not in (priority_A, priority_B):
if A._class_priority > B._class_priority:
return A.__class__
else:
return B.__class__
try:
import numpy
except ImportError:
pass
else:
if isinstance(A, numpy.ndarray):
return B.__class__
if isinstance(B, numpy.ndarray):
return A.__class__
raise TypeError("Incompatible classes %s, %s" % (A.__class__, B.__class__))
|
ad8d0fd0dcaf33f3ee4809a633e6620ecd32d8c280cd1f8e7ba9361853ef3269 | from __future__ import division, print_function
import random
from sympy.core import SympifyError
from sympy.core.basic import Basic
from sympy.core.compatibility import is_sequence, range, reduce
from sympy.core.expr import Expr
from sympy.core.function import count_ops, expand_mul
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.matrices.common import a2idx, classof
from sympy.matrices.matrices import MatrixBase, ShapeError
from sympy.simplify import simplify as _simplify
from sympy.utilities.decorator import doctest_depends_on
from sympy.utilities.misc import filldedent
def _iszero(x):
"""Returns True if x is zero."""
return x.is_zero
def _compare_sequence(a, b):
"""Compares the elements of a list/tuple `a`
and a list/tuple `b`. `_compare_sequence((1,2), [1, 2])`
is True, whereas `(1,2) == [1, 2]` is False"""
if type(a) is type(b):
# if they are the same type, compare directly
return a == b
# there is no overhead for calling `tuple` on a
# tuple
return tuple(a) == tuple(b)
class DenseMatrix(MatrixBase):
is_MatrixExpr = False
_op_priority = 10.01
_class_priority = 4
def __eq__(self, other):
other = sympify(other)
self_shape = getattr(self, 'shape', None)
other_shape = getattr(other, 'shape', None)
if None in (self_shape, other_shape):
return False
if self_shape != other_shape:
return False
if isinstance(other, Matrix):
return _compare_sequence(self._mat, other._mat)
elif isinstance(other, MatrixBase):
return _compare_sequence(self._mat, Matrix(other)._mat)
def __getitem__(self, key):
"""Return portion of self defined by key. If the key involves a slice
then a list will be returned (if key is a single slice) or a matrix
(if key was a tuple involving a slice).
Examples
========
>>> from sympy import Matrix, I
>>> m = Matrix([
... [1, 2 + I],
... [3, 4 ]])
If the key is a tuple that doesn't involve a slice then that element
is returned:
>>> m[1, 0]
3
When a tuple key involves a slice, a matrix is returned. Here, the
first column is selected (all rows, column 0):
>>> m[:, 0]
Matrix([
[1],
[3]])
If the slice is not a tuple then it selects from the underlying
list of elements that are arranged in row order and a list is
returned if a slice is involved:
>>> m[0]
1
>>> m[::2]
[1, 3]
"""
if isinstance(key, tuple):
i, j = key
try:
i, j = self.key2ij(key)
return self._mat[i*self.cols + j]
except (TypeError, IndexError):
if (isinstance(i, Expr) and not i.is_number) or (isinstance(j, Expr) and not j.is_number):
if ((j < 0) is True) or ((j >= self.shape[1]) is True) or\
((i < 0) is True) or ((i >= self.shape[0]) is True):
raise ValueError("index out of boundary")
from sympy.matrices.expressions.matexpr import MatrixElement
return MatrixElement(self, i, j)
if isinstance(i, slice):
# XXX remove list() when PY2 support is dropped
i = list(range(self.rows))[i]
elif is_sequence(i):
pass
else:
i = [i]
if isinstance(j, slice):
# XXX remove list() when PY2 support is dropped
j = list(range(self.cols))[j]
elif is_sequence(j):
pass
else:
j = [j]
return self.extract(i, j)
else:
# row-wise decomposition of matrix
if isinstance(key, slice):
return self._mat[key]
return self._mat[a2idx(key)]
def __setitem__(self, key, value):
raise NotImplementedError()
def _cholesky(self, hermitian=True):
"""Helper function of cholesky.
Without the error checks.
To be used privately.
Implements the Cholesky-Banachiewicz algorithm.
Returns L such that L*L.H == self if hermitian flag is True,
or L*L.T == self if hermitian is False.
"""
L = zeros(self.rows, self.rows)
if hermitian:
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / L[j, j])*expand_mul(self[i, j] -
sum(L[i, k]*L[j, k].conjugate() for k in range(j)))
Lii2 = expand_mul(self[i, i] -
sum(L[i, k]*L[i, k].conjugate() for k in range(i)))
if Lii2.is_positive is False:
raise ValueError("Matrix must be positive-definite")
L[i, i] = sqrt(Lii2)
else:
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / L[j, j])*(self[i, j] -
sum(L[i, k]*L[j, k] for k in range(j)))
L[i, i] = sqrt(self[i, i] -
sum(L[i, k]**2 for k in range(i)))
return self._new(L)
def _diagonal_solve(self, rhs):
"""Helper function of function diagonal_solve,
without the error checks, to be used privately.
"""
return self._new(rhs.rows, rhs.cols, lambda i, j: rhs[i, j] / self[i, i])
def _eval_add(self, other):
# we assume both arguments are dense matrices since
# sparse matrices have a higher priority
mat = [a + b for a,b in zip(self._mat, other._mat)]
return classof(self, other)._new(self.rows, self.cols, mat, copy=False)
def _eval_extract(self, rowsList, colsList):
mat = self._mat
cols = self.cols
indices = (i * cols + j for i in rowsList for j in colsList)
return self._new(len(rowsList), len(colsList),
list(mat[i] for i in indices), copy=False)
def _eval_matrix_mul(self, other):
from sympy import Add
# cache attributes for faster access
self_rows, self_cols = self.rows, self.cols
other_rows, other_cols = other.rows, other.cols
other_len = other_rows * other_cols
new_mat_rows = self.rows
new_mat_cols = other.cols
# preallocate the array
new_mat = [S.Zero]*new_mat_rows*new_mat_cols
# if we multiply an n x 0 with a 0 x m, the
# expected behavior is to produce an n x m matrix of zeros
if self.cols != 0 and other.rows != 0:
# cache self._mat and other._mat for performance
mat = self._mat
other_mat = other._mat
for i in range(len(new_mat)):
row, col = i // new_mat_cols, i % new_mat_cols
row_indices = range(self_cols*row, self_cols*(row+1))
col_indices = range(col, other_len, other_cols)
vec = (mat[a]*other_mat[b] for a,b in zip(row_indices, col_indices))
try:
new_mat[i] = Add(*vec)
except (TypeError, SympifyError):
# Block matrices don't work with `sum` or `Add` (ISSUE #11599)
# They don't work with `sum` because `sum` tries to add `0`
# initially, and for a matrix, that is a mix of a scalar and
# a matrix, which raises a TypeError. Fall back to a
# block-matrix-safe way to multiply if the `sum` fails.
vec = (mat[a]*other_mat[b] for a,b in zip(row_indices, col_indices))
new_mat[i] = reduce(lambda a,b: a + b, vec)
return classof(self, other)._new(new_mat_rows, new_mat_cols, new_mat, copy=False)
def _eval_matrix_mul_elementwise(self, other):
mat = [a*b for a,b in zip(self._mat, other._mat)]
return classof(self, other)._new(self.rows, self.cols, mat, copy=False)
def _eval_inverse(self, **kwargs):
"""Return the matrix inverse using the method indicated (default
is Gauss elimination).
kwargs
======
method : ('GE', 'LU', or 'ADJ')
iszerofunc
try_block_diag
Notes
=====
According to the ``method`` keyword, it calls the appropriate method:
GE .... inverse_GE(); default
LU .... inverse_LU()
ADJ ... inverse_ADJ()
According to the ``try_block_diag`` keyword, it will try to form block
diagonal matrices using the method get_diag_blocks(), invert these
individually, and then reconstruct the full inverse matrix.
Note, the GE and LU methods may require the matrix to be simplified
before it is inverted in order to properly detect zeros during
pivoting. In difficult cases a custom zero detection function can
be provided by setting the ``iszerosfunc`` argument to a function that
should return True if its argument is zero. The ADJ routine computes
the determinant and uses that to detect singular matrices in addition
to testing for zeros on the diagonal.
See Also
========
inverse_LU
inverse_GE
inverse_ADJ
"""
from sympy.matrices import diag
method = kwargs.get('method', 'GE')
iszerofunc = kwargs.get('iszerofunc', _iszero)
if kwargs.get('try_block_diag', False):
blocks = self.get_diag_blocks()
r = []
for block in blocks:
r.append(block.inv(method=method, iszerofunc=iszerofunc))
return diag(*r)
M = self.as_mutable()
if method == "GE":
rv = M.inverse_GE(iszerofunc=iszerofunc)
elif method == "LU":
rv = M.inverse_LU(iszerofunc=iszerofunc)
elif method == "ADJ":
rv = M.inverse_ADJ(iszerofunc=iszerofunc)
else:
# make sure to add an invertibility check (as in inverse_LU)
# if a new method is added.
raise ValueError("Inversion method unrecognized")
return self._new(rv)
def _eval_scalar_mul(self, other):
mat = [other*a for a in self._mat]
return self._new(self.rows, self.cols, mat, copy=False)
def _eval_scalar_rmul(self, other):
mat = [a*other for a in self._mat]
return self._new(self.rows, self.cols, mat, copy=False)
def _eval_tolist(self):
mat = list(self._mat)
cols = self.cols
return [mat[i*cols:(i + 1)*cols] for i in range(self.rows)]
def _LDLdecomposition(self, hermitian=True):
"""Helper function of LDLdecomposition.
Without the error checks.
To be used privately.
Returns L and D such that L*D*L.H == self if hermitian flag is True,
or L*D*L.T == self if hermitian is False.
"""
# https://en.wikipedia.org/wiki/Cholesky_decomposition#LDL_decomposition_2
D = zeros(self.rows, self.rows)
L = eye(self.rows)
if hermitian:
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / D[j, j])*expand_mul(self[i, j] - sum(
L[i, k]*L[j, k].conjugate()*D[k, k] for k in range(j)))
D[i, i] = expand_mul(self[i, i] -
sum(L[i, k]*L[i, k].conjugate()*D[k, k] for k in range(i)))
if D[i, i].is_positive is False:
raise ValueError("Matrix must be positive-definite")
else:
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / D[j, j])*(self[i, j] - sum(
L[i, k]*L[j, k]*D[k, k] for k in range(j)))
D[i, i] = self[i, i] - sum(L[i, k]**2*D[k, k] for k in range(i))
return self._new(L), self._new(D)
def _lower_triangular_solve(self, rhs):
"""Helper function of function lower_triangular_solve.
Without the error checks.
To be used privately.
"""
X = zeros(self.rows, rhs.cols)
for j in range(rhs.cols):
for i in range(self.rows):
if self[i, i] == 0:
raise TypeError("Matrix must be non-singular.")
X[i, j] = (rhs[i, j] - sum(self[i, k]*X[k, j]
for k in range(i))) / self[i, i]
return self._new(X)
def _upper_triangular_solve(self, rhs):
"""Helper function of function upper_triangular_solve.
Without the error checks, to be used privately. """
X = zeros(self.rows, rhs.cols)
for j in range(rhs.cols):
for i in reversed(range(self.rows)):
if self[i, i] == 0:
raise ValueError("Matrix must be non-singular.")
X[i, j] = (rhs[i, j] - sum(self[i, k]*X[k, j]
for k in range(i + 1, self.rows))) / self[i, i]
return self._new(X)
def as_immutable(self):
"""Returns an Immutable version of this Matrix
"""
from .immutable import ImmutableDenseMatrix as cls
if self.rows and self.cols:
return cls._new(self.tolist())
return cls._new(self.rows, self.cols, [])
def as_mutable(self):
"""Returns a mutable version of this matrix
Examples
========
>>> from sympy import ImmutableMatrix
>>> X = ImmutableMatrix([[1, 2], [3, 4]])
>>> Y = X.as_mutable()
>>> Y[1, 1] = 5 # Can set values in Y
>>> Y
Matrix([
[1, 2],
[3, 5]])
"""
return Matrix(self)
def equals(self, other, failing_expression=False):
"""Applies ``equals`` to corresponding elements of the matrices,
trying to prove that the elements are equivalent, returning True
if they are, False if any pair is not, and None (or the first
failing expression if failing_expression is True) if it cannot
be decided if the expressions are equivalent or not. This is, in
general, an expensive operation.
Examples
========
>>> from sympy.matrices import Matrix
>>> from sympy.abc import x
>>> from sympy import cos
>>> A = Matrix([x*(x - 1), 0])
>>> B = Matrix([x**2 - x, 0])
>>> A == B
False
>>> A.simplify() == B.simplify()
True
>>> A.equals(B)
True
>>> A.equals(2)
False
See Also
========
sympy.core.expr.equals
"""
self_shape = getattr(self, 'shape', None)
other_shape = getattr(other, 'shape', None)
if None in (self_shape, other_shape):
return False
if self_shape != other_shape:
return False
rv = True
for i in range(self.rows):
for j in range(self.cols):
ans = self[i, j].equals(other[i, j], failing_expression)
if ans is False:
return False
elif ans is not True and rv is True:
rv = ans
return rv
def _force_mutable(x):
"""Return a matrix as a Matrix, otherwise return x."""
if getattr(x, 'is_Matrix', False):
return x.as_mutable()
elif isinstance(x, Basic):
return x
elif hasattr(x, '__array__'):
a = x.__array__()
if len(a.shape) == 0:
return sympify(a)
return Matrix(x)
return x
class MutableDenseMatrix(DenseMatrix, MatrixBase):
def __new__(cls, *args, **kwargs):
return cls._new(*args, **kwargs)
@classmethod
def _new(cls, *args, **kwargs):
# if the `copy` flag is set to False, the input
# was rows, cols, [list]. It should be used directly
# without creating a copy.
if kwargs.get('copy', True) is False:
if len(args) != 3:
raise TypeError("'copy=False' requires a matrix be initialized as rows,cols,[list]")
rows, cols, flat_list = args
else:
rows, cols, flat_list = cls._handle_creation_inputs(*args, **kwargs)
flat_list = list(flat_list) # create a shallow copy
self = object.__new__(cls)
self.rows = rows
self.cols = cols
self._mat = flat_list
return self
def __setitem__(self, key, value):
"""
Examples
========
>>> from sympy import Matrix, I, zeros, ones
>>> m = Matrix(((1, 2+I), (3, 4)))
>>> m
Matrix([
[1, 2 + I],
[3, 4]])
>>> m[1, 0] = 9
>>> m
Matrix([
[1, 2 + I],
[9, 4]])
>>> m[1, 0] = [[0, 1]]
To replace row r you assign to position r*m where m
is the number of columns:
>>> M = zeros(4)
>>> m = M.cols
>>> M[3*m] = ones(1, m)*2; M
Matrix([
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[2, 2, 2, 2]])
And to replace column c you can assign to position c:
>>> M[2] = ones(m, 1)*4; M
Matrix([
[0, 0, 4, 0],
[0, 0, 4, 0],
[0, 0, 4, 0],
[2, 2, 4, 2]])
"""
rv = self._setitem(key, value)
if rv is not None:
i, j, value = rv
self._mat[i*self.cols + j] = value
def as_mutable(self):
return self.copy()
def col_del(self, i):
"""Delete the given column.
Examples
========
>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.col_del(1)
>>> M
Matrix([
[1, 0],
[0, 0],
[0, 1]])
See Also
========
col
row_del
"""
if i < -self.cols or i >= self.cols:
raise IndexError("Index out of range: 'i=%s', valid -%s <= i < %s"
% (i, self.cols, self.cols))
for j in range(self.rows - 1, -1, -1):
del self._mat[i + j*self.cols]
self.cols -= 1
def col_op(self, j, f):
"""In-place operation on col j using two-arg functor whose args are
interpreted as (self[i, j], i).
Examples
========
>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.col_op(1, lambda v, i: v + 2*M[i, 0]); M
Matrix([
[1, 2, 0],
[0, 1, 0],
[0, 0, 1]])
See Also
========
col
row_op
"""
self._mat[j::self.cols] = [f(*t) for t in list(zip(self._mat[j::self.cols], list(range(self.rows))))]
def col_swap(self, i, j):
"""Swap the two given columns of the matrix in-place.
Examples
========
>>> from sympy.matrices import Matrix
>>> M = Matrix([[1, 0], [1, 0]])
>>> M
Matrix([
[1, 0],
[1, 0]])
>>> M.col_swap(0, 1)
>>> M
Matrix([
[0, 1],
[0, 1]])
See Also
========
col
row_swap
"""
for k in range(0, self.rows):
self[k, i], self[k, j] = self[k, j], self[k, i]
def copyin_list(self, key, value):
"""Copy in elements from a list.
Parameters
==========
key : slice
The section of this matrix to replace.
value : iterable
The iterable to copy values from.
Examples
========
>>> from sympy.matrices import eye
>>> I = eye(3)
>>> I[:2, 0] = [1, 2] # col
>>> I
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
>>> I[1, :2] = [[3, 4]]
>>> I
Matrix([
[1, 0, 0],
[3, 4, 0],
[0, 0, 1]])
See Also
========
copyin_matrix
"""
if not is_sequence(value):
raise TypeError("`value` must be an ordered iterable, not %s." % type(value))
return self.copyin_matrix(key, Matrix(value))
def copyin_matrix(self, key, value):
"""Copy in values from a matrix into the given bounds.
Parameters
==========
key : slice
The section of this matrix to replace.
value : Matrix
The matrix to copy values from.
Examples
========
>>> from sympy.matrices import Matrix, eye
>>> M = Matrix([[0, 1], [2, 3], [4, 5]])
>>> I = eye(3)
>>> I[:3, :2] = M
>>> I
Matrix([
[0, 1, 0],
[2, 3, 0],
[4, 5, 1]])
>>> I[0, 1] = M
>>> I
Matrix([
[0, 0, 1],
[2, 2, 3],
[4, 4, 5]])
See Also
========
copyin_list
"""
rlo, rhi, clo, chi = self.key2bounds(key)
shape = value.shape
dr, dc = rhi - rlo, chi - clo
if shape != (dr, dc):
raise ShapeError(filldedent("The Matrix `value` doesn't have the "
"same dimensions "
"as the in sub-Matrix given by `key`."))
for i in range(value.rows):
for j in range(value.cols):
self[i + rlo, j + clo] = value[i, j]
def fill(self, value):
"""Fill the matrix with the scalar value.
See Also
========
zeros
ones
"""
self._mat = [value]*len(self)
def row_del(self, i):
"""Delete the given row.
Examples
========
>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.row_del(1)
>>> M
Matrix([
[1, 0, 0],
[0, 0, 1]])
See Also
========
row
col_del
"""
if i < -self.rows or i >= self.rows:
raise IndexError("Index out of range: 'i = %s', valid -%s <= i"
" < %s" % (i, self.rows, self.rows))
if i < 0:
i += self.rows
del self._mat[i*self.cols:(i+1)*self.cols]
self.rows -= 1
def row_op(self, i, f):
"""In-place operation on row ``i`` using two-arg functor whose args are
interpreted as ``(self[i, j], j)``.
Examples
========
>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.row_op(1, lambda v, j: v + 2*M[0, j]); M
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
See Also
========
row
zip_row_op
col_op
"""
i0 = i*self.cols
ri = self._mat[i0: i0 + self.cols]
self._mat[i0: i0 + self.cols] = [f(x, j) for x, j in zip(ri, list(range(self.cols)))]
def row_swap(self, i, j):
"""Swap the two given rows of the matrix in-place.
Examples
========
>>> from sympy.matrices import Matrix
>>> M = Matrix([[0, 1], [1, 0]])
>>> M
Matrix([
[0, 1],
[1, 0]])
>>> M.row_swap(0, 1)
>>> M
Matrix([
[1, 0],
[0, 1]])
See Also
========
row
col_swap
"""
for k in range(0, self.cols):
self[i, k], self[j, k] = self[j, k], self[i, k]
def simplify(self, ratio=1.7, measure=count_ops, rational=False, inverse=False):
"""Applies simplify to the elements of a matrix in place.
This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure))
See Also
========
sympy.simplify.simplify.simplify
"""
for i in range(len(self._mat)):
self._mat[i] = _simplify(self._mat[i], ratio=ratio, measure=measure,
rational=rational, inverse=inverse)
def zip_row_op(self, i, k, f):
"""In-place operation on row ``i`` using two-arg functor whose args are
interpreted as ``(self[i, j], self[k, j])``.
Examples
========
>>> from sympy.matrices import eye
>>> M = eye(3)
>>> M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
Matrix([
[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
See Also
========
row
row_op
col_op
"""
i0 = i*self.cols
k0 = k*self.cols
ri = self._mat[i0: i0 + self.cols]
rk = self._mat[k0: k0 + self.cols]
self._mat[i0: i0 + self.cols] = [f(x, y) for x, y in zip(ri, rk)]
# Utility functions
MutableMatrix = Matrix = MutableDenseMatrix
###########
# Numpy Utility Functions:
# list2numpy, matrix2numpy, symmarray, rot_axis[123]
###########
def list2numpy(l, dtype=object): # pragma: no cover
"""Converts python list of SymPy expressions to a NumPy array.
See Also
========
matrix2numpy
"""
from numpy import empty
a = empty(len(l), dtype)
for i, s in enumerate(l):
a[i] = s
return a
def matrix2numpy(m, dtype=object): # pragma: no cover
"""Converts SymPy's matrix to a NumPy array.
See Also
========
list2numpy
"""
from numpy import empty
a = empty(m.shape, dtype)
for i in range(m.rows):
for j in range(m.cols):
a[i, j] = m[i, j]
return a
def rot_axis3(theta):
"""Returns a rotation matrix for a rotation of theta (in radians) about
the 3-axis.
Examples
========
>>> from sympy import pi
>>> from sympy.matrices import rot_axis3
A rotation of pi/3 (60 degrees):
>>> theta = pi/3
>>> rot_axis3(theta)
Matrix([
[ 1/2, sqrt(3)/2, 0],
[-sqrt(3)/2, 1/2, 0],
[ 0, 0, 1]])
If we rotate by pi/2 (90 degrees):
>>> rot_axis3(pi/2)
Matrix([
[ 0, 1, 0],
[-1, 0, 0],
[ 0, 0, 1]])
See Also
========
rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
about the 1-axis
rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
about the 2-axis
"""
ct = cos(theta)
st = sin(theta)
lil = ((ct, st, 0),
(-st, ct, 0),
(0, 0, 1))
return Matrix(lil)
def rot_axis2(theta):
"""Returns a rotation matrix for a rotation of theta (in radians) about
the 2-axis.
Examples
========
>>> from sympy import pi
>>> from sympy.matrices import rot_axis2
A rotation of pi/3 (60 degrees):
>>> theta = pi/3
>>> rot_axis2(theta)
Matrix([
[ 1/2, 0, -sqrt(3)/2],
[ 0, 1, 0],
[sqrt(3)/2, 0, 1/2]])
If we rotate by pi/2 (90 degrees):
>>> rot_axis2(pi/2)
Matrix([
[0, 0, -1],
[0, 1, 0],
[1, 0, 0]])
See Also
========
rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
about the 1-axis
rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
about the 3-axis
"""
ct = cos(theta)
st = sin(theta)
lil = ((ct, 0, -st),
(0, 1, 0),
(st, 0, ct))
return Matrix(lil)
def rot_axis1(theta):
"""Returns a rotation matrix for a rotation of theta (in radians) about
the 1-axis.
Examples
========
>>> from sympy import pi
>>> from sympy.matrices import rot_axis1
A rotation of pi/3 (60 degrees):
>>> theta = pi/3
>>> rot_axis1(theta)
Matrix([
[1, 0, 0],
[0, 1/2, sqrt(3)/2],
[0, -sqrt(3)/2, 1/2]])
If we rotate by pi/2 (90 degrees):
>>> rot_axis1(pi/2)
Matrix([
[1, 0, 0],
[0, 0, 1],
[0, -1, 0]])
See Also
========
rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
about the 2-axis
rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
about the 3-axis
"""
ct = cos(theta)
st = sin(theta)
lil = ((1, 0, 0),
(0, ct, st),
(0, -st, ct))
return Matrix(lil)
@doctest_depends_on(modules=('numpy',))
def symarray(prefix, shape, **kwargs): # pragma: no cover
r"""Create a numpy ndarray of symbols (as an object array).
The created symbols are named ``prefix_i1_i2_``... You should thus provide a
non-empty prefix if you want your symbols to be unique for different output
arrays, as SymPy symbols with identical names are the same object.
Parameters
----------
prefix : string
A prefix prepended to the name of every symbol.
shape : int or tuple
Shape of the created array. If an int, the array is one-dimensional; for
more than one dimension the shape must be a tuple.
\*\*kwargs : dict
keyword arguments passed on to Symbol
Examples
========
These doctests require numpy.
>>> from sympy import symarray
>>> symarray('', 3)
[_0 _1 _2]
If you want multiple symarrays to contain distinct symbols, you *must*
provide unique prefixes:
>>> a = symarray('', 3)
>>> b = symarray('', 3)
>>> a[0] == b[0]
True
>>> a = symarray('a', 3)
>>> b = symarray('b', 3)
>>> a[0] == b[0]
False
Creating symarrays with a prefix:
>>> symarray('a', 3)
[a_0 a_1 a_2]
For more than one dimension, the shape must be given as a tuple:
>>> symarray('a', (2, 3))
[[a_0_0 a_0_1 a_0_2]
[a_1_0 a_1_1 a_1_2]]
>>> symarray('a', (2, 3, 2))
[[[a_0_0_0 a_0_0_1]
[a_0_1_0 a_0_1_1]
[a_0_2_0 a_0_2_1]]
<BLANKLINE>
[[a_1_0_0 a_1_0_1]
[a_1_1_0 a_1_1_1]
[a_1_2_0 a_1_2_1]]]
For setting assumptions of the underlying Symbols:
>>> [s.is_real for s in symarray('a', 2, real=True)]
[True, True]
"""
from numpy import empty, ndindex
arr = empty(shape, dtype=object)
for index in ndindex(shape):
arr[index] = Symbol('%s_%s' % (prefix, '_'.join(map(str, index))),
**kwargs)
return arr
###############
# Functions
###############
def casoratian(seqs, n, zero=True):
"""Given linear difference operator L of order 'k' and homogeneous
equation Ly = 0 we want to compute kernel of L, which is a set
of 'k' sequences: a(n), b(n), ... z(n).
Solutions of L are linearly independent iff their Casoratian,
denoted as C(a, b, ..., z), do not vanish for n = 0.
Casoratian is defined by k x k determinant::
+ a(n) b(n) . . . z(n) +
| a(n+1) b(n+1) . . . z(n+1) |
| . . . . |
| . . . . |
| . . . . |
+ a(n+k-1) b(n+k-1) . . . z(n+k-1) +
It proves very useful in rsolve_hyper() where it is applied
to a generating set of a recurrence to factor out linearly
dependent solutions and return a basis:
>>> from sympy import Symbol, casoratian, factorial
>>> n = Symbol('n', integer=True)
Exponential and factorial are linearly independent:
>>> casoratian([2**n, factorial(n)], n) != 0
True
"""
seqs = list(map(sympify, seqs))
if not zero:
f = lambda i, j: seqs[j].subs(n, n + i)
else:
f = lambda i, j: seqs[j].subs(n, i)
k = len(seqs)
return Matrix(k, k, f).det()
def eye(*args, **kwargs):
"""Create square identity matrix n x n
See Also
========
diag
zeros
ones
"""
return Matrix.eye(*args, **kwargs)
def diag(*values, **kwargs):
"""Returns a matrix with the provided values placed on the
diagonal. If non-square matrices are included, they will
produce a block-diagonal matrix.
Examples
========
This version of diag is a thin wrapper to Matrix.diag that differs
in that it treats all lists like matrices -- even when a single list
is given. If this is not desired, either put a `*` before the list or
set `unpack=True`.
>>> from sympy import diag
>>> diag([1, 2, 3], unpack=True) # = diag(1,2,3) or diag(*[1,2,3])
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
>>> diag([1, 2, 3]) # a column vector
Matrix([
[1],
[2],
[3]])
See Also
========
.common.MatrixCommon.eye
.common.MatrixCommon.diagonal - to extract a diagonal
.common.MatrixCommon.diag
.expressions.blockmatrix.BlockMatrix
"""
# Extract any setting so we don't duplicate keywords sent
# as named parameters:
kw = kwargs.copy()
strict = kw.pop('strict', True) # lists will be converted to Matrices
unpack = kw.pop('unpack', False)
return Matrix.diag(*values, strict=strict, unpack=unpack, **kw)
def GramSchmidt(vlist, orthonormal=False):
"""Apply the Gram-Schmidt process to a set of vectors.
Parameters
==========
vlist : List of Matrix
Vectors to be orthogonalized for.
orthonormal : Bool, optional
If true, return an orthonormal basis.
Returns
=======
vlist : List of Matrix
Orthogonalized vectors
Notes
=====
This routine is mostly duplicate from ``Matrix.orthogonalize``,
except for some difference that this always raises error when
linearly dependent vectors are found, and the keyword ``normalize``
has been named as ``orthonormal`` in this function.
See Also
========
.matrices.MatrixSubspaces.orthogonalize
References
==========
.. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
"""
return MutableDenseMatrix.orthogonalize(
*vlist, normalize=orthonormal, rankcheck=True
)
def hessian(f, varlist, constraints=[]):
"""Compute Hessian matrix for a function f wrt parameters in varlist
which may be given as a sequence or a row/column vector. A list of
constraints may optionally be given.
Examples
========
>>> from sympy import Function, hessian, pprint
>>> from sympy.abc import x, y
>>> f = Function('f')(x, y)
>>> g1 = Function('g')(x, y)
>>> g2 = x**2 + 3*y
>>> pprint(hessian(f, (x, y), [g1, g2]))
[ d d ]
[ 0 0 --(g(x, y)) --(g(x, y)) ]
[ dx dy ]
[ ]
[ 0 0 2*x 3 ]
[ ]
[ 2 2 ]
[d d d ]
[--(g(x, y)) 2*x ---(f(x, y)) -----(f(x, y))]
[dx 2 dy dx ]
[ dx ]
[ ]
[ 2 2 ]
[d d d ]
[--(g(x, y)) 3 -----(f(x, y)) ---(f(x, y)) ]
[dy dy dx 2 ]
[ dy ]
References
==========
https://en.wikipedia.org/wiki/Hessian_matrix
See Also
========
sympy.matrices.mutable.Matrix.jacobian
wronskian
"""
# f is the expression representing a function f, return regular matrix
if isinstance(varlist, MatrixBase):
if 1 not in varlist.shape:
raise ShapeError("`varlist` must be a column or row vector.")
if varlist.cols == 1:
varlist = varlist.T
varlist = varlist.tolist()[0]
if is_sequence(varlist):
n = len(varlist)
if not n:
raise ShapeError("`len(varlist)` must not be zero.")
else:
raise ValueError("Improper variable list in hessian function")
if not getattr(f, 'diff'):
# check differentiability
raise ValueError("Function `f` (%s) is not differentiable" % f)
m = len(constraints)
N = m + n
out = zeros(N)
for k, g in enumerate(constraints):
if not getattr(g, 'diff'):
# check differentiability
raise ValueError("Function `f` (%s) is not differentiable" % f)
for i in range(n):
out[k, i + m] = g.diff(varlist[i])
for i in range(n):
for j in range(i, n):
out[i + m, j + m] = f.diff(varlist[i]).diff(varlist[j])
for i in range(N):
for j in range(i + 1, N):
out[j, i] = out[i, j]
return out
def jordan_cell(eigenval, n):
"""
Create a Jordan block:
Examples
========
>>> from sympy.matrices import jordan_cell
>>> from sympy.abc import x
>>> jordan_cell(x, 4)
Matrix([
[x, 1, 0, 0],
[0, x, 1, 0],
[0, 0, x, 1],
[0, 0, 0, x]])
"""
return Matrix.jordan_block(size=n, eigenvalue=eigenval)
def matrix_multiply_elementwise(A, B):
"""Return the Hadamard product (elementwise product) of A and B
>>> from sympy.matrices import matrix_multiply_elementwise
>>> from sympy.matrices import Matrix
>>> A = Matrix([[0, 1, 2], [3, 4, 5]])
>>> B = Matrix([[1, 10, 100], [100, 10, 1]])
>>> matrix_multiply_elementwise(A, B)
Matrix([
[ 0, 10, 200],
[300, 40, 5]])
See Also
========
__mul__
"""
return A.multiply_elementwise(B)
def ones(*args, **kwargs):
"""Returns a matrix of ones with ``rows`` rows and ``cols`` columns;
if ``cols`` is omitted a square matrix will be returned.
See Also
========
zeros
eye
diag
"""
if 'c' in kwargs:
kwargs['cols'] = kwargs.pop('c')
return Matrix.ones(*args, **kwargs)
def randMatrix(r, c=None, min=0, max=99, seed=None, symmetric=False,
percent=100, prng=None):
"""Create random matrix with dimensions ``r`` x ``c``. If ``c`` is omitted
the matrix will be square. If ``symmetric`` is True the matrix must be
square. If ``percent`` is less than 100 then only approximately the given
percentage of elements will be non-zero.
The pseudo-random number generator used to generate matrix is chosen in the
following way.
* If ``prng`` is supplied, it will be used as random number generator.
It should be an instance of :class:`random.Random`, or at least have
``randint`` and ``shuffle`` methods with same signatures.
* if ``prng`` is not supplied but ``seed`` is supplied, then new
:class:`random.Random` with given ``seed`` will be created;
* otherwise, a new :class:`random.Random` with default seed will be used.
Examples
========
>>> from sympy.matrices import randMatrix
>>> randMatrix(3) # doctest:+SKIP
[25, 45, 27]
[44, 54, 9]
[23, 96, 46]
>>> randMatrix(3, 2) # doctest:+SKIP
[87, 29]
[23, 37]
[90, 26]
>>> randMatrix(3, 3, 0, 2) # doctest:+SKIP
[0, 2, 0]
[2, 0, 1]
[0, 0, 1]
>>> randMatrix(3, symmetric=True) # doctest:+SKIP
[85, 26, 29]
[26, 71, 43]
[29, 43, 57]
>>> A = randMatrix(3, seed=1)
>>> B = randMatrix(3, seed=2)
>>> A == B # doctest:+SKIP
False
>>> A == randMatrix(3, seed=1)
True
>>> randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP
[77, 70, 0],
[70, 0, 0],
[ 0, 0, 88]
"""
if c is None:
c = r
# Note that ``Random()`` is equivalent to ``Random(None)``
prng = prng or random.Random(seed)
if not symmetric:
m = Matrix._new(r, c, lambda i, j: prng.randint(min, max))
if percent == 100:
return m
z = int(r*c*(100 - percent) // 100)
m._mat[:z] = [S.Zero]*z
prng.shuffle(m._mat)
return m
# Symmetric case
if r != c:
raise ValueError('For symmetric matrices, r must equal c, but %i != %i' % (r, c))
m = zeros(r)
ij = [(i, j) for i in range(r) for j in range(i, r)]
if percent != 100:
ij = prng.sample(ij, int(len(ij)*percent // 100))
for i, j in ij:
value = prng.randint(min, max)
m[i, j] = m[j, i] = value
return m
def wronskian(functions, var, method='bareiss'):
"""
Compute Wronskian for [] of functions
::
| f1 f2 ... fn |
| f1' f2' ... fn' |
| . . . . |
W(f1, ..., fn) = | . . . . |
| . . . . |
| (n) (n) (n) |
| D (f1) D (f2) ... D (fn) |
see: https://en.wikipedia.org/wiki/Wronskian
See Also
========
sympy.matrices.mutable.Matrix.jacobian
hessian
"""
for index in range(0, len(functions)):
functions[index] = sympify(functions[index])
n = len(functions)
if n == 0:
return 1
W = Matrix(n, n, lambda i, j: functions[i].diff(var, j))
return W.det(method)
def zeros(*args, **kwargs):
"""Returns a matrix of zeros with ``rows`` rows and ``cols`` columns;
if ``cols`` is omitted a square matrix will be returned.
See Also
========
ones
eye
diag
"""
if 'c' in kwargs:
kwargs['cols'] = kwargs.pop('c')
return Matrix.zeros(*args, **kwargs)
|
854106cb72929cb79e5cac68c05bae797b78807d7fc9991351f03591b08ee4db | from __future__ import division, print_function
import copy
from collections import defaultdict
from sympy.core.compatibility import Callable, as_int, is_sequence, range
from sympy.core.containers import Dict
from sympy.core.expr import Expr
from sympy.core.singleton import S
from sympy.functions import Abs
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.utilities.iterables import uniq
from sympy.utilities.misc import filldedent
from .common import a2idx
from .dense import Matrix
from .matrices import MatrixBase, ShapeError
class SparseMatrix(MatrixBase):
"""
A sparse matrix (a matrix with a large number of zero elements).
Examples
========
>>> from sympy.matrices import SparseMatrix, ones
>>> SparseMatrix(2, 2, range(4))
Matrix([
[0, 1],
[2, 3]])
>>> SparseMatrix(2, 2, {(1, 1): 2})
Matrix([
[0, 0],
[0, 2]])
A SparseMatrix can be instantiated from a ragged list of lists:
>>> SparseMatrix([[1, 2, 3], [1, 2], [1]])
Matrix([
[1, 2, 3],
[1, 2, 0],
[1, 0, 0]])
For safety, one may include the expected size and then an error
will be raised if the indices of any element are out of range or
(for a flat list) if the total number of elements does not match
the expected shape:
>>> SparseMatrix(2, 2, [1, 2])
Traceback (most recent call last):
...
ValueError: List length (2) != rows*columns (4)
Here, an error is not raised because the list is not flat and no
element is out of range:
>>> SparseMatrix(2, 2, [[1, 2]])
Matrix([
[1, 2],
[0, 0]])
But adding another element to the first (and only) row will cause
an error to be raised:
>>> SparseMatrix(2, 2, [[1, 2, 3]])
Traceback (most recent call last):
...
ValueError: The location (0, 2) is out of designated range: (1, 1)
To autosize the matrix, pass None for rows:
>>> SparseMatrix(None, [[1, 2, 3]])
Matrix([[1, 2, 3]])
>>> SparseMatrix(None, {(1, 1): 1, (3, 3): 3})
Matrix([
[0, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 3]])
Values that are themselves a Matrix are automatically expanded:
>>> SparseMatrix(4, 4, {(1, 1): ones(2)})
Matrix([
[0, 0, 0, 0],
[0, 1, 1, 0],
[0, 1, 1, 0],
[0, 0, 0, 0]])
A ValueError is raised if the expanding matrix tries to overwrite
a different element already present:
>>> SparseMatrix(3, 3, {(0, 0): ones(2), (1, 1): 2})
Traceback (most recent call last):
...
ValueError: collision at (1, 1)
See Also
========
sympy.matrices.common.diag
sympy.matrices.dense.Matrix
"""
def __new__(cls, *args, **kwargs):
self = object.__new__(cls)
if len(args) == 1 and isinstance(args[0], SparseMatrix):
self.rows = args[0].rows
self.cols = args[0].cols
self._smat = dict(args[0]._smat)
return self
self._smat = {}
# autosizing
if len(args) == 2 and args[0] is None:
args = (None,) + args
if len(args) == 3:
r, c = args[:2]
if r is c is None:
self.rows = self.cols = None
elif None in (r, c):
raise ValueError(
'Pass rows=None and no cols for autosizing.')
else:
self.rows, self.cols = map(as_int, args[:2])
if isinstance(args[2], Callable):
op = args[2]
for i in range(self.rows):
for j in range(self.cols):
value = self._sympify(
op(self._sympify(i), self._sympify(j)))
if value:
self._smat[i, j] = value
elif isinstance(args[2], (dict, Dict)):
def update(i, j, v):
# update self._smat and make sure there are
# no collisions
if v:
if (i, j) in self._smat and v != self._smat[i, j]:
raise ValueError('collision at %s' % ((i, j),))
self._smat[i, j] = v
# manual copy, copy.deepcopy() doesn't work
for key, v in args[2].items():
r, c = key
if isinstance(v, SparseMatrix):
for (i, j), vij in v._smat.items():
update(r + i, c + j, vij)
else:
if isinstance(v, (Matrix, list, tuple)):
v = SparseMatrix(v)
for i, j in v._smat:
update(r + i, c + j, v[i, j])
else:
v = self._sympify(v)
update(r, c, self._sympify(v))
elif is_sequence(args[2]):
flat = not any(is_sequence(i) for i in args[2])
if not flat:
s = SparseMatrix(args[2])
self._smat = s._smat
else:
if len(args[2]) != self.rows*self.cols:
raise ValueError(
'Flat list length (%s) != rows*columns (%s)' %
(len(args[2]), self.rows*self.cols))
flat_list = args[2]
for i in range(self.rows):
for j in range(self.cols):
value = self._sympify(flat_list[i*self.cols + j])
if value:
self._smat[i, j] = value
if self.rows is None: # autosizing
k = self._smat.keys()
self.rows = max([i[0] for i in k]) + 1 if k else 0
self.cols = max([i[1] for i in k]) + 1 if k else 0
else:
for i, j in self._smat.keys():
if i and i >= self.rows or j and j >= self.cols:
r, c = self.shape
raise ValueError(filldedent('''
The location %s is out of designated
range: %s''' % ((i, j), (r - 1, c - 1))))
else:
if (len(args) == 1 and isinstance(args[0], (list, tuple))):
# list of values or lists
v = args[0]
c = 0
for i, row in enumerate(v):
if not isinstance(row, (list, tuple)):
row = [row]
for j, vij in enumerate(row):
if vij:
self._smat[i, j] = self._sympify(vij)
c = max(c, len(row))
self.rows = len(v) if c else 0
self.cols = c
else:
# handle full matrix forms with _handle_creation_inputs
r, c, _list = Matrix._handle_creation_inputs(*args)
self.rows = r
self.cols = c
for i in range(self.rows):
for j in range(self.cols):
value = _list[self.cols*i + j]
if value:
self._smat[i, j] = value
return self
def __eq__(self, other):
self_shape = getattr(self, 'shape', None)
other_shape = getattr(other, 'shape', None)
if None in (self_shape, other_shape):
return False
if self_shape != other_shape:
return False
if isinstance(other, SparseMatrix):
return self._smat == other._smat
elif isinstance(other, MatrixBase):
return self._smat == MutableSparseMatrix(other)._smat
def __getitem__(self, key):
if isinstance(key, tuple):
i, j = key
try:
i, j = self.key2ij(key)
return self._smat.get((i, j), S.Zero)
except (TypeError, IndexError):
if isinstance(i, slice):
# XXX remove list() when PY2 support is dropped
i = list(range(self.rows))[i]
elif is_sequence(i):
pass
elif isinstance(i, Expr) and not i.is_number:
from sympy.matrices.expressions.matexpr import MatrixElement
return MatrixElement(self, i, j)
else:
if i >= self.rows:
raise IndexError('Row index out of bounds')
i = [i]
if isinstance(j, slice):
# XXX remove list() when PY2 support is dropped
j = list(range(self.cols))[j]
elif is_sequence(j):
pass
elif isinstance(j, Expr) and not j.is_number:
from sympy.matrices.expressions.matexpr import MatrixElement
return MatrixElement(self, i, j)
else:
if j >= self.cols:
raise IndexError('Col index out of bounds')
j = [j]
return self.extract(i, j)
# check for single arg, like M[:] or M[3]
if isinstance(key, slice):
lo, hi = key.indices(len(self))[:2]
L = []
for i in range(lo, hi):
m, n = divmod(i, self.cols)
L.append(self._smat.get((m, n), S.Zero))
return L
i, j = divmod(a2idx(key, len(self)), self.cols)
return self._smat.get((i, j), S.Zero)
def __setitem__(self, key, value):
raise NotImplementedError()
def _cholesky_solve(self, rhs):
# for speed reasons, this is not uncommented, but if you are
# having difficulties, try uncommenting to make sure that the
# input matrix is symmetric
#assert self.is_symmetric()
L = self._cholesky_sparse()
Y = L._lower_triangular_solve(rhs)
rv = L.T._upper_triangular_solve(Y)
return rv
def _cholesky_sparse(self):
"""Algorithm for numeric Cholesky factorization of a sparse matrix."""
Crowstruc = self.row_structure_symbolic_cholesky()
C = self.zeros(self.rows)
for i in range(len(Crowstruc)):
for j in Crowstruc[i]:
if i != j:
C[i, j] = self[i, j]
summ = 0
for p1 in Crowstruc[i]:
if p1 < j:
for p2 in Crowstruc[j]:
if p2 < j:
if p1 == p2:
summ += C[i, p1]*C[j, p1]
else:
break
else:
break
C[i, j] -= summ
C[i, j] /= C[j, j]
else:
C[j, j] = self[j, j]
summ = 0
for k in Crowstruc[j]:
if k < j:
summ += C[j, k]**2
else:
break
C[j, j] -= summ
C[j, j] = sqrt(C[j, j])
return C
def _diagonal_solve(self, rhs):
"Diagonal solve."
return self._new(self.rows, 1, lambda i, j: rhs[i, 0] / self[i, i])
def _eval_inverse(self, **kwargs):
"""Return the matrix inverse using Cholesky or LDL (default)
decomposition as selected with the ``method`` keyword: 'CH' or 'LDL',
respectively.
Examples
========
>>> from sympy import SparseMatrix, Matrix
>>> A = SparseMatrix([
... [ 2, -1, 0],
... [-1, 2, -1],
... [ 0, 0, 2]])
>>> A.inv('CH')
Matrix([
[2/3, 1/3, 1/6],
[1/3, 2/3, 1/3],
[ 0, 0, 1/2]])
>>> A.inv(method='LDL') # use of 'method=' is optional
Matrix([
[2/3, 1/3, 1/6],
[1/3, 2/3, 1/3],
[ 0, 0, 1/2]])
>>> A * _
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
"""
sym = self.is_symmetric()
M = self.as_mutable()
I = M.eye(M.rows)
if not sym:
t = M.T
r1 = M[0, :]
M = t*M
I = t*I
method = kwargs.get('method', 'LDL')
if method in "LDL":
solve = M._LDL_solve
elif method == "CH":
solve = M._cholesky_solve
else:
raise NotImplementedError(
'Method may be "CH" or "LDL", not %s.' % method)
rv = M.hstack(*[solve(I[:, i]) for i in range(I.cols)])
if not sym:
scale = (r1*rv[:, 0])[0, 0]
rv /= scale
return self._new(rv)
def _eval_Abs(self):
return self.applyfunc(lambda x: Abs(x))
def _eval_add(self, other):
"""If `other` is a SparseMatrix, add efficiently. Otherwise,
do standard addition."""
if not isinstance(other, SparseMatrix):
return self + self._new(other)
smat = {}
zero = self._sympify(0)
for key in set().union(self._smat.keys(), other._smat.keys()):
sum = self._smat.get(key, zero) + other._smat.get(key, zero)
if sum != 0:
smat[key] = sum
return self._new(self.rows, self.cols, smat)
def _eval_col_insert(self, icol, other):
if not isinstance(other, SparseMatrix):
other = SparseMatrix(other)
new_smat = {}
# make room for the new rows
for key, val in self._smat.items():
row, col = key
if col >= icol:
col += other.cols
new_smat[row, col] = val
# add other's keys
for key, val in other._smat.items():
row, col = key
new_smat[row, col + icol] = val
return self._new(self.rows, self.cols + other.cols, new_smat)
def _eval_conjugate(self):
smat = {key: val.conjugate() for key,val in self._smat.items()}
return self._new(self.rows, self.cols, smat)
def _eval_extract(self, rowsList, colsList):
urow = list(uniq(rowsList))
ucol = list(uniq(colsList))
smat = {}
if len(urow)*len(ucol) < len(self._smat):
# there are fewer elements requested than there are elements in the matrix
for i, r in enumerate(urow):
for j, c in enumerate(ucol):
smat[i, j] = self._smat.get((r, c), 0)
else:
# most of the request will be zeros so check all of self's entries,
# keeping only the ones that are desired
for rk, ck in self._smat:
if rk in urow and ck in ucol:
smat[urow.index(rk), ucol.index(ck)] = self._smat[rk, ck]
rv = self._new(len(urow), len(ucol), smat)
# rv is nominally correct but there might be rows/cols
# which require duplication
if len(rowsList) != len(urow):
for i, r in enumerate(rowsList):
i_previous = rowsList.index(r)
if i_previous != i:
rv = rv.row_insert(i, rv.row(i_previous))
if len(colsList) != len(ucol):
for i, c in enumerate(colsList):
i_previous = colsList.index(c)
if i_previous != i:
rv = rv.col_insert(i, rv.col(i_previous))
return rv
@classmethod
def _eval_eye(cls, rows, cols):
entries = {(i,i): S.One for i in range(min(rows, cols))}
return cls._new(rows, cols, entries)
def _eval_has(self, *patterns):
# if the matrix has any zeros, see if S.Zero
# has the pattern. If _smat is full length,
# the matrix has no zeros.
zhas = S.Zero.has(*patterns)
if len(self._smat) == self.rows*self.cols:
zhas = False
return any(self[key].has(*patterns) for key in self._smat) or zhas
def _eval_is_Identity(self):
if not all(self[i, i] == 1 for i in range(self.rows)):
return False
return len(self._smat) == self.rows
def _eval_is_symmetric(self, simpfunc):
diff = (self - self.T).applyfunc(simpfunc)
return len(diff.values()) == 0
def _eval_matrix_mul(self, other):
"""Fast multiplication exploiting the sparsity of the matrix."""
if not isinstance(other, SparseMatrix):
return self*self._new(other)
# if we made it here, we're both sparse matrices
# create quick lookups for rows and cols
row_lookup = defaultdict(dict)
for (i,j), val in self._smat.items():
row_lookup[i][j] = val
col_lookup = defaultdict(dict)
for (i,j), val in other._smat.items():
col_lookup[j][i] = val
smat = {}
for row in row_lookup.keys():
for col in col_lookup.keys():
# find the common indices of non-zero entries.
# these are the only things that need to be multiplied.
indices = set(col_lookup[col].keys()) & set(row_lookup[row].keys())
if indices:
val = sum(row_lookup[row][k]*col_lookup[col][k] for k in indices)
smat[row, col] = val
return self._new(self.rows, other.cols, smat)
def _eval_row_insert(self, irow, other):
if not isinstance(other, SparseMatrix):
other = SparseMatrix(other)
new_smat = {}
# make room for the new rows
for key, val in self._smat.items():
row, col = key
if row >= irow:
row += other.rows
new_smat[row, col] = val
# add other's keys
for key, val in other._smat.items():
row, col = key
new_smat[row + irow, col] = val
return self._new(self.rows + other.rows, self.cols, new_smat)
def _eval_scalar_mul(self, other):
return self.applyfunc(lambda x: x*other)
def _eval_scalar_rmul(self, other):
return self.applyfunc(lambda x: other*x)
def _eval_transpose(self):
"""Returns the transposed SparseMatrix of this SparseMatrix.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> a = SparseMatrix(((1, 2), (3, 4)))
>>> a
Matrix([
[1, 2],
[3, 4]])
>>> a.T
Matrix([
[1, 3],
[2, 4]])
"""
smat = {(j,i): val for (i,j),val in self._smat.items()}
return self._new(self.cols, self.rows, smat)
def _eval_values(self):
return [v for k,v in self._smat.items() if not v.is_zero]
@classmethod
def _eval_zeros(cls, rows, cols):
return cls._new(rows, cols, {})
def _LDL_solve(self, rhs):
# for speed reasons, this is not uncommented, but if you are
# having difficulties, try uncommenting to make sure that the
# input matrix is symmetric
#assert self.is_symmetric()
L, D = self._LDL_sparse()
Z = L._lower_triangular_solve(rhs)
Y = D._diagonal_solve(Z)
return L.T._upper_triangular_solve(Y)
def _LDL_sparse(self):
"""Algorithm for numeric LDL factization, exploiting sparse structure.
"""
Lrowstruc = self.row_structure_symbolic_cholesky()
L = self.eye(self.rows)
D = self.zeros(self.rows, self.cols)
for i in range(len(Lrowstruc)):
for j in Lrowstruc[i]:
if i != j:
L[i, j] = self[i, j]
summ = 0
for p1 in Lrowstruc[i]:
if p1 < j:
for p2 in Lrowstruc[j]:
if p2 < j:
if p1 == p2:
summ += L[i, p1]*L[j, p1]*D[p1, p1]
else:
break
else:
break
L[i, j] -= summ
L[i, j] /= D[j, j]
else: # i == j
D[i, i] = self[i, i]
summ = 0
for k in Lrowstruc[i]:
if k < i:
summ += L[i, k]**2*D[k, k]
else:
break
D[i, i] -= summ
return L, D
def _lower_triangular_solve(self, rhs):
"""Fast algorithm for solving a lower-triangular system,
exploiting the sparsity of the given matrix.
"""
rows = [[] for i in range(self.rows)]
for i, j, v in self.row_list():
if i > j:
rows[i].append((j, v))
X = rhs.copy()
for i in range(self.rows):
for j, v in rows[i]:
X[i, 0] -= v*X[j, 0]
X[i, 0] /= self[i, i]
return self._new(X)
@property
def _mat(self):
"""Return a list of matrix elements. Some routines
in DenseMatrix use `_mat` directly to speed up operations."""
return list(self)
def _upper_triangular_solve(self, rhs):
"""Fast algorithm for solving an upper-triangular system,
exploiting the sparsity of the given matrix.
"""
rows = [[] for i in range(self.rows)]
for i, j, v in self.row_list():
if i < j:
rows[i].append((j, v))
X = rhs.copy()
for i in range(self.rows - 1, -1, -1):
rows[i].reverse()
for j, v in rows[i]:
X[i, 0] -= v*X[j, 0]
X[i, 0] /= self[i, i]
return self._new(X)
def applyfunc(self, f):
"""Apply a function to each element of the matrix.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> m = SparseMatrix(2, 2, lambda i, j: i*2+j)
>>> m
Matrix([
[0, 1],
[2, 3]])
>>> m.applyfunc(lambda i: 2*i)
Matrix([
[0, 2],
[4, 6]])
"""
if not callable(f):
raise TypeError("`f` must be callable.")
out = self.copy()
for k, v in self._smat.items():
fv = f(v)
if fv:
out._smat[k] = fv
else:
out._smat.pop(k, None)
return out
def as_immutable(self):
"""Returns an Immutable version of this Matrix."""
from .immutable import ImmutableSparseMatrix
return ImmutableSparseMatrix(self)
def as_mutable(self):
"""Returns a mutable version of this matrix.
Examples
========
>>> from sympy import ImmutableMatrix
>>> X = ImmutableMatrix([[1, 2], [3, 4]])
>>> Y = X.as_mutable()
>>> Y[1, 1] = 5 # Can set values in Y
>>> Y
Matrix([
[1, 2],
[3, 5]])
"""
return MutableSparseMatrix(self)
def cholesky(self):
"""
Returns the Cholesky decomposition L of a matrix A
such that L * L.T = A
A must be a square, symmetric, positive-definite
and non-singular matrix
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> A = SparseMatrix(((25,15,-5),(15,18,0),(-5,0,11)))
>>> A.cholesky()
Matrix([
[ 5, 0, 0],
[ 3, 3, 0],
[-1, 1, 3]])
>>> A.cholesky() * A.cholesky().T == A
True
"""
from sympy.core.numbers import nan, oo
if not self.is_symmetric():
raise ValueError('Cholesky decomposition applies only to '
'symmetric matrices.')
M = self.as_mutable()._cholesky_sparse()
if M.has(nan) or M.has(oo):
raise ValueError('Cholesky decomposition applies only to '
'positive-definite matrices')
return self._new(M)
def col_list(self):
"""Returns a column-sorted list of non-zero elements of the matrix.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> a=SparseMatrix(((1, 2), (3, 4)))
>>> a
Matrix([
[1, 2],
[3, 4]])
>>> a.CL
[(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)]
See Also
========
col_op
row_list
"""
return [tuple(k + (self[k],)) for k in sorted(list(self._smat.keys()), key=lambda k: list(reversed(k)))]
def copy(self):
return self._new(self.rows, self.cols, self._smat)
def LDLdecomposition(self):
"""
Returns the LDL Decomposition (matrices ``L`` and ``D``) of matrix
``A``, such that ``L * D * L.T == A``. ``A`` must be a square,
symmetric, positive-definite and non-singular.
This method eliminates the use of square root and ensures that all
the diagonal entries of L are 1.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
>>> L, D = A.LDLdecomposition()
>>> L
Matrix([
[ 1, 0, 0],
[ 3/5, 1, 0],
[-1/5, 1/3, 1]])
>>> D
Matrix([
[25, 0, 0],
[ 0, 9, 0],
[ 0, 0, 9]])
>>> L * D * L.T == A
True
"""
from sympy.core.numbers import nan, oo
if not self.is_symmetric():
raise ValueError('LDL decomposition applies only to '
'symmetric matrices.')
L, D = self.as_mutable()._LDL_sparse()
if L.has(nan) or L.has(oo) or D.has(nan) or D.has(oo):
raise ValueError('LDL decomposition applies only to '
'positive-definite matrices')
return self._new(L), self._new(D)
def liupc(self):
"""Liu's algorithm, for pre-determination of the Elimination Tree of
the given matrix, used in row-based symbolic Cholesky factorization.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> S = SparseMatrix([
... [1, 0, 3, 2],
... [0, 0, 1, 0],
... [4, 0, 0, 5],
... [0, 6, 7, 0]])
>>> S.liupc()
([[0], [], [0], [1, 2]], [4, 3, 4, 4])
References
==========
Symbolic Sparse Cholesky Factorization using Elimination Trees,
Jeroen Van Grondelle (1999)
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582
"""
# Algorithm 2.4, p 17 of reference
# get the indices of the elements that are non-zero on or below diag
R = [[] for r in range(self.rows)]
for r, c, _ in self.row_list():
if c <= r:
R[r].append(c)
inf = len(R) # nothing will be this large
parent = [inf]*self.rows
virtual = [inf]*self.rows
for r in range(self.rows):
for c in R[r][:-1]:
while virtual[c] < r:
t = virtual[c]
virtual[c] = r
c = t
if virtual[c] == inf:
parent[c] = virtual[c] = r
return R, parent
def nnz(self):
"""Returns the number of non-zero elements in Matrix."""
return len(self._smat)
def row_list(self):
"""Returns a row-sorted list of non-zero elements of the matrix.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> a = SparseMatrix(((1, 2), (3, 4)))
>>> a
Matrix([
[1, 2],
[3, 4]])
>>> a.RL
[(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)]
See Also
========
row_op
col_list
"""
return [tuple(k + (self[k],)) for k in
sorted(list(self._smat.keys()), key=lambda k: list(k))]
def row_structure_symbolic_cholesky(self):
"""Symbolic cholesky factorization, for pre-determination of the
non-zero structure of the Cholesky factororization.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> S = SparseMatrix([
... [1, 0, 3, 2],
... [0, 0, 1, 0],
... [4, 0, 0, 5],
... [0, 6, 7, 0]])
>>> S.row_structure_symbolic_cholesky()
[[0], [], [0], [1, 2]]
References
==========
Symbolic Sparse Cholesky Factorization using Elimination Trees,
Jeroen Van Grondelle (1999)
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582
"""
R, parent = self.liupc()
inf = len(R) # this acts as infinity
Lrow = copy.deepcopy(R)
for k in range(self.rows):
for j in R[k]:
while j != inf and j != k:
Lrow[k].append(j)
j = parent[j]
Lrow[k] = list(sorted(set(Lrow[k])))
return Lrow
def scalar_multiply(self, scalar):
"Scalar element-wise multiplication"
M = self.zeros(*self.shape)
if scalar:
for i in self._smat:
v = scalar*self._smat[i]
if v:
M._smat[i] = v
else:
M._smat.pop(i, None)
return M
def solve_least_squares(self, rhs, method='LDL'):
"""Return the least-square fit to the data.
By default the cholesky_solve routine is used (method='CH'); other
methods of matrix inversion can be used. To find out which are
available, see the docstring of the .inv() method.
Examples
========
>>> from sympy.matrices import SparseMatrix, Matrix, ones
>>> A = Matrix([1, 2, 3])
>>> B = Matrix([2, 3, 4])
>>> S = SparseMatrix(A.row_join(B))
>>> S
Matrix([
[1, 2],
[2, 3],
[3, 4]])
If each line of S represent coefficients of Ax + By
and x and y are [2, 3] then S*xy is:
>>> r = S*Matrix([2, 3]); r
Matrix([
[ 8],
[13],
[18]])
But let's add 1 to the middle value and then solve for the
least-squares value of xy:
>>> xy = S.solve_least_squares(Matrix([8, 14, 18])); xy
Matrix([
[ 5/3],
[10/3]])
The error is given by S*xy - r:
>>> S*xy - r
Matrix([
[1/3],
[1/3],
[1/3]])
>>> _.norm().n(2)
0.58
If a different xy is used, the norm will be higher:
>>> xy += ones(2, 1)/10
>>> (S*xy - r).norm().n(2)
1.5
"""
t = self.T
return (t*self).inv(method=method)*t*rhs
def solve(self, rhs, method='LDL'):
"""Return solution to self*soln = rhs using given inversion method.
For a list of possible inversion methods, see the .inv() docstring.
"""
if not self.is_square:
if self.rows < self.cols:
raise ValueError('Under-determined system.')
elif self.rows > self.cols:
raise ValueError('For over-determined system, M, having '
'more rows than columns, try M.solve_least_squares(rhs).')
else:
return self.inv(method=method)*rhs
RL = property(row_list, None, None, "Alternate faster representation")
CL = property(col_list, None, None, "Alternate faster representation")
class MutableSparseMatrix(SparseMatrix, MatrixBase):
@classmethod
def _new(cls, *args, **kwargs):
return cls(*args)
def __setitem__(self, key, value):
"""Assign value to position designated by key.
Examples
========
>>> from sympy.matrices import SparseMatrix, ones
>>> M = SparseMatrix(2, 2, {})
>>> M[1] = 1; M
Matrix([
[0, 1],
[0, 0]])
>>> M[1, 1] = 2; M
Matrix([
[0, 1],
[0, 2]])
>>> M = SparseMatrix(2, 2, {})
>>> M[:, 1] = [1, 1]; M
Matrix([
[0, 1],
[0, 1]])
>>> M = SparseMatrix(2, 2, {})
>>> M[1, :] = [[1, 1]]; M
Matrix([
[0, 0],
[1, 1]])
To replace row r you assign to position r*m where m
is the number of columns:
>>> M = SparseMatrix(4, 4, {})
>>> m = M.cols
>>> M[3*m] = ones(1, m)*2; M
Matrix([
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[2, 2, 2, 2]])
And to replace column c you can assign to position c:
>>> M[2] = ones(m, 1)*4; M
Matrix([
[0, 0, 4, 0],
[0, 0, 4, 0],
[0, 0, 4, 0],
[2, 2, 4, 2]])
"""
rv = self._setitem(key, value)
if rv is not None:
i, j, value = rv
if value:
self._smat[i, j] = value
elif (i, j) in self._smat:
del self._smat[i, j]
def as_mutable(self):
return self.copy()
__hash__ = None
def col_del(self, k):
"""Delete the given column of the matrix.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> M = SparseMatrix([[0, 0], [0, 1]])
>>> M
Matrix([
[0, 0],
[0, 1]])
>>> M.col_del(0)
>>> M
Matrix([
[0],
[1]])
See Also
========
row_del
"""
newD = {}
k = a2idx(k, self.cols)
for (i, j) in self._smat:
if j == k:
pass
elif j > k:
newD[i, j - 1] = self._smat[i, j]
else:
newD[i, j] = self._smat[i, j]
self._smat = newD
self.cols -= 1
def col_join(self, other):
"""Returns B augmented beneath A (row-wise joining)::
[A]
[B]
Examples
========
>>> from sympy import SparseMatrix, Matrix, ones
>>> A = SparseMatrix(ones(3))
>>> A
Matrix([
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])
>>> B = SparseMatrix.eye(3)
>>> B
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> C = A.col_join(B); C
Matrix([
[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> C == A.col_join(Matrix(B))
True
Joining along columns is the same as appending rows at the end
of the matrix:
>>> C == A.row_insert(A.rows, Matrix(B))
True
"""
# A null matrix can always be stacked (see #10770)
if self.rows == 0 and self.cols != other.cols:
return self._new(0, other.cols, []).col_join(other)
A, B = self, other
if not A.cols == B.cols:
raise ShapeError()
A = A.copy()
if not isinstance(B, SparseMatrix):
k = 0
b = B._mat
for i in range(B.rows):
for j in range(B.cols):
v = b[k]
if v:
A._smat[i + A.rows, j] = v
k += 1
else:
for (i, j), v in B._smat.items():
A._smat[i + A.rows, j] = v
A.rows += B.rows
return A
def col_op(self, j, f):
"""In-place operation on col j using two-arg functor whose args are
interpreted as (self[i, j], i) for i in range(self.rows).
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> M = SparseMatrix.eye(3)*2
>>> M[1, 0] = -1
>>> M.col_op(1, lambda v, i: v + 2*M[i, 0]); M
Matrix([
[ 2, 4, 0],
[-1, 0, 0],
[ 0, 0, 2]])
"""
for i in range(self.rows):
v = self._smat.get((i, j), S.Zero)
fv = f(v, i)
if fv:
self._smat[i, j] = fv
elif v:
self._smat.pop((i, j))
def col_swap(self, i, j):
"""Swap, in place, columns i and j.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> S = SparseMatrix.eye(3); S[2, 1] = 2
>>> S.col_swap(1, 0); S
Matrix([
[0, 1, 0],
[1, 0, 0],
[2, 0, 1]])
"""
if i > j:
i, j = j, i
rows = self.col_list()
temp = []
for ii, jj, v in rows:
if jj == i:
self._smat.pop((ii, jj))
temp.append((ii, v))
elif jj == j:
self._smat.pop((ii, jj))
self._smat[ii, i] = v
elif jj > j:
break
for k, v in temp:
self._smat[k, j] = v
def copyin_list(self, key, value):
if not is_sequence(value):
raise TypeError("`value` must be of type list or tuple.")
self.copyin_matrix(key, Matrix(value))
def copyin_matrix(self, key, value):
# include this here because it's not part of BaseMatrix
rlo, rhi, clo, chi = self.key2bounds(key)
shape = value.shape
dr, dc = rhi - rlo, chi - clo
if shape != (dr, dc):
raise ShapeError(
"The Matrix `value` doesn't have the same dimensions "
"as the in sub-Matrix given by `key`.")
if not isinstance(value, SparseMatrix):
for i in range(value.rows):
for j in range(value.cols):
self[i + rlo, j + clo] = value[i, j]
else:
if (rhi - rlo)*(chi - clo) < len(self):
for i in range(rlo, rhi):
for j in range(clo, chi):
self._smat.pop((i, j), None)
else:
for i, j, v in self.row_list():
if rlo <= i < rhi and clo <= j < chi:
self._smat.pop((i, j), None)
for k, v in value._smat.items():
i, j = k
self[i + rlo, j + clo] = value[i, j]
def fill(self, value):
"""Fill self with the given value.
Notes
=====
Unless many values are going to be deleted (i.e. set to zero)
this will create a matrix that is slower than a dense matrix in
operations.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> M = SparseMatrix.zeros(3); M
Matrix([
[0, 0, 0],
[0, 0, 0],
[0, 0, 0]])
>>> M.fill(1); M
Matrix([
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])
"""
if not value:
self._smat = {}
else:
v = self._sympify(value)
self._smat = {(i, j): v
for i in range(self.rows) for j in range(self.cols)}
def row_del(self, k):
"""Delete the given row of the matrix.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> M = SparseMatrix([[0, 0], [0, 1]])
>>> M
Matrix([
[0, 0],
[0, 1]])
>>> M.row_del(0)
>>> M
Matrix([[0, 1]])
See Also
========
col_del
"""
newD = {}
k = a2idx(k, self.rows)
for (i, j) in self._smat:
if i == k:
pass
elif i > k:
newD[i - 1, j] = self._smat[i, j]
else:
newD[i, j] = self._smat[i, j]
self._smat = newD
self.rows -= 1
def row_join(self, other):
"""Returns B appended after A (column-wise augmenting)::
[A B]
Examples
========
>>> from sympy import SparseMatrix, Matrix
>>> A = SparseMatrix(((1, 0, 1), (0, 1, 0), (1, 1, 0)))
>>> A
Matrix([
[1, 0, 1],
[0, 1, 0],
[1, 1, 0]])
>>> B = SparseMatrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
>>> B
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> C = A.row_join(B); C
Matrix([
[1, 0, 1, 1, 0, 0],
[0, 1, 0, 0, 1, 0],
[1, 1, 0, 0, 0, 1]])
>>> C == A.row_join(Matrix(B))
True
Joining at row ends is the same as appending columns at the end
of the matrix:
>>> C == A.col_insert(A.cols, B)
True
"""
# A null matrix can always be stacked (see #10770)
if self.cols == 0 and self.rows != other.rows:
return self._new(other.rows, 0, []).row_join(other)
A, B = self, other
if not A.rows == B.rows:
raise ShapeError()
A = A.copy()
if not isinstance(B, SparseMatrix):
k = 0
b = B._mat
for i in range(B.rows):
for j in range(B.cols):
v = b[k]
if v:
A._smat[i, j + A.cols] = v
k += 1
else:
for (i, j), v in B._smat.items():
A._smat[i, j + A.cols] = v
A.cols += B.cols
return A
def row_op(self, i, f):
"""In-place operation on row ``i`` using two-arg functor whose args are
interpreted as ``(self[i, j], j)``.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> M = SparseMatrix.eye(3)*2
>>> M[0, 1] = -1
>>> M.row_op(1, lambda v, j: v + 2*M[0, j]); M
Matrix([
[2, -1, 0],
[4, 0, 0],
[0, 0, 2]])
See Also
========
row
zip_row_op
col_op
"""
for j in range(self.cols):
v = self._smat.get((i, j), S.Zero)
fv = f(v, j)
if fv:
self._smat[i, j] = fv
elif v:
self._smat.pop((i, j))
def row_swap(self, i, j):
"""Swap, in place, columns i and j.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> S = SparseMatrix.eye(3); S[2, 1] = 2
>>> S.row_swap(1, 0); S
Matrix([
[0, 1, 0],
[1, 0, 0],
[0, 2, 1]])
"""
if i > j:
i, j = j, i
rows = self.row_list()
temp = []
for ii, jj, v in rows:
if ii == i:
self._smat.pop((ii, jj))
temp.append((jj, v))
elif ii == j:
self._smat.pop((ii, jj))
self._smat[i, jj] = v
elif ii > j:
break
for k, v in temp:
self._smat[j, k] = v
def zip_row_op(self, i, k, f):
"""In-place operation on row ``i`` using two-arg functor whose args are
interpreted as ``(self[i, j], self[k, j])``.
Examples
========
>>> from sympy.matrices import SparseMatrix
>>> M = SparseMatrix.eye(3)*2
>>> M[0, 1] = -1
>>> M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
Matrix([
[2, -1, 0],
[4, 0, 0],
[0, 0, 2]])
See Also
========
row
row_op
col_op
"""
self.row_op(i, lambda v, j: f(v, self[k, j]))
|
9825a62caeba45a5e5907cf1b26040ead4e9c7cd85809b14fbd25e2e83bf16a2 | from __future__ import division, print_function
from types import FunctionType
from mpmath.libmp.libmpf import prec_to_dps
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.compatibility import (
Callable, NotIterable, as_int, default_sort_key, is_sequence, range,
reduce, string_types)
from sympy.core.decorators import deprecated
from sympy.core.expr import Expr
from sympy.core.function import expand_mul
from sympy.core.numbers import Float, Integer, mod_inverse
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import Dummy, Symbol, _uniquely_named_symbol, symbols
from sympy.core.sympify import sympify
from sympy.functions import exp, factorial
from sympy.functions.elementary.miscellaneous import Max, Min, sqrt
from sympy.polys import PurePoly, cancel, roots
from sympy.printing import sstr
from sympy.simplify import nsimplify
from sympy.simplify import simplify as _simplify
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import flatten, numbered_symbols
from sympy.utilities.misc import filldedent
from .common import (
MatrixCommon, MatrixError, NonSquareMatrixError, ShapeError)
def _iszero(x):
"""Returns True if x is zero."""
return getattr(x, 'is_zero', None)
def _is_zero_after_expand_mul(x):
"""Tests by expand_mul only, suitable for polynomials and rational
functions."""
return expand_mul(x) == 0
class DeferredVector(Symbol, NotIterable):
"""A vector whose components are deferred (e.g. for use with lambdify)
Examples
========
>>> from sympy import DeferredVector, lambdify
>>> X = DeferredVector( 'X' )
>>> X
X
>>> expr = (X[0] + 2, X[2] + 3)
>>> func = lambdify( X, expr)
>>> func( [1, 2, 3] )
(3, 6)
"""
def __getitem__(self, i):
if i == -0:
i = 0
if i < 0:
raise IndexError('DeferredVector index out of range')
component_name = '%s[%d]' % (self.name, i)
return Symbol(component_name)
def __str__(self):
return sstr(self)
def __repr__(self):
return "DeferredVector('%s')" % self.name
class MatrixDeterminant(MatrixCommon):
"""Provides basic matrix determinant operations.
Should not be instantiated directly."""
def _eval_berkowitz_toeplitz_matrix(self):
"""Return (A,T) where T the Toeplitz matrix used in the Berkowitz algorithm
corresponding to ``self`` and A is the first principal submatrix."""
# the 0 x 0 case is trivial
if self.rows == 0 and self.cols == 0:
return self._new(1,1, [S.One])
#
# Partition self = [ a_11 R ]
# [ C A ]
#
a, R = self[0,0], self[0, 1:]
C, A = self[1:, 0], self[1:,1:]
#
# The Toeplitz matrix looks like
#
# [ 1 ]
# [ -a 1 ]
# [ -RC -a 1 ]
# [ -RAC -RC -a 1 ]
# [ -RA**2C -RAC -RC -a 1 ]
# etc.
# Compute the diagonal entries.
# Because multiplying matrix times vector is so much
# more efficient than matrix times matrix, recursively
# compute -R * A**n * C.
diags = [C]
for i in range(self.rows - 2):
diags.append(A * diags[i])
diags = [(-R*d)[0, 0] for d in diags]
diags = [S.One, -a] + diags
def entry(i,j):
if j > i:
return S.Zero
return diags[i - j]
toeplitz = self._new(self.cols + 1, self.rows, entry)
return (A, toeplitz)
def _eval_berkowitz_vector(self):
""" Run the Berkowitz algorithm and return a vector whose entries
are the coefficients of the characteristic polynomial of ``self``.
Given N x N matrix, efficiently compute
coefficients of characteristic polynomials of ``self``
without division in the ground domain.
This method is particularly useful for computing determinant,
principal minors and characteristic polynomial when ``self``
has complicated coefficients e.g. polynomials. Semi-direct
usage of this algorithm is also important in computing
efficiently sub-resultant PRS.
Assuming that M is a square matrix of dimension N x N and
I is N x N identity matrix, then the Berkowitz vector is
an N x 1 vector whose entries are coefficients of the
polynomial
charpoly(M) = det(t*I - M)
As a consequence, all polynomials generated by Berkowitz
algorithm are monic.
For more information on the implemented algorithm refer to:
[1] S.J. Berkowitz, On computing the determinant in small
parallel time using a small number of processors, ACM,
Information Processing Letters 18, 1984, pp. 147-150
[2] M. Keber, Division-Free computation of sub-resultants
using Bezout matrices, Tech. Report MPI-I-2006-1-006,
Saarbrucken, 2006
"""
# handle the trivial cases
if self.rows == 0 and self.cols == 0:
return self._new(1, 1, [S.One])
elif self.rows == 1 and self.cols == 1:
return self._new(2, 1, [S.One, -self[0,0]])
submat, toeplitz = self._eval_berkowitz_toeplitz_matrix()
return toeplitz * submat._eval_berkowitz_vector()
def _eval_det_bareiss(self, iszerofunc=_is_zero_after_expand_mul):
"""Compute matrix determinant using Bareiss' fraction-free
algorithm which is an extension of the well known Gaussian
elimination method. This approach is best suited for dense
symbolic matrices and will result in a determinant with
minimal number of fractions. It means that less term
rewriting is needed on resulting formulae.
TODO: Implement algorithm for sparse matrices (SFF),
http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.
"""
# Recursively implemented Bareiss' algorithm as per Deanna Richelle Leggett's
# thesis http://www.math.usm.edu/perry/Research/Thesis_DRL.pdf
def bareiss(mat, cumm=1):
if mat.rows == 0:
return S.One
elif mat.rows == 1:
return mat[0, 0]
# find a pivot and extract the remaining matrix
# With the default iszerofunc, _find_reasonable_pivot slows down
# the computation by the factor of 2.5 in one test.
# Relevant issues: #10279 and #13877.
pivot_pos, pivot_val, _, _ = _find_reasonable_pivot(mat[:, 0],
iszerofunc=iszerofunc)
if pivot_pos is None:
return S.Zero
# if we have a valid pivot, we'll do a "row swap", so keep the
# sign of the det
sign = (-1) ** (pivot_pos % 2)
# we want every row but the pivot row and every column
rows = list(i for i in range(mat.rows) if i != pivot_pos)
cols = list(range(mat.cols))
tmp_mat = mat.extract(rows, cols)
def entry(i, j):
ret = (pivot_val*tmp_mat[i, j + 1] - mat[pivot_pos, j + 1]*tmp_mat[i, 0]) / cumm
if not ret.is_Atom:
return cancel(ret)
return ret
return sign*bareiss(self._new(mat.rows - 1, mat.cols - 1, entry), pivot_val)
return cancel(bareiss(self))
def _eval_det_berkowitz(self):
""" Use the Berkowitz algorithm to compute the determinant."""
berk_vector = self._eval_berkowitz_vector()
return (-1)**(len(berk_vector) - 1) * berk_vector[-1]
def _eval_det_lu(self, iszerofunc=_iszero, simpfunc=None):
""" Computes the determinant of a matrix from its LU decomposition.
This function uses the LU decomposition computed by
LUDecomposition_Simple().
The keyword arguments iszerofunc and simpfunc are passed to
LUDecomposition_Simple().
iszerofunc is a callable that returns a boolean indicating if its
input is zero, or None if it cannot make the determination.
simpfunc is a callable that simplifies its input.
The default is simpfunc=None, which indicate that the pivot search
algorithm should not attempt to simplify any candidate pivots.
If simpfunc fails to simplify its input, then it must return its input
instead of a copy."""
if self.rows == 0:
return S.One
# sympy/matrices/tests/test_matrices.py contains a test that
# suggests that the determinant of a 0 x 0 matrix is one, by
# convention.
lu, row_swaps = self.LUdecomposition_Simple(iszerofunc=iszerofunc, simpfunc=None)
# P*A = L*U => det(A) = det(L)*det(U)/det(P) = det(P)*det(U).
# Lower triangular factor L encoded in lu has unit diagonal => det(L) = 1.
# P is a permutation matrix => det(P) in {-1, 1} => 1/det(P) = det(P).
# LUdecomposition_Simple() returns a list of row exchange index pairs, rather
# than a permutation matrix, but det(P) = (-1)**len(row_swaps).
# Avoid forming the potentially time consuming product of U's diagonal entries
# if the product is zero.
# Bottom right entry of U is 0 => det(A) = 0.
# It may be impossible to determine if this entry of U is zero when it is symbolic.
if iszerofunc(lu[lu.rows-1, lu.rows-1]):
return S.Zero
# Compute det(P)
det = -S.One if len(row_swaps)%2 else S.One
# Compute det(U) by calculating the product of U's diagonal entries.
# The upper triangular portion of lu is the upper triangular portion of the
# U factor in the LU decomposition.
for k in range(lu.rows):
det *= lu[k, k]
# return det(P)*det(U)
return det
def _eval_determinant(self):
"""Assumed to exist by matrix expressions; If we subclass
MatrixDeterminant, we can fully evaluate determinants."""
return self.det()
def adjugate(self, method="berkowitz"):
"""Returns the adjugate, or classical adjoint, of
a matrix. That is, the transpose of the matrix of cofactors.
https://en.wikipedia.org/wiki/Adjugate
See Also
========
cofactor_matrix
transpose
"""
return self.cofactor_matrix(method).transpose()
def charpoly(self, x='lambda', simplify=_simplify):
"""Computes characteristic polynomial det(x*I - self) where I is
the identity matrix.
A PurePoly is returned, so using different variables for ``x`` does
not affect the comparison or the polynomials:
Examples
========
>>> from sympy import Matrix
>>> from sympy.abc import x, y
>>> A = Matrix([[1, 3], [2, 0]])
>>> A.charpoly(x) == A.charpoly(y)
True
Specifying ``x`` is optional; a symbol named ``lambda`` is used by
default (which looks good when pretty-printed in unicode):
>>> A.charpoly().as_expr()
lambda**2 - lambda - 6
And if ``x`` clashes with an existing symbol, underscores will
be preppended to the name to make it unique:
>>> A = Matrix([[1, 2], [x, 0]])
>>> A.charpoly(x).as_expr()
_x**2 - _x - 2*x
Whether you pass a symbol or not, the generator can be obtained
with the gen attribute since it may not be the same as the symbol
that was passed:
>>> A.charpoly(x).gen
_x
>>> A.charpoly(x).gen == x
False
Notes
=====
The Samuelson-Berkowitz algorithm is used to compute
the characteristic polynomial efficiently and without any
division operations. Thus the characteristic polynomial over any
commutative ring without zero divisors can be computed.
See Also
========
det
"""
if not self.is_square:
raise NonSquareMatrixError()
berk_vector = self._eval_berkowitz_vector()
x = _uniquely_named_symbol(x, berk_vector)
return PurePoly([simplify(a) for a in berk_vector], x)
def cofactor(self, i, j, method="berkowitz"):
"""Calculate the cofactor of an element.
See Also
========
cofactor_matrix
minor
minor_submatrix
"""
if not self.is_square or self.rows < 1:
raise NonSquareMatrixError()
return (-1)**((i + j) % 2) * self.minor(i, j, method)
def cofactor_matrix(self, method="berkowitz"):
"""Return a matrix containing the cofactor of each element.
See Also
========
cofactor
minor
minor_submatrix
adjugate
"""
if not self.is_square or self.rows < 1:
raise NonSquareMatrixError()
return self._new(self.rows, self.cols,
lambda i, j: self.cofactor(i, j, method))
def det(self, method="bareiss", iszerofunc=None):
"""Computes the determinant of a matrix.
Parameters
==========
method : string, optional
Specifies the algorithm used for computing the matrix determinant.
If the matrix is at most 3x3, a hard-coded formula is used and the
specified method is ignored. Otherwise, it defaults to
``'bareiss'``.
If it is set to ``'bareiss'``, Bareiss' fraction-free algorithm will
be used.
If it is set to ``'berkowitz'``, Berkowitz' algorithm will be used.
Otherwise, if it is set to ``'lu'``, LU decomposition will be used.
.. note::
For backward compatibility, legacy keys like "bareis" and
"det_lu" can still be used to indicate the corresponding
methods.
And the keys are also case-insensitive for now. However, it is
suggested to use the precise keys for specifying the method.
iszerofunc : FunctionType or None, optional
If it is set to ``None``, it will be defaulted to ``_iszero`` if the
method is set to ``'bareiss'``, and ``_is_zero_after_expand_mul`` if
the method is set to ``'lu'``.
It can also accept any user-specified zero testing function, if it
is formatted as a function which accepts a single symbolic argument
and returns ``True`` if it is tested as zero and ``False`` if it
tested as non-zero, and also ``None`` if it is undecidable.
Returns
=======
det : Basic
Result of determinant.
Raises
======
ValueError
If unrecognized keys are given for ``method`` or ``iszerofunc``.
NonSquareMatrixError
If attempted to calculate determinant from a non-square matrix.
"""
# sanitize `method`
method = method.lower()
if method == "bareis":
method = "bareiss"
if method == "det_lu":
method = "lu"
if method not in ("bareiss", "berkowitz", "lu"):
raise ValueError("Determinant method '%s' unrecognized" % method)
if iszerofunc is None:
if method == "bareiss":
iszerofunc = _is_zero_after_expand_mul
elif method == "lu":
iszerofunc = _iszero
elif not isinstance(iszerofunc, FunctionType):
raise ValueError("Zero testing method '%s' unrecognized" % iszerofunc)
# if methods were made internal and all determinant calculations
# passed through here, then these lines could be factored out of
# the method routines
if not self.is_square:
raise NonSquareMatrixError()
n = self.rows
if n == 0:
return S.One
elif n == 1:
return self[0,0]
elif n == 2:
return self[0, 0] * self[1, 1] - self[0, 1] * self[1, 0]
elif n == 3:
return (self[0, 0] * self[1, 1] * self[2, 2]
+ self[0, 1] * self[1, 2] * self[2, 0]
+ self[0, 2] * self[1, 0] * self[2, 1]
- self[0, 2] * self[1, 1] * self[2, 0]
- self[0, 0] * self[1, 2] * self[2, 1]
- self[0, 1] * self[1, 0] * self[2, 2])
if method == "bareiss":
return self._eval_det_bareiss(iszerofunc=iszerofunc)
elif method == "berkowitz":
return self._eval_det_berkowitz()
elif method == "lu":
return self._eval_det_lu(iszerofunc=iszerofunc)
def minor(self, i, j, method="berkowitz"):
"""Return the (i,j) minor of ``self``. That is,
return the determinant of the matrix obtained by deleting
the `i`th row and `j`th column from ``self``.
See Also
========
minor_submatrix
cofactor
det
"""
if not self.is_square or self.rows < 1:
raise NonSquareMatrixError()
return self.minor_submatrix(i, j).det(method=method)
def minor_submatrix(self, i, j):
"""Return the submatrix obtained by removing the `i`th row
and `j`th column from ``self``.
See Also
========
minor
cofactor
"""
if i < 0:
i += self.rows
if j < 0:
j += self.cols
if not 0 <= i < self.rows or not 0 <= j < self.cols:
raise ValueError("`i` and `j` must satisfy 0 <= i < ``self.rows`` "
"(%d)" % self.rows + "and 0 <= j < ``self.cols`` (%d)." % self.cols)
rows = [a for a in range(self.rows) if a != i]
cols = [a for a in range(self.cols) if a != j]
return self.extract(rows, cols)
class MatrixReductions(MatrixDeterminant):
"""Provides basic matrix row/column operations.
Should not be instantiated directly."""
def _eval_col_op_swap(self, col1, col2):
def entry(i, j):
if j == col1:
return self[i, col2]
elif j == col2:
return self[i, col1]
return self[i, j]
return self._new(self.rows, self.cols, entry)
def _eval_col_op_multiply_col_by_const(self, col, k):
def entry(i, j):
if j == col:
return k * self[i, j]
return self[i, j]
return self._new(self.rows, self.cols, entry)
def _eval_col_op_add_multiple_to_other_col(self, col, k, col2):
def entry(i, j):
if j == col:
return self[i, j] + k * self[i, col2]
return self[i, j]
return self._new(self.rows, self.cols, entry)
def _eval_row_op_swap(self, row1, row2):
def entry(i, j):
if i == row1:
return self[row2, j]
elif i == row2:
return self[row1, j]
return self[i, j]
return self._new(self.rows, self.cols, entry)
def _eval_row_op_multiply_row_by_const(self, row, k):
def entry(i, j):
if i == row:
return k * self[i, j]
return self[i, j]
return self._new(self.rows, self.cols, entry)
def _eval_row_op_add_multiple_to_other_row(self, row, k, row2):
def entry(i, j):
if i == row:
return self[i, j] + k * self[row2, j]
return self[i, j]
return self._new(self.rows, self.cols, entry)
def _eval_echelon_form(self, iszerofunc, simpfunc):
"""Returns (mat, swaps) where ``mat`` is a row-equivalent matrix
in echelon form and ``swaps`` is a list of row-swaps performed."""
reduced, pivot_cols, swaps = self._row_reduce(iszerofunc, simpfunc,
normalize_last=True,
normalize=False,
zero_above=False)
return reduced, pivot_cols, swaps
def _eval_is_echelon(self, iszerofunc):
if self.rows <= 0 or self.cols <= 0:
return True
zeros_below = all(iszerofunc(t) for t in self[1:, 0])
if iszerofunc(self[0, 0]):
return zeros_below and self[:, 1:]._eval_is_echelon(iszerofunc)
return zeros_below and self[1:, 1:]._eval_is_echelon(iszerofunc)
def _eval_rref(self, iszerofunc, simpfunc, normalize_last=True):
reduced, pivot_cols, swaps = self._row_reduce(iszerofunc, simpfunc,
normalize_last, normalize=True,
zero_above=True)
return reduced, pivot_cols
def _normalize_op_args(self, op, col, k, col1, col2, error_str="col"):
"""Validate the arguments for a row/column operation. ``error_str``
can be one of "row" or "col" depending on the arguments being parsed."""
if op not in ["n->kn", "n<->m", "n->n+km"]:
raise ValueError("Unknown {} operation '{}'. Valid col operations "
"are 'n->kn', 'n<->m', 'n->n+km'".format(error_str, op))
# normalize and validate the arguments
if op == "n->kn":
col = col if col is not None else col1
if col is None or k is None:
raise ValueError("For a {0} operation 'n->kn' you must provide the "
"kwargs `{0}` and `k`".format(error_str))
if not 0 <= col <= self.cols:
raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col))
if op == "n<->m":
# we need two cols to swap. It doesn't matter
# how they were specified, so gather them together and
# remove `None`
cols = set((col, k, col1, col2)).difference([None])
if len(cols) > 2:
# maybe the user left `k` by mistake?
cols = set((col, col1, col2)).difference([None])
if len(cols) != 2:
raise ValueError("For a {0} operation 'n<->m' you must provide the "
"kwargs `{0}1` and `{0}2`".format(error_str))
col1, col2 = cols
if not 0 <= col1 <= self.cols:
raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col1))
if not 0 <= col2 <= self.cols:
raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col2))
if op == "n->n+km":
col = col1 if col is None else col
col2 = col1 if col2 is None else col2
if col is None or col2 is None or k is None:
raise ValueError("For a {0} operation 'n->n+km' you must provide the "
"kwargs `{0}`, `k`, and `{0}2`".format(error_str))
if col == col2:
raise ValueError("For a {0} operation 'n->n+km' `{0}` and `{0}2` must "
"be different.".format(error_str))
if not 0 <= col <= self.cols:
raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col))
if not 0 <= col2 <= self.cols:
raise ValueError("This matrix doesn't have a {} '{}'".format(error_str, col2))
return op, col, k, col1, col2
def _permute_complexity_right(self, iszerofunc):
"""Permute columns with complicated elements as
far right as they can go. Since the ``sympy`` row reduction
algorithms start on the left, having complexity right-shifted
speeds things up.
Returns a tuple (mat, perm) where perm is a permutation
of the columns to perform to shift the complex columns right, and mat
is the permuted matrix."""
def complexity(i):
# the complexity of a column will be judged by how many
# element's zero-ness cannot be determined
return sum(1 if iszerofunc(e) is None else 0 for e in self[:, i])
complex = [(complexity(i), i) for i in range(self.cols)]
perm = [j for (i, j) in sorted(complex)]
return (self.permute(perm, orientation='cols'), perm)
def _row_reduce(self, iszerofunc, simpfunc, normalize_last=True,
normalize=True, zero_above=True):
"""Row reduce ``self`` and return a tuple (rref_matrix,
pivot_cols, swaps) where pivot_cols are the pivot columns
and swaps are any row swaps that were used in the process
of row reduction.
Parameters
==========
iszerofunc : determines if an entry can be used as a pivot
simpfunc : used to simplify elements and test if they are
zero if ``iszerofunc`` returns `None`
normalize_last : indicates where all row reduction should
happen in a fraction-free manner and then the rows are
normalized (so that the pivots are 1), or whether
rows should be normalized along the way (like the naive
row reduction algorithm)
normalize : whether pivot rows should be normalized so that
the pivot value is 1
zero_above : whether entries above the pivot should be zeroed.
If ``zero_above=False``, an echelon matrix will be returned.
"""
rows, cols = self.rows, self.cols
mat = list(self)
def get_col(i):
return mat[i::cols]
def row_swap(i, j):
mat[i*cols:(i + 1)*cols], mat[j*cols:(j + 1)*cols] = \
mat[j*cols:(j + 1)*cols], mat[i*cols:(i + 1)*cols]
def cross_cancel(a, i, b, j):
"""Does the row op row[i] = a*row[i] - b*row[j]"""
q = (j - i)*cols
for p in range(i*cols, (i + 1)*cols):
mat[p] = a*mat[p] - b*mat[p + q]
piv_row, piv_col = 0, 0
pivot_cols = []
swaps = []
# use a fraction free method to zero above and below each pivot
while piv_col < cols and piv_row < rows:
pivot_offset, pivot_val, \
assumed_nonzero, newly_determined = _find_reasonable_pivot(
get_col(piv_col)[piv_row:], iszerofunc, simpfunc)
# _find_reasonable_pivot may have simplified some things
# in the process. Let's not let them go to waste
for (offset, val) in newly_determined:
offset += piv_row
mat[offset*cols + piv_col] = val
if pivot_offset is None:
piv_col += 1
continue
pivot_cols.append(piv_col)
if pivot_offset != 0:
row_swap(piv_row, pivot_offset + piv_row)
swaps.append((piv_row, pivot_offset + piv_row))
# if we aren't normalizing last, we normalize
# before we zero the other rows
if normalize_last is False:
i, j = piv_row, piv_col
mat[i*cols + j] = S.One
for p in range(i*cols + j + 1, (i + 1)*cols):
mat[p] = mat[p] / pivot_val
# after normalizing, the pivot value is 1
pivot_val = S.One
# zero above and below the pivot
for row in range(rows):
# don't zero our current row
if row == piv_row:
continue
# don't zero above the pivot unless we're told.
if zero_above is False and row < piv_row:
continue
# if we're already a zero, don't do anything
val = mat[row*cols + piv_col]
if iszerofunc(val):
continue
cross_cancel(pivot_val, row, val, piv_row)
piv_row += 1
# normalize each row
if normalize_last is True and normalize is True:
for piv_i, piv_j in enumerate(pivot_cols):
pivot_val = mat[piv_i*cols + piv_j]
mat[piv_i*cols + piv_j] = S.One
for p in range(piv_i*cols + piv_j + 1, (piv_i + 1)*cols):
mat[p] = mat[p] / pivot_val
return self._new(self.rows, self.cols, mat), tuple(pivot_cols), tuple(swaps)
def echelon_form(self, iszerofunc=_iszero, simplify=False, with_pivots=False):
"""Returns a matrix row-equivalent to ``self`` that is
in echelon form. Note that echelon form of a matrix
is *not* unique, however, properties like the row
space and the null space are preserved."""
simpfunc = simplify if isinstance(
simplify, FunctionType) else _simplify
mat, pivots, swaps = self._eval_echelon_form(iszerofunc, simpfunc)
if with_pivots:
return mat, pivots
return mat
def elementary_col_op(self, op="n->kn", col=None, k=None, col1=None, col2=None):
"""Performs the elementary column operation `op`.
`op` may be one of
* "n->kn" (column n goes to k*n)
* "n<->m" (swap column n and column m)
* "n->n+km" (column n goes to column n + k*column m)
Parameters
==========
op : string; the elementary row operation
col : the column to apply the column operation
k : the multiple to apply in the column operation
col1 : one column of a column swap
col2 : second column of a column swap or column "m" in the column operation
"n->n+km"
"""
op, col, k, col1, col2 = self._normalize_op_args(op, col, k, col1, col2, "col")
# now that we've validated, we're all good to dispatch
if op == "n->kn":
return self._eval_col_op_multiply_col_by_const(col, k)
if op == "n<->m":
return self._eval_col_op_swap(col1, col2)
if op == "n->n+km":
return self._eval_col_op_add_multiple_to_other_col(col, k, col2)
def elementary_row_op(self, op="n->kn", row=None, k=None, row1=None, row2=None):
"""Performs the elementary row operation `op`.
`op` may be one of
* "n->kn" (row n goes to k*n)
* "n<->m" (swap row n and row m)
* "n->n+km" (row n goes to row n + k*row m)
Parameters
==========
op : string; the elementary row operation
row : the row to apply the row operation
k : the multiple to apply in the row operation
row1 : one row of a row swap
row2 : second row of a row swap or row "m" in the row operation
"n->n+km"
"""
op, row, k, row1, row2 = self._normalize_op_args(op, row, k, row1, row2, "row")
# now that we've validated, we're all good to dispatch
if op == "n->kn":
return self._eval_row_op_multiply_row_by_const(row, k)
if op == "n<->m":
return self._eval_row_op_swap(row1, row2)
if op == "n->n+km":
return self._eval_row_op_add_multiple_to_other_row(row, k, row2)
@property
def is_echelon(self, iszerofunc=_iszero):
"""Returns `True` if the matrix is in echelon form.
That is, all rows of zeros are at the bottom, and below
each leading non-zero in a row are exclusively zeros."""
return self._eval_is_echelon(iszerofunc)
def rank(self, iszerofunc=_iszero, simplify=False):
"""
Returns the rank of a matrix
>>> from sympy import Matrix
>>> from sympy.abc import x
>>> m = Matrix([[1, 2], [x, 1 - 1/x]])
>>> m.rank()
2
>>> n = Matrix(3, 3, range(1, 10))
>>> n.rank()
2
"""
simpfunc = simplify if isinstance(
simplify, FunctionType) else _simplify
# for small matrices, we compute the rank explicitly
# if is_zero on elements doesn't answer the question
# for small matrices, we fall back to the full routine.
if self.rows <= 0 or self.cols <= 0:
return 0
if self.rows <= 1 or self.cols <= 1:
zeros = [iszerofunc(x) for x in self]
if False in zeros:
return 1
if self.rows == 2 and self.cols == 2:
zeros = [iszerofunc(x) for x in self]
if not False in zeros and not None in zeros:
return 0
det = self.det()
if iszerofunc(det) and False in zeros:
return 1
if iszerofunc(det) is False:
return 2
mat, _ = self._permute_complexity_right(iszerofunc=iszerofunc)
echelon_form, pivots, swaps = mat._eval_echelon_form(iszerofunc=iszerofunc, simpfunc=simpfunc)
return len(pivots)
def rref(self, iszerofunc=_iszero, simplify=False, pivots=True, normalize_last=True):
"""Return reduced row-echelon form of matrix and indices of pivot vars.
Parameters
==========
iszerofunc : Function
A function used for detecting whether an element can
act as a pivot. ``lambda x: x.is_zero`` is used by default.
simplify : Function
A function used to simplify elements when looking for a pivot.
By default SymPy's ``simplify`` is used.
pivots : True or False
If ``True``, a tuple containing the row-reduced matrix and a tuple
of pivot columns is returned. If ``False`` just the row-reduced
matrix is returned.
normalize_last : True or False
If ``True``, no pivots are normalized to `1` until after all
entries above and below each pivot are zeroed. This means the row
reduction algorithm is fraction free until the very last step.
If ``False``, the naive row reduction procedure is used where
each pivot is normalized to be `1` before row operations are
used to zero above and below the pivot.
Notes
=====
The default value of ``normalize_last=True`` can provide significant
speedup to row reduction, especially on matrices with symbols. However,
if you depend on the form row reduction algorithm leaves entries
of the matrix, set ``noramlize_last=False``
Examples
========
>>> from sympy import Matrix
>>> from sympy.abc import x
>>> m = Matrix([[1, 2], [x, 1 - 1/x]])
>>> m.rref()
(Matrix([
[1, 0],
[0, 1]]), (0, 1))
>>> rref_matrix, rref_pivots = m.rref()
>>> rref_matrix
Matrix([
[1, 0],
[0, 1]])
>>> rref_pivots
(0, 1)
"""
simpfunc = simplify if isinstance(
simplify, FunctionType) else _simplify
ret, pivot_cols = self._eval_rref(iszerofunc=iszerofunc,
simpfunc=simpfunc,
normalize_last=normalize_last)
if pivots:
ret = (ret, pivot_cols)
return ret
class MatrixSubspaces(MatrixReductions):
"""Provides methods relating to the fundamental subspaces
of a matrix. Should not be instantiated directly."""
def columnspace(self, simplify=False):
"""Returns a list of vectors (Matrix objects) that span columnspace of ``self``
Examples
========
>>> from sympy.matrices import Matrix
>>> m = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6])
>>> m
Matrix([
[ 1, 3, 0],
[-2, -6, 0],
[ 3, 9, 6]])
>>> m.columnspace()
[Matrix([
[ 1],
[-2],
[ 3]]), Matrix([
[0],
[0],
[6]])]
See Also
========
nullspace
rowspace
"""
reduced, pivots = self.echelon_form(simplify=simplify, with_pivots=True)
return [self.col(i) for i in pivots]
def nullspace(self, simplify=False, iszerofunc=_iszero):
"""Returns list of vectors (Matrix objects) that span nullspace of ``self``
Examples
========
>>> from sympy.matrices import Matrix
>>> m = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6])
>>> m
Matrix([
[ 1, 3, 0],
[-2, -6, 0],
[ 3, 9, 6]])
>>> m.nullspace()
[Matrix([
[-3],
[ 1],
[ 0]])]
See Also
========
columnspace
rowspace
"""
reduced, pivots = self.rref(iszerofunc=iszerofunc, simplify=simplify)
free_vars = [i for i in range(self.cols) if i not in pivots]
basis = []
for free_var in free_vars:
# for each free variable, we will set it to 1 and all others
# to 0. Then, we will use back substitution to solve the system
vec = [S.Zero]*self.cols
vec[free_var] = S.One
for piv_row, piv_col in enumerate(pivots):
vec[piv_col] -= reduced[piv_row, free_var]
basis.append(vec)
return [self._new(self.cols, 1, b) for b in basis]
def rowspace(self, simplify=False):
"""Returns a list of vectors that span the row space of ``self``."""
reduced, pivots = self.echelon_form(simplify=simplify, with_pivots=True)
return [reduced.row(i) for i in range(len(pivots))]
@classmethod
def orthogonalize(cls, *vecs, **kwargs):
"""Apply the Gram-Schmidt orthogonalization procedure
to vectors supplied in ``vecs``.
Parameters
==========
vecs
vectors to be made orthogonal
normalize : bool
If ``True``, return an orthonormal basis.
rankcheck : bool
If ``True``, the computation does not stop when encountering
linearly dependent vectors.
If ``False``, it will raise ``ValueError`` when any zero
or linearly dependent vectors are found.
Returns
=======
list
List of orthogonal (or orthonormal) basis vectors.
See Also
========
MatrixBase.QRdecomposition
References
==========
.. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
"""
normalize = kwargs.get('normalize', False)
rankcheck = kwargs.get('rankcheck', False)
def project(a, b):
return b * (a.dot(b) / b.dot(b))
def perp_to_subspace(vec, basis):
"""projects vec onto the subspace given
by the orthogonal basis ``basis``"""
components = [project(vec, b) for b in basis]
if len(basis) == 0:
return vec
return vec - reduce(lambda a, b: a + b, components)
ret = []
# make sure we start with a non-zero vector
vecs = list(vecs)
while len(vecs) > 0 and vecs[0].is_zero:
if rankcheck is False:
del vecs[0]
else:
raise ValueError(
"GramSchmidt: vector set not linearly independent")
for vec in vecs:
perp = perp_to_subspace(vec, ret)
if not perp.is_zero:
ret.append(perp)
elif rankcheck is True:
raise ValueError(
"GramSchmidt: vector set not linearly independent")
if normalize:
ret = [vec / vec.norm() for vec in ret]
return ret
class MatrixEigen(MatrixSubspaces):
"""Provides basic matrix eigenvalue/vector operations.
Should not be instantiated directly."""
@property
def _cache_is_diagonalizable(self):
SymPyDeprecationWarning(
feature='_cache_is_diagonalizable',
deprecated_since_version="1.4",
issue=15887
).warn()
return None
@property
def _cache_eigenvects(self):
SymPyDeprecationWarning(
feature='_cache_eigenvects',
deprecated_since_version="1.4",
issue=15887
).warn()
return None
def diagonalize(self, reals_only=False, sort=False, normalize=False):
"""
Return (P, D), where D is diagonal and
D = P^-1 * M * P
where M is current matrix.
Parameters
==========
reals_only : bool. Whether to throw an error if complex numbers are need
to diagonalize. (Default: False)
sort : bool. Sort the eigenvalues along the diagonal. (Default: False)
normalize : bool. If True, normalize the columns of P. (Default: False)
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
>>> m
Matrix([
[1, 2, 0],
[0, 3, 0],
[2, -4, 2]])
>>> (P, D) = m.diagonalize()
>>> D
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
>>> P
Matrix([
[-1, 0, -1],
[ 0, 0, -1],
[ 2, 1, 2]])
>>> P.inv() * m * P
Matrix([
[1, 0, 0],
[0, 2, 0],
[0, 0, 3]])
See Also
========
is_diagonal
is_diagonalizable
"""
if not self.is_square:
raise NonSquareMatrixError()
if not self.is_diagonalizable(reals_only=reals_only):
raise MatrixError("Matrix is not diagonalizable")
eigenvecs = self.eigenvects(simplify=True)
if sort:
eigenvecs = sorted(eigenvecs, key=default_sort_key)
p_cols, diag = [], []
for val, mult, basis in eigenvecs:
diag += [val] * mult
p_cols += basis
if normalize:
p_cols = [v / v.norm() for v in p_cols]
return self.hstack(*p_cols), self.diag(*diag)
def eigenvals(self, error_when_incomplete=True, **flags):
r"""Return eigenvalues using the Berkowitz agorithm to compute
the characteristic polynomial.
Parameters
==========
error_when_incomplete : bool, optional
If it is set to ``True``, it will raise an error if not all
eigenvalues are computed. This is caused by ``roots`` not returning
a full list of eigenvalues.
simplify : bool or function, optional
If it is set to ``True``, it attempts to return the most
simplified form of expressions returned by applying default
simplification method in every routine.
If it is set to ``False``, it will skip simplification in this
particular routine to save computation resources.
If a function is passed to, it will attempt to apply
the particular function as simplification method.
rational : bool, optional
If it is set to ``True``, every floating point numbers would be
replaced with rationals before computation. It can solve some
issues of ``roots`` routine not working well with floats.
multiple : bool, optional
If it is set to ``True``, the result will be in the form of a
list.
If it is set to ``False``, the result will be in the form of a
dictionary.
Returns
=======
eigs : list or dict
Eigenvalues of a matrix. The return format would be specified by
the key ``multiple``.
Raises
======
MatrixError
If not enough roots had got computed.
NonSquareMatrixError
If attempted to compute eigenvalues from a non-square matrix.
See Also
========
MatrixDeterminant.charpoly
eigenvects
Notes
=====
Eigenvalues of a matrix `A` can be computed by solving a matrix
equation `\det(A - \lambda I) = 0`
"""
simplify = flags.get('simplify', False) # Collect simplify flag before popped up, to reuse later in the routine.
multiple = flags.get('multiple', False) # Collect multiple flag to decide whether return as a dict or list.
rational = flags.pop('rational', True)
mat = self
if not mat:
return {}
if rational:
mat = mat.applyfunc(
lambda x: nsimplify(x, rational=True) if x.has(Float) else x)
if mat.is_upper or mat.is_lower:
if not self.is_square:
raise NonSquareMatrixError()
diagonal_entries = [mat[i, i] for i in range(mat.rows)]
if multiple:
eigs = diagonal_entries
else:
eigs = {}
for diagonal_entry in diagonal_entries:
if diagonal_entry not in eigs:
eigs[diagonal_entry] = 0
eigs[diagonal_entry] += 1
else:
flags.pop('simplify', None) # pop unsupported flag
if isinstance(simplify, FunctionType):
eigs = roots(mat.charpoly(x=Dummy('x'), simplify=simplify), **flags)
else:
eigs = roots(mat.charpoly(x=Dummy('x')), **flags)
# make sure the algebraic multiplicty sums to the
# size of the matrix
if error_when_incomplete and (sum(eigs.values()) if
isinstance(eigs, dict) else len(eigs)) != self.cols:
raise MatrixError("Could not compute eigenvalues for {}".format(self))
# Since 'simplify' flag is unsupported in roots()
# simplify() function will be applied once at the end of the routine.
if not simplify:
return eigs
if not isinstance(simplify, FunctionType):
simplify = _simplify
# With 'multiple' flag set true, simplify() will be mapped for the list
# Otherwise, simplify() will be mapped for the keys of the dictionary
if not multiple:
return {simplify(key): value for key, value in eigs.items()}
else:
return [simplify(value) for value in eigs]
def eigenvects(self, error_when_incomplete=True, iszerofunc=_iszero, **flags):
"""Return list of triples (eigenval, multiplicity, eigenspace).
Parameters
==========
error_when_incomplete : bool, optional
Raise an error when not all eigenvalues are computed. This is
caused by ``roots`` not returning a full list of eigenvalues.
iszerofunc : function, optional
Specifies a zero testing function to be used in ``rref``.
Default value is ``_iszero``, which uses SymPy's naive and fast
default assumption handler.
It can also accept any user-specified zero testing function, if it
is formatted as a function which accepts a single symbolic argument
and returns ``True`` if it is tested as zero and ``False`` if it
is tested as non-zero, and ``None`` if it is undecidable.
simplify : bool or function, optional
If ``True``, ``as_content_primitive()`` will be used to tidy up
normalization artifacts.
It will also be used by the ``nullspace`` routine.
chop : bool or positive number, optional
If the matrix contains any Floats, they will be changed to Rationals
for computation purposes, but the answers will be returned after
being evaluated with evalf. The ``chop`` flag is passed to ``evalf``.
When ``chop=True`` a default precision will be used; a number will
be interpreted as the desired level of precision.
Returns
=======
ret : [(eigenval, multiplicity, eigenspace), ...]
A ragged list containing tuples of data obtained by ``eigenvals``
and ``nullspace``.
``eigenspace`` is a list containing the ``eigenvector`` for each
eigenvalue.
``eigenvector`` is a vector in the form of a ``Matrix``. e.g.
a vector of length 3 is returned as ``Matrix([a_1, a_2, a_3])``.
Raises
======
NotImplementedError
If failed to compute nullspace.
See Also
========
eigenvals
MatrixSubspaces.nullspace
"""
from sympy.matrices import eye
simplify = flags.get('simplify', True)
if not isinstance(simplify, FunctionType):
simpfunc = _simplify if simplify else lambda x: x
primitive = flags.get('simplify', False)
chop = flags.pop('chop', False)
flags.pop('multiple', None) # remove this if it's there
mat = self
# roots doesn't like Floats, so replace them with Rationals
has_floats = self.has(Float)
if has_floats:
mat = mat.applyfunc(lambda x: nsimplify(x, rational=True))
def eigenspace(eigenval):
"""Get a basis for the eigenspace for a particular eigenvalue"""
m = mat - self.eye(mat.rows) * eigenval
ret = m.nullspace(iszerofunc=iszerofunc)
# the nullspace for a real eigenvalue should be
# non-trivial. If we didn't find an eigenvector, try once
# more a little harder
if len(ret) == 0 and simplify:
ret = m.nullspace(iszerofunc=iszerofunc, simplify=True)
if len(ret) == 0:
raise NotImplementedError(
"Can't evaluate eigenvector for eigenvalue %s" % eigenval)
return ret
eigenvals = mat.eigenvals(rational=False,
error_when_incomplete=error_when_incomplete,
**flags)
ret = [(val, mult, eigenspace(val)) for val, mult in
sorted(eigenvals.items(), key=default_sort_key)]
if primitive:
# if the primitive flag is set, get rid of any common
# integer denominators
def denom_clean(l):
from sympy import gcd
return [(v / gcd(list(v))).applyfunc(simpfunc) for v in l]
ret = [(val, mult, denom_clean(es)) for val, mult, es in ret]
if has_floats:
# if we had floats to start with, turn the eigenvectors to floats
ret = [(val.evalf(chop=chop), mult, [v.evalf(chop=chop) for v in es]) for val, mult, es in ret]
return ret
def is_diagonalizable(self, reals_only=False, **kwargs):
"""Returns true if a matrix is diagonalizable.
Parameters
==========
reals_only : bool. If reals_only=True, determine whether the matrix can be
diagonalized without complex numbers. (Default: False)
kwargs
======
clear_cache : bool. If True, clear the result of any computations when finished.
(Default: True)
Examples
========
>>> from sympy import Matrix
>>> m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
>>> m
Matrix([
[1, 2, 0],
[0, 3, 0],
[2, -4, 2]])
>>> m.is_diagonalizable()
True
>>> m = Matrix(2, 2, [0, 1, 0, 0])
>>> m
Matrix([
[0, 1],
[0, 0]])
>>> m.is_diagonalizable()
False
>>> m = Matrix(2, 2, [0, 1, -1, 0])
>>> m
Matrix([
[ 0, 1],
[-1, 0]])
>>> m.is_diagonalizable()
True
>>> m.is_diagonalizable(reals_only=True)
False
See Also
========
is_diagonal
diagonalize
"""
if 'clear_cache' in kwargs:
SymPyDeprecationWarning(
feature='clear_cache',
deprecated_since_version=1.4,
issue=15887
).warn()
if 'clear_subproducts' in kwargs:
SymPyDeprecationWarning(
feature='clear_subproducts',
deprecated_since_version=1.4,
issue=15887
).warn()
if not self.is_square:
return False
if all(e.is_real for e in self) and self.is_symmetric():
# every real symmetric matrix is real diagonalizable
return True
eigenvecs = self.eigenvects(simplify=True)
ret = True
for val, mult, basis in eigenvecs:
# if we have a complex eigenvalue
if reals_only and not val.is_real:
ret = False
# if the geometric multiplicity doesn't equal the algebraic
if mult != len(basis):
ret = False
return ret
def jordan_form(self, calc_transform=True, **kwargs):
"""Return ``(P, J)`` where `J` is a Jordan block
matrix and `P` is a matrix such that
``self == P*J*P**-1``
Parameters
==========
calc_transform : bool
If ``False``, then only `J` is returned.
chop : bool
All matrices are convered to exact types when computing
eigenvalues and eigenvectors. As a result, there may be
approximation errors. If ``chop==True``, these errors
will be truncated.
Examples
========
>>> from sympy import Matrix
>>> m = Matrix([[ 6, 5, -2, -3], [-3, -1, 3, 3], [ 2, 1, -2, -3], [-1, 1, 5, 5]])
>>> P, J = m.jordan_form()
>>> J
Matrix([
[2, 1, 0, 0],
[0, 2, 0, 0],
[0, 0, 2, 1],
[0, 0, 0, 2]])
See Also
========
jordan_block
"""
if not self.is_square:
raise NonSquareMatrixError("Only square matrices have Jordan forms")
chop = kwargs.pop('chop', False)
mat = self
has_floats = self.has(Float)
if has_floats:
try:
max_prec = max(term._prec for term in self._mat if isinstance(term, Float))
except ValueError:
# if no term in the matrix is explicitly a Float calling max()
# will throw a error so setting max_prec to default value of 53
max_prec = 53
# setting minimum max_dps to 15 to prevent loss of precision in
# matrix containing non evaluated expressions
max_dps = max(prec_to_dps(max_prec), 15)
def restore_floats(*args):
"""If ``has_floats`` is `True`, cast all ``args`` as
matrices of floats."""
if has_floats:
args = [m.evalf(prec=max_dps, chop=chop) for m in args]
if len(args) == 1:
return args[0]
return args
# cache calculations for some speedup
mat_cache = {}
def eig_mat(val, pow):
"""Cache computations of ``(self - val*I)**pow`` for quick
retrieval"""
if (val, pow) in mat_cache:
return mat_cache[(val, pow)]
if (val, pow - 1) in mat_cache:
mat_cache[(val, pow)] = mat_cache[(val, pow - 1)] * mat_cache[(val, 1)]
else:
mat_cache[(val, pow)] = (mat - val*self.eye(self.rows))**pow
return mat_cache[(val, pow)]
# helper functions
def nullity_chain(val, algebraic_multiplicity):
"""Calculate the sequence [0, nullity(E), nullity(E**2), ...]
until it is constant where ``E = self - val*I``"""
# mat.rank() is faster than computing the null space,
# so use the rank-nullity theorem
cols = self.cols
ret = [0]
nullity = cols - eig_mat(val, 1).rank()
i = 2
while nullity != ret[-1]:
ret.append(nullity)
if nullity == algebraic_multiplicity:
break
nullity = cols - eig_mat(val, i).rank()
i += 1
# Due to issues like #7146 and #15872, SymPy sometimes
# gives the wrong rank. In this case, raise an error
# instead of returning an incorrect matrix
if nullity < ret[-1] or nullity > algebraic_multiplicity:
raise MatrixError(
"SymPy had encountered an inconsistent "
"result while computing Jordan block: "
"{}".format(self))
return ret
def blocks_from_nullity_chain(d):
"""Return a list of the size of each Jordan block.
If d_n is the nullity of E**n, then the number
of Jordan blocks of size n is
2*d_n - d_(n-1) - d_(n+1)"""
# d[0] is always the number of columns, so skip past it
mid = [2*d[n] - d[n - 1] - d[n + 1] for n in range(1, len(d) - 1)]
# d is assumed to plateau with "d[ len(d) ] == d[-1]", so
# 2*d_n - d_(n-1) - d_(n+1) == d_n - d_(n-1)
end = [d[-1] - d[-2]] if len(d) > 1 else [d[0]]
return mid + end
def pick_vec(small_basis, big_basis):
"""Picks a vector from big_basis that isn't in
the subspace spanned by small_basis"""
if len(small_basis) == 0:
return big_basis[0]
for v in big_basis:
_, pivots = self.hstack(*(small_basis + [v])).echelon_form(with_pivots=True)
if pivots[-1] == len(small_basis):
return v
# roots doesn't like Floats, so replace them with Rationals
if has_floats:
mat = mat.applyfunc(lambda x: nsimplify(x, rational=True))
# first calculate the jordan block structure
eigs = mat.eigenvals()
# make sure that we found all the roots by counting
# the algebraic multiplicity
if sum(m for m in eigs.values()) != mat.cols:
raise MatrixError("Could not compute eigenvalues for {}".format(mat))
# most matrices have distinct eigenvalues
# and so are diagonalizable. In this case, don't
# do extra work!
if len(eigs.keys()) == mat.cols:
blocks = list(sorted(eigs.keys(), key=default_sort_key))
jordan_mat = mat.diag(*blocks)
if not calc_transform:
return restore_floats(jordan_mat)
jordan_basis = [eig_mat(eig, 1).nullspace()[0] for eig in blocks]
basis_mat = mat.hstack(*jordan_basis)
return restore_floats(basis_mat, jordan_mat)
block_structure = []
for eig in sorted(eigs.keys(), key=default_sort_key):
algebraic_multiplicity = eigs[eig]
chain = nullity_chain(eig, algebraic_multiplicity)
block_sizes = blocks_from_nullity_chain(chain)
# if block_sizes == [a, b, c, ...], then the number of
# Jordan blocks of size 1 is a, of size 2 is b, etc.
# create an array that has (eig, block_size) with one
# entry for each block
size_nums = [(i+1, num) for i, num in enumerate(block_sizes)]
# we expect larger Jordan blocks to come earlier
size_nums.reverse()
block_structure.extend(
(eig, size) for size, num in size_nums for _ in range(num))
jordan_form_size = sum(size for eig, size in block_structure)
if jordan_form_size != self.rows:
raise MatrixError(
"SymPy had encountered an inconsistent result while "
"computing Jordan block. : {}".format(self))
blocks = (mat.jordan_block(size=size, eigenvalue=eig) for eig, size in block_structure)
jordan_mat = mat.diag(*blocks)
if not calc_transform:
return restore_floats(jordan_mat)
# For each generalized eigenspace, calculate a basis.
# We start by looking for a vector in null( (A - eig*I)**n )
# which isn't in null( (A - eig*I)**(n-1) ) where n is
# the size of the Jordan block
#
# Ideally we'd just loop through block_structure and
# compute each generalized eigenspace. However, this
# causes a lot of unneeded computation. Instead, we
# go through the eigenvalues separately, since we know
# their generalized eigenspaces must have bases that
# are linearly independent.
jordan_basis = []
for eig in sorted(eigs.keys(), key=default_sort_key):
eig_basis = []
for block_eig, size in block_structure:
if block_eig != eig:
continue
null_big = (eig_mat(eig, size)).nullspace()
null_small = (eig_mat(eig, size - 1)).nullspace()
# we want to pick something that is in the big basis
# and not the small, but also something that is independent
# of any other generalized eigenvectors from a different
# generalized eigenspace sharing the same eigenvalue.
vec = pick_vec(null_small + eig_basis, null_big)
new_vecs = [(eig_mat(eig, i))*vec for i in range(size)]
eig_basis.extend(new_vecs)
jordan_basis.extend(reversed(new_vecs))
basis_mat = mat.hstack(*jordan_basis)
return restore_floats(basis_mat, jordan_mat)
def left_eigenvects(self, **flags):
"""Returns left eigenvectors and eigenvalues.
This function returns the list of triples (eigenval, multiplicity,
basis) for the left eigenvectors. Options are the same as for
eigenvects(), i.e. the ``**flags`` arguments gets passed directly to
eigenvects().
Examples
========
>>> from sympy import Matrix
>>> M = Matrix([[0, 1, 1], [1, 0, 0], [1, 1, 1]])
>>> M.eigenvects()
[(-1, 1, [Matrix([
[-1],
[ 1],
[ 0]])]), (0, 1, [Matrix([
[ 0],
[-1],
[ 1]])]), (2, 1, [Matrix([
[2/3],
[1/3],
[ 1]])])]
>>> M.left_eigenvects()
[(-1, 1, [Matrix([[-2, 1, 1]])]), (0, 1, [Matrix([[-1, -1, 1]])]), (2,
1, [Matrix([[1, 1, 1]])])]
"""
eigs = self.transpose().eigenvects(**flags)
return [(val, mult, [l.transpose() for l in basis]) for val, mult, basis in eigs]
def singular_values(self):
"""Compute the singular values of a Matrix
Examples
========
>>> from sympy import Matrix, Symbol
>>> x = Symbol('x', real=True)
>>> A = Matrix([[0, 1, 0], [0, x, 0], [-1, 0, 0]])
>>> A.singular_values()
[sqrt(x**2 + 1), 1, 0]
See Also
========
condition_number
"""
mat = self
if self.rows >= self.cols:
valmultpairs = (mat.H * mat).eigenvals()
else:
valmultpairs = (mat * mat.H).eigenvals()
# Expands result from eigenvals into a simple list
vals = []
for k, v in valmultpairs.items():
vals += [sqrt(k)] * v # dangerous! same k in several spots!
# Pad with zeros if singular values are computed in reverse way,
# to give consistent format.
if len(vals) < self.cols:
vals += [S.Zero] * (self.cols - len(vals))
# sort them in descending order
vals.sort(reverse=True, key=default_sort_key)
return vals
class MatrixCalculus(MatrixCommon):
"""Provides calculus-related matrix operations."""
def diff(self, *args, **kwargs):
"""Calculate the derivative of each element in the matrix.
``args`` will be passed to the ``integrate`` function.
Examples
========
>>> from sympy.matrices import Matrix
>>> from sympy.abc import x, y
>>> M = Matrix([[x, y], [1, 0]])
>>> M.diff(x)
Matrix([
[1, 0],
[0, 0]])
See Also
========
integrate
limit
"""
# XXX this should be handled here rather than in Derivative
from sympy import Derivative
kwargs.setdefault('evaluate', True)
deriv = Derivative(self, *args, evaluate=True)
if not isinstance(self, Basic):
return deriv.as_mutable()
else:
return deriv
def _eval_derivative(self, arg):
return self.applyfunc(lambda x: x.diff(arg))
def _accept_eval_derivative(self, s):
return s._visit_eval_derivative_array(self)
def _visit_eval_derivative_scalar(self, base):
# Types are (base: scalar, self: matrix)
return self.applyfunc(lambda x: base.diff(x))
def _visit_eval_derivative_array(self, base):
# Types are (base: array/matrix, self: matrix)
from sympy import derive_by_array
return derive_by_array(base, self)
def integrate(self, *args):
"""Integrate each element of the matrix. ``args`` will
be passed to the ``integrate`` function.
Examples
========
>>> from sympy.matrices import Matrix
>>> from sympy.abc import x, y
>>> M = Matrix([[x, y], [1, 0]])
>>> M.integrate((x, ))
Matrix([
[x**2/2, x*y],
[ x, 0]])
>>> M.integrate((x, 0, 2))
Matrix([
[2, 2*y],
[2, 0]])
See Also
========
limit
diff
"""
return self.applyfunc(lambda x: x.integrate(*args))
def jacobian(self, X):
"""Calculates the Jacobian matrix (derivative of a vector-valued function).
Parameters
==========
``self`` : vector of expressions representing functions f_i(x_1, ..., x_n).
X : set of x_i's in order, it can be a list or a Matrix
Both ``self`` and X can be a row or a column matrix in any order
(i.e., jacobian() should always work).
Examples
========
>>> from sympy import sin, cos, Matrix
>>> from sympy.abc import rho, phi
>>> X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])
>>> Y = Matrix([rho, phi])
>>> X.jacobian(Y)
Matrix([
[cos(phi), -rho*sin(phi)],
[sin(phi), rho*cos(phi)],
[ 2*rho, 0]])
>>> X = Matrix([rho*cos(phi), rho*sin(phi)])
>>> X.jacobian(Y)
Matrix([
[cos(phi), -rho*sin(phi)],
[sin(phi), rho*cos(phi)]])
See Also
========
hessian
wronskian
"""
if not isinstance(X, MatrixBase):
X = self._new(X)
# Both X and ``self`` can be a row or a column matrix, so we need to make
# sure all valid combinations work, but everything else fails:
if self.shape[0] == 1:
m = self.shape[1]
elif self.shape[1] == 1:
m = self.shape[0]
else:
raise TypeError("``self`` must be a row or a column matrix")
if X.shape[0] == 1:
n = X.shape[1]
elif X.shape[1] == 1:
n = X.shape[0]
else:
raise TypeError("X must be a row or a column matrix")
# m is the number of functions and n is the number of variables
# computing the Jacobian is now easy:
return self._new(m, n, lambda j, i: self[j].diff(X[i]))
def limit(self, *args):
"""Calculate the limit of each element in the matrix.
``args`` will be passed to the ``limit`` function.
Examples
========
>>> from sympy.matrices import Matrix
>>> from sympy.abc import x, y
>>> M = Matrix([[x, y], [1, 0]])
>>> M.limit(x, 2)
Matrix([
[2, y],
[1, 0]])
See Also
========
integrate
diff
"""
return self.applyfunc(lambda x: x.limit(*args))
# https://github.com/sympy/sympy/pull/12854
class MatrixDeprecated(MatrixCommon):
"""A class to house deprecated matrix methods."""
def _legacy_array_dot(self, b):
"""Compatibility function for deprecated behavior of ``matrix.dot(vector)``
"""
from .dense import Matrix
if not isinstance(b, MatrixBase):
if is_sequence(b):
if len(b) != self.cols and len(b) != self.rows:
raise ShapeError(
"Dimensions incorrect for dot product: %s, %s" % (
self.shape, len(b)))
return self.dot(Matrix(b))
else:
raise TypeError(
"`b` must be an ordered iterable or Matrix, not %s." %
type(b))
mat = self
if mat.cols == b.rows:
if b.cols != 1:
mat = mat.T
b = b.T
prod = flatten((mat * b).tolist())
return prod
if mat.cols == b.cols:
return mat.dot(b.T)
elif mat.rows == b.rows:
return mat.T.dot(b)
else:
raise ShapeError("Dimensions incorrect for dot product: %s, %s" % (
self.shape, b.shape))
def berkowitz_charpoly(self, x=Dummy('lambda'), simplify=_simplify):
return self.charpoly(x=x)
def berkowitz_det(self):
"""Computes determinant using Berkowitz method.
See Also
========
det
berkowitz
"""
return self.det(method='berkowitz')
def berkowitz_eigenvals(self, **flags):
"""Computes eigenvalues of a Matrix using Berkowitz method.
See Also
========
berkowitz
"""
return self.eigenvals(**flags)
def berkowitz_minors(self):
"""Computes principal minors using Berkowitz method.
See Also
========
berkowitz
"""
sign, minors = S.One, []
for poly in self.berkowitz():
minors.append(sign * poly[-1])
sign = -sign
return tuple(minors)
def berkowitz(self):
from sympy.matrices import zeros
berk = ((1,),)
if not self:
return berk
if not self.is_square:
raise NonSquareMatrixError()
A, N = self, self.rows
transforms = [0] * (N - 1)
for n in range(N, 1, -1):
T, k = zeros(n + 1, n), n - 1
R, C = -A[k, :k], A[:k, k]
A, a = A[:k, :k], -A[k, k]
items = [C]
for i in range(0, n - 2):
items.append(A * items[i])
for i, B in enumerate(items):
items[i] = (R * B)[0, 0]
items = [S.One, a] + items
for i in range(n):
T[i:, i] = items[:n - i + 1]
transforms[k - 1] = T
polys = [self._new([S.One, -A[0, 0]])]
for i, T in enumerate(transforms):
polys.append(T * polys[i])
return berk + tuple(map(tuple, polys))
def cofactorMatrix(self, method="berkowitz"):
return self.cofactor_matrix(method=method)
def det_bareis(self):
return self.det(method='bareiss')
def det_bareiss(self):
"""Compute matrix determinant using Bareiss' fraction-free
algorithm which is an extension of the well known Gaussian
elimination method. This approach is best suited for dense
symbolic matrices and will result in a determinant with
minimal number of fractions. It means that less term
rewriting is needed on resulting formulae.
TODO: Implement algorithm for sparse matrices (SFF),
http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.
See Also
========
det
berkowitz_det
"""
return self.det(method='bareiss')
def det_LU_decomposition(self):
"""Compute matrix determinant using LU decomposition
Note that this method fails if the LU decomposition itself
fails. In particular, if the matrix has no inverse this method
will fail.
TODO: Implement algorithm for sparse matrices (SFF),
http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.
See Also
========
det
det_bareiss
berkowitz_det
"""
return self.det(method='lu')
def jordan_cell(self, eigenval, n):
return self.jordan_block(size=n, eigenvalue=eigenval)
def jordan_cells(self, calc_transformation=True):
P, J = self.jordan_form()
return P, J.get_diag_blocks()
def minorEntry(self, i, j, method="berkowitz"):
return self.minor(i, j, method=method)
def minorMatrix(self, i, j):
return self.minor_submatrix(i, j)
def permuteBkwd(self, perm):
"""Permute the rows of the matrix with the given permutation in reverse."""
return self.permute_rows(perm, direction='backward')
def permuteFwd(self, perm):
"""Permute the rows of the matrix with the given permutation."""
return self.permute_rows(perm, direction='forward')
class MatrixBase(MatrixDeprecated,
MatrixCalculus,
MatrixEigen,
MatrixCommon):
"""Base class for matrix objects."""
# Added just for numpy compatibility
__array_priority__ = 11
is_Matrix = True
_class_priority = 3
_sympify = staticmethod(sympify)
__hash__ = None # Mutable
# Defined here the same as on Basic.
# We don't define _repr_png_ here because it would add a large amount of
# data to any notebook containing SymPy expressions, without adding
# anything useful to the notebook. It can still enabled manually, e.g.,
# for the qtconsole, with init_printing().
def _repr_latex_(self):
"""
IPython/Jupyter LaTeX printing
To change the behavior of this (e.g., pass in some settings to LaTeX),
use init_printing(). init_printing() will also enable LaTeX printing
for built in numeric types like ints and container types that contain
SymPy objects, like lists and dictionaries of expressions.
"""
from sympy.printing.latex import latex
s = latex(self, mode='plain')
return "$\\displaystyle %s$" % s
_repr_latex_orig = _repr_latex_
def __array__(self, dtype=object):
from .dense import matrix2numpy
return matrix2numpy(self, dtype=dtype)
def __getattr__(self, attr):
if attr in ('diff', 'integrate', 'limit'):
def doit(*args):
item_doit = lambda item: getattr(item, attr)(*args)
return self.applyfunc(item_doit)
return doit
else:
raise AttributeError(
"%s has no attribute %s." % (self.__class__.__name__, attr))
def __len__(self):
"""Return the number of elements of ``self``.
Implemented mainly so bool(Matrix()) == False.
"""
return self.rows * self.cols
def __mathml__(self):
mml = ""
for i in range(self.rows):
mml += "<matrixrow>"
for j in range(self.cols):
mml += self[i, j].__mathml__()
mml += "</matrixrow>"
return "<matrix>" + mml + "</matrix>"
# needed for python 2 compatibility
def __ne__(self, other):
return not self == other
def _matrix_pow_by_jordan_blocks(self, num):
from sympy.matrices import diag, MutableMatrix
from sympy import binomial
def jordan_cell_power(jc, n):
N = jc.shape[0]
l = jc[0, 0]
if l == 0 and (n < N - 1) != False:
raise ValueError("Matrix det == 0; not invertible")
elif l == 0 and N > 1 and n % 1 != 0:
raise ValueError("Non-integer power cannot be evaluated")
for i in range(N):
for j in range(N-i):
bn = binomial(n, i)
if isinstance(bn, binomial):
bn = bn._eval_expand_func()
jc[j, i+j] = l**(n-i)*bn
P, J = self.jordan_form()
jordan_cells = J.get_diag_blocks()
# Make sure jordan_cells matrices are mutable:
jordan_cells = [MutableMatrix(j) for j in jordan_cells]
for j in jordan_cells:
jordan_cell_power(j, num)
return self._new(P*diag(*jordan_cells)*P.inv())
def __repr__(self):
return sstr(self)
def __str__(self):
if self.rows == 0 or self.cols == 0:
return 'Matrix(%s, %s, [])' % (self.rows, self.cols)
return "Matrix(%s)" % str(self.tolist())
def _diagonalize_clear_subproducts(self):
del self._is_symbolic
del self._is_symmetric
del self._eigenvects
def _format_str(self, printer=None):
if not printer:
from sympy.printing.str import StrPrinter
printer = StrPrinter()
# Handle zero dimensions:
if self.rows == 0 or self.cols == 0:
return 'Matrix(%s, %s, [])' % (self.rows, self.cols)
if self.rows == 1:
return "Matrix([%s])" % self.table(printer, rowsep=',\n')
return "Matrix([\n%s])" % self.table(printer, rowsep=',\n')
@classmethod
def irregular(cls, ntop, *matrices, **kwargs):
"""Return a matrix filled by the given matrices which
are listed in order of appearance from left to right, top to
bottom as they first appear in the matrix. They must fill the
matrix completely.
Examples
========
>>> from sympy import ones, Matrix
>>> Matrix.irregular(3, ones(2,1), ones(3,3)*2, ones(2,2)*3,
... ones(1,1)*4, ones(2,2)*5, ones(1,2)*6, ones(1,2)*7)
Matrix([
[1, 2, 2, 2, 3, 3],
[1, 2, 2, 2, 3, 3],
[4, 2, 2, 2, 5, 5],
[6, 6, 7, 7, 5, 5]])
"""
from sympy.core.compatibility import as_int
ntop = as_int(ntop)
# make sure we are working with explicit matrices
b = [i.as_explicit() if hasattr(i, 'as_explicit') else i
for i in matrices]
q = list(range(len(b)))
dat = [i.rows for i in b]
active = [q.pop(0) for _ in range(ntop)]
cols = sum([b[i].cols for i in active])
rows = []
while any(dat):
r = []
for a, j in enumerate(active):
r.extend(b[j][-dat[j], :])
dat[j] -= 1
if dat[j] == 0 and q:
active[a] = q.pop(0)
if len(r) != cols:
raise ValueError(filldedent('''
Matrices provided do not appear to fill
the space completely.'''))
rows.append(r)
return cls._new(rows)
@classmethod
def _handle_creation_inputs(cls, *args, **kwargs):
"""Return the number of rows, cols and flat matrix elements.
Examples
========
>>> from sympy import Matrix, I
Matrix can be constructed as follows:
* from a nested list of iterables
>>> Matrix( ((1, 2+I), (3, 4)) )
Matrix([
[1, 2 + I],
[3, 4]])
* from un-nested iterable (interpreted as a column)
>>> Matrix( [1, 2] )
Matrix([
[1],
[2]])
* from un-nested iterable with dimensions
>>> Matrix(1, 2, [1, 2] )
Matrix([[1, 2]])
* from no arguments (a 0 x 0 matrix)
>>> Matrix()
Matrix(0, 0, [])
* from a rule
>>> Matrix(2, 2, lambda i, j: i/(j + 1) )
Matrix([
[0, 0],
[1, 1/2]])
See Also
========
irregular - filling a matrix with irregular blocks
"""
from sympy.matrices.sparse import SparseMatrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.utilities.iterables import reshape
flat_list = None
if len(args) == 1:
# Matrix(SparseMatrix(...))
if isinstance(args[0], SparseMatrix):
return args[0].rows, args[0].cols, flatten(args[0].tolist())
# Matrix(Matrix(...))
elif isinstance(args[0], MatrixBase):
return args[0].rows, args[0].cols, args[0]._mat
# Matrix(MatrixSymbol('X', 2, 2))
elif isinstance(args[0], Basic) and args[0].is_Matrix:
return args[0].rows, args[0].cols, args[0].as_explicit()._mat
# Matrix(numpy.ones((2, 2)))
elif hasattr(args[0], "__array__"):
# NumPy array or matrix or some other object that implements
# __array__. So let's first use this method to get a
# numpy.array() and then make a python list out of it.
arr = args[0].__array__()
if len(arr.shape) == 2:
rows, cols = arr.shape[0], arr.shape[1]
flat_list = [cls._sympify(i) for i in arr.ravel()]
return rows, cols, flat_list
elif len(arr.shape) == 1:
rows, cols = arr.shape[0], 1
flat_list = [S.Zero] * rows
for i in range(len(arr)):
flat_list[i] = cls._sympify(arr[i])
return rows, cols, flat_list
else:
raise NotImplementedError(
"SymPy supports just 1D and 2D matrices")
# Matrix([1, 2, 3]) or Matrix([[1, 2], [3, 4]])
elif is_sequence(args[0]) \
and not isinstance(args[0], DeferredVector):
dat = list(args[0])
ismat = lambda i: isinstance(i, MatrixBase) and (
evaluate or
isinstance(i, BlockMatrix) or
isinstance(i, MatrixSymbol))
raw = lambda i: is_sequence(i) and not ismat(i)
evaluate = kwargs.get('evaluate', True)
if evaluate:
def do(x):
# make Block and Symbol explicit
if isinstance(x, (list, tuple)):
return type(x)([do(i) for i in x])
if isinstance(x, BlockMatrix) or \
isinstance(x, MatrixSymbol) and \
all(_.is_Integer for _ in x.shape):
return x.as_explicit()
return x
dat = do(dat)
if dat == [] or dat == [[]]:
rows = cols = 0
flat_list = []
elif not any(raw(i) or ismat(i) for i in dat):
# a column as a list of values
flat_list = [cls._sympify(i) for i in dat]
rows = len(flat_list)
cols = 1 if rows else 0
elif evaluate and all(ismat(i) for i in dat):
# a column as a list of matrices
ncol = set(i.cols for i in dat if any(i.shape))
if ncol:
if len(ncol) != 1:
raise ValueError('mismatched dimensions')
flat_list = [_ for i in dat for r in i.tolist() for _ in r]
cols = ncol.pop()
rows = len(flat_list)//cols
else:
rows = cols = 0
flat_list = []
elif evaluate and any(ismat(i) for i in dat):
ncol = set()
flat_list = []
for i in dat:
if ismat(i):
flat_list.extend(
[k for j in i.tolist() for k in j])
if any(i.shape):
ncol.add(i.cols)
elif raw(i):
if i:
ncol.add(len(i))
flat_list.extend(i)
else:
ncol.add(1)
flat_list.append(i)
if len(ncol) > 1:
raise ValueError('mismatched dimensions')
cols = ncol.pop()
rows = len(flat_list)//cols
else:
# list of lists; each sublist is a logical row
# which might consist of many rows if the values in
# the row are matrices
flat_list = []
ncol = set()
rows = cols = 0
for row in dat:
if not is_sequence(row) and \
not getattr(row, 'is_Matrix', False):
raise ValueError('expecting list of lists')
if not row:
continue
if evaluate and all(ismat(i) for i in row):
r, c, flatT = cls._handle_creation_inputs(
[i.T for i in row])
T = reshape(flatT, [c])
flat = [T[i][j] for j in range(c) for i in range(r)]
r, c = c, r
else:
r = 1
if getattr(row, 'is_Matrix', False):
c = 1
flat = [row]
else:
c = len(row)
flat = [cls._sympify(i) for i in row]
ncol.add(c)
if len(ncol) > 1:
raise ValueError('mismatched dimensions')
flat_list.extend(flat)
rows += r
cols = ncol.pop() if ncol else 0
elif len(args) == 3:
rows = as_int(args[0])
cols = as_int(args[1])
if rows < 0 or cols < 0:
raise ValueError("Cannot create a {} x {} matrix. "
"Both dimensions must be positive".format(rows, cols))
# Matrix(2, 2, lambda i, j: i+j)
if len(args) == 3 and isinstance(args[2], Callable):
op = args[2]
flat_list = []
for i in range(rows):
flat_list.extend(
[cls._sympify(op(cls._sympify(i), cls._sympify(j)))
for j in range(cols)])
# Matrix(2, 2, [1, 2, 3, 4])
elif len(args) == 3 and is_sequence(args[2]):
flat_list = args[2]
if len(flat_list) != rows * cols:
raise ValueError(
'List length should be equal to rows*columns')
flat_list = [cls._sympify(i) for i in flat_list]
# Matrix()
elif len(args) == 0:
# Empty Matrix
rows = cols = 0
flat_list = []
if flat_list is None:
raise TypeError(filldedent('''
Data type not understood; expecting list of lists
or lists of values.'''))
return rows, cols, flat_list
def _setitem(self, key, value):
"""Helper to set value at location given by key.
Examples
========
>>> from sympy import Matrix, I, zeros, ones
>>> m = Matrix(((1, 2+I), (3, 4)))
>>> m
Matrix([
[1, 2 + I],
[3, 4]])
>>> m[1, 0] = 9
>>> m
Matrix([
[1, 2 + I],
[9, 4]])
>>> m[1, 0] = [[0, 1]]
To replace row r you assign to position r*m where m
is the number of columns:
>>> M = zeros(4)
>>> m = M.cols
>>> M[3*m] = ones(1, m)*2; M
Matrix([
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[2, 2, 2, 2]])
And to replace column c you can assign to position c:
>>> M[2] = ones(m, 1)*4; M
Matrix([
[0, 0, 4, 0],
[0, 0, 4, 0],
[0, 0, 4, 0],
[2, 2, 4, 2]])
"""
from .dense import Matrix
is_slice = isinstance(key, slice)
i, j = key = self.key2ij(key)
is_mat = isinstance(value, MatrixBase)
if type(i) is slice or type(j) is slice:
if is_mat:
self.copyin_matrix(key, value)
return
if not isinstance(value, Expr) and is_sequence(value):
self.copyin_list(key, value)
return
raise ValueError('unexpected value: %s' % value)
else:
if (not is_mat and
not isinstance(value, Basic) and is_sequence(value)):
value = Matrix(value)
is_mat = True
if is_mat:
if is_slice:
key = (slice(*divmod(i, self.cols)),
slice(*divmod(j, self.cols)))
else:
key = (slice(i, i + value.rows),
slice(j, j + value.cols))
self.copyin_matrix(key, value)
else:
return i, j, self._sympify(value)
return
def add(self, b):
"""Return self + b """
return self + b
def cholesky_solve(self, rhs):
"""Solves ``Ax = B`` using Cholesky decomposition,
for a general square non-singular matrix.
For a non-square matrix with rows > cols,
the least squares solution is returned.
See Also
========
lower_triangular_solve
upper_triangular_solve
gauss_jordan_solve
diagonal_solve
LDLsolve
LUsolve
QRsolve
pinv_solve
"""
hermitian = True
if self.is_symmetric():
hermitian = False
L = self._cholesky(hermitian=hermitian)
elif self.is_hermitian:
L = self._cholesky(hermitian=hermitian)
elif self.rows >= self.cols:
L = (self.H * self)._cholesky(hermitian=hermitian)
rhs = self.H * rhs
else:
raise NotImplementedError('Under-determined System. '
'Try M.gauss_jordan_solve(rhs)')
Y = L._lower_triangular_solve(rhs)
if hermitian:
return (L.H)._upper_triangular_solve(Y)
else:
return (L.T)._upper_triangular_solve(Y)
def cholesky(self, hermitian=True):
"""Returns the Cholesky-type decomposition L of a matrix A
such that L * L.H == A if hermitian flag is True,
or L * L.T == A if hermitian is False.
A must be a Hermitian positive-definite matrix if hermitian is True,
or a symmetric matrix if it is False.
Examples
========
>>> from sympy.matrices import Matrix
>>> A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
>>> A.cholesky()
Matrix([
[ 5, 0, 0],
[ 3, 3, 0],
[-1, 1, 3]])
>>> A.cholesky() * A.cholesky().T
Matrix([
[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]])
The matrix can have complex entries:
>>> from sympy import I
>>> A = Matrix(((9, 3*I), (-3*I, 5)))
>>> A.cholesky()
Matrix([
[ 3, 0],
[-I, 2]])
>>> A.cholesky() * A.cholesky().H
Matrix([
[ 9, 3*I],
[-3*I, 5]])
Non-hermitian Cholesky-type decomposition may be useful when the
matrix is not positive-definite.
>>> A = Matrix([[1, 2], [2, 1]])
>>> L = A.cholesky(hermitian=False)
>>> L
Matrix([
[1, 0],
[2, sqrt(3)*I]])
>>> L*L.T == A
True
See Also
========
LDLdecomposition
LUdecomposition
QRdecomposition
"""
if not self.is_square:
raise NonSquareMatrixError("Matrix must be square.")
if hermitian and not self.is_hermitian:
raise ValueError("Matrix must be Hermitian.")
if not hermitian and not self.is_symmetric():
raise ValueError("Matrix must be symmetric.")
return self._cholesky(hermitian=hermitian)
def condition_number(self):
"""Returns the condition number of a matrix.
This is the maximum singular value divided by the minimum singular value
Examples
========
>>> from sympy import Matrix, S
>>> A = Matrix([[1, 0, 0], [0, 10, 0], [0, 0, S.One/10]])
>>> A.condition_number()
100
See Also
========
singular_values
"""
if not self:
return S.Zero
singularvalues = self.singular_values()
return Max(*singularvalues) / Min(*singularvalues)
def copy(self):
"""
Returns the copy of a matrix.
Examples
========
>>> from sympy import Matrix
>>> A = Matrix(2, 2, [1, 2, 3, 4])
>>> A.copy()
Matrix([
[1, 2],
[3, 4]])
"""
return self._new(self.rows, self.cols, self._mat)
def cross(self, b):
r"""
Return the cross product of ``self`` and ``b`` relaxing the condition
of compatible dimensions: if each has 3 elements, a matrix of the
same type and shape as ``self`` will be returned. If ``b`` has the same
shape as ``self`` then common identities for the cross product (like
`a \times b = - b \times a`) will hold.
Parameters
==========
b : 3x1 or 1x3 Matrix
See Also
========
dot
multiply
multiply_elementwise
"""
if not is_sequence(b):
raise TypeError(
"`b` must be an ordered iterable or Matrix, not %s." %
type(b))
if not (self.rows * self.cols == b.rows * b.cols == 3):
raise ShapeError("Dimensions incorrect for cross product: %s x %s" %
((self.rows, self.cols), (b.rows, b.cols)))
else:
return self._new(self.rows, self.cols, (
(self[1] * b[2] - self[2] * b[1]),
(self[2] * b[0] - self[0] * b[2]),
(self[0] * b[1] - self[1] * b[0])))
@property
def D(self):
"""Return Dirac conjugate (if ``self.rows == 4``).
Examples
========
>>> from sympy import Matrix, I, eye
>>> m = Matrix((0, 1 + I, 2, 3))
>>> m.D
Matrix([[0, 1 - I, -2, -3]])
>>> m = (eye(4) + I*eye(4))
>>> m[0, 3] = 2
>>> m.D
Matrix([
[1 - I, 0, 0, 0],
[ 0, 1 - I, 0, 0],
[ 0, 0, -1 + I, 0],
[ 2, 0, 0, -1 + I]])
If the matrix does not have 4 rows an AttributeError will be raised
because this property is only defined for matrices with 4 rows.
>>> Matrix(eye(2)).D
Traceback (most recent call last):
...
AttributeError: Matrix has no attribute D.
See Also
========
conjugate: By-element conjugation
H: Hermite conjugation
"""
from sympy.physics.matrices import mgamma
if self.rows != 4:
# In Python 3.2, properties can only return an AttributeError
# so we can't raise a ShapeError -- see commit which added the
# first line of this inline comment. Also, there is no need
# for a message since MatrixBase will raise the AttributeError
raise AttributeError
return self.H * mgamma(0)
def diagonal_solve(self, rhs):
"""Solves ``Ax = B`` efficiently, where A is a diagonal Matrix,
with non-zero diagonal entries.
Examples
========
>>> from sympy.matrices import Matrix, eye
>>> A = eye(2)*2
>>> B = Matrix([[1, 2], [3, 4]])
>>> A.diagonal_solve(B) == B/2
True
See Also
========
lower_triangular_solve
upper_triangular_solve
gauss_jordan_solve
cholesky_solve
LDLsolve
LUsolve
QRsolve
pinv_solve
"""
if not self.is_diagonal():
raise TypeError("Matrix should be diagonal")
if rhs.rows != self.rows:
raise TypeError("Size mis-match")
return self._diagonal_solve(rhs)
def dot(self, b, hermitian=None, conjugate_convention=None):
"""Return the dot or inner product of two vectors of equal length.
Here ``self`` must be a ``Matrix`` of size 1 x n or n x 1, and ``b``
must be either a matrix of size 1 x n, n x 1, or a list/tuple of length n.
A scalar is returned.
By default, ``dot`` does not conjugate ``self`` or ``b``, even if there are
complex entries. Set ``hermitian=True`` (and optionally a ``conjugate_convention``)
to compute the hermitian inner product.
Possible kwargs are ``hermitian`` and ``conjugate_convention``.
If ``conjugate_convention`` is ``"left"``, ``"math"`` or ``"maths"``,
the conjugate of the first vector (``self``) is used. If ``"right"``
or ``"physics"`` is specified, the conjugate of the second vector ``b`` is used.
Examples
========
>>> from sympy import Matrix
>>> M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> v = Matrix([1, 1, 1])
>>> M.row(0).dot(v)
6
>>> M.col(0).dot(v)
12
>>> v = [3, 2, 1]
>>> M.row(0).dot(v)
10
>>> from sympy import I
>>> q = Matrix([1*I, 1*I, 1*I])
>>> q.dot(q, hermitian=False)
-3
>>> q.dot(q, hermitian=True)
3
>>> q1 = Matrix([1, 1, 1*I])
>>> q.dot(q1, hermitian=True, conjugate_convention="maths")
1 - 2*I
>>> q.dot(q1, hermitian=True, conjugate_convention="physics")
1 + 2*I
See Also
========
cross
multiply
multiply_elementwise
"""
from .dense import Matrix
if not isinstance(b, MatrixBase):
if is_sequence(b):
if len(b) != self.cols and len(b) != self.rows:
raise ShapeError(
"Dimensions incorrect for dot product: %s, %s" % (
self.shape, len(b)))
return self.dot(Matrix(b))
else:
raise TypeError(
"`b` must be an ordered iterable or Matrix, not %s." %
type(b))
mat = self
if (1 not in mat.shape) or (1 not in b.shape) :
SymPyDeprecationWarning(
feature="Dot product of non row/column vectors",
issue=13815,
deprecated_since_version="1.2",
useinstead="* to take matrix products").warn()
return mat._legacy_array_dot(b)
if len(mat) != len(b):
raise ShapeError("Dimensions incorrect for dot product: %s, %s" % (self.shape, b.shape))
n = len(mat)
if mat.shape != (1, n):
mat = mat.reshape(1, n)
if b.shape != (n, 1):
b = b.reshape(n, 1)
# Now ``mat`` is a row vector and ``b`` is a column vector.
# If it so happens that only conjugate_convention is passed
# then automatically set hermitian to True. If only hermitian
# is true but no conjugate_convention is not passed then
# automatically set it to ``"maths"``
if conjugate_convention is not None and hermitian is None:
hermitian = True
if hermitian and conjugate_convention is None:
conjugate_convention = "maths"
if hermitian == True:
if conjugate_convention in ("maths", "left", "math"):
mat = mat.conjugate()
elif conjugate_convention in ("physics", "right"):
b = b.conjugate()
else:
raise ValueError("Unknown conjugate_convention was entered."
" conjugate_convention must be one of the"
" following: math, maths, left, physics or right.")
return (mat * b)[0]
def dual(self):
"""Returns the dual of a matrix, which is:
``(1/2)*levicivita(i, j, k, l)*M(k, l)`` summed over indices `k` and `l`
Since the levicivita method is anti_symmetric for any pairwise
exchange of indices, the dual of a symmetric matrix is the zero
matrix. Strictly speaking the dual defined here assumes that the
'matrix' `M` is a contravariant anti_symmetric second rank tensor,
so that the dual is a covariant second rank tensor.
"""
from sympy import LeviCivita
from sympy.matrices import zeros
M, n = self[:, :], self.rows
work = zeros(n)
if self.is_symmetric():
return work
for i in range(1, n):
for j in range(1, n):
acum = 0
for k in range(1, n):
acum += LeviCivita(i, j, 0, k) * M[0, k]
work[i, j] = acum
work[j, i] = -acum
for l in range(1, n):
acum = 0
for a in range(1, n):
for b in range(1, n):
acum += LeviCivita(0, l, a, b) * M[a, b]
acum /= 2
work[0, l] = -acum
work[l, 0] = acum
return work
def exp(self):
"""Return the exponentiation of a square matrix."""
if not self.is_square:
raise NonSquareMatrixError(
"Exponentiation is valid only for square matrices")
try:
P, J = self.jordan_form()
cells = J.get_diag_blocks()
except MatrixError:
raise NotImplementedError(
"Exponentiation is implemented only for matrices for which the Jordan normal form can be computed")
def _jblock_exponential(b):
# This function computes the matrix exponential for one single Jordan block
nr = b.rows
l = b[0, 0]
if nr == 1:
res = exp(l)
else:
from sympy import eye
# extract the diagonal part
d = b[0, 0] * eye(nr)
# and the nilpotent part
n = b - d
# compute its exponential
nex = eye(nr)
for i in range(1, nr):
nex = nex + n ** i / factorial(i)
# combine the two parts
res = exp(b[0, 0]) * nex
return (res)
blocks = list(map(_jblock_exponential, cells))
from sympy.matrices import diag
from sympy import re
eJ = diag(*blocks)
# n = self.rows
ret = P * eJ * P.inv()
if all(value.is_real for value in self.values()):
return type(self)(re(ret))
else:
return type(self)(ret)
def gauss_jordan_solve(self, B, freevar=False):
"""
Solves ``Ax = B`` using Gauss Jordan elimination.
There may be zero, one, or infinite solutions. If one solution
exists, it will be returned. If infinite solutions exist, it will
be returned parametrically. If no solutions exist, It will throw
ValueError.
Parameters
==========
B : Matrix
The right hand side of the equation to be solved for. Must have
the same number of rows as matrix A.
freevar : List
If the system is underdetermined (e.g. A has more columns than
rows), infinite solutions are possible, in terms of arbitrary
values of free variables. Then the index of the free variables
in the solutions (column Matrix) will be returned by freevar, if
the flag `freevar` is set to `True`.
Returns
=======
x : Matrix
The matrix that will satisfy ``Ax = B``. Will have as many rows as
matrix A has columns, and as many columns as matrix B.
params : Matrix
If the system is underdetermined (e.g. A has more columns than
rows), infinite solutions are possible, in terms of arbitrary
parameters. These arbitrary parameters are returned as params
Matrix.
Examples
========
>>> from sympy import Matrix
>>> A = Matrix([[1, 2, 1, 1], [1, 2, 2, -1], [2, 4, 0, 6]])
>>> B = Matrix([7, 12, 4])
>>> sol, params = A.gauss_jordan_solve(B)
>>> sol
Matrix([
[-2*tau0 - 3*tau1 + 2],
[ tau0],
[ 2*tau1 + 5],
[ tau1]])
>>> params
Matrix([
[tau0],
[tau1]])
>>> taus_zeroes = { tau:0 for tau in params }
>>> sol_unique = sol.xreplace(taus_zeroes)
>>> sol_unique
Matrix([
[2],
[0],
[5],
[0]])
>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
>>> B = Matrix([3, 6, 9])
>>> sol, params = A.gauss_jordan_solve(B)
>>> sol
Matrix([
[-1],
[ 2],
[ 0]])
>>> params
Matrix(0, 1, [])
>>> A = Matrix([[2, -7], [-1, 4]])
>>> B = Matrix([[-21, 3], [12, -2]])
>>> sol, params = A.gauss_jordan_solve(B)
>>> sol
Matrix([
[0, -2],
[3, -1]])
>>> params
Matrix(0, 2, [])
See Also
========
lower_triangular_solve
upper_triangular_solve
cholesky_solve
diagonal_solve
LDLsolve
LUsolve
QRsolve
pinv
References
==========
.. [1] https://en.wikipedia.org/wiki/Gaussian_elimination
"""
from sympy.matrices import Matrix, zeros
aug = self.hstack(self.copy(), B.copy())
B_cols = B.cols
row, col = aug[:, :-B_cols].shape
# solve by reduced row echelon form
A, pivots = aug.rref(simplify=True)
A, v = A[:, :-B_cols], A[:, -B_cols:]
pivots = list(filter(lambda p: p < col, pivots))
rank = len(pivots)
# Bring to block form
permutation = Matrix(range(col)).T
for i, c in enumerate(pivots):
permutation.col_swap(i, c)
# check for existence of solutions
# rank of aug Matrix should be equal to rank of coefficient matrix
if not v[rank:, :].is_zero:
raise ValueError("Linear system has no solution")
# Get index of free symbols (free parameters)
free_var_index = permutation[
len(pivots):] # non-pivots columns are free variables
# Free parameters
# what are current unnumbered free symbol names?
name = _uniquely_named_symbol('tau', aug,
compare=lambda i: str(i).rstrip('1234567890')).name
gen = numbered_symbols(name)
tau = Matrix([next(gen) for k in range((col - rank)*B_cols)]).reshape(
col - rank, B_cols)
# Full parametric solution
V = A[:rank,:]
for c in reversed(pivots):
V.col_del(c)
vt = v[:rank, :]
free_sol = tau.vstack(vt - V * tau, tau)
# Undo permutation
sol = zeros(col, B_cols)
for k in range(col):
sol[permutation[k], :] = free_sol[k,:]
if freevar:
return sol, tau, free_var_index
else:
return sol, tau
def inv_mod(self, m):
r"""
Returns the inverse of the matrix `K` (mod `m`), if it exists.
Method to find the matrix inverse of `K` (mod `m`) implemented in this function:
* Compute `\mathrm{adj}(K) = \mathrm{cof}(K)^t`, the adjoint matrix of `K`.
* Compute `r = 1/\mathrm{det}(K) \pmod m`.
* `K^{-1} = r\cdot \mathrm{adj}(K) \pmod m`.
Examples
========
>>> from sympy import Matrix
>>> A = Matrix(2, 2, [1, 2, 3, 4])
>>> A.inv_mod(5)
Matrix([
[3, 1],
[4, 2]])
>>> A.inv_mod(3)
Matrix([
[1, 1],
[0, 1]])
"""
if not self.is_square:
raise NonSquareMatrixError()
N = self.cols
det_K = self.det()
det_inv = None
try:
det_inv = mod_inverse(det_K, m)
except ValueError:
raise ValueError('Matrix is not invertible (mod %d)' % m)
K_adj = self.adjugate()
K_inv = self.__class__(N, N,
[det_inv * K_adj[i, j] % m for i in range(N) for
j in range(N)])
return K_inv
def inverse_ADJ(self, iszerofunc=_iszero):
"""Calculates the inverse using the adjugate matrix and a determinant.
See Also
========
inv
inverse_LU
inverse_GE
"""
if not self.is_square:
raise NonSquareMatrixError("A Matrix must be square to invert.")
d = self.det(method='berkowitz')
zero = d.equals(0)
if zero is None:
# if equals() can't decide, will rref be able to?
ok = self.rref(simplify=True)[0]
zero = any(iszerofunc(ok[j, j]) for j in range(ok.rows))
if zero:
raise ValueError("Matrix det == 0; not invertible.")
return self.adjugate() / d
def inverse_GE(self, iszerofunc=_iszero):
"""Calculates the inverse using Gaussian elimination.
See Also
========
inv
inverse_LU
inverse_ADJ
"""
from .dense import Matrix
if not self.is_square:
raise NonSquareMatrixError("A Matrix must be square to invert.")
big = Matrix.hstack(self.as_mutable(), Matrix.eye(self.rows))
red = big.rref(iszerofunc=iszerofunc, simplify=True)[0]
if any(iszerofunc(red[j, j]) for j in range(red.rows)):
raise ValueError("Matrix det == 0; not invertible.")
return self._new(red[:, big.rows:])
def inverse_LU(self, iszerofunc=_iszero):
"""Calculates the inverse using LU decomposition.
See Also
========
inv
inverse_GE
inverse_ADJ
"""
if not self.is_square:
raise NonSquareMatrixError()
ok = self.rref(simplify=True)[0]
if any(iszerofunc(ok[j, j]) for j in range(ok.rows)):
raise ValueError("Matrix det == 0; not invertible.")
return self.LUsolve(self.eye(self.rows), iszerofunc=_iszero)
def inv(self, method=None, **kwargs):
"""
Return the inverse of a matrix.
CASE 1: If the matrix is a dense matrix.
Return the matrix inverse using the method indicated (default
is Gauss elimination).
Parameters
==========
method : ('GE', 'LU', or 'ADJ')
Notes
=====
According to the ``method`` keyword, it calls the appropriate method:
GE .... inverse_GE(); default
LU .... inverse_LU()
ADJ ... inverse_ADJ()
See Also
========
inverse_LU
inverse_GE
inverse_ADJ
Raises
------
ValueError
If the determinant of the matrix is zero.
CASE 2: If the matrix is a sparse matrix.
Return the matrix inverse using Cholesky or LDL (default).
kwargs
======
method : ('CH', 'LDL')
Notes
=====
According to the ``method`` keyword, it calls the appropriate method:
LDL ... inverse_LDL(); default
CH .... inverse_CH()
Raises
------
ValueError
If the determinant of the matrix is zero.
"""
if not self.is_square:
raise NonSquareMatrixError()
if method is not None:
kwargs['method'] = method
return self._eval_inverse(**kwargs)
def is_nilpotent(self):
"""Checks if a matrix is nilpotent.
A matrix B is nilpotent if for some integer k, B**k is
a zero matrix.
Examples
========
>>> from sympy import Matrix
>>> a = Matrix([[0, 0, 0], [1, 0, 0], [1, 1, 0]])
>>> a.is_nilpotent()
True
>>> a = Matrix([[1, 0, 1], [1, 0, 0], [1, 1, 0]])
>>> a.is_nilpotent()
False
"""
if not self:
return True
if not self.is_square:
raise NonSquareMatrixError(
"Nilpotency is valid only for square matrices")
x = _uniquely_named_symbol('x', self)
p = self.charpoly(x)
if p.args[0] == x ** self.rows:
return True
return False
def key2bounds(self, keys):
"""Converts a key with potentially mixed types of keys (integer and slice)
into a tuple of ranges and raises an error if any index is out of ``self``'s
range.
See Also
========
key2ij
"""
from sympy.matrices.common import a2idx as a2idx_ # Remove this line after deprecation of a2idx from matrices.py
islice, jslice = [isinstance(k, slice) for k in keys]
if islice:
if not self.rows:
rlo = rhi = 0
else:
rlo, rhi = keys[0].indices(self.rows)[:2]
else:
rlo = a2idx_(keys[0], self.rows)
rhi = rlo + 1
if jslice:
if not self.cols:
clo = chi = 0
else:
clo, chi = keys[1].indices(self.cols)[:2]
else:
clo = a2idx_(keys[1], self.cols)
chi = clo + 1
return rlo, rhi, clo, chi
def key2ij(self, key):
"""Converts key into canonical form, converting integers or indexable
items into valid integers for ``self``'s range or returning slices
unchanged.
See Also
========
key2bounds
"""
from sympy.matrices.common import a2idx as a2idx_ # Remove this line after deprecation of a2idx from matrices.py
if is_sequence(key):
if not len(key) == 2:
raise TypeError('key must be a sequence of length 2')
return [a2idx_(i, n) if not isinstance(i, slice) else i
for i, n in zip(key, self.shape)]
elif isinstance(key, slice):
return key.indices(len(self))[:2]
else:
return divmod(a2idx_(key, len(self)), self.cols)
def LDLdecomposition(self, hermitian=True):
"""Returns the LDL Decomposition (L, D) of matrix A,
such that L * D * L.H == A if hermitian flag is True, or
L * D * L.T == A if hermitian is False.
This method eliminates the use of square root.
Further this ensures that all the diagonal entries of L are 1.
A must be a Hermitian positive-definite matrix if hermitian is True,
or a symmetric matrix otherwise.
Examples
========
>>> from sympy.matrices import Matrix, eye
>>> A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
>>> L, D = A.LDLdecomposition()
>>> L
Matrix([
[ 1, 0, 0],
[ 3/5, 1, 0],
[-1/5, 1/3, 1]])
>>> D
Matrix([
[25, 0, 0],
[ 0, 9, 0],
[ 0, 0, 9]])
>>> L * D * L.T * A.inv() == eye(A.rows)
True
The matrix can have complex entries:
>>> from sympy import I
>>> A = Matrix(((9, 3*I), (-3*I, 5)))
>>> L, D = A.LDLdecomposition()
>>> L
Matrix([
[ 1, 0],
[-I/3, 1]])
>>> D
Matrix([
[9, 0],
[0, 4]])
>>> L*D*L.H == A
True
See Also
========
cholesky
LUdecomposition
QRdecomposition
"""
if not self.is_square:
raise NonSquareMatrixError("Matrix must be square.")
if hermitian and not self.is_hermitian:
raise ValueError("Matrix must be Hermitian.")
if not hermitian and not self.is_symmetric():
raise ValueError("Matrix must be symmetric.")
return self._LDLdecomposition(hermitian=hermitian)
def LDLsolve(self, rhs):
"""Solves ``Ax = B`` using LDL decomposition,
for a general square and non-singular matrix.
For a non-square matrix with rows > cols,
the least squares solution is returned.
Examples
========
>>> from sympy.matrices import Matrix, eye
>>> A = eye(2)*2
>>> B = Matrix([[1, 2], [3, 4]])
>>> A.LDLsolve(B) == B/2
True
See Also
========
LDLdecomposition
lower_triangular_solve
upper_triangular_solve
gauss_jordan_solve
cholesky_solve
diagonal_solve
LUsolve
QRsolve
pinv_solve
"""
hermitian = True
if self.is_symmetric():
hermitian = False
L, D = self.LDLdecomposition(hermitian=hermitian)
elif self.is_hermitian:
L, D = self.LDLdecomposition(hermitian=hermitian)
elif self.rows >= self.cols:
L, D = (self.H * self).LDLdecomposition(hermitian=hermitian)
rhs = self.H * rhs
else:
raise NotImplementedError('Under-determined System. '
'Try M.gauss_jordan_solve(rhs)')
Y = L._lower_triangular_solve(rhs)
Z = D._diagonal_solve(Y)
if hermitian:
return (L.H)._upper_triangular_solve(Z)
else:
return (L.T)._upper_triangular_solve(Z)
def lower_triangular_solve(self, rhs):
"""Solves ``Ax = B``, where A is a lower triangular matrix.
See Also
========
upper_triangular_solve
gauss_jordan_solve
cholesky_solve
diagonal_solve
LDLsolve
LUsolve
QRsolve
pinv_solve
"""
if not self.is_square:
raise NonSquareMatrixError("Matrix must be square.")
if rhs.rows != self.rows:
raise ShapeError("Matrices size mismatch.")
if not self.is_lower:
raise ValueError("Matrix must be lower triangular.")
return self._lower_triangular_solve(rhs)
def LUdecomposition(self,
iszerofunc=_iszero,
simpfunc=None,
rankcheck=False):
"""Returns (L, U, perm) where L is a lower triangular matrix with unit
diagonal, U is an upper triangular matrix, and perm is a list of row
swap index pairs. If A is the original matrix, then
A = (L*U).permuteBkwd(perm), and the row permutation matrix P such
that P*A = L*U can be computed by P=eye(A.row).permuteFwd(perm).
See documentation for LUCombined for details about the keyword argument
rankcheck, iszerofunc, and simpfunc.
Examples
========
>>> from sympy import Matrix
>>> a = Matrix([[4, 3], [6, 3]])
>>> L, U, _ = a.LUdecomposition()
>>> L
Matrix([
[ 1, 0],
[3/2, 1]])
>>> U
Matrix([
[4, 3],
[0, -3/2]])
See Also
========
cholesky
LDLdecomposition
QRdecomposition
LUdecomposition_Simple
LUdecompositionFF
LUsolve
"""
combined, p = self.LUdecomposition_Simple(iszerofunc=iszerofunc,
simpfunc=simpfunc,
rankcheck=rankcheck)
# L is lower triangular ``self.rows x self.rows``
# U is upper triangular ``self.rows x self.cols``
# L has unit diagonal. For each column in combined, the subcolumn
# below the diagonal of combined is shared by L.
# If L has more columns than combined, then the remaining subcolumns
# below the diagonal of L are zero.
# The upper triangular portion of L and combined are equal.
def entry_L(i, j):
if i < j:
# Super diagonal entry
return S.Zero
elif i == j:
return S.One
elif j < combined.cols:
return combined[i, j]
# Subdiagonal entry of L with no corresponding
# entry in combined
return S.Zero
def entry_U(i, j):
return S.Zero if i > j else combined[i, j]
L = self._new(combined.rows, combined.rows, entry_L)
U = self._new(combined.rows, combined.cols, entry_U)
return L, U, p
def LUdecomposition_Simple(self,
iszerofunc=_iszero,
simpfunc=None,
rankcheck=False):
"""Compute an lu decomposition of m x n matrix A, where P*A = L*U
* L is m x m lower triangular with unit diagonal
* U is m x n upper triangular
* P is an m x m permutation matrix
Returns an m x n matrix lu, and an m element list perm where each
element of perm is a pair of row exchange indices.
The factors L and U are stored in lu as follows:
The subdiagonal elements of L are stored in the subdiagonal elements
of lu, that is lu[i, j] = L[i, j] whenever i > j.
The elements on the diagonal of L are all 1, and are not explicitly
stored.
U is stored in the upper triangular portion of lu, that is
lu[i ,j] = U[i, j] whenever i <= j.
The output matrix can be visualized as:
Matrix([
[u, u, u, u],
[l, u, u, u],
[l, l, u, u],
[l, l, l, u]])
where l represents a subdiagonal entry of the L factor, and u
represents an entry from the upper triangular entry of the U
factor.
perm is a list row swap index pairs such that if A is the original
matrix, then A = (L*U).permuteBkwd(perm), and the row permutation
matrix P such that ``P*A = L*U`` can be computed by
``P=eye(A.row).permuteFwd(perm)``.
The keyword argument rankcheck determines if this function raises a
ValueError when passed a matrix whose rank is strictly less than
min(num rows, num cols). The default behavior is to decompose a rank
deficient matrix. Pass rankcheck=True to raise a
ValueError instead. (This mimics the previous behavior of this function).
The keyword arguments iszerofunc and simpfunc are used by the pivot
search algorithm.
iszerofunc is a callable that returns a boolean indicating if its
input is zero, or None if it cannot make the determination.
simpfunc is a callable that simplifies its input.
The default is simpfunc=None, which indicate that the pivot search
algorithm should not attempt to simplify any candidate pivots.
If simpfunc fails to simplify its input, then it must return its input
instead of a copy.
When a matrix contains symbolic entries, the pivot search algorithm
differs from the case where every entry can be categorized as zero or
nonzero.
The algorithm searches column by column through the submatrix whose
top left entry coincides with the pivot position.
If it exists, the pivot is the first entry in the current search
column that iszerofunc guarantees is nonzero.
If no such candidate exists, then each candidate pivot is simplified
if simpfunc is not None.
The search is repeated, with the difference that a candidate may be
the pivot if ``iszerofunc()`` cannot guarantee that it is nonzero.
In the second search the pivot is the first candidate that
iszerofunc can guarantee is nonzero.
If no such candidate exists, then the pivot is the first candidate
for which iszerofunc returns None.
If no such candidate exists, then the search is repeated in the next
column to the right.
The pivot search algorithm differs from the one in ``rref()``, which
relies on ``_find_reasonable_pivot()``.
Future versions of ``LUdecomposition_simple()`` may use
``_find_reasonable_pivot()``.
See Also
========
LUdecomposition
LUdecompositionFF
LUsolve
"""
if rankcheck:
# https://github.com/sympy/sympy/issues/9796
pass
if self.rows == 0 or self.cols == 0:
# Define LU decomposition of a matrix with no entries as a matrix
# of the same dimensions with all zero entries.
return self.zeros(self.rows, self.cols), []
lu = self.as_mutable()
row_swaps = []
pivot_col = 0
for pivot_row in range(0, lu.rows - 1):
# Search for pivot. Prefer entry that iszeropivot determines
# is nonzero, over entry that iszeropivot cannot guarantee
# is zero.
# XXX ``_find_reasonable_pivot`` uses slow zero testing. Blocked by bug #10279
# Future versions of LUdecomposition_simple can pass iszerofunc and simpfunc
# to _find_reasonable_pivot().
# In pass 3 of _find_reasonable_pivot(), the predicate in ``if x.equals(S.Zero):``
# calls sympy.simplify(), and not the simplification function passed in via
# the keyword argument simpfunc.
iszeropivot = True
while pivot_col != self.cols and iszeropivot:
sub_col = (lu[r, pivot_col] for r in range(pivot_row, self.rows))
pivot_row_offset, pivot_value, is_assumed_non_zero, ind_simplified_pairs =\
_find_reasonable_pivot_naive(sub_col, iszerofunc, simpfunc)
iszeropivot = pivot_value is None
if iszeropivot:
# All candidate pivots in this column are zero.
# Proceed to next column.
pivot_col += 1
if rankcheck and pivot_col != pivot_row:
# All entries including and below the pivot position are
# zero, which indicates that the rank of the matrix is
# strictly less than min(num rows, num cols)
# Mimic behavior of previous implementation, by throwing a
# ValueError.
raise ValueError("Rank of matrix is strictly less than"
" number of rows or columns."
" Pass keyword argument"
" rankcheck=False to compute"
" the LU decomposition of this matrix.")
candidate_pivot_row = None if pivot_row_offset is None else pivot_row + pivot_row_offset
if candidate_pivot_row is None and iszeropivot:
# If candidate_pivot_row is None and iszeropivot is True
# after pivot search has completed, then the submatrix
# below and to the right of (pivot_row, pivot_col) is
# all zeros, indicating that Gaussian elimination is
# complete.
return lu, row_swaps
# Update entries simplified during pivot search.
for offset, val in ind_simplified_pairs:
lu[pivot_row + offset, pivot_col] = val
if pivot_row != candidate_pivot_row:
# Row swap book keeping:
# Record which rows were swapped.
# Update stored portion of L factor by multiplying L on the
# left and right with the current permutation.
# Swap rows of U.
row_swaps.append([pivot_row, candidate_pivot_row])
# Update L.
lu[pivot_row, 0:pivot_row], lu[candidate_pivot_row, 0:pivot_row] = \
lu[candidate_pivot_row, 0:pivot_row], lu[pivot_row, 0:pivot_row]
# Swap pivot row of U with candidate pivot row.
lu[pivot_row, pivot_col:lu.cols], lu[candidate_pivot_row, pivot_col:lu.cols] = \
lu[candidate_pivot_row, pivot_col:lu.cols], lu[pivot_row, pivot_col:lu.cols]
# Introduce zeros below the pivot by adding a multiple of the
# pivot row to a row under it, and store the result in the
# row under it.
# Only entries in the target row whose index is greater than
# start_col may be nonzero.
start_col = pivot_col + 1
for row in range(pivot_row + 1, lu.rows):
# Store factors of L in the subcolumn below
# (pivot_row, pivot_row).
lu[row, pivot_row] =\
lu[row, pivot_col]/lu[pivot_row, pivot_col]
# Form the linear combination of the pivot row and the current
# row below the pivot row that zeros the entries below the pivot.
# Employing slicing instead of a loop here raises
# NotImplementedError: Cannot add Zero to MutableSparseMatrix
# in sympy/matrices/tests/test_sparse.py.
# c = pivot_row + 1 if pivot_row == pivot_col else pivot_col
for c in range(start_col, lu.cols):
lu[row, c] = lu[row, c] - lu[row, pivot_row]*lu[pivot_row, c]
if pivot_row != pivot_col:
# matrix rank < min(num rows, num cols),
# so factors of L are not stored directly below the pivot.
# These entries are zero by construction, so don't bother
# computing them.
for row in range(pivot_row + 1, lu.rows):
lu[row, pivot_col] = S.Zero
pivot_col += 1
if pivot_col == lu.cols:
# All candidate pivots are zero implies that Gaussian
# elimination is complete.
return lu, row_swaps
if rankcheck:
if iszerofunc(
lu[Min(lu.rows, lu.cols) - 1, Min(lu.rows, lu.cols) - 1]):
raise ValueError("Rank of matrix is strictly less than"
" number of rows or columns."
" Pass keyword argument"
" rankcheck=False to compute"
" the LU decomposition of this matrix.")
return lu, row_swaps
def LUdecompositionFF(self):
"""Compute a fraction-free LU decomposition.
Returns 4 matrices P, L, D, U such that PA = L D**-1 U.
If the elements of the matrix belong to some integral domain I, then all
elements of L, D and U are guaranteed to belong to I.
**Reference**
- W. Zhou & D.J. Jeffrey, "Fraction-free matrix factors: new forms
for LU and QR factors". Frontiers in Computer Science in China,
Vol 2, no. 1, pp. 67-80, 2008.
See Also
========
LUdecomposition
LUdecomposition_Simple
LUsolve
"""
from sympy.matrices import SparseMatrix
zeros = SparseMatrix.zeros
eye = SparseMatrix.eye
n, m = self.rows, self.cols
U, L, P = self.as_mutable(), eye(n), eye(n)
DD = zeros(n, n)
oldpivot = 1
for k in range(n - 1):
if U[k, k] == 0:
for kpivot in range(k + 1, n):
if U[kpivot, k]:
break
else:
raise ValueError("Matrix is not full rank")
U[k, k:], U[kpivot, k:] = U[kpivot, k:], U[k, k:]
L[k, :k], L[kpivot, :k] = L[kpivot, :k], L[k, :k]
P[k, :], P[kpivot, :] = P[kpivot, :], P[k, :]
L[k, k] = Ukk = U[k, k]
DD[k, k] = oldpivot * Ukk
for i in range(k + 1, n):
L[i, k] = Uik = U[i, k]
for j in range(k + 1, m):
U[i, j] = (Ukk * U[i, j] - U[k, j] * Uik) / oldpivot
U[i, k] = 0
oldpivot = Ukk
DD[n - 1, n - 1] = oldpivot
return P, L, DD, U
def LUsolve(self, rhs, iszerofunc=_iszero):
"""Solve the linear system ``Ax = rhs`` for ``x`` where ``A = self``.
This is for symbolic matrices, for real or complex ones use
mpmath.lu_solve or mpmath.qr_solve.
See Also
========
lower_triangular_solve
upper_triangular_solve
gauss_jordan_solve
cholesky_solve
diagonal_solve
LDLsolve
QRsolve
pinv_solve
LUdecomposition
"""
if rhs.rows != self.rows:
raise ShapeError(
"``self`` and ``rhs`` must have the same number of rows.")
m = self.rows
n = self.cols
if m < n:
raise NotImplementedError("Underdetermined systems not supported.")
try:
A, perm = self.LUdecomposition_Simple(
iszerofunc=_iszero, rankcheck=True)
except ValueError:
raise NotImplementedError("Underdetermined systems not supported.")
b = rhs.permute_rows(perm).as_mutable()
# forward substitution, all diag entries are scaled to 1
for i in range(m):
for j in range(min(i, n)):
scale = A[i, j]
b.zip_row_op(i, j, lambda x, y: x - y * scale)
# consistency check for overdetermined systems
if m > n:
for i in range(n, m):
for j in range(b.cols):
if not iszerofunc(b[i, j]):
raise ValueError("The system is inconsistent.")
b = b[0:n, :] # truncate zero rows if consistent
# backward substitution
for i in range(n - 1, -1, -1):
for j in range(i + 1, n):
scale = A[i, j]
b.zip_row_op(i, j, lambda x, y: x - y * scale)
scale = A[i, i]
b.row_op(i, lambda x, _: x / scale)
return rhs.__class__(b)
def multiply(self, b):
"""Returns ``self*b``
See Also
========
dot
cross
multiply_elementwise
"""
return self * b
def normalized(self, iszerofunc=_iszero):
"""Return the normalized version of ``self``.
Parameters
==========
iszerofunc : Function, optional
A function to determine whether ``self`` is a zero vector.
The default ``_iszero`` tests to see if each element is
exactly zero.
Returns
=======
Matrix
Normalized vector form of ``self``.
It has the same length as a unit vector. However, a zero vector
will be returned for a vector with norm 0.
Raises
======
ShapeError
If the matrix is not in a vector form.
See Also
========
norm
"""
if self.rows != 1 and self.cols != 1:
raise ShapeError("A Matrix must be a vector to normalize.")
norm = self.norm()
if iszerofunc(norm):
out = self.zeros(self.rows, self.cols)
else:
out = self.applyfunc(lambda i: i / norm)
return out
def norm(self, ord=None):
"""Return the Norm of a Matrix or Vector.
In the simplest case this is the geometric size of the vector
Other norms can be specified by the ord parameter
===== ============================ ==========================
ord norm for matrices norm for vectors
===== ============================ ==========================
None Frobenius norm 2-norm
'fro' Frobenius norm - does not exist
inf maximum row sum max(abs(x))
-inf -- min(abs(x))
1 maximum column sum as below
-1 -- as below
2 2-norm (largest sing. value) as below
-2 smallest singular value as below
other - does not exist sum(abs(x)**ord)**(1./ord)
===== ============================ ==========================
Examples
========
>>> from sympy import Matrix, Symbol, trigsimp, cos, sin, oo
>>> x = Symbol('x', real=True)
>>> v = Matrix([cos(x), sin(x)])
>>> trigsimp( v.norm() )
1
>>> v.norm(10)
(sin(x)**10 + cos(x)**10)**(1/10)
>>> A = Matrix([[1, 1], [1, 1]])
>>> A.norm(1) # maximum sum of absolute values of A is 2
2
>>> A.norm(2) # Spectral norm (max of |Ax|/|x| under 2-vector-norm)
2
>>> A.norm(-2) # Inverse spectral norm (smallest singular value)
0
>>> A.norm() # Frobenius Norm
2
>>> A.norm(oo) # Infinity Norm
2
>>> Matrix([1, -2]).norm(oo)
2
>>> Matrix([-1, 2]).norm(-oo)
1
See Also
========
normalized
"""
# Row or Column Vector Norms
vals = list(self.values()) or [0]
if self.rows == 1 or self.cols == 1:
if ord == 2 or ord is None: # Common case sqrt(<x, x>)
return sqrt(Add(*(abs(i) ** 2 for i in vals)))
elif ord == 1: # sum(abs(x))
return Add(*(abs(i) for i in vals))
elif ord == S.Infinity: # max(abs(x))
return Max(*[abs(i) for i in vals])
elif ord == S.NegativeInfinity: # min(abs(x))
return Min(*[abs(i) for i in vals])
# Otherwise generalize the 2-norm, Sum(x_i**ord)**(1/ord)
# Note that while useful this is not mathematically a norm
try:
return Pow(Add(*(abs(i) ** ord for i in vals)), S(1) / ord)
except (NotImplementedError, TypeError):
raise ValueError("Expected order to be Number, Symbol, oo")
# Matrix Norms
else:
if ord == 1: # Maximum column sum
m = self.applyfunc(abs)
return Max(*[sum(m.col(i)) for i in range(m.cols)])
elif ord == 2: # Spectral Norm
# Maximum singular value
return Max(*self.singular_values())
elif ord == -2:
# Minimum singular value
return Min(*self.singular_values())
elif ord == S.Infinity: # Infinity Norm - Maximum row sum
m = self.applyfunc(abs)
return Max(*[sum(m.row(i)) for i in range(m.rows)])
elif (ord is None or isinstance(ord,
string_types) and ord.lower() in
['f', 'fro', 'frobenius', 'vector']):
# Reshape as vector and send back to norm function
return self.vec().norm(ord=2)
else:
raise NotImplementedError("Matrix Norms under development")
def pinv_solve(self, B, arbitrary_matrix=None):
"""Solve ``Ax = B`` using the Moore-Penrose pseudoinverse.
There may be zero, one, or infinite solutions. If one solution
exists, it will be returned. If infinite solutions exist, one will
be returned based on the value of arbitrary_matrix. If no solutions
exist, the least-squares solution is returned.
Parameters
==========
B : Matrix
The right hand side of the equation to be solved for. Must have
the same number of rows as matrix A.
arbitrary_matrix : Matrix
If the system is underdetermined (e.g. A has more columns than
rows), infinite solutions are possible, in terms of an arbitrary
matrix. This parameter may be set to a specific matrix to use
for that purpose; if so, it must be the same shape as x, with as
many rows as matrix A has columns, and as many columns as matrix
B. If left as None, an appropriate matrix containing dummy
symbols in the form of ``wn_m`` will be used, with n and m being
row and column position of each symbol.
Returns
=======
x : Matrix
The matrix that will satisfy ``Ax = B``. Will have as many rows as
matrix A has columns, and as many columns as matrix B.
Examples
========
>>> from sympy import Matrix
>>> A = Matrix([[1, 2, 3], [4, 5, 6]])
>>> B = Matrix([7, 8])
>>> A.pinv_solve(B)
Matrix([
[ _w0_0/6 - _w1_0/3 + _w2_0/6 - 55/18],
[-_w0_0/3 + 2*_w1_0/3 - _w2_0/3 + 1/9],
[ _w0_0/6 - _w1_0/3 + _w2_0/6 + 59/18]])
>>> A.pinv_solve(B, arbitrary_matrix=Matrix([0, 0, 0]))
Matrix([
[-55/18],
[ 1/9],
[ 59/18]])
See Also
========
lower_triangular_solve
upper_triangular_solve
gauss_jordan_solve
cholesky_solve
diagonal_solve
LDLsolve
LUsolve
QRsolve
pinv
Notes
=====
This may return either exact solutions or least squares solutions.
To determine which, check ``A * A.pinv() * B == B``. It will be
True if exact solutions exist, and False if only a least-squares
solution exists. Be aware that the left hand side of that equation
may need to be simplified to correctly compare to the right hand
side.
References
==========
.. [1] https://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse#Obtaining_all_solutions_of_a_linear_system
"""
from sympy.matrices import eye
A = self
A_pinv = self.pinv()
if arbitrary_matrix is None:
rows, cols = A.cols, B.cols
w = symbols('w:{0}_:{1}'.format(rows, cols), cls=Dummy)
arbitrary_matrix = self.__class__(cols, rows, w).T
return A_pinv * B + (eye(A.cols) - A_pinv * A) * arbitrary_matrix
def pinv(self):
"""Calculate the Moore-Penrose pseudoinverse of the matrix.
The Moore-Penrose pseudoinverse exists and is unique for any matrix.
If the matrix is invertible, the pseudoinverse is the same as the
inverse.
Examples
========
>>> from sympy import Matrix
>>> Matrix([[1, 2, 3], [4, 5, 6]]).pinv()
Matrix([
[-17/18, 4/9],
[ -1/9, 1/9],
[ 13/18, -2/9]])
See Also
========
inv
pinv_solve
References
==========
.. [1] https://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse
"""
A = self
AH = self.H
# Trivial case: pseudoinverse of all-zero matrix is its transpose.
if A.is_zero:
return AH
try:
if self.rows >= self.cols:
return (AH * A).inv() * AH
else:
return AH * (A * AH).inv()
except ValueError:
# Matrix is not full rank, so A*AH cannot be inverted.
pass
try:
# However, A*AH is Hermitian, so we can diagonalize it.
if self.rows >= self.cols:
P, D = (AH * A).diagonalize(normalize=True)
D_pinv = D.applyfunc(lambda x: 0 if _iszero(x) else 1 / x)
return P * D_pinv * P.H * AH
else:
P, D = (A * AH).diagonalize(normalize=True)
D_pinv = D.applyfunc(lambda x: 0 if _iszero(x) else 1 / x)
return AH * P * D_pinv * P.H
except MatrixError:
raise NotImplementedError('pinv for rank-deficient matrices where diagonalization '
'of A.H*A fails is not supported yet.')
def print_nonzero(self, symb="X"):
"""Shows location of non-zero entries for fast shape lookup.
Examples
========
>>> from sympy.matrices import Matrix, eye
>>> m = Matrix(2, 3, lambda i, j: i*3+j)
>>> m
Matrix([
[0, 1, 2],
[3, 4, 5]])
>>> m.print_nonzero()
[ XX]
[XXX]
>>> m = eye(4)
>>> m.print_nonzero("x")
[x ]
[ x ]
[ x ]
[ x]
"""
s = []
for i in range(self.rows):
line = []
for j in range(self.cols):
if self[i, j] == 0:
line.append(" ")
else:
line.append(str(symb))
s.append("[%s]" % ''.join(line))
print('\n'.join(s))
def project(self, v):
"""Return the projection of ``self`` onto the line containing ``v``.
Examples
========
>>> from sympy import Matrix, S, sqrt
>>> V = Matrix([sqrt(3)/2, S.Half])
>>> x = Matrix([[1, 0]])
>>> V.project(x)
Matrix([[sqrt(3)/2, 0]])
>>> V.project(-x)
Matrix([[sqrt(3)/2, 0]])
"""
return v * (self.dot(v) / v.dot(v))
def QRdecomposition(self):
"""Return Q, R where A = Q*R, Q is orthogonal and R is upper triangular.
Examples
========
This is the example from wikipedia:
>>> from sympy import Matrix
>>> A = Matrix([[12, -51, 4], [6, 167, -68], [-4, 24, -41]])
>>> Q, R = A.QRdecomposition()
>>> Q
Matrix([
[ 6/7, -69/175, -58/175],
[ 3/7, 158/175, 6/175],
[-2/7, 6/35, -33/35]])
>>> R
Matrix([
[14, 21, -14],
[ 0, 175, -70],
[ 0, 0, 35]])
>>> A == Q*R
True
QR factorization of an identity matrix:
>>> A = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
>>> Q, R = A.QRdecomposition()
>>> Q
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> R
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
See Also
========
cholesky
LDLdecomposition
LUdecomposition
QRsolve
"""
cls = self.__class__
mat = self.as_mutable()
n = mat.rows
m = mat.cols
ranked = list()
# Pad with additional rows to make wide matrices square
# nOrig keeps track of original size so zeros can be trimmed from Q
if n < m:
nOrig = n
n = m
mat = mat.col_join(mat.zeros(n - nOrig, m))
else:
nOrig = n
Q, R = mat.zeros(n, m), mat.zeros(m)
for j in range(m): # for each column vector
tmp = mat[:, j] # take original v
for i in range(j):
# subtract the project of mat on new vector
R[i, j] = Q[:, i].dot(mat[:, j])
tmp -= Q[:, i] * R[i, j]
tmp.expand()
# normalize it
R[j, j] = tmp.norm()
if not R[j, j].is_zero:
ranked.append(j)
Q[:, j] = tmp / R[j, j]
if len(ranked) != 0:
return (
cls(Q.extract(range(nOrig), ranked)),
cls(R.extract(ranked, range(R.cols)))
)
else:
# Trivial case handling for zero-rank matrix
# Force Q as matrix containing standard basis vectors
for i in range(Min(nOrig, m)):
Q[i, i] = 1
return (
cls(Q.extract(range(nOrig), range(Min(nOrig, m)))),
cls(R.extract(range(Min(nOrig, m)), range(R.cols)))
)
def QRsolve(self, b):
"""Solve the linear system ``Ax = b``.
``self`` is the matrix ``A``, the method argument is the vector
``b``. The method returns the solution vector ``x``. If ``b`` is a
matrix, the system is solved for each column of ``b`` and the
return value is a matrix of the same shape as ``b``.
This method is slower (approximately by a factor of 2) but
more stable for floating-point arithmetic than the LUsolve method.
However, LUsolve usually uses an exact arithmetic, so you don't need
to use QRsolve.
This is mainly for educational purposes and symbolic matrices, for real
(or complex) matrices use mpmath.qr_solve.
See Also
========
lower_triangular_solve
upper_triangular_solve
gauss_jordan_solve
cholesky_solve
diagonal_solve
LDLsolve
LUsolve
pinv_solve
QRdecomposition
"""
Q, R = self.as_mutable().QRdecomposition()
y = Q.T * b
# back substitution to solve R*x = y:
# We build up the result "backwards" in the vector 'x' and reverse it
# only in the end.
x = []
n = R.rows
for j in range(n - 1, -1, -1):
tmp = y[j, :]
for k in range(j + 1, n):
tmp -= R[j, k] * x[n - 1 - k]
x.append(tmp / R[j, j])
return self._new([row._mat for row in reversed(x)])
def rank_decomposition(self, iszerofunc=_iszero, simplify=False):
r"""Returns a pair of matrices (`C`, `F`) with matching rank
such that `A = C F`.
Parameters
==========
iszerofunc : Function, optional
A function used for detecting whether an element can
act as a pivot. ``lambda x: x.is_zero`` is used by default.
simplify : Bool or Function, optional
A function used to simplify elements when looking for a
pivot. By default SymPy's ``simplify`` is used.
Returns
=======
(C, F) : Matrices
`C` and `F` are full-rank matrices with rank as same as `A`,
whose product gives `A`.
See Notes for additional mathematical details.
Examples
========
>>> from sympy.matrices import Matrix
>>> A = Matrix([
... [1, 3, 1, 4],
... [2, 7, 3, 9],
... [1, 5, 3, 1],
... [1, 2, 0, 8]
... ])
>>> C, F = A.rank_decomposition()
>>> C
Matrix([
[1, 3, 4],
[2, 7, 9],
[1, 5, 1],
[1, 2, 8]])
>>> F
Matrix([
[1, 0, -2, 0],
[0, 1, 1, 0],
[0, 0, 0, 1]])
>>> C * F == A
True
Notes
=====
Obtaining `F`, an RREF of `A`, is equivalent to creating a
product
.. math::
E_n E_{n-1} ... E_1 A = F
where `E_n, E_{n-1}, ... , E_1` are the elimination matrices or
permutation matrices equivalent to each row-reduction step.
The inverse of the same product of elimination matrices gives
`C`:
.. math::
C = (E_n E_{n-1} ... E_1)^{-1}
It is not necessary, however, to actually compute the inverse:
the columns of `C` are those from the original matrix with the
same column indices as the indices of the pivot columns of `F`.
References
==========
.. [1] https://en.wikipedia.org/wiki/Rank_factorization
.. [2] Piziak, R.; Odell, P. L. (1 June 1999).
"Full Rank Factorization of Matrices".
Mathematics Magazine. 72 (3): 193. doi:10.2307/2690882
See Also
========
rref
"""
(F, pivot_cols) = self.rref(
simplify=simplify, iszerofunc=iszerofunc, pivots=True)
rank = len(pivot_cols)
C = self.extract(range(self.rows), pivot_cols)
F = F[:rank, :]
return (C, F)
def solve_least_squares(self, rhs, method='CH'):
"""Return the least-square fit to the data.
Parameters
==========
rhs : Matrix
Vector representing the right hand side of the linear equation.
method : string or boolean, optional
If set to ``'CH'``, ``cholesky_solve`` routine will be used.
If set to ``'LDL'``, ``LDLsolve`` routine will be used.
If set to ``'QR'``, ``QRsolve`` routine will be used.
If set to ``'PINV'``, ``pinv_solve`` routine will be used.
Otherwise, the conjugate of ``self`` will be used to create a system
of equations that is passed to ``solve`` along with the hint
defined by ``method``.
Returns
=======
solutions : Matrix
Vector representing the solution.
Examples
========
>>> from sympy.matrices import Matrix, ones
>>> A = Matrix([1, 2, 3])
>>> B = Matrix([2, 3, 4])
>>> S = Matrix(A.row_join(B))
>>> S
Matrix([
[1, 2],
[2, 3],
[3, 4]])
If each line of S represent coefficients of Ax + By
and x and y are [2, 3] then S*xy is:
>>> r = S*Matrix([2, 3]); r
Matrix([
[ 8],
[13],
[18]])
But let's add 1 to the middle value and then solve for the
least-squares value of xy:
>>> xy = S.solve_least_squares(Matrix([8, 14, 18])); xy
Matrix([
[ 5/3],
[10/3]])
The error is given by S*xy - r:
>>> S*xy - r
Matrix([
[1/3],
[1/3],
[1/3]])
>>> _.norm().n(2)
0.58
If a different xy is used, the norm will be higher:
>>> xy += ones(2, 1)/10
>>> (S*xy - r).norm().n(2)
1.5
"""
if method == 'CH':
return self.cholesky_solve(rhs)
elif method == 'QR':
return self.QRsolve(rhs)
elif method == 'LDL':
return self.LDLsolve(rhs)
elif method == 'PINV':
return self.pinv_solve(rhs)
else:
t = self.H
return (t * self).solve(t * rhs, method=method)
def solve(self, rhs, method='GJ'):
"""Solves linear equation where the unique solution exists.
Parameters
==========
rhs : Matrix
Vector representing the right hand side of the linear equation.
method : string, optional
If set to ``'GJ'``, the Gauss-Jordan elimination will be used, which
is implemented in the routine ``gauss_jordan_solve``.
If set to ``'LU'``, ``LUsolve`` routine will be used.
If set to ``'QR'``, ``QRsolve`` routine will be used.
If set to ``'PINV'``, ``pinv_solve`` routine will be used.
It also supports the methods available for special linear systems
For positive definite systems:
If set to ``'CH'``, ``cholesky_solve`` routine will be used.
If set to ``'LDL'``, ``LDLsolve`` routine will be used.
To use a different method and to compute the solution via the
inverse, use a method defined in the .inv() docstring.
Returns
=======
solutions : Matrix
Vector representing the solution.
Raises
======
ValueError
If there is not a unique solution then a ``ValueError`` will be
raised.
If ``self`` is not square, a ``ValueError`` and a different routine
for solving the system will be suggested.
"""
if method == 'GJ':
try:
soln, param = self.gauss_jordan_solve(rhs)
if param:
raise ValueError("Matrix det == 0; not invertible. "
"Try ``self.gauss_jordan_solve(rhs)`` to obtain a parametric solution.")
except ValueError:
# raise same error as in inv:
self.zeros(1).inv()
return soln
elif method == 'LU':
return self.LUsolve(rhs)
elif method == 'CH':
return self.cholesky_solve(rhs)
elif method == 'QR':
return self.QRsolve(rhs)
elif method == 'LDL':
return self.LDLsolve(rhs)
elif method == 'PINV':
return self.pinv_solve(rhs)
else:
return self.inv(method=method)*rhs
def table(self, printer, rowstart='[', rowend=']', rowsep='\n',
colsep=', ', align='right'):
r"""
String form of Matrix as a table.
``printer`` is the printer to use for on the elements (generally
something like StrPrinter())
``rowstart`` is the string used to start each row (by default '[').
``rowend`` is the string used to end each row (by default ']').
``rowsep`` is the string used to separate rows (by default a newline).
``colsep`` is the string used to separate columns (by default ', ').
``align`` defines how the elements are aligned. Must be one of 'left',
'right', or 'center'. You can also use '<', '>', and '^' to mean the
same thing, respectively.
This is used by the string printer for Matrix.
Examples
========
>>> from sympy import Matrix
>>> from sympy.printing.str import StrPrinter
>>> M = Matrix([[1, 2], [-33, 4]])
>>> printer = StrPrinter()
>>> M.table(printer)
'[ 1, 2]\n[-33, 4]'
>>> print(M.table(printer))
[ 1, 2]
[-33, 4]
>>> print(M.table(printer, rowsep=',\n'))
[ 1, 2],
[-33, 4]
>>> print('[%s]' % M.table(printer, rowsep=',\n'))
[[ 1, 2],
[-33, 4]]
>>> print(M.table(printer, colsep=' '))
[ 1 2]
[-33 4]
>>> print(M.table(printer, align='center'))
[ 1 , 2]
[-33, 4]
>>> print(M.table(printer, rowstart='{', rowend='}'))
{ 1, 2}
{-33, 4}
"""
# Handle zero dimensions:
if self.rows == 0 or self.cols == 0:
return '[]'
# Build table of string representations of the elements
res = []
# Track per-column max lengths for pretty alignment
maxlen = [0] * self.cols
for i in range(self.rows):
res.append([])
for j in range(self.cols):
s = printer._print(self[i, j])
res[-1].append(s)
maxlen[j] = max(len(s), maxlen[j])
# Patch strings together
align = {
'left': 'ljust',
'right': 'rjust',
'center': 'center',
'<': 'ljust',
'>': 'rjust',
'^': 'center',
}[align]
for i, row in enumerate(res):
for j, elem in enumerate(row):
row[j] = getattr(elem, align)(maxlen[j])
res[i] = rowstart + colsep.join(row) + rowend
return rowsep.join(res)
def upper_triangular_solve(self, rhs):
"""Solves ``Ax = B``, where A is an upper triangular matrix.
See Also
========
lower_triangular_solve
gauss_jordan_solve
cholesky_solve
diagonal_solve
LDLsolve
LUsolve
QRsolve
pinv_solve
"""
if not self.is_square:
raise NonSquareMatrixError("Matrix must be square.")
if rhs.rows != self.rows:
raise TypeError("Matrix size mismatch.")
if not self.is_upper:
raise TypeError("Matrix is not upper triangular.")
return self._upper_triangular_solve(rhs)
def vech(self, diagonal=True, check_symmetry=True):
"""Return the unique elements of a symmetric Matrix as a one column matrix
by stacking the elements in the lower triangle.
Arguments:
diagonal -- include the diagonal cells of ``self`` or not
check_symmetry -- checks symmetry of ``self`` but not completely reliably
Examples
========
>>> from sympy import Matrix
>>> m=Matrix([[1, 2], [2, 3]])
>>> m
Matrix([
[1, 2],
[2, 3]])
>>> m.vech()
Matrix([
[1],
[2],
[3]])
>>> m.vech(diagonal=False)
Matrix([[2]])
See Also
========
vec
"""
from sympy.matrices import zeros
c = self.cols
if c != self.rows:
raise ShapeError("Matrix must be square")
if check_symmetry:
self.simplify()
if self != self.transpose():
raise ValueError(
"Matrix appears to be asymmetric; consider check_symmetry=False")
count = 0
if diagonal:
v = zeros(c * (c + 1) // 2, 1)
for j in range(c):
for i in range(j, c):
v[count] = self[i, j]
count += 1
else:
v = zeros(c * (c - 1) // 2, 1)
for j in range(c):
for i in range(j + 1, c):
v[count] = self[i, j]
count += 1
return v
@deprecated(
issue=15109,
useinstead="from sympy.matrices.common import classof",
deprecated_since_version="1.3")
def classof(A, B):
from sympy.matrices.common import classof as classof_
return classof_(A, B)
@deprecated(
issue=15109,
deprecated_since_version="1.3",
useinstead="from sympy.matrices.common import a2idx")
def a2idx(j, n=None):
from sympy.matrices.common import a2idx as a2idx_
return a2idx_(j, n)
def _find_reasonable_pivot(col, iszerofunc=_iszero, simpfunc=_simplify):
""" Find the lowest index of an item in ``col`` that is
suitable for a pivot. If ``col`` consists only of
Floats, the pivot with the largest norm is returned.
Otherwise, the first element where ``iszerofunc`` returns
False is used. If ``iszerofunc`` doesn't return false,
items are simplified and retested until a suitable
pivot is found.
Returns a 4-tuple
(pivot_offset, pivot_val, assumed_nonzero, newly_determined)
where pivot_offset is the index of the pivot, pivot_val is
the (possibly simplified) value of the pivot, assumed_nonzero
is True if an assumption that the pivot was non-zero
was made without being proved, and newly_determined are
elements that were simplified during the process of pivot
finding."""
newly_determined = []
col = list(col)
# a column that contains a mix of floats and integers
# but at least one float is considered a numerical
# column, and so we do partial pivoting
if all(isinstance(x, (Float, Integer)) for x in col) and any(
isinstance(x, Float) for x in col):
col_abs = [abs(x) for x in col]
max_value = max(col_abs)
if iszerofunc(max_value):
# just because iszerofunc returned True, doesn't
# mean the value is numerically zero. Make sure
# to replace all entries with numerical zeros
if max_value != 0:
newly_determined = [(i, 0) for i, x in enumerate(col) if x != 0]
return (None, None, False, newly_determined)
index = col_abs.index(max_value)
return (index, col[index], False, newly_determined)
# PASS 1 (iszerofunc directly)
possible_zeros = []
for i, x in enumerate(col):
is_zero = iszerofunc(x)
# is someone wrote a custom iszerofunc, it may return
# BooleanFalse or BooleanTrue instead of True or False,
# so use == for comparison instead of `is`
if is_zero == False:
# we found something that is definitely not zero
return (i, x, False, newly_determined)
possible_zeros.append(is_zero)
# by this point, we've found no certain non-zeros
if all(possible_zeros):
# if everything is definitely zero, we have
# no pivot
return (None, None, False, newly_determined)
# PASS 2 (iszerofunc after simplify)
# we haven't found any for-sure non-zeros, so
# go through the elements iszerofunc couldn't
# make a determination about and opportunistically
# simplify to see if we find something
for i, x in enumerate(col):
if possible_zeros[i] is not None:
continue
simped = simpfunc(x)
is_zero = iszerofunc(simped)
if is_zero == True or is_zero == False:
newly_determined.append((i, simped))
if is_zero == False:
return (i, simped, False, newly_determined)
possible_zeros[i] = is_zero
# after simplifying, some things that were recognized
# as zeros might be zeros
if all(possible_zeros):
# if everything is definitely zero, we have
# no pivot
return (None, None, False, newly_determined)
# PASS 3 (.equals(0))
# some expressions fail to simplify to zero, but
# ``.equals(0)`` evaluates to True. As a last-ditch
# attempt, apply ``.equals`` to these expressions
for i, x in enumerate(col):
if possible_zeros[i] is not None:
continue
if x.equals(S.Zero):
# ``.iszero`` may return False with
# an implicit assumption (e.g., ``x.equals(0)``
# when ``x`` is a symbol), so only treat it
# as proved when ``.equals(0)`` returns True
possible_zeros[i] = True
newly_determined.append((i, S.Zero))
if all(possible_zeros):
return (None, None, False, newly_determined)
# at this point there is nothing that could definitely
# be a pivot. To maintain compatibility with existing
# behavior, we'll assume that an illdetermined thing is
# non-zero. We should probably raise a warning in this case
i = possible_zeros.index(None)
return (i, col[i], True, newly_determined)
def _find_reasonable_pivot_naive(col, iszerofunc=_iszero, simpfunc=None):
"""
Helper that computes the pivot value and location from a
sequence of contiguous matrix column elements. As a side effect
of the pivot search, this function may simplify some of the elements
of the input column. A list of these simplified entries and their
indices are also returned.
This function mimics the behavior of _find_reasonable_pivot(),
but does less work trying to determine if an indeterminate candidate
pivot simplifies to zero. This more naive approach can be much faster,
with the trade-off that it may erroneously return a pivot that is zero.
``col`` is a sequence of contiguous column entries to be searched for
a suitable pivot.
``iszerofunc`` is a callable that returns a Boolean that indicates
if its input is zero, or None if no such determination can be made.
``simpfunc`` is a callable that simplifies its input. It must return
its input if it does not simplify its input. Passing in
``simpfunc=None`` indicates that the pivot search should not attempt
to simplify any candidate pivots.
Returns a 4-tuple:
(pivot_offset, pivot_val, assumed_nonzero, newly_determined)
``pivot_offset`` is the sequence index of the pivot.
``pivot_val`` is the value of the pivot.
pivot_val and col[pivot_index] are equivalent, but will be different
when col[pivot_index] was simplified during the pivot search.
``assumed_nonzero`` is a boolean indicating if the pivot cannot be
guaranteed to be zero. If assumed_nonzero is true, then the pivot
may or may not be non-zero. If assumed_nonzero is false, then
the pivot is non-zero.
``newly_determined`` is a list of index-value pairs of pivot candidates
that were simplified during the pivot search.
"""
# indeterminates holds the index-value pairs of each pivot candidate
# that is neither zero or non-zero, as determined by iszerofunc().
# If iszerofunc() indicates that a candidate pivot is guaranteed
# non-zero, or that every candidate pivot is zero then the contents
# of indeterminates are unused.
# Otherwise, the only viable candidate pivots are symbolic.
# In this case, indeterminates will have at least one entry,
# and all but the first entry are ignored when simpfunc is None.
indeterminates = []
for i, col_val in enumerate(col):
col_val_is_zero = iszerofunc(col_val)
if col_val_is_zero == False:
# This pivot candidate is non-zero.
return i, col_val, False, []
elif col_val_is_zero is None:
# The candidate pivot's comparison with zero
# is indeterminate.
indeterminates.append((i, col_val))
if len(indeterminates) == 0:
# All candidate pivots are guaranteed to be zero, i.e. there is
# no pivot.
return None, None, False, []
if simpfunc is None:
# Caller did not pass in a simplification function that might
# determine if an indeterminate pivot candidate is guaranteed
# to be nonzero, so assume the first indeterminate candidate
# is non-zero.
return indeterminates[0][0], indeterminates[0][1], True, []
# newly_determined holds index-value pairs of candidate pivots
# that were simplified during the search for a non-zero pivot.
newly_determined = []
for i, col_val in indeterminates:
tmp_col_val = simpfunc(col_val)
if id(col_val) != id(tmp_col_val):
# simpfunc() simplified this candidate pivot.
newly_determined.append((i, tmp_col_val))
if iszerofunc(tmp_col_val) == False:
# Candidate pivot simplified to a guaranteed non-zero value.
return i, tmp_col_val, False, newly_determined
return indeterminates[0][0], indeterminates[0][1], True, newly_determined
|
b8996c9b47ae25dce5aa6ac7ce8215a06ef88700a0084c248afdf102a4f95bb3 | from __future__ import division, print_function
from sympy.core.compatibility import range, as_int
from sympy.utilities.iterables import is_sequence
from sympy.utilities.misc import filldedent
from .sparse import SparseMatrix
def _doktocsr(dok):
"""Converts a sparse matrix to Compressed Sparse Row (CSR) format.
Parameters
==========
A : contains non-zero elements sorted by key (row, column)
JA : JA[i] is the column corresponding to A[i]
IA : IA[i] contains the index in A for the first non-zero element
of row[i]. Thus IA[i+1] - IA[i] gives number of non-zero
elements row[i]. The length of IA is always 1 more than the
number of rows in the matrix.
"""
row, JA, A = [list(i) for i in zip(*dok.row_list())]
IA = [0]*((row[0] if row else 0) + 1)
for i, r in enumerate(row):
IA.extend([i]*(r - row[i - 1])) # if i = 0 nothing is extended
IA.extend([len(A)]*(dok.rows - len(IA) + 1))
shape = [dok.rows, dok.cols]
return [A, JA, IA, shape]
def _csrtodok(csr):
"""Converts a CSR representation to DOK representation"""
smat = {}
A, JA, IA, shape = csr
for i in range(len(IA) - 1):
indices = slice(IA[i], IA[i + 1])
for l, m in zip(A[indices], JA[indices]):
smat[i, m] = l
return SparseMatrix(*(shape + [smat]))
def banded(*args, **kwargs):
"""Returns a SparseMatrix from the given dictionary describing
the diagonals of the matrix. The keys are positive for upper
diagonals and negative for those below the main diagonal. The
values may be:
* expressions or single-argument functions,
* lists or tuples of values,
* matrices
Unless dimensions are given, the size of the returned matrix will
be large enough to contain the largest non-zero value provided.
kwargs
======
rows : rows of the resulting matrix; computed if
not given.
cols : columns of the resulting matrix; computed if
not given.
Examples
========
>>> from sympy import banded, ones, Matrix
>>> from sympy.abc import x
If explicit values are given in tuples,
the matrix will autosize to contain all values, otherwise
a single value is filled onto the entire diagonal:
>>> banded({1: (1, 2, 3), -1: (4, 5, 6), 0: x})
Matrix([
[x, 1, 0, 0],
[4, x, 2, 0],
[0, 5, x, 3],
[0, 0, 6, x]])
A function accepting a single argument can be used to fill the
diagonal as a function of diagonal index (which starts at 0).
The size (or shape) of the matrix must be given to obtain more
than a 1x1 matrix:
>>> s = lambda d: (1 + d)**2
>>> banded(5, {0: s, 2: s, -2: 2})
Matrix([
[1, 0, 1, 0, 0],
[0, 4, 0, 4, 0],
[2, 0, 9, 0, 9],
[0, 2, 0, 16, 0],
[0, 0, 2, 0, 25]])
The diagonal of matrices placed on a diagonal will coincide
with the indicated diagonal:
>>> vert = Matrix([1, 2, 3])
>>> banded({0: vert}, cols=3)
Matrix([
[1, 0, 0],
[2, 1, 0],
[3, 2, 1],
[0, 3, 2],
[0, 0, 3]])
>>> banded(4, {0: ones(2)})
Matrix([
[1, 1, 0, 0],
[1, 1, 0, 0],
[0, 0, 1, 1],
[0, 0, 1, 1]])
Errors are raised if the designated size will not hold
all values an integral number of times. Here, the rows
are designated as odd (but an even number is required to
hold the off-diagonal 2x2 ones):
>>> banded({0: 2, 1: ones(2)}, rows=5)
Traceback (most recent call last):
...
ValueError:
sequence does not fit an integral number of times in the matrix
And here, an even number of rows is given...but the square
matrix has an even number of columns, too. As we saw
in the previous example, an odd number is required:
>>> banded(4, {0: 2, 1: ones(2)}) # trying to make 4x4 and cols must be odd
Traceback (most recent call last):
...
ValueError:
sequence does not fit an integral number of times in the matrix
A way around having to count rows is to enclosing matrix elements
in a tuple and indicate the desired number of them to the right:
>>> banded({0: 2, 2: (ones(2),)*3})
Matrix([
[2, 0, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 1, 0, 0, 0, 0],
[0, 0, 2, 0, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 0, 2, 0, 1, 1],
[0, 0, 0, 0, 0, 2, 1, 1]])
An error will be raised if more than one value
is written to a given entry. Here, the ones overlap
with the main diagonal if they are placed on the
first diagonal:
>>> banded({0: (2,)*5, 1: (ones(2),)*3})
Traceback (most recent call last):
...
ValueError: collision at (1, 1)
By placing a 0 at the bottom left of the 2x2 matrix of
ones, the collision is avoided:
>>> u2 = Matrix([
... [1, 1],
... [0, 1]])
>>> banded({0: [2]*5, 1: [u2]*3})
Matrix([
[2, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 0, 0, 0, 0],
[0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 0, 0],
[0, 0, 0, 0, 2, 1, 1],
[0, 0, 0, 0, 0, 0, 1]])
"""
from sympy import Dict, Dummy, SparseMatrix
try:
if len(args) not in (1, 2, 3):
raise TypeError
if not isinstance(args[-1], (dict, Dict)):
raise TypeError
if len(args) == 1:
rows = kwargs.get('rows', None)
cols = kwargs.get('cols', None)
if rows is not None:
rows = as_int(rows)
if cols is not None:
cols = as_int(cols)
elif len(args) == 2:
rows = cols = as_int(args[0])
else:
rows, cols = map(as_int, args[:2])
# fails with ValueError if any keys are not ints
_ = all(as_int(k) for k in args[-1])
except (ValueError, TypeError):
raise TypeError(filldedent(
'''unrecognized input to banded:
expecting [[row,] col,] {int: value}'''))
def rc(d):
# return row,col coord of diagonal start
r = -d if d < 0 else 0
c = 0 if r else d
return r, c
smat = {}
undone = []
tba = Dummy()
# first handle objects with size
for d, v in args[-1].items():
r, c = rc(d)
# note: only list and tuple are recognized since this
# will allow other Basic objects like Tuple
# into the matrix if so desired
if isinstance(v, (list, tuple)):
extra = 0
for i, vi in enumerate(v):
i += extra
if is_sequence(vi):
vi = SparseMatrix(vi)
smat[r + i, c + i] = vi
extra += min(vi.shape) - 1
else:
smat[r + i, c + i] = vi
elif is_sequence(v):
v = SparseMatrix(v)
rv, cv = v.shape
if rows and cols:
nr, xr = divmod(rows - r, rv)
nc, xc = divmod(cols - c, cv)
x = xr or xc
do = min(nr, nc)
elif rows:
do, x = divmod(rows - r, rv)
elif cols:
do, x = divmod(cols - c, cv)
else:
do = 1
x = 0
if x:
raise ValueError(filldedent('''
sequence does not fit an integral number of times
in the matrix'''))
j = min(v.shape)
for i in range(do):
smat[r, c] = v
r += j
c += j
elif v:
smat[r, c] = tba
undone.append((d, v))
s = SparseMatrix(None, smat) # to expand matrices
smat = s._smat
# check for dim errors here
if rows is not None and rows < s.rows:
raise ValueError('Designated rows %s < needed %s' % (rows, s.rows))
if cols is not None and cols < s.cols:
raise ValueError('Designated cols %s < needed %s' % (cols, s.cols))
if rows is cols is None:
rows = s.rows
cols = s.cols
elif rows is not None and cols is None:
cols = max(rows, s.cols)
elif cols is not None and rows is None:
rows = max(cols, s.rows)
def update(i, j, v):
# update smat and make sure there are
# no collisions
if v:
if (i, j) in smat and smat[i, j] not in (tba, v):
raise ValueError('collision at %s' % ((i, j),))
smat[i, j] = v
if undone:
for d, vi in undone:
r, c = rc(d)
v = vi if callable(vi) else lambda _: vi
i = 0
while r + i < rows and c + i < cols:
update(r + i, c + i, v(i))
i += 1
return SparseMatrix(rows, cols, smat)
|
ca79d859bba5a73c807f89f08b5aac95e6b64d6e8f310bf5734b71d887f0af9e | from .plot import plot_backends
from .plot_implicit import plot_implicit
from .textplot import textplot
from .pygletplot import PygletPlot
from .plot import PlotGrid
from .plot import (plot, plot_parametric, plot3d, plot3d_parametric_surface,
plot3d_parametric_line)
|
936783de5c2f5df8ceedbcc5c7d8935819bce9ab21ec21974695f463fdc2e721 | """Plotting module for Sympy.
A plot is represented by the ``Plot`` class that contains a reference to the
backend and a list of the data series to be plotted. The data series are
instances of classes meant to simplify getting points and meshes from sympy
expressions. ``plot_backends`` is a dictionary with all the backends.
This module gives only the essential. For all the fancy stuff use directly
the backend. You can get the backend wrapper for every plot from the
``_backend`` attribute. Moreover the data series classes have various useful
methods like ``get_points``, ``get_segments``, ``get_meshes``, etc, that may
be useful if you wish to use another plotting library.
Especially if you need publication ready graphs and this module is not enough
for you - just get the ``_backend`` attribute and add whatever you want
directly to it. In the case of matplotlib (the common way to graph data in
python) just copy ``_backend.fig`` which is the figure and ``_backend.ax``
which is the axis and work on them as you would on any other matplotlib object.
Simplicity of code takes much greater importance than performance. Don't use it
if you care at all about performance. A new backend instance is initialized
every time you call ``show()`` and the old one is left to the garbage collector.
"""
from __future__ import print_function, division
import warnings
from sympy import sympify, Expr, Tuple, Dummy, Symbol
from sympy.external import import_module
from sympy.core.function import arity
from sympy.core.compatibility import range, Callable
from sympy.utilities.iterables import is_sequence
from .experimental_lambdify import (vectorized_lambdify, lambdify)
# N.B.
# When changing the minimum module version for matplotlib, please change
# the same in the `SymPyDocTestFinder`` in `sympy/utilities/runtests.py`
# Backend specific imports - textplot
from sympy.plotting.textplot import textplot
# Global variable
# Set to False when running tests / doctests so that the plots don't show.
_show = True
def unset_show():
"""
Disable show(). For use in the tests.
"""
global _show
_show = False
##############################################################################
# The public interface
##############################################################################
class Plot(object):
"""The central class of the plotting module.
For interactive work the function ``plot`` is better suited.
This class permits the plotting of sympy expressions using numerous
backends (matplotlib, textplot, the old pyglet module for sympy, Google
charts api, etc).
The figure can contain an arbitrary number of plots of sympy expressions,
lists of coordinates of points, etc. Plot has a private attribute _series that
contains all data series to be plotted (expressions for lines or surfaces,
lists of points, etc (all subclasses of BaseSeries)). Those data series are
instances of classes not imported by ``from sympy import *``.
The customization of the figure is on two levels. Global options that
concern the figure as a whole (eg title, xlabel, scale, etc) and
per-data series options (eg name) and aesthetics (eg. color, point shape,
line type, etc.).
The difference between options and aesthetics is that an aesthetic can be
a function of the coordinates (or parameters in a parametric plot). The
supported values for an aesthetic are:
- None (the backend uses default values)
- a constant
- a function of one variable (the first coordinate or parameter)
- a function of two variables (the first and second coordinate or
parameters)
- a function of three variables (only in nonparametric 3D plots)
Their implementation depends on the backend so they may not work in some
backends.
If the plot is parametric and the arity of the aesthetic function permits
it the aesthetic is calculated over parameters and not over coordinates.
If the arity does not permit calculation over parameters the calculation is
done over coordinates.
Only cartesian coordinates are supported for the moment, but you can use
the parametric plots to plot in polar, spherical and cylindrical
coordinates.
The arguments for the constructor Plot must be subclasses of BaseSeries.
Any global option can be specified as a keyword argument.
The global options for a figure are:
- title : str
- xlabel : str
- ylabel : str
- legend : bool
- xscale : {'linear', 'log'}
- yscale : {'linear', 'log'}
- axis : bool
- axis_center : tuple of two floats or {'center', 'auto'}
- xlim : tuple of two floats
- ylim : tuple of two floats
- aspect_ratio : tuple of two floats or {'auto'}
- autoscale : bool
- margin : float in [0, 1]
The per data series options and aesthetics are:
There are none in the base series. See below for options for subclasses.
Some data series support additional aesthetics or options:
ListSeries, LineOver1DRangeSeries, Parametric2DLineSeries,
Parametric3DLineSeries support the following:
Aesthetics:
- line_color : function which returns a float.
options:
- label : str
- steps : bool
- integers_only : bool
SurfaceOver2DRangeSeries, ParametricSurfaceSeries support the following:
aesthetics:
- surface_color : function which returns a float.
"""
def __init__(self, *args, **kwargs):
super(Plot, self).__init__()
# Options for the graph as a whole.
# The possible values for each option are described in the docstring of
# Plot. They are based purely on convention, no checking is done.
self.title = None
self.xlabel = None
self.ylabel = None
self.aspect_ratio = 'auto'
self.xlim = None
self.ylim = None
self.axis_center = 'auto'
self.axis = True
self.xscale = 'linear'
self.yscale = 'linear'
self.legend = False
self.autoscale = True
self.margin = 0
# Contains the data objects to be plotted. The backend should be smart
# enough to iterate over this list.
self._series = []
self._series.extend(args)
# The backend type. On every show() a new backend instance is created
# in self._backend which is tightly coupled to the Plot instance
# (thanks to the parent attribute of the backend).
self.backend = DefaultBackend
# The keyword arguments should only contain options for the plot.
for key, val in kwargs.items():
if hasattr(self, key):
setattr(self, key, val)
def show(self):
# TODO move this to the backend (also for save)
if hasattr(self, '_backend'):
self._backend.close()
self._backend = self.backend(self)
self._backend.show()
def save(self, path):
if hasattr(self, '_backend'):
self._backend.close()
self._backend = self.backend(self)
self._backend.save(path)
def __str__(self):
series_strs = [('[%d]: ' % i) + str(s)
for i, s in enumerate(self._series)]
return 'Plot object containing:\n' + '\n'.join(series_strs)
def __getitem__(self, index):
return self._series[index]
def __setitem__(self, index, *args):
if len(args) == 1 and isinstance(args[0], BaseSeries):
self._series[index] = args
def __delitem__(self, index):
del self._series[index]
def append(self, arg):
"""Adds an element from a plot's series to an existing plot.
Examples
========
Consider two ``Plot`` objects, ``p1`` and ``p2``. To add the
second plot's first series object to the first, use the
``append`` method, like so:
.. plot::
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
>>> p1 = plot(x*x, show=False)
>>> p2 = plot(x, show=False)
>>> p1.append(p2[0])
>>> p1
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
[1]: cartesian line: x for x over (-10.0, 10.0)
>>> p1.show()
See Also
========
extend
"""
if isinstance(arg, BaseSeries):
self._series.append(arg)
else:
raise TypeError('Must specify element of plot to append.')
def extend(self, arg):
"""Adds all series from another plot.
Examples
========
Consider two ``Plot`` objects, ``p1`` and ``p2``. To add the
second plot to the first, use the ``extend`` method, like so:
.. plot::
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
>>> p1 = plot(x**2, show=False)
>>> p2 = plot(x, -x, show=False)
>>> p1.extend(p2)
>>> p1
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
[1]: cartesian line: x for x over (-10.0, 10.0)
[2]: cartesian line: -x for x over (-10.0, 10.0)
>>> p1.show()
"""
if isinstance(arg, Plot):
self._series.extend(arg._series)
elif is_sequence(arg):
self._series.extend(arg)
else:
raise TypeError('Expecting Plot or sequence of BaseSeries')
class PlotGrid(object):
"""This class helps to plot subplots from already created sympy plots
in a single figure.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot, plot3d, PlotGrid
>>> x, y = symbols('x, y')
>>> p1 = plot(x, x**2, x**3, (x, -5, 5))
>>> p2 = plot((x**2, (x, -6, 6)), (x, (x, -5, 5)))
>>> p3 = plot(x**3, (x, -5, 5))
>>> p4 = plot3d(x*y, (x, -5, 5), (y, -5, 5))
Plotting vertically in a single line:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> PlotGrid(2, 1 , p1, p2)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
Plotting horizontally in a single line:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> PlotGrid(1, 3 , p2, p3, p4)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[2]:Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
Plotting in a grid form:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> PlotGrid(2, 2, p1, p2 ,p3, p4)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
Plot[2]:Plot object containing:
[0]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[3]:Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
"""
def __init__(self, nrows, ncolumns, *args, **kwargs):
"""
Parameters
==========
nrows : The number of rows that should be in the grid of the
required subplot
ncolumns : The number of columns that should be in the grid
of the required subplot
nrows and ncolumns together define the required grid
Arguments
=========
A list of predefined plot objects entered in a row-wise sequence
i.e. plot objects which are to be in the top row of the required
grid are written first, then the second row objects and so on
Keyword arguments
=================
show : Boolean
The default value is set to ``True``. Set show to ``False`` and
the function will not display the subplot. The returned instance
of the ``PlotGrid`` class can then be used to save or display the
plot by calling the ``save()`` and ``show()`` methods
respectively.
"""
self.nrows = nrows
self.ncolumns = ncolumns
self._series = []
self.args = args
for arg in args:
self._series.append(arg._series)
self.backend = DefaultBackend
show = kwargs.pop('show', True)
if show:
self.show()
def show(self):
if hasattr(self, '_backend'):
self._backend.close()
self._backend = self.backend(self)
self._backend.show()
def save(self, path):
if hasattr(self, '_backend'):
self._backend.close()
self._backend = self.backend(self)
self._backend.save(path)
def __str__(self):
plot_strs = [('Plot[%d]:' % i) + str(plot)
for i, plot in enumerate(self.args)]
return 'PlotGrid object containing:\n' + '\n'.join(plot_strs)
##############################################################################
# Data Series
##############################################################################
#TODO more general way to calculate aesthetics (see get_color_array)
### The base class for all series
class BaseSeries(object):
"""Base class for the data objects containing stuff to be plotted.
The backend should check if it supports the data series that it's given.
(eg TextBackend supports only LineOver1DRange).
It's the backend responsibility to know how to use the class of
data series that it's given.
Some data series classes are grouped (using a class attribute like is_2Dline)
according to the api they present (based only on convention). The backend is
not obliged to use that api (eg. The LineOver1DRange belongs to the
is_2Dline group and presents the get_points method, but the
TextBackend does not use the get_points method).
"""
# Some flags follow. The rationale for using flags instead of checking base
# classes is that setting multiple flags is simpler than multiple
# inheritance.
is_2Dline = False
# Some of the backends expect:
# - get_points returning 1D np.arrays list_x, list_y
# - get_segments returning np.array (done in Line2DBaseSeries)
# - get_color_array returning 1D np.array (done in Line2DBaseSeries)
# with the colors calculated at the points from get_points
is_3Dline = False
# Some of the backends expect:
# - get_points returning 1D np.arrays list_x, list_y, list_y
# - get_segments returning np.array (done in Line2DBaseSeries)
# - get_color_array returning 1D np.array (done in Line2DBaseSeries)
# with the colors calculated at the points from get_points
is_3Dsurface = False
# Some of the backends expect:
# - get_meshes returning mesh_x, mesh_y, mesh_z (2D np.arrays)
# - get_points an alias for get_meshes
is_contour = False
# Some of the backends expect:
# - get_meshes returning mesh_x, mesh_y, mesh_z (2D np.arrays)
# - get_points an alias for get_meshes
is_implicit = False
# Some of the backends expect:
# - get_meshes returning mesh_x (1D array), mesh_y(1D array,
# mesh_z (2D np.arrays)
# - get_points an alias for get_meshes
#Different from is_contour as the colormap in backend will be
#different
is_parametric = False
# The calculation of aesthetics expects:
# - get_parameter_points returning one or two np.arrays (1D or 2D)
# used for calculation aesthetics
def __init__(self):
super(BaseSeries, self).__init__()
@property
def is_3D(self):
flags3D = [
self.is_3Dline,
self.is_3Dsurface
]
return any(flags3D)
@property
def is_line(self):
flagslines = [
self.is_2Dline,
self.is_3Dline
]
return any(flagslines)
### 2D lines
class Line2DBaseSeries(BaseSeries):
"""A base class for 2D lines.
- adding the label, steps and only_integers options
- making is_2Dline true
- defining get_segments and get_color_array
"""
is_2Dline = True
_dim = 2
def __init__(self):
super(Line2DBaseSeries, self).__init__()
self.label = None
self.steps = False
self.only_integers = False
self.line_color = None
def get_segments(self):
np = import_module('numpy')
points = self.get_points()
if self.steps is True:
x = np.array((points[0], points[0])).T.flatten()[1:]
y = np.array((points[1], points[1])).T.flatten()[:-1]
points = (x, y)
points = np.ma.array(points).T.reshape(-1, 1, self._dim)
return np.ma.concatenate([points[:-1], points[1:]], axis=1)
def get_color_array(self):
np = import_module('numpy')
c = self.line_color
if hasattr(c, '__call__'):
f = np.vectorize(c)
nargs = arity(c)
if nargs == 1 and self.is_parametric:
x = self.get_parameter_points()
return f(centers_of_segments(x))
else:
variables = list(map(centers_of_segments, self.get_points()))
if nargs == 1:
return f(variables[0])
elif nargs == 2:
return f(*variables[:2])
else: # only if the line is 3D (otherwise raises an error)
return f(*variables)
else:
return c*np.ones(self.nb_of_points)
class List2DSeries(Line2DBaseSeries):
"""Representation for a line consisting of list of points."""
def __init__(self, list_x, list_y):
np = import_module('numpy')
super(List2DSeries, self).__init__()
self.list_x = np.array(list_x)
self.list_y = np.array(list_y)
self.label = 'list'
def __str__(self):
return 'list plot'
def get_points(self):
return (self.list_x, self.list_y)
class LineOver1DRangeSeries(Line2DBaseSeries):
"""Representation for a line consisting of a SymPy expression over a range."""
def __init__(self, expr, var_start_end, **kwargs):
super(LineOver1DRangeSeries, self).__init__()
self.expr = sympify(expr)
self.label = str(self.expr)
self.var = sympify(var_start_end[0])
self.start = float(var_start_end[1])
self.end = float(var_start_end[2])
self.nb_of_points = kwargs.get('nb_of_points', 300)
self.adaptive = kwargs.get('adaptive', True)
self.depth = kwargs.get('depth', 12)
self.line_color = kwargs.get('line_color', None)
self.xscale=kwargs.get('xscale','linear')
self.flag=0
def __str__(self):
return 'cartesian line: %s for %s over %s' % (
str(self.expr), str(self.var), str((self.start, self.end)))
def get_segments(self):
"""
Adaptively gets segments for plotting.
The adaptive sampling is done by recursively checking if three
points are almost collinear. If they are not collinear, then more
points are added between those points.
References
==========
[1] Adaptive polygonal approximation of parametric curves,
Luiz Henrique de Figueiredo.
"""
if self.only_integers or not self.adaptive:
return super(LineOver1DRangeSeries, self).get_segments()
else:
f = lambdify([self.var], self.expr)
list_segments = []
np=import_module('numpy')
def sample(p, q, depth):
""" Samples recursively if three points are almost collinear.
For depth < 6, points are added irrespective of whether they
satisfy the collinearity condition or not. The maximum depth
allowed is 12.
"""
np = import_module('numpy')
#Randomly sample to avoid aliasing.
random = 0.45 + np.random.rand() * 0.1
xnew = p[0] + random * (q[0] - p[0])
ynew = f(xnew)
new_point = np.array([xnew, ynew])
if self.flag==1:
return
#Maximum depth
if depth > self.depth:
if p[1] is None or q[1] is None:
self.flag=1
return
list_segments.append([p, q])
#Sample irrespective of whether the line is flat till the
#depth of 6. We are not using linspace to avoid aliasing.
elif depth < 6:
sample(p, new_point, depth + 1)
sample(new_point, q, depth + 1)
#Sample ten points if complex values are encountered
#at both ends. If there is a real value in between, then
#sample those points further.
elif p[1] is None and q[1] is None:
if self.xscale is 'log':
xarray = np.logspace(p[0],q[0], 10)
else:
xarray = np.linspace(p[0], q[0], 10)
yarray = list(map(f, xarray))
if any(y is not None for y in yarray):
for i in range(len(yarray) - 1):
if yarray[i] is not None or yarray[i + 1] is not None:
sample([xarray[i], yarray[i]],
[xarray[i + 1], yarray[i + 1]], depth + 1)
#Sample further if one of the end points in None( i.e. a complex
#value) or the three points are not almost collinear.
elif (p[1] is None or q[1] is None or new_point[1] is None
or not flat(p, new_point, q)):
sample(p, new_point, depth + 1)
sample(new_point, q, depth + 1)
else:
list_segments.append([p, q])
if self.xscale is 'log':
self.start=np.log10(self.start)
self.end=np.log10(self.end)
f_start = f(self.start)
f_end = f(self.end)
sample([self.start, f_start], [self.end, f_end], 0)
return list_segments
def get_points(self):
np = import_module('numpy')
if self.only_integers is True:
if self.xscale is 'log':
list_x = np.logspace(int(self.start), int(self.end),
num=int(self.end) - int(self.start) + 1)
else:
list_x = np.linspace(int(self.start), int(self.end),
num=int(self.end) - int(self.start) + 1)
else:
if self.xscale is 'log':
list_x = np.logspace(self.start, self.end, num=self.nb_of_points)
else:
list_x = np.linspace(self.start, self.end, num=self.nb_of_points)
f = vectorized_lambdify([self.var], self.expr)
list_y = f(list_x)
return (list_x, list_y)
class Parametric2DLineSeries(Line2DBaseSeries):
"""Representation for a line consisting of two parametric sympy expressions
over a range."""
is_parametric = True
def __init__(self, expr_x, expr_y, var_start_end, **kwargs):
super(Parametric2DLineSeries, self).__init__()
self.expr_x = sympify(expr_x)
self.expr_y = sympify(expr_y)
self.label = "(%s, %s)" % (str(self.expr_x), str(self.expr_y))
self.var = sympify(var_start_end[0])
self.start = float(var_start_end[1])
self.end = float(var_start_end[2])
self.nb_of_points = kwargs.get('nb_of_points', 300)
self.adaptive = kwargs.get('adaptive', True)
self.depth = kwargs.get('depth', 12)
self.line_color = kwargs.get('line_color', None)
def __str__(self):
return 'parametric cartesian line: (%s, %s) for %s over %s' % (
str(self.expr_x), str(self.expr_y), str(self.var),
str((self.start, self.end)))
def get_parameter_points(self):
np = import_module('numpy')
return np.linspace(self.start, self.end, num=self.nb_of_points)
def get_points(self):
param = self.get_parameter_points()
fx = vectorized_lambdify([self.var], self.expr_x)
fy = vectorized_lambdify([self.var], self.expr_y)
list_x = fx(param)
list_y = fy(param)
return (list_x, list_y)
def get_segments(self):
"""
Adaptively gets segments for plotting.
The adaptive sampling is done by recursively checking if three
points are almost collinear. If they are not collinear, then more
points are added between those points.
References
==========
[1] Adaptive polygonal approximation of parametric curves,
Luiz Henrique de Figueiredo.
"""
if not self.adaptive:
return super(Parametric2DLineSeries, self).get_segments()
f_x = lambdify([self.var], self.expr_x)
f_y = lambdify([self.var], self.expr_y)
list_segments = []
def sample(param_p, param_q, p, q, depth):
""" Samples recursively if three points are almost collinear.
For depth < 6, points are added irrespective of whether they
satisfy the collinearity condition or not. The maximum depth
allowed is 12.
"""
#Randomly sample to avoid aliasing.
np = import_module('numpy')
random = 0.45 + np.random.rand() * 0.1
param_new = param_p + random * (param_q - param_p)
xnew = f_x(param_new)
ynew = f_y(param_new)
new_point = np.array([xnew, ynew])
#Maximum depth
if depth > self.depth:
list_segments.append([p, q])
#Sample irrespective of whether the line is flat till the
#depth of 6. We are not using linspace to avoid aliasing.
elif depth < 6:
sample(param_p, param_new, p, new_point, depth + 1)
sample(param_new, param_q, new_point, q, depth + 1)
#Sample ten points if complex values are encountered
#at both ends. If there is a real value in between, then
#sample those points further.
elif ((p[0] is None and q[1] is None) or
(p[1] is None and q[1] is None)):
param_array = np.linspace(param_p, param_q, 10)
x_array = list(map(f_x, param_array))
y_array = list(map(f_y, param_array))
if any(x is not None and y is not None
for x, y in zip(x_array, y_array)):
for i in range(len(y_array) - 1):
if ((x_array[i] is not None and y_array[i] is not None) or
(x_array[i + 1] is not None and y_array[i + 1] is not None)):
point_a = [x_array[i], y_array[i]]
point_b = [x_array[i + 1], y_array[i + 1]]
sample(param_array[i], param_array[i], point_a,
point_b, depth + 1)
#Sample further if one of the end points in None( ie a complex
#value) or the three points are not almost collinear.
elif (p[0] is None or p[1] is None
or q[1] is None or q[0] is None
or not flat(p, new_point, q)):
sample(param_p, param_new, p, new_point, depth + 1)
sample(param_new, param_q, new_point, q, depth + 1)
else:
list_segments.append([p, q])
f_start_x = f_x(self.start)
f_start_y = f_y(self.start)
start = [f_start_x, f_start_y]
f_end_x = f_x(self.end)
f_end_y = f_y(self.end)
end = [f_end_x, f_end_y]
sample(self.start, self.end, start, end, 0)
return list_segments
### 3D lines
class Line3DBaseSeries(Line2DBaseSeries):
"""A base class for 3D lines.
Most of the stuff is derived from Line2DBaseSeries."""
is_2Dline = False
is_3Dline = True
_dim = 3
def __init__(self):
super(Line3DBaseSeries, self).__init__()
class Parametric3DLineSeries(Line3DBaseSeries):
"""Representation for a 3D line consisting of two parametric sympy
expressions and a range."""
def __init__(self, expr_x, expr_y, expr_z, var_start_end, **kwargs):
super(Parametric3DLineSeries, self).__init__()
self.expr_x = sympify(expr_x)
self.expr_y = sympify(expr_y)
self.expr_z = sympify(expr_z)
self.label = "(%s, %s)" % (str(self.expr_x), str(self.expr_y))
self.var = sympify(var_start_end[0])
self.start = float(var_start_end[1])
self.end = float(var_start_end[2])
self.nb_of_points = kwargs.get('nb_of_points', 300)
self.line_color = kwargs.get('line_color', None)
def __str__(self):
return '3D parametric cartesian line: (%s, %s, %s) for %s over %s' % (
str(self.expr_x), str(self.expr_y), str(self.expr_z),
str(self.var), str((self.start, self.end)))
def get_parameter_points(self):
np = import_module('numpy')
return np.linspace(self.start, self.end, num=self.nb_of_points)
def get_points(self):
param = self.get_parameter_points()
fx = vectorized_lambdify([self.var], self.expr_x)
fy = vectorized_lambdify([self.var], self.expr_y)
fz = vectorized_lambdify([self.var], self.expr_z)
list_x = fx(param)
list_y = fy(param)
list_z = fz(param)
return (list_x, list_y, list_z)
### Surfaces
class SurfaceBaseSeries(BaseSeries):
"""A base class for 3D surfaces."""
is_3Dsurface = True
def __init__(self):
super(SurfaceBaseSeries, self).__init__()
self.surface_color = None
def get_color_array(self):
np = import_module('numpy')
c = self.surface_color
if isinstance(c, Callable):
f = np.vectorize(c)
nargs = arity(c)
if self.is_parametric:
variables = list(map(centers_of_faces, self.get_parameter_meshes()))
if nargs == 1:
return f(variables[0])
elif nargs == 2:
return f(*variables)
variables = list(map(centers_of_faces, self.get_meshes()))
if nargs == 1:
return f(variables[0])
elif nargs == 2:
return f(*variables[:2])
else:
return f(*variables)
else:
return c*np.ones(self.nb_of_points)
class SurfaceOver2DRangeSeries(SurfaceBaseSeries):
"""Representation for a 3D surface consisting of a sympy expression and 2D
range."""
def __init__(self, expr, var_start_end_x, var_start_end_y, **kwargs):
super(SurfaceOver2DRangeSeries, self).__init__()
self.expr = sympify(expr)
self.var_x = sympify(var_start_end_x[0])
self.start_x = float(var_start_end_x[1])
self.end_x = float(var_start_end_x[2])
self.var_y = sympify(var_start_end_y[0])
self.start_y = float(var_start_end_y[1])
self.end_y = float(var_start_end_y[2])
self.nb_of_points_x = kwargs.get('nb_of_points_x', 50)
self.nb_of_points_y = kwargs.get('nb_of_points_y', 50)
self.surface_color = kwargs.get('surface_color', None)
def __str__(self):
return ('cartesian surface: %s for'
' %s over %s and %s over %s') % (
str(self.expr),
str(self.var_x),
str((self.start_x, self.end_x)),
str(self.var_y),
str((self.start_y, self.end_y)))
def get_meshes(self):
np = import_module('numpy')
mesh_x, mesh_y = np.meshgrid(np.linspace(self.start_x, self.end_x,
num=self.nb_of_points_x),
np.linspace(self.start_y, self.end_y,
num=self.nb_of_points_y))
f = vectorized_lambdify((self.var_x, self.var_y), self.expr)
return (mesh_x, mesh_y, f(mesh_x, mesh_y))
class ParametricSurfaceSeries(SurfaceBaseSeries):
"""Representation for a 3D surface consisting of three parametric sympy
expressions and a range."""
is_parametric = True
def __init__(
self, expr_x, expr_y, expr_z, var_start_end_u, var_start_end_v,
**kwargs):
super(ParametricSurfaceSeries, self).__init__()
self.expr_x = sympify(expr_x)
self.expr_y = sympify(expr_y)
self.expr_z = sympify(expr_z)
self.var_u = sympify(var_start_end_u[0])
self.start_u = float(var_start_end_u[1])
self.end_u = float(var_start_end_u[2])
self.var_v = sympify(var_start_end_v[0])
self.start_v = float(var_start_end_v[1])
self.end_v = float(var_start_end_v[2])
self.nb_of_points_u = kwargs.get('nb_of_points_u', 50)
self.nb_of_points_v = kwargs.get('nb_of_points_v', 50)
self.surface_color = kwargs.get('surface_color', None)
def __str__(self):
return ('parametric cartesian surface: (%s, %s, %s) for'
' %s over %s and %s over %s') % (
str(self.expr_x),
str(self.expr_y),
str(self.expr_z),
str(self.var_u),
str((self.start_u, self.end_u)),
str(self.var_v),
str((self.start_v, self.end_v)))
def get_parameter_meshes(self):
np = import_module('numpy')
return np.meshgrid(np.linspace(self.start_u, self.end_u,
num=self.nb_of_points_u),
np.linspace(self.start_v, self.end_v,
num=self.nb_of_points_v))
def get_meshes(self):
mesh_u, mesh_v = self.get_parameter_meshes()
fx = vectorized_lambdify((self.var_u, self.var_v), self.expr_x)
fy = vectorized_lambdify((self.var_u, self.var_v), self.expr_y)
fz = vectorized_lambdify((self.var_u, self.var_v), self.expr_z)
return (fx(mesh_u, mesh_v), fy(mesh_u, mesh_v), fz(mesh_u, mesh_v))
### Contours
class ContourSeries(BaseSeries):
"""Representation for a contour plot."""
# The code is mostly repetition of SurfaceOver2DRange.
# Presently used in contour_plot function
is_contour = True
def __init__(self, expr, var_start_end_x, var_start_end_y):
super(ContourSeries, self).__init__()
self.nb_of_points_x = 50
self.nb_of_points_y = 50
self.expr = sympify(expr)
self.var_x = sympify(var_start_end_x[0])
self.start_x = float(var_start_end_x[1])
self.end_x = float(var_start_end_x[2])
self.var_y = sympify(var_start_end_y[0])
self.start_y = float(var_start_end_y[1])
self.end_y = float(var_start_end_y[2])
self.get_points = self.get_meshes
def __str__(self):
return ('contour: %s for '
'%s over %s and %s over %s') % (
str(self.expr),
str(self.var_x),
str((self.start_x, self.end_x)),
str(self.var_y),
str((self.start_y, self.end_y)))
def get_meshes(self):
np = import_module('numpy')
mesh_x, mesh_y = np.meshgrid(np.linspace(self.start_x, self.end_x,
num=self.nb_of_points_x),
np.linspace(self.start_y, self.end_y,
num=self.nb_of_points_y))
f = vectorized_lambdify((self.var_x, self.var_y), self.expr)
return (mesh_x, mesh_y, f(mesh_x, mesh_y))
##############################################################################
# Backends
##############################################################################
class BaseBackend(object):
def __init__(self, parent):
super(BaseBackend, self).__init__()
self.parent = parent
## don't have to check for the success of importing matplotlib in each case;
## we will only be using this backend if we can successfully import matploblib
class MatplotlibBackend(BaseBackend):
def __init__(self, parent):
super(MatplotlibBackend, self).__init__(parent)
self.matplotlib = import_module('matplotlib',
__import__kwargs={'fromlist': ['pyplot', 'cm', 'collections']},
min_module_version='1.1.0', catch=(RuntimeError,))
self.plt = self.matplotlib.pyplot
self.cm = self.matplotlib.cm
self.LineCollection = self.matplotlib.collections.LineCollection
if isinstance(self.parent, Plot):
nrows, ncolumns = 1, 1
series_list = [self.parent._series]
elif isinstance(self.parent, PlotGrid):
nrows, ncolumns = self.parent.nrows, self.parent.ncolumns
series_list = self.parent._series
self.ax = []
self.fig = self.plt.figure()
for i, series in enumerate(series_list):
are_3D = [s.is_3D for s in series]
if any(are_3D) and not all(are_3D):
raise ValueError('The matplotlib backend can not mix 2D and 3D.')
elif all(are_3D):
# mpl_toolkits.mplot3d is necessary for
# projection='3d'
mpl_toolkits = import_module('mpl_toolkits',
__import__kwargs={'fromlist': ['mplot3d']})
self.ax.append(self.fig.add_subplot(nrows, ncolumns, i + 1, projection='3d'))
elif not any(are_3D):
self.ax.append(self.fig.add_subplot(nrows, ncolumns, i + 1))
self.ax[i].spines['left'].set_position('zero')
self.ax[i].spines['right'].set_color('none')
self.ax[i].spines['bottom'].set_position('zero')
self.ax[i].spines['top'].set_color('none')
self.ax[i].spines['left'].set_smart_bounds(True)
self.ax[i].spines['bottom'].set_smart_bounds(False)
self.ax[i].xaxis.set_ticks_position('bottom')
self.ax[i].yaxis.set_ticks_position('left')
def _process_series(self, series, ax, parent):
for s in series:
# Create the collections
if s.is_2Dline:
collection = self.LineCollection(s.get_segments())
ax.add_collection(collection)
elif s.is_contour:
ax.contour(*s.get_meshes())
elif s.is_3Dline:
# TODO too complicated, I blame matplotlib
mpl_toolkits = import_module('mpl_toolkits',
__import__kwargs={'fromlist': ['mplot3d']})
art3d = mpl_toolkits.mplot3d.art3d
collection = art3d.Line3DCollection(s.get_segments())
ax.add_collection(collection)
x, y, z = s.get_points()
ax.set_xlim((min(x), max(x)))
ax.set_ylim((min(y), max(y)))
ax.set_zlim((min(z), max(z)))
elif s.is_3Dsurface:
x, y, z = s.get_meshes()
collection = ax.plot_surface(x, y, z,
cmap=getattr(self.cm, 'viridis', self.cm.jet),
rstride=1, cstride=1, linewidth=0.1)
elif s.is_implicit:
# Smart bounds have to be set to False for implicit plots.
ax.spines['left'].set_smart_bounds(False)
ax.spines['bottom'].set_smart_bounds(False)
points = s.get_raster()
if len(points) == 2:
# interval math plotting
x, y = _matplotlib_list(points[0])
ax.fill(x, y, facecolor=s.line_color, edgecolor='None')
else:
# use contourf or contour depending on whether it is
# an inequality or equality.
# XXX: ``contour`` plots multiple lines. Should be fixed.
ListedColormap = self.matplotlib.colors.ListedColormap
colormap = ListedColormap(["white", s.line_color])
xarray, yarray, zarray, plot_type = points
if plot_type == 'contour':
ax.contour(xarray, yarray, zarray, cmap=colormap)
else:
ax.contourf(xarray, yarray, zarray, cmap=colormap)
else:
raise ValueError('The matplotlib backend supports only '
'is_2Dline, is_3Dline, is_3Dsurface and '
'is_contour objects.')
# Customise the collections with the corresponding per-series
# options.
if hasattr(s, 'label'):
collection.set_label(s.label)
if s.is_line and s.line_color:
if isinstance(s.line_color, (float, int)) or isinstance(s.line_color, Callable):
color_array = s.get_color_array()
collection.set_array(color_array)
else:
collection.set_color(s.line_color)
if s.is_3Dsurface and s.surface_color:
if self.matplotlib.__version__ < "1.2.0": # TODO in the distant future remove this check
warnings.warn('The version of matplotlib is too old to use surface coloring.')
elif isinstance(s.surface_color, (float, int)) or isinstance(s.surface_color, Callable):
color_array = s.get_color_array()
color_array = color_array.reshape(color_array.size)
collection.set_array(color_array)
else:
collection.set_color(s.surface_color)
# Set global options.
# TODO The 3D stuff
# XXX The order of those is important.
mpl_toolkits = import_module('mpl_toolkits',
__import__kwargs={'fromlist': ['mplot3d']})
Axes3D = mpl_toolkits.mplot3d.Axes3D
if parent.xscale and not isinstance(ax, Axes3D):
ax.set_xscale(parent.xscale)
if parent.yscale and not isinstance(ax, Axes3D):
ax.set_yscale(parent.yscale)
if parent.xlim:
from sympy.core.basic import Basic
xlim = parent.xlim
if any(isinstance(i,Basic) and not i.is_real for i in xlim):
raise ValueError(
"All numbers from xlim={} must be real".format(xlim))
if any(isinstance(i,Basic) and not i.is_finite for i in xlim):
raise ValueError(
"All numbers from xlim={} must be finite".format(xlim))
xlim = (float(i) for i in xlim)
ax.set_xlim(xlim)
else:
if all(isinstance(s, LineOver1DRangeSeries) for s in parent._series):
starts = [s.start for s in parent._series]
ends = [s.end for s in parent._series]
ax.set_xlim(min(starts), max(ends))
if parent.ylim:
from sympy.core.basic import Basic
ylim = parent.ylim
if any(isinstance(i,Basic) and not i.is_real for i in ylim):
raise ValueError(
"All numbers from ylim={} must be real".format(ylim))
if any(isinstance(i,Basic) and not i.is_finite for i in ylim):
raise ValueError(
"All numbers from ylim={} must be finite".format(ylim))
ylim = (float(i) for i in ylim)
ax.set_ylim(ylim)
if not isinstance(ax, Axes3D) or self.matplotlib.__version__ >= '1.2.0': # XXX in the distant future remove this check
ax.set_autoscale_on(parent.autoscale)
if parent.axis_center:
val = parent.axis_center
if isinstance(ax, Axes3D):
pass
elif val == 'center':
ax.spines['left'].set_position('center')
ax.spines['bottom'].set_position('center')
elif val == 'auto':
xl, xh = ax.get_xlim()
yl, yh = ax.get_ylim()
pos_left = ('data', 0) if xl*xh <= 0 else 'center'
pos_bottom = ('data', 0) if yl*yh <= 0 else 'center'
ax.spines['left'].set_position(pos_left)
ax.spines['bottom'].set_position(pos_bottom)
else:
ax.spines['left'].set_position(('data', val[0]))
ax.spines['bottom'].set_position(('data', val[1]))
if not parent.axis:
ax.set_axis_off()
if parent.legend:
if ax.legend():
ax.legend_.set_visible(parent.legend)
if parent.margin:
ax.set_xmargin(parent.margin)
ax.set_ymargin(parent.margin)
if parent.title:
ax.set_title(parent.title)
if parent.xlabel:
ax.set_xlabel(parent.xlabel, position=(1, 0))
if parent.ylabel:
ax.set_ylabel(parent.ylabel, position=(0, 1))
def process_series(self):
"""
Iterates over every ``Plot`` object and further calls
_process_series()
"""
parent = self.parent
if isinstance(parent, Plot):
series_list = [parent._series]
else:
series_list = parent._series
for i, (series, ax) in enumerate(zip(series_list, self.ax)):
if isinstance(self.parent, PlotGrid):
parent = self.parent.args[i]
self._process_series(series, ax, parent)
def show(self):
self.process_series()
#TODO after fixing https://github.com/ipython/ipython/issues/1255
# you can uncomment the next line and remove the pyplot.show() call
#self.fig.show()
if _show:
self.fig.tight_layout()
self.plt.show()
else:
self.close()
def save(self, path):
self.process_series()
self.fig.savefig(path)
def close(self):
self.plt.close(self.fig)
class TextBackend(BaseBackend):
def __init__(self, parent):
super(TextBackend, self).__init__(parent)
def show(self):
if not _show:
return
if len(self.parent._series) != 1:
raise ValueError(
'The TextBackend supports only one graph per Plot.')
elif not isinstance(self.parent._series[0], LineOver1DRangeSeries):
raise ValueError(
'The TextBackend supports only expressions over a 1D range')
else:
ser = self.parent._series[0]
textplot(ser.expr, ser.start, ser.end)
def close(self):
pass
class DefaultBackend(BaseBackend):
def __new__(cls, parent):
matplotlib = import_module('matplotlib', min_module_version='1.1.0', catch=(RuntimeError,))
if matplotlib:
return MatplotlibBackend(parent)
else:
return TextBackend(parent)
plot_backends = {
'matplotlib': MatplotlibBackend,
'text': TextBackend,
'default': DefaultBackend
}
##############################################################################
# Finding the centers of line segments or mesh faces
##############################################################################
def centers_of_segments(array):
np = import_module('numpy')
return np.mean(np.vstack((array[:-1], array[1:])), 0)
def centers_of_faces(array):
np = import_module('numpy')
return np.mean(np.dstack((array[:-1, :-1],
array[1:, :-1],
array[:-1, 1: ],
array[:-1, :-1],
)), 2)
def flat(x, y, z, eps=1e-3):
"""Checks whether three points are almost collinear"""
np = import_module('numpy')
# Workaround plotting piecewise (#8577):
# workaround for `lambdify` in `.experimental_lambdify` fails
# to return numerical values in some cases. Lower-level fix
# in `lambdify` is possible.
vector_a = (x - y).astype(np.float)
vector_b = (z - y).astype(np.float)
dot_product = np.dot(vector_a, vector_b)
vector_a_norm = np.linalg.norm(vector_a)
vector_b_norm = np.linalg.norm(vector_b)
cos_theta = dot_product / (vector_a_norm * vector_b_norm)
return abs(cos_theta + 1) < eps
def _matplotlib_list(interval_list):
"""
Returns lists for matplotlib ``fill`` command from a list of bounding
rectangular intervals
"""
xlist = []
ylist = []
if len(interval_list):
for intervals in interval_list:
intervalx = intervals[0]
intervaly = intervals[1]
xlist.extend([intervalx.start, intervalx.start,
intervalx.end, intervalx.end, None])
ylist.extend([intervaly.start, intervaly.end,
intervaly.end, intervaly.start, None])
else:
#XXX Ugly hack. Matplotlib does not accept empty lists for ``fill``
xlist.extend([None, None, None, None])
ylist.extend([None, None, None, None])
return xlist, ylist
####New API for plotting module ####
# TODO: Add color arrays for plots.
# TODO: Add more plotting options for 3d plots.
# TODO: Adaptive sampling for 3D plots.
def plot(*args, **kwargs):
"""
Plots a function of a single variable and returns an instance of
the ``Plot`` class (also, see the description of the
``show`` keyword argument below).
The plotting uses an adaptive algorithm which samples recursively to
accurately plot the plot. The adaptive algorithm uses a random point near
the midpoint of two points that has to be further sampled. Hence the same
plots can appear slightly different.
Usage
=====
Single Plot
``plot(expr, range, **kwargs)``
If the range is not specified, then a default range of (-10, 10) is used.
Multiple plots with same range.
``plot(expr1, expr2, ..., range, **kwargs)``
If the range is not specified, then a default range of (-10, 10) is used.
Multiple plots with different ranges.
``plot((expr1, range), (expr2, range), ..., **kwargs)``
Range has to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
``expr`` : Expression representing the function of single variable
``range``: (x, 0, 5), A 3-tuple denoting the range of the free variable.
Keyword Arguments
=================
Arguments for ``plot`` function:
``show``: Boolean. The default value is set to ``True``. Set show to
``False`` and the function will not display the plot. The returned
instance of the ``Plot`` class can then be used to save or display
the plot by calling the ``save()`` and ``show()`` methods
respectively.
Arguments for ``LineOver1DRangeSeries`` class:
``adaptive``: Boolean. The default value is set to True. Set adaptive to False and
specify ``nb_of_points`` if uniform sampling is required.
``depth``: int Recursion depth of the adaptive algorithm. A depth of value ``n``
samples a maximum of `2^{n}` points.
``nb_of_points``: int. Used when the ``adaptive`` is set to False. The function
is uniformly sampled at ``nb_of_points`` number of points.
Aesthetics options:
``line_color``: float. Specifies the color for the plot.
See ``Plot`` to see how to set color for the plots.
If there are multiple plots, then the same series series are applied to
all the plots. If you want to set these options separately, you can index
the ``Plot`` object returned and set it.
Arguments for ``Plot`` class:
``title`` : str. Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
``xlabel`` : str. Label for the x-axis.
``ylabel`` : str. Label for the y-axis.
``xscale``: {'linear', 'log'} Sets the scaling of the x-axis.
``yscale``: {'linear', 'log'} Sets the scaling if the y-axis.
``axis_center``: tuple of two floats denoting the coordinates of the center or
{'center', 'auto'}
``xlim`` : tuple of two floats, denoting the x-axis limits.
``ylim`` : tuple of two floats, denoting the y-axis limits.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
Single Plot
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot(x**2, (x, -5, 5))
Plot object containing:
[0]: cartesian line: x**2 for x over (-5.0, 5.0)
Multiple plots with single range.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot(x, x**2, x**3, (x, -5, 5))
Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)
Multiple plots with different ranges.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot((x**2, (x, -6, 6)), (x, (x, -5, 5)))
Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
No adaptive sampling.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot(x**2, adaptive=False, nb_of_points=400)
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
See Also
========
Plot, LineOver1DRangeSeries.
"""
args = list(map(sympify, args))
free = set()
for a in args:
if isinstance(a, Expr):
free |= a.free_symbols
if len(free) > 1:
raise ValueError(
'The same variable should be used in all '
'univariate expressions being plotted.')
x = free.pop() if free else Symbol('x')
kwargs.setdefault('xlabel', x.name)
kwargs.setdefault('ylabel', 'f(%s)' % x.name)
show = kwargs.pop('show', True)
series = []
plot_expr = check_arguments(args, 1, 1)
series = [LineOver1DRangeSeries(*arg, **kwargs) for arg in plot_expr]
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot_parametric(*args, **kwargs):
"""
Plots a 2D parametric plot.
The plotting uses an adaptive algorithm which samples recursively to
accurately plot the plot. The adaptive algorithm uses a random point near
the midpoint of two points that has to be further sampled. Hence the same
plots can appear slightly different.
Usage
=====
Single plot.
``plot_parametric(expr_x, expr_y, range, **kwargs)``
If the range is not specified, then a default range of (-10, 10) is used.
Multiple plots with same range.
``plot_parametric((expr1_x, expr1_y), (expr2_x, expr2_y), range, **kwargs)``
If the range is not specified, then a default range of (-10, 10) is used.
Multiple plots with different ranges.
``plot_parametric((expr_x, expr_y, range), ..., **kwargs)``
Range has to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
``expr_x`` : Expression representing the function along x.
``expr_y`` : Expression representing the function along y.
``range``: (u, 0, 5), A 3-tuple denoting the range of the parameter
variable.
Keyword Arguments
=================
Arguments for ``Parametric2DLineSeries`` class:
``adaptive``: Boolean. The default value is set to True. Set adaptive to
False and specify ``nb_of_points`` if uniform sampling is required.
``depth``: int Recursion depth of the adaptive algorithm. A depth of
value ``n`` samples a maximum of `2^{n}` points.
``nb_of_points``: int. Used when the ``adaptive`` is set to False. The
function is uniformly sampled at ``nb_of_points`` number of points.
Aesthetics
----------
``line_color``: function which returns a float. Specifies the color for the
plot. See ``sympy.plotting.Plot`` for more details.
If there are multiple plots, then the same Series arguments are applied to
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class:
``xlabel`` : str. Label for the x-axis.
``ylabel`` : str. Label for the y-axis.
``xscale``: {'linear', 'log'} Sets the scaling of the x-axis.
``yscale``: {'linear', 'log'} Sets the scaling if the y-axis.
``axis_center``: tuple of two floats denoting the coordinates of the center
or {'center', 'auto'}
``xlim`` : tuple of two floats, denoting the x-axis limits.
``ylim`` : tuple of two floats, denoting the y-axis limits.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import symbols, cos, sin
>>> from sympy.plotting import plot_parametric
>>> u = symbols('u')
Single Parametric plot
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_parametric(cos(u), sin(u), (u, -5, 5))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-5.0, 5.0)
Multiple parametric plot with single range.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_parametric((cos(u), sin(u)), (u, cos(u)))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-10.0, 10.0)
[1]: parametric cartesian line: (u, cos(u)) for u over (-10.0, 10.0)
Multiple parametric plots.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_parametric((cos(u), sin(u), (u, -5, 5)),
... (cos(u), u, (u, -5, 5)))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-5.0, 5.0)
[1]: parametric cartesian line: (cos(u), u) for u over (-5.0, 5.0)
See Also
========
Plot, Parametric2DLineSeries
"""
args = list(map(sympify, args))
show = kwargs.pop('show', True)
series = []
plot_expr = check_arguments(args, 2, 1)
series = [Parametric2DLineSeries(*arg, **kwargs) for arg in plot_expr]
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot3d_parametric_line(*args, **kwargs):
"""
Plots a 3D parametric line plot.
Usage
=====
Single plot:
``plot3d_parametric_line(expr_x, expr_y, expr_z, range, **kwargs)``
If the range is not specified, then a default range of (-10, 10) is used.
Multiple plots.
``plot3d_parametric_line((expr_x, expr_y, expr_z, range), ..., **kwargs)``
Ranges have to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
``expr_x`` : Expression representing the function along x.
``expr_y`` : Expression representing the function along y.
``expr_z`` : Expression representing the function along z.
``range``: ``(u, 0, 5)``, A 3-tuple denoting the range of the parameter
variable.
Keyword Arguments
=================
Arguments for ``Parametric3DLineSeries`` class.
``nb_of_points``: The range is uniformly sampled at ``nb_of_points``
number of points.
Aesthetics:
``line_color``: function which returns a float. Specifies the color for the
plot. See ``sympy.plotting.Plot`` for more details.
If there are multiple plots, then the same series arguments are applied to
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class.
``title`` : str. Title of the plot.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import symbols, cos, sin
>>> from sympy.plotting import plot3d_parametric_line
>>> u = symbols('u')
Single plot.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d_parametric_line(cos(u), sin(u), u, (u, -5, 5))
Plot object containing:
[0]: 3D parametric cartesian line: (cos(u), sin(u), u) for u over (-5.0, 5.0)
Multiple plots.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d_parametric_line((cos(u), sin(u), u, (u, -5, 5)),
... (sin(u), u**2, u, (u, -5, 5)))
Plot object containing:
[0]: 3D parametric cartesian line: (cos(u), sin(u), u) for u over (-5.0, 5.0)
[1]: 3D parametric cartesian line: (sin(u), u**2, u) for u over (-5.0, 5.0)
See Also
========
Plot, Parametric3DLineSeries
"""
args = list(map(sympify, args))
show = kwargs.pop('show', True)
series = []
plot_expr = check_arguments(args, 3, 1)
series = [Parametric3DLineSeries(*arg, **kwargs) for arg in plot_expr]
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot3d(*args, **kwargs):
"""
Plots a 3D surface plot.
Usage
=====
Single plot
``plot3d(expr, range_x, range_y, **kwargs)``
If the ranges are not specified, then a default range of (-10, 10) is used.
Multiple plot with the same range.
``plot3d(expr1, expr2, range_x, range_y, **kwargs)``
If the ranges are not specified, then a default range of (-10, 10) is used.
Multiple plots with different ranges.
``plot3d((expr1, range_x, range_y), (expr2, range_x, range_y), ..., **kwargs)``
Ranges have to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
``expr`` : Expression representing the function along x.
``range_x``: (x, 0, 5), A 3-tuple denoting the range of the x
variable.
``range_y``: (y, 0, 5), A 3-tuple denoting the range of the y
variable.
Keyword Arguments
=================
Arguments for ``SurfaceOver2DRangeSeries`` class:
``nb_of_points_x``: int. The x range is sampled uniformly at
``nb_of_points_x`` of points.
``nb_of_points_y``: int. The y range is sampled uniformly at
``nb_of_points_y`` of points.
Aesthetics:
``surface_color``: Function which returns a float. Specifies the color for
the surface of the plot. See ``sympy.plotting.Plot`` for more details.
If there are multiple plots, then the same series arguments are applied to
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class:
``title`` : str. Title of the plot.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot3d
>>> x, y = symbols('x y')
Single plot
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d(x*y, (x, -5, 5), (y, -5, 5))
Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
Multiple plots with same range
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d(x*y, -x*y, (x, -5, 5), (y, -5, 5))
Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
[1]: cartesian surface: -x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
Multiple plots with different ranges.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d((x**2 + y**2, (x, -5, 5), (y, -5, 5)),
... (x*y, (x, -3, 3), (y, -3, 3)))
Plot object containing:
[0]: cartesian surface: x**2 + y**2 for x over (-5.0, 5.0) and y over (-5.0, 5.0)
[1]: cartesian surface: x*y for x over (-3.0, 3.0) and y over (-3.0, 3.0)
See Also
========
Plot, SurfaceOver2DRangeSeries
"""
args = list(map(sympify, args))
show = kwargs.pop('show', True)
series = []
plot_expr = check_arguments(args, 1, 2)
series = [SurfaceOver2DRangeSeries(*arg, **kwargs) for arg in plot_expr]
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot3d_parametric_surface(*args, **kwargs):
"""
Plots a 3D parametric surface plot.
Usage
=====
Single plot.
``plot3d_parametric_surface(expr_x, expr_y, expr_z, range_u, range_v, **kwargs)``
If the ranges is not specified, then a default range of (-10, 10) is used.
Multiple plots.
``plot3d_parametric_surface((expr_x, expr_y, expr_z, range_u, range_v), ..., **kwargs)``
Ranges have to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
``expr_x``: Expression representing the function along ``x``.
``expr_y``: Expression representing the function along ``y``.
``expr_z``: Expression representing the function along ``z``.
``range_u``: ``(u, 0, 5)``, A 3-tuple denoting the range of the ``u``
variable.
``range_v``: ``(v, 0, 5)``, A 3-tuple denoting the range of the v
variable.
Keyword Arguments
=================
Arguments for ``ParametricSurfaceSeries`` class:
``nb_of_points_u``: int. The ``u`` range is sampled uniformly at
``nb_of_points_v`` of points
``nb_of_points_y``: int. The ``v`` range is sampled uniformly at
``nb_of_points_y`` of points
Aesthetics:
``surface_color``: Function which returns a float. Specifies the color for
the surface of the plot. See ``sympy.plotting.Plot`` for more details.
If there are multiple plots, then the same series arguments are applied for
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class:
``title`` : str. Title of the plot.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import symbols, cos, sin
>>> from sympy.plotting import plot3d_parametric_surface
>>> u, v = symbols('u v')
Single plot.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d_parametric_surface(cos(u + v), sin(u - v), u - v,
... (u, -5, 5), (v, -5, 5))
Plot object containing:
[0]: parametric cartesian surface: (cos(u + v), sin(u - v), u - v) for u over (-5.0, 5.0) and v over (-5.0, 5.0)
See Also
========
Plot, ParametricSurfaceSeries
"""
args = list(map(sympify, args))
show = kwargs.pop('show', True)
series = []
plot_expr = check_arguments(args, 3, 2)
series = [ParametricSurfaceSeries(*arg, **kwargs) for arg in plot_expr]
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot_contour(*args, **kwargs):
"""
Draws contour plot of a function
Usage
=====
Single plot
``plot_contour(expr, range_x, range_y, **kwargs)``
If the ranges are not specified, then a default range of (-10, 10) is used.
Multiple plot with the same range.
``plot_contour(expr1, expr2, range_x, range_y, **kwargs)``
If the ranges are not specified, then a default range of (-10, 10) is used.
Multiple plots with different ranges.
``plot_contour((expr1, range_x, range_y), (expr2, range_x, range_y), ..., **kwargs)``
Ranges have to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
``expr`` : Expression representing the function along x.
``range_x``: (x, 0, 5), A 3-tuple denoting the range of the x
variable.
``range_y``: (y, 0, 5), A 3-tuple denoting the range of the y
variable.
Keyword Arguments
=================
Arguments for ``ContourSeries`` class:
``nb_of_points_x``: int. The x range is sampled uniformly at
``nb_of_points_x`` of points.
``nb_of_points_y``: int. The y range is sampled uniformly at
``nb_of_points_y`` of points.
Aesthetics:
``surface_color``: Function which returns a float. Specifies the color for
the surface of the plot. See ``sympy.plotting.Plot`` for more details.
If there are multiple plots, then the same series arguments are applied to
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class:
``title`` : str. Title of the plot.
See Also
========
Plot, ContourSeries
"""
args = list(map(sympify, args))
show = kwargs.pop('show', True)
plot_expr = check_arguments(args, 1, 2)
series = [ContourSeries(*arg) for arg in plot_expr]
plot_contours = Plot(*series, **kwargs)
if len(plot_expr[0].free_symbols) > 2:
raise ValueError('Contour Plot cannot Plot for more than two variables.')
if show:
plot_contours.show()
return plot_contours
def check_arguments(args, expr_len, nb_of_free_symbols):
"""
Checks the arguments and converts into tuples of the
form (exprs, ranges)
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import plot, cos, sin, symbols
>>> from sympy.plotting.plot import check_arguments
>>> x = symbols('x')
>>> check_arguments([cos(x), sin(x)], 2, 1)
[(cos(x), sin(x), (x, -10, 10))]
>>> check_arguments([x, x**2], 1, 1)
[(x, (x, -10, 10)), (x**2, (x, -10, 10))]
"""
if expr_len > 1 and isinstance(args[0], Expr):
# Multiple expressions same range.
# The arguments are tuples when the expression length is
# greater than 1.
if len(args) < expr_len:
raise ValueError("len(args) should not be less than expr_len")
for i in range(len(args)):
if isinstance(args[i], Tuple):
break
else:
i = len(args) + 1
exprs = Tuple(*args[:i])
free_symbols = list(set().union(*[e.free_symbols for e in exprs]))
if len(args) == expr_len + nb_of_free_symbols:
#Ranges given
plots = [exprs + Tuple(*args[expr_len:])]
else:
default_range = Tuple(-10, 10)
ranges = []
for symbol in free_symbols:
ranges.append(Tuple(symbol) + default_range)
for i in range(len(free_symbols) - nb_of_free_symbols):
ranges.append(Tuple(Dummy()) + default_range)
plots = [exprs + Tuple(*ranges)]
return plots
if isinstance(args[0], Expr) or (isinstance(args[0], Tuple) and
len(args[0]) == expr_len and
expr_len != 3):
# Cannot handle expressions with number of expression = 3. It is
# not possible to differentiate between expressions and ranges.
#Series of plots with same range
for i in range(len(args)):
if isinstance(args[i], Tuple) and len(args[i]) != expr_len:
break
if not isinstance(args[i], Tuple):
args[i] = Tuple(args[i])
else:
i = len(args) + 1
exprs = args[:i]
assert all(isinstance(e, Expr) for expr in exprs for e in expr)
free_symbols = list(set().union(*[e.free_symbols for expr in exprs
for e in expr]))
if len(free_symbols) > nb_of_free_symbols:
raise ValueError("The number of free_symbols in the expression "
"is greater than %d" % nb_of_free_symbols)
if len(args) == i + nb_of_free_symbols and isinstance(args[i], Tuple):
ranges = Tuple(*[range_expr for range_expr in args[
i:i + nb_of_free_symbols]])
plots = [expr + ranges for expr in exprs]
return plots
else:
#Use default ranges.
default_range = Tuple(-10, 10)
ranges = []
for symbol in free_symbols:
ranges.append(Tuple(symbol) + default_range)
for i in range(nb_of_free_symbols - len(free_symbols)):
ranges.append(Tuple(Dummy()) + default_range)
ranges = Tuple(*ranges)
plots = [expr + ranges for expr in exprs]
return plots
elif isinstance(args[0], Tuple) and len(args[0]) == expr_len + nb_of_free_symbols:
#Multiple plots with different ranges.
for arg in args:
for i in range(expr_len):
if not isinstance(arg[i], Expr):
raise ValueError("Expected an expression, given %s" %
str(arg[i]))
for i in range(nb_of_free_symbols):
if not len(arg[i + expr_len]) == 3:
raise ValueError("The ranges should be a tuple of "
"length 3, got %s" % str(arg[i + expr_len]))
return args
|
7ac9ef38e7669c7b24f8d47baa616f02c2d48eab524908b882edfef6938ecda8 | from sympy import symbols, pi, oo, S, exp, sqrt, besselk, Indexed
from sympy.stats import density
from sympy.stats.joint_rv import marginal_distribution
from sympy.stats.joint_rv_types import JointRV
from sympy.stats.crv_types import Normal
from sympy.utilities.pytest import raises, XFAIL
from sympy.integrals.integrals import integrate
from sympy.matrices import Matrix
x, y, z, a, b = symbols('x y z a b')
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
assert density(m)(1, 2) == 1/(2*pi)
raises (ValueError, lambda:m[2])
raises (ValueError,\
lambda: Normal('M',[1, 2], [[0, 0], [0, 1]]))
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1/(2*pi)
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
N = Normal('N', [1, 2], [[x, 0], [0, y]])
assert density(N)(0, 0) == exp(-2/y - 1/(2*x))/(2*pi*sqrt(x*y))
raises (ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
def test_MultivariateTDist():
from sympy.stats.joint_rv_types import MultivariateT
t1 = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
assert(density(t1))(1, 1) == 1/(8*pi)
assert integrate(density(t1)(x, y), (x, -oo, oo), \
(y, -oo, oo)).evalf() == 1
raises(ValueError, lambda: MultivariateT('T', [1, 2], [[1, 1], [1, -1]], 1))
t2 = MultivariateT('t2', [1, 2], [[x, 0], [0, y]], 1)
assert density(t2)(1, 2) == 1/(2*pi*sqrt(x*y))
def test_multivariate_laplace():
from sympy.stats.crv_types import Laplace
raises(ValueError, lambda: Laplace('T', [1, 2], [[1, 2], [2, 1]]))
L = Laplace('L', [1, 0], [[1, 2], [0, 1]])
assert density(L)(2, 3) == exp(2)*besselk(0, sqrt(3))/pi
L1 = Laplace('L1', [1, 2], [[x, 0], [0, y]])
assert density(L1)(0, 1) == \
exp(2/y)*besselk(0, sqrt((2 + 4/y + 1/x)/y))/(pi*sqrt(x*y))
def test_NormalGamma():
from sympy.stats.joint_rv_types import NormalGamma
from sympy import gamma
ng = NormalGamma('G', 1, 2, 3, 4)
assert density(ng)(1, 1) == 32*exp(-4)/sqrt(pi)
raises(ValueError, lambda:NormalGamma('G', 1, 2, 3, -1))
assert marginal_distribution(ng, 0)(1) == \
3*sqrt(10)*gamma(S(7)/4)/(10*sqrt(pi)*gamma(S(5)/4))
assert marginal_distribution(ng, y)(1) == exp(-S(1)/4)/128
def test_JointPSpace_margial_distribution():
from sympy.stats.joint_rv_types import MultivariateT
from sympy import polar_lift
T = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
assert marginal_distribution(T, T[1])(x) == sqrt(2)*(x**2 + 2)/(
8*polar_lift(x**2/2 + 1)**(S(5)/2))
assert integrate(marginal_distribution(T, 1)(x), (x, -oo, oo)) == 1
t = MultivariateT('T', [0, 0, 0], [[1, 0, 0], [0, 1, 0], [0, 0, 1]], 3)
assert marginal_distribution(t, 0)(1).evalf().round(1) == 0.2
def test_JointRV():
from sympy.stats.joint_rv import JointDistributionHandmade
x1, x2 = (Indexed('x', i) for i in (1, 2))
pdf = exp(-x1**2/2 + x1 - x2**2/2 - S(1)/2)/(2*pi)
X = JointRV('x', pdf)
assert density(X)(1, 2) == exp(-2)/(2*pi)
assert isinstance(X.pspace.distribution, JointDistributionHandmade)
assert marginal_distribution(X, 0)(2) == sqrt(2)*exp(-S(1)/2)/(2*sqrt(pi))
def test_expectation():
from sympy import simplify
from sympy.stats import E
m = Normal('A', [x, y], [[1, 0], [0, 1]])
assert simplify(E(m[1])) == y
@XFAIL
def test_joint_vector_expectation():
from sympy.stats import E
m = Normal('A', [x, y], [[1, 0], [0, 1]])
assert E(m) == (x, y)
|
9d303d25295d4f41ee329b9464ba2abf2d23d3dc6548bdcc8dbcbf9289a3e518 | from sympy import (Symbol, Abs, exp, S, N, pi, simplify, Interval, erf, erfc, Ne,
Eq, log, lowergamma, uppergamma, Sum, symbols, sqrt, And, gamma, beta,
Piecewise, Integral, sin, cos, atan, besseli, factorial, binomial,
floor, expand_func, Rational, I, re, im, lambdify, hyper, diff, Or, Mul)
from sympy.core.compatibility import range
from sympy.external import import_module
from sympy.stats import (P, E, where, density, variance, covariance, skewness,
given, pspace, cdf, characteristic_function, ContinuousRV, sample,
Arcsin, Benini, Beta, BetaPrime, Cauchy,
Chi, ChiSquared,
ChiNoncentral, Dagum, Erlang, Exponential,
FDistribution, FisherZ, Frechet, Gamma, GammaInverse,
Gompertz, Gumbel, Kumaraswamy, Laplace, Logistic,
LogNormal, Maxwell, Nakagami, Normal, Pareto,
QuadraticU, RaisedCosine, Rayleigh, ShiftedGompertz,
StudentT, Trapezoidal, Triangular, Uniform, UniformSum,
VonMises, Weibull, WignerSemicircle, correlation,
moment, cmoment, smoment)
from sympy.stats.crv_types import NormalDistribution
from sympy.stats.joint_rv import JointPSpace
from sympy.utilities.pytest import raises, XFAIL, slow, skip
from sympy.utilities.randtest import verify_numerically as tn
oo = S.Infinity
x, y, z = map(Symbol, 'xyz')
def test_single_normal():
mu = Symbol('mu', real=True, finite=True)
sigma = Symbol('sigma', real=True, positive=True, finite=True)
X = Normal('x', 0, 1)
Y = X*sigma + mu
assert simplify(E(Y)) == mu
assert simplify(variance(Y)) == sigma**2
pdf = density(Y)
x = Symbol('x')
assert (pdf(x) ==
2**S.Half*exp(-(mu - x)**2/(2*sigma**2))/(2*pi**S.Half*sigma))
assert P(X**2 < 1) == erf(2**S.Half/2)
assert E(X, Eq(X, mu)) == mu
@XFAIL
def test_conditional_1d():
X = Normal('x', 0, 1)
Y = given(X, X >= 0)
assert density(Y) == 2 * density(X)
assert Y.pspace.domain.set == Interval(0, oo)
assert E(Y) == sqrt(2) / sqrt(pi)
assert E(X**2) == E(Y**2)
def test_ContinuousDomain():
X = Normal('x', 0, 1)
assert where(X**2 <= 1).set == Interval(-1, 1)
assert where(X**2 <= 1).symbol == X.symbol
where(And(X**2 <= 1, X >= 0)).set == Interval(0, 1)
raises(ValueError, lambda: where(sin(X) > 1))
Y = given(X, X >= 0)
assert Y.pspace.domain.set == Interval(0, oo)
@slow
def test_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2*X + Y, -X) == -2*variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X*X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == S.Half
def test_symbolic():
mu1, mu2 = symbols('mu1 mu2', real=True, finite=True)
s1, s2 = symbols('sigma1 sigma2', real=True, finite=True, positive=True)
rate = Symbol('lambda', real=True, positive=True, finite=True)
X = Normal('x', mu1, s1)
Y = Normal('y', mu2, s2)
Z = Exponential('z', rate)
a, b, c = symbols('a b c', real=True, finite=True)
assert E(X) == mu1
assert E(X + Y) == mu1 + mu2
assert E(a*X + b) == a*E(X) + b
assert variance(X) == s1**2
assert simplify(variance(X + a*Y + b)) == variance(X) + a**2*variance(Y)
assert E(Z) == 1/rate
assert E(a*Z + b) == a*E(Z) + b
assert E(X + a*Z + b) == mu1 + a/rate + b
def test_cdf():
X = Normal('x', 0, 1)
d = cdf(X)
assert P(X < 1) == d(1).rewrite(erfc)
assert d(0) == S.Half
d = cdf(X, X > 0) # given X>0
assert d(0) == 0
Y = Exponential('y', 10)
d = cdf(Y)
assert d(-5) == 0
assert P(Y > 3) == 1 - d(3)
raises(ValueError, lambda: cdf(X + Y))
Z = Exponential('z', 1)
f = cdf(Z)
z = Symbol('z')
assert f(z) == Piecewise((1 - exp(-z), z >= 0), (0, True))
def test_characteristic_function():
X = Uniform('x', 0, 1)
cf = characteristic_function(X)
assert cf(1) == -I*(-1 + exp(I))
Y = Normal('y', 1, 1)
cf = characteristic_function(Y)
assert cf(0) == 1
assert simplify(cf(1)) == exp(I - S(1)/2)
Z = Exponential('z', 5)
cf = characteristic_function(Z)
assert cf(0) == 1
assert simplify(cf(1)) == S(25)/26 + 5*I/26
def test_sample_continuous():
z = Symbol('z')
Z = ContinuousRV(z, exp(-z), set=Interval(0, oo))
assert sample(Z) in Z.pspace.domain.set
sym, val = list(Z.pspace.sample().items())[0]
assert sym == Z and val in Interval(0, oo)
assert density(Z)(-1) == 0
def test_ContinuousRV():
x = Symbol('x')
pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution
# X and Y should be equivalent
X = ContinuousRV(x, pdf)
Y = Normal('y', 0, 1)
assert variance(X) == variance(Y)
assert P(X > 0) == P(Y > 0)
def test_arcsin():
from sympy import asin
a = Symbol("a", real=True)
b = Symbol("b", real=True)
X = Arcsin('x', a, b)
assert density(X)(x) == 1/(pi*sqrt((-x + b)*(x - a)))
assert cdf(X)(x) == Piecewise((0, a > x),
(2*asin(sqrt((-a + x)/(-a + b)))/pi, b >= x),
(1, True))
def test_benini():
alpha = Symbol("alpha", positive=True)
beta = Symbol("beta", positive=True)
sigma = Symbol("sigma", positive=True)
X = Benini('x', alpha, beta, sigma)
assert density(X)(x) == ((alpha/x + 2*beta*log(x/sigma)/x)
*exp(-alpha*log(x/sigma) - beta*log(x/sigma)**2))
alpha = Symbol("alpha", positive=False)
raises(ValueError, lambda: Benini('x', alpha, beta, sigma))
beta = Symbol("beta", positive=False)
raises(ValueError, lambda: Benini('x', alpha, beta, sigma))
alpha = Symbol("alpha", positive=True)
raises(ValueError, lambda: Benini('x', alpha, beta, sigma))
beta = Symbol("beta", positive=True)
sigma = Symbol("sigma", positive=False)
raises(ValueError, lambda: Benini('x', alpha, beta, sigma))
def test_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta('x', a, b)
assert pspace(B).domain.set == Interval(0, 1)
dens = density(B)
x = Symbol('x')
assert dens(x) == x**(a - 1)*(1 - x)**(b - 1) / beta(a, b)
assert simplify(E(B)) == a / (a + b)
assert simplify(variance(B)) == a*b / (a**3 + 3*a**2*b + a**2 + 3*a*b**2 + 2*a*b + b**3 + b**2)
# Full symbolic solution is too much, test with numeric version
a, b = 1, 2
B = Beta('x', a, b)
assert expand_func(E(B)) == a / S(a + b)
assert expand_func(variance(B)) == (a*b) / S((a + b)**2 * (a + b + 1))
def test_betaprime():
alpha = Symbol("alpha", positive=True)
betap = Symbol("beta", positive=True)
X = BetaPrime('x', alpha, betap)
assert density(X)(x) == x**(alpha - 1)*(x + 1)**(-alpha - betap)/beta(alpha, betap)
alpha = Symbol("alpha", positive=False)
raises(ValueError, lambda: BetaPrime('x', alpha, betap))
alpha = Symbol("alpha", positive=True)
betap = Symbol("beta", positive=False)
raises(ValueError, lambda: BetaPrime('x', alpha, betap))
def test_cauchy():
x0 = Symbol("x0")
gamma = Symbol("gamma", positive=True)
X = Cauchy('x', x0, gamma)
assert density(X)(x) == 1/(pi*gamma*(1 + (x - x0)**2/gamma**2))
assert cdf(X)(x) == atan((x - x0)/gamma)/pi + S.Half
assert diff(cdf(X)(x), x) == density(X)(x)
gamma = Symbol("gamma", positive=False)
raises(ValueError, lambda: Cauchy('x', x0, gamma))
def test_chi():
k = Symbol("k", integer=True)
X = Chi('x', k)
assert density(X)(x) == 2**(-k/2 + 1)*x**(k - 1)*exp(-x**2/2)/gamma(k/2)
k = Symbol("k", integer=True, positive=False)
raises(ValueError, lambda: Chi('x', k))
k = Symbol("k", integer=False, positive=True)
raises(ValueError, lambda: Chi('x', k))
def test_chi_noncentral():
k = Symbol("k", integer=True)
l = Symbol("l")
X = ChiNoncentral("x", k, l)
assert density(X)(x) == (x**k*l*(x*l)**(-k/2)*
exp(-x**2/2 - l**2/2)*besseli(k/2 - 1, x*l))
k = Symbol("k", integer=True, positive=False)
raises(ValueError, lambda: ChiNoncentral('x', k, l))
k = Symbol("k", integer=True, positive=True)
l = Symbol("l", positive=False)
raises(ValueError, lambda: ChiNoncentral('x', k, l))
k = Symbol("k", integer=False)
l = Symbol("l", positive=True)
raises(ValueError, lambda: ChiNoncentral('x', k, l))
def test_chi_squared():
k = Symbol("k", integer=True)
X = ChiSquared('x', k)
assert density(X)(x) == 2**(-k/2)*x**(k/2 - 1)*exp(-x/2)/gamma(k/2)
assert cdf(X)(x) == Piecewise((lowergamma(k/2, x/2)/gamma(k/2), x >= 0), (0, True))
assert E(X) == k
assert variance(X) == 2*k
X = ChiSquared('x', 15)
assert cdf(X)(3) == -14873*sqrt(6)*exp(-S(3)/2)/(5005*sqrt(pi)) + erf(sqrt(6)/2)
k = Symbol("k", integer=True, positive=False)
raises(ValueError, lambda: ChiSquared('x', k))
k = Symbol("k", integer=False, positive=True)
raises(ValueError, lambda: ChiSquared('x', k))
def test_dagum():
p = Symbol("p", positive=True)
b = Symbol("b", positive=True)
a = Symbol("a", positive=True)
X = Dagum('x', p, a, b)
assert density(X)(x) == a*p*(x/b)**(a*p)*((x/b)**a + 1)**(-p - 1)/x
assert cdf(X)(x) == Piecewise(((1 + (x/b)**(-a))**(-p), x >= 0),
(0, True))
p = Symbol("p", positive=False)
raises(ValueError, lambda: Dagum('x', p, a, b))
p = Symbol("p", positive=True)
b = Symbol("b", positive=False)
raises(ValueError, lambda: Dagum('x', p, a, b))
b = Symbol("b", positive=True)
a = Symbol("a", positive=False)
raises(ValueError, lambda: Dagum('x', p, a, b))
def test_erlang():
k = Symbol("k", integer=True, positive=True)
l = Symbol("l", positive=True)
X = Erlang("x", k, l)
assert density(X)(x) == x**(k - 1)*l**k*exp(-x*l)/gamma(k)
assert cdf(X)(x) == Piecewise((lowergamma(k, l*x)/gamma(k), x > 0),
(0, True))
def test_exponential():
rate = Symbol('lambda', positive=True, real=True, finite=True)
X = Exponential('x', rate)
assert E(X) == 1/rate
assert variance(X) == 1/rate**2
assert skewness(X) == 2
assert skewness(X) == smoment(X, 3)
assert smoment(2*X, 4) == smoment(X, 4)
assert moment(X, 3) == 3*2*1/rate**3
assert P(X > 0) == S(1)
assert P(X > 1) == exp(-rate)
assert P(X > 10) == exp(-10*rate)
assert where(X <= 1).set == Interval(0, 1)
def test_f_distribution():
d1 = Symbol("d1", positive=True)
d2 = Symbol("d2", positive=True)
X = FDistribution("x", d1, d2)
assert density(X)(x) == (d2**(d2/2)*sqrt((d1*x)**d1*(d1*x + d2)**(-d1 - d2))
/(x*beta(d1/2, d2/2)))
d1 = Symbol("d1", positive=False)
raises(ValueError, lambda: FDistribution('x', d1, d1))
d1 = Symbol("d1", positive=True, integer=False)
raises(ValueError, lambda: FDistribution('x', d1, d1))
d1 = Symbol("d1", positive=True)
d2 = Symbol("d2", positive=False)
raises(ValueError, lambda: FDistribution('x', d1, d2))
d2 = Symbol("d2", positive=True, integer=False)
raises(ValueError, lambda: FDistribution('x', d1, d2))
def test_fisher_z():
d1 = Symbol("d1", positive=True)
d2 = Symbol("d2", positive=True)
X = FisherZ("x", d1, d2)
assert density(X)(x) == (2*d1**(d1/2)*d2**(d2/2)*(d1*exp(2*x) + d2)
**(-d1/2 - d2/2)*exp(d1*x)/beta(d1/2, d2/2))
def test_frechet():
a = Symbol("a", positive=True)
s = Symbol("s", positive=True)
m = Symbol("m", real=True)
X = Frechet("x", a, s=s, m=m)
assert density(X)(x) == a*((x - m)/s)**(-a - 1)*exp(-((x - m)/s)**(-a))/s
assert cdf(X)(x) == Piecewise((exp(-((-m + x)/s)**(-a)), m <= x), (0, True))
def test_gamma():
k = Symbol("k", positive=True)
theta = Symbol("theta", positive=True)
X = Gamma('x', k, theta)
assert density(X)(x) == x**(k - 1)*theta**(-k)*exp(-x/theta)/gamma(k)
assert cdf(X, meijerg=True)(z) == Piecewise(
(-k*lowergamma(k, 0)/gamma(k + 1) +
k*lowergamma(k, z/theta)/gamma(k + 1), z >= 0),
(0, True))
# assert simplify(variance(X)) == k*theta**2 # handled numerically below
assert E(X) == moment(X, 1)
k, theta = symbols('k theta', real=True, finite=True, positive=True)
X = Gamma('x', k, theta)
assert E(X) == k*theta
assert variance(X) == k*theta**2
assert simplify(skewness(X)) == 2/sqrt(k)
def test_gamma_inverse():
a = Symbol("a", positive=True)
b = Symbol("b", positive=True)
X = GammaInverse("x", a, b)
assert density(X)(x) == x**(-a - 1)*b**a*exp(-b/x)/gamma(a)
assert cdf(X)(x) == Piecewise((uppergamma(a, b/x)/gamma(a), x > 0), (0, True))
def test_sampling_gamma_inverse():
scipy = import_module('scipy')
if not scipy:
skip('Scipy not installed. Abort tests for sampling of gamma inverse.')
X = GammaInverse("x", 1, 1)
assert sample(X) in X.pspace.domain.set
def test_gompertz():
b = Symbol("b", positive=True)
eta = Symbol("eta", positive=True)
X = Gompertz("x", b, eta)
assert density(X)(x) == b*eta*exp(eta)*exp(b*x)*exp(-eta*exp(b*x))
assert cdf(X)(x) == 1 - exp(eta)*exp(-eta*exp(b*x))
assert diff(cdf(X)(x), x) == density(X)(x)
def test_gumbel():
beta = Symbol("beta", positive=True)
mu = Symbol("mu")
x = Symbol("x")
X = Gumbel("x", beta, mu)
assert str(density(X)(x)) == 'exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta'
assert cdf(X)(x) == exp(-exp((mu - x)/beta))
def test_kumaraswamy():
a = Symbol("a", positive=True)
b = Symbol("b", positive=True)
X = Kumaraswamy("x", a, b)
assert density(X)(x) == x**(a - 1)*a*b*(-x**a + 1)**(b - 1)
assert cdf(X)(x) == Piecewise((0, x < 0),
(-(-x**a + 1)**b + 1, x <= 1),
(1, True))
def test_laplace():
mu = Symbol("mu")
b = Symbol("b", positive=True)
X = Laplace('x', mu, b)
assert density(X)(x) == exp(-Abs(x - mu)/b)/(2*b)
assert cdf(X)(x) == Piecewise((exp((-mu + x)/b)/2, mu > x),
(-exp((mu - x)/b)/2 + 1, True))
def test_logistic():
mu = Symbol("mu", real=True)
s = Symbol("s", positive=True)
X = Logistic('x', mu, s)
assert density(X)(x) == exp((-x + mu)/s)/(s*(exp((-x + mu)/s) + 1)**2)
assert cdf(X)(x) == 1/(exp((mu - x)/s) + 1)
def test_lognormal():
mean = Symbol('mu', real=True, finite=True)
std = Symbol('sigma', positive=True, real=True, finite=True)
X = LogNormal('x', mean, std)
# The sympy integrator can't do this too well
#assert E(X) == exp(mean+std**2/2)
#assert variance(X) == (exp(std**2)-1) * exp(2*mean + std**2)
# Right now, only density function and sampling works
# Test sampling: Only e^mean in sample std of 0
for i in range(3):
X = LogNormal('x', i, 0)
assert S(sample(X)) == N(exp(i))
# The sympy integrator can't do this too well
#assert E(X) ==
mu = Symbol("mu", real=True)
sigma = Symbol("sigma", positive=True)
X = LogNormal('x', mu, sigma)
assert density(X)(x) == (sqrt(2)*exp(-(-mu + log(x))**2
/(2*sigma**2))/(2*x*sqrt(pi)*sigma))
X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1
assert density(X)(x) == sqrt(2)*exp(-log(x)**2/2)/(2*x*sqrt(pi))
def test_maxwell():
a = Symbol("a", positive=True)
X = Maxwell('x', a)
assert density(X)(x) == (sqrt(2)*x**2*exp(-x**2/(2*a**2))/
(sqrt(pi)*a**3))
assert E(X) == 2*sqrt(2)*a/sqrt(pi)
assert simplify(variance(X)) == a**2*(-8 + 3*pi)/pi
assert cdf(X)(x) == erf(sqrt(2)*x/(2*a)) - sqrt(2)*x*exp(-x**2/(2*a**2))/(sqrt(pi)*a)
assert diff(cdf(X)(x), x) == density(X)(x)
def test_nakagami():
mu = Symbol("mu", positive=True)
omega = Symbol("omega", positive=True)
X = Nakagami('x', mu, omega)
assert density(X)(x) == (2*x**(2*mu - 1)*mu**mu*omega**(-mu)
*exp(-x**2*mu/omega)/gamma(mu))
assert simplify(E(X)) == (sqrt(mu)*sqrt(omega)
*gamma(mu + S.Half)/gamma(mu + 1))
assert simplify(variance(X)) == (
omega - omega*gamma(mu + S(1)/2)**2/(gamma(mu)*gamma(mu + 1)))
assert cdf(X)(x) == Piecewise(
(lowergamma(mu, mu*x**2/omega)/gamma(mu), x > 0),
(0, True))
def test_pareto():
xm, beta = symbols('xm beta', positive=True, finite=True)
alpha = beta + 5
X = Pareto('x', xm, alpha)
dens = density(X)
x = Symbol('x')
assert dens(x) == x**(-(alpha + 1))*xm**(alpha)*(alpha)
assert simplify(E(X)) == alpha*xm/(alpha-1)
# computation of taylor series for MGF still too slow
#assert simplify(variance(X)) == xm**2*alpha / ((alpha-1)**2*(alpha-2))
def test_pareto_numeric():
xm, beta = 3, 2
alpha = beta + 5
X = Pareto('x', xm, alpha)
assert E(X) == alpha*xm/S(alpha - 1)
assert variance(X) == xm**2*alpha / S(((alpha - 1)**2*(alpha - 2)))
# Skewness tests too slow. Try shortcutting function?
def test_raised_cosine():
mu = Symbol("mu", real=True)
s = Symbol("s", positive=True)
X = RaisedCosine("x", mu, s)
assert density(X)(x) == (Piecewise(((cos(pi*(x - mu)/s) + 1)/(2*s),
And(x <= mu + s, mu - s <= x)), (0, True)))
def test_rayleigh():
sigma = Symbol("sigma", positive=True)
X = Rayleigh('x', sigma)
assert density(X)(x) == x*exp(-x**2/(2*sigma**2))/sigma**2
assert E(X) == sqrt(2)*sqrt(pi)*sigma/2
assert variance(X) == -pi*sigma**2/2 + 2*sigma**2
assert cdf(X)(x) == 1 - exp(-x**2/(2*sigma**2))
assert diff(cdf(X)(x), x) == density(X)(x)
def test_shiftedgompertz():
b = Symbol("b", positive=True)
eta = Symbol("eta", positive=True)
X = ShiftedGompertz("x", b, eta)
assert density(X)(x) == b*(eta*(1 - exp(-b*x)) + 1)*exp(-b*x)*exp(-eta*exp(-b*x))
def test_studentt():
nu = Symbol("nu", positive=True)
X = StudentT('x', nu)
assert density(X)(x) == (1 + x**2/nu)**(-nu/2 - S(1)/2)/(sqrt(nu)*beta(S(1)/2, nu/2))
assert cdf(X)(x) == S(1)/2 + x*gamma(nu/2 + S(1)/2)*hyper((S(1)/2, nu/2 + S(1)/2),
(S(3)/2,), -x**2/nu)/(sqrt(pi)*sqrt(nu)*gamma(nu/2))
def test_trapezoidal():
a = Symbol("a", real=True)
b = Symbol("b", real=True)
c = Symbol("c", real=True)
d = Symbol("d", real=True)
X = Trapezoidal('x', a, b, c, d)
assert density(X)(x) == Piecewise(((-2*a + 2*x)/((-a + b)*(-a - b + c + d)), (a <= x) & (x < b)),
(2/(-a - b + c + d), (b <= x) & (x < c)),
((2*d - 2*x)/((-c + d)*(-a - b + c + d)), (c <= x) & (x <= d)),
(0, True))
X = Trapezoidal('x', 0, 1, 2, 3)
assert E(X) == S(3)/2
assert variance(X) == S(5)/12
assert P(X < 2) == S(3)/4
@XFAIL
def test_triangular():
a = Symbol("a")
b = Symbol("b")
c = Symbol("c")
X = Triangular('x', a, b, c)
assert density(X)(x) == Piecewise(
((2*x - 2*a)/((-a + b)*(-a + c)), And(a <= x, x < c)),
(2/(-a + b), x == c),
((-2*x + 2*b)/((-a + b)*(b - c)), And(x <= b, c < x)),
(0, True))
def test_quadratic_u():
a = Symbol("a", real=True)
b = Symbol("b", real=True)
X = QuadraticU("x", a, b)
assert density(X)(x) == (Piecewise((12*(x - a/2 - b/2)**2/(-a + b)**3,
And(x <= b, a <= x)), (0, True)))
def test_uniform():
l = Symbol('l', real=True, finite=True)
w = Symbol('w', positive=True, finite=True)
X = Uniform('x', l, l + w)
assert simplify(E(X)) == l + w/2
assert simplify(variance(X)) == w**2/12
# With numbers all is well
X = Uniform('x', 3, 5)
assert P(X < 3) == 0 and P(X > 5) == 0
assert P(X < 4) == P(X > 4) == S.Half
z = Symbol('z')
p = density(X)(z)
assert p.subs(z, 3.7) == S(1)/2
assert p.subs(z, -1) == 0
assert p.subs(z, 6) == 0
c = cdf(X)
assert c(2) == 0 and c(3) == 0
assert c(S(7)/2) == S(1)/4
assert c(5) == 1 and c(6) == 1
def test_uniform_P():
""" This stopped working because SingleContinuousPSpace.compute_density no
longer calls integrate on a DiracDelta but rather just solves directly.
integrate used to call UniformDistribution.expectation which special-cased
subsed out the Min and Max terms that Uniform produces
I decided to regress on this class for general cleanliness (and I suspect
speed) of the algorithm.
"""
l = Symbol('l', real=True, finite=True)
w = Symbol('w', positive=True, finite=True)
X = Uniform('x', l, l + w)
assert P(X < l) == 0 and P(X > l + w) == 0
@XFAIL
def test_uniformsum():
n = Symbol("n", integer=True)
_k = Symbol("k")
X = UniformSum('x', n)
assert density(X)(x) == (Sum((-1)**_k*(-_k + x)**(n - 1)
*binomial(n, _k), (_k, 0, floor(x)))/factorial(n - 1))
def test_von_mises():
mu = Symbol("mu")
k = Symbol("k", positive=True)
X = VonMises("x", mu, k)
assert density(X)(x) == exp(k*cos(x - mu))/(2*pi*besseli(0, k))
def test_weibull():
a, b = symbols('a b', positive=True)
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(a**2 * gamma(1 + 2/b) - E(X)**2)
assert simplify(skewness(X)) == (2*gamma(1 + 1/b)**3 - 3*gamma(1 + 1/b)*gamma(1 + 2/b) + gamma(1 + 3/b))/(-gamma(1 + 1/b)**2 + gamma(1 + 2/b))**(S(3)/2)
def test_weibull_numeric():
# Test for integers and rationals
a = 1
bvals = [S.Half, 1, S(3)/2, 5]
for b in bvals:
X = Weibull('x', a, b)
assert simplify(E(X)) == expand_func(a * gamma(1 + 1/S(b)))
assert simplify(variance(X)) == simplify(
a**2 * gamma(1 + 2/S(b)) - E(X)**2)
# Not testing Skew... it's slow with int/frac values > 3/2
def test_wignersemicircle():
R = Symbol("R", positive=True)
X = WignerSemicircle('x', R)
assert density(X)(x) == 2*sqrt(-x**2 + R**2)/(pi*R**2)
assert E(X) == 0
def test_prefab_sampling():
N = Normal('X', 0, 1)
L = LogNormal('L', 0, 1)
E = Exponential('Ex', 1)
P = Pareto('P', 1, 3)
W = Weibull('W', 1, 1)
U = Uniform('U', 0, 1)
B = Beta('B', 2, 5)
G = Gamma('G', 1, 3)
variables = [N, L, E, P, W, U, B, G]
niter = 10
for var in variables:
for i in range(niter):
assert sample(var) in var.pspace.domain.set
def test_input_value_assertions():
a, b = symbols('a b')
p, q = symbols('p q', positive=True)
m, n = symbols('m n', positive=False, real=True)
raises(ValueError, lambda: Normal('x', 3, 0))
raises(ValueError, lambda: Normal('x', m, n))
Normal('X', a, p) # No error raised
raises(ValueError, lambda: Exponential('x', m))
Exponential('Ex', p) # No error raised
for fn in [Pareto, Weibull, Beta, Gamma]:
raises(ValueError, lambda: fn('x', m, p))
raises(ValueError, lambda: fn('x', p, n))
fn('x', p, q) # No error raised
@XFAIL
def test_unevaluated():
X = Normal('x', 0, 1)
assert E(X, evaluate=False) == (
Integral(sqrt(2)*x*exp(-x**2/2)/(2*sqrt(pi)), (x, -oo, oo)))
assert E(X + 1, evaluate=False) == (
Integral(sqrt(2)*x*exp(-x**2/2)/(2*sqrt(pi)), (x, -oo, oo)) + 1)
assert P(X > 0, evaluate=False) == (
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)), (x, 0, oo)))
assert P(X > 0, X**2 < 1, evaluate=False) == (
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)*
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)),
(x, -1, 1))), (x, 0, 1)))
def test_probability_unevaluated():
T = Normal('T', 30, 3)
assert type(P(T > 33, evaluate=False)) == Integral
def test_density_unevaluated():
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 2)
assert isinstance(density(X+Y, evaluate=False)(z), Integral)
def test_NormalDistribution():
nd = NormalDistribution(0, 1)
x = Symbol('x')
assert nd.cdf(x) == erf(sqrt(2)*x/2)/2 + S.One/2
assert isinstance(nd.sample(), float) or nd.sample().is_Number
assert nd.expectation(1, x) == 1
assert nd.expectation(x, x) == 0
assert nd.expectation(x**2, x) == 1
def test_random_parameters():
mu = Normal('mu', 2, 3)
meas = Normal('T', mu, 1)
assert density(meas, evaluate=False)(z)
assert isinstance(pspace(meas), JointPSpace)
#assert density(meas, evaluate=False)(z) == Integral(mu.pspace.pdf *
# meas.pspace.pdf, (mu.symbol, -oo, oo)).subs(meas.symbol, z)
def test_random_parameters_given():
mu = Normal('mu', 2, 3)
meas = Normal('T', mu, 1)
assert given(meas, Eq(mu, 5)) == Normal('T', 5, 1)
def test_conjugate_priors():
mu = Normal('mu', 2, 3)
x = Normal('x', mu, 1)
assert isinstance(simplify(density(mu, Eq(x, y), evaluate=False)(z)),
Mul)
def test_difficult_univariate():
""" Since using solve in place of deltaintegrate we're able to perform
substantially more complex density computations on single continuous random
variables """
x = Normal('x', 0, 1)
assert density(x**3)
assert density(exp(x**2))
assert density(log(x))
def test_issue_10003():
X = Exponential('x', 3)
G = Gamma('g', 1, 2)
assert P(X < -1) == S.Zero
assert P(G < -1) == S.Zero
@slow
def test_precomputed_cdf():
x = symbols("x", real=True, finite=True)
mu = symbols("mu", real=True, finite=True)
sigma, xm, alpha = symbols("sigma xm alpha", positive=True, finite=True)
n = symbols("n", integer=True, positive=True, finite=True)
distribs = [
Normal("X", mu, sigma),
Pareto("P", xm, alpha),
ChiSquared("C", n),
Exponential("E", sigma),
# LogNormal("L", mu, sigma),
]
for X in distribs:
compdiff = cdf(X)(x) - simplify(X.pspace.density.compute_cdf()(x))
compdiff = simplify(compdiff.rewrite(erfc))
assert compdiff == 0
@slow
def test_precomputed_characteristic_functions():
import mpmath
def test_cf(dist, support_lower_limit, support_upper_limit):
pdf = density(dist)
t = Symbol('t')
x = Symbol('x')
# first function is the hardcoded CF of the distribution
cf1 = lambdify([t], characteristic_function(dist)(t), 'mpmath')
# second function is the Fourier transform of the density function
f = lambdify([x, t], pdf(x)*exp(I*x*t), 'mpmath')
cf2 = lambda t: mpmath.quad(lambda x: f(x, t), [support_lower_limit, support_upper_limit], maxdegree=10)
# compare the two functions at various points
for test_point in [2, 5, 8, 11]:
n1 = cf1(test_point)
n2 = cf2(test_point)
assert abs(re(n1) - re(n2)) < 1e-12
assert abs(im(n1) - im(n2)) < 1e-12
test_cf(Beta('b', 1, 2), 0, 1)
test_cf(Chi('c', 3), 0, mpmath.inf)
test_cf(ChiSquared('c', 2), 0, mpmath.inf)
test_cf(Exponential('e', 6), 0, mpmath.inf)
test_cf(Logistic('l', 1, 2), -mpmath.inf, mpmath.inf)
test_cf(Normal('n', -1, 5), -mpmath.inf, mpmath.inf)
test_cf(RaisedCosine('r', 3, 1), 2, 4)
test_cf(Rayleigh('r', 0.5), 0, mpmath.inf)
test_cf(Uniform('u', -1, 1), -1, 1)
test_cf(WignerSemicircle('w', 3), -3, 3)
def test_long_precomputed_cdf():
x = symbols("x", real=True, finite=True)
distribs = [
Arcsin("A", -5, 9),
Dagum("D", 4, 10, 3),
Erlang("E", 14, 5),
Frechet("F", 2, 6, -3),
Gamma("G", 2, 7),
GammaInverse("GI", 3, 5),
Kumaraswamy("K", 6, 8),
Laplace("LA", -5, 4),
Logistic("L", -6, 7),
Nakagami("N", 2, 7),
StudentT("S", 4)
]
for distr in distribs:
for _ in range(5):
assert tn(diff(cdf(distr)(x), x), density(distr)(x), x, a=0, b=0, c=1, d=0)
US = UniformSum("US", 5)
pdf01 = density(US)(x).subs(floor(x), 0).doit() # pdf on (0, 1)
cdf01 = cdf(US, evaluate=False)(x).subs(floor(x), 0).doit() # cdf on (0, 1)
assert tn(diff(cdf01, x), pdf01, x, a=0, b=0, c=1, d=0)
def test_issue_13324():
X = Uniform('X', 0, 1)
assert E(X, X > Rational(1, 2)) == Rational(3, 4)
assert E(X, X > 0) == Rational(1, 2)
def test_FiniteSet_prob():
x = symbols('x')
E = Exponential('E', 3)
N = Normal('N', 5, 7)
assert P(Eq(E, 1)) is S.Zero
assert P(Eq(N, 2)) is S.Zero
assert P(Eq(N, x)) is S.Zero
def test_prob_neq():
E = Exponential('E', 4)
X = ChiSquared('X', 4)
x = symbols('x')
assert P(Ne(E, 2)) == 1
assert P(Ne(X, 4)) == 1
assert P(Ne(X, 4)) == 1
assert P(Ne(X, 5)) == 1
assert P(Ne(E, x)) == 1
def test_union():
N = Normal('N', 3, 2)
assert simplify(P(N**2 - N > 2)) == \
-erf(sqrt(2))/2 - erfc(sqrt(2)/4)/2 + S(3)/2
assert simplify(P(N**2 - 4 > 0)) == \
-erf(5*sqrt(2)/4)/2 - erfc(sqrt(2)/4)/2 + S(3)/2
def test_Or():
N = Normal('N', 0, 1)
assert simplify(P(Or(N > 2, N < 1))) == \
-erf(sqrt(2))/2 - erfc(sqrt(2)/2)/2 + S(3)/2
assert P(Or(N < 0, N < 1)) == P(N < 1)
assert P(Or(N > 0, N < 0)) == 1
def test_conditional_eq():
E = Exponential('E', 1)
assert P(Eq(E, 1), Eq(E, 1)) == 1
assert P(Eq(E, 1), Eq(E, 2)) == 0
assert P(E > 1, Eq(E, 2)) == 1
assert P(E < 1, Eq(E, 2)) == 0
|
981498310288bc3ce46bb6b79877daf35e95bc99004ecb325c091187b7385c78 | from sympy import S
from sympy.combinatorics.fp_groups import (FpGroup, low_index_subgroups,
reidemeister_presentation, FpSubgroup,
simplify_presentation)
from sympy.combinatorics.free_groups import (free_group, FreeGroup)
from sympy.utilities.pytest import slow
"""
References
==========
[1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
[2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490.
"Implementation and Analysis of the Todd-Coxeter Algorithm"
[3] PROC. SECOND INTERNAT. CONF. THEORY OF GROUPS, CANBERRA 1973,
pp. 347-356. "A Reidemeister-Schreier program" by George Havas.
http://staff.itee.uq.edu.au/havas/1973cdhw.pdf
"""
def test_low_index_subgroups():
F, x, y = free_group("x, y")
# Example 5.10 from [1] Pg. 194
f = FpGroup(F, [x**2, y**3, (x*y)**4])
L = low_index_subgroups(f, 4)
t1 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]],
[[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]],
[[1, 1, 0, 0], [0, 0, 1, 1]]]
for i in range(len(t1)):
assert L[i].table == t1[i]
f = FpGroup(F, [x**2, y**3, (x*y)**7])
L = low_index_subgroups(f, 15)
t2 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[4, 4, 5, 3], [6, 6, 3, 4], [5, 5, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [5, 5, 3, 4], [4, 4, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [9, 9, 6, 8], [12, 12, 11, 7], [8, 8, 7, 10],
[10, 10, 13, 14], [14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [12, 12, 6, 8], [10, 10, 11, 7], [8, 8, 7, 10],
[9, 9, 13, 14], [14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [12, 12, 6, 8], [13, 13, 11, 7], [8, 8, 7, 10],
[9, 9, 12, 12], [10, 10, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 3, 3], [2, 2, 5, 6]
, [7, 7, 6, 4], [8, 8, 4, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [11, 11, 12, 10], [13, 13, 10, 11],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 3, 3], [2, 2, 5, 6]
, [7, 7, 6, 4], [8, 8, 4, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [13, 13, 12, 10], [12, 12, 10, 11],
[11, 11, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 4, 4]
, [7, 7, 6, 3], [8, 8, 3, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [13, 13, 12, 10], [12, 12, 10, 11],
[11, 11, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [5, 5, 6, 3], [9, 9, 3, 5], [10, 10, 8, 4], [8, 8, 4, 7],
[6, 6, 10, 11], [7, 7, 11, 9], [12, 12, 9, 10], [11, 11, 13, 14],
[14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [6, 6, 6, 3], [5, 5, 3, 5], [8, 8, 8, 4], [7, 7, 4, 7]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [6, 6, 3, 5], [10, 10, 8, 4], [11, 11, 4, 7],
[5, 5, 10, 12], [7, 7, 12, 9], [8, 8, 11, 11], [13, 13, 9, 10],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [6, 6, 3, 5], [10, 10, 8, 4], [11, 11, 4, 7],
[5, 5, 12, 11], [7, 7, 10, 10], [8, 8, 9, 12], [13, 13, 11, 9],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [7, 7, 8, 4], [11, 11, 4, 7],
[5, 5, 9, 9], [6, 6, 11, 12], [8, 8, 12, 10], [13, 13, 10, 11],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [7, 7, 8, 4], [11, 11, 4, 7],
[5, 5, 12, 11], [6, 6, 10, 10], [8, 8, 9, 12], [13, 13, 11, 9],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [11, 11, 8, 4], [12, 12, 4, 7],
[5, 5, 9, 9], [6, 6, 12, 13], [7, 7, 11, 11], [8, 8, 13, 10],
[13, 13, 10, 12]],
[[1, 1, 0, 0], [0, 0, 2, 3], [4, 4, 3, 1], [5, 5, 1, 2], [2, 2, 4, 4]
, [3, 3, 6, 7], [7, 7, 7, 5], [6, 6, 5, 6]]]
for i in range(len(t2)):
assert L[i].table == t2[i]
f = FpGroup(F, [x**2, y**3, (x*y)**7])
L = low_index_subgroups(f, 10, [x])
t3 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5], [4, 4, 5, 3],
[6, 6, 3, 4], [5, 5, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5], [6, 6, 5, 3],
[5, 5, 3, 4], [4, 4, 6, 6]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8],
[6, 6, 6, 3], [5, 5, 3, 5], [8, 8, 8, 4], [7, 7, 4, 7]]]
for i in range(len(t3)):
assert L[i].table == t3[i]
def test_subgroup_presentations():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
p1 = reidemeister_presentation(f, H)
assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"
H = f.subgroup(H)
assert (H.generators, H.relators) == p1
f = FpGroup(F, [x**3, y**3, (x*y)**3])
H = [x*y, x*y**-1]
p2 = reidemeister_presentation(f, H)
assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"
f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
H = [x]
p3 = reidemeister_presentation(f, H)
assert str(p3) == "((x_0,), (x_0**4,))"
f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
H = [x]
p4 = reidemeister_presentation(f, H)
assert str(p4) == "((x_0,), (x_0**6,))"
# this presentation can be improved, the most simplified form
# of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
# See [2] Pg 474 group PSL_2(11)
# This is the group PSL_2(11)
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
H = [a, b, c**2]
gens, rels = reidemeister_presentation(f, H)
assert str(gens) == "(b_1, c_3)"
assert len(rels) == 18
@slow
def test_order():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert f.order() == 8
f = FpGroup(F, [x*y*x**-1*y**-1, y**2])
assert f.order() == S.Infinity
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**250, b**2, c*b*c**-1*b, c**4, c**-1*a**-1*c*a, a**-1*b**-1*a*b])
assert f.order() == 2000
F, x = free_group("x")
f = FpGroup(F, [])
assert f.order() == S.Infinity
f = FpGroup(free_group('')[0], [])
assert f.order() == 1
def test_fp_subgroup():
def _test_subgroup(K, T, S):
_gens = T(K.generators)
assert all(elem in S for elem in _gens)
assert T.is_injective()
assert T.image().order() == S.order()
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
S = FpSubgroup(f, [x*y])
assert (x*y)**-3 in S
K, T = f.subgroup([x*y], homomorphism=True)
assert T(K.generators) == [y*x**-1]
_test_subgroup(K, T, S)
S = FpSubgroup(f, [x**-1*y*x])
assert x**-1*y**4*x in S
assert x**-1*y**4*x**2 not in S
K, T = f.subgroup([x**-1*y*x], homomorphism=True)
assert T(K.generators[0]**3) == y**3
_test_subgroup(K, T, S)
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
K, T = f.subgroup(H, homomorphism=True)
S = FpSubgroup(f, H)
_test_subgroup(K, T, S)
def test_permutation_methods():
from sympy.combinatorics.fp_groups import FpSubgroup
F, x, y = free_group("x, y")
# DihedralGroup(8)
G = FpGroup(F, [x**2, y**8, x*y*x**-1*y])
T = G._to_perm_group()[1]
assert T.is_isomorphism()
assert G.center() == [y**4]
# DiheadralGroup(4)
G = FpGroup(F, [x**2, y**4, x*y*x**-1*y])
S = FpSubgroup(G, G.normal_closure([x]))
assert x in S
assert y**-1*x*y in S
# Z_5xZ_4
G = FpGroup(F, [x*y*x**-1*y**-1, y**5, x**4])
assert G.is_abelian
assert G.is_solvable
# AlternatingGroup(5)
G = FpGroup(F, [x**3, y**2, (x*y)**5])
assert not G.is_solvable
# AlternatingGroup(4)
G = FpGroup(F, [x**3, y**2, (x*y)**3])
assert len(G.derived_series()) == 3
S = FpSubgroup(G, G.derived_subgroup())
assert S.order() == 4
def test_simplify_presentation():
# ref #16083
G = simplify_presentation(FpGroup(FreeGroup([]), []))
assert not G.generators
assert not G.relators
def test_cyclic():
F, x, y = free_group("x, y")
f = FpGroup(F, [x*y, x**-1*y**-1*x*y*x])
assert f.is_cyclic
f = FpGroup(F, [x*y, x*y**-1])
assert f.is_cyclic
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert not f.is_cyclic
|
b648081919c7461ee7e43de8d04da79c121b3f0d96a66181099d0ef537bab0ab | from sympy.core.compatibility import range
from sympy.combinatorics.perm_groups import (PermutationGroup,
_orbit_transversal)
from sympy.combinatorics.named_groups import SymmetricGroup, CyclicGroup,\
DihedralGroup, AlternatingGroup, AbelianGroup, RubikGroup
from sympy.combinatorics.permutations import Permutation
from sympy.utilities.pytest import skip, XFAIL
from sympy.combinatorics.generators import rubik_cube_generators
from sympy.combinatorics.polyhedron import tetrahedron as Tetra, cube
from sympy.combinatorics.testutil import _verify_bsgs, _verify_centralizer,\
_verify_normal_closure
from sympy.utilities.pytest import raises, slow
rmul = Permutation.rmul
def test_has():
a = Permutation([1, 0])
G = PermutationGroup([a])
assert G.is_abelian
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
G = PermutationGroup([a, b])
assert not G.is_abelian
G = PermutationGroup([a])
assert G.has(a)
assert not G.has(b)
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([0, 2, 1, 3, 4])
assert PermutationGroup(a, b).degree == \
PermutationGroup(a, b).degree == 6
def test_generate():
a = Permutation([1, 0])
g = list(PermutationGroup([a]).generate())
assert g == [Permutation([0, 1]), Permutation([1, 0])]
assert len(list(PermutationGroup(Permutation((0, 1))).generate())) == 1
g = PermutationGroup([a]).generate(method='dimino')
assert list(g) == [Permutation([0, 1]), Permutation([1, 0])]
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
G = PermutationGroup([a, b])
g = G.generate()
v1 = [p.array_form for p in list(g)]
v1.sort()
assert v1 == [[0, 1, 2], [0, 2, 1], [1, 0, 2], [1, 2, 0], [2, 0,
1], [2, 1, 0]]
v2 = list(G.generate(method='dimino', af=True))
assert v1 == sorted(v2)
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
g = PermutationGroup([a, b]).generate(af=True)
assert len(list(g)) == 360
def test_order():
a = Permutation([2, 0, 1, 3, 4, 5, 6, 7, 8, 9])
b = Permutation([2, 1, 3, 4, 5, 6, 7, 8, 9, 0])
g = PermutationGroup([a, b])
assert g.order() == 1814400
assert PermutationGroup().order() == 1
def test_equality():
p_1 = Permutation(0, 1, 3)
p_2 = Permutation(0, 2, 3)
p_3 = Permutation(0, 1, 2)
p_4 = Permutation(0, 1, 3)
g_1 = PermutationGroup(p_1, p_2)
g_2 = PermutationGroup(p_3, p_4)
g_3 = PermutationGroup(p_2, p_1)
assert g_1 == g_2
assert g_1.generators != g_2.generators
assert g_1 == g_3
def test_stabilizer():
S = SymmetricGroup(2)
H = S.stabilizer(0)
assert H.generators == [Permutation(1)]
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
G = PermutationGroup([a, b])
G0 = G.stabilizer(0)
assert G0.order() == 60
gens_cube = [[1, 3, 5, 7, 0, 2, 4, 6], [1, 3, 0, 2, 5, 7, 4, 6]]
gens = [Permutation(p) for p in gens_cube]
G = PermutationGroup(gens)
G2 = G.stabilizer(2)
assert G2.order() == 6
G2_1 = G2.stabilizer(1)
v = list(G2_1.generate(af=True))
assert v == [[0, 1, 2, 3, 4, 5, 6, 7], [3, 1, 2, 0, 7, 5, 6, 4]]
gens = (
(1, 2, 0, 4, 5, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19),
(0, 1, 2, 3, 4, 5, 19, 6, 8, 9, 10, 11, 12, 13, 14,
15, 16, 7, 17, 18),
(0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 16, 11, 12, 13, 14, 15, 8, 17, 10, 19))
gens = [Permutation(p) for p in gens]
G = PermutationGroup(gens)
G2 = G.stabilizer(2)
assert G2.order() == 181440
S = SymmetricGroup(3)
assert [G.order() for G in S.basic_stabilizers] == [6, 2]
def test_center():
# the center of the dihedral group D_n is of order 2 for even n
for i in (4, 6, 10):
D = DihedralGroup(i)
assert (D.center()).order() == 2
# the center of the dihedral group D_n is of order 1 for odd n>2
for i in (3, 5, 7):
D = DihedralGroup(i)
assert (D.center()).order() == 1
# the center of an abelian group is the group itself
for i in (2, 3, 5):
for j in (1, 5, 7):
for k in (1, 1, 11):
G = AbelianGroup(i, j, k)
assert G.center().is_subgroup(G)
# the center of a nonabelian simple group is trivial
for i in(1, 5, 9):
A = AlternatingGroup(i)
assert (A.center()).order() == 1
# brute-force verifications
D = DihedralGroup(5)
A = AlternatingGroup(3)
C = CyclicGroup(4)
G.is_subgroup(D*A*C)
assert _verify_centralizer(G, G)
def test_centralizer():
# the centralizer of the trivial group is the entire group
S = SymmetricGroup(2)
assert S.centralizer(Permutation(list(range(2)))).is_subgroup(S)
A = AlternatingGroup(5)
assert A.centralizer(Permutation(list(range(5)))).is_subgroup(A)
# a centralizer in the trivial group is the trivial group itself
triv = PermutationGroup([Permutation([0, 1, 2, 3])])
D = DihedralGroup(4)
assert triv.centralizer(D).is_subgroup(triv)
# brute-force verifications for centralizers of groups
for i in (4, 5, 6):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
D = DihedralGroup(i)
for gp in (S, A, C, D):
for gp2 in (S, A, C, D):
if not gp2.is_subgroup(gp):
assert _verify_centralizer(gp, gp2)
# verify the centralizer for all elements of several groups
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_centralizer(S, element)
A = AlternatingGroup(5)
elements = list(A.generate_dimino())
for element in elements:
assert _verify_centralizer(A, element)
D = DihedralGroup(7)
elements = list(D.generate_dimino())
for element in elements:
assert _verify_centralizer(D, element)
# verify centralizers of small groups within small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp.degree == gp2.degree:
assert _verify_centralizer(gp, gp2)
def test_coset_rank():
gens_cube = [[1, 3, 5, 7, 0, 2, 4, 6], [1, 3, 0, 2, 5, 7, 4, 6]]
gens = [Permutation(p) for p in gens_cube]
G = PermutationGroup(gens)
i = 0
for h in G.generate(af=True):
rk = G.coset_rank(h)
assert rk == i
h1 = G.coset_unrank(rk, af=True)
assert h == h1
i += 1
assert G.coset_unrank(48) == None
assert G.coset_unrank(G.coset_rank(gens[0])) == gens[0]
def test_coset_factor():
a = Permutation([0, 2, 1])
G = PermutationGroup([a])
c = Permutation([2, 1, 0])
assert not G.coset_factor(c)
assert G.coset_rank(c) is None
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
g = PermutationGroup([a, b])
assert g.order() == 360
d = Permutation([1, 0, 2, 3, 4, 5])
assert not g.coset_factor(d.array_form)
assert not g.contains(d)
assert Permutation(2) in G
c = Permutation([1, 0, 2, 3, 5, 4])
v = g.coset_factor(c, True)
tr = g.basic_transversals
p = Permutation.rmul(*[tr[i][v[i]] for i in range(len(g.base))])
assert p == c
v = g.coset_factor(c)
p = Permutation.rmul(*v)
assert p == c
assert g.contains(c)
G = PermutationGroup([Permutation([2, 1, 0])])
p = Permutation([1, 0, 2])
assert G.coset_factor(p) == []
def test_orbits():
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
g = PermutationGroup([a, b])
assert g.orbit(0) == {0, 1, 2}
assert g.orbits() == [{0, 1, 2}]
assert g.is_transitive() and g.is_transitive(strict=False)
assert g.orbit_transversal(0) == \
[Permutation(
[0, 1, 2]), Permutation([2, 0, 1]), Permutation([1, 2, 0])]
assert g.orbit_transversal(0, True) == \
[(0, Permutation([0, 1, 2])), (2, Permutation([2, 0, 1])),
(1, Permutation([1, 2, 0]))]
G = DihedralGroup(6)
transversal, slps = _orbit_transversal(G.degree, G.generators, 0, True, slp=True)
for i, t in transversal:
slp = slps[i]
w = G.identity
for s in slp:
w = G.generators[s]*w
assert w == t
a = Permutation(list(range(1, 100)) + [0])
G = PermutationGroup([a])
assert [min(o) for o in G.orbits()] == [0]
G = PermutationGroup(rubik_cube_generators())
assert [min(o) for o in G.orbits()] == [0, 1]
assert not G.is_transitive() and not G.is_transitive(strict=False)
G = PermutationGroup([Permutation(0, 1, 3), Permutation(3)(0, 1)])
assert not G.is_transitive() and G.is_transitive(strict=False)
assert PermutationGroup(
Permutation(3)).is_transitive(strict=False) is False
def test_is_normal():
gens_s5 = [Permutation(p) for p in [[1, 2, 3, 4, 0], [2, 1, 4, 0, 3]]]
G1 = PermutationGroup(gens_s5)
assert G1.order() == 120
gens_a5 = [Permutation(p) for p in [[1, 0, 3, 2, 4], [2, 1, 4, 3, 0]]]
G2 = PermutationGroup(gens_a5)
assert G2.order() == 60
assert G2.is_normal(G1)
gens3 = [Permutation(p) for p in [[2, 1, 3, 0, 4], [1, 2, 0, 3, 4]]]
G3 = PermutationGroup(gens3)
assert not G3.is_normal(G1)
assert G3.order() == 12
G4 = G1.normal_closure(G3.generators)
assert G4.order() == 60
gens5 = [Permutation(p) for p in [[1, 2, 3, 0, 4], [1, 2, 0, 3, 4]]]
G5 = PermutationGroup(gens5)
assert G5.order() == 24
G6 = G1.normal_closure(G5.generators)
assert G6.order() == 120
assert G1.is_subgroup(G6)
assert not G1.is_subgroup(G4)
assert G2.is_subgroup(G4)
I5 = PermutationGroup(Permutation(4))
assert I5.is_normal(G5)
assert I5.is_normal(G6, strict=False)
p1 = Permutation([1, 0, 2, 3, 4])
p2 = Permutation([0, 1, 2, 4, 3])
p3 = Permutation([3, 4, 2, 1, 0])
id_ = Permutation([0, 1, 2, 3, 4])
H = PermutationGroup([p1, p3])
H_n1 = PermutationGroup([p1, p2])
H_n2_1 = PermutationGroup(p1)
H_n2_2 = PermutationGroup(p2)
H_id = PermutationGroup(id_)
assert H_n1.is_normal(H)
assert H_n2_1.is_normal(H_n1)
assert H_n2_2.is_normal(H_n1)
assert H_id.is_normal(H_n2_1)
assert H_id.is_normal(H_n1)
assert H_id.is_normal(H)
assert not H_n2_1.is_normal(H)
assert not H_n2_2.is_normal(H)
def test_eq():
a = [[1, 2, 0, 3, 4, 5], [1, 0, 2, 3, 4, 5], [2, 1, 0, 3, 4, 5], [
1, 2, 0, 3, 4, 5]]
a = [Permutation(p) for p in a + [[1, 2, 3, 4, 5, 0]]]
g = Permutation([1, 2, 3, 4, 5, 0])
G1, G2, G3 = [PermutationGroup(x) for x in [a[:2], a[2:4], [g, g**2]]]
assert G1.order() == G2.order() == G3.order() == 6
assert G1.is_subgroup(G2)
assert not G1.is_subgroup(G3)
G4 = PermutationGroup([Permutation([0, 1])])
assert not G1.is_subgroup(G4)
assert G4.is_subgroup(G1, 0)
assert PermutationGroup(g, g).is_subgroup(PermutationGroup(g))
assert SymmetricGroup(3).is_subgroup(SymmetricGroup(4), 0)
assert SymmetricGroup(3).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)
assert not CyclicGroup(5).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)
assert CyclicGroup(3).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)
def test_derived_subgroup():
a = Permutation([1, 0, 2, 4, 3])
b = Permutation([0, 1, 3, 2, 4])
G = PermutationGroup([a, b])
C = G.derived_subgroup()
assert C.order() == 3
assert C.is_normal(G)
assert C.is_subgroup(G, 0)
assert not G.is_subgroup(C, 0)
gens_cube = [[1, 3, 5, 7, 0, 2, 4, 6], [1, 3, 0, 2, 5, 7, 4, 6]]
gens = [Permutation(p) for p in gens_cube]
G = PermutationGroup(gens)
C = G.derived_subgroup()
assert C.order() == 12
def test_is_solvable():
a = Permutation([1, 2, 0])
b = Permutation([1, 0, 2])
G = PermutationGroup([a, b])
assert G.is_solvable
G = PermutationGroup([a])
assert G.is_solvable
a = Permutation([1, 2, 3, 4, 0])
b = Permutation([1, 0, 2, 3, 4])
G = PermutationGroup([a, b])
assert not G.is_solvable
P = SymmetricGroup(10)
S = P.sylow_subgroup(3)
assert S.is_solvable
def test_rubik1():
gens = rubik_cube_generators()
gens1 = [gens[-1]] + [p**2 for p in gens[1:]]
G1 = PermutationGroup(gens1)
assert G1.order() == 19508428800
gens2 = [p**2 for p in gens]
G2 = PermutationGroup(gens2)
assert G2.order() == 663552
assert G2.is_subgroup(G1, 0)
C1 = G1.derived_subgroup()
assert C1.order() == 4877107200
assert C1.is_subgroup(G1, 0)
assert not G2.is_subgroup(C1, 0)
G = RubikGroup(2)
assert G.order() == 3674160
@XFAIL
def test_rubik():
skip('takes too much time')
G = PermutationGroup(rubik_cube_generators())
assert G.order() == 43252003274489856000
G1 = PermutationGroup(G[:3])
assert G1.order() == 170659735142400
assert not G1.is_normal(G)
G2 = G.normal_closure(G1.generators)
assert G2.is_subgroup(G)
def test_direct_product():
C = CyclicGroup(4)
D = DihedralGroup(4)
G = C*C*C
assert G.order() == 64
assert G.degree == 12
assert len(G.orbits()) == 3
assert G.is_abelian is True
H = D*C
assert H.order() == 32
assert H.is_abelian is False
def test_orbit_rep():
G = DihedralGroup(6)
assert G.orbit_rep(1, 3) in [Permutation([2, 3, 4, 5, 0, 1]),
Permutation([4, 3, 2, 1, 0, 5])]
H = CyclicGroup(4)*G
assert H.orbit_rep(1, 5) is False
def test_schreier_vector():
G = CyclicGroup(50)
v = [0]*50
v[23] = -1
assert G.schreier_vector(23) == v
H = DihedralGroup(8)
assert H.schreier_vector(2) == [0, 1, -1, 0, 0, 1, 0, 0]
L = SymmetricGroup(4)
assert L.schreier_vector(1) == [1, -1, 0, 0]
def test_random_pr():
D = DihedralGroup(6)
r = 11
n = 3
_random_prec_n = {}
_random_prec_n[0] = {'s': 7, 't': 3, 'x': 2, 'e': -1}
_random_prec_n[1] = {'s': 5, 't': 5, 'x': 1, 'e': -1}
_random_prec_n[2] = {'s': 3, 't': 4, 'x': 2, 'e': 1}
D._random_pr_init(r, n, _random_prec_n=_random_prec_n)
assert D._random_gens[11] == [0, 1, 2, 3, 4, 5]
_random_prec = {'s': 2, 't': 9, 'x': 1, 'e': -1}
assert D.random_pr(_random_prec=_random_prec) == \
Permutation([0, 5, 4, 3, 2, 1])
def test_is_alt_sym():
G = DihedralGroup(10)
assert G.is_alt_sym() is False
S = SymmetricGroup(10)
N_eps = 10
_random_prec = {'N_eps': N_eps,
0: Permutation([[2], [1, 4], [0, 6, 7, 8, 9, 3, 5]]),
1: Permutation([[1, 8, 7, 6, 3, 5, 2, 9], [0, 4]]),
2: Permutation([[5, 8], [4, 7], [0, 1, 2, 3, 6, 9]]),
3: Permutation([[3], [0, 8, 2, 7, 4, 1, 6, 9, 5]]),
4: Permutation([[8], [4, 7, 9], [3, 6], [0, 5, 1, 2]]),
5: Permutation([[6], [0, 2, 4, 5, 1, 8, 3, 9, 7]]),
6: Permutation([[6, 9, 8], [4, 5], [1, 3, 7], [0, 2]]),
7: Permutation([[4], [0, 2, 9, 1, 3, 8, 6, 5, 7]]),
8: Permutation([[1, 5, 6, 3], [0, 2, 7, 8, 4, 9]]),
9: Permutation([[8], [6, 7], [2, 3, 4, 5], [0, 1, 9]])}
assert S.is_alt_sym(_random_prec=_random_prec) is True
A = AlternatingGroup(10)
_random_prec = {'N_eps': N_eps,
0: Permutation([[1, 6, 4, 2, 7, 8, 5, 9, 3], [0]]),
1: Permutation([[1], [0, 5, 8, 4, 9, 2, 3, 6, 7]]),
2: Permutation([[1, 9, 8, 3, 2, 5], [0, 6, 7, 4]]),
3: Permutation([[6, 8, 9], [4, 5], [1, 3, 7, 2], [0]]),
4: Permutation([[8], [5], [4], [2, 6, 9, 3], [1], [0, 7]]),
5: Permutation([[3, 6], [0, 8, 1, 7, 5, 9, 4, 2]]),
6: Permutation([[5], [2, 9], [1, 8, 3], [0, 4, 7, 6]]),
7: Permutation([[1, 8, 4, 7, 2, 3], [0, 6, 9, 5]]),
8: Permutation([[5, 8, 7], [3], [1, 4, 2, 6], [0, 9]]),
9: Permutation([[4, 9, 6], [3, 8], [1, 2], [0, 5, 7]])}
assert A.is_alt_sym(_random_prec=_random_prec) is False
def test_minimal_block():
D = DihedralGroup(6)
block_system = D.minimal_block([0, 3])
for i in range(3):
assert block_system[i] == block_system[i + 3]
S = SymmetricGroup(6)
assert S.minimal_block([0, 1]) == [0, 0, 0, 0, 0, 0]
assert Tetra.pgroup.minimal_block([0, 1]) == [0, 0, 0, 0]
P1 = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5))
P2 = PermutationGroup(Permutation(0, 1, 2, 3, 4, 5), Permutation(1, 5)(2, 4))
assert P1.minimal_block([0, 2]) == [0, 1, 0, 1, 0, 1]
assert P2.minimal_block([0, 2]) == [0, 1, 0, 1, 0, 1]
def test_minimal_blocks():
P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5))
assert P.minimal_blocks() == [[0, 1, 0, 1, 0, 1], [0, 1, 2, 0, 1, 2]]
P = SymmetricGroup(5)
assert P.minimal_blocks() == [[0]*5]
P = PermutationGroup(Permutation(0, 3))
assert P.minimal_blocks() == False
def test_max_div():
S = SymmetricGroup(10)
assert S.max_div == 5
def test_is_primitive():
S = SymmetricGroup(5)
assert S.is_primitive() is True
C = CyclicGroup(7)
assert C.is_primitive() is True
def test_random_stab():
S = SymmetricGroup(5)
_random_el = Permutation([1, 3, 2, 0, 4])
_random_prec = {'rand': _random_el}
g = S.random_stab(2, _random_prec=_random_prec)
assert g == Permutation([1, 3, 2, 0, 4])
h = S.random_stab(1)
assert h(1) == 1
def test_transitivity_degree():
perm = Permutation([1, 2, 0])
C = PermutationGroup([perm])
assert C.transitivity_degree == 1
gen1 = Permutation([1, 2, 0, 3, 4])
gen2 = Permutation([1, 2, 3, 4, 0])
# alternating group of degree 5
Alt = PermutationGroup([gen1, gen2])
assert Alt.transitivity_degree == 3
def test_schreier_sims_random():
assert sorted(Tetra.pgroup.base) == [0, 1]
S = SymmetricGroup(3)
base = [0, 1]
strong_gens = [Permutation([1, 2, 0]), Permutation([1, 0, 2]),
Permutation([0, 2, 1])]
assert S.schreier_sims_random(base, strong_gens, 5) == (base, strong_gens)
D = DihedralGroup(3)
_random_prec = {'g': [Permutation([2, 0, 1]), Permutation([1, 2, 0]),
Permutation([1, 0, 2])]}
base = [0, 1]
strong_gens = [Permutation([1, 2, 0]), Permutation([2, 1, 0]),
Permutation([0, 2, 1])]
assert D.schreier_sims_random([], D.generators, 2,
_random_prec=_random_prec) == (base, strong_gens)
def test_baseswap():
S = SymmetricGroup(4)
S.schreier_sims()
base = S.base
strong_gens = S.strong_gens
assert base == [0, 1, 2]
deterministic = S.baseswap(base, strong_gens, 1, randomized=False)
randomized = S.baseswap(base, strong_gens, 1)
assert deterministic[0] == [0, 2, 1]
assert _verify_bsgs(S, deterministic[0], deterministic[1]) is True
assert randomized[0] == [0, 2, 1]
assert _verify_bsgs(S, randomized[0], randomized[1]) is True
def test_schreier_sims_incremental():
identity = Permutation([0, 1, 2, 3, 4])
TrivialGroup = PermutationGroup([identity])
base, strong_gens = TrivialGroup.schreier_sims_incremental(base=[0, 1, 2])
assert _verify_bsgs(TrivialGroup, base, strong_gens) is True
S = SymmetricGroup(5)
base, strong_gens = S.schreier_sims_incremental(base=[0, 1, 2])
assert _verify_bsgs(S, base, strong_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental(base=[1])
assert _verify_bsgs(D, base, strong_gens) is True
A = AlternatingGroup(7)
gens = A.generators[:]
gen0 = gens[0]
gen1 = gens[1]
gen1 = rmul(gen1, ~gen0)
gen0 = rmul(gen0, gen1)
gen1 = rmul(gen0, gen1)
base, strong_gens = A.schreier_sims_incremental(base=[0, 1], gens=gens)
assert _verify_bsgs(A, base, strong_gens) is True
C = CyclicGroup(11)
gen = C.generators[0]
base, strong_gens = C.schreier_sims_incremental(gens=[gen**3])
assert _verify_bsgs(C, base, strong_gens) is True
def _subgroup_search(i, j, k):
prop_true = lambda x: True
prop_fix_points = lambda x: [x(point) for point in points] == points
prop_comm_g = lambda x: rmul(x, g) == rmul(g, x)
prop_even = lambda x: x.is_even
for i in range(i, j, k):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
Sym = S.subgroup_search(prop_true)
assert Sym.is_subgroup(S)
Alt = S.subgroup_search(prop_even)
assert Alt.is_subgroup(A)
Sym = S.subgroup_search(prop_true, init_subgroup=C)
assert Sym.is_subgroup(S)
points = [7]
assert S.stabilizer(7).is_subgroup(S.subgroup_search(prop_fix_points))
points = [3, 4]
assert S.stabilizer(3).stabilizer(4).is_subgroup(
S.subgroup_search(prop_fix_points))
points = [3, 5]
fix35 = A.subgroup_search(prop_fix_points)
points = [5]
fix5 = A.subgroup_search(prop_fix_points)
assert A.subgroup_search(prop_fix_points, init_subgroup=fix35
).is_subgroup(fix5)
base, strong_gens = A.schreier_sims_incremental()
g = A.generators[0]
comm_g = \
A.subgroup_search(prop_comm_g, base=base, strong_gens=strong_gens)
assert _verify_bsgs(comm_g, base, comm_g.generators) is True
assert [prop_comm_g(gen) is True for gen in comm_g.generators]
def test_subgroup_search():
_subgroup_search(10, 15, 2)
@XFAIL
def test_subgroup_search2():
skip('takes too much time')
_subgroup_search(16, 17, 1)
def test_normal_closure():
# the normal closure of the trivial group is trivial
S = SymmetricGroup(3)
identity = Permutation([0, 1, 2])
closure = S.normal_closure(identity)
assert closure.is_trivial
# the normal closure of the entire group is the entire group
A = AlternatingGroup(4)
assert A.normal_closure(A).is_subgroup(A)
# brute-force verifications for subgroups
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
C = CyclicGroup(i)
for gp in (A, D, C):
assert _verify_normal_closure(S, gp)
# brute-force verifications for all elements of a group
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_normal_closure(S, element)
# small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp2.is_subgroup(gp, 0) and gp2.degree == gp.degree:
assert _verify_normal_closure(gp, gp2)
def test_derived_series():
# the derived series of the trivial group consists only of the trivial group
triv = PermutationGroup([Permutation([0, 1, 2])])
assert triv.derived_series()[0].is_subgroup(triv)
# the derived series for a simple group consists only of the group itself
for i in (5, 6, 7):
A = AlternatingGroup(i)
assert A.derived_series()[0].is_subgroup(A)
# the derived series for S_4 is S_4 > A_4 > K_4 > triv
S = SymmetricGroup(4)
series = S.derived_series()
assert series[1].is_subgroup(AlternatingGroup(4))
assert series[2].is_subgroup(DihedralGroup(2))
assert series[3].is_trivial
def test_lower_central_series():
# the lower central series of the trivial group consists of the trivial
# group
triv = PermutationGroup([Permutation([0, 1, 2])])
assert triv.lower_central_series()[0].is_subgroup(triv)
# the lower central series of a simple group consists of the group itself
for i in (5, 6, 7):
A = AlternatingGroup(i)
assert A.lower_central_series()[0].is_subgroup(A)
# GAP-verified example
S = SymmetricGroup(6)
series = S.lower_central_series()
assert len(series) == 2
assert series[1].is_subgroup(AlternatingGroup(6))
def test_commutator():
# the commutator of the trivial group and the trivial group is trivial
S = SymmetricGroup(3)
triv = PermutationGroup([Permutation([0, 1, 2])])
assert S.commutator(triv, triv).is_subgroup(triv)
# the commutator of the trivial group and any other group is again trivial
A = AlternatingGroup(3)
assert S.commutator(triv, A).is_subgroup(triv)
# the commutator is commutative
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
assert S.commutator(A, D).is_subgroup(S.commutator(D, A))
# the commutator of an abelian group is trivial
S = SymmetricGroup(7)
A1 = AbelianGroup(2, 5)
A2 = AbelianGroup(3, 4)
triv = PermutationGroup([Permutation([0, 1, 2, 3, 4, 5, 6])])
assert S.commutator(A1, A1).is_subgroup(triv)
assert S.commutator(A2, A2).is_subgroup(triv)
# examples calculated by hand
S = SymmetricGroup(3)
A = AlternatingGroup(3)
assert S.commutator(A, S).is_subgroup(A)
def test_is_nilpotent():
# every abelian group is nilpotent
for i in (1, 2, 3):
C = CyclicGroup(i)
Ab = AbelianGroup(i, i + 2)
assert C.is_nilpotent
assert Ab.is_nilpotent
Ab = AbelianGroup(5, 7, 10)
assert Ab.is_nilpotent
# A_5 is not solvable and thus not nilpotent
assert AlternatingGroup(5).is_nilpotent is False
def test_is_trivial():
for i in range(5):
triv = PermutationGroup([Permutation(list(range(i)))])
assert triv.is_trivial
def test_pointwise_stabilizer():
S = SymmetricGroup(2)
stab = S.pointwise_stabilizer([0])
assert stab.generators == [Permutation(1)]
S = SymmetricGroup(5)
points = []
stab = S
for point in (2, 0, 3, 4, 1):
stab = stab.stabilizer(point)
points.append(point)
assert S.pointwise_stabilizer(points).is_subgroup(stab)
def test_make_perm():
assert cube.pgroup.make_perm(5, seed=list(range(5))) == \
Permutation([4, 7, 6, 5, 0, 3, 2, 1])
assert cube.pgroup.make_perm(7, seed=list(range(7))) == \
Permutation([6, 7, 3, 2, 5, 4, 0, 1])
def test_elements():
p = Permutation(2, 3)
assert PermutationGroup(p).elements == {Permutation(3), Permutation(2, 3)}
def test_is_group():
assert PermutationGroup(Permutation(1,2), Permutation(2,4)).is_group == True
assert SymmetricGroup(4).is_group == True
def test_PermutationGroup():
assert PermutationGroup() == PermutationGroup(Permutation())
assert (PermutationGroup() == 0) is False
def test_coset_transvesal():
G = AlternatingGroup(5)
H = PermutationGroup(Permutation(0,1,2),Permutation(1,2)(3,4))
assert G.coset_transversal(H) == \
[Permutation(4), Permutation(2, 3, 4), Permutation(2, 4, 3),
Permutation(1, 2, 4), Permutation(4)(1, 2, 3), Permutation(1, 3)(2, 4),
Permutation(0, 1, 2, 3, 4), Permutation(0, 1, 2, 4, 3),
Permutation(0, 1, 3, 2, 4), Permutation(0, 2, 4, 1, 3)]
def test_coset_table():
G = PermutationGroup(Permutation(0,1,2,3), Permutation(0,1,2),
Permutation(0,4,2,7), Permutation(5,6), Permutation(0,7));
H = PermutationGroup(Permutation(0,1,2,3), Permutation(0,7))
assert G.coset_table(H) == \
[[0, 0, 0, 0, 1, 2, 3, 3, 0, 0], [4, 5, 2, 5, 6, 0, 7, 7, 1, 1],
[5, 4, 5, 1, 0, 6, 8, 8, 6, 6], [3, 3, 3, 3, 7, 8, 0, 0, 3, 3],
[2, 1, 4, 4, 4, 4, 9, 9, 4, 4], [1, 2, 1, 2, 5, 5, 10, 10, 5, 5],
[6, 6, 6, 6, 2, 1, 11, 11, 2, 2], [9, 10, 8, 10, 11, 3, 1, 1, 7, 7],
[10, 9, 10, 7, 3, 11, 2, 2, 11, 11], [8, 7, 9, 9, 9, 9, 4, 4, 9, 9],
[7, 8, 7, 8, 10, 10, 5, 5, 10, 10], [11, 11, 11, 11, 8, 7, 6, 6, 8, 8]]
def test_subgroup():
G = PermutationGroup(Permutation(0,1,2), Permutation(0,2,3))
H = G.subgroup([Permutation(0,1,3)])
assert H.is_subgroup(G)
def test_generator_product():
G = SymmetricGroup(5)
p = Permutation(0, 2, 3)(1, 4)
gens = G.generator_product(p)
assert all(g in G.strong_gens for g in gens)
w = G.identity
for g in gens:
w = g*w
assert w == p
def test_sylow_subgroup():
P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5))
S = P.sylow_subgroup(2)
assert S.order() == 4
P = DihedralGroup(12)
S = P.sylow_subgroup(3)
assert S.order() == 3
P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5), Permutation(0, 2))
S = P.sylow_subgroup(3)
assert S.order() == 9
S = P.sylow_subgroup(2)
assert S.order() == 8
P = SymmetricGroup(10)
S = P.sylow_subgroup(2)
assert S.order() == 256
S = P.sylow_subgroup(3)
assert S.order() == 81
S = P.sylow_subgroup(5)
assert S.order() == 25
# the length of the lower central series
# of a p-Sylow subgroup of Sym(n) grows with
# the highest exponent exp of p such
# that n >= p**exp
exp = 1
length = 0
for i in range(2, 9):
P = SymmetricGroup(i)
S = P.sylow_subgroup(2)
ls = S.lower_central_series()
if i // 2**exp > 0:
# length increases with exponent
assert len(ls) > length
length = len(ls)
exp += 1
else:
assert len(ls) == length
G = SymmetricGroup(100)
S = G.sylow_subgroup(3)
assert G.order() % S.order() == 0
assert G.order()/S.order() % 3 > 0
G = AlternatingGroup(100)
S = G.sylow_subgroup(2)
assert G.order() % S.order() == 0
assert G.order()/S.order() % 2 > 0
@slow
def test_presentation():
def _test(P):
G = P.presentation()
return G.order() == P.order()
def _strong_test(P):
G = P.strong_presentation()
chk = len(G.generators) == len(P.strong_gens)
return chk and G.order() == P.order()
P = PermutationGroup(Permutation(0,1,5,2)(3,7,4,6), Permutation(0,3,5,4)(1,6,2,7))
assert _test(P)
P = AlternatingGroup(5)
assert _test(P)
P = SymmetricGroup(5)
assert _test(P)
P = PermutationGroup([Permutation(0,3,1,2), Permutation(3)(0,1), Permutation(0,1)(2,3)])
G = P.strong_presentation()
assert _strong_test(P)
P = DihedralGroup(6)
G = P.strong_presentation()
assert _strong_test(P)
a = Permutation(0,1)(2,3)
b = Permutation(0,2)(3,1)
c = Permutation(4,5)
P = PermutationGroup(c, a, b)
assert _strong_test(P)
def test_polycyclic():
a = Permutation([0, 1, 2])
b = Permutation([2, 1, 0])
G = PermutationGroup([a, b])
assert G.is_polycyclic == True
a = Permutation([1, 2, 3, 4, 0])
b = Permutation([1, 0, 2, 3, 4])
G = PermutationGroup([a, b])
assert G.is_polycyclic == False
def test_elementary():
a = Permutation([1, 5, 2, 0, 3, 6, 4])
G = PermutationGroup([a])
assert G.is_elementary(7) == False
a = Permutation(0, 1)(2, 3)
b = Permutation(0, 2)(3, 1)
G = PermutationGroup([a, b])
assert G.is_elementary(2) == True
c = Permutation(4, 5, 6)
G = PermutationGroup([a, b, c])
assert G.is_elementary(2) == False
G = SymmetricGroup(4).sylow_subgroup(2)
assert G.is_elementary(2) == False
H = AlternatingGroup(4).sylow_subgroup(2)
assert H.is_elementary(2) == True
def test_perfect():
G = AlternatingGroup(3)
assert G.is_perfect == False
G = AlternatingGroup(5)
assert G.is_perfect == True
def test_index():
G = PermutationGroup(Permutation(0,1,2), Permutation(0,2,3))
H = G.subgroup([Permutation(0,1,3)])
assert G.index(H) == 4
def test_cyclic():
G = SymmetricGroup(2)
assert G.is_cyclic
G = AbelianGroup(3, 7)
assert G.is_cyclic
G = AbelianGroup(7, 7)
assert not G.is_cyclic
G = AlternatingGroup(3)
assert G.is_cyclic
G = AlternatingGroup(4)
assert not G.is_cyclic
|
befb58cba163d92643a1f87311efb27e7a25fac3323689c082bb2be9298572c3 | from sympy.core.compatibility import range, ordered
from sympy.combinatorics.partitions import (Partition, IntegerPartition,
RGS_enum, RGS_unrank, RGS_rank,
random_integer_partition)
from sympy.utilities.pytest import raises
from sympy.utilities.iterables import default_sort_key, partitions
from sympy.sets.sets import Set
def test_partition():
from sympy.abc import x
raises(ValueError, lambda: Partition(*list(range(3))))
raises(ValueError, lambda: Partition([1, 1, 2]))
a = Partition([1, 2, 3], [4])
b = Partition([1, 2], [3, 4])
c = Partition([x])
l = [a, b, c]
l.sort(key=default_sort_key)
assert l == [c, a, b]
l.sort(key=lambda w: default_sort_key(w, order='rev-lex'))
assert l == [c, a, b]
assert (a == b) is False
assert a <= b
assert (a > b) is False
assert a != b
assert a < b
assert (a + 2).partition == [[1, 2], [3, 4]]
assert (b - 1).partition == [[1, 2, 4], [3]]
assert (a - 1).partition == [[1, 2, 3, 4]]
assert (a + 1).partition == [[1, 2, 4], [3]]
assert (b + 1).partition == [[1, 2], [3], [4]]
assert a.rank == 1
assert b.rank == 3
assert a.RGS == (0, 0, 0, 1)
assert b.RGS == (0, 0, 1, 1)
def test_integer_partition():
# no zeros in partition
raises(ValueError, lambda: IntegerPartition(list(range(3))))
# check fails since 1 + 2 != 100
raises(ValueError, lambda: IntegerPartition(100, list(range(1, 3))))
a = IntegerPartition(8, [1, 3, 4])
b = a.next_lex()
c = IntegerPartition([1, 3, 4])
d = IntegerPartition(8, {1: 3, 3: 1, 2: 1})
assert a == c
assert a.integer == d.integer
assert a.conjugate == [3, 2, 2, 1]
assert (a == b) is False
assert a <= b
assert (a > b) is False
assert a != b
for i in range(1, 11):
next = set()
prev = set()
a = IntegerPartition([i])
ans = {IntegerPartition(p) for p in partitions(i)}
n = len(ans)
for j in range(n):
next.add(a)
a = a.next_lex()
IntegerPartition(i, a.partition) # check it by giving i
for j in range(n):
prev.add(a)
a = a.prev_lex()
IntegerPartition(i, a.partition) # check it by giving i
assert next == ans
assert prev == ans
assert IntegerPartition([1, 2, 3]).as_ferrers() == '###\n##\n#'
assert IntegerPartition([1, 1, 3]).as_ferrers('o') == 'ooo\no\no'
assert str(IntegerPartition([1, 1, 3])) == '[3, 1, 1]'
assert IntegerPartition([1, 1, 3]).partition == [3, 1, 1]
raises(ValueError, lambda: random_integer_partition(-1))
assert random_integer_partition(1) == [1]
assert random_integer_partition(10, seed=[1, 3, 2, 1, 5, 1]
) == [5, 2, 1, 1, 1]
def test_rgs():
raises(ValueError, lambda: RGS_unrank(-1, 3))
raises(ValueError, lambda: RGS_unrank(3, 0))
raises(ValueError, lambda: RGS_unrank(10, 1))
raises(ValueError, lambda: Partition.from_rgs(list(range(3)), list(range(2))))
raises(ValueError, lambda: Partition.from_rgs(list(range(1, 3)), list(range(2))))
assert RGS_enum(-1) == 0
assert RGS_enum(1) == 1
assert RGS_unrank(7, 5) == [0, 0, 1, 0, 2]
assert RGS_unrank(23, 14) == [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2]
assert RGS_rank(RGS_unrank(40, 100)) == 40
def test_ordered_partition_9608():
a = Partition([1, 2, 3], [4])
b = Partition([1, 2], [3, 4])
assert list(ordered([a,b], Set._infimum_key))
|
b2765295297f70ab91898dc44de7c1a7f116f37465883fed3acc1a954e0a886e | from sympy import (
symbols, powsimp, symbols, MatrixSymbol, sqrt, pi, Mul, gamma, Function,
S, I, exp, simplify, sin, E, log, hyper, Symbol, Dummy, powdenest, root,
Rational, oo)
from sympy.abc import x, y, z, t, a, b, c, d, e, f, g, h, i, k
def test_powsimp():
x, y, z, n = symbols('x,y,z,n')
f = Function('f')
assert powsimp( 4**x * 2**(-x) * 2**(-x) ) == 1
assert powsimp( (-4)**x * (-2)**(-x) * 2**(-x) ) == 1
assert powsimp(
f(4**x * 2**(-x) * 2**(-x)) ) == f(4**x * 2**(-x) * 2**(-x))
assert powsimp( f(4**x * 2**(-x) * 2**(-x)), deep=True ) == f(1)
assert exp(x)*exp(y) == exp(x)*exp(y)
assert powsimp(exp(x)*exp(y)) == exp(x + y)
assert powsimp(exp(x)*exp(y)*2**x*2**y) == (2*E)**(x + y)
assert powsimp(exp(x)*exp(y)*2**x*2**y, combine='exp') == \
exp(x + y)*2**(x + y)
assert powsimp(exp(x)*exp(y)*exp(2)*sin(x) + sin(y) + 2**x*2**y) == \
exp(2 + x + y)*sin(x) + sin(y) + 2**(x + y)
assert powsimp(sin(exp(x)*exp(y))) == sin(exp(x)*exp(y))
assert powsimp(sin(exp(x)*exp(y)), deep=True) == sin(exp(x + y))
assert powsimp(x**2*x**y) == x**(2 + y)
# This should remain factored, because 'exp' with deep=True is supposed
# to act like old automatic exponent combining.
assert powsimp((1 + E*exp(E))*exp(-E), combine='exp', deep=True) == \
(1 + exp(1 + E))*exp(-E)
assert powsimp((1 + E*exp(E))*exp(-E), deep=True) == \
(1 + exp(1 + E))*exp(-E)
assert powsimp((1 + E*exp(E))*exp(-E)) == (1 + exp(1 + E))*exp(-E)
assert powsimp((1 + E*exp(E))*exp(-E), combine='exp') == \
(1 + exp(1 + E))*exp(-E)
assert powsimp((1 + E*exp(E))*exp(-E), combine='base') == \
(1 + E*exp(E))*exp(-E)
x, y = symbols('x,y', nonnegative=True)
n = Symbol('n', real=True)
assert powsimp(y**n * (y/x)**(-n)) == x**n
assert powsimp(x**(x**(x*y)*y**(x*y))*y**(x**(x*y)*y**(x*y)), deep=True) \
== (x*y)**(x*y)**(x*y)
assert powsimp(2**(2**(2*x)*x), deep=False) == 2**(2**(2*x)*x)
assert powsimp(2**(2**(2*x)*x), deep=True) == 2**(x*4**x)
assert powsimp(
exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \
exp(-x + exp(-x)*exp(-x*log(x)))
assert powsimp(
exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \
exp(-x + exp(-x)*exp(-x*log(x)))
assert powsimp((x + y)/(3*z), deep=False, combine='exp') == (x + y)/(3*z)
assert powsimp((x/3 + y/3)/z, deep=True, combine='exp') == (x/3 + y/3)/z
assert powsimp(exp(x)/(1 + exp(x)*exp(y)), deep=True) == \
exp(x)/(1 + exp(x + y))
assert powsimp(x*y**(z**x*z**y), deep=True) == x*y**(z**(x + y))
assert powsimp((z**x*z**y)**x, deep=True) == (z**(x + y))**x
assert powsimp(x*(z**x*z**y)**x, deep=True) == x*(z**(x + y))**x
p = symbols('p', positive=True)
assert powsimp((1/x)**log(2)/x) == (1/x)**(1 + log(2))
assert powsimp((1/p)**log(2)/p) == p**(-1 - log(2))
# coefficient of exponent can only be simplified for positive bases
assert powsimp(2**(2*x)) == 4**x
assert powsimp((-1)**(2*x)) == (-1)**(2*x)
i = symbols('i', integer=True)
assert powsimp((-1)**(2*i)) == 1
assert powsimp((-1)**(-x)) != (-1)**x # could be 1/((-1)**x), but is not
# force=True overrides assumptions
assert powsimp((-1)**(2*x), force=True) == 1
# rational exponents allow combining of negative terms
w, n, m = symbols('w n m', negative=True)
e = i/a # not a rational exponent if `a` is unknown
ex = w**e*n**e*m**e
assert powsimp(ex) == m**(i/a)*n**(i/a)*w**(i/a)
e = i/3
ex = w**e*n**e*m**e
assert powsimp(ex) == (-1)**i*(-m*n*w)**(i/3)
e = (3 + i)/i
ex = w**e*n**e*m**e
assert powsimp(ex) == (-1)**(3*e)*(-m*n*w)**e
eq = x**(2*a/3)
# eq != (x**a)**(2/3) (try x = -1 and a = 3 to see)
assert powsimp(eq).exp == eq.exp == 2*a/3
# powdenest goes the other direction
assert powsimp(2**(2*x)) == 4**x
assert powsimp(exp(p/2)) == exp(p/2)
# issue 6368
eq = Mul(*[sqrt(Dummy(imaginary=True)) for i in range(3)])
assert powsimp(eq) == eq and eq.is_Mul
assert all(powsimp(e) == e for e in (sqrt(x**a), sqrt(x**2)))
# issue 8836
assert str( powsimp(exp(I*pi/3)*root(-1,3)) ) == '(-1)**(2/3)'
# issue 9183
assert powsimp(-0.1**x) == -0.1**x
# issue 10095
assert powsimp((1/(2*E))**oo) == (exp(-1)/2)**oo
# PR 13131
eq = sin(2*x)**2*sin(2.0*x)**2
assert powsimp(eq) == eq
# issue 14615
assert powsimp(x**2*y**3*(x*y**2)**(S(3)/2)
) == x*y*(x*y**2)**(S(5)/2)
def test_powsimp_negated_base():
assert powsimp((-x + y)/sqrt(x - y)) == -sqrt(x - y)
assert powsimp((-x + y)*(-z + y)/sqrt(x - y)/sqrt(z - y)) == sqrt(x - y)*sqrt(z - y)
p = symbols('p', positive=True)
assert powsimp((-p)**a/p**a) == (-1)**a
n = symbols('n', negative=True)
assert powsimp((-n)**a/n**a) == (-1)**a
# if x is 0 then the lhs is 0**a*oo**a which is not (-1)**a
assert powsimp((-x)**a/x**a) != (-1)**a
def test_powsimp_nc():
x, y, z = symbols('x,y,z')
A, B, C = symbols('A B C', commutative=False)
assert powsimp(A**x*A**y, combine='all') == A**(x + y)
assert powsimp(A**x*A**y, combine='base') == A**x*A**y
assert powsimp(A**x*A**y, combine='exp') == A**(x + y)
assert powsimp(A**x*B**x, combine='all') == A**x*B**x
assert powsimp(A**x*B**x, combine='base') == A**x*B**x
assert powsimp(A**x*B**x, combine='exp') == A**x*B**x
assert powsimp(B**x*A**x, combine='all') == B**x*A**x
assert powsimp(B**x*A**x, combine='base') == B**x*A**x
assert powsimp(B**x*A**x, combine='exp') == B**x*A**x
assert powsimp(A**x*A**y*A**z, combine='all') == A**(x + y + z)
assert powsimp(A**x*A**y*A**z, combine='base') == A**x*A**y*A**z
assert powsimp(A**x*A**y*A**z, combine='exp') == A**(x + y + z)
assert powsimp(A**x*B**x*C**x, combine='all') == A**x*B**x*C**x
assert powsimp(A**x*B**x*C**x, combine='base') == A**x*B**x*C**x
assert powsimp(A**x*B**x*C**x, combine='exp') == A**x*B**x*C**x
assert powsimp(B**x*A**x*C**x, combine='all') == B**x*A**x*C**x
assert powsimp(B**x*A**x*C**x, combine='base') == B**x*A**x*C**x
assert powsimp(B**x*A**x*C**x, combine='exp') == B**x*A**x*C**x
def test_issue_6440():
assert powsimp(16*2**a*8**b) == 2**(a + 3*b + 4)
def test_powdenest():
from sympy import powdenest
from sympy.abc import x, y, z, a, b
p, q = symbols('p q', positive=True)
i, j = symbols('i,j', integer=True)
assert powdenest(x) == x
assert powdenest(x + 2*(x**(2*a/3))**(3*x)) == (x + 2*(x**(2*a/3))**(3*x))
assert powdenest((exp(2*a/3))**(3*x)) # -X-> (exp(a/3))**(6*x)
assert powdenest((x**(2*a/3))**(3*x)) == ((x**(2*a/3))**(3*x))
assert powdenest(exp(3*x*log(2))) == 2**(3*x)
assert powdenest(sqrt(p**2)) == p
i, j = symbols('i,j', integer=True)
eq = p**(2*i)*q**(4*i)
assert powdenest(eq) == (p*q**2)**(2*i)
# -X-> (x**x)**i*(x**x)**j == x**(x*(i + j))
assert powdenest((x**x)**(i + j))
assert powdenest(exp(3*y*log(x))) == x**(3*y)
assert powdenest(exp(y*(log(a) + log(b)))) == (a*b)**y
assert powdenest(exp(3*(log(a) + log(b)))) == a**3*b**3
assert powdenest(((x**(2*i))**(3*y))**x) == ((x**(2*i))**(3*y))**x
assert powdenest(((x**(2*i))**(3*y))**x, force=True) == x**(6*i*x*y)
assert powdenest(((x**(2*a/3))**(3*y/i))**x) == \
(((x**(2*a/3))**(3*y/i))**x)
assert powdenest((x**(2*i)*y**(4*i))**z, force=True) == (x*y**2)**(2*i*z)
assert powdenest((p**(2*i)*q**(4*i))**j) == (p*q**2)**(2*i*j)
e = ((p**(2*a))**(3*y))**x
assert powdenest(e) == e
e = ((x**2*y**4)**a)**(x*y)
assert powdenest(e) == e
e = (((x**2*y**4)**a)**(x*y))**3
assert powdenest(e) == ((x**2*y**4)**a)**(3*x*y)
assert powdenest((((x**2*y**4)**a)**(x*y)), force=True) == \
(x*y**2)**(2*a*x*y)
assert powdenest((((x**2*y**4)**a)**(x*y))**3, force=True) == \
(x*y**2)**(6*a*x*y)
assert powdenest((x**2*y**6)**i) != (x*y**3)**(2*i)
x, y = symbols('x,y', positive=True)
assert powdenest((x**2*y**6)**i) == (x*y**3)**(2*i)
assert powdenest((x**(2*i/3)*y**(i/2))**(2*i)) == (x**(S(4)/3)*y)**(i**2)
assert powdenest(sqrt(x**(2*i)*y**(6*i))) == (x*y**3)**i
assert powdenest(4**x) == 2**(2*x)
assert powdenest((4**x)**y) == 2**(2*x*y)
assert powdenest(4**x*y) == 2**(2*x)*y
def test_powdenest_polar():
x, y, z = symbols('x y z', polar=True)
a, b, c = symbols('a b c')
assert powdenest((x*y*z)**a) == x**a*y**a*z**a
assert powdenest((x**a*y**b)**c) == x**(a*c)*y**(b*c)
assert powdenest(((x**a)**b*y**c)**c) == x**(a*b*c)*y**(c**2)
def test_issue_5805():
arg = ((gamma(x)*hyper((), (), x))*pi)**2
assert powdenest(arg) == (pi*gamma(x)*hyper((), (), x))**2
assert arg.is_positive is None
def test_issue_9324_powsimp_on_matrix_symbol():
M = MatrixSymbol('M', 10, 10)
expr = powsimp(M, deep=True)
assert expr == M
assert expr.args[0] == Symbol('M')
def test_issue_6367():
z = -5*sqrt(2)/(2*sqrt(2*sqrt(29) + 29)) + sqrt(-sqrt(29)/29 + S(1)/2)
assert Mul(*[powsimp(a) for a in Mul.make_args(z.normal())]) == 0
assert powsimp(z.normal()) == 0
assert simplify(z) == 0
assert powsimp(sqrt(2 + sqrt(3))*sqrt(2 - sqrt(3)) + 1) == 2
assert powsimp(z) != 0
def test_powsimp_polar():
from sympy import polar_lift, exp_polar
x, y, z = symbols('x y z')
p, q, r = symbols('p q r', polar=True)
assert (polar_lift(-1))**(2*x) == exp_polar(2*pi*I*x)
assert powsimp(p**x * q**x) == (p*q)**x
assert p**x * (1/p)**x == 1
assert (1/p)**x == p**(-x)
assert exp_polar(x)*exp_polar(y) == exp_polar(x)*exp_polar(y)
assert powsimp(exp_polar(x)*exp_polar(y)) == exp_polar(x + y)
assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y) == \
(p*exp_polar(1))**(x + y)
assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y, combine='exp') == \
exp_polar(x + y)*p**(x + y)
assert powsimp(
exp_polar(x)*exp_polar(y)*exp_polar(2)*sin(x) + sin(y) + p**x*p**y) \
== p**(x + y) + sin(x)*exp_polar(2 + x + y) + sin(y)
assert powsimp(sin(exp_polar(x)*exp_polar(y))) == \
sin(exp_polar(x)*exp_polar(y))
assert powsimp(sin(exp_polar(x)*exp_polar(y)), deep=True) == \
sin(exp_polar(x + y))
def test_issue_5728():
b = x*sqrt(y)
a = sqrt(b)
c = sqrt(sqrt(x)*y)
assert powsimp(a*b) == sqrt(b)**3
assert powsimp(a*b**2*sqrt(y)) == sqrt(y)*a**5
assert powsimp(a*x**2*c**3*y) == c**3*a**5
assert powsimp(a*x*c**3*y**2) == c**7*a
assert powsimp(x*c**3*y**2) == c**7
assert powsimp(x*c**3*y) == x*y*c**3
assert powsimp(sqrt(x)*c**3*y) == c**5
assert powsimp(sqrt(x)*a**3*sqrt(y)) == sqrt(x)*sqrt(y)*a**3
assert powsimp(Mul(sqrt(x)*c**3*sqrt(y), y, evaluate=False)) == \
sqrt(x)*sqrt(y)**3*c**3
assert powsimp(a**2*a*x**2*y) == a**7
# symbolic powers work, too
b = x**y*y
a = b*sqrt(b)
assert a.is_Mul is True
assert powsimp(a) == sqrt(b)**3
# as does exp
a = x*exp(2*y/3)
assert powsimp(a*sqrt(a)) == sqrt(a)**3
assert powsimp(a**2*sqrt(a)) == sqrt(a)**5
assert powsimp(a**2*sqrt(sqrt(a))) == sqrt(sqrt(a))**9
def test_issue_from_PR1599():
n1, n2, n3, n4 = symbols('n1 n2 n3 n4', negative=True)
assert (powsimp(sqrt(n1)*sqrt(n2)*sqrt(n3)) ==
-I*sqrt(-n1)*sqrt(-n2)*sqrt(-n3))
assert (powsimp(root(n1, 3)*root(n2, 3)*root(n3, 3)*root(n4, 3)) ==
-(-1)**(S(1)/3)*
(-n1)**(S(1)/3)*(-n2)**(S(1)/3)*(-n3)**(S(1)/3)*(-n4)**(S(1)/3))
def test_issue_10195():
a = Symbol('a', integer=True)
l = Symbol('l', even=True, nonzero=True)
n = Symbol('n', odd=True)
e_x = (-1)**(n/2 - Rational(1, 2)) - (-1)**(3*n/2 - Rational(1, 2))
assert powsimp((-1)**(l/2)) == I**l
assert powsimp((-1)**(n/2)) == I**n
assert powsimp((-1)**(3*n/2)) == -I**n
assert powsimp(e_x) == (-1)**(n/2 - Rational(1, 2)) + (-1)**(3*n/2 +
Rational(1,2))
assert powsimp((-1)**(3*a/2)) == (-I)**a
def test_issue_15709():
assert powsimp(2*3**x/3) == 2*3**(x-1)
def test_issue_11981():
x, y = symbols('x y', commutative=False)
assert powsimp((x*y)**2 * (y*x)**2) == (x*y)**2 * (y*x)**2
|
83c4a92bc6750a726576c53ea05c1a876acdfec9a8c3dfe6e596fb84a10ae8b5 | from sympy import (
Abs, acos, Add, asin, atan, Basic, binomial, besselsimp,
collect,cos, cosh, cot, coth, count_ops, csch, Derivative, diff, E,
Eq, erf, exp, exp_polar, expand, expand_multinomial, factor,
factorial, Float, fraction, Function, gamma, GoldenRatio, hyper,
hypersimp, I, Integral, integrate, log, logcombine, Lt, Matrix,
MatrixSymbol, Mul, nsimplify, O, oo, pi, Piecewise, posify, rad,
Rational, root, S, separatevars, signsimp, simplify, sign, sin,
sinc, sinh, solve, sqrt, Sum, Symbol, symbols, sympify, tan, tanh,
zoo)
from sympy.core.mul import _keep_coeff
from sympy.simplify.simplify import nthroot, inversecombine
from sympy.utilities.pytest import XFAIL, slow
from sympy.core.compatibility import range
from sympy.abc import x, y, z, t, a, b, c, d, e, f, g, h, i, k
def test_issue_7263():
assert abs((simplify(30.8**2 - 82.5**2 * sin(rad(11.6))**2)).evalf() - \
673.447451402970) < 1e-12
@XFAIL
def test_factorial_simplify():
# There are more tests in test_factorials.py. These are just to
# ensure that simplify() calls factorial_simplify correctly
from sympy.specfun.factorials import factorial
x = Symbol('x')
assert simplify(factorial(x)/x) == factorial(x - 1)
assert simplify(factorial(factorial(x))) == factorial(factorial(x))
def test_simplify_expr():
x, y, z, k, n, m, w, s, A = symbols('x,y,z,k,n,m,w,s,A')
f = Function('f')
assert all(simplify(tmp) == tmp for tmp in [I, E, oo, x, -x, -oo, -E, -I])
e = 1/x + 1/y
assert e != (x + y)/(x*y)
assert simplify(e) == (x + y)/(x*y)
e = A**2*s**4/(4*pi*k*m**3)
assert simplify(e) == e
e = (4 + 4*x - 2*(2 + 2*x))/(2 + 2*x)
assert simplify(e) == 0
e = (-4*x*y**2 - 2*y**3 - 2*x**2*y)/(x + y)**2
assert simplify(e) == -2*y
e = -x - y - (x + y)**(-1)*y**2 + (x + y)**(-1)*x**2
assert simplify(e) == -2*y
e = (x + x*y)/x
assert simplify(e) == 1 + y
e = (f(x) + y*f(x))/f(x)
assert simplify(e) == 1 + y
e = (2 * (1/n - cos(n * pi)/n))/pi
assert simplify(e) == (-cos(pi*n) + 1)/(pi*n)*2
e = integrate(1/(x**3 + 1), x).diff(x)
assert simplify(e) == 1/(x**3 + 1)
e = integrate(x/(x**2 + 3*x + 1), x).diff(x)
assert simplify(e) == x/(x**2 + 3*x + 1)
f = Symbol('f')
A = Matrix([[2*k - m*w**2, -k], [-k, k - m*w**2]]).inv()
assert simplify((A*Matrix([0, f]))[1]) == \
-f*(2*k - m*w**2)/(k**2 - (k - m*w**2)*(2*k - m*w**2))
f = -x + y/(z + t) + z*x/(z + t) + z*a/(z + t) + t*x/(z + t)
assert simplify(f) == (y + a*z)/(z + t)
# issue 10347
expr = -x*(y**2 - 1)*(2*y**2*(x**2 - 1)/(a*(x**2 - y**2)**2) + (x**2 - 1)
/(a*(x**2 - y**2)))/(a*(x**2 - y**2)) + x*(-2*x**2*sqrt(-x**2*y**2 + x**2
+ y**2 - 1)*sin(z)/(a*(x**2 - y**2)**2) - x**2*sqrt(-x**2*y**2 + x**2 +
y**2 - 1)*sin(z)/(a*(x**2 - 1)*(x**2 - y**2)) + (x**2*sqrt((-x**2 + 1)*
(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(x**2 - 1) + sqrt(
(-x**2 + 1)*(y**2 - 1))*(x*(-x*y**2 + x)/sqrt(-x**2*y**2 + x**2 + y**2 -
1) + sqrt(-x**2*y**2 + x**2 + y**2 - 1))*sin(z))/(a*sqrt((-x**2 + 1)*(
y**2 - 1))*(x**2 - y**2)))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(a*
(x**2 - y**2)) + x*(-2*x**2*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*
(x**2 - y**2)**2) - x**2*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*
(x**2 - 1)*(x**2 - y**2)) + (x**2*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2
*y**2 + x**2 + y**2 - 1)*cos(z)/(x**2 - 1) + x*sqrt((-x**2 + 1)*(y**2 -
1))*(-x*y**2 + x)*cos(z)/sqrt(-x**2*y**2 + x**2 + y**2 - 1) + sqrt((-x**2
+ 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z))/(a*sqrt((-x**2
+ 1)*(y**2 - 1))*(x**2 - y**2)))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(
z)/(a*(x**2 - y**2)) - y*sqrt((-x**2 + 1)*(y**2 - 1))*(-x*y*sqrt(-x**2*
y**2 + x**2 + y**2 - 1)*sin(z)/(a*(x**2 - y**2)*(y**2 - 1)) + 2*x*y*sqrt(
-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(a*(x**2 - y**2)**2) + (x*y*sqrt((
-x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(y**2 -
1) + x*sqrt((-x**2 + 1)*(y**2 - 1))*(-x**2*y + y)*sin(z)/sqrt(-x**2*y**2
+ x**2 + y**2 - 1))/(a*sqrt((-x**2 + 1)*(y**2 - 1))*(x**2 - y**2)))*sin(
z)/(a*(x**2 - y**2)) + y*(x**2 - 1)*(-2*x*y*(x**2 - 1)/(a*(x**2 - y**2)
**2) + 2*x*y/(a*(x**2 - y**2)))/(a*(x**2 - y**2)) + y*(x**2 - 1)*(y**2 -
1)*(-x*y*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*(x**2 - y**2)*(y**2
- 1)) + 2*x*y*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*(x**2 - y**2)
**2) + (x*y*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 -
1)*cos(z)/(y**2 - 1) + x*sqrt((-x**2 + 1)*(y**2 - 1))*(-x**2*y + y)*cos(
z)/sqrt(-x**2*y**2 + x**2 + y**2 - 1))/(a*sqrt((-x**2 + 1)*(y**2 - 1)
)*(x**2 - y**2)))*cos(z)/(a*sqrt((-x**2 + 1)*(y**2 - 1))*(x**2 - y**2)
) - x*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(
z)**2/(a**2*(x**2 - 1)*(x**2 - y**2)*(y**2 - 1)) - x*sqrt((-x**2 + 1)*(
y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)**2/(a**2*(x**2 - 1)*(
x**2 - y**2)*(y**2 - 1))
assert simplify(expr) == 2*x/(a**2*(x**2 - y**2))
A, B = symbols('A,B', commutative=False)
assert simplify(A*B - B*A) == A*B - B*A
assert simplify(A/(1 + y/x)) == x*A/(x + y)
assert simplify(A*(1/x + 1/y)) == A/x + A/y #(x + y)*A/(x*y)
assert simplify(log(2) + log(3)) == log(6)
assert simplify(log(2*x) - log(2)) == log(x)
assert simplify(hyper([], [], x)) == exp(x)
def test_issue_3557():
f_1 = x*a + y*b + z*c - 1
f_2 = x*d + y*e + z*f - 1
f_3 = x*g + y*h + z*i - 1
solutions = solve([f_1, f_2, f_3], x, y, z, simplify=False)
assert simplify(solutions[y]) == \
(a*i + c*d + f*g - a*f - c*g - d*i)/ \
(a*e*i + b*f*g + c*d*h - a*f*h - b*d*i - c*e*g)
def test_simplify_other():
assert simplify(sin(x)**2 + cos(x)**2) == 1
assert simplify(gamma(x + 1)/gamma(x)) == x
assert simplify(sin(x)**2 + cos(x)**2 + factorial(x)/gamma(x)) == 1 + x
assert simplify(
Eq(sin(x)**2 + cos(x)**2, factorial(x)/gamma(x))) == Eq(x, 1)
nc = symbols('nc', commutative=False)
assert simplify(x + x*nc) == x*(1 + nc)
# issue 6123
# f = exp(-I*(k*sqrt(t) + x/(2*sqrt(t)))**2)
# ans = integrate(f, (k, -oo, oo), conds='none')
ans = I*(-pi*x*exp(-3*I*pi/4 + I*x**2/(4*t))*erf(x*exp(-3*I*pi/4)/
(2*sqrt(t)))/(2*sqrt(t)) + pi*x*exp(-3*I*pi/4 + I*x**2/(4*t))/
(2*sqrt(t)))*exp(-I*x**2/(4*t))/(sqrt(pi)*x) - I*sqrt(pi) * \
(-erf(x*exp(I*pi/4)/(2*sqrt(t))) + 1)*exp(I*pi/4)/(2*sqrt(t))
assert simplify(ans) == -(-1)**(S(3)/4)*sqrt(pi)/sqrt(t)
# issue 6370
assert simplify(2**(2 + x)/4) == 2**x
def test_simplify_complex():
cosAsExp = cos(x)._eval_rewrite_as_exp(x)
tanAsExp = tan(x)._eval_rewrite_as_exp(x)
assert simplify(cosAsExp*tanAsExp) == sin(x) # issue 4341
# issue 10124
assert simplify(exp(Matrix([[0, -1], [1, 0]]))) == Matrix([[cos(1),
-sin(1)], [sin(1), cos(1)]])
def test_simplify_ratio():
# roots of x**3-3*x+5
roots = ['(1/2 - sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3) + 1/((1/2 - '
'sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3))',
'1/((1/2 + sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3)) + '
'(1/2 + sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3)',
'-(sqrt(21)/2 + 5/2)**(1/3) - 1/(sqrt(21)/2 + 5/2)**(1/3)']
for r in roots:
r = S(r)
assert count_ops(simplify(r, ratio=1)) <= count_ops(r)
# If ratio=oo, simplify() is always applied:
assert simplify(r, ratio=oo) is not r
def test_simplify_measure():
measure1 = lambda expr: len(str(expr))
measure2 = lambda expr: -count_ops(expr)
# Return the most complicated result
expr = (x + 1)/(x + sin(x)**2 + cos(x)**2)
assert measure1(simplify(expr, measure=measure1)) <= measure1(expr)
assert measure2(simplify(expr, measure=measure2)) <= measure2(expr)
expr2 = Eq(sin(x)**2 + cos(x)**2, 1)
assert measure1(simplify(expr2, measure=measure1)) <= measure1(expr2)
assert measure2(simplify(expr2, measure=measure2)) <= measure2(expr2)
def test_simplify_rational():
expr = 2**x*2.**y
assert simplify(expr, rational = True) == 2**(x+y)
assert simplify(expr, rational = None) == 2.0**(x+y)
assert simplify(expr, rational = False) == expr
def test_simplify_issue_1308():
assert simplify(exp(-Rational(1, 2)) + exp(-Rational(3, 2))) == \
(1 + E)*exp(-Rational(3, 2))
def test_issue_5652():
assert simplify(E + exp(-E)) == exp(-E) + E
n = symbols('n', commutative=False)
assert simplify(n + n**(-n)) == n + n**(-n)
def test_simplify_fail1():
x = Symbol('x')
y = Symbol('y')
e = (x + y)**2/(-4*x*y**2 - 2*y**3 - 2*x**2*y)
assert simplify(e) == 1 / (-2*y)
def test_nthroot():
assert nthroot(90 + 34*sqrt(7), 3) == sqrt(7) + 3
q = 1 + sqrt(2) - 2*sqrt(3) + sqrt(6) + sqrt(7)
assert nthroot(expand_multinomial(q**3), 3) == q
assert nthroot(41 + 29*sqrt(2), 5) == 1 + sqrt(2)
assert nthroot(-41 - 29*sqrt(2), 5) == -1 - sqrt(2)
expr = 1320*sqrt(10) + 4216 + 2576*sqrt(6) + 1640*sqrt(15)
assert nthroot(expr, 5) == 1 + sqrt(6) + sqrt(15)
q = 1 + sqrt(2) + sqrt(3) + sqrt(5)
assert expand_multinomial(nthroot(expand_multinomial(q**5), 5)) == q
q = 1 + sqrt(2) + 7*sqrt(6) + 2*sqrt(10)
assert nthroot(expand_multinomial(q**5), 5, 8) == q
q = 1 + sqrt(2) - 2*sqrt(3) + 1171*sqrt(6)
assert nthroot(expand_multinomial(q**3), 3) == q
assert nthroot(expand_multinomial(q**6), 6) == q
def test_nthroot1():
q = 1 + sqrt(2) + sqrt(3) + S(1)/10**20
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
q = 1 + sqrt(2) + sqrt(3) + S(1)/10**30
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
def test_separatevars():
x, y, z, n = symbols('x,y,z,n')
assert separatevars(2*n*x*z + 2*x*y*z) == 2*x*z*(n + y)
assert separatevars(x*z + x*y*z) == x*z*(1 + y)
assert separatevars(pi*x*z + pi*x*y*z) == pi*x*z*(1 + y)
assert separatevars(x*y**2*sin(x) + x*sin(x)*sin(y)) == \
x*(sin(y) + y**2)*sin(x)
assert separatevars(x*exp(x + y) + x*exp(x)) == x*(1 + exp(y))*exp(x)
assert separatevars((x*(y + 1))**z).is_Pow # != x**z*(1 + y)**z
assert separatevars(1 + x + y + x*y) == (x + 1)*(y + 1)
assert separatevars(y/pi*exp(-(z - x)/cos(n))) == \
y*exp(x/cos(n))*exp(-z/cos(n))/pi
assert separatevars((x + y)*(x - y) + y**2 + 2*x + 1) == (x + 1)**2
# issue 4858
p = Symbol('p', positive=True)
assert separatevars(sqrt(p**2 + x*p**2)) == p*sqrt(1 + x)
assert separatevars(sqrt(y*(p**2 + x*p**2))) == p*sqrt(y*(1 + x))
assert separatevars(sqrt(y*(p**2 + x*p**2)), force=True) == \
p*sqrt(y)*sqrt(1 + x)
# issue 4865
assert separatevars(sqrt(x*y)).is_Pow
assert separatevars(sqrt(x*y), force=True) == sqrt(x)*sqrt(y)
# issue 4957
# any type sequence for symbols is fine
assert separatevars(((2*x + 2)*y), dict=True, symbols=()) == \
{'coeff': 1, x: 2*x + 2, y: y}
# separable
assert separatevars(((2*x + 2)*y), dict=True, symbols=[x]) == \
{'coeff': y, x: 2*x + 2}
assert separatevars(((2*x + 2)*y), dict=True, symbols=[]) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True, symbols=None) == \
{'coeff': y*(2*x + 2)}
# not separable
assert separatevars(3, dict=True) is None
assert separatevars(2*x + y, dict=True, symbols=()) is None
assert separatevars(2*x + y, dict=True) is None
assert separatevars(2*x + y, dict=True, symbols=None) == {'coeff': 2*x + y}
# issue 4808
n, m = symbols('n,m', commutative=False)
assert separatevars(m + n*m) == (1 + n)*m
assert separatevars(x + x*n) == x*(1 + n)
# issue 4910
f = Function('f')
assert separatevars(f(x) + x*f(x)) == f(x) + x*f(x)
# a noncommutable object present
eq = x*(1 + hyper((), (), y*z))
assert separatevars(eq) == eq
def test_separatevars_advanced_factor():
x, y, z = symbols('x,y,z')
assert separatevars(1 + log(x)*log(y) + log(x) + log(y)) == \
(log(x) + 1)*(log(y) + 1)
assert separatevars(1 + x - log(z) - x*log(z) - exp(y)*log(z) -
x*exp(y)*log(z) + x*exp(y) + exp(y)) == \
-((x + 1)*(log(z) - 1)*(exp(y) + 1))
x, y = symbols('x,y', positive=True)
assert separatevars(1 + log(x**log(y)) + log(x*y)) == \
(log(x) + 1)*(log(y) + 1)
def test_hypersimp():
n, k = symbols('n,k', integer=True)
assert hypersimp(factorial(k), k) == k + 1
assert hypersimp(factorial(k**2), k) is None
assert hypersimp(1/factorial(k), k) == 1/(k + 1)
assert hypersimp(2**k/factorial(k)**2, k) == 2/(k + 1)**2
assert hypersimp(binomial(n, k), k) == (n - k)/(k + 1)
assert hypersimp(binomial(n + 1, k), k) == (n - k + 1)/(k + 1)
term = (4*k + 1)*factorial(k)/factorial(2*k + 1)
assert hypersimp(term, k) == (S(1)/2)*((4*k + 5)/(3 + 14*k + 8*k**2))
term = 1/((2*k - 1)*factorial(2*k + 1))
assert hypersimp(term, k) == (k - S(1)/2)/((k + 1)*(2*k + 1)*(2*k + 3))
term = binomial(n, k)*(-1)**k/factorial(k)
assert hypersimp(term, k) == (k - n)/(k + 1)**2
def test_nsimplify():
x = Symbol("x")
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1 + x) == 1 + x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1 - GoldenRatio) == (1 - sqrt(5))/2
assert nsimplify((1 + sqrt(5))/4, [GoldenRatio]) == GoldenRatio/2
assert nsimplify(2/GoldenRatio, [GoldenRatio]) == 2*GoldenRatio - 2
assert nsimplify(exp(5*pi*I/3, evaluate=False)) == \
sympify('1/2 - sqrt(3)*I/2')
assert nsimplify(sin(3*pi/5, evaluate=False)) == \
sympify('sqrt(sqrt(5)/8 + 5/8)')
assert nsimplify(sqrt(atan('1', evaluate=False))*(2 + I), [pi]) == \
sqrt(pi) + sqrt(pi)/2*I
assert nsimplify(2 + exp(2*atan('1/4')*I)) == sympify('49/17 + 8*I/17')
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0**(1/3.), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == \
2**Rational(1, 3)
assert nsimplify(x + .5, rational=True) == Rational(1, 2) + x
assert nsimplify(1/.3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == \
sympify('109861228866811/100000000000000')
assert nsimplify(Float(0.272198261287950), [pi, log(2)]) == pi*log(2)/8
assert nsimplify(Float(0.272198261287950).n(3), [pi, log(2)]) == \
-pi/4 - log(2) + S(7)/4
assert nsimplify(x/7.0) == x/7
assert nsimplify(pi/1e2) == pi/100
assert nsimplify(pi/1e2, rational=False) == pi/100.0
assert nsimplify(pi/1e-7) == 10000000*pi
assert not nsimplify(
factor(-3.0*z**2*(z**2)**(-2.5) + 3*(z**2)**(-1.5))).atoms(Float)
e = x**0.0
assert e.is_Pow and nsimplify(x**0.0) == 1
assert nsimplify(3.333333, tolerance=0.1, rational=True) == Rational(10, 3)
assert nsimplify(3.333333, tolerance=0.01, rational=True) == Rational(10, 3)
assert nsimplify(3.666666, tolerance=0.1, rational=True) == Rational(11, 3)
assert nsimplify(3.666666, tolerance=0.01, rational=True) == Rational(11, 3)
assert nsimplify(33, tolerance=10, rational=True) == Rational(33)
assert nsimplify(33.33, tolerance=10, rational=True) == Rational(30)
assert nsimplify(37.76, tolerance=10, rational=True) == Rational(40)
assert nsimplify(-203.1) == -S(2031)/10
assert nsimplify(.2, tolerance=0) == S.One/5
assert nsimplify(-.2, tolerance=0) == -S.One/5
assert nsimplify(.2222, tolerance=0) == S(1111)/5000
assert nsimplify(-.2222, tolerance=0) == -S(1111)/5000
# issue 7211, PR 4112
assert nsimplify(S(2e-8)) == S(1)/50000000
# issue 7322 direct test
assert nsimplify(1e-42, rational=True) != 0
# issue 10336
inf = Float('inf')
infs = (-oo, oo, inf, -inf)
for i in infs:
ans = sign(i)*oo
assert nsimplify(i) == ans
assert nsimplify(i + x) == x + ans
assert nsimplify(0.33333333, rational=True, rational_conversion='exact') == Rational(0.33333333)
# Make sure nsimplify on expressions uses full precision
assert nsimplify(pi.evalf(100)*x, rational_conversion='exact').evalf(100) == pi.evalf(100)*x
def test_issue_9448():
tmp = sympify("1/(1 - (-1)**(2/3) - (-1)**(1/3)) + 1/(1 + (-1)**(2/3) + (-1)**(1/3))")
assert nsimplify(tmp) == S(1)/2
def test_extract_minus_sign():
x = Symbol("x")
y = Symbol("y")
a = Symbol("a")
b = Symbol("b")
assert simplify(-x/-y) == x/y
assert simplify(-x/y) == -x/y
assert simplify(x/y) == x/y
assert simplify(x/-y) == -x/y
assert simplify(-x/0) == zoo*x
assert simplify(S(-5)/0) == zoo
assert simplify(-a*x/(-y - b)) == a*x/(b + y)
def test_diff():
x = Symbol("x")
y = Symbol("y")
f = Function("f")
g = Function("g")
assert simplify(g(x).diff(x)*f(x).diff(x) - f(x).diff(x)*g(x).diff(x)) == 0
assert simplify(2*f(x)*f(x).diff(x) - diff(f(x)**2, x)) == 0
assert simplify(diff(1/f(x), x) + f(x).diff(x)/f(x)**2) == 0
assert simplify(f(x).diff(x, y) - f(x).diff(y, x)) == 0
def test_logcombine_1():
x, y = symbols("x,y")
a = Symbol("a")
z, w = symbols("z,w", positive=True)
b = Symbol("b", real=True)
assert logcombine(log(x) + 2*log(y)) == log(x) + 2*log(y)
assert logcombine(log(x) + 2*log(y), force=True) == log(x*y**2)
assert logcombine(a*log(w) + log(z)) == a*log(w) + log(z)
assert logcombine(b*log(z) + b*log(x)) == log(z**b) + b*log(x)
assert logcombine(b*log(z) - log(w)) == log(z**b/w)
assert logcombine(log(x)*log(z)) == log(x)*log(z)
assert logcombine(log(w)*log(x)) == log(w)*log(x)
assert logcombine(cos(-2*log(z) + b*log(w))) in [cos(log(w**b/z**2)),
cos(log(z**2/w**b))]
assert logcombine(log(log(x) - log(y)) - log(z), force=True) == \
log(log(x/y)/z)
assert logcombine((2 + I)*log(x), force=True) == (2 + I)*log(x)
assert logcombine((x**2 + log(x) - log(y))/(x*y), force=True) == \
(x**2 + log(x/y))/(x*y)
# the following could also give log(z*x**log(y**2)), what we
# are testing is that a canonical result is obtained
assert logcombine(log(x)*2*log(y) + log(z), force=True) == \
log(z*y**log(x**2))
assert logcombine((x*y + sqrt(x**4 + y**4) + log(x) - log(y))/(pi*x**Rational(2, 3)*
sqrt(y)**3), force=True) == (
x*y + sqrt(x**4 + y**4) + log(x/y))/(pi*x**(S(2)/3)*y**(S(3)/2))
assert logcombine(gamma(-log(x/y))*acos(-log(x/y)), force=True) == \
acos(-log(x/y))*gamma(-log(x/y))
assert logcombine(2*log(z)*log(w)*log(x) + log(z) + log(w)) == \
log(z**log(w**2))*log(x) + log(w*z)
assert logcombine(3*log(w) + 3*log(z)) == log(w**3*z**3)
assert logcombine(x*(y + 1) + log(2) + log(3)) == x*(y + 1) + log(6)
assert logcombine((x + y)*log(w) + (-x - y)*log(3)) == (x + y)*log(w/3)
# a single unknown can combine
assert logcombine(log(x) + log(2)) == log(2*x)
eq = log(abs(x)) + log(abs(y))
assert logcombine(eq) == eq
reps = {x: 0, y: 0}
assert log(abs(x)*abs(y)).subs(reps) != eq.subs(reps)
def test_logcombine_complex_coeff():
i = Integral((sin(x**2) + cos(x**3))/x, x)
assert logcombine(i, force=True) == i
assert logcombine(i + 2*log(x), force=True) == \
i + log(x**2)
def test_issue_5950():
x, y = symbols("x,y", positive=True)
assert logcombine(log(3) - log(2)) == log(Rational(3,2), evaluate=False)
assert logcombine(log(x) - log(y)) == log(x/y)
assert logcombine(log(Rational(3,2), evaluate=False) - log(2)) == \
log(Rational(3,4), evaluate=False)
def test_posify():
from sympy.abc import x
assert str(posify(
x +
Symbol('p', positive=True) +
Symbol('n', negative=True))) == '(_x + n + p, {_x: x})'
eq, rep = posify(1/x)
assert log(eq).expand().subs(rep) == -log(x)
assert str(posify([x, 1 + x])) == '([_x, _x + 1], {_x: x})'
x = symbols('x')
p = symbols('p', positive=True)
n = symbols('n', negative=True)
orig = [x, n, p]
modified, reps = posify(orig)
assert str(modified) == '[_x, n, p]'
assert [w.subs(reps) for w in modified] == orig
assert str(Integral(posify(1/x + y)[0], (y, 1, 3)).expand()) == \
'Integral(1/_x, (y, 1, 3)) + Integral(_y, (y, 1, 3))'
assert str(Sum(posify(1/x**n)[0], (n,1,3)).expand()) == \
'Sum(_x**(-n), (n, 1, 3))'
# issue 16438
k = Symbol('k', finite=True)
eq, rep = posify(k)
assert eq.assumptions0 == {'positive': True, 'zero': False, 'imaginary': False,
'nonpositive': False, 'commutative': True, 'hermitian': True, 'real': True, 'nonzero': True,
'nonnegative': True, 'negative': False, 'complex': True, 'finite': True, 'infinite': False}
def test_issue_4194():
# simplify should call cancel
from sympy.abc import x, y
f = Function('f')
assert simplify((4*x + 6*f(y))/(2*x + 3*f(y))) == 2
@XFAIL
def test_simplify_float_vs_integer():
# Test for issue 4473:
# https://github.com/sympy/sympy/issues/4473
assert simplify(x**2.0 - x**2) == 0
assert simplify(x**2 - x**2.0) == 0
def test_as_content_primitive():
assert (x/2 + y).as_content_primitive() == (S.Half, x + 2*y)
assert (x/2 + y).as_content_primitive(clear=False) == (S.One, x/2 + y)
assert (y*(x/2 + y)).as_content_primitive() == (S.Half, y*(x + 2*y))
assert (y*(x/2 + y)).as_content_primitive(clear=False) == (S.One, y*(x/2 + y))
# although the _as_content_primitive methods do not alter the underlying structure,
# the as_content_primitive function will touch up the expression and join
# bases that would otherwise have not been joined.
assert ((x*(2 + 2*x)*(3*x + 3)**2)).as_content_primitive() == \
(18, x*(x + 1)**3)
assert (2 + 2*x + 2*y*(3 + 3*y)).as_content_primitive() == \
(2, x + 3*y*(y + 1) + 1)
assert ((2 + 6*x)**2).as_content_primitive() == \
(4, (3*x + 1)**2)
assert ((2 + 6*x)**(2*y)).as_content_primitive() == \
(1, (_keep_coeff(S(2), (3*x + 1)))**(2*y))
assert (5 + 10*x + 2*y*(3 + 3*y)).as_content_primitive() == \
(1, 10*x + 6*y*(y + 1) + 5)
assert ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() == \
(11, x*(y + 1))
assert ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive() == \
(121, x**2*(y + 1)**2)
assert (y**2).as_content_primitive() == \
(1, y**2)
assert (S.Infinity).as_content_primitive() == (1, oo)
eq = x**(2 + y)
assert (eq).as_content_primitive() == (1, eq)
assert (S.Half**(2 + x)).as_content_primitive() == (S(1)/4, 2**-x)
assert ((-S.Half)**(2 + x)).as_content_primitive() == \
(S(1)/4, (-S.Half)**x)
assert ((-S.Half)**(2 + x)).as_content_primitive() == \
(S(1)/4, (-S.Half)**x)
assert (4**((1 + y)/2)).as_content_primitive() == (2, 4**(y/2))
assert (3**((1 + y)/2)).as_content_primitive() == \
(1, 3**(Mul(S(1)/2, 1 + y, evaluate=False)))
assert (5**(S(3)/4)).as_content_primitive() == (1, 5**(S(3)/4))
assert (5**(S(7)/4)).as_content_primitive() == (5, 5**(S(3)/4))
assert Add(5*z/7, 0.5*x, 3*y/2, evaluate=False).as_content_primitive() == \
(S(1)/14, 7.0*x + 21*y + 10*z)
assert (2**(S(3)/4) + 2**(S(1)/4)*sqrt(3)).as_content_primitive(radical=True) == \
(1, 2**(S(1)/4)*(sqrt(2) + sqrt(3)))
def test_signsimp():
e = x*(-x + 1) + x*(x - 1)
assert signsimp(Eq(e, 0)) is S.true
assert Abs(x - 1) == Abs(1 - x)
assert signsimp(y - x) == y - x
assert signsimp(y - x, evaluate=False) == Mul(-1, x - y, evaluate=False)
def test_besselsimp():
from sympy import besselj, besseli, exp_polar, cosh, cosine_transform
assert besselsimp(exp(-I*pi*y/2)*besseli(y, z*exp_polar(I*pi/2))) == \
besselj(y, z)
assert besselsimp(exp(-I*pi*a/2)*besseli(a, 2*sqrt(x)*exp_polar(I*pi/2))) == \
besselj(a, 2*sqrt(x))
assert besselsimp(sqrt(2)*sqrt(pi)*x**(S(1)/4)*exp(I*pi/4)*exp(-I*pi*a/2) *
besseli(-S(1)/2, sqrt(x)*exp_polar(I*pi/2)) *
besseli(a, sqrt(x)*exp_polar(I*pi/2))/2) == \
besselj(a, sqrt(x)) * cos(sqrt(x))
assert besselsimp(besseli(S(-1)/2, z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(a, z*exp_polar(-I*pi/2))) == \
exp(-I*pi*a/2)*besselj(a, z)
assert cosine_transform(1/t*sin(a/t), t, y) == \
sqrt(2)*sqrt(pi)*besselj(0, 2*sqrt(a)*sqrt(y))/2
def test_Piecewise():
e1 = x*(x + y) - y*(x + y)
e2 = sin(x)**2 + cos(x)**2
e3 = expand((x + y)*y/x)
s1 = simplify(e1)
s2 = simplify(e2)
s3 = simplify(e3)
assert simplify(Piecewise((e1, x < e2), (e3, True))) == \
Piecewise((s1, x < s2), (s3, True))
def test_polymorphism():
class A(Basic):
def _eval_simplify(x, **kwargs):
return 1
a = A(5, 2)
assert simplify(a) == 1
def test_issue_from_PR1599():
n1, n2, n3, n4 = symbols('n1 n2 n3 n4', negative=True)
assert simplify(I*sqrt(n1)) == -sqrt(-n1)
def test_issue_6811():
eq = (x + 2*y)*(2*x + 2)
assert simplify(eq) == (x + 1)*(x + 2*y)*2
# reject the 2-arg Mul -- these are a headache for test writing
assert simplify(eq.expand()) == \
2*x**2 + 4*x*y + 2*x + 4*y
def test_issue_6920():
e = [cos(x) + I*sin(x), cos(x) - I*sin(x),
cosh(x) - sinh(x), cosh(x) + sinh(x)]
ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)]
# wrap in f to show that the change happens wherever ei occurs
f = Function('f')
assert [simplify(f(ei)).args[0] for ei in e] == ok
def test_issue_7001():
from sympy.abc import r, R
assert simplify(-(r*Piecewise((4*pi/3, r <= R),
(-8*pi*R**3/(3*r**3), True)) + 2*Piecewise((4*pi*r/3, r <= R),
(4*pi*R**3/(3*r**2), True)))/(4*pi*r)) == \
Piecewise((-1, r <= R), (0, True))
def test_inequality_no_auto_simplify():
# no simplify on creation but can be simplified
lhs = cos(x)**2 + sin(x)**2
rhs = 2
e = Lt(lhs, rhs, evaluate=False)
assert e is not S.true
assert simplify(e)
def test_issue_9398():
from sympy import Number, cancel
assert cancel(1e-14) != 0
assert cancel(1e-14*I) != 0
assert simplify(1e-14) != 0
assert simplify(1e-14*I) != 0
assert (I*Number(1.)*Number(10)**Number(-14)).simplify() != 0
assert cancel(1e-20) != 0
assert cancel(1e-20*I) != 0
assert simplify(1e-20) != 0
assert simplify(1e-20*I) != 0
assert cancel(1e-100) != 0
assert cancel(1e-100*I) != 0
assert simplify(1e-100) != 0
assert simplify(1e-100*I) != 0
f = Float("1e-1000")
assert cancel(f) != 0
assert cancel(f*I) != 0
assert simplify(f) != 0
assert simplify(f*I) != 0
def test_issue_9324_simplify():
M = MatrixSymbol('M', 10, 10)
e = M[0, 0] + M[5, 4] + 1304
assert simplify(e) == e
def test_issue_13474():
x = Symbol('x')
assert simplify(x + csch(sinc(1))) == x + csch(sinc(1))
def test_simplify_function_inverse():
# "inverse" attribute does not guarantee that f(g(x)) is x
# so this simplification should not happen automatically.
# See issue #12140
x, y = symbols('x, y')
g = Function('g')
class f(Function):
def inverse(self, argindex=1):
return g
assert simplify(f(g(x))) == f(g(x))
assert inversecombine(f(g(x))) == x
assert simplify(f(g(x)), inverse=True) == x
assert simplify(f(g(sin(x)**2 + cos(x)**2)), inverse=True) == 1
assert simplify(f(g(x, y)), inverse=True) == f(g(x, y))
assert simplify(2*asin(sin(3*x)), inverse=True) == 6*x
assert simplify(log(exp(x))) == log(exp(x))
assert simplify(log(exp(x)), inverse=True) == x
assert simplify(log(exp(x), 2), inverse=True) == x/log(2)
assert simplify(log(exp(x), 2, evaluate=False), inverse=True) == x/log(2)
def test_clear_coefficients():
from sympy.simplify.simplify import clear_coefficients
assert clear_coefficients(4*y*(6*x + 3)) == (y*(2*x + 1), 0)
assert clear_coefficients(4*y*(6*x + 3) - 2) == (y*(2*x + 1), S(1)/6)
assert clear_coefficients(4*y*(6*x + 3) - 2, x) == (y*(2*x + 1), x/12 + S(1)/6)
assert clear_coefficients(sqrt(2) - 2) == (sqrt(2), 2)
assert clear_coefficients(4*sqrt(2) - 2) == (sqrt(2), S.Half)
assert clear_coefficients(S(3), x) == (0, x - 3)
assert clear_coefficients(S.Infinity, x) == (S.Infinity, x)
assert clear_coefficients(-S.Pi, x) == (S.Pi, -x)
assert clear_coefficients(2 - S.Pi/3, x) == (pi, -3*x + 6)
def test_nc_simplify():
from sympy.simplify.simplify import nc_simplify
from sympy.matrices.expressions import (MatrixExpr, MatAdd, MatMul,
MatPow, Identity)
from sympy.core import Pow
from functools import reduce
a, b, c, d = symbols('a b c d', commutative = False)
x = Symbol('x')
A = MatrixSymbol("A", x, x)
B = MatrixSymbol("B", x, x)
C = MatrixSymbol("C", x, x)
D = MatrixSymbol("D", x, x)
subst = {a: A, b: B, c: C, d:D}
funcs = {Add: lambda x,y: x+y, Mul: lambda x,y: x*y }
def _to_matrix(expr):
if expr in subst:
return subst[expr]
if isinstance(expr, Pow):
return MatPow(_to_matrix(expr.args[0]), expr.args[1])
elif isinstance(expr, (Add, Mul)):
return reduce(funcs[expr.func],[_to_matrix(a) for a in expr.args])
else:
return expr*Identity(x)
def _check(expr, simplified, deep=True, matrix=True):
assert nc_simplify(expr, deep=deep) == simplified
assert expand(expr) == expand(simplified)
if matrix:
m_simp = _to_matrix(simplified).doit(inv_expand=False)
assert nc_simplify(_to_matrix(expr), deep=deep) == m_simp
_check(a*b*a*b*a*b*c*(a*b)**3*c, ((a*b)**3*c)**2)
_check(a*b*(a*b)**-2*a*b, 1)
_check(a**2*b*a*b*a*b*(a*b)**-1, a*(a*b)**2, matrix=False)
_check(b*a*b**2*a*b**2*a*b**2, b*(a*b**2)**3)
_check(a*b*a**2*b*a**2*b*a**3, (a*b*a)**3*a**2)
_check(a**2*b*a**4*b*a**4*b*a**2, (a**2*b*a**2)**3)
_check(a**3*b*a**4*b*a**4*b*a, a**3*(b*a**4)**3*a**-3)
_check(a*b*a*b + a*b*c*x*a*b*c, (a*b)**2 + x*(a*b*c)**2)
_check(a*b*a*b*c*a*b*a*b*c, ((a*b)**2*c)**2)
_check(b**-1*a**-1*(a*b)**2, a*b)
_check(a**-1*b*c**-1, (c*b**-1*a)**-1)
expr = a**3*b*a**4*b*a**4*b*a**2*b*a**2*(b*a**2)**2*b*a**2*b*a**2
for i in range(10):
expr *= a*b
_check(expr, a**3*(b*a**4)**2*(b*a**2)**6*(a*b)**10)
_check((a*b*a*b)**2, (a*b*a*b)**2, deep=False)
_check(a*b*(c*d)**2, a*b*(c*d)**2)
expr = b**-1*(a**-1*b**-1 - a**-1*c*b**-1)**-1*a**-1
assert nc_simplify(expr) == (1-c)**-1
# commutative expressions should be returned without an error
assert nc_simplify(2*x**2) == 2*x**2
|
1d824f418b2e77d807646a0caa9b7eb1d24b20fb564ea7608fd1d7c1ad86f7b5 | from sympy import symbols, IndexedBase, HadamardProduct, Identity, cos
from sympy.codegen.array_utils import (CodegenArrayContraction,
CodegenArrayTensorProduct, CodegenArrayDiagonal,
CodegenArrayPermuteDims, CodegenArrayElementwiseAdd,
_codegen_array_parse, _recognize_matrix_expression, _RecognizeMatOp,
_RecognizeMatMulLines, _unfold_recognized_expr,
parse_indexed_expression, recognize_matrix_expression,
_parse_matrix_expression)
from sympy import (MatrixSymbol, Sum)
from sympy.combinatorics import Permutation
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.expressions.diagonal import DiagonalizeVector
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.matrices import (Trace, MatAdd, MatMul, Transpose)
from sympy.utilities.pytest import raises, XFAIL
A, B = symbols("A B", cls=IndexedBase)
i, j, k, l, m, n = symbols("i j k l m n")
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
P = MatrixSymbol("P", k, k)
Q = MatrixSymbol("Q", k, k)
def test_codegen_array_contraction_construction():
cg = CodegenArrayContraction(A)
assert cg == A
s = Sum(A[i]*B[i], (i, 0, 3))
cg = parse_indexed_expression(s)
assert cg == CodegenArrayContraction(CodegenArrayTensorProduct(A, B), (0, 1))
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, B), (1, 0))
assert cg == CodegenArrayContraction(CodegenArrayTensorProduct(A, B), (0, 1))
expr = M*N
result = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2))
assert CodegenArrayContraction.from_MatMul(expr) == result
elem = expr[i, j]
assert parse_indexed_expression(elem) == result
expr = M*N*M
result = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, M), (1, 2), (3, 4))
assert CodegenArrayContraction.from_MatMul(expr) == result
elem = expr[i, j]
result = CodegenArrayContraction(CodegenArrayTensorProduct(M, M, N), (1, 4), (2, 5))
cg = parse_indexed_expression(elem)
cg = cg.sort_args_by_name()
assert cg == result
def test_codegen_array_contraction_indices_types():
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 1))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 0), (0, 1)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(0, 1)]
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 1), (1, 0)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(1, 2)]
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, M, N), (1, 4), (2, 5))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 1), (2, 0)], [(1, 0), (2, 1)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(1, 4), (2, 5)]
def test_codegen_array_recognize_matrix_mul_lines():
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M), (0, 1))
assert recognize_matrix_expression(cg) == Trace(M)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 1), (2, 3))
assert recognize_matrix_expression(cg) == [Trace(M), Trace(N)]
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 3), (1, 2))
assert recognize_matrix_expression(cg) == Trace(M*N)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 2), (1, 3))
assert recognize_matrix_expression(cg) == Trace(M*N.T)
cg = parse_indexed_expression((M*N*P)[i,j])
assert recognize_matrix_expression(cg) == M*N*P
cg = CodegenArrayContraction.from_MatMul(M*N*P)
assert recognize_matrix_expression(cg) == M*N*P
cg = parse_indexed_expression((M*N.T*P)[i,j])
assert recognize_matrix_expression(cg) == M*N.T*P
cg = CodegenArrayContraction.from_MatMul(M*N.T*P)
assert recognize_matrix_expression(cg) == M*N.T*P
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M,N,P,Q), (1, 2), (5, 6))
assert recognize_matrix_expression(cg) == [M*N, P*Q]
expr = -2*M*N
elem = expr[i, j]
cg = parse_indexed_expression(elem)
assert recognize_matrix_expression(cg) == -2*M*N
def test_codegen_array_flatten():
# Flatten nested CodegenArrayTensorProduct objects:
expr1 = CodegenArrayTensorProduct(M, N)
expr2 = CodegenArrayTensorProduct(P, Q)
expr = CodegenArrayTensorProduct(expr1, expr2)
assert expr == CodegenArrayTensorProduct(M, N, P, Q)
assert expr.args == (M, N, P, Q)
# Flatten mixed CodegenArrayTensorProduct and CodegenArrayContraction objects:
cg1 = CodegenArrayContraction(expr1, (1, 2))
cg2 = CodegenArrayContraction(expr2, (0, 3))
expr = CodegenArrayTensorProduct(cg1, cg2)
assert expr == CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (1, 2), (4, 7))
expr = CodegenArrayTensorProduct(M, cg1)
assert expr == CodegenArrayContraction(CodegenArrayTensorProduct(M, M, N), (3, 4))
# Flatten nested CodegenArrayContraction objects:
cgnested = CodegenArrayContraction(cg1, (0, 1))
assert cgnested == CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 3), (1, 2))
cgnested = CodegenArrayContraction(CodegenArrayTensorProduct(cg1, cg2), (0, 3))
assert cgnested == CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (0, 6), (1, 2), (4, 7))
cg3 = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (1, 3), (2, 4))
cgnested = CodegenArrayContraction(cg3, (0, 1))
assert cgnested == CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (0, 5), (1, 3), (2, 4))
cgnested = CodegenArrayContraction(cg3, (0, 3), (1, 2))
assert cgnested == CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (0, 7), (1, 3), (2, 4), (5, 6))
cg4 = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (1, 5), (3, 7))
cgnested = CodegenArrayContraction(cg4, (0, 1))
assert cgnested == CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (0, 2), (1, 5), (3, 7))
cgnested = CodegenArrayContraction(cg4, (0, 1), (2, 3))
assert cgnested == CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (0, 2), (1, 5), (3, 7), (4, 6))
# Flatten nested CodegenArrayDiagonal objects:
cg1 = CodegenArrayDiagonal(expr1, (1, 2))
cg2 = CodegenArrayDiagonal(expr2, (0, 3))
cg3 = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P, Q), (1, 3), (2, 4))
cg4 = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P, Q), (1, 5), (3, 7))
cgnested = CodegenArrayDiagonal(cg1, (0, 1))
assert cgnested == CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N), (1, 2), (0, 3))
cgnested = CodegenArrayDiagonal(cg3, (1, 2))
assert cgnested == CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P, Q), (1, 3), (2, 4), (5, 6))
cgnested = CodegenArrayDiagonal(cg4, (1, 2))
assert cgnested == CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P, Q), (1, 5), (3, 7), (2, 4))
def test_codegen_array_parse():
expr = M[i, j]
assert _codegen_array_parse(expr) == (M, (i, j))
expr = M[i, j]*N[k, l]
assert _codegen_array_parse(expr) == (CodegenArrayTensorProduct(M, N), (i, j, k, l))
expr = M[i, j]*N[j, k]
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N), (1, 2)), (i, k, j))
expr = Sum(M[i, j]*N[j, k], (j, 0, k-1))
assert _codegen_array_parse(expr) == (CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2)), (i, k))
expr = M[i, j] + N[i, j]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(M, N), (i, j))
expr = M[i, j] + N[j, i]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(M, CodegenArrayPermuteDims(N, Permutation([1,0]))), (i, j))
expr = M[i, j] + M[j, i]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(M, CodegenArrayPermuteDims(M, Permutation([1,0]))), (i, j))
expr = (M*N*P)[i, j]
assert _codegen_array_parse(expr) == (CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P), (1, 2), (3, 4)), (i, j))
expr = expr.function # Disregard summation in previous expression
ret1, ret2 = _codegen_array_parse(expr)
assert ret1 == CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P), (1, 2), (3, 4))
assert str(ret2) == "(i, j, _i_1, _i_2)"
expr = KroneckerDelta(i, j)*M[i, k]
assert _codegen_array_parse(expr) == (M, ({i, j}, k))
expr = KroneckerDelta(i, j)*KroneckerDelta(j, k)*M[i, l]
assert _codegen_array_parse(expr) == (M, ({i, j, k}, l))
expr = KroneckerDelta(j, k)*(M[i, j]*N[k, l] + N[i, j]*M[k, l])
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(M, N),
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), Permutation(0, 2)(1, 3))
), (1, 2)), (i, l, frozenset({j, k})))
expr = KroneckerDelta(j, m)*KroneckerDelta(m, k)*(M[i, j]*N[k, l] + N[i, j]*M[k, l])
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(M, N),
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), Permutation(0, 2)(1, 3))
), (1, 2)), (i, l, frozenset({j, m, k})))
expr = KroneckerDelta(i, j)*KroneckerDelta(j, k)*KroneckerDelta(k,m)*M[i, 0]*KroneckerDelta(m, n)
assert _codegen_array_parse(expr) == (M, ({i,j,k,m,n}, 0))
expr = M[i, i]
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(M, (0, 1)), (i,))
def test_codegen_array_diagonal():
cg = CodegenArrayDiagonal(M, (1, 0))
assert cg == CodegenArrayDiagonal(M, (0, 1))
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P), (4, 1), (2, 0))
assert cg == CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P), (1, 4), (0, 2))
def test_codegen_recognize_matrix_expression():
expr = CodegenArrayElementwiseAdd(M, CodegenArrayPermuteDims(M, [1, 0]))
rec = _recognize_matrix_expression(expr)
assert rec == _RecognizeMatOp(MatAdd, [M, _RecognizeMatOp(Transpose, [M])])
assert _unfold_recognized_expr(rec) == M + Transpose(M)
expr = M[i,j] + N[i,j]
p1, p2 = _codegen_array_parse(expr)
rec = _recognize_matrix_expression(p1)
assert rec == _RecognizeMatOp(MatAdd, [M, N])
assert _unfold_recognized_expr(rec) == M + N
expr = M[i,j] + N[j,i]
p1, p2 = _codegen_array_parse(expr)
rec = _recognize_matrix_expression(p1)
assert rec == _RecognizeMatOp(MatAdd, [M, _RecognizeMatOp(Transpose, [N])])
assert _unfold_recognized_expr(rec) == M + N.T
expr = M[i,j]*N[k,l] + N[i,j]*M[k,l]
p1, p2 = _codegen_array_parse(expr)
rec = _recognize_matrix_expression(p1)
assert rec == _RecognizeMatOp(MatAdd, [_RecognizeMatMulLines([M, N]), _RecognizeMatMulLines([N, M])])
#assert _unfold_recognized_expr(rec) == TensorProduct(M, N) + TensorProduct(N, M) maybe?
expr = (M*N*P)[i, j]
p1, p2 = _codegen_array_parse(expr)
rec = _recognize_matrix_expression(p1)
assert rec == _RecognizeMatMulLines([_RecognizeMatOp(MatMul, [M, N, P])])
assert _unfold_recognized_expr(rec) == M*N*P
expr = Sum(M[i,j]*(N*P)[j,m], (j, 0, k-1))
p1, p2 = _codegen_array_parse(expr)
rec = _recognize_matrix_expression(p1)
assert rec == _RecognizeMatOp(MatMul, [M, N, P])
assert _unfold_recognized_expr(rec) == M*N*P
expr = Sum((P[j, m] + P[m, j])*(M[i,j]*N[m,n] + N[i,j]*M[m,n]), (j, 0, k-1), (m, 0, k-1))
p1, p2 = _codegen_array_parse(expr)
rec = _recognize_matrix_expression(p1)
assert rec == _RecognizeMatOp(MatAdd, [
_RecognizeMatOp(MatMul, [M, _RecognizeMatOp(MatAdd, [P, _RecognizeMatOp(Transpose, [P])]), N]),
_RecognizeMatOp(MatMul, [N, _RecognizeMatOp(MatAdd, [P, _RecognizeMatOp(Transpose, [P])]), M])
])
assert _unfold_recognized_expr(rec) == M*(P + P.T)*N + N*(P + P.T)*M
def test_codegen_array_shape():
expr = CodegenArrayTensorProduct(M, N, P, Q)
assert expr.shape == (k, k, k, k, k, k, k, k)
Z = MatrixSymbol("Z", m, n)
expr = CodegenArrayTensorProduct(M, Z)
assert expr.shape == (k, k, m, n)
expr2 = CodegenArrayContraction(expr, (0, 1))
assert expr2.shape == (m, n)
expr2 = CodegenArrayDiagonal(expr, (0, 1))
assert expr2.shape == (m, n, k)
exprp = CodegenArrayPermuteDims(expr, [2, 1, 3, 0])
assert exprp.shape == (m, k, n, k)
expr3 = CodegenArrayTensorProduct(N, Z)
expr2 = CodegenArrayElementwiseAdd(expr, expr3)
assert expr2.shape == (k, k, m, n)
# Contraction along axes with discordant dimensions:
raises(ValueError, lambda: CodegenArrayContraction(expr, (1, 2)))
# Also diagonal needs the same dimensions:
raises(ValueError, lambda: CodegenArrayDiagonal(expr, (1, 2)))
def test_codegen_array_parse_out_of_bounds():
expr = Sum(M[i, i], (i, 0, 4))
raises(ValueError, lambda: parse_indexed_expression(expr))
expr = Sum(M[i, i], (i, 0, k))
raises(ValueError, lambda: parse_indexed_expression(expr))
expr = Sum(M[i, i], (i, 1, k-1))
raises(ValueError, lambda: parse_indexed_expression(expr))
expr = Sum(M[i, j]*N[j,m], (j, 0, 4))
raises(ValueError, lambda: parse_indexed_expression(expr))
expr = Sum(M[i, j]*N[j,m], (j, 0, k))
raises(ValueError, lambda: parse_indexed_expression(expr))
expr = Sum(M[i, j]*N[j,m], (j, 1, k-1))
raises(ValueError, lambda: parse_indexed_expression(expr))
def test_codegen_permutedims_sink():
cg = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), [0, 1, 3, 2])
sunk = cg.nest_permutation()
assert sunk == CodegenArrayTensorProduct(M, CodegenArrayPermuteDims(N, [1, 0]))
assert recognize_matrix_expression(sunk) == [M, N.T]
cg = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), [1, 0, 3, 2])
sunk = cg.nest_permutation()
assert sunk == CodegenArrayTensorProduct(CodegenArrayPermuteDims(M, [1, 0]), CodegenArrayPermuteDims(N, [1, 0]))
assert recognize_matrix_expression(sunk) == [M.T, N.T]
cg = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), [3, 2, 1, 0])
sunk = cg.nest_permutation()
assert sunk == CodegenArrayTensorProduct(CodegenArrayPermuteDims(N, [1, 0]), CodegenArrayPermuteDims(M, [1, 0]))
assert recognize_matrix_expression(sunk) == [N.T, M.T]
cg = CodegenArrayPermuteDims(CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2)), [1, 0])
sunk = cg.nest_permutation()
assert sunk == CodegenArrayContraction(CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), [[0, 3]]), (1, 2))
cg = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), [1, 0, 3, 2])
sunk = cg.nest_permutation()
assert sunk == CodegenArrayTensorProduct(CodegenArrayPermuteDims(M, [1, 0]), CodegenArrayPermuteDims(N, [1, 0]))
cg = CodegenArrayPermuteDims(CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P), (1, 2), (3, 4)), [1, 0])
sunk = cg.nest_permutation()
assert sunk == CodegenArrayContraction(CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N, P), [[0, 5]]), (1, 2), (3, 4))
sunk2 = sunk.expr.nest_permutation()
def test_parsing_of_matrix_expressions():
expr = M*N
assert _parse_matrix_expression(expr) == CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2))
expr = Transpose(M)
assert _parse_matrix_expression(expr) == CodegenArrayPermuteDims(M, [1, 0])
expr = M*Transpose(N)
assert _parse_matrix_expression(expr) == CodegenArrayContraction(CodegenArrayTensorProduct(M, CodegenArrayPermuteDims(N, [1, 0])), (1, 2))
def test_special_matrices():
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
expr = a.T*b
elem = expr[0, 0]
cg = parse_indexed_expression(elem)
assert cg == CodegenArrayContraction(CodegenArrayTensorProduct(a, b), (0, 2))
assert recognize_matrix_expression(cg) == a.T*b
def test_recognize_diagonalized_vectors():
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a), (1, 2))
assert cg.split_multiple_contractions() == cg
assert recognize_matrix_expression(cg) == A*a
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, B), (1, 2, 4))
assert cg.split_multiple_contractions() == CodegenArrayContraction(CodegenArrayTensorProduct(A, DiagonalizeVector(a), B), (1, 2), (3, 4))
assert recognize_matrix_expression(cg) == A*DiagonalizeVector(a)*B
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, B), (0, 2, 4))
assert cg.split_multiple_contractions() == CodegenArrayContraction(CodegenArrayTensorProduct(A, DiagonalizeVector(a), B), (0, 2), (3, 4))
assert recognize_matrix_expression(cg) == A.T*DiagonalizeVector(a)*B
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, b, a.T, B), (0, 2, 4, 7, 9))
assert cg.split_multiple_contractions() == CodegenArrayContraction(CodegenArrayTensorProduct(A, DiagonalizeVector(a), DiagonalizeVector(b),
DiagonalizeVector(a), B),
(0, 2), (3, 4), (5, 7), (6, 9))
assert recognize_matrix_expression(cg).doit() == A.T*DiagonalizeVector(a)*DiagonalizeVector(b)*DiagonalizeVector(a)*B.T
I1 = Identity(1)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(I1, I1, I1), (1, 2, 4))
assert cg.split_multiple_contractions() == CodegenArrayContraction(CodegenArrayTensorProduct(I1, I1, I1), (1, 2), (3, 4))
assert recognize_matrix_expression(cg).doit() == Identity(1)
I = Identity(k)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(I, I, I, I, A), (1, 2, 8), (5, 6, 9))
assert recognize_matrix_expression(cg.split_multiple_contractions()).doit() == A
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, C, a, B), (1, 2, 4), (5, 6, 8))
assert cg.split_multiple_contractions() == CodegenArrayContraction(CodegenArrayTensorProduct(A, DiagonalizeVector(a), C, DiagonalizeVector(a), B), (1, 2), (3, 4), (5, 6), (7, 8))
assert recognize_matrix_expression(cg) == A*DiagonalizeVector(a)*C*DiagonalizeVector(a)*B
cg = CodegenArrayContraction(CodegenArrayTensorProduct(a, I1, b, I1, (a.T*b).applyfunc(cos)), (1, 2, 8), (5, 6, 9))
assert cg.split_multiple_contractions() == CodegenArrayContraction(CodegenArrayTensorProduct(a, I1, b, I1, (a.T*b).applyfunc(cos)), (1, 2), (3, 8), (5, 6), (7, 9))
assert recognize_matrix_expression(cg) == MatMul(a, I1, (a.T*b).applyfunc(cos), Transpose(I1), b.T)
# Check no overlap of lines:
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, C, a, B), (1, 2, 4), (5, 6, 8), (3, 7))
raises(ValueError, lambda: cg.split_multiple_contractions())
cg = CodegenArrayContraction(CodegenArrayTensorProduct(a, b, A), (0, 2, 4), (1, 3))
raises(ValueError, lambda: cg.split_multiple_contractions())
|
7f50948772ce0ed0efbe511dfd257e4abd9acdbfafedd2d7a5954d467991b892 | from sympy.core import symbols
from sympy.core.compatibility import range
from sympy.crypto.crypto import (cycle_list,
encipher_shift, encipher_affine, encipher_substitution,
check_and_join, encipher_vigenere, decipher_vigenere,
encipher_hill, decipher_hill, encipher_bifid5, encipher_bifid6,
bifid5_square, bifid6_square, bifid5, bifid6, bifid10,
decipher_bifid5, decipher_bifid6, encipher_kid_rsa,
decipher_kid_rsa, kid_rsa_private_key, kid_rsa_public_key,
decipher_rsa, rsa_private_key, rsa_public_key, encipher_rsa,
lfsr_connection_polynomial, lfsr_autocorrelation, lfsr_sequence,
encode_morse, decode_morse, elgamal_private_key, elgamal_public_key,
encipher_elgamal, decipher_elgamal, dh_private_key, dh_public_key,
dh_shared_key, decipher_shift, decipher_affine, encipher_bifid,
decipher_bifid, bifid_square, padded_key, uniq, decipher_gm,
encipher_gm, gm_public_key, gm_private_key, encipher_bg, decipher_bg,
bg_private_key, bg_public_key, encipher_rot13, decipher_rot13,
encipher_atbash, decipher_atbash)
from sympy.matrices import Matrix
from sympy.ntheory import isprime, is_primitive_root
from sympy.polys.domains import FF
from sympy.utilities.pytest import raises, slow, warns_deprecated_sympy
from random import randrange
def test_cycle_list():
assert cycle_list(3, 4) == [3, 0, 1, 2]
assert cycle_list(-1, 4) == [3, 0, 1, 2]
assert cycle_list(1, 4) == [1, 2, 3, 0]
def test_encipher_shift():
assert encipher_shift("ABC", 0) == "ABC"
assert encipher_shift("ABC", 1) == "BCD"
assert encipher_shift("ABC", -1) == "ZAB"
assert decipher_shift("ZAB", -1) == "ABC"
def test_encipher_rot13():
assert encipher_rot13("ABC") == "NOP"
assert encipher_rot13("NOP") == "ABC"
assert decipher_rot13("ABC") == "NOP"
assert decipher_rot13("NOP") == "ABC"
def test_encipher_affine():
assert encipher_affine("ABC", (1, 0)) == "ABC"
assert encipher_affine("ABC", (1, 1)) == "BCD"
assert encipher_affine("ABC", (-1, 0)) == "AZY"
assert encipher_affine("ABC", (-1, 1), symbols="ABCD") == "BAD"
assert encipher_affine("123", (-1, 1), symbols="1234") == "214"
assert encipher_affine("ABC", (3, 16)) == "QTW"
assert decipher_affine("QTW", (3, 16)) == "ABC"
def test_encipher_atbash():
assert encipher_atbash("ABC") == "ZYX"
assert encipher_atbash("ZYX") == "ABC"
assert decipher_atbash("ABC") == "ZYX"
assert decipher_atbash("ZYX") == "ABC"
def test_encipher_substitution():
assert encipher_substitution("ABC", "BAC", "ABC") == "BAC"
assert encipher_substitution("123", "1243", "1234") == "124"
def test_check_and_join():
assert check_and_join("abc") == "abc"
assert check_and_join(uniq("aaabc")) == "abc"
assert check_and_join("ab c".split()) == "abc"
assert check_and_join("abc", "a", filter=True) == "a"
raises(ValueError, lambda: check_and_join('ab', 'a'))
def test_encipher_vigenere():
assert encipher_vigenere("ABC", "ABC") == "ACE"
assert encipher_vigenere("ABC", "ABC", symbols="ABCD") == "ACA"
assert encipher_vigenere("ABC", "AB", symbols="ABCD") == "ACC"
assert encipher_vigenere("AB", "ABC", symbols="ABCD") == "AC"
assert encipher_vigenere("A", "ABC", symbols="ABCD") == "A"
def test_decipher_vigenere():
assert decipher_vigenere("ABC", "ABC") == "AAA"
assert decipher_vigenere("ABC", "ABC", symbols="ABCD") == "AAA"
assert decipher_vigenere("ABC", "AB", symbols="ABCD") == "AAC"
assert decipher_vigenere("AB", "ABC", symbols="ABCD") == "AA"
assert decipher_vigenere("A", "ABC", symbols="ABCD") == "A"
def test_encipher_hill():
A = Matrix(2, 2, [1, 2, 3, 5])
assert encipher_hill("ABCD", A) == "CFIV"
A = Matrix(2, 2, [1, 0, 0, 1])
assert encipher_hill("ABCD", A) == "ABCD"
assert encipher_hill("ABCD", A, symbols="ABCD") == "ABCD"
A = Matrix(2, 2, [1, 2, 3, 5])
assert encipher_hill("ABCD", A, symbols="ABCD") == "CBAB"
assert encipher_hill("AB", A, symbols="ABCD") == "CB"
# message length, n, does not need to be a multiple of k;
# it is padded
assert encipher_hill("ABA", A) == "CFGC"
assert encipher_hill("ABA", A, pad="Z") == "CFYV"
def test_decipher_hill():
A = Matrix(2, 2, [1, 2, 3, 5])
assert decipher_hill("CFIV", A) == "ABCD"
A = Matrix(2, 2, [1, 0, 0, 1])
assert decipher_hill("ABCD", A) == "ABCD"
assert decipher_hill("ABCD", A, symbols="ABCD") == "ABCD"
A = Matrix(2, 2, [1, 2, 3, 5])
assert decipher_hill("CBAB", A, symbols="ABCD") == "ABCD"
assert decipher_hill("CB", A, symbols="ABCD") == "AB"
# n does not need to be a multiple of k
assert decipher_hill("CFA", A) == "ABAA"
def test_encipher_bifid5():
assert encipher_bifid5("AB", "AB") == "AB"
assert encipher_bifid5("AB", "CD") == "CO"
assert encipher_bifid5("ab", "c") == "CH"
assert encipher_bifid5("a bc", "b") == "BAC"
def test_bifid5_square():
A = bifid5
f = lambda i, j: symbols(A[5*i + j])
M = Matrix(5, 5, f)
assert bifid5_square("") == M
def test_decipher_bifid5():
assert decipher_bifid5("AB", "AB") == "AB"
assert decipher_bifid5("CO", "CD") == "AB"
assert decipher_bifid5("ch", "c") == "AB"
assert decipher_bifid5("b ac", "b") == "ABC"
def test_encipher_bifid6():
assert encipher_bifid6("AB", "AB") == "AB"
assert encipher_bifid6("AB", "CD") == "CP"
assert encipher_bifid6("ab", "c") == "CI"
assert encipher_bifid6("a bc", "b") == "BAC"
def test_decipher_bifid6():
assert decipher_bifid6("AB", "AB") == "AB"
assert decipher_bifid6("CP", "CD") == "AB"
assert decipher_bifid6("ci", "c") == "AB"
assert decipher_bifid6("b ac", "b") == "ABC"
def test_bifid6_square():
A = bifid6
f = lambda i, j: symbols(A[6*i + j])
M = Matrix(6, 6, f)
assert bifid6_square("") == M
def test_rsa_public_key():
assert rsa_public_key(2, 3, 1) == (6, 1)
assert rsa_public_key(5, 3, 3) == (15, 3)
assert rsa_public_key(8, 8, 8) is False
with warns_deprecated_sympy():
assert rsa_public_key(2, 2, 1) == (4, 1)
def test_rsa_private_key():
assert rsa_private_key(2, 3, 1) == (6, 1)
assert rsa_private_key(5, 3, 3) == (15, 3)
assert rsa_private_key(23,29,5) == (667,493)
assert rsa_private_key(8, 8, 8) is False
with warns_deprecated_sympy():
assert rsa_private_key(2, 2, 1) == (4, 1)
def test_rsa_large_key():
# Sample from
# http://www.herongyang.com/Cryptography/JCE-Public-Key-RSA-Private-Public-Key-Pair-Sample.html
p = int('101565610013301240713207239558950144682174355406589305284428666'\
'903702505233009')
q = int('894687191887545488935455605955948413812376003053143521429242133'\
'12069293984003')
e = int('65537')
d = int('893650581832704239530398858744759129594796235440844479456143566'\
'6999402846577625762582824202269399672579058991442587406384754958587'\
'400493169361356902030209')
assert rsa_public_key(p, q, e) == (p*q, e)
assert rsa_private_key(p, q, e) == (p*q, d)
def test_encipher_rsa():
puk = rsa_public_key(2, 3, 1)
assert encipher_rsa(2, puk) == 2
puk = rsa_public_key(5, 3, 3)
assert encipher_rsa(2, puk) == 8
with warns_deprecated_sympy():
puk = rsa_public_key(2, 2, 1)
assert encipher_rsa(2, puk) == 2
def test_decipher_rsa():
prk = rsa_private_key(2, 3, 1)
assert decipher_rsa(2, prk) == 2
prk = rsa_private_key(5, 3, 3)
assert decipher_rsa(8, prk) == 2
with warns_deprecated_sympy():
prk = rsa_private_key(2, 2, 1)
assert decipher_rsa(2, prk) == 2
def test_kid_rsa_public_key():
assert kid_rsa_public_key(1, 2, 1, 1) == (5, 2)
assert kid_rsa_public_key(1, 2, 2, 1) == (8, 3)
assert kid_rsa_public_key(1, 2, 1, 2) == (7, 2)
def test_kid_rsa_private_key():
assert kid_rsa_private_key(1, 2, 1, 1) == (5, 3)
assert kid_rsa_private_key(1, 2, 2, 1) == (8, 3)
assert kid_rsa_private_key(1, 2, 1, 2) == (7, 4)
def test_encipher_kid_rsa():
assert encipher_kid_rsa(1, (5, 2)) == 2
assert encipher_kid_rsa(1, (8, 3)) == 3
assert encipher_kid_rsa(1, (7, 2)) == 2
def test_decipher_kid_rsa():
assert decipher_kid_rsa(2, (5, 3)) == 1
assert decipher_kid_rsa(3, (8, 3)) == 1
assert decipher_kid_rsa(2, (7, 4)) == 1
def test_encode_morse():
assert encode_morse('ABC') == '.-|-...|-.-.'
assert encode_morse('SMS ') == '...|--|...||'
assert encode_morse('SMS\n') == '...|--|...||'
assert encode_morse('') == ''
assert encode_morse(' ') == '||'
assert encode_morse(' ', sep='`') == '``'
assert encode_morse(' ', sep='``') == '````'
assert encode_morse('!@#$%^&*()_+') == '-.-.--|.--.-.|...-..-|-.--.|-.--.-|..--.-|.-.-.'
def test_decode_morse():
assert decode_morse('-.-|.|-.--') == 'KEY'
assert decode_morse('.-.|..-|-.||') == 'RUN'
raises(KeyError, lambda: decode_morse('.....----'))
def test_lfsr_sequence():
raises(TypeError, lambda: lfsr_sequence(1, [1], 1))
raises(TypeError, lambda: lfsr_sequence([1], 1, 1))
F = FF(2)
assert lfsr_sequence([F(1)], [F(1)], 2) == [F(1), F(1)]
assert lfsr_sequence([F(0)], [F(1)], 2) == [F(1), F(0)]
F = FF(3)
assert lfsr_sequence([F(1)], [F(1)], 2) == [F(1), F(1)]
assert lfsr_sequence([F(0)], [F(2)], 2) == [F(2), F(0)]
assert lfsr_sequence([F(1)], [F(2)], 2) == [F(2), F(2)]
def test_lfsr_autocorrelation():
raises(TypeError, lambda: lfsr_autocorrelation(1, 2, 3))
F = FF(2)
s = lfsr_sequence([F(1), F(0)], [F(0), F(1)], 5)
assert lfsr_autocorrelation(s, 2, 0) == 1
assert lfsr_autocorrelation(s, 2, 1) == -1
def test_lfsr_connection_polynomial():
F = FF(2)
x = symbols("x")
s = lfsr_sequence([F(1), F(0)], [F(0), F(1)], 5)
assert lfsr_connection_polynomial(s) == x**2 + 1
s = lfsr_sequence([F(1), F(1)], [F(0), F(1)], 5)
assert lfsr_connection_polynomial(s) == x**2 + x + 1
def test_elgamal_private_key():
a, b, _ = elgamal_private_key(digit=100)
assert isprime(a)
assert is_primitive_root(b, a)
assert len(bin(a)) >= 102
def test_elgamal():
dk = elgamal_private_key(5)
ek = elgamal_public_key(dk)
P = ek[0]
assert P - 1 == decipher_elgamal(encipher_elgamal(P - 1, ek), dk)
raises(ValueError, lambda: encipher_elgamal(P, dk))
raises(ValueError, lambda: encipher_elgamal(-1, dk))
def test_dh_private_key():
p, g, _ = dh_private_key(digit = 100)
assert isprime(p)
assert is_primitive_root(g, p)
assert len(bin(p)) >= 102
def test_dh_public_key():
p1, g1, a = dh_private_key(digit = 100)
p2, g2, ga = dh_public_key((p1, g1, a))
assert p1 == p2
assert g1 == g2
assert ga == pow(g1, a, p1)
def test_dh_shared_key():
prk = dh_private_key(digit = 100)
p, _, ga = dh_public_key(prk)
b = randrange(2, p)
sk = dh_shared_key((p, _, ga), b)
assert sk == pow(ga, b, p)
raises(ValueError, lambda: dh_shared_key((1031, 14, 565), 2000))
def test_padded_key():
assert padded_key('b', 'ab') == 'ba'
raises(ValueError, lambda: padded_key('ab', 'ace'))
raises(ValueError, lambda: padded_key('ab', 'abba'))
def test_bifid():
raises(ValueError, lambda: encipher_bifid('abc', 'b', 'abcde'))
assert encipher_bifid('abc', 'b', 'abcd') == 'bdb'
raises(ValueError, lambda: decipher_bifid('bdb', 'b', 'abcde'))
assert encipher_bifid('bdb', 'b', 'abcd') == 'abc'
raises(ValueError, lambda: bifid_square('abcde'))
assert bifid5_square("B") == \
bifid5_square('BACDEFGHIKLMNOPQRSTUVWXYZ')
assert bifid6_square('B0') == \
bifid6_square('B0ACDEFGHIJKLMNOPQRSTUVWXYZ123456789')
def test_encipher_decipher_gm():
ps = [131, 137, 139, 149, 151, 157, 163, 167,
173, 179, 181, 191, 193, 197, 199]
qs = [89, 97, 101, 103, 107, 109, 113, 127,
131, 137, 139, 149, 151, 157, 47]
messages = [
0, 32855, 34303, 14805, 1280, 75859, 38368,
724, 60356, 51675, 76697, 61854, 18661,
]
for p, q in zip(ps, qs):
pri = gm_private_key(p, q)
for msg in messages:
pub = gm_public_key(p, q)
enc = encipher_gm(msg, pub)
dec = decipher_gm(enc, pri)
assert dec == msg
def test_gm_private_key():
raises(ValueError, lambda: gm_public_key(13, 15))
raises(ValueError, lambda: gm_public_key(0, 0))
raises(ValueError, lambda: gm_public_key(0, 5))
assert 17, 19 == gm_public_key(17, 19)
def test_gm_public_key():
assert 323 == gm_public_key(17, 19)[1]
assert 15 == gm_public_key(3, 5)[1]
raises(ValueError, lambda: gm_public_key(15, 19))
def test_encipher_decipher_bg():
ps = [67, 7, 71, 103, 11, 43, 107, 47,
79, 19, 83, 23, 59, 127, 31]
qs = qs = [7, 71, 103, 11, 43, 107, 47,
79, 19, 83, 23, 59, 127, 31, 67]
messages = [
0, 328, 343, 148, 1280, 758, 383,
724, 603, 516, 766, 618, 186,
]
for p, q in zip(ps, qs):
pri = bg_private_key(p, q)
for msg in messages:
pub = bg_public_key(p, q)
enc = encipher_bg(msg, pub)
dec = decipher_bg(enc, pri)
assert dec == msg
def test_bg_private_key():
raises(ValueError, lambda: bg_private_key(8, 16))
raises(ValueError, lambda: bg_private_key(8, 8))
raises(ValueError, lambda: bg_private_key(13, 17))
assert 23, 31 == bg_private_key(23, 31)
def test_bg_public_key():
assert 5293 == bg_public_key(67, 79)
assert 713 == bg_public_key(23, 31)
raises(ValueError, lambda: bg_private_key(13, 17))
|
c079abd5b4e775b685482931fa19e9601619abb9f7b274b53b853898dacd3d67 | """
This module contains query handlers responsible for calculus queries:
infinitesimal, bounded, etc.
"""
from __future__ import print_function, division
from sympy.logic.boolalg import conjuncts
from sympy.assumptions import Q, ask
from sympy.assumptions.handlers import CommonHandler, test_closed_group
from sympy.matrices.expressions import MatMul, MatrixExpr
from sympy.core.logic import fuzzy_and
from sympy.utilities.iterables import sift
from sympy.core import Basic
from functools import partial
def _Factorization(predicate, expr, assumptions):
if predicate in expr.predicates:
return True
class AskSquareHandler(CommonHandler):
"""
Handler for key 'square'
"""
@staticmethod
def MatrixExpr(expr, assumptions):
return expr.shape[0] == expr.shape[1]
class AskSymmetricHandler(CommonHandler):
"""
Handler for key 'symmetric'
"""
@staticmethod
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if all(ask(Q.symmetric(arg), assumptions) for arg in mmul.args):
return True
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if len(mmul.args) >= 2 and mmul.args[0] == mmul.args[-1].T:
if len(mmul.args) == 2:
return True
return ask(Q.symmetric(MatMul(*mmul.args[1:-1])), assumptions)
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.symmetric(base), assumptions)
return None
@staticmethod
def MatAdd(expr, assumptions):
return all(ask(Q.symmetric(arg), assumptions) for arg in expr.args)
@staticmethod
def MatrixSymbol(expr, assumptions):
if not expr.is_square:
return False
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if Q.symmetric(expr) in conjuncts(assumptions):
return True
@staticmethod
def ZeroMatrix(expr, assumptions):
return ask(Q.square(expr), assumptions)
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.symmetric(expr.arg), assumptions)
Inverse = Transpose
@staticmethod
def MatrixSlice(expr, assumptions):
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if not expr.on_diag:
return None
else:
return ask(Q.symmetric(expr.parent), assumptions)
Identity = staticmethod(CommonHandler.AlwaysTrue)
class AskInvertibleHandler(CommonHandler):
"""
Handler for key 'invertible'
"""
@staticmethod
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if all(ask(Q.invertible(arg), assumptions) for arg in mmul.args):
return True
if any(ask(Q.invertible(arg), assumptions) is False
for arg in mmul.args):
return False
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
if exp.is_negative == False:
return ask(Q.invertible(base), assumptions)
return None
@staticmethod
def MatAdd(expr, assumptions):
return None
@staticmethod
def MatrixSymbol(expr, assumptions):
if not expr.is_square:
return False
if Q.invertible(expr) in conjuncts(assumptions):
return True
Identity, Inverse = [staticmethod(CommonHandler.AlwaysTrue)]*2
ZeroMatrix = staticmethod(CommonHandler.AlwaysFalse)
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.invertible(expr.arg), assumptions)
@staticmethod
def MatrixSlice(expr, assumptions):
if not expr.on_diag:
return None
else:
return ask(Q.invertible(expr.parent), assumptions)
class AskOrthogonalHandler(CommonHandler):
"""
Handler for key 'orthogonal'
"""
predicate = Q.orthogonal
@staticmethod
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if (all(ask(Q.orthogonal(arg), assumptions) for arg in mmul.args) and
factor == 1):
return True
if any(ask(Q.invertible(arg), assumptions) is False
for arg in mmul.args):
return False
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if int_exp:
return ask(Q.orthogonal(base), assumptions)
return None
@staticmethod
def MatAdd(expr, assumptions):
if (len(expr.args) == 1 and
ask(Q.orthogonal(expr.args[0]), assumptions)):
return True
@staticmethod
def MatrixSymbol(expr, assumptions):
if (not expr.is_square or
ask(Q.invertible(expr), assumptions) is False):
return False
if Q.orthogonal(expr) in conjuncts(assumptions):
return True
Identity = staticmethod(CommonHandler.AlwaysTrue)
ZeroMatrix = staticmethod(CommonHandler.AlwaysFalse)
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.orthogonal(expr.arg), assumptions)
Inverse = Transpose
@staticmethod
def MatrixSlice(expr, assumptions):
if not expr.on_diag:
return None
else:
return ask(Q.orthogonal(expr.parent), assumptions)
Factorization = staticmethod(partial(_Factorization, Q.orthogonal))
class AskUnitaryHandler(CommonHandler):
"""
Handler for key 'unitary'
"""
predicate = Q.unitary
@staticmethod
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if (all(ask(Q.unitary(arg), assumptions) for arg in mmul.args) and
abs(factor) == 1):
return True
if any(ask(Q.invertible(arg), assumptions) is False
for arg in mmul.args):
return False
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if int_exp:
return ask(Q.unitary(base), assumptions)
return None
@staticmethod
def MatrixSymbol(expr, assumptions):
if (not expr.is_square or
ask(Q.invertible(expr), assumptions) is False):
return False
if Q.unitary(expr) in conjuncts(assumptions):
return True
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.unitary(expr.arg), assumptions)
Inverse = Transpose
@staticmethod
def MatrixSlice(expr, assumptions):
if not expr.on_diag:
return None
else:
return ask(Q.unitary(expr.parent), assumptions)
@staticmethod
def DFT(expr, assumptions):
return True
Factorization = staticmethod(partial(_Factorization, Q.unitary))
Identity = staticmethod(CommonHandler.AlwaysTrue)
ZeroMatrix = staticmethod(CommonHandler.AlwaysFalse)
class AskFullRankHandler(CommonHandler):
"""
Handler for key 'fullrank'
"""
@staticmethod
def MatMul(expr, assumptions):
if all(ask(Q.fullrank(arg), assumptions) for arg in expr.args):
return True
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if int_exp and ask(~Q.negative(exp), assumptions):
return ask(Q.fullrank(base), assumptions)
return None
Identity = staticmethod(CommonHandler.AlwaysTrue)
ZeroMatrix = staticmethod(CommonHandler.AlwaysFalse)
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.fullrank(expr.arg), assumptions)
Inverse = Transpose
@staticmethod
def MatrixSlice(expr, assumptions):
if ask(Q.orthogonal(expr.parent), assumptions):
return True
class AskPositiveDefiniteHandler(CommonHandler):
"""
Handler for key 'positive_definite'
"""
@staticmethod
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if (all(ask(Q.positive_definite(arg), assumptions)
for arg in mmul.args) and factor > 0):
return True
if (len(mmul.args) >= 2
and mmul.args[0] == mmul.args[-1].T
and ask(Q.fullrank(mmul.args[0]), assumptions)):
return ask(Q.positive_definite(
MatMul(*mmul.args[1:-1])), assumptions)
@staticmethod
def MatPow(expr, assumptions):
# a power of a positive definite matrix is positive definite
if ask(Q.positive_definite(expr.args[0]), assumptions):
return True
@staticmethod
def MatAdd(expr, assumptions):
if all(ask(Q.positive_definite(arg), assumptions)
for arg in expr.args):
return True
@staticmethod
def MatrixSymbol(expr, assumptions):
if not expr.is_square:
return False
if Q.positive_definite(expr) in conjuncts(assumptions):
return True
Identity = staticmethod(CommonHandler.AlwaysTrue)
ZeroMatrix = staticmethod(CommonHandler.AlwaysFalse)
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.positive_definite(expr.arg), assumptions)
Inverse = Transpose
@staticmethod
def MatrixSlice(expr, assumptions):
if not expr.on_diag:
return None
else:
return ask(Q.positive_definite(expr.parent), assumptions)
class AskUpperTriangularHandler(CommonHandler):
"""
Handler for key 'upper_triangular'
"""
@staticmethod
def MatMul(expr, assumptions):
factor, matrices = expr.as_coeff_matrices()
if all(ask(Q.upper_triangular(m), assumptions) for m in matrices):
return True
@staticmethod
def MatAdd(expr, assumptions):
if all(ask(Q.upper_triangular(arg), assumptions) for arg in expr.args):
return True
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.upper_triangular(base), assumptions)
return None
@staticmethod
def MatrixSymbol(expr, assumptions):
if Q.upper_triangular(expr) in conjuncts(assumptions):
return True
Identity, ZeroMatrix = [staticmethod(CommonHandler.AlwaysTrue)]*2
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.lower_triangular(expr.arg), assumptions)
@staticmethod
def Inverse(expr, assumptions):
return ask(Q.upper_triangular(expr.arg), assumptions)
@staticmethod
def MatrixSlice(expr, assumptions):
if not expr.on_diag:
return None
else:
return ask(Q.upper_triangular(expr.parent), assumptions)
Factorization = staticmethod(partial(_Factorization, Q.upper_triangular))
class AskLowerTriangularHandler(CommonHandler):
"""
Handler for key 'lower_triangular'
"""
@staticmethod
def MatMul(expr, assumptions):
factor, matrices = expr.as_coeff_matrices()
if all(ask(Q.lower_triangular(m), assumptions) for m in matrices):
return True
@staticmethod
def MatAdd(expr, assumptions):
if all(ask(Q.lower_triangular(arg), assumptions) for arg in expr.args):
return True
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.lower_triangular(base), assumptions)
return None
@staticmethod
def MatrixSymbol(expr, assumptions):
if Q.lower_triangular(expr) in conjuncts(assumptions):
return True
Identity, ZeroMatrix = [staticmethod(CommonHandler.AlwaysTrue)]*2
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.upper_triangular(expr.arg), assumptions)
@staticmethod
def Inverse(expr, assumptions):
return ask(Q.lower_triangular(expr.arg), assumptions)
@staticmethod
def MatrixSlice(expr, assumptions):
if not expr.on_diag:
return None
else:
return ask(Q.lower_triangular(expr.parent), assumptions)
Factorization = staticmethod(partial(_Factorization, Q.lower_triangular))
class AskDiagonalHandler(CommonHandler):
"""
Handler for key 'diagonal'
"""
@staticmethod
def _is_empty_or_1x1(expr):
return expr.shape == (0, 0) or expr.shape == (1, 1)
@staticmethod
def MatMul(expr, assumptions):
if AskDiagonalHandler._is_empty_or_1x1(expr):
return True
factor, matrices = expr.as_coeff_matrices()
if all(ask(Q.diagonal(m), assumptions) for m in matrices):
return True
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.diagonal(base), assumptions)
return None
@staticmethod
def MatAdd(expr, assumptions):
if all(ask(Q.diagonal(arg), assumptions) for arg in expr.args):
return True
@staticmethod
def MatrixSymbol(expr, assumptions):
if AskDiagonalHandler._is_empty_or_1x1(expr):
return True
if Q.diagonal(expr) in conjuncts(assumptions):
return True
Identity, ZeroMatrix = [staticmethod(CommonHandler.AlwaysTrue)]*2
@staticmethod
def Transpose(expr, assumptions):
return ask(Q.diagonal(expr.arg), assumptions)
@staticmethod
def Inverse(expr, assumptions):
return ask(Q.diagonal(expr.arg), assumptions)
@staticmethod
def MatrixSlice(expr, assumptions):
if AskDiagonalHandler._is_empty_or_1x1(expr):
return True
if not expr.on_diag:
return None
else:
return ask(Q.diagonal(expr.parent), assumptions)
@staticmethod
def DiagonalMatrix(expr, assumptions):
return True
@staticmethod
def DiagonalizeVector(expr, assumptions):
return True
@staticmethod
def Identity(expr, assumptions):
return True
Factorization = staticmethod(partial(_Factorization, Q.diagonal))
def BM_elements(predicate, expr, assumptions):
""" Block Matrix elements """
return all(ask(predicate(b), assumptions) for b in expr.blocks)
def MS_elements(predicate, expr, assumptions):
""" Matrix Slice elements """
return ask(predicate(expr.parent), assumptions)
def MatMul_elements(matrix_predicate, scalar_predicate, expr, assumptions):
d = sift(expr.args, lambda x: isinstance(x, MatrixExpr))
factors, matrices = d[False], d[True]
return fuzzy_and([
test_closed_group(Basic(*factors), assumptions, scalar_predicate),
test_closed_group(Basic(*matrices), assumptions, matrix_predicate)])
class AskIntegerElementsHandler(CommonHandler):
@staticmethod
def MatAdd(expr, assumptions):
return test_closed_group(expr, assumptions, Q.integer_elements)
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
if exp.is_negative == False:
return ask(Q.integer_elements(base), assumptions)
return None
HadamardProduct, Determinant, Trace, Transpose = [MatAdd]*4
ZeroMatrix, Identity = [staticmethod(CommonHandler.AlwaysTrue)]*2
MatMul = staticmethod(partial(MatMul_elements, Q.integer_elements,
Q.integer))
MatrixSlice = staticmethod(partial(MS_elements, Q.integer_elements))
BlockMatrix = staticmethod(partial(BM_elements, Q.integer_elements))
class AskRealElementsHandler(CommonHandler):
@staticmethod
def MatAdd(expr, assumptions):
return test_closed_group(expr, assumptions, Q.real_elements)
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.real_elements(base), assumptions)
return None
HadamardProduct, Determinant, Trace, Transpose, \
Factorization = [MatAdd]*5
MatMul = staticmethod(partial(MatMul_elements, Q.real_elements, Q.real))
MatrixSlice = staticmethod(partial(MS_elements, Q.real_elements))
BlockMatrix = staticmethod(partial(BM_elements, Q.real_elements))
class AskComplexElementsHandler(CommonHandler):
@staticmethod
def MatAdd(expr, assumptions):
return test_closed_group(expr, assumptions, Q.complex_elements)
@staticmethod
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.complex_elements(base), assumptions)
return None
HadamardProduct, Determinant, Trace, Transpose, Inverse, \
Factorization = [MatAdd]*6
MatMul = staticmethod(partial(MatMul_elements, Q.complex_elements,
Q.complex))
MatrixSlice = staticmethod(partial(MS_elements, Q.complex_elements))
BlockMatrix = staticmethod(partial(BM_elements, Q.complex_elements))
DFT = staticmethod(CommonHandler.AlwaysTrue)
|
1ee44f503b1c8a0b05e043d88275f9b22e1c981f41217b50c1cb3bf076782ee7 | """
Handlers for predicates related to set membership: integer, rational, etc.
"""
from __future__ import print_function, division
from sympy.assumptions import Q, ask
from sympy.assumptions.handlers import CommonHandler, test_closed_group
from sympy.core.numbers import pi
from sympy.functions.elementary.exponential import exp, log
from sympy import I
class AskIntegerHandler(CommonHandler):
"""
Handler for Q.integer
Test that an expression belongs to the field of integer numbers
"""
@staticmethod
def Expr(expr, assumptions):
return expr.is_integer
@staticmethod
def _number(expr, assumptions):
# helper method
try:
i = int(expr.round())
if not (expr - i).equals(0):
raise TypeError
return True
except TypeError:
return False
@staticmethod
def Add(expr, assumptions):
"""
Integer + Integer -> Integer
Integer + !Integer -> !Integer
!Integer + !Integer -> ?
"""
if expr.is_number:
return AskIntegerHandler._number(expr, assumptions)
return test_closed_group(expr, assumptions, Q.integer)
@staticmethod
def Mul(expr, assumptions):
"""
Integer*Integer -> Integer
Integer*Irrational -> !Integer
Odd/Even -> !Integer
Integer*Rational -> ?
"""
if expr.is_number:
return AskIntegerHandler._number(expr, assumptions)
_output = True
for arg in expr.args:
if not ask(Q.integer(arg), assumptions):
if arg.is_Rational:
if arg.q == 2:
return ask(Q.even(2*expr), assumptions)
if ~(arg.q & 1):
return None
elif ask(Q.irrational(arg), assumptions):
if _output:
_output = False
else:
return
else:
return
return _output
Pow = Add
int, Integer = [staticmethod(CommonHandler.AlwaysTrue)]*2
Pi, Exp1, GoldenRatio, TribonacciConstant, Infinity, NegativeInfinity, ImaginaryUnit = \
[staticmethod(CommonHandler.AlwaysFalse)]*7
@staticmethod
def Rational(expr, assumptions):
# rationals with denominator one get
# evaluated to Integers
return False
@staticmethod
def Abs(expr, assumptions):
return ask(Q.integer(expr.args[0]), assumptions)
@staticmethod
def MatrixElement(expr, assumptions):
return ask(Q.integer_elements(expr.args[0]), assumptions)
Determinant = Trace = MatrixElement
class AskRationalHandler(CommonHandler):
"""
Handler for Q.rational
Test that an expression belongs to the field of rational numbers
"""
@staticmethod
def Expr(expr, assumptions):
return expr.is_rational
@staticmethod
def Add(expr, assumptions):
"""
Rational + Rational -> Rational
Rational + !Rational -> !Rational
!Rational + !Rational -> ?
"""
if expr.is_number:
if expr.as_real_imag()[1]:
return False
return test_closed_group(expr, assumptions, Q.rational)
Mul = Add
@staticmethod
def Pow(expr, assumptions):
"""
Rational ** Integer -> Rational
Irrational ** Rational -> Irrational
Rational ** Irrational -> ?
"""
if ask(Q.integer(expr.exp), assumptions):
return ask(Q.rational(expr.base), assumptions)
elif ask(Q.rational(expr.exp), assumptions):
if ask(Q.prime(expr.base), assumptions):
return False
Rational = staticmethod(CommonHandler.AlwaysTrue)
Float = staticmethod(CommonHandler.AlwaysNone)
ImaginaryUnit, Infinity, NegativeInfinity, Pi, Exp1, GoldenRatio, TribonacciConstant = \
[staticmethod(CommonHandler.AlwaysFalse)]*7
@staticmethod
def exp(expr, assumptions):
x = expr.args[0]
if ask(Q.rational(x), assumptions):
return ask(~Q.nonzero(x), assumptions)
@staticmethod
def cot(expr, assumptions):
x = expr.args[0]
if ask(Q.rational(x), assumptions):
return False
@staticmethod
def log(expr, assumptions):
x = expr.args[0]
if ask(Q.rational(x), assumptions):
return ask(~Q.nonzero(x - 1), assumptions)
sin, cos, tan, asin, atan = [exp]*5
acos, acot = log, cot
class AskIrrationalHandler(CommonHandler):
@staticmethod
def Expr(expr, assumptions):
return expr.is_irrational
@staticmethod
def Basic(expr, assumptions):
_real = ask(Q.real(expr), assumptions)
if _real:
_rational = ask(Q.rational(expr), assumptions)
if _rational is None:
return None
return not _rational
else:
return _real
class AskRealHandler(CommonHandler):
"""
Handler for Q.real
Test that an expression belongs to the field of real numbers
"""
@staticmethod
def Expr(expr, assumptions):
return expr.is_real
@staticmethod
def _number(expr, assumptions):
# let as_real_imag() work first since the expression may
# be simpler to evaluate
i = expr.as_real_imag()[1].evalf(2)
if i._prec != 1:
return not i
# allow None to be returned if we couldn't show for sure
# that i was 0
@staticmethod
def Add(expr, assumptions):
"""
Real + Real -> Real
Real + (Complex & !Real) -> !Real
"""
if expr.is_number:
return AskRealHandler._number(expr, assumptions)
return test_closed_group(expr, assumptions, Q.real)
@staticmethod
def Mul(expr, assumptions):
"""
Real*Real -> Real
Real*Imaginary -> !Real
Imaginary*Imaginary -> Real
"""
if expr.is_number:
return AskRealHandler._number(expr, assumptions)
result = True
for arg in expr.args:
if ask(Q.real(arg), assumptions):
pass
elif ask(Q.imaginary(arg), assumptions):
result = result ^ True
else:
break
else:
return result
@staticmethod
def Pow(expr, assumptions):
"""
Real**Integer -> Real
Positive**Real -> Real
Real**(Integer/Even) -> Real if base is nonnegative
Real**(Integer/Odd) -> Real
Imaginary**(Integer/Even) -> Real
Imaginary**(Integer/Odd) -> not Real
Imaginary**Real -> ? since Real could be 0 (giving real) or 1 (giving imaginary)
b**Imaginary -> Real if log(b) is imaginary and b != 0 and exponent != integer multiple of I*pi/log(b)
Real**Real -> ? e.g. sqrt(-1) is imaginary and sqrt(2) is not
"""
if expr.is_number:
return AskRealHandler._number(expr, assumptions)
if expr.base.func == exp:
if ask(Q.imaginary(expr.base.args[0]), assumptions):
if ask(Q.imaginary(expr.exp), assumptions):
return True
# If the i = (exp's arg)/(I*pi) is an integer or half-integer
# multiple of I*pi then 2*i will be an integer. In addition,
# exp(i*I*pi) = (-1)**i so the overall realness of the expr
# can be determined by replacing exp(i*I*pi) with (-1)**i.
i = expr.base.args[0]/I/pi
if ask(Q.integer(2*i), assumptions):
return ask(Q.real(((-1)**i)**expr.exp), assumptions)
return
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
odd = ask(Q.odd(expr.exp), assumptions)
if odd is not None:
return not odd
return
if ask(Q.imaginary(expr.exp), assumptions):
imlog = ask(Q.imaginary(log(expr.base)), assumptions)
if imlog is not None:
# I**i -> real, log(I) is imag;
# (2*I)**i -> complex, log(2*I) is not imag
return imlog
if ask(Q.real(expr.base), assumptions):
if ask(Q.real(expr.exp), assumptions):
if expr.exp.is_Rational and \
ask(Q.even(expr.exp.q), assumptions):
return ask(Q.positive(expr.base), assumptions)
elif ask(Q.integer(expr.exp), assumptions):
return True
elif ask(Q.positive(expr.base), assumptions):
return True
elif ask(Q.negative(expr.base), assumptions):
return False
Rational, Float, Pi, Exp1, GoldenRatio, TribonacciConstant, Abs, re, im = \
[staticmethod(CommonHandler.AlwaysTrue)]*9
ImaginaryUnit, Infinity, NegativeInfinity = \
[staticmethod(CommonHandler.AlwaysFalse)]*3
@staticmethod
def sin(expr, assumptions):
if ask(Q.real(expr.args[0]), assumptions):
return True
cos = sin
@staticmethod
def exp(expr, assumptions):
return ask(Q.integer(expr.args[0]/I/pi) | Q.real(expr.args[0]), assumptions)
@staticmethod
def log(expr, assumptions):
return ask(Q.positive(expr.args[0]), assumptions)
@staticmethod
def MatrixElement(expr, assumptions):
return ask(Q.real_elements(expr.args[0]), assumptions)
Determinant = Trace = MatrixElement
class AskExtendedRealHandler(AskRealHandler):
"""
Handler for Q.extended_real
Test that an expression belongs to the field of extended real numbers,
that is real numbers union {Infinity, -Infinity}
"""
@staticmethod
def Add(expr, assumptions):
return test_closed_group(expr, assumptions, Q.extended_real)
Mul, Pow = [Add]*2
Infinity, NegativeInfinity = [staticmethod(CommonHandler.AlwaysTrue)]*2
class AskHermitianHandler(AskRealHandler):
"""
Handler for Q.hermitian
Test that an expression belongs to the field of Hermitian operators
"""
@staticmethod
def Add(expr, assumptions):
"""
Hermitian + Hermitian -> Hermitian
Hermitian + !Hermitian -> !Hermitian
"""
if expr.is_number:
return AskRealHandler._number(expr, assumptions)
return test_closed_group(expr, assumptions, Q.hermitian)
@staticmethod
def Mul(expr, assumptions):
"""
As long as there is at most only one noncommutative term:
Hermitian*Hermitian -> Hermitian
Hermitian*Antihermitian -> !Hermitian
Antihermitian*Antihermitian -> Hermitian
"""
if expr.is_number:
return AskRealHandler._number(expr, assumptions)
nccount = 0
result = True
for arg in expr.args:
if ask(Q.antihermitian(arg), assumptions):
result = result ^ True
elif not ask(Q.hermitian(arg), assumptions):
break
if ask(~Q.commutative(arg), assumptions):
nccount += 1
if nccount > 1:
break
else:
return result
@staticmethod
def Pow(expr, assumptions):
"""
Hermitian**Integer -> Hermitian
"""
if expr.is_number:
return AskRealHandler._number(expr, assumptions)
if ask(Q.hermitian(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
return True
@staticmethod
def sin(expr, assumptions):
if ask(Q.hermitian(expr.args[0]), assumptions):
return True
cos, exp = [sin]*2
class AskComplexHandler(CommonHandler):
"""
Handler for Q.complex
Test that an expression belongs to the field of complex numbers
"""
@staticmethod
def Expr(expr, assumptions):
return expr.is_complex
@staticmethod
def Add(expr, assumptions):
return test_closed_group(expr, assumptions, Q.complex)
Mul, Pow = [Add]*2
Number, sin, cos, log, exp, re, im, NumberSymbol, Abs, ImaginaryUnit = \
[staticmethod(CommonHandler.AlwaysTrue)]*10 # they are all complex functions or expressions
Infinity, NegativeInfinity = [staticmethod(CommonHandler.AlwaysFalse)]*2
@staticmethod
def MatrixElement(expr, assumptions):
return ask(Q.complex_elements(expr.args[0]), assumptions)
Determinant = Trace = MatrixElement
class AskImaginaryHandler(CommonHandler):
"""
Handler for Q.imaginary
Test that an expression belongs to the field of imaginary numbers,
that is, numbers in the form x*I, where x is real
"""
@staticmethod
def Expr(expr, assumptions):
return expr.is_imaginary
@staticmethod
def _number(expr, assumptions):
# let as_real_imag() work first since the expression may
# be simpler to evaluate
r = expr.as_real_imag()[0].evalf(2)
if r._prec != 1:
return not r
# allow None to be returned if we couldn't show for sure
# that r was 0
@staticmethod
def Add(expr, assumptions):
"""
Imaginary + Imaginary -> Imaginary
Imaginary + Complex -> ?
Imaginary + Real -> !Imaginary
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
reals = 0
for arg in expr.args:
if ask(Q.imaginary(arg), assumptions):
pass
elif ask(Q.real(arg), assumptions):
reals += 1
else:
break
else:
if reals == 0:
return True
if reals == 1 or (len(expr.args) == reals):
# two reals could sum 0 thus giving an imaginary
return False
@staticmethod
def Mul(expr, assumptions):
"""
Real*Imaginary -> Imaginary
Imaginary*Imaginary -> Real
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
result = False
reals = 0
for arg in expr.args:
if ask(Q.imaginary(arg), assumptions):
result = result ^ True
elif not ask(Q.real(arg), assumptions):
break
else:
if reals == len(expr.args):
return False
return result
@staticmethod
def Pow(expr, assumptions):
"""
Imaginary**Odd -> Imaginary
Imaginary**Even -> Real
b**Imaginary -> !Imaginary if exponent is an integer multiple of I*pi/log(b)
Imaginary**Real -> ?
Positive**Real -> Real
Negative**Integer -> Real
Negative**(Integer/2) -> Imaginary
Negative**Real -> not Imaginary if exponent is not Rational
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
if expr.base.func == exp:
if ask(Q.imaginary(expr.base.args[0]), assumptions):
if ask(Q.imaginary(expr.exp), assumptions):
return False
i = expr.base.args[0]/I/pi
if ask(Q.integer(2*i), assumptions):
return ask(Q.imaginary(((-1)**i)**expr.exp), assumptions)
if ask(Q.imaginary(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
odd = ask(Q.odd(expr.exp), assumptions)
if odd is not None:
return odd
return
if ask(Q.imaginary(expr.exp), assumptions):
imlog = ask(Q.imaginary(log(expr.base)), assumptions)
if imlog is not None:
return False # I**i -> real; (2*I)**i -> complex ==> not imaginary
if ask(Q.real(expr.base) & Q.real(expr.exp), assumptions):
if ask(Q.positive(expr.base), assumptions):
return False
else:
rat = ask(Q.rational(expr.exp), assumptions)
if not rat:
return rat
if ask(Q.integer(expr.exp), assumptions):
return False
else:
half = ask(Q.integer(2*expr.exp), assumptions)
if half:
return ask(Q.negative(expr.base), assumptions)
return half
@staticmethod
def log(expr, assumptions):
if ask(Q.real(expr.args[0]), assumptions):
if ask(Q.positive(expr.args[0]), assumptions):
return False
return
# XXX it should be enough to do
# return ask(Q.nonpositive(expr.args[0]), assumptions)
# but ask(Q.nonpositive(exp(x)), Q.imaginary(x)) -> None;
# it should return True since exp(x) will be either 0 or complex
if expr.args[0].func == exp:
if expr.args[0].args[0] in [I, -I]:
return True
im = ask(Q.imaginary(expr.args[0]), assumptions)
if im is False:
return False
@staticmethod
def exp(expr, assumptions):
a = expr.args[0]/I/pi
return ask(Q.integer(2*a) & ~Q.integer(a), assumptions)
@staticmethod
def Number(expr, assumptions):
return not (expr.as_real_imag()[1] == 0)
NumberSymbol = Number
ImaginaryUnit = staticmethod(CommonHandler.AlwaysTrue)
class AskAntiHermitianHandler(AskImaginaryHandler):
"""
Handler for Q.antihermitian
Test that an expression belongs to the field of anti-Hermitian operators,
that is, operators in the form x*I, where x is Hermitian
"""
@staticmethod
def Add(expr, assumptions):
"""
Antihermitian + Antihermitian -> Antihermitian
Antihermitian + !Antihermitian -> !Antihermitian
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
return test_closed_group(expr, assumptions, Q.antihermitian)
@staticmethod
def Mul(expr, assumptions):
"""
As long as there is at most only one noncommutative term:
Hermitian*Hermitian -> !Antihermitian
Hermitian*Antihermitian -> Antihermitian
Antihermitian*Antihermitian -> !Antihermitian
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
nccount = 0
result = False
for arg in expr.args:
if ask(Q.antihermitian(arg), assumptions):
result = result ^ True
elif not ask(Q.hermitian(arg), assumptions):
break
if ask(~Q.commutative(arg), assumptions):
nccount += 1
if nccount > 1:
break
else:
return result
@staticmethod
def Pow(expr, assumptions):
"""
Hermitian**Integer -> !Antihermitian
Antihermitian**Even -> !Antihermitian
Antihermitian**Odd -> Antihermitian
"""
if expr.is_number:
return AskImaginaryHandler._number(expr, assumptions)
if ask(Q.hermitian(expr.base), assumptions):
if ask(Q.integer(expr.exp), assumptions):
return False
elif ask(Q.antihermitian(expr.base), assumptions):
if ask(Q.even(expr.exp), assumptions):
return False
elif ask(Q.odd(expr.exp), assumptions):
return True
class AskAlgebraicHandler(CommonHandler):
"""Handler for Q.algebraic key. """
@staticmethod
def Add(expr, assumptions):
return test_closed_group(expr, assumptions, Q.algebraic)
@staticmethod
def Mul(expr, assumptions):
return test_closed_group(expr, assumptions, Q.algebraic)
@staticmethod
def Pow(expr, assumptions):
return expr.exp.is_Rational and ask(
Q.algebraic(expr.base), assumptions)
@staticmethod
def Rational(expr, assumptions):
return expr.q != 0
Float, GoldenRatio, TribonacciConstant, ImaginaryUnit, AlgebraicNumber = \
[staticmethod(CommonHandler.AlwaysTrue)]*5
Infinity, NegativeInfinity, ComplexInfinity, Pi, Exp1 = \
[staticmethod(CommonHandler.AlwaysFalse)]*5
@staticmethod
def exp(expr, assumptions):
x = expr.args[0]
if ask(Q.algebraic(x), assumptions):
return ask(~Q.nonzero(x), assumptions)
@staticmethod
def cot(expr, assumptions):
x = expr.args[0]
if ask(Q.algebraic(x), assumptions):
return False
@staticmethod
def log(expr, assumptions):
x = expr.args[0]
if ask(Q.algebraic(x), assumptions):
return ask(~Q.nonzero(x - 1), assumptions)
sin, cos, tan, asin, atan = [exp]*5
acos, acot = log, cot
|
0dc387c780cb5191fc8d59bd647f75d0da5436cf7ccf40ea4ae35cedea794be1 | from sympy.assumptions.satask import satask
from sympy import symbols, Q, assuming, Implies, MatrixSymbol, I, pi, Rational
from sympy.utilities.pytest import raises, XFAIL, slow
x, y, z = symbols('x y z')
def test_satask():
# No relevant facts
assert satask(Q.real(x), Q.real(x)) is True
assert satask(Q.real(x), ~Q.real(x)) is False
assert satask(Q.real(x)) is None
assert satask(Q.real(x), Q.positive(x)) is True
assert satask(Q.positive(x), Q.real(x)) is None
assert satask(Q.real(x), ~Q.positive(x)) is None
assert satask(Q.positive(x), ~Q.real(x)) is False
raises(ValueError, lambda: satask(Q.real(x), Q.real(x) & ~Q.real(x)))
with assuming(Q.positive(x)):
assert satask(Q.real(x)) is True
assert satask(~Q.positive(x)) is False
raises(ValueError, lambda: satask(Q.real(x), ~Q.positive(x)))
assert satask(Q.zero(x), Q.nonzero(x)) is False
assert satask(Q.positive(x), Q.zero(x)) is False
assert satask(Q.real(x), Q.zero(x)) is True
assert satask(Q.zero(x), Q.zero(x*y)) is None
assert satask(Q.zero(x*y), Q.zero(x))
def test_zero():
"""
Everything in this test doesn't work with the ask handlers, and most
things would be very difficult or impossible to make work under that
model.
"""
assert satask(Q.zero(x) | Q.zero(y), Q.zero(x*y)) is True
assert satask(Q.zero(x*y), Q.zero(x) | Q.zero(y)) is True
assert satask(Implies(Q.zero(x), Q.zero(x*y))) is True
# This one in particular requires computing the fixed-point of the
# relevant facts, because going from Q.nonzero(x*y) -> ~Q.zero(x*y) and
# Q.zero(x*y) -> Equivalent(Q.zero(x*y), Q.zero(x) | Q.zero(y)) takes two
# steps.
assert satask(Q.zero(x) | Q.zero(y), Q.nonzero(x*y)) is False
assert satask(Q.zero(x), Q.zero(x**2)) is True
def test_zero_positive():
assert satask(Q.zero(x + y), Q.positive(x) & Q.positive(y)) is False
assert satask(Q.positive(x) & Q.positive(y), Q.zero(x + y)) is False
assert satask(Q.nonzero(x + y), Q.positive(x) & Q.positive(y)) is True
assert satask(Q.positive(x) & Q.positive(y), Q.nonzero(x + y)) is None
# This one requires several levels of forward chaining
assert satask(Q.zero(x*(x + y)), Q.positive(x) & Q.positive(y)) is False
assert satask(Q.positive(pi*x*y + 1), Q.positive(x) & Q.positive(y)) is True
assert satask(Q.positive(pi*x*y - 5), Q.positive(x) & Q.positive(y)) is None
def test_zero_pow():
assert satask(Q.zero(x**y), Q.zero(x) & Q.positive(y)) is True
assert satask(Q.zero(x**y), Q.nonzero(x) & Q.zero(y)) is False
assert satask(Q.zero(x), Q.zero(x**y)) is True
assert satask(Q.zero(x**y), Q.zero(x)) is None
@XFAIL
# Requires correct Q.square calculation first
def test_invertible():
A = MatrixSymbol('A', 5, 5)
B = MatrixSymbol('B', 5, 5)
assert satask(Q.invertible(A*B), Q.invertible(A) & Q.invertible(B)) is True
assert satask(Q.invertible(A), Q.invertible(A*B)) is True
assert satask(Q.invertible(A) & Q.invertible(B), Q.invertible(A*B)) is True
def test_prime():
assert satask(Q.prime(5)) is True
assert satask(Q.prime(6)) is False
assert satask(Q.prime(-5)) is False
assert satask(Q.prime(x*y), Q.integer(x) & Q.integer(y)) is None
assert satask(Q.prime(x*y), Q.prime(x) & Q.prime(y)) is False
def test_old_assump():
assert satask(Q.positive(1)) is True
assert satask(Q.positive(-1)) is False
assert satask(Q.positive(0)) is False
assert satask(Q.positive(I)) is False
assert satask(Q.positive(pi)) is True
assert satask(Q.negative(1)) is False
assert satask(Q.negative(-1)) is True
assert satask(Q.negative(0)) is False
assert satask(Q.negative(I)) is False
assert satask(Q.negative(pi)) is False
assert satask(Q.zero(1)) is False
assert satask(Q.zero(-1)) is False
assert satask(Q.zero(0)) is True
assert satask(Q.zero(I)) is False
assert satask(Q.zero(pi)) is False
assert satask(Q.nonzero(1)) is True
assert satask(Q.nonzero(-1)) is True
assert satask(Q.nonzero(0)) is False
assert satask(Q.nonzero(I)) is False
assert satask(Q.nonzero(pi)) is True
assert satask(Q.nonpositive(1)) is False
assert satask(Q.nonpositive(-1)) is True
assert satask(Q.nonpositive(0)) is True
assert satask(Q.nonpositive(I)) is False
assert satask(Q.nonpositive(pi)) is False
assert satask(Q.nonnegative(1)) is True
assert satask(Q.nonnegative(-1)) is False
assert satask(Q.nonnegative(0)) is True
assert satask(Q.nonnegative(I)) is False
assert satask(Q.nonnegative(pi)) is True
def test_rational_irrational():
assert satask(Q.irrational(2)) is False
assert satask(Q.rational(2)) is True
assert satask(Q.irrational(pi)) is True
assert satask(Q.rational(pi)) is False
assert satask(Q.irrational(I)) is False
assert satask(Q.rational(I)) is False
assert satask(Q.irrational(x*y*z), Q.irrational(x) & Q.irrational(y) &
Q.rational(z)) is None
assert satask(Q.irrational(x*y*z), Q.irrational(x) & Q.rational(y) &
Q.rational(z)) is True
assert satask(Q.irrational(pi*x*y), Q.rational(x) & Q.rational(y)) is True
assert satask(Q.irrational(x + y + z), Q.irrational(x) & Q.irrational(y) &
Q.rational(z)) is None
assert satask(Q.irrational(x + y + z), Q.irrational(x) & Q.rational(y) &
Q.rational(z)) is True
assert satask(Q.irrational(pi + x + y), Q.rational(x) & Q.rational(y)) is True
assert satask(Q.irrational(x*y*z), Q.rational(x) & Q.rational(y) &
Q.rational(z)) is False
assert satask(Q.rational(x*y*z), Q.rational(x) & Q.rational(y) &
Q.rational(z)) is True
assert satask(Q.irrational(x + y + z), Q.rational(x) & Q.rational(y) &
Q.rational(z)) is False
assert satask(Q.rational(x + y + z), Q.rational(x) & Q.rational(y) &
Q.rational(z)) is True
def test_even_satask():
assert satask(Q.even(2)) is True
assert satask(Q.even(3)) is False
assert satask(Q.even(x*y), Q.even(x) & Q.odd(y)) is True
assert satask(Q.even(x*y), Q.even(x) & Q.integer(y)) is True
assert satask(Q.even(x*y), Q.even(x) & Q.even(y)) is True
assert satask(Q.even(x*y), Q.odd(x) & Q.odd(y)) is False
assert satask(Q.even(x*y), Q.even(x)) is None
assert satask(Q.even(x*y), Q.odd(x) & Q.integer(y)) is None
assert satask(Q.even(x*y), Q.odd(x) & Q.odd(y)) is False
assert satask(Q.even(abs(x)), Q.even(x)) is True
assert satask(Q.even(abs(x)), Q.odd(x)) is False
assert satask(Q.even(x), Q.even(abs(x))) is None # x could be complex
def test_odd_satask():
assert satask(Q.odd(2)) is False
assert satask(Q.odd(3)) is True
assert satask(Q.odd(x*y), Q.even(x) & Q.odd(y)) is False
assert satask(Q.odd(x*y), Q.even(x) & Q.integer(y)) is False
assert satask(Q.odd(x*y), Q.even(x) & Q.even(y)) is False
assert satask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) is True
assert satask(Q.odd(x*y), Q.even(x)) is None
assert satask(Q.odd(x*y), Q.odd(x) & Q.integer(y)) is None
assert satask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) is True
assert satask(Q.odd(abs(x)), Q.even(x)) is False
assert satask(Q.odd(abs(x)), Q.odd(x)) is True
assert satask(Q.odd(x), Q.odd(abs(x))) is None # x could be complex
def test_integer():
assert satask(Q.integer(1)) is True
assert satask(Q.integer(Rational(1, 2))) is False
assert satask(Q.integer(x + y), Q.integer(x) & Q.integer(y)) is True
assert satask(Q.integer(x + y), Q.integer(x)) is None
assert satask(Q.integer(x + y), Q.integer(x) & ~Q.integer(y)) is False
assert satask(Q.integer(x + y + z), Q.integer(x) & Q.integer(y) &
~Q.integer(z)) is False
assert satask(Q.integer(x + y + z), Q.integer(x) & ~Q.integer(y) &
~Q.integer(z)) is None
assert satask(Q.integer(x + y + z), Q.integer(x) & ~Q.integer(y)) is None
assert satask(Q.integer(x + y), Q.integer(x) & Q.irrational(y)) is False
assert satask(Q.integer(x*y), Q.integer(x) & Q.integer(y)) is True
assert satask(Q.integer(x*y), Q.integer(x)) is None
assert satask(Q.integer(x*y), Q.integer(x) & ~Q.integer(y)) is None
assert satask(Q.integer(x*y), Q.integer(x) & ~Q.rational(y)) is False
assert satask(Q.integer(x*y*z), Q.integer(x) & Q.integer(y) &
~Q.rational(z)) is False
assert satask(Q.integer(x*y*z), Q.integer(x) & ~Q.rational(y) &
~Q.rational(z)) is None
assert satask(Q.integer(x*y*z), Q.integer(x) & ~Q.rational(y)) is None
assert satask(Q.integer(x*y), Q.integer(x) & Q.irrational(y)) is False
def test_abs():
assert satask(Q.nonnegative(abs(x))) is True
assert satask(Q.positive(abs(x)), ~Q.zero(x)) is True
assert satask(Q.zero(x), ~Q.zero(abs(x))) is False
assert satask(Q.zero(x), Q.zero(abs(x))) is True
assert satask(Q.nonzero(x), ~Q.zero(abs(x))) is None # x could be complex
assert satask(Q.zero(abs(x)), Q.zero(x)) is True
def test_imaginary():
assert satask(Q.imaginary(2*I)) is True
assert satask(Q.imaginary(x*y), Q.imaginary(x)) is None
assert satask(Q.imaginary(x*y), Q.imaginary(x) & Q.real(y)) is True
assert satask(Q.imaginary(x), Q.real(x)) is False
assert satask(Q.imaginary(1)) is False
assert satask(Q.imaginary(x*y), Q.real(x) & Q.real(y)) is False
assert satask(Q.imaginary(x + y), Q.real(x) & Q.real(y)) is False
def test_real():
assert satask(Q.real(x*y), Q.real(x) & Q.real(y)) is True
assert satask(Q.real(x + y), Q.real(x) & Q.real(y)) is True
assert satask(Q.real(x*y*z), Q.real(x) & Q.real(y) & Q.real(z)) is True
assert satask(Q.real(x*y*z), Q.real(x) & Q.real(y)) is None
assert satask(Q.real(x*y*z), Q.real(x) & Q.real(y) & Q.imaginary(z)) is False
assert satask(Q.real(x + y + z), Q.real(x) & Q.real(y) & Q.real(z)) is True
assert satask(Q.real(x + y + z), Q.real(x) & Q.real(y)) is None
def test_pos_neg():
assert satask(~Q.positive(x), Q.negative(x)) is True
assert satask(~Q.negative(x), Q.positive(x)) is True
assert satask(Q.positive(x + y), Q.positive(x) & Q.positive(y)) is True
assert satask(Q.negative(x + y), Q.negative(x) & Q.negative(y)) is True
assert satask(Q.positive(x + y), Q.negative(x) & Q.negative(y)) is False
assert satask(Q.negative(x + y), Q.positive(x) & Q.positive(y)) is False
@slow
def test_pow_pos_neg():
assert satask(Q.nonnegative(x**2), Q.positive(x)) is True
assert satask(Q.nonpositive(x**2), Q.positive(x)) is False
assert satask(Q.positive(x**2), Q.positive(x)) is True
assert satask(Q.negative(x**2), Q.positive(x)) is False
assert satask(Q.real(x**2), Q.positive(x)) is True
assert satask(Q.nonnegative(x**2), Q.negative(x)) is True
assert satask(Q.nonpositive(x**2), Q.negative(x)) is False
assert satask(Q.positive(x**2), Q.negative(x)) is True
assert satask(Q.negative(x**2), Q.negative(x)) is False
assert satask(Q.real(x**2), Q.negative(x)) is True
assert satask(Q.nonnegative(x**2), Q.nonnegative(x)) is True
assert satask(Q.nonpositive(x**2), Q.nonnegative(x)) is None
assert satask(Q.positive(x**2), Q.nonnegative(x)) is None
assert satask(Q.negative(x**2), Q.nonnegative(x)) is False
assert satask(Q.real(x**2), Q.nonnegative(x)) is True
assert satask(Q.nonnegative(x**2), Q.nonpositive(x)) is True
assert satask(Q.nonpositive(x**2), Q.nonpositive(x)) is None
assert satask(Q.positive(x**2), Q.nonpositive(x)) is None
assert satask(Q.negative(x**2), Q.nonpositive(x)) is False
assert satask(Q.real(x**2), Q.nonpositive(x)) is True
assert satask(Q.nonnegative(x**3), Q.positive(x)) is True
assert satask(Q.nonpositive(x**3), Q.positive(x)) is False
assert satask(Q.positive(x**3), Q.positive(x)) is True
assert satask(Q.negative(x**3), Q.positive(x)) is False
assert satask(Q.real(x**3), Q.positive(x)) is True
assert satask(Q.nonnegative(x**3), Q.negative(x)) is False
assert satask(Q.nonpositive(x**3), Q.negative(x)) is True
assert satask(Q.positive(x**3), Q.negative(x)) is False
assert satask(Q.negative(x**3), Q.negative(x)) is True
assert satask(Q.real(x**3), Q.negative(x)) is True
assert satask(Q.nonnegative(x**3), Q.nonnegative(x)) is True
assert satask(Q.nonpositive(x**3), Q.nonnegative(x)) is None
assert satask(Q.positive(x**3), Q.nonnegative(x)) is None
assert satask(Q.negative(x**3), Q.nonnegative(x)) is False
assert satask(Q.real(x**3), Q.nonnegative(x)) is True
assert satask(Q.nonnegative(x**3), Q.nonpositive(x)) is None
assert satask(Q.nonpositive(x**3), Q.nonpositive(x)) is True
assert satask(Q.positive(x**3), Q.nonpositive(x)) is False
assert satask(Q.negative(x**3), Q.nonpositive(x)) is None
assert satask(Q.real(x**3), Q.nonpositive(x)) is True
# If x is zero, x**negative is not real.
assert satask(Q.nonnegative(x**-2), Q.nonpositive(x)) is None
assert satask(Q.nonpositive(x**-2), Q.nonpositive(x)) is None
assert satask(Q.positive(x**-2), Q.nonpositive(x)) is None
assert satask(Q.negative(x**-2), Q.nonpositive(x)) is None
assert satask(Q.real(x**-2), Q.nonpositive(x)) is None
# We could deduce things for negative powers if x is nonzero, but it
# isn't implemented yet.
|
97cacfb0b168e559fdea6e711e3b6a4324c4cfae22a36f675c1f8d323054cd93 | from sympy import Q, ask, Symbol, DiagonalizeVector, DiagonalMatrix
from sympy.matrices.expressions import (MatrixSymbol, Identity, ZeroMatrix,
Trace, MatrixSlice, Determinant)
from sympy.matrices.expressions.factorizations import LofLU
from sympy.utilities.pytest import XFAIL
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 3)
Z = MatrixSymbol('Z', 2, 2)
A1x1 = MatrixSymbol('A1x1', 1, 1)
B1x1 = MatrixSymbol('B1x1', 1, 1)
C0x0 = MatrixSymbol('C0x0', 0, 0)
V1 = MatrixSymbol('V1', 2, 1)
V2 = MatrixSymbol('V2', 2, 1)
def test_square():
assert ask(Q.square(X))
assert not ask(Q.square(Y))
assert ask(Q.square(Y*Y.T))
def test_invertible():
assert ask(Q.invertible(X), Q.invertible(X))
assert ask(Q.invertible(Y)) is False
assert ask(Q.invertible(X*Y), Q.invertible(X)) is False
assert ask(Q.invertible(X*Z), Q.invertible(X)) is None
assert ask(Q.invertible(X*Z), Q.invertible(X) & Q.invertible(Z)) is True
assert ask(Q.invertible(X.T)) is None
assert ask(Q.invertible(X.T), Q.invertible(X)) is True
assert ask(Q.invertible(X.I)) is True
assert ask(Q.invertible(Identity(3))) is True
assert ask(Q.invertible(ZeroMatrix(3, 3))) is False
assert ask(Q.invertible(X), Q.fullrank(X) & Q.square(X))
def test_singular():
assert ask(Q.singular(X)) is None
assert ask(Q.singular(X), Q.invertible(X)) is False
assert ask(Q.singular(X), ~Q.invertible(X)) is True
@XFAIL
def test_invertible_fullrank():
assert ask(Q.invertible(X), Q.fullrank(X)) is True
def test_symmetric():
assert ask(Q.symmetric(X), Q.symmetric(X))
assert ask(Q.symmetric(X*Z), Q.symmetric(X)) is None
assert ask(Q.symmetric(X*Z), Q.symmetric(X) & Q.symmetric(Z)) is True
assert ask(Q.symmetric(X + Z), Q.symmetric(X) & Q.symmetric(Z)) is True
assert ask(Q.symmetric(Y)) is False
assert ask(Q.symmetric(Y*Y.T)) is True
assert ask(Q.symmetric(Y.T*X*Y)) is None
assert ask(Q.symmetric(Y.T*X*Y), Q.symmetric(X)) is True
assert ask(Q.symmetric(X**10), Q.symmetric(X)) is True
assert ask(Q.symmetric(A1x1)) is True
assert ask(Q.symmetric(A1x1 + B1x1)) is True
assert ask(Q.symmetric(A1x1 * B1x1)) is True
assert ask(Q.symmetric(V1.T*V1)) is True
assert ask(Q.symmetric(V1.T*(V1 + V2))) is True
assert ask(Q.symmetric(V1.T*(V1 + V2) + A1x1)) is True
assert ask(Q.symmetric(MatrixSlice(Y, (0, 1), (1, 2)))) is True
def _test_orthogonal_unitary(predicate):
assert ask(predicate(X), predicate(X))
assert ask(predicate(X.T), predicate(X)) is True
assert ask(predicate(X.I), predicate(X)) is True
assert ask(predicate(X**2), predicate(X))
assert ask(predicate(Y)) is False
assert ask(predicate(X)) is None
assert ask(predicate(X), ~Q.invertible(X)) is False
assert ask(predicate(X*Z*X), predicate(X) & predicate(Z)) is True
assert ask(predicate(Identity(3))) is True
assert ask(predicate(ZeroMatrix(3, 3))) is False
assert ask(Q.invertible(X), predicate(X))
assert not ask(predicate(X + Z), predicate(X) & predicate(Z))
def test_orthogonal():
_test_orthogonal_unitary(Q.orthogonal)
def test_unitary():
_test_orthogonal_unitary(Q.unitary)
assert ask(Q.unitary(X), Q.orthogonal(X))
def test_fullrank():
assert ask(Q.fullrank(X), Q.fullrank(X))
assert ask(Q.fullrank(X**2), Q.fullrank(X))
assert ask(Q.fullrank(X.T), Q.fullrank(X)) is True
assert ask(Q.fullrank(X)) is None
assert ask(Q.fullrank(Y)) is None
assert ask(Q.fullrank(X*Z), Q.fullrank(X) & Q.fullrank(Z)) is True
assert ask(Q.fullrank(Identity(3))) is True
assert ask(Q.fullrank(ZeroMatrix(3, 3))) is False
assert ask(Q.invertible(X), ~Q.fullrank(X)) == False
def test_positive_definite():
assert ask(Q.positive_definite(X), Q.positive_definite(X))
assert ask(Q.positive_definite(X.T), Q.positive_definite(X)) is True
assert ask(Q.positive_definite(X.I), Q.positive_definite(X)) is True
assert ask(Q.positive_definite(Y)) is False
assert ask(Q.positive_definite(X)) is None
assert ask(Q.positive_definite(X**3), Q.positive_definite(X))
assert ask(Q.positive_definite(X*Z*X),
Q.positive_definite(X) & Q.positive_definite(Z)) is True
assert ask(Q.positive_definite(X), Q.orthogonal(X))
assert ask(Q.positive_definite(Y.T*X*Y),
Q.positive_definite(X) & Q.fullrank(Y)) is True
assert not ask(Q.positive_definite(Y.T*X*Y), Q.positive_definite(X))
assert ask(Q.positive_definite(Identity(3))) is True
assert ask(Q.positive_definite(ZeroMatrix(3, 3))) is False
assert ask(Q.positive_definite(X + Z), Q.positive_definite(X) &
Q.positive_definite(Z)) is True
assert not ask(Q.positive_definite(-X), Q.positive_definite(X))
assert ask(Q.positive(X[1, 1]), Q.positive_definite(X))
def test_triangular():
assert ask(Q.upper_triangular(X + Z.T + Identity(2)), Q.upper_triangular(X) &
Q.lower_triangular(Z)) is True
assert ask(Q.upper_triangular(X*Z.T), Q.upper_triangular(X) &
Q.lower_triangular(Z)) is True
assert ask(Q.lower_triangular(Identity(3))) is True
assert ask(Q.lower_triangular(ZeroMatrix(3, 3))) is True
assert ask(Q.triangular(X), Q.unit_triangular(X))
assert ask(Q.upper_triangular(X**3), Q.upper_triangular(X))
assert ask(Q.lower_triangular(X**3), Q.lower_triangular(X))
def test_diagonal():
assert ask(Q.diagonal(X + Z.T + Identity(2)), Q.diagonal(X) &
Q.diagonal(Z)) is True
assert ask(Q.diagonal(ZeroMatrix(3, 3)))
assert ask(Q.lower_triangular(X) & Q.upper_triangular(X), Q.diagonal(X))
assert ask(Q.diagonal(X), Q.lower_triangular(X) & Q.upper_triangular(X))
assert ask(Q.symmetric(X), Q.diagonal(X))
assert ask(Q.triangular(X), Q.diagonal(X))
assert ask(Q.diagonal(C0x0))
assert ask(Q.diagonal(A1x1))
assert ask(Q.diagonal(A1x1 + B1x1))
assert ask(Q.diagonal(A1x1*B1x1))
assert ask(Q.diagonal(V1.T*V2))
assert ask(Q.diagonal(V1.T*(X + Z)*V1))
assert ask(Q.diagonal(MatrixSlice(Y, (0, 1), (1, 2)))) is True
assert ask(Q.diagonal(V1.T*(V1 + V2))) is True
assert ask(Q.diagonal(X**3), Q.diagonal(X))
assert ask(Q.diagonal(Identity(3)))
assert ask(Q.diagonal(DiagonalizeVector(V1)))
assert ask(Q.diagonal(DiagonalMatrix(X)))
def test_non_atoms():
assert ask(Q.real(Trace(X)), Q.positive(Trace(X)))
@XFAIL
def test_non_trivial_implies():
X = MatrixSymbol('X', 3, 3)
Y = MatrixSymbol('Y', 3, 3)
assert ask(Q.lower_triangular(X+Y), Q.lower_triangular(X) &
Q.lower_triangular(Y)) is True
assert ask(Q.triangular(X), Q.lower_triangular(X)) is True
assert ask(Q.triangular(X+Y), Q.lower_triangular(X) &
Q.lower_triangular(Y)) is True
def test_MatrixSlice():
X = MatrixSymbol('X', 4, 4)
B = MatrixSlice(X, (1, 3), (1, 3))
C = MatrixSlice(X, (0, 3), (1, 3))
assert ask(Q.symmetric(B), Q.symmetric(X))
assert ask(Q.invertible(B), Q.invertible(X))
assert ask(Q.diagonal(B), Q.diagonal(X))
assert ask(Q.orthogonal(B), Q.orthogonal(X))
assert ask(Q.upper_triangular(B), Q.upper_triangular(X))
assert not ask(Q.symmetric(C), Q.symmetric(X))
assert not ask(Q.invertible(C), Q.invertible(X))
assert not ask(Q.diagonal(C), Q.diagonal(X))
assert not ask(Q.orthogonal(C), Q.orthogonal(X))
assert not ask(Q.upper_triangular(C), Q.upper_triangular(X))
def test_det_trace_positive():
X = MatrixSymbol('X', 4, 4)
assert ask(Q.positive(Trace(X)), Q.positive_definite(X))
assert ask(Q.positive(Determinant(X)), Q.positive_definite(X))
def test_field_assumptions():
X = MatrixSymbol('X', 4, 4)
Y = MatrixSymbol('Y', 4, 4)
assert ask(Q.real_elements(X), Q.real_elements(X))
assert not ask(Q.integer_elements(X), Q.real_elements(X))
assert ask(Q.complex_elements(X), Q.real_elements(X))
assert ask(Q.complex_elements(X**2), Q.real_elements(X))
assert ask(Q.real_elements(X**2), Q.integer_elements(X))
assert ask(Q.real_elements(X+Y), Q.real_elements(X)) is None
assert ask(Q.real_elements(X+Y), Q.real_elements(X) & Q.real_elements(Y))
from sympy.matrices.expressions.hadamard import HadamardProduct
assert ask(Q.real_elements(HadamardProduct(X, Y)),
Q.real_elements(X) & Q.real_elements(Y))
assert ask(Q.complex_elements(X+Y), Q.real_elements(X) & Q.complex_elements(Y))
assert ask(Q.real_elements(X.T), Q.real_elements(X))
assert ask(Q.real_elements(X.I), Q.real_elements(X) & Q.invertible(X))
assert ask(Q.real_elements(Trace(X)), Q.real_elements(X))
assert ask(Q.integer_elements(Determinant(X)), Q.integer_elements(X))
assert not ask(Q.integer_elements(X.I), Q.integer_elements(X))
alpha = Symbol('alpha')
assert ask(Q.real_elements(alpha*X), Q.real_elements(X) & Q.real(alpha))
assert ask(Q.real_elements(LofLU(X)), Q.real_elements(X))
e = Symbol('e', integer=True, negative=True)
assert ask(Q.real_elements(X**e), Q.real_elements(X) & Q.invertible(X))
assert ask(Q.real_elements(X**e), Q.real_elements(X)) is None
def test_matrix_element_sets():
X = MatrixSymbol('X', 4, 4)
assert ask(Q.real(X[1, 2]), Q.real_elements(X))
assert ask(Q.integer(X[1, 2]), Q.integer_elements(X))
assert ask(Q.complex(X[1, 2]), Q.complex_elements(X))
assert ask(Q.integer_elements(Identity(3)))
assert ask(Q.integer_elements(ZeroMatrix(3, 3)))
from sympy.matrices.expressions.fourier import DFT
assert ask(Q.complex_elements(DFT(3)))
def test_matrix_element_sets_slices_blocks():
from sympy.matrices.expressions import BlockMatrix
X = MatrixSymbol('X', 4, 4)
assert ask(Q.integer_elements(X[:, 3]), Q.integer_elements(X))
assert ask(Q.integer_elements(BlockMatrix([[X], [X]])),
Q.integer_elements(X))
def test_matrix_element_sets_determinant_trace():
assert ask(Q.integer(Determinant(X)), Q.integer_elements(X))
assert ask(Q.integer(Trace(X)), Q.integer_elements(X))
|
fadb4128e7b348b7c5390743bc9238814e389e247b43cde6e5715725b060b2f6 | """
This module implements some special functions that commonly appear in
combinatorial contexts (e.g. in power series); in particular,
sequences of rational numbers such as Bernoulli and Fibonacci numbers.
Factorials, binomial coefficients and related functions are located in
the separate 'factorials' module.
"""
from __future__ import print_function, division
from sympy.core import S, Symbol, Rational, Integer, Add, Dummy
from sympy.core.cache import cacheit
from sympy.core.compatibility import as_int, SYMPY_INTS, range
from sympy.core.function import Function, expand_mul
from sympy.core.logic import fuzzy_not
from sympy.core.numbers import E, pi
from sympy.core.relational import LessThan, StrictGreaterThan
from sympy.functions.combinatorial.factorials import binomial, factorial
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt, cbrt
from sympy.functions.elementary.trigonometric import sin, cos, cot
from sympy.ntheory import isprime
from sympy.ntheory.primetest import is_square
from sympy.utilities.memoization import recurrence_memo
from mpmath import bernfrac, workprec
from mpmath.libmp import ifib as _ifib
def _product(a, b):
p = 1
for k in range(a, b + 1):
p *= k
return p
# Dummy symbol used for computing polynomial sequences
_sym = Symbol('x')
#----------------------------------------------------------------------------#
# #
# Carmichael numbers #
# #
#----------------------------------------------------------------------------#
class carmichael(Function):
"""
Carmichael Numbers:
Certain cryptographic algorithms make use of big prime numbers.
However, checking whether a big number is prime is not so easy.
Randomized prime number checking tests exist that offer a high degree of confidence of
accurate determination at low cost, such as the Fermat test.
Let 'a' be a random number between 2 and n - 1, where n is the number whose primality we are testing.
Then, n is probably prime if it satisfies the modular arithmetic congruence relation :
a^(n-1) = 1(mod n).
(where mod refers to the modulo operation)
If a number passes the Fermat test several times, then it is prime with a
high probability.
Unfortunately, certain composite numbers (non-primes) still pass the Fermat test
with every number smaller than themselves.
These numbers are called Carmichael numbers.
A Carmichael number will pass a Fermat primality test to every base b relatively prime to the number,
even though it is not actually prime. This makes tests based on Fermat's Little Theorem less effective than
strong probable prime tests such as the Baillie-PSW primality test and the Miller-Rabin primality test.
mr functions given in sympy/sympy/ntheory/primetest.py will produce wrong results for each and every
carmichael number.
Examples
========
>>> from sympy import carmichael
>>> carmichael.find_first_n_carmichaels(5)
[561, 1105, 1729, 2465, 2821]
>>> carmichael.is_prime(2465)
False
>>> carmichael.is_prime(1729)
False
>>> carmichael.find_carmichael_numbers_in_range(0, 562)
[561]
>>> carmichael.find_carmichael_numbers_in_range(0,1000)
[561]
>>> carmichael.find_carmichael_numbers_in_range(0,2000)
[561, 1105, 1729]
References
==========
.. [1] https://en.wikipedia.org/wiki/Carmichael_number
.. [2] https://en.wikipedia.org/wiki/Fermat_primality_test
.. [3] https://www.jstor.org/stable/23248683?seq=1#metadata_info_tab_contents
"""
@staticmethod
def is_perfect_square(n):
return is_square(n)
@staticmethod
def divides(p, n):
return n % p == 0
@staticmethod
def is_prime(n):
return isprime(n)
@staticmethod
def is_carmichael(n):
if n >= 0:
if (n == 1) or (carmichael.is_prime(n)) or (n % 2 == 0):
return False
divisors = list([1, n])
# get divisors
for i in range(3, n // 2 + 1, 2):
if n % i == 0:
divisors.append(i)
for i in divisors:
if carmichael.is_perfect_square(i) and i != 1:
return False
if carmichael.is_prime(i):
if not carmichael.divides(i - 1, n - 1):
return False
return True
else:
raise ValueError('The provided number must be greater than or equal to 0')
@staticmethod
def find_carmichael_numbers_in_range(x, y):
if 0 <= x <= y:
if x % 2 == 0:
return list([i for i in range(x + 1, y, 2) if carmichael.is_carmichael(i)])
else:
return list([i for i in range(x, y, 2) if carmichael.is_carmichael(i)])
else:
raise ValueError('The provided range is not valid. x and y must be non-negative integers and x <= y')
@staticmethod
def find_first_n_carmichaels(n):
i = 1
carmichaels = list()
while len(carmichaels) < n:
if carmichael.is_carmichael(i):
carmichaels.append(i)
i += 2
return carmichaels
#----------------------------------------------------------------------------#
# #
# Fibonacci numbers #
# #
#----------------------------------------------------------------------------#
class fibonacci(Function):
r"""
Fibonacci numbers / Fibonacci polynomials
The Fibonacci numbers are the integer sequence defined by the
initial terms `F_0 = 0`, `F_1 = 1` and the two-term recurrence
relation `F_n = F_{n-1} + F_{n-2}`. This definition
extended to arbitrary real and complex arguments using
the formula
.. math :: F_z = \frac{\phi^z - \cos(\pi z) \phi^{-z}}{\sqrt 5}
The Fibonacci polynomials are defined by `F_1(x) = 1`,
`F_2(x) = x`, and `F_n(x) = x*F_{n-1}(x) + F_{n-2}(x)` for `n > 2`.
For all positive integers `n`, `F_n(1) = F_n`.
* ``fibonacci(n)`` gives the `n^{th}` Fibonacci number, `F_n`
* ``fibonacci(n, x)`` gives the `n^{th}` Fibonacci polynomial in `x`, `F_n(x)`
Examples
========
>>> from sympy import fibonacci, Symbol
>>> [fibonacci(x) for x in range(11)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> fibonacci(5, Symbol('t'))
t**4 + 3*t**2 + 1
See Also
========
bell, bernoulli, catalan, euler, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Fibonacci_number
.. [2] http://mathworld.wolfram.com/FibonacciNumber.html
"""
@staticmethod
def _fib(n):
return _ifib(n)
@staticmethod
@recurrence_memo([None, S.One, _sym])
def _fibpoly(n, prev):
return (prev[-2] + _sym*prev[-1]).expand()
@classmethod
def eval(cls, n, sym=None):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
n = int(n)
if n < 0:
return S.NegativeOne**(n + 1) * fibonacci(-n)
if sym is None:
return Integer(cls._fib(n))
else:
if n < 1:
raise ValueError("Fibonacci polynomials are defined "
"only for positive integer indices.")
return cls._fibpoly(n).subs(_sym, sym)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
return 2**(-n)*sqrt(5)*((1 + sqrt(5))**n - (-sqrt(5) + 1)**n) / 5
def _eval_rewrite_as_GoldenRatio(self,n, **kwargs):
return (S.GoldenRatio**n - 1/(-S.GoldenRatio)**n)/(2*S.GoldenRatio-1)
#----------------------------------------------------------------------------#
# #
# Lucas numbers #
# #
#----------------------------------------------------------------------------#
class lucas(Function):
"""
Lucas numbers
Lucas numbers satisfy a recurrence relation similar to that of
the Fibonacci sequence, in which each term is the sum of the
preceding two. They are generated by choosing the initial
values `L_0 = 2` and `L_1 = 1`.
* ``lucas(n)`` gives the `n^{th}` Lucas number
Examples
========
>>> from sympy import lucas
>>> [lucas(x) for x in range(11)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Lucas_number
.. [2] http://mathworld.wolfram.com/LucasNumber.html
"""
@classmethod
def eval(cls, n):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
return fibonacci(n + 1) + fibonacci(n - 1)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
return 2**(-n)*((1 + sqrt(5))**n + (-sqrt(5) + 1)**n)
#----------------------------------------------------------------------------#
# #
# Tribonacci numbers #
# #
#----------------------------------------------------------------------------#
class tribonacci(Function):
r"""
Tribonacci numbers / Tribonacci polynomials
The Tribonacci numbers are the integer sequence defined by the
initial terms `T_0 = 0`, `T_1 = 1`, `T_2 = 1` and the three-term
recurrence relation `T_n = T_{n-1} + T_{n-2} + T_{n-3}`.
The Tribonacci polynomials are defined by `T_0(x) = 0`, `T_1(x) = 1`,
`T_2(x) = x^2`, and `T_n(x) = x^2 T_{n-1}(x) + x T_{n-2}(x) + T_{n-3}(x)`
for `n > 2`. For all positive integers `n`, `T_n(1) = T_n`.
* ``tribonacci(n)`` gives the `n^{th}` Tribonacci number, `T_n`
* ``tribonacci(n, x)`` gives the `n^{th}` Tribonacci polynomial in `x`, `T_n(x)`
Examples
========
>>> from sympy import tribonacci, Symbol
>>> [tribonacci(x) for x in range(11)]
[0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]
>>> tribonacci(5, Symbol('t'))
t**8 + 3*t**5 + 3*t**2
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition
References
==========
.. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers
.. [2] http://mathworld.wolfram.com/TribonacciNumber.html
.. [3] https://oeis.org/A000073
"""
@staticmethod
@recurrence_memo([S.Zero, S.One, S.One])
def _trib(n, prev):
return (prev[-3] + prev[-2] + prev[-1])
@staticmethod
@recurrence_memo([S.Zero, S.One, _sym**2])
def _tribpoly(n, prev):
return (prev[-3] + _sym*prev[-2] + _sym**2*prev[-1]).expand()
@classmethod
def eval(cls, n, sym=None):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
n = int(n)
if n < 0:
raise ValueError("Tribonacci polynomials are defined "
"only for non-negative integer indices.")
if sym is None:
return Integer(cls._trib(n))
else:
return cls._tribpoly(n).subs(_sym, sym)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2
a = (1 + cbrt(19 + 3*sqrt(33)) + cbrt(19 - 3*sqrt(33))) / 3
b = (1 + w*cbrt(19 + 3*sqrt(33)) + w**2*cbrt(19 - 3*sqrt(33))) / 3
c = (1 + w**2*cbrt(19 + 3*sqrt(33)) + w*cbrt(19 - 3*sqrt(33))) / 3
Tn = (a**(n + 1)/((a - b)*(a - c))
+ b**(n + 1)/((b - a)*(b - c))
+ c**(n + 1)/((c - a)*(c - b)))
return Tn
def _eval_rewrite_as_TribonacciConstant(self, n, **kwargs):
b = cbrt(586 + 102*sqrt(33))
Tn = 3 * b * S.TribonacciConstant**n / (b**2 - 2*b + 4)
return floor(Tn + S.Half)
#----------------------------------------------------------------------------#
# #
# Bernoulli numbers #
# #
#----------------------------------------------------------------------------#
class bernoulli(Function):
r"""
Bernoulli numbers / Bernoulli polynomials
The Bernoulli numbers are a sequence of rational numbers
defined by `B_0 = 1` and the recursive relation (`n > 0`):
.. math :: 0 = \sum_{k=0}^n \binom{n+1}{k} B_k
They are also commonly defined by their exponential generating
function, which is `\frac{x}{e^x - 1}`. For odd indices > 1, the
Bernoulli numbers are zero.
The Bernoulli polynomials satisfy the analogous formula:
.. math :: B_n(x) = \sum_{k=0}^n \binom{n}{k} B_k x^{n-k}
Bernoulli numbers and Bernoulli polynomials are related as
`B_n(0) = B_n`.
We compute Bernoulli numbers using Ramanujan's formula:
.. math :: B_n = \frac{A(n) - S(n)}{\binom{n+3}{n}}
where:
.. math :: A(n) = \begin{cases} \frac{n+3}{3} &
n \equiv 0\ \text{or}\ 2 \pmod{6} \\
-\frac{n+3}{6} & n \equiv 4 \pmod{6} \end{cases}
and:
.. math :: S(n) = \sum_{k=1}^{[n/6]} \binom{n+3}{n-6k} B_{n-6k}
This formula is similar to the sum given in the definition, but
cuts 2/3 of the terms. For Bernoulli polynomials, we use the
formula in the definition.
* ``bernoulli(n)`` gives the nth Bernoulli number, `B_n`
* ``bernoulli(n, x)`` gives the nth Bernoulli polynomial in `x`, `B_n(x)`
Examples
========
>>> from sympy import bernoulli
>>> [bernoulli(n) for n in range(11)]
[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
>>> bernoulli(1000001)
0
See Also
========
bell, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Bernoulli_number
.. [2] https://en.wikipedia.org/wiki/Bernoulli_polynomial
.. [3] http://mathworld.wolfram.com/BernoulliNumber.html
.. [4] http://mathworld.wolfram.com/BernoulliPolynomial.html
"""
# Calculates B_n for positive even n
@staticmethod
def _calc_bernoulli(n):
s = 0
a = int(binomial(n + 3, n - 6))
for j in range(1, n//6 + 1):
s += a * bernoulli(n - 6*j)
# Avoid computing each binomial coefficient from scratch
a *= _product(n - 6 - 6*j + 1, n - 6*j)
a //= _product(6*j + 4, 6*j + 9)
if n % 6 == 4:
s = -Rational(n + 3, 6) - s
else:
s = Rational(n + 3, 3) - s
return s / binomial(n + 3, n)
# We implement a specialized memoization scheme to handle each
# case modulo 6 separately
_cache = {0: S.One, 2: Rational(1, 6), 4: Rational(-1, 30)}
_highest = {0: 0, 2: 2, 4: 4}
@classmethod
def eval(cls, n, sym=None):
if n.is_Number:
if n.is_Integer and n.is_nonnegative:
if n is S.Zero:
return S.One
elif n is S.One:
if sym is None:
return -S.Half
else:
return sym - S.Half
# Bernoulli numbers
elif sym is None:
if n.is_odd:
return S.Zero
n = int(n)
# Use mpmath for enormous Bernoulli numbers
if n > 500:
p, q = bernfrac(n)
return Rational(int(p), int(q))
case = n % 6
highest_cached = cls._highest[case]
if n <= highest_cached:
return cls._cache[n]
# To avoid excessive recursion when, say, bernoulli(1000) is
# requested, calculate and cache the entire sequence ... B_988,
# B_994, B_1000 in increasing order
for i in range(highest_cached + 6, n + 6, 6):
b = cls._calc_bernoulli(i)
cls._cache[i] = b
cls._highest[case] = i
return b
# Bernoulli polynomials
else:
n, result = int(n), []
for k in range(n + 1):
result.append(binomial(n, k)*cls(k)*sym**(n - k))
return Add(*result)
else:
raise ValueError("Bernoulli numbers are defined only"
" for nonnegative integer indices.")
if sym is None:
if n.is_odd and (n - 1).is_positive:
return S.Zero
#----------------------------------------------------------------------------#
# #
# Bell numbers #
# #
#----------------------------------------------------------------------------#
class bell(Function):
r"""
Bell numbers / Bell polynomials
The Bell numbers satisfy `B_0 = 1` and
.. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.
They are also given by:
.. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.
The Bell polynomials are given by `B_0(x) = 1` and
.. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).
The second kind of Bell polynomials (are sometimes called "partial" Bell
polynomials or incomplete Bell polynomials) are defined as
.. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
\sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
\frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
\left(\frac{x_1}{1!} \right)^{j_1}
\left(\frac{x_2}{2!} \right)^{j_2} \dotsb
\left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.
* ``bell(n)`` gives the `n^{th}` Bell number, `B_n`.
* ``bell(n, x)`` gives the `n^{th}` Bell polynomial, `B_n(x)`.
* ``bell(n, k, (x1, x2, ...))`` gives Bell polynomials of the second kind,
`B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`.
Notes
=====
Not to be confused with Bernoulli numbers and Bernoulli polynomials,
which use the same notation.
Examples
========
>>> from sympy import bell, Symbol, symbols
>>> [bell(n) for n in range(11)]
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]
>>> bell(30)
846749014511809332450147
>>> bell(4, Symbol('t'))
t**4 + 6*t**3 + 7*t**2 + t
>>> bell(6, 2, symbols('x:6')[1:])
6*x1*x5 + 15*x2*x4 + 10*x3**2
See Also
========
bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Bell_number
.. [2] http://mathworld.wolfram.com/BellNumber.html
.. [3] http://mathworld.wolfram.com/BellPolynomial.html
"""
@staticmethod
@recurrence_memo([1, 1])
def _bell(n, prev):
s = 1
a = 1
for k in range(1, n):
a = a * (n - k) // k
s += a * prev[k]
return s
@staticmethod
@recurrence_memo([S.One, _sym])
def _bell_poly(n, prev):
s = 1
a = 1
for k in range(2, n + 1):
a = a * (n - k + 1) // (k - 1)
s += a * prev[k - 1]
return expand_mul(_sym * s)
@staticmethod
def _bell_incomplete_poly(n, k, symbols):
r"""
The second kind of Bell polynomials (incomplete Bell polynomials).
Calculated by recurrence formula:
.. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) =
\sum_{m=1}^{n-k+1}
\x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k})
where
`B_{0,0} = 1;`
`B_{n,0} = 0; for n \ge 1`
`B_{0,k} = 0; for k \ge 1`
"""
if (n == 0) and (k == 0):
return S.One
elif (n == 0) or (k == 0):
return S.Zero
s = S.Zero
a = S.One
for m in range(1, n - k + 2):
s += a * bell._bell_incomplete_poly(
n - m, k - 1, symbols) * symbols[m - 1]
a = a * (n - m) / m
return expand_mul(s)
@classmethod
def eval(cls, n, k_sym=None, symbols=None):
if n is S.Infinity:
if k_sym is None:
return S.Infinity
else:
raise ValueError("Bell polynomial is not defined")
if n.is_negative or n.is_integer is False:
raise ValueError("a non-negative integer expected")
if n.is_Integer and n.is_nonnegative:
if k_sym is None:
return Integer(cls._bell(int(n)))
elif symbols is None:
return cls._bell_poly(int(n)).subs(_sym, k_sym)
else:
r = cls._bell_incomplete_poly(int(n), int(k_sym), symbols)
return r
def _eval_rewrite_as_Sum(self, n, k_sym=None, symbols=None, **kwargs):
from sympy import Sum
if (k_sym is not None) or (symbols is not None):
return self
# Dobinski's formula
if not n.is_nonnegative:
return self
k = Dummy('k', integer=True, nonnegative=True)
return 1 / E * Sum(k**n / factorial(k), (k, 0, S.Infinity))
#----------------------------------------------------------------------------#
# #
# Harmonic numbers #
# #
#----------------------------------------------------------------------------#
class harmonic(Function):
r"""
Harmonic numbers
The nth harmonic number is given by `\operatorname{H}_{n} =
1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}`.
More generally:
.. math:: \operatorname{H}_{n,m} = \sum_{k=1}^{n} \frac{1}{k^m}
As `n \rightarrow \infty`, `\operatorname{H}_{n,m} \rightarrow \zeta(m)`,
the Riemann zeta function.
* ``harmonic(n)`` gives the nth harmonic number, `\operatorname{H}_n`
* ``harmonic(n, m)`` gives the nth generalized harmonic number
of order `m`, `\operatorname{H}_{n,m}`, where
``harmonic(n) == harmonic(n, 1)``
Examples
========
>>> from sympy import harmonic, oo
>>> [harmonic(n) for n in range(6)]
[0, 1, 3/2, 11/6, 25/12, 137/60]
>>> [harmonic(n, 2) for n in range(6)]
[0, 1, 5/4, 49/36, 205/144, 5269/3600]
>>> harmonic(oo, 2)
pi**2/6
>>> from sympy import Symbol, Sum
>>> n = Symbol("n")
>>> harmonic(n).rewrite(Sum)
Sum(1/_k, (_k, 1, n))
We can evaluate harmonic numbers for all integral and positive
rational arguments:
>>> from sympy import S, expand_func, simplify
>>> harmonic(8)
761/280
>>> harmonic(11)
83711/27720
>>> H = harmonic(1/S(3))
>>> H
harmonic(1/3)
>>> He = expand_func(H)
>>> He
-log(6) - sqrt(3)*pi/6 + 2*Sum(log(sin(_k*pi/3))*cos(2*_k*pi/3), (_k, 1, 1))
+ 3*Sum(1/(3*_k + 1), (_k, 0, 0))
>>> He.doit()
-log(6) - sqrt(3)*pi/6 - log(sqrt(3)/2) + 3
>>> H = harmonic(25/S(7))
>>> He = simplify(expand_func(H).doit())
>>> He
log(sin(pi/7)**(-2*cos(pi/7))*sin(2*pi/7)**(2*cos(16*pi/7))*cos(pi/14)**(-2*sin(pi/14))/14)
+ pi*tan(pi/14)/2 + 30247/9900
>>> He.n(40)
1.983697455232980674869851942390639915940
>>> harmonic(25/S(7)).n(40)
1.983697455232980674869851942390639915940
We can rewrite harmonic numbers in terms of polygamma functions:
>>> from sympy import digamma, polygamma
>>> m = Symbol("m")
>>> harmonic(n).rewrite(digamma)
polygamma(0, n + 1) + EulerGamma
>>> harmonic(n).rewrite(polygamma)
polygamma(0, n + 1) + EulerGamma
>>> harmonic(n,3).rewrite(polygamma)
polygamma(2, n + 1)/2 - polygamma(2, 1)/2
>>> harmonic(n,m).rewrite(polygamma)
(-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)
Integer offsets in the argument can be pulled out:
>>> from sympy import expand_func
>>> expand_func(harmonic(n+4))
harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
>>> expand_func(harmonic(n-4))
harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
Some limits can be computed as well:
>>> from sympy import limit, oo
>>> limit(harmonic(n), n, oo)
oo
>>> limit(harmonic(n, 2), n, oo)
pi**2/6
>>> limit(harmonic(n, 3), n, oo)
-polygamma(2, 1)/2
However we can not compute the general relation yet:
>>> limit(harmonic(n, m), n, oo)
harmonic(oo, m)
which equals ``zeta(m)`` for ``m > 1``.
See Also
========
bell, bernoulli, catalan, euler, fibonacci, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Harmonic_number
.. [2] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber/
.. [3] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/
"""
# Generate one memoized Harmonic number-generating function for each
# order and store it in a dictionary
_functions = {}
@classmethod
def eval(cls, n, m=None):
from sympy import zeta
if m is S.One:
return cls(n)
if m is None:
m = S.One
if m.is_zero:
return n
if n is S.Infinity and m.is_Number:
# TODO: Fix for symbolic values of m
if m.is_negative:
return S.NaN
elif LessThan(m, S.One):
return S.Infinity
elif StrictGreaterThan(m, S.One):
return zeta(m)
else:
return cls
if n == 0:
return S.Zero
if n.is_Integer and n.is_nonnegative and m.is_Integer:
if not m in cls._functions:
@recurrence_memo([0])
def f(n, prev):
return prev[-1] + S.One / n**m
cls._functions[m] = f
return cls._functions[m](int(n))
def _eval_rewrite_as_polygamma(self, n, m=1, **kwargs):
from sympy.functions.special.gamma_functions import polygamma
return S.NegativeOne**m/factorial(m - 1) * (polygamma(m - 1, 1) - polygamma(m - 1, n + 1))
def _eval_rewrite_as_digamma(self, n, m=1, **kwargs):
from sympy.functions.special.gamma_functions import polygamma
return self.rewrite(polygamma)
def _eval_rewrite_as_trigamma(self, n, m=1, **kwargs):
from sympy.functions.special.gamma_functions import polygamma
return self.rewrite(polygamma)
def _eval_rewrite_as_Sum(self, n, m=None, **kwargs):
from sympy import Sum
k = Dummy("k", integer=True)
if m is None:
m = S.One
return Sum(k**(-m), (k, 1, n))
def _eval_expand_func(self, **hints):
from sympy import Sum
n = self.args[0]
m = self.args[1] if len(self.args) == 2 else 1
if m == S.One:
if n.is_Add:
off = n.args[0]
nnew = n - off
if off.is_Integer and off.is_positive:
result = [S.One/(nnew + i) for i in range(off, 0, -1)] + [harmonic(nnew)]
return Add(*result)
elif off.is_Integer and off.is_negative:
result = [-S.One/(nnew + i) for i in range(0, off, -1)] + [harmonic(nnew)]
return Add(*result)
if n.is_Rational:
# Expansions for harmonic numbers at general rational arguments (u + p/q)
# Split n as u + p/q with p < q
p, q = n.as_numer_denom()
u = p // q
p = p - u * q
if u.is_nonnegative and p.is_positive and q.is_positive and p < q:
k = Dummy("k")
t1 = q * Sum(1 / (q * k + p), (k, 0, u))
t2 = 2 * Sum(cos((2 * pi * p * k) / S(q)) *
log(sin((pi * k) / S(q))),
(k, 1, floor((q - 1) / S(2))))
t3 = (pi / 2) * cot((pi * p) / q) + log(2 * q)
return t1 + t2 - t3
return self
def _eval_rewrite_as_tractable(self, n, m=1, **kwargs):
from sympy import polygamma
return self.rewrite(polygamma).rewrite("tractable", deep=True)
def _eval_evalf(self, prec):
from sympy import polygamma
if all(i.is_number for i in self.args):
return self.rewrite(polygamma)._eval_evalf(prec)
#----------------------------------------------------------------------------#
# #
# Euler numbers #
# #
#----------------------------------------------------------------------------#
class euler(Function):
r"""
Euler numbers / Euler polynomials
The Euler numbers are given by:
.. math:: E_{2n} = I \sum_{k=1}^{2n+1} \sum_{j=0}^k \binom{k}{j}
\frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k}
.. math:: E_{2n+1} = 0
Euler numbers and Euler polynomials are related by
.. math:: E_n = 2^n E_n\left(\frac{1}{2}\right).
We compute symbolic Euler polynomials using [5]_
.. math:: E_n(x) = \sum_{k=0}^n \binom{n}{k} \frac{E_k}{2^k}
\left(x - \frac{1}{2}\right)^{n-k}.
However, numerical evaluation of the Euler polynomial is computed
more efficiently (and more accurately) using the mpmath library.
* ``euler(n)`` gives the `n^{th}` Euler number, `E_n`.
* ``euler(n, x)`` gives the `n^{th}` Euler polynomial, `E_n(x)`.
Examples
========
>>> from sympy import Symbol, S
>>> from sympy.functions import euler
>>> [euler(n) for n in range(10)]
[1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]
>>> n = Symbol("n")
>>> euler(n + 2*n)
euler(3*n)
>>> x = Symbol("x")
>>> euler(n, x)
euler(n, x)
>>> euler(0, x)
1
>>> euler(1, x)
x - 1/2
>>> euler(2, x)
x**2 - x
>>> euler(3, x)
x**3 - 3*x**2/2 + 1/4
>>> euler(4, x)
x**4 - 2*x**3 + x
>>> euler(12, S.Half)
2702765/4096
>>> euler(12)
2702765
See Also
========
bell, bernoulli, catalan, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Euler_numbers
.. [2] http://mathworld.wolfram.com/EulerNumber.html
.. [3] https://en.wikipedia.org/wiki/Alternating_permutation
.. [4] http://mathworld.wolfram.com/AlternatingPermutation.html
.. [5] http://dlmf.nist.gov/24.2#ii
"""
@classmethod
def eval(cls, m, sym=None):
if m.is_Number:
if m.is_Integer and m.is_nonnegative:
# Euler numbers
if sym is None:
if m.is_odd:
return S.Zero
from mpmath import mp
m = m._to_mpmath(mp.prec)
res = mp.eulernum(m, exact=True)
return Integer(res)
# Euler polynomial
else:
from sympy.core.evalf import pure_complex
reim = pure_complex(sym, or_real=True)
# Evaluate polynomial numerically using mpmath
if reim and all(a.is_Float or a.is_Integer for a in reim) \
and any(a.is_Float for a in reim):
from mpmath import mp
from sympy import Expr
m = int(m)
# XXX ComplexFloat (#12192) would be nice here, above
prec = min([a._prec for a in reim if a.is_Float])
with workprec(prec):
res = mp.eulerpoly(m, sym)
return Expr._from_mpmath(res, prec)
# Construct polynomial symbolically from definition
m, result = int(m), []
for k in range(m + 1):
result.append(binomial(m, k)*cls(k)/(2**k)*(sym - S.Half)**(m - k))
return Add(*result).expand()
else:
raise ValueError("Euler numbers are defined only"
" for nonnegative integer indices.")
if sym is None:
if m.is_odd and m.is_positive:
return S.Zero
def _eval_rewrite_as_Sum(self, n, x=None, **kwargs):
from sympy import Sum
if x is None and n.is_even:
k = Dummy("k", integer=True)
j = Dummy("j", integer=True)
n = n / 2
Em = (S.ImaginaryUnit * Sum(Sum(binomial(k, j) * ((-1)**j * (k - 2*j)**(2*n + 1)) /
(2**k*S.ImaginaryUnit**k * k), (j, 0, k)), (k, 1, 2*n + 1)))
return Em
if x:
k = Dummy("k", integer=True)
return Sum(binomial(n, k)*euler(k)/2**k*(x-S.Half)**(n-k), (k, 0, n))
def _eval_evalf(self, prec):
m, x = (self.args[0], None) if len(self.args) == 1 else self.args
if x is None and m.is_Integer and m.is_nonnegative:
from mpmath import mp
from sympy import Expr
m = m._to_mpmath(prec)
with workprec(prec):
res = mp.eulernum(m)
return Expr._from_mpmath(res, prec)
if x and x.is_number and m.is_Integer and m.is_nonnegative:
from mpmath import mp
from sympy import Expr
m = int(m)
x = x._to_mpmath(prec)
with workprec(prec):
res = mp.eulerpoly(m, x)
return Expr._from_mpmath(res, prec)
#----------------------------------------------------------------------------#
# #
# Catalan numbers #
# #
#----------------------------------------------------------------------------#
class catalan(Function):
r"""
Catalan numbers
The `n^{th}` catalan number is given by:
.. math :: C_n = \frac{1}{n+1} \binom{2n}{n}
* ``catalan(n)`` gives the `n^{th}` Catalan number, `C_n`
Examples
========
>>> from sympy import (Symbol, binomial, gamma, hyper, polygamma,
... catalan, diff, combsimp, Rational, I)
>>> [catalan(i) for i in range(1,10)]
[1, 2, 5, 14, 42, 132, 429, 1430, 4862]
>>> n = Symbol("n", integer=True)
>>> catalan(n)
catalan(n)
Catalan numbers can be transformed into several other, identical
expressions involving other mathematical functions
>>> catalan(n).rewrite(binomial)
binomial(2*n, n)/(n + 1)
>>> catalan(n).rewrite(gamma)
4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))
>>> catalan(n).rewrite(hyper)
hyper((1 - n, -n), (2,), 1)
For some non-integer values of n we can get closed form
expressions by rewriting in terms of gamma functions:
>>> catalan(Rational(1,2)).rewrite(gamma)
8/(3*pi)
We can differentiate the Catalan numbers C(n) interpreted as a
continuous real function in n:
>>> diff(catalan(n), n)
(polygamma(0, n + 1/2) - polygamma(0, n + 2) + log(4))*catalan(n)
As a more advanced example consider the following ratio
between consecutive numbers:
>>> combsimp((catalan(n + 1)/catalan(n)).rewrite(binomial))
2*(2*n + 1)/(n + 2)
The Catalan numbers can be generalized to complex numbers:
>>> catalan(I).rewrite(gamma)
4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))
and evaluated with arbitrary precision:
>>> catalan(I).evalf(20)
0.39764993382373624267 - 0.020884341620842555705*I
See Also
========
bell, bernoulli, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
sympy.functions.combinatorial.factorials.binomial
References
==========
.. [1] https://en.wikipedia.org/wiki/Catalan_number
.. [2] http://mathworld.wolfram.com/CatalanNumber.html
.. [3] http://functions.wolfram.com/GammaBetaErf/CatalanNumber/
.. [4] http://geometer.org/mathcircles/catalan.pdf
"""
@classmethod
def eval(cls, n):
from sympy import gamma
if (n.is_Integer and n.is_nonnegative) or \
(n.is_noninteger and n.is_negative):
return 4**n*gamma(n + S.Half)/(gamma(S.Half)*gamma(n + 2))
if (n.is_integer and n.is_negative):
if (n + 1).is_negative:
return S.Zero
if (n + 1).is_zero:
return -S.Half
def fdiff(self, argindex=1):
from sympy import polygamma, log
n = self.args[0]
return catalan(n)*(polygamma(0, n + Rational(1, 2)) - polygamma(0, n + 2) + log(4))
def _eval_rewrite_as_binomial(self, n, **kwargs):
return binomial(2*n, n)/(n + 1)
def _eval_rewrite_as_factorial(self, n, **kwargs):
return factorial(2*n) / (factorial(n+1) * factorial(n))
def _eval_rewrite_as_gamma(self, n, **kwargs):
from sympy import gamma
# The gamma function allows to generalize Catalan numbers to complex n
return 4**n*gamma(n + S.Half)/(gamma(S.Half)*gamma(n + 2))
def _eval_rewrite_as_hyper(self, n, **kwargs):
from sympy import hyper
return hyper([1 - n, -n], [2], 1)
def _eval_rewrite_as_Product(self, n, **kwargs):
from sympy import Product
if not (n.is_integer and n.is_nonnegative):
return self
k = Dummy('k', integer=True, positive=True)
return Product((n + k) / k, (k, 2, n))
def _eval_is_integer(self):
if self.args[0].is_integer and self.args[0].is_nonnegative:
return True
def _eval_is_positive(self):
if self.args[0].is_nonnegative:
return True
def _eval_is_composite(self):
if self.args[0].is_integer and (self.args[0] - 3).is_positive:
return True
def _eval_evalf(self, prec):
from sympy import gamma
if self.args[0].is_number:
return self.rewrite(gamma)._eval_evalf(prec)
#----------------------------------------------------------------------------#
# #
# Genocchi numbers #
# #
#----------------------------------------------------------------------------#
class genocchi(Function):
r"""
Genocchi numbers
The Genocchi numbers are a sequence of integers `G_n` that satisfy the
relation:
.. math:: \frac{2t}{e^t + 1} = \sum_{n=1}^\infty \frac{G_n t^n}{n!}
Examples
========
>>> from sympy import Symbol
>>> from sympy.functions import genocchi
>>> [genocchi(n) for n in range(1, 9)]
[1, -1, 0, 1, 0, -3, 0, 17]
>>> n = Symbol('n', integer=True, positive=True)
>>> genocchi(2*n + 1)
0
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Genocchi_number
.. [2] http://mathworld.wolfram.com/GenocchiNumber.html
"""
@classmethod
def eval(cls, n):
if n.is_Number:
if (not n.is_Integer) or n.is_nonpositive:
raise ValueError("Genocchi numbers are defined only for " +
"positive integers")
return 2 * (1 - S(2) ** n) * bernoulli(n)
if n.is_odd and (n - 1).is_positive:
return S.Zero
if (n - 1).is_zero:
return S.One
def _eval_rewrite_as_bernoulli(self, n, **kwargs):
if n.is_integer and n.is_nonnegative:
return (1 - S(2) ** n) * bernoulli(n) * 2
def _eval_is_integer(self):
if self.args[0].is_integer and self.args[0].is_positive:
return True
def _eval_is_negative(self):
n = self.args[0]
if n.is_integer and n.is_positive:
if n.is_odd:
return False
return (n / 2).is_odd
def _eval_is_positive(self):
n = self.args[0]
if n.is_integer and n.is_positive:
if n.is_odd:
return fuzzy_not((n - 1).is_positive)
return (n / 2).is_even
def _eval_is_even(self):
n = self.args[0]
if n.is_integer and n.is_positive:
if n.is_even:
return False
return (n - 1).is_positive
def _eval_is_odd(self):
n = self.args[0]
if n.is_integer and n.is_positive:
if n.is_even:
return True
return fuzzy_not((n - 1).is_positive)
def _eval_is_prime(self):
n = self.args[0]
# only G_6 = -3 and G_8 = 17 are prime,
# but SymPy does not consider negatives as prime
# so only n=8 is tested
return (n - 8).is_zero
#----------------------------------------------------------------------------#
# #
# Partition numbers #
# #
#----------------------------------------------------------------------------#
_npartition = [1, 1]
class partition(Function):
r"""
Partition numbers
The Partition numbers are a sequence of integers `p_n` that represent the
number of distinct ways of representing `n` as a sum of natural numbers
(with order irrelevant). The generating function for `p_n` is given by:
.. math:: \sum_{n=0}^\infty p_n x^n = \prod_{k=1}^\infty (1 - x^k)^{-1}
Examples
========
>>> from sympy import Symbol
>>> from sympy.functions import partition
>>> [partition(n) for n in range(9)]
[1, 1, 2, 3, 5, 7, 11, 15, 22]
>>> n = Symbol('n', integer=True, negative=True)
>>> partition(n)
0
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Partition_(number_theory%29
.. [2] https://en.wikipedia.org/wiki/Pentagonal_number_theorem
"""
@staticmethod
def _partition(n):
L = len(_npartition)
if n < L:
return _npartition[n]
# lengthen cache
for _n in range(L, n + 1):
v, p, i = 0, 0, 0
while 1:
s = 0
p += 3*i + 1 # p = pentagonal number: 1, 5, 12, ...
if _n >= p:
s += _npartition[_n - p]
i += 1
gp = p + i # gp = generalized pentagonal: 2, 7, 15, ...
if _n >= gp:
s += _npartition[_n - gp]
if s == 0:
break
else:
v += s if i%2 == 1 else -s
_npartition.append(v)
return v
@classmethod
def eval(cls, n):
is_int = n.is_integer
if is_int == False:
raise ValueError("Partition numbers are defined only for "
"integers")
elif is_int:
if n.is_negative:
return S.Zero
if n.is_zero or (n - 1).is_zero:
return S.One
if n.is_Integer:
return Integer(cls._partition(n))
def _eval_is_integer(self):
if self.args[0].is_integer:
return True
def _eval_is_negative(self):
if self.args[0].is_integer:
return False
def _eval_is_positive(self):
n = self.args[0]
if n.is_nonnegative and n.is_integer:
return True
#######################################################################
###
### Functions for enumerating partitions, permutations and combinations
###
#######################################################################
class _MultisetHistogram(tuple):
pass
_N = -1
_ITEMS = -2
_M = slice(None, _ITEMS)
def _multiset_histogram(n):
"""Return tuple used in permutation and combination counting. Input
is a dictionary giving items with counts as values or a sequence of
items (which need not be sorted).
The data is stored in a class deriving from tuple so it is easily
recognized and so it can be converted easily to a list.
"""
if isinstance(n, dict): # item: count
if not all(isinstance(v, int) and v >= 0 for v in n.values()):
raise ValueError
tot = sum(n.values())
items = sum(1 for k in n if n[k] > 0)
return _MultisetHistogram([n[k] for k in n if n[k] > 0] + [items, tot])
else:
n = list(n)
s = set(n)
if len(s) == len(n):
n = [1]*len(n)
n.extend([len(n), len(n)])
return _MultisetHistogram(n)
m = dict(zip(s, range(len(s))))
d = dict(zip(range(len(s)), [0]*len(s)))
for i in n:
d[m[i]] += 1
return _multiset_histogram(d)
def nP(n, k=None, replacement=False):
"""Return the number of permutations of ``n`` items taken ``k`` at a time.
Possible values for ``n``::
integer - set of length ``n``
sequence - converted to a multiset internally
multiset - {element: multiplicity}
If ``k`` is None then the total of all permutations of length 0
through the number of items represented by ``n`` will be returned.
If ``replacement`` is True then a given item can appear more than once
in the ``k`` items. (For example, for 'ab' permutations of 2 would
include 'aa', 'ab', 'ba' and 'bb'.) The multiplicity of elements in
``n`` is ignored when ``replacement`` is True but the total number
of elements is considered since no element can appear more times than
the number of elements in ``n``.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nP
>>> from sympy.utilities.iterables import multiset_permutations, multiset
>>> nP(3, 2)
6
>>> nP('abc', 2) == nP(multiset('abc'), 2) == 6
True
>>> nP('aab', 2)
3
>>> nP([1, 2, 2], 2)
3
>>> [nP(3, i) for i in range(4)]
[1, 3, 6, 6]
>>> nP(3) == sum(_)
True
When ``replacement`` is True, each item can have multiplicity
equal to the length represented by ``n``:
>>> nP('aabc', replacement=True)
121
>>> [len(list(multiset_permutations('aaaabbbbcccc', i))) for i in range(5)]
[1, 3, 9, 27, 81]
>>> sum(_)
121
See Also
========
sympy.utilities.iterables.multiset_permutations
References
==========
.. [1] https://en.wikipedia.org/wiki/Permutation
"""
try:
n = as_int(n)
except ValueError:
return Integer(_nP(_multiset_histogram(n), k, replacement))
return Integer(_nP(n, k, replacement))
@cacheit
def _nP(n, k=None, replacement=False):
from sympy.functions.combinatorial.factorials import factorial
from sympy.core.mul import prod
if k == 0:
return 1
if isinstance(n, SYMPY_INTS): # n different items
# assert n >= 0
if k is None:
return sum(_nP(n, i, replacement) for i in range(n + 1))
elif replacement:
return n**k
elif k > n:
return 0
elif k == n:
return factorial(k)
elif k == 1:
return n
else:
# assert k >= 0
return _product(n - k + 1, n)
elif isinstance(n, _MultisetHistogram):
if k is None:
return sum(_nP(n, i, replacement) for i in range(n[_N] + 1))
elif replacement:
return n[_ITEMS]**k
elif k == n[_N]:
return factorial(k)/prod([factorial(i) for i in n[_M] if i > 1])
elif k > n[_N]:
return 0
elif k == 1:
return n[_ITEMS]
else:
# assert k >= 0
tot = 0
n = list(n)
for i in range(len(n[_M])):
if not n[i]:
continue
n[_N] -= 1
if n[i] == 1:
n[i] = 0
n[_ITEMS] -= 1
tot += _nP(_MultisetHistogram(n), k - 1)
n[_ITEMS] += 1
n[i] = 1
else:
n[i] -= 1
tot += _nP(_MultisetHistogram(n), k - 1)
n[i] += 1
n[_N] += 1
return tot
@cacheit
def _AOP_product(n):
"""for n = (m1, m2, .., mk) return the coefficients of the polynomial,
prod(sum(x**i for i in range(nj + 1)) for nj in n); i.e. the coefficients
of the product of AOPs (all-one polynomials) or order given in n. The
resulting coefficient corresponding to x**r is the number of r-length
combinations of sum(n) elements with multiplicities given in n.
The coefficients are given as a default dictionary (so if a query is made
for a key that is not present, 0 will be returned).
Examples
========
>>> from sympy.functions.combinatorial.numbers import _AOP_product
>>> from sympy.abc import x
>>> n = (2, 2, 3) # e.g. aabbccc
>>> prod = ((x**2 + x + 1)*(x**2 + x + 1)*(x**3 + x**2 + x + 1)).expand()
>>> c = _AOP_product(n); dict(c)
{0: 1, 1: 3, 2: 6, 3: 8, 4: 8, 5: 6, 6: 3, 7: 1}
>>> [c[i] for i in range(8)] == [prod.coeff(x, i) for i in range(8)]
True
The generating poly used here is the same as that listed in
http://tinyurl.com/cep849r, but in a refactored form.
"""
from collections import defaultdict
n = list(n)
ord = sum(n)
need = (ord + 2)//2
rv = [1]*(n.pop() + 1)
rv.extend([0]*(need - len(rv)))
rv = rv[:need]
while n:
ni = n.pop()
N = ni + 1
was = rv[:]
for i in range(1, min(N, len(rv))):
rv[i] += rv[i - 1]
for i in range(N, need):
rv[i] += rv[i - 1] - was[i - N]
rev = list(reversed(rv))
if ord % 2:
rv = rv + rev
else:
rv[-1:] = rev
d = defaultdict(int)
for i in range(len(rv)):
d[i] = rv[i]
return d
def nC(n, k=None, replacement=False):
"""Return the number of combinations of ``n`` items taken ``k`` at a time.
Possible values for ``n``::
integer - set of length ``n``
sequence - converted to a multiset internally
multiset - {element: multiplicity}
If ``k`` is None then the total of all combinations of length 0
through the number of items represented in ``n`` will be returned.
If ``replacement`` is True then a given item can appear more than once
in the ``k`` items. (For example, for 'ab' sets of 2 would include 'aa',
'ab', and 'bb'.) The multiplicity of elements in ``n`` is ignored when
``replacement`` is True but the total number of elements is considered
since no element can appear more times than the number of elements in
``n``.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nC
>>> from sympy.utilities.iterables import multiset_combinations
>>> nC(3, 2)
3
>>> nC('abc', 2)
3
>>> nC('aab', 2)
2
When ``replacement`` is True, each item can have multiplicity
equal to the length represented by ``n``:
>>> nC('aabc', replacement=True)
35
>>> [len(list(multiset_combinations('aaaabbbbcccc', i))) for i in range(5)]
[1, 3, 6, 10, 15]
>>> sum(_)
35
If there are ``k`` items with multiplicities ``m_1, m_2, ..., m_k``
then the total of all combinations of length 0 through ``k`` is the
product, ``(m_1 + 1)*(m_2 + 1)*...*(m_k + 1)``. When the multiplicity
of each item is 1 (i.e., k unique items) then there are 2**k
combinations. For example, if there are 4 unique items, the total number
of combinations is 16:
>>> sum(nC(4, i) for i in range(5))
16
See Also
========
sympy.utilities.iterables.multiset_combinations
References
==========
.. [1] https://en.wikipedia.org/wiki/Combination
.. [2] http://tinyurl.com/cep849r
"""
from sympy.functions.combinatorial.factorials import binomial
from sympy.core.mul import prod
if isinstance(n, SYMPY_INTS):
if k is None:
if not replacement:
return 2**n
return sum(nC(n, i, replacement) for i in range(n + 1))
if k < 0:
raise ValueError("k cannot be negative")
if replacement:
return binomial(n + k - 1, k)
return binomial(n, k)
if isinstance(n, _MultisetHistogram):
N = n[_N]
if k is None:
if not replacement:
return prod(m + 1 for m in n[_M])
return sum(nC(n, i, replacement) for i in range(N + 1))
elif replacement:
return nC(n[_ITEMS], k, replacement)
# assert k >= 0
elif k in (1, N - 1):
return n[_ITEMS]
elif k in (0, N):
return 1
return _AOP_product(tuple(n[_M]))[k]
else:
return nC(_multiset_histogram(n), k, replacement)
@cacheit
def _stirling1(n, k):
if n == k == 0:
return S.One
if 0 in (n, k):
return S.Zero
n1 = n - 1
# some special values
if n == k:
return S.One
elif k == 1:
return factorial(n1)
elif k == n1:
return binomial(n, 2)
elif k == n - 2:
return (3*n - 1)*binomial(n, 3)/4
elif k == n - 3:
return binomial(n, 2)*binomial(n, 4)
# general recurrence
return n1*_stirling1(n1, k) + _stirling1(n1, k - 1)
@cacheit
def _stirling2(n, k):
if n == k == 0:
return S.One
if 0 in (n, k):
return S.Zero
n1 = n - 1
# some special values
if k == n1:
return binomial(n, 2)
elif k == 2:
return 2**n1 - 1
# general recurrence
return k*_stirling2(n1, k) + _stirling2(n1, k - 1)
def stirling(n, k, d=None, kind=2, signed=False):
r"""Return Stirling number `S(n, k)` of the first or second (default) kind.
The sum of all Stirling numbers of the second kind for `k = 1`
through `n` is ``bell(n)``. The recurrence relationship for these numbers
is:
.. math :: {0 \brace 0} = 1; {n \brace 0} = {0 \brace k} = 0;
.. math :: {{n+1} \brace k} = j {n \brace k} + {n \brace {k-1}}
where `j` is:
`n` for Stirling numbers of the first kind
`-n` for signed Stirling numbers of the first kind
`k` for Stirling numbers of the second kind
The first kind of Stirling number counts the number of permutations of
``n`` distinct items that have ``k`` cycles; the second kind counts the
ways in which ``n`` distinct items can be partitioned into ``k`` parts.
If ``d`` is given, the "reduced Stirling number of the second kind" is
returned: ``S^{d}(n, k) = S(n - d + 1, k - d + 1)`` with ``n >= k >= d``.
(This counts the ways to partition ``n`` consecutive integers into
``k`` groups with no pairwise difference less than ``d``. See example
below.)
To obtain the signed Stirling numbers of the first kind, use keyword
``signed=True``. Using this keyword automatically sets ``kind`` to 1.
Examples
========
>>> from sympy.functions.combinatorial.numbers import stirling, bell
>>> from sympy.combinatorics import Permutation
>>> from sympy.utilities.iterables import multiset_partitions, permutations
First kind (unsigned by default):
>>> [stirling(6, i, kind=1) for i in range(7)]
[0, 120, 274, 225, 85, 15, 1]
>>> perms = list(permutations(range(4)))
>>> [sum(Permutation(p).cycles == i for p in perms) for i in range(5)]
[0, 6, 11, 6, 1]
>>> [stirling(4, i, kind=1) for i in range(5)]
[0, 6, 11, 6, 1]
First kind (signed):
>>> [stirling(4, i, signed=True) for i in range(5)]
[0, -6, 11, -6, 1]
Second kind:
>>> [stirling(10, i) for i in range(12)]
[0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1, 0]
>>> sum(_) == bell(10)
True
>>> len(list(multiset_partitions(range(4), 2))) == stirling(4, 2)
True
Reduced second kind:
>>> from sympy import subsets, oo
>>> def delta(p):
... if len(p) == 1:
... return oo
... return min(abs(i[0] - i[1]) for i in subsets(p, 2))
>>> parts = multiset_partitions(range(5), 3)
>>> d = 2
>>> sum(1 for p in parts if all(delta(i) >= d for i in p))
7
>>> stirling(5, 3, 2)
7
See Also
========
sympy.utilities.iterables.multiset_partitions
References
==========
.. [1] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
.. [2] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
"""
# TODO: make this a class like bell()
n = as_int(n)
k = as_int(k)
if n < 0:
raise ValueError('n must be nonnegative')
if k > n:
return S.Zero
if d:
# assert k >= d
# kind is ignored -- only kind=2 is supported
return _stirling2(n - d + 1, k - d + 1)
elif signed:
# kind is ignored -- only kind=1 is supported
return (-1)**(n - k)*_stirling1(n, k)
if kind == 1:
return _stirling1(n, k)
elif kind == 2:
return _stirling2(n, k)
else:
raise ValueError('kind must be 1 or 2, not %s' % k)
@cacheit
def _nT(n, k):
"""Return the partitions of ``n`` items into ``k`` parts. This
is used by ``nT`` for the case when ``n`` is an integer."""
# really quick exits
if k > n or k < 0:
return 0
if k == n or k == 1:
return 1
if k == 0:
return 0
# exits that could be done below but this is quicker
if k == 2:
return n//2
d = n - k
if d <= 3:
return d
# quick exit
if 3*k >= n: # or, equivalently, 2*k >= d
# all the information needed in this case
# will be in the cache needed to calculate
# partition(d), so...
# update cache
tot = partition._partition(d)
# and correct for values not needed
if d - k > 0:
tot -= sum(_npartition[:d - k])
return tot
# regular exit
# nT(n, k) = Sum(nT(n - k, m), (m, 1, k));
# calculate needed nT(i, j) values
p = [1]*d
for i in range(2, k + 1):
for m in range(i + 1, d):
p[m] += p[m - i]
d -= 1
# if p[0] were appended to the end of p then the last
# k values of p are the nT(n, j) values for 0 < j < k in reverse
# order p[-1] = nT(n, 1), p[-2] = nT(n, 2), etc.... Instead of
# putting the 1 from p[0] there, however, it is simply added to
# the sum below which is valid for 1 < k <= n//2
return (1 + sum(p[1 - k:]))
def nT(n, k=None):
"""Return the number of ``k``-sized partitions of ``n`` items.
Possible values for ``n``::
integer - ``n`` identical items
sequence - converted to a multiset internally
multiset - {element: multiplicity}
Note: the convention for ``nT`` is different than that of ``nC`` and
``nP`` in that
here an integer indicates ``n`` *identical* items instead of a set of
length ``n``; this is in keeping with the ``partitions`` function which
treats its integer-``n`` input like a list of ``n`` 1s. One can use
``range(n)`` for ``n`` to indicate ``n`` distinct items.
If ``k`` is None then the total number of ways to partition the elements
represented in ``n`` will be returned.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nT
Partitions of the given multiset:
>>> [nT('aabbc', i) for i in range(1, 7)]
[1, 8, 11, 5, 1, 0]
>>> nT('aabbc') == sum(_)
True
>>> [nT("mississippi", i) for i in range(1, 12)]
[1, 74, 609, 1521, 1768, 1224, 579, 197, 50, 9, 1]
Partitions when all items are identical:
>>> [nT(5, i) for i in range(1, 6)]
[1, 2, 2, 1, 1]
>>> nT('1'*5) == sum(_)
True
When all items are different:
>>> [nT(range(5), i) for i in range(1, 6)]
[1, 15, 25, 10, 1]
>>> nT(range(5)) == sum(_)
True
Partitions of an integer expressed as a sum of positive integers:
>>> from sympy.functions.combinatorial.numbers import partition
>>> partition(4)
5
>>> nT(4, 1) + nT(4, 2) + nT(4, 3) + nT(4, 4)
5
>>> nT('1'*4)
5
See Also
========
sympy.utilities.iterables.partitions
sympy.utilities.iterables.multiset_partitions
sympy.functions.combinatorial.numbers.partition
References
==========
.. [1] http://undergraduate.csse.uwa.edu.au/units/CITS7209/partition.pdf
"""
from sympy.utilities.enumerative import MultisetPartitionTraverser
if isinstance(n, SYMPY_INTS):
# n identical items
if k is None:
return partition(n)
if isinstance(k, SYMPY_INTS):
n = as_int(n)
k = as_int(k)
return Integer(_nT(n, k))
if not isinstance(n, _MultisetHistogram):
try:
# if n contains hashable items there is some
# quick handling that can be done
u = len(set(n))
if u <= 1:
return nT(len(n), k)
elif u == len(n):
n = range(u)
raise TypeError
except TypeError:
n = _multiset_histogram(n)
N = n[_N]
if k is None and N == 1:
return 1
if k in (1, N):
return 1
if k == 2 or N == 2 and k is None:
m, r = divmod(N, 2)
rv = sum(nC(n, i) for i in range(1, m + 1))
if not r:
rv -= nC(n, m)//2
if k is None:
rv += 1 # for k == 1
return rv
if N == n[_ITEMS]:
# all distinct
if k is None:
return bell(N)
return stirling(N, k)
m = MultisetPartitionTraverser()
if k is None:
return m.count_partitions(n[_M])
# MultisetPartitionTraverser does not have a range-limited count
# method, so need to enumerate and count
tot = 0
for discard in m.enum_range(n[_M], k-1, k):
tot += 1
return tot
|
3fa6c05137ab3366a565e59130c526690353512076df442ef61a984b241a3f32 | from __future__ import print_function, division
from sympy.core.add import Add
from sympy.core.basic import sympify, cacheit
from sympy.core.compatibility import range
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.logic import fuzzy_not, fuzzy_or
from sympy.core.numbers import igcdex, Rational, pi
from sympy.core.relational import Ne
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.factorials import factorial, RisingFactorial
from sympy.functions.elementary.exponential import log, exp
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.hyperbolic import (acoth, asinh, atanh, cosh,
coth, HyperbolicFunction, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.sets.sets import FiniteSet
from sympy.utilities.iterables import numbered_symbols
###############################################################################
########################## TRIGONOMETRIC FUNCTIONS ############################
###############################################################################
class TrigonometricFunction(Function):
"""Base class for trigonometric functions. """
unbranched = True
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational and fuzzy_not(s.args[0].is_zero):
return False
else:
return s.is_rational
def _eval_is_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if fuzzy_not(self.args[0].is_zero) and self.args[0].is_algebraic:
return False
pi_coeff = _pi_coeff(self.args[0])
if pi_coeff is not None and pi_coeff.is_rational:
return True
else:
return s.is_algebraic
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=deep, **hints)
return re_part + im_part*S.ImaginaryUnit
def _as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.args[0].expand(deep, **hints), S.Zero)
else:
return (self.args[0], S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (re, im)
def _period(self, general_period, symbol=None):
f = self.args[0]
if symbol is None:
symbol = tuple(f.free_symbols)[0]
if not f.has(symbol):
return S.Zero
if f == symbol:
return general_period
if symbol in f.free_symbols:
if f.is_Mul:
g, h = f.as_independent(symbol)
if h == symbol:
return general_period/abs(g)
if f.is_Add:
a, h = f.as_independent(symbol)
g, h = h.as_independent(symbol, as_Add=False)
if h == symbol:
return general_period/abs(g)
raise NotImplementedError("Use the periodicity function instead.")
def _peeloff_pi(arg):
"""
Split ARG into two parts, a "rest" and a multiple of pi/2.
This assumes ARG to be an Add.
The multiple of pi returned in the second position is always a Rational.
Examples
========
>>> from sympy.functions.elementary.trigonometric import _peeloff_pi as peel
>>> from sympy import pi
>>> from sympy.abc import x, y
>>> peel(x + pi/2)
(x, pi/2)
>>> peel(x + 2*pi/3 + pi*y)
(x + pi*y + pi/6, pi/2)
"""
for a in Add.make_args(arg):
if a is S.Pi:
K = S.One
break
elif a.is_Mul:
K, p = a.as_two_terms()
if p is S.Pi and K.is_Rational:
break
else:
return arg, S.Zero
m1 = (K % S.Half) * S.Pi
m2 = K*S.Pi - m1
return arg - m2, m2
def _pi_coeff(arg, cycles=1):
"""
When arg is a Number times pi (e.g. 3*pi/2) then return the Number
normalized to be in the range [0, 2], else None.
When an even multiple of pi is encountered, if it is multiplying
something with known parity then the multiple is returned as 0 otherwise
as 2.
Examples
========
>>> from sympy.functions.elementary.trigonometric import _pi_coeff as coeff
>>> from sympy import pi, Dummy
>>> from sympy.abc import x, y
>>> coeff(3*x*pi)
3*x
>>> coeff(11*pi/7)
11/7
>>> coeff(-11*pi/7)
3/7
>>> coeff(4*pi)
0
>>> coeff(5*pi)
1
>>> coeff(5.0*pi)
1
>>> coeff(5.5*pi)
3/2
>>> coeff(2 + pi)
>>> coeff(2*Dummy(integer=True)*pi)
2
>>> coeff(2*Dummy(even=True)*pi)
0
"""
arg = sympify(arg)
if arg is S.Pi:
return S.One
elif not arg:
return S.Zero
elif arg.is_Mul:
cx = arg.coeff(S.Pi)
if cx:
c, x = cx.as_coeff_Mul() # pi is not included as coeff
if c.is_Float:
# recast exact binary fractions to Rationals
f = abs(c) % 1
if f != 0:
p = -int(round(log(f, 2).evalf()))
m = 2**p
cm = c*m
i = int(cm)
if i == cm:
c = Rational(i, m)
cx = c*x
else:
c = Rational(int(c))
cx = c*x
if x.is_integer:
c2 = c % 2
if c2 == 1:
return x
elif not c2:
if x.is_even is not None: # known parity
return S.Zero
return S(2)
else:
return c2*x
return cx
class sin(TrigonometricFunction):
"""
The sine function.
Returns the sine of x (measured in radians).
Notes
=====
This function will evaluate automatically in the
case x/pi is some rational number [4]_. For example,
if x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.
Examples
========
>>> from sympy import sin, pi
>>> from sympy.abc import x
>>> sin(x**2).diff(x)
2*x*cos(x**2)
>>> sin(1).diff(x)
0
>>> sin(pi)
0
>>> sin(pi/2)
1
>>> sin(pi/6)
1/2
>>> sin(pi/12)
-sqrt(2)/4 + sqrt(6)/4
See Also
========
csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Sin
.. [4] http://mathworld.wolfram.com/TrigonometryAngles.html
"""
def period(self, symbol=None):
return self._period(2*pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return cos(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.calculus import AccumBounds
from sympy.sets.setexpr import SetExpr
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.Infinity or arg is S.NegativeInfinity:
return AccumBounds(-1, 1)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
min, max = arg.min, arg.max
d = floor(min/(2*S.Pi))
if min is not S.NegativeInfinity:
min = min - d*2*S.Pi
if max is not S.Infinity:
max = max - d*2*S.Pi
if AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 5*S.Pi/2)) \
is not S.EmptySet and \
AccumBounds(min, max).intersection(FiniteSet(3*S.Pi/2,
7*S.Pi/2)) is not S.EmptySet:
return AccumBounds(-1, 1)
elif AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 5*S.Pi/2)) \
is not S.EmptySet:
return AccumBounds(Min(sin(min), sin(max)), 1)
elif AccumBounds(min, max).intersection(FiniteSet(3*S.Pi/2, 8*S.Pi/2)) \
is not S.EmptySet:
return AccumBounds(-1, Max(sin(min), sin(max)))
else:
return AccumBounds(Min(sin(min), sin(max)),
Max(sin(min), sin(max)))
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * sinh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.Zero
if (2*pi_coeff).is_integer:
if pi_coeff.is_even:
return S.Zero
elif pi_coeff.is_even is False:
return S.NegativeOne**(pi_coeff - S.Half)
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
# https://github.com/sympy/sympy/issues/6048
# transform a sine to a cosine, to avoid redundant code
if pi_coeff.is_Rational:
x = pi_coeff % 2
if x > 1:
return -cls((x % 1)*S.Pi)
if 2*x > 1:
return cls((1 - x)*S.Pi)
narg = ((pi_coeff + Rational(3, 2)) % 2)*S.Pi
result = cos(narg)
if not isinstance(result, cos):
return result
if pi_coeff*S.Pi != arg:
return cls(pi_coeff*S.Pi)
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
return sin(m)*cos(x) + cos(m)*sin(x)
if isinstance(arg, asin):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return x / sqrt(1 + x**2)
if isinstance(arg, atan2):
y, x = arg.args
return y / sqrt(x**2 + y**2)
if isinstance(arg, acos):
x = arg.args[0]
return sqrt(1 - x**2)
if isinstance(arg, acot):
x = arg.args[0]
return 1 / (sqrt(1 + 1 / x**2) * x)
if isinstance(arg, acsc):
x = arg.args[0]
return 1 / x
if isinstance(arg, asec):
x = arg.args[0]
return sqrt(1 - 1 / x**2)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p * x**2 / (n*(n - 1))
else:
return (-1)**(n//2) * x**(n)/factorial(n)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):
arg = arg.func(arg.args[0]).rewrite(exp)
return (exp(arg*I) - exp(-arg*I)) / (2*I)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return I*x**-I / 2 - I*x**I /2
def _eval_rewrite_as_cos(self, arg, **kwargs):
return cos(arg - S.Pi / 2, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
tan_half = tan(S.Half*arg)
return 2*tan_half/(1 + tan_half**2)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)*cos(arg)/cos(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(S.Half*arg)
return 2*cot_half/(1 + cot_half**2)
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self.rewrite(cos).rewrite(pow)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
return self.rewrite(cos).rewrite(sqrt)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return 1/csc(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1 / sec(arg - S.Pi / 2, evaluate=False)
def _eval_rewrite_as_sinc(self, arg, **kwargs):
return arg*sinc(arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
return (sin(re)*cosh(im), cos(re)*sinh(im))
def _eval_expand_trig(self, **hints):
from sympy import expand_mul
from sympy.functions.special.polynomials import chebyshevt, chebyshevu
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
# TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return sx*cy + sy*cx
else:
n, x = arg.as_coeff_Mul(rational=True)
if n.is_Integer: # n will be positive because of .eval
# canonicalization
# See http://mathworld.wolfram.com/Multiple-AngleFormulas.html
if n.is_odd:
return (-1)**((n - 1)/2)*chebyshevt(n, sin(x))
else:
return expand_mul((-1)**(n/2 - 1)*cos(x)*chebyshevu(n -
1, sin(x)), deep=False)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return sin(arg)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
if self.args[0].is_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real:
return True
class cos(TrigonometricFunction):
"""
The cosine function.
Returns the cosine of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import cos, pi
>>> from sympy.abc import x
>>> cos(x**2).diff(x)
-2*x*sin(x**2)
>>> cos(1).diff(x)
0
>>> cos(pi)
-1
>>> cos(pi/2)
0
>>> cos(2*pi/3)
-1/2
>>> cos(pi/12)
sqrt(2)/4 + sqrt(6)/4
See Also
========
sin, csc, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Cos
"""
def period(self, symbol=None):
return self._period(2*pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return -sin(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.functions.special.polynomials import chebyshevt
from sympy.calculus.util import AccumBounds
from sympy.sets.setexpr import SetExpr
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.One
elif arg is S.Infinity or arg is S.NegativeInfinity:
# In this case it is better to return AccumBounds(-1, 1)
# rather than returning S.NaN, since AccumBounds(-1, 1)
# preserves the information that sin(oo) is between
# -1 and 1, where S.NaN does not do that.
return AccumBounds(-1, 1)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
return sin(arg + S.Pi/2)
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.could_extract_minus_sign():
return cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return cosh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return (S.NegativeOne)**pi_coeff
if (2*pi_coeff).is_integer:
if pi_coeff.is_even:
return (S.NegativeOne)**(pi_coeff/2)
elif pi_coeff.is_even is False:
return S.Zero
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
# cosine formula #####################
# https://github.com/sympy/sympy/issues/6048
# explicit calculations are preformed for
# cos(k pi/n) for n = 8,10,12,15,20,24,30,40,60,120
# Some other exact values like cos(k pi/240) can be
# calculated using a partial-fraction decomposition
# by calling cos( X ).rewrite(sqrt)
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1)/4,
}
if pi_coeff.is_Rational:
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*S.Pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*S.Pi
return -cls(narg)
# If nested sqrt's are worse than un-evaluation
# you can require q to be in (1, 2, 3, 4, 6, 12)
# q <= 12, q=15, q=20, q=24, q=30, q=40, q=60, q=120 return
# expressions with 2 or fewer sqrt nestings.
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
if q in table2:
a, b = p*S.Pi/table2[q][0], p*S.Pi/table2[q][1]
nvala, nvalb = cls(a), cls(b)
if None == nvala or None == nvalb:
return None
return nvala*nvalb + cls(S.Pi/2 - a)*cls(S.Pi/2 - b)
if q > 12:
return None
if q in cst_table_some:
cts = cst_table_some[pi_coeff.q]
return chebyshevt(pi_coeff.p, cts).expand()
if 0 == q % 2:
narg = (pi_coeff*2)*S.Pi
nval = cls(narg)
if None == nval:
return None
x = (2*pi_coeff + 1)/2
sign_cos = (-1)**((-1 if x < 0 else 1)*int(abs(x)))
return sign_cos*sqrt( (1 + nval)/2 )
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
return cos(m)*cos(x) - sin(m)*sin(x)
if isinstance(arg, acos):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if isinstance(arg, atan2):
y, x = arg.args
return x / sqrt(x**2 + y**2)
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x ** 2)
if isinstance(arg, acot):
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
if isinstance(arg, acsc):
x = arg.args[0]
return sqrt(1 - 1 / x**2)
if isinstance(arg, asec):
x = arg.args[0]
return 1 / x
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p * x**2 / (n*(n - 1))
else:
return (-1)**(n//2)*x**(n)/factorial(n)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):
arg = arg.func(arg.args[0]).rewrite(exp)
return (exp(arg*I) + exp(-arg*I)) / 2
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return x**I/2 + x**-I/2
def _eval_rewrite_as_sin(self, arg, **kwargs):
return sin(arg + S.Pi / 2, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
tan_half = tan(S.Half*arg)**2
return (1 - tan_half)/(1 + tan_half)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)*cos(arg)/sin(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(S.Half*arg)**2
return (cot_half - 1)/(cot_half + 1)
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self._eval_rewrite_as_sqrt(arg)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
from sympy.functions.special.polynomials import chebyshevt
def migcdex(x):
# recursive calcuation of gcd and linear combination
# for a sequence of integers.
# Given (x1, x2, x3)
# Returns (y1, y1, y3, g)
# such that g is the gcd and x1*y1+x2*y2+x3*y3 - g = 0
# Note, that this is only one such linear combination.
if len(x) == 1:
return (1, x[0])
if len(x) == 2:
return igcdex(x[0], x[-1])
g = migcdex(x[1:])
u, v, h = igcdex(x[0], g[-1])
return tuple([u] + [v*i for i in g[0:-1] ] + [h])
def ipartfrac(r, factors=None):
from sympy.ntheory import factorint
if isinstance(r, int):
return r
if not isinstance(r, Rational):
raise TypeError("r is not rational")
n = r.q
if 2 > r.q*r.q:
return r.q
if None == factors:
a = [n//x**y for x, y in factorint(r.q).items()]
else:
a = [n//x for x in factors]
if len(a) == 1:
return [ r ]
h = migcdex(a)
ans = [ r.p*Rational(i*j, r.q) for i, j in zip(h[:-1], a) ]
assert r == sum(ans)
return ans
pi_coeff = _pi_coeff(arg)
if pi_coeff is None:
return None
if pi_coeff.is_integer:
# it was unevaluated
return self.func(pi_coeff*S.Pi)
if not pi_coeff.is_Rational:
return None
def _cospi257():
""" Express cos(pi/257) explicitly as a function of radicals
Based upon the equations in
http://math.stackexchange.com/questions/516142/how-does-cos2-pi-257-look-like-in-real-radicals
See also http://www.susqu.edu/brakke/constructions/257-gon.m.txt
"""
def f1(a, b):
return (a + sqrt(a**2 + b))/2, (a - sqrt(a**2 + b))/2
def f2(a, b):
return (a - sqrt(a**2 + b))/2
t1, t2 = f1(-1, 256)
z1, z3 = f1(t1, 64)
z2, z4 = f1(t2, 64)
y1, y5 = f1(z1, 4*(5 + t1 + 2*z1))
y6, y2 = f1(z2, 4*(5 + t2 + 2*z2))
y3, y7 = f1(z3, 4*(5 + t1 + 2*z3))
y8, y4 = f1(z4, 4*(5 + t2 + 2*z4))
x1, x9 = f1(y1, -4*(t1 + y1 + y3 + 2*y6))
x2, x10 = f1(y2, -4*(t2 + y2 + y4 + 2*y7))
x3, x11 = f1(y3, -4*(t1 + y3 + y5 + 2*y8))
x4, x12 = f1(y4, -4*(t2 + y4 + y6 + 2*y1))
x5, x13 = f1(y5, -4*(t1 + y5 + y7 + 2*y2))
x6, x14 = f1(y6, -4*(t2 + y6 + y8 + 2*y3))
x15, x7 = f1(y7, -4*(t1 + y7 + y1 + 2*y4))
x8, x16 = f1(y8, -4*(t2 + y8 + y2 + 2*y5))
v1 = f2(x1, -4*(x1 + x2 + x3 + x6))
v2 = f2(x2, -4*(x2 + x3 + x4 + x7))
v3 = f2(x8, -4*(x8 + x9 + x10 + x13))
v4 = f2(x9, -4*(x9 + x10 + x11 + x14))
v5 = f2(x10, -4*(x10 + x11 + x12 + x15))
v6 = f2(x16, -4*(x16 + x1 + x2 + x5))
u1 = -f2(-v1, -4*(v2 + v3))
u2 = -f2(-v4, -4*(v5 + v6))
w1 = -2*f2(-u1, -4*u2)
return sqrt(sqrt(2)*sqrt(w1 + 4)/8 + S.Half)
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1)/4,
17: sqrt((15 + sqrt(17))/32 + sqrt(2)*(sqrt(17 - sqrt(17)) +
sqrt(sqrt(2)*(-8*sqrt(17 + sqrt(17)) - (1 - sqrt(17))
*sqrt(17 - sqrt(17))) + 6*sqrt(17) + 34))/32),
257: _cospi257()
# 65537 is the only other known Fermat prime and the very
# large expression is intentionally omitted from SymPy; see
# http://www.susqu.edu/brakke/constructions/65537-gon.m.txt
}
def _fermatCoords(n):
# if n can be factored in terms of Fermat primes with
# multiplicity of each being 1, return those primes, else
# False
primes = []
for p_i in cst_table_some:
quotient, remainder = divmod(n, p_i)
if remainder == 0:
n = quotient
primes.append(p_i)
if n == 1:
return tuple(primes)
return False
if pi_coeff.q in cst_table_some:
rv = chebyshevt(pi_coeff.p, cst_table_some[pi_coeff.q])
if pi_coeff.q < 257:
rv = rv.expand()
return rv
if not pi_coeff.q % 2: # recursively remove factors of 2
pico2 = pi_coeff*2
nval = cos(pico2*S.Pi).rewrite(sqrt)
x = (pico2 + 1)/2
sign_cos = -1 if int(x) % 2 else 1
return sign_cos*sqrt( (1 + nval)/2 )
FC = _fermatCoords(pi_coeff.q)
if FC:
decomp = ipartfrac(pi_coeff, FC)
X = [(x[1], x[0]*S.Pi) for x in zip(decomp, numbered_symbols('z'))]
pcls = cos(sum([x[0] for x in X]))._eval_expand_trig().subs(X)
return pcls.rewrite(sqrt)
else:
decomp = ipartfrac(pi_coeff)
X = [(x[1], x[0]*S.Pi) for x in zip(decomp, numbered_symbols('z'))]
pcls = cos(sum([x[0] for x in X]))._eval_expand_trig().subs(X)
return pcls
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1/sec(arg)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return 1 / sec(arg)._eval_rewrite_as_csc(arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
return (cos(re)*cosh(im), -sin(re)*sinh(im))
def _eval_expand_trig(self, **hints):
from sympy.functions.special.polynomials import chebyshevt
arg = self.args[0]
x = None
if arg.is_Add: # TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return cx*cy - sx*sy
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer:
return chebyshevt(coeff, cos(terms))
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return cos(arg)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return S.One
else:
return self.func(arg)
def _eval_is_real(self):
if self.args[0].is_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real:
return True
class tan(TrigonometricFunction):
"""
The tangent function.
Returns the tangent of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import tan, pi
>>> from sympy.abc import x
>>> tan(x**2).diff(x)
2*x*(tan(x**2)**2 + 1)
>>> tan(1).diff(x)
0
>>> tan(pi/8).expand()
-1 + sqrt(2)
See Also
========
sin, csc, cos, sec, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Tan
"""
def period(self, symbol=None):
return self._period(pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return S.One + self**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return atan
@classmethod
def eval(cls, arg):
from sympy.calculus.util import AccumBounds
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.Infinity or arg is S.NegativeInfinity:
return AccumBounds(S.NegativeInfinity, S.Infinity)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
min, max = arg.min, arg.max
d = floor(min/S.Pi)
if min is not S.NegativeInfinity:
min = min - d*S.Pi
if max is not S.Infinity:
max = max - d*S.Pi
if AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 3*S.Pi/2)):
return AccumBounds(S.NegativeInfinity, S.Infinity)
else:
return AccumBounds(tan(min), tan(max))
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * tanh(i_coeff)
pi_coeff = _pi_coeff(arg, 2)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.Zero
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
if pi_coeff.is_Rational:
if not pi_coeff.q % 2:
narg = pi_coeff*S.Pi*2
cresult, sresult = cos(narg), cos(narg - S.Pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if sresult == 0:
return S.ComplexInfinity
return 1/sresult - cresult/sresult
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
q = pi_coeff.q
p = pi_coeff.p % q
if q in table2:
nvala, nvalb = cls(p*S.Pi/table2[q][0]), cls(p*S.Pi/table2[q][1])
if None == nvala or None == nvalb:
return None
return (nvala - nvalb)/(1 + nvala*nvalb)
narg = ((pi_coeff + S.Half) % 1 - S.Half)*S.Pi
# see cos() to specify which expressions should be
# expanded automatically in terms of radicals
cresult, sresult = cos(narg), cos(narg - S.Pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if cresult == 0:
return S.ComplexInfinity
return (sresult/cresult)
if narg != arg:
return cls(narg)
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
tanm = tan(m)
if tanm is S.ComplexInfinity:
return -cot(x)
else: # tanm == 0
return tan(x)
if isinstance(arg, atan):
return arg.args[0]
if isinstance(arg, atan2):
y, x = arg.args
return y/x
if isinstance(arg, asin):
x = arg.args[0]
return x / sqrt(1 - x**2)
if isinstance(arg, acos):
x = arg.args[0]
return sqrt(1 - x**2) / x
if isinstance(arg, acot):
x = arg.args[0]
return 1 / x
if isinstance(arg, acsc):
x = arg.args[0]
return 1 / (sqrt(1 - 1 / x**2) * x)
if isinstance(arg, asec):
x = arg.args[0]
return sqrt(1 - 1 / x**2) * x
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a, b = ((n - 1)//2), 2**(n + 1)
B = bernoulli(n + 1)
F = factorial(n + 1)
return (-1)**a * b*(b - 1) * B/F * x**n
def _eval_nseries(self, x, n, logx):
i = self.args[0].limit(x, 0)*2/S.Pi
if i and i.is_Integer:
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
return Function._eval_nseries(self, x, n=n, logx=logx)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return I*(x**-I - x**I)/(x**-I + x**I)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
if im:
denom = cos(2*re) + cosh(2*im)
return (sin(2*re)/denom, sinh(2*im)/denom)
else:
return (self.func(re), S.Zero)
def _eval_expand_trig(self, **hints):
from sympy import im, re
arg = self.args[0]
x = None
if arg.is_Add:
from sympy import symmetric_poly
n = len(arg.args)
TX = []
for x in arg.args:
tx = tan(x, evaluate=False)._eval_expand_trig()
TX.append(tx)
Yg = numbered_symbols('Y')
Y = [ next(Yg) for i in range(n) ]
p = [0, 0]
for i in range(n + 1):
p[1 - i % 2] += symmetric_poly(i, Y)*(-1)**((i % 4)//2)
return (p[0]/p[1]).subs(list(zip(Y, TX)))
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = Symbol('dummy', real=True)
P = ((1 + I*z)**coeff).expand()
return (im(P)/re(P)).subs([(z, tan(terms))])
return tan(arg)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):
arg = arg.func(arg.args[0]).rewrite(exp)
neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
return I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
def _eval_rewrite_as_sin(self, x, **kwargs):
return 2*sin(x)**2/sin(2*x)
def _eval_rewrite_as_cos(self, x, **kwargs):
return cos(x - S.Pi / 2, evaluate=False) / cos(x)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)/cos(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
return 1/cot(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
sin_in_sec_form = sin(arg)._eval_rewrite_as_sec(arg)
cos_in_sec_form = cos(arg)._eval_rewrite_as_sec(arg)
return sin_in_sec_form / cos_in_sec_form
def _eval_rewrite_as_csc(self, arg, **kwargs):
sin_in_csc_form = sin(arg)._eval_rewrite_as_csc(arg)
cos_in_csc_form = cos(arg)._eval_rewrite_as_csc(arg)
return sin_in_csc_form / cos_in_csc_form
def _eval_rewrite_as_pow(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(pow)
if y.has(cos):
return None
return y
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(sqrt)
if y.has(cos):
return None
return y
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
return self.args[0].is_real
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_imaginary:
return True
class cot(TrigonometricFunction):
"""
The cotangent function.
Returns the cotangent of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import cot, pi
>>> from sympy.abc import x
>>> cot(x**2).diff(x)
2*x*(-cot(x**2)**2 - 1)
>>> cot(1).diff(x)
0
>>> cot(pi/12)
sqrt(3) + 2
See Also
========
sin, csc, cos, sec, tan
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Cot
"""
def period(self, symbol=None):
return self._period(pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return S.NegativeOne - self**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return acot
@classmethod
def eval(cls, arg):
from sympy.calculus.util import AccumBounds
if arg.is_Number:
if arg is S.NaN:
return S.NaN
if arg is S.Zero:
return S.ComplexInfinity
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
return -tan(arg + S.Pi/2)
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return -S.ImaginaryUnit * coth(i_coeff)
pi_coeff = _pi_coeff(arg, 2)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.ComplexInfinity
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
if pi_coeff.is_Rational:
if pi_coeff.q > 2 and not pi_coeff.q % 2:
narg = pi_coeff*S.Pi*2
cresult, sresult = cos(narg), cos(narg - S.Pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
return (1 + cresult)/sresult
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
q = pi_coeff.q
p = pi_coeff.p % q
if q in table2:
nvala, nvalb = cls(p*S.Pi/table2[q][0]), cls(p*S.Pi/table2[q][1])
if None == nvala or None == nvalb:
return None
return (1 + nvala*nvalb)/(nvalb - nvala)
narg = (((pi_coeff + S.Half) % 1) - S.Half)*S.Pi
# see cos() to specify which expressions should be
# expanded automatically in terms of radicals
cresult, sresult = cos(narg), cos(narg - S.Pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if sresult == 0:
return S.ComplexInfinity
return cresult / sresult
if narg != arg:
return cls(narg)
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
cotm = cot(m)
if cotm is S.ComplexInfinity:
return cot(x)
else: # cotm == 0
return -tan(x)
if isinstance(arg, acot):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1 / x
if isinstance(arg, atan2):
y, x = arg.args
return x/y
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x**2) / x
if isinstance(arg, acos):
x = arg.args[0]
return x / sqrt(1 - x**2)
if isinstance(arg, acsc):
x = arg.args[0]
return sqrt(1 - 1 / x**2) * x
if isinstance(arg, asec):
x = arg.args[0]
return 1 / (sqrt(1 - 1 / x**2) * x)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return (-1)**((n + 1)//2) * 2**(n + 1) * B/F * x**n
def _eval_nseries(self, x, n, logx):
i = self.args[0].limit(x, 0)/S.Pi
if i and i.is_Integer:
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
return self.rewrite(tan)._eval_nseries(x, n=n, logx=logx)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
if im:
denom = cos(2*re) - cosh(2*im)
return (-sin(2*re)/denom, -sinh(2*im)/denom)
else:
return (self.func(re), S.Zero)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):
arg = arg.func(arg.args[0]).rewrite(exp)
neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
return I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return -I*(x**-I + x**I)/(x**-I - x**I)
def _eval_rewrite_as_sin(self, x, **kwargs):
return sin(2*x)/(2*(sin(x)**2))
def _eval_rewrite_as_cos(self, x, **kwargs):
return cos(x) / cos(x - S.Pi / 2, evaluate=False)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return cos(arg)/sin(arg)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return 1/tan(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
cos_in_sec_form = cos(arg)._eval_rewrite_as_sec(arg)
sin_in_sec_form = sin(arg)._eval_rewrite_as_sec(arg)
return cos_in_sec_form / sin_in_sec_form
def _eval_rewrite_as_csc(self, arg, **kwargs):
cos_in_csc_form = cos(arg)._eval_rewrite_as_csc(arg)
sin_in_csc_form = sin(arg)._eval_rewrite_as_csc(arg)
return cos_in_csc_form / sin_in_csc_form
def _eval_rewrite_as_pow(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(pow)
if y.has(cos):
return None
return y
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(sqrt)
if y.has(cos):
return None
return y
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return 1/arg
else:
return self.func(arg)
def _eval_is_real(self):
return self.args[0].is_real
def _eval_expand_trig(self, **hints):
from sympy import im, re
arg = self.args[0]
x = None
if arg.is_Add:
from sympy import symmetric_poly
n = len(arg.args)
CX = []
for x in arg.args:
cx = cot(x, evaluate=False)._eval_expand_trig()
CX.append(cx)
Yg = numbered_symbols('Y')
Y = [ next(Yg) for i in range(n) ]
p = [0, 0]
for i in range(n, -1, -1):
p[(n - i) % 2] += symmetric_poly(i, Y)*(-1)**(((n - i) % 4)//2)
return (p[0]/p[1]).subs(list(zip(Y, CX)))
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = Symbol('dummy', real=True)
P = ((z + I)**coeff).expand()
return (re(P)/im(P)).subs([(z, cot(terms))])
return cot(arg)
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_imaginary:
return True
def _eval_subs(self, old, new):
if self == old:
return new
arg = self.args[0]
argnew = arg.subs(old, new)
if arg != argnew and (argnew/S.Pi).is_integer:
return S.ComplexInfinity
return cot(argnew)
class ReciprocalTrigonometricFunction(TrigonometricFunction):
"""Base class for reciprocal functions of trigonometric functions. """
_reciprocal_of = None # mandatory, to be defined in subclass
# _is_even and _is_odd are used for correct evaluation of csc(-x), sec(-x)
# TODO refactor into TrigonometricFunction common parts of
# trigonometric functions eval() like even/odd, func(x+2*k*pi), etc.
_is_even = None # optional, to be defined in subclass
_is_odd = None # optional, to be defined in subclass
@classmethod
def eval(cls, arg):
if arg.could_extract_minus_sign():
if cls._is_even:
return cls(-arg)
if cls._is_odd:
return -cls(-arg)
pi_coeff = _pi_coeff(arg)
if (pi_coeff is not None
and not (2*pi_coeff).is_integer
and pi_coeff.is_Rational):
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*S.Pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*S.Pi
if cls._is_odd:
return cls(narg)
elif cls._is_even:
return -cls(narg)
if hasattr(arg, 'inverse') and arg.inverse() == cls:
return arg.args[0]
t = cls._reciprocal_of.eval(arg)
if t is None:
return t
elif any(isinstance(i, cos) for i in (t, -t)):
return (1/t).rewrite(sec)
elif any(isinstance(i, sin) for i in (t, -t)):
return (1/t).rewrite(csc)
else:
return 1/t
def _call_reciprocal(self, method_name, *args, **kwargs):
# Calls method_name on _reciprocal_of
o = self._reciprocal_of(self.args[0])
return getattr(o, method_name)(*args, **kwargs)
def _calculate_reciprocal(self, method_name, *args, **kwargs):
# If calling method_name on _reciprocal_of returns a value != None
# then return the reciprocal of that value
t = self._call_reciprocal(method_name, *args, **kwargs)
return 1/t if t is not None else t
def _rewrite_reciprocal(self, method_name, arg):
# Special handling for rewrite functions. If reciprocal rewrite returns
# unmodified expression, then return None
t = self._call_reciprocal(method_name, arg)
if t is not None and t != self._reciprocal_of(arg):
return 1/t
def _period(self, symbol):
f = self.args[0]
return self._reciprocal_of(f).period(symbol)
def fdiff(self, argindex=1):
return -self._calculate_reciprocal("fdiff", argindex)/self**2
def _eval_rewrite_as_exp(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_Pow", arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_sin", arg)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_cos", arg)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_tan", arg)
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_pow", arg)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_sqrt", arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
return (1/self._reciprocal_of(self.args[0])).as_real_imag(deep,
**hints)
def _eval_expand_trig(self, **hints):
return self._calculate_reciprocal("_eval_expand_trig", **hints)
def _eval_is_real(self):
return self._reciprocal_of(self.args[0])._eval_is_real()
def _eval_as_leading_term(self, x):
return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x)
def _eval_is_finite(self):
return (1/self._reciprocal_of(self.args[0])).is_finite
def _eval_nseries(self, x, n, logx):
return (1/self._reciprocal_of(self.args[0]))._eval_nseries(x, n, logx)
class sec(ReciprocalTrigonometricFunction):
"""
The secant function.
Returns the secant of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import sec
>>> from sympy.abc import x
>>> sec(x**2).diff(x)
2*x*tan(x**2)*sec(x**2)
>>> sec(1).diff(x)
0
See Also
========
sin, csc, cos, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Sec
"""
_reciprocal_of = cos
_is_even = True
def period(self, symbol=None):
return self._period(symbol)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half_sq = cot(arg/2)**2
return (cot_half_sq + 1)/(cot_half_sq - 1)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return (1/cos(arg))
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)/(cos(arg)*sin(arg))
def _eval_rewrite_as_sin(self, arg, **kwargs):
return (1 / cos(arg)._eval_rewrite_as_sin(arg))
def _eval_rewrite_as_tan(self, arg, **kwargs):
return (1 / cos(arg)._eval_rewrite_as_tan(arg))
def _eval_rewrite_as_csc(self, arg, **kwargs):
return csc(pi / 2 - arg, evaluate=False)
def fdiff(self, argindex=1):
if argindex == 1:
return tan(self.args[0])*sec(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
# Reference Formula:
# http://functions.wolfram.com/ElementaryFunctions/Sec/06/01/02/01/
from sympy.functions.combinatorial.numbers import euler
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
k = n//2
return (-1)**k*euler(2*k)/factorial(2*k)*x**(2*k)
class csc(ReciprocalTrigonometricFunction):
"""
The cosecant function.
Returns the cosecant of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import csc
>>> from sympy.abc import x
>>> csc(x**2).diff(x)
-2*x*cot(x**2)*csc(x**2)
>>> csc(1).diff(x)
0
See Also
========
sin, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Csc
"""
_reciprocal_of = sin
_is_odd = True
def period(self, symbol=None):
return self._period(symbol)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return (1/sin(arg))
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return cos(arg)/(sin(arg)*cos(arg))
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(arg/2)
return (1 + cot_half**2)/(2*cot_half)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return (1 / sin(arg)._eval_rewrite_as_cos(arg))
def _eval_rewrite_as_sec(self, arg, **kwargs):
return sec(pi / 2 - arg, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return (1 / sin(arg)._eval_rewrite_as_tan(arg))
def fdiff(self, argindex=1):
if argindex == 1:
return -cot(self.args[0])*csc(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = n//2 + 1
return ((-1)**(k - 1)*2*(2**(2*k - 1) - 1)*
bernoulli(2*k)*x**(2*k - 1)/factorial(2*k))
class sinc(Function):
r"""Represents unnormalized sinc function
Examples
========
>>> from sympy import sinc, oo, jn, Product, Symbol
>>> from sympy.abc import x
>>> sinc(x)
sinc(x)
* Automated Evaluation
>>> sinc(0)
1
>>> sinc(oo)
0
* Differentiation
>>> sinc(x).diff()
(x*cos(x) - sin(x))/x**2
* Series Expansion
>>> sinc(x).series()
1 - x**2/6 + x**4/120 + O(x**6)
* As zero'th order spherical Bessel Function
>>> sinc(x).rewrite(jn)
jn(0, x)
References
==========
.. [1] https://en.wikipedia.org/wiki/Sinc_function
"""
def fdiff(self, argindex=1):
x = self.args[0]
if argindex == 1:
return (x*cos(x) - sin(x)) / x**2
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.One
if arg.is_Number:
if arg in [S.Infinity, -S.Infinity]:
return S.Zero
elif arg is S.NaN:
return S.NaN
if arg is S.ComplexInfinity:
return S.NaN
if arg.could_extract_minus_sign():
return cls(-arg)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
if fuzzy_not(arg.is_zero):
return S.Zero
elif (2*pi_coeff).is_integer:
return S.NegativeOne**(pi_coeff - S.Half) / arg
def _eval_nseries(self, x, n, logx):
x = self.args[0]
return (sin(x)/x)._eval_nseries(x, n, logx)
def _eval_rewrite_as_jn(self, arg, **kwargs):
from sympy.functions.special.bessel import jn
return jn(0, arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return Piecewise((sin(arg)/arg, Ne(arg, 0)), (1, True))
###############################################################################
########################### TRIGONOMETRIC INVERSES ############################
###############################################################################
class InverseTrigonometricFunction(Function):
"""Base class for inverse trigonometric functions."""
pass
class asin(InverseTrigonometricFunction):
"""
The inverse sine function.
Returns the arcsine of x in radians.
Notes
=====
asin(x) will evaluate automatically in the cases oo, -oo, 0, 1,
-1 and for some instances when the result is a rational multiple
of pi (see the eval class method).
Examples
========
>>> from sympy import asin, oo, pi
>>> asin(1)
pi/2
>>> asin(-1)
-pi/2
See Also
========
sin, csc, cos, sec, tan, cot
acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSin
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self._eval_is_real() and self.args[0].is_positive
def _eval_is_negative(self):
return self._eval_is_real() and self.args[0].is_negative
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.NegativeInfinity * S.ImaginaryUnit
elif arg is S.NegativeInfinity:
return S.Infinity * S.ImaginaryUnit
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Pi / 2
elif arg is S.NegativeOne:
return -S.Pi / 2
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
cst_table = {
sqrt(3)/2: 3,
-sqrt(3)/2: -3,
sqrt(2)/2: 4,
-sqrt(2)/2: -4,
1/sqrt(2): 4,
-1/sqrt(2): -4,
sqrt((5 - sqrt(5))/8): 5,
-sqrt((5 - sqrt(5))/8): -5,
S.Half: 6,
-S.Half: -6,
sqrt(2 - sqrt(2))/2: 8,
-sqrt(2 - sqrt(2))/2: -8,
(sqrt(5) - 1)/4: 10,
(1 - sqrt(5))/4: -10,
(sqrt(3) - 1)/sqrt(2**3): 12,
(1 - sqrt(3))/sqrt(2**3): -12,
(sqrt(5) + 1)/4: S(10)/3,
-(sqrt(5) + 1)/4: -S(10)/3
}
if arg in cst_table:
return S.Pi / cst_table[arg]
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * asinh(i_coeff)
if isinstance(arg, sin):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to (-pi,pi]
ang = pi - ang
# restrict to [-pi/2,pi/2]
if ang > pi/2:
ang = pi - ang
if ang < -pi/2:
ang = -pi - ang
return ang
if isinstance(arg, cos): # acos(x) + asin(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - acos(arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n - 2)**2/(n*(n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return R / F * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_rewrite_as_acos(self, x, **kwargs):
return S.Pi/2 - acos(x)
def _eval_rewrite_as_atan(self, x, **kwargs):
return 2*atan(x/(1 + sqrt(1 - x**2)))
def _eval_rewrite_as_log(self, x, **kwargs):
return -S.ImaginaryUnit*log(S.ImaginaryUnit*x + sqrt(1 - x**2))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return 2*acot((1 + sqrt(1 - arg**2))/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return S.Pi/2 - asec(1/arg)
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return acsc(1/arg)
def _eval_is_real(self):
x = self.args[0]
return x.is_real and (1 - abs(x)).is_nonnegative
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sin
class acos(InverseTrigonometricFunction):
"""
The inverse cosine function.
Returns the arc cosine of x (measured in radians).
Notes
=====
``acos(x)`` will evaluate automatically in the cases
``oo``, ``-oo``, ``0``, ``1``, ``-1``.
``acos(zoo)`` evaluates to ``zoo``
(see note in :py:class`sympy.functions.elementary.trigonometric.asec`)
Examples
========
>>> from sympy import acos, oo, pi
>>> acos(1)
0
>>> acos(0)
pi/2
>>> acos(oo)
oo*I
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCos
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1/sqrt(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity * S.ImaginaryUnit
elif arg is S.NegativeInfinity:
return S.NegativeInfinity * S.ImaginaryUnit
elif arg is S.Zero:
return S.Pi / 2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.is_number:
cst_table = {
S.Half: S.Pi/3,
-S.Half: 2*S.Pi/3,
sqrt(2)/2: S.Pi/4,
-sqrt(2)/2: 3*S.Pi/4,
1/sqrt(2): S.Pi/4,
-1/sqrt(2): 3*S.Pi/4,
sqrt(3)/2: S.Pi/6,
-sqrt(3)/2: 5*S.Pi/6,
}
if arg in cst_table:
return cst_table[arg]
if isinstance(arg, cos):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to [0,pi]
ang = 2*pi - ang
return ang
if isinstance(arg, sin): # acos(x) + asin(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - asin(arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi / 2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n - 2)**2/(n*(n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return -R / F * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
x = self.args[0]
return x.is_real and (1 - abs(x)).is_nonnegative
def _eval_is_nonnegative(self):
return self._eval_is_real()
def _eval_nseries(self, x, n, logx):
return self._eval_rewrite_as_log(self.args[0])._eval_nseries(x, n, logx)
def _eval_rewrite_as_log(self, x, **kwargs):
return S.Pi/2 + S.ImaginaryUnit * \
log(S.ImaginaryUnit * x + sqrt(1 - x**2))
def _eval_rewrite_as_asin(self, x, **kwargs):
return S.Pi/2 - asin(x)
def _eval_rewrite_as_atan(self, x, **kwargs):
return atan(sqrt(1 - x**2)/x) + (S.Pi/2)*(1 - x*sqrt(1/x**2))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cos
def _eval_rewrite_as_acot(self, arg, **kwargs):
return S.Pi/2 - 2*acot((1 + sqrt(1 - arg**2))/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return asec(1/arg)
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return S.Pi/2 - acsc(1/arg)
def _eval_conjugate(self):
z = self.args[0]
r = self.func(self.args[0].conjugate())
if z.is_real is False:
return r
elif z.is_real and (z + 1).is_nonnegative and (z - 1).is_nonpositive:
return r
class atan(InverseTrigonometricFunction):
"""
The inverse tangent function.
Returns the arc tangent of x (measured in radians).
Notes
=====
atan(x) will evaluate automatically in the cases
oo, -oo, 0, 1, -1.
Examples
========
>>> from sympy import atan, oo, pi
>>> atan(0)
0
>>> atan(1)
pi/4
>>> atan(oo)
pi/2
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 + self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self.args[0].is_positive
def _eval_is_nonnegative(self):
return self.args[0].is_nonnegative
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Pi / 2
elif arg is S.NegativeInfinity:
return -S.Pi / 2
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Pi / 4
elif arg is S.NegativeOne:
return -S.Pi / 4
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return AccumBounds(-S.Pi/2, S.Pi/2)
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
cst_table = {
sqrt(3)/3: 6,
-sqrt(3)/3: -6,
1/sqrt(3): 6,
-1/sqrt(3): -6,
sqrt(3): 3,
-sqrt(3): -3,
(1 + sqrt(2)): S(8)/3,
-(1 + sqrt(2)): S(8)/3,
(sqrt(2) - 1): 8,
(1 - sqrt(2)): -8,
sqrt((5 + 2*sqrt(5))): S(5)/2,
-sqrt((5 + 2*sqrt(5))): -S(5)/2,
(2 - sqrt(3)): 12,
-(2 - sqrt(3)): -12
}
if arg in cst_table:
return S.Pi / cst_table[arg]
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * atanh(i_coeff)
if isinstance(arg, tan):
ang = arg.args[0]
if ang.is_comparable:
ang %= pi # restrict to [0,pi)
if ang > pi/2: # restrict to [-pi/2,pi/2]
ang -= pi
return ang
if isinstance(arg, cot): # atan(x) + acot(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
ang = pi/2 - acot(arg)
if ang > pi/2: # restrict to [-pi/2,pi/2]
ang -= pi
return ang
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return (-1)**((n - 1)//2) * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
return self.args[0].is_real
def _eval_rewrite_as_log(self, x, **kwargs):
return S.ImaginaryUnit/2 * (log(S(1) - S.ImaginaryUnit * x)
- log(S(1) + S.ImaginaryUnit * x))
def _eval_aseries(self, n, args0, x, logx):
if args0[0] == S.Infinity:
return (S.Pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
elif args0[0] == S.NegativeInfinity:
return (-S.Pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
else:
return super(atan, self)._eval_aseries(n, args0, x, logx)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return tan
def _eval_rewrite_as_asin(self, arg, **kwargs):
return sqrt(arg**2)/arg*(S.Pi/2 - asin(1/sqrt(1 + arg**2)))
def _eval_rewrite_as_acos(self, arg, **kwargs):
return sqrt(arg**2)/arg*acos(1/sqrt(1 + arg**2))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return acot(1/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return sqrt(arg**2)/arg*asec(sqrt(1 + arg**2))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return sqrt(arg**2)/arg*(S.Pi/2 - acsc(sqrt(1 + arg**2)))
class acot(InverseTrigonometricFunction):
"""
The inverse cotangent function.
Returns the arc cotangent of x (measured in radians).
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCot
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1 / (1 + self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self.args[0].is_nonnegative
def _eval_is_negative(self):
return self.args[0].is_negative
def _eval_is_real(self):
return self.args[0].is_real
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.Zero:
return S.Pi/ 2
elif arg is S.One:
return S.Pi / 4
elif arg is S.NegativeOne:
return -S.Pi / 4
if arg is S.ComplexInfinity:
return S.Zero
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
cst_table = {
sqrt(3)/3: 3,
-sqrt(3)/3: -3,
1/sqrt(3): 3,
-1/sqrt(3): -3,
sqrt(3): 6,
-sqrt(3): -6,
(1 + sqrt(2)): 8,
-(1 + sqrt(2)): -8,
(1 - sqrt(2)): -S(8)/3,
(sqrt(2) - 1): S(8)/3,
sqrt(5 + 2*sqrt(5)): 10,
-sqrt(5 + 2*sqrt(5)): -10,
(2 + sqrt(3)): 12,
-(2 + sqrt(3)): -12,
(2 - sqrt(3)): S(12)/5,
-(2 - sqrt(3)): -S(12)/5,
}
if arg in cst_table:
return S.Pi / cst_table[arg]
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return -S.ImaginaryUnit * acoth(i_coeff)
if isinstance(arg, cot):
ang = arg.args[0]
if ang.is_comparable:
ang %= pi # restrict to [0,pi)
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi;
return ang
if isinstance(arg, tan): # atan(x) + acot(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
ang = pi/2 - atan(arg)
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi
return ang
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi / 2 # FIX THIS
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return (-1)**((n + 1)//2) * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_aseries(self, n, args0, x, logx):
if args0[0] == S.Infinity:
return (S.Pi/2 - acot(1/self.args[0]))._eval_nseries(x, n, logx)
elif args0[0] == S.NegativeInfinity:
return (3*S.Pi/2 - acot(1/self.args[0]))._eval_nseries(x, n, logx)
else:
return super(atan, self)._eval_aseries(n, args0, x, logx)
def _eval_rewrite_as_log(self, x, **kwargs):
return S.ImaginaryUnit/2 * (log(1 - S.ImaginaryUnit/x)
- log(1 + S.ImaginaryUnit/x))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cot
def _eval_rewrite_as_asin(self, arg, **kwargs):
return (arg*sqrt(1/arg**2)*
(S.Pi/2 - asin(sqrt(-arg**2)/sqrt(-arg**2 - 1))))
def _eval_rewrite_as_acos(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*acos(sqrt(-arg**2)/sqrt(-arg**2 - 1))
def _eval_rewrite_as_atan(self, arg, **kwargs):
return atan(1/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*asec(sqrt((1 + arg**2)/arg**2))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*(S.Pi/2 - acsc(sqrt((1 + arg**2)/arg**2)))
class asec(InverseTrigonometricFunction):
r"""
The inverse secant function.
Returns the arc secant of x (measured in radians).
Notes
=====
``asec(x)`` will evaluate automatically in the cases
``oo``, ``-oo``, ``0``, ``1``, ``-1``.
``asec(x)`` has branch cut in the interval [-1, 1]. For complex arguments,
it can be defined [4]_ as
.. math::
sec^{-1}(z) = -i*(log(\sqrt{1 - z^2} + 1) / z)
At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For
negative branch cut, the limit
.. math::
\lim_{z \to 0}-i*(log(-\sqrt{1 - z^2} + 1) / z)
simplifies to :math:`-i*log(z/2 + O(z^3))` which ultimately evaluates to
``zoo``.
As ``asex(x)`` = ``asec(1/x)``, a similar argument can be given for
``acos(x)``.
Examples
========
>>> from sympy import asec, oo, pi
>>> asec(1)
0
>>> asec(-1)
pi
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec
.. [4] http://reference.wolfram.com/language/ref/ArcSec.html
"""
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.ComplexInfinity
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return S.Pi/2
if isinstance(arg, sec):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to [0,pi]
ang = 2*pi - ang
return ang
if isinstance(arg, csc): # asec(x) + acsc(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - acsc(arg)
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sec
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if Order(1,x).contains(arg):
return log(arg)
else:
return self.func(arg)
def _eval_is_real(self):
x = self.args[0]
if x.is_real is False:
return False
return fuzzy_or(((x - 1).is_nonnegative, (-x - 1).is_nonnegative))
def _eval_rewrite_as_log(self, arg, **kwargs):
return S.Pi/2 + S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))
def _eval_rewrite_as_asin(self, arg, **kwargs):
return S.Pi/2 - asin(1/arg)
def _eval_rewrite_as_acos(self, arg, **kwargs):
return acos(1/arg)
def _eval_rewrite_as_atan(self, arg, **kwargs):
return sqrt(arg**2)/arg*(-S.Pi/2 + 2*atan(arg + sqrt(arg**2 - 1)))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return sqrt(arg**2)/arg*(-S.Pi/2 + 2*acot(arg - sqrt(arg**2 - 1)))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return S.Pi/2 - acsc(arg)
class acsc(InverseTrigonometricFunction):
"""
The inverse cosecant function.
Returns the arc cosecant of x (measured in radians).
Notes
=====
acsc(x) will evaluate automatically in the cases
oo, -oo, 0, 1, -1.
Examples
========
>>> from sympy import acsc, oo, pi
>>> acsc(1)
pi/2
>>> acsc(-1)
-pi/2
See Also
========
sin, csc, cos, sec, tan, cot
asin, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsc
"""
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.One:
return S.Pi/2
elif arg is S.NegativeOne:
return -S.Pi/2
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return S.Zero
if isinstance(arg, csc):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to (-pi,pi]
ang = pi - ang
# restrict to [-pi/2,pi/2]
if ang > pi/2:
ang = pi - ang
if ang < -pi/2:
ang = -pi - ang
return ang
if isinstance(arg, sec): # asec(x) + acsc(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - asec(arg)
def fdiff(self, argindex=1):
if argindex == 1:
return -1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return csc
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if Order(1,x).contains(arg):
return log(arg)
else:
return self.func(arg)
def _eval_rewrite_as_log(self, arg, **kwargs):
return -S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))
def _eval_rewrite_as_asin(self, arg, **kwargs):
return asin(1/arg)
def _eval_rewrite_as_acos(self, arg, **kwargs):
return S.Pi/2 - acos(1/arg)
def _eval_rewrite_as_atan(self, arg, **kwargs):
return sqrt(arg**2)/arg*(S.Pi/2 - atan(sqrt(arg**2 - 1)))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return sqrt(arg**2)/arg*(S.Pi/2 - acot(1/sqrt(arg**2 - 1)))
def _eval_rewrite_as_asec(self, arg, **kwargs):
return S.Pi/2 - asec(arg)
class atan2(InverseTrigonometricFunction):
r"""
The function ``atan2(y, x)`` computes `\operatorname{atan}(y/x)` taking
two arguments `y` and `x`. Signs of both `y` and `x` are considered to
determine the appropriate quadrant of `\operatorname{atan}(y/x)`.
The range is `(-\pi, \pi]`. The complete definition reads as follows:
.. math::
\operatorname{atan2}(y, x) =
\begin{cases}
\arctan\left(\frac y x\right) & \qquad x > 0 \\
\arctan\left(\frac y x\right) + \pi& \qquad y \ge 0 , x < 0 \\
\arctan\left(\frac y x\right) - \pi& \qquad y < 0 , x < 0 \\
+\frac{\pi}{2} & \qquad y > 0 , x = 0 \\
-\frac{\pi}{2} & \qquad y < 0 , x = 0 \\
\text{undefined} & \qquad y = 0, x = 0
\end{cases}
Attention: Note the role reversal of both arguments. The `y`-coordinate
is the first argument and the `x`-coordinate the second.
Examples
========
Going counter-clock wise around the origin we find the
following angles:
>>> from sympy import atan2
>>> atan2(0, 1)
0
>>> atan2(1, 1)
pi/4
>>> atan2(1, 0)
pi/2
>>> atan2(1, -1)
3*pi/4
>>> atan2(0, -1)
pi
>>> atan2(-1, -1)
-3*pi/4
>>> atan2(-1, 0)
-pi/2
>>> atan2(-1, 1)
-pi/4
which are all correct. Compare this to the results of the ordinary
`\operatorname{atan}` function for the point `(x, y) = (-1, 1)`
>>> from sympy import atan, S
>>> atan(S(1) / -1)
-pi/4
>>> atan2(1, -1)
3*pi/4
where only the `\operatorname{atan2}` function reurns what we expect.
We can differentiate the function with respect to both arguments:
>>> from sympy import diff
>>> from sympy.abc import x, y
>>> diff(atan2(y, x), x)
-y/(x**2 + y**2)
>>> diff(atan2(y, x), y)
x/(x**2 + y**2)
We can express the `\operatorname{atan2}` function in terms of
complex logarithms:
>>> from sympy import log
>>> atan2(y, x).rewrite(log)
-I*log((x + I*y)/sqrt(x**2 + y**2))
and in terms of `\operatorname(atan)`:
>>> from sympy import atan
>>> atan2(y, x).rewrite(atan)
Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, x < 0), (0, x > 0), (nan, True))
but note that this form is undefined on the negative real axis.
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://en.wikipedia.org/wiki/Atan2
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan2
"""
@classmethod
def eval(cls, y, x):
from sympy import Heaviside, im, re
if x is S.NegativeInfinity:
if y.is_zero:
# Special case y = 0 because we define Heaviside(0) = 1/2
return S.Pi
return 2*S.Pi*(Heaviside(re(y))) - S.Pi
elif x is S.Infinity:
return S.Zero
elif x.is_imaginary and y.is_imaginary and x.is_number and y.is_number:
x = im(x)
y = im(y)
if x.is_real and y.is_real:
if x.is_positive:
return atan(y / x)
elif x.is_negative:
if y.is_negative:
return atan(y / x) - S.Pi
elif y.is_nonnegative:
return atan(y / x) + S.Pi
elif x.is_zero:
if y.is_positive:
return S.Pi/2
elif y.is_negative:
return -S.Pi/2
elif y.is_zero:
return S.NaN
if y.is_zero and x.is_real and fuzzy_not(x.is_zero):
return S.Pi * (S.One - Heaviside(x))
if x.is_number and y.is_number:
return -S.ImaginaryUnit*log(
(x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2))
def _eval_rewrite_as_log(self, y, x, **kwargs):
return -S.ImaginaryUnit*log((x + S.ImaginaryUnit*y) / sqrt(x**2 + y**2))
def _eval_rewrite_as_atan(self, y, x, **kwargs):
return Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, x < 0), (0, x > 0), (S.NaN, True))
def _eval_rewrite_as_arg(self, y, x, **kwargs):
from sympy import arg
if x.is_real and y.is_real:
return arg(x + y*S.ImaginaryUnit)
I = S.ImaginaryUnit
n = x + I*y
d = x**2 + y**2
return arg(n/sqrt(d)) - I*log(abs(n)/sqrt(abs(d)))
def _eval_is_real(self):
return self.args[0].is_real and self.args[1].is_real
def _eval_conjugate(self):
return self.func(self.args[0].conjugate(), self.args[1].conjugate())
def fdiff(self, argindex):
y, x = self.args
if argindex == 1:
# Diff wrt y
return x/(x**2 + y**2)
elif argindex == 2:
# Diff wrt x
return -y/(x**2 + y**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_evalf(self, prec):
y, x = self.args
if x.is_real and y.is_real:
super(atan2, self)._eval_evalf(prec)
|
7ae8bfe882a06995a50eff7282b6db42ec585f464541261f5c41ef47d604f8bd | from __future__ import print_function, division
from sympy.core import Function, S, sympify
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.operations import LatticeOp, ShortCircuit
from sympy.core.function import (Application, Lambda,
ArgumentIndexError)
from sympy.core.expr import Expr
from sympy.core.mod import Mod
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.power import Pow
from sympy.core.relational import Eq, Relational
from sympy.core.singleton import Singleton
from sympy.core.symbol import Dummy
from sympy.core.rules import Transform
from sympy.core.compatibility import with_metaclass, range
from sympy.core.logic import fuzzy_and, fuzzy_or, _torf
from sympy.logic.boolalg import And, Or
def _minmax_as_Piecewise(op, *args):
# helper for Min/Max rewrite as Piecewise
from sympy.functions.elementary.piecewise import Piecewise
ec = []
for i, a in enumerate(args):
c = []
for j in range(i + 1, len(args)):
c.append(Relational(a, args[j], op))
ec.append((a, And(*c)))
return Piecewise(*ec)
class IdentityFunction(with_metaclass(Singleton, Lambda)):
"""
The identity function
Examples
========
>>> from sympy import Id, Symbol
>>> x = Symbol('x')
>>> Id(x)
x
"""
def __new__(cls):
from sympy.sets.sets import FiniteSet
x = Dummy('x')
#construct "by hand" to avoid infinite loop
obj = Expr.__new__(cls, Tuple(x), x)
obj.nargs = FiniteSet(1)
return obj
Id = S.IdentityFunction
###############################################################################
############################# ROOT and SQUARE ROOT FUNCTION ###################
###############################################################################
def sqrt(arg, evaluate=None):
"""The square root function
sqrt(x) -> Returns the principal square root of x.
The parameter evaluate determines if the expression should be evaluated.
If None, its value is taken from global_evaluate
Examples
========
>>> from sympy import sqrt, Symbol
>>> x = Symbol('x')
>>> sqrt(x)
sqrt(x)
>>> sqrt(x)**2
x
Note that sqrt(x**2) does not simplify to x.
>>> sqrt(x**2)
sqrt(x**2)
This is because the two are not equal to each other in general.
For example, consider x == -1:
>>> from sympy import Eq
>>> Eq(sqrt(x**2), x).subs(x, -1)
False
This is because sqrt computes the principal square root, so the square may
put the argument in a different branch. This identity does hold if x is
positive:
>>> y = Symbol('y', positive=True)
>>> sqrt(y**2)
y
You can force this simplification by using the powdenest() function with
the force option set to True:
>>> from sympy import powdenest
>>> sqrt(x**2)
sqrt(x**2)
>>> powdenest(sqrt(x**2), force=True)
x
To get both branches of the square root you can use the rootof function:
>>> from sympy import rootof
>>> [rootof(x**2-3,i) for i in (0,1)]
[-sqrt(3), sqrt(3)]
See Also
========
sympy.polys.rootoftools.rootof, root, real_root
References
==========
.. [1] https://en.wikipedia.org/wiki/Square_root
.. [2] https://en.wikipedia.org/wiki/Principal_value
"""
# arg = sympify(arg) is handled by Pow
return Pow(arg, S.Half, evaluate=evaluate)
def cbrt(arg, evaluate=None):
"""This function computes the principal cube root of `arg`, so
it's just a shortcut for `arg**Rational(1, 3)`.
The parameter evaluate determines if the expression should be evaluated.
If None, its value is taken from global_evaluate.
Examples
========
>>> from sympy import cbrt, Symbol
>>> x = Symbol('x')
>>> cbrt(x)
x**(1/3)
>>> cbrt(x)**3
x
Note that cbrt(x**3) does not simplify to x.
>>> cbrt(x**3)
(x**3)**(1/3)
This is because the two are not equal to each other in general.
For example, consider `x == -1`:
>>> from sympy import Eq
>>> Eq(cbrt(x**3), x).subs(x, -1)
False
This is because cbrt computes the principal cube root, this
identity does hold if `x` is positive:
>>> y = Symbol('y', positive=True)
>>> cbrt(y**3)
y
See Also
========
sympy.polys.rootoftools.rootof, root, real_root
References
==========
* https://en.wikipedia.org/wiki/Cube_root
* https://en.wikipedia.org/wiki/Principal_value
"""
return Pow(arg, Rational(1, 3), evaluate=evaluate)
def root(arg, n, k=0, evaluate=None):
"""root(x, n, k) -> Returns the k-th n-th root of x, defaulting to the
principal root (k=0).
The parameter evaluate determines if the expression should be evaluated.
If None, its value is taken from global_evaluate.
Examples
========
>>> from sympy import root, Rational
>>> from sympy.abc import x, n
>>> root(x, 2)
sqrt(x)
>>> root(x, 3)
x**(1/3)
>>> root(x, n)
x**(1/n)
>>> root(x, -Rational(2, 3))
x**(-3/2)
To get the k-th n-th root, specify k:
>>> root(-2, 3, 2)
-(-1)**(2/3)*2**(1/3)
To get all n n-th roots you can use the rootof function.
The following examples show the roots of unity for n
equal 2, 3 and 4:
>>> from sympy import rootof, I
>>> [rootof(x**2 - 1, i) for i in range(2)]
[-1, 1]
>>> [rootof(x**3 - 1,i) for i in range(3)]
[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]
>>> [rootof(x**4 - 1,i) for i in range(4)]
[-1, 1, -I, I]
SymPy, like other symbolic algebra systems, returns the
complex root of negative numbers. This is the principal
root and differs from the text-book result that one might
be expecting. For example, the cube root of -8 does not
come back as -2:
>>> root(-8, 3)
2*(-1)**(1/3)
The real_root function can be used to either make the principal
result real (or simply to return the real root directly):
>>> from sympy import real_root
>>> real_root(_)
-2
>>> real_root(-32, 5)
-2
Alternatively, the n//2-th n-th root of a negative number can be
computed with root:
>>> root(-32, 5, 5//2)
-2
See Also
========
sympy.polys.rootoftools.rootof
sympy.core.power.integer_nthroot
sqrt, real_root
References
==========
* https://en.wikipedia.org/wiki/Square_root
* https://en.wikipedia.org/wiki/Real_root
* https://en.wikipedia.org/wiki/Root_of_unity
* https://en.wikipedia.org/wiki/Principal_value
* http://mathworld.wolfram.com/CubeRoot.html
"""
n = sympify(n)
if k:
return Mul(Pow(arg, S.One/n, evaluate=evaluate), S.NegativeOne**(2*k/n), evaluate=evaluate)
return Pow(arg, 1/n, evaluate=evaluate)
def real_root(arg, n=None, evaluate=None):
"""Return the real nth-root of arg if possible. If n is omitted then
all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this
will only create a real root of a principal root -- the presence of
other factors may cause the result to not be real.
The parameter evaluate determines if the expression should be evaluated.
If None, its value is taken from global_evaluate.
Examples
========
>>> from sympy import root, real_root, Rational
>>> from sympy.abc import x, n
>>> real_root(-8, 3)
-2
>>> root(-8, 3)
2*(-1)**(1/3)
>>> real_root(_)
-2
If one creates a non-principal root and applies real_root, the
result will not be real (so use with caution):
>>> root(-8, 3, 2)
-2*(-1)**(2/3)
>>> real_root(_)
-2*(-1)**(2/3)
See Also
========
sympy.polys.rootoftools.rootof
sympy.core.power.integer_nthroot
root, sqrt
"""
from sympy.functions.elementary.complexes import Abs, im, sign
from sympy.functions.elementary.piecewise import Piecewise
if n is not None:
return Piecewise(
(root(arg, n, evaluate=evaluate), Or(Eq(n, S.One), Eq(n, S.NegativeOne))),
(Mul(sign(arg), root(Abs(arg), n, evaluate=evaluate), evaluate=evaluate),
And(Eq(im(arg), S.Zero), Eq(Mod(n, 2), S.One))),
(root(arg, n, evaluate=evaluate), True))
rv = sympify(arg)
n1pow = Transform(lambda x: -(-x.base)**x.exp,
lambda x:
x.is_Pow and
x.base.is_negative and
x.exp.is_Rational and
x.exp.p == 1 and x.exp.q % 2)
return rv.xreplace(n1pow)
###############################################################################
############################# MINIMUM and MAXIMUM #############################
###############################################################################
class MinMaxBase(Expr, LatticeOp):
def __new__(cls, *args, **assumptions):
evaluate = assumptions.pop('evaluate', True)
args = (sympify(arg) for arg in args)
# first standard filter, for cls.zero and cls.identity
# also reshape Max(a, Max(b, c)) to Max(a, b, c)
if evaluate:
try:
args = frozenset(cls._new_args_filter(args))
except ShortCircuit:
return cls.zero
else:
args = frozenset(args)
if evaluate:
# remove redundant args that are easily identified
args = cls._collapse_arguments(args, **assumptions)
# find local zeros
args = cls._find_localzeros(args, **assumptions)
if not args:
return cls.identity
if len(args) == 1:
return list(args).pop()
# base creation
_args = frozenset(args)
obj = Expr.__new__(cls, _args, **assumptions)
obj._argset = _args
return obj
@classmethod
def _collapse_arguments(cls, args, **assumptions):
"""Remove redundant args.
Examples
========
>>> from sympy import Min, Max
>>> from sympy.abc import a, b, c, d, e
Any arg in parent that appears in any
parent-like function in any of the flat args
of parent can be removed from that sub-arg:
>>> Min(a, Max(b, Min(a, c, d)))
Min(a, Max(b, Min(c, d)))
If the arg of parent appears in an opposite-than parent
function in any of the flat args of parent that function
can be replaced with the arg:
>>> Min(a, Max(b, Min(c, d, Max(a, e))))
Min(a, Max(b, Min(a, c, d)))
"""
from sympy.utilities.iterables import ordered
from sympy.simplify.simplify import walk
if not args:
return args
args = list(ordered(args))
if cls == Min:
other = Max
else:
other = Min
# find global comparable max of Max and min of Min if a new
# value is being introduced in these args at position 0 of
# the ordered args
if args[0].is_number:
sifted = mins, maxs = [], []
for i in args:
for v in walk(i, Min, Max):
if v.args[0].is_comparable:
sifted[isinstance(v, Max)].append(v)
small = Min.identity
for i in mins:
v = i.args[0]
if v.is_number and (v < small) == True:
small = v
big = Max.identity
for i in maxs:
v = i.args[0]
if v.is_number and (v > big) == True:
big = v
# at the point when this function is called from __new__,
# there may be more than one numeric arg present since
# local zeros have not been handled yet, so look through
# more than the first arg
if cls == Min:
for i in range(len(args)):
if not args[i].is_number:
break
if (args[i] < small) == True:
small = args[i]
elif cls == Max:
for i in range(len(args)):
if not args[i].is_number:
break
if (args[i] > big) == True:
big = args[i]
T = None
if cls == Min:
if small != Min.identity:
other = Max
T = small
elif big != Max.identity:
other = Min
T = big
if T is not None:
# remove numerical redundancy
for i in range(len(args)):
a = args[i]
if isinstance(a, other):
a0 = a.args[0]
if ((a0 > T) if other == Max else (a0 < T)) == True:
args[i] = cls.identity
# remove redundant symbolic args
def do(ai, a):
if not isinstance(ai, (Min, Max)):
return ai
cond = a in ai.args
if not cond:
return ai.func(*[do(i, a) for i in ai.args],
evaluate=False)
if isinstance(ai, cls):
return ai.func(*[do(i, a) for i in ai.args if i != a],
evaluate=False)
return a
for i, a in enumerate(args):
args[i + 1:] = [do(ai, a) for ai in args[i + 1:]]
# factor out common elements as for
# Min(Max(x, y), Max(x, z)) -> Max(x, Min(y, z))
# and vice versa when swapping Min/Max -- do this only for the
# easy case where all functions contain something in common;
# trying to find some optimal subset of args to modify takes
# too long
if len(args) > 1:
common = None
remove = []
sets = []
for i in range(len(args)):
a = args[i]
if not isinstance(a, other):
continue
s = set(a.args)
common = s if common is None else (common & s)
if not common:
break
sets.append(s)
remove.append(i)
if common:
sets = filter(None, [s - common for s in sets])
sets = [other(*s, evaluate=False) for s in sets]
for i in reversed(remove):
args.pop(i)
oargs = [cls(*sets)] if sets else []
oargs.extend(common)
args.append(other(*oargs, evaluate=False))
return args
@classmethod
def _new_args_filter(cls, arg_sequence):
"""
Generator filtering args.
first standard filter, for cls.zero and cls.identity.
Also reshape Max(a, Max(b, c)) to Max(a, b, c),
and check arguments for comparability
"""
for arg in arg_sequence:
# pre-filter, checking comparability of arguments
if not isinstance(arg, Expr) or arg.is_real is False or (
arg.is_number and
not arg.is_comparable):
raise ValueError("The argument '%s' is not comparable." % arg)
if arg == cls.zero:
raise ShortCircuit(arg)
elif arg == cls.identity:
continue
elif arg.func == cls:
for x in arg.args:
yield x
else:
yield arg
@classmethod
def _find_localzeros(cls, values, **options):
"""
Sequentially allocate values to localzeros.
When a value is identified as being more extreme than another member it
replaces that member; if this is never true, then the value is simply
appended to the localzeros.
"""
localzeros = set()
for v in values:
is_newzero = True
localzeros_ = list(localzeros)
for z in localzeros_:
if id(v) == id(z):
is_newzero = False
else:
con = cls._is_connected(v, z)
if con:
is_newzero = False
if con is True or con == cls:
localzeros.remove(z)
localzeros.update([v])
if is_newzero:
localzeros.update([v])
return localzeros
@classmethod
def _is_connected(cls, x, y):
"""
Check if x and y are connected somehow.
"""
from sympy.core.exprtools import factor_terms
def hit(v, t, f):
if not v.is_Relational:
return t if v else f
for i in range(2):
if x == y:
return True
r = hit(x >= y, Max, Min)
if r is not None:
return r
r = hit(y <= x, Max, Min)
if r is not None:
return r
r = hit(x <= y, Min, Max)
if r is not None:
return r
r = hit(y >= x, Min, Max)
if r is not None:
return r
# simplification can be expensive, so be conservative
# in what is attempted
x = factor_terms(x - y)
y = S.Zero
return False
def _eval_derivative(self, s):
# f(x).diff(s) -> x.diff(s) * f.fdiff(1)(s)
i = 0
l = []
for a in self.args:
i += 1
da = a.diff(s)
if da is S.Zero:
continue
try:
df = self.fdiff(i)
except ArgumentIndexError:
df = Function.fdiff(self, i)
l.append(df * da)
return Add(*l)
def _eval_rewrite_as_Abs(self, *args, **kwargs):
from sympy.functions.elementary.complexes import Abs
s = (args[0] + self.func(*args[1:]))/2
d = abs(args[0] - self.func(*args[1:]))/2
return (s + d if isinstance(self, Max) else s - d).rewrite(Abs)
def evalf(self, prec=None, **options):
return self.func(*[a.evalf(prec, **options) for a in self.args])
n = evalf
_eval_is_algebraic = lambda s: _torf(i.is_algebraic for i in s.args)
_eval_is_antihermitian = lambda s: _torf(i.is_antihermitian for i in s.args)
_eval_is_commutative = lambda s: _torf(i.is_commutative for i in s.args)
_eval_is_complex = lambda s: _torf(i.is_complex for i in s.args)
_eval_is_composite = lambda s: _torf(i.is_composite for i in s.args)
_eval_is_even = lambda s: _torf(i.is_even for i in s.args)
_eval_is_finite = lambda s: _torf(i.is_finite for i in s.args)
_eval_is_hermitian = lambda s: _torf(i.is_hermitian for i in s.args)
_eval_is_imaginary = lambda s: _torf(i.is_imaginary for i in s.args)
_eval_is_infinite = lambda s: _torf(i.is_infinite for i in s.args)
_eval_is_integer = lambda s: _torf(i.is_integer for i in s.args)
_eval_is_irrational = lambda s: _torf(i.is_irrational for i in s.args)
_eval_is_negative = lambda s: _torf(i.is_negative for i in s.args)
_eval_is_noninteger = lambda s: _torf(i.is_noninteger for i in s.args)
_eval_is_nonnegative = lambda s: _torf(i.is_nonnegative for i in s.args)
_eval_is_nonpositive = lambda s: _torf(i.is_nonpositive for i in s.args)
_eval_is_nonzero = lambda s: _torf(i.is_nonzero for i in s.args)
_eval_is_odd = lambda s: _torf(i.is_odd for i in s.args)
_eval_is_polar = lambda s: _torf(i.is_polar for i in s.args)
_eval_is_positive = lambda s: _torf(i.is_positive for i in s.args)
_eval_is_prime = lambda s: _torf(i.is_prime for i in s.args)
_eval_is_rational = lambda s: _torf(i.is_rational for i in s.args)
_eval_is_real = lambda s: _torf(i.is_real for i in s.args)
_eval_is_transcendental = lambda s: _torf(i.is_transcendental for i in s.args)
_eval_is_zero = lambda s: _torf(i.is_zero for i in s.args)
class Max(MinMaxBase, Application):
"""
Return, if possible, the maximum value of the list.
When number of arguments is equal one, then
return this argument.
When number of arguments is equal two, then
return, if possible, the value from (a, b) that is >= the other.
In common case, when the length of list greater than 2, the task
is more complicated. Return only the arguments, which are greater
than others, if it is possible to determine directional relation.
If is not possible to determine such a relation, return a partially
evaluated result.
Assumptions are used to make the decision too.
Also, only comparable arguments are permitted.
It is named ``Max`` and not ``max`` to avoid conflicts
with the built-in function ``max``.
Examples
========
>>> from sympy import Max, Symbol, oo
>>> from sympy.abc import x, y
>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)
>>> Max(x, -2) #doctest: +SKIP
Max(x, -2)
>>> Max(x, -2).subs(x, 3)
3
>>> Max(p, -2)
p
>>> Max(x, y)
Max(x, y)
>>> Max(x, y) == Max(y, x)
True
>>> Max(x, Max(y, z)) #doctest: +SKIP
Max(x, y, z)
>>> Max(n, 8, p, 7, -oo) #doctest: +SKIP
Max(8, p)
>>> Max (1, x, oo)
oo
* Algorithm
The task can be considered as searching of supremums in the
directed complete partial orders [1]_.
The source values are sequentially allocated by the isolated subsets
in which supremums are searched and result as Max arguments.
If the resulted supremum is single, then it is returned.
The isolated subsets are the sets of values which are only the comparable
with each other in the current set. E.g. natural numbers are comparable with
each other, but not comparable with the `x` symbol. Another example: the
symbol `x` with negative assumption is comparable with a natural number.
Also there are "least" elements, which are comparable with all others,
and have a zero property (maximum or minimum for all elements). E.g. `oo`.
In case of it the allocation operation is terminated and only this value is
returned.
Assumption:
- if A > B > C then A > C
- if A == B then B can be removed
References
==========
.. [1] https://en.wikipedia.org/wiki/Directed_complete_partial_order
.. [2] https://en.wikipedia.org/wiki/Lattice_%28order%29
See Also
========
Min : find minimum values
"""
zero = S.Infinity
identity = S.NegativeInfinity
def fdiff( self, argindex ):
from sympy import Heaviside
n = len(self.args)
if 0 < argindex and argindex <= n:
argindex -= 1
if n == 2:
return Heaviside(self.args[argindex] - self.args[1 - argindex])
newargs = tuple([self.args[i] for i in range(n) if i != argindex])
return Heaviside(self.args[argindex] - Max(*newargs))
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_Heaviside(self, *args, **kwargs):
from sympy import Heaviside
return Add(*[j*Mul(*[Heaviside(j - i) for i in args if i!=j]) \
for j in args])
def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
return _minmax_as_Piecewise('>=', *args)
def _eval_is_positive(self):
return fuzzy_or(a.is_positive for a in self.args)
def _eval_is_nonnegative(self):
return fuzzy_or(a.is_nonnegative for a in self.args)
def _eval_is_negative(self):
return fuzzy_and(a.is_negative for a in self.args)
class Min(MinMaxBase, Application):
"""
Return, if possible, the minimum value of the list.
It is named ``Min`` and not ``min`` to avoid conflicts
with the built-in function ``min``.
Examples
========
>>> from sympy import Min, Symbol, oo
>>> from sympy.abc import x, y
>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)
>>> Min(x, -2) #doctest: +SKIP
Min(x, -2)
>>> Min(x, -2).subs(x, 3)
-2
>>> Min(p, -3)
-3
>>> Min(x, y) #doctest: +SKIP
Min(x, y)
>>> Min(n, 8, p, -7, p, oo) #doctest: +SKIP
Min(n, -7)
See Also
========
Max : find maximum values
"""
zero = S.NegativeInfinity
identity = S.Infinity
def fdiff( self, argindex ):
from sympy import Heaviside
n = len(self.args)
if 0 < argindex and argindex <= n:
argindex -= 1
if n == 2:
return Heaviside( self.args[1-argindex] - self.args[argindex] )
newargs = tuple([ self.args[i] for i in range(n) if i != argindex])
return Heaviside( Min(*newargs) - self.args[argindex] )
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_Heaviside(self, *args, **kwargs):
from sympy import Heaviside
return Add(*[j*Mul(*[Heaviside(i-j) for i in args if i!=j]) \
for j in args])
def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
return _minmax_as_Piecewise('<=', *args)
def _eval_is_positive(self):
return fuzzy_and(a.is_positive for a in self.args)
def _eval_is_nonnegative(self):
return fuzzy_and(a.is_nonnegative for a in self.args)
def _eval_is_negative(self):
return fuzzy_or(a.is_negative for a in self.args)
|
67851ad069e8f0fd99615fe194a06077f853004eb091b0659c5dfc7d859e9be1 | from __future__ import print_function, division
from sympy.core import Basic, S, Function, diff, Tuple, Dummy, Symbol
from sympy.core.basic import as_Basic
from sympy.core.compatibility import range
from sympy.core.numbers import Rational, NumberSymbol
from sympy.core.relational import (Equality, Unequality, Relational,
_canonical)
from sympy.functions.elementary.miscellaneous import Max, Min
from sympy.logic.boolalg import (And, Boolean, distribute_and_over_or,
true, false, Or, ITE, simplify_logic)
from sympy.utilities.iterables import uniq, ordered, product, sift
from sympy.utilities.misc import filldedent, func_name
Undefined = S.NaN # Piecewise()
class ExprCondPair(Tuple):
"""Represents an expression, condition pair."""
def __new__(cls, expr, cond):
expr = as_Basic(expr)
if cond == True:
return Tuple.__new__(cls, expr, true)
elif cond == False:
return Tuple.__new__(cls, expr, false)
elif isinstance(cond, Basic) and cond.has(Piecewise):
cond = piecewise_fold(cond)
if isinstance(cond, Piecewise):
cond = cond.rewrite(ITE)
if not isinstance(cond, Boolean):
raise TypeError(filldedent('''
Second argument must be a Boolean,
not `%s`''' % func_name(cond)))
return Tuple.__new__(cls, expr, cond)
@property
def expr(self):
"""
Returns the expression of this pair.
"""
return self.args[0]
@property
def cond(self):
"""
Returns the condition of this pair.
"""
return self.args[1]
@property
def is_commutative(self):
return self.expr.is_commutative
def __iter__(self):
yield self.expr
yield self.cond
def _eval_simplify(self, ratio, measure, rational, inverse):
return self.func(*[a.simplify(
ratio=ratio,
measure=measure,
rational=rational,
inverse=inverse) for a in self.args])
class Piecewise(Function):
"""
Represents a piecewise function.
Usage:
Piecewise( (expr,cond), (expr,cond), ... )
- Each argument is a 2-tuple defining an expression and condition
- The conds are evaluated in turn returning the first that is True.
If any of the evaluated conds are not determined explicitly False,
e.g. x < 1, the function is returned in symbolic form.
- If the function is evaluated at a place where all conditions are False,
nan will be returned.
- Pairs where the cond is explicitly False, will be removed.
Examples
========
>>> from sympy import Piecewise, log, ITE, piecewise_fold
>>> from sympy.abc import x, y
>>> f = x**2
>>> g = log(x)
>>> p = Piecewise((0, x < -1), (f, x <= 1), (g, True))
>>> p.subs(x,1)
1
>>> p.subs(x,5)
log(5)
Booleans can contain Piecewise elements:
>>> cond = (x < y).subs(x, Piecewise((2, x < 0), (3, True))); cond
Piecewise((2, x < 0), (3, True)) < y
The folded version of this results in a Piecewise whose
expressions are Booleans:
>>> folded_cond = piecewise_fold(cond); folded_cond
Piecewise((2 < y, x < 0), (3 < y, True))
When a Boolean containing Piecewise (like cond) or a Piecewise
with Boolean expressions (like folded_cond) is used as a condition,
it is converted to an equivalent ITE object:
>>> Piecewise((1, folded_cond))
Piecewise((1, ITE(x < 0, y > 2, y > 3)))
When a condition is an ITE, it will be converted to a simplified
Boolean expression:
>>> piecewise_fold(_)
Piecewise((1, ((x >= 0) | (y > 2)) & ((y > 3) | (x < 0))))
See Also
========
piecewise_fold, ITE
"""
nargs = None
is_Piecewise = True
def __new__(cls, *args, **options):
if len(args) == 0:
raise TypeError("At least one (expr, cond) pair expected.")
# (Try to) sympify args first
newargs = []
for ec in args:
# ec could be a ExprCondPair or a tuple
pair = ExprCondPair(*getattr(ec, 'args', ec))
cond = pair.cond
if cond is false:
continue
newargs.append(pair)
if cond is true:
break
if options.pop('evaluate', True):
r = cls.eval(*newargs)
else:
r = None
if r is None:
return Basic.__new__(cls, *newargs, **options)
else:
return r
@classmethod
def eval(cls, *_args):
"""Either return a modified version of the args or, if no
modifications were made, return None.
Modifications that are made here:
1) relationals are made canonical
2) any False conditions are dropped
3) any repeat of a previous condition is ignored
3) any args past one with a true condition are dropped
If there are no args left, nan will be returned.
If there is a single arg with a True condition, its
corresponding expression will be returned.
"""
if not _args:
return Undefined
if len(_args) == 1 and _args[0][-1] == True:
return _args[0][0]
newargs = [] # the unevaluated conditions
current_cond = set() # the conditions up to a given e, c pair
# make conditions canonical
args = []
for e, c in _args:
if not c.is_Atom and not isinstance(c, Relational):
free = c.free_symbols
if len(free) == 1:
funcs = [i for i in c.atoms(Function)
if not isinstance(i, Boolean)]
if len(funcs) == 1 and len(
c.xreplace({list(funcs)[0]: Dummy()}
).free_symbols) == 1:
# we can treat function like a symbol
free = funcs
_c = c
x = free.pop()
try:
c = c.as_set().as_relational(x)
except NotImplementedError:
pass
else:
reps = {}
for i in c.atoms(Relational):
ic = i.canonical
if ic.rhs in (S.Infinity, S.NegativeInfinity):
if not _c.has(ic.rhs):
# don't accept introduction of
# new Relationals with +/-oo
reps[i] = S.true
elif ('=' not in ic.rel_op and
c.xreplace({x: i.rhs}) !=
_c.xreplace({x: i.rhs})):
reps[i] = Relational(
i.lhs, i.rhs, i.rel_op + '=')
c = c.xreplace(reps)
args.append((e, _canonical(c)))
for expr, cond in args:
# Check here if expr is a Piecewise and collapse if one of
# the conds in expr matches cond. This allows the collapsing
# of Piecewise((Piecewise((x,x<0)),x<0)) to Piecewise((x,x<0)).
# This is important when using piecewise_fold to simplify
# multiple Piecewise instances having the same conds.
# Eventually, this code should be able to collapse Piecewise's
# having different intervals, but this will probably require
# using the new assumptions.
if isinstance(expr, Piecewise):
unmatching = []
for i, (e, c) in enumerate(expr.args):
if c in current_cond:
# this would already have triggered
continue
if c == cond:
if c != True:
# nothing past this condition will ever
# trigger and only those args before this
# that didn't match a previous condition
# could possibly trigger
if unmatching:
expr = Piecewise(*(
unmatching + [(e, c)]))
else:
expr = e
break
else:
unmatching.append((e, c))
# check for condition repeats
got = False
# -- if an And contains a condition that was
# already encountered, then the And will be
# False: if the previous condition was False
# then the And will be False and if the previous
# condition is True then then we wouldn't get to
# this point. In either case, we can skip this condition.
for i in ([cond] +
(list(cond.args) if isinstance(cond, And) else
[])):
if i in current_cond:
got = True
break
if got:
continue
# -- if not(c) is already in current_cond then c is
# a redundant condition in an And. This does not
# apply to Or, however: (e1, c), (e2, Or(~c, d))
# is not (e1, c), (e2, d) because if c and d are
# both False this would give no results when the
# true answer should be (e2, True)
if isinstance(cond, And):
nonredundant = []
for c in cond.args:
if (isinstance(c, Relational) and
c.negated.canonical in current_cond):
continue
nonredundant.append(c)
cond = cond.func(*nonredundant)
elif isinstance(cond, Relational):
if cond.negated.canonical in current_cond:
cond = S.true
current_cond.add(cond)
# collect successive e,c pairs when exprs or cond match
if newargs:
if newargs[-1].expr == expr:
orcond = Or(cond, newargs[-1].cond)
if isinstance(orcond, (And, Or)):
orcond = distribute_and_over_or(orcond)
newargs[-1] = ExprCondPair(expr, orcond)
continue
elif newargs[-1].cond == cond:
orexpr = Or(expr, newargs[-1].expr)
if isinstance(orexpr, (And, Or)):
orexpr = distribute_and_over_or(orexpr)
newargs[-1] == ExprCondPair(orexpr, cond)
continue
newargs.append(ExprCondPair(expr, cond))
# some conditions may have been redundant
missing = len(newargs) != len(_args)
# some conditions may have changed
same = all(a == b for a, b in zip(newargs, _args))
# if either change happened we return the expr with the
# updated args
if not newargs:
raise ValueError(filldedent('''
There are no conditions (or none that
are not trivially false) to define an
expression.'''))
if missing or not same:
return cls(*newargs)
def doit(self, **hints):
"""
Evaluate this piecewise function.
"""
newargs = []
for e, c in self.args:
if hints.get('deep', True):
if isinstance(e, Basic):
e = e.doit(**hints)
if isinstance(c, Basic):
c = c.doit(**hints)
newargs.append((e, c))
return self.func(*newargs)
def _eval_simplify(self, ratio, measure, rational, inverse):
args = [a._eval_simplify(ratio, measure, rational, inverse)
for a in self.args]
_blessed = lambda e: getattr(e.lhs, '_diff_wrt', False) and (
getattr(e.rhs, '_diff_wrt', None) or
isinstance(e.rhs, (Rational, NumberSymbol)))
for i, (expr, cond) in enumerate(args):
# try to simplify conditions and the expression for
# equalities that are part of the condition, e.g.
# Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
# -> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Equality), binary=True)
elif isinstance(cond, Equality):
eqs, other = [cond], []
else:
eqs = other = []
if eqs:
eqs = list(ordered(eqs))
for j, e in enumerate(eqs):
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if _blessed(e):
expr = expr.subs(*e.args)
eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
other = [ei.subs(*e.args) for ei in other]
cond = And(*(eqs + other))
args[i] = args[i].func(expr, cond)
# See if expressions valid for an Equal expression happens to evaluate
# to the same function as in the next piecewise segment, see:
# https://github.com/sympy/sympy/issues/8458
prevexpr = None
for i, (expr, cond) in reversed(list(enumerate(args))):
if prevexpr is not None:
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Equality), binary=True)
elif isinstance(cond, Equality):
eqs, other = [cond], []
else:
eqs = other = []
_prevexpr = prevexpr
_expr = expr
if eqs and not other:
eqs = list(ordered(eqs))
for e in eqs:
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if _blessed(e):
_prevexpr = _prevexpr.subs(*e.args)
_expr = _expr.subs(*e.args)
# Did it evaluate to the same?
if _prevexpr == _expr:
# Set the expression for the Not equal section to the same
# as the next. These will be merged when creating the new
# Piecewise
args[i] = args[i].func(args[i+1][0], cond)
else:
# Update the expression that we compare against
prevexpr = expr
else:
prevexpr = expr
return self.func(*args)
def _eval_as_leading_term(self, x):
for e, c in self.args:
if c == True or c.subs(x, 0) == True:
return e.as_leading_term(x)
def _eval_adjoint(self):
return self.func(*[(e.adjoint(), c) for e, c in self.args])
def _eval_conjugate(self):
return self.func(*[(e.conjugate(), c) for e, c in self.args])
def _eval_derivative(self, x):
return self.func(*[(diff(e, x), c) for e, c in self.args])
def _eval_evalf(self, prec):
return self.func(*[(e._evalf(prec), c) for e, c in self.args])
def piecewise_integrate(self, x, **kwargs):
"""Return the Piecewise with each expression being
replaced with its antiderivative. To obtain a continuous
antiderivative, use the `integrate` function or method.
Examples
========
>>> from sympy import Piecewise
>>> from sympy.abc import x
>>> p = Piecewise((0, x < 0), (1, x < 1), (2, True))
>>> p.piecewise_integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x, True))
Note that this does not give a continuous function, e.g.
at x = 1 the 3rd condition applies and the antiderivative
there is 2*x so the value of the antiderivative is 2:
>>> anti = _
>>> anti.subs(x, 1)
2
The continuous derivative accounts for the integral *up to*
the point of interest, however:
>>> p.integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x - 1, True))
>>> _.subs(x, 1)
1
See Also
========
Piecewise._eval_integral
"""
from sympy.integrals import integrate
return self.func(*[(integrate(e, x, **kwargs), c) for e, c in self.args])
def _handle_irel(self, x, handler):
"""Return either None (if the conditions of self depend only on x) else
a Piecewise expression whose expressions (handled by the handler that
was passed) are paired with the governing x-independent relationals,
e.g. Piecewise((A, a(x) & b(y)), (B, c(x) | c(y)) ->
Piecewise(
(handler(Piecewise((A, a(x) & True), (B, c(x) | True)), b(y) & c(y)),
(handler(Piecewise((A, a(x) & True), (B, c(x) | False)), b(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | True)), c(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | False)), True))
"""
# identify governing relationals
rel = self.atoms(Relational)
irel = list(ordered([r for r in rel if x not in r.free_symbols
and r not in (S.true, S.false)]))
if irel:
args = {}
exprinorder = []
for truth in product((1, 0), repeat=len(irel)):
reps = dict(zip(irel, truth))
# only store the true conditions since the false are implied
# when they appear lower in the Piecewise args
if 1 not in truth:
cond = None # flag this one so it doesn't get combined
else:
andargs = Tuple(*[i for i in reps if reps[i]])
free = list(andargs.free_symbols)
if len(free) == 1:
from sympy.solvers.inequalities import (
reduce_inequalities, _solve_inequality)
try:
t = reduce_inequalities(andargs, free[0])
# ValueError when there are potentially
# nonvanishing imaginary parts
except (ValueError, NotImplementedError):
# at least isolate free symbol on left
t = And(*[_solve_inequality(
a, free[0], linear=True)
for a in andargs])
else:
t = And(*andargs)
if t is S.false:
continue # an impossible combination
cond = t
expr = handler(self.xreplace(reps))
if isinstance(expr, self.func) and len(expr.args) == 1:
expr, econd = expr.args[0]
cond = And(econd, True if cond is None else cond)
# the ec pairs are being collected since all possibilities
# are being enumerated, but don't put the last one in since
# its expr might match a previous expression and it
# must appear last in the args
if cond is not None:
args.setdefault(expr, []).append(cond)
# but since we only store the true conditions we must maintain
# the order so that the expression with the most true values
# comes first
exprinorder.append(expr)
# convert collected conditions as args of Or
for k in args:
args[k] = Or(*args[k])
# take them in the order obtained
args = [(e, args[e]) for e in uniq(exprinorder)]
# add in the last arg
args.append((expr, True))
# if any condition reduced to True, it needs to go last
# and there should only be one of them or else the exprs
# should agree
trues = [i for i in range(len(args)) if args[i][1] is S.true]
if not trues:
# make the last one True since all cases were enumerated
e, c = args[-1]
args[-1] = (e, S.true)
else:
assert len(set([e for e, c in [args[i] for i in trues]])) == 1
args.append(args.pop(trues.pop()))
while trues:
args.pop(trues.pop())
return Piecewise(*args)
def _eval_integral(self, x, _first=True, **kwargs):
"""Return the indefinite integral of the
Piecewise such that subsequent substitution of x with a
value will give the value of the integral (not including
the constant of integration) up to that point. To only
integrate the individual parts of Piecewise, use the
`piecewise_integrate` method.
Examples
========
>>> from sympy import Piecewise
>>> from sympy.abc import x
>>> p = Piecewise((0, x < 0), (1, x < 1), (2, True))
>>> p.integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x - 1, True))
>>> p.piecewise_integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x, True))
See Also
========
Piecewise.piecewise_integrate
"""
from sympy.integrals.integrals import integrate
if _first:
def handler(ipw):
if isinstance(ipw, self.func):
return ipw._eval_integral(x, _first=False, **kwargs)
else:
return ipw.integrate(x, **kwargs)
irv = self._handle_irel(x, handler)
if irv is not None:
return irv
# handle a Piecewise from -oo to oo with and no x-independent relationals
# -----------------------------------------------------------------------
try:
abei = self._intervals(x)
except NotImplementedError:
from sympy import Integral
return Integral(self, x) # unevaluated
pieces = [(a, b) for a, b, _, _ in abei]
oo = S.Infinity
done = [(-oo, oo, -1)]
for k, p in enumerate(pieces):
if p == (-oo, oo):
# all undone intervals will get this key
for j, (a, b, i) in enumerate(done):
if i == -1:
done[j] = a, b, k
break # nothing else to consider
N = len(done) - 1
for j, (a, b, i) in enumerate(reversed(done)):
if i == -1:
j = N - j
done[j: j + 1] = _clip(p, (a, b), k)
done = [(a, b, i) for a, b, i in done if a != b]
# append an arg if there is a hole so a reference to
# argument -1 will give Undefined
if any(i == -1 for (a, b, i) in done):
abei.append((-oo, oo, Undefined, -1))
# return the sum of the intervals
args = []
sum = None
for a, b, i in done:
anti = integrate(abei[i][-2], x, **kwargs)
if sum is None:
sum = anti
else:
sum = sum.subs(x, a)
if sum == Undefined:
sum = 0
sum += anti._eval_interval(x, a, x)
# see if we know whether b is contained in original
# condition
if b is S.Infinity:
cond = True
elif self.args[abei[i][-1]].cond.subs(x, b) == False:
cond = (x < b)
else:
cond = (x <= b)
args.append((sum, cond))
return Piecewise(*args)
def _eval_interval(self, sym, a, b, _first=True):
"""Evaluates the function along the sym in a given interval [a, b]"""
# FIXME: Currently complex intervals are not supported. A possible
# replacement algorithm, discussed in issue 5227, can be found in the
# following papers;
# http://portal.acm.org/citation.cfm?id=281649
# http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.4127&rep=rep1&type=pdf
from sympy.core.symbol import Dummy
if a is None or b is None:
# In this case, it is just simple substitution
return super(Piecewise, self)._eval_interval(sym, a, b)
else:
x, lo, hi = map(as_Basic, (sym, a, b))
if _first: # get only x-dependent relationals
def handler(ipw):
if isinstance(ipw, self.func):
return ipw._eval_interval(x, lo, hi, _first=None)
else:
return ipw._eval_interval(x, lo, hi)
irv = self._handle_irel(x, handler)
if irv is not None:
return irv
if (lo < hi) is S.false or (
lo is S.Infinity or hi is S.NegativeInfinity):
rv = self._eval_interval(x, hi, lo, _first=False)
if isinstance(rv, Piecewise):
rv = Piecewise(*[(-e, c) for e, c in rv.args])
else:
rv = -rv
return rv
if (lo < hi) is S.true or (
hi is S.Infinity or lo is S.NegativeInfinity):
pass
else:
_a = Dummy('lo')
_b = Dummy('hi')
a = lo if lo.is_comparable else _a
b = hi if hi.is_comparable else _b
pos = self._eval_interval(x, a, b, _first=False)
if a == _a and b == _b:
# it's purely symbolic so just swap lo and hi and
# change the sign to get the value for when lo > hi
neg, pos = (-pos.xreplace({_a: hi, _b: lo}),
pos.xreplace({_a: lo, _b: hi}))
else:
# at least one of the bounds was comparable, so allow
# _eval_interval to use that information when computing
# the interval with lo and hi reversed
neg, pos = (-self._eval_interval(x, hi, lo, _first=False),
pos.xreplace({_a: lo, _b: hi}))
# allow simplification based on ordering of lo and hi
p = Dummy('', positive=True)
if lo.is_Symbol:
pos = pos.xreplace({lo: hi - p}).xreplace({p: hi - lo})
neg = neg.xreplace({lo: hi + p}).xreplace({p: lo - hi})
elif hi.is_Symbol:
pos = pos.xreplace({hi: lo + p}).xreplace({p: hi - lo})
neg = neg.xreplace({hi: lo - p}).xreplace({p: lo - hi})
# assemble return expression; make the first condition be Lt
# b/c then the first expression will look the same whether
# the lo or hi limit is symbolic
if a == _a: # the lower limit was symbolic
rv = Piecewise(
(pos,
lo < hi),
(neg,
True))
else:
rv = Piecewise(
(neg,
hi < lo),
(pos,
True))
if rv == Undefined:
raise ValueError("Can't integrate across undefined region.")
if any(isinstance(i, Piecewise) for i in (pos, neg)):
rv = piecewise_fold(rv)
return rv
# handle a Piecewise with lo <= hi and no x-independent relationals
# -----------------------------------------------------------------
try:
abei = self._intervals(x)
except NotImplementedError:
from sympy import Integral
# not being able to do the interval of f(x) can
# be stated as not being able to do the integral
# of f'(x) over the same range
return Integral(self.diff(x), (x, lo, hi)) # unevaluated
pieces = [(a, b) for a, b, _, _ in abei]
done = [(lo, hi, -1)]
oo = S.Infinity
for k, p in enumerate(pieces):
if p[:2] == (-oo, oo):
# all undone intervals will get this key
for j, (a, b, i) in enumerate(done):
if i == -1:
done[j] = a, b, k
break # nothing else to consider
N = len(done) - 1
for j, (a, b, i) in enumerate(reversed(done)):
if i == -1:
j = N - j
done[j: j + 1] = _clip(p, (a, b), k)
done = [(a, b, i) for a, b, i in done if a != b]
# return the sum of the intervals
sum = S.Zero
upto = None
for a, b, i in done:
if i == -1:
if upto is None:
return Undefined
# TODO simplify hi <= upto
return Piecewise((sum, hi <= upto), (Undefined, True))
sum += abei[i][-2]._eval_interval(x, a, b)
upto = b
return sum
def _intervals(self, sym):
"""Return a list of unique tuples, (a, b, e, i), where a and b
are the lower and upper bounds in which the expression e of
argument i in self is defined and a < b (when involving
numbers) or a <= b when involving symbols.
If there are any relationals not involving sym, or any
relational cannot be solved for sym, NotImplementedError is
raised. The calling routine should have removed such
relationals before calling this routine.
The evaluated conditions will be returned as ranges.
Discontinuous ranges will be returned separately with
identical expressions. The first condition that evaluates to
True will be returned as the last tuple with a, b = -oo, oo.
"""
from sympy.solvers.inequalities import _solve_inequality
from sympy.logic.boolalg import to_cnf, distribute_or_over_and
assert isinstance(self, Piecewise)
def _solve_relational(r):
if sym not in r.free_symbols:
nonsymfail(r)
rv = _solve_inequality(r, sym)
if isinstance(rv, Relational):
free = rv.args[1].free_symbols
if rv.args[0] != sym or sym in free:
raise NotImplementedError(filldedent('''
Unable to solve relational
%s for %s.''' % (r, sym)))
if rv.rel_op == '==':
# this equality has been affirmed to have the form
# Eq(sym, rhs) where rhs is sym-free; it represents
# a zero-width interval which will be ignored
# whether it is an isolated condition or contained
# within an And or an Or
rv = S.false
elif rv.rel_op == '!=':
try:
rv = Or(sym < rv.rhs, sym > rv.rhs)
except TypeError:
# e.g. x != I ==> all real x satisfy
rv = S.true
elif rv == (S.NegativeInfinity < sym) & (sym < S.Infinity):
rv = S.true
return rv
def nonsymfail(cond):
raise NotImplementedError(filldedent('''
A condition not involving
%s appeared: %s''' % (sym, cond)))
# make self canonical wrt Relationals
reps = dict([
(r, _solve_relational(r)) for r in self.atoms(Relational)])
# process args individually so if any evaluate, their position
# in the original Piecewise will be known
args = [i.xreplace(reps) for i in self.args]
# precondition args
expr_cond = []
default = idefault = None
for i, (expr, cond) in enumerate(args):
if cond is S.false:
continue
elif cond is S.true:
default = expr
idefault = i
break
cond = to_cnf(cond)
if isinstance(cond, And):
cond = distribute_or_over_and(cond)
if isinstance(cond, Or):
expr_cond.extend(
[(i, expr, o) for o in cond.args
if not isinstance(o, Equality)])
elif cond is not S.false:
expr_cond.append((i, expr, cond))
# determine intervals represented by conditions
int_expr = []
for iarg, expr, cond in expr_cond:
if isinstance(cond, And):
lower = S.NegativeInfinity
upper = S.Infinity
for cond2 in cond.args:
if isinstance(cond2, Equality):
lower = upper # ignore
break
elif cond2.lts == sym:
upper = Min(cond2.gts, upper)
elif cond2.gts == sym:
lower = Max(cond2.lts, lower)
else:
nonsymfail(cond2) # should never get here
elif isinstance(cond, Relational):
lower, upper = cond.lts, cond.gts # part 1: initialize with givens
if cond.lts == sym: # part 1a: expand the side ...
lower = S.NegativeInfinity # e.g. x <= 0 ---> -oo <= 0
elif cond.gts == sym: # part 1a: ... that can be expanded
upper = S.Infinity # e.g. x >= 0 ---> oo >= 0
else:
nonsymfail(cond)
else:
raise NotImplementedError(
'unrecognized condition: %s' % cond)
lower, upper = lower, Max(lower, upper)
if (lower >= upper) is not S.true:
int_expr.append((lower, upper, expr, iarg))
if default is not None:
int_expr.append(
(S.NegativeInfinity, S.Infinity, default, idefault))
return list(uniq(int_expr))
def _eval_nseries(self, x, n, logx):
args = [(ec.expr._eval_nseries(x, n, logx), ec.cond) for ec in self.args]
return self.func(*args)
def _eval_power(self, s):
return self.func(*[(e**s, c) for e, c in self.args])
def _eval_subs(self, old, new):
# this is strictly not necessary, but we can keep track
# of whether True or False conditions arise and be
# somewhat more efficient by avoiding other substitutions
# and avoiding invalid conditions that appear after a
# True condition
args = list(self.args)
args_exist = False
for i, (e, c) in enumerate(args):
c = c._subs(old, new)
if c != False:
args_exist = True
e = e._subs(old, new)
args[i] = (e, c)
if c == True:
break
if not args_exist:
args = ((Undefined, True),)
return self.func(*args)
def _eval_transpose(self):
return self.func(*[(e.transpose(), c) for e, c in self.args])
def _eval_template_is_attr(self, is_attr):
b = None
for expr, _ in self.args:
a = getattr(expr, is_attr)
if a is None:
return
if b is None:
b = a
elif b is not a:
return
return b
_eval_is_finite = lambda self: self._eval_template_is_attr(
'is_finite')
_eval_is_complex = lambda self: self._eval_template_is_attr('is_complex')
_eval_is_even = lambda self: self._eval_template_is_attr('is_even')
_eval_is_imaginary = lambda self: self._eval_template_is_attr(
'is_imaginary')
_eval_is_integer = lambda self: self._eval_template_is_attr('is_integer')
_eval_is_irrational = lambda self: self._eval_template_is_attr(
'is_irrational')
_eval_is_negative = lambda self: self._eval_template_is_attr('is_negative')
_eval_is_nonnegative = lambda self: self._eval_template_is_attr(
'is_nonnegative')
_eval_is_nonpositive = lambda self: self._eval_template_is_attr(
'is_nonpositive')
_eval_is_nonzero = lambda self: self._eval_template_is_attr(
'is_nonzero')
_eval_is_odd = lambda self: self._eval_template_is_attr('is_odd')
_eval_is_polar = lambda self: self._eval_template_is_attr('is_polar')
_eval_is_positive = lambda self: self._eval_template_is_attr('is_positive')
_eval_is_real = lambda self: self._eval_template_is_attr('is_real')
_eval_is_zero = lambda self: self._eval_template_is_attr(
'is_zero')
@classmethod
def __eval_cond(cls, cond):
"""Return the truth value of the condition."""
if cond == True:
return True
if isinstance(cond, Equality):
try:
diff = cond.lhs - cond.rhs
if diff.is_commutative:
return diff.is_zero
except TypeError:
pass
def as_expr_set_pairs(self, domain=S.Reals):
"""Return tuples for each argument of self that give
the expression and the interval in which it is valid
which is contained within the given domain.
If a condition cannot be converted to a set, an error
will be raised. The variable of the conditions is
assumed to be real; sets of real values are returned.
Examples
========
>>> from sympy import Piecewise, Interval
>>> from sympy.abc import x
>>> p = Piecewise(
... (1, x < 2),
... (2,(x > 0) & (x < 4)),
... (3, True))
>>> p.as_expr_set_pairs()
[(1, Interval.open(-oo, 2)),
(2, Interval.Ropen(2, 4)),
(3, Interval(4, oo))]
>>> p.as_expr_set_pairs(Interval(0, 3))
[(1, Interval.Ropen(0, 2)),
(2, Interval(2, 3)), (3, EmptySet())]
"""
exp_sets = []
U = domain
complex = not domain.is_subset(S.Reals)
for expr, cond in self.args:
if complex:
for i in cond.atoms(Relational):
if not isinstance(i, (Equality, Unequality)):
raise ValueError(filldedent('''
Inequalities in the complex domain are
not supported. Try the real domain by
setting domain=S.Reals'''))
cond_int = U.intersect(cond.as_set())
U = U - cond_int
exp_sets.append((expr, cond_int))
return exp_sets
def _eval_rewrite_as_ITE(self, *args, **kwargs):
byfree = {}
args = list(args)
default = any(c == True for b, c in args)
for i, (b, c) in enumerate(args):
if not isinstance(b, Boolean) and b != True:
raise TypeError(filldedent('''
Expecting Boolean or bool but got `%s`
''' % func_name(b)))
if c == True:
break
# loop over independent conditions for this b
for c in c.args if isinstance(c, Or) else [c]:
free = c.free_symbols
x = free.pop()
try:
byfree[x] = byfree.setdefault(
x, S.EmptySet).union(c.as_set())
except NotImplementedError:
if not default:
raise NotImplementedError(filldedent('''
A method to determine whether a multivariate
conditional is consistent with a complete coverage
of all variables has not been implemented so the
rewrite is being stopped after encountering `%s`.
This error would not occur if a default expression
like `(foo, True)` were given.
''' % c))
if byfree[x] in (S.UniversalSet, S.Reals):
# collapse the ith condition to True and break
args[i] = list(args[i])
c = args[i][1] = True
break
if c == True:
break
if c != True:
raise ValueError(filldedent('''
Conditions must cover all reals or a final default
condition `(foo, True)` must be given.
'''))
last, _ = args[i] # ignore all past ith arg
for a, c in reversed(args[:i]):
last = ITE(c, a, last)
return _canonical(last)
def piecewise_fold(expr):
"""
Takes an expression containing a piecewise function and returns the
expression in piecewise form. In addition, any ITE conditions are
rewritten in negation normal form and simplified.
Examples
========
>>> from sympy import Piecewise, piecewise_fold, sympify as S
>>> from sympy.abc import x
>>> p = Piecewise((x, x < 1), (1, S(1) <= x))
>>> piecewise_fold(x*p)
Piecewise((x**2, x < 1), (x, True))
See Also
========
Piecewise
"""
if not isinstance(expr, Basic) or not expr.has(Piecewise):
return expr
new_args = []
if isinstance(expr, (ExprCondPair, Piecewise)):
for e, c in expr.args:
if not isinstance(e, Piecewise):
e = piecewise_fold(e)
# we don't keep Piecewise in condition because
# it has to be checked to see that it's complete
# and we convert it to ITE at that time
assert not c.has(Piecewise) # pragma: no cover
if isinstance(c, ITE):
c = c.to_nnf()
c = simplify_logic(c, form='cnf')
if isinstance(e, Piecewise):
new_args.extend([(piecewise_fold(ei), And(ci, c))
for ei, ci in e.args])
else:
new_args.append((e, c))
else:
from sympy.utilities.iterables import cartes, sift, common_prefix
# Given
# P1 = Piecewise((e11, c1), (e12, c2), A)
# P2 = Piecewise((e21, c1), (e22, c2), B)
# ...
# the folding of f(P1, P2) is trivially
# Piecewise(
# (f(e11, e21), c1),
# (f(e12, e22), c2),
# (f(Piecewise(A), Piecewise(B)), True))
# Certain objects end up rewriting themselves as thus, so
# we do that grouping before the more generic folding.
# The following applies this idea when f = Add or f = Mul
# (and the expression is commutative).
if expr.is_Add or expr.is_Mul and expr.is_commutative:
p, args = sift(expr.args, lambda x: x.is_Piecewise, binary=True)
pc = sift(p, lambda x: tuple([c for e,c in x.args]))
for c in list(ordered(pc)):
if len(pc[c]) > 1:
pargs = [list(i.args) for i in pc[c]]
# the first one is the same; there may be more
com = common_prefix(*[
[i.cond for i in j] for j in pargs])
n = len(com)
collected = []
for i in range(n):
collected.append((
expr.func(*[ai[i].expr for ai in pargs]),
com[i]))
remains = []
for a in pargs:
if n == len(a): # no more args
continue
if a[n].cond == True: # no longer Piecewise
remains.append(a[n].expr)
else: # restore the remaining Piecewise
remains.append(
Piecewise(*a[n:], evaluate=False))
if remains:
collected.append((expr.func(*remains), True))
args.append(Piecewise(*collected, evaluate=False))
continue
args.extend(pc[c])
else:
args = expr.args
# fold
folded = list(map(piecewise_fold, args))
for ec in cartes(*[
(i.args if isinstance(i, Piecewise) else
[(i, true)]) for i in folded]):
e, c = zip(*ec)
new_args.append((expr.func(*e), And(*c)))
return Piecewise(*new_args)
def _clip(A, B, k):
"""Return interval B as intervals that are covered by A (keyed
to k) and all other intervals of B not covered by A keyed to -1.
The reference point of each interval is the rhs; if the lhs is
greater than the rhs then an interval of zero width interval will
result, e.g. (4, 1) is treated like (1, 1).
Examples
========
>>> from sympy.functions.elementary.piecewise import _clip
>>> from sympy import Tuple
>>> A = Tuple(1, 3)
>>> B = Tuple(2, 4)
>>> _clip(A, B, 0)
[(2, 3, 0), (3, 4, -1)]
Interpretation: interval portion (2, 3) of interval (2, 4) is
covered by interval (1, 3) and is keyed to 0 as requested;
interval (3, 4) was not covered by (1, 3) and is keyed to -1.
"""
a, b = B
c, d = A
c, d = Min(Max(c, a), b), Min(Max(d, a), b)
a, b = Min(a, b), b
p = []
if a != c:
p.append((a, c, -1))
else:
pass
if c != d:
p.append((c, d, k))
else:
pass
if b != d:
if d == c and p and p[-1][-1] == -1:
p[-1] = p[-1][0], b, -1
else:
p.append((d, b, -1))
else:
pass
return p
|
ac07bcf7e11fc9145f39465fea9495dcd221434fadd889af982c839af0f88810 | import string
from sympy import (
Symbol, symbols, Dummy, S, Sum, Rational, oo, pi, I,
expand_func, diff, EulerGamma, cancel, re, im, Product, carmichael)
from sympy.functions import (
bernoulli, harmonic, bell, fibonacci, tribonacci, lucas, euler, catalan,
genocchi, partition, binomial, gamma, sqrt, cbrt, hyper, log, digamma,
trigamma, polygamma, factorial, sin, cos, cot, zeta)
from sympy.functions.combinatorial.numbers import _nT
from sympy.core.compatibility import range
from sympy.utilities.pytest import XFAIL, raises
from sympy.core.numbers import GoldenRatio
x = Symbol('x')
def test_carmichael():
assert carmichael.find_carmichael_numbers_in_range(0, 561) == []
assert carmichael.find_carmichael_numbers_in_range(561, 562) == [561]
assert carmichael.find_carmichael_numbers_in_range(561, 1105) == carmichael.find_carmichael_numbers_in_range(561,
562)
assert carmichael.find_first_n_carmichaels(5) == [561, 1105, 1729, 2465, 2821]
assert carmichael.is_prime(2821) == False
assert carmichael.is_prime(2465) == False
assert carmichael.is_prime(1729) == False
assert carmichael.is_prime(1105) == False
assert carmichael.is_prime(561) == False
def test_bernoulli():
assert bernoulli(0) == 1
assert bernoulli(1) == Rational(-1, 2)
assert bernoulli(2) == Rational(1, 6)
assert bernoulli(3) == 0
assert bernoulli(4) == Rational(-1, 30)
assert bernoulli(5) == 0
assert bernoulli(6) == Rational(1, 42)
assert bernoulli(7) == 0
assert bernoulli(8) == Rational(-1, 30)
assert bernoulli(10) == Rational(5, 66)
assert bernoulli(1000001) == 0
assert bernoulli(0, x) == 1
assert bernoulli(1, x) == x - Rational(1, 2)
assert bernoulli(2, x) == x**2 - x + Rational(1, 6)
assert bernoulli(3, x) == x**3 - (3*x**2)/2 + x/2
# Should be fast; computed with mpmath
b = bernoulli(1000)
assert b.p % 10**10 == 7950421099
assert b.q == 342999030
b = bernoulli(10**6, evaluate=False).evalf()
assert str(b) == '-2.23799235765713e+4767529'
# Issue #8527
l = Symbol('l', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
n = Symbol('n', integer=True, positive=True)
assert isinstance(bernoulli(2 * l + 1), bernoulli)
assert isinstance(bernoulli(2 * m + 1), bernoulli)
assert bernoulli(2 * n + 1) == 0
def test_fibonacci():
assert [fibonacci(n) for n in range(-3, 5)] == [2, -1, 1, 0, 1, 1, 2, 3]
assert fibonacci(100) == 354224848179261915075
assert [lucas(n) for n in range(-3, 5)] == [-4, 3, -1, 2, 1, 3, 4, 7]
assert lucas(100) == 792070839848372253127
assert fibonacci(1, x) == 1
assert fibonacci(2, x) == x
assert fibonacci(3, x) == x**2 + 1
assert fibonacci(4, x) == x**3 + 2*x
# issue #8800
n = Dummy('n')
assert fibonacci(n).limit(n, S.Infinity) == S.Infinity
assert lucas(n).limit(n, S.Infinity) == S.Infinity
assert fibonacci(n).rewrite(sqrt) == \
2**(-n)*sqrt(5)*((1 + sqrt(5))**n - (-sqrt(5) + 1)**n) / 5
assert fibonacci(n).rewrite(sqrt).subs(n, 10).expand() == fibonacci(10)
assert fibonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \
fibonacci(10)
assert lucas(n).rewrite(sqrt) == \
(fibonacci(n-1).rewrite(sqrt) + fibonacci(n+1).rewrite(sqrt)).simplify()
assert lucas(n).rewrite(sqrt).subs(n, 10).expand() == lucas(10)
def test_tribonacci():
assert [tribonacci(n) for n in range(8)] == [0, 1, 1, 2, 4, 7, 13, 24]
assert tribonacci(100) == 98079530178586034536500564
assert tribonacci(0, x) == 0
assert tribonacci(1, x) == 1
assert tribonacci(2, x) == x**2
assert tribonacci(3, x) == x**4 + x
assert tribonacci(4, x) == x**6 + 2*x**3 + 1
assert tribonacci(5, x) == x**8 + 3*x**5 + 3*x**2
n = Dummy('n')
assert tribonacci(n).limit(n, S.Infinity) == S.Infinity
w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2
a = (1 + cbrt(19 + 3*sqrt(33)) + cbrt(19 - 3*sqrt(33))) / 3
b = (1 + w*cbrt(19 + 3*sqrt(33)) + w**2*cbrt(19 - 3*sqrt(33))) / 3
c = (1 + w**2*cbrt(19 + 3*sqrt(33)) + w*cbrt(19 - 3*sqrt(33))) / 3
assert tribonacci(n).rewrite(sqrt) == \
(a**(n + 1)/((a - b)*(a - c))
+ b**(n + 1)/((b - a)*(b - c))
+ c**(n + 1)/((c - a)*(c - b)))
assert tribonacci(n).rewrite(sqrt).subs(n, 4).simplify() == tribonacci(4)
assert tribonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \
tribonacci(10)
raises(ValueError, lambda: tribonacci(-1, x))
def test_bell():
assert [bell(n) for n in range(8)] == [1, 1, 2, 5, 15, 52, 203, 877]
assert bell(0, x) == 1
assert bell(1, x) == x
assert bell(2, x) == x**2 + x
assert bell(5, x) == x**5 + 10*x**4 + 25*x**3 + 15*x**2 + x
assert bell(oo) == S.Infinity
raises(ValueError, lambda: bell(oo, x))
raises(ValueError, lambda: bell(-1))
raises(ValueError, lambda: bell(S(1)/2))
X = symbols('x:6')
# X = (x0, x1, .. x5)
# at the same time: X[1] = x1, X[2] = x2 for standard readablity.
# but we must supply zero-based indexed object X[1:] = (x1, .. x5)
assert bell(6, 2, X[1:]) == 6*X[5]*X[1] + 15*X[4]*X[2] + 10*X[3]**2
assert bell(
6, 3, X[1:]) == 15*X[4]*X[1]**2 + 60*X[3]*X[2]*X[1] + 15*X[2]**3
X = (1, 10, 100, 1000, 10000)
assert bell(6, 2, X) == (6 + 15 + 10)*10000
X = (1, 2, 3, 3, 5)
assert bell(6, 2, X) == 6*5 + 15*3*2 + 10*3**2
X = (1, 2, 3, 5)
assert bell(6, 3, X) == 15*5 + 60*3*2 + 15*2**3
# Dobinski's formula
n = Symbol('n', integer=True, nonnegative=True)
# For large numbers, this is too slow
# For nonintegers, there are significant precision errors
for i in [0, 2, 3, 7, 13, 42, 55]:
assert bell(i).evalf() == bell(n).rewrite(Sum).evalf(subs={n: i})
# issue 9184
n = Dummy('n')
assert bell(n).limit(n, S.Infinity) == S.Infinity
def test_harmonic():
n = Symbol("n")
m = Symbol("m")
assert harmonic(n, 0) == n
assert harmonic(n).evalf() == harmonic(n)
assert harmonic(n, 1) == harmonic(n)
assert harmonic(1, n).evalf() == harmonic(1, n)
assert harmonic(0, 1) == 0
assert harmonic(1, 1) == 1
assert harmonic(2, 1) == Rational(3, 2)
assert harmonic(3, 1) == Rational(11, 6)
assert harmonic(4, 1) == Rational(25, 12)
assert harmonic(0, 2) == 0
assert harmonic(1, 2) == 1
assert harmonic(2, 2) == Rational(5, 4)
assert harmonic(3, 2) == Rational(49, 36)
assert harmonic(4, 2) == Rational(205, 144)
assert harmonic(0, 3) == 0
assert harmonic(1, 3) == 1
assert harmonic(2, 3) == Rational(9, 8)
assert harmonic(3, 3) == Rational(251, 216)
assert harmonic(4, 3) == Rational(2035, 1728)
assert harmonic(oo, -1) == S.NaN
assert harmonic(oo, 0) == oo
assert harmonic(oo, S.Half) == oo
assert harmonic(oo, 1) == oo
assert harmonic(oo, 2) == (pi**2)/6
assert harmonic(oo, 3) == zeta(3)
assert harmonic(0, m) == 0
def test_harmonic_rational():
ne = S(6)
no = S(5)
pe = S(8)
po = S(9)
qe = S(10)
qo = S(13)
Heee = harmonic(ne + pe/qe)
Aeee = (-log(10) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ pi*(1/S(4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + 5/S(8)))
+ 13944145/S(4720968))
Heeo = harmonic(ne + pe/qo)
Aeeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13)
+ 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13)
- 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2 - 2*log(sin(pi/13))*cos(3*pi/13)
+ 2422020029/S(702257080))
Heoe = harmonic(ne + po/qe)
Aeoe = (-log(20) + 2*(1/S(4) + sqrt(5)/4)*log(-1/S(4) + sqrt(5)/4)
+ 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 + 1/S(4))*log(1/S(4) + sqrt(5)/4)
+ 11818877030/S(4286604231) + pi*(sqrt(5)/8 + 5/S(8))/sqrt(-sqrt(5)/8 + 5/S(8)))
Heoo = harmonic(ne + po/qo)
Aeoo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13)
+ 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13)
- 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2
- 2*log(sin(2*pi/13))*cos(3*pi/13) + 11669332571/S(3628714320))
Hoee = harmonic(no + pe/qe)
Aoee = (-log(10) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ pi*(1/S(4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + 5/S(8)))
+ 779405/S(277704))
Hoeo = harmonic(no + pe/qo)
Aoeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13)
+ 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13)
- 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2
- 2*log(sin(pi/13))*cos(3*pi/13) + 53857323/S(16331560))
Hooe = harmonic(no + po/qe)
Aooe = (-log(20) + 2*(1/S(4) + sqrt(5)/4)*log(-1/S(4) + sqrt(5)/4)
+ 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 + 1/S(4))*log(1/S(4) + sqrt(5)/4)
+ 486853480/S(186374097) + pi*(sqrt(5)/8 + 5/S(8))/sqrt(-sqrt(5)/8 + 5/S(8)))
Hooo = harmonic(no + po/qo)
Aooo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13)
+ 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13)
- 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2
- 2*log(sin(2*pi/13))*cos(3*pi/13) + 383693479/S(125128080))
H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo]
A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo]
for h, a in zip(H, A):
e = expand_func(h).doit()
assert cancel(e/a) == 1
assert abs(h.n() - a.n()) < 1e-12
def test_harmonic_evalf():
assert str(harmonic(1.5).evalf(n=10)) == '1.280372306'
assert str(harmonic(1.5, 2).evalf(n=10)) == '1.154576311' # issue 7443
def test_harmonic_rewrite_polygamma():
n = Symbol("n")
m = Symbol("m")
assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)
assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)
@XFAIL
def test_harmonic_limit_fail():
n = Symbol("n")
m = Symbol("m")
# For m > 1:
assert limit(harmonic(n, m), n, oo) == zeta(m)
@XFAIL
def test_harmonic_rewrite_sum_fail():
n = Symbol("n")
m = Symbol("m")
_k = Dummy("k")
assert harmonic(n).rewrite(Sum) == Sum(1/_k, (_k, 1, n))
assert harmonic(n, m).rewrite(Sum) == Sum(_k**(-m), (_k, 1, n))
def replace_dummy(expr, sym):
dum = expr.atoms(Dummy)
if not dum:
return expr
assert len(dum) == 1
return expr.xreplace({dum.pop(): sym})
def test_harmonic_rewrite_sum():
n = Symbol("n")
m = Symbol("m")
_k = Dummy("k")
assert replace_dummy(harmonic(n).rewrite(Sum), _k) == Sum(1/_k, (_k, 1, n))
assert replace_dummy(harmonic(n, m).rewrite(Sum), _k) == Sum(_k**(-m), (_k, 1, n))
def test_euler():
assert euler(0) == 1
assert euler(1) == 0
assert euler(2) == -1
assert euler(3) == 0
assert euler(4) == 5
assert euler(6) == -61
assert euler(8) == 1385
assert euler(20, evaluate=False) != 370371188237525
n = Symbol('n', integer=True)
assert euler(n) != -1
assert euler(n).subs(n, 2) == -1
raises(ValueError, lambda: euler(-2))
raises(ValueError, lambda: euler(-3))
raises(ValueError, lambda: euler(2.3))
assert euler(20).evalf() == 370371188237525.0
assert euler(20, evaluate=False).evalf() == 370371188237525.0
assert euler(n).rewrite(Sum) == euler(n)
# XXX: Not sure what the guy who wrote this test was trying to do with the _j and _k stuff
n = Symbol('n', integer=True, nonnegative=True)
assert euler(2*n + 1).rewrite(Sum) == 0
@XFAIL
def test_euler_failing():
# depends on dummy variables being implemented https://github.com/sympy/sympy/issues/5665
assert euler(2*n).rewrite(Sum) == I*Sum(Sum((-1)**_j*2**(-_k)*I**(-_k)*(-2*_j + _k)**(2*n + 1)*binomial(_k, _j)/_k, (_j, 0, _k)), (_k, 1, 2*n + 1))
def test_euler_odd():
n = Symbol('n', odd=True, positive=True)
assert euler(n) == 0
n = Symbol('n', odd=True)
assert euler(n) != 0
def test_euler_polynomials():
assert euler(0, x) == 1
assert euler(1, x) == x - Rational(1, 2)
assert euler(2, x) == x**2 - x
assert euler(3, x) == x**3 - (3*x**2)/2 + Rational(1, 4)
m = Symbol('m')
assert isinstance(euler(m, x), euler)
from sympy import Float
A = Float('-0.46237208575048694923364757452876131e8') # from Maple
B = euler(19, S.Pi.evalf(32))
assert abs((A - B)/A) < 1e-31 # expect low relative error
C = euler(19, S.Pi, evaluate=False).evalf(32)
assert abs((A - C)/A) < 1e-31
def test_euler_polynomial_rewrite():
m = Symbol('m')
A = euler(m, x).rewrite('Sum');
assert A.subs({m:3, x:5}).doit() == euler(3, 5)
def test_catalan():
n = Symbol('n', integer=True)
m = Symbol('m', integer=True, positive=True)
k = Symbol('k', integer=True, nonnegative=True)
p = Symbol('p', nonnegative=True)
catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]
for i, c in enumerate(catalans):
assert catalan(i) == c
assert catalan(n).rewrite(factorial).subs(n, i) == c
assert catalan(n).rewrite(Product).subs(n, i).doit() == c
assert catalan(x) == catalan(x)
assert catalan(2*x).rewrite(binomial) == binomial(4*x, 2*x)/(2*x + 1)
assert catalan(Rational(1, 2)).rewrite(gamma) == 8/(3*pi)
assert catalan(Rational(1, 2)).rewrite(factorial).rewrite(gamma) ==\
8 / (3 * pi)
assert catalan(3*x).rewrite(gamma) == 4**(
3*x)*gamma(3*x + Rational(1, 2))/(sqrt(pi)*gamma(3*x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2,), 1)
assert catalan(n).rewrite(factorial) == factorial(2*n) / (factorial(n + 1)
* factorial(n))
assert isinstance(catalan(n).rewrite(Product), catalan)
assert isinstance(catalan(m).rewrite(Product), Product)
assert diff(catalan(x), x) == (polygamma(
0, x + Rational(1, 2)) - polygamma(0, x + 2) + log(4))*catalan(x)
assert catalan(x).evalf() == catalan(x)
c = catalan(S.Half).evalf()
assert str(c) == '0.848826363156775'
c = catalan(I).evalf(3)
assert str((re(c), im(c))) == '(0.398, -0.0209)'
# Assumptions
assert catalan(p).is_positive is True
assert catalan(k).is_integer is True
assert catalan(m+3).is_composite is True
def test_genocchi():
genocchis = [1, -1, 0, 1, 0, -3, 0, 17]
for n, g in enumerate(genocchis):
assert genocchi(n + 1) == g
m = Symbol('m', integer=True)
n = Symbol('n', integer=True, positive=True)
assert genocchi(m) == genocchi(m)
assert genocchi(n).rewrite(bernoulli) == (1 - 2 ** n) * bernoulli(n) * 2
assert genocchi(2 * n).is_odd
assert genocchi(4 * n).is_positive
# these are the only 2 prime Genocchi numbers
assert genocchi(6, evaluate=False).is_prime == S(-3).is_prime
assert genocchi(8, evaluate=False).is_prime
assert genocchi(4 * n + 2).is_negative
assert genocchi(4 * n - 2).is_negative
def test_partition():
partition_nums = [1, 1, 2, 3, 5, 7, 11, 15, 22]
for n, p in enumerate(partition_nums):
assert partition(n) == p
x = Symbol('x')
y = Symbol('y', real=True)
m = Symbol('m', integer=True)
n = Symbol('n', integer=True, negative=True)
p = Symbol('p', integer=True, nonnegative=True)
assert partition(m).is_integer
assert not partition(m).is_negative
assert partition(m).is_nonnegative
assert partition(n).is_zero
assert partition(p).is_positive
assert partition(x).subs(x, 7) == 15
assert partition(y).subs(y, 8) == 22
raises(ValueError, lambda: partition(S(5)/4))
def test__nT():
assert [_nT(i, j) for i in range(5) for j in range(i + 2)] == [
1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 0]
check = [_nT(10, i) for i in range(11)]
assert check == [0, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1]
assert all(type(i) is int for i in check)
assert _nT(10, 5) == 7
assert _nT(100, 98) == 2
assert _nT(100, 100) == 1
assert _nT(10, 3) == 8
def test_nC_nP_nT():
from sympy.utilities.iterables import (
multiset_permutations, multiset_combinations, multiset_partitions,
partitions, subsets, permutations)
from sympy.functions.combinatorial.numbers import (
nP, nC, nT, stirling, _multiset_histogram, _AOP_product)
from sympy.combinatorics.permutations import Permutation
from sympy.core.numbers import oo
from random import choice
c = string.ascii_lowercase
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nP(s, i)
tot += check
assert len(list(multiset_permutations(s, i))) == check
if u:
assert nP(len(s), i) == check
assert nP(s) == tot
except AssertionError:
print(s, i, 'failed perm test')
raise ValueError()
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nC(s, i)
tot += check
assert len(list(multiset_combinations(s, i))) == check
if u:
assert nC(len(s), i) == check
assert nC(s) == tot
if u:
assert nC(len(s)) == tot
except AssertionError:
print(s, i, 'failed combo test')
raise ValueError()
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(i, j)
assert check.is_Integer
tot += check
assert sum(1 for p in partitions(i, j, size=True) if p[0] == j) == check
assert nT(i) == tot
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(range(i), j)
tot += check
assert len(list(multiset_partitions(list(range(i)), j))) == check
assert nT(range(i)) == tot
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(1, 8):
check = nT(s, i)
tot += check
assert len(list(multiset_partitions(s, i))) == check
if u:
assert nT(range(len(s)), i) == check
if u:
assert nT(range(len(s))) == tot
assert nT(s) == tot
except AssertionError:
print(s, i, 'failed partition test')
raise ValueError()
# tests for Stirling numbers of the first kind that are not tested in the
# above
assert [stirling(9, i, kind=1) for i in range(11)] == [
0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0]
perms = list(permutations(range(4)))
assert [sum(1 for p in perms if Permutation(p).cycles == i)
for i in range(5)] == [0, 6, 11, 6, 1] == [
stirling(4, i, kind=1) for i in range(5)]
# http://oeis.org/A008275
assert [stirling(n, k, signed=1)
for n in range(10) for k in range(1, n + 1)] == [
1, -1,
1, 2, -3,
1, -6, 11, -6,
1, 24, -50, 35, -10,
1, -120, 274, -225, 85, -15,
1, 720, -1764, 1624, -735, 175, -21,
1, -5040, 13068, -13132, 6769, -1960, 322, -28,
1, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
assert [stirling(n, k, kind=1)
for n in range(10) for k in range(n+1)] == [
1,
0, 1,
0, 1, 1,
0, 2, 3, 1,
0, 6, 11, 6, 1,
0, 24, 50, 35, 10, 1,
0, 120, 274, 225, 85, 15, 1,
0, 720, 1764, 1624, 735, 175, 21, 1,
0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1,
0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
assert [stirling(n, k, kind=2)
for n in range(10) for k in range(n+1)] == [
1,
0, 1,
0, 1, 1,
0, 1, 3, 1,
0, 1, 7, 6, 1,
0, 1, 15, 25, 10, 1,
0, 1, 31, 90, 65, 15, 1,
0, 1, 63, 301, 350, 140, 21, 1,
0, 1, 127, 966, 1701, 1050, 266, 28, 1,
0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1]
assert stirling(3, 4, kind=1) == stirling(3, 4, kind=1) == 0
raises(ValueError, lambda: stirling(-2, 2))
def delta(p):
if len(p) == 1:
return oo
return min(abs(i[0] - i[1]) for i in subsets(p, 2))
parts = multiset_partitions(range(5), 3)
d = 2
assert (sum(1 for p in parts if all(delta(i) >= d for i in p)) ==
stirling(5, 3, d=d) == 7)
# other coverage tests
assert nC('abb', 2) == nC('aab', 2) == 2
assert nP(3, 3, replacement=True) == nP('aabc', 3, replacement=True) == 27
assert nP(3, 4) == 0
assert nP('aabc', 5) == 0
assert nC(4, 2, replacement=True) == nC('abcdd', 2, replacement=True) == \
len(list(multiset_combinations('aabbccdd', 2))) == 10
assert nC('abcdd') == sum(nC('abcdd', i) for i in range(6)) == 24
assert nC(list('abcdd'), 4) == 4
assert nT('aaaa') == nT(4) == len(list(partitions(4))) == 5
assert nT('aaab') == len(list(multiset_partitions('aaab'))) == 7
assert nC('aabb'*3, 3) == 4 # aaa, bbb, abb, baa
assert dict(_AOP_product((4,1,1,1))) == {
0: 1, 1: 4, 2: 7, 3: 8, 4: 8, 5: 7, 6: 4, 7: 1}
# the following was the first t that showed a problem in a previous form of
# the function, so it's not as random as it may appear
t = (3, 9, 4, 6, 6, 5, 5, 2, 10, 4)
assert sum(_AOP_product(t)[i] for i in range(55)) == 58212000
raises(ValueError, lambda: _multiset_histogram({1:'a'}))
def test_PR_14617():
from sympy.functions.combinatorial.numbers import nT
for n in (0, []):
for k in (-1, 0, 1):
if k == 0:
assert nT(n, k) == 1
else:
assert nT(n, k) == 0
def test_issue_8496():
n = Symbol("n")
k = Symbol("k")
raises(TypeError, lambda: catalan(n, k))
def test_issue_8601():
n = Symbol('n', integer=True, negative=True)
assert catalan(n - 1) == S.Zero
assert catalan(-S.Half) == S.ComplexInfinity
assert catalan(-S.One) == -S.Half
c1 = catalan(-5.6).evalf()
assert str(c1) == '6.93334070531408e-5'
c2 = catalan(-35.4).evalf()
assert str(c2) == '-4.14189164517449e-24'
|
7b9c9162d7e6395b7585ac59ebbcf3fc1cf514974114dfa057f8593c5af94744 | from sympy import (symbols, Symbol, nan, oo, zoo, I, sinh, sin, pi, atan,
acos, Rational, sqrt, asin, acot, coth, E, S, tan, tanh, cos,
cosh, atan2, exp, log, asinh, acoth, atanh, O, cancel, Matrix, re, im,
Float, Pow, gcd, sec, csc, cot, diff, simplify, Heaviside, arg,
conjugate, series, FiniteSet, asec, acsc, Mul, sinc, jn, Product,
AccumBounds)
from sympy.core.compatibility import range
from sympy.utilities.pytest import XFAIL, slow, raises
from sympy.core.relational import Ne, Eq
from sympy.functions.elementary.piecewise import Piecewise
x, y, z = symbols('x y z')
r = Symbol('r', real=True)
k = Symbol('k', integer=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
a = Symbol('a', algebraic=True)
na = Symbol('na', nonzero=True, algebraic=True)
def test_sin():
x, y = symbols('x y')
assert sin.nargs == FiniteSet(1)
assert sin(nan) == nan
assert sin(zoo) == nan
assert sin(oo) == AccumBounds(-1, 1)
assert sin(oo) - sin(oo) == AccumBounds(-2, 2)
assert sin(oo*I) == oo*I
assert sin(-oo*I) == -oo*I
assert 0*sin(oo) == S.Zero
assert 0/sin(oo) == S.Zero
assert 0 + sin(oo) == AccumBounds(-1, 1)
assert 5 + sin(oo) == AccumBounds(4, 6)
assert sin(0) == 0
assert sin(asin(x)) == x
assert sin(atan(x)) == x / sqrt(1 + x**2)
assert sin(acos(x)) == sqrt(1 - x**2)
assert sin(acot(x)) == 1 / (sqrt(1 + 1 / x**2) * x)
assert sin(acsc(x)) == 1 / x
assert sin(asec(x)) == sqrt(1 - 1 / x**2)
assert sin(atan2(y, x)) == y / sqrt(x**2 + y**2)
assert sin(pi*I) == sinh(pi)*I
assert sin(-pi*I) == -sinh(pi)*I
assert sin(-2*I) == -sinh(2)*I
assert sin(pi) == 0
assert sin(-pi) == 0
assert sin(2*pi) == 0
assert sin(-2*pi) == 0
assert sin(-3*10**73*pi) == 0
assert sin(7*10**103*pi) == 0
assert sin(pi/2) == 1
assert sin(-pi/2) == -1
assert sin(5*pi/2) == 1
assert sin(7*pi/2) == -1
ne = symbols('ne', integer=True, even=False)
e = symbols('e', even=True)
assert sin(pi*ne/2) == (-1)**(ne/2 - S.Half)
assert sin(pi*k/2).func == sin
assert sin(pi*e/2) == 0
assert sin(pi*k) == 0
assert sin(pi*k).subs(k, 3) == sin(pi*k/2).subs(k, 6) # issue 8298
assert sin(pi/3) == S.Half*sqrt(3)
assert sin(-2*pi/3) == -S.Half*sqrt(3)
assert sin(pi/4) == S.Half*sqrt(2)
assert sin(-pi/4) == -S.Half*sqrt(2)
assert sin(17*pi/4) == S.Half*sqrt(2)
assert sin(-3*pi/4) == -S.Half*sqrt(2)
assert sin(pi/6) == S.Half
assert sin(-pi/6) == -S.Half
assert sin(7*pi/6) == -S.Half
assert sin(-5*pi/6) == -S.Half
assert sin(1*pi/5) == sqrt((5 - sqrt(5)) / 8)
assert sin(2*pi/5) == sqrt((5 + sqrt(5)) / 8)
assert sin(3*pi/5) == sin(2*pi/5)
assert sin(4*pi/5) == sin(1*pi/5)
assert sin(6*pi/5) == -sin(1*pi/5)
assert sin(8*pi/5) == -sin(2*pi/5)
assert sin(-1273*pi/5) == -sin(2*pi/5)
assert sin(pi/8) == sqrt((2 - sqrt(2))/4)
assert sin(pi/10) == -S(1)/4 + sqrt(5)/4
assert sin(pi/12) == -sqrt(2)/4 + sqrt(6)/4
assert sin(5*pi/12) == sqrt(2)/4 + sqrt(6)/4
assert sin(-7*pi/12) == -sqrt(2)/4 - sqrt(6)/4
assert sin(-11*pi/12) == sqrt(2)/4 - sqrt(6)/4
assert sin(104*pi/105) == sin(pi/105)
assert sin(106*pi/105) == -sin(pi/105)
assert sin(-104*pi/105) == -sin(pi/105)
assert sin(-106*pi/105) == sin(pi/105)
assert sin(x*I) == sinh(x)*I
assert sin(k*pi) == 0
assert sin(17*k*pi) == 0
assert sin(k*pi*I) == sinh(k*pi)*I
assert sin(r).is_real is True
assert sin(0, evaluate=False).is_algebraic
assert sin(a).is_algebraic is None
assert sin(na).is_algebraic is False
q = Symbol('q', rational=True)
assert sin(pi*q).is_algebraic
qn = Symbol('qn', rational=True, nonzero=True)
assert sin(qn).is_rational is False
assert sin(q).is_rational is None # issue 8653
assert isinstance(sin( re(x) - im(y)), sin) is True
assert isinstance(sin(-re(x) + im(y)), sin) is False
for d in list(range(1, 22)) + [60, 85]:
for n in range(0, d*2 + 1):
x = n*pi/d
e = abs( float(sin(x)) - sin(float(x)) )
assert e < 1e-12
def test_sin_cos():
for d in [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 24, 30, 40, 60, 120]: # list is not exhaustive...
for n in range(-2*d, d*2):
x = n*pi/d
assert sin(x + pi/2) == cos(x), "fails for %d*pi/%d" % (n, d)
assert sin(x - pi/2) == -cos(x), "fails for %d*pi/%d" % (n, d)
assert sin(x) == cos(x - pi/2), "fails for %d*pi/%d" % (n, d)
assert -sin(x) == cos(x + pi/2), "fails for %d*pi/%d" % (n, d)
def test_sin_series():
assert sin(x).series(x, 0, 9) == \
x - x**3/6 + x**5/120 - x**7/5040 + O(x**9)
def test_sin_rewrite():
assert sin(x).rewrite(exp) == -I*(exp(I*x) - exp(-I*x))/2
assert sin(x).rewrite(tan) == 2*tan(x/2)/(1 + tan(x/2)**2)
assert sin(x).rewrite(cot) == 2*cot(x/2)/(1 + cot(x/2)**2)
assert sin(sinh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sinh(3)).n()
assert sin(cosh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cosh(3)).n()
assert sin(tanh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tanh(3)).n()
assert sin(coth(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, coth(3)).n()
assert sin(sin(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sin(3)).n()
assert sin(cos(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cos(3)).n()
assert sin(tan(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tan(3)).n()
assert sin(cot(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cot(3)).n()
assert sin(log(x)).rewrite(Pow) == I*x**-I / 2 - I*x**I /2
assert sin(x).rewrite(csc) == 1/csc(x)
assert sin(x).rewrite(cos) == cos(x - pi / 2, evaluate=False)
assert sin(x).rewrite(sec) == 1 / sec(x - pi / 2, evaluate=False)
def test_sin_expansion():
# Note: these formulas are not unique. The ones here come from the
# Chebyshev formulas.
assert sin(x + y).expand(trig=True) == sin(x)*cos(y) + cos(x)*sin(y)
assert sin(x - y).expand(trig=True) == sin(x)*cos(y) - cos(x)*sin(y)
assert sin(y - x).expand(trig=True) == cos(x)*sin(y) - sin(x)*cos(y)
assert sin(2*x).expand(trig=True) == 2*sin(x)*cos(x)
assert sin(3*x).expand(trig=True) == -4*sin(x)**3 + 3*sin(x)
assert sin(4*x).expand(trig=True) == -8*sin(x)**3*cos(x) + 4*sin(x)*cos(x)
assert sin(2).expand(trig=True) == 2*sin(1)*cos(1)
assert sin(3).expand(trig=True) == -4*sin(1)**3 + 3*sin(1)
def test_sin_AccumBounds():
assert sin(AccumBounds(-oo, oo)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, oo)) == AccumBounds(-1, 1)
assert sin(AccumBounds(-oo, 0)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, 3*S.Pi/4)) == AccumBounds(0, 1)
assert sin(AccumBounds(3*S.Pi/4, 7*S.Pi/4)) == AccumBounds(-1, sin(3*S.Pi/4))
assert sin(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(sin(S.Pi/4), sin(S.Pi/3))
assert sin(AccumBounds(3*S.Pi/4, 5*S.Pi/6)) == AccumBounds(sin(5*S.Pi/6), sin(3*S.Pi/4))
def test_trig_symmetry():
assert sin(-x) == -sin(x)
assert cos(-x) == cos(x)
assert tan(-x) == -tan(x)
assert cot(-x) == -cot(x)
assert sin(x + pi) == -sin(x)
assert sin(x + 2*pi) == sin(x)
assert sin(x + 3*pi) == -sin(x)
assert sin(x + 4*pi) == sin(x)
assert sin(x - 5*pi) == -sin(x)
assert cos(x + pi) == -cos(x)
assert cos(x + 2*pi) == cos(x)
assert cos(x + 3*pi) == -cos(x)
assert cos(x + 4*pi) == cos(x)
assert cos(x - 5*pi) == -cos(x)
assert tan(x + pi) == tan(x)
assert tan(x - 3*pi) == tan(x)
assert cot(x + pi) == cot(x)
assert cot(x - 3*pi) == cot(x)
assert sin(pi/2 - x) == cos(x)
assert sin(3*pi/2 - x) == -cos(x)
assert sin(5*pi/2 - x) == cos(x)
assert cos(pi/2 - x) == sin(x)
assert cos(3*pi/2 - x) == -sin(x)
assert cos(5*pi/2 - x) == sin(x)
assert tan(pi/2 - x) == cot(x)
assert tan(3*pi/2 - x) == cot(x)
assert tan(5*pi/2 - x) == cot(x)
assert cot(pi/2 - x) == tan(x)
assert cot(3*pi/2 - x) == tan(x)
assert cot(5*pi/2 - x) == tan(x)
assert sin(pi/2 + x) == cos(x)
assert cos(pi/2 + x) == -sin(x)
assert tan(pi/2 + x) == -cot(x)
assert cot(pi/2 + x) == -tan(x)
def test_cos():
x, y = symbols('x y')
assert cos.nargs == FiniteSet(1)
assert cos(nan) == nan
assert cos(oo) == AccumBounds(-1, 1)
assert cos(oo) - cos(oo) == AccumBounds(-2, 2)
assert cos(oo*I) == oo
assert cos(-oo*I) == oo
assert cos(zoo) == nan
assert cos(0) == 1
assert cos(acos(x)) == x
assert cos(atan(x)) == 1 / sqrt(1 + x**2)
assert cos(asin(x)) == sqrt(1 - x**2)
assert cos(acot(x)) == 1 / sqrt(1 + 1 / x**2)
assert cos(acsc(x)) == sqrt(1 - 1 / x**2)
assert cos(asec(x)) == 1 / x
assert cos(atan2(y, x)) == x / sqrt(x**2 + y**2)
assert cos(pi*I) == cosh(pi)
assert cos(-pi*I) == cosh(pi)
assert cos(-2*I) == cosh(2)
assert cos(pi/2) == 0
assert cos(-pi/2) == 0
assert cos(pi/2) == 0
assert cos(-pi/2) == 0
assert cos((-3*10**73 + 1)*pi/2) == 0
assert cos((7*10**103 + 1)*pi/2) == 0
n = symbols('n', integer=True, even=False)
e = symbols('e', even=True)
assert cos(pi*n/2) == 0
assert cos(pi*e/2) == (-1)**(e/2)
assert cos(pi) == -1
assert cos(-pi) == -1
assert cos(2*pi) == 1
assert cos(5*pi) == -1
assert cos(8*pi) == 1
assert cos(pi/3) == S.Half
assert cos(-2*pi/3) == -S.Half
assert cos(pi/4) == S.Half*sqrt(2)
assert cos(-pi/4) == S.Half*sqrt(2)
assert cos(11*pi/4) == -S.Half*sqrt(2)
assert cos(-3*pi/4) == -S.Half*sqrt(2)
assert cos(pi/6) == S.Half*sqrt(3)
assert cos(-pi/6) == S.Half*sqrt(3)
assert cos(7*pi/6) == -S.Half*sqrt(3)
assert cos(-5*pi/6) == -S.Half*sqrt(3)
assert cos(1*pi/5) == (sqrt(5) + 1)/4
assert cos(2*pi/5) == (sqrt(5) - 1)/4
assert cos(3*pi/5) == -cos(2*pi/5)
assert cos(4*pi/5) == -cos(1*pi/5)
assert cos(6*pi/5) == -cos(1*pi/5)
assert cos(8*pi/5) == cos(2*pi/5)
assert cos(-1273*pi/5) == -cos(2*pi/5)
assert cos(pi/8) == sqrt((2 + sqrt(2))/4)
assert cos(pi/12) == sqrt(2)/4 + sqrt(6)/4
assert cos(5*pi/12) == -sqrt(2)/4 + sqrt(6)/4
assert cos(7*pi/12) == sqrt(2)/4 - sqrt(6)/4
assert cos(11*pi/12) == -sqrt(2)/4 - sqrt(6)/4
assert cos(104*pi/105) == -cos(pi/105)
assert cos(106*pi/105) == -cos(pi/105)
assert cos(-104*pi/105) == -cos(pi/105)
assert cos(-106*pi/105) == -cos(pi/105)
assert cos(x*I) == cosh(x)
assert cos(k*pi*I) == cosh(k*pi)
assert cos(r).is_real is True
assert cos(0, evaluate=False).is_algebraic
assert cos(a).is_algebraic is None
assert cos(na).is_algebraic is False
q = Symbol('q', rational=True)
assert cos(pi*q).is_algebraic
assert cos(2*pi/7).is_algebraic
assert cos(k*pi) == (-1)**k
assert cos(2*k*pi) == 1
for d in list(range(1, 22)) + [60, 85]:
for n in range(0, 2*d + 1):
x = n*pi/d
e = abs( float(cos(x)) - cos(float(x)) )
assert e < 1e-12
def test_issue_6190():
c = Float('123456789012345678901234567890.25', '')
for cls in [sin, cos, tan, cot]:
assert cls(c*pi) == cls(pi/4)
assert cls(4.125*pi) == cls(pi/8)
assert cls(4.7*pi) == cls((4.7 % 2)*pi)
def test_cos_series():
assert cos(x).series(x, 0, 9) == \
1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 + O(x**9)
def test_cos_rewrite():
assert cos(x).rewrite(exp) == exp(I*x)/2 + exp(-I*x)/2
assert cos(x).rewrite(tan) == (1 - tan(x/2)**2)/(1 + tan(x/2)**2)
assert cos(x).rewrite(cot) == -(1 - cot(x/2)**2)/(1 + cot(x/2)**2)
assert cos(sinh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sinh(3)).n()
assert cos(cosh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cosh(3)).n()
assert cos(tanh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tanh(3)).n()
assert cos(coth(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, coth(3)).n()
assert cos(sin(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sin(3)).n()
assert cos(cos(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cos(3)).n()
assert cos(tan(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tan(3)).n()
assert cos(cot(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cot(3)).n()
assert cos(log(x)).rewrite(Pow) == x**I/2 + x**-I/2
assert cos(x).rewrite(sec) == 1/sec(x)
assert cos(x).rewrite(sin) == sin(x + pi/2, evaluate=False)
assert cos(x).rewrite(csc) == 1/csc(-x + pi/2, evaluate=False)
def test_cos_expansion():
assert cos(x + y).expand(trig=True) == cos(x)*cos(y) - sin(x)*sin(y)
assert cos(x - y).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y)
assert cos(y - x).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y)
assert cos(2*x).expand(trig=True) == 2*cos(x)**2 - 1
assert cos(3*x).expand(trig=True) == 4*cos(x)**3 - 3*cos(x)
assert cos(4*x).expand(trig=True) == 8*cos(x)**4 - 8*cos(x)**2 + 1
assert cos(2).expand(trig=True) == 2*cos(1)**2 - 1
assert cos(3).expand(trig=True) == 4*cos(1)**3 - 3*cos(1)
def test_cos_AccumBounds():
assert cos(AccumBounds(-oo, oo)) == AccumBounds(-1, 1)
assert cos(AccumBounds(0, oo)) == AccumBounds(-1, 1)
assert cos(AccumBounds(-oo, 0)) == AccumBounds(-1, 1)
assert cos(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1)
assert cos(AccumBounds(-S.Pi/3, S.Pi/4)) == AccumBounds(cos(-S.Pi/3), 1)
assert cos(AccumBounds(3*S.Pi/4, 5*S.Pi/4)) == AccumBounds(-1, cos(3*S.Pi/4))
assert cos(AccumBounds(5*S.Pi/4, 4*S.Pi/3)) == AccumBounds(cos(5*S.Pi/4), cos(4*S.Pi/3))
assert cos(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(cos(S.Pi/3), cos(S.Pi/4))
def test_tan():
assert tan(nan) == nan
assert tan(zoo) == nan
assert tan(oo) == AccumBounds(-oo, oo)
assert tan(oo) - tan(oo) == AccumBounds(-oo, oo)
assert tan.nargs == FiniteSet(1)
assert tan(oo*I) == I
assert tan(-oo*I) == -I
assert tan(0) == 0
assert tan(atan(x)) == x
assert tan(asin(x)) == x / sqrt(1 - x**2)
assert tan(acos(x)) == sqrt(1 - x**2) / x
assert tan(acot(x)) == 1 / x
assert tan(acsc(x)) == 1 / (sqrt(1 - 1 / x**2) * x)
assert tan(asec(x)) == sqrt(1 - 1 / x**2) * x
assert tan(atan2(y, x)) == y/x
assert tan(pi*I) == tanh(pi)*I
assert tan(-pi*I) == -tanh(pi)*I
assert tan(-2*I) == -tanh(2)*I
assert tan(pi) == 0
assert tan(-pi) == 0
assert tan(2*pi) == 0
assert tan(-2*pi) == 0
assert tan(-3*10**73*pi) == 0
assert tan(pi/2) == zoo
assert tan(3*pi/2) == zoo
assert tan(pi/3) == sqrt(3)
assert tan(-2*pi/3) == sqrt(3)
assert tan(pi/4) == S.One
assert tan(-pi/4) == -S.One
assert tan(17*pi/4) == S.One
assert tan(-3*pi/4) == S.One
assert tan(pi/6) == 1/sqrt(3)
assert tan(-pi/6) == -1/sqrt(3)
assert tan(7*pi/6) == 1/sqrt(3)
assert tan(-5*pi/6) == 1/sqrt(3)
assert tan(pi/8).expand() == -1 + sqrt(2)
assert tan(3*pi/8).expand() == 1 + sqrt(2)
assert tan(5*pi/8).expand() == -1 - sqrt(2)
assert tan(7*pi/8).expand() == 1 - sqrt(2)
assert tan(pi/12) == -sqrt(3) + 2
assert tan(5*pi/12) == sqrt(3) + 2
assert tan(7*pi/12) == -sqrt(3) - 2
assert tan(11*pi/12) == sqrt(3) - 2
assert tan(pi/24).radsimp() == -2 - sqrt(3) + sqrt(2) + sqrt(6)
assert tan(5*pi/24).radsimp() == -2 + sqrt(3) - sqrt(2) + sqrt(6)
assert tan(7*pi/24).radsimp() == 2 - sqrt(3) - sqrt(2) + sqrt(6)
assert tan(11*pi/24).radsimp() == 2 + sqrt(3) + sqrt(2) + sqrt(6)
assert tan(13*pi/24).radsimp() == -2 - sqrt(3) - sqrt(2) - sqrt(6)
assert tan(17*pi/24).radsimp() == -2 + sqrt(3) + sqrt(2) - sqrt(6)
assert tan(19*pi/24).radsimp() == 2 - sqrt(3) + sqrt(2) - sqrt(6)
assert tan(23*pi/24).radsimp() == 2 + sqrt(3) - sqrt(2) - sqrt(6)
assert 1 == (tan(8*pi/15)*cos(8*pi/15)/sin(8*pi/15)).ratsimp()
assert tan(x*I) == tanh(x)*I
assert tan(k*pi) == 0
assert tan(17*k*pi) == 0
assert tan(k*pi*I) == tanh(k*pi)*I
assert tan(r).is_real is True
assert tan(0, evaluate=False).is_algebraic
assert tan(a).is_algebraic is None
assert tan(na).is_algebraic is False
assert tan(10*pi/7) == tan(3*pi/7)
assert tan(11*pi/7) == -tan(3*pi/7)
assert tan(-11*pi/7) == tan(3*pi/7)
assert tan(15*pi/14) == tan(pi/14)
assert tan(-15*pi/14) == -tan(pi/14)
def test_tan_series():
assert tan(x).series(x, 0, 9) == \
x + x**3/3 + 2*x**5/15 + 17*x**7/315 + O(x**9)
def test_tan_rewrite():
neg_exp, pos_exp = exp(-x*I), exp(x*I)
assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
assert tan(x).rewrite(cos) == cos(x - S.Pi/2, evaluate=False)/cos(x)
assert tan(x).rewrite(cot) == 1/cot(x)
assert tan(sinh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
assert tan(cosh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
assert tan(tanh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
assert tan(coth(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
assert tan(sin(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
assert tan(cos(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
assert tan(tan(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
assert tan(cot(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
assert 0 == (cos(pi/34)*tan(pi/34) - sin(pi/34)).rewrite(pow)
assert 0 == (cos(pi/17)*tan(pi/17) - sin(pi/17)).rewrite(pow)
assert tan(pi/19).rewrite(pow) == tan(pi/19)
assert tan(8*pi/19).rewrite(sqrt) == tan(8*pi/19)
assert tan(x).rewrite(sec) == sec(x)/sec(x - pi/2, evaluate=False)
assert tan(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)/csc(x)
def test_tan_subs():
assert tan(x).subs(tan(x), y) == y
assert tan(x).subs(x, y) == tan(y)
assert tan(x).subs(x, S.Pi/2) == zoo
assert tan(x).subs(x, 3*S.Pi/2) == zoo
def test_tan_expansion():
assert tan(x + y).expand(trig=True) == ((tan(x) + tan(y))/(1 - tan(x)*tan(y))).expand()
assert tan(x - y).expand(trig=True) == ((tan(x) - tan(y))/(1 + tan(x)*tan(y))).expand()
assert tan(x + y + z).expand(trig=True) == (
(tan(x) + tan(y) + tan(z) - tan(x)*tan(y)*tan(z))/
(1 - tan(x)*tan(y) - tan(x)*tan(z) - tan(y)*tan(z))).expand()
assert 0 == tan(2*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 7))])*24 - 7
assert 0 == tan(3*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*55 - 37
assert 0 == tan(4*x - pi/4).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*239 - 1
def test_tan_AccumBounds():
assert tan(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo)
assert tan(AccumBounds(S.Pi/3, 2*S.Pi/3)) == AccumBounds(-oo, oo)
assert tan(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(tan(S.Pi/6), tan(S.Pi/3))
def test_cot():
assert cot(nan) == nan
assert cot.nargs == FiniteSet(1)
assert cot(oo*I) == -I
assert cot(-oo*I) == I
assert cot(zoo) == nan
assert cot(0) == zoo
assert cot(2*pi) == zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(acsc(x)) == sqrt(1 - 1 / x**2) * x
assert cot(asec(x)) == 1 / (sqrt(1 - 1 / x**2) * x)
assert cot(atan2(y, x)) == x/y
assert cot(pi*I) == -coth(pi)*I
assert cot(-pi*I) == coth(pi)*I
assert cot(-2*I) == coth(2)*I
assert cot(pi) == cot(2*pi) == cot(3*pi)
assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
assert cot(pi/2) == 0
assert cot(-pi/2) == 0
assert cot(5*pi/2) == 0
assert cot(7*pi/2) == 0
assert cot(pi/3) == 1/sqrt(3)
assert cot(-2*pi/3) == 1/sqrt(3)
assert cot(pi/4) == S.One
assert cot(-pi/4) == -S.One
assert cot(17*pi/4) == S.One
assert cot(-3*pi/4) == S.One
assert cot(pi/6) == sqrt(3)
assert cot(-pi/6) == -sqrt(3)
assert cot(7*pi/6) == sqrt(3)
assert cot(-5*pi/6) == sqrt(3)
assert cot(pi/8).expand() == 1 + sqrt(2)
assert cot(3*pi/8).expand() == -1 + sqrt(2)
assert cot(5*pi/8).expand() == 1 - sqrt(2)
assert cot(7*pi/8).expand() == -1 - sqrt(2)
assert cot(pi/12) == sqrt(3) + 2
assert cot(5*pi/12) == -sqrt(3) + 2
assert cot(7*pi/12) == sqrt(3) - 2
assert cot(11*pi/12) == -sqrt(3) - 2
assert cot(pi/24).radsimp() == sqrt(2) + sqrt(3) + 2 + sqrt(6)
assert cot(5*pi/24).radsimp() == -sqrt(2) - sqrt(3) + 2 + sqrt(6)
assert cot(7*pi/24).radsimp() == -sqrt(2) + sqrt(3) - 2 + sqrt(6)
assert cot(11*pi/24).radsimp() == sqrt(2) - sqrt(3) - 2 + sqrt(6)
assert cot(13*pi/24).radsimp() == -sqrt(2) + sqrt(3) + 2 - sqrt(6)
assert cot(17*pi/24).radsimp() == sqrt(2) - sqrt(3) + 2 - sqrt(6)
assert cot(19*pi/24).radsimp() == sqrt(2) + sqrt(3) - 2 - sqrt(6)
assert cot(23*pi/24).radsimp() == -sqrt(2) - sqrt(3) - 2 - sqrt(6)
assert 1 == (cot(4*pi/15)*sin(4*pi/15)/cos(4*pi/15)).ratsimp()
assert cot(x*I) == -coth(x)*I
assert cot(k*pi*I) == -coth(k*pi)*I
assert cot(r).is_real is True
assert cot(a).is_algebraic is None
assert cot(na).is_algebraic is False
assert cot(10*pi/7) == cot(3*pi/7)
assert cot(11*pi/7) == -cot(3*pi/7)
assert cot(-11*pi/7) == cot(3*pi/7)
assert cot(39*pi/34) == cot(5*pi/34)
assert cot(-41*pi/34) == -cot(7*pi/34)
assert cot(x).is_finite is None
assert cot(r).is_finite is None
i = Symbol('i', imaginary=True)
assert cot(i).is_finite is True
assert cot(x).subs(x, 3*pi) == zoo
def test_cot_series():
assert cot(x).series(x, 0, 9) == \
1/x - x/3 - x**3/45 - 2*x**5/945 - x**7/4725 + O(x**9)
# issue 6210
assert cot(x**4 + x**5).series(x, 0, 1) == \
x**(-4) - 1/x**3 + x**(-2) - 1/x + 1 + O(x)
def test_cot_rewrite():
neg_exp, pos_exp = exp(-x*I), exp(x*I)
assert cot(x).rewrite(exp) == I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
assert cot(x).rewrite(sin) == sin(2*x)/(2*(sin(x)**2))
assert cot(x).rewrite(cos) == cos(x)/cos(x - pi/2, evaluate=False)
assert cot(x).rewrite(tan) == 1/tan(x)
assert cot(sinh(x)).rewrite(
exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, sinh(3)).n()
assert cot(cosh(x)).rewrite(
exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, cosh(3)).n()
assert cot(tanh(x)).rewrite(
exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, tanh(3)).n()
assert cot(coth(x)).rewrite(
exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, coth(3)).n()
assert cot(sin(x)).rewrite(
exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, sin(3)).n()
assert cot(tan(x)).rewrite(
exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, tan(3)).n()
assert cot(log(x)).rewrite(Pow) == -I*(x**-I + x**I)/(x**-I - x**I)
assert cot(4*pi/34).rewrite(pow).ratsimp() == (cos(4*pi/34)/sin(4*pi/34)).rewrite(pow).ratsimp()
assert cot(4*pi/17).rewrite(pow) == (cos(4*pi/17)/sin(4*pi/17)).rewrite(pow)
assert cot(pi/19).rewrite(pow) == cot(pi/19)
assert cot(pi/19).rewrite(sqrt) == cot(pi/19)
assert cot(x).rewrite(sec) == sec(x - pi / 2, evaluate=False) / sec(x)
assert cot(x).rewrite(csc) == csc(x) / csc(- x + pi / 2, evaluate=False)
def test_cot_subs():
assert cot(x).subs(cot(x), y) == y
assert cot(x).subs(x, y) == cot(y)
assert cot(x).subs(x, 0) == zoo
assert cot(x).subs(x, S.Pi) == zoo
def test_cot_expansion():
assert cot(x + y).expand(trig=True) == ((cot(x)*cot(y) - 1)/(cot(x) + cot(y))).expand()
assert cot(x - y).expand(trig=True) == (-(cot(x)*cot(y) + 1)/(cot(x) - cot(y))).expand()
assert cot(x + y + z).expand(trig=True) == (
(cot(x)*cot(y)*cot(z) - cot(x) - cot(y) - cot(z))/
(-1 + cot(x)*cot(y) + cot(x)*cot(z) + cot(y)*cot(z))).expand()
assert cot(3*x).expand(trig=True) == ((cot(x)**3 - 3*cot(x))/(3*cot(x)**2 - 1)).expand()
assert 0 == cot(2*x).expand(trig=True).rewrite(cot).subs([(cot(x), Rational(1, 3))])*3 + 4
assert 0 == cot(3*x).expand(trig=True).rewrite(cot).subs([(cot(x), Rational(1, 5))])*55 - 37
assert 0 == cot(4*x - pi/4).expand(trig=True).rewrite(cot).subs([(cot(x), Rational(1, 7))])*863 + 191
def test_cot_AccumBounds():
assert cot(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo)
assert cot(AccumBounds(-S.Pi/3, S.Pi/3)) == AccumBounds(-oo, oo)
assert cot(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(cot(S.Pi/3), cot(S.Pi/6))
def test_sinc():
assert isinstance(sinc(x), sinc)
s = Symbol('s', zero=True)
assert sinc(s) == S.One
assert sinc(S.Infinity) == S.Zero
assert sinc(-S.Infinity) == S.Zero
assert sinc(S.NaN) == S.NaN
assert sinc(S.ComplexInfinity) == S.NaN
n = Symbol('n', integer=True, nonzero=True)
assert sinc(n*pi) == S.Zero
assert sinc(-n*pi) == S.Zero
assert sinc(pi/2) == 2 / pi
assert sinc(-pi/2) == 2 / pi
assert sinc(5*pi/2) == 2 / (5*pi)
assert sinc(7*pi/2) == -2 / (7*pi)
assert sinc(-x) == sinc(x)
assert sinc(x).diff() == (x*cos(x) - sin(x)) / x**2
assert sinc(x).series() == 1 - x**2/6 + x**4/120 + O(x**6)
assert sinc(x).rewrite(jn) == jn(0, x)
assert sinc(x).rewrite(sin) == Piecewise((sin(x)/x, Ne(x, 0)), (1, True))
def test_asin():
assert asin(nan) == nan
assert asin.nargs == FiniteSet(1)
assert asin(oo) == -I*oo
assert asin(-oo) == I*oo
assert asin(zoo) == zoo
# Note: asin(-x) = - asin(x)
assert asin(0) == 0
assert asin(1) == pi/2
assert asin(-1) == -pi/2
assert asin(sqrt(3)/2) == pi/3
assert asin(-sqrt(3)/2) == -pi/3
assert asin(sqrt(2)/2) == pi/4
assert asin(-sqrt(2)/2) == -pi/4
assert asin(sqrt((5 - sqrt(5))/8)) == pi/5
assert asin(-sqrt((5 - sqrt(5))/8)) == -pi/5
assert asin(Rational(1, 2)) == pi/6
assert asin(-Rational(1, 2)) == -pi/6
assert asin((sqrt(2 - sqrt(2)))/2) == pi/8
assert asin(-(sqrt(2 - sqrt(2)))/2) == -pi/8
assert asin((sqrt(5) - 1)/4) == pi/10
assert asin(-(sqrt(5) - 1)/4) == -pi/10
assert asin((sqrt(3) - 1)/sqrt(2**3)) == pi/12
assert asin(-(sqrt(3) - 1)/sqrt(2**3)) == -pi/12
assert asin(x).diff(x) == 1/sqrt(1 - x**2)
assert asin(0.2).is_real is True
assert asin(-2).is_real is False
assert asin(r).is_real is None
assert asin(-2*I) == -I*asinh(2)
assert asin(Rational(1, 7), evaluate=False).is_positive is True
assert asin(Rational(-1, 7), evaluate=False).is_positive is False
assert asin(p).is_positive is None
def test_asin_series():
assert asin(x).series(x, 0, 9) == \
x + x**3/6 + 3*x**5/40 + 5*x**7/112 + O(x**9)
t5 = asin(x).taylor_term(5, x)
assert t5 == 3*x**5/40
assert asin(x).taylor_term(7, x, t5, 0) == 5*x**7/112
def test_asin_rewrite():
assert asin(x).rewrite(log) == -I*log(I*x + sqrt(1 - x**2))
assert asin(x).rewrite(atan) == 2*atan(x/(1 + sqrt(1 - x**2)))
assert asin(x).rewrite(acos) == S.Pi/2 - acos(x)
assert asin(x).rewrite(acot) == 2*acot((sqrt(-x**2 + 1) + 1)/x)
assert asin(x).rewrite(asec) == -asec(1/x) + pi/2
assert asin(x).rewrite(acsc) == acsc(1/x)
def test_acos():
assert acos(nan) == nan
assert acos(zoo) == zoo
assert acos.nargs == FiniteSet(1)
assert acos(oo) == I*oo
assert acos(-oo) == -I*oo
# Note: acos(-x) = pi - acos(x)
assert acos(0) == pi/2
assert acos(Rational(1, 2)) == pi/3
assert acos(-Rational(1, 2)) == (2*pi)/3
assert acos(1) == 0
assert acos(-1) == pi
assert acos(sqrt(2)/2) == pi/4
assert acos(-sqrt(2)/2) == (3*pi)/4
assert acos(x).diff(x) == -1/sqrt(1 - x**2)
assert acos(0.2).is_real is True
assert acos(-2).is_real is False
assert acos(r).is_real is None
assert acos(Rational(1, 7), evaluate=False).is_positive is True
assert acos(Rational(-1, 7), evaluate=False).is_positive is True
assert acos(Rational(3, 2), evaluate=False).is_positive is False
assert acos(p).is_positive is None
assert acos(2 + p).conjugate() != acos(10 + p)
assert acos(-3 + n).conjugate() != acos(-3 + n)
assert acos(S.One/3).conjugate() == acos(S.One/3)
assert acos(-S.One/3).conjugate() == acos(-S.One/3)
assert acos(p + n*I).conjugate() == acos(p - n*I)
assert acos(z).conjugate() != acos(conjugate(z))
def test_acos_series():
assert acos(x).series(x, 0, 8) == \
pi/2 - x - x**3/6 - 3*x**5/40 - 5*x**7/112 + O(x**8)
assert acos(x).series(x, 0, 8) == pi/2 - asin(x).series(x, 0, 8)
t5 = acos(x).taylor_term(5, x)
assert t5 == -3*x**5/40
assert acos(x).taylor_term(7, x, t5, 0) == -5*x**7/112
def test_acos_rewrite():
assert acos(x).rewrite(log) == pi/2 + I*log(I*x + sqrt(1 - x**2))
assert acos(x).rewrite(atan) == \
atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2))
assert acos(0).rewrite(atan) == S.Pi/2
assert acos(0.5).rewrite(atan) == acos(0.5).rewrite(log)
assert acos(x).rewrite(asin) == S.Pi/2 - asin(x)
assert acos(x).rewrite(acot) == -2*acot((sqrt(-x**2 + 1) + 1)/x) + pi/2
assert acos(x).rewrite(asec) == asec(1/x)
assert acos(x).rewrite(acsc) == -acsc(1/x) + pi/2
def test_atan():
assert atan(nan) == nan
assert atan.nargs == FiniteSet(1)
assert atan(oo) == pi/2
assert atan(-oo) == -pi/2
assert atan(zoo) == AccumBounds(-pi/2, pi/2)
assert atan(0) == 0
assert atan(1) == pi/4
assert atan(sqrt(3)) == pi/3
assert atan(oo) == pi/2
assert atan(x).diff(x) == 1/(1 + x**2)
assert atan(r).is_real is True
assert atan(-2*I) == -I*atanh(2)
assert atan(p).is_positive is True
assert atan(n).is_positive is False
assert atan(x).is_positive is None
def test_atan_rewrite():
assert atan(x).rewrite(log) == I*(log(1 - I*x)-log(1 + I*x))/2
assert atan(x).rewrite(asin) == (-asin(1/sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
assert atan(x).rewrite(acos) == sqrt(x**2)*acos(1/sqrt(x**2 + 1))/x
assert atan(x).rewrite(acot) == acot(1/x)
assert atan(x).rewrite(asec) == sqrt(x**2)*asec(sqrt(x**2 + 1))/x
assert atan(x).rewrite(acsc) == (-acsc(sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
assert atan(-5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:-5*I})
assert atan(5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:5*I})
def test_atan2():
assert atan2.nargs == FiniteSet(2)
assert atan2(0, 0) == S.NaN
assert atan2(0, 1) == 0
assert atan2(1, 1) == pi/4
assert atan2(1, 0) == pi/2
assert atan2(1, -1) == 3*pi/4
assert atan2(0, -1) == pi
assert atan2(-1, -1) == -3*pi/4
assert atan2(-1, 0) == -pi/2
assert atan2(-1, 1) == -pi/4
i = symbols('i', imaginary=True)
r = symbols('r', real=True)
eq = atan2(r, i)
ans = -I*log((i + I*r)/sqrt(i**2 + r**2))
reps = ((r, 2), (i, I))
assert eq.subs(reps) == ans.subs(reps)
x = Symbol('x', negative=True)
y = Symbol('y', negative=True)
assert atan2(y, x) == atan(y/x) - pi
y = Symbol('y', nonnegative=True)
assert atan2(y, x) == atan(y/x) + pi
y = Symbol('y')
assert atan2(y, x) == atan2(y, x, evaluate=False)
u = Symbol("u", positive=True)
assert atan2(0, u) == 0
u = Symbol("u", negative=True)
assert atan2(0, u) == pi
assert atan2(y, oo) == 0
assert atan2(y, -oo)== 2*pi*Heaviside(re(y)) - pi
assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2))
assert atan2(0, 0).rewrite(atan) == S.NaN
w = Symbol('w')
assert atan2(0, w).rewrite(atan) == Piecewise((pi, w < 0), (0, w > 0), (S.NaN, True))
ex = atan2(y, x) - arg(x + I*y)
assert ex.subs({x:2, y:3}).rewrite(arg) == 0
assert ex.subs({x:2, y:3*I}).rewrite(arg) == -pi - I*log(sqrt(5)*I/5)
assert ex.subs({x:2*I, y:3}).rewrite(arg) == -pi/2 - I*log(sqrt(5)*I)
assert ex.subs({x:2*I, y:3*I}).rewrite(arg) == -pi + atan(2/S(3)) + atan(3/S(2))
i = symbols('i', imaginary=True)
r = symbols('r', real=True)
e = atan2(i, r)
rewrite = e.rewrite(arg)
reps = {i: I, r: -2}
assert rewrite == -I*log(abs(I*i + r)/sqrt(abs(i**2 + r**2))) + arg((I*i + r)/sqrt(i**2 + r**2))
assert (e - rewrite).subs(reps).equals(0)
assert conjugate(atan2(x, y)) == atan2(conjugate(x), conjugate(y))
assert diff(atan2(y, x), x) == -y/(x**2 + y**2)
assert diff(atan2(y, x), y) == x/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), y)) == x/(x**2 + y**2)
def test_acot():
assert acot(nan) == nan
assert acot.nargs == FiniteSet(1)
assert acot(-oo) == 0
assert acot(oo) == 0
assert acot(zoo) == 0
assert acot(1) == pi/4
assert acot(0) == pi/2
assert acot(sqrt(3)/3) == pi/3
assert acot(1/sqrt(3)) == pi/3
assert acot(-1/sqrt(3)) == -pi/3
assert acot(x).diff(x) == -1/(1 + x**2)
assert acot(r).is_real is True
assert acot(I*pi) == -I*acoth(pi)
assert acot(-2*I) == I*acoth(2)
assert acot(x).is_positive is None
assert acot(n).is_positive is False
assert acot(p).is_positive is True
assert acot(I).is_positive is False
def test_acot_rewrite():
assert acot(x).rewrite(log) == I*(log(1 - I/x)-log(1 + I/x))/2
assert acot(x).rewrite(asin) == x*(-asin(sqrt(-x**2)/sqrt(-x**2 - 1)) + pi/2)*sqrt(x**(-2))
assert acot(x).rewrite(acos) == x*sqrt(x**(-2))*acos(sqrt(-x**2)/sqrt(-x**2 - 1))
assert acot(x).rewrite(atan) == atan(1/x)
assert acot(x).rewrite(asec) == x*sqrt(x**(-2))*asec(sqrt((x**2 + 1)/x**2))
assert acot(x).rewrite(acsc) == x*(-acsc(sqrt((x**2 + 1)/x**2)) + pi/2)*sqrt(x**(-2))
assert acot(-I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:-I/5})
assert acot(I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:I/5})
def test_attributes():
assert sin(x).args == (x,)
def test_sincos_rewrite():
assert sin(pi/2 - x) == cos(x)
assert sin(pi - x) == sin(x)
assert cos(pi/2 - x) == sin(x)
assert cos(pi - x) == -cos(x)
def _check_even_rewrite(func, arg):
"""Checks that the expr has been rewritten using f(-x) -> f(x)
arg : -x
"""
return func(arg).args[0] == -arg
def _check_odd_rewrite(func, arg):
"""Checks that the expr has been rewritten using f(-x) -> -f(x)
arg : -x
"""
return func(arg).func.is_Mul
def _check_no_rewrite(func, arg):
"""Checks that the expr is not rewritten"""
return func(arg).args[0] == arg
def test_evenodd_rewrite():
a = cos(2) # negative
b = sin(1) # positive
even = [cos]
odd = [sin, tan, cot, asin, atan, acot]
with_minus = [-1, -2**1024 * E, -pi/105, -x*y, -x - y]
for func in even:
for expr in with_minus:
assert _check_even_rewrite(func, expr)
assert _check_no_rewrite(func, a*b)
assert func(
x - y) == func(y - x) # it doesn't matter which form is canonical
for func in odd:
for expr in with_minus:
assert _check_odd_rewrite(func, expr)
assert _check_no_rewrite(func, a*b)
assert func(
x - y) == -func(y - x) # it doesn't matter which form is canonical
def test_issue_4547():
assert sin(x).rewrite(cot) == 2*cot(x/2)/(1 + cot(x/2)**2)
assert cos(x).rewrite(cot) == -(1 - cot(x/2)**2)/(1 + cot(x/2)**2)
assert tan(x).rewrite(cot) == 1/cot(x)
assert cot(x).fdiff() == -1 - cot(x)**2
def test_as_leading_term_issue_5272():
assert sin(x).as_leading_term(x) == x
assert cos(x).as_leading_term(x) == 1
assert tan(x).as_leading_term(x) == x
assert cot(x).as_leading_term(x) == 1/x
assert asin(x).as_leading_term(x) == x
assert acos(x).as_leading_term(x) == x
assert atan(x).as_leading_term(x) == x
assert acot(x).as_leading_term(x) == x
def test_leading_terms():
for func in [sin, cos, tan, cot, asin, acos, atan, acot]:
for arg in (1/x, S.Half):
eq = func(arg)
assert eq.as_leading_term(x) == eq
def test_atan2_expansion():
assert cancel(atan2(x**2, x + 1).diff(x) - atan(x**2/(x + 1)).diff(x)) == 0
assert cancel(atan(y/x).series(y, 0, 5) - atan2(y, x).series(y, 0, 5)
+ atan2(0, x) - atan(0)) == O(y**5)
assert cancel(atan(y/x).series(x, 1, 4) - atan2(y, x).series(x, 1, 4)
+ atan2(y, 1) - atan(y)) == O((x - 1)**4, (x, 1))
assert cancel(atan((y + x)/x).series(x, 1, 3) - atan2(y + x, x).series(x, 1, 3)
+ atan2(1 + y, 1) - atan(1 + y)) == O((x - 1)**3, (x, 1))
assert Matrix([atan2(y, x)]).jacobian([y, x]) == \
Matrix([[x/(y**2 + x**2), -y/(y**2 + x**2)]])
def test_aseries():
def t(n, v, d, e):
assert abs(
n(1/v).evalf() - n(1/x).series(x, dir=d).removeO().subs(x, v)) < e
t(atan, 0.1, '+', 1e-5)
t(atan, -0.1, '-', 1e-5)
t(acot, 0.1, '+', 1e-5)
t(acot, -0.1, '-', 1e-5)
def test_issue_4420():
i = Symbol('i', integer=True)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
# unknown parity for variable
assert cos(4*i*pi) == 1
assert sin(4*i*pi) == 0
assert tan(4*i*pi) == 0
assert cot(4*i*pi) == zoo
assert cos(3*i*pi) == cos(pi*i) # +/-1
assert sin(3*i*pi) == 0
assert tan(3*i*pi) == 0
assert cot(3*i*pi) == zoo
assert cos(4.0*i*pi) == 1
assert sin(4.0*i*pi) == 0
assert tan(4.0*i*pi) == 0
assert cot(4.0*i*pi) == zoo
assert cos(3.0*i*pi) == cos(pi*i) # +/-1
assert sin(3.0*i*pi) == 0
assert tan(3.0*i*pi) == 0
assert cot(3.0*i*pi) == zoo
assert cos(4.5*i*pi) == cos(0.5*pi*i)
assert sin(4.5*i*pi) == sin(0.5*pi*i)
assert tan(4.5*i*pi) == tan(0.5*pi*i)
assert cot(4.5*i*pi) == cot(0.5*pi*i)
# parity of variable is known
assert cos(4*e*pi) == 1
assert sin(4*e*pi) == 0
assert tan(4*e*pi) == 0
assert cot(4*e*pi) == zoo
assert cos(3*e*pi) == 1
assert sin(3*e*pi) == 0
assert tan(3*e*pi) == 0
assert cot(3*e*pi) == zoo
assert cos(4.0*e*pi) == 1
assert sin(4.0*e*pi) == 0
assert tan(4.0*e*pi) == 0
assert cot(4.0*e*pi) == zoo
assert cos(3.0*e*pi) == 1
assert sin(3.0*e*pi) == 0
assert tan(3.0*e*pi) == 0
assert cot(3.0*e*pi) == zoo
assert cos(4.5*e*pi) == cos(0.5*pi*e)
assert sin(4.5*e*pi) == sin(0.5*pi*e)
assert tan(4.5*e*pi) == tan(0.5*pi*e)
assert cot(4.5*e*pi) == cot(0.5*pi*e)
assert cos(4*o*pi) == 1
assert sin(4*o*pi) == 0
assert tan(4*o*pi) == 0
assert cot(4*o*pi) == zoo
assert cos(3*o*pi) == -1
assert sin(3*o*pi) == 0
assert tan(3*o*pi) == 0
assert cot(3*o*pi) == zoo
assert cos(4.0*o*pi) == 1
assert sin(4.0*o*pi) == 0
assert tan(4.0*o*pi) == 0
assert cot(4.0*o*pi) == zoo
assert cos(3.0*o*pi) == -1
assert sin(3.0*o*pi) == 0
assert tan(3.0*o*pi) == 0
assert cot(3.0*o*pi) == zoo
assert cos(4.5*o*pi) == cos(0.5*pi*o)
assert sin(4.5*o*pi) == sin(0.5*pi*o)
assert tan(4.5*o*pi) == tan(0.5*pi*o)
assert cot(4.5*o*pi) == cot(0.5*pi*o)
# x could be imaginary
assert cos(4*x*pi) == cos(4*pi*x)
assert sin(4*x*pi) == sin(4*pi*x)
assert tan(4*x*pi) == tan(4*pi*x)
assert cot(4*x*pi) == cot(4*pi*x)
assert cos(3*x*pi) == cos(3*pi*x)
assert sin(3*x*pi) == sin(3*pi*x)
assert tan(3*x*pi) == tan(3*pi*x)
assert cot(3*x*pi) == cot(3*pi*x)
assert cos(4.0*x*pi) == cos(4.0*pi*x)
assert sin(4.0*x*pi) == sin(4.0*pi*x)
assert tan(4.0*x*pi) == tan(4.0*pi*x)
assert cot(4.0*x*pi) == cot(4.0*pi*x)
assert cos(3.0*x*pi) == cos(3.0*pi*x)
assert sin(3.0*x*pi) == sin(3.0*pi*x)
assert tan(3.0*x*pi) == tan(3.0*pi*x)
assert cot(3.0*x*pi) == cot(3.0*pi*x)
assert cos(4.5*x*pi) == cos(4.5*pi*x)
assert sin(4.5*x*pi) == sin(4.5*pi*x)
assert tan(4.5*x*pi) == tan(4.5*pi*x)
assert cot(4.5*x*pi) == cot(4.5*pi*x)
def test_inverses():
raises(AttributeError, lambda: sin(x).inverse())
raises(AttributeError, lambda: cos(x).inverse())
assert tan(x).inverse() == atan
assert cot(x).inverse() == acot
raises(AttributeError, lambda: csc(x).inverse())
raises(AttributeError, lambda: sec(x).inverse())
assert asin(x).inverse() == sin
assert acos(x).inverse() == cos
assert atan(x).inverse() == tan
assert acot(x).inverse() == cot
def test_real_imag():
a, b = symbols('a b', real=True)
z = a + b*I
for deep in [True, False]:
assert sin(
z).as_real_imag(deep=deep) == (sin(a)*cosh(b), cos(a)*sinh(b))
assert cos(
z).as_real_imag(deep=deep) == (cos(a)*cosh(b), -sin(a)*sinh(b))
assert tan(z).as_real_imag(deep=deep) == (sin(2*a)/(cos(2*a) +
cosh(2*b)), sinh(2*b)/(cos(2*a) + cosh(2*b)))
assert cot(z).as_real_imag(deep=deep) == (-sin(2*a)/(cos(2*a) -
cosh(2*b)), -sinh(2*b)/(cos(2*a) - cosh(2*b)))
assert sin(a).as_real_imag(deep=deep) == (sin(a), 0)
assert cos(a).as_real_imag(deep=deep) == (cos(a), 0)
assert tan(a).as_real_imag(deep=deep) == (tan(a), 0)
assert cot(a).as_real_imag(deep=deep) == (cot(a), 0)
@XFAIL
def test_sin_cos_with_infinity():
# Test for issue 5196
# https://github.com/sympy/sympy/issues/5196
assert sin(oo) == S.NaN
assert cos(oo) == S.NaN
@slow
def test_sincos_rewrite_sqrt():
# equivalent to testing rewrite(pow)
for p in [1, 3, 5, 17]:
for t in [1, 8]:
n = t*p
# The vertices `exp(i*pi/n)` of a regular `n`-gon can
# be expressed by means of nested square roots if and
# only if `n` is a product of Fermat primes, `p`, and
# powers of 2, `t'. The code aims to check all vertices
# not belonging to an `m`-gon for `m < n`(`gcd(i, n) == 1`).
# For large `n` this makes the test too slow, therefore
# the vertices are limited to those of index `i < 10`.
for i in range(1, min((n + 1)//2 + 1, 10)):
if 1 == gcd(i, n):
x = i*pi/n
s1 = sin(x).rewrite(sqrt)
c1 = cos(x).rewrite(sqrt)
assert not s1.has(cos, sin), "fails for %d*pi/%d" % (i, n)
assert not c1.has(cos, sin), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs(sin(x.evalf(5)) - s1.evalf(2)), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs(cos(x.evalf(5)) - c1.evalf(2)), "fails for %d*pi/%d" % (i, n)
assert cos(pi/14).rewrite(sqrt) == sqrt(cos(pi/7)/2 + S.Half)
assert cos(pi/257).rewrite(sqrt).evalf(64) == cos(pi/257).evalf(64)
assert cos(-15*pi/2/11, evaluate=False).rewrite(
sqrt) == -sqrt(-cos(4*pi/11)/2 + S.Half)
assert cos(Mul(2, pi, S.Half, evaluate=False), evaluate=False).rewrite(
sqrt) == -1
e = cos(pi/3/17) # don't use pi/15 since that is caught at instantiation
a = (
-3*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17) + 17)/64 -
3*sqrt(34)*sqrt(sqrt(17) + 17)/128 - sqrt(sqrt(17) +
17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17)
+ sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 - sqrt(-sqrt(17)
+ 17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 - S(1)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 +
3*sqrt(2)*sqrt(sqrt(17) + 17)/128 + sqrt(34)*sqrt(-sqrt(17) + 17)/128
+ 13*sqrt(2)*sqrt(-sqrt(17) + 17)/128 + sqrt(17)*sqrt(-sqrt(17) +
17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17)
+ sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 + 5*sqrt(17)/32
+ sqrt(3)*sqrt(-sqrt(2)*sqrt(sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + S(15)/32)/8 -
5*sqrt(2)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
S(15)/32)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 -
3*sqrt(2)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + S(15)/32)/32
+ sqrt(34)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
S(15)/32)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 +
sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + S(15)/32)/2 +
S.Half + sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) +
17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) -
sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) +
6*sqrt(17) + 34)/32 + S(15)/32)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) -
sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) +
6*sqrt(17) + 34)/32 + sqrt(34)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
S(15)/32)/32)/2)
assert e.rewrite(sqrt) == a
assert e.n() == a.n()
# coverage of fermatCoords: multiplicity > 1; the following could be
# different but that portion of the code should be tested in some way
assert cos(pi/9/17).rewrite(sqrt) == \
sin(pi/9)*sin(2*pi/17) + cos(pi/9)*cos(2*pi/17)
@slow
def test_tancot_rewrite_sqrt():
# equivalent to testing rewrite(pow)
for p in [1, 3, 5, 17]:
for t in [1, 8]:
n = t*p
for i in range(1, min((n + 1)//2 + 1, 10)):
if 1 == gcd(i, n):
x = i*pi/n
if 2*i != n and 3*i != 2*n:
t1 = tan(x).rewrite(sqrt)
assert not t1.has(cot, tan), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs( tan(x.evalf(7)) - t1.evalf(4) ), "fails for %d*pi/%d" % (i, n)
if i != 0 and i != n:
c1 = cot(x).rewrite(sqrt)
assert not c1.has(cot, tan), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs( cot(x.evalf(7)) - c1.evalf(4) ), "fails for %d*pi/%d" % (i, n)
def test_sec():
x = symbols('x', real=True)
z = symbols('z')
assert sec.nargs == FiniteSet(1)
assert sec(zoo) == nan
assert sec(0) == 1
assert sec(pi) == -1
assert sec(pi/2) == zoo
assert sec(-pi/2) == zoo
assert sec(pi/6) == 2*sqrt(3)/3
assert sec(pi/3) == 2
assert sec(5*pi/2) == zoo
assert sec(9*pi/7) == -sec(2*pi/7)
assert sec(3*pi/4) == -sqrt(2) # issue 8421
assert sec(I) == 1/cosh(1)
assert sec(x*I) == 1/cosh(x)
assert sec(-x) == sec(x)
assert sec(asec(x)) == x
assert sec(z).conjugate() == sec(conjugate(z))
assert (sec(z).as_real_imag() ==
(cos(re(z))*cosh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
cos(re(z))**2*cosh(im(z))**2),
sin(re(z))*sinh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
cos(re(z))**2*cosh(im(z))**2)))
assert sec(x).expand(trig=True) == 1/cos(x)
assert sec(2*x).expand(trig=True) == 1/(2*cos(x)**2 - 1)
assert sec(x).is_real == True
assert sec(z).is_real == None
assert sec(a).is_algebraic is None
assert sec(na).is_algebraic is False
assert sec(x).as_leading_term() == sec(x)
assert sec(0).is_finite == True
assert sec(x).is_finite == None
assert sec(pi/2).is_finite == False
assert series(sec(x), x, x0=0, n=6) == 1 + x**2/2 + 5*x**4/24 + O(x**6)
# https://github.com/sympy/sympy/issues/7166
assert series(sqrt(sec(x))) == 1 + x**2/4 + 7*x**4/96 + O(x**6)
# https://github.com/sympy/sympy/issues/7167
assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
1/sqrt(x - 3*pi/2) + (x - 3*pi/2)**(S(3)/2)/12 +
(x - 3*pi/2)**(S(7)/2)/160 + O((x - 3*pi/2)**4, (x, 3*pi/2)))
assert sec(x).diff(x) == tan(x)*sec(x)
# Taylor Term checks
assert sec(z).taylor_term(4, z) == 5*z**4/24
assert sec(z).taylor_term(6, z) == 61*z**6/720
assert sec(z).taylor_term(5, z) == 0
def test_sec_rewrite():
assert sec(x).rewrite(exp) == 1/(exp(I*x)/2 + exp(-I*x)/2)
assert sec(x).rewrite(cos) == 1/cos(x)
assert sec(x).rewrite(tan) == (tan(x/2)**2 + 1)/(-tan(x/2)**2 + 1)
assert sec(x).rewrite(pow) == sec(x)
assert sec(x).rewrite(sqrt) == sec(x)
assert sec(z).rewrite(cot) == (cot(z/2)**2 + 1)/(cot(z/2)**2 - 1)
assert sec(x).rewrite(sin) == 1 / sin(x + pi / 2, evaluate=False)
assert sec(x).rewrite(tan) == (tan(x / 2)**2 + 1) / (-tan(x / 2)**2 + 1)
assert sec(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)
def test_csc():
x = symbols('x', real=True)
z = symbols('z')
# https://github.com/sympy/sympy/issues/6707
cosecant = csc('x')
alternate = 1/sin('x')
assert cosecant.equals(alternate) == True
assert alternate.equals(cosecant) == True
assert csc.nargs == FiniteSet(1)
assert csc(0) == zoo
assert csc(pi) == zoo
assert csc(zoo) == nan
assert csc(pi/2) == 1
assert csc(-pi/2) == -1
assert csc(pi/6) == 2
assert csc(pi/3) == 2*sqrt(3)/3
assert csc(5*pi/2) == 1
assert csc(9*pi/7) == -csc(2*pi/7)
assert csc(3*pi/4) == sqrt(2) # issue 8421
assert csc(I) == -I/sinh(1)
assert csc(x*I) == -I/sinh(x)
assert csc(-x) == -csc(x)
assert csc(acsc(x)) == x
assert csc(z).conjugate() == csc(conjugate(z))
assert (csc(z).as_real_imag() ==
(sin(re(z))*cosh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
cos(re(z))**2*sinh(im(z))**2),
-cos(re(z))*sinh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
cos(re(z))**2*sinh(im(z))**2)))
assert csc(x).expand(trig=True) == 1/sin(x)
assert csc(2*x).expand(trig=True) == 1/(2*sin(x)*cos(x))
assert csc(x).is_real == True
assert csc(z).is_real == None
assert csc(a).is_algebraic is None
assert csc(na).is_algebraic is False
assert csc(x).as_leading_term() == csc(x)
assert csc(0).is_finite == False
assert csc(x).is_finite == None
assert csc(pi/2).is_finite == True
assert series(csc(x), x, x0=pi/2, n=6) == \
1 + (x - pi/2)**2/2 + 5*(x - pi/2)**4/24 + O((x - pi/2)**6, (x, pi/2))
assert series(csc(x), x, x0=0, n=6) == \
1/x + x/6 + 7*x**3/360 + 31*x**5/15120 + O(x**6)
assert csc(x).diff(x) == -cot(x)*csc(x)
assert csc(x).taylor_term(2, x) == 0
assert csc(x).taylor_term(3, x) == 7*x**3/360
assert csc(x).taylor_term(5, x) == 31*x**5/15120
def test_asec():
z = Symbol('z', zero=True)
assert asec(z) == zoo
assert asec(nan) == nan
assert asec(1) == 0
assert asec(-1) == pi
assert asec(oo) == pi/2
assert asec(-oo) == pi/2
assert asec(zoo) == pi/2
assert asec(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2))
assert asec(x).as_leading_term(x) == log(x)
assert asec(x).rewrite(log) == I*log(sqrt(1 - 1/x**2) + I/x) + pi/2
assert asec(x).rewrite(asin) == -asin(1/x) + pi/2
assert asec(x).rewrite(acos) == acos(1/x)
assert asec(x).rewrite(atan) == (2*atan(x + sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x
assert asec(x).rewrite(acot) == (2*acot(x - sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x
assert asec(x).rewrite(acsc) == -acsc(x) + pi/2
def test_asec_is_real():
assert asec(S(1)/2).is_real is False
n = Symbol('n', positive=True, integer=True)
assert asec(n).is_real is True
assert asec(x).is_real is None
assert asec(r).is_real is None
t = Symbol('t', real=False)
assert asec(t).is_real is False
def test_acsc():
assert acsc(nan) == nan
assert acsc(1) == pi/2
assert acsc(-1) == -pi/2
assert acsc(oo) == 0
assert acsc(-oo) == 0
assert acsc(zoo) == 0
assert acsc(x).diff(x) == -1/(x**2*sqrt(1 - 1/x**2))
assert acsc(x).as_leading_term(x) == log(x)
assert acsc(x).rewrite(log) == -I*log(sqrt(1 - 1/x**2) + I/x)
assert acsc(x).rewrite(asin) == asin(1/x)
assert acsc(x).rewrite(acos) == -acos(1/x) + pi/2
assert acsc(x).rewrite(atan) == (-atan(sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
assert acsc(x).rewrite(acot) == (-acot(1/sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
assert acsc(x).rewrite(asec) == -asec(x) + pi/2
def test_csc_rewrite():
assert csc(x).rewrite(pow) == csc(x)
assert csc(x).rewrite(sqrt) == csc(x)
assert csc(x).rewrite(exp) == 2*I/(exp(I*x) - exp(-I*x))
assert csc(x).rewrite(sin) == 1/sin(x)
assert csc(x).rewrite(tan) == (tan(x/2)**2 + 1)/(2*tan(x/2))
assert csc(x).rewrite(cot) == (cot(x/2)**2 + 1)/(2*cot(x/2))
assert csc(x).rewrite(cos) == 1/cos(x - pi/2, evaluate=False)
assert csc(x).rewrite(sec) == sec(-x + pi/2, evaluate=False)
def test_issue_8653():
n = Symbol('n', integer=True)
assert sin(n).is_irrational is None
assert cos(n).is_irrational is None
assert tan(n).is_irrational is None
def test_issue_9157():
n = Symbol('n', integer=True, positive=True)
atan(n - 1).is_nonnegative is True
def test_trig_period():
x, y = symbols('x, y')
assert sin(x).period() == 2*pi
assert cos(x).period() == 2*pi
assert tan(x).period() == pi
assert cot(x).period() == pi
assert sec(x).period() == 2*pi
assert csc(x).period() == 2*pi
assert sin(2*x).period() == pi
assert cot(4*x - 6).period() == pi/4
assert cos((-3)*x).period() == 2*pi/3
assert cos(x*y).period(x) == 2*pi/abs(y)
assert sin(3*x*y + 2*pi).period(y) == 2*pi/abs(3*x)
assert tan(3*x).period(y) == S.Zero
raises(NotImplementedError, lambda: sin(x**2).period(x))
def test_issue_7171():
assert sin(x).rewrite(sqrt) == sin(x)
assert sin(x).rewrite(pow) == sin(x)
def test_issue_11864():
w, k = symbols('w, k', real=True)
F = Piecewise((1, Eq(2*pi*k, 0)), (sin(pi*k)/(pi*k), True))
soln = Piecewise((1, Eq(2*pi*k, 0)), (sinc(pi*k), True))
assert F.rewrite(sinc) == soln
def test_real_assumptions():
z = Symbol('z', real=False)
assert sin(z).is_real is None
assert cos(z).is_real is None
assert tan(z).is_real is False
assert sec(z).is_real is None
assert csc(z).is_real is None
assert cot(z).is_real is False
assert asin(p).is_real is None
assert asin(n).is_real is None
assert asec(p).is_real is None
assert asec(n).is_real is None
assert acos(p).is_real is None
assert acos(n).is_real is None
assert acsc(p).is_real is None
assert acsc(n).is_real is None
assert atan(p).is_positive is True
assert atan(n).is_negative is True
assert acot(p).is_positive is True
assert acot(n).is_negative is True
def test_issue_14320():
assert asin(sin(2)) == -2 + pi and (-pi/2 <= -2 + pi <= pi/2) and sin(2) == sin(-2 + pi)
assert asin(cos(2)) == -2 + pi/2 and (-pi/2 <= -2 + pi/2 <= pi/2) and cos(2) == sin(-2 + pi/2)
assert acos(sin(2)) == -pi/2 + 2 and (0 <= -pi/2 + 2 <= pi) and sin(2) == cos(-pi/2 + 2)
assert acos(cos(20)) == -6*pi + 20 and (0 <= -6*pi + 20 <= pi) and cos(20) == cos(-6*pi + 20)
assert acos(cos(30)) == -30 + 10*pi and (0 <= -30 + 10*pi <= pi) and cos(30) == cos(-30 + 10*pi)
assert atan(tan(17)) == -5*pi + 17 and (-pi/2 < -5*pi + 17 < pi/2) and tan(17) == tan(-5*pi + 17)
assert atan(tan(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 < pi/2) and tan(15) == tan(-5*pi + 15)
assert atan(cot(12)) == -12 + 7*pi/2 and (-pi/2 < -12 + 7*pi/2 < pi/2) and cot(12) == tan(-12 + 7*pi/2)
assert acot(cot(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 <= pi/2) and cot(15) == cot(-5*pi + 15)
assert acot(tan(19)) == -19 + 13*pi/2 and (-pi/2 < -19 + 13*pi/2 <= pi/2) and tan(19) == cot(-19 + 13*pi/2)
assert asec(sec(11)) == -11 + 4*pi and (0 <= -11 + 4*pi <= pi) and cos(11) == cos(-11 + 4*pi)
assert asec(csc(13)) == -13 + 9*pi/2 and (0 <= -13 + 9*pi/2 <= pi) and sin(13) == cos(-13 + 9*pi/2)
assert acsc(csc(14)) == -4*pi + 14 and (-pi/2 <= -4*pi + 14 <= pi/2) and sin(14) == sin(-4*pi + 14)
assert acsc(sec(10)) == -7*pi/2 + 10 and (-pi/2 <= -7*pi/2 + 10 <= pi/2) and cos(10) == sin(-7*pi/2 + 10)
def test_issue_14543():
assert sec(2*pi + 11) == sec(11)
assert sec(2*pi - 11) == sec(11)
assert sec(pi + 11) == -sec(11)
assert sec(pi - 11) == -sec(11)
assert csc(2*pi + 17) == csc(17)
assert csc(2*pi - 17) == -csc(17)
assert csc(pi + 17) == -csc(17)
assert csc(pi - 17) == csc(17)
x = Symbol('x')
assert csc(pi/2 + x) == sec(x)
assert csc(pi/2 - x) == sec(x)
assert csc(3*pi/2 + x) == -sec(x)
assert csc(3*pi/2 - x) == -sec(x)
assert sec(pi/2 - x) == csc(x)
assert sec(pi/2 + x) == -csc(x)
assert sec(3*pi/2 + x) == csc(x)
assert sec(3*pi/2 - x) == -csc(x)
def test_issue_15959():
assert tan(3*pi/8) == 1 + sqrt(2)
|
db6842f4637520d8ae271693f3a06587598ab969103e498010fd104c978d61fb | from sympy import (
adjoint, And, Basic, conjugate, diff, expand, Eq, Function, I, ITE,
Integral, integrate, Interval, lambdify, log, Max, Min, oo, Or, pi,
Piecewise, piecewise_fold, Rational, solve, symbols, transpose,
cos, sin, exp, Abs, Ne, Not, Symbol, S, sqrt, Tuple, zoo,
factor_terms, DiracDelta, Heaviside, Add, Mul, factorial, Ge)
from sympy.printing import srepr
from sympy.utilities.pytest import raises, slow
from sympy.functions.elementary.piecewise import Undefined
a, b, c, d, x, y = symbols('a:d, x, y')
z = symbols('z', nonzero=True)
def test_piecewise():
# Test canonicalization
assert Piecewise((x, x < 1), (0, True)) == Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, True), (1, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, False), (-1, 1 > 2)) == \
Piecewise((x, x < 1))
assert Piecewise((x, x < 1), (0, x < 1), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, x < 2), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (0, True)) == \
Piecewise((x, Or(x < 1, x < 2)), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (x, True)) == x
assert Piecewise((x, True)) == x
# Explicitly constructed empty Piecewise not accepted
raises(TypeError, lambda: Piecewise())
# False condition is never retained
assert Piecewise((2*x, x < 0), (x, False)) == \
Piecewise((2*x, x < 0), (x, False), evaluate=False) == \
Piecewise((2*x, x < 0))
assert Piecewise((x, False)) == Undefined
raises(TypeError, lambda: Piecewise(x))
assert Piecewise((x, 1)) == x # 1 and 0 are accepted as True/False
raises(TypeError, lambda: Piecewise((x, 2)))
raises(TypeError, lambda: Piecewise((x, x**2)))
raises(TypeError, lambda: Piecewise(([1], True)))
assert Piecewise(((1, 2), True)) == Tuple(1, 2)
cond = (Piecewise((1, x < 0), (2, True)) < y)
assert Piecewise((1, cond)
) == Piecewise((1, ITE(x < 0, y > 1, y > 2)))
assert Piecewise((1, x > 0), (2, And(x <= 0, x > -1))
) == Piecewise((1, x > 0), (2, x > -1))
# Test subs
p = Piecewise((-1, x < -1), (x**2, x < 0), (log(x), x >= 0))
p_x2 = Piecewise((-1, x**2 < -1), (x**4, x**2 < 0), (log(x**2), x**2 >= 0))
assert p.subs(x, x**2) == p_x2
assert p.subs(x, -5) == -1
assert p.subs(x, -1) == 1
assert p.subs(x, 1) == log(1)
# More subs tests
p2 = Piecewise((1, x < pi), (-1, x < 2*pi), (0, x > 2*pi))
p3 = Piecewise((1, Eq(x, 0)), (1/x, True))
p4 = Piecewise((1, Eq(x, 0)), (2, 1/x>2))
assert p2.subs(x, 2) == 1
assert p2.subs(x, 4) == -1
assert p2.subs(x, 10) == 0
assert p3.subs(x, 0.0) == 1
assert p4.subs(x, 0.0) == 1
f, g, h = symbols('f,g,h', cls=Function)
pf = Piecewise((f(x), x < -1), (f(x) + h(x) + 2, x <= 1))
pg = Piecewise((g(x), x < -1), (g(x) + h(x) + 2, x <= 1))
assert pg.subs(g, f) == pf
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 0) == 1
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 1) == 0
assert Piecewise((1, Eq(x, y)), (0, True)).subs(x, y) == 1
assert Piecewise((1, Eq(x, z)), (0, True)).subs(x, z) == 1
assert Piecewise((1, Eq(exp(x), cos(z))), (0, True)).subs(x, z) == \
Piecewise((1, Eq(exp(z), cos(z))), (0, True))
p5 = Piecewise( (0, Eq(cos(x) + y, 0)), (1, True))
assert p5.subs(y, 0) == Piecewise( (0, Eq(cos(x), 0)), (1, True))
assert Piecewise((-1, y < 1), (0, x < 0), (1, Eq(x, 0)), (2, True)
).subs(x, 1) == Piecewise((-1, y < 1), (2, True))
assert Piecewise((1, Eq(x**2, -1)), (2, x < 0)).subs(x, I) == 1
p6 = Piecewise((x, x > 0))
n = symbols('n', negative=True)
assert p6.subs(x, n) == Undefined
# Test evalf
assert p.evalf() == p
assert p.evalf(subs={x: -2}) == -1
assert p.evalf(subs={x: -1}) == 1
assert p.evalf(subs={x: 1}) == log(1)
assert p6.evalf(subs={x: -5}) == Undefined
# Test doit
f_int = Piecewise((Integral(x, (x, 0, 1)), x < 1))
assert f_int.doit() == Piecewise( (S(1)/2, x < 1) )
# Test differentiation
f = x
fp = x*p
dp = Piecewise((0, x < -1), (2*x, x < 0), (1/x, x >= 0))
fp_dx = x*dp + p
assert diff(p, x) == dp
assert diff(f*p, x) == fp_dx
# Test simple arithmetic
assert x*p == fp
assert x*p + p == p + x*p
assert p + f == f + p
assert p + dp == dp + p
assert p - dp == -(dp - p)
# Test power
dp2 = Piecewise((0, x < -1), (4*x**2, x < 0), (1/x**2, x >= 0))
assert dp**2 == dp2
# Test _eval_interval
f1 = x*y + 2
f2 = x*y**2 + 3
peval = Piecewise((f1, x < 0), (f2, x > 0))
peval_interval = f1.subs(
x, 0) - f1.subs(x, -1) + f2.subs(x, 1) - f2.subs(x, 0)
assert peval._eval_interval(x, 0, 0) == 0
assert peval._eval_interval(x, -1, 1) == peval_interval
peval2 = Piecewise((f1, x < 0), (f2, True))
assert peval2._eval_interval(x, 0, 0) == 0
assert peval2._eval_interval(x, 1, -1) == -peval_interval
assert peval2._eval_interval(x, -1, -2) == f1.subs(x, -2) - f1.subs(x, -1)
assert peval2._eval_interval(x, -1, 1) == peval_interval
assert peval2._eval_interval(x, None, 0) == peval2.subs(x, 0)
assert peval2._eval_interval(x, -1, None) == -peval2.subs(x, -1)
# Test integration
assert p.integrate() == Piecewise(
(-x, x < -1),
(x**3/3 + S(4)/3, x < 0),
(x*log(x) - x + S(4)/3, True))
p = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
assert integrate(p, (x, -2, 2)) == 5/6.0
assert integrate(p, (x, 2, -2)) == -5/6.0
p = Piecewise((0, x < 0), (1, x < 1), (0, x < 2), (1, x < 3), (0, True))
assert integrate(p, (x, -oo, oo)) == 2
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
assert integrate(p, (x, -2, 2)) == Undefined
# Test commutativity
assert isinstance(p, Piecewise) and p.is_commutative is True
def test_piecewise_free_symbols():
f = Piecewise((x, a < 0), (y, True))
assert f.free_symbols == {x, y, a}
def test_piecewise_integrate1():
x, y = symbols('x y', real=True, finite=True)
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert integrate(f, (x, -2, 2)) == Rational(14, 3)
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == Rational(43, 6)
assert g == Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
g = Piecewise(((x - 5)**5, 2 <= x), (f, x < 2))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == -Rational(701, 6)
assert g == Piecewise(((x - 5)**5, 2 <= x), (f, True))
g = Piecewise(((x - 5)**5, 2 <= x), (2*f, True))
assert integrate(g, (x, -2, 2)) == 2 * Rational(14, 3)
assert integrate(g, (x, -2, 5)) == -Rational(673, 6)
def test_piecewise_integrate1b():
g = Piecewise((1, x > 0), (0, Eq(x, 0)), (-1, x < 0))
assert integrate(g, (x, -1, 1)) == 0
g = Piecewise((1, x - y < 0), (0, True))
assert integrate(g, (y, -oo, 0)) == -Min(0, x)
assert g.subs(x, -3).integrate((y, -oo, 0)) == 3
assert integrate(g, (y, 0, -oo)) == Min(0, x)
assert integrate(g, (y, 0, oo)) == -Max(0, x) + oo
assert integrate(g, (y, -oo, 42)) == -Min(42, x) + 42
assert integrate(g, (y, -oo, oo)) == -x + oo
g = Piecewise((0, x < 0), (x, x <= 1), (1, True))
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
for yy in (-1, S.Half, 2):
assert g.integrate((x, yy, 1)) == gy1.subs(y, yy)
assert g.integrate((x, 1, yy)) == g1y.subs(y, yy)
assert gy1 == Piecewise(
(-Min(1, Max(0, y))**2/2 + S(1)/2, y < 1),
(-y + 1, True))
assert g1y == Piecewise(
(Min(1, Max(0, y))**2/2 - S(1)/2, y < 1),
(y - 1, True))
@slow
def test_piecewise_integrate1ca():
y = symbols('y', real=True)
g = Piecewise(
(1 - x, Interval(0, 1).contains(x)),
(1 + x, Interval(-1, 0).contains(x)),
(0, True)
)
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
# XXX Make test pass without simplify
assert g.integrate((x, 2, 1)) == gy1.subs(y, 2).simplify()
assert g.integrate((x, 1, 2)) == g1y.subs(y, 2).simplify()
assert piecewise_fold(gy1.rewrite(Piecewise)) == \
Piecewise(
(1, y <= -1),
(-y**2/2 - y + S(1)/2, y <= 0),
(y**2/2 - y + S(1)/2, y < 1),
(0, True))
assert piecewise_fold(g1y.rewrite(Piecewise)) == \
Piecewise(
(-1, y <= -1),
(y**2/2 + y - S(1)/2, y <= 0),
(-y**2/2 + y - S(1)/2, y < 1),
(0, True))
# g1y and gy1 should simplify if the condition that y < 1
# is applied, e.g. Min(1, Max(-1, y)) --> Max(-1, y)
# XXX Make test pass without simplify
assert gy1.simplify() == Piecewise(
(
-Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
Min(1, Max(0, y))**2 + S(1)/2, y < 1),
(0, True)
)
assert g1y.simplify() == Piecewise(
(
Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
Min(1, Max(0, y))**2 - S(1)/2, y < 1),
(0, True))
@slow
def test_piecewise_integrate1cb():
y = symbols('y', real=True)
g = Piecewise(
(0, Or(x <= -1, x >= 1)),
(1 - x, x > 0),
(1 + x, True)
)
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
assert g.integrate((x, 2, 1)) == gy1.subs(y, 2)
assert g.integrate((x, 1, 2)) == g1y.subs(y, 2)
assert piecewise_fold(gy1.rewrite(Piecewise)) == \
Piecewise(
(1, y <= -1),
(-y**2/2 - y + S(1)/2, y <= 0),
(y**2/2 - y + S(1)/2, y < 1),
(0, True))
assert piecewise_fold(g1y.rewrite(Piecewise)) == \
Piecewise(
(-1, y <= -1),
(y**2/2 + y - S(1)/2, y <= 0),
(-y**2/2 + y - S(1)/2, y < 1),
(0, True))
# g1y and gy1 should simplify if the condition that y < 1
# is applied, e.g. Min(1, Max(-1, y)) --> Max(-1, y)
assert gy1 == Piecewise(
(
-Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
Min(1, Max(0, y))**2 + S(1)/2, y < 1),
(0, True)
)
assert g1y == Piecewise(
(
Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
Min(1, Max(0, y))**2 - S(1)/2, y < 1),
(0, True))
def test_piecewise_integrate2():
from itertools import permutations
lim = Tuple(x, c, d)
p = Piecewise((1, x < a), (2, x > b), (3, True))
q = p.integrate(lim)
assert q == Piecewise(
(-c + 2*d - 2*Min(d, Max(a, c)) + Min(d, Max(a, b, c)), c < d),
(-2*c + d + 2*Min(c, Max(a, d)) - Min(c, Max(a, b, d)), True))
for v in permutations((1, 2, 3, 4)):
r = dict(zip((a, b, c, d), v))
assert p.subs(r).integrate(lim.subs(r)) == q.subs(r)
def test_meijer_bypass():
# totally bypass meijerg machinery when dealing
# with Piecewise in integrate
assert Piecewise((1, x < 4), (0, True)).integrate((x, oo, 1)) == -3
def test_piecewise_integrate3_inequality_conditions():
from sympy.utilities.iterables import cartes
lim = (x, 0, 5)
# set below includes two pts below range, 2 pts in range,
# 2 pts above range, and the boundaries
N = (-2, -1, 0, 1, 2, 5, 6, 7)
p = Piecewise((1, x > a), (2, x > b), (0, True))
ans = p.integrate(lim)
for i, j in cartes(N, repeat=2):
reps = dict(zip((a, b), (i, j)))
assert ans.subs(reps) == p.subs(reps).integrate(lim)
assert ans.subs(a, 4).subs(b, 1) == 0 + 2*3 + 1
p = Piecewise((1, x > a), (2, x < b), (0, True))
ans = p.integrate(lim)
for i, j in cartes(N, repeat=2):
reps = dict(zip((a, b), (i, j)))
assert ans.subs(reps) == p.subs(reps).integrate(lim)
# delete old tests that involved c1 and c2 since those
# reduce to the above except that a value of 0 was used
# for two expressions whereas the above uses 3 different
# values
@slow
def test_piecewise_integrate4_symbolic_conditions():
a = Symbol('a', real=True, finite=True)
b = Symbol('b', real=True, finite=True)
x = Symbol('x', real=True, finite=True)
y = Symbol('y', real=True, finite=True)
p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
p1 = Piecewise((0, x < a), (0, x > b), (1, True))
p2 = Piecewise((0, x > b), (0, x < a), (1, True))
p3 = Piecewise((0, x < a), (1, x < b), (0, True))
p4 = Piecewise((0, x > b), (1, x > a), (0, True))
p5 = Piecewise((1, And(a < x, x < b)), (0, True))
# check values of a=1, b=3 (and reversed) with values
# of y of 0, 1, 2, 3, 4
lim = Tuple(x, -oo, y)
for p in (p0, p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a: 3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
lim = Tuple(x, y, oo)
for p in (p0, p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a:3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
ans = Piecewise(
(0, x <= Min(a, b)),
(x - Min(a, b), x <= b),
(b - Min(a, b), True))
for i in (p0, p1, p2, p4):
assert i.integrate(x) == ans
assert p3.integrate(x) == Piecewise(
(0, x < a),
(-a + x, x <= Max(a, b)),
(-a + Max(a, b), True))
assert p5.integrate(x) == Piecewise(
(0, x <= a),
(-a + x, x <= Max(a, b)),
(-a + Max(a, b), True))
p1 = Piecewise((0, x < a), (0.5, x > b), (1, True))
p2 = Piecewise((0.5, x > b), (0, x < a), (1, True))
p3 = Piecewise((0, x < a), (1, x < b), (0.5, True))
p4 = Piecewise((0.5, x > b), (1, x > a), (0, True))
p5 = Piecewise((1, And(a < x, x < b)), (0.5, x > b), (0, True))
# check values of a=1, b=3 (and reversed) with values
# of y of 0, 1, 2, 3, 4
lim = Tuple(x, -oo, y)
for p in (p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a: 3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
def test_piecewise_integrate5_independent_conditions():
p = Piecewise((0, Eq(y, 0)), (x*y, True))
assert integrate(p, (x, 1, 3)) == Piecewise((0, Eq(y, 0)), (4*y, True))
def test_piecewise_simplify():
p = Piecewise(((x**2 + 1)/x**2, Eq(x*(1 + x) - x**2, 0)),
((-1)**x*(-1), True))
assert p.simplify() == \
Piecewise((zoo, Eq(x, 0)), ((-1)**(x + 1), True))
# simplify when there are Eq in conditions
assert Piecewise(
(a, And(Eq(a, 0), Eq(a + b, 0))), (1, True)).simplify(
) == Piecewise(
(0, And(Eq(a, 0), Eq(b, 0))), (1, True))
assert Piecewise((2*x*factorial(a)/(factorial(y)*factorial(-y + a)),
Eq(y, 0) & Eq(-y + a, 0)), (2*factorial(a)/(factorial(y)*factorial(-y
+ a)), Eq(y, 0) & Eq(-y + a, 1)), (0, True)).simplify(
) == Piecewise(
(2*x, And(Eq(a, 0), Eq(y, 0))),
(2, And(Eq(a, 1), Eq(y, 0))),
(0, True))
args = (2, And(Eq(x, 2), Ge(y ,0))), (x, True)
assert Piecewise(*args).simplify() == Piecewise(*args)
args = (1, Eq(x, 0)), (sin(x)/x, True)
assert Piecewise(*args).simplify() == Piecewise(*args)
assert Piecewise((2 + y, And(Eq(x, 2), Eq(y, 0))), (x, True)
).simplify() == x
# check that x or f(x) are recognized as being Symbol-like for lhs
args = Tuple((1, Eq(x, 0)), (sin(x) + 1 + x, True))
ans = x + sin(x) + 1
f = Function('f')
assert Piecewise(*args).simplify() == ans
assert Piecewise(*args.subs(x, f(x))).simplify() == ans.subs(x, f(x))
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs(x, x - 2)
assert solve(f, x) == [2]
assert solve(f - 1, x) == [1, 3]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [2]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2),
(-x + 2, x - 2 <= 0), (x - 2, x - 2 > 0))
assert solve(g, x) == [5]
# if no symbol is given the piecewise detection must still work
assert solve(Piecewise((x - 2, x > 2), (2 - x, True)) - 3) == [-1, 5]
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
raises(NotImplementedError, lambda: solve(f, x))
def nona(ans):
return list(filter(lambda x: x is not S.NaN, ans))
p = Piecewise((x**2 - 4, x < y), (x - 2, True))
ans = solve(p, x)
assert nona([i.subs(y, -2) for i in ans]) == [2]
assert nona([i.subs(y, 2) for i in ans]) == [-2, 2]
assert nona([i.subs(y, 3) for i in ans]) == [-2, 2]
assert ans == [
Piecewise((-2, y > -2), (S.NaN, True)),
Piecewise((2, y <= 2), (S.NaN, True)),
Piecewise((2, y > 2), (S.NaN, True))]
# issue 6060
absxm3 = Piecewise(
(x - 3, S(0) <= x - 3),
(3 - x, S(0) > x - 3)
)
assert solve(absxm3 - y, x) == [
Piecewise((-y + 3, -y < 0), (S.NaN, True)),
Piecewise((y + 3, y >= 0), (S.NaN, True))]
p = Symbol('p', positive=True)
assert solve(absxm3 - p, x) == [-p + 3, p + 3]
# issue 6989
f = Function('f')
assert solve(Eq(-f(x), Piecewise((1, x > 0), (0, True))), f(x)) == \
[Piecewise((-1, x > 0), (0, True))]
# issue 8587
f = Piecewise((2*x**2, And(S(0) < x, x < 1)), (2, True))
assert solve(f - 1) == [1/sqrt(2)]
def test_piecewise_fold():
p = Piecewise((x, x < 1), (1, 1 <= x))
assert piecewise_fold(x*p) == Piecewise((x**2, x < 1), (x, 1 <= x))
assert piecewise_fold(p + p) == Piecewise((2*x, x < 1), (2, 1 <= x))
assert piecewise_fold(Piecewise((1, x < 0), (2, True))
+ Piecewise((10, x < 0), (-10, True))) == \
Piecewise((11, x < 0), (-8, True))
p1 = Piecewise((0, x < 0), (x, x <= 1), (0, True))
p2 = Piecewise((0, x < 0), (1 - x, x <= 1), (0, True))
p = 4*p1 + 2*p2
assert integrate(
piecewise_fold(p), (x, -oo, oo)) == integrate(2*x + 2, (x, 0, 1))
assert piecewise_fold(
Piecewise((1, y <= 0), (-Piecewise((2, y >= 0)), True)
)) == Piecewise((1, y <= 0), (-2, y >= 0))
assert piecewise_fold(Piecewise((x, ITE(x > 0, y < 1, y > 1)))
) == Piecewise((x, ((x <= 0) | (y < 1)) & ((x > 0) | (y > 1))))
a, b = (Piecewise((2, Eq(x, 0)), (0, True)),
Piecewise((x, Eq(-x + y, 0)), (1, Eq(-x + y, 1)), (0, True)))
assert piecewise_fold(Mul(a, b, evaluate=False)
) == piecewise_fold(Mul(b, a, evaluate=False))
def test_piecewise_fold_piecewise_in_cond():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (p1 / Abs(p1), True))
p3 = piecewise_fold(p2)
assert(p2.subs(x, -pi/2) == 0.0)
assert(p2.subs(x, 1) == 0.0)
assert(p2.subs(x, -pi/4) == 1.0)
p4 = Piecewise((0, Eq(p1, 0)), (1,True))
ans = piecewise_fold(p4)
for i in range(-1, 1):
assert ans.subs(x, i) == p4.subs(x, i)
r1 = 1 < Piecewise((1, x < 1), (3, True))
ans = piecewise_fold(r1)
for i in range(2):
assert ans.subs(x, i) == r1.subs(x, i)
p5 = Piecewise((1, x < 0), (3, True))
p6 = Piecewise((1, x < 1), (3, True))
p7 = Piecewise((1, p5 < p6), (0, True))
ans = piecewise_fold(p7)
for i in range(-1, 2):
assert ans.subs(x, i) == p7.subs(x, i)
def test_piecewise_fold_piecewise_in_cond_2():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (1 / p1, True))
p3 = Piecewise(
(0, (x >= 0) | Eq(cos(x), 0)),
(1/cos(x), x < 0),
(zoo, True)) # redundant b/c all x are already covered
assert(piecewise_fold(p2) == p3)
def test_piecewise_fold_expand():
p1 = Piecewise((1, Interval(0, 1, False, True).contains(x)), (0, True))
p2 = piecewise_fold(expand((1 - x)*p1))
assert p2 == Piecewise((1 - x, (x >= 0) & (x < 1)), (0, True))
assert p2 == expand(piecewise_fold((1 - x)*p1))
def test_piecewise_duplicate():
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
assert p == Piecewise(*p.args)
def test_doit():
p1 = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
p2 = Piecewise((x, x < 1), (Integral(2 * x), -1 <= x), (x, 3 < x))
assert p2.doit() == p1
assert p2.doit(deep=False) == p2
def test_piecewise_interval():
p1 = Piecewise((x, Interval(0, 1).contains(x)), (0, True))
assert p1.subs(x, -0.5) == 0
assert p1.subs(x, 0.5) == 0.5
assert p1.diff(x) == Piecewise((1, Interval(0, 1).contains(x)), (0, True))
assert integrate(p1, x) == Piecewise(
(0, x <= 0),
(x**2/2, x <= 1),
(S(1)/2, True))
def test_piecewise_collapse():
assert Piecewise((x, True)) == x
a = x < 1
assert Piecewise((x, a), (x + 1, a)) == Piecewise((x, a))
assert Piecewise((x, a), (x + 1, a.reversed)) == Piecewise((x, a))
b = x < 5
def canonical(i):
if isinstance(i, Piecewise):
return Piecewise(*i.args)
return i
for args in [
((1, a), (Piecewise((2, a), (3, b)), b)),
((1, a), (Piecewise((2, a), (3, b.reversed)), b)),
((1, a), (Piecewise((2, a), (3, b)), b), (4, True)),
((1, a), (Piecewise((2, a), (3, b), (4, True)), b)),
((1, a), (Piecewise((2, a), (3, b), (4, True)), b), (5, True))]:
for i in (0, 2, 10):
assert canonical(
Piecewise(*args, evaluate=False).subs(x, i)
) == canonical(Piecewise(*args).subs(x, i))
r1, r2, r3, r4 = symbols('r1:5')
a = x < r1
b = x < r2
c = x < r3
d = x < r4
assert Piecewise((1, a), (Piecewise(
(2, a), (3, b), (4, c)), b), (5, c)
) == Piecewise((1, a), (3, b), (5, c))
assert Piecewise((1, a), (Piecewise(
(2, a), (3, b), (4, c), (6, True)), c), (5, d)
) == Piecewise((1, a), (Piecewise(
(3, b), (4, c)), c), (5, d))
assert Piecewise((1, Or(a, d)), (Piecewise(
(2, d), (3, b), (4, c)), b), (5, c)
) == Piecewise((1, Or(a, d)), (Piecewise(
(2, d), (3, b)), b), (5, c))
assert Piecewise((1, c), (2, ~c), (3, S.true)
) == Piecewise((1, c), (2, S.true))
assert Piecewise((1, c), (2, And(~c, b)), (3,True)
) == Piecewise((1, c), (2, b), (3, True))
assert Piecewise((1, c), (2, Or(~c, b)), (3,True)
).subs(dict(zip((r1, r2, r3, r4, x), (1, 2, 3, 4, 3.5)))) == 2
assert Piecewise((1, c), (2, ~c)) == Piecewise((1, c), (2, True))
def test_piecewise_lambdify():
p = Piecewise(
(x**2, x < 0),
(x, Interval(0, 1, False, True).contains(x)),
(2 - x, x >= 1),
(0, True)
)
f = lambdify(x, p)
assert f(-2.0) == 4.0
assert f(0.0) == 0.0
assert f(0.5) == 0.5
assert f(2.0) == 0.0
def test_piecewise_series():
from sympy import sin, cos, O
p1 = Piecewise((sin(x), x < 0), (cos(x), x > 0))
p2 = Piecewise((x + O(x**2), x < 0), (1 + O(x**2), x > 0))
assert p1.nseries(x, n=2) == p2
def test_piecewise_as_leading_term():
p1 = Piecewise((1/x, x > 1), (0, True))
p2 = Piecewise((x, x > 1), (0, True))
p3 = Piecewise((1/x, x > 1), (x, True))
p4 = Piecewise((x, x > 1), (1/x, True))
p5 = Piecewise((1/x, x > 1), (x, True))
p6 = Piecewise((1/x, x < 1), (x, True))
p7 = Piecewise((x, x < 1), (1/x, True))
p8 = Piecewise((x, x > 1), (1/x, True))
assert p1.as_leading_term(x) == 0
assert p2.as_leading_term(x) == 0
assert p3.as_leading_term(x) == x
assert p4.as_leading_term(x) == 1/x
assert p5.as_leading_term(x) == x
assert p6.as_leading_term(x) == 1/x
assert p7.as_leading_term(x) == x
assert p8.as_leading_term(x) == 1/x
def test_piecewise_complex():
p1 = Piecewise((2, x < 0), (1, 0 <= x))
p2 = Piecewise((2*I, x < 0), (I, 0 <= x))
p3 = Piecewise((I*x, x > 1), (1 + I, True))
p4 = Piecewise((-I*conjugate(x), x > 1), (1 - I, True))
assert conjugate(p1) == p1
assert conjugate(p2) == piecewise_fold(-p2)
assert conjugate(p3) == p4
assert p1.is_imaginary is False
assert p1.is_real is True
assert p2.is_imaginary is True
assert p2.is_real is False
assert p3.is_imaginary is None
assert p3.is_real is None
assert p1.as_real_imag() == (p1, 0)
assert p2.as_real_imag() == (0, -I*p2)
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
p = Piecewise((A*B**2, x > 0), (A**2*B, True))
assert p.adjoint() == \
Piecewise((adjoint(A*B**2), x > 0), (adjoint(A**2*B), True))
assert p.conjugate() == \
Piecewise((conjugate(A*B**2), x > 0), (conjugate(A**2*B), True))
assert p.transpose() == \
Piecewise((transpose(A*B**2), x > 0), (transpose(A**2*B), True))
def test_piecewise_evaluate():
assert Piecewise((x, True)) == x
assert Piecewise((x, True), evaluate=True) == x
p = Piecewise((x, True), evaluate=False)
assert p != x
assert p.is_Piecewise
assert all(isinstance(i, Basic) for i in p.args)
assert Piecewise((1, Eq(1, x))).args == ((1, Eq(x, 1)),)
assert Piecewise((1, Eq(1, x)), evaluate=False).args == (
(1, Eq(1, x)),)
def test_as_expr_set_pairs():
assert Piecewise((x, x > 0), (-x, x <= 0)).as_expr_set_pairs() == \
[(x, Interval(0, oo, True, True)), (-x, Interval(-oo, 0))]
assert Piecewise(((x - 2)**2, x >= 0), (0, True)).as_expr_set_pairs() == \
[((x - 2)**2, Interval(0, oo)), (0, Interval(-oo, 0, True, True))]
def test_S_srepr_is_identity():
p = Piecewise((10, Eq(x, 0)), (12, True))
q = S(srepr(p))
assert p == q
def test_issue_12587():
# sort holes into intervals
p = Piecewise((1, x > 4), (2, Not((x <= 3) & (x > -1))), (3, True))
assert p.integrate((x, -5, 5)) == 23
p = Piecewise((1, x > 1), (2, x < y), (3, True))
lim = x, -3, 3
ans = p.integrate(lim)
for i in range(-1, 3):
assert ans.subs(y, i) == p.subs(y, i).integrate(lim)
def test_issue_11045():
assert integrate(1/(x*sqrt(x**2 - 1)), (x, 1, 2)) == pi/3
# handle And with Or arguments
assert Piecewise((1, And(Or(x < 1, x > 3), x < 2)), (0, True)
).integrate((x, 0, 3)) == 1
# hidden false
assert Piecewise((1, x > 1), (2, x > x + 1), (3, True)
).integrate((x, 0, 3)) == 5
# targetcond is Eq
assert Piecewise((1, x > 1), (2, Eq(1, x)), (3, True)
).integrate((x, 0, 4)) == 6
# And has Relational needing to be solved
assert Piecewise((1, And(2*x > x + 1, x < 2)), (0, True)
).integrate((x, 0, 3)) == 1
# Or has Relational needing to be solved
assert Piecewise((1, Or(2*x > x + 2, x < 1)), (0, True)
).integrate((x, 0, 3)) == 2
# ignore hidden false (handled in canonicalization)
assert Piecewise((1, x > 1), (2, x > x + 1), (3, True)
).integrate((x, 0, 3)) == 5
# watch for hidden True Piecewise
assert Piecewise((2, Eq(1 - x, x*(1/x - 1))), (0, True)
).integrate((x, 0, 3)) == 6
# overlapping conditions of targetcond are recognized and ignored;
# the condition x > 3 will be pre-empted by the first condition
assert Piecewise((1, Or(x < 1, x > 2)), (2, x > 3), (3, True)
).integrate((x, 0, 4)) == 6
# convert Ne to Or
assert Piecewise((1, Ne(x, 0)), (2, True)
).integrate((x, -1, 1)) == 2
# no default but well defined
assert Piecewise((x, (x > 1) & (x < 3)), (1, (x < 4))
).integrate((x, 1, 4)) == 5
p = Piecewise((x, (x > 1) & (x < 3)), (1, (x < 4)))
nan = Undefined
i = p.integrate((x, 1, y))
assert i == Piecewise(
(y - 1, y < 1),
(Min(3, y)**2/2 - Min(3, y) + Min(4, y) - S(1)/2,
y <= Min(4, y)),
(nan, True))
assert p.integrate((x, 1, -1)) == i.subs(y, -1)
assert p.integrate((x, 1, 4)) == 5
assert p.integrate((x, 1, 5)) == nan
# handle Not
p = Piecewise((1, x > 1), (2, Not(And(x > 1, x< 3))), (3, True))
assert p.integrate((x, 0, 3)) == 4
# handle updating of int_expr when there is overlap
p = Piecewise(
(1, And(5 > x, x > 1)),
(2, Or(x < 3, x > 7)),
(4, x < 8))
assert p.integrate((x, 0, 10)) == 20
# And with Eq arg handling
assert Piecewise((1, x < 1), (2, And(Eq(x, 3), x > 1))
).integrate((x, 0, 3)) == S.NaN
assert Piecewise((1, x < 1), (2, And(Eq(x, 3), x > 1)), (3, True)
).integrate((x, 0, 3)) == 7
assert Piecewise((1, x < 0), (2, And(Eq(x, 3), x < 1)), (3, True)
).integrate((x, -1, 1)) == 4
# middle condition doesn't matter: it's a zero width interval
assert Piecewise((1, x < 1), (2, Eq(x, 3) & (y < x)), (3, True)
).integrate((x, 0, 3)) == 7
def test_holes():
nan = Undefined
assert Piecewise((1, x < 2)).integrate(x) == Piecewise(
(x, x < 2), (nan, True))
assert Piecewise((1, And(x > 1, x < 2))).integrate(x) == Piecewise(
(nan, x < 1), (x - 1, x < 2), (nan, True))
assert Piecewise((1, And(x > 1, x < 2))).integrate((x, 0, 3)) == nan
assert Piecewise((1, And(x > 0, x < 4))).integrate((x, 1, 3)) == 2
# this also tests that the integrate method is used on non-Piecwise
# arguments in _eval_integral
A, B = symbols("A B")
a, b = symbols('a b', finite=True)
assert Piecewise((A, And(x < 0, a < 1)), (B, Or(x < 1, a > 2))
).integrate(x) == Piecewise(
(B*x, a > 2),
(Piecewise((A*x, x < 0), (B*x, x < 1), (nan, True)), a < 1),
(Piecewise((B*x, x < 1), (nan, True)), True))
def test_issue_11922():
def f(x):
return Piecewise((0, x < -1), (1 - x**2, x < 1), (0, True))
autocorr = lambda k: (
f(x) * f(x + k)).integrate((x, -1, 1))
assert autocorr(1.9) > 0
k = symbols('k')
good_autocorr = lambda k: (
(1 - x**2) * f(x + k)).integrate((x, -1, 1))
a = good_autocorr(k)
assert a.subs(k, 3) == 0
k = symbols('k', positive=True)
a = good_autocorr(k)
assert a.subs(k, 3) == 0
assert Piecewise((0, x < 1), (10, (x >= 1))
).integrate() == Piecewise((0, x < 1), (10*x - 10, True))
def test_issue_5227():
f = 0.0032513612725229*Piecewise((0, x < -80.8461538461539),
(-0.0160799238820171*x + 1.33215984776403, x < 2),
(Piecewise((0.3, x > 123), (0.7, True)) +
Piecewise((0.4, x > 2), (0.6, True)), x <=
123), (-0.00817409766454352*x + 2.10541401273885, x <
380.571428571429), (0, True))
i = integrate(f, (x, -oo, oo))
assert i == Integral(f, (x, -oo, oo)).doit()
assert str(i) == '1.00195081676351'
assert Piecewise((1, x - y < 0), (0, True)
).integrate(y) == Piecewise((0, y <= x), (-x + y, True))
def test_issue_10137():
a = Symbol('a', real=True, finite=True)
b = Symbol('b', real=True, finite=True)
x = Symbol('x', real=True, finite=True)
y = Symbol('y', real=True, finite=True)
p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
p1 = Piecewise((0, Or(a > x, b < x)), (1, True))
assert integrate(p0, (x, y, oo)) == integrate(p1, (x, y, oo))
p3 = Piecewise((1, And(0 < x, x < a)), (0, True))
p4 = Piecewise((1, And(a > x, x > 0)), (0, True))
ip3 = integrate(p3, x)
assert ip3 == Piecewise(
(0, x <= 0),
(x, x <= Max(0, a)),
(Max(0, a), True))
ip4 = integrate(p4, x)
assert ip4 == ip3
assert p3.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
assert p4.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
def test_stackoverflow_43852159():
f = lambda x: Piecewise((1 , (x >= -1) & (x <= 1)) , (0, True))
Conv = lambda x: integrate(f(x - y)*f(y), (y, -oo, +oo))
cx = Conv(x)
assert cx.subs(x, -1.5) == cx.subs(x, 1.5)
assert cx.subs(x, 3) == 0
assert piecewise_fold(f(x - y)*f(y)) == Piecewise(
(1, (y >= -1) & (y <= 1) & (x - y >= -1) & (x - y <= 1)),
(0, True))
def test_issue_12557():
'''
# 3200 seconds to compute the fourier part of issue
import sympy as sym
x,y,z,t = sym.symbols('x y z t')
k = sym.symbols("k", integer=True)
fourier = sym.fourier_series(sym.cos(k*x)*sym.sqrt(x**2),
(x, -sym.pi, sym.pi))
assert fourier == FourierSeries(
sqrt(x**2)*cos(k*x), (x, -pi, pi), (Piecewise((pi**2,
Eq(k, 0)), (2*(-1)**k/k**2 - 2/k**2, True))/(2*pi),
SeqFormula(Piecewise((pi**2, (Eq(_n, 0) & Eq(k, 0)) | (Eq(_n, 0) &
Eq(_n, k) & Eq(k, 0)) | (Eq(_n, 0) & Eq(k, 0) & Eq(_n, -k)) | (Eq(_n,
0) & Eq(_n, k) & Eq(k, 0) & Eq(_n, -k))), (pi**2/2, Eq(_n, k) | Eq(_n,
-k) | (Eq(_n, 0) & Eq(_n, k)) | (Eq(_n, k) & Eq(k, 0)) | (Eq(_n, 0) &
Eq(_n, -k)) | (Eq(_n, k) & Eq(_n, -k)) | (Eq(k, 0) & Eq(_n, -k)) |
(Eq(_n, 0) & Eq(_n, k) & Eq(_n, -k)) | (Eq(_n, k) & Eq(k, 0) & Eq(_n,
-k))), ((-1)**k*pi**2*_n**3*sin(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi**2*_n**3*sin(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2
- pi*k**4) + (-1)**k*pi*_n**2*cos(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi*_n**2*cos(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) - (-1)**k*pi**2*_n*k**2*sin(pi*_n)/(pi*_n**4 -
2*pi*_n**2*k**2 + pi*k**4) +
(-1)**k*pi**2*_n*k**2*sin(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) + (-1)**k*pi*k**2*cos(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi*k**2*cos(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) - (2*_n**2 + 2*k**2)/(_n**4 - 2*_n**2*k**2 + k**4),
True))*cos(_n*x)/pi, (_n, 1, oo)), SeqFormula(0, (_k, 1, oo))))
'''
x = symbols("x", real=True)
k = symbols('k', integer=True, finite=True)
abs2 = lambda x: Piecewise((-x, x <= 0), (x, x > 0))
assert integrate(abs2(x), (x, -pi, pi)) == pi**2
func = cos(k*x)*sqrt(x**2)
assert integrate(func, (x, -pi, pi)) == Piecewise(
(2*(-1)**k/k**2 - 2/k**2, Ne(k, 0)), (pi**2, True))
def test_issue_6900():
from itertools import permutations
t0, t1, T, t = symbols('t0, t1 T t')
f = Piecewise((0, t < t0), (x, And(t0 <= t, t < t1)), (0, t >= t1))
g = f.integrate(t)
assert g == Piecewise(
(0, t <= t0),
(t*x - t0*x, t <= Max(t0, t1)),
(-t0*x + x*Max(t0, t1), True))
for i in permutations(range(2)):
reps = dict(zip((t0,t1), i))
for tt in range(-1,3):
assert (g.xreplace(reps).subs(t,tt) ==
f.xreplace(reps).integrate(t).subs(t,tt))
lim = Tuple(t, t0, T)
g = f.integrate(lim)
ans = Piecewise(
(-t0*x + x*Min(T, Max(t0, t1)), T > t0),
(0, True))
for i in permutations(range(3)):
reps = dict(zip((t0,t1,T), i))
tru = f.xreplace(reps).integrate(lim.xreplace(reps))
assert tru == ans.xreplace(reps)
assert g == ans
def test_issue_10122():
assert solve(abs(x) + abs(x - 1) - 1 > 0, x
) == Or(And(-oo < x, x < 0), And(S.One < x, x < oo))
def test_issue_4313():
u = Piecewise((0, x <= 0), (1, x >= a), (x/a, True))
e = (u - u.subs(x, y))**2/(x - y)**2
M = Max(0, a)
assert integrate(e, x).expand() == Piecewise(
(Piecewise(
(0, x <= 0),
(-y**2/(a**2*x - a**2*y) + x/a**2 - 2*y*log(-y)/a**2 +
2*y*log(x - y)/a**2 - y/a**2, x <= M),
(-y**2/(-a**2*y + a**2*M) + 1/(-y + M) -
1/(x - y) - 2*y*log(-y)/a**2 + 2*y*log(-y +
M)/a**2 - y/a**2 + M/a**2, True)),
((a <= y) & (y <= 0)) | ((y <= 0) & (y > -oo))),
(Piecewise(
(-1/(x - y), x <= 0),
(-a**2/(a**2*x - a**2*y) + 2*a*y/(a**2*x - a**2*y) -
y**2/(a**2*x - a**2*y) + 2*log(-y)/a - 2*log(x - y)/a +
2/a + x/a**2 - 2*y*log(-y)/a**2 + 2*y*log(x - y)/a**2 -
y/a**2, x <= M),
(-a**2/(-a**2*y + a**2*M) + 2*a*y/(-a**2*y +
a**2*M) - y**2/(-a**2*y + a**2*M) +
2*log(-y)/a - 2*log(-y + M)/a + 2/a -
2*y*log(-y)/a**2 + 2*y*log(-y + M)/a**2 -
y/a**2 + M/a**2, True)),
a <= y),
(Piecewise(
(-y**2/(a**2*x - a**2*y), x <= 0),
(x/a**2 + y/a**2, x <= M),
(a**2/(-a**2*y + a**2*M) -
a**2/(a**2*x - a**2*y) - 2*a*y/(-a**2*y + a**2*M) +
2*a*y/(a**2*x - a**2*y) + y**2/(-a**2*y + a**2*M) -
y**2/(a**2*x - a**2*y) + y/a**2 + M/a**2, True)),
True))
def test__intervals():
assert Piecewise((x + 2, Eq(x, 3)))._intervals(x) == []
assert Piecewise(
(1, x > x + 1),
(Piecewise((1, x < x + 1)), 2*x < 2*x + 1),
(1, True))._intervals(x) == [(-oo, oo, 1, 1)]
assert Piecewise((1, Ne(x, I)), (0, True))._intervals(x) == [
(-oo, oo, 1, 0)]
assert Piecewise((-cos(x), sin(x) >= 0), (cos(x), True)
)._intervals(x) == [(0, pi, -cos(x), 0), (-oo, oo, cos(x), 1)]
# the following tests that duplicates are removed and that non-Eq
# generated zero-width intervals are removed
assert Piecewise((1, Abs(x**(-2)) > 1), (0, True)
)._intervals(x) == [(-1, 0, 1, 0), (0, 1, 1, 0), (-oo, oo, 0, 1)]
def test_containment():
a, b, c, d, e = [1, 2, 3, 4, 5]
p = (Piecewise((d, x > 1), (e, True))*
Piecewise((a, Abs(x - 1) < 1), (b, Abs(x - 2) < 2), (c, True)))
assert p.integrate(x).diff(x) == Piecewise(
(c*e, x <= 0),
(a*e, x <= 1),
(a*d, x < 2), # this is what we want to get right
(b*d, x < 4),
(c*d, True))
def test_piecewise_with_DiracDelta():
d1 = DiracDelta(x - 1)
assert integrate(d1, (x, -oo, oo)) == 1
assert integrate(d1, (x, 0, 2)) == 1
assert Piecewise((d1, Eq(x, 2)), (0, True)).integrate(x) == 0
assert Piecewise((d1, x < 2), (0, True)).integrate(x) == Piecewise(
(Heaviside(x - 1), x < 2), (1, True))
# TODO raise error if function is discontinuous at limit of
# integration, e.g. integrate(d1, (x, -2, 1)) or Piecewise(
# (d1, Eq(x ,1)
def test_issue_10258():
assert Piecewise((0, x < 1), (1, True)).is_zero is None
assert Piecewise((-1, x < 1), (1, True)).is_zero is False
a = Symbol('a', zero=True)
assert Piecewise((0, x < 1), (a, True)).is_zero
assert Piecewise((1, x < 1), (a, x < 3)).is_zero is None
a = Symbol('a')
assert Piecewise((0, x < 1), (a, True)).is_zero is None
assert Piecewise((0, x < 1), (1, True)).is_nonzero is None
assert Piecewise((1, x < 1), (2, True)).is_nonzero
assert Piecewise((0, x < 1), (oo, True)).is_finite is None
assert Piecewise((0, x < 1), (1, True)).is_finite
b = Basic()
assert Piecewise((b, x < 1)).is_finite is None
# 10258
c = Piecewise((1, x < 0), (2, True)) < 3
assert c != True
assert piecewise_fold(c) == True
def test_issue_10087():
a, b = Piecewise((x, x > 1), (2, True)), Piecewise((x, x > 3), (3, True))
m = a*b
f = piecewise_fold(m)
for i in (0, 2, 4):
assert m.subs(x, i) == f.subs(x, i)
m = a + b
f = piecewise_fold(m)
for i in (0, 2, 4):
assert m.subs(x, i) == f.subs(x, i)
def test_issue_8919():
c = symbols('c:5')
x = symbols("x")
f1 = Piecewise((c[1], x < 1), (c[2], True))
f2 = Piecewise((c[3], x < S(1)/3), (c[4], True))
assert integrate(f1*f2, (x, 0, 2)
) == c[1]*c[3]/3 + 2*c[1]*c[4]/3 + c[2]*c[4]
f1 = Piecewise((0, x < 1), (2, True))
f2 = Piecewise((3, x < 2), (0, True))
assert integrate(f1*f2, (x, 0, 3)) == 6
y = symbols("y", positive=True)
a, b, c, x, z = symbols("a,b,c,x,z", real=True)
I = Integral(Piecewise(
(0, (x >= y) | (x < 0) | (b > c)),
(a, True)), (x, 0, z))
ans = I.doit()
assert ans == Piecewise((0, b > c), (a*Min(y, z) - a*Min(0, z), True))
for cond in (True, False):
for yy in range(1, 3):
for zz in range(-yy, 0, yy):
reps = [(b > c, cond), (y, yy), (z, zz)]
assert ans.subs(reps) == I.subs(reps).doit()
def test_unevaluated_integrals():
f = Function('f')
p = Piecewise((1, Eq(f(x) - 1, 0)), (2, x - 10 < 0), (0, True))
assert p.integrate(x) == Integral(p, x)
assert p.integrate((x, 0, 5)) == Integral(p, (x, 0, 5))
# test it by replacing f(x) with x%2 which will not
# affect the answer: the integrand is essentially 2 over
# the domain of integration
assert Integral(p, (x, 0, 5)).subs(f(x), x%2).n() == 10
# this is a test of using _solve_inequality when
# solve_univariate_inequality fails
assert p.integrate(y) == Piecewise(
(y, Eq(f(x), 1) | ((x < 10) & Eq(f(x), 1))),
(2*y, (x >= -oo) & (x < 10)), (0, True))
def test_conditions_as_alternate_booleans():
a, b, c = symbols('a:c')
assert Piecewise((x, Piecewise((y < 1, x > 0), (y > 1, True)))
) == Piecewise((x, ITE(x > 0, y < 1, y > 1)))
def test_Piecewise_rewrite_as_ITE():
a, b, c, d = symbols('a:d')
def _ITE(*args):
return Piecewise(*args).rewrite(ITE)
assert _ITE((a, x < 1), (b, x >= 1)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, x < oo)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, Or(y < 1, x < oo)), (c, y > 0)
) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, True)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, x < 2), (c, True)
) == ITE(x < 1, a, ITE(x < 2, b, c))
assert _ITE((a, x < 1), (b, y < 2), (c, True)
) == ITE(x < 1, a, ITE(y < 2, b, c))
assert _ITE((a, x < 1), (b, x < oo), (c, y < 1)
) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (c, y < 1), (b, x < oo), (d, True)
) == ITE(x < 1, a, ITE(y < 1, c, b))
assert _ITE((a, x < 0), (b, Or(x < oo, y < 1))
) == ITE(x < 0, a, b)
raises(TypeError, lambda: _ITE((x + 1, x < 1), (x, True)))
# if `a` in the following were replaced with y then the coverage
# is complete but something other than as_set would need to be
# used to detect this
raises(NotImplementedError, lambda: _ITE((x, x < y), (y, x >= a)))
raises(ValueError, lambda: _ITE((a, x < 2), (b, x > 3)))
def test_issue_14052():
assert integrate(abs(sin(x)), (x, 0, 2*pi)) == 4
def test_issue_14240():
assert piecewise_fold(
Piecewise((1, a), (2, b), (4, True)) +
Piecewise((8, a), (16, True))
) == Piecewise((9, a), (18, b), (20, True))
assert piecewise_fold(
Piecewise((2, a), (3, b), (5, True)) *
Piecewise((7, a), (11, True))
) == Piecewise((14, a), (33, b), (55, True))
# these will hang if naive folding is used
assert piecewise_fold(Add(*[
Piecewise((i, a), (0, True)) for i in range(40)])
) == Piecewise((780, a), (0, True))
assert piecewise_fold(Mul(*[
Piecewise((i, a), (0, True)) for i in range(1, 41)])
) == Piecewise((factorial(40), a), (0, True))
def test_issue_14787():
x = Symbol('x')
f = Piecewise((x, x < 1), ((S(58) / 7), True))
assert str(f.evalf()) == "Piecewise((x, x < 1), (8.28571428571429, True))"
def test_issue_8458():
x, y = symbols('x y')
# Original issue
p1 = Piecewise((0, Eq(x, 0)), (sin(x), True))
assert p1.simplify() == sin(x)
# Slightly larger variant
p2 = Piecewise((x, Eq(x, 0)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
assert p2.simplify() == sin(x)
# Test for problem highlighted during review
p3 = Piecewise((x+1, Eq(x, -1)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
assert p3.simplify() == Piecewise((0, Eq(x, -1)), (sin(x), True))
|
6b4ef07f121c8f2e67c5c2fa773fa45d4fcf80f8d697f7ce63ae74a4d11aea2b | import itertools as it
from sympy.core.function import Function
from sympy.core.numbers import I, oo, Rational
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.elementary.miscellaneous import (sqrt, cbrt, root, Min,
Max, real_root)
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.integers import floor, ceiling
from sympy.functions.special.delta_functions import Heaviside
from sympy.utilities.lambdify import lambdify
from sympy.utilities.pytest import raises, skip, ignore_warnings
from sympy.external import import_module
def test_Min():
from sympy.abc import x, y, z
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
nn = Symbol('nn', nonnegative=True)
nn_ = Symbol('nn_', nonnegative=True)
p = Symbol('p', positive=True)
p_ = Symbol('p_', positive=True)
np = Symbol('np', nonpositive=True)
np_ = Symbol('np_', nonpositive=True)
r = Symbol('r', real=True)
assert Min(5, 4) == 4
assert Min(-oo, -oo) == -oo
assert Min(-oo, n) == -oo
assert Min(n, -oo) == -oo
assert Min(-oo, np) == -oo
assert Min(np, -oo) == -oo
assert Min(-oo, 0) == -oo
assert Min(0, -oo) == -oo
assert Min(-oo, nn) == -oo
assert Min(nn, -oo) == -oo
assert Min(-oo, p) == -oo
assert Min(p, -oo) == -oo
assert Min(-oo, oo) == -oo
assert Min(oo, -oo) == -oo
assert Min(n, n) == n
assert Min(n, np) == Min(n, np)
assert Min(np, n) == Min(np, n)
assert Min(n, 0) == n
assert Min(0, n) == n
assert Min(n, nn) == n
assert Min(nn, n) == n
assert Min(n, p) == n
assert Min(p, n) == n
assert Min(n, oo) == n
assert Min(oo, n) == n
assert Min(np, np) == np
assert Min(np, 0) == np
assert Min(0, np) == np
assert Min(np, nn) == np
assert Min(nn, np) == np
assert Min(np, p) == np
assert Min(p, np) == np
assert Min(np, oo) == np
assert Min(oo, np) == np
assert Min(0, 0) == 0
assert Min(0, nn) == 0
assert Min(nn, 0) == 0
assert Min(0, p) == 0
assert Min(p, 0) == 0
assert Min(0, oo) == 0
assert Min(oo, 0) == 0
assert Min(nn, nn) == nn
assert Min(nn, p) == Min(nn, p)
assert Min(p, nn) == Min(p, nn)
assert Min(nn, oo) == nn
assert Min(oo, nn) == nn
assert Min(p, p) == p
assert Min(p, oo) == p
assert Min(oo, p) == p
assert Min(oo, oo) == oo
assert Min(n, n_).func is Min
assert Min(nn, nn_).func is Min
assert Min(np, np_).func is Min
assert Min(p, p_).func is Min
# lists
assert Min() == S.Infinity
assert Min(x) == x
assert Min(x, y) == Min(y, x)
assert Min(x, y, z) == Min(z, y, x)
assert Min(x, Min(y, z)) == Min(z, y, x)
assert Min(x, Max(y, -oo)) == Min(x, y)
assert Min(p, oo, n, p, p, p_) == n
assert Min(p_, n_, p) == n_
assert Min(n, oo, -7, p, p, 2) == Min(n, -7)
assert Min(2, x, p, n, oo, n_, p, 2, -2, -2) == Min(-2, x, n, n_)
assert Min(0, x, 1, y) == Min(0, x, y)
assert Min(1000, 100, -100, x, p, n) == Min(n, x, -100)
assert Min(cos(x), sin(x)) == Min(cos(x), sin(x))
assert Min(cos(x), sin(x)).subs(x, 1) == cos(1)
assert Min(cos(x), sin(x)).subs(x, S(1)/2) == sin(S(1)/2)
raises(ValueError, lambda: Min(cos(x), sin(x)).subs(x, I))
raises(ValueError, lambda: Min(I))
raises(ValueError, lambda: Min(I, x))
raises(ValueError, lambda: Min(S.ComplexInfinity, x))
assert Min(1, x).diff(x) == Heaviside(1 - x)
assert Min(x, 1).diff(x) == Heaviside(1 - x)
assert Min(0, -x, 1 - 2*x).diff(x) == -Heaviside(x + Min(0, -2*x + 1)) \
- 2*Heaviside(2*x + Min(0, -x) - 1)
# issue 7619
f = Function('f')
assert Min(1, 2*Min(f(1), 2)) # doesn't fail
# issue 7233
e = Min(0, x)
assert e.evalf == e.n
assert e.n().args == (0, x)
# issue 8643
m = Min(n, p_, n_, r)
assert m.is_positive is False
assert m.is_nonnegative is False
assert m.is_negative is True
m = Min(p, p_)
assert m.is_positive is True
assert m.is_nonnegative is True
assert m.is_negative is False
m = Min(p, nn_, p_)
assert m.is_positive is None
assert m.is_nonnegative is True
assert m.is_negative is False
m = Min(nn, p, r)
assert m.is_positive is None
assert m.is_nonnegative is None
assert m.is_negative is None
def test_Max():
from sympy.abc import x, y, z
n = Symbol('n', negative=True)
n_ = Symbol('n_', negative=True)
nn = Symbol('nn', nonnegative=True)
nn_ = Symbol('nn_', nonnegative=True)
p = Symbol('p', positive=True)
p_ = Symbol('p_', positive=True)
np = Symbol('np', nonpositive=True)
np_ = Symbol('np_', nonpositive=True)
r = Symbol('r', real=True)
assert Max(5, 4) == 5
# lists
assert Max() == S.NegativeInfinity
assert Max(x) == x
assert Max(x, y) == Max(y, x)
assert Max(x, y, z) == Max(z, y, x)
assert Max(x, Max(y, z)) == Max(z, y, x)
assert Max(x, Min(y, oo)) == Max(x, y)
assert Max(n, -oo, n_, p, 2) == Max(p, 2)
assert Max(n, -oo, n_, p) == p
assert Max(2, x, p, n, -oo, S.NegativeInfinity, n_, p, 2) == Max(2, x, p)
assert Max(0, x, 1, y) == Max(1, x, y)
assert Max(r, r + 1, r - 1) == 1 + r
assert Max(1000, 100, -100, x, p, n) == Max(p, x, 1000)
assert Max(cos(x), sin(x)) == Max(sin(x), cos(x))
assert Max(cos(x), sin(x)).subs(x, 1) == sin(1)
assert Max(cos(x), sin(x)).subs(x, S(1)/2) == cos(S(1)/2)
raises(ValueError, lambda: Max(cos(x), sin(x)).subs(x, I))
raises(ValueError, lambda: Max(I))
raises(ValueError, lambda: Max(I, x))
raises(ValueError, lambda: Max(S.ComplexInfinity, 1))
assert Max(n, -oo, n_, p, 2) == Max(p, 2)
assert Max(n, -oo, n_, p, 1000) == Max(p, 1000)
assert Max(1, x).diff(x) == Heaviside(x - 1)
assert Max(x, 1).diff(x) == Heaviside(x - 1)
assert Max(x**2, 1 + x, 1).diff(x) == \
2*x*Heaviside(x**2 - Max(1, x + 1)) \
+ Heaviside(x - Max(1, x**2) + 1)
e = Max(0, x)
assert e.evalf == e.n
assert e.n().args == (0, x)
# issue 8643
m = Max(p, p_, n, r)
assert m.is_positive is True
assert m.is_nonnegative is True
assert m.is_negative is False
m = Max(n, n_)
assert m.is_positive is False
assert m.is_nonnegative is False
assert m.is_negative is True
m = Max(n, n_, r)
assert m.is_positive is None
assert m.is_nonnegative is None
assert m.is_negative is None
m = Max(n, nn, r)
assert m.is_positive is None
assert m.is_nonnegative is True
assert m.is_negative is False
def test_minmax_assumptions():
r = Symbol('r', real=True)
a = Symbol('a', real=True, algebraic=True)
t = Symbol('t', real=True, transcendental=True)
q = Symbol('q', rational=True)
p = Symbol('p', real=True, rational=False)
n = Symbol('n', rational=True, integer=False)
i = Symbol('i', integer=True)
o = Symbol('o', odd=True)
e = Symbol('e', even=True)
k = Symbol('k', prime=True)
reals = [r, a, t, q, p, n, i, o, e, k]
for ext in (Max, Min):
for x, y in it.product(reals, repeat=2):
# Must be real
assert ext(x, y).is_real
# Algebraic?
if x.is_algebraic and y.is_algebraic:
assert ext(x, y).is_algebraic
elif x.is_transcendental and y.is_transcendental:
assert ext(x, y).is_transcendental
else:
assert ext(x, y).is_algebraic is None
# Rational?
if x.is_rational and y.is_rational:
assert ext(x, y).is_rational
elif x.is_irrational and y.is_irrational:
assert ext(x, y).is_irrational
else:
assert ext(x, y).is_rational is None
# Integer?
if x.is_integer and y.is_integer:
assert ext(x, y).is_integer
elif x.is_noninteger and y.is_noninteger:
assert ext(x, y).is_noninteger
else:
assert ext(x, y).is_integer is None
# Odd?
if x.is_odd and y.is_odd:
assert ext(x, y).is_odd
elif x.is_odd is False and y.is_odd is False:
assert ext(x, y).is_odd is False
else:
assert ext(x, y).is_odd is None
# Even?
if x.is_even and y.is_even:
assert ext(x, y).is_even
elif x.is_even is False and y.is_even is False:
assert ext(x, y).is_even is False
else:
assert ext(x, y).is_even is None
# Prime?
if x.is_prime and y.is_prime:
assert ext(x, y).is_prime
elif x.is_prime is False and y.is_prime is False:
assert ext(x, y).is_prime is False
else:
assert ext(x, y).is_prime is None
def test_issue_8413():
x = Symbol('x', real=True)
# we can't evaluate in general because non-reals are not
# comparable: Min(floor(3.2 + I), 3.2 + I) -> ValueError
assert Min(floor(x), x) == floor(x)
assert Min(ceiling(x), x) == x
assert Max(floor(x), x) == x
assert Max(ceiling(x), x) == ceiling(x)
def test_root():
from sympy.abc import x
n = Symbol('n', integer=True)
k = Symbol('k', integer=True)
assert root(2, 2) == sqrt(2)
assert root(2, 1) == 2
assert root(2, 3) == 2**Rational(1, 3)
assert root(2, 3) == cbrt(2)
assert root(2, -5) == 2**Rational(4, 5)/2
assert root(-2, 1) == -2
assert root(-2, 2) == sqrt(2)*I
assert root(-2, 1) == -2
assert root(x, 2) == sqrt(x)
assert root(x, 1) == x
assert root(x, 3) == x**Rational(1, 3)
assert root(x, 3) == cbrt(x)
assert root(x, -5) == x**Rational(-1, 5)
assert root(x, n) == x**(1/n)
assert root(x, -n) == x**(-1/n)
assert root(x, n, k) == (-1)**(2*k/n)*x**(1/n)
def test_real_root():
assert real_root(-8, 3) == -2
assert real_root(-16, 4) == root(-16, 4)
r = root(-7, 4)
assert real_root(r) == r
r1 = root(-1, 3)
r2 = r1**2
r3 = root(-1, 4)
assert real_root(r1 + r2 + r3) == -1 + r2 + r3
assert real_root(root(-2, 3)) == -root(2, 3)
assert real_root(-8., 3) == -2
x = Symbol('x')
n = Symbol('n')
g = real_root(x, n)
assert g.subs(dict(x=-8, n=3)) == -2
assert g.subs(dict(x=8, n=3)) == 2
# give principle root if there is no real root -- if this is not desired
# then maybe a Root class is needed to raise an error instead
assert g.subs(dict(x=I, n=3)) == cbrt(I)
assert g.subs(dict(x=-8, n=2)) == sqrt(-8)
assert g.subs(dict(x=I, n=2)) == sqrt(I)
def test_issue_11463():
numpy = import_module('numpy')
if not numpy:
skip("numpy not installed.")
x = Symbol('x')
f = lambdify(x, real_root((log(x/(x-2))), 3), 'numpy')
# numpy.select evaluates all options before considering conditions,
# so it raises a warning about root of negative number which does
# not affect the outcome. This warning is suppressed here
with ignore_warnings(RuntimeWarning):
assert f(numpy.array(-1)) < -1
def test_rewrite_MaxMin_as_Heaviside():
from sympy.abc import x
assert Max(0, x).rewrite(Heaviside) == x*Heaviside(x)
assert Max(3, x).rewrite(Heaviside) == x*Heaviside(x - 3) + \
3*Heaviside(-x + 3)
assert Max(0, x+2, 2*x).rewrite(Heaviside) == \
2*x*Heaviside(2*x)*Heaviside(x - 2) + \
(x + 2)*Heaviside(-x + 2)*Heaviside(x + 2)
assert Min(0, x).rewrite(Heaviside) == x*Heaviside(-x)
assert Min(3, x).rewrite(Heaviside) == x*Heaviside(-x + 3) + \
3*Heaviside(x - 3)
assert Min(x, -x, -2).rewrite(Heaviside) == \
x*Heaviside(-2*x)*Heaviside(-x - 2) - \
x*Heaviside(2*x)*Heaviside(x - 2) \
- 2*Heaviside(-x + 2)*Heaviside(x + 2)
def test_rewrite_MaxMin_as_Piecewise():
from sympy import symbols, Piecewise
x, y, z, a, b = symbols('x y z a b', real=True)
vx, vy, va = symbols('vx vy va')
assert Max(a, b).rewrite(Piecewise) == Piecewise((a, a >= b), (b, True))
assert Max(x, y, z).rewrite(Piecewise) == Piecewise((x, (x >= y) & (x >= z)), (y, y >= z), (z, True))
assert Max(x, y, a, b).rewrite(Piecewise) == Piecewise((a, (a >= b) & (a >= x) & (a >= y)),
(b, (b >= x) & (b >= y)), (x, x >= y), (y, True))
assert Min(a, b).rewrite(Piecewise) == Piecewise((a, a <= b), (b, True))
assert Min(x, y, z).rewrite(Piecewise) == Piecewise((x, (x <= y) & (x <= z)), (y, y <= z), (z, True))
assert Min(x, y, a, b).rewrite(Piecewise) == Piecewise((a, (a <= b) & (a <= x) & (a <= y)),
(b, (b <= x) & (b <= y)), (x, x <= y), (y, True))
# Piecewise rewriting of Min/Max does also takes place for not explicitly real arguments
assert Max(vx, vy).rewrite(Piecewise) == Piecewise((vx, vx >= vy), (vy, True))
assert Min(va, vx, vy).rewrite(Piecewise) == Piecewise((va, (va <= vx) & (va <= vy)), (vx, vx <= vy), (vy, True))
def test_issue_11099():
from sympy.abc import x, y
# some fixed value tests
fixed_test_data = {x: -2, y: 3}
assert Min(x, y).evalf(subs=fixed_test_data) == \
Min(x, y).subs(fixed_test_data).evalf()
assert Max(x, y).evalf(subs=fixed_test_data) == \
Max(x, y).subs(fixed_test_data).evalf()
# randomly generate some test data
from random import randint
for i in range(20):
random_test_data = {x: randint(-100, 100), y: randint(-100, 100)}
assert Min(x, y).evalf(subs=random_test_data) == \
Min(x, y).subs(random_test_data).evalf()
assert Max(x, y).evalf(subs=random_test_data) == \
Max(x, y).subs(random_test_data).evalf()
def test_issue_12638():
from sympy.abc import a, b, c, d
assert Min(a, b, c, Max(a, b)) == Min(a, b, c)
assert Min(a, b, Max(a, b, c)) == Min(a, b)
assert Min(a, b, Max(a, c)) == Min(a, b)
def test_instantiation_evaluation():
from sympy.abc import v, w, x, y, z
assert Min(1, Max(2, x)) == 1
assert Max(3, Min(2, x)) == 3
assert Min(Max(x, y), Max(x, z)) == Max(x, Min(y, z))
assert set(Min(Max(w, x), Max(y, z)).args) == set(
[Max(w, x), Max(y, z)])
assert Min(Max(x, y), Max(x, z), w) == Min(
w, Max(x, Min(y, z)))
A, B = Min, Max
for i in range(2):
assert A(x, B(x, y)) == x
assert A(x, B(y, A(x, w, z))) == A(x, B(y, A(w, z)))
A, B = B, A
assert Min(w, Max(x, y), Max(v, x, z)) == Min(
w, Max(x, Min(y, Max(v, z))))
def test_rewrite_as_Abs():
from itertools import permutations
from sympy.functions.elementary.complexes import Abs
from sympy.abc import x, y, z, w
def test(e):
free = e.free_symbols
a = e.rewrite(Abs)
assert not a.has(Min, Max)
for i in permutations(range(len(free))):
reps = dict(zip(free, i))
assert a.xreplace(reps) == e.xreplace(reps)
test(Min(x, y))
test(Max(x, y))
test(Min(x, y, z))
test(Min(Max(w, x), Max(y, z)))
def test_issue_14000():
assert isinstance(sqrt(4, evaluate=False), Pow) == True
assert isinstance(cbrt(3.5, evaluate=False), Pow) == True
assert isinstance(root(16, 4, evaluate=False), Pow) == True
assert sqrt(4, evaluate=False) == Pow(4, S.Half, evaluate=False)
assert cbrt(3.5, evaluate=False) == Pow(3.5, Rational(1, 3), evaluate=False)
assert root(4, 2, evaluate=False) == Pow(4, Rational(1, 2), evaluate=False)
assert root(16, 4, 2, evaluate=False).has(Pow) == True
assert real_root(-8, 3, evaluate=False).has(Pow) == True
|
6421836dc0f995ee4a87199a7d88f497de8e33ee05b83b120d48c7c36f42a871 | from sympy.core.containers import Tuple
from sympy.core.function import (Function, Lambda, nfloat)
from sympy.core.mod import Mod
from sympy.core.numbers import (E, I, Rational, oo, pi)
from sympy.core.relational import (Eq, Gt,
Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.functions.elementary.complexes import (Abs, arg, im, re, sign)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (HyperbolicFunction,
atanh, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (
TrigonometricFunction, acos, acot, acsc, asec, asin, atan, atan2,
cos, cot, csc, sec, sin, tan)
from sympy.functions.special.error_functions import (erf, erfc,
erfcinv, erfinv)
from sympy.logic.boolalg import And
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.polys.polytools import Poly
from sympy.polys.rootoftools import CRootOf
from sympy.sets.contains import Contains
from sympy.sets.conditionset import ConditionSet
from sympy.sets.fancysets import ImageSet
from sympy.sets.sets import (Complement, EmptySet, FiniteSet,
Intersection, Interval, Union, imageset)
from sympy.tensor.indexed import Indexed
from sympy.utilities.iterables import numbered_symbols
from sympy.utilities.pytest import XFAIL, raises, skip, slow, SKIP
from sympy.utilities.randtest import verify_numerically as tn
from sympy.physics.units import cm
from sympy.core.containers import Dict
from sympy.solvers.solveset import (
solveset_real, domain_check, solveset_complex, linear_eq_to_matrix,
linsolve, _is_function_class_equation, invert_real, invert_complex,
solveset, solve_decomposition, substitution, nonlinsolve, solvify,
_is_finite_with_finite_vars, _transolve, _is_exponential,
_solve_exponential, _is_logarithmic,
_solve_logarithm, _term_factors)
a = Symbol('a', real=True)
b = Symbol('b', real=True)
c = Symbol('c', real=True)
x = Symbol('x', real=True)
y = Symbol('y', real=True)
z = Symbol('z', real=True)
q = Symbol('q', real=True)
m = Symbol('m', real=True)
n = Symbol('n', real=True)
def test_invert_real():
x = Symbol('x', real=True)
y = Symbol('y')
n = Symbol('n')
def ireal(x, s=S.Reals):
return Intersection(s, x)
# issue 14223
assert invert_real(x, 0, x, Interval(1, 2)) == (x, S.EmptySet)
assert invert_real(exp(x), y, x) == (x, ireal(FiniteSet(log(y))))
y = Symbol('y', positive=True)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, ireal(FiniteSet(log(y - 3))))
assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
assert invert_real(Abs(x), y, x) == (x, FiniteSet(y, -y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2)))
assert invert_real(2**exp(x), y, x) == (x, ireal(FiniteSet(log(log(y)/log(2)))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**Rational(1, 2), y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
lhs = x**31 + x
conditions = Contains(y, Interval(0, oo), evaluate=False)
base_values = FiniteSet(y - 1, -y - 1)
assert invert_real(Abs(x**31 + x + 1), y, x) == (lhs, base_values)
assert invert_real(sin(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers))
assert invert_real(sin(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers))
assert invert_real(csc(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers))
assert invert_real(csc(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers))
assert invert_real(cos(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers)))
assert invert_real(cos(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + Mod(acos(y), 2*pi))), S.Integers), \
imageset(Lambda(n, log(2*n*pi + Mod(-acos(y), 2*pi))), S.Integers)))
assert invert_real(sec(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers)))
assert invert_real(sec(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + Mod(asec(y), 2*pi))), S.Integers), \
imageset(Lambda(n, log(2*n*pi + Mod(-asec(y), 2*pi))), S.Integers)))
assert invert_real(tan(x), y, x) == \
(x, imageset(Lambda(n, n*pi + atan(y) % pi), S.Integers))
assert invert_real(tan(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + atan(y) % pi)), S.Integers))
assert invert_real(cot(x), y, x) == \
(x, imageset(Lambda(n, n*pi + acot(y) % pi), S.Integers))
assert invert_real(cot(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + acot(y) % pi)), S.Integers))
assert invert_real(tan(tan(x)), y, x) == \
(tan(x), imageset(Lambda(n, n*pi + atan(y) % pi), S.Integers))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
def test_invert_complex():
assert invert_complex(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_complex(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_complex(exp(x), y, x) == \
(x, imageset(Lambda(n, I*(2*pi*n + arg(y)) + log(Abs(y))), S.Integers))
assert invert_complex(log(x), y, x) == (x, FiniteSet(exp(y)))
raises(ValueError, lambda: invert_real(1, y, x))
raises(ValueError, lambda: invert_complex(x, x, x))
raises(ValueError, lambda: invert_complex(x, x, 1))
# https://github.com/skirpichev/omg/issues/16
assert invert_complex(sinh(x), 0, x) != (x, FiniteSet(0))
def test_domain_check():
assert domain_check(1/(1 + (1/(x+1))**2), x, -1) is False
assert domain_check(x**2, x, 0) is True
assert domain_check(x, x, oo) is False
assert domain_check(0, x, oo) is False
def test_issue_11536():
assert solveset(0**x - 100, x, S.Reals) == S.EmptySet
assert solveset(0**x - 1, x, S.Reals) == FiniteSet(0)
def test_is_function_class_equation():
from sympy.abc import x, a
assert _is_function_class_equation(TrigonometricFunction,
tan(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x) - a, x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x + a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x*a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
a*tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**2 + sin(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + x, x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2) + sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(sin(x)) + sin(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x) - a, x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x + a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x*a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
a*tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**2 + sinh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + x, x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2) + sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(sinh(x)) + sinh(x), x) is False
def test_garbage_input():
raises(ValueError, lambda: solveset_real([x], x))
assert solveset_real(x, 1) == S.EmptySet
assert solveset_real(x - 1, 1) == FiniteSet(x)
assert solveset_real(x, pi) == S.EmptySet
assert solveset_real(x, x**2) == S.EmptySet
raises(ValueError, lambda: solveset_complex([x], x))
assert solveset_complex(x, pi) == S.EmptySet
raises(ValueError, lambda: solveset((x, y), x))
raises(ValueError, lambda: solveset(x + 1, S.Reals))
raises(ValueError, lambda: solveset(x + 1, x, 2))
def test_solve_mul():
assert solveset_real((a*x + b)*(exp(x) - 3), x) == \
FiniteSet(-b/a, log(3))
assert solveset_real((2*x + 8)*(8 + exp(x)), x) == FiniteSet(S(-4))
assert solveset_real(x/log(x), x) == EmptySet()
def test_solve_invert():
assert solveset_real(exp(x) - 3, x) == FiniteSet(log(3))
assert solveset_real(log(x) - 3, x) == FiniteSet(exp(3))
assert solveset_real(3**(x + 2), x) == FiniteSet()
assert solveset_real(3**(2 - x), x) == FiniteSet()
assert solveset_real(y - b*exp(a/x), x) == Intersection(
S.Reals, FiniteSet(a/log(y/b)))
# issue 4504
assert solveset_real(2**x - 10, x) == FiniteSet(1 + log(5)/log(2))
def test_errorinverses():
assert solveset_real(erf(x) - S.One/2, x) == \
FiniteSet(erfinv(S.One/2))
assert solveset_real(erfinv(x) - 2, x) == \
FiniteSet(erf(2))
assert solveset_real(erfc(x) - S.One, x) == \
FiniteSet(erfcinv(S.One))
assert solveset_real(erfcinv(x) - 2, x) == FiniteSet(erfc(2))
def test_solve_polynomial():
assert solveset_real(3*x - 2, x) == FiniteSet(Rational(2, 3))
assert solveset_real(x**2 - 1, x) == FiniteSet(-S(1), S(1))
assert solveset_real(x - y**3, x) == FiniteSet(y ** 3)
a11, a12, a21, a22, b1, b2 = symbols('a11, a12, a21, a22, b1, b2')
assert solveset_real(x**3 - 15*x - 4, x) == FiniteSet(
-2 + 3 ** Rational(1, 2),
S(4),
-2 - 3 ** Rational(1, 2))
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert len(solveset_real(x**5 + x**3 + 1, x)) == 1
assert len(solveset_real(-2*x**3 + 4*x**2 - 2*x + 6, x)) > 0
assert solveset_real(x**6 + x**4 + I, x) == ConditionSet(x,
Eq(x**6 + x**4 + I, 0), S.Reals)
def test_return_root_of():
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get CRootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(list(solveset_complex(x**5 + 3*x**3 + 7, x))[0],
exponent=False) == CRootOf(x**5 + 3*x**3 + 7, 0).n()
sol = list(solveset_complex(x**6 - 2*x + 2, x))
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
s = x**5 + 4*x**3 + 3*x**2 + S(7)/4
assert solveset_complex(s, x) == \
FiniteSet(*Poly(s*4, domain='ZZ').all_roots())
# Refer issue #7876
eq = x*(x - 1)**2*(x + 1)*(x**6 - x + 1)
assert solveset_complex(eq, x) == \
FiniteSet(-1, 0, 1, CRootOf(x**6 - x + 1, 0),
CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2),
CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4),
CRootOf(x**6 - x + 1, 5))
def test__has_rational_power():
from sympy.solvers.solveset import _has_rational_power
assert _has_rational_power(sqrt(2), x)[0] is False
assert _has_rational_power(x*sqrt(2), x)[0] is False
assert _has_rational_power(x**2*sqrt(x), x) == (True, 2)
assert _has_rational_power(sqrt(2)*x**(S(1)/3), x) == (True, 3)
assert _has_rational_power(sqrt(x)*x**(S(1)/3), x) == (True, 6)
def test_solveset_sqrt_1():
assert solveset_real(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S(1), S(2))
assert solveset_real(sqrt(x - 1) - x + 7, x) == FiniteSet(10)
assert solveset_real(sqrt(x - 2) - 5, x) == FiniteSet(27)
assert solveset_real(sqrt(x) - 2 - 5, x) == FiniteSet(49)
assert solveset_real(sqrt(x**3), x) == FiniteSet(0)
assert solveset_real(sqrt(x - 1), x) == FiniteSet(1)
def test_solveset_sqrt_2():
# http://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solveset_real(sqrt(2*x - 1) - sqrt(x - 4) - 2, x) == \
FiniteSet(S(5), S(13))
assert solveset_real(sqrt(x + 7) + 2 - sqrt(3 - x), x) == \
FiniteSet(-6)
# http://www.purplemath.com/modules/solverad.htm
assert solveset_real(sqrt(17*x - sqrt(x**2 - 5)) - 7, x) == \
FiniteSet(3)
eq = x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4)
assert solveset_real(eq, x) == FiniteSet(-S(1)/2, -S(1)/3)
eq = sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)
assert solveset_real(eq, x) == FiniteSet(0)
eq = sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)
assert solveset_real(eq, x) == FiniteSet(5)
eq = sqrt(x)*sqrt(x - 7) - 12
assert solveset_real(eq, x) == FiniteSet(16)
eq = sqrt(x - 3) + sqrt(x) - 3
assert solveset_real(eq, x) == FiniteSet(4)
eq = sqrt(2*x**2 - 7) - (3 - x)
assert solveset_real(eq, x) == FiniteSet(-S(8), S(2))
# others
eq = sqrt(9*x**2 + 4) - (3*x + 2)
assert solveset_real(eq, x) == FiniteSet(0)
assert solveset_real(sqrt(x - 3) - sqrt(x) - 3, x) == FiniteSet()
eq = (2*x - 5)**Rational(1, 3) - 3
assert solveset_real(eq, x) == FiniteSet(16)
assert solveset_real(sqrt(x) + sqrt(sqrt(x)) - 4, x) == \
FiniteSet((-S.Half + sqrt(17)/2)**4)
eq = sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))
assert solveset_real(eq, x) == FiniteSet()
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
ans = solveset_real(eq, x)
ra = S('''-1484/375 - 4*(-1/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(1/3) - 172564/(140625*(-1/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(1/3))''')
rb = S(4)/5
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
set([i.n(chop=True) for i in ans]) == \
set([i.n(chop=True) for i in (ra, rb)])
assert solveset_real(sqrt(x) + x**Rational(1, 3) +
x**Rational(1, 4), x) == FiniteSet(0)
assert solveset_real(x/sqrt(x**2 + 1), x) == FiniteSet(0)
eq = (x - y**3)/((y**2)*sqrt(1 - y**2))
assert solveset_real(eq, x) == FiniteSet(y**3)
# issue 4497
assert solveset_real(1/(5 + x)**(S(1)/5) - 9, x) == \
FiniteSet(-295244/S(59049))
@XFAIL
def test_solve_sqrt_fail():
# this only works if we check real_root(eq.subs(x, S(1)/3))
# but checksol doesn't work like that
eq = (x**3 - 3*x**2)**Rational(1, 3) + 1 - x
assert solveset_real(eq, x) == FiniteSet(S(1)/3)
@slow
def test_solve_sqrt_3():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solveset_complex(eq, R)
fset = [S(5)/3 + 4*sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3,
-sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 +
40*re(1/((-S(1)/2 - sqrt(3)*I/2)*(S(251)/27 + sqrt(111)*I/9)**(S(1)/3)))/9 +
sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 + S(5)/3 +
I*(-sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
40*im(1/((-S(1)/2 - sqrt(3)*I/2)*(S(251)/27 + sqrt(111)*I/9)**(S(1)/3)))/9)]
cset = [40*re(1/((-S(1)/2 + sqrt(3)*I/2)*(S(251)/27 + sqrt(111)*I/9)**(S(1)/3)))/9 -
sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 - sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 +
S(5)/3 +
I*(40*im(1/((-S(1)/2 + sqrt(3)*I/2)*(S(251)/27 + sqrt(111)*I/9)**(S(1)/3)))/9 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3)]
assert sol._args[0] == FiniteSet(*fset)
assert sol._args[1] == ConditionSet(
R,
Eq(sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1), 0),
FiniteSet(*cset))
# the number of real roots will depend on the value of m: for m=1 there are 4
# and for m=-1 there are none.
eq = -sqrt((m - q)**2 + (-m/(2*q) + S(1)/2)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2)
unsolved_object = ConditionSet(q, Eq(sqrt((m - q)**2 + (-m/(2*q) + S(1)/2)**2) -
sqrt((-m**2/2 - sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2 + (m**2/2 - m -
sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2), 0), S.Reals)
assert solveset_real(eq, q) == unsolved_object
def test_solve_polynomial_symbolic_param():
assert solveset_complex((x**2 - 1)**2 - a, x) == \
FiniteSet(sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a)))
# issue 4507
assert solveset_complex(y - b/(1 + a*x), x) == \
FiniteSet((b/y - 1)/a) - FiniteSet(-1/a)
# issue 4508
assert solveset_complex(y - b*x/(a + x), x) == \
FiniteSet(-a*y/(y - b)) - FiniteSet(-a)
def test_solve_rational():
assert solveset_real(1/x + 1, x) == FiniteSet(-S.One)
assert solveset_real(1/exp(x) - 1, x) == FiniteSet(0)
assert solveset_real(x*(1 - 5/x), x) == FiniteSet(5)
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
assert solveset_real((x**2/(7 - x)).diff(x), x) == \
FiniteSet(S(0), S(14))
def test_solveset_real_gen_is_pow():
assert solveset_real(sqrt(1) + 1, x) == EmptySet()
def test_no_sol():
assert solveset(1 - oo*x) == EmptySet()
assert solveset(oo*x, x) == EmptySet()
assert solveset(oo*x - oo, x) == EmptySet()
assert solveset_real(4, x) == EmptySet()
assert solveset_real(exp(x), x) == EmptySet()
assert solveset_real(x**2 + 1, x) == EmptySet()
assert solveset_real(-3*a/sqrt(x), x) == EmptySet()
assert solveset_real(1/x, x) == EmptySet()
assert solveset_real(-(1 + x)/(2 + x)**2 + 1/(2 + x), x) == \
EmptySet()
def test_sol_zero_real():
assert solveset_real(0, x) == S.Reals
assert solveset(0, x, Interval(1, 2)) == Interval(1, 2)
assert solveset_real(-x**2 - 2*x + (x + 1)**2 - 1, x) == S.Reals
def test_no_sol_rational_extragenous():
assert solveset_real((x/(x + 1) + 3)**(-2), x) == EmptySet()
assert solveset_real((x - 1)/(1 + 1/(x - 1)), x) == EmptySet()
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to
a polynomial equation using the change of variable y -> x**Rational(p, q)
"""
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert solveset_real(x*(x**(S(1) / 3) - 3), x) == \
FiniteSet(S(0), S(27))
def test_solveset_real_rational():
"""Test solveset_real for rational functions"""
assert solveset_real((x - y**3) / ((y**2)*sqrt(1 - y**2)), x) \
== FiniteSet(y**3)
# issue 4486
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
def test_solveset_real_log():
assert solveset_real(log((x-1)*(x+1)), x) == \
FiniteSet(sqrt(2), -sqrt(2))
def test_poly_gens():
assert solveset_real(4**(2*(x**2) + 2*x) - 8, x) == \
FiniteSet(-Rational(3, 2), S.Half)
def test_solve_abs():
x = Symbol('x')
n = Dummy('n')
raises(ValueError, lambda: solveset(Abs(x) - 1, x))
assert solveset(Abs(x) - n, x, S.Reals) == ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n})
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x) + 2, x) is S.EmptySet
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
sol = ConditionSet(
x,
And(
Contains(b, Interval(0, oo)),
Contains(a + b, Interval(0, oo)),
Contains(a - b, Interval(0, oo))),
FiniteSet(-a - b - 3, -a + b - 3, a - b - 3, a + b - 3))
eq = Abs(Abs(x + 3) - a) - b
assert invert_real(eq, 0, x)[1] == sol
reps = {a: 3, b: 1}
eqab = eq.subs(reps)
for i in sol.subs(reps):
assert not eqab.subs(x, i)
assert solveset(Eq(sin(Abs(x)), 1), x, domain=S.Reals) == Union(
Intersection(Interval(0, oo),
ImageSet(Lambda(n, (-1)**n*pi/2 + n*pi), S.Integers)),
Intersection(Interval(-oo, 0),
ImageSet(Lambda(n, n*pi - (-1)**(-n)*pi/2), S.Integers)))
def test_issue_9565():
assert solveset_real(Abs((x - 1)/(x - 5)) <= S(1)/3, x) == Interval(-1, 2)
def test_issue_10069():
eq = abs(1/(x - 1)) - 1 > 0
u = Union(Interval.open(0, 1), Interval.open(1, 2))
assert solveset_real(eq, x) == u
@XFAIL
def test_rewrite_trigh():
# if this import passes then the test below should also pass
from sympy import sech
assert solveset_real(sinh(x) + sech(x), x) == FiniteSet(
2*atanh(-S.Half + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-S.Half + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2),
2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half))
def test_real_imag_splitting():
a, b = symbols('a b', real=True, finite=True)
assert solveset_real(sqrt(a**2 - b**2) - 3, a) == \
FiniteSet(-sqrt(b**2 + 9), sqrt(b**2 + 9))
assert solveset_real(sqrt(a**2 + b**2) - 3, a) != \
S.EmptySet
def test_units():
assert solveset_real(1/x - 1/(2*cm), x) == FiniteSet(2*cm)
def test_solve_only_exp_1():
y = Symbol('y', positive=True, finite=True)
assert solveset_real(exp(x) - y, x) == FiniteSet(log(y))
assert solveset_real(exp(x) + exp(-x) - 4, x) == \
FiniteSet(log(-sqrt(3) + 2), log(sqrt(3) + 2))
assert solveset_real(exp(x) + exp(-x) - y, x) != S.EmptySet
def test_atan2():
# The .inverse() method on atan2 works only if x.is_real is True and the
# second argument is a real constant
assert solveset_real(atan2(x, 2) - pi/3, x) == FiniteSet(2*sqrt(3))
def test_piecewise_solveset():
eq = Piecewise((x - 2, Gt(x, 2)), (2 - x, True)) - 3
assert set(solveset_real(eq, x)) == set(FiniteSet(-1, 5))
absxm3 = Piecewise(
(x - 3, S(0) <= x - 3),
(3 - x, S(0) > x - 3))
y = Symbol('y', positive=True)
assert solveset_real(absxm3 - y, x) == FiniteSet(-y + 3, y + 3)
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solveset(f, x, domain=S.Reals) == Union(FiniteSet(2), Interval(-oo, 0, True, True))
assert solveset(
Piecewise((x + 1, x > 0), (I, True)) - I, x, S.Reals
) == Interval(-oo, 0)
assert solveset(Piecewise((x - 1, Ne(x, I)), (x, True)), x) == FiniteSet(1)
def test_solveset_complex_polynomial():
from sympy.abc import x, a, b, c
assert solveset_complex(a*x**2 + b*x + c, x) == \
FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a),
-b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))
assert solveset_complex(x - y**3, y) == FiniteSet(
(-x**Rational(1, 3))/2 + I*sqrt(3)*x**Rational(1, 3)/2,
x**Rational(1, 3),
(-x**Rational(1, 3))/2 - I*sqrt(3)*x**Rational(1, 3)/2)
assert solveset_complex(x + 1/x - 1, x) == \
FiniteSet(Rational(1, 2) + I*sqrt(3)/2, Rational(1, 2) - I*sqrt(3)/2)
def test_sol_zero_complex():
assert solveset_complex(0, x) == S.Complexes
def test_solveset_complex_rational():
assert solveset_complex((x - 1)*(x - I)/(x - 3), x) == \
FiniteSet(1, I)
assert solveset_complex((x - y**3)/((y**2)*sqrt(1 - y**2)), x) == \
FiniteSet(y**3)
assert solveset_complex(-x**2 - I, x) == \
FiniteSet(-sqrt(2)/2 + sqrt(2)*I/2, sqrt(2)/2 - sqrt(2)*I/2)
def test_solve_quintics():
skip("This test is too slow")
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
f = x**5 + 15*x + 12
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
def test_solveset_complex_exp():
from sympy.abc import x, n
assert solveset_complex(exp(x) - 1, x) == \
imageset(Lambda(n, I*2*n*pi), S.Integers)
assert solveset_complex(exp(x) - I, x) == \
imageset(Lambda(n, I*(2*n*pi + pi/2)), S.Integers)
assert solveset_complex(1/exp(x), x) == S.EmptySet
assert solveset_complex(sinh(x).rewrite(exp), x) == \
imageset(Lambda(n, n*pi*I), S.Integers)
def test_solveset_real_exp():
from sympy.abc import x, y
assert solveset(Eq((-2)**x, 4), x, S.Reals) == FiniteSet(2)
assert solveset(Eq(-2**x, 4), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**x, 27), x, S.Reals) == S.EmptySet
assert solveset(Eq((-5)**(x+1), 625), x, S.Reals) == FiniteSet(3)
assert solveset(Eq(2**(x-3), -16), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**(x - 3), -3**39), x, S.Reals) == FiniteSet(42)
assert solveset(Eq(2**x, y), x, S.Reals) == Intersection(S.Reals, FiniteSet(log(y)/log(2)))
assert invert_real((-2)**(2*x) - 16, 0, x) == (x, FiniteSet(2))
def test_solve_complex_log():
assert solveset_complex(log(x), x) == FiniteSet(1)
assert solveset_complex(1 - log(a + 4*x**2), x) == \
FiniteSet(-sqrt(-a + E)/2, sqrt(-a + E)/2)
def test_solve_complex_sqrt():
assert solveset_complex(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S(1), S(2))
assert solveset_complex(sqrt(5*x + 6) - (2 + 2*I) - x, x) == \
FiniteSet(-S(2), 3 - 4*I)
assert solveset_complex(4*x*(1 - a * sqrt(x)), x) == \
FiniteSet(S(0), 1 / a ** 2)
def test_solveset_complex_tan():
s = solveset_complex(tan(x).rewrite(exp), x)
assert s == imageset(Lambda(n, pi*n), S.Integers) - \
imageset(Lambda(n, pi*n + pi/2), S.Integers)
def test_solve_trig():
from sympy.abc import n
assert solveset_real(sin(x), x) == \
Union(imageset(Lambda(n, 2*pi*n), S.Integers),
imageset(Lambda(n, 2*pi*n + pi), S.Integers))
assert solveset_real(sin(x) - 1, x) == \
imageset(Lambda(n, 2*pi*n + pi/2), S.Integers)
assert solveset_real(cos(x), x) == \
Union(imageset(Lambda(n, 2*pi*n + pi/2), S.Integers),
imageset(Lambda(n, 2*pi*n + 3*pi/2), S.Integers))
assert solveset_real(sin(x) + cos(x), x) == \
Union(imageset(Lambda(n, 2*n*pi + 3*pi/4), S.Integers),
imageset(Lambda(n, 2*n*pi + 7*pi/4), S.Integers))
assert solveset_real(sin(x)**2 + cos(x)**2, x) == S.EmptySet
assert solveset_complex(cos(x) - S.Half, x) == \
Union(imageset(Lambda(n, 2*n*pi + 5*pi/3), S.Integers),
imageset(Lambda(n, 2*n*pi + pi/3), S.Integers))
y, a = symbols('y,a')
assert solveset(sin(y + a) - sin(y), a, domain=S.Reals) == \
imageset(Lambda(n, 2*n*pi), S.Integers)
assert solveset_real(sin(2*x)*cos(x) + cos(2*x)*sin(x)-1, x) == \
ImageSet(Lambda(n, 2*n*pi/3 + pi/6), S.Integers)
# Tests for _solve_trig2() function
assert solveset_real(2*cos(x)*cos(2*x) - 1, x) == \
Union(ImageSet(Lambda(n, 2*n*pi + 2*atan(sqrt(-2*2**(S(1)/3)*(67 +
9*sqrt(57))**(S(2)/3) + 8*2**(S(2)/3) + 11*(67 +
9*sqrt(57))**(S(1)/3))/(3*(67 + 9*sqrt(57))**(S(1)/6)))), S.Integers),
ImageSet(Lambda(n, 2*n*pi - 2*atan(sqrt(-2*2**(S(1)/3)*(67 +
9*sqrt(57))**(S(2)/3) + 8*2**(S(2)/3) + 11*(67 +
9*sqrt(57))**(S(1)/3))/(3*(67 + 9*sqrt(57))**(S(1)/6))) +
2*pi), S.Integers))
assert solveset_real(2*tan(x)*sin(x) + 1, x) == Union(
ImageSet(Lambda(n, 2*n*pi + atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(-sqrt(17) + 1)) + pi), S.Integers),
ImageSet(Lambda(n, 2*n*pi - atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(-sqrt(17) + 1)) + pi), S.Integers))
assert solveset_real(cos(2*x)*cos(4*x) - 1, x) == \
ImageSet(Lambda(n, n*pi), S.Integers)
def test_solve_invalid_sol():
assert 0 not in solveset_real(sin(x)/x, x)
assert 0 not in solveset_complex((exp(x) - 1)/x, x)
@XFAIL
def test_solve_trig_simplified():
from sympy.abc import n
assert solveset_real(sin(x), x) == \
imageset(Lambda(n, n*pi), S.Integers)
assert solveset_real(cos(x), x) == \
imageset(Lambda(n, n*pi + pi/2), S.Integers)
assert solveset_real(cos(x) + sin(x), x) == \
imageset(Lambda(n, n*pi - pi/4), S.Integers)
@XFAIL
def test_solve_lambert():
assert solveset_real(x*exp(x) - 1, x) == FiniteSet(LambertW(1))
assert solveset_real(exp(x) + x, x) == FiniteSet(-LambertW(1))
assert solveset_real(x + 2**x, x) == \
FiniteSet(-LambertW(log(2))/log(2))
# issue 4739
ans = solveset_real(3*x + 5 + 2**(-5*x + 3), x)
assert ans == FiniteSet(-Rational(5, 3) +
LambertW(-10240*2**(S(1)/3)*log(2)/3)/(5*log(2)))
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solveset_real(eq, x)
ans = FiniteSet((log(2401) +
5*LambertW(-log(7**(7*3**Rational(1, 5)/5))))/(3*log(7))/-1)
assert result == ans
assert solveset_real(eq.expand(), x) == result
assert solveset_real(5*x - 1 + 3*exp(2 - 7*x), x) == \
FiniteSet(Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7)
assert solveset_real(2*x + 5 + log(3*x - 2), x) == \
FiniteSet(Rational(2, 3) + LambertW(2*exp(-Rational(19, 3))/3)/2)
assert solveset_real(3*x + log(4*x), x) == \
FiniteSet(LambertW(Rational(3, 4))/3)
assert solveset_real(x**x - 2) == FiniteSet(exp(LambertW(log(2))))
a = Symbol('a')
assert solveset_real(-a*x + 2*x*log(x), x) == FiniteSet(exp(a/2))
a = Symbol('a', real=True)
assert solveset_real(a/x + exp(x/2), x) == \
FiniteSet(2*LambertW(-a/2))
assert solveset_real((a/x + exp(x/2)).diff(x), x) == \
FiniteSet(4*LambertW(sqrt(2)*sqrt(a)/4))
# coverage test
assert solveset_real(tanh(x + 3)*tanh(x - 3) - 1, x) == EmptySet()
assert solveset_real((x**2 - 2*x + 1).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*S.Exp1)/3)
assert solveset_real((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) == \
FiniteSet(LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3)
assert solveset_real((x**2 - 2*x - 2).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*exp(1 + sqrt(3)))/3, LambertW(3*exp(-sqrt(3) + 1))/3)
assert solveset_real(x*log(x) + 3*x + 1, x) == \
FiniteSet(exp(-3 + LambertW(-exp(3))))
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solveset_real(eq, x) == \
FiniteSet(LambertW(3*exp(-LambertW(3))))
assert solveset_real(3*log(a**(3*x + 5)) + a**(3*x + 5), x) == \
FiniteSet(-((log(a**5) + LambertW(S(1)/3))/(3*log(a))))
p = symbols('p', positive=True)
assert solveset_real(3*log(p**(3*x + 5)) + p**(3*x + 5), x) == \
FiniteSet(
log((-3**(S(1)/3) - 3**(S(5)/6)*I)*LambertW(S(1)/3)**(S(1)/3)/(2*p**(S(5)/3)))/log(p),
log((-3**(S(1)/3) + 3**(S(5)/6)*I)*LambertW(S(1)/3)**(S(1)/3)/(2*p**(S(5)/3)))/log(p),
log((3*LambertW(S(1)/3)/p**5)**(1/(3*log(p)))),) # checked numerically
# check collection
b = Symbol('b')
eq = 3*log(a**(3*x + 5)) + b*log(a**(3*x + 5)) + a**(3*x + 5)
assert solveset_real(eq, x) == FiniteSet(
-((log(a**5) + LambertW(1/(b + 3)))/(3*log(a))))
# issue 4271
assert solveset_real((a/x + exp(x/2)).diff(x, 2), x) == FiniteSet(
6*LambertW((-1)**(S(1)/3)*a**(S(1)/3)/3))
assert solveset_real(x**3 - 3**x, x) == \
FiniteSet(-3/log(3)*LambertW(-log(3)/3))
assert solveset_real(3**cos(x) - cos(x)**3) == FiniteSet(
acos(-3*LambertW(-log(3)/3)/log(3)))
assert solveset_real(x**2 - 2**x, x) == \
solveset_real(-x**2 + 2**x, x)
assert solveset_real(3*log(x) - x*log(3)) == FiniteSet(
-3*LambertW(-log(3)/3)/log(3),
-3*LambertW(-log(3)/3, -1)/log(3))
assert solveset_real(LambertW(2*x) - y) == FiniteSet(
y*exp(y)/2)
@XFAIL
def test_other_lambert():
a = S(6)/5
assert solveset_real(x**a - a**x, x) == FiniteSet(
a, -a*LambertW(-log(a)/a)/log(a))
def test_solveset():
x = Symbol('x')
f = Function('f')
raises(ValueError, lambda: solveset(x + y))
assert solveset(x, 1) == S.EmptySet
assert solveset(f(1)**2 + y + 1, f(1)
) == FiniteSet(-sqrt(-y - 1), sqrt(-y - 1))
assert solveset(f(1)**2 - 1, f(1), S.Reals) == FiniteSet(-1, 1)
assert solveset(f(1)**2 + 1, f(1)) == FiniteSet(-I, I)
assert solveset(x - 1, 1) == FiniteSet(x)
assert solveset(sin(x) - cos(x), sin(x)) == FiniteSet(cos(x))
assert solveset(0, domain=S.Reals) == S.Reals
assert solveset(1) == S.EmptySet
assert solveset(True, domain=S.Reals) == S.Reals # issue 10197
assert solveset(False, domain=S.Reals) == S.EmptySet
assert solveset(exp(x) - 1, domain=S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, x, S.Reals) == FiniteSet(0)
assert solveset(Eq(exp(x), 1), x, S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, exp(x), S.Reals) == FiniteSet(1)
A = Indexed('A', x)
assert solveset(A - 1, A, S.Reals) == FiniteSet(1)
assert solveset(x - 1 >= 0, x, S.Reals) == Interval(1, oo)
assert solveset(exp(x) - 1 >= 0, x, S.Reals) == Interval(0, oo)
assert solveset(exp(x) - 1, x) == imageset(Lambda(n, 2*I*pi*n), S.Integers)
assert solveset(Eq(exp(x), 1), x) == imageset(Lambda(n, 2*I*pi*n),
S.Integers)
# issue 13825
assert solveset(x**2 + f(0) + 1, x) == {-sqrt(-f(0) - 1), sqrt(-f(0) - 1)}
def test_conditionset():
assert solveset(Eq(sin(x)**2 + cos(x)**2, 1), x, domain=S.Reals) == \
ConditionSet(x, True, S.Reals)
assert solveset(Eq(x**2 + x*sin(x), 1), x, domain=S.Reals
) == ConditionSet(x, Eq(x**2 + x*sin(x) - 1, 0), S.Reals)
assert solveset(Eq(-I*(exp(I*x) - exp(-I*x))/2, 1), x
) == imageset(Lambda(n, 2*n*pi + pi/2), S.Integers)
assert solveset(x + sin(x) > 1, x, domain=S.Reals
) == ConditionSet(x, x + sin(x) > 1, S.Reals)
assert solveset(Eq(sin(Abs(x)), x), x, domain=S.Reals
) == ConditionSet(x, Eq(-x + sin(Abs(x)), 0), S.Reals)
assert solveset(y**x-z, x, S.Reals) == \
ConditionSet(x, Eq(y**x - z, 0), S.Reals)
@XFAIL
def test_conditionset_equality():
''' Checking equality of different representations of ConditionSet'''
assert solveset(Eq(tan(x), y), x) == ConditionSet(x, Eq(tan(x), y), S.Complexes)
def test_solveset_domain():
x = Symbol('x')
assert solveset(x**2 - x - 6, x, Interval(0, oo)) == FiniteSet(3)
assert solveset(x**2 - 1, x, Interval(0, oo)) == FiniteSet(1)
assert solveset(x**4 - 16, x, Interval(0, 10)) == FiniteSet(2)
def test_improve_coverage():
from sympy.solvers.solveset import _has_rational_power
x = Symbol('x')
solution = solveset(exp(x) + sin(x), x, S.Reals)
unsolved_object = ConditionSet(x, Eq(exp(x) + sin(x), 0), S.Reals)
assert solution == unsolved_object
assert _has_rational_power(sin(x)*exp(x) + 1, x) == (False, S.One)
assert _has_rational_power((sin(x)**2)*(exp(x) + 1)**3, x) == (False, S.One)
def test_issue_9522():
x = Symbol('x')
expr1 = Eq(1/(x**2 - 4) + x, 1/(x**2 - 4) + 2)
expr2 = Eq(1/x + x, 1/x)
assert solveset(expr1, x, S.Reals) == EmptySet()
assert solveset(expr2, x, S.Reals) == EmptySet()
def test_solvify():
x = Symbol('x')
assert solvify(x**2 + 10, x, S.Reals) == []
assert solvify(x**3 + 1, x, S.Complexes) == [-1, S(1)/2 - sqrt(3)*I/2,
S(1)/2 + sqrt(3)*I/2]
assert solvify(log(x), x, S.Reals) == [1]
assert solvify(cos(x), x, S.Reals) == [pi/2, 3*pi/2]
assert solvify(sin(x) + 1, x, S.Reals) == [3*pi/2]
raises(NotImplementedError, lambda: solvify(sin(exp(x)), x, S.Complexes))
def test_abs_invert_solvify():
assert solvify(sin(Abs(x)), x, S.Reals) is None
def test_linear_eq_to_matrix():
x, y, z = symbols('x, y, z')
a, b, c, d, e, f, g, h, i, j, k, l = symbols('a:l')
eqns1 = [2*x + y - 2*z - 3, x - y - z, x + y + 3*z - 12]
eqns2 = [Eq(3*x + 2*y - z, 1), Eq(2*x - 2*y + 4*z, -2), -2*x + y - 2*z]
A, B = linear_eq_to_matrix(eqns1, x, y, z)
assert A == Matrix([[2, 1, -2], [1, -1, -1], [1, 1, 3]])
assert B == Matrix([[3], [0], [12]])
A, B = linear_eq_to_matrix(eqns2, x, y, z)
assert A == Matrix([[3, 2, -1], [2, -2, 4], [-2, 1, -2]])
assert B == Matrix([[1], [-2], [0]])
# Pure symbolic coefficients
eqns3 = [a*b*x + b*y + c*z - d, e*x + d*x + f*y + g*z - h, i*x + j*y + k*z - l]
A, B = linear_eq_to_matrix(eqns3, x, y, z)
assert A == Matrix([[a*b, b, c], [d + e, f, g], [i, j, k]])
assert B == Matrix([[d], [h], [l]])
# raise ValueError if
# 1) no symbols are given
raises(ValueError, lambda: linear_eq_to_matrix(eqns3))
# 2) there are duplicates
raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, x, y]))
# 3) there are non-symbols
raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, 1/a, y]))
# 4) a nonlinear term is detected in the original expression
raises(ValueError, lambda: linear_eq_to_matrix(Eq(1/x + x, 1/x)))
assert linear_eq_to_matrix(1, x) == (Matrix([[0]]), Matrix([[-1]]))
# issue 15195
assert linear_eq_to_matrix(x + y*(z*(3*x + 2) + 3), x) == (
Matrix([[3*y*z + 1]]), Matrix([[-y*(2*z + 3)]]))
assert linear_eq_to_matrix(Matrix(
[[a*x + b*y - 7], [5*x + 6*y - c]]), x, y) == (
Matrix([[a, b], [5, 6]]), Matrix([[7], [c]]))
# issue 15312
assert linear_eq_to_matrix(Eq(x + 2, 1), x) == (
Matrix([[1]]), Matrix([[-1]]))
def test_issue_16577():
assert linear_eq_to_matrix(Eq(a*(2*x + 3*y) + 4*y, 5), x, y) == (
Matrix([[2*a, 3*a + 4]]), Matrix([[5]]))
def test_linsolve():
x, y, z, u, v, w = symbols("x, y, z, u, v, w")
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system1 = A, b = M[:, :-1], M[:, -1]
Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12,
2*x1 + 4*x2 + 6*x4 - 4]
sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, *(x1, x2, x3, x4)) == sol
assert linsolve(system1, (x1, x2, x3, x4)) == sol
assert linsolve(system1, *(x1, x2, x3, x4)) == sol
# issue 9667 - symbols can be Dummy symbols
x1, x2, x3, x4 = symbols('x:4', cls=Dummy)
assert linsolve(system1, x1, x2, x3, x4) == FiniteSet(
(-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns))
raises(ValueError, lambda: linsolve(x1))
raises(ValueError, lambda: linsolve(x1, x2))
raises(ValueError, lambda: linsolve((A,), x1, x2))
raises(ValueError, lambda: linsolve(A, b, x1, x2))
#raise ValueError if equations are non-linear in given variables
raises(ValueError, lambda: linsolve([x + y - 1, x ** 2 + y - 3], [x, y]))
raises(ValueError, lambda: linsolve([cos(x) + y, x + y], [x, y]))
assert linsolve([x + z - 1, x ** 2 + y - 3], [z, y]) == {(-x + 1, -x**2 + 3)}
# Fully symbolic test
a, b, c, d, e, f = symbols('a, b, c, d, e, f')
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [f]])
system2 = (A, B)
sol = FiniteSet(((-b*f + d*e)/(a*d - b*c), (a*f - c*e)/(a*d - b*c)))
assert linsolve(system2, [x, y]) == sol
# No solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 1])
assert linsolve((A, b), (x, y, z)) == EmptySet()
# Issue #10056
A, B, J1, J2 = symbols('A B J1 J2')
Augmatrix = Matrix([
[2*I*J1, 2*I*J2, -2/J1],
[-2*I*J2, -2*I*J1, 2/J2],
[0, 2, 2*I/(J1*J2)],
[2, 0, 0],
])
assert linsolve(Augmatrix, A, B) == FiniteSet((0, I/(J1*J2)))
# Issue #10121 - Assignment of free variables
a, b, c, d, e = symbols('a, b, c, d, e')
Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))
raises(IndexError, lambda: linsolve(Augmatrix, a, b, c))
x0, x1, x2, _x0 = symbols('tau0 tau1 tau2 _tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau1')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
# symbols can be given as generators
x0, x2, x4 = symbols('x0, x2, x4')
assert linsolve(Augmatrix, numbered_symbols('x')
) == FiniteSet((x0, 0, x2, 0, x4))
Augmatrix[-1, -1] = x0
# use Dummy to avoid clash; the names may clash but the symbols
# will not
Augmatrix[-1, -1] = symbols('_x0')
assert len(linsolve(
Augmatrix, numbered_symbols('x', cls=Dummy)).free_symbols) == 4
# Issue #12604
f = Function('f')
assert linsolve([f(x) - 5], f(x)) == FiniteSet((5,))
# Issue #14860
from sympy.physics.units import meter, newton, kilo
Eqns = [8*kilo*newton + x + y, 28*kilo*newton*meter + 3*x*meter]
assert linsolve(Eqns, x, y) == {(-28000*newton/3, 4000*newton/3)}
# linsolve fully expands expressions, so removable singularities
# and other nonlinearity does not raise an error
assert linsolve([Eq(x, x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(1/x, 1/x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(y/x, y/x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(x*(x + 1), x**2 + y)], [x, y]) == {(y, y)}
def test_solve_decomposition():
x = Symbol('x')
n = Dummy('n')
f1 = exp(3*x) - 6*exp(2*x) + 11*exp(x) - 6
f2 = sin(x)**2 - 2*sin(x) + 1
f3 = sin(x)**2 - sin(x)
f4 = sin(x + 1)
f5 = exp(x + 2) - 1
f6 = 1/log(x)
f7 = 1/x
s1 = ImageSet(Lambda(n, 2*n*pi), S.Integers)
s2 = ImageSet(Lambda(n, 2*n*pi + pi), S.Integers)
s3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
s4 = ImageSet(Lambda(n, 2*n*pi - 1), S.Integers)
s5 = ImageSet(Lambda(n, 2*n*pi - 1 + pi), S.Integers)
assert solve_decomposition(f1, x, S.Reals) == FiniteSet(0, log(2), log(3))
assert solve_decomposition(f2, x, S.Reals) == s3
assert solve_decomposition(f3, x, S.Reals) == Union(s1, s2, s3)
assert solve_decomposition(f4, x, S.Reals) == Union(s4, s5)
assert solve_decomposition(f5, x, S.Reals) == FiniteSet(-2)
assert solve_decomposition(f6, x, S.Reals) == S.EmptySet
assert solve_decomposition(f7, x, S.Reals) == S.EmptySet
assert solve_decomposition(x, x, Interval(1, 2)) == S.EmptySet
# nonlinsolve testcases
def test_nonlinsolve_basic():
assert nonlinsolve([],[]) == S.EmptySet
assert nonlinsolve([],[x, y]) == S.EmptySet
system = [x, y - x - 5]
assert nonlinsolve([x],[x, y]) == FiniteSet((0, y))
assert nonlinsolve(system, [y]) == FiniteSet((x + 5,))
soln = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
assert nonlinsolve([sin(x) - 1], [x]) == FiniteSet(tuple(soln))
assert nonlinsolve([x**2 - 1], [x]) == FiniteSet((-1,), (1,))
soln = FiniteSet((y, y))
assert nonlinsolve([x - y, 0], x, y) == soln
assert nonlinsolve([0, x - y], x, y) == soln
assert nonlinsolve([x - y, x - y], x, y) == soln
assert nonlinsolve([x, 0], x, y) == FiniteSet((0, y))
f = Function('f')
assert nonlinsolve([f(x), 0], f(x), y) == FiniteSet((0, y))
assert nonlinsolve([f(x), 0], f(x), f(y)) == FiniteSet((0, f(y)))
A = Indexed('A', x)
assert nonlinsolve([A, 0], A, y) == FiniteSet((0, y))
assert nonlinsolve([x**2 -1], [sin(x)]) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], sin(x)) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], 1) == FiniteSet((x**2,))
assert nonlinsolve([x**2 -1], x + y) == FiniteSet((S.EmptySet,))
def test_nonlinsolve_abs():
soln = FiniteSet((x, Abs(x)))
assert nonlinsolve([Abs(x) - y], x, y) == soln
def test_raise_exception_nonlinsolve():
raises(IndexError, lambda: nonlinsolve([x**2 -1], []))
raises(ValueError, lambda: nonlinsolve([x**2 -1]))
raises(NotImplementedError, lambda: nonlinsolve([(x+y)**2 - 9, x**2 - y**2 - 0.75], (x, y)))
def test_trig_system():
# TODO: add more simple testcases when solveset returns
# simplified soln for Trig eq
assert nonlinsolve([sin(x) - 1, cos(x) -1 ], x) == S.EmptySet
soln1 = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
soln = FiniteSet(soln1)
assert nonlinsolve([sin(x) - 1, cos(x)], x) == soln
@XFAIL
def test_trig_system_fail():
# fails because solveset trig solver is not much smart.
sys = [x + y - pi/2, sin(x) + sin(y) - 1]
# solveset returns conditonset for sin(x) + sin(y) - 1
soln_1 = (ImageSet(Lambda(n, n*pi + pi/2), S.Integers),
ImageSet(Lambda(n, n*pi)), S.Integers)
soln_1 = FiniteSet(soln_1)
soln_2 = (ImageSet(Lambda(n, n*pi), S.Integers),
ImageSet(Lambda(n, n*pi+ pi/2), S.Integers))
soln_2 = FiniteSet(soln_2)
soln = soln_1 + soln_2
assert nonlinsolve(sys, [x, y]) == soln
# Add more cases from here
# http://www.vitutor.com/geometry/trigonometry/equations_systems.html#uno
sys = [sin(x) + sin(y) - (sqrt(3)+1)/2, sin(x) - sin(y) - (sqrt(3) - 1)/2]
soln_x = Union(ImageSet(Lambda(n, 2*n*pi + pi/3), S.Integers),
ImageSet(Lambda(n, 2*n*pi + 2*pi/3), S.Integers))
soln_y = Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + 5*pi/6), S.Integers))
assert nonlinsolve(sys, [x, y]) ==FiniteSet((soln_x, soln_y))
def test_nonlinsolve_positive_dimensional():
x, y, z, a, b, c, d = symbols('x, y, z, a, b, c, d', real = True)
assert nonlinsolve([x*y, x*y - x], [x, y]) == FiniteSet((0, y))
system = [a**2 + a*c, a - b]
assert nonlinsolve(system, [a, b]) == FiniteSet((0, 0), (-c, -c))
# here (a= 0, b = 0) is independent soln so both is printed.
# if symbols = [a, b, c] then only {a : -c ,b : -c}
eq1 = a + b + c + d
eq2 = a*b + b*c + c*d + d*a
eq3 = a*b*c + b*c*d + c*d*a + d*a*b
eq4 = a*b*c*d - 1
system = [eq1, eq2, eq3, eq4]
sol1 = (-1/d, -d, 1/d, FiniteSet(d) - FiniteSet(0))
sol2 = (1/d, -d, -1/d, FiniteSet(d) - FiniteSet(0))
soln = FiniteSet(sol1, sol2)
assert nonlinsolve(system, [a, b, c, d]) == soln
def test_nonlinsolve_polysys():
x, y, z = symbols('x, y, z', real = True)
assert nonlinsolve([x**2 + y - 2, x**2 + y], [x, y]) == S.EmptySet
s = (-y + 2, y)
assert nonlinsolve([(x + y)**2 - 4, x + y - 2], [x, y]) == FiniteSet(s)
system = [x**2 - y**2]
soln_real = FiniteSet((-y, y), (y, y))
soln_complex = FiniteSet((-Abs(y), y), (Abs(y), y))
soln =soln_real + soln_complex
assert nonlinsolve(system, [x, y]) == soln
system = [x**2 - y**2]
soln_real= FiniteSet((y, -y), (y, y))
soln_complex = FiniteSet((y, -Abs(y)), (y, Abs(y)))
soln = soln_real + soln_complex
assert nonlinsolve(system, [y, x]) == soln
system = [x**2 + y - 3, x - y - 4]
assert nonlinsolve(system, (x, y)) != nonlinsolve(system, (y, x))
def test_nonlinsolve_using_substitution():
x, y, z, n = symbols('x, y, z, n', real = True)
system = [(x + y)*n - y**2 + 2]
s_x = (n*y - y**2 + 2)/n
soln = (-s_x, y)
assert nonlinsolve(system, [x, y]) == FiniteSet(soln)
system = [z**2*x**2 - z**2*y**2/exp(x)]
soln_real_1 = (y, x, 0)
soln_real_2 = (-exp(x/2)*Abs(x), x, z)
soln_real_3 = (exp(x/2)*Abs(x), x, z)
soln_complex_1 = (-x*exp(x/2), x, z)
soln_complex_2 = (x*exp(x/2), x, z)
syms = [y, x, z]
soln = FiniteSet(soln_real_1, soln_complex_1, soln_complex_2,\
soln_real_2, soln_real_3)
assert nonlinsolve(system,syms) == soln
def test_nonlinsolve_complex():
x, y, z = symbols('x, y, z')
n = Dummy('n')
real_soln = (log(sin(S(1)/3)), S(1)/3)
img_lamda = Lambda(n, 2*n*I*pi + Mod(log(sin(S(1)/3)), 2*I*pi))
complex_soln = (ImageSet(img_lamda, S.Integers), S(1)/3)
soln = FiniteSet(real_soln, complex_soln)
assert nonlinsolve([exp(x) - sin(y), 1/y - 3], [x, y]) == soln
system = [exp(x) - sin(y), 1/exp(y) - 3]
soln_x = ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sin(log(3)))), S.Integers)
soln_real = FiniteSet((soln_x, -log(S(3))))
# Mod(-log(3), 2*I*pi) is equal to -log(3).
expr_x = I*(2*n*pi + arg(sin(2*n*I*pi + Mod(-log(3), 2*I*pi)))) + \
log(Abs(sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
soln_x = ImageSet(Lambda(n, expr_x), S.Integers)
expr_y = 2*n*I*pi + Mod(-log(3), 2*I*pi)
soln_y = ImageSet(Lambda(n, expr_y), S.Integers)
soln_complex = FiniteSet((soln_x, soln_y))
soln = soln_real + soln_complex
assert nonlinsolve(system, [x, y]) == soln
system = [exp(x) - sin(y), y**2 - 4]
s1 = (log(sin(2)), 2)
s2 = (ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sin(2))), S.Integers), -2 )
img = ImageSet(Lambda(n, 2*n*I*pi + Mod(log(sin(2)), 2*I*pi)), S.Integers)
s3 = (img, 2)
assert nonlinsolve(system, [x, y]) == FiniteSet(s1, s2, s3)
@XFAIL
def test_solve_nonlinear_trans():
# After the transcendental equation solver these will work
x, y, z = symbols('x, y, z', real=True)
soln1 = FiniteSet((2*LambertW(y/2), y))
soln2 = FiniteSet((-x*sqrt(exp(x)), y), (x*sqrt(exp(x)), y))
soln3 = FiniteSet((x*exp(x/2), x))
soln4 = FiniteSet(2*LambertW(y/2), y)
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln1
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln2
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln3
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln4
def test_issue_5132_1():
system = [sqrt(x**2 + y**2) - sqrt(10), x + y - 4]
assert nonlinsolve(system, [x, y]) == FiniteSet((1, 3), (3, 1))
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + Mod(-log(3), 2*I*pi))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2)
)
soln = soln_real + soln_complex
assert nonlinsolve(eqs, [y, z]) == soln
def test_issue_5132_2():
x, y = symbols('x, y', real=True)
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
n = Dummy('n')
soln_real = (log(-z**2 + sin(y))/2, z)
lam = Lambda( n, I*(2*n*pi + arg(-z**2 + sin(y)))/2 + log(Abs(z**2 - sin(y)))/2)
img = ImageSet(lam, S.Integers)
# not sure about the complex soln. But it looks correct.
soln_complex = (img, z)
soln = FiniteSet(soln_real, soln_complex)
assert nonlinsolve(eqs, [x, z]) == soln
r, t = symbols('r, t')
system = [r - x**2 - y**2, tan(t) - y/x]
s_x = sqrt(r/(tan(t)**2 + 1))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x, s_y), (-s_x, -s_y))
assert nonlinsolve(system, [x, y]) == soln
def test_issue_6752():
a,b,c,d = symbols('a, b, c, d', real=True)
assert nonlinsolve([a**2 + a, a - b], [a, b]) == {(-1, -1), (0, 0)}
@SKIP("slow")
def test_issue_5114_solveset():
# slow testcase
a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('a:r')
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = [a, b, c, f, h, k, n]
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(nonlinsolve(eqs, syms)) == 1
@SKIP("Hangs")
def _test_issue_5335():
# Not able to check zero dimensional system.
# is_zero_dimensional Hangs
lam, a0, conc = symbols('lam a0 conc')
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions but only two are valid
assert len(nonlinsolve(eqs, sym)) == 2
# float
lam, a0, conc = symbols('lam a0 conc')
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
assert len(nonlinsolve(eqs, sym)) == 2
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = 191/S(20), 3*sqrt(391)/20
ans = {(a, -b), (a, b)}
assert nonlinsolve((e1, e2), (x, y)) == ans
assert nonlinsolve((e1, e2/(x - a)), (x, y)) == S.EmptySet
# make the 2nd circle's radius be -3
e2 += 6
assert nonlinsolve((e1, e2), (x, y)) == S.EmptySet
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = [x, y, z]
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x2 - x)**2 + (y2 - y)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = [f1, f2, f3]
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = [g1, g2, g3]
# both soln same
A = nonlinsolve(F, v)
B = nonlinsolve(G, v)
assert A == B
def test_nonlinsolve_conditionset():
# when solveset failed to solve all the eq
# return conditionset
f = Function('f')
f1 = f(x) - pi/2
f2 = f(y) - 3*pi/2
intermediate_system = FiniteSet(2*f(x) - pi, 2*f(y) - 3*pi)
symbols = Tuple(x, y)
soln = ConditionSet(
symbols,
intermediate_system,
S.Complexes)
assert nonlinsolve([f1, f2], [x, y]) == soln
def test_substitution_basic():
assert substitution([], [x, y]) == S.EmptySet
assert substitution([], []) == S.EmptySet
system = [2*x**2 + 3*y**2 - 30, 3*x**2 - 2*y**2 - 19]
soln = FiniteSet((-3, -2), (-3, 2), (3, -2), (3, 2))
assert substitution(system, [x, y]) == soln
soln = FiniteSet((-1, 1))
assert substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y]) == soln
assert substitution(
[x + y], [x], [{y: 1}], [y],
set([x + 1]), [y, x]) == S.EmptySet
def test_issue_5132_substitution():
x, y, z, r, t = symbols('x, y, z, r, t', real=True)
system = [r - x**2 - y**2, tan(t) - y/x]
s_x_1 = Complement(FiniteSet(-sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_x_2 = Complement(FiniteSet(sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x_2, s_y)) + FiniteSet((s_x_1, -s_y))
assert substitution(system, [x, y]) == soln
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + Mod(-log(3), 2*I*pi))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2)
)
soln = soln_real + soln_complex
assert substitution(eqs, [y, z]) == soln
def test_raises_substitution():
raises(ValueError, lambda: substitution([x**2 -1], []))
raises(TypeError, lambda: substitution([x**2 -1]))
raises(ValueError, lambda: substitution([x**2 -1], [sin(x)]))
raises(TypeError, lambda: substitution([x**2 -1], x))
raises(TypeError, lambda: substitution([x**2 -1], 1))
# end of tests for nonlinsolve
def test_issue_9556():
x = Symbol('x')
b = Symbol('b', positive=True)
assert solveset(Abs(x) + 1, x, S.Reals) == EmptySet()
assert solveset(Abs(x) + b, x, S.Reals) == EmptySet()
assert solveset(Eq(b, -1), b, S.Reals) == EmptySet()
def test_issue_9611():
x = Symbol('x')
a = Symbol('a')
y = Symbol('y')
assert solveset(Eq(x - x + a, a), x, S.Reals) == S.Reals
assert solveset(Eq(y - y + a, a), y) == S.Complexes
def test_issue_9557():
x = Symbol('x')
a = Symbol('a')
assert solveset(x**2 + a, x, S.Reals) == Intersection(S.Reals,
FiniteSet(-sqrt(-a), sqrt(-a)))
def test_issue_9778():
assert solveset(x**3 + 1, x, S.Reals) == FiniteSet(-1)
assert solveset(x**(S(3)/5) + 1, x, S.Reals) == S.EmptySet
assert solveset(x**3 + y, x, S.Reals) == \
FiniteSet(-Abs(y)**(S(1)/3)*sign(y))
def test_issue_10214():
assert solveset(x**(S(3)/2) + 4, x, S.Reals) == S.EmptySet
assert solveset(x**(S(-3)/2) + 4, x, S.Reals) == S.EmptySet
ans = FiniteSet(-2**(S(2)/3))
assert solveset(x**(S(3)) + 4, x, S.Reals) == ans
assert (x**(S(3)) + 4).subs(x,list(ans)[0]) == 0 # substituting ans and verifying the result.
assert (x**(S(3)) + 4).subs(x,-(-2)**(2/S(3))) == 0
def test_issue_9849():
assert solveset(Abs(sin(x)) + 1, x, S.Reals) == S.EmptySet
def test_issue_9953():
assert linsolve([ ], x) == S.EmptySet
def test_issue_9913():
assert solveset(2*x + 1/(x - 10)**2, x, S.Reals) == \
FiniteSet(-(3*sqrt(24081)/4 + S(4027)/4)**(S(1)/3)/3 - 100/
(3*(3*sqrt(24081)/4 + S(4027)/4)**(S(1)/3)) + S(20)/3)
def test_issue_10397():
assert solveset(sqrt(x), x, S.Complexes) == FiniteSet(0)
def test_issue_14987():
raises(ValueError, lambda: linear_eq_to_matrix(
[x**2], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(-3/x + 1) + 2*y - a], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x**2 - 3*x)/(x - 3) - 3], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)**3 - x**3 - 3*x**2 + 7], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(1/x + 1) + y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)*y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(1/x, 1/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(y/x, y/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(x*(x + 1), x**2 + y)], [x, y]))
def test_simplification():
eq = x + (a - b)/(-2*a + 2*b)
assert solveset(eq, x) == FiniteSet(S.Half)
assert solveset(eq, x, S.Reals) == FiniteSet(S.Half)
def test_issue_10555():
f = Function('f')
g = Function('g')
assert solveset(f(x) - pi/2, x, S.Reals) == \
ConditionSet(x, Eq(f(x) - pi/2, 0), S.Reals)
assert solveset(f(g(x)) - pi/2, g(x), S.Reals) == \
ConditionSet(g(x), Eq(f(g(x)) - pi/2, 0), S.Reals)
def test_issue_8715():
eq = x + 1/x > -2 + 1/x
assert solveset(eq, x, S.Reals) == \
(Interval.open(-2, oo) - FiniteSet(0))
assert solveset(eq.subs(x,log(x)), x, S.Reals) == \
Interval.open(exp(-2), oo) - FiniteSet(1)
def test_issue_11174():
r, t = symbols('r t')
eq = z**2 + exp(2*x) - sin(y)
soln = Intersection(S.Reals, FiniteSet(log(-z**2 + sin(y))/2))
assert solveset(eq, x, S.Reals) == soln
eq = sqrt(r)*Abs(tan(t))/sqrt(tan(t)**2 + 1) + x*tan(t)
s = -sqrt(r)*Abs(tan(t))/(sqrt(tan(t)**2 + 1)*tan(t))
soln = Intersection(S.Reals, FiniteSet(s))
assert solveset(eq, x, S.Reals) == soln
def test_issue_11534():
# eq and eq2 should give the same solution as a Complement
eq = -y + x/sqrt(-x**2 + 1)
eq2 = -y**2 + x**2/(-x**2 + 1)
soln = Complement(FiniteSet(-y/sqrt(y**2 + 1), y/sqrt(y**2 + 1)), FiniteSet(-1, 1))
assert solveset(eq, x, S.Reals) == soln
assert solveset(eq2, x, S.Reals) == soln
def test_issue_10477():
assert solveset((x**2 + 4*x - 3)/x < 2, x, S.Reals) == \
Union(Interval.open(-oo, -3), Interval.open(0, 1))
def test_issue_10671():
assert solveset(sin(y), y, Interval(0, pi)) == FiniteSet(0, pi)
i = Interval(1, 10)
assert solveset((1/x).diff(x) < 0, x, i) == i
def test_issue_11064():
eq = x + sqrt(x**2 - 5)
assert solveset(eq > 0, x, S.Reals) == \
Interval(sqrt(5), oo)
assert solveset(eq < 0, x, S.Reals) == \
Interval(-oo, -sqrt(5))
assert solveset(eq > sqrt(5), x, S.Reals) == \
Interval.Lopen(sqrt(5), oo)
def test_issue_12478():
eq = sqrt(x - 2) + 2
soln = solveset_real(eq, x)
assert soln is S.EmptySet
assert solveset(eq < 0, x, S.Reals) is S.EmptySet
assert solveset(eq > 0, x, S.Reals) == Interval(2, oo)
def test_issue_12429():
eq = solveset(log(x)/x <= 0, x, S.Reals)
sol = Interval.Lopen(0, 1)
assert eq == sol
def test_solveset_arg():
assert solveset(arg(x), x, S.Reals) == Interval.open(0, oo)
assert solveset(arg(4*x -3), x) == Interval.open(S(3)/4, oo)
def test__is_finite_with_finite_vars():
f = _is_finite_with_finite_vars
# issue 12482
assert all(f(1/x) is None for x in (
Dummy(), Dummy(real=True), Dummy(complex=True)))
assert f(1/Dummy(real=False)) is True # b/c it's finite but not 0
def test_issue_13550():
assert solveset(x**2 - 2*x - 15, symbol = x, domain = Interval(-oo, 0)) == FiniteSet(-3)
def test_issue_13849():
t = symbols('t')
assert nonlinsolve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) == EmptySet()
def test_issue_14223():
x = Symbol('x')
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
S.Reals) == FiniteSet(-1, 1)
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
Interval(0, 2)) == FiniteSet(1)
def test_issue_10158():
x = Symbol('x')
dom = S.Reals
assert solveset(x*Max(x, 15) - 10, x, dom) == FiniteSet(2/S(3))
assert solveset(x*Min(x, 15) - 10, x, dom) == FiniteSet(-sqrt(10), sqrt(10))
assert solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom) == FiniteSet(-1, 1)
assert solveset(Abs(x - 1) - Abs(y), x, dom) == FiniteSet(-Abs(y) + 1, Abs(y) + 1)
assert solveset(Abs(x + 4*Abs(x + 1)), x, dom) == FiniteSet(-4/S(3), -4/S(5))
assert solveset(2*Abs(x + Abs(x + Max(3, x))) - 2, x, S.Reals) == FiniteSet(-1, -2)
dom = S.Complexes
raises(ValueError, lambda: solveset(x*Max(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(x*Min(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom))
raises(ValueError, lambda: solveset(Abs(x - 1) - Abs(y), x, dom))
raises(ValueError, lambda: solveset(Abs(x + 4*Abs(x + 1)), x, dom))
def test_issue_14300():
x, y, n = symbols('x y n')
f = 1 - exp(-18000000*x) - y
a1 = FiniteSet(-log(-y + 1)/18000000)
assert solveset(f, x, S.Reals) == \
Intersection(S.Reals, a1)
assert solveset(f, x) == \
ImageSet(Lambda(n, -I*(2*n*pi + arg(-y + 1))/18000000 -
log(Abs(y - 1))/18000000), S.Integers)
def test_issue_14454():
x = Symbol('x')
number = CRootOf(x**4 + x - 1, 2)
raises(ValueError, lambda: invert_real(number, 0, x, S.Reals))
assert invert_real(x**2, number, x, S.Reals) # no error
def test_term_factors():
assert list(_term_factors(3**x - 2)) == [-2, 3**x]
expr = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
assert set(_term_factors(expr)) == set([
3**(x + 2), 4**(x + 2), 3**(x + 3), 4**(x - 1), -1, 4**(x + 1)])
#################### tests for transolve and its helpers ###############
def test_transolve():
assert _transolve(3**x, x, S.Reals) == S.EmptySet
assert _transolve(3**x - 9**(x + 5), x, S.Reals) == FiniteSet(-10)
# exponential tests
def test_exponential_real():
from sympy.abc import x, y, z
e1 = 3**(2*x) - 2**(x + 3)
e2 = 4**(5 - 9*x) - 8**(2 - x)
e3 = 2**x + 4**x
e4 = exp(log(5)*x) - 2**x
e5 = exp(x/y)*exp(-z/y) - 2
e6 = 5**(x/2) - 2**(x/3)
e7 = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
e8 = -9*exp(-2*x + 5) + 4*exp(3*x + 1)
e9 = 2**x + 4**x + 8**x - 84
assert solveset(e1, x, S.Reals) == FiniteSet(
-3*log(2)/(-2*log(3) + log(2)))
assert solveset(e2, x, S.Reals) == FiniteSet(4/S(15))
assert solveset(e3, x, S.Reals) == S.EmptySet
assert solveset(e4, x, S.Reals) == FiniteSet(0)
assert solveset(e5, x, S.Reals) == Intersection(
S.Reals, FiniteSet(y*log(2*exp(z/y))))
assert solveset(e6, x, S.Reals) == FiniteSet(0)
assert solveset(e7, x, S.Reals) == FiniteSet(2)
assert solveset(e8, x, S.Reals) == FiniteSet(-2*log(2)/5 + 2*log(3)/5 + S(4)/5)
assert solveset(e9, x, S.Reals) == FiniteSet(2)
assert solveset_real(-9*exp(-2*x + 5) + 2**(x + 1), x) == FiniteSet(
-((-5 - 2*log(3) + log(2))/(log(2) + 2)))
assert solveset_real(4**(x/2) - 2**(x/3), x) == FiniteSet(0)
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solveset_real(5**(x/2) - 2**(3/x), x) == FiniteSet(-b, b)
# coverage test
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solveset_real(C1 + C2/x**2 - exp(-f(x)), f(x)) == Intersection(
S.Reals, FiniteSet(-log(C1 + C2/x**2)))
y = symbols('y', positive=True)
assert solveset_real(x**2 - y**2/exp(x), y) == Intersection(
S.Reals, FiniteSet(-sqrt(x**2*exp(x)), sqrt(x**2*exp(x))))
p = Symbol('p', positive=True)
assert solveset_real((1/p + 1)**(p + 1), p) == EmptySet()
@XFAIL
def test_exponential_complex():
from sympy.abc import x
from sympy import Dummy
n = Dummy('n')
assert solveset_complex(2**x + 4**x, x) == imageset(
Lambda(n, I*(2*n*pi + pi)/log(2)), S.Integers)
assert solveset_complex(x**z*y**z - 2, z) == FiniteSet(
log(2)/(log(x) + log(y)))
assert solveset_complex(4**(x/2) - 2**(x/3), x) == imageset(
Lambda(n, 3*n*I*pi/log(2)), S.Integers)
assert solveset(2**x + 32, x) == imageset(
Lambda(n, (I*(2*n*pi + pi) + 5*log(2))/log(2)), S.Integers)
eq = (2**exp(y**2/x) + 2)/(x**2 + 15)
a = sqrt(x)*sqrt(-log(log(2)) + log(log(2) + 2*n*I*pi))
assert solveset_complex(eq, y) == FiniteSet(-a, a)
union1 = imageset(Lambda(n, I*(2*n*pi - 2*pi/3)/log(2)), S.Integers)
union2 = imageset(Lambda(n, I*(2*n*pi + 2*pi/3)/log(2)), S.Integers)
assert solveset(2**x + 4**x + 8**x, x) == Union(union1, union2)
eq = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
res = solveset(eq, x)
num = 2*n*I*pi - 4*log(2) + 2*log(3)
den = -2*log(2) + log(3)
ans = imageset(Lambda(n, num/den), S.Integers)
assert res == ans
def test_expo_conditionset():
from sympy.abc import x, y
f1 = (exp(x) + 1)**x - 2
f2 = (x + 2)**y*x - 3
f3 = 2**x - exp(x) - 3
f4 = log(x) - exp(x)
f5 = 2**x + 3**x - 5**x
assert solveset(f1, x, S.Reals) == ConditionSet(
x, Eq((exp(x) + 1)**x - 2, 0), S.Reals)
assert solveset(f2, x, S.Reals) == ConditionSet(
x, Eq(x*(x + 2)**y - 3, 0), S.Reals)
assert solveset(f3, x, S.Reals) == ConditionSet(
x, Eq(2**x - exp(x) - 3, 0), S.Reals)
assert solveset(f4, x, S.Reals) == ConditionSet(
x, Eq(-exp(x) + log(x), 0), S.Reals)
assert solveset(f5, x, S.Reals) == ConditionSet(
x, Eq(2**x + 3**x - 5**x, 0), S.Reals)
def test_exponential_symbols():
x, y, z = symbols('x y z', positive=True)
from sympy import simplify
assert solveset(z**x - y, x, S.Reals) == Intersection(
S.Reals, FiniteSet(log(y)/log(z)))
w = symbols('w')
f1 = 2*x**w - 4*y**w
f2 = (x/y)**w - 2
ans1 = solveset(f1, w, S.Reals)
ans2 = solveset(f2, w, S.Reals)
assert ans1 == simplify(ans2)
assert solveset(x**x, x, S.Reals) == S.EmptySet
assert solveset(x**y - 1, y, S.Reals) == FiniteSet(0)
assert solveset(exp(x/y)*exp(-z/y) - 2, y, S.Reals) == FiniteSet(
(x - z)/log(2)) - FiniteSet(0)
a, b, x, y = symbols('a b x y')
assert solveset_real(a**x - b**x, x) == ConditionSet(
x, (a > 0) & (b > 0), FiniteSet(0))
assert solveset(a**x - b**x, x) == ConditionSet(
x, Ne(a, 0) & Ne(b, 0), FiniteSet(0))
@XFAIL
def test_issue_10864():
assert solveset(x**(y*z) - x, x, S.Reals) == FiniteSet(1)
@XFAIL
def test_solve_only_exp_2():
assert solveset_real(sqrt(exp(x)) + sqrt(exp(-x)) - 4, x) == \
FiniteSet(2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2))
def test_is_exponential():
x, y, z = symbols('x y z')
assert _is_exponential(y, x) is False
assert _is_exponential(3**x - 2, x) is True
assert _is_exponential(5**x - 7**(2 - x), x) is True
assert _is_exponential(sin(2**x) - 4*x, x) is False
assert _is_exponential(x**y - z, y) is True
assert _is_exponential(x**y - z, x) is False
assert _is_exponential(2**x + 4**x - 1, x) is True
assert _is_exponential(x**(y*z) - x, x) is False
assert _is_exponential(x**(2*x) - 3**x, x) is False
assert _is_exponential(x**y - y*z, y) is False
assert _is_exponential(x**y - x*z, y) is True
def test_solve_exponential():
assert _solve_exponential(3**(2*x) - 2**(x + 3), 0, x, S.Reals) == \
FiniteSet(-3*log(2)/(-2*log(3) + log(2)))
assert _solve_exponential(2**y + 4**y, 1, y, S.Reals) == \
FiniteSet(log(-S(1)/2 + sqrt(5)/2)/log(2))
assert _solve_exponential(2**y + 4**y, 0, y, S.Reals) == \
S.EmptySet
assert _solve_exponential(2**x + 3**x - 5**x, 0, x, S.Reals) == \
ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), S.Reals)
# end of exponential tests
# logarithmic tests
def test_logarithmic():
assert solveset_real(log(x - 3) + log(x + 3), x) == FiniteSet(
-sqrt(10), sqrt(10))
assert solveset_real(log(x + 1) - log(2*x - 1), x) == FiniteSet(2)
assert solveset_real(log(x + 3) + log(1 + 3/x) - 3, x) == FiniteSet(
-3 + sqrt(-12 + exp(3))*exp(S(3)/2)/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(S(3)/2)/2 - 3 + exp(3)/2)
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solveset_real(eq, x) == \
Intersection(S.Reals, FiniteSet(-sqrt(y**2 - y*exp(z)),
sqrt(y**2 - y*exp(z)))) - \
Intersection(S.Reals, FiniteSet(-sqrt(y**2), sqrt(y**2)))
assert solveset_real(
log(3*x) - log(-x + 1) - log(4*x + 1), x) == FiniteSet(-S(1)/2, S(1)/2)
assert solveset(log(x**y) - y*log(x), x, S.Reals) == S.Reals
@XFAIL
def test_uselogcombine_2():
eq = log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)
assert solveset_real(eq, x) == EmptySet()
eq = log(8*x) - log(sqrt(x) + 1) - 2
assert solveset_real(eq, x) == EmptySet()
def test_is_logarithmic():
assert _is_logarithmic(y, x) is False
assert _is_logarithmic(log(x), x) is True
assert _is_logarithmic(log(x) - 3, x) is True
assert _is_logarithmic(log(x)*log(y), x) is True
assert _is_logarithmic(log(x)**2, x) is False
assert _is_logarithmic(log(x - 3) + log(x + 3), x) is True
assert _is_logarithmic(log(x**y) - y*log(x), x) is True
assert _is_logarithmic(sin(log(x)), x) is False
assert _is_logarithmic(x + y, x) is False
assert _is_logarithmic(log(3*x) - log(1 - x) + 4, x) is True
assert _is_logarithmic(log(x) + log(y) + x, x) is False
assert _is_logarithmic(log(log(x - 3)) + log(x - 3), x) is True
assert _is_logarithmic(log(log(3) + x) + log(x), x) is True
assert _is_logarithmic(log(x)*(y + 3) + log(x), y) is False
def test_solve_logarithm():
y = Symbol('y')
assert _solve_logarithm(log(x**y) - y*log(x), 0, x, S.Reals) == S.Reals
y = Symbol('y', positive=True)
assert _solve_logarithm(log(x)*log(y), 0, x, S.Reals) == FiniteSet(1)
# end of logarithmic tests
def test_linear_coeffs():
from sympy.solvers.solveset import linear_coeffs
assert linear_coeffs(0, x) == [0, 0]
assert all(i is S.Zero for i in linear_coeffs(0, x))
assert linear_coeffs(x + 2*y + 3, x, y) == [1, 2, 3]
assert linear_coeffs(x + 2*y + 3, y, x) == [2, 1, 3]
assert linear_coeffs(x + 2*x**2 + 3, x, x**2) == [1, 2, 3]
raises(ValueError, lambda:
linear_coeffs(x + 2*x**2 + x**3, x, x**2))
raises(ValueError, lambda:
linear_coeffs(1/x*(x - 1) + 1/x, x))
assert linear_coeffs(a*(x + y), x, y) == [a, a, 0]
|
4a691aeb2483ea57b17ae33c0cbde0599cfbaa92dbbaca2311676440038a91b4 | from sympy import (Eq, Matrix, pi, sin, sqrt, Symbol, Integral, Piecewise,
symbols, Float, I, Rational)
from mpmath import mnorm, mpf
from sympy.solvers import nsolve
from sympy.utilities.lambdify import lambdify
from sympy.utilities.pytest import raises, XFAIL
from sympy.utilities.decorator import conserve_mpmath_dps
@XFAIL
def test_nsolve_fail():
x = symbols('x')
# Sometimes it is better to use the numerator (issue 4829)
# but sometimes it is not (issue 11768) so leave this to
# the discretion of the user
ans = nsolve(x**2/(1 - x)/(1 - 2*x)**2 - 100, x, 0)
assert ans > 0.46 and ans < 0.47
def test_nsolve_denominator():
x = symbols('x')
# Test that nsolve uses the full expression (numerator and denominator).
ans = nsolve((x**2 + 3*x + 2)/(x + 2), -2.1)
# The root -2 was divided out, so make sure we don't find it.
assert ans == -1.0
def test_nsolve():
# onedimensional
x = Symbol('x')
assert nsolve(sin(x), 2) - pi.evalf() < 1e-15
assert nsolve(Eq(2*x, 2), x, -10) == nsolve(2*x - 2, -10)
# Testing checks on number of inputs
raises(TypeError, lambda: nsolve(Eq(2*x, 2)))
raises(TypeError, lambda: nsolve(Eq(2*x, 2), x, 1, 2))
# multidimensional
x1 = Symbol('x1')
x2 = Symbol('x2')
f1 = 3 * x1**2 - 2 * x2**2 - 1
f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8
f = Matrix((f1, f2)).T
F = lambdify((x1, x2), f.T, modules='mpmath')
for x0 in [(-1, 1), (1, -2), (4, 4), (-4, -4)]:
x = nsolve(f, (x1, x2), x0, tol=1.e-8)
assert mnorm(F(*x), 1) <= 1.e-10
# The Chinese mathematician Zhu Shijie was the very first to solve this
# nonlinear system 700 years ago (z was added to make it 3-dimensional)
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
f1 = -x + 2*y
f2 = (x**2 + x*(y**2 - 2) - 4*y) / (x + 4)
f3 = sqrt(x**2 + y**2)*z
f = Matrix((f1, f2, f3)).T
F = lambdify((x, y, z), f.T, modules='mpmath')
def getroot(x0):
root = nsolve(f, (x, y, z), x0)
assert mnorm(F(*root), 1) <= 1.e-8
return root
assert list(map(round, getroot((1, 1, 1)))) == [2.0, 1.0, 0.0]
assert nsolve([Eq(
f1, 0), Eq(f2, 0), Eq(f3, 0)], [x, y, z], (1, 1, 1)) # just see that it works
a = Symbol('a')
assert abs(nsolve(1/(0.001 + a)**3 - 6/(0.9 - a)**3, a, 0.3) -
mpf('0.31883011387318591')) < 1e-15
def test_issue_6408():
x = Symbol('x')
assert nsolve(Piecewise((x, x < 1), (x**2, True)), x, 2) == 0.0
@XFAIL
def test_issue_6408_fail():
x, y = symbols('x y')
assert nsolve(Integral(x*y, (x, 0, 5)), y, 2) == 0.0
@conserve_mpmath_dps
def test_increased_dps():
# Issue 8564
import mpmath
mpmath.mp.dps = 128
x = Symbol('x')
e1 = x**2 - pi
q = nsolve(e1, x, 3.0)
assert abs(sqrt(pi).evalf(128) - q) < 1e-128
def test_nsolve_precision():
x, y = symbols('x y')
sol = nsolve(x**2 - pi, x, 3, prec=128)
assert abs(sqrt(pi).evalf(128) - sol) < 1e-128
assert isinstance(sol, Float)
sols = nsolve((y**2 - x, x**2 - pi), (x, y), (3, 3), prec=128)
assert isinstance(sols, Matrix)
assert sols.shape == (2, 1)
assert abs(sqrt(pi).evalf(128) - sols[0]) < 1e-128
assert abs(sqrt(sqrt(pi)).evalf(128) - sols[1]) < 1e-128
assert all(isinstance(i, Float) for i in sols)
def test_nsolve_complex():
x, y = symbols('x y')
assert nsolve(x**2 + 2, 1j) == sqrt(2.)*I
assert nsolve(x**2 + 2, I) == sqrt(2.)*I
assert nsolve([x**2 + 2, y**2 + 2], [x, y], [I, I]) == Matrix([sqrt(2.)*I, sqrt(2.)*I])
assert nsolve([x**2 + 2, y**2 + 2], [x, y], [I, I]) == Matrix([sqrt(2.)*I, sqrt(2.)*I])
def test_nsolve_dict_kwarg():
x, y = symbols('x y')
# one variable
assert nsolve(x**2 - 2, 1, dict = True) == \
[{x: sqrt(2.)}]
# one variable with complex solution
assert nsolve(x**2 + 2, I, dict = True) == \
[{x: sqrt(2.)*I}]
# two variables
assert nsolve([x**2 + y**2 - 5, x**2 - y**2 + 1], [x, y], [1, 1], dict = True) == \
[{x: sqrt(2.), y: sqrt(3.)}]
def test_nsolve_rational():
x = symbols('x')
assert nsolve(x - Rational(1, 3), 0, prec=100) == Rational(1, 3).evalf(100)
def test_issue_14950():
x = Matrix(symbols('t s'))
x0 = Matrix([17, 23])
eqn = x + x0
assert nsolve(eqn, x, x0) == -x0
assert nsolve(eqn.T, x.T, x0.T) == -x0
|
163d753ea1102817c7eb2ec3e7f5007940b9decbe690094171d9fb516232e2c4 | from sympy import (Derivative as D, Eq, exp, sin,
Function, Symbol, symbols, cos, log)
from sympy.core import S
from sympy.solvers.pde import (pde_separate, pde_separate_add, pde_separate_mul,
pdsolve, classify_pde, checkpdesol)
from sympy.utilities.pytest import raises
a, b, c, x, y = symbols('a b c x y')
def test_pde_separate_add():
x, y, z, t = symbols("x,y,z,t")
F, T, X, Y, Z, u = map(Function, 'FTXYZu')
eq = Eq(D(u(x, t), x), D(u(x, t), t)*exp(u(x, t)))
res = pde_separate_add(eq, u(x, t), [X(x), T(t)])
assert res == [D(X(x), x)*exp(-X(x)), D(T(t), t)*exp(T(t))]
def test_pde_separate():
x, y, z, t = symbols("x,y,z,t")
F, T, X, Y, Z, u = map(Function, 'FTXYZu')
eq = Eq(D(u(x, t), x), D(u(x, t), t)*exp(u(x, t)))
raises(ValueError, lambda: pde_separate(eq, u(x, t), [X(x), T(t)], 'div'))
def test_pde_separate_mul():
x, y, z, t = symbols("x,y,z,t")
c = Symbol("C", real=True)
Phi = Function('Phi')
F, R, T, X, Y, Z, u = map(Function, 'FRTXYZu')
r, theta, z = symbols('r,theta,z')
# Something simple :)
eq = Eq(D(F(x, y, z), x) + D(F(x, y, z), y) + D(F(x, y, z), z), 0)
# Duplicate arguments in functions
raises(
ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(x), u(z, z)]))
# Wrong number of arguments
raises(ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(x), Y(y)]))
# Wrong variables: [x, y] -> [x, z]
raises(
ValueError, lambda: pde_separate_mul(eq, F(x, y, z), [X(t), Y(x, y)]))
assert pde_separate_mul(eq, F(x, y, z), [Y(y), u(x, z)]) == \
[D(Y(y), y)/Y(y), -D(u(x, z), x)/u(x, z) - D(u(x, z), z)/u(x, z)]
assert pde_separate_mul(eq, F(x, y, z), [X(x), Y(y), Z(z)]) == \
[D(X(x), x)/X(x), -D(Z(z), z)/Z(z) - D(Y(y), y)/Y(y)]
# wave equation
wave = Eq(D(u(x, t), t, t), c**2*D(u(x, t), x, x))
res = pde_separate_mul(wave, u(x, t), [X(x), T(t)])
assert res == [D(X(x), x, x)/X(x), D(T(t), t, t)/(c**2*T(t))]
# Laplace equation in cylindrical coords
eq = Eq(1/r * D(Phi(r, theta, z), r) + D(Phi(r, theta, z), r, 2) +
1/r**2 * D(Phi(r, theta, z), theta, 2) + D(Phi(r, theta, z), z, 2), 0)
# Separate z
res = pde_separate_mul(eq, Phi(r, theta, z), [Z(z), u(theta, r)])
assert res == [D(Z(z), z, z)/Z(z),
-D(u(theta, r), r, r)/u(theta, r) -
D(u(theta, r), r)/(r*u(theta, r)) -
D(u(theta, r), theta, theta)/(r**2*u(theta, r))]
# Lets use the result to create a new equation...
eq = Eq(res[1], c)
# ...and separate theta...
res = pde_separate_mul(eq, u(theta, r), [T(theta), R(r)])
assert res == [D(T(theta), theta, theta)/T(theta),
-r*D(R(r), r)/R(r) - r**2*D(R(r), r, r)/R(r) - c*r**2]
# ...or r...
res = pde_separate_mul(eq, u(theta, r), [R(r), T(theta)])
assert res == [r*D(R(r), r)/R(r) + r**2*D(R(r), r, r)/R(r) + c*r**2,
-D(T(theta), theta, theta)/T(theta)]
def test_issue_11726():
x, t = symbols("x t")
f = symbols("f", cls=Function)
X, T = symbols("X T", cls=Function)
u = f(x, t)
eq = u.diff(x, 2) - u.diff(t, 2)
res = pde_separate(eq, u, [T(x), X(t)])
assert res == [D(T(x), x, x)/T(x),D(X(t), t, t)/X(t)]
def test_pde_classify():
# When more number of hints are added, add tests for classifying here.
f = Function('f')
eq1 = a*f(x,y) + b*f(x,y).diff(x) + c*f(x,y).diff(y)
eq2 = 3*f(x,y) + 2*f(x,y).diff(x) + f(x,y).diff(y)
eq3 = a*f(x,y) + b*f(x,y).diff(x) + 2*f(x,y).diff(y)
eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y)
eq5 = x**2*f(x,y) + x*f(x,y).diff(x) + x*y*f(x,y).diff(y)
eq6 = y*x**2*f(x,y) + y*f(x,y).diff(x) + f(x,y).diff(y)
for eq in [eq1, eq2, eq3]:
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',)
for eq in [eq4, eq5, eq6]:
assert classify_pde(eq) == ('1st_linear_variable_coeff',)
def test_checkpdesol():
f, F = map(Function, ['f', 'F'])
eq1 = a*f(x,y) + b*f(x,y).diff(x) + c*f(x,y).diff(y)
eq2 = 3*f(x,y) + 2*f(x,y).diff(x) + f(x,y).diff(y)
eq3 = a*f(x,y) + b*f(x,y).diff(x) + 2*f(x,y).diff(y)
for eq in [eq1, eq2, eq3]:
assert checkpdesol(eq, pdsolve(eq))[0]
eq4 = x*f(x,y) + f(x,y).diff(x) + 3*f(x,y).diff(y)
eq5 = 2*f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y)
eq6 = f(x,y) + 1*f(x,y).diff(x) + 3*f(x,y).diff(y)
assert checkpdesol(eq4, [pdsolve(eq5), pdsolve(eq6)]) == [
(False, (x - 2)*F(3*x - y)*exp(-x/S(5) - 3*y/S(5))),
(False, (x - 1)*F(3*x - y)*exp(-x/S(10) - 3*y/S(10)))]
for eq in [eq4, eq5, eq6]:
assert checkpdesol(eq, pdsolve(eq))[0]
sol = pdsolve(eq4)
sol4 = Eq(sol.lhs - sol.rhs, 0)
raises(NotImplementedError, lambda:
checkpdesol(eq4, sol4, solve_for_func=False))
def test_solvefun():
f, F, G, H = map(Function, ['f', 'F', 'G', 'H'])
eq1 = f(x,y) + f(x,y).diff(x) + f(x,y).diff(y)
assert pdsolve(eq1) == Eq(f(x, y), F(x - y)*exp(-x/2 - y/2))
assert pdsolve(eq1, solvefun=G) == Eq(f(x, y), G(x - y)*exp(-x/2 - y/2))
assert pdsolve(eq1, solvefun=H) == Eq(f(x, y), H(x - y)*exp(-x/2 - y/2))
def test_pde_1st_linear_constant_coeff_homogeneous():
f, F = map(Function, ['f', 'F'])
u = f(x, y)
eq = 2*u + u.diff(x) + u.diff(y)
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',)
sol = pdsolve(eq)
assert sol == Eq(u, F(x - y)*exp(-x - y))
assert checkpdesol(eq, sol)[0]
eq = 4 + (3*u.diff(x)/u) + (2*u.diff(y)/u)
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',)
sol = pdsolve(eq)
assert sol == Eq(u, F(2*x - 3*y)*exp(-S(12)*x/13 - S(8)*y/13))
assert checkpdesol(eq, sol)[0]
eq = u + (6*u.diff(x)) + (7*u.diff(y))
assert classify_pde(eq) == ('1st_linear_constant_coeff_homogeneous',)
sol = pdsolve(eq)
assert sol == Eq(u, F(7*x - 6*y)*exp(-6*x/S(85) - 7*y/S(85)))
assert checkpdesol(eq, sol)[0]
eq = a*u + b*u.diff(x) + c*u.diff(y)
sol = pdsolve(eq)
assert checkpdesol(eq, sol)[0]
def test_pde_1st_linear_constant_coeff():
f, F = map(Function, ['f', 'F'])
u = f(x,y)
eq = -2*u.diff(x) + 4*u.diff(y) + 5*u - exp(x + 3*y)
sol = pdsolve(eq)
assert sol == Eq(f(x,y),
(F(4*x + 2*y) + exp(x/S(2) + 4*y)/S(15))*exp(x/S(2) - y))
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
assert checkpdesol(eq, sol)[0]
eq = (u.diff(x)/u) + (u.diff(y)/u) + 1 - (exp(x + y)/u)
sol = pdsolve(eq)
assert sol == Eq(f(x, y), F(x - y)*exp(-x/2 - y/2) + exp(x + y)/S(3))
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
assert checkpdesol(eq, sol)[0]
eq = 2*u + -u.diff(x) + 3*u.diff(y) + sin(x)
sol = pdsolve(eq)
assert sol == Eq(f(x, y),
F(3*x + y)*exp(x/S(5) - 3*y/S(5)) - 2*sin(x)/S(5) - cos(x)/S(5))
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
assert checkpdesol(eq, sol)[0]
eq = u + u.diff(x) + u.diff(y) + x*y
sol = pdsolve(eq)
assert sol == Eq(f(x, y),
-x*y + x + y + F(x - y)*exp(-x/S(2) - y/S(2)) - 2)
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
assert checkpdesol(eq, sol)[0]
eq = u + u.diff(x) + u.diff(y) + log(x)
assert classify_pde(eq) == ('1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral')
def test_pdsolve_all():
f, F = map(Function, ['f', 'F'])
u = f(x,y)
eq = u + u.diff(x) + u.diff(y) + x**2*y
sol = pdsolve(eq, hint = 'all')
keys = ['1st_linear_constant_coeff',
'1st_linear_constant_coeff_Integral', 'default', 'order']
assert sorted(sol.keys()) == keys
assert sol['order'] == 1
assert sol['default'] == '1st_linear_constant_coeff'
assert sol['1st_linear_constant_coeff'] == Eq(f(x, y),
-x**2*y + x**2 + 2*x*y - 4*x - 2*y + F(x - y)*exp(-x/S(2) - y/S(2)) + 6)
def test_pdsolve_variable_coeff():
f, F = map(Function, ['f', 'F'])
u = f(x, y)
eq = x*(u.diff(x)) - y*(u.diff(y)) + y**2*u - y**2
sol = pdsolve(eq, hint="1st_linear_variable_coeff")
assert sol == Eq(u, F(x*y)*exp(y**2/2) + 1)
assert checkpdesol(eq, sol)[0]
eq = x**2*u + x*u.diff(x) + x*y*u.diff(y)
sol = pdsolve(eq, hint='1st_linear_variable_coeff')
assert sol == Eq(u, F(y*exp(-x))*exp(-x**2/2))
assert checkpdesol(eq, sol)[0]
eq = y*x**2*u + y*u.diff(x) + u.diff(y)
sol = pdsolve(eq, hint='1st_linear_variable_coeff')
assert sol == Eq(u, F(-2*x + y**2)*exp(-x**3/3))
assert checkpdesol(eq, sol)[0]
eq = exp(x)**2*(u.diff(x)) + y
sol = pdsolve(eq, hint='1st_linear_variable_coeff')
assert sol == Eq(u, y*exp(-2*x)/2 + F(y))
assert checkpdesol(eq, sol)[0]
eq = exp(2*x)*(u.diff(y)) + y*u - u
sol = pdsolve(eq, hint='1st_linear_variable_coeff')
assert sol == Eq(u, exp((-y**2 + 2*y + 2*F(x))*exp(-2*x)/2))
|
4629be4c956fff91d6cc9549875e14fd64e1439d90f22b56b988edb3bd020099 | from sympy import (acos, acosh, asinh, atan, cos, Derivative, diff, dsolve,
Dummy, Eq, Ne, erf, erfi, exp, Function, I, Integral, LambertW, log, O, pi,
Rational, rootof, S, simplify, sin, sqrt, Subs, Symbol, tan, asin, sinh,
Piecewise, symbols, Poly, sec, Ei, re, im)
from sympy.solvers.ode import (_undetermined_coefficients_match,
checkodesol, classify_ode, classify_sysode, constant_renumber,
constantsimp, homogeneous_order, infinitesimals, checkinfsol,
checksysodesol, solve_ics, dsolve, get_numbered_constants)
from sympy.solvers.deutils import ode_order
from sympy.utilities.pytest import XFAIL, skip, raises, slow, ON_TRAVIS
C0, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C0:11')
u, x, y, z = symbols('u,x:z', real=True)
f = Function('f')
g = Function('g')
h = Function('h')
# Note: the tests below may fail (but still be correct) if ODE solver,
# the integral engine, solve(), or even simplify() changes. Also, in
# differently formatted solutions, the arbitrary constants might not be
# equal. Using specific hints in tests can help to avoid this.
# Tests of order higher than 1 should run the solutions through
# constant_renumber because it will normalize it (constant_renumber causes
# dsolve() to return different results on different machines)
def test_linear_2eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
k, l, m, n = symbols('k, l, m, n', Integer=True)
t = Symbol('t')
x0, y0 = symbols('x0, y0', cls=Function)
eq1 = (Eq(diff(x(t),t), 9*y(t)), Eq(diff(y(t),t), 12*x(t)))
sol1 = [Eq(x(t), 9*C1*exp(6*sqrt(3)*t) + 9*C2*exp(-6*sqrt(3)*t)), \
Eq(y(t), 6*sqrt(3)*C1*exp(6*sqrt(3)*t) - 6*sqrt(3)*C2*exp(-6*sqrt(3)*t))]
assert checksysodesol(eq1, sol1) == (True, [0, 0])
eq2 = (Eq(diff(x(t),t), 2*x(t) + 4*y(t)), Eq(diff(y(t),t), 12*x(t) + 41*y(t)))
sol2 = [Eq(x(t), 4*C1*exp(t*(sqrt(1713)/2 + S(43)/2)) + 4*C2*exp(t*(-sqrt(1713)/2 + S(43)/2))), \
Eq(y(t), C1*(S(39)/2 + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + S(43)/2)) + \
C2*(-sqrt(1713)/2 + S(39)/2)*exp(t*(-sqrt(1713)/2 + S(43)/2)))]
assert checksysodesol(eq2, sol2) == (True, [0, 0])
eq3 = (Eq(diff(x(t),t), x(t) + y(t)), Eq(diff(y(t),t), -2*x(t) + 2*y(t)))
sol3 = [Eq(x(t), (C1*cos(sqrt(7)*t/2) + C2*sin(sqrt(7)*t/2))*exp(3*t/2)), \
Eq(y(t), (C1*(-sqrt(7)*sin(sqrt(7)*t/2)/2 + cos(sqrt(7)*t/2)/2) + \
C2*(sin(sqrt(7)*t/2)/2 + sqrt(7)*cos(sqrt(7)*t/2)/2))*exp(3*t/2))]
assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol4 = [Eq(x(t), C1*exp(t*(sqrt(6) + 3)) + C2*exp(t*(-sqrt(6) + 3)) - S(22)/3), \
Eq(y(t), C1*(2 + sqrt(6))*exp(t*(sqrt(6) + 3)) + C2*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) - S(5)/3)]
assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
sol5 = [Eq(x(t), (C1*cos(sqrt(2)*t) + C2*sin(sqrt(2)*t))*exp(t) - S(58)/3), \
Eq(y(t), (-sqrt(2)*C1*sin(sqrt(2)*t) + sqrt(2)*C2*cos(sqrt(2)*t))*exp(t) - S(185)/3)]
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol6 = [Eq(x(t), (C1*exp(2*t) + C2*exp(-2*t))*exp(S(5)/2*t**2)), \
Eq(y(t), (C1*exp(2*t) - C2*exp(-2*t))*exp(S(5)/2*t**2))]
s = dsolve(eq6)
assert checksysodesol(eq6, sol6) == (True, [0, 0])
eq7 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
sol7 = [Eq(x(t), (C1*cos((t**3)/3) + C2*sin((t**3)/3))*exp(S(5)/2*t**2)), \
Eq(y(t), (-C1*sin((t**3)/3) + C2*cos((t**3)/3))*exp(S(5)/2*t**2))]
assert checksysodesol(eq7, sol7) == (True, [0, 0])
eq8 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t)))
sol8 = [Eq(x(t), (C1*exp((sqrt(77)/2 + S(9)/2)*(t**3)/3) + \
C2*exp((-sqrt(77)/2 + S(9)/2)*(t**3)/3))*exp(S(5)/2*t**2)), \
Eq(y(t), (C1*(sqrt(77)/2 + S(9)/2)*exp((sqrt(77)/2 + S(9)/2)*(t**3)/3) + \
C2*(-sqrt(77)/2 + S(9)/2)*exp((-sqrt(77)/2 + S(9)/2)*(t**3)/3))*exp(S(5)/2*t**2))]
assert checksysodesol(eq8, sol8) == (True, [0, 0])
eq10 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), (1-t**2)*x(t) + (5*t+9*t**2)*y(t)))
sol10 = [Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)), \
Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + \
exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))]
s = dsolve(eq10)
assert s == sol10 # too complicated to test with subs and simplify
# assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this one fails
def test_linear_2eq_order1_nonhomog_linear():
e = [Eq(diff(f(x), x), f(x) + g(x) + 5*x),
Eq(diff(g(x), x), f(x) - g(x))]
raises(NotImplementedError, lambda: dsolve(e))
def test_linear_2eq_order1_nonhomog():
# Note: once implemented, add some tests esp. with resonance
e = [Eq(diff(f(x), x), f(x) + exp(x)),
Eq(diff(g(x), x), f(x) + g(x) + x*exp(x))]
raises(NotImplementedError, lambda: dsolve(e))
def test_linear_2eq_order1_type2_degen():
e = [Eq(diff(f(x), x), f(x) + 5),
Eq(diff(g(x), x), f(x) + 7)]
s1 = [Eq(f(x), C1*exp(x) - 5), Eq(g(x), C1*exp(x) - C2 + 2*x - 5)]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_dsolve_linear_2eq_order1_diag_triangular():
e = [Eq(diff(f(x), x), f(x)),
Eq(diff(g(x), x), g(x))]
s1 = [Eq(f(x), C1*exp(x)), Eq(g(x), C2*exp(x))]
assert checksysodesol(e, s1) == (True, [0, 0])
e = [Eq(diff(f(x), x), 2*f(x)),
Eq(diff(g(x), x), 3*f(x) + 7*g(x))]
s1 = [Eq(f(x), -5*C2*exp(2*x)),
Eq(g(x), 5*C1*exp(7*x) + 3*C2*exp(2*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_type1_D_lt_0():
e = [Eq(diff(f(x), x), -9*I*f(x) - 4*g(x)),
Eq(diff(g(x), x), -4*I*g(x))]
s1 = [Eq(f(x), -4*C1*exp(-4*I*x) - 4*C2*exp(-9*I*x)), \
Eq(g(x), 5*I*C1*exp(-4*I*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_type1_D_lt_0_b_eq_0():
e = [Eq(diff(f(x), x), -9*I*f(x)),
Eq(diff(g(x), x), -4*I*g(x))]
s1 = [Eq(f(x), -5*I*C2*exp(-9*I*x)), Eq(g(x), 5*I*C1*exp(-4*I*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_many_zeros():
t = Symbol('t')
corner_cases = [(0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0),
(0, 0, 1, 0), (0, 0, 0, 1), (1, 0, 0, I),
(I, 0, 0, -I), (0, I, 0, 0), (0, I, I, 0)]
s1 = [[Eq(f(t), C1), Eq(g(t), C2)],
[Eq(f(t), C1*exp(t)), Eq(g(t), -C2)],
[Eq(f(t), C1 + C2*t), Eq(g(t), C2)],
[Eq(f(t), C2), Eq(g(t), C1 + C2*t)],
[Eq(f(t), -C2), Eq(g(t), C1*exp(t))],
[Eq(f(t), C1*(1 - I)*exp(t)), Eq(g(t), C2*(-1 + I)*exp(I*t))],
[Eq(f(t), 2*I*C1*exp(I*t)), Eq(g(t), -2*I*C2*exp(-I*t))],
[Eq(f(t), I*C1 + I*C2*t), Eq(g(t), C2)],
[Eq(f(t), I*C1*exp(I*t) + I*C2*exp(-I*t)), \
Eq(g(t), I*C1*exp(I*t) - I*C2*exp(-I*t))]
]
for r, sol in zip(corner_cases, s1):
eq = [Eq(diff(f(t), t), r[0]*f(t) + r[1]*g(t)),
Eq(diff(g(t), t), r[2]*f(t) + r[3]*g(t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_dsolve_linsystem_symbol_piecewise():
u = Symbol('u') # XXX it's more complicated with real u
eq = (Eq(diff(f(x), x), 2*f(x) + g(x)),
Eq(diff(g(x), x), u*f(x)))
s1 = [Eq(f(x), Piecewise((C1*exp(x*(sqrt(4*u + 4)/2 + 1)) +
C2*exp(x*(-sqrt(4*u + 4)/2 + 1)), Ne(4*u + 4, 0)), ((C1 + C2*(x +
Piecewise((0, Eq(sqrt(4*u + 4)/2 + 1, 2)), (1/(-sqrt(4*u + 4)/2 + 1),
True))))*exp(x*(sqrt(4*u + 4)/2 + 1)), True))), Eq(g(x),
Piecewise((C1*(sqrt(4*u + 4)/2 - 1)*exp(x*(sqrt(4*u + 4)/2 + 1)) +
C2*(-sqrt(4*u + 4)/2 - 1)*exp(x*(-sqrt(4*u + 4)/2 + 1)), Ne(4*u + 4,
0)), ((C1*(sqrt(4*u + 4)/2 - 1) + C2*(x*(sqrt(4*u + 4)/2 - 1) +
Piecewise((1, Eq(sqrt(4*u + 4)/2 + 1, 2)), (0,
True))))*exp(x*(sqrt(4*u + 4)/2 + 1)), True)))]
assert dsolve(eq) == s1
# FIXME: assert checksysodesol(eq, s) == (True, [0, 0])
# Remove lines below when checksysodesol works
s = [(l.lhs, l.rhs) for l in s1]
for v in [0, 7, -42, 5*I, 3 + 4*I]:
assert eq[0].subs(s).subs(u, v).doit().simplify()
assert eq[1].subs(s).subs(u, v).doit().simplify()
# example from https://groups.google.com/d/msg/sympy/xmzoqW6tWaE/sf0bgQrlCgAJ
i, r1, c1, r2, c2, t = symbols('i, r1, c1, r2, c2, t')
x1 = Function('x1')
x2 = Function('x2')
eq1 = r1*c1*Derivative(x1(t), t) + x1(t) - x2(t) - r1*i
eq2 = r2*c1*Derivative(x1(t), t) + r2*c2*Derivative(x2(t), t) + x2(t) - r2*i
sol = dsolve((eq1, eq2))
# FIXME: assert checksysodesol(eq, sol) == (True, [0, 0])
# Remove line below when checksysodesol works
assert all(s.has(Piecewise) for s in sol)
@slow
def test_linear_2eq_order2():
x, y, z = symbols('x, y, z', cls=Function)
k, l, m, n = symbols('k, l, m, n', Integer=True)
t, l = symbols('t, l')
x0, y0 = symbols('x0, y0', cls=Function)
eq1 = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t)))
sol1 = [Eq(x(t), 43*C1*exp(t*rootof(l**4 - 14*l**2 + 2, 0)) + 43*C2*exp(t*rootof(l**4 - 14*l**2 + 2, 1)) + \
43*C3*exp(t*rootof(l**4 - 14*l**2 + 2, 2)) + 43*C4*exp(t*rootof(l**4 - 14*l**2 + 2, 3))), \
Eq(y(t), C1*(rootof(l**4 - 14*l**2 + 2, 0)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 0)) + \
C2*(rootof(l**4 - 14*l**2 + 2, 1)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 1)) + \
C3*(rootof(l**4 - 14*l**2 + 2, 2)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 2)) + \
C4*(rootof(l**4 - 14*l**2 + 2, 3)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 3)))]
assert dsolve(eq1) == sol1
# FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0]) # this one fails
eq2 = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12))
sol2 = [Eq(x(t), 3*C1*exp(t*rootof(l**4 - 15*l**2 + 29, 0)) + 3*C2*exp(t*rootof(l**4 - 15*l**2 + 29, 1)) + \
3*C3*exp(t*rootof(l**4 - 15*l**2 + 29, 2)) + 3*C4*exp(t*rootof(l**4 - 15*l**2 + 29, 3)) - S(181)/29), \
Eq(y(t), C1*(rootof(l**4 - 15*l**2 + 29, 0)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 0)) + \
C2*(rootof(l**4 - 15*l**2 + 29, 1)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 1)) + \
C3*(rootof(l**4 - 15*l**2 + 29, 2)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 2)) + \
C4*(rootof(l**4 - 15*l**2 + 29, 3)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 3)) + S(183)/29)]
assert dsolve(eq2) == sol2
# FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0]) # this one fails
eq3 = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0))
sol3 = [Eq(x(t), C1*cos(t*(S(9)/2 + sqrt(109)/2)) + C2*sin(t*(S(9)/2 + sqrt(109)/2)) + C3*cos(t*(-sqrt(109)/2 + S(9)/2)) + \
C4*sin(t*(-sqrt(109)/2 + S(9)/2))), Eq(y(t), -C1*sin(t*(S(9)/2 + sqrt(109)/2)) + C2*cos(t*(S(9)/2 + sqrt(109)/2)) - \
C3*sin(t*(-sqrt(109)/2 + S(9)/2)) + C4*cos(t*(-sqrt(109)/2 + S(9)/2)))]
assert dsolve(eq3) == sol3
assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t)))
sol4 = [Eq(x(t), C3*t + t*Integral((9*C1*exp(3*sqrt(7)*t**2/2) + 9*C2*exp(-3*sqrt(7)*t**2/2))/t**2, t)), \
Eq(y(t), C4*t + t*Integral((3*sqrt(7)*C1*exp(3*sqrt(7)*t**2/2) - 3*sqrt(7)*C2*exp(-3*sqrt(7)*t**2/2))/t**2, t))]
assert dsolve(eq4) == sol4
assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(diff(x(t),t,t), (log(t)+t**2)*diff(x(t),t)+(log(t)+t**2)*3*diff(y(t),t)), Eq(diff(y(t),t,t), \
(log(t)+t**2)*2*diff(x(t),t)+(log(t)+t**2)*9*diff(y(t),t)))
sol5 = [Eq(x(t), -sqrt(22)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C2 - \
C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) - C4 - \
(sqrt(22) + 5)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C2) + \
(-sqrt(22) + 5)*(C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C4))/88), \
Eq(y(t), -sqrt(22)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + \
C2 - C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) - C4)/44)]
assert dsolve(eq5) == sol5
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t,t), log(t)*t*diff(y(t),t) - log(t)*y(t)), Eq(diff(y(t),t,t), log(t)*t*diff(x(t),t) - log(t)*x(t)))
sol6 = [Eq(x(t), C3*t + t*Integral((C1*exp(Integral(t*log(t), t)) + \
C2*exp(-Integral(t*log(t), t)))/t**2, t)), Eq(y(t), C4*t + t*Integral((C1*exp(Integral(t*log(t), t)) - \
C2*exp(-Integral(t*log(t), t)))/t**2, t))]
assert dsolve(eq6) == sol6
assert checksysodesol(eq6, sol6) == (True, [0, 0])
eq7 = (Eq(diff(x(t),t,t), log(t)*(t*diff(x(t),t) - x(t)) + exp(t)*(t*diff(y(t),t) - y(t))), \
Eq(diff(y(t),t,t), (t**2)*(t*diff(x(t),t) - x(t)) + (t)*(t*diff(y(t),t) - y(t))))
sol7 = [Eq(x(t), C3*t + t*Integral((C1*x0(t) + C2*x0(t)*Integral(t*exp(t)*exp(Integral(t**2, t))*\
exp(Integral(t*log(t), t))/x0(t)**2, t))/t**2, t)), Eq(y(t), C4*t + t*Integral((C1*y0(t) + \
C2*(y0(t)*Integral(t*exp(t)*exp(Integral(t**2, t))*exp(Integral(t*log(t), t))/x0(t)**2, t) + \
exp(Integral(t**2, t))*exp(Integral(t*log(t), t))/x0(t)))/t**2, t))]
assert dsolve(eq7) == sol7
# FIXME: assert checksysodesol(eq7, sol7) == (True, [0, 0])
eq8 = (Eq(diff(x(t),t,t), t*(4*x(t) + 9*y(t))), Eq(diff(y(t),t,t), t*(12*x(t) - 6*y(t))))
sol8 = ("[Eq(x(t), -sqrt(133)*((-sqrt(133) - 1)*(C2*(133*t**8/24 - t**3/6 + sqrt(133)*t**3/2 + 1) + "
"C1*t*(sqrt(133)*t**4/6 - t**3/12 + 1) + O(t**6)) - (-1 + sqrt(133))*(C2*(-sqrt(133)*t**3/6 - t**3/6 + 1) + "
"C1*t*(-sqrt(133)*t**3/12 - t**3/12 + 1) + O(t**6)) - 4*C2*(133*t**8/24 - t**3/6 + sqrt(133)*t**3/2 + 1) + "
"4*C2*(-sqrt(133)*t**3/6 - t**3/6 + 1) - 4*C1*t*(sqrt(133)*t**4/6 - t**3/12 + 1) + "
"4*C1*t*(-sqrt(133)*t**3/12 - t**3/12 + 1) + O(t**6))/3192), Eq(y(t), -sqrt(133)*(-C2*(133*t**8/24 - t**3/6 + "
"sqrt(133)*t**3/2 + 1) + C2*(-sqrt(133)*t**3/6 - t**3/6 + 1) - C1*t*(sqrt(133)*t**4/6 - t**3/12 + 1) + "
"C1*t*(-sqrt(133)*t**3/12 - t**3/12 + 1) + O(t**6))/266)]")
assert str(dsolve(eq8)) == sol8
# FIXME: assert checksysodesol(eq8, sol8) == (True, [0, 0])
eq9 = (Eq(diff(x(t),t,t), t*(4*diff(x(t),t) + 9*diff(y(t),t))), Eq(diff(y(t),t,t), t*(12*diff(x(t),t) - 6*diff(y(t),t))))
sol9 = [Eq(x(t), -sqrt(133)*(4*C1*Integral(exp((-sqrt(133) - 1)*Integral(t, t)), t) + 4*C2 - \
4*C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) - 4*C4 - (-1 + sqrt(133))*(C1*Integral(exp((-sqrt(133) - \
1)*Integral(t, t)), t) + C2) + (-sqrt(133) - 1)*(C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) + \
C4))/3192), Eq(y(t), -sqrt(133)*(C1*Integral(exp((-sqrt(133) - 1)*Integral(t, t)), t) + C2 - \
C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) - C4)/266)]
assert dsolve(eq9) == sol9
assert checksysodesol(eq9, sol9) == (True, [0, 0])
eq10 = (t**2*diff(x(t),t,t) + 3*t*diff(x(t),t) + 4*t*diff(y(t),t) + 12*x(t) + 9*y(t), \
t**2*diff(y(t),t,t) + 2*t*diff(x(t),t) - 5*t*diff(y(t),t) + 15*x(t) + 8*y(t))
sol10 = [Eq(x(t), -C1*(-2*sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 13 + 2*sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + \
346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))))*exp((-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)))/2)*log(t)) - \
C2*(-2*sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
13 - 2*sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))))*exp((-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*log(t)) - C3*t**(1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*(2*sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 13 + 2*sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))) - C4*t**(-sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2 + 1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2)*(-2*sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))) + 2*sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 13)), Eq(y(t), C1*(-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 14 + (-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)**2 + sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))))*exp((-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*log(t)) + C2*(-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 14 - sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))) + (-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)**2)*exp((-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*log(t)) + C3*t**(1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*(sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))) + 14 + (1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2 + sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)**2) + C4*t**(-sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + \
346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)))/2 + 1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2)*(-sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + \
8 + 346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))) + (-sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + \
346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)))/2 + 1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2)**2 + sqrt(-346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 14))]
assert dsolve(eq10) == sol10
# FIXME: assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this hangs or at least takes a while...
def test_linear_3eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq1 = (Eq(diff(x(t),t), 21*x(t)), Eq(diff(y(t),t), 17*x(t)+3*y(t)), Eq(diff(z(t),t), 5*x(t)+7*y(t)+9*z(t)))
sol1 = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \
Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))]
assert checksysodesol(eq1, sol1) == (True, [0, 0, 0])
eq2 = (Eq(diff(x(t),t),3*y(t)-11*z(t)),Eq(diff(y(t),t),7*z(t)-3*x(t)),Eq(diff(z(t),t),11*x(t)-7*y(t)))
sol2 = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \
Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \
Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))]
assert checksysodesol(eq2, sol2) == (True, [0, 0, 0])
eq3 = (Eq(3*diff(x(t),t),4*5*(y(t)-z(t))),Eq(4*diff(y(t),t),3*5*(z(t)-x(t))),Eq(5*diff(z(t),t),3*4*(x(t)-y(t))))
sol3 = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \
Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \
Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))]
assert checksysodesol(eq3, sol3) == (True, [0, 0, 0])
f = t**3 + log(t)
g = t**2 + sin(t)
eq4 = (Eq(diff(x(t),t),(4*f+g)*x(t)-f*y(t)-2*f*z(t)), Eq(diff(y(t),t),2*f*x(t)+(f+g)*y(t)-2*f*z(t)), Eq(diff(z(t),t),5*f*x(t)+f*y(t)+(-3*f+g)*z(t)))
sol4 = [Eq(x(t), (C1*exp(-2*Integral(t**3 + log(t), t)) + C2*(sqrt(3)*sin(sqrt(3)*Integral(t**3 + log(t), t))/6 \
+ cos(sqrt(3)*Integral(t**3 + log(t), t))/2) + C3*(sin(sqrt(3)*Integral(t**3 + log(t), t))/2 - \
sqrt(3)*cos(sqrt(3)*Integral(t**3 + log(t), t))/6))*exp(Integral(-t**2 - sin(t), t))), Eq(y(t), \
(C2*(sqrt(3)*sin(sqrt(3)*Integral(t**3 + log(t), t))/6 + cos(sqrt(3)*Integral(t**3 + log(t), t))/2) + \
C3*(sin(sqrt(3)*Integral(t**3 + log(t), t))/2 - sqrt(3)*cos(sqrt(3)*Integral(t**3 + log(t), t))/6))*\
exp(Integral(-t**2 - sin(t), t))), Eq(z(t), (C1*exp(-2*Integral(t**3 + log(t), t)) + C2*cos(sqrt(3)*\
Integral(t**3 + log(t), t)) + C3*sin(sqrt(3)*Integral(t**3 + log(t), t)))*exp(Integral(-t**2 - sin(t), t)))]
assert dsolve(eq4) == sol4
# FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0, 0]) # this one fails
eq5 = (Eq(diff(x(t),t),4*x(t) - z(t)),Eq(diff(y(t),t),2*x(t)+2*y(t)-z(t)),Eq(diff(z(t),t),3*x(t)+y(t)))
sol5 = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \
Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \
Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))]
assert checksysodesol(eq5, sol5) == (True, [0, 0, 0])
eq6 = (Eq(diff(x(t),t),4*x(t) - y(t) - 2*z(t)),Eq(diff(y(t),t),2*x(t) + y(t)- 2*z(t)),Eq(diff(z(t),t),5*x(t)-3*z(t)))
sol6 = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t)/5 + 3*cos(t)/5) + C3*(3*sin(t)/5 + cos(t)/5)),
Eq(y(t), C2*(-sin(t)/5 + 3*cos(t)/5) + C3*(3*sin(t)/5 + cos(t)/5)),
Eq(z(t), C1*exp(2*t) + C2*cos(t) + C3*sin(t))]
assert checksysodesol(eq5, sol5) == (True, [0, 0, 0])
def test_linear_3eq_order1_nonhomog():
e = [Eq(diff(f(x), x), -9*f(x) - 4*g(x)),
Eq(diff(g(x), x), -4*g(x)),
Eq(diff(h(x), x), h(x) + exp(x))]
raises(NotImplementedError, lambda: dsolve(e))
@XFAIL
def test_linear_3eq_order1_diagonal():
# code makes assumptions about coefficients being nonzero, breaks when assumptions are not true
e = [Eq(diff(f(x), x), f(x)),
Eq(diff(g(x), x), g(x)),
Eq(diff(h(x), x), h(x))]
s1 = [Eq(f(x), C1*exp(x)), Eq(g(x), C2*exp(x)), Eq(h(x), C3*exp(x))]
s = dsolve(e)
assert s == s1
assert checksysodesol(e, s1) == (True, [0, 0, 0])
def test_nonlinear_2eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq1 = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol1 = [
Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(-S(1)/4))),
Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert dsolve(eq1) == sol1
assert checksysodesol(eq1, sol1) == (True, [0, 0])
eq2 = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
sol2 = [
Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**(S(1)/4))/3),
Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**(S(1)/4))/3),
Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**(S(1)/4))/3),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**(S(1)/4))/3),
Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert dsolve(eq2) == sol2
assert checksysodesol(eq2, sol2) == (True, [0, 0])
eq3 = (Eq(diff(x(t),t), y(t)*x(t)), Eq(diff(y(t),t), x(t)**3))
tt = S(2)/3
sol3 = [
Eq(x(t), 6**tt/(6*(-sinh(sqrt(C1)*(C2 + t)/2)/sqrt(C1))**tt)),
Eq(y(t), sqrt(C1 + C1/sinh(sqrt(C1)*(C2 + t)/2)**2)/3)]
assert dsolve(eq3) == sol3
# FIXME: assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t),x(t)*y(t)*sin(t)**2), Eq(diff(y(t),t),y(t)**2*sin(t)**2))
sol4 = set([Eq(x(t), -2*exp(C1)/(C2*exp(C1) + t - sin(2*t)/2)), Eq(y(t), -2/(C1 + t - sin(2*t)/2))])
assert dsolve(eq4) == sol4
# FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2))
sol5 = set([Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)])
assert dsolve(eq5) == sol5
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t),x(t)**2*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol6 = [
Eq(x(t), 1/(C1 - 1/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), 1/(C1 + (-1/(4*C2 + 4*t))**(-S(1)/4))),
Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), 1/(C1 + I/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), 1/(C1 - I/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert dsolve(eq6) == sol6
assert checksysodesol(eq6, sol6) == (True, [0, 0])
def test_checksysodesol():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq = (Eq(diff(x(t),t), 9*y(t)), Eq(diff(y(t),t), 12*x(t)))
sol = [Eq(x(t), 9*C1*exp(-6*sqrt(3)*t) + 9*C2*exp(6*sqrt(3)*t)), \
Eq(y(t), -6*sqrt(3)*C1*exp(-6*sqrt(3)*t) + 6*sqrt(3)*C2*exp(6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 2*x(t) + 4*y(t)), Eq(diff(y(t),t), 12*x(t) + 41*y(t)))
sol = [Eq(x(t), 4*C1*exp(t*(-sqrt(1713)/2 + S(43)/2)) + 4*C2*exp(t*(sqrt(1713)/2 + \
S(43)/2))), Eq(y(t), C1*(-sqrt(1713)/2 + S(39)/2)*exp(t*(-sqrt(1713)/2 + \
S(43)/2)) + C2*(S(39)/2 + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + S(43)/2)))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t)), Eq(diff(y(t),t), -2*x(t) + 2*y(t)))
sol = [Eq(x(t), (C1*sin(sqrt(7)*t/2) + C2*cos(sqrt(7)*t/2))*exp(3*t/2)), \
Eq(y(t), ((C1/2 - sqrt(7)*C2/2)*sin(sqrt(7)*t/2) + (sqrt(7)*C1/2 + \
C2/2)*cos(sqrt(7)*t/2))*exp(3*t/2))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol = [Eq(x(t), C1*exp(t*(-sqrt(6) + 3)) + C2*exp(t*(sqrt(6) + 3)) - \
S(22)/3), Eq(y(t), C1*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) + C2*(2 + \
sqrt(6))*exp(t*(sqrt(6) + 3)) - S(5)/3)]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
sol = [Eq(x(t), (C1*sin(sqrt(2)*t) + C2*cos(sqrt(2)*t))*exp(t) - S(58)/3), \
Eq(y(t), (sqrt(2)*C1*cos(sqrt(2)*t) - sqrt(2)*C2*sin(sqrt(2)*t))*exp(t) - S(185)/3)]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol = [Eq(x(t), (C1*exp((Integral(2, t).doit())) + C2*exp(-(Integral(2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (C1*exp((Integral(2, t)).doit()) - \
C2*exp(-(Integral(2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
sol = [Eq(x(t), (C1*cos((Integral(t**2, t)).doit()) + C2*sin((Integral(t**2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (-C1*sin((Integral(t**2, t)).doit()) + \
C2*cos((Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t)))
sol = [Eq(x(t), (C1*exp((-sqrt(77)/2 + S(9)/2)*(Integral(t**2, t)).doit()) + \
C2*exp((sqrt(77)/2 + S(9)/2)*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit())), \
Eq(y(t), (C1*(-sqrt(77)/2 + S(9)/2)*exp((-sqrt(77)/2 + S(9)/2)*(Integral(t**2, t)).doit()) + \
C2*(sqrt(77)/2 + S(9)/2)*exp((sqrt(77)/2 + S(9)/2)*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t)))
root0 = -sqrt(-sqrt(47) + 7)
root1 = sqrt(-sqrt(47) + 7)
root2 = -sqrt(sqrt(47) + 7)
root3 = sqrt(sqrt(47) + 7)
sol = [Eq(x(t), 43*C1*exp(t*root0) + 43*C2*exp(t*root1) + 43*C3*exp(t*root2) + 43*C4*exp(t*root3)), \
Eq(y(t), C1*(root0**2 - 5)*exp(t*root0) + C2*(root1**2 - 5)*exp(t*root1) + \
C3*(root2**2 - 5)*exp(t*root2) + C4*(root3**2 - 5)*exp(t*root3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12))
root0 = -sqrt(-sqrt(109)/2 + S(15)/2)
root1 = sqrt(-sqrt(109)/2 + S(15)/2)
root2 = -sqrt(sqrt(109)/2 + S(15)/2)
root3 = sqrt(sqrt(109)/2 + S(15)/2)
sol = [Eq(x(t), 3*C1*exp(t*root0) + 3*C2*exp(t*root1) + 3*C3*exp(t*root2) + 3*C4*exp(t*root3) - S(181)/29), \
Eq(y(t), C1*(root0**2 - 8)*exp(t*root0) + C2*(root1**2 - 8)*exp(t*root1) + \
C3*(root2**2 - 8)*exp(t*root2) + C4*(root3**2 - 8)*exp(t*root3) + S(183)/29)]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0))
sol = [Eq(x(t), C1*cos(t*(S(9)/2 + sqrt(109)/2)) + C2*sin(t*(S(9)/2 + sqrt(109)/2)) + \
C3*cos(t*(-sqrt(109)/2 + S(9)/2)) + C4*sin(t*(-sqrt(109)/2 + S(9)/2))), Eq(y(t), -C1*sin(t*(S(9)/2 + sqrt(109)/2)) \
+ C2*cos(t*(S(9)/2 + sqrt(109)/2)) - C3*sin(t*(-sqrt(109)/2 + S(9)/2)) + C4*cos(t*(-sqrt(109)/2 + S(9)/2)))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t)))
I1 = sqrt(6)*7**(S(1)/4)*sqrt(pi)*erfi(sqrt(6)*7**(S(1)/4)*t/2)/2 - exp(3*sqrt(7)*t**2/2)/t
I2 = -sqrt(6)*7**(S(1)/4)*sqrt(pi)*erf(sqrt(6)*7**(S(1)/4)*t/2)/2 - exp(-3*sqrt(7)*t**2/2)/t
sol = [Eq(x(t), C3*t + t*(9*C1*I1 + 9*C2*I2)), Eq(y(t), C4*t + t*(3*sqrt(7)*C1*I1 - 3*sqrt(7)*C2*I2))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 21*x(t)), Eq(diff(y(t),t), 17*x(t)+3*y(t)), Eq(diff(z(t),t), 5*x(t)+7*y(t)+9*z(t)))
sol = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \
Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),3*y(t)-11*z(t)),Eq(diff(y(t),t),7*z(t)-3*x(t)),Eq(diff(z(t),t),11*x(t)-7*y(t)))
sol = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \
Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \
Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(3*diff(x(t),t),4*5*(y(t)-z(t))),Eq(4*diff(y(t),t),3*5*(z(t)-x(t))),Eq(5*diff(z(t),t),3*4*(x(t)-y(t))))
sol = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \
Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \
Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),4*x(t) - z(t)),Eq(diff(y(t),t),2*x(t)+2*y(t)-z(t)),Eq(diff(z(t),t),3*x(t)+y(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \
Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \
Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),4*x(t) - y(t) - 2*z(t)),Eq(diff(y(t),t),2*x(t) + y(t)- 2*z(t)),Eq(diff(z(t),t),5*x(t)-3*z(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), \
Eq(y(t), C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), Eq(z(t), C1*exp(2*t) + 5*C2*cos(t) + 5*C3*sin(t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol = [Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(-S(1)/4))), Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**(S(1)/4))), Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**(S(1)/4))), Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**(S(1)/4))), Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
sol = [Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**(S(1)/4))/3), Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**(S(1)/4))/3), Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**(S(1)/4))/3), Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**(S(1)/4))/3), Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2))
sol = set([Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)])
assert checksysodesol(eq, sol) == (True, [0, 0])
@slow
def test_nonlinear_3eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t, u = symbols('t u')
eq1 = (4*diff(x(t),t) + 2*y(t)*z(t), 3*diff(y(t),t) - z(t)*x(t), 5*diff(z(t),t) - x(t)*y(t))
sol1 = [Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)), (u, x(t))),
C3 - sqrt(15)*t/15), Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)),
(u, y(t))), C3 + sqrt(5)*t/10), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)*
sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*t/6)]
assert [i.dummy_eq(j) for i, j in zip(dsolve(eq1), sol1)]
# FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0, 0])
eq2 = (4*diff(x(t),t) + 2*y(t)*z(t)*sin(t), 3*diff(y(t),t) - z(t)*x(t)*sin(t), 5*diff(z(t),t) - x(t)*y(t)*sin(t))
sol2 = [Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)), (u, x(t))), C3 +
sqrt(5)*cos(t)/10), Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)),
(u, y(t))), C3 - sqrt(15)*cos(t)/15), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)*
sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*cos(t)/6)]
assert [i.dummy_eq(j) for i, j in zip(dsolve(eq2), sol2)]
# FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0, 0])
@slow
def test_checkodesol():
from sympy import Ei
# For the most part, checkodesol is well tested in the tests below.
# These tests only handle cases not checked below.
raises(ValueError, lambda: checkodesol(f(x, y).diff(x), Eq(f(x, y), x)))
raises(ValueError, lambda: checkodesol(f(x).diff(x), Eq(f(x, y),
x), f(x, y)))
assert checkodesol(f(x).diff(x), Eq(f(x, y), x)) == \
(False, -f(x).diff(x) + f(x, y).diff(x) - 1)
assert checkodesol(f(x).diff(x), Eq(f(x), x)) is not True
assert checkodesol(f(x).diff(x), Eq(f(x), x)) == (False, 1)
sol1 = Eq(f(x)**5 + 11*f(x) - 2*f(x) + x, 0)
assert checkodesol(diff(sol1.lhs, x), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x)*exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 2), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 2)*exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3)*exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3), Eq(f(x), x*log(x))) == \
(False, 60*x**4*((log(x) + 1)**2 + log(x))*(
log(x) + 1)*log(x)**2 - 5*x**4*log(x)**4 - 9)
assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x, 0)) == \
(True, 0)
assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x, 0),
solve_for_func=False) == (True, 0)
assert checkodesol(f(x).diff(x, 2), [Eq(f(x), C1 + C2*x),
Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)]) == \
[(True, 0), (True, 0), (False, C2)]
assert checkodesol(f(x).diff(x, 2), set([Eq(f(x), C1 + C2*x),
Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)])) == \
set([(True, 0), (True, 0), (False, C2)])
assert checkodesol(f(x).diff(x) - 1/f(x)/2, Eq(f(x)**2, x)) == \
[(True, 0), (True, 0)]
assert checkodesol(f(x).diff(x) - f(x), Eq(C1*exp(x), f(x))) == (True, 0)
# Based on test_1st_homogeneous_coeff_ode2_eq3sol. Make sure that
# checkodesol tries back substituting f(x) when it can.
eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x)
sol3 = Eq(f(x), log(log(C1/x)**(-x)))
assert not checkodesol(eq3, sol3)[1].has(f(x))
# This case was failing intermittently depending on hash-seed:
eqn = Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x))
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
@slow
def test_dsolve_options():
eq = x*f(x).diff(x) + f(x)
a = dsolve(eq, hint='all')
b = dsolve(eq, hint='all', simplify=False)
c = dsolve(eq, hint='all_Integral')
keys = ['1st_exact', '1st_exact_Integral', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear',
'1st_linear_Integral', 'almost_linear', 'almost_linear_Integral',
'best', 'best_hint', 'default', 'lie_group',
'nth_linear_euler_eq_homogeneous', 'order',
'separable', 'separable_Integral']
Integral_keys = ['1st_exact_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear_Integral',
'almost_linear_Integral', 'best', 'best_hint', 'default',
'nth_linear_euler_eq_homogeneous',
'order', 'separable_Integral']
assert sorted(a.keys()) == keys
assert a['order'] == ode_order(eq, f(x))
assert a['best'] == Eq(f(x), C1/x)
assert dsolve(eq, hint='best') == Eq(f(x), C1/x)
assert a['default'] == 'separable'
assert a['best_hint'] == 'separable'
assert not a['1st_exact'].has(Integral)
assert not a['separable'].has(Integral)
assert not a['1st_homogeneous_coeff_best'].has(Integral)
assert not a['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
assert not a['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
assert not a['1st_linear'].has(Integral)
assert a['1st_linear_Integral'].has(Integral)
assert a['1st_exact_Integral'].has(Integral)
assert a['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
assert a['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
assert a['separable_Integral'].has(Integral)
assert sorted(b.keys()) == keys
assert b['order'] == ode_order(eq, f(x))
assert b['best'] == Eq(f(x), C1/x)
assert dsolve(eq, hint='best', simplify=False) == Eq(f(x), C1/x)
assert b['default'] == 'separable'
assert b['best_hint'] == '1st_linear'
assert a['separable'] != b['separable']
assert a['1st_homogeneous_coeff_subs_dep_div_indep'] != \
b['1st_homogeneous_coeff_subs_dep_div_indep']
assert a['1st_homogeneous_coeff_subs_indep_div_dep'] != \
b['1st_homogeneous_coeff_subs_indep_div_dep']
assert not b['1st_exact'].has(Integral)
assert not b['separable'].has(Integral)
assert not b['1st_homogeneous_coeff_best'].has(Integral)
assert not b['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
assert not b['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
assert not b['1st_linear'].has(Integral)
assert b['1st_linear_Integral'].has(Integral)
assert b['1st_exact_Integral'].has(Integral)
assert b['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
assert b['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
assert b['separable_Integral'].has(Integral)
assert sorted(c.keys()) == Integral_keys
raises(ValueError, lambda: dsolve(eq, hint='notarealhint'))
raises(ValueError, lambda: dsolve(eq, hint='Liouville'))
assert dsolve(f(x).diff(x) - 1/f(x)**2, hint='all')['best'] == \
dsolve(f(x).diff(x) - 1/f(x)**2, hint='best')
assert dsolve(f(x) + f(x).diff(x) + sin(x).diff(x) + 1, f(x),
hint="1st_linear_Integral") == \
Eq(f(x), (C1 + Integral((-sin(x).diff(x) - 1)*
exp(Integral(1, x)), x))*exp(-Integral(1, x)))
def test_classify_ode():
assert classify_ode(f(x).diff(x, 2), f(x)) == \
(
'nth_algebraic',
'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'Liouville',
'2nd_power_series_ordinary',
'nth_algebraic_Integral',
'Liouville_Integral',
)
assert classify_ode(f(x), f(x)) == ()
assert classify_ode(Eq(f(x).diff(x), 0), f(x)) == (
'nth_algebraic',
'separable',
'1st_linear', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral',
'separable_Integral',
'1st_linear_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
assert classify_ode(f(x).diff(x)**2, f(x)) == (
'nth_algebraic',
'lie_group',
'nth_algebraic_Integral')
# issue 4749: f(x) should be cleared from highest derivative before classifying
a = classify_ode(Eq(f(x).diff(x) + f(x), x), f(x))
b = classify_ode(f(x).diff(x)*f(x) + f(x)*f(x) - x*f(x), f(x))
c = classify_ode(f(x).diff(x)/f(x) + f(x)/f(x) - x/f(x), f(x))
assert a == ('1st_linear',
'Bernoulli',
'almost_linear',
'1st_power_series', "lie_group",
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'1st_linear_Integral',
'Bernoulli_Integral',
'almost_linear_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
assert b == c != ()
assert classify_ode(
2*x*f(x)*f(x).diff(x) + (1 + x)*f(x)**2 - exp(x), f(x)
) == ('Bernoulli', 'almost_linear', 'lie_group',
'Bernoulli_Integral', 'almost_linear_Integral')
assert 'Riccati_special_minus2' in \
classify_ode(2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2), f(x))
raises(ValueError, lambda: classify_ode(x + f(x, y).diff(x).diff(
y), f(x, y)))
# issue 5176
k = Symbol('k')
assert classify_ode(f(x).diff(x)/(k*f(x) + k*x*f(x)) + 2*f(x)/(k*f(x) +
k*x*f(x)) + x*f(x).diff(x)/(k*f(x) + k*x*f(x)) + z, f(x)) == \
('separable', '1st_exact', '1st_power_series', 'lie_group',
'separable_Integral', '1st_exact_Integral')
# preprocessing
ans = ('nth_algebraic', 'separable', '1st_exact', '1st_linear', 'Bernoulli',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
'nth_algebraic_Integral',
'separable_Integral', '1st_exact_Integral',
'1st_linear_Integral',
'Bernoulli_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral')
# w/o f(x) given
assert classify_ode(diff(f(x) + x, x) + diff(f(x), x)) == ans
# w/ f(x) and prep=True
assert classify_ode(diff(f(x) + x, x) + diff(f(x), x), f(x),
prep=True) == ans
assert classify_ode(Eq(2*x**3*f(x).diff(x), 0), f(x)) == \
('nth_algebraic', 'separable', '1st_linear', '1st_power_series',
'lie_group', 'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral', 'separable_Integral',
'1st_linear_Integral')
assert classify_ode(Eq(2*f(x)**3*f(x).diff(x), 0), f(x)) == \
('nth_algebraic', 'separable', '1st_power_series', 'lie_group',
'nth_algebraic_Integral', 'separable_Integral')
# test issue 13864
assert classify_ode(Eq(diff(f(x), x) - f(x)**x, 0), f(x)) == \
('1st_power_series', 'lie_group')
assert isinstance(classify_ode(Eq(f(x), 5), f(x), dict=True), dict)
def test_classify_ode_ics():
# Dummy
eq = f(x).diff(x, x) - f(x)
# Not f(0) or f'(0)
ics = {x: 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
############################
# f(0) type (AppliedUndef) #
############################
# Wrong function
ics = {g(0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(0, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(0): f(1)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(0): 1}
classify_ode(eq, f(x), ics=ics)
#####################
# f'(0) type (Subs) #
#####################
# Wrong function
ics = {g(x).diff(x).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(y).diff(y).subs(y, x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Wrong variable
ics = {f(y).diff(y).subs(y, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(x, y).diff(x).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Derivative wrt wrong vars
ics = {Derivative(f(x), x, y).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(x).diff(x).subs(x, 0): f(0)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(x).diff(x).subs(x, 0): 1}
classify_ode(eq, f(x), ics=ics)
###########################
# f'(y) type (Derivative) #
###########################
# Wrong function
ics = {g(x).diff(x).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(y).diff(y).subs(y, x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(x, y).diff(x).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Derivative wrt wrong vars
ics = {Derivative(f(x), x, z).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(x).diff(x).subs(x, y): f(0)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(x).diff(x).subs(x, y): 1}
classify_ode(eq, f(x), ics=ics)
def test_classify_sysode():
# Here x is assumed to be x(t) and y as y(t) for simplicity.
# Similarly diff(x,t) and diff(y,y) is assumed to be x1 and y1 respectively.
k, l, m, n = symbols('k, l, m, n', Integer=True)
k1, k2, k3, l1, l2, l3, m1, m2, m3 = symbols('k1, k2, k3, l1, l2, l3, m1, m2, m3', Integer=True)
P, Q, R, p, q, r = symbols('P, Q, R, p, q, r', cls=Function)
P1, P2, P3, Q1, Q2, R1, R2 = symbols('P1, P2, P3, Q1, Q2, R1, R2', cls=Function)
x, y, z = symbols('x, y, z', cls=Function)
t = symbols('t')
x1 = diff(x(t),t) ; y1 = diff(y(t),t) ; z1 = diff(z(t),t)
x2 = diff(x(t),t,t) ; y2 = diff(y(t),t,t) ; z2 = diff(z(t),t,t)
eq1 = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol1 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -5*t, (1, x(t), 1): 0, (0, x(t), 1): 1, \
(1, y(t), 0): -5*t, (1, x(t), 0): -2, (0, y(t), 1): 0, (0, y(t), 0): -2, (1, y(t), 1): 1}, \
'type_of_equation': 'type3', 'func': [x(t), y(t)], 'is_linear': True, 'eq': [-5*t*x(t) - 2*y(t) + \
Derivative(x(t), t), -5*t*y(t) - 2*x(t) + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq1) == sol1
eq2 = (Eq(x2, k*x(t) - l*y1), Eq(y2, l*x1 + k*y(t)))
sol2 = {'order': {y(t): 2, x(t): 2}, 'type_of_equation': 'type3', 'is_linear': True, 'eq': \
[-k*x(t) + l*Derivative(y(t), t) + Derivative(x(t), t, t), -k*y(t) - l*Derivative(x(t), t) + \
Derivative(y(t), t, t)], 'no_of_equation': 2, 'func_coeff': {(0, y(t), 0): 0, (0, x(t), 2): 1, \
(1, y(t), 1): 0, (1, y(t), 2): 1, (1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -k, (1, x(t), 1): \
-l, (0, x(t), 1): 0, (0, y(t), 1): l, (1, x(t), 0): 0, (1, y(t), 0): -k}, 'func': [x(t), y(t)]}
assert classify_sysode(eq2) == sol2
eq3 = (Eq(x2+4*x1+3*y1+9*x(t)+7*y(t), 11*exp(I*t)), Eq(y2+5*x1+8*y1+3*x(t)+12*y(t), 2*exp(I*t)))
sol3 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): 9, \
(1, x(t), 1): 5, (0, x(t), 1): 4, (0, y(t), 1): 3, (1, x(t), 0): 3, (1, y(t), 0): 12, (0, y(t), 0): 7, \
(0, x(t), 2): 1, (1, y(t), 2): 1, (1, y(t), 1): 8}, 'type_of_equation': 'type4', 'func': [x(t), y(t)], \
'is_linear': True, 'eq': [9*x(t) + 7*y(t) - 11*exp(I*t) + 4*Derivative(x(t), t) + 3*Derivative(y(t), t) + \
Derivative(x(t), t, t), 3*x(t) + 12*y(t) - 2*exp(I*t) + 5*Derivative(x(t), t) + 8*Derivative(y(t), t) + \
Derivative(y(t), t, t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq3) == sol3
eq4 = (Eq((4*t**2 + 7*t + 1)**2*x2, 5*x(t) + 35*y(t)), Eq((4*t**2 + 7*t + 1)**2*y2, x(t) + 9*y(t)))
sol4 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -5, \
(1, x(t), 1): 0, (0, x(t), 1): 0, (0, y(t), 1): 0, (1, x(t), 0): -1, (1, y(t), 0): -9, (0, y(t), 0): -35, \
(0, x(t), 2): 16*t**4 + 56*t**3 + 57*t**2 + 14*t + 1, (1, y(t), 2): 16*t**4 + 56*t**3 + 57*t**2 + 14*t + 1, \
(1, y(t), 1): 0}, 'type_of_equation': 'type10', 'func': [x(t), y(t)], 'is_linear': True, \
'eq': [(4*t**2 + 7*t + 1)**2*Derivative(x(t), t, t) - 5*x(t) - 35*y(t), (4*t**2 + 7*t + 1)**2*Derivative(y(t), t, t)\
- x(t) - 9*y(t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq4) == sol4
eq5 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol5 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -1, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): -5, \
(1, x(t), 0): -2, (0, y(t), 1): 0, (0, y(t), 0): -1, (1, y(t), 1): 1}, 'type_of_equation': 'type2', \
'func': [x(t), y(t)], 'is_linear': True, 'eq': [-x(t) - y(t) + Derivative(x(t), t) - 9, -2*x(t) - 5*y(t) + \
Derivative(y(t), t) - 23], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq5) == sol5
eq6 = (Eq(x1, exp(k*x(t))*P(x(t),y(t))), Eq(y1,r(y(t))*P(x(t),y(t))))
sol6 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \
(1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': 'type2', 'func': \
[x(t), y(t)], 'is_linear': False, 'eq': [-P(x(t), y(t))*exp(k*x(t)) + Derivative(x(t), t), -P(x(t), \
y(t))*r(y(t)) + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq6) == sol6
eq7 = (Eq(x1, x(t)**2+y(t)/x(t)), Eq(y1, x(t)/y(t)))
sol7 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \
(1, x(t), 0): -1/y(t), (0, y(t), 1): 0, (0, y(t), 0): -1/x(t), (1, y(t), 1): 1}, 'type_of_equation': 'type3', \
'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)**2 + Derivative(x(t), t) - y(t)/x(t), -x(t)/y(t) + \
Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq7) == sol7
eq8 = (Eq(x1, P1(x(t))*Q1(y(t))*R(x(t),y(t),t)), Eq(y1, P1(x(t))*Q1(y(t))*R(x(t),y(t),t)))
sol8 = {'func': [x(t), y(t)], 'is_linear': False, 'type_of_equation': 'type4', 'eq': \
[-P1(x(t))*Q1(y(t))*R(x(t), y(t), t) + Derivative(x(t), t), -P1(x(t))*Q1(y(t))*R(x(t), y(t), t) + \
Derivative(y(t), t)], 'func_coeff': {(0, y(t), 1): 0, (1, y(t), 1): 1, (1, x(t), 1): 0, (0, y(t), 0): 0, \
(1, x(t), 0): 0, (0, x(t), 0): 0, (1, y(t), 0): 0, (0, x(t), 1): 1}, 'order': {y(t): 1, x(t): 1}, 'no_of_equation': 2}
assert classify_sysode(eq8) == sol8
eq9 = (Eq(x1,3*y(t)-11*z(t)),Eq(y1,7*z(t)-3*x(t)),Eq(z1,11*x(t)-7*y(t)))
sol9 = {'no_of_equation': 3, 'func_coeff': {(1, y(t), 0): 0, (2, y(t), 1): 0, (2, z(t), 1): 1, \
(0, x(t), 0): 0, (2, x(t), 1): 0, (1, x(t), 1): 0, (2, y(t), 0): 7, (0, x(t), 1): 1, (1, z(t), 1): 0, \
(0, y(t), 1): 0, (1, x(t), 0): 3, (0, z(t), 0): 11, (0, y(t), 0): -3, (1, z(t), 0): -7, (0, z(t), 1): 0, \
(2, x(t), 0): -11, (2, z(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': 'type2', 'func': [x(t), y(t), z(t)], \
'is_linear': True, 'eq': [-3*y(t) + 11*z(t) + Derivative(x(t), t), 3*x(t) - 7*z(t) + Derivative(y(t), t), \
-11*x(t) + 7*y(t) + Derivative(z(t), t)], 'order': {z(t): 1, y(t): 1, x(t): 1}}
assert classify_sysode(eq9) == sol9
eq10 = (x2 + log(t)*(t*x1 - x(t)) + exp(t)*(t*y1 - y(t)), y2 + (t**2)*(t*x1 - x(t)) + (t)*(t*y1 - y(t)))
sol10 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -log(t), \
(1, x(t), 1): t**3, (0, x(t), 1): t*log(t), (0, y(t), 1): t*exp(t), (1, x(t), 0): -t**2, (1, y(t), 0): -t, \
(0, y(t), 0): -exp(t), (0, x(t), 2): 1, (1, y(t), 2): 1, (1, y(t), 1): t**2}, 'type_of_equation': 'type11', \
'func': [x(t), y(t)], 'is_linear': True, 'eq': [(t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) - \
y(t))*exp(t) + Derivative(x(t), t, t), t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t), t) - y(t)) \
+ Derivative(y(t), t, t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq10) == sol10
eq11 = (Eq(x1,x(t)*y(t)**3), Eq(y1,y(t)**5))
sol11 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -y(t)**3, (1, x(t), 1): 0, (0, x(t), 1): 1, \
(1, y(t), 0): 0, (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': \
'type1', 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)*y(t)**3 + Derivative(x(t), t), \
-y(t)**5 + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq11) == sol11
eq12 = (Eq(x1, y(t)), Eq(y1, x(t)))
sol12 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \
(1, x(t), 0): -1, (0, y(t), 1): 0, (0, y(t), 0): -1, (1, y(t), 1): 1}, 'type_of_equation': 'type1', 'func': \
[x(t), y(t)], 'is_linear': True, 'eq': [-y(t) + Derivative(x(t), t), -x(t) + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq12) == sol12
eq13 = (Eq(x1,x(t)*y(t)*sin(t)**2), Eq(y1,y(t)**2*sin(t)**2))
sol13 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -y(t)*sin(t)**2, (1, x(t), 1): 0, (0, x(t), 1): 1, \
(1, y(t), 0): 0, (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): -x(t)*sin(t)**2, (1, y(t), 1): 1}, \
'type_of_equation': 'type4', 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)*y(t)*sin(t)**2 + \
Derivative(x(t), t), -y(t)**2*sin(t)**2 + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq13) == sol13
eq14 = (Eq(x1, 21*x(t)), Eq(y1, 17*x(t)+3*y(t)), Eq(z1, 5*x(t)+7*y(t)+9*z(t)))
sol14 = {'no_of_equation': 3, 'func_coeff': {(1, y(t), 0): -3, (2, y(t), 1): 0, (2, z(t), 1): 1, \
(0, x(t), 0): -21, (2, x(t), 1): 0, (1, x(t), 1): 0, (2, y(t), 0): -7, (0, x(t), 1): 1, (1, z(t), 1): 0, \
(0, y(t), 1): 0, (1, x(t), 0): -17, (0, z(t), 0): 0, (0, y(t), 0): 0, (1, z(t), 0): 0, (0, z(t), 1): 0, \
(2, x(t), 0): -5, (2, z(t), 0): -9, (1, y(t), 1): 1}, 'type_of_equation': 'type1', 'func': [x(t), y(t), z(t)], \
'is_linear': True, 'eq': [-21*x(t) + Derivative(x(t), t), -17*x(t) - 3*y(t) + Derivative(y(t), t), -5*x(t) - \
7*y(t) - 9*z(t) + Derivative(z(t), t)], 'order': {z(t): 1, y(t): 1, x(t): 1}}
assert classify_sysode(eq14) == sol14
eq15 = (Eq(x1,4*x(t)+5*y(t)+2*z(t)),Eq(y1,x(t)+13*y(t)+9*z(t)),Eq(z1,32*x(t)+41*y(t)+11*z(t)))
sol15 = {'no_of_equation': 3, 'func_coeff': {(1, y(t), 0): -13, (2, y(t), 1): 0, (2, z(t), 1): 1, \
(0, x(t), 0): -4, (2, x(t), 1): 0, (1, x(t), 1): 0, (2, y(t), 0): -41, (0, x(t), 1): 1, (1, z(t), 1): 0, \
(0, y(t), 1): 0, (1, x(t), 0): -1, (0, z(t), 0): -2, (0, y(t), 0): -5, (1, z(t), 0): -9, (0, z(t), 1): 0, \
(2, x(t), 0): -32, (2, z(t), 0): -11, (1, y(t), 1): 1}, 'type_of_equation': 'type6', 'func': \
[x(t), y(t), z(t)], 'is_linear': True, 'eq': [-4*x(t) - 5*y(t) - 2*z(t) + Derivative(x(t), t), -x(t) - 13*y(t) - \
9*z(t) + Derivative(y(t), t), -32*x(t) - 41*y(t) - 11*z(t) + Derivative(z(t), t)], 'order': {z(t): 1, y(t): 1, x(t): 1}}
assert classify_sysode(eq15) == sol15
eq16 = (Eq(3*x1,4*5*(y(t)-z(t))),Eq(4*y1,3*5*(z(t)-x(t))),Eq(5*z1,3*4*(x(t)-y(t))))
sol16 = {'no_of_equation': 3, 'func_coeff': {(1, y(t), 0): 0, (2, y(t), 1): 0, (2, z(t), 1): 5, \
(0, x(t), 0): 0, (2, x(t), 1): 0, (1, x(t), 1): 0, (2, y(t), 0): 12, (0, x(t), 1): 3, (1, z(t), 1): 0, \
(0, y(t), 1): 0, (1, x(t), 0): 15, (0, z(t), 0): 20, (0, y(t), 0): -20, (1, z(t), 0): -15, (0, z(t), 1): 0, \
(2, x(t), 0): -12, (2, z(t), 0): 0, (1, y(t), 1): 4}, 'type_of_equation': 'type3', 'func': [x(t), y(t), z(t)], \
'is_linear': True, 'eq': [-20*y(t) + 20*z(t) + 3*Derivative(x(t), t), 15*x(t) - 15*z(t) + 4*Derivative(y(t), t), \
-12*x(t) + 12*y(t) + 5*Derivative(z(t), t)], 'order': {z(t): 1, y(t): 1, x(t): 1}}
assert classify_sysode(eq16) == sol16
# issue 8193: funcs parameter for classify_sysode has to actually work
assert classify_sysode(eq1, funcs=[x(t), y(t)]) == sol1
def test_solve_ics():
# Basic tests that things work from dsolve.
assert dsolve(f(x).diff(x) - 1/f(x), f(x), ics={f(1): 2}) == \
Eq(f(x), sqrt(2 * x + 2))
assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(0): 1}) == Eq(f(x), exp(x))
assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(x).diff(x).subs(x, 0): 1}) == Eq(f(x), exp(x))
assert dsolve(f(x).diff(x, x) + f(x), f(x), ics={f(0): 1,
f(x).diff(x).subs(x, 0): 1}) == Eq(f(x), sin(x) + cos(x))
assert dsolve([f(x).diff(x) - f(x) + g(x), g(x).diff(x) - g(x) - f(x)],
[f(x), g(x)], ics={f(0): 1, g(0): 0}) == [Eq(f(x), exp(x)*cos(x)),
Eq(g(x), exp(x)*sin(x))]
# Test cases where dsolve returns two solutions.
eq = (x**2*f(x)**2 - x).diff(x)
assert dsolve(eq, f(x), ics={f(1): 0}) == [Eq(f(x),
-sqrt(x - 1)/x), Eq(f(x), sqrt(x - 1)/x)]
assert dsolve(eq, f(x), ics={f(x).diff(x).subs(x, 1): 0}) == [Eq(f(x),
-sqrt(x - S(1)/2)/x), Eq(f(x), sqrt(x - S(1)/2)/x)]
eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
assert dsolve(eq, f(x),
ics={f(0):1}, hint='1st_exact', simplify=False) == Eq(x*cos(f(x)) + f(x)**3/3, S(1)/3)
assert dsolve(eq, f(x),
ics={f(0):1}, hint='1st_exact', simplify=True) == Eq(x*cos(f(x)) + f(x)**3/3, S(1)/3)
assert solve_ics([Eq(f(x), C1*exp(x))], [f(x)], [C1], {f(0): 1}) == {C1: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2],
{f(0): 1, f(pi/2): 1}) == {C1: 1, C2: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2],
{f(0): 1, f(x).diff(x).subs(x, 0): 1}) == {C1: 1, C2: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1}) == \
{C2: 1}
# Some more complicated tests Refer to PR #16098
assert dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x, 1):0}) == \
[Eq(f(x), 0), Eq(f(x), x ** 3 / 6 - x / 2)]
assert dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0}) == \
[Eq(f(x), 0), Eq(f(x), C2*x + x**3/6)]
K, r, f0 = symbols('K r f0')
sol = Eq(f(x), -K*f0*exp(r*x)/((K - f0)*(-f0*exp(r*x)/(K - f0) - 1)))
assert (dsolve(Eq(f(x).diff(x), r * f(x) * (1 - f(x) / K)), f(x), ics={f(0): f0})) == sol
#Order dependent issues Refer to PR #16098
assert dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(x).diff(x).subs(x,0):0, f(0):0}) == \
[Eq(f(x), 0), Eq(f(x), x ** 3 / 6)]
assert dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x,0):0}) == \
[Eq(f(x), 0), Eq(f(x), x ** 3 / 6)]
# XXX: Ought to be ValueError
raises(ValueError, lambda: solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1, f(pi): 1}))
# Degenerate case. f'(0) is identically 0.
raises(ValueError, lambda: solve_ics([Eq(f(x), sqrt(C1 - x**2))], [f(x)], [C1], {f(x).diff(x).subs(x, 0): 0}))
EI, q, L = symbols('EI q L')
# eq = Eq(EI*diff(f(x), x, 4), q)
sols = [Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3 + q*x**4/(24*EI))]
funcs = [f(x)]
constants = [C1, C2, C3, C4]
# Test both cases, Derivative (the default from f(x).diff(x).subs(x, L)),
# and Subs
ics1 = {f(0): 0,
f(x).diff(x).subs(x, 0): 0,
f(L).diff(L, 2): 0,
f(L).diff(L, 3): 0}
ics2 = {f(0): 0,
f(x).diff(x).subs(x, 0): 0,
Subs(f(x).diff(x, 2), x, L): 0,
Subs(f(x).diff(x, 3), x, L): 0}
solved_constants1 = solve_ics(sols, funcs, constants, ics1)
solved_constants2 = solve_ics(sols, funcs, constants, ics2)
assert solved_constants1 == solved_constants2 == {
C1: 0,
C2: 0,
C3: L**2*q/(4*EI),
C4: -L*q/(6*EI)}
def test_ode_order():
f = Function('f')
g = Function('g')
x = Symbol('x')
assert ode_order(3*x*exp(f(x)), f(x)) == 0
assert ode_order(x*diff(f(x), x) + 3*x*f(x) - sin(x)/x, f(x)) == 1
assert ode_order(x**2*f(x).diff(x, x) + x*diff(f(x), x) - f(x), f(x)) == 2
assert ode_order(diff(x*exp(f(x)), x, x), f(x)) == 2
assert ode_order(diff(x*diff(x*exp(f(x)), x, x), x), f(x)) == 3
assert ode_order(diff(f(x), x, x), g(x)) == 0
assert ode_order(diff(f(x), x, x)*diff(g(x), x), f(x)) == 2
assert ode_order(diff(f(x), x, x)*diff(g(x), x), g(x)) == 1
assert ode_order(diff(x*diff(x*exp(f(x)), x, x), x), g(x)) == 0
# issue 5835: ode_order has to also work for unevaluated derivatives
# (ie, without using doit()).
assert ode_order(Derivative(x*f(x), x), f(x)) == 1
assert ode_order(x*sin(Derivative(x*f(x)**2, x, x)), f(x)) == 2
assert ode_order(Derivative(x*Derivative(x*exp(f(x)), x, x), x), g(x)) == 0
assert ode_order(Derivative(f(x), x, x), g(x)) == 0
assert ode_order(Derivative(x*exp(f(x)), x, x), f(x)) == 2
assert ode_order(Derivative(f(x), x, x)*Derivative(g(x), x), g(x)) == 1
assert ode_order(Derivative(x*Derivative(f(x), x, x), x), f(x)) == 3
assert ode_order(
x*sin(Derivative(x*Derivative(f(x), x)**2, x, x)), f(x)) == 3
# In all tests below, checkodesol has the order option set to prevent
# superfluous calls to ode_order(), and the solve_for_func flag set to False
# because dsolve() already tries to solve for the function, unless the
# simplify=False option is set.
def test_old_ode_tests():
# These are simple tests from the old ode module
eq1 = Eq(f(x).diff(x), 0)
eq2 = Eq(3*f(x).diff(x) - 5, 0)
eq3 = Eq(3*f(x).diff(x), 5)
eq4 = Eq(9*f(x).diff(x, x) + f(x), 0)
eq5 = Eq(9*f(x).diff(x, x), f(x))
# Type: a(x)f'(x)+b(x)*f(x)+c(x)=0
eq6 = Eq(x**2*f(x).diff(x) + 3*x*f(x) - sin(x)/x, 0)
eq7 = Eq(f(x).diff(x, x) - 3*diff(f(x), x) + 2*f(x), 0)
# Type: 2nd order, constant coefficients (two real different roots)
eq8 = Eq(f(x).diff(x, x) - 4*diff(f(x), x) + 4*f(x), 0)
# Type: 2nd order, constant coefficients (two real equal roots)
eq9 = Eq(f(x).diff(x, x) + 2*diff(f(x), x) + 3*f(x), 0)
# Type: 2nd order, constant coefficients (two complex roots)
eq10 = Eq(3*f(x).diff(x) - 1, 0)
eq11 = Eq(x*f(x).diff(x) - 1, 0)
sol1 = Eq(f(x), C1)
sol2 = Eq(f(x), C1 + 5*x/3)
sol3 = Eq(f(x), C1 + 5*x/3)
sol4 = Eq(f(x), C1*sin(x/3) + C2*cos(x/3))
sol5 = Eq(f(x), C1*exp(-x/3) + C2*exp(x/3))
sol6 = Eq(f(x), (C1 - cos(x))/x**3)
sol7 = Eq(f(x), (C1 + C2*exp(x))*exp(x))
sol8 = Eq(f(x), (C1 + C2*x)*exp(2*x))
sol9 = Eq(f(x), (C1*sin(x*sqrt(2)) + C2*cos(x*sqrt(2)))*exp(-x))
sol10 = Eq(f(x), C1 + x/3)
sol11 = Eq(f(x), C1 + log(x))
assert dsolve(eq1) == sol1
assert dsolve(eq1.lhs) == sol1
assert dsolve(eq2) == sol2
assert dsolve(eq3) == sol3
assert dsolve(eq4) == sol4
assert dsolve(eq5) == sol5
assert dsolve(eq6) == sol6
assert dsolve(eq7) == sol7
assert dsolve(eq8) == sol8
assert dsolve(eq9) == sol9
assert dsolve(eq10) == sol10
assert dsolve(eq11) == sol11
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=2, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=1, solve_for_func=False)[0]
assert checkodesol(eq11, sol11, order=1, solve_for_func=False)[0]
def test_1st_linear():
# Type: first order linear form f'(x)+p(x)f(x)=q(x)
eq = Eq(f(x).diff(x) + x*f(x), x**2)
sol = Eq(f(x), (C1 + x*exp(x**2/2)
- sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2)/2)*exp(-x**2/2))
assert dsolve(eq, hint='1st_linear') == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_Bernoulli():
# Type: Bernoulli, f'(x) + p(x)*f(x) == q(x)*f(x)**n
eq = Eq(x*f(x).diff(x) + f(x) - f(x)**2, 0)
sol = dsolve(eq, f(x), hint='Bernoulli')
assert sol == Eq(f(x), 1/(x*(C1 + 1/x)))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_Riccati_special_minus2():
# Type: Riccati special alpha = -2, a*dy/dx + b*y**2 + c*y/x +d/x**2
eq = 2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2)
sol = dsolve(eq, f(x), hint='Riccati_special_minus2')
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
@slow
def test_1st_exact1():
# Type: Exact differential equation, p(x,f) + q(x,f)*f' == 0,
# where dp/df == dq/dx
eq1 = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x)
eq2 = (2*x*f(x) + 1)/f(x) + (f(x) - x)/f(x)**2*f(x).diff(x)
eq3 = 2*x + f(x)*cos(x) + (2*f(x) + sin(x) - sin(f(x)))*f(x).diff(x)
eq4 = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
eq5 = 2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x)
sol1 = [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
sol2 = Eq(f(x), exp(C1 - x**2 + LambertW(-x*exp(-C1 + x**2))))
sol2b = Eq(log(f(x)) + x/f(x) + x**2, C1)
sol3 = Eq(f(x)*sin(x) + cos(f(x)) + x**2 + f(x)**2, C1)
sol4 = Eq(x*cos(f(x)) + f(x)**3/3, C1)
sol5 = Eq(x**2*f(x) + f(x)**3/3, C1)
assert dsolve(eq1, f(x), hint='1st_exact') == sol1
assert dsolve(eq2, f(x), hint='1st_exact') == sol2
assert dsolve(eq3, f(x), hint='1st_exact') == sol3
assert dsolve(eq4, hint='1st_exact') == sol4
assert dsolve(eq5, hint='1st_exact', simplify=False) == sol5
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
# issue 5080 blocks the testing of this solution
# FIXME: assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2b, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=1, solve_for_func=False)[0]
@slow
@XFAIL
def test_1st_exact2():
"""
This is an exact equation that fails under the exact engine. It is caught
by first order homogeneous albeit with a much contorted solution. The
exact engine fails because of a poorly simplified integral of q(0,y)dy,
where q is the function multiplying f'. The solutions should be
Eq(sqrt(x**2+f(x)**2)**3+y**3, C1). The equation below is
equivalent, but it is so complex that checkodesol fails, and takes a long
time to do so.
"""
if ON_TRAVIS:
skip("Too slow for travis.")
eq = (x*sqrt(x**2 + f(x)**2) - (x**2*f(x)/(f(x) -
sqrt(x**2 + f(x)**2)))*f(x).diff(x))
sol = dsolve(eq)
assert sol == Eq(log(x),
C1 - 9*sqrt(1 + f(x)**2/x**2)*asinh(f(x)/x)/(-27*f(x)/x +
27*sqrt(1 + f(x)**2/x**2)) - 9*sqrt(1 + f(x)**2/x**2)*
log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2)) +
9*asinh(f(x)/x)*f(x)/(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))) +
9*f(x)*log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_separable1():
# test_separable1-5 are from Ordinary Differential Equations, Tenenbaum and
# Pollard, pg. 55
eq1 = f(x).diff(x) - f(x)
eq2 = x*f(x).diff(x) - f(x)
eq3 = f(x).diff(x) + sin(x)
eq4 = f(x)**2 + 1 - (x**2 + 1)*f(x).diff(x)
eq5 = f(x).diff(x)/tan(x) - f(x) - 2
eq6 = f(x).diff(x) * (1 - sin(f(x))) - 1
sol1 = Eq(f(x), C1*exp(x))
sol2 = Eq(f(x), C1*x)
sol3 = Eq(f(x), C1 + cos(x))
sol4 = Eq(f(x), tan(C1 + atan(x)))
sol5 = Eq(f(x), C1/cos(x) - 2)
sol6 = Eq(-x + f(x) + cos(f(x)), C1)
assert dsolve(eq1, hint='separable') == sol1
assert dsolve(eq2, hint='separable') == sol2
assert dsolve(eq3, hint='separable') == sol3
assert dsolve(eq4, hint='separable') == sol4
assert dsolve(eq5, hint='separable') == sol5
assert dsolve(eq6, hint='separable') == sol6
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=1, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
@slow
def test_separable2():
a = Symbol('a')
eq6 = f(x)*x**2*f(x).diff(x) - f(x)**3 - 2*x**2*f(x).diff(x)
eq7 = f(x)**2 - 1 - (2*f(x) + x*f(x))*f(x).diff(x)
eq8 = x*log(x)*f(x).diff(x) + sqrt(1 + f(x)**2)
eq9 = exp(x + 1)*tan(f(x)) + cos(f(x))*f(x).diff(x)
eq10 = (x*cos(f(x)) + x**2*sin(f(x))*f(x).diff(x) -
a**2*sin(f(x))*f(x).diff(x))
sol6 = Eq(Integral((u - 2)/u**3, (u, f(x))),
C1 + Integral(x**(-2), x))
sol7 = Eq(-log(-1 + f(x)**2)/2, C1 - log(2 + x))
sol8 = Eq(asinh(f(x)), C1 - log(log(x)))
# integrate cannot handle the integral on the lhs (cos/tan)
sol9 = Eq(Integral(cos(u)/tan(u), (u, f(x))),
C1 + Integral(-exp(1)*exp(x), x))
sol10 = Eq(-log(cos(f(x))), C1 - log(- a**2 + x**2)/2)
assert dsolve(eq6, hint='separable_Integral').dummy_eq(sol6)
assert dsolve(eq7, hint='separable', simplify=False) == sol7
assert dsolve(eq8, hint='separable', simplify=False) == sol8
assert dsolve(eq9, hint='separable_Integral').dummy_eq(sol9)
assert dsolve(eq10, hint='separable', simplify=False) == sol10
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=1, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=1, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=1, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=1, solve_for_func=False)[0]
def test_separable3():
eq11 = f(x).diff(x) - f(x)*tan(x)
eq12 = (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x))
eq13 = f(x).diff(x) - f(x)*log(f(x))/tan(x)
sol11 = Eq(f(x), C1/cos(x))
sol12 = Eq(log(sin(f(x))), C1 + 2*x + 2*log(x - 1))
sol13 = Eq(log(log(f(x))), C1 + log(sin(x)))
assert dsolve(eq11, hint='separable') == sol11
assert dsolve(eq12, hint='separable', simplify=False) == sol12
assert dsolve(eq13, hint='separable', simplify=False) == sol13
assert checkodesol(eq11, sol11, order=1, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=1, solve_for_func=False)[0]
assert checkodesol(eq13, sol13, order=1, solve_for_func=False)[0]
def test_separable4():
# This has a slow integral (1/((1 + y**2)*atan(y))), so we isolate it.
eq14 = x*f(x).diff(x) + (1 + f(x)**2)*atan(f(x))
sol14 = Eq(log(atan(f(x))), C1 - log(x))
assert dsolve(eq14, hint='separable', simplify=False) == sol14
assert checkodesol(eq14, sol14, order=1, solve_for_func=False)[0]
def test_separable5():
eq15 = f(x).diff(x) + x*(f(x) + 1)
eq16 = exp(f(x)**2)*(x**2 + 2*x + 1) + (x*f(x) + f(x))*f(x).diff(x)
eq17 = f(x).diff(x) + f(x)
eq18 = sin(x)*cos(2*f(x)) + cos(x)*sin(2*f(x))*f(x).diff(x)
eq19 = (1 - x)*f(x).diff(x) - x*(f(x) + 1)
eq20 = f(x)*diff(f(x), x) + x - 3*x*f(x)**2
eq21 = f(x).diff(x) - exp(x + f(x))
sol15 = Eq(f(x), -1 + C1*exp(-x**2/2))
sol16 = Eq(-exp(-f(x)**2)/2, C1 - x - x**2/2)
sol17 = Eq(f(x), C1*exp(-x))
sol18 = Eq(-log(cos(2*f(x)))/2, C1 + log(cos(x)))
sol19 = Eq(f(x), (C1*exp(-x) - x + 1)/(x - 1))
sol20 = Eq(log(-1 + 3*f(x)**2)/6, C1 + x**2/2)
sol21 = Eq(-exp(-f(x)), C1 + exp(x))
assert dsolve(eq15, hint='separable') == sol15
assert dsolve(eq16, hint='separable', simplify=False) == sol16
assert dsolve(eq17, hint='separable') == sol17
assert dsolve(eq18, hint='separable', simplify=False) == sol18
assert dsolve(eq19, hint='separable') == sol19
assert dsolve(eq20, hint='separable', simplify=False) == sol20
assert dsolve(eq21, hint='separable', simplify=False) == sol21
assert checkodesol(eq15, sol15, order=1, solve_for_func=False)[0]
assert checkodesol(eq16, sol16, order=1, solve_for_func=False)[0]
assert checkodesol(eq17, sol17, order=1, solve_for_func=False)[0]
assert checkodesol(eq18, sol18, order=1, solve_for_func=False)[0]
assert checkodesol(eq19, sol19, order=1, solve_for_func=False)[0]
assert checkodesol(eq20, sol20, order=1, solve_for_func=False)[0]
assert checkodesol(eq21, sol21, order=1, solve_for_func=False)[0]
def test_separable_1_5_checkodesol():
eq12 = (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x))
sol12 = Eq(-log(1 - cos(f(x))**2)/2, C1 - 2*x - 2*log(1 - x))
assert checkodesol(eq12, sol12, order=1, solve_for_func=False)[0]
def test_homogeneous_order():
assert homogeneous_order(exp(y/x) + tan(y/x), x, y) == 0
assert homogeneous_order(x**2 + sin(x)*cos(y), x, y) is None
assert homogeneous_order(x - y - x*sin(y/x), x, y) == 1
assert homogeneous_order((x*y + sqrt(x**4 + y**4) + x**2*(log(x) - log(y)))/
(pi*x**Rational(2, 3)*sqrt(y)**3), x, y) == Rational(-1, 6)
assert homogeneous_order(y/x*cos(y/x) - x/y*sin(y/x) + cos(y/x), x, y) == 0
assert homogeneous_order(f(x), x, f(x)) == 1
assert homogeneous_order(f(x)**2, x, f(x)) == 2
assert homogeneous_order(x*y*z, x, y) == 2
assert homogeneous_order(x*y*z, x, y, z) == 3
assert homogeneous_order(x**2*f(x)/sqrt(x**2 + f(x)**2), f(x)) is None
assert homogeneous_order(f(x, y)**2, x, f(x, y), y) == 2
assert homogeneous_order(f(x, y)**2, x, f(x), y) is None
assert homogeneous_order(f(x, y)**2, x, f(x, y)) is None
assert homogeneous_order(f(y, x)**2, x, y, f(x, y)) is None
assert homogeneous_order(f(y), f(x), x) is None
assert homogeneous_order(-f(x)/x + 1/sin(f(x)/ x), f(x), x) == 0
assert homogeneous_order(log(1/y) + log(x**2), x, y) is None
assert homogeneous_order(log(1/y) + log(x), x, y) == 0
assert homogeneous_order(log(x/y), x, y) == 0
assert homogeneous_order(2*log(1/y) + 2*log(x), x, y) == 0
a = Symbol('a')
assert homogeneous_order(a*log(1/y) + a*log(x), x, y) == 0
assert homogeneous_order(f(x).diff(x), x, y) is None
assert homogeneous_order(-f(x).diff(x) + x, x, y) is None
assert homogeneous_order(O(x), x, y) is None
assert homogeneous_order(x + O(x**2), x, y) is None
assert homogeneous_order(x**pi, x) == pi
assert homogeneous_order(x**x, x) is None
raises(ValueError, lambda: homogeneous_order(x*y))
@slow
def test_1st_homogeneous_coeff_ode():
# Type: First order homogeneous, y'=f(y/x)
eq1 = f(x)/x*cos(f(x)/x) - (x/f(x)*sin(f(x)/x) + cos(f(x)/x))*f(x).diff(x)
eq2 = x*f(x).diff(x) - f(x) - x*sin(f(x)/x)
eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x)
eq4 = 2*f(x)*exp(x/f(x)) + f(x)*f(x).diff(x) - 2*x*exp(x/f(x))*f(x).diff(x)
eq5 = 2*x**2*f(x) + f(x)**3 + (x*f(x)**2 - 2*x**3)*f(x).diff(x)
eq6 = x*exp(f(x)/x) - f(x)*sin(f(x)/x) + x*sin(f(x)/x)*f(x).diff(x)
eq7 = (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x)
eq8 = x + f(x) - (x - f(x))*f(x).diff(x)
sol1 = Eq(log(x), C1 - log(f(x)*sin(f(x)/x)/x))
sol2 = Eq(log(x), log(C1) + log(cos(f(x)/x) - 1)/2 - log(cos(f(x)/x) + 1)/2)
sol3 = Eq(f(x), -exp(C1)*LambertW(-x*exp(-C1 + 1)))
sol4 = Eq(log(f(x)), C1 - 2*exp(x/f(x)))
sol5 = Eq(f(x), exp(2*C1 + LambertW(-2*x**4*exp(-4*C1))/2)/x)
sol6 = Eq(log(x), C1 + exp(-f(x)/x)*sin(f(x)/x)/2 + exp(-f(x)/x)*cos(f(x)/x)/2)
sol7 = Eq(log(f(x)), C1 - 2*sqrt(-x/f(x) + 1))
sol8 = Eq(log(x), C1 - log(sqrt(1 + f(x)**2/x**2)) + atan(f(x)/x))
# indep_div_dep actually has a simpler solution for eq2,
# but it runs too slow
assert dsolve(eq1, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol1
assert dsolve(eq2, hint='1st_homogeneous_coeff_subs_dep_div_indep', simplify=False) == sol2
assert dsolve(eq3, hint='1st_homogeneous_coeff_best') == sol3
assert dsolve(eq4, hint='1st_homogeneous_coeff_best') == sol4
assert dsolve(eq5, hint='1st_homogeneous_coeff_best') == sol5
assert dsolve(eq6, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol6
assert dsolve(eq7, hint='1st_homogeneous_coeff_best') == sol7
assert dsolve(eq8, hint='1st_homogeneous_coeff_best') == sol8
# FIXME: sol3 and sol5 don't work with checkodesol (because of LambertW?)
# previous code was testing with these other solutions:
sol3b = Eq(-f(x)/(1 + log(x/f(x))), C1)
sol5b = Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3b, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
assert checkodesol(eq5, sol5b, order=1, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode_check2():
eq2 = x*f(x).diff(x) - f(x) - x*sin(f(x)/x)
sol2 = Eq(x/tan(f(x)/(2*x)), C1)
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
@XFAIL
def test_1st_homogeneous_coeff_ode_check3():
skip('This is a known issue.')
# checker cannot determine that the following expression is zero:
# (False,
# x*(log(exp(-LambertW(C1*x))) +
# LambertW(C1*x))*exp(-LambertW(C1*x) + 1))
# This is blocked by issue 5080.
eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x)
sol3a = Eq(f(x), x*exp(1 - LambertW(C1*x)))
assert checkodesol(eq3, sol3a, solve_for_func=True)[0]
# Checker can't verify this form either
# (False,
# C1*(log(C1*LambertW(C2*x)/x) + LambertW(C2*x) - 1)*LambertW(C2*x))
# It is because a = W(a)*exp(W(a)), so log(a) == log(W(a)) + W(a) and C2 =
# -E/C1 (which can be verified by solving with simplify=False).
sol3b = Eq(f(x), C1*LambertW(C2*x))
assert checkodesol(eq3, sol3b, solve_for_func=True)[0]
def test_1st_homogeneous_coeff_ode_check7():
eq7 = (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x)
sol7 = Eq(log(C1*f(x)) + 2*sqrt(1 - x/f(x)), 0)
assert checkodesol(eq7, sol7, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode2():
eq1 = f(x).diff(x) - f(x)/x + 1/sin(f(x)/x)
eq2 = x**2 + f(x)**2 - 2*x*f(x)*f(x).diff(x)
eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x)
sol1 = [Eq(f(x), x*(-acos(C1 + log(x)) + 2*pi)), Eq(f(x), x*acos(C1 + log(x)))]
sol2 = Eq(log(f(x)), log(C1) + log(x/f(x)) - log(x**2/f(x)**2 - 1))
sol3 = Eq(f(x), log((1/(C1 - log(x)))**x))
# specific hints are applied for speed reasons
assert dsolve(eq1, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol1
assert dsolve(eq2, hint='1st_homogeneous_coeff_best', simplify=False) == sol2
assert dsolve(eq3, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol3
# FIXME: sol3 doesn't work with checkodesol (because of **x?)
# previous code was testing with this other solution:
sol3b = Eq(f(x), log(log(C1/x)**(-x)))
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3b, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode_check9():
_u2 = Dummy('u2')
__a = Dummy('a')
eq9 = f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x)
sol9 = Eq(-Integral(-1/(-(1 - sqrt(1 - _u2**2))*_u2 + _u2), (_u2, __a,
x/f(x))) + log(C1*f(x)), 0)
assert checkodesol(eq9, sol9, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode3():
# The standard integration engine cannot handle one of the integrals
# involved (see issue 4551). meijerg code comes up with an answer, but in
# unconventional form.
# checkodesol fails for this equation, so its test is in
# test_1st_homogeneous_coeff_ode_check9 above. It has to compare string
# expressions because u2 is a dummy variable.
eq = f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x)
sol = Eq(log(f(x)), C1 + Piecewise(
(acosh(f(x)/x), abs(f(x)**2)/x**2 > 1),
(-I*asin(f(x)/x), True)))
assert dsolve(eq, hint='1st_homogeneous_coeff_subs_indep_div_dep') == sol
def test_1st_homogeneous_coeff_corner_case():
eq1 = f(x).diff(x) - f(x)/x
c1 = classify_ode(eq1, f(x))
eq2 = x*f(x).diff(x) - f(x)
c2 = classify_ode(eq2, f(x))
sdi = "1st_homogeneous_coeff_subs_dep_div_indep"
sid = "1st_homogeneous_coeff_subs_indep_div_dep"
assert sid not in c1 and sdi not in c1
assert sid not in c2 and sdi not in c2
@slow
def test_nth_linear_constant_coeff_homogeneous():
# From Exercise 20, in Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 220
a = Symbol('a', positive=True)
k = Symbol('k', real=True)
eq1 = f(x).diff(x, 2) + 2*f(x).diff(x)
eq2 = f(x).diff(x, 2) - 3*f(x).diff(x) + 2*f(x)
eq3 = f(x).diff(x, 2) - f(x)
eq4 = f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x)
eq5 = 6*f(x).diff(x, 2) - 11*f(x).diff(x) + 4*f(x)
eq6 = Eq(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), 0)
eq7 = diff(f(x), x, 3) + diff(f(x), x, 2) - 10*diff(f(x), x) - 6*f(x)
eq8 = f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
4*f(x).diff(x)
eq9 = f(x).diff(x, 4) + 4*f(x).diff(x, 3) + f(x).diff(x, 2) - \
4*f(x).diff(x) - 2*f(x)
eq10 = f(x).diff(x, 4) - a**2*f(x)
eq11 = f(x).diff(x, 2) - 2*k*f(x).diff(x) - 2*f(x)
eq12 = f(x).diff(x, 2) + 4*k*f(x).diff(x) - 12*k**2*f(x)
eq13 = f(x).diff(x, 4)
eq14 = f(x).diff(x, 2) + 4*f(x).diff(x) + 4*f(x)
eq15 = 3*f(x).diff(x, 3) + 5*f(x).diff(x, 2) + f(x).diff(x) - f(x)
eq16 = f(x).diff(x, 3) - 6*f(x).diff(x, 2) + 12*f(x).diff(x) - 8*f(x)
eq17 = f(x).diff(x, 2) - 2*a*f(x).diff(x) + a**2*f(x)
eq18 = f(x).diff(x, 4) + 3*f(x).diff(x, 3)
eq19 = f(x).diff(x, 4) - 2*f(x).diff(x, 2)
eq20 = f(x).diff(x, 4) + 2*f(x).diff(x, 3) - 11*f(x).diff(x, 2) - \
12*f(x).diff(x) + 36*f(x)
eq21 = 36*f(x).diff(x, 4) - 37*f(x).diff(x, 2) + 4*f(x).diff(x) + 5*f(x)
eq22 = f(x).diff(x, 4) - 8*f(x).diff(x, 2) + 16*f(x)
eq23 = f(x).diff(x, 2) - 2*f(x).diff(x) + 5*f(x)
eq24 = f(x).diff(x, 2) - f(x).diff(x) + f(x)
eq25 = f(x).diff(x, 4) + 5*f(x).diff(x, 2) + 6*f(x)
eq26 = f(x).diff(x, 2) - 4*f(x).diff(x) + 20*f(x)
eq27 = f(x).diff(x, 4) + 4*f(x).diff(x, 2) + 4*f(x)
eq28 = f(x).diff(x, 3) + 8*f(x)
eq29 = f(x).diff(x, 4) + 4*f(x).diff(x, 2)
eq30 = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x)
eq31 = f(x).diff(x, 4) + f(x).diff(x, 2) + f(x)
eq32 = f(x).diff(x, 4) + 4*f(x).diff(x, 2) + f(x)
sol1 = Eq(f(x), C1 + C2*exp(-2*x))
sol2 = Eq(f(x), (C1 + C2*exp(x))*exp(x))
sol3 = Eq(f(x), C1*exp(x) + C2*exp(-x))
sol4 = Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))
sol5 = Eq(f(x), C1*exp(x/2) + C2*exp(4*x/3))
sol6 = Eq(f(x), C1*exp(x*(-1 + sqrt(2))) + C2*exp(x*(-sqrt(2) - 1)))
sol7 = Eq(f(x),
C1*exp(3*x) + C2*exp(x*(-2 - sqrt(2))) + C3*exp(x*(-2 + sqrt(2))))
sol8 = Eq(f(x), C1 + C2*exp(x) + C3*exp(-2*x) + C4*exp(2*x))
sol9 = Eq(f(x),
C1*exp(x) + C2*exp(-x) + C3*exp(x*(-2 + sqrt(2))) +
C4*exp(x*(-2 - sqrt(2))))
sol10 = Eq(f(x),
C1*sin(x*sqrt(a)) + C2*cos(x*sqrt(a)) + C3*exp(x*sqrt(a)) +
C4*exp(-x*sqrt(a)))
sol11 = Eq(f(x),
C1*exp(x*(k - sqrt(k**2 + 2))) + C2*exp(x*(k + sqrt(k**2 + 2))))
sol12 = Eq(f(x), C1*exp(-6*k*x) + C2*exp(2*k*x))
sol13 = Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3)
sol14 = Eq(f(x), (C1 + C2*x)*exp(-2*x))
sol15 = Eq(f(x), (C1 + C2*x)*exp(-x) + C3*exp(x/3))
sol16 = Eq(f(x), (C1 + C2*x + C3*x**2)*exp(2*x))
sol17 = Eq(f(x), (C1 + C2*x)*exp(a*x))
sol18 = Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))
sol19 = Eq(f(x), C1 + C2*x + C3*exp(x*sqrt(2)) + C4*exp(-x*sqrt(2)))
sol20 = Eq(f(x), (C1 + C2*x)*exp(-3*x) + (C3 + C4*x)*exp(2*x))
sol21 = Eq(f(x), C1*exp(x/2) + C2*exp(-x) + C3*exp(-x/3) + C4*exp(5*x/6))
sol22 = Eq(f(x), (C1 + C2*x)*exp(-2*x) + (C3 + C4*x)*exp(2*x))
sol23 = Eq(f(x), (C1*sin(2*x) + C2*cos(2*x))*exp(x))
sol24 = Eq(f(x), (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(x/2))
sol25 = Eq(f(x),
C1*cos(x*sqrt(3)) + C2*sin(x*sqrt(3)) + C3*sin(x*sqrt(2)) +
C4*cos(x*sqrt(2)))
sol26 = Eq(f(x), (C1*sin(4*x) + C2*cos(4*x))*exp(2*x))
sol27 = Eq(f(x), (C1 + C2*x)*sin(x*sqrt(2)) + (C3 + C4*x)*cos(x*sqrt(2)))
sol28 = Eq(f(x),
(C1*sin(x*sqrt(3)) + C2*cos(x*sqrt(3)))*exp(x) + C3*exp(-2*x))
sol29 = Eq(f(x), C1 + C2*sin(2*x) + C3*cos(2*x) + C4*x)
sol30 = Eq(f(x), C1 + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))
sol31 = Eq(f(x), (C1*sin(sqrt(3)*x/2) + C2*cos(sqrt(3)*x/2))/sqrt(exp(x))
+ (C3*sin(sqrt(3)*x/2) + C4*cos(sqrt(3)*x/2))*sqrt(exp(x)))
sol32 = Eq(f(x), C1*sin(x*sqrt(-sqrt(3) + 2)) + C2*sin(x*sqrt(sqrt(3) + 2))
+ C3*cos(x*sqrt(-sqrt(3) + 2)) + C4*cos(x*sqrt(sqrt(3) + 2)))
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
sol6s = constant_renumber(sol6)
sol7s = constant_renumber(sol7)
sol8s = constant_renumber(sol8)
sol9s = constant_renumber(sol9)
sol10s = constant_renumber(sol10)
sol11s = constant_renumber(sol11)
sol12s = constant_renumber(sol12)
sol13s = constant_renumber(sol13)
sol14s = constant_renumber(sol14)
sol15s = constant_renumber(sol15)
sol16s = constant_renumber(sol16)
sol17s = constant_renumber(sol17)
sol18s = constant_renumber(sol18)
sol19s = constant_renumber(sol19)
sol20s = constant_renumber(sol20)
sol21s = constant_renumber(sol21)
sol22s = constant_renumber(sol22)
sol23s = constant_renumber(sol23)
sol24s = constant_renumber(sol24)
sol25s = constant_renumber(sol25)
sol26s = constant_renumber(sol26)
sol27s = constant_renumber(sol27)
sol28s = constant_renumber(sol28)
sol29s = constant_renumber(sol29)
sol30s = constant_renumber(sol30)
assert dsolve(eq1) in (sol1, sol1s)
assert dsolve(eq2) in (sol2, sol2s)
assert dsolve(eq3) in (sol3, sol3s)
assert dsolve(eq4) in (sol4, sol4s)
assert dsolve(eq5) in (sol5, sol5s)
assert dsolve(eq6) in (sol6, sol6s)
assert dsolve(eq7) in (sol7, sol7s)
assert dsolve(eq8) in (sol8, sol8s)
assert dsolve(eq9) in (sol9, sol9s)
assert dsolve(eq10) in (sol10, sol10s)
assert dsolve(eq11) in (sol11, sol11s)
assert dsolve(eq12) in (sol12, sol12s)
assert dsolve(eq13) in (sol13, sol13s)
assert dsolve(eq14) in (sol14, sol14s)
assert dsolve(eq15) in (sol15, sol15s)
assert dsolve(eq16) in (sol16, sol16s)
assert dsolve(eq17) in (sol17, sol17s)
assert dsolve(eq18) in (sol18, sol18s)
assert dsolve(eq19) in (sol19, sol19s)
assert dsolve(eq20) in (sol20, sol20s)
assert dsolve(eq21) in (sol21, sol21s)
assert dsolve(eq22) in (sol22, sol22s)
assert dsolve(eq23) in (sol23, sol23s)
assert dsolve(eq24) in (sol24, sol24s)
assert dsolve(eq25) in (sol25, sol25s)
assert dsolve(eq26) in (sol26, sol26s)
assert dsolve(eq27) in (sol27, sol27s)
assert dsolve(eq28) in (sol28, sol28s)
assert dsolve(eq29) in (sol29, sol29s)
assert dsolve(eq30) in (sol30, sol30s)
assert dsolve(eq31) in (sol31,)
assert dsolve(eq32) in (sol32,)
assert checkodesol(eq1, sol1, order=2, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=2, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=3, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=3, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=4, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=4, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=4, solve_for_func=False)[0]
assert checkodesol(eq11, sol11, order=2, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=2, solve_for_func=False)[0]
assert checkodesol(eq13, sol13, order=4, solve_for_func=False)[0]
assert checkodesol(eq14, sol14, order=2, solve_for_func=False)[0]
assert checkodesol(eq15, sol15, order=3, solve_for_func=False)[0]
assert checkodesol(eq16, sol16, order=3, solve_for_func=False)[0]
assert checkodesol(eq17, sol17, order=2, solve_for_func=False)[0]
assert checkodesol(eq18, sol18, order=4, solve_for_func=False)[0]
assert checkodesol(eq19, sol19, order=4, solve_for_func=False)[0]
assert checkodesol(eq20, sol20, order=4, solve_for_func=False)[0]
assert checkodesol(eq21, sol21, order=4, solve_for_func=False)[0]
assert checkodesol(eq22, sol22, order=4, solve_for_func=False)[0]
assert checkodesol(eq23, sol23, order=2, solve_for_func=False)[0]
assert checkodesol(eq24, sol24, order=2, solve_for_func=False)[0]
assert checkodesol(eq25, sol25, order=4, solve_for_func=False)[0]
assert checkodesol(eq26, sol26, order=2, solve_for_func=False)[0]
assert checkodesol(eq27, sol27, order=4, solve_for_func=False)[0]
assert checkodesol(eq28, sol28, order=3, solve_for_func=False)[0]
assert checkodesol(eq29, sol29, order=4, solve_for_func=False)[0]
assert checkodesol(eq30, sol30, order=5, solve_for_func=False)[0]
assert checkodesol(eq31, sol31, order=4, solve_for_func=False)[0]
assert checkodesol(eq32, sol32, order=4, solve_for_func=False)[0]
# Issue #15237
eqn = Derivative(x*f(x), x, x, x)
hint = 'nth_linear_constant_coeff_homogeneous'
raises(ValueError, lambda: dsolve(eqn, f(x), hint, prep=True))
raises(ValueError, lambda: dsolve(eqn, f(x), hint, prep=False))
def test_nth_linear_constant_coeff_homogeneous_rootof():
# One real root, two complex conjugate pairs
eq = f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 + 11*x - 2, n) for n in range(5)]
sol = Eq(f(x),
C5*exp(r1*x)
+ exp(re(r2)*x) * (C1*sin(im(r2)*x) + C2*cos(im(r2)*x))
+ exp(re(r4)*x) * (C3*sin(im(r4)*x) + C4*cos(im(r4)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Three real roots, one complex conjugate pair
eq = f(x).diff(x,5) - 3*f(x).diff(x) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - 3*x + 1, n) for n in range(5)]
sol = Eq(f(x),
C3*exp(r1*x) + C4*exp(r2*x) + C5*exp(r3*x)
+ exp(re(r4)*x) * (C1*sin(im(r4)*x) + C2*cos(im(r4)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Five distinct real roots
eq = f(x).diff(x,5) - 100*f(x).diff(x,3) + 1000*f(x).diff(x) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - 100*x**3 + 1000*x + 1, n) for n in range(5)]
sol = Eq(f(x), C1*exp(r1*x) + C2*exp(r2*x) + C3*exp(r3*x) + C4*exp(r4*x) + C5*exp(r5*x))
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Rational root and unsolvable quintic
eq = f(x).diff(x, 6) - 6*f(x).diff(x, 5) + 5*f(x).diff(x, 4) + 10*f(x).diff(x) - 50 * f(x)
r2, r3, r4, r5, r6 = [rootof(x**5 - x**4 + 10, n) for n in range(5)]
sol = Eq(f(x),
C5*exp(5*x)
+ C6*exp(x*r2)
+ exp(re(r3)*x) * (C1*sin(im(r3)*x) + C2*cos(im(r3)*x))
+ exp(re(r5)*x) * (C3*sin(im(r5)*x) + C4*cos(im(r5)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Five double roots (this is (x**5 - x + 1)**2)
eq = f(x).diff(x, 10) - 2*f(x).diff(x, 6) + 2*f(x).diff(x, 5) + f(x).diff(x, 2) - 2*f(x).diff(x, 1) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - x + 1, n) for n in range(5)]
sol = Eq(f(x),
(C1 + C2 *x)*exp(r1*x)
+ exp(re(r2)*x) * ((C3 + C4*x)*sin(im(r2)*x) + (C5 + C6 *x)*cos(im(r2)*x))
+ exp(re(r4)*x) * ((C7 + C8*x)*sin(im(r4)*x) + (C9 + C10*x)*cos(im(r4)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
def test_nth_linear_constant_coeff_homogeneous_irrational():
our_hint='nth_linear_constant_coeff_homogeneous'
eq = Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(2**(S(3)/4)*x/2) + C3*cos(2**(S(3)/4)*x/2))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
E = exp(1)
eq = Eq(E * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(x/sqrt(E)) + C3*cos(x/sqrt(E)))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
eq = Eq(pi * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(x/sqrt(pi)) + C3*cos(x/sqrt(pi)))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
eq = Eq(I * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*exp(-sqrt(I)*x) + C3*exp(sqrt(I)*x))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
@XFAIL
@slow
def test_nth_linear_constant_coeff_homogeneous_rootof_sol():
if ON_TRAVIS:
skip("Too slow for travis.")
eq = f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x)
sol = Eq(f(x),
C1*exp(x*rootof(x**5 + 11*x - 2, 0)) +
C2*exp(x*rootof(x**5 + 11*x - 2, 1)) +
C3*exp(x*rootof(x**5 + 11*x - 2, 2)) +
C4*exp(x*rootof(x**5 + 11*x - 2, 3)) +
C5*exp(x*rootof(x**5 + 11*x - 2, 4)))
assert checkodesol(eq, sol, order=5, solve_for_func=False)[0]
@XFAIL
def test_noncircularized_real_imaginary_parts():
# If this passes, lines numbered 3878-3882 (at the time of this commit)
# of sympy/solvers/ode.py for nth_linear_constant_coeff_homogeneous
# should be removed.
y = sqrt(1+x)
i, r = im(y), re(y)
assert not (i.has(atan2) and r.has(atan2))
@XFAIL
def test_collect_respecting_exponentials():
# If this test passes, lines 1306-1311 (at the time of this commit)
# of sympy/solvers/ode.py should be removed.
sol = 1 + exp(x/2)
assert sol == collect( sol, exp(x/3))
def test_undetermined_coefficients_match():
assert _undetermined_coefficients_match(g(x), x) == {'test': False}
assert _undetermined_coefficients_match(sin(2*x + sqrt(5)), x) == \
{'test': True, 'trialset':
set([cos(2*x + sqrt(5)), sin(2*x + sqrt(5))])}
assert _undetermined_coefficients_match(sin(x)*cos(x), x) == \
{'test': False}
s = set([cos(x), x*cos(x), x**2*cos(x), x**2*sin(x), x*sin(x), sin(x)])
assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': s}
assert _undetermined_coefficients_match(
sin(x)*x**2 + sin(x)*x + sin(x), x) == {'test': True, 'trialset': s}
assert _undetermined_coefficients_match(
exp(2*x)*sin(x)*(x**2 + x + 1), x
) == {
'test': True, 'trialset': set([exp(2*x)*sin(x), x**2*exp(2*x)*sin(x),
cos(x)*exp(2*x), x**2*cos(x)*exp(2*x), x*cos(x)*exp(2*x),
x*exp(2*x)*sin(x)])}
assert _undetermined_coefficients_match(1/sin(x), x) == {'test': False}
assert _undetermined_coefficients_match(log(x), x) == {'test': False}
assert _undetermined_coefficients_match(2**(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': set([2**x, x*2**x, x**2*2**x])}
assert _undetermined_coefficients_match(x**y, x) == {'test': False}
assert _undetermined_coefficients_match(exp(x)*exp(2*x + 1), x) == \
{'test': True, 'trialset': set([exp(1 + 3*x)])}
assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': set([x*cos(x), x*sin(x), x**2*cos(x),
x**2*sin(x), cos(x), sin(x)])}
assert _undetermined_coefficients_match(sin(x)*(x + sin(x)), x) == \
{'test': False}
assert _undetermined_coefficients_match(sin(x)*(x + sin(2*x)), x) == \
{'test': False}
assert _undetermined_coefficients_match(sin(x)*tan(x), x) == \
{'test': False}
assert _undetermined_coefficients_match(
x**2*sin(x)*exp(x) + x*sin(x) + x, x
) == {
'test': True, 'trialset': set([x**2*cos(x)*exp(x), x, cos(x), S(1),
exp(x)*sin(x), sin(x), x*exp(x)*sin(x), x*cos(x), x*cos(x)*exp(x),
x*sin(x), cos(x)*exp(x), x**2*exp(x)*sin(x)])}
assert _undetermined_coefficients_match(4*x*sin(x - 2), x) == {
'trialset': set([x*cos(x - 2), x*sin(x - 2), cos(x - 2), sin(x - 2)]),
'test': True,
}
assert _undetermined_coefficients_match(2**x*x, x) == \
{'test': True, 'trialset': set([2**x, x*2**x])}
assert _undetermined_coefficients_match(2**x*exp(2*x), x) == \
{'test': True, 'trialset': set([2**x*exp(2*x)])}
assert _undetermined_coefficients_match(exp(-x)/x, x) == \
{'test': False}
# Below are from Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 231
assert _undetermined_coefficients_match(S(4), x) == \
{'test': True, 'trialset': set([S(1)])}
assert _undetermined_coefficients_match(12*exp(x), x) == \
{'test': True, 'trialset': set([exp(x)])}
assert _undetermined_coefficients_match(exp(I*x), x) == \
{'test': True, 'trialset': set([exp(I*x)])}
assert _undetermined_coefficients_match(sin(x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x)])}
assert _undetermined_coefficients_match(cos(x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x)])}
assert _undetermined_coefficients_match(8 + 6*exp(x) + 2*sin(x), x) == \
{'test': True, 'trialset': set([S(1), cos(x), sin(x), exp(x)])}
assert _undetermined_coefficients_match(x**2, x) == \
{'test': True, 'trialset': set([S(1), x, x**2])}
assert _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(x), exp(x), exp(-x)])}
assert _undetermined_coefficients_match(2*exp(2*x)*sin(x), x) == \
{'test': True, 'trialset': set([exp(2*x)*sin(x), cos(x)*exp(2*x)])}
assert _undetermined_coefficients_match(x - sin(x), x) == \
{'test': True, 'trialset': set([S(1), x, cos(x), sin(x)])}
assert _undetermined_coefficients_match(x**2 + 2*x, x) == \
{'test': True, 'trialset': set([S(1), x, x**2])}
assert _undetermined_coefficients_match(4*x*sin(x), x) == \
{'test': True, 'trialset': set([x*cos(x), x*sin(x), cos(x), sin(x)])}
assert _undetermined_coefficients_match(x*sin(2*x), x) == \
{'test': True, 'trialset':
set([x*cos(2*x), x*sin(2*x), cos(2*x), sin(2*x)])}
assert _undetermined_coefficients_match(x**2*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), x**2*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(2*exp(-x) - x**2*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), x**2*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(exp(-2*x) + x**2, x) == \
{'test': True, 'trialset': set([S(1), x, x**2, exp(-2*x)])}
assert _undetermined_coefficients_match(x*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(x + exp(2*x), x) == \
{'test': True, 'trialset': set([S(1), x, exp(2*x)])}
assert _undetermined_coefficients_match(sin(x) + exp(-x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x), exp(-x)])}
assert _undetermined_coefficients_match(exp(x), x) == \
{'test': True, 'trialset': set([exp(x)])}
# converted from sin(x)**2
assert _undetermined_coefficients_match(S(1)/2 - cos(2*x)/2, x) == \
{'test': True, 'trialset': set([S(1), cos(2*x), sin(2*x)])}
# converted from exp(2*x)*sin(x)**2
assert _undetermined_coefficients_match(
exp(2*x)*(S(1)/2 + cos(2*x)/2), x
) == {
'test': True, 'trialset': set([exp(2*x)*sin(2*x), cos(2*x)*exp(2*x),
exp(2*x)])}
assert _undetermined_coefficients_match(2*x + sin(x) + cos(x), x) == \
{'test': True, 'trialset': set([S(1), x, cos(x), sin(x)])}
# converted from sin(2*x)*sin(x)
assert _undetermined_coefficients_match(cos(x)/2 - cos(3*x)/2, x) == \
{'test': True, 'trialset': set([cos(x), cos(3*x), sin(x), sin(3*x)])}
assert _undetermined_coefficients_match(cos(x**2), x) == {'test': False}
assert _undetermined_coefficients_match(2**(x**2), x) == {'test': False}
@slow
def test_nth_linear_constant_coeff_undetermined_coefficients():
hint = 'nth_linear_constant_coeff_undetermined_coefficients'
g = exp(-x)
f2 = f(x).diff(x, 2)
c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
eq1 = c - x*g
eq2 = c - g
# 3-27 below are from Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 231
eq3 = f2 + 3*f(x).diff(x) + 2*f(x) - 4
eq4 = f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x)
eq5 = f2 + 3*f(x).diff(x) + 2*f(x) - exp(I*x)
eq6 = f2 + 3*f(x).diff(x) + 2*f(x) - sin(x)
eq7 = f2 + 3*f(x).diff(x) + 2*f(x) - cos(x)
eq8 = f2 + 3*f(x).diff(x) + 2*f(x) - (8 + 6*exp(x) + 2*sin(x))
eq9 = f2 + f(x).diff(x) + f(x) - x**2
eq10 = f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x)
eq11 = f2 - 3*f(x).diff(x) - 2*exp(2*x)*sin(x)
eq12 = f(x).diff(x, 4) - 2*f2 + f(x) - x + sin(x)
eq13 = f2 + f(x).diff(x) - x**2 - 2*x
eq14 = f2 + f(x).diff(x) - x - sin(2*x)
eq15 = f2 + f(x) - 4*x*sin(x)
eq16 = f2 + 4*f(x) - x*sin(2*x)
eq17 = f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x)
eq18 = f(x).diff(x, 3) + 3*f2 + 3*f(x).diff(x) + f(x) - 2*exp(-x) + \
x**2*exp(-x)
eq19 = f2 + 3*f(x).diff(x) + 2*f(x) - exp(-2*x) - x**2
eq20 = f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x)
eq21 = f2 + f(x).diff(x) - 6*f(x) - x - exp(2*x)
eq22 = f2 + f(x) - sin(x) - exp(-x)
eq23 = f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x)
# sin(x)**2
eq24 = f2 + f(x) - S(1)/2 - cos(2*x)/2
# exp(2*x)*sin(x)**2
eq25 = f(x).diff(x, 3) - f(x).diff(x) - exp(2*x)*(S(1)/2 - cos(2*x)/2)
eq26 = (f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x -
sin(x) - cos(x))
# sin(2*x)*sin(x), skip 3127 for now, match bug
eq27 = f2 + f(x) - cos(x)/2 + cos(3*x)/2
eq28 = f(x).diff(x) - 1
sol1 = Eq(f(x),
-1 - x + (C1 + C2*x - 3*x**2/32 - x**3/24)*exp(-x) + C3*exp(x/3))
sol2 = Eq(f(x), -1 - x + (C1 + C2*x - x**2/8)*exp(-x) + C3*exp(x/3))
sol3 = Eq(f(x), 2 + C1*exp(-x) + C2*exp(-2*x))
sol4 = Eq(f(x), 2*exp(x) + C1*exp(-x) + C2*exp(-2*x))
sol5 = Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + exp(I*x)/10 - 3*I*exp(I*x)/10)
sol6 = Eq(f(x), -3*cos(x)/10 + sin(x)/10 + C1*exp(-x) + C2*exp(-2*x))
sol7 = Eq(f(x), cos(x)/10 + 3*sin(x)/10 + C1*exp(-x) + C2*exp(-2*x))
sol8 = Eq(f(x),
4 - 3*cos(x)/5 + sin(x)/5 + exp(x) + C1*exp(-x) + C2*exp(-2*x))
sol9 = Eq(f(x),
-2*x + x**2 + (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(-x/2))
sol10 = Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))
sol11 = Eq(f(x), C1 + C2*exp(3*x) + (-3*sin(x) - cos(x))*exp(2*x)/5)
sol12 = Eq(f(x), x - sin(x)/4 + (C1 + C2*x)*exp(-x) + (C3 + C4*x)*exp(x))
sol13 = Eq(f(x), C1 + x**3/3 + C2*exp(-x))
sol14 = Eq(f(x), C1 - x - sin(2*x)/5 - cos(2*x)/10 + x**2/2 + C2*exp(-x))
sol15 = Eq(f(x), (C1 + x)*sin(x) + (C2 - x**2)*cos(x))
sol16 = Eq(f(x), (C1 + x/16)*sin(2*x) + (C2 - x**2/8)*cos(2*x))
sol17 = Eq(f(x), (C1 + C2*x + x**4/12)*exp(-x))
sol18 = Eq(f(x), (C1 + C2*x + C3*x**2 - x**5/60 + x**3/3)*exp(-x))
sol19 = Eq(f(x), S(7)/4 - 3*x/2 + x**2/2 + C1*exp(-x) + (C2 - x)*exp(-2*x))
sol20 = Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)
sol21 = Eq(f(x), -S(1)/36 - x/6 + C1*exp(-3*x) + (C2 + x/5)*exp(2*x))
sol22 = Eq(f(x), C1*sin(x) + (C2 - x/2)*cos(x) + exp(-x)/2)
sol23 = Eq(f(x), (C1 + C2*x + C3*x**2 + x**3/6)*exp(x))
sol24 = Eq(f(x), S(1)/2 - cos(2*x)/6 + C1*sin(x) + C2*cos(x))
sol25 = Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) +
(-21*sin(2*x) + 27*cos(2*x) + 130)*exp(2*x)/1560)
sol26 = Eq(f(x),
C1 + (C2 + C3*x - x**2/8)*sin(x) + (C4 + C5*x + x**2/8)*cos(x) + x**2)
sol27 = Eq(f(x), cos(3*x)/16 + C1*cos(x) + (C2 + x/4)*sin(x))
sol28 = Eq(f(x), C1 + x)
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
sol6s = constant_renumber(sol6)
sol7s = constant_renumber(sol7)
sol8s = constant_renumber(sol8)
sol9s = constant_renumber(sol9)
sol10s = constant_renumber(sol10)
sol11s = constant_renumber(sol11)
sol12s = constant_renumber(sol12)
sol13s = constant_renumber(sol13)
sol14s = constant_renumber(sol14)
sol15s = constant_renumber(sol15)
sol16s = constant_renumber(sol16)
sol17s = constant_renumber(sol17)
sol18s = constant_renumber(sol18)
sol19s = constant_renumber(sol19)
sol20s = constant_renumber(sol20)
sol21s = constant_renumber(sol21)
sol22s = constant_renumber(sol22)
sol23s = constant_renumber(sol23)
sol24s = constant_renumber(sol24)
sol25s = constant_renumber(sol25)
sol26s = constant_renumber(sol26)
sol27s = constant_renumber(sol27)
assert dsolve(eq1, hint=hint) in (sol1, sol1s)
assert dsolve(eq2, hint=hint) in (sol2, sol2s)
assert dsolve(eq3, hint=hint) in (sol3, sol3s)
assert dsolve(eq4, hint=hint) in (sol4, sol4s)
assert dsolve(eq5, hint=hint) in (sol5, sol5s)
assert dsolve(eq6, hint=hint) in (sol6, sol6s)
assert dsolve(eq7, hint=hint) in (sol7, sol7s)
assert dsolve(eq8, hint=hint) in (sol8, sol8s)
assert dsolve(eq9, hint=hint) in (sol9, sol9s)
assert dsolve(eq10, hint=hint) in (sol10, sol10s)
assert dsolve(eq11, hint=hint) in (sol11, sol11s)
assert dsolve(eq12, hint=hint) in (sol12, sol12s)
assert dsolve(eq13, hint=hint) in (sol13, sol13s)
assert dsolve(eq14, hint=hint) in (sol14, sol14s)
assert dsolve(eq15, hint=hint) in (sol15, sol15s)
assert dsolve(eq16, hint=hint) in (sol16, sol16s)
assert dsolve(eq17, hint=hint) in (sol17, sol17s)
assert dsolve(eq18, hint=hint) in (sol18, sol18s)
assert dsolve(eq19, hint=hint) in (sol19, sol19s)
assert dsolve(eq20, hint=hint) in (sol20, sol20s)
assert dsolve(eq21, hint=hint) in (sol21, sol21s)
assert dsolve(eq22, hint=hint) in (sol22, sol22s)
assert dsolve(eq23, hint=hint) in (sol23, sol23s)
assert dsolve(eq24, hint=hint) in (sol24, sol24s)
assert dsolve(eq25, hint=hint) in (sol25, sol25s)
assert dsolve(eq26, hint=hint) in (sol26, sol26s)
assert dsolve(eq27, hint=hint) in (sol27, sol27s)
assert dsolve(eq28, hint=hint) == sol28
assert checkodesol(eq1, sol1, order=3, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=3, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=2, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=2, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=2, solve_for_func=False)[0]
assert checkodesol(eq11, sol11, order=2, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=4, solve_for_func=False)[0]
assert checkodesol(eq13, sol13, order=2, solve_for_func=False)[0]
assert checkodesol(eq14, sol14, order=2, solve_for_func=False)[0]
assert checkodesol(eq15, sol15, order=2, solve_for_func=False)[0]
assert checkodesol(eq16, sol16, order=2, solve_for_func=False)[0]
assert checkodesol(eq17, sol17, order=2, solve_for_func=False)[0]
assert checkodesol(eq18, sol18, order=3, solve_for_func=False)[0]
assert checkodesol(eq19, sol19, order=2, solve_for_func=False)[0]
assert checkodesol(eq20, sol20, order=2, solve_for_func=False)[0]
assert checkodesol(eq21, sol21, order=2, solve_for_func=False)[0]
assert checkodesol(eq22, sol22, order=2, solve_for_func=False)[0]
assert checkodesol(eq23, sol23, order=3, solve_for_func=False)[0]
assert checkodesol(eq24, sol24, order=2, solve_for_func=False)[0]
assert checkodesol(eq25, sol25, order=3, solve_for_func=False)[0]
assert checkodesol(eq26, sol26, order=5, solve_for_func=False)[0]
assert checkodesol(eq27, sol27, order=2, solve_for_func=False)[0]
assert checkodesol(eq28, sol28, order=1, solve_for_func=False)[0]
def test_issue_5787():
# This test case is to show the classification of imaginary constants under
# nth_linear_constant_coeff_undetermined_coefficients
eq = Eq(diff(f(x), x), I*f(x) + S(1)/2 - I)
our_hint = 'nth_linear_constant_coeff_undetermined_coefficients'
assert our_hint in classify_ode(eq)
@XFAIL
def test_nth_linear_constant_coeff_undetermined_coefficients_imaginary_exp():
# Equivalent to eq26 in
# test_nth_linear_constant_coeff_undetermined_coefficients above.
# This fails because the algorithm for undetermined coefficients
# doesn't know to multiply exp(I*x) by sufficient x because it is linearly
# dependent on sin(x) and cos(x).
hint = 'nth_linear_constant_coeff_undetermined_coefficients'
eq26a = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
sol26 = Eq(f(x),
C1 + (C2 + C3*x - x**2/8)*sin(x) + (C4 + C5*x + x**2/8)*cos(x) + x**2)
assert dsolve(eq26a, hint=hint) == sol26
assert checkodesol(eq26a, sol26, order=5, solve_for_func=False)[0]
@slow
def test_nth_linear_constant_coeff_variation_of_parameters():
hint = 'nth_linear_constant_coeff_variation_of_parameters'
g = exp(-x)
f2 = f(x).diff(x, 2)
c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
eq1 = c - x*g
eq2 = c - g
eq3 = f(x).diff(x) - 1
eq4 = f2 + 3*f(x).diff(x) + 2*f(x) - 4
eq5 = f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x)
eq6 = f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x)
eq7 = f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x)
eq8 = f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x)
eq9 = f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x)
eq10 = f2 + 2*f(x).diff(x) + f(x) - exp(-x)/x
eq11 = f2 + f(x) - 1/sin(x)*1/cos(x)
eq12 = f(x).diff(x, 4) - 1/x
sol1 = Eq(f(x),
-1 - x + (C1 + C2*x - 3*x**2/32 - x**3/24)*exp(-x) + C3*exp(x/3))
sol2 = Eq(f(x), -1 - x + (C1 + C2*x - x**2/8)*exp(-x) + C3*exp(x/3))
sol3 = Eq(f(x), C1 + x)
sol4 = Eq(f(x), 2 + C1*exp(-x) + C2*exp(-2*x))
sol5 = Eq(f(x), 2*exp(x) + C1*exp(-x) + C2*exp(-2*x))
sol6 = Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))
sol7 = Eq(f(x), (C1 + C2*x + x**4/12)*exp(-x))
sol8 = Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)
sol9 = Eq(f(x), (C1 + C2*x + C3*x**2 + x**3/6)*exp(x))
sol10 = Eq(f(x), (C1 + x*(C2 + log(x)))*exp(-x))
sol11 = Eq(f(x), cos(x)*(C2 - Integral(1/cos(x), x)) + sin(x)*(C1 +
Integral(1/sin(x), x)))
sol12 = Eq(f(x), C1 + C2*x + x**3*(C3 + log(x)/6) + C4*x**2)
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
sol6s = constant_renumber(sol6)
sol7s = constant_renumber(sol7)
sol8s = constant_renumber(sol8)
sol9s = constant_renumber(sol9)
sol10s = constant_renumber(sol10)
sol11s = constant_renumber(sol11)
sol12s = constant_renumber(sol12)
assert dsolve(eq1, hint=hint) in (sol1, sol1s)
assert dsolve(eq2, hint=hint) in (sol2, sol2s)
assert dsolve(eq3, hint=hint) in (sol3, sol3s)
assert dsolve(eq4, hint=hint) in (sol4, sol4s)
assert dsolve(eq5, hint=hint) in (sol5, sol5s)
assert dsolve(eq6, hint=hint) in (sol6, sol6s)
assert dsolve(eq7, hint=hint) in (sol7, sol7s)
assert dsolve(eq8, hint=hint) in (sol8, sol8s)
assert dsolve(eq9, hint=hint) in (sol9, sol9s)
assert dsolve(eq10, hint=hint) in (sol10, sol10s)
assert dsolve(eq11, hint=hint + '_Integral') in (sol11, sol11s)
assert dsolve(eq12, hint=hint) in (sol12, sol12s)
assert checkodesol(eq1, sol1, order=3, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=3, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=3, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=2, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=4, solve_for_func=False)[0]
@slow
def test_nth_linear_constant_coeff_variation_of_parameters_simplify_False():
# solve_variation_of_parameters shouldn't attempt to simplify the
# Wronskian if simplify=False. If wronskian() ever gets good enough
# to simplify the result itself, this test might fail.
our_hint = 'nth_linear_constant_coeff_variation_of_parameters_Integral'
eq = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
sol_simp = dsolve(eq, f(x), hint=our_hint, simplify=True)
sol_nsimp = dsolve(eq, f(x), hint=our_hint, simplify=False)
assert sol_simp != sol_nsimp
assert checkodesol(eq, sol_simp, order=5, solve_for_func=False)[0]
assert checkodesol(eq, sol_nsimp, order=5, solve_for_func=False)[0]
def test_Liouville_ODE():
hint = 'Liouville'
# The first part here used to be test_ODE_1() from test_solvers.py
eq1 = diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2
eq1a = diff(x*exp(-f(x)), x, x)
# compare to test_unexpanded_Liouville_ODE() below
eq2 = (eq1*exp(-f(x))/exp(f(x))).expand()
eq3 = diff(f(x), x, x) + 1/f(x)*(diff(f(x), x))**2 + 1/x*diff(f(x), x)
eq4 = x*diff(f(x), x, x) + x/f(x)*diff(f(x), x)**2 + x*diff(f(x), x)
eq5 = Eq((x*exp(f(x))).diff(x, x), 0)
sol1 = Eq(f(x), log(x/(C1 + C2*x)))
sol1a = Eq(C1 + C2/x - exp(-f(x)), 0)
sol2 = sol1
sol3 = set(
[Eq(f(x), -sqrt(C1 + C2*log(x))),
Eq(f(x), sqrt(C1 + C2*log(x)))])
sol4 = set([Eq(f(x), sqrt(C1 + C2*exp(x))*exp(-x/2)),
Eq(f(x), -sqrt(C1 + C2*exp(x))*exp(-x/2))])
sol5 = Eq(f(x), log(C1 + C2/x))
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
assert dsolve(eq1, hint=hint) in (sol1, sol1s)
assert dsolve(eq1a, hint=hint) in (sol1, sol1s)
assert dsolve(eq2, hint=hint) in (sol2, sol2s)
assert set(dsolve(eq3, hint=hint)) in (sol3, sol3s)
assert set(dsolve(eq4, hint=hint)) in (sol4, sol4s)
assert dsolve(eq5, hint=hint) in (sol5, sol5s)
assert checkodesol(eq1, sol1, order=2, solve_for_func=False)[0]
assert checkodesol(eq1a, sol1a, order=2, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=2, solve_for_func=False) == {(True, 0)}
assert checkodesol(eq4, sol4, order=2, solve_for_func=False) == {(True, 0)}
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
not_Liouville1 = classify_ode(diff(f(x), x)/x + f(x)*diff(f(x), x, x)/2 -
diff(f(x), x)**2/2, f(x))
not_Liouville2 = classify_ode(diff(f(x), x)/x + diff(f(x), x, x)/2 -
x*diff(f(x), x)**2/2, f(x))
assert hint not in not_Liouville1
assert hint not in not_Liouville2
assert hint + '_Integral' not in not_Liouville1
assert hint + '_Integral' not in not_Liouville2
def test_unexpanded_Liouville_ODE():
# This is the same as eq1 from test_Liouville_ODE() above.
eq1 = diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2
eq2 = eq1*exp(-f(x))/exp(f(x))
sol2 = Eq(f(x), log(x/(C1 + C2*x)))
sol2s = constant_renumber(sol2)
assert dsolve(eq2) in (sol2, sol2s)
assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
def test_issue_4785():
from sympy.abc import A
eq = x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2
assert classify_ode(eq, f(x)) == ('1st_linear', 'almost_linear',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'1st_linear_Integral', 'almost_linear_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
# issue 4864
eq = (x**2 + f(x)**2)*f(x).diff(x) - 2*x*f(x)
assert classify_ode(eq, f(x)) == ('1st_exact',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series',
'lie_group', '1st_exact_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
def test_issue_4825():
raises(ValueError, lambda: dsolve(f(x, y).diff(x) - y*f(x, y), f(x)))
assert classify_ode(f(x, y).diff(x) - y*f(x, y), f(x), dict=True) == \
{'default': None, 'order': 0}
# See also issue 3793, test Z13.
raises(ValueError, lambda: dsolve(f(x).diff(x), f(y)))
assert classify_ode(f(x).diff(x), f(y), dict=True) == \
{'default': None, 'order': 0}
def test_constant_renumber_order_issue_5308():
from sympy.utilities.iterables import variations
eq = f(x).diff(x) - y - x
assert constant_renumber(C1*x + C2*y) == \
constant_renumber(C1*y + C2*x) == \
C1*x + C2*y
e = C1*(C2 + x)*(C3 + y)
for a, b, c in variations([C1, C2, C3], 3):
assert constant_renumber(a*(b + x)*(c + y)) == e
def test_issue_5770():
k = Symbol("k", real=True)
t = Symbol('t')
w = Function('w')
sol = dsolve(w(t).diff(t, 6) - k**6*w(t), w(t))
assert len([s for s in sol.free_symbols if s.name.startswith('C')]) == 6
assert constantsimp((C1*cos(x) + C2*cos(x))*exp(x), set([C1, C2])) == \
C1*cos(x)*exp(x)
assert constantsimp(C1*cos(x) + C2*cos(x) + C3*sin(x), set([C1, C2, C3])) == \
C1*cos(x) + C3*sin(x)
assert constantsimp(exp(C1 + x), set([C1])) == C1*exp(x)
assert constantsimp(x + C1 + y, set([C1, y])) == C1 + x
assert constantsimp(x + C1 + Integral(x, (x, 1, 2)), set([C1])) == C1 + x
def test_issue_5112_5430():
assert homogeneous_order(-log(x) + acosh(x), x) is None
assert homogeneous_order(y - log(x), x, y) is None
def test_nth_order_linear_euler_eq_homogeneous():
x, t, a, b, c = symbols('x t a b c')
y = Function('y')
our_hint = "nth_linear_euler_eq_homogeneous"
eq = diff(f(t), t, 4)*t**4 - 13*diff(f(t), t, 2)*t**2 + 36*f(t)
assert our_hint in classify_ode(eq)
eq = a*y(t) + b*t*diff(y(t), t) + c*t**2*diff(y(t), t, 2)
assert our_hint in classify_ode(eq)
eq = Eq(-3*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0)
sol = C1 + C2*x**Rational(5, 2)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(3*f(x) - 5*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0)
sol = C1*sqrt(x) + C2*x**3
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(4*f(x) + 5*diff(f(x), x)*x + x**2*diff(f(x), x, x), 0)
sol = (C1 + C2*log(x))/x**2
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(6*f(x) - 6*diff(f(x), x)*x + 1*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0)
sol = dsolve(eq, f(x), hint=our_hint)
sol = C1/x**2 + C2*x + C3*x**3
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(-125*f(x) + 61*diff(f(x), x)*x - 12*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0)
sol = x**5*(C1 + C2*log(x) + C3*log(x)**2)
sols = [sol, constant_renumber(sol)]
sols += [sols[-1].expand()]
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in sols
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = t**2*diff(y(t), t, 2) + t*diff(y(t), t) - 9*y(t)
sol = C1*t**3 + C2*t**-3
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, y(t), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = sin(x)*x**2*f(x).diff(x, 2) + sin(x)*x*f(x).diff(x) + sin(x)*f(x)
sol = C1*sin(log(x)) + C2*cos(log(x))
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
def test_nth_order_linear_euler_eq_nonhomogeneous_undetermined_coefficients():
x, t = symbols('x t')
a, b, c, d = symbols('a b c d', integer=True)
our_hint = "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients"
eq = x**4*diff(f(x), x, 4) - 13*x**2*diff(f(x), x, 2) + 36*f(x) + x
assert our_hint in classify_ode(eq, f(x))
eq = a*x**2*diff(f(x), x, 2) + b*x*diff(f(x), x) + c*f(x) + d*log(x)
assert our_hint in classify_ode(eq, f(x))
eq = Eq(x**2*diff(f(x), x, x) + x*diff(f(x), x), 1)
sol = C1 + C2*log(x) + log(x)**2/2
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq, f(x))
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**2*diff(f(x), x, x) - 2*x*diff(f(x), x) + 2*f(x), x**3)
sol = x*(C1 + C2*x + Rational(1, 2)*x**2)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq, f(x))
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**2*diff(f(x), x, x) - x*diff(f(x), x) - 3*f(x), log(x)/x)
sol = C1/x + C2*x**3 - Rational(1, 16)*log(x)/x - Rational(1, 8)*log(x)**2/x
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq, f(x))
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**2*diff(f(x), x, x) + 3*x*diff(f(x), x) - 8*f(x), log(x)**3 - log(x))
sol = C1/x**4 + C2*x**2 - Rational(1,8)*log(x)**3 - Rational(3,32)*log(x)**2 - Rational(1,64)*log(x) - Rational(7, 256)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**3*diff(f(x), x, x, x) - 3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), log(x))
sol = C1*x + C2*x**2 + C3*x**3 - Rational(1, 6)*log(x) - Rational(11, 36)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
def test_nth_order_linear_euler_eq_nonhomogeneous_variation_of_parameters():
x, t = symbols('x, t')
a, b, c, d = symbols('a, b, c, d', integer=True)
our_hint = "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters"
eq = Eq(x**2*diff(f(x),x,2) - 8*x*diff(f(x),x) + 12*f(x), x**2)
assert our_hint in classify_ode(eq, f(x))
eq = Eq(a*x**3*diff(f(x),x,3) + b*x**2*diff(f(x),x,2) + c*x*diff(f(x),x) + d*f(x), x*log(x))
assert our_hint in classify_ode(eq, f(x))
eq = Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4)
sol = C1*x + C2*x**2 + x**4/6
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), x**3*exp(x))
sol = C1/x**2 + C2*x + x*exp(x)/3 - 4*exp(x)/3 + 8*exp(x)/(3*x) - 8*exp(x)/(3*x**2)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4*exp(x))
sol = C1*x + C2*x**2 + x**2*exp(x) - 2*x*exp(x)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x)
sol = C1*x + C2*x**2 + log(x)/2 + S(3)/4
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
def test_issue_5095():
f = Function('f')
raises(ValueError, lambda: dsolve(f(x).diff(x)**2, f(x), 'separable'))
raises(ValueError, lambda: dsolve(f(x).diff(x)**2, f(x), 'fdsjf'))
def test_almost_linear():
from sympy import Ei
A = Symbol('A', positive=True)
our_hint = 'almost_linear'
f = Function('f')
d = f(x).diff(x)
eq = x**2*f(x)**2*d + f(x)**3 + 1
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol[0].rhs == (C1*exp(3/x) - 1)**(S(1)/3)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = x*f(x)*d + 2*x*f(x)**2 + 1
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol[0].rhs == -sqrt(C1 - 2*Ei(4*x))*exp(-2*x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = x*d + x*f(x) + 1
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol.rhs == (C1 - Ei(x))*exp(-x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
assert our_hint in classify_ode(eq, f(x))
eq = x*exp(f(x))*d + exp(f(x)) + 3*x
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol.rhs == log(C1/x - 3*x/2)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol.rhs == (C1 + Piecewise(
(x, Eq(A + 1, 0)), ((-A*x + A - x - 1)*exp(x)/(A + 1), True)))*exp(-x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_exact_enhancement():
f = Function('f')(x)
df = Derivative(f, x)
eq = f/x**2 + ((f*x - 1)/x)*df
sol = [Eq(f, (i*sqrt(C1*x**2 + 1) + 1)/x) for i in (-1, 1)]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
eq = (x*f - 1) + df*(x**2 - x*f)
sol = [Eq(f, x - sqrt(C1 + x**2 - 2*log(x))),
Eq(f, x + sqrt(C1 + x**2 - 2*log(x)))]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
eq = (x + 2)*sin(f) + df*x*cos(f)
sol = [Eq(f, -asin(C1*exp(-x)/x**2) + pi),
Eq(f, asin(C1*exp(-x)/x**2))]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
@slow
def test_separable_reduced():
f = Function('f')
x = Symbol('x')
df = f(x).diff(x)
eq = (x / f(x))*df + tan(x**2*f(x) / (x**2*f(x) - 1))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
eq = x* df + f(x)* (1 / (x**2*f(x) - 1))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
sol = dsolve(eq, hint = 'separable_reduced', simplify=False)
assert sol.lhs == log(x**2*f(x))/3 + log(x**2*f(x) - S(3)/2)/6
assert sol.rhs == C1 + log(x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = f(x).diff(x) + (f(x) / (x**4*f(x) - x))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
sol = dsolve(eq, hint = 'separable_reduced')
# FIXME: This one hangs
#assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0)] * 4
assert len(sol) == 4
eq = x*df + f(x)*(x**2*f(x))
sol = dsolve(eq, hint = 'separable_reduced', simplify=False)
assert sol == Eq(log(x**2*f(x))/2 - log(x**2*f(x) - 2)/2, C1 + log(x))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_homogeneous_function():
f = Function('f')
eq1 = tan(x + f(x))
eq2 = sin((3*x)/(4*f(x)))
eq3 = cos(3*x/4*f(x))
eq4 = log((3*x + 4*f(x))/(5*f(x) + 7*x))
eq5 = exp((2*x**2)/(3*f(x)**2))
eq6 = log((3*x + 4*f(x))/(5*f(x) + 7*x) + exp((2*x**2)/(3*f(x)**2)))
eq7 = sin((3*x)/(5*f(x) + x**2))
assert homogeneous_order(eq1, x, f(x)) == None
assert homogeneous_order(eq2, x, f(x)) == 0
assert homogeneous_order(eq3, x, f(x)) == None
assert homogeneous_order(eq4, x, f(x)) == 0
assert homogeneous_order(eq5, x, f(x)) == 0
assert homogeneous_order(eq6, x, f(x)) == 0
assert homogeneous_order(eq7, x, f(x)) == None
def test_linear_coeff_match():
from sympy.solvers.ode import _linear_coeff_match
n, d = z*(2*x + 3*f(x) + 5), z*(7*x + 9*f(x) + 11)
rat = n/d
eq1 = sin(rat) + cos(rat.expand())
eq2 = rat
eq3 = log(sin(rat))
ans = (4, -S(13)/3)
assert _linear_coeff_match(eq1, f(x)) == ans
assert _linear_coeff_match(eq2, f(x)) == ans
assert _linear_coeff_match(eq3, f(x)) == ans
# no c
eq4 = (3*x)/f(x)
# not x and f(x)
eq5 = (3*x + 2)/x
# denom will be zero
eq6 = (3*x + 2*f(x) + 1)/(3*x + 2*f(x) + 5)
# not rational coefficient
eq7 = (3*x + 2*f(x) + sqrt(2))/(3*x + 2*f(x) + 5)
assert _linear_coeff_match(eq4, f(x)) is None
assert _linear_coeff_match(eq5, f(x)) is None
assert _linear_coeff_match(eq6, f(x)) is None
assert _linear_coeff_match(eq7, f(x)) is None
def test_linear_coefficients():
f = Function('f')
sol = Eq(f(x), C1/(x**2 + 6*x + 9) - S(3)/2)
eq = f(x).diff(x) + (3 + 2*f(x))/(x + 3)
assert dsolve(eq, hint='linear_coefficients') == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_constantsimp_take_problem():
c = exp(C1) + 2
assert len(Poly(constantsimp(exp(C1) + c + c*x, [C1])).gens) == 2
def test_issue_6879():
f = Function('f')
eq = Eq(Derivative(f(x), x, 2) - 2*Derivative(f(x), x) + f(x), sin(x))
sol = (C1 + C2*x)*exp(x) + cos(x)/2
assert dsolve(eq).rhs == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_issue_6989():
f = Function('f')
k = Symbol('k')
eq = f(x).diff(x) - x*exp(-k*x)
sol = Eq(f(x), C1 + Piecewise(
((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),
(x**2/2, True)
))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = -f(x).diff(x) + x*exp(-k*x)
sol = Eq(f(x), C1 + Piecewise(
((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),
(+x**2/2, True)
))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_heuristic1():
y, a, b, c, a4, a3, a2, a1, a0 = symbols("y a b c a4 a3 a2 a1 a0")
y = Symbol('y')
f = Function('f')
xi = Function('xi')
eta = Function('eta')
df = f(x).diff(x)
eq = Eq(df, x**2*f(x))
eq1 = f(x).diff(x) + a*f(x) - c*exp(b*x)
eq2 = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2)
eq3 = (1 + 2*x)*df + 2 - 4*exp(-f(x))
eq4 = f(x).diff(x) - (a4*x**4 + a3*x**3 + a2*x**2 + a1*x + a0)**(S(-1)/2)
eq5 = x**2*df - f(x) + x**2*exp(x - (1/x))
eqlist = [eq, eq1, eq2, eq3, eq4, eq5]
i = infinitesimals(eq, hint='abaco1_simple')
assert i == [{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0},
{eta(x, f(x)): f(x), xi(x, f(x)): 0},
{eta(x, f(x)): 0, xi(x, f(x)): x**(-2)}]
i1 = infinitesimals(eq1, hint='abaco1_simple')
assert i1 == [{eta(x, f(x)): exp(-a*x), xi(x, f(x)): 0}]
i2 = infinitesimals(eq2, hint='abaco1_simple')
assert i2 == [{eta(x, f(x)): exp(-x**2), xi(x, f(x)): 0}]
i3 = infinitesimals(eq3, hint='abaco1_simple')
assert i3 == [{eta(x, f(x)): 0, xi(x, f(x)): 2*x + 1},
{eta(x, f(x)): 0, xi(x, f(x)): 1/(exp(f(x)) - 2)}]
i4 = infinitesimals(eq4, hint='abaco1_simple')
assert i4 == [{eta(x, f(x)): 1, xi(x, f(x)): 0},
{eta(x, f(x)): 0,
xi(x, f(x)): sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4)}]
i5 = infinitesimals(eq5, hint='abaco1_simple')
assert i5 == [{xi(x, f(x)): 0, eta(x, f(x)): exp(-1/x)}]
ilist = [i, i1, i2, i3, i4, i5]
for eq, i in (zip(eqlist, ilist)):
check = checkinfsol(eq, i)
assert check[0]
def test_issue_6247():
eq = x**2*f(x)**2 + x*Derivative(f(x), x)
sol = Eq(f(x), 2*C1/(C1*x**2 - 1))
assert dsolve(eq, hint = 'separable_reduced') == sol
assert checkodesol(eq, sol, order=1)[0]
eq = f(x).diff(x, x) + 4*f(x)
sol = Eq(f(x), C1*sin(2*x) + C2*cos(2*x))
assert dsolve(eq) == sol
assert checkodesol(eq, sol, order=1)[0]
def test_heuristic2():
y = Symbol('y')
xi = Function('xi')
eta = Function('eta')
df = f(x).diff(x)
# This ODE can be solved by the Lie Group method, when there are
# better assumptions
eq = df - (f(x)/x)*(x*log(x**2/f(x)) + 2)
i = infinitesimals(eq, hint='abaco1_product')
assert i == [{eta(x, f(x)): f(x)*exp(-x), xi(x, f(x)): 0}]
assert checkinfsol(eq, i)[0]
@slow
def test_heuristic3():
y = Symbol('y')
xi = Function('xi')
eta = Function('eta')
a, b = symbols("a b")
df = f(x).diff(x)
eq = x**2*df + x*f(x) + f(x)**2 + x**2
i = infinitesimals(eq, hint='bivariate')
assert i == [{eta(x, f(x)): f(x), xi(x, f(x)): x}]
assert checkinfsol(eq, i)[0]
eq = x**2*(-f(x)**2 + df)- a*x**2*f(x) + 2 - a*x
i = infinitesimals(eq, hint='bivariate')
assert checkinfsol(eq, i)[0]
def test_heuristic_4():
y, a = symbols("y a")
xi = Function('xi')
eta = Function('eta')
eq = x*(f(x).diff(x)) + 1 - f(x)**2
i = infinitesimals(eq, hint='chi')
assert checkinfsol(eq, i)[0]
def test_heuristic_function_sum():
xi = Function('xi')
eta = Function('eta')
eq = f(x).diff(x) - (3*(1 + x**2/f(x)**2)*atan(f(x)/x) + (1 - 2*f(x))/x +
(1 - 3*f(x))*(x/f(x)**2))
i = infinitesimals(eq, hint='function_sum')
assert i == [{eta(x, f(x)): f(x)**(-2) + x**(-2), xi(x, f(x)): 0}]
assert checkinfsol(eq, i)[0]
def test_heuristic_abaco2_similar():
xi = Function('xi')
eta = Function('eta')
F = Function('F')
a, b = symbols("a b")
eq = f(x).diff(x) - F(a*x + b*f(x))
i = infinitesimals(eq, hint='abaco2_similar')
assert i == [{eta(x, f(x)): -a/b, xi(x, f(x)): 1}]
assert checkinfsol(eq, i)[0]
eq = f(x).diff(x) - (f(x)**2 / (sin(f(x) - x) - x**2 + 2*x*f(x)))
i = infinitesimals(eq, hint='abaco2_similar')
assert i == [{eta(x, f(x)): f(x)**2, xi(x, f(x)): f(x)**2}]
assert checkinfsol(eq, i)[0]
def test_heuristic_abaco2_unique_unknown():
xi = Function('xi')
eta = Function('eta')
F = Function('F')
a, b = symbols("a b")
x = Symbol("x", positive=True)
eq = f(x).diff(x) - x**(a - 1)*(f(x)**(1 - b))*F(x**a/a + f(x)**b/b)
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert i == [{eta(x, f(x)): -f(x)*f(x)**(-b), xi(x, f(x)): x*x**(-a)}]
assert checkinfsol(eq, i)[0]
eq = f(x).diff(x) + tan(F(x**2 + f(x)**2) + atan(x/f(x)))
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert i == [{eta(x, f(x)): x, xi(x, f(x)): -f(x)}]
assert checkinfsol(eq, i)[0]
eq = (x*f(x).diff(x) + f(x) + 2*x)**2 -4*x*f(x) -4*x**2 -4*a
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert checkinfsol(eq, i)[0]
def test_heuristic_linear():
xi = Function('xi')
eta = Function('eta')
F = Function('F')
a, b, m, n = symbols("a b m n")
eq = x**(n*(m + 1) - m)*(f(x).diff(x)) - a*f(x)**n -b*x**(n*(m + 1))
i = infinitesimals(eq, hint='linear')
assert checkinfsol(eq, i)[0]
@XFAIL
def test_kamke():
a, b, alpha, c = symbols("a b alpha c")
eq = x**2*(a*f(x)**2+(f(x).diff(x))) + b*x**alpha + c
i = infinitesimals(eq, hint='sum_function')
assert checkinfsol(eq, i)[0]
def test_series():
# FIXME: Maybe there should be a way to check series solutions
# checkodesol doesn't work with them.
C1 = Symbol("C1")
eq = f(x).diff(x) - f(x)
assert dsolve(eq, hint='1st_power_series') == Eq(f(x),
C1 + C1*x + C1*x**2/2 + C1*x**3/6 + C1*x**4/24 +
C1*x**5/120 + O(x**6))
eq = f(x).diff(x) - x*f(x)
assert dsolve(eq, hint='1st_power_series') == Eq(f(x),
C1*x**4/8 + C1*x**2/2 + C1 + O(x**6))
eq = f(x).diff(x) - sin(x*f(x))
sol = Eq(f(x), (x - 2)**2*(1+ sin(4))*cos(4) + (x - 2)*sin(4) + 2 + O(x**3))
assert dsolve(eq, hint='1st_power_series', ics={f(2): 2}, n=3) == sol
@slow
def test_lie_group():
C1 = Symbol("C1")
x = Symbol("x") # assuming x is real generates an error!
a, b, c = symbols("a b c")
eq = f(x).diff(x)**2
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol)[0]
eq = Eq(f(x).diff(x), x**2*f(x))
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), C1*exp(x**3)**(S(1)/3))
assert checkodesol(eq, sol)[0]
eq = f(x).diff(x) + a*f(x) - c*exp(b*x)
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol)[0]
eq = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2)
sol = dsolve(eq, f(x), hint='lie_group')
actual_sol = Eq(f(x), (C1 + x**2/2)*exp(-x**2))
errstr = str(eq)+' : '+str(sol)+' == '+str(actual_sol)
assert sol == actual_sol, errstr
assert checkodesol(eq, sol)[0]
eq = (1 + 2*x)*(f(x).diff(x)) + 2 - 4*exp(-f(x))
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), log(C1/(2*x + 1) + 2))
assert checkodesol(eq, sol)[0]
eq = x**2*(f(x).diff(x)) - f(x) + x**2*exp(x - (1/x))
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol)[0]
eq = x**2*f(x)**2 + x*Derivative(f(x), x)
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), 2/(C1 + x**2))
assert checkodesol(eq, sol)[0]
@XFAIL
def test_lie_group_issue15219():
eqn = exp(f(x).diff(x)-f(x))
assert 'lie_group' not in classify_ode(eqn, f(x))
def test_user_infinitesimals():
x = Symbol("x") # assuming x is real generates an error
eq = x*(f(x).diff(x)) + 1 - f(x)**2
sol = Eq(f(x), (C1 + x**2)/(C1 - x**2))
infinitesimals = {'xi':sqrt(f(x) - 1)/sqrt(f(x) + 1), 'eta':0}
assert dsolve(eq, hint='lie_group', **infinitesimals) == sol
assert checkodesol(eq, sol) == (True, 0)
raises(ValueError, lambda: dsolve(eq, hint='lie_group', xi=0, eta=f(x)))
def test_issue_7081():
eq = x*(f(x).diff(x)) + 1 - f(x)**2
s = Eq(f(x), -1/(-C1 + x**2)*(C1 + x**2))
assert dsolve(eq) == s
assert checkodesol(eq, s) == (True, 0)
@slow
def test_2nd_power_series_ordinary():
# FIXME: Maybe there should be a way to check series solutions
# checkodesol doesn't work with them.
C1, C2 = symbols("C1 C2")
eq = f(x).diff(x, 2) - x*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq) == Eq(f(x),
C2*(x**3/6 + 1) + C1*x*(x**3/12 + 1) + O(x**6))
assert dsolve(eq, x0=-2) == Eq(f(x),
C2*((x + 2)**4/6 + (x + 2)**3/6 - (x + 2)**2 + 1)
+ C1*(x + (x + 2)**4/12 - (x + 2)**3/3 + S(2))
+ O(x**6))
assert dsolve(eq, n=2) == Eq(f(x), C2*x + C1 + O(x**2))
eq = (1 + x**2)*(f(x).diff(x, 2)) + 2*x*(f(x).diff(x)) -2*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq) == Eq(f(x), C2*(-x**4/3 + x**2 + 1) + C1*x
+ O(x**6))
eq = f(x).diff(x, 2) + x*(f(x).diff(x)) + f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq) == Eq(f(x), C2*(
x**4/8 - x**2/2 + 1) + C1*x*(-x**2/3 + 1) + O(x**6))
eq = f(x).diff(x, 2) + f(x).diff(x) - x*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq) == Eq(f(x), C2*(
-x**4/24 + x**3/6 + 1) + C1*x*(x**3/24 + x**2/6 - x/2
+ 1) + O(x**6))
eq = f(x).diff(x, 2) + x*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq, n=7) == Eq(f(x), C2*(
x**6/180 - x**3/6 + 1) + C1*x*(-x**3/12 + 1) + O(x**7))
def test_2nd_power_series_regular():
# FIXME: Maybe there should be a way to check series solutions
# checkodesol doesn't work with them.
C1, C2 = symbols("C1 C2")
eq = x**2*(f(x).diff(x, 2)) - 3*x*(f(x).diff(x)) + (4*x + 4)*f(x)
assert dsolve(eq) == Eq(f(x), C1*x**2*(-16*x**3/9 +
4*x**2 - 4*x + 1) + O(x**6))
eq = 4*x**2*(f(x).diff(x, 2)) -8*x**2*(f(x).diff(x)) + (4*x**2 +
1)*f(x)
assert dsolve(eq) == Eq(f(x), C1*sqrt(x)*(
x**4/24 + x**3/6 + x**2/2 + x + 1) + O(x**6))
eq = x**2*(f(x).diff(x, 2)) - x**2*(f(x).diff(x)) + (
x**2 - 2)*f(x)
assert dsolve(eq) == Eq(f(x), C1*(-x**6/720 - 3*x**5/80 - x**4/8 +
x**2/2 + x/2 + 1)/x + C2*x**2*(-x**3/60 + x**2/20 + x/2 + 1)
+ O(x**6))
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - S(1)/4)*f(x)
assert dsolve(eq) == Eq(f(x), C1*(x**4/24 - x**2/2 + 1)/sqrt(x) +
C2*sqrt(x)*(x**4/120 - x**2/6 + 1) + O(x**6))
eq = x*(f(x).diff(x, 2)) - f(x).diff(x) + 4*x**3*f(x)
assert dsolve(eq) == Eq(f(x), C2*(-x**4/2 + 1) + C1*x**2 + O(x**6))
def test_issue_7093():
x = Symbol("x") # assuming x is real leads to an error
sol = [Eq(f(x), C1 - 2*x*sqrt(x**3)/5),
Eq(f(x), C1 + 2*x*sqrt(x**3)/5)]
eq = Derivative(f(x), x)**2 - x**3
assert set(dsolve(eq)) == set(sol)
assert checkodesol(eq, sol) == [(True, 0)] * 2
def test_dsolve_linsystem_symbol():
eps = Symbol('epsilon', positive=True)
eq1 = (Eq(diff(f(x), x), -eps*g(x)), Eq(diff(g(x), x), eps*f(x)))
sol1 = [Eq(f(x), -C1*eps*cos(eps*x) - C2*eps*sin(eps*x)),
Eq(g(x), -C1*eps*sin(eps*x) + C2*eps*cos(eps*x))]
assert checksysodesol(eq1, sol1) == (True, [0, 0])
def test_C1_function_9239():
t = Symbol('t')
C1 = Function('C1')
C2 = Function('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
eq = (Eq(diff(C1(t), t), 9*C2(t)), Eq(diff(C2(t), t), 12*C1(t)))
sol = [Eq(C1(t), 9*C3*exp(6*sqrt(3)*t) + 9*C4*exp(-6*sqrt(3)*t)),
Eq(C2(t), 6*sqrt(3)*C3*exp(6*sqrt(3)*t) - 6*sqrt(3)*C4*exp(-6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_issue_15056():
t = Symbol('t')
C3 = Symbol('C3')
assert get_numbered_constants(Symbol('C1') * Function('C2')(t)) == C3
def test_issue_10379():
t,y = symbols('t,y')
eq = f(t).diff(t)-(1-51.05*y*f(t))
sol = Eq(f(t), (0.019588638589618*exp(y*(C1 - 51.05*t)) + 0.019588638589618)/y)
dsolve_sol = dsolve(eq, rational=False)
assert str(dsolve_sol) == str(sol)
assert checkodesol(eq, dsolve_sol)[0]
def test_issue_10867():
x = Symbol('x')
eq = Eq(g(x).diff(x).diff(x), (x-2)**2 + (x-3)**3)
sol = Eq(g(x), C1 + C2*x + x**5/20 - 2*x**4/3 + 23*x**3/6 - 23*x**2/2)
assert dsolve(eq, g(x)) == sol
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
def test_issue_11290():
eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
sol_1 = dsolve(eq, f(x), simplify=False, hint='1st_exact_Integral')
sol_0 = dsolve(eq, f(x), simplify=False, hint='1st_exact')
assert sol_1.dummy_eq(Eq(Subs(
Integral(u**2 - x*sin(u) - Integral(-sin(u), x), u) +
Integral(cos(u), x), u, f(x)), C1))
assert sol_1.doit() == sol_0
assert checkodesol(eq, sol_0, order=1, solve_for_func=False)
assert checkodesol(eq, sol_1, order=1, solve_for_func=False)
def test_issue_4838():
# Issue #15999
eq = f(x).diff(x) - C1*f(x)
sol = Eq(f(x), C2*exp(C1*x))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False) == (True, 0)
# Issue #13691
eq = f(x).diff(x) - C1*g(x).diff(x)
sol = Eq(f(x), C2 + C1*g(x))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, f(x), order=1, solve_for_func=False) == (True, 0)
# Issue #4838
eq = f(x).diff(x) - 3*C1 - 3*x**2
sol = Eq(f(x), C2 + 3*C1*x + x**3)
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False) == (True, 0)
@slow
def test_issue_14395():
eq = Derivative(f(x), x, x) + 9*f(x) - sec(x)
sol = Eq(f(x), (C1 - x/3 + sin(2*x)/3)*sin(3*x) + (C2 + log(cos(x))
- 2*log(cos(x)**2)/3 + 2*cos(x)**2/3)*cos(3*x))
assert dsolve(eq, f(x)) == sol
# FIXME: assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
def test_sysode_linear_neq_order1():
from sympy.abc import t
Z0 = Function('Z0')
Z1 = Function('Z1')
Z2 = Function('Z2')
Z3 = Function('Z3')
k01, k10, k20, k21, k23, k30 = symbols('k01 k10 k20 k21 k23 k30')
eq = (Eq(Derivative(Z0(t), t), -k01*Z0(t) + k10*Z1(t) + k20*Z2(t) + k30*Z3(t)), Eq(Derivative(Z1(t), t),
k01*Z0(t) - k10*Z1(t) + k21*Z2(t)), Eq(Derivative(Z2(t), t), -(k20 + k21 + k23)*Z2(t)), Eq(Derivative(Z3(t),
t), k23*Z2(t) - k30*Z3(t)))
sols_eq = [Eq(Z0(t), C1*k10/k01 + C2*(-k10 + k30)*exp(-k30*t)/(k01 + k10 - k30) - C3*exp(t*(-
k01 - k10)) + C4*(k10*k20 + k10*k21 - k10*k30 - k20**2 - k20*k21 - k20*k23 + k20*k30 +
k23*k30)*exp(t*(-k20 - k21 - k23))/(k23*(k01 + k10 - k20 - k21 - k23))),
Eq(Z1(t), C1 - C2*k01*exp(-k30*t)/(k01 + k10 - k30) + C3*exp(t*(-k01 - k10)) + C4*(k01*k20 + k01*k21
- k01*k30 - k20*k21 - k21**2 - k21*k23 + k21*k30)*exp(t*(-k20 - k21 - k23))/(k23*(k01 + k10 - k20 -
k21 - k23))),
Eq(Z2(t), C4*(-k20 - k21 - k23 + k30)*exp(t*(-k20 - k21 - k23))/k23),
Eq(Z3(t), C2*exp(-k30*t) + C4*exp(t*(-k20 - k21 - k23)))]
assert dsolve(eq, simplify=False) == sols_eq
assert checksysodesol(eq, sols_eq) == (True, [0, 0, 0, 0])
def test_nth_order_reducible():
from sympy.solvers.ode import _nth_order_reducible_match
eqn = Eq(x*Derivative(f(x), x)**2 + Derivative(f(x), x, 2), 0)
sol = Eq(f(x),
C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) + sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert sol == dsolve(eqn, f(x), hint='nth_order_reducible')
assert sol == dsolve(eqn, f(x))
F = lambda eq: _nth_order_reducible_match(eq, f(x))
D = Derivative
assert F(D(y*f(x), x, y) + D(f(x), x)) is None
assert F(D(y*f(y), y, y) + D(f(y), y)) is None
assert F(f(x)*D(f(x), x) + D(f(x), x, 2)) is None
assert F(D(x*f(y), y, 2) + D(u*y*f(x), x, 3)) is None # no simplification by design
assert F(D(f(y), y, 2) + D(f(y), y, 3) + D(f(x), x, 4)) is None
assert F(D(f(x), x, 2) + D(f(x), x, 3)) == dict(n=2)
eqn = -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert sol == dsolve(eqn, f(x))
assert sol == dsolve(eqn, f(x), hint='nth_order_reducible')
eqn = Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(2**(S(3)/4)*x/2) + C3*cos(2**(S(3)/4)*x/2))
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert sol == dsolve(eqn, f(x))
assert sol == dsolve(eqn, f(x), hint='nth_order_reducible')
eqn = f(x).diff(x, 2) + 2*f(x).diff(x)
sol = Eq(f(x), C1 + C2*exp(-2*x))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='nth_order_reducible') in (sol, sols)
eqn = f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x)
sol = Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='nth_order_reducible') in (sol, sols)
eqn = f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
4*f(x).diff(x)
sol = Eq(f(x), C1 + C2*exp(x) + C3*exp(-2*x) + C4*exp(2*x))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='nth_order_reducible') in (sol, sols)
eqn = f(x).diff(x, 4) + 3*f(x).diff(x, 3)
sol = Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='nth_order_reducible') in (sol, sols)
eqn = f(x).diff(x, 4) - 2*f(x).diff(x, 2)
sol = Eq(f(x), C1 + C2*x + C3*exp(x*sqrt(2)) + C4*exp(-x*sqrt(2)))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='nth_order_reducible') in (sol, sols)
eqn = f(x).diff(x, 4) + 4*f(x).diff(x, 2)
sol = Eq(f(x), C1 + C2*sin(2*x) + C3*cos(2*x) + C4*x)
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='nth_order_reducible') in (sol, sols)
eqn = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x)
# These are equivalent:
sol1 = Eq(f(x), C1 + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))
sol2 = Eq(f(x), C1 + C2*(x*sin(x) + cos(x)) + C3*(-x*cos(x) + sin(x)) + C4*sin(x) + C5*cos(x))
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
assert checkodesol(eqn, sol1, order=2, solve_for_func=False) == (True, 0)
assert checkodesol(eqn, sol2, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol1, sol1s)
assert dsolve(eqn, f(x), hint='nth_order_reducible') in (sol2, sol2s)
# In this case the reduced ODE has two distinct solutions
eqn = f(x).diff(x, 2) - f(x).diff(x)**3
sol = [Eq(f(x), C2 - sqrt(2)*I*(C1 + x)*sqrt(1/(C1 + x))),
Eq(f(x), C2 + sqrt(2)*I*(C1 + x)*sqrt(1/(C1 + x)))]
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == [(True, 0), (True, 0)]
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='nth_order_reducible') in (sol, sols)
def test_nth_algebraic():
eqn = Eq(Derivative(f(x), x), Derivative(g(x), x))
sol = Eq(f(x), C1 + g(x))
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = (diff(f(x)) - x)*(diff(f(x)) + x)
sol = [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)]
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = (1 - sin(f(x))) * f(x).diff(x)
sol = Eq(f(x), C1)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
M, m, r, t = symbols('M m r t')
phi = Function('phi')
eqn = Eq(-M * phi(t).diff(t),
Rational(3, 2) * m * r**2 * phi(t).diff(t) * phi(t).diff(t,t))
solns = [Eq(phi(t), C1), Eq(phi(t), C1 + C2*t - M*t**2/(3*m*r**2))]
assert checkodesol(eqn, solns[0], order=2, solve_for_func=False)[0]
assert checkodesol(eqn, solns[1], order=2, solve_for_func=False)[0]
assert set(solns) == set(dsolve(eqn, phi(t), hint='nth_algebraic'))
assert set(solns) == set(dsolve(eqn, phi(t)))
eqn = f(x) * f(x).diff(x) * f(x).diff(x, x)
sol = Eq(f(x), C1 + C2*x)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1)
sol = Eq(f(x), C1 + C2*x)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1) * (f(x).diff(x) - x)
solns = [Eq(f(x), C1 + x**2/2), Eq(f(x), C1 + C2*x)]
assert checkodesol(eqn, solns[0], order=2, solve_for_func=False)[0]
assert checkodesol(eqn, solns[1], order=2, solve_for_func=False)[0]
assert set(solns) == set(dsolve(eqn, f(x), hint='nth_algebraic'))
assert set(solns) == set(dsolve(eqn, f(x)))
def test_nth_algebraic_issue15999():
eqn = f(x).diff(x) - C1
sol = Eq(f(x), C1*x + C2) # Correct solution
assert checkodesol(eqn, sol, order=1, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x), hint='nth_algebraic') == sol
assert dsolve(eqn, f(x)) == sol
def test_nth_algebraic_redundant_solutions():
# This one has a redundant solution that should be removed
eqn = f(x)*f(x).diff(x)
soln = Eq(f(x), C1)
assert checkodesol(eqn, soln, order=1, solve_for_func=False)[0]
assert soln == dsolve(eqn, f(x), hint='nth_algebraic')
assert soln == dsolve(eqn, f(x))
# This has two integral solutions and no algebraic solutions
eqn = (diff(f(x)) - x)*(diff(f(x)) + x)
sol = [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)]
assert all(c[0] for c in checkodesol(eqn, sol, order=1, solve_for_func=False))
assert set(sol) == set(dsolve(eqn, f(x), hint='nth_algebraic'))
assert set(sol) == set(dsolve(eqn, f(x)))
# This one doesn't work with dsolve at the time of writing but the
# redundancy checking code should not remove the algebraic solution.
from sympy.solvers.ode import _nth_algebraic_remove_redundant_solutions
eqn = f(x) + f(x)*f(x).diff(x)
solns = [Eq(f(x), 0),
Eq(f(x), C1 - x)]
solns_final = _nth_algebraic_remove_redundant_solutions(eqn, solns, 1, x)
assert all(c[0] for c in checkodesol(eqn, solns, order=1, solve_for_func=False))
assert set(solns) == set(solns_final)
solns = [Eq(f(x), exp(x)),
Eq(f(x), C1*exp(C2*x))]
solns_final = _nth_algebraic_remove_redundant_solutions(eqn, solns, 2, x)
assert solns_final == [Eq(f(x), C1*exp(C2*x))]
# This one needs a substitution f' = g.
eqn = -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x))
#
# These tests can be combined with the above test if they get fixed
# so that dsolve actually works in all these cases.
#
# Fails due to division by f(x) eliminating the solution before nth_algebraic
# is called.
@XFAIL
def test_nth_algebraic_find_multiple1():
eqn = f(x) + f(x)*f(x).diff(x)
solns = [Eq(f(x), 0),
Eq(f(x), C1 - x)]
assert all(c[0] for c in checkodesol(eqn, solns, order=1, solve_for_func=False))
assert set(solns) == set(dsolve(eqn, f(x)))
# prep = True breaks this
def test_nth_algebraic_noprep1():
eqn = Derivative(x*f(x), x, x, x)
sol = Eq(f(x), (C1 + C2*x + C3*x**2) / x)
assert checkodesol(eqn, sol, order=3, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=False, hint='nth_algebraic')
@XFAIL
def test_nth_algebraic_prep1():
eqn = Derivative(x*f(x), x, x, x)
sol = Eq(f(x), (C1 + C2*x + C3*x**2) / x)
assert checkodesol(eqn, sol, order=3, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=True, hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
# prep = True breaks this
def test_nth_algebraic_noprep2():
eqn = Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x))
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=False, hint='nth_algebraic')
@XFAIL
def test_nth_algebraic_prep2():
eqn = Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x))
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=True, hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
# This needs a combination of solutions from nth_algebraic and some other
# method from dsolve
@XFAIL
def test_nth_algebraic_find_multiple2():
eqn = f(x)**2 + f(x)*f(x).diff(x)
solns = [Eq(f(x), 0),
Eq(f(x), C1*exp(-x))]
assert all(c[0] for c in checkodesol(eqn, solns, order=1, solve_for_func=False))
assert set(solns) == dsolve(eqn, f(x))
# Needs to be a way to know how to combine derivatives in the expression
@XFAIL
def test_factoring_ode():
eqn = Derivative(x*f(x), x, x, x) + Derivative(f(x), x, x, x)
soln = Eq(f(x), (C1*x**2/2 + C2*x + C3 - x)/(1 + x))
assert checkodesol(eqn, soln, order=2, solve_for_func=False)[0]
assert soln == dsolve(eqn, f(x))
def test_issue_15913():
eq = -C1/x - 2*x*f(x) - f(x) + Derivative(f(x), x)
sol = C2*exp(x**2 + x) + exp(x**2 + x)*Integral(C1*exp(-x**2 - x)/x, x)
assert checkodesol(eq, sol) == (True, 0)
sol = C1 + C2*exp(-x*y)
eq = Derivative(y*f(x), x) + f(x).diff(x, 2)
assert checkodesol(eq, sol, f(x)) == (True, 0)
def test_issue_16146():
raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x), g(x), h(x)]))
raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x)]))
|
a60e83217d1ef6b348daf6545b35bc29e0b3f4b663e650acea8fa7f7f48007ea | from sympy import Symbol, Function, Derivative as D, Eq, cos, sin
from sympy.utilities.pytest import raises
from sympy.calculus.euler import euler_equations as euler
def test_euler_interface():
x = Function('x')
y = Symbol('y')
t = Symbol('t')
raises(TypeError, lambda: euler())
raises(TypeError, lambda: euler(D(x(t), t)*y(t), [x(t), y]))
raises(ValueError, lambda: euler(D(x(t), t)*x(y), [x(t), x(y)]))
raises(TypeError, lambda: euler(D(x(t), t)**2, x(0)))
assert euler(D(x(t), t)**2/2, {x(t)}) == [Eq(-D(x(t), t, t), 0)]
assert euler(D(x(t), t)**2/2, x(t), {t}) == [Eq(-D(x(t), t, t), 0)]
def test_euler_pendulum():
x = Function('x')
t = Symbol('t')
L = D(x(t), t)**2/2 + cos(x(t))
assert euler(L, x(t), t) == [Eq(-sin(x(t)) - D(x(t), t, t), 0)]
def test_euler_henonheiles():
x = Function('x')
y = Function('y')
t = Symbol('t')
L = sum(D(z(t), t)**2/2 - z(t)**2/2 for z in [x, y])
L += -x(t)**2*y(t) + y(t)**3/3
assert euler(L, [x(t), y(t)], t) == [Eq(-2*x(t)*y(t) - x(t) -
D(x(t), t, t), 0),
Eq(-x(t)**2 + y(t)**2 -
y(t) - D(y(t), t, t), 0)]
def test_euler_sineg():
psi = Function('psi')
t = Symbol('t')
x = Symbol('x')
L = D(psi(t, x), t)**2/2 - D(psi(t, x), x)**2/2 + cos(psi(t, x))
assert euler(L, psi(t, x), [t, x]) == [Eq(-sin(psi(t, x)) -
D(psi(t, x), t, t) +
D(psi(t, x), x, x), 0)]
def test_euler_high_order():
# an example from hep-th/0309038
m = Symbol('m')
k = Symbol('k')
x = Function('x')
y = Function('y')
t = Symbol('t')
L = (m*D(x(t), t)**2/2 + m*D(y(t), t)**2/2 -
k*D(x(t), t)*D(y(t), t, t) + k*D(y(t), t)*D(x(t), t, t))
assert euler(L, [x(t), y(t)]) == [Eq(2*k*D(y(t), t, t, t) -
m*D(x(t), t, t), 0),
Eq(-2*k*D(x(t), t, t, t) -
m*D(y(t), t, t), 0)]
w = Symbol('w')
L = D(x(t, w), t, w)**2/2
assert euler(L) == [Eq(D(x(t, w), t, t, w, w), 0)]
|
5415e6f445c0a15b3366150e8f5dc19aa3580e911168646320ef629488e93403 | from __future__ import (absolute_import, division, print_function)
import os
import shutil
import subprocess
import sys
import tempfile
import warnings
from distutils.errors import CompileError
from distutils.sysconfig import get_config_var
from .util import (
get_abspath, make_dirs, copy, Glob, ArbitraryDepthGlob,
glob_at_depth, import_module_from_file, pyx_is_cplus,
sha256_of_string, sha256_of_file
)
from .runners import (
CCompilerRunner,
CppCompilerRunner,
FortranCompilerRunner
)
sharedext = get_config_var('EXT_SUFFIX' if sys.version_info >= (3, 3) else 'SO')
if os.name == 'posix':
objext = '.o'
elif os.name == 'nt':
objext = '.obj'
else:
warnings.warn("Unknown os.name: {}".format(os.name))
objext = '.o'
def compile_sources(files, Runner=None, destdir=None, cwd=None, keep_dir_struct=False,
per_file_kwargs=None, **kwargs):
""" Compile source code files to object files.
Parameters
==========
files : iterable of str
Paths to source files, if ``cwd`` is given, the paths are taken as relative.
Runner: CompilerRunner subclass (optional)
Could be e.g. ``FortranCompilerRunner``. Will be inferred from filename
extensions if missing.
destdir: str
Output directory, if cwd is given, the path is taken as relative.
cwd: str
Working directory. Specify to have compiler run in other directory.
also used as root of relative paths.
keep_dir_struct: bool
Reproduce directory structure in `destdir`. default: ``False``
per_file_kwargs: dict
Dict mapping instances in ``files`` to keyword arguments.
\\*\\*kwargs: dict
Default keyword arguments to pass to ``Runner``.
"""
_per_file_kwargs = {}
if per_file_kwargs is not None:
for k, v in per_file_kwargs.items():
if isinstance(k, Glob):
for path in glob.glob(k.pathname):
_per_file_kwargs[path] = v
elif isinstance(k, ArbitraryDepthGlob):
for path in glob_at_depth(k.filename, cwd):
_per_file_kwargs[path] = v
else:
_per_file_kwargs[k] = v
# Set up destination directory
destdir = destdir or '.'
if not os.path.isdir(destdir):
if os.path.exists(destdir):
raise IOError("{} is not a directory".format(destdir))
else:
make_dirs(destdir)
if cwd is None:
cwd = '.'
for f in files:
copy(f, destdir, only_update=True, dest_is_dir=True)
# Compile files and return list of paths to the objects
dstpaths = []
for f in files:
if keep_dir_struct:
name, ext = os.path.splitext(f)
else:
name, ext = os.path.splitext(os.path.basename(f))
file_kwargs = kwargs.copy()
file_kwargs.update(_per_file_kwargs.get(f, {}))
dstpaths.append(src2obj(f, Runner, cwd=cwd, **file_kwargs))
return dstpaths
def get_mixed_fort_c_linker(vendor=None, cplus=False, cwd=None):
vendor = vendor or os.environ.get('SYMPY_COMPILER_VENDOR', 'gnu')
if vendor.lower() == 'intel':
if cplus:
return (FortranCompilerRunner,
{'flags': ['-nofor_main', '-cxxlib']}, vendor)
else:
return (FortranCompilerRunner,
{'flags': ['-nofor_main']}, vendor)
elif vendor.lower() == 'gnu' or 'llvm':
if cplus:
return (CppCompilerRunner,
{'lib_options': ['fortran']}, vendor)
else:
return (FortranCompilerRunner,
{}, vendor)
else:
raise ValueError("No vendor found.")
def link(obj_files, out_file=None, shared=False, Runner=None,
cwd=None, cplus=False, fort=False, **kwargs):
""" Link object files.
Parameters
==========
obj_files: iterable of str
Paths to object files.
out_file: str (optional)
Path to executable/shared library, if ``None`` it will be
deduced from the last item in obj_files.
shared: bool
Generate a shared library?
Runner: CompilerRunner subclass (optional)
If not given the ``cplus`` and ``fort`` flags will be inspected
(fallback is the C compiler).
cwd: str
Path to the root of relative paths and working directory for compiler.
cplus: bool
C++ objects? default: ``False``.
fort: bool
Fortran objects? default: ``False``.
\\*\\*kwargs: dict
Keyword arguments passed to ``Runner``.
Returns
=======
The absolute path to the generated shared object / executable.
"""
if out_file is None:
out_file, ext = os.path.splitext(os.path.basename(obj_files[-1]))
if shared:
out_file += sharedext
if not Runner:
if fort:
Runner, extra_kwargs, vendor = \
get_mixed_fort_c_linker(
vendor=kwargs.get('vendor', None),
cplus=cplus,
cwd=cwd,
)
for k, v in extra_kwargs.items():
if k in kwargs:
kwargs[k].expand(v)
else:
kwargs[k] = v
else:
if cplus:
Runner = CppCompilerRunner
else:
Runner = CCompilerRunner
flags = kwargs.pop('flags', [])
if shared:
if '-shared' not in flags:
flags.append('-shared')
run_linker = kwargs.pop('run_linker', True)
if not run_linker:
raise ValueError("run_linker was set to False (nonsensical).")
out_file = get_abspath(out_file, cwd=cwd)
runner = Runner(obj_files, out_file, flags, cwd=cwd, **kwargs)
runner.run()
return out_file
def link_py_so(obj_files, so_file=None, cwd=None, libraries=None,
cplus=False, fort=False, **kwargs):
""" Link python extension module (shared object) for importing
Parameters
==========
obj_files: iterable of str
Paths to object files to be linked.
so_file: str
Name (path) of shared object file to create. If not specified it will
have the basname of the last object file in `obj_files` but with the
extension '.so' (Unix).
cwd: path string
Root of relative paths and working directory of linker.
libraries: iterable of strings
Libraries to link against, e.g. ['m'].
cplus: bool
Any C++ objects? default: ``False``.
fort: bool
Any Fortran objects? default: ``False``.
kwargs**: dict
Keyword arguments passed to ``link(...)``.
Returns
=======
Absolute path to the generate shared object.
"""
libraries = libraries or []
include_dirs = kwargs.pop('include_dirs', [])
library_dirs = kwargs.pop('library_dirs', [])
# from distutils/command/build_ext.py:
if sys.platform == "win32":
warnings.warn("Windows not yet supported.")
elif sys.platform == 'darwin':
# Don't use the default code below
pass
elif sys.platform[:3] == 'aix':
# Don't use the default code below
pass
else:
from distutils import sysconfig
if sysconfig.get_config_var('Py_ENABLE_SHARED'):
ABIFLAGS = sysconfig.get_config_var('ABIFLAGS')
pythonlib = 'python{}.{}{}'.format(
sys.hexversion >> 24, (sys.hexversion >> 16) & 0xff,
ABIFLAGS or '')
libraries += [pythonlib]
else:
pass
flags = kwargs.pop('flags', [])
needed_flags = ('-pthread',)
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
return link(obj_files, shared=True, flags=flags, cwd=cwd,
cplus=cplus, fort=fort, include_dirs=include_dirs,
libraries=libraries, library_dirs=library_dirs, **kwargs)
def simple_cythonize(src, destdir=None, cwd=None, **cy_kwargs):
""" Generates a C file from a Cython source file.
Parameters
==========
src: str
Path to Cython source.
destdir: str (optional)
Path to output directory (default: '.').
cwd: path string (optional)
Root of relative paths (default: '.').
**cy_kwargs:
Second argument passed to cy_compile. Generates a .cpp file if ``cplus=True`` in ``cy_kwargs``,
else a .c file.
"""
from Cython.Compiler.Main import (
default_options, CompilationOptions
)
from Cython.Compiler.Main import compile as cy_compile
assert src.lower().endswith('.pyx') or src.lower().endswith('.py')
cwd = cwd or '.'
destdir = destdir or '.'
ext = '.cpp' if cy_kwargs.get('cplus', False) else '.c'
c_name = os.path.splitext(os.path.basename(src))[0] + ext
dstfile = os.path.join(destdir, c_name)
if cwd:
ori_dir = os.getcwd()
else:
ori_dir = '.'
os.chdir(cwd)
try:
cy_options = CompilationOptions(default_options)
cy_options.__dict__.update(cy_kwargs)
cy_result = cy_compile([src], cy_options)
if cy_result.num_errors > 0:
raise ValueError("Cython compilation failed.")
if os.path.abspath(os.path.dirname(src)) != os.path.abspath(destdir):
if os.path.exists(dstfile):
os.unlink(dstfile)
shutil.move(os.path.join(os.path.dirname(src), c_name), destdir)
finally:
os.chdir(ori_dir)
return dstfile
extension_mapping = {
'.c': (CCompilerRunner, None),
'.cpp': (CppCompilerRunner, None),
'.cxx': (CppCompilerRunner, None),
'.f': (FortranCompilerRunner, None),
'.for': (FortranCompilerRunner, None),
'.ftn': (FortranCompilerRunner, None),
'.f90': (FortranCompilerRunner, None), # ifort only knows about .f90
'.f95': (FortranCompilerRunner, 'f95'),
'.f03': (FortranCompilerRunner, 'f2003'),
'.f08': (FortranCompilerRunner, 'f2008'),
}
def src2obj(srcpath, Runner=None, objpath=None, cwd=None, inc_py=False, **kwargs):
""" Compiles a source code file to an object file.
Files ending with '.pyx' assumed to be cython files and
are dispatched to pyx2obj.
Parameters
==========
srcpath: str
Path to source file.
Runner: CompilerRunner subclass (optional)
If ``None``: deduced from extension of srcpath.
objpath : str (optional)
Path to generated object. If ``None``: deduced from ``srcpath``.
cwd: str (optional)
Working directory and root of relative paths. If ``None``: current dir.
inc_py: bool
Add Python include path to kwarg "include_dirs". Default: False
\\*\\*kwargs: dict
keyword arguments passed to Runner or pyx2obj
"""
name, ext = os.path.splitext(os.path.basename(srcpath))
if objpath is None:
if os.path.isabs(srcpath):
objpath = '.'
else:
objpath = os.path.dirname(srcpath)
objpath = objpath or '.' # avoid objpath == ''
if os.path.isdir(objpath):
objpath = os.path.join(objpath, name+objext)
include_dirs = kwargs.pop('include_dirs', [])
if inc_py:
from distutils.sysconfig import get_python_inc
py_inc_dir = get_python_inc()
if py_inc_dir not in include_dirs:
include_dirs.append(py_inc_dir)
if ext.lower() == '.pyx':
return pyx2obj(srcpath, objpath=objpath, include_dirs=include_dirs, cwd=cwd,
**kwargs)
if Runner is None:
Runner, std = extension_mapping[ext.lower()]
if 'std' not in kwargs:
kwargs['std'] = std
flags = kwargs.pop('flags', [])
needed_flags = ('-fPIC',)
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
# src2obj implies not running the linker...
run_linker = kwargs.pop('run_linker', False)
if run_linker:
raise CompileError("src2obj called with run_linker=True")
runner = Runner([srcpath], objpath, include_dirs=include_dirs,
run_linker=run_linker, cwd=cwd, flags=flags, **kwargs)
runner.run()
return objpath
def pyx2obj(pyxpath, objpath=None, destdir=None, cwd=None,
include_dirs=None, cy_kwargs=None, cplus=None, **kwargs):
"""
Convenience function
If cwd is specified, pyxpath and dst are taken to be relative
If only_update is set to `True` the modification time is checked
and compilation is only run if the source is newer than the
destination
Parameters
==========
pyxpath: str
Path to Cython source file.
objpath: str (optional)
Path to object file to generate.
destdir: str (optional)
Directory to put generated C file. When ``None``: directory of ``objpath``.
cwd: str (optional)
Working directory and root of relative paths.
include_dirs: iterable of path strings (optional)
Passed onto src2obj and via cy_kwargs['include_path']
to simple_cythonize.
cy_kwargs: dict (optional)
Keyword arguments passed onto `simple_cythonize`
cplus: bool (optional)
Indicate whether C++ is used. default: auto-detect using ``.util.pyx_is_cplus``.
compile_kwargs: dict
keyword arguments passed onto src2obj
Returns
=======
Absolute path of generated object file.
"""
assert pyxpath.endswith('.pyx')
cwd = cwd or '.'
objpath = objpath or '.'
destdir = destdir or os.path.dirname(objpath)
abs_objpath = get_abspath(objpath, cwd=cwd)
if os.path.isdir(abs_objpath):
pyx_fname = os.path.basename(pyxpath)
name, ext = os.path.splitext(pyx_fname)
objpath = os.path.join(objpath, name+objext)
cy_kwargs = cy_kwargs or {}
cy_kwargs['output_dir'] = cwd
if cplus is None:
cplus = pyx_is_cplus(pyxpath)
cy_kwargs['cplus'] = cplus
interm_c_file = simple_cythonize(pyxpath, destdir=destdir, cwd=cwd, **cy_kwargs)
include_dirs = include_dirs or []
flags = kwargs.pop('flags', [])
needed_flags = ('-fwrapv', '-pthread', '-fPIC')
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
options = kwargs.pop('options', [])
if kwargs.pop('strict_aliasing', False):
raise CompileError("Cython requires strict aliasing to be disabled.")
# Let's be explicit about standard
if cplus:
std = kwargs.pop('std', 'c++98')
else:
std = kwargs.pop('std', 'c99')
return src2obj(interm_c_file, objpath=objpath, cwd=cwd,
include_dirs=include_dirs, flags=flags, std=std,
options=options, inc_py=True, strict_aliasing=False,
**kwargs)
def _any_X(srcs, cls):
for src in srcs:
name, ext = os.path.splitext(src)
key = ext.lower()
if key in extension_mapping:
if extension_mapping[key][0] == cls:
return True
return False
def any_fortran_src(srcs):
return _any_X(srcs, FortranCompilerRunner)
def any_cplus_src(srcs):
return _any_X(srcs, CppCompilerRunner)
def compile_link_import_py_ext(sources, extname=None, build_dir='.', compile_kwargs=None,
link_kwargs=None):
""" Compiles sources to a shared object (python extension) and imports it
Sources in ``sources`` which is imported. If shared object is newer than the sources, they
are not recompiled but instead it is imported.
Parameters
==========
sources : string
List of paths to sources.
extname : string
Name of extension (default: ``None``).
If ``None``: taken from the last file in ``sources`` without extension.
build_dir: str
Path to directory in which objects files etc. are generated.
compile_kwargs: dict
keyword arguments passed to ``compile_sources``
link_kwargs: dict
keyword arguments passed to ``link_py_so``
Returns
=======
The imported module from of the python extension.
Examples
========
>>> mod = compile_link_import_py_ext(['fft.f90', 'conv.cpp', '_fft.pyx']) # doctest: +SKIP
>>> Aprim = mod.fft(A) # doctest: +SKIP
"""
if extname is None:
extname = os.path.splitext(os.path.basename(sources[-1]))[0]
compile_kwargs = compile_kwargs or {}
link_kwargs = link_kwargs or {}
try:
mod = import_module_from_file(os.path.join(build_dir, extname), sources)
except ImportError:
objs = compile_sources(list(map(get_abspath, sources)), destdir=build_dir,
cwd=build_dir, **compile_kwargs)
so = link_py_so(objs, cwd=build_dir, fort=any_fortran_src(sources),
cplus=any_cplus_src(sources), **link_kwargs)
mod = import_module_from_file(so)
return mod
def _write_sources_to_build_dir(sources, build_dir):
build_dir = build_dir or tempfile.mkdtemp()
if not os.path.isdir(build_dir):
raise OSError("Non-existent directory: ", build_dir)
source_files = []
for name, src in sources:
dest = os.path.join(build_dir, name)
differs = True
sha256_in_mem = sha256_of_string(src.encode('utf-8')).hexdigest()
if os.path.exists(dest):
if os.path.exists(dest+'.sha256'):
sha256_on_disk = open(dest+'.sha256', 'rt').read()
else:
sha256_on_disk = sha256_of_file(dest).hexdigest()
differs = sha256_on_disk != sha256_in_mem
if differs:
with open(dest, 'wt') as fh:
fh.write(src)
open(dest+'.sha256', 'wt').write(sha256_in_mem)
source_files.append(dest)
return source_files, build_dir
def compile_link_import_strings(sources, build_dir=None, **kwargs):
""" Compiles, links and imports extension module from source.
Parameters
==========
sources : iterable of name/source pair tuples
build_dir : string (default: None)
Path. ``None`` implies use a temporary directory.
**kwargs:
Keyword arguments passed onto `compile_link_import_py_ext`.
Returns
=======
mod : module
The compiled and imported extension module.
info : dict
Containing ``build_dir`` as 'build_dir'.
"""
source_files, build_dir = _write_sources_to_build_dir(sources, build_dir)
mod = compile_link_import_py_ext(source_files, build_dir=build_dir, **kwargs)
info = dict(build_dir=build_dir)
return mod, info
def compile_run_strings(sources, build_dir=None, clean=False, compile_kwargs=None, link_kwargs=None):
""" Compiles, links and runs a program built from sources.
Parameters
==========
sources : iterable of name/source pair tuples
build_dir : string (default: None)
Path. ``None`` implies use a temporary directory.
clean : bool
Whether to remove build_dir after use. This will only have an
effect if ``build_dir`` is ``None`` (which creates a temporary directory).
Passing ``clean == True`` and ``build_dir != None`` raises a ``ValueError``.
This will also set ``build_dir`` in returned info dictionary to ``None``.
compile_kwargs: dict
Keyword arguments passed onto ``compile_sources``
link_kwargs: dict
Keyword arguments passed onto ``link``
Returns
=======
(stdout, stderr): pair of strings
info: dict
Containing exit status as 'exit_status' and ``build_dir`` as 'build_dir'
"""
if clean and build_dir is not None:
raise ValueError("Automatic removal of build_dir is only available for temporary directory.")
try:
source_files, build_dir = _write_sources_to_build_dir(sources, build_dir)
objs = compile_sources(list(map(get_abspath, source_files)), destdir=build_dir,
cwd=build_dir, **(compile_kwargs or {}))
prog = link(objs, cwd=build_dir,
fort=any_fortran_src(source_files),
cplus=any_cplus_src(source_files), **(link_kwargs or {}))
p = subprocess.Popen([prog], stdout=subprocess.PIPE, stderr=subprocess.PIPE)
exit_status = p.wait()
stdout, stderr = [txt.decode('utf-8') for txt in p.communicate()]
finally:
if clean and os.path.isdir(build_dir):
shutil.rmtree(build_dir)
build_dir = None
info = dict(exit_status=exit_status, build_dir=build_dir)
return (stdout, stderr), info
|
f4f8a3ff4846cdf6b07126a0042a1e5a7f30679f8e2e9e947f3c17cb45ca7f17 | from __future__ import print_function, division, absolute_import
from collections import OrderedDict
from distutils.errors import CompileError
import os
import re
import subprocess
import sys
from .util import (
find_binary_of_command, unique_list
)
from sympy.core.compatibility import string_types
class CompilerRunner(object):
""" CompilerRunner base class.
Parameters
==========
sources : list of str
Paths to sources.
out : str
flags : iterable of str
Compiler flags.
run_linker : bool
compiler_name_exe : (str, str) tuple
Tuple of compiler name & command to call.
cwd : str
Path of root of relative paths.
include_dirs : list of str
Include directories.
libraries : list of str
Libraries to link against.
library_dirs : list of str
Paths to search for shared libraries.
std : str
Standard string, e.g. ``'c++11'``, ``'c99'``, ``'f2003'``.
define: iterable of strings
macros to define
undef : iterable of strings
macros to undefine
preferred_vendor : string
name of preferred vendor e.g. 'gnu' or 'intel'
Methods
=======
run():
Invoke compilation as a subprocess.
"""
compiler_dict = None # Subclass to vendor/binary dict
# Standards should be a tuple of supported standards
# (first one will be the default)
standards = None
std_formater = None # Subclass to dict of binary/formater-callback
# subclass to be e.g. {'gcc': 'gnu', ...}
compiler_name_vendor_mapping = None
def __init__(self, sources, out, flags=None, run_linker=True, compiler=None, cwd='.',
include_dirs=None, libraries=None, library_dirs=None, std=None, define=None,
undef=None, strict_aliasing=None, preferred_vendor=None, **kwargs):
if isinstance(sources, string_types):
raise ValueError("Expected argument sources to be a list of strings.")
self.sources = list(sources)
self.out = out
self.flags = flags or []
self.cwd = cwd
if compiler:
self.compiler_name, self.compiler_binary = compiler
else:
# Find a compiler
if preferred_vendor is None:
preferred_vendor = os.environ.get('SYMPY_COMPILER_VENDOR', None)
self.compiler_name, self.compiler_binary, self.compiler_vendor = self.find_compiler(preferred_vendor)
if self.compiler_binary is None:
raise ValueError("No compiler found (searched: {0})".format(', '.join(self.compiler_dict.values())))
self.define = define or []
self.undef = undef or []
self.include_dirs = include_dirs or []
self.libraries = libraries or []
self.library_dirs = library_dirs or []
self.std = std or self.standards[0]
self.run_linker = run_linker
if self.run_linker:
# both gnu and intel compilers use '-c' for disabling linker
self.flags = list(filter(lambda x: x != '-c', self.flags))
else:
if '-c' not in self.flags:
self.flags.append('-c')
if self.std:
self.flags.append(self.std_formater[
self.compiler_name](self.std))
self.linkline = []
if strict_aliasing is not None:
nsa_re = re.compile("no-strict-aliasing$")
sa_re = re.compile("strict-aliasing$")
if strict_aliasing is True:
if any(map(nsa_re.match, flags)):
raise CompileError("Strict aliasing cannot be both enforced and disabled")
elif any(map(sa_re.match, flags)):
pass # already enforced
else:
flags.append('-fstrict-aliasing')
elif strict_aliasing is False:
if any(map(nsa_re.match, flags)):
pass # already disabled
else:
if any(map(sa_re.match, flags)):
raise CompileError("Strict aliasing cannot be both enforced and disabled")
else:
flags.append('-fno-strict-aliasing')
else:
msg = "Expected argument strict_aliasing to be True/False, got {}"
raise ValueError(msg.format(strict_aliasing))
@classmethod
def find_compiler(cls, preferred_vendor=None):
""" Identify a suitable C/fortran/other compiler. """
candidates = list(cls.compiler_dict.keys())
if preferred_vendor:
if preferred_vendor in candidates:
candidates = [preferred_vendor]+candidates
else:
raise ValueError("Unknown vendor {}".format(preferred_vendor))
name, path = find_binary_of_command([cls.compiler_dict[x] for x in candidates])
return name, path, cls.compiler_name_vendor_mapping[name]
def cmd(self):
""" List of arguments (str) to be passed to e.g. ``subprocess.Popen``. """
cmd = (
[self.compiler_binary] +
self.flags +
['-U'+x for x in self.undef] +
['-D'+x for x in self.define] +
['-I'+x for x in self.include_dirs] +
self.sources
)
if self.run_linker:
cmd += (['-L'+x for x in self.library_dirs] +
['-l'+x for x in self.libraries] +
self.linkline)
counted = []
for envvar in re.findall(r'\$\{(\w+)\}', ' '.join(cmd)):
if os.getenv(envvar) is None:
if envvar not in counted:
counted.append(envvar)
msg = "Environment variable '{}' undefined.".format(envvar)
raise CompileError(msg)
return cmd
def run(self):
self.flags = unique_list(self.flags)
# Append output flag and name to tail of flags
self.flags.extend(['-o', self.out])
env = os.environ.copy()
env['PWD'] = self.cwd
# NOTE: intel compilers seems to need shell=True
p = subprocess.Popen(' '.join(self.cmd()),
shell=True,
cwd=self.cwd,
stdin=subprocess.PIPE,
stdout=subprocess.PIPE,
stderr=subprocess.STDOUT,
env=env)
comm = p.communicate()
if sys.version_info[0] == 2:
self.cmd_outerr = comm[0]
else:
try:
self.cmd_outerr = comm[0].decode('utf-8')
except UnicodeDecodeError:
self.cmd_outerr = comm[0].decode('iso-8859-1') # win32
self.cmd_returncode = p.returncode
# Error handling
if self.cmd_returncode != 0:
msg = "Error executing '{0}' in {1} (exited status {2}):\n {3}\n".format(
' '.join(self.cmd()), self.cwd, str(self.cmd_returncode), self.cmd_outerr
)
raise CompileError(msg)
return self.cmd_outerr, self.cmd_returncode
class CCompilerRunner(CompilerRunner):
compiler_dict = OrderedDict([
('gnu', 'gcc'),
('intel', 'icc'),
('llvm', 'clang'),
])
standards = ('c89', 'c90', 'c99', 'c11') # First is default
std_formater = {
'gcc': '-std={}'.format,
'icc': '-std={}'.format,
'clang': '-std={}'.format,
}
compiler_name_vendor_mapping = {
'gcc': 'gnu',
'icc': 'intel',
'clang': 'llvm'
}
def _mk_flag_filter(cmplr_name): # helper for class initialization
not_welcome = {'g++': ("Wimplicit-interface",)} # "Wstrict-prototypes",)}
if cmplr_name in not_welcome:
def fltr(x):
for nw in not_welcome[cmplr_name]:
if nw in x:
return False
return True
else:
def fltr(x):
return True
return fltr
class CppCompilerRunner(CompilerRunner):
compiler_dict = OrderedDict([
('gnu', 'g++'),
('intel', 'icpc'),
('llvm', 'clang++'),
])
# First is the default, c++0x == c++11
standards = ('c++98', 'c++0x')
std_formater = {
'g++': '-std={}'.format,
'icpc': '-std={}'.format,
'clang++': '-std={}'.format,
}
compiler_name_vendor_mapping = {
'g++': 'gnu',
'icpc': 'intel',
'clang++': 'llvm'
}
class FortranCompilerRunner(CompilerRunner):
standards = (None, 'f77', 'f95', 'f2003', 'f2008')
std_formater = {
'gfortran': lambda x: '-std=gnu' if x is None else '-std=legacy' if x == 'f77' else '-std={}'.format(x),
'ifort': lambda x: '-stand f08' if x is None else '-stand f{}'.format(x[-2:]), # f2008 => f08
}
compiler_dict = OrderedDict([
('gnu', 'gfortran'),
('intel', 'ifort'),
])
compiler_name_vendor_mapping = {
'gfortran': 'gnu',
'ifort': 'intel',
}
|
04c58f3b17649e9e8b38c761d4fb75242c143dba719cfc8ce2bcd89ee4673e43 | from __future__ import (absolute_import, division, print_function)
from collections import namedtuple
from hashlib import sha256
import os
import shutil
import sys
import tempfile
from sympy.utilities.pytest import XFAIL
def may_xfail(func):
if sys.platform.lower() == 'darwin' or os.name == 'nt':
# sympy.utilities._compilation needs more testing on Windows and macOS
# once those two platforms are reliably supported this xfail decorator
# may be removed.
return XFAIL(func)
else:
return func
if sys.version_info[0] == 2:
class FileNotFoundError(IOError):
pass
class TemporaryDirectory(object):
def __init__(self):
self.path = tempfile.mkdtemp()
def __enter__(self):
return self.path
def __exit__(self, exc, value, tb):
shutil.rmtree(self.path)
else:
FileNotFoundError = FileNotFoundError
TemporaryDirectory = tempfile.TemporaryDirectory
class CompilerNotFoundError(FileNotFoundError):
pass
def get_abspath(path, cwd='.'):
""" Returns the aboslute path.
Parameters
==========
path : str
(relative) path.
cwd : str
Path to root of relative path.
"""
if os.path.isabs(path):
return path
else:
if not os.path.isabs(cwd):
cwd = os.path.abspath(cwd)
return os.path.abspath(
os.path.join(cwd, path)
)
def make_dirs(path):
""" Create directories (equivalent of ``mkdir -p``). """
if path[-1] == '/':
parent = os.path.dirname(path[:-1])
else:
parent = os.path.dirname(path)
if len(parent) > 0:
if not os.path.exists(parent):
make_dirs(parent)
if not os.path.exists(path):
os.mkdir(path, 0o777)
else:
assert os.path.isdir(path)
def copy(src, dst, only_update=False, copystat=True, cwd=None,
dest_is_dir=False, create_dest_dirs=False):
""" Variation of ``shutil.copy`` with extra options.
Parameters
==========
src : str
Path to source file.
dst : str
Path to destination.
only_update : bool
Only copy if source is newer than destination
(returns None if it was newer), default: ``False``.
copystat : bool
See ``shutil.copystat``. default: ``True``.
cwd : str
Path to working directory (root of relative paths).
dest_is_dir : bool
Ensures that dst is treated as a directory. default: ``False``
create_dest_dirs : bool
Creates directories if needed.
Returns
=======
Path to the copied file.
"""
if cwd: # Handle working directory
if not os.path.isabs(src):
src = os.path.join(cwd, src)
if not os.path.isabs(dst):
dst = os.path.join(cwd, dst)
if not os.path.exists(src): # Make sure source file extists
raise FileNotFoundError("Source: `{}` does not exist".format(src))
# We accept both (re)naming destination file _or_
# passing a (possible non-existant) destination directory
if dest_is_dir:
if not dst[-1] == '/':
dst = dst+'/'
else:
if os.path.exists(dst) and os.path.isdir(dst):
dest_is_dir = True
if dest_is_dir:
dest_dir = dst
dest_fname = os.path.basename(src)
dst = os.path.join(dest_dir, dest_fname)
else:
dest_dir = os.path.dirname(dst)
dest_fname = os.path.basename(dst)
if not os.path.exists(dest_dir):
if create_dest_dirs:
make_dirs(dest_dir)
else:
raise FileNotFoundError("You must create directory first.")
if only_update:
if not missing_or_other_newer(dst, src):
return
if os.path.islink(dst):
_cwd = os.path.dirname(dst)
dst = os.path.abspath(os.path.realpath(dst), cwd=cwd)
shutil.copy(src, dst)
if copystat:
shutil.copystat(src, dst)
return dst
Glob = namedtuple('Glob', 'pathname')
ArbitraryDepthGlob = namedtuple('ArbitraryDepthGlob', 'filename')
def glob_at_depth(filename_glob, cwd=None):
if cwd is not None:
cwd = '.'
globbed = []
for root, dirs, filenames in os.walk(cwd):
for fn in filenames:
if fnmatch.fnmatch(fn, filename_glob):
globbed.append(os.path.join(root, fn))
return globbed
def sha256_of_file(path, nblocks=128):
""" Computes the SHA256 hash of a file.
Parameters
==========
path : string
Path to file to compute hash of.
nblocks : int
Number of blocks to read per iteration.
Returns
=======
hashlib sha256 hash object. Use ``.digest()`` or ``.hexdigest()``
on returned object to get binary or hex encoded string.
"""
sh = sha256()
with open(path, 'rb') as f:
for chunk in iter(lambda: f.read(nblocks*sh.block_size), b''):
sh.update(chunk)
return sh
def sha256_of_string(string):
""" Computes the SHA256 hash of a string. """
sh = sha256()
sh.update(string)
return sh
def pyx_is_cplus(path):
"""
Inspect a Cython source file (.pyx) and look for comment line like:
# distutils: language = c++
Returns True if such a file is present in the file, else False.
"""
for line in open(path, 'rt'):
if line.startswith('#') and '=' in line:
splitted = line.split('=')
if len(splitted) != 2:
continue
lhs, rhs = splitted
if lhs.strip().split()[-1].lower() == 'language' and \
rhs.strip().split()[0].lower() == 'c++':
return True
return False
def import_module_from_file(filename, only_if_newer_than=None):
""" Imports python extension (from shared object file)
Provide a list of paths in `only_if_newer_than` to check
timestamps of dependencies. import_ raises an ImportError
if any is newer.
Word of warning: The OS may cache shared objects which makes
reimporting same path of an shared object file very problematic.
It will not detect the new time stamp, nor new checksum, but will
instead silently use old module. Use unique names for this reason.
Parameters
==========
filename : str
Path to shared object.
only_if_newer_than : iterable of strings
Paths to dependencies of the shared object.
Raises
======
``ImportError`` if any of the files specified in ``only_if_newer_than`` are newer
than the file given by filename.
"""
path, name = os.path.split(filename)
name, ext = os.path.splitext(name)
name = name.split('.')[0]
if sys.version_info[0] == 2:
from imp import find_module, load_module
fobj, filename, data = find_module(name, [path])
if only_if_newer_than:
for dep in only_if_newer_than:
if os.path.getmtime(filename) < os.path.getmtime(dep):
raise ImportError("{} is newer than {}".format(dep, filename))
mod = load_module(name, fobj, filename, data)
else:
import importlib.util
spec = importlib.util.spec_from_file_location(name, filename)
if spec is None:
raise ImportError("Failed to import: '%s'" % filename)
mod = importlib.util.module_from_spec(spec)
spec.loader.exec_module(mod)
return mod
def find_binary_of_command(candidates):
""" Finds binary first matching name among candidates.
Calls `find_executable` from distuils for provided candidates and returns
first hit.
Parameters
==========
candidates : iterable of str
Names of candidate commands
Raises
======
CompilerNotFoundError if no candidates match.
"""
from distutils.spawn import find_executable
for c in candidates:
binary_path = find_executable(c)
if c and binary_path:
return c, binary_path
raise CompilerNotFoundError('No binary located for candidates: {}'.format(candidates))
def unique_list(l):
""" Uniquify a list (skip duplicate items). """
result = []
for x in l:
if x not in result:
result.append(x)
return result
|
46ead10b2c87368b6dc478fe309f313e5a38345ffd1d9b4cd7afce5ce5874647 | """ Tests from Michael Wester's 1999 paper "Review of CAS mathematical
capabilities".
http://www.math.unm.edu/~wester/cas/book/Wester.pdf
See also http://math.unm.edu/~wester/cas_review.html for detailed output of
each tested system.
"""
from sympy import (Rational, symbols, Dummy, factorial, sqrt, log, exp, oo, zoo,
product, binomial, rf, pi, gamma, igcd, factorint, radsimp, combsimp,
npartitions, totient, primerange, factor, simplify, gcd, resultant, expand,
I, trigsimp, tan, sin, cos, cot, diff, nan, limit, EulerGamma, polygamma,
bernoulli, hyper, hyperexpand, besselj, asin, assoc_legendre, Function, re,
im, DiracDelta, chebyshevt, legendre_poly, polylog, series, O,
atan, sinh, cosh, tanh, floor, ceiling, solve, asinh, acot, csc, sec,
LambertW, N, apart, sqrtdenest, factorial2, powdenest, Mul, S, ZZ,
Poly, expand_func, E, Q, And, Or, Ne, Eq, Le, Lt, Min,
ask, refine, AlgebraicNumber, continued_fraction_iterator as cf_i,
continued_fraction_periodic as cf_p, continued_fraction_convergents as cf_c,
continued_fraction_reduce as cf_r, FiniteSet, elliptic_e, elliptic_f,
powsimp, hessian, wronskian, fibonacci, sign, Lambda, Piecewise, Subs,
residue, Derivative, logcombine, Symbol, Intersection, Union,
EmptySet, Interval, Integral, idiff, ImageSet, acos, Max, MatMul)
import mpmath
from sympy.functions.combinatorial.numbers import stirling
from sympy.functions.special.delta_functions import Heaviside
from sympy.functions.special.error_functions import Ci, Si, erf
from sympy.functions.special.zeta_functions import zeta
from sympy.integrals.deltafunctions import deltaintegrate
from sympy.utilities.pytest import XFAIL, slow, SKIP, skip, ON_TRAVIS
from sympy.utilities.iterables import partitions
from mpmath import mpi, mpc
from sympy.matrices import Matrix, GramSchmidt, eye
from sympy.matrices.expressions.blockmatrix import BlockMatrix, block_collapse
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
from sympy.physics.quantum import Commutator
from sympy.assumptions import assuming
from sympy.polys.rings import vring
from sympy.polys.fields import vfield
from sympy.polys.solvers import solve_lin_sys
from sympy.concrete import Sum
from sympy.concrete.products import Product
from sympy.integrals import integrate
from sympy.integrals.transforms import laplace_transform,\
inverse_laplace_transform, LaplaceTransform, fourier_transform,\
mellin_transform
from sympy.solvers.recurr import rsolve
from sympy.solvers.solveset import solveset, solveset_real, linsolve
from sympy.solvers.ode import dsolve
from sympy.core.relational import Equality
from sympy.core.compatibility import range, PY3
from itertools import islice, takewhile
from sympy.series.formal import fps
from sympy.series.fourier import fourier_series
R = Rational
x, y, z = symbols('x y z')
i, j, k, l, m, n = symbols('i j k l m n', integer=True)
f = Function('f')
g = Function('g')
# A. Boolean Logic and Quantifier Elimination
# Not implemented.
# B. Set Theory
def test_B1():
assert (FiniteSet(i, j, j, k, k, k) | FiniteSet(l, k, j) |
FiniteSet(j, m, j)) == FiniteSet(i, j, k, l, m)
def test_B2():
a, b, c = FiniteSet(j), FiniteSet(m), FiniteSet(j, k)
d, e = FiniteSet(i), FiniteSet(j, k, l)
assert (FiniteSet(i, j, j, k, k, k) & FiniteSet(l, k, j) &
FiniteSet(j, m, j)) == Union(a, Intersection(b, Union(c, Intersection(d, FiniteSet(l)))))
# {j} U Intersection({m}, {j, k} U Intersection({i}, {l}))
def test_B3():
assert (FiniteSet(i, j, k, l, m) - FiniteSet(j) ==
FiniteSet(i, k, l, m))
def test_B4():
assert (FiniteSet(*(FiniteSet(i, j)*FiniteSet(k, l))) ==
FiniteSet((i, k), (i, l), (j, k), (j, l)))
# C. Numbers
def test_C1():
assert (factorial(50) ==
30414093201713378043612608166064768844377641568960512000000000000)
def test_C2():
assert (factorint(factorial(50)) == {2: 47, 3: 22, 5: 12, 7: 8,
11: 4, 13: 3, 17: 2, 19: 2, 23: 2, 29: 1, 31: 1, 37: 1,
41: 1, 43: 1, 47: 1})
def test_C3():
assert (factorial2(10), factorial2(9)) == (3840, 945)
# Base conversions; not really implemented by sympy
# Whatever. Take credit!
def test_C4():
assert 0xABC == 2748
def test_C5():
assert 123 == int('234', 7)
def test_C6():
assert int('677', 8) == int('1BF', 16) == 447
def test_C7():
assert log(32768, 8) == 5
def test_C8():
# Modular multiplicative inverse. Would be nice if divmod could do this.
assert ZZ.invert(5, 7) == 3
assert ZZ.invert(5, 6) == 5
def test_C9():
assert igcd(igcd(1776, 1554), 5698) == 74
def test_C10():
x = 0
for n in range(2, 11):
x += R(1, n)
assert x == R(4861, 2520)
def test_C11():
assert R(1, 7) == S('0.[142857]')
def test_C12():
assert R(7, 11) * R(22, 7) == 2
def test_C13():
test = R(10, 7) * (1 + R(29, 1000)) ** R(1, 3)
good = 3 ** R(1, 3)
assert test == good
def test_C14():
assert sqrtdenest(sqrt(2*sqrt(3) + 4)) == 1 + sqrt(3)
def test_C15():
test = sqrtdenest(sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2))))))
good = sqrt(2) + 3
assert test == good
def test_C16():
test = sqrtdenest(sqrt(10 + 2*sqrt(6) + 2*sqrt(10) + 2*sqrt(15)))
good = sqrt(2) + sqrt(3) + sqrt(5)
assert test == good
def test_C17():
test = radsimp((sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2)))
good = 5 + 2*sqrt(6)
assert test == good
def test_C18():
assert simplify((sqrt(-2 + sqrt(-5)) * sqrt(-2 - sqrt(-5))).expand(complex=True)) == 3
@XFAIL
def test_C19():
assert radsimp(simplify((90 + 34*sqrt(7)) ** R(1, 3))) == 3 + sqrt(7)
def test_C20():
inside = (135 + 78*sqrt(3))
test = AlgebraicNumber((inside**R(2, 3) + 3) * sqrt(3) / inside**R(1, 3))
assert simplify(test) == AlgebraicNumber(12)
def test_C21():
assert simplify(AlgebraicNumber((41 + 29*sqrt(2)) ** R(1, 5))) == \
AlgebraicNumber(1 + sqrt(2))
@XFAIL
def test_C22():
test = simplify(((6 - 4*sqrt(2))*log(3 - 2*sqrt(2)) + (3 - 2*sqrt(2))*log(17
- 12*sqrt(2)) + 32 - 24*sqrt(2)) / (48*sqrt(2) - 72))
good = sqrt(2)/3 - log(sqrt(2) - 1)/3
assert test == good
def test_C23():
assert 2 * oo - 3 == oo
@XFAIL
def test_C24():
raise NotImplementedError("2**aleph_null == aleph_1")
# D. Numerical Analysis
def test_D1():
assert 0.0 / sqrt(2) == 0.0
def test_D2():
assert str(exp(-1000000).evalf()) == '3.29683147808856e-434295'
def test_D3():
assert exp(pi*sqrt(163)).evalf(50).num.ae(262537412640768744)
def test_D4():
assert floor(R(-5, 3)) == -2
assert ceiling(R(-5, 3)) == -1
@XFAIL
def test_D5():
raise NotImplementedError("cubic_spline([1, 2, 4, 5], [1, 4, 2, 3], x)(3) == 27/8")
@XFAIL
def test_D6():
raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to FORTRAN")
@XFAIL
def test_D7():
raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to C")
@XFAIL
def test_D8():
# One way is to cheat by converting the sum to a string,
# and replacing the '[' and ']' with ''.
# E.g., horner(S(str(_).replace('[','').replace(']','')))
raise NotImplementedError("apply Horner's rule to sum(a[i]*x**i, (i,1,5))")
@XFAIL
def test_D9():
raise NotImplementedError("translate D8 to FORTRAN")
@XFAIL
def test_D10():
raise NotImplementedError("translate D8 to C")
@XFAIL
def test_D11():
#Is there a way to use count_ops?
raise NotImplementedError("flops(sum(product(f[i][k], (i,1,k)), (k,1,n)))")
@XFAIL
def test_D12():
assert (mpi(-4, 2) * x + mpi(1, 3)) ** 2 == mpi(-8, 16)*x**2 + mpi(-24, 12)*x + mpi(1, 9)
@XFAIL
def test_D13():
raise NotImplementedError("discretize a PDE: diff(f(x,t),t) == diff(diff(f(x,t),x),x)")
# E. Statistics
# See scipy; all of this is numerical.
# F. Combinatorial Theory.
def test_F1():
assert rf(x, 3) == x*(1 + x)*(2 + x)
def test_F2():
assert expand_func(binomial(n, 3)) == n*(n - 1)*(n - 2)/6
@XFAIL
def test_F3():
assert combsimp(2**n * factorial(n) * factorial2(2*n - 1)) == factorial(2*n)
@XFAIL
def test_F4():
assert combsimp((2**n * factorial(n) * product(2*k - 1, (k, 1, n)))) == factorial(2*n)
@XFAIL
def test_F5():
assert gamma(n + R(1, 2)) / sqrt(pi) / factorial(n) == factorial(2*n)/2**(2*n)/factorial(n)**2
def test_F6():
partTest = [p.copy() for p in partitions(4)]
partDesired = [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2:1}, {1: 4}]
assert partTest == partDesired
def test_F7():
assert npartitions(4) == 5
def test_F8():
assert stirling(5, 2, signed=True) == -50 # if signed, then kind=1
def test_F9():
assert totient(1776) == 576
# G. Number Theory
def test_G1():
assert list(primerange(999983, 1000004)) == [999983, 1000003]
@XFAIL
def test_G2():
raise NotImplementedError("find the primitive root of 191 == 19")
@XFAIL
def test_G3():
raise NotImplementedError("(a+b)**p mod p == a**p + b**p mod p; p prime")
# ... G14 Modular equations are not implemented.
def test_G15():
assert Rational(sqrt(3).evalf()).limit_denominator(15) == Rational(26, 15)
assert list(takewhile(lambda x: x.q <= 15, cf_c(cf_i(sqrt(3)))))[-1] == \
Rational(26, 15)
def test_G16():
assert list(islice(cf_i(pi),10)) == [3, 7, 15, 1, 292, 1, 1, 1, 2, 1]
def test_G17():
assert cf_p(0, 1, 23) == [4, [1, 3, 1, 8]]
def test_G18():
assert cf_p(1, 2, 5) == [[1]]
assert cf_r([[1]]) == S.Half + sqrt(5)/2
@XFAIL
def test_G19():
s = symbols('s', integer=True, positive=True)
it = cf_i((exp(1/s) - 1)/(exp(1/s) + 1))
assert list(islice(it, 5)) == [0, 2*s, 6*s, 10*s, 14*s]
def test_G20():
s = symbols('s', integer=True, positive=True)
# Wester erroneously has this as -s + sqrt(s**2 + 1)
assert cf_r([[2*s]]) == s + sqrt(s**2 + 1)
@XFAIL
def test_G20b():
s = symbols('s', integer=True, positive=True)
assert cf_p(s, 1, s**2 + 1) == [[2*s]]
# H. Algebra
def test_H1():
assert simplify(2*2**n) == simplify(2**(n + 1))
assert powdenest(2*2**n) == simplify(2**(n + 1))
def test_H2():
assert powsimp(4 * 2**n) == 2**(n + 2)
def test_H3():
assert (-1)**(n*(n + 1)) == 1
def test_H4():
expr = factor(6*x - 10)
assert type(expr) is Mul
assert expr.args[0] == 2
assert expr.args[1] == 3*x - 5
p1 = 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81
p2 = 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81
q = 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86
def test_H5():
assert gcd(p1, p2, x) == 1
def test_H6():
assert gcd(expand(p1 * q), expand(p2 * q)) == q
def test_H7():
p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
assert gcd(p1, p2, x, y, z) == 1
def test_H8():
p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
q = 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8
assert gcd(p1 * q, p2 * q, x, y, z) == q
def test_H9():
p1 = 2*x**(n + 4) - x**(n + 2)
p2 = 4*x**(n + 1) + 3*x**n
assert gcd(p1, p2) == x**n
def test_H10():
p1 = 3*x**4 + 3*x**3 + x**2 - x - 2
p2 = x**3 - 3*x**2 + x + 5
assert resultant(p1, p2, x) == 0
def test_H11():
assert resultant(p1 * q, p2 * q, x) == 0
def test_H12():
num = x**2 - 4
den = x**2 + 4*x + 4
assert simplify(num/den) == (x - 2)/(x + 2)
@XFAIL
def test_H13():
assert simplify((exp(x) - 1) / (exp(x/2) + 1)) == exp(x/2) - 1
def test_H14():
p = (x + 1) ** 20
ep = expand(p)
assert ep == (1 + 20*x + 190*x**2 + 1140*x**3 + 4845*x**4 + 15504*x**5
+ 38760*x**6 + 77520*x**7 + 125970*x**8 + 167960*x**9 + 184756*x**10
+ 167960*x**11 + 125970*x**12 + 77520*x**13 + 38760*x**14 + 15504*x**15
+ 4845*x**16 + 1140*x**17 + 190*x**18 + 20*x**19 + x**20)
dep = diff(ep, x)
assert dep == (20 + 380*x + 3420*x**2 + 19380*x**3 + 77520*x**4
+ 232560*x**5 + 542640*x**6 + 1007760*x**7 + 1511640*x**8 + 1847560*x**9
+ 1847560*x**10 + 1511640*x**11 + 1007760*x**12 + 542640*x**13
+ 232560*x**14 + 77520*x**15 + 19380*x**16 + 3420*x**17 + 380*x**18
+ 20*x**19)
assert factor(dep) == 20*(1 + x)**19
def test_H15():
assert simplify((Mul(*[x - r for r in solveset(x**3 + x**2 - 7)]))) == x**3 + x**2 - 7
def test_H16():
assert factor(x**100 - 1) == ((x - 1)*(x + 1)*(x**2 + 1)*(x**4 - x**3
+ x**2 - x + 1)*(x**4 + x**3 + x**2 + x + 1)*(x**8 - x**6 + x**4
- x**2 + 1)*(x**20 - x**15 + x**10 - x**5 + 1)*(x**20 + x**15 + x**10
+ x**5 + 1)*(x**40 - x**30 + x**20 - x**10 + 1))
def test_H17():
assert simplify(factor(expand(p1 * p2)) - p1*p2) == 0
@XFAIL
def test_H18():
# Factor over complex rationals.
test = factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153)
good = (2*x + 3*I)*(2*x - 3*I)*(x + 1 - 4*I)*(x + 1 + 4*I)
assert test == good
def test_H19():
a = symbols('a')
# The idea is to let a**2 == 2, then solve 1/(a-1). Answer is a+1")
assert Poly(a - 1).invert(Poly(a**2 - 2)) == a + 1
@XFAIL
def test_H20():
raise NotImplementedError("let a**2==2; (x**3 + (a-2)*x**2 - "
+ "(2*a+3)*x - 3*a) / (x**2-2) = (x**2 - 2*x - 3) / (x-a)")
@XFAIL
def test_H21():
raise NotImplementedError("evaluate (b+c)**4 assuming b**3==2, c**2==3. \
Answer is 2*b + 8*c + 18*b**2 + 12*b*c + 9")
def test_H22():
assert factor(x**4 - 3*x**2 + 1, modulus=5) == (x - 2)**2 * (x + 2)**2
def test_H23():
f = x**11 + x + 1
g = (x**2 + x + 1) * (x**9 - x**8 + x**6 - x**5 + x**3 - x**2 + 1)
assert factor(f, modulus=65537) == g
def test_H24():
phi = AlgebraicNumber(S.GoldenRatio.expand(func=True), alias='phi')
assert factor(x**4 - 3*x**2 + 1, extension=phi) == \
(x - phi)*(x + 1 - phi)*(x - 1 + phi)*(x + phi)
def test_H25():
e = (x - 2*y**2 + 3*z**3) ** 20
assert factor(expand(e)) == e
def test_H26():
g = expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20)
assert factor(g, expand=False) == (-sin(x) + 2*cos(y)**2 - 3*tan(z)**3)**20
def test_H27():
f = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
g = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
h = -2*z*y**7 \
*(6*x**9*y**9*z**3 + 10*x**7*z**6 + 17*y*x**5*z**12 + 40*y**7) \
*(3*x**22 + 47*x**17*y**5*z**8 - 6*x**15*y**9*z**2 - 24*x*y**19*z**8 - 5)
assert factor(expand(f*g)) == h
@XFAIL
def test_H28():
raise NotImplementedError("expand ((1 - c**2)**5 * (1 - s**2)**5 * "
+ "(c**2 + s**2)**10) with c**2 + s**2 = 1. Answer is c**10*s**10.")
@XFAIL
def test_H29():
assert factor(4*x**2 - 21*x*y + 20*y**2, modulus=3) == (x + y)*(x - y)
def test_H30():
test = factor(x**3 + y**3, extension=sqrt(-3))
answer = (x + y)*(x + y*(-R(1, 2) - sqrt(3)/2*I))*(x + y*(-R(1, 2) + sqrt(3)/2*I))
assert answer == test
def test_H31():
f = (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2)
g = 2 / (x + 1)**2 - 2 / (x + 1) + 3 / (x + 2)
assert apart(f) == g
@XFAIL
def test_H32(): # issue 6558
raise NotImplementedError("[A*B*C - (A*B*C)**(-1)]*A*C*B (product \
of a non-commuting product and its inverse)")
def test_H33():
A, B, C = symbols('A, B, C', commutative=False)
assert (Commutator(A, Commutator(B, C))
+ Commutator(B, Commutator(C, A))
+ Commutator(C, Commutator(A, B))).doit().expand() == 0
# I. Trigonometry
@XFAIL
def test_I1():
assert tan(7*pi/10) == -sqrt(1 + 2/sqrt(5))
@XFAIL
def test_I2():
assert sqrt((1 + cos(6))/2) == -cos(3)
def test_I3():
assert cos(n*pi) + sin((4*n - 1)*pi/2) == (-1)**n - 1
def test_I4():
assert refine(cos(pi*cos(n*pi)) + sin(pi/2*cos(n*pi)), Q.integer(n)) == (-1)**n - 1
@XFAIL
def test_I5():
assert sin((n**5/5 + n**4/2 + n**3/3 - n/30) * pi) == 0
@XFAIL
def test_I6():
raise NotImplementedError("assuming -3*pi<x<-5*pi/2, abs(cos(x)) == -cos(x), abs(sin(x)) == -sin(x)")
@XFAIL
def test_I7():
assert cos(3*x)/cos(x) == cos(x)**2 - 3*sin(x)**2
@XFAIL
def test_I8():
assert cos(3*x)/cos(x) == 2*cos(2*x) - 1
@XFAIL
def test_I9():
# Supposed to do this with rewrite rules.
assert cos(3*x)/cos(x) == cos(x)**2 - 3*sin(x)**2
def test_I10():
assert trigsimp((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1)) == nan
@SKIP("hangs")
@XFAIL
def test_I11():
assert limit((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1), x, 0) != 0
@XFAIL
def test_I12():
try:
# This should fail or return nan or something.
diff((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1), x)
except:
assert True
else:
assert False, "taking the derivative with a fraction equivalent to 0/0 should fail"
# J. Special functions.
def test_J1():
assert bernoulli(16) == R(-3617, 510)
def test_J2():
assert diff(elliptic_e(x, y**2), y) == (elliptic_e(x, y**2) - elliptic_f(x, y**2))/y
@XFAIL
def test_J3():
raise NotImplementedError("Jacobi elliptic functions: diff(dn(u,k), u) == -k**2*sn(u,k)*cn(u,k)")
def test_J4():
assert gamma(R(-1, 2)) == -2*sqrt(pi)
def test_J5():
assert polygamma(0, R(1, 3)) == -log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))
def test_J6():
assert mpmath.besselj(2, 1 + 1j).ae(mpc('0.04157988694396212', '0.24739764151330632'))
def test_J7():
assert simplify(besselj(R(-5,2), pi/2)) == 12/(pi**2)
def test_J8():
p = besselj(R(3,2), z)
q = (sin(z)/z - cos(z))/sqrt(pi*z/2)
assert simplify(expand_func(p) -q) == 0
def test_J9():
assert besselj(0, z).diff(z) == - besselj(1, z)
def test_J10():
mu, nu = symbols('mu, nu', integer=True)
assert assoc_legendre(nu, mu, 0) == 2**mu*sqrt(pi)/gamma((nu - mu)/2 + 1)/gamma((-nu - mu + 1)/2)
def test_J11():
assert simplify(assoc_legendre(3, 1, x)) == simplify(-R(3, 2)*sqrt(1 - x**2)*(5*x**2 - 1))
@slow
def test_J12():
assert simplify(chebyshevt(1008, x) - 2*x*chebyshevt(1007, x) + chebyshevt(1006, x)) == 0
def test_J13():
a = symbols('a', integer=True, negative=False)
assert chebyshevt(a, -1) == (-1)**a
def test_J14():
p = hyper([S(1)/2, S(1)/2], [S(3)/2], z**2)
assert hyperexpand(p) == asin(z)/z
@XFAIL
def test_J15():
raise NotImplementedError("F((n+2)/2,-(n-2)/2,R(3,2),sin(z)**2) == sin(n*z)/(n*sin(z)*cos(z)); F(.) is hypergeometric function")
@XFAIL
def test_J16():
raise NotImplementedError("diff(zeta(x), x) @ x=0 == -log(2*pi)/2")
def test_J17():
assert integrate(f((x + 2)/5)*DiracDelta((x - 2)/3) - g(x)*diff(DiracDelta(x - 1), x), (x, 0, 3)) == 3*f(S(4)/5) + Subs(Derivative(g(x), x), x, 1)
@XFAIL
def test_J18():
raise NotImplementedError("define an antisymmetric function")
# K. The Complex Domain
def test_K1():
z1, z2 = symbols('z1, z2', complex=True)
assert re(z1 + I*z2) == -im(z2) + re(z1)
assert im(z1 + I*z2) == im(z1) + re(z2)
def test_K2():
assert abs(3 - sqrt(7) + I*sqrt(6*sqrt(7) - 15)) == 1
@XFAIL
def test_K3():
a, b = symbols('a, b', real=True)
assert simplify(abs(1/(a + I/a + I*b))) == 1/sqrt(a**2 + (I/a + b)**2)
def test_K4():
assert log(3 + 4*I).expand(complex=True) == log(5) + I*atan(R(4, 3))
def test_K5():
x, y = symbols('x, y', real=True)
assert tan(x + I*y).expand(complex=True) == (sin(2*x)/(cos(2*x) +
cosh(2*y)) + I*sinh(2*y)/(cos(2*x) + cosh(2*y)))
def test_K6():
assert sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z)) == sqrt(x*y)/sqrt(x)
assert sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z)) != sqrt(y)
def test_K7():
y = symbols('y', real=True, negative=False)
expr = sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z))
sexpr = simplify(expr)
assert sexpr == sqrt(y)
@XFAIL
def test_K8():
z = symbols('z', complex=True)
assert simplify(sqrt(1/z) - 1/sqrt(z)) != 0 # Passes
z = symbols('z', complex=True, negative=False)
assert simplify(sqrt(1/z) - 1/sqrt(z)) == 0 # Fails
def test_K9():
z = symbols('z', real=True, positive=True)
assert simplify(sqrt(1/z) - 1/sqrt(z)) == 0
def test_K10():
z = symbols('z', real=True, negative=True)
assert simplify(sqrt(1/z) + 1/sqrt(z)) == 0
# This goes up to K25
# L. Determining Zero Equivalence
def test_L1():
assert sqrt(997) - (997**3)**R(1, 6) == 0
def test_L2():
assert sqrt(999983) - (999983**3)**R(1, 6) == 0
def test_L3():
assert simplify((2**R(1, 3) + 4**R(1, 3))**3 - 6*(2**R(1, 3) + 4**R(1, 3)) - 6) == 0
def test_L4():
assert trigsimp(cos(x)**3 + cos(x)*sin(x)**2 - cos(x)) == 0
@XFAIL
def test_L5():
assert log(tan(R(1, 2)*x + pi/4)) - asinh(tan(x)) == 0
def test_L6():
assert (log(tan(x/2 + pi/4)) - asinh(tan(x))).diff(x).subs({x: 0}) == 0
@XFAIL
def test_L7():
assert simplify(log((2*sqrt(x) + 1)/(sqrt(4*x + 4*sqrt(x) + 1)))) == 0
@XFAIL
def test_L8():
assert simplify((4*x + 4*sqrt(x) + 1)**(sqrt(x)/(2*sqrt(x) + 1)) \
*(2*sqrt(x) + 1)**(1/(2*sqrt(x) + 1)) - 2*sqrt(x) - 1) == 0
@XFAIL
def test_L9():
z = symbols('z', complex=True)
assert simplify(2**(1 - z)*gamma(z)*zeta(z)*cos(z*pi/2) - pi**2*zeta(1 - z)) == 0
# M. Equations
@XFAIL
def test_M1():
assert Equality(x, 2)/2 + Equality(1, 1) == Equality(x/2 + 1, 2)
def test_M2():
# The roots of this equation should all be real. Note that this
# doesn't test that they are correct.
sol = solveset(3*x**3 - 18*x**2 + 33*x - 19, x)
assert all(s.expand(complex=True).is_real for s in sol)
@XFAIL
def test_M5():
assert solveset(x**6 - 9*x**4 - 4*x**3 + 27*x**2 - 36*x - 23, x) == FiniteSet(2**(1/3) + sqrt(3), 2**(1/3) - sqrt(3), +sqrt(3) - 1/2**(2/3) + I*sqrt(3)/2**(2/3), +sqrt(3) - 1/2**(2/3) - I*sqrt(3)/2**(2/3), -sqrt(3) - 1/2**(2/3) + I*sqrt(3)/2**(2/3), -sqrt(3) - 1/2**(2/3) - I*sqrt(3)/2**(2/3))
def test_M6():
assert set(solveset(x**7 - 1, x)) == \
{cos(n*2*pi/7) + I*sin(n*2*pi/7) for n in range(0, 7)}
# The paper asks for exp terms, but sin's and cos's may be acceptable;
# if the results are simplified, exp terms appear for all but
# -sin(pi/14) - I*cos(pi/14) and -sin(pi/14) + I*cos(pi/14) which
# will simplify if you apply the transformation foo.rewrite(exp).expand()
def test_M7():
# TODO: Replace solve with solveset, as of now test fails for solveset
sol = solve(x**8 - 8*x**7 + 34*x**6 - 92*x**5 + 175*x**4 - 236*x**3 +
226*x**2 - 140*x + 46, x)
assert [s.simplify() for s in sol] == [
1 - sqrt(-6 - 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(-6 - 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 - sqrt(-6 + 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(-6 + 2*I*sqrt(3 + 4*sqrt (3)))/2,
1 - sqrt(-6 + 2*sqrt(-3 + 4*sqrt(3)))/2,
1 + sqrt(-6 + 2*sqrt(-3 + 4*sqrt(3)))/2,
1 - sqrt(-6 - 2*sqrt(-3 + 4*sqrt(3)))/2,
1 + sqrt(-6 - 2*sqrt(-3 + 4*sqrt(3)))/2]
@XFAIL # There are an infinite number of solutions.
def test_M8():
x = Symbol('x')
z = symbols('z', complex=True)
assert solveset(exp(2*x) + 2*exp(x) + 1 - z, x, S.Reals) == \
FiniteSet(log(1 + z - 2*sqrt(z))/2, log(1 + z + 2*sqrt(z))/2)
# This one could be simplified better (the 1/2 could be pulled into the log
# as a sqrt, and the function inside the log can be factored as a square,
# giving [log(sqrt(z) - 1), log(sqrt(z) + 1)]). Also, there should be an
# infinite number of solutions.
# x = {log(sqrt(z) - 1), log(sqrt(z) + 1) + i pi} [+ n 2 pi i, + n 2 pi i]
# where n is an arbitrary integer. See url of detailed output above.
@XFAIL
def test_M9():
x = symbols('x')
raise NotImplementedError("solveset(exp(2-x**2)-exp(-x),x) has complex solutions.")
def test_M10():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(exp(x) - x, x) == [-LambertW(-1)]
@XFAIL
def test_M11():
assert solveset(x**x - x, x) == FiniteSet(-1, 1)
def test_M12():
# TODO: x = [-1, 2*(+/-asinh(1)*I + n*pi}, 3*(pi/6 + n*pi/3)]
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve((x + 1)*(sin(x)**2 + 1)**2*cos(3*x)**3, x) == [
-1, pi/6, pi/2,
- I*log(1 + sqrt(2)), I*log(1 + sqrt(2)),
pi - I*log(1 + sqrt(2)), pi + I*log(1 + sqrt(2)),
]
@XFAIL
def test_M13():
n = Dummy('n')
assert solveset_real(sin(x) - cos(x), x) == ImageSet(Lambda(n, n*pi - 7*pi/4), S.Integers)
@XFAIL
def test_M14():
n = Dummy('n')
assert solveset_real(tan(x) - 1, x) == ImageSet(Lambda(n, n*pi + pi/4), S.Integers)
def test_M15():
if PY3:
n = Dummy('n')
assert solveset(sin(x) - S.Half) in (Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + 5*pi/6), S.Integers)),
Union(ImageSet(Lambda(n, 2*n*pi + 5*pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers)))
@XFAIL
def test_M16():
n = Dummy('n')
assert solveset(sin(x) - tan(x), x) == ImageSet(Lambda(n, n*pi), S.Integers)
@XFAIL
def test_M17():
assert solveset_real(asin(x) - atan(x), x) == FiniteSet(0)
@XFAIL
def test_M18():
assert solveset_real(acos(x) - atan(x), x) == FiniteSet(sqrt((sqrt(5) - 1)/2))
def test_M19():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve((x - 2)/x**R(1, 3), x) == [2]
def test_M20():
assert solveset(sqrt(x**2 + 1) - x + 2, x) == EmptySet()
def test_M21():
assert solveset(x + sqrt(x) - 2) == FiniteSet(1)
def test_M22():
assert solveset(2*sqrt(x) + 3*x**R(1, 4) - 2) == FiniteSet(R(1, 16))
def test_M23():
x = symbols('x', complex=True)
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(x - 1/sqrt(1 + x**2)) == [
-I*sqrt(S.Half + sqrt(5)/2), sqrt(-S.Half + sqrt(5)/2)]
def test_M24():
# TODO: Replace solve with solveset, as of now test fails for solveset
solution = solve(1 - binomial(m, 2)*2**k, k)
answer = log(2/(m*(m - 1)), 2)
assert solution[0].expand() == answer.expand()
def test_M25():
a, b, c, d = symbols(':d', positive=True)
x = symbols('x')
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(a*b**x - c*d**x, x)[0].expand() == (log(c/a)/log(b/d)).expand()
def test_M26():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(sqrt(log(x)) - log(sqrt(x))) == [1, exp(4)]
@XFAIL
def test_M27():
x = symbols('x', real=True)
b = symbols('b', real=True)
with assuming(Q.is_true(sin(cos(1/E**2) + 1) + b > 0)):
# TODO: Replace solve with solveset
solve(log(acos(asin(x**R(2, 3) - b) - 1)) + 2, x) == [-b - sin(1 + cos(1/e**2))**R(3/2), b + sin(1 + cos(1/e**2))**R(3/2)]
@XFAIL
def test_M28():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
assert solve(5*x + exp((x - 5)/2) - 8*x**3, x, assume=Q.real(x)) == [-0.784966, -0.016291, 0.802557]
def test_M29():
x = symbols('x')
assert solveset(abs(x - 1) - 2, domain=S.Reals) == FiniteSet(-1, 3)
def test_M30():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# assert solve(abs(2*x + 5) - abs(x - 2),x, assume=Q.real(x)) == [-1, -7]
assert solveset_real(abs(2*x + 5) - abs(x - 2), x) == FiniteSet(-1, -7)
def test_M31():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# assert solve(1 - abs(x) - max(-x - 2, x - 2),x, assume=Q.real(x)) == [-3/2, 3/2]
assert solveset_real(1 - abs(x) - Max(-x - 2, x - 2), x) == FiniteSet(-S(3)/2, S(3)/2)
@XFAIL
def test_M32():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
assert solveset_real(Max(2 - x**2, x)- Max(-x, (x**3)/9), x) == FiniteSet(-1, 3)
@XFAIL
def test_M33():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# Second answer can be written in another form. The second answer is the root of x**3 + 9*x**2 - 18 = 0 in the interval (-2, -1).
assert solveset_real(Max(2 - x**2, x) - x**3/9, x) == FiniteSet(-3, -1.554894, 3)
@XFAIL
def test_M34():
z = symbols('z', complex=True)
assert solveset((1 + I) * z + (2 - I) * conjugate(z) + 3*I, z) == FiniteSet(2 + 3*I)
def test_M35():
x, y = symbols('x y', real=True)
assert linsolve((3*x - 2*y - I*y + 3*I).as_real_imag(), y, x) == FiniteSet((3, 2))
def test_M36():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports solving for function
# assert solve(f**2 + f - 2, x) == [Eq(f(x), 1), Eq(f(x), -2)]
assert solveset(f(x)**2 + f(x) - 2, f(x)) == FiniteSet(-2, 1)
def test_M37():
assert linsolve([x + y + z - 6, 2*x + y + 2*z - 10, x + 3*y + z - 10 ], x, y, z) == \
FiniteSet((-z + 4, 2, z))
def test_M38():
variables = vring("k1:50", vfield("a,b,c", ZZ).to_domain())
system = [
-b*k8/a + c*k8/a, -b*k11/a + c*k11/a, -b*k10/a + c*k10/a + k2, -k3 - b*k9/a + c*k9/a,
-b*k14/a + c*k14/a, -b*k15/a + c*k15/a, -b*k18/a + c*k18/a - k2, -b*k17/a + c*k17/a,
-b*k16/a + c*k16/a + k4, -b*k13/a + c*k13/a - b*k21/a + c*k21/a + b*k5/a - c*k5/a,
b*k44/a - c*k44/a, -b*k45/a + c*k45/a, -b*k20/a + c*k20/a, -b*k44/a + c*k44/a,
b*k46/a - c*k46/a, b**2*k47/a**2 - 2*b*c*k47/a**2 + c**2*k47/a**2, k3, -k4,
-b*k12/a + c*k12/a - a*k6/b + c*k6/b, -b*k19/a + c*k19/a + a*k7/c - b*k7/c,
b*k45/a - c*k45/a, -b*k46/a + c*k46/a, -k48 + c*k48/a + c*k48/b - c**2*k48/(a*b),
-k49 + b*k49/a + b*k49/c - b**2*k49/(a*c), a*k1/b - c*k1/b, a*k4/b - c*k4/b,
a*k3/b - c*k3/b + k9, -k10 + a*k2/b - c*k2/b, a*k7/b - c*k7/b, -k9, k11,
b*k12/a - c*k12/a + a*k6/b - c*k6/b, a*k15/b - c*k15/b, k10 + a*k18/b - c*k18/b,
-k11 + a*k17/b - c*k17/b, a*k16/b - c*k16/b, -a*k13/b + c*k13/b + a*k21/b - c*k21/b + a*k5/b - c*k5/b,
-a*k44/b + c*k44/b, a*k45/b - c*k45/b, a*k14/c - b*k14/c + a*k20/b - c*k20/b,
a*k44/b - c*k44/b, -a*k46/b + c*k46/b, -k47 + c*k47/a + c*k47/b - c**2*k47/(a*b),
a*k19/b - c*k19/b, -a*k45/b + c*k45/b, a*k46/b - c*k46/b, a**2*k48/b**2 - 2*a*c*k48/b**2 + c**2*k48/b**2,
-k49 + a*k49/b + a*k49/c - a**2*k49/(b*c), k16, -k17, -a*k1/c + b*k1/c,
-k16 - a*k4/c + b*k4/c, -a*k3/c + b*k3/c, k18 - a*k2/c + b*k2/c, b*k19/a - c*k19/a - a*k7/c + b*k7/c,
-a*k6/c + b*k6/c, -a*k8/c + b*k8/c, -a*k11/c + b*k11/c + k17, -a*k10/c + b*k10/c - k18,
-a*k9/c + b*k9/c, -a*k14/c + b*k14/c - a*k20/b + c*k20/b, -a*k13/c + b*k13/c + a*k21/c - b*k21/c - a*k5/c + b*k5/c,
a*k44/c - b*k44/c, -a*k45/c + b*k45/c, -a*k44/c + b*k44/c, a*k46/c - b*k46/c,
-k47 + b*k47/a + b*k47/c - b**2*k47/(a*c), -a*k12/c + b*k12/c, a*k45/c - b*k45/c,
-a*k46/c + b*k46/c, -k48 + a*k48/b + a*k48/c - a**2*k48/(b*c),
a**2*k49/c**2 - 2*a*b*k49/c**2 + b**2*k49/c**2, k8, k11, -k15, k10 - k18,
-k17, k9, -k16, -k29, k14 - k32, -k21 + k23 - k31, -k24 - k30, -k35, k44,
-k45, k36, k13 - k23 + k39, -k20 + k38, k25 + k37, b*k26/a - c*k26/a - k34 + k42,
-2*k44, k45, k46, b*k47/a - c*k47/a, k41, k44, -k46, -b*k47/a + c*k47/a,
k12 + k24, -k19 - k25, -a*k27/b + c*k27/b - k33, k45, -k46, -a*k48/b + c*k48/b,
a*k28/c - b*k28/c + k40, -k45, k46, a*k48/b - c*k48/b, a*k49/c - b*k49/c,
-a*k49/c + b*k49/c, -k1, -k4, -k3, k15, k18 - k2, k17, k16, k22, k25 - k7,
k24 + k30, k21 + k23 - k31, k28, -k44, k45, -k30 - k6, k20 + k32, k27 + b*k33/a - c*k33/a,
k44, -k46, -b*k47/a + c*k47/a, -k36, k31 - k39 - k5, -k32 - k38, k19 - k37,
k26 - a*k34/b + c*k34/b - k42, k44, -2*k45, k46, a*k48/b - c*k48/b,
a*k35/c - b*k35/c - k41, -k44, k46, b*k47/a - c*k47/a, -a*k49/c + b*k49/c,
-k40, k45, -k46, -a*k48/b + c*k48/b, a*k49/c - b*k49/c, k1, k4, k3, -k8,
-k11, -k10 + k2, -k9, k37 + k7, -k14 - k38, -k22, -k25 - k37, -k24 + k6,
-k13 - k23 + k39, -k28 + b*k40/a - c*k40/a, k44, -k45, -k27, -k44, k46,
b*k47/a - c*k47/a, k29, k32 + k38, k31 - k39 + k5, -k12 + k30, k35 - a*k41/b + c*k41/b,
-k44, k45, -k26 + k34 + a*k42/c - b*k42/c, k44, k45, -2*k46, -b*k47/a + c*k47/a,
-a*k48/b + c*k48/b, a*k49/c - b*k49/c, k33, -k45, k46, a*k48/b - c*k48/b,
-a*k49/c + b*k49/c
]
solution = {
k49: 0, k48: 0, k47: 0, k46: 0, k45: 0, k44: 0, k41: 0, k40: 0,
k38: 0, k37: 0, k36: 0, k35: 0, k33: 0, k32: 0, k30: 0, k29: 0,
k28: 0, k27: 0, k25: 0, k24: 0, k22: 0, k21: 0, k20: 0, k19: 0,
k18: 0, k17: 0, k16: 0, k15: 0, k14: 0, k13: 0, k12: 0, k11: 0,
k10: 0, k9: 0, k8: 0, k7: 0, k6: 0, k5: 0, k4: 0, k3: 0,
k2: 0, k1: 0,
k34: b/c*k42, k31: k39, k26: a/c*k42, k23: k39
}
assert solve_lin_sys(system, variables) == solution
def test_M39():
x, y, z = symbols('x y z', complex=True)
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports non-linear multivariate
assert solve([x**2*y + 3*y*z - 4, -3*x**2*z + 2*y**2 + 1, 2*y*z**2 - z**2 - 1 ]) ==\
[{y: 1, z: 1, x: -1}, {y: 1, z: 1, x: 1},\
{y: sqrt(2)*I, z: R(1,3) - sqrt(2)*I/3, x: -sqrt(-1 - sqrt(2)*I)},\
{y: sqrt(2)*I, z: R(1,3) - sqrt(2)*I/3, x: sqrt(-1 - sqrt(2)*I)},\
{y: -sqrt(2)*I, z: R(1,3) + sqrt(2)*I/3, x: -sqrt(-1 + sqrt(2)*I)},\
{y: -sqrt(2)*I, z: R(1,3) + sqrt(2)*I/3, x: sqrt(-1 + sqrt(2)*I)}]
# N. Inequalities
def test_N1():
assert ask(Q.is_true(E**pi > pi**E))
@XFAIL
def test_N2():
x = symbols('x', real=True)
assert ask(Q.is_true(x**4 - x + 1 > 0)) is True
assert ask(Q.is_true(x**4 - x + 1 > 1)) is False
@XFAIL
def test_N3():
x = symbols('x', real=True)
assert ask(Q.is_true(And(Lt(-1, x), Lt(x, 1))), Q.is_true(abs(x) < 1 ))
@XFAIL
def test_N4():
x, y = symbols('x y', real=True)
assert ask(Q.is_true(2*x**2 > 2*y**2), Q.is_true((x > y) & (y > 0))) is True
@XFAIL
def test_N5():
x, y, k = symbols('x y k', real=True)
assert ask(Q.is_true(k*x**2 > k*y**2), Q.is_true((x > y) & (y > 0) & (k > 0))) is True
@XFAIL
def test_N6():
x, y, k, n = symbols('x y k n', real=True)
assert ask(Q.is_true(k*x**n > k*y**n), Q.is_true((x > y) & (y > 0) & (k > 0) & (n > 0))) is True
@XFAIL
def test_N7():
x, y = symbols('x y', real=True)
assert ask(Q.is_true(y > 0), Q.is_true((x > 1) & (y >= x - 1))) is True
@XFAIL
def test_N8():
x, y, z = symbols('x y z', real=True)
assert ask(Q.is_true((x == y) & (y == z)),
Q.is_true((x >= y) & (y >= z) & (z >= x)))
def test_N9():
x = Symbol('x')
assert solveset(abs(x - 1) > 2, domain=S.Reals) == Union(Interval(-oo, -1, False, True),
Interval(3, oo, True))
def test_N10():
x = Symbol('x')
p = (x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)
assert solveset(expand(p) < 0, domain=S.Reals) == Union(Interval(-oo, 1, True, True),
Interval(2, 3, True, True),
Interval(4, 5, True, True))
def test_N11():
x = Symbol('x')
assert solveset(6/(x - 3) <= 3, domain=S.Reals) == Union(Interval(-oo, 3, True, True), Interval(5, oo))
def test_N12():
x = Symbol('x')
assert solveset(sqrt(x) < 2, domain=S.Reals) == Interval(0, 4, False, True)
def test_N13():
x = Symbol('x')
assert solveset(sin(x) < 2, domain=S.Reals) == S.Reals
@XFAIL
def test_N14():
x = Symbol('x')
# Gives 'Union(Interval(Integer(0), Mul(Rational(1, 2), pi), false, true),
# Interval(Mul(Rational(1, 2), pi), Mul(Integer(2), pi), true, false))'
# which is not the correct answer, but the provided also seems wrong.
assert solveset(sin(x) < 1, x, domain=S.Reals) == Union(Interval(-oo, pi/2, True, True),
Interval(pi/2, oo, True, True))
def test_N15():
r, t = symbols('r t')
# raises NotImplementedError: only univariate inequalities are supported
solveset(abs(2*r*(cos(t) - 1) + 1) <= 1, r, S.Reals)
def test_N16():
r, t = symbols('r t')
solveset((r**2)*((cos(t) - 4)**2)*sin(t)**2 < 9, r, S.Reals)
@XFAIL
def test_N17():
# currently only univariate inequalities are supported
assert solveset((x + y > 0, x - y < 0), (x, y)) == (abs(x) < y)
def test_O1():
M = Matrix((1 + I, -2, 3*I))
assert sqrt(expand(M.dot(M.H))) == sqrt(15)
def test_O2():
assert Matrix((2, 2, -3)).cross(Matrix((1, 3, 1))) == Matrix([[11],
[-5],
[4]])
# The vector module has no way of representing vectors symbolically (without
# respect to a basis)
@XFAIL
def test_O3():
assert (va ^ vb) | (vc ^ vd) == -(va | vc)*(vb | vd) + (va | vd)*(vb | vc)
def test_O4():
from sympy.vector import CoordSys3D, Del
N = CoordSys3D("N")
delop = Del()
i, j, k = N.base_vectors()
x, y, z = N.base_scalars()
F = i*(x*y*z) + j*((x*y*z)**2) + k*((y**2)*(z**3))
assert delop.cross(F).doit() == (-2*x**2*y**2*z + 2*y*z**3)*i + x*y*j + (2*x*y**2*z**2 - x*z)*k
# The vector module has no way of representing vectors symbolically (without
# respect to a basis)
@XFAIL
def test_O5():
assert grad|(f^g)-g|(grad^f)+f|(grad^g) == 0
#testO8-O9 MISSING!!
def test_O10():
L = [Matrix([2, 3, 5]), Matrix([3, 6, 2]), Matrix([8, 3, 6])]
assert GramSchmidt(L) == [Matrix([
[2],
[3],
[5]]),
Matrix([
[S(23)/19],
[S(63)/19],
[S(-47)/19]]),
Matrix([
[S(1692)/353],
[S(-1551)/706],
[S(-423)/706]])]
def test_P1():
assert Matrix(3, 3, lambda i, j: j - i).diagonal(-1) == Matrix(
1, 2, [-1, -1])
def test_P2():
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
M.row_del(1)
M.col_del(2)
assert M == Matrix([[1, 2],
[7, 8]])
def test_P3():
A = Matrix([
[11, 12, 13, 14],
[21, 22, 23, 24],
[31, 32, 33, 34],
[41, 42, 43, 44]])
A11 = A[0:3, 1:4]
A12 = A[(0, 1, 3), (2, 0, 3)]
A21 = A
A221 = -A[0:2, 2:4]
A222 = -A[(3, 0), (2, 1)]
A22 = BlockMatrix([[A221, A222]]).T
rows = [[-A11, A12], [A21, A22]]
from sympy.utilities.pytest import raises
raises(ValueError, lambda: BlockMatrix(rows))
B = Matrix(rows)
assert B == Matrix([
[-12, -13, -14, 13, 11, 14],
[-22, -23, -24, 23, 21, 24],
[-32, -33, -34, 43, 41, 44],
[11, 12, 13, 14, -13, -23],
[21, 22, 23, 24, -14, -24],
[31, 32, 33, 34, -43, -13],
[41, 42, 43, 44, -42, -12]])
@XFAIL
def test_P4():
raise NotImplementedError("Block matrix diagonalization not supported")
def test_P5():
M = Matrix([[7, 11],
[3, 8]])
assert M % 2 == Matrix([[1, 1],
[1, 0]])
def test_P6():
M = Matrix([[cos(x), sin(x)],
[-sin(x), cos(x)]])
assert M.diff(x, 2) == Matrix([[-cos(x), -sin(x)],
[sin(x), -cos(x)]])
def test_P7():
M = Matrix([[x, y]])*(
z*Matrix([[1, 3, 5],
[2, 4, 6]]) + Matrix([[7, -9, 11],
[-8, 10, -12]]))
assert M == Matrix([[x*(z + 7) + y*(2*z - 8), x*(3*z - 9) + y*(4*z + 10),
x*(5*z + 11) + y*(6*z - 12)]])
def test_P8():
M = Matrix([[1, -2*I],
[-3*I, 4]])
assert M.norm(ord=S.Infinity) == 7
def test_P9():
a, b, c = symbols('a b c', real=True)
M = Matrix([[a/(b*c), 1/c, 1/b],
[1/c, b/(a*c), 1/a],
[1/b, 1/a, c/(a*b)]])
assert factor(M.norm('fro')) == (a**2 + b**2 + c**2)/(abs(a)*abs(b)*abs(c))
@XFAIL
def test_P10():
M = Matrix([[1, 2 + 3*I],
[f(4 - 5*I), 6]])
# conjugate(f(4 - 5*i)) is not simplified to f(4+5*I)
assert M.H == Matrix([[1, f(4 + 5*I)],
[2 + 3*I, 6]])
@XFAIL
def test_P11():
# raises NotImplementedError("Matrix([[x,y],[1,x*y]]).inv()
# not simplifying to extract common factor")
assert Matrix([[x, y],
[1, x*y]]).inv() == (1/(x**2 - 1))*Matrix([[x, -1],
[-1/y, x/y]])
def test_P11_workaround():
M = Matrix([[x, y], [1, x*y]]).inv()
c = gcd(tuple(M))
assert MatMul(c, M/c, evaluate=False) == MatMul(c, Matrix([
[-x*y, y],
[ 1, -x]]), evaluate=False)
def test_P12():
A11 = MatrixSymbol('A11', n, n)
A12 = MatrixSymbol('A12', n, n)
A22 = MatrixSymbol('A22', n, n)
B = BlockMatrix([[A11, A12],
[ZeroMatrix(n, n), A22]])
assert block_collapse(B.I) == BlockMatrix([[A11.I, (-1)*A11.I*A12*A22.I],
[ZeroMatrix(n, n), A22.I]])
def test_P13():
M = Matrix([[1, x - 2, x - 3],
[x - 1, x**2 - 3*x + 6, x**2 - 3*x - 2],
[x - 2, x**2 - 8, 2*(x**2) - 12*x + 14]])
L, U, _ = M.LUdecomposition()
assert simplify(L) == Matrix([[1, 0, 0],
[x - 1, 1, 0],
[x - 2, x - 3, 1]])
assert simplify(U) == Matrix([[1, x - 2, x - 3],
[0, 4, x - 5],
[0, 0, x - 7]])
def test_P14():
M = Matrix([[1, 2, 3, 1, 3],
[3, 2, 1, 1, 7],
[0, 2, 4, 1, 1],
[1, 1, 1, 1, 4]])
R, _ = M.rref()
assert R == Matrix([[1, 0, -1, 0, 2],
[0, 1, 2, 0, -1],
[0, 0, 0, 1, 3],
[0, 0, 0, 0, 0]])
def test_P15():
M = Matrix([[-1, 3, 7, -5],
[4, -2, 1, 3],
[2, 4, 15, -7]])
assert M.rank() == 2
def test_P16():
M = Matrix([[2*sqrt(2), 8],
[6*sqrt(6), 24*sqrt(3)]])
assert M.rank() == 1
def test_P17():
t = symbols('t', real=True)
M=Matrix([
[sin(2*t), cos(2*t)],
[2*(1 - (cos(t)**2))*cos(t), (1 - 2*(sin(t)**2))*sin(t)]])
assert M.rank() == 1
def test_P18():
M = Matrix([[1, 0, -2, 0],
[-2, 1, 0, 3],
[-1, 2, -6, 6]])
assert M.nullspace() == [Matrix([[2],
[4],
[1],
[0]]),
Matrix([[0],
[-3],
[0],
[1]])]
def test_P19():
w = symbols('w')
M = Matrix([[1, 1, 1, 1],
[w, x, y, z],
[w**2, x**2, y**2, z**2],
[w**3, x**3, y**3, z**3]])
assert M.det() == (w**3*x**2*y - w**3*x**2*z - w**3*x*y**2 + w**3*x*z**2
+ w**3*y**2*z - w**3*y*z**2 - w**2*x**3*y + w**2*x**3*z
+ w**2*x*y**3 - w**2*x*z**3 - w**2*y**3*z + w**2*y*z**3
+ w*x**3*y**2 - w*x**3*z**2 - w*x**2*y**3 + w*x**2*z**3
+ w*y**3*z**2 - w*y**2*z**3 - x**3*y**2*z + x**3*y*z**2
+ x**2*y**3*z - x**2*y*z**3 - x*y**3*z**2 + x*y**2*z**3
)
@XFAIL
def test_P20():
raise NotImplementedError("Matrix minimal polynomial not supported")
def test_P21():
M = Matrix([[5, -3, -7],
[-2, 1, 2],
[2, -3, -4]])
assert M.charpoly(x).as_expr() == x**3 - 2*x**2 - 5*x + 6
def test_P22():
d = 100
M = (2 - x)*eye(d)
assert M.eigenvals() == {-x + 2: d}
def test_P23():
M = Matrix([
[2, 1, 0, 0, 0],
[1, 2, 1, 0, 0],
[0, 1, 2, 1, 0],
[0, 0, 1, 2, 1],
[0, 0, 0, 1, 2]])
assert M.eigenvals() == {
S('1'): 1,
S('2'): 1,
S('3'): 1,
S('sqrt(3) + 2'): 1,
S('-sqrt(3) + 2'): 1}
def test_P24():
M = Matrix([[611, 196, -192, 407, -8, -52, -49, 29],
[196, 899, 113, -192, -71, -43, -8, -44],
[-192, 113, 899, 196, 61, 49, 8, 52],
[ 407, -192, 196, 611, 8, 44, 59, -23],
[ -8, -71, 61, 8, 411, -599, 208, 208],
[ -52, -43, 49, 44, -599, 411, 208, 208],
[ -49, -8, 8, 59, 208, 208, 99, -911],
[ 29, -44, 52, -23, 208, 208, -911, 99]])
assert M.eigenvals() == {
S('0'): 1,
S('10*sqrt(10405)'): 1,
S('100*sqrt(26) + 510'): 1,
S('1000'): 2,
S('-100*sqrt(26) + 510'): 1,
S('-10*sqrt(10405)'): 1,
S('1020'): 1}
def test_P25():
MF = N(Matrix([[ 611, 196, -192, 407, -8, -52, -49, 29],
[ 196, 899, 113, -192, -71, -43, -8, -44],
[-192, 113, 899, 196, 61, 49, 8, 52],
[ 407, -192, 196, 611, 8, 44, 59, -23],
[ -8, -71, 61, 8, 411, -599, 208, 208],
[ -52, -43, 49, 44, -599, 411, 208, 208],
[ -49, -8, 8, 59, 208, 208, 99, -911],
[ 29, -44, 52, -23, 208, 208, -911, 99]]))
assert (Matrix(sorted(MF.eigenvals())) - Matrix(
[-1020.0490184299969, 0.0, 0.09804864072151699, 1000.0,
1019.9019513592784, 1020.0, 1020.0490184299969])).norm() < 1e-13
def test_P26():
a0, a1, a2, a3, a4 = symbols('a0 a1 a2 a3 a4')
M = Matrix([[-a4, -a3, -a2, -a1, -a0, 0, 0, 0, 0],
[ 1, 0, 0, 0, 0, 0, 0, 0, 0],
[ 0, 1, 0, 0, 0, 0, 0, 0, 0],
[ 0, 0, 1, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, -1, -1, 0, 0],
[ 0, 0, 0, 0, 0, 1, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 1, -1, -1],
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]])
assert M.eigenvals(error_when_incomplete=False) == {
S('-1/2 - sqrt(3)*I/2'): 2,
S('-1/2 + sqrt(3)*I/2'): 2}
def test_P27():
a = symbols('a')
M = Matrix([[a, 0, 0, 0, 0],
[0, 0, 0, 0, 1],
[0, 0, a, 0, 0],
[0, 0, 0, a, 0],
[0, -2, 0, 0, 2]])
assert M.eigenvects() == [(a, 3, [Matrix([[1],
[0],
[0],
[0],
[0]]),
Matrix([[0],
[0],
[1],
[0],
[0]]),
Matrix([[0],
[0],
[0],
[1],
[0]])]),
(1 - I, 1, [Matrix([[ 0],
[-1/(-1 + I)],
[ 0],
[ 0],
[ 1]])]),
(1 + I, 1, [Matrix([[ 0],
[-1/(-1 - I)],
[ 0],
[ 0],
[ 1]])])]
@XFAIL
def test_P28():
raise NotImplementedError("Generalized eigenvectors not supported \
https://github.com/sympy/sympy/issues/5293")
@XFAIL
def test_P29():
raise NotImplementedError("Generalized eigenvectors not supported \
https://github.com/sympy/sympy/issues/5293")
def test_P30():
M = Matrix([[1, 0, 0, 1, -1],
[0, 1, -2, 3, -3],
[0, 0, -1, 2, -2],
[1, -1, 1, 0, 1],
[1, -1, 1, -1, 2]])
_, J = M.jordan_form()
assert J == Matrix([[-1, 0, 0, 0, 0],
[0, 1, 1, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]])
@XFAIL
def test_P31():
raise NotImplementedError("Smith normal form not implemented")
def test_P32():
M = Matrix([[1, -2],
[2, 1]])
assert exp(M).rewrite(cos).simplify() == Matrix([[E*cos(2), -E*sin(2)],
[E*sin(2), E*cos(2)]])
def test_P33():
w, t = symbols('w t')
M = Matrix([[0, 1, 0, 0],
[0, 0, 0, 2*w],
[0, 0, 0, 1],
[0, -2*w, 3*w**2, 0]])
assert exp(M*t).rewrite(cos).expand() == Matrix([
[1, -3*t + 4*sin(t*w)/w, 6*t*w - 6*sin(t*w), -2*cos(t*w)/w + 2/w],
[0, 4*cos(t*w) - 3, -6*w*cos(t*w) + 6*w, 2*sin(t*w)],
[0, 2*cos(t*w)/w - 2/w, -3*cos(t*w) + 4, sin(t*w)/w],
[0, -2*sin(t*w), 3*w*sin(t*w), cos(t*w)]])
@XFAIL
def test_P34():
a, b, c = symbols('a b c', real=True)
M = Matrix([[a, 1, 0, 0, 0, 0],
[0, a, 0, 0, 0, 0],
[0, 0, b, 0, 0, 0],
[0, 0, 0, c, 1, 0],
[0, 0, 0, 0, c, 1],
[0, 0, 0, 0, 0, c]])
# raises exception, sin(M) not supported. exp(M*I) also not supported
# https://github.com/sympy/sympy/issues/6218
assert sin(M) == Matrix([[sin(a), cos(a), 0, 0, 0, 0],
[0, sin(a), 0, 0, 0, 0],
[0, 0, sin(b), 0, 0, 0],
[0, 0, 0, sin(c), cos(c), -sin(c)/2],
[0, 0, 0, 0, sin(c), cos(c)],
[0, 0, 0, 0, 0, sin(c)]])
@XFAIL
def test_P35():
M = pi/2*Matrix([[2, 1, 1],
[2, 3, 2],
[1, 1, 2]])
# raises exception, sin(M) not supported. exp(M*I) also not supported
# https://github.com/sympy/sympy/issues/6218
assert sin(M) == eye(3)
@XFAIL
def test_P36():
M = Matrix([[10, 7],
[7, 17]])
assert sqrt(M) == Matrix([[3, 1],
[1, 4]])
def test_P37():
M = Matrix([[1, 1, 0],
[0, 1, 0],
[0, 0, 1]])
assert M**Rational(1, 2) == Matrix([[1, R(1, 2), 0],
[0, 1, 0],
[0, 0, 1]])
@XFAIL
def test_P38():
M=Matrix([[0, 1, 0],
[0, 0, 0],
[0, 0, 0]])
#raises ValueError: Matrix det == 0; not invertible
M**Rational(1,2)
@XFAIL
def test_P39():
"""
M=Matrix([
[1, 1],
[2, 2],
[3, 3]])
M.SVD()
"""
raise NotImplementedError("Singular value decomposition not implemented")
def test_P40():
r, t = symbols('r t', real=True)
M = Matrix([r*cos(t), r*sin(t)])
assert M.jacobian(Matrix([r, t])) == Matrix([[cos(t), -r*sin(t)],
[sin(t), r*cos(t)]])
def test_P41():
r, t = symbols('r t', real=True)
assert hessian(r**2*sin(t),(r,t)) == Matrix([[ 2*sin(t), 2*r*cos(t)],
[2*r*cos(t), -r**2*sin(t)]])
def test_P42():
assert wronskian([cos(x), sin(x)], x).simplify() == 1
def test_P43():
def __my_jacobian(M, Y):
return Matrix([M.diff(v).T for v in Y]).T
r, t = symbols('r t', real=True)
M = Matrix([r*cos(t), r*sin(t)])
assert __my_jacobian(M,[r,t]) == Matrix([[cos(t), -r*sin(t)],
[sin(t), r*cos(t)]])
def test_P44():
def __my_hessian(f, Y):
V = Matrix([diff(f, v) for v in Y])
return Matrix([V.T.diff(v) for v in Y])
r, t = symbols('r t', real=True)
assert __my_hessian(r**2*sin(t), (r, t)) == Matrix([
[ 2*sin(t), 2*r*cos(t)],
[2*r*cos(t), -r**2*sin(t)]])
def test_P45():
def __my_wronskian(Y, v):
M = Matrix([Matrix(Y).T.diff(x, n) for n in range(0, len(Y))])
return M.det()
assert __my_wronskian([cos(x), sin(x)], x).simplify() == 1
# Q1-Q6 Tensor tests missing
@XFAIL
def test_R1():
i, j, n = symbols('i j n', integer=True, positive=True)
xn = MatrixSymbol('xn', n, 1)
Sm = Sum((xn[i, 0] - Sum(xn[j, 0], (j, 0, n - 1))/n)**2, (i, 0, n - 1))
# sum does not calculate
# Unknown result
Sm.doit()
raise NotImplementedError('Unknown result')
@XFAIL
def test_R2():
m, b = symbols('m b')
i, n = symbols('i n', integer=True, positive=True)
xn = MatrixSymbol('xn', n, 1)
yn = MatrixSymbol('yn', n, 1)
f = Sum((yn[i, 0] - m*xn[i, 0] - b)**2, (i, 0, n - 1))
f1 = diff(f, m)
f2 = diff(f, b)
# raises TypeError: solveset() takes at most 2 arguments (3 given)
solveset((f1, f2), (m, b), domain=S.Reals)
@XFAIL
def test_R3():
n, k = symbols('n k', integer=True, positive=True)
sk = ((-1)**k) * (binomial(2*n, k))**2
Sm = Sum(sk, (k, 1, oo))
T = Sm.doit()
T2 = T.combsimp()
# returns -((-1)**n*factorial(2*n)
# - (factorial(n))**2)*exp_polar(-I*pi)/(factorial(n))**2
assert T2 == (-1)**n*binomial(2*n, n)
@XFAIL
def test_R4():
# Macsyma indefinite sum test case:
#(c15) /* Check whether the full Gosper algorithm is implemented
# => 1/2^(n + 1) binomial(n, k - 1) */
#closedform(indefsum(binomial(n, k)/2^n - binomial(n + 1, k)/2^(n + 1), k));
#Time= 2690 msecs
# (- n + k - 1) binomial(n + 1, k)
#(d15) - --------------------------------
# n
# 2 2 (n + 1)
#
#(c16) factcomb(makefact(%));
#Time= 220 msecs
# n!
#(d16) ----------------
# n
# 2 k! 2 (n - k)!
# Might be possible after fixing https://github.com/sympy/sympy/pull/1879
raise NotImplementedError("Indefinite sum not supported")
@XFAIL
def test_R5():
a, b, c, n, k = symbols('a b c n k', integer=True, positive=True)
sk = ((-1)**k)*(binomial(a + b, a + k)
*binomial(b + c, b + k)*binomial(c + a, c + k))
Sm = Sum(sk, (k, 1, oo))
T = Sm.doit() # hypergeometric series not calculated
assert T == factorial(a+b+c)/(factorial(a)*factorial(b)*factorial(c))
def test_R6():
n, k = symbols('n k', integer=True, positive=True)
gn = MatrixSymbol('gn', n + 2, 1)
Sm = Sum(gn[k, 0] - gn[k - 1, 0], (k, 1, n + 1))
assert Sm.doit() == -gn[0, 0] + gn[n + 1, 0]
def test_R7():
n, k = symbols('n k', integer=True, positive=True)
T = Sum(k**3,(k,1,n)).doit()
assert T.factor() == n**2*(n + 1)**2/4
@XFAIL
def test_R8():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(k**2*binomial(n, k), (k, 1, n))
T = Sm.doit() #returns Piecewise function
assert T.combsimp() == n*(n + 1)*2**(n - 2)
def test_R9():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n, k - 1)/k, (k, 1, n + 1))
assert Sm.doit().simplify() == (2**(n + 1) - 1)/(n + 1)
@XFAIL
def test_R10():
n, m, r, k = symbols('n m r k', integer=True, positive=True)
Sm = Sum(binomial(n, k)*binomial(m, r - k), (k, 0, r))
T = Sm.doit()
T2 = T.combsimp().rewrite(factorial)
assert T2 == factorial(m + n)/(factorial(r)*factorial(m + n - r))
assert T2 == binomial(m + n, r).rewrite(factorial)
# rewrite(binomial) is not working.
# https://github.com/sympy/sympy/issues/7135
T3 = T2.rewrite(binomial)
assert T3 == binomial(m + n, r)
@XFAIL
def test_R11():
n, k = symbols('n k', integer=True, positive=True)
sk = binomial(n, k)*fibonacci(k)
Sm = Sum(sk, (k, 0, n))
T = Sm.doit()
# Fibonacci simplification not implemented
# https://github.com/sympy/sympy/issues/7134
assert T == fibonacci(2*n)
@XFAIL
def test_R12():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(fibonacci(k)**2, (k, 0, n))
T = Sm.doit()
assert T == fibonacci(n)*fibonacci(n + 1)
@XFAIL
def test_R13():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(sin(k*x), (k, 1, n))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == cot(x/2)/2 - cos(x*(2*n + 1)/2)/(2*sin(x/2))
@XFAIL
def test_R14():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(sin((2*k - 1)*x), (k, 1, n))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == sin(n*x)**2/sin(x)
@XFAIL
def test_R15():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n - k, k), (k, 0, floor(n/2)))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == fibonacci(n + 1)
def test_R16():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/k**2 + 1/k**3, (k, 1, oo))
assert Sm.doit() == zeta(3) + pi**2/6
def test_R17():
k = symbols('k', integer=True, positive=True)
assert abs(float(Sum(1/k**2 + 1/k**3, (k, 1, oo)))
- 2.8469909700078206) < 1e-15
def test_R18():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/(2**k*k**2), (k, 1, oo))
T = Sm.doit()
assert T.simplify() == -log(2)**2/2 + pi**2/12
@slow
@XFAIL
def test_R19():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/((3*k + 1)*(3*k + 2)*(3*k + 3)), (k, 0, oo))
T = Sm.doit()
# assert fails, T not simplified
assert T.simplify() == -log(3)/4 + sqrt(3)*pi/12
@XFAIL
def test_R20():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n, 4*k), (k, 0, oo))
T = Sm.doit()
# assert fails, T not simplified
assert T.simplify() == 2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2
@XFAIL
def test_R21():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/(sqrt(k*(k + 1)) * (sqrt(k) + sqrt(k + 1))), (k, 1, oo))
T = Sm.doit() # Sum not calculated
assert T.simplify() == 1
# test_R22 answer not available in Wester samples
# Sum(Sum(binomial(n, k)*binomial(n - k, n - 2*k)*x**n*y**(n - 2*k),
# (k, 0, floor(n/2))), (n, 0, oo)) with abs(x*y)<1?
@XFAIL
def test_R23():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(Sum((factorial(n)/(factorial(k)**2*factorial(n - 2*k)))*
(x/y)**k*(x*y)**(n - k), (n, 2*k, oo)), (k, 0, oo))
# Missing how to express constraint abs(x*y)<1?
T = Sm.doit() # Sum not calculated
assert T == -1/sqrt(x**2*y**2 - 4*x**2 - 2*x*y + 1)
def test_R24():
m, k = symbols('m k', integer=True, positive=True)
Sm = Sum(Product(k/(2*k - 1), (k, 1, m)), (m, 2, oo))
assert Sm.doit() == pi/2
def test_S1():
k = symbols('k', integer=True, positive=True)
Pr = Product(gamma(k/3), (k, 1, 8))
assert Pr.doit().simplify() == 640*sqrt(3)*pi**3/6561
def test_S2():
n, k = symbols('n k', integer=True, positive=True)
assert Product(k, (k, 1, n)).doit() == factorial(n)
def test_S3():
n, k = symbols('n k', integer=True, positive=True)
assert Product(x**k, (k, 1, n)).doit().simplify() == x**(n*(n + 1)/2)
def test_S4():
n, k = symbols('n k', integer=True, positive=True)
assert Product(1 + 1/k, (k, 1, n -1)).doit().simplify() == n
def test_S5():
n, k = symbols('n k', integer=True, positive=True)
assert (Product((2*k - 1)/(2*k), (k, 1, n)).doit().gammasimp() ==
gamma(n + Rational(1, 2))/(sqrt(pi)*gamma(n + 1)))
@XFAIL
def test_S6():
n, k = symbols('n k', integer=True, positive=True)
# Product does not evaluate
assert (Product(x**2 -2*x*cos(k*pi/n) + 1, (k, 1, n - 1)).doit().simplify()
== (x**(2*n) - 1)/(x**2 - 1))
@XFAIL
def test_S7():
k = symbols('k', integer=True, positive=True)
Pr = Product((k**3 - 1)/(k**3 + 1), (k, 2, oo))
T = Pr.doit() # Product does not evaluate
assert T.simplify() == Rational(2, 3)
@XFAIL
def test_S8():
k = symbols('k', integer=True, positive=True)
Pr = Product(1 - 1/(2*k)**2, (k, 1, oo))
T = Pr.doit()
# Product does not evaluate
assert T.simplify() == 2/pi
@XFAIL
def test_S9():
k = symbols('k', integer=True, positive=True)
Pr = Product(1 + (-1)**(k + 1)/(2*k - 1), (k, 1, oo))
T = Pr.doit()
# Product produces 0
# https://github.com/sympy/sympy/issues/7133
assert T.simplify() == sqrt(2)
@XFAIL
def test_S10():
k = symbols('k', integer=True, positive=True)
Pr = Product((k*(k + 1) + 1 + I)/(k*(k + 1) + 1 - I), (k, 0, oo))
T = Pr.doit()
# Product does not evaluate
assert T.simplify() == -1
def test_T1():
assert limit((1 + 1/n)**n, n, oo) == E
assert limit((1 - cos(x))/x**2, x, 0) == Rational(1, 2)
def test_T2():
assert limit((3**x + 5**x)**(1/x), x, oo) == 5
def test_T3():
assert limit(log(x)/(log(x) + sin(x)), x, oo) == 1
def test_T4():
assert limit((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1))))
- exp(x))/x, x, oo) == -exp(2)
def test_T5():
assert limit(x*log(x)*log(x*exp(x) - x**2)**2/log(log(x**2
+ 2*exp(exp(3*x**3*log(x))))), x, oo) == Rational(1, 3)
def test_T6():
assert limit(1/n * factorial(n)**(1/n), n, oo) == exp(-1)
def test_T7():
limit(1/n * gamma(n + 1)**(1/n), n, oo)
def test_T8():
a, z = symbols('a z', real=True, positive=True)
assert limit(gamma(z + a)/gamma(z)*exp(-a*log(z)), z, oo) == 1
@XFAIL
def test_T9():
z, k = symbols('z k', real=True, positive=True)
# raises NotImplementedError:
# Don't know how to calculate the mrv of '(1, k)'
assert limit(hyper((1, k), (1,), z/k), k, oo) == exp(z)
@XFAIL
def test_T10():
# No longer raises PoleError, but should return euler-mascheroni constant
assert limit(zeta(x) - 1/(x - 1), x, 1) == integrate(-1/x + 1/floor(x), (x, 1, oo))
@XFAIL
def test_T11():
n, k = symbols('n k', integer=True, positive=True)
# evaluates to 0
assert limit(n**x/(x*product((1 + x/k), (k, 1, n))), n, oo) == gamma(x)
@XFAIL
def test_T12():
x, t = symbols('x t', real=True)
# Does not evaluate the limit but returns an expression with erf
assert limit(x * integrate(exp(-t**2), (t, 0, x))/(1 - exp(-x**2)),
x, 0) == 1
def test_T13():
x = symbols('x', real=True)
assert [limit(x/abs(x), x, 0, dir='-'),
limit(x/abs(x), x, 0, dir='+')] == [-1, 1]
def test_T14():
x = symbols('x', real=True)
assert limit(atan(-log(x)), x, 0, dir='+') == pi/2
def test_U1():
x = symbols('x', real=True)
assert diff(abs(x), x) == sign(x)
def test_U2():
f = Lambda(x, Piecewise((-x, x < 0), (x, x >= 0)))
assert diff(f(x), x) == Piecewise((-1, x < 0), (1, x >= 0))
def test_U3():
f = Lambda(x, Piecewise((x**2 - 1, x == 1), (x**3, x != 1)))
f1 = Lambda(x, diff(f(x), x))
assert f1(x) == 3*x**2
assert f1(1) == 3
@XFAIL
def test_U4():
n = symbols('n', integer=True, positive=True)
x = symbols('x', real=True)
d = diff(x**n, x, n)
assert d.rewrite(factorial) == factorial(n)
def test_U5():
# issue 6681
t = symbols('t')
ans = (
Derivative(f(g(t)), g(t))*Derivative(g(t), (t, 2)) +
Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t)**2)
assert f(g(t)).diff(t, 2) == ans
assert ans.doit() == ans
def test_U6():
h = Function('h')
T = integrate(f(y), (y, h(x), g(x)))
assert T.diff(x) == (
f(g(x))*Derivative(g(x), x) - f(h(x))*Derivative(h(x), x))
@XFAIL
def test_U7():
p, t = symbols('p t', real=True)
# Exact differential => d(V(P, T)) => dV/dP DP + dV/dT DT
# raises ValueError: Since there is more than one variable in the
# expression, the variable(s) of differentiation must be supplied to
# differentiate f(p,t)
diff(f(p, t))
def test_U8():
x, y = symbols('x y', real=True)
eq = cos(x*y) + x
# If SymPy had implicit_diff() function this hack could be avoided
# TODO: Replace solve with solveset, current test fails for solveset
assert idiff(y - eq, y, x) == (-y*sin(x*y) + 1)/(x*sin(x*y) + 1)
def test_U9():
# Wester sample case for Maple:
# O29 := diff(f(x, y), x) + diff(f(x, y), y);
# /d \ /d \
# |-- f(x, y)| + |-- f(x, y)|
# \dx / \dy /
#
# O30 := factor(subs(f(x, y) = g(x^2 + y^2), %));
# 2 2
# 2 D(g)(x + y ) (x + y)
x, y = symbols('x y', real=True)
su = diff(f(x, y), x) + diff(f(x, y), y)
s2 = su.subs(f(x, y), g(x**2 + y**2))
s3 = s2.doit().factor()
# Subs not performed, s3 = 2*(x + y)*Subs(Derivative(
# g(_xi_1), _xi_1), _xi_1, x**2 + y**2)
# Derivative(g(x*2 + y**2), x**2 + y**2) is not valid in SymPy,
# and probably will remain that way. You can take derivatives with respect
# to other expressions only if they are atomic, like a symbol or a
# function.
# D operator should be added to SymPy
# See https://github.com/sympy/sympy/issues/4719.
assert s3 == (x + y)*Subs(Derivative(g(x), x), x, x**2 + y**2)*2
def test_U10():
# see issue 2519:
assert residue((z**3 + 5)/((z**4 - 1)*(z + 1)), z, -1) == Rational(-9, 4)
@XFAIL
def test_U11():
assert (2*dx + dz) ^ (3*dx + dy + dz) ^ (dx + dy + 4*dz) == 8*dx ^ dy ^dz
@XFAIL
def test_U12():
# Wester sample case:
# (c41) /* d(3 x^5 dy /\ dz + 5 x y^2 dz /\ dx + 8 z dx /\ dy)
# => (15 x^4 + 10 x y + 8) dx /\ dy /\ dz */
# factor(ext_diff(3*x^5 * dy ~ dz + 5*x*y^2 * dz ~ dx + 8*z * dx ~ dy));
# 4
# (d41) (10 x y + 15 x + 8) dx dy dz
raise NotImplementedError(
"External diff of differential form not supported")
@XFAIL
def test_U13():
#assert minimize(x**4 - x + 1, x)== -3*2**Rational(1,3)/8 + 1
raise NotImplementedError("minimize() not supported")
@XFAIL
def test_U14():
#f = 1/(x**2 + y**2 + 1)
#assert [minimize(f), maximize(f)] == [0,1]
raise NotImplementedError("minimize(), maximize() not supported")
@XFAIL
def test_U15():
raise NotImplementedError("minimize() not supported and also solve does \
not support multivariate inequalities")
@XFAIL
def test_U16():
raise NotImplementedError("minimize() not supported in SymPy and also \
solve does not support multivariate inequalities")
@XFAIL
def test_U17():
raise NotImplementedError("Linear programming, symbolic simplex not \
supported in SymPy")
def test_V1():
x = symbols('x', real=True)
assert integrate(abs(x), x) == Piecewise((-x**2/2, x <= 0), (x**2/2, True))
def test_V2():
assert integrate(Piecewise((-x, x < 0), (x, x >= 0)), x
) == Piecewise((-x**2/2, x < 0), (x**2/2, True))
def test_V3():
assert integrate(1/(x**3 + 2),x).diff().simplify() == 1/(x**3 + 2)
def test_V4():
assert integrate(2**x/sqrt(1 + 4**x), x) == asinh(2**x)/log(2)
@XFAIL
def test_V5():
# Returns (-45*x**2 + 80*x - 41)/(5*sqrt(2*x - 1)*(4*x**2 - 4*x + 1))
assert (integrate((3*x - 5)**2/(2*x - 1)**(Rational(7, 2)), x).simplify() ==
(-41 + 80*x - 45*x**2)/(5*(2*x - 1)**Rational(5, 2)))
@XFAIL
def test_V6():
# returns RootSum(40*_z**2 - 1, Lambda(_i, _i*log(-4*_i + exp(-m*x))))/m
assert (integrate(1/(2*exp(m*x) - 5*exp(-m*x)), x) == sqrt(10)*(
log(2*exp(m*x) - sqrt(10)) - log(2*exp(m*x) + sqrt(10)))/(20*m))
def test_V7():
r1 = integrate(sinh(x)**4/cosh(x)**2)
assert r1.simplify() == -3*x/2 + sinh(x)**3/(2*cosh(x)) + 3*tanh(x)/2
@XFAIL
def test_V8_V9():
#Macsyma test case:
#(c27) /* This example involves several symbolic parameters
# => 1/sqrt(b^2 - a^2) log([sqrt(b^2 - a^2) tan(x/2) + a + b]/
# [sqrt(b^2 - a^2) tan(x/2) - a - b]) (a^2 < b^2)
# [Gradshteyn and Ryzhik 2.553(3)] */
#assume(b^2 > a^2)$
#(c28) integrate(1/(a + b*cos(x)), x);
#(c29) trigsimp(ratsimp(diff(%, x)));
# 1
#(d29) ------------
# b cos(x) + a
raise NotImplementedError(
"Integrate with assumption not supported")
def test_V10():
assert integrate(1/(3 + 3*cos(x) + 4*sin(x)), x) == log(tan(x/2) + Rational(3, 4))/4
def test_V11():
r1 = integrate(1/(4 + 3*cos(x) + 4*sin(x)), x)
r2 = factor(r1)
assert (logcombine(r2, force=True) ==
log(((tan(x/2) + 1)/(tan(x/2) + 7))**Rational(1, 3)))
@XFAIL
def test_V12():
r1 = integrate(1/(5 + 3*cos(x) + 4*sin(x)), x)
# Correct result in python2.7.4, wrong result in python3.5
# https://github.com/sympy/sympy/issues/7157
assert r1 == -1/(tan(x/2) + 2)
@XFAIL
def test_V13():
r1 = integrate(1/(6 + 3*cos(x) + 4*sin(x)), x)
# expression not simplified, returns: -sqrt(11)*I*log(tan(x/2) + 4/3
# - sqrt(11)*I/3)/11 + sqrt(11)*I*log(tan(x/2) + 4/3 + sqrt(11)*I/3)/11
assert r1.simplify() == 2*sqrt(11)*atan(sqrt(11)*(3*tan(x/2) + 4)/11)/11
@slow
@XFAIL
def test_V14():
r1 = integrate(log(abs(x**2 - y**2)), x)
# Piecewise result does not simplify to the desired result.
assert (r1.simplify() == x*log(abs(x**2 - y**2))
+ y*log(x + y) - y*log(x - y) - 2*x)
def test_V15():
r1 = integrate(x*acot(x/y), x)
assert simplify(r1 - (x*y + (x**2 + y**2)*acot(x/y))/2) == 0
@XFAIL
def test_V16():
# Integral not calculated
assert integrate(cos(5*x)*Ci(2*x), x) == Ci(2*x)*sin(5*x)/5 - (Si(3*x) + Si(7*x))/10
@XFAIL
def test_V17():
r1 = integrate((diff(f(x), x)*g(x)
- f(x)*diff(g(x), x))/(f(x)**2 - g(x)**2), x)
# integral not calculated
assert simplify(r1 - (f(x) - g(x))/(f(x) + g(x))/2) == 0
@XFAIL
def test_W1():
# The function has a pole at y.
# The integral has a Cauchy principal value of zero but SymPy returns -I*pi
# https://github.com/sympy/sympy/issues/7159
assert integrate(1/(x - y), (x, y - 1, y + 1)) == 0
@XFAIL
def test_W2():
# The function has a pole at y.
# The integral is divergent but SymPy returns -2
# https://github.com/sympy/sympy/issues/7160
# Test case in Macsyma:
# (c6) errcatch(integrate(1/(x - a)^2, x, a - 1, a + 1));
# Integral is divergent
assert integrate(1/(x - y)**2, (x, y - 1, y + 1)) == zoo
@XFAIL
def test_W3():
# integral is not calculated
# https://github.com/sympy/sympy/issues/7161
assert integrate(sqrt(x + 1/x - 2), (x, 0, 1)) == S(4)/3
@XFAIL
def test_W4():
# integral is not calculated
assert integrate(sqrt(x + 1/x - 2), (x, 1, 2)) == -2*sqrt(2)/3 + S(4)/3
@XFAIL
def test_W5():
# integral is not calculated
assert integrate(sqrt(x + 1/x - 2), (x, 0, 2)) == -2*sqrt(2)/3 + S(8)/3
@XFAIL
@slow
def test_W6():
# integral is not calculated
assert integrate(sqrt(2 - 2*cos(2*x))/2, (x, -3*pi/4, -pi/4)) == sqrt(2)
def test_W7():
a = symbols('a', real=True, positive=True)
r1 = integrate(cos(x)/(x**2 + a**2), (x, -oo, oo))
assert r1.simplify() == pi*exp(-a)/a
@XFAIL
def test_W8():
# Test case in Mathematica:
# In[19]:= Integrate[t^(a - 1)/(1 + t), {t, 0, Infinity},
# Assumptions -> 0 < a < 1]
# Out[19]= Pi Csc[a Pi]
raise NotImplementedError(
"Integrate with assumption 0 < a < 1 not supported")
@XFAIL
def test_W9():
# Integrand with a residue at infinity => -2 pi [sin(pi/5) + sin(2pi/5)]
# (principal value) [Levinson and Redheffer, p. 234] *)
r1 = integrate(5*x**3/(1 + x + x**2 + x**3 + x**4), (x, -oo, oo))
r2 = r1.doit()
assert r2 == -2*pi*(sqrt(-sqrt(5)/8 + 5/8) + sqrt(sqrt(5)/8 + 5/8))
@XFAIL
def test_W10():
# integrate(1/[1 + x + x^2 + ... + x^(2 n)], x = -infinity..infinity) =
# 2 pi/(2 n + 1) [1 + cos(pi/[2 n + 1])] csc(2 pi/[2 n + 1])
# [Levinson and Redheffer, p. 255] => 2 pi/5 [1 + cos(pi/5)] csc(2 pi/5) */
r1 = integrate(x/(1 + x + x**2 + x**4), (x, -oo, oo))
r2 = r1.doit()
assert r2 == 2*pi*(sqrt(5)/4 + 5/4)*csc(2*pi/5)/5
@XFAIL
def test_W11():
# integral not calculated
assert (integrate(sqrt(1 - x**2)/(1 + x**2), (x, -1, 1)) ==
pi*(-1 + sqrt(2)))
def test_W12():
p = symbols('p', real=True, positive=True)
q = symbols('q', real=True)
r1 = integrate(x*exp(-p*x**2 + 2*q*x), (x, -oo, oo))
assert r1.simplify() == sqrt(pi)*q*exp(q**2/p)/p**Rational(3, 2)
@XFAIL
def test_W13():
# Integral not calculated. Expected result is 2*(Euler_mascheroni_constant)
r1 = integrate(1/log(x) + 1/(1 - x) - log(log(1/x)), (x, 0, 1))
assert r1 == 2*EulerGamma
def test_W14():
assert integrate(sin(x)/x*exp(2*I*x), (x, -oo, oo)) == 0
@XFAIL
def test_W15():
# integral not calculated
assert integrate(log(gamma(x))*cos(6*pi*x), (x, 0, 1)) == S(1)/12
def test_W16():
assert integrate((1 + x)**3*legendre_poly(1, x)*legendre_poly(2, x),
(x, -1, 1)) == S(36)/35
def test_W17():
a, b = symbols('a b', real=True, positive=True)
assert integrate(exp(-a*x)*besselj(0, b*x),
(x, 0, oo)) == 1/(b*sqrt(a**2/b**2 + 1))
def test_W18():
assert integrate((besselj(1, x)/x)**2, (x, 0, oo)) == 4/(3*pi)
@XFAIL
def test_W19():
# Integral not calculated
# Expected result is (cos 7 - 1)/7 [Gradshteyn and Ryzhik 6.782(3)]
assert integrate(Ci(x)*besselj(0, 2*sqrt(7*x)), (x, 0, oo)) == (cos(7) - 1)/7
@XFAIL
def test_W20():
# integral not calculated
assert (integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)) ==
-pi**2/36 - S(17)/108 + zeta(3)/4 +
(-pi**2/2 - 4*log(2) + log(2)**2 + 35/3)*log(2)/9)
def test_W21():
assert abs(N(integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)))
- 0.210882859565594) < 1e-15
def test_W22():
t, u = symbols('t u', real=True)
s = Lambda(x, Piecewise((1, And(x >= 1, x <= 2)), (0, True)))
assert integrate(s(t)*cos(t), (t, 0, u)) == Piecewise(
(0, u < 0),
(-sin(Min(1, u)) + sin(Min(2, u)), True))
@slow
def test_W23():
a, b = symbols('a b', real=True, positive=True)
r1 = integrate(integrate(x/(x**2 + y**2), (x, a, b)), (y, -oo, oo))
assert r1.collect(pi) == pi*(-a + b)
def test_W23b():
# like W23 but limits are reversed
x = symbols('x', real=True, positive=True)
y = symbols('y', real=True)
a, b = symbols('a b', real=True, positive=True)
r2 = integrate(integrate(x/(x**2 + y**2), (y, -oo, oo)), (x, a, b))
assert r2.collect(pi) == pi*(-a + b)
@XFAIL
@slow
def test_W24():
if ON_TRAVIS:
skip("Too slow for travis.")
# Not that slow, but does not fully evaluate so simplify is slow.
# Maybe also require doit()
x, y = symbols('x y', real=True)
r1 = integrate(integrate(sqrt(x**2 + y**2), (x, 0, 1)), (y, 0, 1))
assert (r1 - (sqrt(2) + asinh(1))/3).simplify() == 0
@XFAIL
@slow
def test_W25():
if ON_TRAVIS:
skip("Too slow for travis.")
a, x, y = symbols('a x y', real=True)
i1 = integrate(
sin(a)*sin(y)/sqrt(1 - sin(a)**2*sin(x)**2*sin(y)**2),
(x, 0, pi/2))
i2 = integrate(i1, (y, 0, pi/2))
assert (i2 - pi*a/2).simplify() == 0
def test_W26():
x, y = symbols('x y', real=True)
assert integrate(integrate(abs(y - x**2), (y, 0, 2)),
(x, -1, 1)) == S(46)/15
def test_W27():
a, b, c = symbols('a b c')
assert integrate(integrate(integrate(1, (z, 0, c*(1 - x/a - y/b))),
(y, 0, b*(1 - x/a))),
(x, 0, a)) == a*b*c/6
def test_X1():
v, c = symbols('v c', real=True)
assert (series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8) ==
5*v**6/(16*c**6) + 3*v**4/(8*c**4) + v**2/(2*c**2) + 1 + O(v**8))
def test_X2():
v, c = symbols('v c', real=True)
s1 = series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8)
assert (1/s1**2).series(v, x0=0, n=8) == -v**2/c**2 + 1 + O(v**8)
def test_X3():
s1 = (sin(x).series()/cos(x).series()).series()
s2 = tan(x).series()
assert s2 == x + x**3/3 + 2*x**5/15 + O(x**6)
assert s1 == s2
def test_X4():
s1 = log(sin(x)/x).series()
assert s1 == -x**2/6 - x**4/180 + O(x**6)
assert log(series(sin(x)/x)).series() == s1
@XFAIL
def test_X5():
# test case in Mathematica syntax:
# In[21]:= (* => [a f'(a d) + g(b d) + integrate(h(c y), y = 0..d)]
# + [a^2 f''(a d) + b g'(b d) + h(c d)] (x - d) *)
# In[22]:= D[f[a*x], x] + g[b*x] + Integrate[h[c*y], {y, 0, x}]
# Out[22]= g[b x] + Integrate[h[c y], {y, 0, x}] + a f'[a x]
# In[23]:= Series[%, {x, d, 1}]
# Out[23]= (g[b d] + Integrate[h[c y], {y, 0, d}] + a f'[a d]) +
# 2 2
# (h[c d] + b g'[b d] + a f''[a d]) (-d + x) + O[-d + x]
h = Function('h')
a, b, c, d = symbols('a b c d', real=True)
# series() raises NotImplementedError:
# The _eval_nseries method should be added to <class
# 'sympy.core.function.Subs'> to give terms up to O(x**n) at x=0
series(diff(f(a*x), x) + g(b*x) + integrate(h(c*y), (y, 0, x)),
x, x0=d, n=2)
# assert missing, until exception is removed
def test_X6():
# Taylor series of nonscalar objects (noncommutative multiplication)
# expected result => (B A - A B) t^2/2 + O(t^3) [Stanly Steinberg]
a, b = symbols('a b', commutative=False, scalar=False)
assert (series(exp((a + b)*x) - exp(a*x) * exp(b*x), x, x0=0, n=3) ==
x**2*(-a*b/2 + b*a/2) + O(x**3))
def test_X7():
# => sum( Bernoulli[k]/k! x^(k - 2), k = 1..infinity )
# = 1/x^2 - 1/(2 x) + 1/12 - x^2/720 + x^4/30240 + O(x^6)
# [Levinson and Redheffer, p. 173]
assert (series(1/(x*(exp(x) - 1)), x, 0, 7) == x**(-2) - 1/(2*x) +
S(1)/12 - x**2/720 + x**4/30240 - x**6/1209600 + O(x**7))
def test_X8():
# Puiseux series (terms with fractional degree):
# => 1/sqrt(x - 3/2 pi) + (x - 3/2 pi)^(3/2) / 12 + O([x - 3/2 pi]^(7/2))
# see issue 7167:
x = symbols('x', real=True)
assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
1/sqrt(x - 3*pi/2) + (x - 3*pi/2)**(S(3)/2)/12 +
(x - 3*pi/2)**(S(7)/2)/160 + O((x - 3*pi/2)**4, (x, 3*pi/2)))
def test_X9():
assert (series(x**x, x, x0=0, n=4) == 1 + x*log(x) + x**2*log(x)**2/2 +
x**3*log(x)**3/6 + O(x**4*log(x)**4))
def test_X10():
z, w = symbols('z w')
assert (series(log(sinh(z)) + log(cosh(z + w)), z, x0=0, n=2) ==
log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2))
def test_X11():
z, w = symbols('z w')
assert (series(log(sinh(z) * cosh(z + w)), z, x0=0, n=2) ==
log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2))
@XFAIL
def test_X12():
# Look at the generalized Taylor series around x = 1
# Result => (x - 1)^a/e^b [1 - (a + 2 b) (x - 1) / 2 + O((x - 1)^2)]
a, b, x = symbols('a b x', real=True)
# series returns O(log(x-1)**2)
# https://github.com/sympy/sympy/issues/7168
assert (series(log(x)**a*exp(-b*x), x, x0=1, n=2) ==
(x - 1)**a/exp(b)*(1 - (a + 2*b)*(x - 1)/2 + O((x - 1)**2)))
def test_X13():
assert series(sqrt(2*x**2 + 1), x, x0=oo, n=1) == sqrt(2)*x + O(1/x, (x, oo))
@XFAIL
def test_X14():
# Wallis' product => 1/sqrt(pi n) + ... [Knopp, p. 385]
assert series(1/2**(2*n)*binomial(2*n, n),
n, x==oo, n=1) == 1/(sqrt(pi)*sqrt(n)) + O(1/x, (x, oo))
@SKIP("https://github.com/sympy/sympy/issues/7164")
def test_X15():
# => 0!/x - 1!/x^2 + 2!/x^3 - 3!/x^4 + O(1/x^5) [Knopp, p. 544]
x, t = symbols('x t', real=True)
# raises RuntimeError: maximum recursion depth exceeded
# https://github.com/sympy/sympy/issues/7164
# 2019-02-17: Raises
# PoleError:
# Asymptotic expansion of Ei around [-oo] is not implemented.
e1 = integrate(exp(-t)/t, (t, x, oo))
assert (series(e1, x, x0=oo, n=5) ==
6/x**4 + 2/x**3 - 1/x**2 + 1/x + O(x**(-5), (x, oo)))
def test_X16():
# Multivariate Taylor series expansion => 1 - (x^2 + 2 x y + y^2)/2 + O(x^4)
assert (series(cos(x + y), x + y, x0=0, n=4) == 1 - (x + y)**2/2 +
O(x**4 + x**3*y + x**2*y**2 + x*y**3 + y**4, x, y))
@XFAIL
def test_X17():
# Power series (compute the general formula)
# (c41) powerseries(log(sin(x)/x), x, 0);
# /aquarius/data2/opt/local/macsyma_422/library1/trgred.so being loaded.
# inf
# ==== i1 2 i1 2 i1
# \ (- 1) 2 bern(2 i1) x
# (d41) > ------------------------------
# / 2 i1 (2 i1)!
# ====
# i1 = 1
# fps does not calculate
assert fps(log(sin(x)/x)) == \
Sum((-1)**k*2**(2*k - 1)*bernoulli(2*k)*x**(2*k)/(k*factorial(2*k)), (k, 1, oo))
@XFAIL
def test_X18():
# Power series (compute the general formula). Maple FPS:
# > FormalPowerSeries(exp(-x)*sin(x), x = 0);
# infinity
# ----- (1/2 k) k
# \ 2 sin(3/4 k Pi) x
# ) -------------------------
# / k!
# -----
#
# Now, sympy returns
# oo
# _____
# \ `
# \ / k k\
# \ k |I*(-1 - I) I*(-1 + I) |
# \ x *|----------- - -----------|
# / \ 2 2 /
# / ------------------------------
# / k!
# /____,
# k = 0
k = Dummy('k')
assert fps(exp(-x)*sin(x)) == \
Sum(2**(S(1)/2*k)*sin(S(3)/4*k*pi)*x**k/factorial(k), (k, 0, oo))
@XFAIL
def test_X19():
# (c45) /* Derive an explicit Taylor series solution of y as a function of
# x from the following implicit relation:
# y = x - 1 + (x - 1)^2/2 + 2/3 (x - 1)^3 + (x - 1)^4 +
# 17/10 (x - 1)^5 + ...
# */
# x = sin(y) + cos(y);
# Time= 0 msecs
# (d45) x = sin(y) + cos(y)
#
# (c46) taylor_revert(%, y, 7);
raise NotImplementedError("Solve using series not supported. \
Inverse Taylor series expansion also not supported")
@XFAIL
def test_X20():
# Pade (rational function) approximation => (2 - x)/(2 + x)
# > numapprox[pade](exp(-x), x = 0, [1, 1]);
# bytes used=9019816, alloc=3669344, time=13.12
# 1 - 1/2 x
# ---------
# 1 + 1/2 x
# mpmath support numeric Pade approximant but there is
# no symbolic implementation in SymPy
# https://en.wikipedia.org/wiki/Pad%C3%A9_approximant
raise NotImplementedError("Symbolic Pade approximant not supported")
def test_X21():
"""
Test whether `fourier_series` of x periodical on the [-p, p] interval equals
`- (2 p / pi) sum( (-1)^n / n sin(n pi x / p), n = 1..infinity )`.
"""
p = symbols('p', positive=True)
n = symbols('n', positive=True, integer=True)
s = fourier_series(x, (x, -p, p))
# All cosine coefficients are equal to 0
assert s.an.formula == 0
# Check for sine coefficients
assert s.bn.formula.subs(s.bn.variables[0], 0) == 0
assert s.bn.formula.subs(s.bn.variables[0], n) == \
-2*p/pi * (-1)**n / n * sin(n*pi*x/p)
@XFAIL
def test_X22():
# (c52) /* => p / 2
# - (2 p / pi^2) sum( [1 - (-1)^n] cos(n pi x / p) / n^2,
# n = 1..infinity ) */
# fourier_series(abs(x), x, p);
# p
# (e52) a = -
# 0 2
#
# %nn
# (2 (- 1) - 2) p
# (e53) a = ------------------
# %nn 2 2
# %pi %nn
#
# (e54) b = 0
# %nn
#
# Time= 5290 msecs
# inf %nn %pi %nn x
# ==== (2 (- 1) - 2) cos(---------)
# \ p
# p > -------------------------------
# / 2
# ==== %nn
# %nn = 1 p
# (d54) ----------------------------------------- + -
# 2 2
# %pi
raise NotImplementedError("Fourier series not supported")
def test_Y1():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
F, _, _ = laplace_transform(cos((w - 1)*t), t, s)
assert F == s/(s**2 + (w - 1)**2)
def test_Y2():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
f = inverse_laplace_transform(s/(s**2 + (w - 1)**2), s, t)
assert f == cos(t*w - t)
def test_Y3():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
F, _, _ = laplace_transform(sinh(w*t)*cosh(w*t), t, s)
assert F == w/(s**2 - 4*w**2)
def test_Y4():
t = symbols('t', real=True, positive=True)
s = symbols('s')
F, _, _ = laplace_transform(erf(3/sqrt(t)), t, s)
assert F == (1 - exp(-6*sqrt(s)))/s
@XFAIL
def test_Y5_Y6():
# Solve y'' + y = 4 [H(t - 1) - H(t - 2)], y(0) = 1, y'(0) = 0 where H is the
# Heaviside (unit step) function (the RHS describes a pulse of magnitude 4 and
# duration 1). See David A. Sanchez, Richard C. Allen, Jr. and Walter T.
# Kyner, _Differential Equations: An Introduction_, Addison-Wesley Publishing
# Company, 1983, p. 211. First, take the Laplace transform of the ODE
# => s^2 Y(s) - s + Y(s) = 4/s [e^(-s) - e^(-2 s)]
# where Y(s) is the Laplace transform of y(t)
t = symbols('t', real=True, positive=True)
s = symbols('s')
y = Function('y')
F, _, _ = laplace_transform(diff(y(t), t, 2)
+ y(t)
- 4*(Heaviside(t - 1)
- Heaviside(t - 2)), t, s)
# Laplace transform for diff() not calculated
# https://github.com/sympy/sympy/issues/7176
assert (F == s**2*LaplaceTransform(y(t), t, s) - s
+ LaplaceTransform(y(t), t, s) - 4*exp(-s)/s + 4*exp(-2*s)/s)
# TODO implement second part of test case
# Now, solve for Y(s) and then take the inverse Laplace transform
# => Y(s) = s/(s^2 + 1) + 4 [1/s - s/(s^2 + 1)] [e^(-s) - e^(-2 s)]
# => y(t) = cos t + 4 {[1 - cos(t - 1)] H(t - 1) - [1 - cos(t - 2)] H(t - 2)}
@XFAIL
def test_Y7():
# What is the Laplace transform of an infinite square wave?
# => 1/s + 2 sum( (-1)^n e^(- s n a)/s, n = 1..infinity )
# [Sanchez, Allen and Kyner, p. 213]
t = symbols('t', real=True, positive=True)
a = symbols('a', real=True)
s = symbols('s')
F, _, _ = laplace_transform(1 + 2*Sum((-1)**n*Heaviside(t - n*a),
(n, 1, oo)), t, s)
# returns 2*LaplaceTransform(Sum((-1)**n*Heaviside(-a*n + t),
# (n, 1, oo)), t, s) + 1/s
# https://github.com/sympy/sympy/issues/7177
assert F == 2*Sum((-1)**n*exp(-a*n*s)/s, (n, 1, oo)) + 1/s
@XFAIL
def test_Y8():
assert fourier_transform(1, x, z) == DiracDelta(z)
def test_Y9():
assert (fourier_transform(exp(-9*x**2), x, z) ==
sqrt(pi)*exp(-pi**2*z**2/9)/3)
def test_Y10():
assert (fourier_transform(abs(x)*exp(-3*abs(x)), x, z) ==
(-8*pi**2*z**2 + 18)/(16*pi**4*z**4 + 72*pi**2*z**2 + 81))
@SKIP("https://github.com/sympy/sympy/issues/7181")
@slow
def test_Y11():
# => pi cot(pi s) (0 < Re s < 1) [Gradshteyn and Ryzhik 17.43(5)]
x, s = symbols('x s')
# raises RuntimeError: maximum recursion depth exceeded
# https://github.com/sympy/sympy/issues/7181
# Update 2019-02-17 raises:
# TypeError: cannot unpack non-iterable MellinTransform object
F, _, _ = mellin_transform(1/(1 - x), x, s)
assert F == pi*cot(pi*s)
@XFAIL
def test_Y12():
# => 2^(s - 4) gamma(s/2)/gamma(4 - s/2) (0 < Re s < 1)
# [Gradshteyn and Ryzhik 17.43(16)]
x, s = symbols('x s')
# returns Wrong value -2**(s - 4)*gamma(s/2 - 3)/gamma(-s/2 + 1)
# https://github.com/sympy/sympy/issues/7182
F, _, _ = mellin_transform(besselj(3, x)/x**3, x, s)
assert F == -2**(s - 4)*gamma(s/2)/gamma(-s/2 + 4)
@XFAIL
def test_Y13():
# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function) z
raise NotImplementedError("z-transform not supported")
@XFAIL
def test_Y14():
# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function)
raise NotImplementedError("z-transform not supported")
def test_Z1():
r = Function('r')
assert (rsolve(r(n + 2) - 2*r(n + 1) + r(n) - 2, r(n),
{r(0): 1, r(1): m}).simplify() == n**2 + n*(m - 2) + 1)
def test_Z2():
r = Function('r')
assert (rsolve(r(n) - (5*r(n - 1) - 6*r(n - 2)), r(n), {r(0): 0, r(1): 1})
== -2**n + 3**n)
def test_Z3():
# => r(n) = Fibonacci[n + 1] [Cohen, p. 83]
r = Function('r')
# recurrence solution is correct, Wester expects it to be simplified to
# fibonacci(n+1), but that is quite hard
assert (rsolve(r(n) - (r(n - 1) + r(n - 2)), r(n),
{r(1): 1, r(2): 2}).simplify()
== 2**(-n)*((1 + sqrt(5))**n*(sqrt(5) + 5) +
(-sqrt(5) + 1)**n*(-sqrt(5) + 5))/10)
@XFAIL
def test_Z4():
# => [c^(n+1) [c^(n+1) - 2 c - 2] + (n+1) c^2 + 2 c - n] / [(c-1)^3 (c+1)]
# [Joan Z. Yu and Robert Israel in sci.math.symbolic]
r = Function('r')
c = symbols('c')
# raises ValueError: Polynomial or rational function expected,
# got '(c**2 - c**n)/(c - c**n)
s = rsolve(r(n) - ((1 + c - c**(n-1) - c**(n+1))/(1 - c**n)*r(n - 1)
- c*(1 - c**(n-2))/(1 - c**(n-1))*r(n - 2) + 1),
r(n), {r(1): 1, r(2): (2 + 2*c + c**2)/(1 + c)})
assert (s - (c*(n + 1)*(c*(n + 1) - 2*c - 2) +
(n + 1)*c**2 + 2*c - n)/((c-1)**3*(c+1)) == 0)
@XFAIL
def test_Z5():
# Second order ODE with initial conditions---solve directly
# transform: f(t) = sin(2 t)/8 - t cos(2 t)/4
C1, C2 = symbols('C1 C2')
# initial conditions not supported, this is a manual workaround
# https://github.com/sympy/sympy/issues/4720
eq = Derivative(f(x), x, 2) + 4*f(x) - sin(2*x)
sol = dsolve(eq, f(x))
f0 = Lambda(x, sol.rhs)
assert f0(x) == C2*sin(2*x) + (C1 - x/4)*cos(2*x)
f1 = Lambda(x, diff(f0(x), x))
# TODO: Replace solve with solveset, when it works for solveset
const_dict = solve((f0(0), f1(0)))
result = f0(x).subs(C1, const_dict[C1]).subs(C2, const_dict[C2])
assert result == -x*cos(2*x)/4 + sin(2*x)/8
# Result is OK, but ODE solving with initial conditions should be
# supported without all this manual work
raise NotImplementedError('ODE solving with initial conditions \
not supported')
@XFAIL
def test_Z6():
# Second order ODE with initial conditions---solve using Laplace
# transform: f(t) = sin(2 t)/8 - t cos(2 t)/4
t = symbols('t', real=True, positive=True)
s = symbols('s')
eq = Derivative(f(t), t, 2) + 4*f(t) - sin(2*t)
F, _, _ = laplace_transform(eq, t, s)
# Laplace transform for diff() not calculated
# https://github.com/sympy/sympy/issues/7176
assert (F == s**2*LaplaceTransform(f(t), t, s) +
4*LaplaceTransform(f(t), t, s) - 2/(s**2 + 4))
# rest of test case not implemented
|
3afe870777daa5ee0e212abd2581ecaa5925d4ad6004ab0db75930c5aedf2d8d | from __future__ import print_function, division
import itertools
from sympy.core import S
from sympy.core.compatibility import range, string_types
from sympy.core.containers import Tuple
from sympy.core.function import _coeff_isneg
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.power import Pow
from sympy.core.relational import Equality
from sympy.core.symbol import Symbol
from sympy.core.sympify import SympifyError
from sympy.printing.conventions import requires_partial
from sympy.printing.precedence import PRECEDENCE, precedence, precedence_traditional
from sympy.printing.printer import Printer
from sympy.printing.str import sstr
from sympy.utilities import default_sort_key
from sympy.utilities.iterables import has_variety
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.printing.pretty.pretty_symbology import xstr, hobj, vobj, xobj, \
xsym, pretty_symbol, pretty_atom, pretty_use_unicode, greek_unicode, U, \
pretty_try_use_unicode, annotated
# rename for usage from outside
pprint_use_unicode = pretty_use_unicode
pprint_try_use_unicode = pretty_try_use_unicode
class PrettyPrinter(Printer):
"""Printer, which converts an expression into 2D ASCII-art figure."""
printmethod = "_pretty"
_default_settings = {
"order": None,
"full_prec": "auto",
"use_unicode": None,
"wrap_line": True,
"num_columns": None,
"use_unicode_sqrt_char": True,
"root_notation": True,
"mat_symbol_style": "plain",
"imaginary_unit": "i",
}
def __init__(self, settings=None):
Printer.__init__(self, settings)
if not isinstance(self._settings['imaginary_unit'], string_types):
raise TypeError("'imaginary_unit' must a string, not {}".format(self._settings['imaginary_unit']))
elif self._settings['imaginary_unit'] not in ["i", "j"]:
raise ValueError("'imaginary_unit' must be either 'i' or 'j', not '{}'".format(self._settings['imaginary_unit']))
self.emptyPrinter = lambda x: prettyForm(xstr(x))
@property
def _use_unicode(self):
if self._settings['use_unicode']:
return True
else:
return pretty_use_unicode()
def doprint(self, expr):
return self._print(expr).render(**self._settings)
# empty op so _print(stringPict) returns the same
def _print_stringPict(self, e):
return e
def _print_basestring(self, e):
return prettyForm(e)
def _print_atan2(self, e):
pform = prettyForm(*self._print_seq(e.args).parens())
pform = prettyForm(*pform.left('atan2'))
return pform
def _print_Symbol(self, e, bold_name=False):
symb = pretty_symbol(e.name, bold_name)
return prettyForm(symb)
_print_RandomSymbol = _print_Symbol
def _print_MatrixSymbol(self, e):
return self._print_Symbol(e, self._settings['mat_symbol_style'] == "bold")
def _print_Float(self, e):
# we will use StrPrinter's Float printer, but we need to handle the
# full_prec ourselves, according to the self._print_level
full_prec = self._settings["full_prec"]
if full_prec == "auto":
full_prec = self._print_level == 1
return prettyForm(sstr(e, full_prec=full_prec))
def _print_Cross(self, e):
vec1 = e._expr1
vec2 = e._expr2
pform = self._print(vec2)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN'))))
pform = prettyForm(*pform.left(')'))
pform = prettyForm(*pform.left(self._print(vec1)))
pform = prettyForm(*pform.left('('))
return pform
def _print_Curl(self, e):
vec = e._expr
pform = self._print(vec)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN'))))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Divergence(self, e):
vec = e._expr
pform = self._print(vec)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR'))))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Dot(self, e):
vec1 = e._expr1
vec2 = e._expr2
pform = self._print(vec2)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR'))))
pform = prettyForm(*pform.left(')'))
pform = prettyForm(*pform.left(self._print(vec1)))
pform = prettyForm(*pform.left('('))
return pform
def _print_Gradient(self, e):
func = e._expr
pform = self._print(func)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Laplacian(self, e):
func = e._expr
pform = self._print(func)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('INCREMENT'))))
return pform
def _print_Atom(self, e):
try:
# print atoms like Exp1 or Pi
return prettyForm(pretty_atom(e.__class__.__name__, printer=self))
except KeyError:
return self.emptyPrinter(e)
# Infinity inherits from Number, so we have to override _print_XXX order
_print_Infinity = _print_Atom
_print_NegativeInfinity = _print_Atom
_print_EmptySet = _print_Atom
_print_Naturals = _print_Atom
_print_Naturals0 = _print_Atom
_print_Integers = _print_Atom
_print_Complexes = _print_Atom
def _print_Reals(self, e):
if self._use_unicode:
return self._print_Atom(e)
else:
inf_list = ['-oo', 'oo']
return self._print_seq(inf_list, '(', ')')
def _print_subfactorial(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('!'))
return pform
def _print_factorial(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right('!'))
return pform
def _print_factorial2(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right('!!'))
return pform
def _print_binomial(self, e):
n, k = e.args
n_pform = self._print(n)
k_pform = self._print(k)
bar = ' '*max(n_pform.width(), k_pform.width())
pform = prettyForm(*k_pform.above(bar))
pform = prettyForm(*pform.above(n_pform))
pform = prettyForm(*pform.parens('(', ')'))
pform.baseline = (pform.baseline + 1)//2
return pform
def _print_Relational(self, e):
op = prettyForm(' ' + xsym(e.rel_op) + ' ')
l = self._print(e.lhs)
r = self._print(e.rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
def _print_Not(self, e):
from sympy import Equivalent, Implies
if self._use_unicode:
arg = e.args[0]
pform = self._print(arg)
if isinstance(arg, Equivalent):
return self._print_Equivalent(arg, altchar=u"\N{LEFT RIGHT DOUBLE ARROW WITH STROKE}")
if isinstance(arg, Implies):
return self._print_Implies(arg, altchar=u"\N{RIGHTWARDS ARROW WITH STROKE}")
if arg.is_Boolean and not arg.is_Not:
pform = prettyForm(*pform.parens())
return prettyForm(*pform.left(u"\N{NOT SIGN}"))
else:
return self._print_Function(e)
def __print_Boolean(self, e, char, sort=True):
args = e.args
if sort:
args = sorted(e.args, key=default_sort_key)
arg = args[0]
pform = self._print(arg)
if arg.is_Boolean and not arg.is_Not:
pform = prettyForm(*pform.parens())
for arg in args[1:]:
pform_arg = self._print(arg)
if arg.is_Boolean and not arg.is_Not:
pform_arg = prettyForm(*pform_arg.parens())
pform = prettyForm(*pform.right(u' %s ' % char))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_And(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{LOGICAL AND}")
else:
return self._print_Function(e, sort=True)
def _print_Or(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{LOGICAL OR}")
else:
return self._print_Function(e, sort=True)
def _print_Xor(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{XOR}")
else:
return self._print_Function(e, sort=True)
def _print_Nand(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{NAND}")
else:
return self._print_Function(e, sort=True)
def _print_Nor(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{NOR}")
else:
return self._print_Function(e, sort=True)
def _print_Implies(self, e, altchar=None):
if self._use_unicode:
return self.__print_Boolean(e, altchar or u"\N{RIGHTWARDS ARROW}", sort=False)
else:
return self._print_Function(e)
def _print_Equivalent(self, e, altchar=None):
if self._use_unicode:
return self.__print_Boolean(e, altchar or u"\N{LEFT RIGHT DOUBLE ARROW}")
else:
return self._print_Function(e, sort=True)
def _print_conjugate(self, e):
pform = self._print(e.args[0])
return prettyForm( *pform.above( hobj('_', pform.width())) )
def _print_Abs(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('|', '|'))
return pform
_print_Determinant = _print_Abs
def _print_floor(self, e):
if self._use_unicode:
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('lfloor', 'rfloor'))
return pform
else:
return self._print_Function(e)
def _print_ceiling(self, e):
if self._use_unicode:
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('lceil', 'rceil'))
return pform
else:
return self._print_Function(e)
def _print_Derivative(self, deriv):
if requires_partial(deriv) and self._use_unicode:
deriv_symbol = U('PARTIAL DIFFERENTIAL')
else:
deriv_symbol = r'd'
x = None
count_total_deriv = 0
for sym, num in reversed(deriv.variable_count):
s = self._print(sym)
ds = prettyForm(*s.left(deriv_symbol))
count_total_deriv += num
if (not num.is_Integer) or (num > 1):
ds = ds**prettyForm(str(num))
if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
f = prettyForm(
binding=prettyForm.FUNC, *self._print(deriv.expr).parens())
pform = prettyForm(deriv_symbol)
if (count_total_deriv > 1) != False:
pform = pform**prettyForm(str(count_total_deriv))
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
pform.binding = prettyForm.MUL
return pform
def _print_Cycle(self, dc):
from sympy.combinatorics.permutations import Permutation, Cycle
# for Empty Cycle
if dc == Cycle():
cyc = stringPict('')
return prettyForm(*cyc.parens())
dc_list = Permutation(dc.list()).cyclic_form
# for Identity Cycle
if dc_list == []:
cyc = self._print(dc.size - 1)
return prettyForm(*cyc.parens())
cyc = stringPict('')
for i in dc_list:
l = self._print(str(tuple(i)).replace(',', ''))
cyc = prettyForm(*cyc.right(l))
return cyc
def _print_PDF(self, pdf):
lim = self._print(pdf.pdf.args[0])
lim = prettyForm(*lim.right(', '))
lim = prettyForm(*lim.right(self._print(pdf.domain[0])))
lim = prettyForm(*lim.right(', '))
lim = prettyForm(*lim.right(self._print(pdf.domain[1])))
lim = prettyForm(*lim.parens())
f = self._print(pdf.pdf.args[1])
f = prettyForm(*f.right(', '))
f = prettyForm(*f.right(lim))
f = prettyForm(*f.parens())
pform = prettyForm('PDF')
pform = prettyForm(*pform.right(f))
return pform
def _print_Integral(self, integral):
f = integral.function
# Add parentheses if arg involves addition of terms and
# create a pretty form for the argument
prettyF = self._print(f)
# XXX generalize parens
if f.is_Add:
prettyF = prettyForm(*prettyF.parens())
# dx dy dz ...
arg = prettyF
for x in integral.limits:
prettyArg = self._print(x[0])
# XXX qparens (parens if needs-parens)
if prettyArg.width() > 1:
prettyArg = prettyForm(*prettyArg.parens())
arg = prettyForm(*arg.right(' d', prettyArg))
# \int \int \int ...
firstterm = True
s = None
for lim in integral.limits:
x = lim[0]
# Create bar based on the height of the argument
h = arg.height()
H = h + 2
# XXX hack!
ascii_mode = not self._use_unicode
if ascii_mode:
H += 2
vint = vobj('int', H)
# Construct the pretty form with the integral sign and the argument
pform = prettyForm(vint)
pform.baseline = arg.baseline + (
H - h)//2 # covering the whole argument
if len(lim) > 1:
# Create pretty forms for endpoints, if definite integral.
# Do not print empty endpoints.
if len(lim) == 2:
prettyA = prettyForm("")
prettyB = self._print(lim[1])
if len(lim) == 3:
prettyA = self._print(lim[1])
prettyB = self._print(lim[2])
if ascii_mode: # XXX hack
# Add spacing so that endpoint can more easily be
# identified with the correct integral sign
spc = max(1, 3 - prettyB.width())
prettyB = prettyForm(*prettyB.left(' ' * spc))
spc = max(1, 4 - prettyA.width())
prettyA = prettyForm(*prettyA.right(' ' * spc))
pform = prettyForm(*pform.above(prettyB))
pform = prettyForm(*pform.below(prettyA))
if not ascii_mode: # XXX hack
pform = prettyForm(*pform.right(' '))
if firstterm:
s = pform # first term
firstterm = False
else:
s = prettyForm(*s.left(pform))
pform = prettyForm(*arg.left(s))
pform.binding = prettyForm.MUL
return pform
def _print_Product(self, expr):
func = expr.term
pretty_func = self._print(func)
horizontal_chr = xobj('_', 1)
corner_chr = xobj('_', 1)
vertical_chr = xobj('|', 1)
if self._use_unicode:
# use unicode corners
horizontal_chr = xobj('-', 1)
corner_chr = u'\N{BOX DRAWINGS LIGHT DOWN AND HORIZONTAL}'
func_height = pretty_func.height()
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
width = (func_height + 2) * 5 // 3 - 2
sign_lines = [horizontal_chr + corner_chr + (horizontal_chr * (width-2)) + corner_chr + horizontal_chr]
for _ in range(func_height + 1):
sign_lines.append(' ' + vertical_chr + (' ' * (width-2)) + vertical_chr + ' ')
pretty_sign = stringPict('')
pretty_sign = prettyForm(*pretty_sign.stack(*sign_lines))
pretty_upper = self._print(lim[2])
pretty_lower = self._print(Equality(lim[0], lim[1]))
max_upper = max(max_upper, pretty_upper.height())
if first:
sign_height = pretty_sign.height()
pretty_sign = prettyForm(*pretty_sign.above(pretty_upper))
pretty_sign = prettyForm(*pretty_sign.below(pretty_lower))
if first:
pretty_func.baseline = 0
first = False
height = pretty_sign.height()
padding = stringPict('')
padding = prettyForm(*padding.stack(*[' ']*(height - 1)))
pretty_sign = prettyForm(*pretty_sign.right(padding))
pretty_func = prettyForm(*pretty_sign.right(pretty_func))
pretty_func.baseline = max_upper + sign_height//2
pretty_func.binding = prettyForm.MUL
return pretty_func
def _print_Sum(self, expr):
ascii_mode = not self._use_unicode
def asum(hrequired, lower, upper, use_ascii):
def adjust(s, wid=None, how='<^>'):
if not wid or len(s) > wid:
return s
need = wid - len(s)
if how == '<^>' or how == "<" or how not in list('<^>'):
return s + ' '*need
half = need//2
lead = ' '*half
if how == ">":
return " "*need + s
return lead + s + ' '*(need - len(lead))
h = max(hrequired, 2)
d = h//2
w = d + 1
more = hrequired % 2
lines = []
if use_ascii:
lines.append("_"*(w) + ' ')
lines.append(r"\%s`" % (' '*(w - 1)))
for i in range(1, d):
lines.append('%s\\%s' % (' '*i, ' '*(w - i)))
if more:
lines.append('%s)%s' % (' '*(d), ' '*(w - d)))
for i in reversed(range(1, d)):
lines.append('%s/%s' % (' '*i, ' '*(w - i)))
lines.append("/" + "_"*(w - 1) + ',')
return d, h + more, lines, more
else:
w = w + more
d = d + more
vsum = vobj('sum', 4)
lines.append("_"*(w))
for i in range(0, d):
lines.append('%s%s%s' % (' '*i, vsum[2], ' '*(w - i - 1)))
for i in reversed(range(0, d)):
lines.append('%s%s%s' % (' '*i, vsum[4], ' '*(w - i - 1)))
lines.append(vsum[8]*(w))
return d, h + 2*more, lines, more
f = expr.function
prettyF = self._print(f)
if f.is_Add: # add parens
prettyF = prettyForm(*prettyF.parens())
H = prettyF.height() + 2
# \sum \sum \sum ...
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
if len(lim) == 3:
prettyUpper = self._print(lim[2])
prettyLower = self._print(Equality(lim[0], lim[1]))
elif len(lim) == 2:
prettyUpper = self._print("")
prettyLower = self._print(Equality(lim[0], lim[1]))
elif len(lim) == 1:
prettyUpper = self._print("")
prettyLower = self._print(lim[0])
max_upper = max(max_upper, prettyUpper.height())
# Create sum sign based on the height of the argument
d, h, slines, adjustment = asum(
H, prettyLower.width(), prettyUpper.width(), ascii_mode)
prettySign = stringPict('')
prettySign = prettyForm(*prettySign.stack(*slines))
if first:
sign_height = prettySign.height()
prettySign = prettyForm(*prettySign.above(prettyUpper))
prettySign = prettyForm(*prettySign.below(prettyLower))
if first:
# change F baseline so it centers on the sign
prettyF.baseline -= d - (prettyF.height()//2 -
prettyF.baseline)
first = False
# put padding to the right
pad = stringPict('')
pad = prettyForm(*pad.stack(*[' ']*h))
prettySign = prettyForm(*prettySign.right(pad))
# put the present prettyF to the right
prettyF = prettyForm(*prettySign.right(prettyF))
# adjust baseline of ascii mode sigma with an odd height so that it is
# exactly through the center
ascii_adjustment = ascii_mode if not adjustment else 0
prettyF.baseline = max_upper + sign_height//2 + ascii_adjustment
prettyF.binding = prettyForm.MUL
return prettyF
def _print_Limit(self, l):
e, z, z0, dir = l.args
E = self._print(e)
if precedence(e) <= PRECEDENCE["Mul"]:
E = prettyForm(*E.parens('(', ')'))
Lim = prettyForm('lim')
LimArg = self._print(z)
if self._use_unicode:
LimArg = prettyForm(*LimArg.right(u'\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{RIGHTWARDS ARROW}'))
else:
LimArg = prettyForm(*LimArg.right('->'))
LimArg = prettyForm(*LimArg.right(self._print(z0)))
if str(dir) == '+-' or z0 in (S.Infinity, S.NegativeInfinity):
dir = ""
else:
if self._use_unicode:
dir = u'\N{SUPERSCRIPT PLUS SIGN}' if str(dir) == "+" else u'\N{SUPERSCRIPT MINUS}'
LimArg = prettyForm(*LimArg.right(self._print(dir)))
Lim = prettyForm(*Lim.below(LimArg))
Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)
return Lim
def _print_matrix_contents(self, e):
"""
This method factors out what is essentially grid printing.
"""
M = e # matrix
Ms = {} # i,j -> pretty(M[i,j])
for i in range(M.rows):
for j in range(M.cols):
Ms[i, j] = self._print(M[i, j])
# h- and v- spacers
hsep = 2
vsep = 1
# max width for columns
maxw = [-1] * M.cols
for j in range(M.cols):
maxw[j] = max([Ms[i, j].width() for i in range(M.rows)] or [0])
# drawing result
D = None
for i in range(M.rows):
D_row = None
for j in range(M.cols):
s = Ms[i, j]
# reshape s to maxw
# XXX this should be generalized, and go to stringPict.reshape ?
assert s.width() <= maxw[j]
# hcenter it, +0.5 to the right 2
# ( it's better to align formula starts for say 0 and r )
# XXX this is not good in all cases -- maybe introduce vbaseline?
wdelta = maxw[j] - s.width()
wleft = wdelta // 2
wright = wdelta - wleft
s = prettyForm(*s.right(' '*wright))
s = prettyForm(*s.left(' '*wleft))
# we don't need vcenter cells -- this is automatically done in
# a pretty way because when their baselines are taking into
# account in .right()
if D_row is None:
D_row = s # first box in a row
continue
D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer
D_row = prettyForm(*D_row.right(s))
if D is None:
D = D_row # first row in a picture
continue
# v-spacer
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
if D is None:
D = prettyForm('') # Empty Matrix
return D
def _print_MatrixBase(self, e):
D = self._print_matrix_contents(e)
D.baseline = D.height()//2
D = prettyForm(*D.parens('[', ']'))
return D
_print_ImmutableMatrix = _print_MatrixBase
_print_Matrix = _print_MatrixBase
def _print_TensorProduct(self, expr):
# This should somehow share the code with _print_WedgeProduct:
circled_times = "\u2297"
return self._print_seq(expr.args, None, None, circled_times,
parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"])
def _print_WedgeProduct(self, expr):
# This should somehow share the code with _print_TensorProduct:
wedge_symbol = u"\u2227"
return self._print_seq(expr.args, None, None, wedge_symbol,
parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"])
def _print_Trace(self, e):
D = self._print(e.arg)
D = prettyForm(*D.parens('(',')'))
D.baseline = D.height()//2
D = prettyForm(*D.left('\n'*(0) + 'tr'))
return D
def _print_MatrixElement(self, expr):
from sympy.matrices import MatrixSymbol
from sympy import Symbol
if (isinstance(expr.parent, MatrixSymbol)
and expr.i.is_number and expr.j.is_number):
return self._print(
Symbol(expr.parent.name + '_%d%d' % (expr.i, expr.j)))
else:
prettyFunc = self._print(expr.parent)
prettyFunc = prettyForm(*prettyFunc.parens())
prettyIndices = self._print_seq((expr.i, expr.j), delimiter=', '
).parens(left='[', right=']')[0]
pform = prettyForm(binding=prettyForm.FUNC,
*stringPict.next(prettyFunc, prettyIndices))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyIndices
return pform
def _print_MatrixSlice(self, m):
# XXX works only for applied functions
prettyFunc = self._print(m.parent)
def ppslice(x):
x = list(x)
if x[2] == 1:
del x[2]
if x[1] == x[0] + 1:
del x[1]
if x[0] == 0:
x[0] = ''
return prettyForm(*self._print_seq(x, delimiter=':'))
prettyArgs = self._print_seq((ppslice(m.rowslice),
ppslice(m.colslice)), delimiter=', ').parens(left='[', right=']')[0]
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_Transpose(self, expr):
pform = self._print(expr.arg)
from sympy.matrices import MatrixSymbol
if not isinstance(expr.arg, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**(prettyForm('T'))
return pform
def _print_Adjoint(self, expr):
pform = self._print(expr.arg)
if self._use_unicode:
dag = prettyForm(u'\N{DAGGER}')
else:
dag = prettyForm('+')
from sympy.matrices import MatrixSymbol
if not isinstance(expr.arg, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**dag
return pform
def _print_BlockMatrix(self, B):
if B.blocks.shape == (1, 1):
return self._print(B.blocks[0, 0])
return self._print(B.blocks)
def _print_MatAdd(self, expr):
s = None
for item in expr.args:
pform = self._print(item)
if s is None:
s = pform # First element
else:
coeff = item.as_coeff_mmul()[0]
if _coeff_isneg(S(coeff)):
s = prettyForm(*stringPict.next(s, ' '))
pform = self._print(item)
else:
s = prettyForm(*stringPict.next(s, ' + '))
s = prettyForm(*stringPict.next(s, pform))
return s
def _print_MatMul(self, expr):
args = list(expr.args)
from sympy import Add, MatAdd, HadamardProduct, KroneckerProduct
for i, a in enumerate(args):
if (isinstance(a, (Add, MatAdd, HadamardProduct, KroneckerProduct))
and len(expr.args) > 1):
args[i] = prettyForm(*self._print(a).parens())
else:
args[i] = self._print(a)
return prettyForm.__mul__(*args)
def _print_DotProduct(self, expr):
args = list(expr.args)
for i, a in enumerate(args):
args[i] = self._print(a)
return prettyForm.__mul__(*args)
def _print_MatPow(self, expr):
pform = self._print(expr.base)
from sympy.matrices import MatrixSymbol
if not isinstance(expr.base, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**(self._print(expr.exp))
return pform
def _print_HadamardProduct(self, expr):
from sympy import MatAdd, MatMul
if self._use_unicode:
delim = pretty_atom('Ring')
else:
delim = '.*'
return self._print_seq(expr.args, None, None, delim,
parenthesize=lambda x: isinstance(x, (MatAdd, MatMul)))
def _print_KroneckerProduct(self, expr):
from sympy import MatAdd, MatMul
if self._use_unicode:
delim = u' \N{N-ARY CIRCLED TIMES OPERATOR} '
else:
delim = ' x '
return self._print_seq(expr.args, None, None, delim,
parenthesize=lambda x: isinstance(x, (MatAdd, MatMul)))
def _print_FunctionMatrix(self, X):
D = self._print(X.lamda.expr)
D = prettyForm(*D.parens('[', ']'))
return D
def _print_BasisDependent(self, expr):
from sympy.vector import Vector
if not self._use_unicode:
raise NotImplementedError("ASCII pretty printing of BasisDependent is not implemented")
if expr == expr.zero:
return prettyForm(expr.zero._pretty_form)
o1 = []
vectstrs = []
if isinstance(expr, Vector):
items = expr.separate().items()
else:
items = [(0, expr)]
for system, vect in items:
inneritems = list(vect.components.items())
inneritems.sort(key = lambda x: x[0].__str__())
for k, v in inneritems:
#if the coef of the basis vector is 1
#we skip the 1
if v == 1:
o1.append(u"" +
k._pretty_form)
#Same for -1
elif v == -1:
o1.append(u"(-1) " +
k._pretty_form)
#For a general expr
else:
#We always wrap the measure numbers in
#parentheses
arg_str = self._print(
v).parens()[0]
o1.append(arg_str + ' ' + k._pretty_form)
vectstrs.append(k._pretty_form)
#outstr = u("").join(o1)
if o1[0].startswith(u" + "):
o1[0] = o1[0][3:]
elif o1[0].startswith(" "):
o1[0] = o1[0][1:]
#Fixing the newlines
lengths = []
strs = ['']
flag = []
for i, partstr in enumerate(o1):
flag.append(0)
# XXX: What is this hack?
if '\n' in partstr:
tempstr = partstr
tempstr = tempstr.replace(vectstrs[i], '')
if u'\N{right parenthesis extension}' in tempstr: # If scalar is a fraction
for paren in range(len(tempstr)):
flag[i] = 1
if tempstr[paren] == u'\N{right parenthesis extension}':
tempstr = tempstr[:paren] + u'\N{right parenthesis extension}'\
+ ' ' + vectstrs[i] + tempstr[paren + 1:]
break
elif u'\N{RIGHT PARENTHESIS LOWER HOOK}' in tempstr:
flag[i] = 1
tempstr = tempstr.replace(u'\N{RIGHT PARENTHESIS LOWER HOOK}',
u'\N{RIGHT PARENTHESIS LOWER HOOK}'
+ ' ' + vectstrs[i])
else:
tempstr = tempstr.replace(u'\N{RIGHT PARENTHESIS UPPER HOOK}',
u'\N{RIGHT PARENTHESIS UPPER HOOK}'
+ ' ' + vectstrs[i])
o1[i] = tempstr
o1 = [x.split('\n') for x in o1]
n_newlines = max([len(x) for x in o1]) # Width of part in its pretty form
if 1 in flag: # If there was a fractional scalar
for i, parts in enumerate(o1):
if len(parts) == 1: # If part has no newline
parts.insert(0, ' ' * (len(parts[0])))
flag[i] = 1
for i, parts in enumerate(o1):
lengths.append(len(parts[flag[i]]))
for j in range(n_newlines):
if j+1 <= len(parts):
if j >= len(strs):
strs.append(' ' * (sum(lengths[:-1]) +
3*(len(lengths)-1)))
if j == flag[i]:
strs[flag[i]] += parts[flag[i]] + ' + '
else:
strs[j] += parts[j] + ' '*(lengths[-1] -
len(parts[j])+
3)
else:
if j >= len(strs):
strs.append(' ' * (sum(lengths[:-1]) +
3*(len(lengths)-1)))
strs[j] += ' '*(lengths[-1]+3)
return prettyForm(u'\n'.join([s[:-3] for s in strs]))
def _print_NDimArray(self, expr):
from sympy import ImmutableMatrix
if expr.rank() == 0:
return self._print(expr[()])
level_str = [[]] + [[] for i in range(expr.rank())]
shape_ranges = [list(range(i)) for i in expr.shape]
# leave eventual matrix elements unflattened
mat = lambda x: ImmutableMatrix(x, evaluate=False)
for outer_i in itertools.product(*shape_ranges):
level_str[-1].append(expr[outer_i])
even = True
for back_outer_i in range(expr.rank()-1, -1, -1):
if len(level_str[back_outer_i+1]) < expr.shape[back_outer_i]:
break
if even:
level_str[back_outer_i].append(level_str[back_outer_i+1])
else:
level_str[back_outer_i].append(mat(
level_str[back_outer_i+1]))
if len(level_str[back_outer_i + 1]) == 1:
level_str[back_outer_i][-1] = mat(
[[level_str[back_outer_i][-1]]])
even = not even
level_str[back_outer_i+1] = []
out_expr = level_str[0][0]
if expr.rank() % 2 == 1:
out_expr = mat([out_expr])
return self._print(out_expr)
_print_ImmutableDenseNDimArray = _print_NDimArray
_print_ImmutableSparseNDimArray = _print_NDimArray
_print_MutableDenseNDimArray = _print_NDimArray
_print_MutableSparseNDimArray = _print_NDimArray
def _printer_tensor_indices(self, name, indices, index_map={}):
center = stringPict(name)
top = stringPict(" "*center.width())
bot = stringPict(" "*center.width())
last_valence = None
prev_map = None
for i, index in enumerate(indices):
indpic = self._print(index.args[0])
if ((index in index_map) or prev_map) and last_valence == index.is_up:
if index.is_up:
top = prettyForm(*stringPict.next(top, ","))
else:
bot = prettyForm(*stringPict.next(bot, ","))
if index in index_map:
indpic = prettyForm(*stringPict.next(indpic, "="))
indpic = prettyForm(*stringPict.next(indpic, self._print(index_map[index])))
prev_map = True
else:
prev_map = False
if index.is_up:
top = stringPict(*top.right(indpic))
center = stringPict(*center.right(" "*indpic.width()))
bot = stringPict(*bot.right(" "*indpic.width()))
else:
bot = stringPict(*bot.right(indpic))
center = stringPict(*center.right(" "*indpic.width()))
top = stringPict(*top.right(" "*indpic.width()))
last_valence = index.is_up
pict = prettyForm(*center.above(top))
pict = prettyForm(*pict.below(bot))
return pict
def _print_Tensor(self, expr):
name = expr.args[0].name
indices = expr.get_indices()
return self._printer_tensor_indices(name, indices)
def _print_TensorElement(self, expr):
name = expr.expr.args[0].name
indices = expr.expr.get_indices()
index_map = expr.index_map
return self._printer_tensor_indices(name, indices, index_map)
def _print_TensMul(self, expr):
sign, args = expr._get_args_for_traditional_printer()
args = [
prettyForm(*self._print(i).parens()) if
precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i)
for i in args
]
pform = prettyForm.__mul__(*args)
if sign:
return prettyForm(*pform.left(sign))
else:
return pform
def _print_TensAdd(self, expr):
args = [
prettyForm(*self._print(i).parens()) if
precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i)
for i in expr.args
]
return prettyForm.__add__(*args)
def _print_TensorIndex(self, expr):
sym = expr.args[0]
if not expr.is_up:
sym = -sym
return self._print(sym)
def _print_PartialDerivative(self, deriv):
if self._use_unicode:
deriv_symbol = U('PARTIAL DIFFERENTIAL')
else:
deriv_symbol = r'd'
x = None
for variable in reversed(deriv.variables):
s = self._print(variable)
ds = prettyForm(*s.left(deriv_symbol))
if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
f = prettyForm(
binding=prettyForm.FUNC, *self._print(deriv.expr).parens())
pform = prettyForm(deriv_symbol)
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
pform.binding = prettyForm.MUL
return pform
def _print_Piecewise(self, pexpr):
P = {}
for n, ec in enumerate(pexpr.args):
P[n, 0] = self._print(ec.expr)
if ec.cond == True:
P[n, 1] = prettyForm('otherwise')
else:
P[n, 1] = prettyForm(
*prettyForm('for ').right(self._print(ec.cond)))
hsep = 2
vsep = 1
len_args = len(pexpr.args)
# max widths
maxw = [max([P[i, j].width() for i in range(len_args)])
for j in range(2)]
# FIXME: Refactor this code and matrix into some tabular environment.
# drawing result
D = None
for i in range(len_args):
D_row = None
for j in range(2):
p = P[i, j]
assert p.width() <= maxw[j]
wdelta = maxw[j] - p.width()
wleft = wdelta // 2
wright = wdelta - wleft
p = prettyForm(*p.right(' '*wright))
p = prettyForm(*p.left(' '*wleft))
if D_row is None:
D_row = p
continue
D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer
D_row = prettyForm(*D_row.right(p))
if D is None:
D = D_row # first row in a picture
continue
# v-spacer
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
D = prettyForm(*D.parens('{', ''))
D.baseline = D.height()//2
D.binding = prettyForm.OPEN
return D
def _print_ITE(self, ite):
from sympy.functions.elementary.piecewise import Piecewise
return self._print(ite.rewrite(Piecewise))
def _hprint_vec(self, v):
D = None
for a in v:
p = a
if D is None:
D = p
else:
D = prettyForm(*D.right(', '))
D = prettyForm(*D.right(p))
if D is None:
D = stringPict(' ')
return D
def _hprint_vseparator(self, p1, p2):
tmp = prettyForm(*p1.right(p2))
sep = stringPict(vobj('|', tmp.height()), baseline=tmp.baseline)
return prettyForm(*p1.right(sep, p2))
def _print_hyper(self, e):
# FIXME refactor Matrix, Piecewise, and this into a tabular environment
ap = [self._print(a) for a in e.ap]
bq = [self._print(b) for b in e.bq]
P = self._print(e.argument)
P.baseline = P.height()//2
# Drawing result - first create the ap, bq vectors
D = None
for v in [ap, bq]:
D_row = self._hprint_vec(v)
if D is None:
D = D_row # first row in a picture
else:
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
# make sure that the argument `z' is centred vertically
D.baseline = D.height()//2
# insert horizontal separator
P = prettyForm(*P.left(' '))
D = prettyForm(*D.right(' '))
# insert separating `|`
D = self._hprint_vseparator(D, P)
# add parens
D = prettyForm(*D.parens('(', ')'))
# create the F symbol
above = D.height()//2 - 1
below = D.height() - above - 1
sz, t, b, add, img = annotated('F')
F = prettyForm('\n' * (above - t) + img + '\n' * (below - b),
baseline=above + sz)
add = (sz + 1)//2
F = prettyForm(*F.left(self._print(len(e.ap))))
F = prettyForm(*F.right(self._print(len(e.bq))))
F.baseline = above + add
D = prettyForm(*F.right(' ', D))
return D
def _print_meijerg(self, e):
# FIXME refactor Matrix, Piecewise, and this into a tabular environment
v = {}
v[(0, 0)] = [self._print(a) for a in e.an]
v[(0, 1)] = [self._print(a) for a in e.aother]
v[(1, 0)] = [self._print(b) for b in e.bm]
v[(1, 1)] = [self._print(b) for b in e.bother]
P = self._print(e.argument)
P.baseline = P.height()//2
vp = {}
for idx in v:
vp[idx] = self._hprint_vec(v[idx])
for i in range(2):
maxw = max(vp[(0, i)].width(), vp[(1, i)].width())
for j in range(2):
s = vp[(j, i)]
left = (maxw - s.width()) // 2
right = maxw - left - s.width()
s = prettyForm(*s.left(' ' * left))
s = prettyForm(*s.right(' ' * right))
vp[(j, i)] = s
D1 = prettyForm(*vp[(0, 0)].right(' ', vp[(0, 1)]))
D1 = prettyForm(*D1.below(' '))
D2 = prettyForm(*vp[(1, 0)].right(' ', vp[(1, 1)]))
D = prettyForm(*D1.below(D2))
# make sure that the argument `z' is centred vertically
D.baseline = D.height()//2
# insert horizontal separator
P = prettyForm(*P.left(' '))
D = prettyForm(*D.right(' '))
# insert separating `|`
D = self._hprint_vseparator(D, P)
# add parens
D = prettyForm(*D.parens('(', ')'))
# create the G symbol
above = D.height()//2 - 1
below = D.height() - above - 1
sz, t, b, add, img = annotated('G')
F = prettyForm('\n' * (above - t) + img + '\n' * (below - b),
baseline=above + sz)
pp = self._print(len(e.ap))
pq = self._print(len(e.bq))
pm = self._print(len(e.bm))
pn = self._print(len(e.an))
def adjust(p1, p2):
diff = p1.width() - p2.width()
if diff == 0:
return p1, p2
elif diff > 0:
return p1, prettyForm(*p2.left(' '*diff))
else:
return prettyForm(*p1.left(' '*-diff)), p2
pp, pm = adjust(pp, pm)
pq, pn = adjust(pq, pn)
pu = prettyForm(*pm.right(', ', pn))
pl = prettyForm(*pp.right(', ', pq))
ht = F.baseline - above - 2
if ht > 0:
pu = prettyForm(*pu.below('\n'*ht))
p = prettyForm(*pu.below(pl))
F.baseline = above
F = prettyForm(*F.right(p))
F.baseline = above + add
D = prettyForm(*F.right(' ', D))
return D
def _print_ExpBase(self, e):
# TODO should exp_polar be printed differently?
# what about exp_polar(0), exp_polar(1)?
base = prettyForm(pretty_atom('Exp1', 'e'))
return base ** self._print(e.args[0])
def _print_Function(self, e, sort=False, func_name=None):
# optional argument func_name for supplying custom names
# XXX works only for applied functions
func = e.func
args = e.args
if sort:
args = sorted(args, key=default_sort_key)
if not func_name:
func_name = func.__name__
prettyFunc = self._print(Symbol(func_name))
prettyArgs = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
@property
def _special_function_classes(self):
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.functions.special.gamma_functions import gamma, lowergamma
from sympy.functions.special.zeta_functions import lerchphi
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.error_functions import Chi
return {KroneckerDelta: [greek_unicode['delta'], 'delta'],
gamma: [greek_unicode['Gamma'], 'Gamma'],
lerchphi: [greek_unicode['Phi'], 'lerchphi'],
lowergamma: [greek_unicode['gamma'], 'gamma'],
beta: [greek_unicode['Beta'], 'B'],
DiracDelta: [greek_unicode['delta'], 'delta'],
Chi: ['Chi', 'Chi']}
def _print_FunctionClass(self, expr):
for cls in self._special_function_classes:
if issubclass(expr, cls) and expr.__name__ == cls.__name__:
if self._use_unicode:
return prettyForm(self._special_function_classes[cls][0])
else:
return prettyForm(self._special_function_classes[cls][1])
func_name = expr.__name__
return prettyForm(pretty_symbol(func_name))
def _print_GeometryEntity(self, expr):
# GeometryEntity is based on Tuple but should not print like a Tuple
return self.emptyPrinter(expr)
def _print_lerchphi(self, e):
func_name = greek_unicode['Phi'] if self._use_unicode else 'lerchphi'
return self._print_Function(e, func_name=func_name)
def _print_Lambda(self, e):
vars, expr = e.args
if self._use_unicode:
arrow = u" \N{RIGHTWARDS ARROW FROM BAR} "
else:
arrow = " -> "
if len(vars) == 1:
var_form = self._print(vars[0])
else:
var_form = self._print(tuple(vars))
return prettyForm(*stringPict.next(var_form, arrow, self._print(expr)), binding=8)
def _print_Order(self, expr):
pform = self._print(expr.expr)
if (expr.point and any(p != S.Zero for p in expr.point)) or \
len(expr.variables) > 1:
pform = prettyForm(*pform.right("; "))
if len(expr.variables) > 1:
pform = prettyForm(*pform.right(self._print(expr.variables)))
elif len(expr.variables):
pform = prettyForm(*pform.right(self._print(expr.variables[0])))
if self._use_unicode:
pform = prettyForm(*pform.right(u" \N{RIGHTWARDS ARROW} "))
else:
pform = prettyForm(*pform.right(" -> "))
if len(expr.point) > 1:
pform = prettyForm(*pform.right(self._print(expr.point)))
else:
pform = prettyForm(*pform.right(self._print(expr.point[0])))
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left("O"))
return pform
def _print_SingularityFunction(self, e):
if self._use_unicode:
shift = self._print(e.args[0]-e.args[1])
n = self._print(e.args[2])
base = prettyForm("<")
base = prettyForm(*base.right(shift))
base = prettyForm(*base.right(">"))
pform = base**n
return pform
else:
n = self._print(e.args[2])
shift = self._print(e.args[0]-e.args[1])
base = self._print_seq(shift, "<", ">", ' ')
return base**n
def _print_beta(self, e):
func_name = greek_unicode['Beta'] if self._use_unicode else 'B'
return self._print_Function(e, func_name=func_name)
def _print_gamma(self, e):
func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma'
return self._print_Function(e, func_name=func_name)
def _print_uppergamma(self, e):
func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma'
return self._print_Function(e, func_name=func_name)
def _print_lowergamma(self, e):
func_name = greek_unicode['gamma'] if self._use_unicode else 'lowergamma'
return self._print_Function(e, func_name=func_name)
def _print_DiracDelta(self, e):
if self._use_unicode:
if len(e.args) == 2:
a = prettyForm(greek_unicode['delta'])
b = self._print(e.args[1])
b = prettyForm(*b.parens())
c = self._print(e.args[0])
c = prettyForm(*c.parens())
pform = a**b
pform = prettyForm(*pform.right(' '))
pform = prettyForm(*pform.right(c))
return pform
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(greek_unicode['delta']))
return pform
else:
return self._print_Function(e)
def _print_expint(self, e):
from sympy import Function
if e.args[0].is_Integer and self._use_unicode:
return self._print_Function(Function('E_%s' % e.args[0])(e.args[1]))
return self._print_Function(e)
def _print_Chi(self, e):
# This needs a special case since otherwise it comes out as greek
# letter chi...
prettyFunc = prettyForm("Chi")
prettyArgs = prettyForm(*self._print_seq(e.args).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_elliptic_e(self, e):
pforma0 = self._print(e.args[0])
if len(e.args) == 1:
pform = pforma0
else:
pforma1 = self._print(e.args[1])
pform = self._hprint_vseparator(pforma0, pforma1)
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('E'))
return pform
def _print_elliptic_k(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('K'))
return pform
def _print_elliptic_f(self, e):
pforma0 = self._print(e.args[0])
pforma1 = self._print(e.args[1])
pform = self._hprint_vseparator(pforma0, pforma1)
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('F'))
return pform
def _print_elliptic_pi(self, e):
name = greek_unicode['Pi'] if self._use_unicode else 'Pi'
pforma0 = self._print(e.args[0])
pforma1 = self._print(e.args[1])
if len(e.args) == 2:
pform = self._hprint_vseparator(pforma0, pforma1)
else:
pforma2 = self._print(e.args[2])
pforma = self._hprint_vseparator(pforma1, pforma2)
pforma = prettyForm(*pforma.left('; '))
pform = prettyForm(*pforma.left(pforma0))
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(name))
return pform
def _print_GoldenRatio(self, expr):
if self._use_unicode:
return prettyForm(pretty_symbol('phi'))
return self._print(Symbol("GoldenRatio"))
def _print_EulerGamma(self, expr):
if self._use_unicode:
return prettyForm(pretty_symbol('gamma'))
return self._print(Symbol("EulerGamma"))
def _print_Mod(self, expr):
pform = self._print(expr.args[0])
if pform.binding > prettyForm.MUL:
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right(' mod '))
pform = prettyForm(*pform.right(self._print(expr.args[1])))
pform.binding = prettyForm.OPEN
return pform
def _print_Add(self, expr, order=None):
if self.order == 'none':
terms = list(expr.args)
else:
terms = self._as_ordered_terms(expr, order=order)
pforms, indices = [], []
def pretty_negative(pform, index):
"""Prepend a minus sign to a pretty form. """
#TODO: Move this code to prettyForm
if index == 0:
if pform.height() > 1:
pform_neg = '- '
else:
pform_neg = '-'
else:
pform_neg = ' - '
if (pform.binding > prettyForm.NEG
or pform.binding == prettyForm.ADD):
p = stringPict(*pform.parens())
else:
p = pform
p = stringPict.next(pform_neg, p)
# Lower the binding to NEG, even if it was higher. Otherwise, it
# will print as a + ( - (b)), instead of a - (b).
return prettyForm(binding=prettyForm.NEG, *p)
for i, term in enumerate(terms):
if term.is_Mul and _coeff_isneg(term):
coeff, other = term.as_coeff_mul(rational=False)
pform = self._print(Mul(-coeff, *other, evaluate=False))
pforms.append(pretty_negative(pform, i))
elif term.is_Rational and term.q > 1:
pforms.append(None)
indices.append(i)
elif term.is_Number and term < 0:
pform = self._print(-term)
pforms.append(pretty_negative(pform, i))
elif term.is_Relational:
pforms.append(prettyForm(*self._print(term).parens()))
else:
pforms.append(self._print(term))
if indices:
large = True
for pform in pforms:
if pform is not None and pform.height() > 1:
break
else:
large = False
for i in indices:
term, negative = terms[i], False
if term < 0:
term, negative = -term, True
if large:
pform = prettyForm(str(term.p))/prettyForm(str(term.q))
else:
pform = self._print(term)
if negative:
pform = pretty_negative(pform, i)
pforms[i] = pform
return prettyForm.__add__(*pforms)
def _print_Mul(self, product):
from sympy.physics.units import Quantity
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
if self.order not in ('old', 'none'):
args = product.as_ordered_factors()
else:
args = list(product.args)
# If quantities are present append them at the back
args = sorted(args, key=lambda x: isinstance(x, Quantity) or
(isinstance(x, Pow) and isinstance(x.base, Quantity)))
# Gather terms for numerator/denominator
for item in args:
if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative:
if item.exp != -1:
b.append(Pow(item.base, -item.exp, evaluate=False))
else:
b.append(Pow(item.base, -item.exp))
elif item.is_Rational and item is not S.Infinity:
if item.p != 1:
a.append( Rational(item.p) )
if item.q != 1:
b.append( Rational(item.q) )
else:
a.append(item)
from sympy import Integral, Piecewise, Product, Sum
# Convert to pretty forms. Add parens to Add instances if there
# is more than one term in the numer/denom
for i in range(0, len(a)):
if (a[i].is_Add and len(a) > 1) or (i != len(a) - 1 and
isinstance(a[i], (Integral, Piecewise, Product, Sum))):
a[i] = prettyForm(*self._print(a[i]).parens())
elif a[i].is_Relational:
a[i] = prettyForm(*self._print(a[i]).parens())
else:
a[i] = self._print(a[i])
for i in range(0, len(b)):
if (b[i].is_Add and len(b) > 1) or (i != len(b) - 1 and
isinstance(b[i], (Integral, Piecewise, Product, Sum))):
b[i] = prettyForm(*self._print(b[i]).parens())
else:
b[i] = self._print(b[i])
# Construct a pretty form
if len(b) == 0:
return prettyForm.__mul__(*a)
else:
if len(a) == 0:
a.append( self._print(S.One) )
return prettyForm.__mul__(*a)/prettyForm.__mul__(*b)
# A helper function for _print_Pow to print x**(1/n)
def _print_nth_root(self, base, expt):
bpretty = self._print(base)
# In very simple cases, use a single-char root sign
if (self._settings['use_unicode_sqrt_char'] and self._use_unicode
and expt is S.Half and bpretty.height() == 1
and (bpretty.width() == 1
or (base.is_Integer and base.is_nonnegative))):
return prettyForm(*bpretty.left(u'\N{SQUARE ROOT}'))
# Construct root sign, start with the \/ shape
_zZ = xobj('/', 1)
rootsign = xobj('\\', 1) + _zZ
# Make exponent number to put above it
if isinstance(expt, Rational):
exp = str(expt.q)
if exp == '2':
exp = ''
else:
exp = str(expt.args[0])
exp = exp.ljust(2)
if len(exp) > 2:
rootsign = ' '*(len(exp) - 2) + rootsign
# Stack the exponent
rootsign = stringPict(exp + '\n' + rootsign)
rootsign.baseline = 0
# Diagonal: length is one less than height of base
linelength = bpretty.height() - 1
diagonal = stringPict('\n'.join(
' '*(linelength - i - 1) + _zZ + ' '*i
for i in range(linelength)
))
# Put baseline just below lowest line: next to exp
diagonal.baseline = linelength - 1
# Make the root symbol
rootsign = prettyForm(*rootsign.right(diagonal))
# Det the baseline to match contents to fix the height
# but if the height of bpretty is one, the rootsign must be one higher
rootsign.baseline = max(1, bpretty.baseline)
#build result
s = prettyForm(hobj('_', 2 + bpretty.width()))
s = prettyForm(*bpretty.above(s))
s = prettyForm(*s.left(rootsign))
return s
def _print_Pow(self, power):
from sympy.simplify.simplify import fraction
b, e = power.as_base_exp()
if power.is_commutative:
if e is S.NegativeOne:
return prettyForm("1")/self._print(b)
n, d = fraction(e)
if n is S.One and d.is_Atom and not e.is_Integer and self._settings['root_notation']:
return self._print_nth_root(b, e)
if e.is_Rational and e < 0:
return prettyForm("1")/self._print(Pow(b, -e, evaluate=False))
if b.is_Relational:
return prettyForm(*self._print(b).parens()).__pow__(self._print(e))
return self._print(b)**self._print(e)
def _print_UnevaluatedExpr(self, expr):
return self._print(expr.args[0])
def __print_numer_denom(self, p, q):
if q == 1:
if p < 0:
return prettyForm(str(p), binding=prettyForm.NEG)
else:
return prettyForm(str(p))
elif abs(p) >= 10 and abs(q) >= 10:
# If more than one digit in numer and denom, print larger fraction
if p < 0:
return prettyForm(str(p), binding=prettyForm.NEG)/prettyForm(str(q))
# Old printing method:
#pform = prettyForm(str(-p))/prettyForm(str(q))
#return prettyForm(binding=prettyForm.NEG, *pform.left('- '))
else:
return prettyForm(str(p))/prettyForm(str(q))
else:
return None
def _print_Rational(self, expr):
result = self.__print_numer_denom(expr.p, expr.q)
if result is not None:
return result
else:
return self.emptyPrinter(expr)
def _print_Fraction(self, expr):
result = self.__print_numer_denom(expr.numerator, expr.denominator)
if result is not None:
return result
else:
return self.emptyPrinter(expr)
def _print_ProductSet(self, p):
if len(p.sets) > 1 and not has_variety(p.sets):
from sympy import Pow
return self._print(Pow(p.sets[0], len(p.sets), evaluate=False))
else:
prod_char = u"\N{MULTIPLICATION SIGN}" if self._use_unicode else 'x'
return self._print_seq(p.sets, None, None, ' %s ' % prod_char,
parenthesize=lambda set: set.is_Union or
set.is_Intersection or set.is_ProductSet)
def _print_FiniteSet(self, s):
items = sorted(s.args, key=default_sort_key)
return self._print_seq(items, '{', '}', ', ' )
def _print_Range(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
if s.start.is_infinite:
printset = dots, s[-1] - s.step, s[-1]
elif s.stop.is_infinite:
it = iter(s)
printset = next(it), next(it), dots
elif len(s) > 4:
it = iter(s)
printset = next(it), next(it), dots, s[-1]
else:
printset = tuple(s)
return self._print_seq(printset, '{', '}', ', ' )
def _print_Interval(self, i):
if i.start == i.end:
return self._print_seq(i.args[:1], '{', '}')
else:
if i.left_open:
left = '('
else:
left = '['
if i.right_open:
right = ')'
else:
right = ']'
return self._print_seq(i.args[:2], left, right)
def _print_AccumulationBounds(self, i):
left = '<'
right = '>'
return self._print_seq(i.args[:2], left, right)
def _print_Intersection(self, u):
delimiter = ' %s ' % pretty_atom('Intersection', 'n')
return self._print_seq(u.args, None, None, delimiter,
parenthesize=lambda set: set.is_ProductSet or
set.is_Union or set.is_Complement)
def _print_Union(self, u):
union_delimiter = ' %s ' % pretty_atom('Union', 'U')
return self._print_seq(u.args, None, None, union_delimiter,
parenthesize=lambda set: set.is_ProductSet or
set.is_Intersection or set.is_Complement)
def _print_SymmetricDifference(self, u):
if not self._use_unicode:
raise NotImplementedError("ASCII pretty printing of SymmetricDifference is not implemented")
sym_delimeter = ' %s ' % pretty_atom('SymmetricDifference')
return self._print_seq(u.args, None, None, sym_delimeter)
def _print_Complement(self, u):
delimiter = r' \ '
return self._print_seq(u.args, None, None, delimiter,
parenthesize=lambda set: set.is_ProductSet or set.is_Intersection
or set.is_Union)
def _print_ImageSet(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
else:
inn = 'in'
variables = ts.lamda.variables
expr = self._print(ts.lamda.expr)
bar = self._print("|")
sets = [self._print(i) for i in ts.args[1:]]
if len(sets) == 1:
return self._print_seq((expr, bar, variables[0], inn, sets[0]), "{", "}", ' ')
else:
pargs = tuple(j for var, setv in zip(variables, sets) for j in (var, inn, setv, ","))
return self._print_seq((expr, bar) + pargs[:-1], "{", "}", ' ')
def _print_ConditionSet(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
# using _and because and is a keyword and it is bad practice to
# overwrite them
_and = u"\N{LOGICAL AND}"
else:
inn = 'in'
_and = 'and'
variables = self._print_seq(Tuple(ts.sym))
as_expr = getattr(ts.condition, 'as_expr', None)
if as_expr is not None:
cond = self._print(ts.condition.as_expr())
else:
cond = self._print(ts.condition)
if self._use_unicode:
cond = self._print_seq(cond, "(", ")")
bar = self._print("|")
if ts.base_set is S.UniversalSet:
return self._print_seq((variables, bar, cond), "{", "}", ' ')
base = self._print(ts.base_set)
return self._print_seq((variables, bar, variables, inn,
base, _and, cond), "{", "}", ' ')
def _print_ComplexRegion(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
else:
inn = 'in'
variables = self._print_seq(ts.variables)
expr = self._print(ts.expr)
bar = self._print("|")
prodsets = self._print(ts.sets)
return self._print_seq((expr, bar, variables, inn, prodsets), "{", "}", ' ')
def _print_Contains(self, e):
var, set = e.args
if self._use_unicode:
el = u" \N{ELEMENT OF} "
return prettyForm(*stringPict.next(self._print(var),
el, self._print(set)), binding=8)
else:
return prettyForm(sstr(e))
def _print_FourierSeries(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
return self._print_Add(s.truncate()) + self._print(dots)
def _print_FormalPowerSeries(self, s):
return self._print_Add(s.infinite)
def _print_SetExpr(self, se):
pretty_set = prettyForm(*self._print(se.set).parens())
pretty_name = self._print(Symbol("SetExpr"))
return prettyForm(*pretty_name.right(pretty_set))
def _print_SeqFormula(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
if len(s.start.free_symbols) > 0 or len(s.stop.free_symbols) > 0:
raise NotImplementedError("Pretty printing of sequences with symbolic bound not implemented")
if s.start is S.NegativeInfinity:
stop = s.stop
printset = (dots, s.coeff(stop - 3), s.coeff(stop - 2),
s.coeff(stop - 1), s.coeff(stop))
elif s.stop is S.Infinity or s.length > 4:
printset = s[:4]
printset.append(dots)
printset = tuple(printset)
else:
printset = tuple(s)
return self._print_list(printset)
_print_SeqPer = _print_SeqFormula
_print_SeqAdd = _print_SeqFormula
_print_SeqMul = _print_SeqFormula
def _print_seq(self, seq, left=None, right=None, delimiter=', ',
parenthesize=lambda x: False):
s = None
try:
for item in seq:
pform = self._print(item)
if parenthesize(item):
pform = prettyForm(*pform.parens())
if s is None:
# first element
s = pform
else:
# XXX: Under the tests from #15686 this raises:
# AttributeError: 'Fake' object has no attribute 'baseline'
# This is caught below but that is not the right way to
# fix it.
s = prettyForm(*stringPict.next(s, delimiter))
s = prettyForm(*stringPict.next(s, pform))
if s is None:
s = stringPict('')
except AttributeError:
s = None
for item in seq:
pform = self.doprint(item)
if parenthesize(item):
pform = prettyForm(*pform.parens())
if s is None:
# first element
s = pform
else :
s = prettyForm(*stringPict.next(s, delimiter))
s = prettyForm(*stringPict.next(s, pform))
if s is None:
s = stringPict('')
s = prettyForm(*s.parens(left, right, ifascii_nougly=True))
return s
def join(self, delimiter, args):
pform = None
for arg in args:
if pform is None:
pform = arg
else:
pform = prettyForm(*pform.right(delimiter))
pform = prettyForm(*pform.right(arg))
if pform is None:
return prettyForm("")
else:
return pform
def _print_list(self, l):
return self._print_seq(l, '[', ']')
def _print_tuple(self, t):
if len(t) == 1:
ptuple = prettyForm(*stringPict.next(self._print(t[0]), ','))
return prettyForm(*ptuple.parens('(', ')', ifascii_nougly=True))
else:
return self._print_seq(t, '(', ')')
def _print_Tuple(self, expr):
return self._print_tuple(expr)
def _print_dict(self, d):
keys = sorted(d.keys(), key=default_sort_key)
items = []
for k in keys:
K = self._print(k)
V = self._print(d[k])
s = prettyForm(*stringPict.next(K, ': ', V))
items.append(s)
return self._print_seq(items, '{', '}')
def _print_Dict(self, d):
return self._print_dict(d)
def _print_set(self, s):
if not s:
return prettyForm('set()')
items = sorted(s, key=default_sort_key)
pretty = self._print_seq(items)
pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True))
return pretty
def _print_frozenset(self, s):
if not s:
return prettyForm('frozenset()')
items = sorted(s, key=default_sort_key)
pretty = self._print_seq(items)
pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True))
pretty = prettyForm(*pretty.parens('(', ')', ifascii_nougly=True))
pretty = prettyForm(*stringPict.next(type(s).__name__, pretty))
return pretty
def _print_PolyRing(self, ring):
return prettyForm(sstr(ring))
def _print_FracField(self, field):
return prettyForm(sstr(field))
def _print_FreeGroupElement(self, elm):
return prettyForm(str(elm))
def _print_PolyElement(self, poly):
return prettyForm(sstr(poly))
def _print_FracElement(self, frac):
return prettyForm(sstr(frac))
def _print_AlgebraicNumber(self, expr):
if expr.is_aliased:
return self._print(expr.as_poly().as_expr())
else:
return self._print(expr.as_expr())
def _print_ComplexRootOf(self, expr):
args = [self._print_Add(expr.expr, order='lex'), expr.index]
pform = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(*pform.left('CRootOf'))
return pform
def _print_RootSum(self, expr):
args = [self._print_Add(expr.expr, order='lex')]
if expr.fun is not S.IdentityFunction:
args.append(self._print(expr.fun))
pform = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(*pform.left('RootSum'))
return pform
def _print_FiniteField(self, expr):
if self._use_unicode:
form = u'\N{DOUBLE-STRUCK CAPITAL Z}_%d'
else:
form = 'GF(%d)'
return prettyForm(pretty_symbol(form % expr.mod))
def _print_IntegerRing(self, expr):
if self._use_unicode:
return prettyForm(u'\N{DOUBLE-STRUCK CAPITAL Z}')
else:
return prettyForm('ZZ')
def _print_RationalField(self, expr):
if self._use_unicode:
return prettyForm(u'\N{DOUBLE-STRUCK CAPITAL Q}')
else:
return prettyForm('QQ')
def _print_RealField(self, domain):
if self._use_unicode:
prefix = u'\N{DOUBLE-STRUCK CAPITAL R}'
else:
prefix = 'RR'
if domain.has_default_precision:
return prettyForm(prefix)
else:
return self._print(pretty_symbol(prefix + "_" + str(domain.precision)))
def _print_ComplexField(self, domain):
if self._use_unicode:
prefix = u'\N{DOUBLE-STRUCK CAPITAL C}'
else:
prefix = 'CC'
if domain.has_default_precision:
return prettyForm(prefix)
else:
return self._print(pretty_symbol(prefix + "_" + str(domain.precision)))
def _print_PolynomialRing(self, expr):
args = list(expr.symbols)
if not expr.order.is_default:
order = prettyForm(*prettyForm("order=").right(self._print(expr.order)))
args.append(order)
pform = self._print_seq(args, '[', ']')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_FractionField(self, expr):
args = list(expr.symbols)
if not expr.order.is_default:
order = prettyForm(*prettyForm("order=").right(self._print(expr.order)))
args.append(order)
pform = self._print_seq(args, '(', ')')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_PolynomialRingBase(self, expr):
g = expr.symbols
if str(expr.order) != str(expr.default_order):
g = g + ("order=" + str(expr.order),)
pform = self._print_seq(g, '[', ']')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_GroebnerBasis(self, basis):
exprs = [ self._print_Add(arg, order=basis.order)
for arg in basis.exprs ]
exprs = prettyForm(*self.join(", ", exprs).parens(left="[", right="]"))
gens = [ self._print(gen) for gen in basis.gens ]
domain = prettyForm(
*prettyForm("domain=").right(self._print(basis.domain)))
order = prettyForm(
*prettyForm("order=").right(self._print(basis.order)))
pform = self.join(", ", [exprs] + gens + [domain, order])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(basis.__class__.__name__))
return pform
def _print_Subs(self, e):
pform = self._print(e.expr)
pform = prettyForm(*pform.parens())
h = pform.height() if pform.height() > 1 else 2
rvert = stringPict(vobj('|', h), baseline=pform.baseline)
pform = prettyForm(*pform.right(rvert))
b = pform.baseline
pform.baseline = pform.height() - 1
pform = prettyForm(*pform.right(self._print_seq([
self._print_seq((self._print(v[0]), xsym('=='), self._print(v[1])),
delimiter='') for v in zip(e.variables, e.point) ])))
pform.baseline = b
return pform
def _print_euler(self, e):
pform = prettyForm("E")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
if len(e.args) == 1:
return pform
m, x = e.args
# TODO: copy-pasted from _print_Function: can we do better?
prettyFunc = pform
prettyArgs = prettyForm(*self._print_seq([x]).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_catalan(self, e):
pform = prettyForm("C")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_bernoulli(self, e):
pform = prettyForm("B")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
_print_bell = _print_bernoulli
def _print_lucas(self, e):
pform = prettyForm("L")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_fibonacci(self, e):
pform = prettyForm("F")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_tribonacci(self, e):
pform = prettyForm("T")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_KroneckerDelta(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.right((prettyForm(','))))
pform = prettyForm(*pform.right((self._print(e.args[1]))))
if self._use_unicode:
a = stringPict(pretty_symbol('delta'))
else:
a = stringPict('d')
b = pform
top = stringPict(*b.left(' '*a.width()))
bot = stringPict(*a.right(' '*b.width()))
return prettyForm(binding=prettyForm.POW, *bot.below(top))
def _print_RandomDomain(self, d):
if hasattr(d, 'as_boolean'):
pform = self._print('Domain: ')
pform = prettyForm(*pform.right(self._print(d.as_boolean())))
return pform
elif hasattr(d, 'set'):
pform = self._print('Domain: ')
pform = prettyForm(*pform.right(self._print(d.symbols)))
pform = prettyForm(*pform.right(self._print(' in ')))
pform = prettyForm(*pform.right(self._print(d.set)))
return pform
elif hasattr(d, 'symbols'):
pform = self._print('Domain on ')
pform = prettyForm(*pform.right(self._print(d.symbols)))
return pform
else:
return self._print(None)
def _print_DMP(self, p):
try:
if p.ring is not None:
# TODO incorporate order
return self._print(p.ring.to_sympy(p))
except SympifyError:
pass
return self._print(repr(p))
def _print_DMF(self, p):
return self._print_DMP(p)
def _print_Object(self, object):
return self._print(pretty_symbol(object.name))
def _print_Morphism(self, morphism):
arrow = xsym("-->")
domain = self._print(morphism.domain)
codomain = self._print(morphism.codomain)
tail = domain.right(arrow, codomain)[0]
return prettyForm(tail)
def _print_NamedMorphism(self, morphism):
pretty_name = self._print(pretty_symbol(morphism.name))
pretty_morphism = self._print_Morphism(morphism)
return prettyForm(pretty_name.right(":", pretty_morphism)[0])
def _print_IdentityMorphism(self, morphism):
from sympy.categories import NamedMorphism
return self._print_NamedMorphism(
NamedMorphism(morphism.domain, morphism.codomain, "id"))
def _print_CompositeMorphism(self, morphism):
circle = xsym(".")
# All components of the morphism have names and it is thus
# possible to build the name of the composite.
component_names_list = [pretty_symbol(component.name) for
component in morphism.components]
component_names_list.reverse()
component_names = circle.join(component_names_list) + ":"
pretty_name = self._print(component_names)
pretty_morphism = self._print_Morphism(morphism)
return prettyForm(pretty_name.right(pretty_morphism)[0])
def _print_Category(self, category):
return self._print(pretty_symbol(category.name))
def _print_Diagram(self, diagram):
if not diagram.premises:
# This is an empty diagram.
return self._print(S.EmptySet)
pretty_result = self._print(diagram.premises)
if diagram.conclusions:
results_arrow = " %s " % xsym("==>")
pretty_conclusions = self._print(diagram.conclusions)[0]
pretty_result = pretty_result.right(
results_arrow, pretty_conclusions)
return prettyForm(pretty_result[0])
def _print_DiagramGrid(self, grid):
from sympy.matrices import Matrix
from sympy import Symbol
matrix = Matrix([[grid[i, j] if grid[i, j] else Symbol(" ")
for j in range(grid.width)]
for i in range(grid.height)])
return self._print_matrix_contents(matrix)
def _print_FreeModuleElement(self, m):
# Print as row vector for convenience, for now.
return self._print_seq(m, '[', ']')
def _print_SubModule(self, M):
return self._print_seq(M.gens, '<', '>')
def _print_FreeModule(self, M):
return self._print(M.ring)**self._print(M.rank)
def _print_ModuleImplementedIdeal(self, M):
return self._print_seq([x for [x] in M._module.gens], '<', '>')
def _print_QuotientRing(self, R):
return self._print(R.ring) / self._print(R.base_ideal)
def _print_QuotientRingElement(self, R):
return self._print(R.data) + self._print(R.ring.base_ideal)
def _print_QuotientModuleElement(self, m):
return self._print(m.data) + self._print(m.module.killed_module)
def _print_QuotientModule(self, M):
return self._print(M.base) / self._print(M.killed_module)
def _print_MatrixHomomorphism(self, h):
matrix = self._print(h._sympy_matrix())
matrix.baseline = matrix.height() // 2
pform = prettyForm(*matrix.right(' : ', self._print(h.domain),
' %s> ' % hobj('-', 2), self._print(h.codomain)))
return pform
def _print_BaseScalarField(self, field):
string = field._coord_sys._names[field._index]
return self._print(pretty_symbol(string))
def _print_BaseVectorField(self, field):
s = U('PARTIAL DIFFERENTIAL') + '_' + field._coord_sys._names[field._index]
return self._print(pretty_symbol(s))
def _print_Differential(self, diff):
field = diff._form_field
if hasattr(field, '_coord_sys'):
string = field._coord_sys._names[field._index]
return self._print(u'\N{DOUBLE-STRUCK ITALIC SMALL D} ' + pretty_symbol(string))
else:
pform = self._print(field)
pform = prettyForm(*pform.parens())
return prettyForm(*pform.left(u"\N{DOUBLE-STRUCK ITALIC SMALL D}"))
def _print_Tr(self, p):
#TODO: Handle indices
pform = self._print(p.args[0])
pform = prettyForm(*pform.left('%s(' % (p.__class__.__name__)))
pform = prettyForm(*pform.right(')'))
return pform
def _print_primenu(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
if self._use_unicode:
pform = prettyForm(*pform.left(greek_unicode['nu']))
else:
pform = prettyForm(*pform.left('nu'))
return pform
def _print_primeomega(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
if self._use_unicode:
pform = prettyForm(*pform.left(greek_unicode['Omega']))
else:
pform = prettyForm(*pform.left('Omega'))
return pform
def _print_Quantity(self, e):
if e.name.name == 'degree':
pform = self._print(u"\N{DEGREE SIGN}")
return pform
else:
return self.emptyPrinter(e)
def _print_AssignmentBase(self, e):
op = prettyForm(' ' + xsym(e.op) + ' ')
l = self._print(e.lhs)
r = self._print(e.rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
def pretty(expr, **settings):
"""Returns a string containing the prettified form of expr.
For information on keyword arguments see pretty_print function.
"""
pp = PrettyPrinter(settings)
# XXX: this is an ugly hack, but at least it works
use_unicode = pp._settings['use_unicode']
uflag = pretty_use_unicode(use_unicode)
try:
return pp.doprint(expr)
finally:
pretty_use_unicode(uflag)
def pretty_print(expr, wrap_line=True, num_columns=None, use_unicode=None,
full_prec="auto", order=None, use_unicode_sqrt_char=True,
root_notation = True, mat_symbol_style="plain", imaginary_unit="i"):
"""Prints expr in pretty form.
pprint is just a shortcut for this function.
Parameters
==========
expr : expression
The expression to print.
wrap_line : bool, optional (default=True)
Line wrapping enabled/disabled.
num_columns : int or None, optional (default=None)
Number of columns before line breaking (default to None which reads
the terminal width), useful when using SymPy without terminal.
use_unicode : bool or None, optional (default=None)
Use unicode characters, such as the Greek letter pi instead of
the string pi.
full_prec : bool or string, optional (default="auto")
Use full precision.
order : bool or string, optional (default=None)
Set to 'none' for long expressions if slow; default is None.
use_unicode_sqrt_char : bool, optional (default=True)
Use compact single-character square root symbol (when unambiguous).
root_notation : bool, optional (default=True)
Set to 'False' for printing exponents of the form 1/n in fractional form.
By default exponent is printed in root form.
mat_symbol_style : string, optional (default="plain")
Set to "bold" for printing MatrixSymbols using a bold mathematical symbol face.
By default the standard face is used.
imaginary_unit : string, optional (default="i")
Letter to use for imaginary unit when use_unicode is True.
Can be "i" (default) or "j".
"""
print(pretty(expr, wrap_line=wrap_line, num_columns=num_columns,
use_unicode=use_unicode, full_prec=full_prec, order=order,
use_unicode_sqrt_char=use_unicode_sqrt_char,
root_notation=root_notation, mat_symbol_style=mat_symbol_style,
imaginary_unit=imaginary_unit))
pprint = pretty_print
def pager_print(expr, **settings):
"""Prints expr using the pager, in pretty form.
This invokes a pager command using pydoc. Lines are not wrapped
automatically. This routine is meant to be used with a pager that allows
sideways scrolling, like ``less -S``.
Parameters are the same as for ``pretty_print``. If you wish to wrap lines,
pass ``num_columns=None`` to auto-detect the width of the terminal.
"""
from pydoc import pager
from locale import getpreferredencoding
if 'num_columns' not in settings:
settings['num_columns'] = 500000 # disable line wrap
pager(pretty(expr, **settings).encode(getpreferredencoding()))
|
95e10bcdf9a58962404635b441a54842a4c21cf72cc1855fbdee5792d4002378 | from sympy.utilities.pytest import raises
from sympy import (symbols, Function, Integer, Matrix, Abs,
Rational, Float, S, WildFunction, ImmutableDenseMatrix, sin, true, false, ones,
sqrt, root, AlgebraicNumber, Symbol, Dummy, Wild, MatrixSymbol)
from sympy.core.compatibility import exec_
from sympy.geometry import Point, Ellipse
from sympy.printing import srepr
from sympy.polys import ring, field, ZZ, QQ, lex, grlex, Poly
from sympy.polys.polyclasses import DMP
from sympy.polys.agca.extensions import FiniteExtension
x, y = symbols('x,y')
# eval(srepr(expr)) == expr has to succeed in the right environment. The right
# environment is the scope of "from sympy import *" for most cases.
ENV = {}
exec_("from sympy import *", ENV)
def sT(expr, string):
"""
sT := sreprTest
Tests that srepr delivers the expected string and that
the condition eval(srepr(expr))==expr holds.
"""
assert srepr(expr) == string
assert eval(string, ENV) == expr
def test_printmethod():
class R(Abs):
def _sympyrepr(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert srepr(R(x)) == "foo(Symbol('x'))"
def test_Add():
sT(x + y, "Add(Symbol('x'), Symbol('y'))")
assert srepr(x**2 + 1, order='lex') == "Add(Pow(Symbol('x'), Integer(2)), Integer(1))"
assert srepr(x**2 + 1, order='old') == "Add(Integer(1), Pow(Symbol('x'), Integer(2)))"
def test_more_than_255_args_issue_10259():
from sympy import Add, Mul
for op in (Add, Mul):
expr = op(*symbols('x:256'))
assert eval(srepr(expr)) == expr
def test_Function():
sT(Function("f")(x), "Function('f')(Symbol('x'))")
# test unapplied Function
sT(Function('f'), "Function('f')")
sT(sin(x), "sin(Symbol('x'))")
sT(sin, "sin")
def test_Geometry():
sT(Point(0, 0), "Point2D(Integer(0), Integer(0))")
sT(Ellipse(Point(0, 0), 5, 1),
"Ellipse(Point2D(Integer(0), Integer(0)), Integer(5), Integer(1))")
# TODO more tests
def test_Singletons():
sT(S.Catalan, 'Catalan')
sT(S.ComplexInfinity, 'zoo')
sT(S.EulerGamma, 'EulerGamma')
sT(S.Exp1, 'E')
sT(S.GoldenRatio, 'GoldenRatio')
sT(S.TribonacciConstant, 'TribonacciConstant')
sT(S.Half, 'Rational(1, 2)')
sT(S.ImaginaryUnit, 'I')
sT(S.Infinity, 'oo')
sT(S.NaN, 'nan')
sT(S.NegativeInfinity, '-oo')
sT(S.NegativeOne, 'Integer(-1)')
sT(S.One, 'Integer(1)')
sT(S.Pi, 'pi')
sT(S.Zero, 'Integer(0)')
def test_Integer():
sT(Integer(4), "Integer(4)")
def test_list():
sT([x, Integer(4)], "[Symbol('x'), Integer(4)]")
def test_Matrix():
for cls, name in [(Matrix, "MutableDenseMatrix"), (ImmutableDenseMatrix, "ImmutableDenseMatrix")]:
sT(cls([[x**+1, 1], [y, x + y]]),
"%s([[Symbol('x'), Integer(1)], [Symbol('y'), Add(Symbol('x'), Symbol('y'))]])" % name)
sT(cls(), "%s([])" % name)
sT(cls([[x**+1, 1], [y, x + y]]), "%s([[Symbol('x'), Integer(1)], [Symbol('y'), Add(Symbol('x'), Symbol('y'))]])" % name)
def test_empty_Matrix():
sT(ones(0, 3), "MutableDenseMatrix(0, 3, [])")
sT(ones(4, 0), "MutableDenseMatrix(4, 0, [])")
sT(ones(0, 0), "MutableDenseMatrix([])")
def test_Rational():
sT(Rational(1, 3), "Rational(1, 3)")
sT(Rational(-1, 3), "Rational(-1, 3)")
def test_Float():
sT(Float('1.23', dps=3), "Float('1.22998', precision=13)")
sT(Float('1.23456789', dps=9), "Float('1.23456788994', precision=33)")
sT(Float('1.234567890123456789', dps=19),
"Float('1.234567890123456789013', precision=66)")
sT(Float('0.60038617995049726', dps=15),
"Float('0.60038617995049726', precision=53)")
sT(Float('1.23', precision=13), "Float('1.22998', precision=13)")
sT(Float('1.23456789', precision=33),
"Float('1.23456788994', precision=33)")
sT(Float('1.234567890123456789', precision=66),
"Float('1.234567890123456789013', precision=66)")
sT(Float('0.60038617995049726', precision=53),
"Float('0.60038617995049726', precision=53)")
sT(Float('0.60038617995049726', 15),
"Float('0.60038617995049726', precision=53)")
def test_Symbol():
sT(x, "Symbol('x')")
sT(y, "Symbol('y')")
sT(Symbol('x', negative=True), "Symbol('x', negative=True)")
def test_Symbol_two_assumptions():
x = Symbol('x', negative=0, integer=1)
# order could vary
s1 = "Symbol('x', integer=True, negative=False)"
s2 = "Symbol('x', negative=False, integer=True)"
assert srepr(x) in (s1, s2)
assert eval(srepr(x), ENV) == x
def test_Symbol_no_special_commutative_treatment():
sT(Symbol('x'), "Symbol('x')")
sT(Symbol('x', commutative=False), "Symbol('x', commutative=False)")
sT(Symbol('x', commutative=0), "Symbol('x', commutative=False)")
sT(Symbol('x', commutative=True), "Symbol('x', commutative=True)")
sT(Symbol('x', commutative=1), "Symbol('x', commutative=True)")
def test_Wild():
sT(Wild('x', even=True), "Wild('x', even=True)")
def test_Dummy():
d = Dummy('d')
sT(d, "Dummy('d', dummy_index=%s)" % str(d.dummy_index))
def test_Dummy_assumption():
d = Dummy('d', nonzero=True)
assert d == eval(srepr(d))
s1 = "Dummy('d', dummy_index=%s, nonzero=True)" % str(d.dummy_index)
s2 = "Dummy('d', nonzero=True, dummy_index=%s)" % str(d.dummy_index)
assert srepr(d) in (s1, s2)
def test_Dummy_from_Symbol():
# should not get the full dictionary of assumptions
n = Symbol('n', integer=True)
d = n.as_dummy()
assert srepr(d
) == "Dummy('n', dummy_index=%s)" % str(d.dummy_index)
def test_tuple():
sT((x,), "(Symbol('x'),)")
sT((x, y), "(Symbol('x'), Symbol('y'))")
def test_WildFunction():
sT(WildFunction('w'), "WildFunction('w')")
def test_settins():
raises(TypeError, lambda: srepr(x, method="garbage"))
def test_Mul():
sT(3*x**3*y, "Mul(Integer(3), Pow(Symbol('x'), Integer(3)), Symbol('y'))")
assert srepr(3*x**3*y, order='old') == "Mul(Integer(3), Symbol('y'), Pow(Symbol('x'), Integer(3)))"
def test_AlgebraicNumber():
a = AlgebraicNumber(sqrt(2))
sT(a, "AlgebraicNumber(Pow(Integer(2), Rational(1, 2)), [Integer(1), Integer(0)])")
a = AlgebraicNumber(root(-2, 3))
sT(a, "AlgebraicNumber(Pow(Integer(-2), Rational(1, 3)), [Integer(1), Integer(0)])")
def test_PolyRing():
assert srepr(ring("x", ZZ, lex)[0]) == "PolyRing((Symbol('x'),), ZZ, lex)"
assert srepr(ring("x,y", QQ, grlex)[0]) == "PolyRing((Symbol('x'), Symbol('y')), QQ, grlex)"
assert srepr(ring("x,y,z", ZZ["t"], lex)[0]) == "PolyRing((Symbol('x'), Symbol('y'), Symbol('z')), ZZ[t], lex)"
def test_FracField():
assert srepr(field("x", ZZ, lex)[0]) == "FracField((Symbol('x'),), ZZ, lex)"
assert srepr(field("x,y", QQ, grlex)[0]) == "FracField((Symbol('x'), Symbol('y')), QQ, grlex)"
assert srepr(field("x,y,z", ZZ["t"], lex)[0]) == "FracField((Symbol('x'), Symbol('y'), Symbol('z')), ZZ[t], lex)"
def test_PolyElement():
R, x, y = ring("x,y", ZZ)
assert srepr(3*x**2*y + 1) == "PolyElement(PolyRing((Symbol('x'), Symbol('y')), ZZ, lex), [((2, 1), 3), ((0, 0), 1)])"
def test_FracElement():
F, x, y = field("x,y", ZZ)
assert srepr((3*x**2*y + 1)/(x - y**2)) == "FracElement(FracField((Symbol('x'), Symbol('y')), ZZ, lex), [((2, 1), 3), ((0, 0), 1)], [((1, 0), 1), ((0, 2), -1)])"
def test_FractionField():
assert srepr(QQ.frac_field(x)) == \
"FractionField(FracField((Symbol('x'),), QQ, lex))"
assert srepr(QQ.frac_field(x, y, order=grlex)) == \
"FractionField(FracField((Symbol('x'), Symbol('y')), QQ, grlex))"
def test_PolynomialRingBase():
assert srepr(ZZ.old_poly_ring(x)) == \
"GlobalPolynomialRing(ZZ, Symbol('x'))"
assert srepr(ZZ[x].old_poly_ring(y)) == \
"GlobalPolynomialRing(ZZ[x], Symbol('y'))"
assert srepr(QQ.frac_field(x).old_poly_ring(y)) == \
"GlobalPolynomialRing(FractionField(FracField((Symbol('x'),), QQ, lex)), Symbol('y'))"
def test_DMP():
assert srepr(DMP([1, 2], ZZ)) == 'DMP([1, 2], ZZ)'
assert srepr(ZZ.old_poly_ring(x)([1, 2])) == \
"DMP([1, 2], ZZ, ring=GlobalPolynomialRing(ZZ, Symbol('x')))"
def test_FiniteExtension():
assert srepr(FiniteExtension(Poly(x**2 + 1, x))) == \
"FiniteExtension(Poly(x**2 + 1, x, domain='ZZ'))"
def test_ExtensionElement():
A = FiniteExtension(Poly(x**2 + 1, x))
assert srepr(A.generator) == \
"ExtElem(DMP([1, 0], ZZ, ring=GlobalPolynomialRing(ZZ, Symbol('x'))), FiniteExtension(Poly(x**2 + 1, x, domain='ZZ')))"
def test_BooleanAtom():
assert srepr(true) == "true"
assert srepr(false) == "false"
def test_Integers():
sT(S.Integers, "Integers")
def test_Naturals():
sT(S.Naturals, "Naturals")
def test_Naturals0():
sT(S.Naturals0, "Naturals0")
def test_Reals():
sT(S.Reals, "Reals")
def test_matrix_expressions():
n = symbols('n', integer=True)
A = MatrixSymbol("A", n, n)
B = MatrixSymbol("B", n, n)
sT(A, "MatrixSymbol(Symbol('A'), Symbol('n', integer=True), Symbol('n', integer=True))")
sT(A*B, "MatMul(MatrixSymbol(Symbol('A'), Symbol('n', integer=True), Symbol('n', integer=True)), MatrixSymbol(Symbol('B'), Symbol('n', integer=True), Symbol('n', integer=True)))")
sT(A + B, "MatAdd(MatrixSymbol(Symbol('A'), Symbol('n', integer=True), Symbol('n', integer=True)), MatrixSymbol(Symbol('B'), Symbol('n', integer=True), Symbol('n', integer=True)))")
|
a3746b244791e1cad9bea3ed47da2060f6d8539094717cc461aa31f2dc206e4b | from sympy import (
Add, Abs, Chi, Ci, CosineTransform, Dict, Ei, Eq, FallingFactorial,
FiniteSet, Float, FourierTransform, Function, Indexed, IndexedBase, Integral,
Interval, InverseCosineTransform, InverseFourierTransform,
InverseLaplaceTransform, InverseMellinTransform, InverseSineTransform,
Lambda, LaplaceTransform, Limit, Matrix, Max, MellinTransform, Min, Mul,
Order, Piecewise, Poly, ring, field, ZZ, Pow, Product, Range, Rational,
RisingFactorial, rootof, RootSum, S, Shi, Si, SineTransform, Subs,
Sum, Symbol, ImageSet, Tuple, Union, Ynm, Znm, arg, asin, acsc, Mod,
assoc_laguerre, assoc_legendre, beta, binomial, catalan, ceiling, Complement,
chebyshevt, chebyshevu, conjugate, cot, coth, diff, dirichlet_eta, euler,
exp, expint, factorial, factorial2, floor, gamma, gegenbauer, hermite,
hyper, im, jacobi, laguerre, legendre, lerchphi, log,
meijerg, oo, polar_lift, polylog, re, root, sin, sqrt, symbols,
uppergamma, zeta, subfactorial, totient, elliptic_k, elliptic_f,
elliptic_e, elliptic_pi, cos, tan, Wild, true, false, Equivalent, Not,
Contains, divisor_sigma, SymmetricDifference, SeqPer, SeqFormula,
SeqAdd, SeqMul, fourier_series, pi, ConditionSet, ComplexRegion, fps,
AccumBounds, reduced_totient, primenu, primeomega, SingularityFunction,
UnevaluatedExpr, Quaternion, I, KroneckerProduct, Intersection)
from sympy.ntheory.factor_ import udivisor_sigma
from sympy.abc import mu, tau
from sympy.printing.latex import (latex, translate, greek_letters_set,
tex_greek_dictionary)
from sympy.tensor.array import (ImmutableDenseNDimArray,
ImmutableSparseNDimArray,
MutableSparseNDimArray,
MutableDenseNDimArray)
from sympy.tensor.array import tensorproduct
from sympy.utilities.pytest import XFAIL, raises
from sympy.functions import DiracDelta, Heaviside, KroneckerDelta, LeviCivita
from sympy.functions.combinatorial.numbers import bernoulli, bell, lucas, \
fibonacci, tribonacci
from sympy.logic import Implies
from sympy.logic.boolalg import And, Or, Xor
from sympy.physics.quantum import Commutator, Operator
from sympy.physics.units import degree, radian, kg, meter, gibibyte, microgram, second
from sympy.core.trace import Tr
from sympy.core.compatibility import range
from sympy.combinatorics.permutations import Cycle, Permutation
from sympy import MatrixSymbol, ln
from sympy.vector import CoordSys3D, Cross, Curl, Dot, Divergence, Gradient, Laplacian
from sympy.sets.setexpr import SetExpr
import sympy as sym
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
x, y, z, t, a, b, c = symbols('x y z t a b c')
k, m, n = symbols('k m n', integer=True)
def test_printmethod():
class R(Abs):
def _latex(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert latex(R(x)) == "foo(x)"
class R(Abs):
def _latex(self, printer):
return "foo"
assert latex(R(x)) == "foo"
def test_latex_basic():
assert latex(1 + x) == "x + 1"
assert latex(x**2) == "x^{2}"
assert latex(x**(1 + x)) == "x^{x + 1}"
assert latex(x**3 + x + 1 + x**2) == "x^{3} + x^{2} + x + 1"
assert latex(2*x*y) == "2 x y"
assert latex(2*x*y, mul_symbol='dot') == r"2 \cdot x \cdot y"
assert latex(3*x**2*y, mul_symbol='\\,') == r"3\,x^{2}\,y"
assert latex(1.5*3**x, mul_symbol='\\,') == r"1.5 \cdot 3^{x}"
assert latex(1/x) == r"\frac{1}{x}"
assert latex(1/x, fold_short_frac=True) == "1 / x"
assert latex(-S(3)/2) == r"- \frac{3}{2}"
assert latex(-S(3)/2, fold_short_frac=True) == r"- 3 / 2"
assert latex(1/x**2) == r"\frac{1}{x^{2}}"
assert latex(1/(x + y)/2) == r"\frac{1}{2 \left(x + y\right)}"
assert latex(x/2) == r"\frac{x}{2}"
assert latex(x/2, fold_short_frac=True) == "x / 2"
assert latex((x + y)/(2*x)) == r"\frac{x + y}{2 x}"
assert latex((x + y)/(2*x), fold_short_frac=True) == \
r"\left(x + y\right) / 2 x"
assert latex((x + y)/(2*x), long_frac_ratio=0) == \
r"\frac{1}{2 x} \left(x + y\right)"
assert latex((x + y)/x) == r"\frac{x + y}{x}"
assert latex((x + y)/x, long_frac_ratio=3) == r"\frac{x + y}{x}"
assert latex((2*sqrt(2)*x)/3) == r"\frac{2 \sqrt{2} x}{3}"
assert latex((2*sqrt(2)*x)/3, long_frac_ratio=2) == \
r"\frac{2 x}{3} \sqrt{2}"
assert latex(2*Integral(x, x)/3) == r"\frac{2 \int x\, dx}{3}"
assert latex(2*Integral(x, x)/3, fold_short_frac=True) == \
r"\left(2 \int x\, dx\right) / 3"
assert latex(sqrt(x)) == r"\sqrt{x}"
assert latex(x**Rational(1, 3)) == r"\sqrt[3]{x}"
assert latex(x**Rational(1, 3), root_notation=False) == r"x^{\frac{1}{3}}"
assert latex(sqrt(x)**3) == r"x^{\frac{3}{2}}"
assert latex(sqrt(x), itex=True) == r"\sqrt{x}"
assert latex(x**Rational(1, 3), itex=True) == r"\root{3}{x}"
assert latex(sqrt(x)**3, itex=True) == r"x^{\frac{3}{2}}"
assert latex(x**Rational(3, 4)) == r"x^{\frac{3}{4}}"
assert latex(x**Rational(3, 4), fold_frac_powers=True) == "x^{3/4}"
assert latex((x + 1)**Rational(3, 4)) == \
r"\left(x + 1\right)^{\frac{3}{4}}"
assert latex((x + 1)**Rational(3, 4), fold_frac_powers=True) == \
r"\left(x + 1\right)^{3/4}"
assert latex(1.5e20*x) == r"1.5 \cdot 10^{20} x"
assert latex(1.5e20*x, mul_symbol='dot') == r"1.5 \cdot 10^{20} \cdot x"
assert latex(1.5e20*x, mul_symbol='times') == \
r"1.5 \times 10^{20} \times x"
assert latex(1/sin(x)) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**-1) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**Rational(3, 2)) == \
r"\sin^{\frac{3}{2}}{\left(x \right)}"
assert latex(sin(x)**Rational(3, 2), fold_frac_powers=True) == \
r"\sin^{3/2}{\left(x \right)}"
assert latex(~x) == r"\neg x"
assert latex(x & y) == r"x \wedge y"
assert latex(x & y & z) == r"x \wedge y \wedge z"
assert latex(x | y) == r"x \vee y"
assert latex(x | y | z) == r"x \vee y \vee z"
assert latex((x & y) | z) == r"z \vee \left(x \wedge y\right)"
assert latex(Implies(x, y)) == r"x \Rightarrow y"
assert latex(~(x >> ~y)) == r"x \not\Rightarrow \neg y"
assert latex(Implies(Or(x,y), z)) == r"\left(x \vee y\right) \Rightarrow z"
assert latex(Implies(z, Or(x,y))) == r"z \Rightarrow \left(x \vee y\right)"
assert latex(~x, symbol_names={x: "x_i"}) == r"\neg x_i"
assert latex(x & y, symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \wedge y_i"
assert latex(x & y & z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \wedge y_i \wedge z_i"
assert latex(x | y, symbol_names={x: "x_i", y: "y_i"}) == r"x_i \vee y_i"
assert latex(x | y | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \vee y_i \vee z_i"
assert latex((x & y) | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"z_i \vee \left(x_i \wedge y_i\right)"
assert latex(Implies(x, y), symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \Rightarrow y_i"
p = Symbol('p', positive=True)
assert latex(exp(-p)*log(p)) == r"e^{- p} \log{\left(p \right)}"
def test_latex_builtins():
assert latex(True) == r"\text{True}"
assert latex(False) == r"\text{False}"
assert latex(None) == r"\text{None}"
assert latex(true) == r"\text{True}"
assert latex(false) == r'\text{False}'
def test_latex_SingularityFunction():
assert latex(SingularityFunction(x, 4, 5)) == \
r"{\left\langle x - 4 \right\rangle}^{5}"
assert latex(SingularityFunction(x, -3, 4)) == \
r"{\left\langle x + 3 \right\rangle}^{4}"
assert latex(SingularityFunction(x, 0, 4)) == \
r"{\left\langle x \right\rangle}^{4}"
assert latex(SingularityFunction(x, a, n)) == \
r"{\left\langle - a + x \right\rangle}^{n}"
assert latex(SingularityFunction(x, 4, -2)) == \
r"{\left\langle x - 4 \right\rangle}^{-2}"
assert latex(SingularityFunction(x, 4, -1)) == \
r"{\left\langle x - 4 \right\rangle}^{-1}"
def test_latex_cycle():
assert latex(Cycle(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Cycle(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Cycle()) == r"\left( \right)"
def test_latex_permutation():
assert latex(Permutation(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Permutation(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Permutation()) == r"\left( \right)"
assert latex(Permutation(2, 4)*Permutation(5)) == \
r"\left( 2\; 4\right)\left( 5\right)"
assert latex(Permutation(5)) == r"\left( 5\right)"
def test_latex_Float():
assert latex(Float(1.0e100)) == r"1.0 \cdot 10^{100}"
assert latex(Float(1.0e-100)) == r"1.0 \cdot 10^{-100}"
assert latex(Float(1.0e-100), mul_symbol="times") == \
r"1.0 \times 10^{-100}"
assert latex(1.0*oo) == r"\infty"
assert latex(-1.0*oo) == r"- \infty"
def test_latex_vector_expressions():
A = CoordSys3D('A')
assert latex(Cross(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Cross(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}"
assert latex(x*Cross(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}\right)"
assert latex(Cross(x*A.i, A.j)) == \
r'- \mathbf{\hat{j}_{A}} \times \left((x)\mathbf{\hat{i}_{A}}\right)'
assert latex(Curl(3*A.x*A.j)) == \
r"\nabla\times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*A.x*A.j+A.i)) == \
r"\nabla\times \left(\mathbf{\hat{i}_{A}} + (3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*x*A.x*A.j)) == \
r"\nabla\times \left((3 \mathbf{{x}_{A}} x)\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Curl(3*A.x*A.j)) == \
r"x \left(\nabla\times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Divergence(3*A.x*A.j+A.i)) == \
r"\nabla\cdot \left(\mathbf{\hat{i}_{A}} + (3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Divergence(3*A.x*A.j)) == \
r"\nabla\cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Divergence(3*A.x*A.j)) == \
r"x \left(\nabla\cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Dot(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Dot(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}"
assert latex(Dot(x*A.i, A.j)) == \
r"\mathbf{\hat{j}_{A}} \cdot \left((x)\mathbf{\hat{i}_{A}}\right)"
assert latex(x*Dot(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}\right)"
assert latex(Gradient(A.x)) == r"\nabla \mathbf{{x}_{A}}"
assert latex(Gradient(A.x + 3*A.y)) == \
r"\nabla \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Gradient(A.x)) == r"x \left(\nabla \mathbf{{x}_{A}}\right)"
assert latex(Gradient(x*A.x)) == r"\nabla \left(\mathbf{{x}_{A}} x\right)"
assert latex(Laplacian(A.x)) == r"\triangle \mathbf{{x}_{A}}"
assert latex(Laplacian(A.x + 3*A.y)) == \
r"\triangle \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Laplacian(A.x)) == r"x \left(\triangle \mathbf{{x}_{A}}\right)"
assert latex(Laplacian(x*A.x)) == r"\triangle \left(\mathbf{{x}_{A}} x\right)"
def test_latex_symbols():
Gamma, lmbda, rho = symbols('Gamma, lambda, rho')
tau, Tau, TAU, taU = symbols('tau, Tau, TAU, taU')
assert latex(tau) == r"\tau"
assert latex(Tau) == "T"
assert latex(TAU) == r"\tau"
assert latex(taU) == r"\tau"
# Check that all capitalized greek letters are handled explicitly
capitalized_letters = set(l.capitalize() for l in greek_letters_set)
assert len(capitalized_letters - set(tex_greek_dictionary.keys())) == 0
assert latex(Gamma + lmbda) == r"\Gamma + \lambda"
assert latex(Gamma * lmbda) == r"\Gamma \lambda"
assert latex(Symbol('q1')) == r"q_{1}"
assert latex(Symbol('q21')) == r"q_{21}"
assert latex(Symbol('epsilon0')) == r"\epsilon_{0}"
assert latex(Symbol('omega1')) == r"\omega_{1}"
assert latex(Symbol('91')) == r"91"
assert latex(Symbol('alpha_new')) == r"\alpha_{new}"
assert latex(Symbol('C^orig')) == r"C^{orig}"
assert latex(Symbol('x^alpha')) == r"x^{\alpha}"
assert latex(Symbol('beta^alpha')) == r"\beta^{\alpha}"
assert latex(Symbol('e^Alpha')) == r"e^{A}"
assert latex(Symbol('omega_alpha^beta')) == r"\omega^{\beta}_{\alpha}"
assert latex(Symbol('omega') ** Symbol('beta')) == r"\omega^{\beta}"
@XFAIL
def test_latex_symbols_failing():
rho, mass, volume = symbols('rho, mass, volume')
assert latex(
volume * rho == mass) == r"\rho \mathrm{volume} = \mathrm{mass}"
assert latex(volume / mass * rho == 1) == \
r"\rho \mathrm{volume} {\mathrm{mass}}^{(-1)} = 1"
assert latex(mass**3 * volume**3) == \
r"{\mathrm{mass}}^{3} \cdot {\mathrm{volume}}^{3}"
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == r'f{\left(x \right)}'
assert latex(f) == r'f'
g = Function('g')
assert latex(g(x, y)) == r'g{\left(x,y \right)}'
assert latex(g) == r'g'
h = Function('h')
assert latex(h(x, y, z)) == r'h{\left(x,y,z \right)}'
assert latex(h) == r'h'
Li = Function('Li')
assert latex(Li) == r'\operatorname{Li}'
assert latex(Li(x)) == r'\operatorname{Li}{\left(x \right)}'
mybeta = Function('beta')
# not to be confused with the beta function
assert latex(mybeta(x, y, z)) == r"\beta{\left(x,y,z \right)}"
assert latex(beta(x, y)) == r'\operatorname{B}\left(x, y\right)'
assert latex(beta(x, y)**2) == r'\operatorname{B}^{2}\left(x, y\right)'
assert latex(mybeta(x)) == r"\beta{\left(x \right)}"
assert latex(mybeta) == r"\beta"
g = Function('gamma')
# not to be confused with the gamma function
assert latex(g(x, y, z)) == r"\gamma{\left(x,y,z \right)}"
assert latex(g(x)) == r"\gamma{\left(x \right)}"
assert latex(g) == r"\gamma"
a1 = Function('a_1')
assert latex(a1) == r"\operatorname{a_{1}}"
assert latex(a1(x)) == r"\operatorname{a_{1}}{\left(x \right)}"
# issue 5868
omega1 = Function('omega1')
assert latex(omega1) == r"\omega_{1}"
assert latex(omega1(x)) == r"\omega_{1}{\left(x \right)}"
assert latex(sin(x)) == r"\sin{\left(x \right)}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left(x \right)}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(acsc(x), inv_trig_style="full") == \
r"\operatorname{arccsc}{\left(x \right)}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial(k)**2) == r"k!^{2}"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(subfactorial(k)**2) == r"\left(!k\right)^{2}"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(factorial2(k)**2) == r"k!!^{2}"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(binomial(2, k)**2) == r"{\binom{2}{k}}^{2}"
assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{k}"
assert latex(RisingFactorial(3, k)) == r"{3}^{\left(k\right)}"
assert latex(floor(x)) == r"\left\lfloor{x}\right\rfloor"
assert latex(ceiling(x)) == r"\left\lceil{x}\right\rceil"
assert latex(floor(x)**2) == r"\left\lfloor{x}\right\rfloor^{2}"
assert latex(ceiling(x)**2) == r"\left\lceil{x}\right\rceil^{2}"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left|{x}\right|"
assert latex(Abs(x)**2) == r"\left|{x}\right|^{2}"
assert latex(re(x)) == r"\operatorname{re}{\left(x\right)}"
assert latex(re(x + y)) == \
r"\operatorname{re}{\left(x\right)} + \operatorname{re}{\left(y\right)}"
assert latex(im(x)) == r"\operatorname{im}{\left(x\right)}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(conjugate(x)**2) == r"\overline{x}^{2}"
assert latex(conjugate(x**2)) == r"\overline{x}^{2}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
w = Wild('w')
assert latex(gamma(w)) == r"\Gamma\left(w\right)"
assert latex(Order(x)) == r"O\left(x\right)"
assert latex(Order(x, x)) == r"O\left(x\right)"
assert latex(Order(x, (x, 0))) == r"O\left(x\right)"
assert latex(Order(x, (x, oo))) == r"O\left(x; x\rightarrow \infty\right)"
assert latex(Order(x - y, (x, y))) == \
r"O\left(x - y; x\rightarrow y\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, (x, oo), (y, oo))) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( \infty, \ \infty\right)\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(lowergamma(x, y)**2) == r'\gamma^{2}\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(uppergamma(x, y)**2) == r'\Gamma^{2}\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left(x \right)}'
assert latex(coth(x)) == r'\coth{\left(x \right)}'
assert latex(re(x)) == r'\operatorname{re}{\left(x\right)}'
assert latex(im(x)) == r'\operatorname{im}{\left(x\right)}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left(x \right)}'
assert latex(zeta(x)) == r'\zeta\left(x\right)'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(
polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left(x \right)}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left(x \right)}'
assert latex(expint(x, y)) == r'\operatorname{E}_{x}\left(y\right)'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left(x \right)}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left(x \right)}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left(x \right)}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}\left(x\right)'
assert latex(Chi(x)) == r'\operatorname{Chi}\left(x\right)'
assert latex(jacobi(n, a, b, x)) == \
r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(n, a, b, x)**2) == \
r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a, x)) == \
r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(n, a, x)**2) == \
r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n, x)**2) == \
r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n, x)**2) == \
r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a, x)) == \
r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(n, a, x)**2) == \
r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a, x)) == \
r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(n, a, x)**2) == \
r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
assert latex(Ynm(n, m, theta, phi)**3) == \
r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
assert latex(Znm(n, m, theta, phi)**3) == \
r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
# Test latex printing of function names with "_"
assert latex(polar_lift(0)) == \
r"\operatorname{polar\_lift}{\left(0 \right)}"
assert latex(polar_lift(0)**3) == \
r"\operatorname{polar\_lift}^{3}{\left(0 \right)}"
assert latex(totient(n)) == r'\phi\left(n\right)'
assert latex(totient(n) ** 2) == r'\left(\phi\left(n\right)\right)^{2}'
assert latex(reduced_totient(n)) == r'\lambda\left(n\right)'
assert latex(reduced_totient(n) ** 2) == \
r'\left(\lambda\left(n\right)\right)^{2}'
assert latex(divisor_sigma(x)) == r"\sigma\left(x\right)"
assert latex(divisor_sigma(x)**2) == r"\sigma^{2}\left(x\right)"
assert latex(divisor_sigma(x, y)) == r"\sigma_y\left(x\right)"
assert latex(divisor_sigma(x, y)**2) == r"\sigma^{2}_y\left(x\right)"
assert latex(udivisor_sigma(x)) == r"\sigma^*\left(x\right)"
assert latex(udivisor_sigma(x)**2) == r"\sigma^*^{2}\left(x\right)"
assert latex(udivisor_sigma(x, y)) == r"\sigma^*_y\left(x\right)"
assert latex(udivisor_sigma(x, y)**2) == r"\sigma^*^{2}_y\left(x\right)"
assert latex(primenu(n)) == r'\nu\left(n\right)'
assert latex(primenu(n) ** 2) == r'\left(\nu\left(n\right)\right)^{2}'
assert latex(primeomega(n)) == r'\Omega\left(n\right)'
assert latex(primeomega(n) ** 2) == \
r'\left(\Omega\left(n\right)\right)^{2}'
assert latex(Mod(x, 7)) == r'x\bmod{7}'
assert latex(Mod(x + 1, 7)) == r'\left(x + 1\right)\bmod{7}'
assert latex(Mod(2 * x, 7)) == r'2 x\bmod{7}'
assert latex(Mod(x, 7) + 1) == r'\left(x\bmod{7}\right) + 1'
assert latex(2 * Mod(x, 7)) == r'2 \left(x\bmod{7}\right)'
# some unknown function name should get rendered with \operatorname
fjlkd = Function('fjlkd')
assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left(x \right)}'
# even when it is referred to without an argument
assert latex(fjlkd) == r'\operatorname{fjlkd}'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert latex(mygamma) == r"\operatorname{mygamma}"
assert latex(mygamma(x)) == r"\operatorname{mygamma}{\left(x \right)}"
def test_hyper_printing():
from sympy import pi
from sympy.abc import x, z
assert latex(meijerg(Tuple(pi, pi, x), Tuple(1),
(0, 1), Tuple(1, 2, 3/pi), z)) == \
r'{G_{4, 5}^{2, 3}\left(\begin{matrix} \pi, \pi, x & 1 \\0, 1 & 1, 2, '\
r'\frac{3}{\pi} \end{matrix} \middle| {z} \right)}'
assert latex(meijerg(Tuple(), Tuple(1), (0,), Tuple(), z)) == \
r'{G_{1, 1}^{1, 0}\left(\begin{matrix} & 1 \\0 & \end{matrix} \middle| {z} \right)}'
assert latex(hyper((x, 2), (3,), z)) == \
r'{{}_{2}F_{1}\left(\begin{matrix} x, 2 ' \
r'\\ 3 \end{matrix}\middle| {z} \right)}'
assert latex(hyper(Tuple(), Tuple(1), z)) == \
r'{{}_{0}F_{1}\left(\begin{matrix} ' \
r'\\ 1 \end{matrix}\middle| {z} \right)}'
def test_latex_bessel():
from sympy.functions.special.bessel import (besselj, bessely, besseli,
besselk, hankel1, hankel2,
jn, yn, hn1, hn2)
from sympy.abc import z
assert latex(besselj(n, z**2)**k) == r'J^{k}_{n}\left(z^{2}\right)'
assert latex(bessely(n, z)) == r'Y_{n}\left(z\right)'
assert latex(besseli(n, z)) == r'I_{n}\left(z\right)'
assert latex(besselk(n, z)) == r'K_{n}\left(z\right)'
assert latex(hankel1(n, z**2)**2) == \
r'\left(H^{(1)}_{n}\left(z^{2}\right)\right)^{2}'
assert latex(hankel2(n, z)) == r'H^{(2)}_{n}\left(z\right)'
assert latex(jn(n, z)) == r'j_{n}\left(z\right)'
assert latex(yn(n, z)) == r'y_{n}\left(z\right)'
assert latex(hn1(n, z)) == r'h^{(1)}_{n}\left(z\right)'
assert latex(hn2(n, z)) == r'h^{(2)}_{n}\left(z\right)'
def test_latex_fresnel():
from sympy.functions.special.error_functions import (fresnels, fresnelc)
from sympy.abc import z
assert latex(fresnels(z)) == r'S\left(z\right)'
assert latex(fresnelc(z)) == r'C\left(z\right)'
assert latex(fresnels(z)**2) == r'S^{2}\left(z\right)'
assert latex(fresnelc(z)**2) == r'C^{2}\left(z\right)'
def test_latex_brackets():
assert latex((-1)**x) == r"\left(-1\right)^{x}"
def test_latex_indexed():
Psi_symbol = Symbol('Psi_0', complex=True, real=False)
Psi_indexed = IndexedBase(Symbol('Psi', complex=True, real=False))
symbol_latex = latex(Psi_symbol * conjugate(Psi_symbol))
indexed_latex = latex(Psi_indexed[0] * conjugate(Psi_indexed[0]))
# \\overline{{\\Psi}_{0}} {\\Psi}_{0} vs. \\Psi_{0} \\overline{\\Psi_{0}}
assert symbol_latex == '\\Psi_{0} \\overline{\\Psi_{0}}'
assert indexed_latex == '\\overline{{\\Psi}_{0}} {\\Psi}_{0}'
# Symbol('gamma') gives r'\gamma'
assert latex(Indexed('x1', Symbol('i'))) == '{x_{1}}_{i}'
assert latex(IndexedBase('gamma')) == r'\gamma'
assert latex(IndexedBase('a b')) == 'a b'
assert latex(IndexedBase('a_b')) == 'a_{b}'
def test_latex_derivatives():
# regular "d" for ordinary derivatives
assert latex(diff(x**3, x, evaluate=False)) == \
r"\frac{d}{d x} x^{3}"
assert latex(diff(sin(x) + x**2, x, evaluate=False)) == \
r"\frac{d}{d x} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False))\
== \
r"\frac{d^{2}}{d x^{2}} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False), evaluate=False)) == \
r"\frac{d^{3}}{d x^{3}} \left(x^{2} + \sin{\left(x \right)}\right)"
# \partial for partial derivatives
assert latex(diff(sin(x * y), x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \sin{\left(x y \right)}"
assert latex(diff(sin(x * y) + x**2, x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial x^{2}} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial x^{3}} \left(x^{2} + \sin{\left(x y \right)}\right)"
# mixed partial derivatives
f = Function("f")
assert latex(diff(diff(f(x, y), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial y\partial x} " + latex(f(x, y))
assert latex(diff(diff(diff(f(x, y), x, evaluate=False), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial y\partial x^{2}} " + latex(f(x, y))
# use ordinary d when one of the variables has been integrated out
assert latex(diff(Integral(exp(-x*y), (x, 0, oo)), y, evaluate=False)) == \
r"\frac{d}{d y} \int\limits_{0}^{\infty} e^{- x y}\, dx"
# Derivative wrapped in power:
assert latex(diff(x, x, evaluate=False)**2) == \
r"\left(\frac{d}{d x} x\right)^{2}"
assert latex(diff(f(x), x)**2) == \
r"\left(\frac{d}{d x} f{\left(x \right)}\right)^{2}"
assert latex(diff(f(x), (x, n))) == \
r"\frac{d^{n}}{d x^{n}} f{\left(x \right)}"
def test_latex_subs():
assert latex(Subs(x*y, (
x, y), (1, 2))) == r'\left. x y \right|_{\substack{ x=1\\ y=2 }}'
def test_latex_integrals():
assert latex(Integral(log(x), x)) == r"\int \log{\left(x \right)}\, dx"
assert latex(Integral(x**2, (x, 0, 1))) == \
r"\int\limits_{0}^{1} x^{2}\, dx"
assert latex(Integral(x**2, (x, 10, 20))) == \
r"\int\limits_{10}^{20} x^{2}\, dx"
assert latex(Integral(y*x**2, (x, 0, 1), y)) == \
r"\int\int\limits_{0}^{1} x^{2} y\, dx\, dy"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*') == \
r"\begin{equation*}\int\int\limits_{0}^{1} x^{2} y\, dx\, dy\end{equation*}"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*', itex=True) \
== r"$$\int\int_{0}^{1} x^{2} y\, dx\, dy$$"
assert latex(Integral(x, (x, 0))) == r"\int\limits^{0} x\, dx"
assert latex(Integral(x*y, x, y)) == r"\iint x y\, dx\, dy"
assert latex(Integral(x*y*z, x, y, z)) == r"\iiint x y z\, dx\, dy\, dz"
assert latex(Integral(x*y*z*t, x, y, z, t)) == \
r"\iiiint t x y z\, dx\, dy\, dz\, dt"
assert latex(Integral(x, x, x, x, x, x, x)) == \
r"\int\int\int\int\int\int x\, dx\, dx\, dx\, dx\, dx\, dx"
assert latex(Integral(x, x, y, (z, 0, 1))) == \
r"\int\limits_{0}^{1}\int\int x\, dx\, dy\, dz"
# fix issue #10806
assert latex(Integral(z, z)**2) == r"\left(\int z\, dz\right)^{2}"
assert latex(Integral(x + z, z)) == r"\int \left(x + z\right)\, dz"
assert latex(Integral(x+z/2, z)) == \
r"\int \left(x + \frac{z}{2}\right)\, dz"
assert latex(Integral(x**y, z)) == r"\int x^{y}\, dz"
def test_latex_sets():
for s in (frozenset, set):
assert latex(s([x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
s = FiniteSet
assert latex(s(*[x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(*range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(*range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
def test_latex_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
assert latex(se) == r"SetExpr\left(\left[1, 3\right]\right)"
def test_latex_Range():
assert latex(Range(1, 51)) == \
r'\left\{1, 2, \ldots, 50\right\}'
assert latex(Range(1, 4)) == r'\left\{1, 2, 3\right\}'
assert latex(Range(0, 3, 1)) == r'\left\{0, 1, 2\right\}'
assert latex(Range(0, 30, 1)) == r'\left\{0, 1, \ldots, 29\right\}'
assert latex(Range(30, 1, -1)) == r'\left\{30, 29, \ldots, 2\right\}'
assert latex(Range(0, oo, 2)) == r'\left\{0, 2, \ldots\right\}'
assert latex(Range(oo, -2, -2)) == r'\left\{\ldots, 2, 0\right\}'
assert latex(Range(-2, -oo, -1)) == \
r'\left\{-2, -3, \ldots\right\}'
def test_latex_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
latex_str = r'\left[0, 1, 4, 9, \ldots\right]'
assert latex(s1) == latex_str
latex_str = r'\left[1, 2, 1, 2, \ldots\right]'
assert latex(s2) == latex_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
latex_str = r'\left[0, 1, 4\right]'
assert latex(s3) == latex_str
latex_str = r'\left[1, 2, 1\right]'
assert latex(s4) == latex_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
latex_str = r'\left[\ldots, 9, 4, 1, 0\right]'
assert latex(s5) == latex_str
latex_str = r'\left[\ldots, 2, 1, 2, 1\right]'
assert latex(s6) == latex_str
latex_str = r'\left[1, 3, 5, 11, \ldots\right]'
assert latex(SeqAdd(s1, s2)) == latex_str
latex_str = r'\left[1, 3, 5\right]'
assert latex(SeqAdd(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 11, 5, 3, 1\right]'
assert latex(SeqAdd(s5, s6)) == latex_str
latex_str = r'\left[0, 2, 4, 18, \ldots\right]'
assert latex(SeqMul(s1, s2)) == latex_str
latex_str = r'\left[0, 2, 4\right]'
assert latex(SeqMul(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 18, 4, 2, 0\right]'
assert latex(SeqMul(s5, s6)) == latex_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
latex_str = r'\left\{a^{2}\right\}_{a=0}^{x}'
assert latex(s7) == latex_str
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
latex_str = r'\left[0, b, 4 b\right]'
assert latex(s8) == latex_str
def test_latex_FourierSeries():
latex_str = \
r'2 \sin{\left(x \right)} - \sin{\left(2 x \right)} + \frac{2 \sin{\left(3 x \right)}}{3} + \ldots'
assert latex(fourier_series(x, (x, -pi, pi))) == latex_str
def test_latex_FormalPowerSeries():
latex_str = r'\sum_{k=1}^{\infty} - \frac{\left(-1\right)^{- k} x^{k}}{k}'
assert latex(fps(log(1 + x))) == latex_str
def test_latex_intervals():
a = Symbol('a', real=True)
assert latex(Interval(0, 0)) == r"\left\{0\right\}"
assert latex(Interval(0, a)) == r"\left[0, a\right]"
assert latex(Interval(0, a, False, False)) == r"\left[0, a\right]"
assert latex(Interval(0, a, True, False)) == r"\left(0, a\right]"
assert latex(Interval(0, a, False, True)) == r"\left[0, a\right)"
assert latex(Interval(0, a, True, True)) == r"\left(0, a\right)"
def test_latex_AccumuBounds():
a = Symbol('a', real=True)
assert latex(AccumBounds(0, 1)) == r"\left\langle 0, 1\right\rangle"
assert latex(AccumBounds(0, a)) == r"\left\langle 0, a\right\rangle"
assert latex(AccumBounds(a + 1, a + 2)) == \
r"\left\langle a + 1, a + 2\right\rangle"
def test_latex_emptyset():
assert latex(S.EmptySet) == r"\emptyset"
def test_latex_commutator():
A = Operator('A')
B = Operator('B')
comm = Commutator(B, A)
assert latex(comm.doit()) == r"- (A B - B A)"
def test_latex_union():
assert latex(Union(Interval(0, 1), Interval(2, 3))) == \
r"\left[0, 1\right] \cup \left[2, 3\right]"
assert latex(Union(Interval(1, 1), Interval(2, 2), Interval(3, 4))) == \
r"\left\{1, 2\right\} \cup \left[3, 4\right]"
def test_latex_intersection():
assert latex(Intersection(Interval(0, 1), Interval(x, y))) == \
r"\left[0, 1\right] \cap \left[x, y\right]"
def test_latex_symmetric_difference():
assert latex(SymmetricDifference(Interval(2, 5), Interval(4, 7),
evaluate=False)) == \
r'\left[2, 5\right] \triangle \left[4, 7\right]'
def test_latex_Complement():
assert latex(Complement(S.Reals, S.Naturals)) == \
r"\mathbb{R} \setminus \mathbb{N}"
def test_latex_Complexes():
assert latex(S.Complexes) == r"\mathbb{C}"
def test_latex_productset():
line = Interval(0, 1)
bigline = Interval(0, 10)
fset = FiniteSet(1, 2, 3)
assert latex(line**2) == r"%s^{2}" % latex(line)
assert latex(line**10) == r"%s^{10}" % latex(line)
assert latex(line * bigline * fset) == r"%s \times %s \times %s" % (
latex(line), latex(bigline), latex(fset))
def test_latex_Naturals():
assert latex(S.Naturals) == r"\mathbb{N}"
def test_latex_Naturals0():
assert latex(S.Naturals0) == r"\mathbb{N}_0"
def test_latex_Integers():
assert latex(S.Integers) == r"\mathbb{Z}"
def test_latex_ImageSet():
x = Symbol('x')
assert latex(ImageSet(Lambda(x, x**2), S.Naturals)) == \
r"\left\{x^{2}\; |\; x \in \mathbb{N}\right\}"
y = Symbol('y')
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
assert latex(imgset) == \
r"\left\{x + y\; |\; x \in \left\{1, 2, 3\right\}, y \in \left\{3, 4\right\}\right\}"
def test_latex_ConditionSet():
x = Symbol('x')
assert latex(ConditionSet(x, Eq(x**2, 1), S.Reals)) == \
r"\left\{x \mid x \in \mathbb{R} \wedge x^{2} = 1 \right\}"
assert latex(ConditionSet(x, Eq(x**2, 1), S.UniversalSet)) == \
r"\left\{x \mid x^{2} = 1 \right\}"
def test_latex_ComplexRegion():
assert latex(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == \
r"\left\{x + y i\; |\; x, y \in \left[3, 5\right] \times \left[4, 6\right] \right\}"
assert latex(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == \
r"\left\{r \left(i \sin{\left(\theta \right)} + \cos{\left(\theta "\
r"\right)}\right)\; |\; r, \theta \in \left[0, 1\right] \times \left[0, 2 \pi\right) \right\}"
def test_latex_Contains():
x = Symbol('x')
assert latex(Contains(x, S.Naturals)) == r"x \in \mathbb{N}"
def test_latex_sum():
assert latex(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\sum_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Sum(x**2, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} x^{2}"
assert latex(Sum(x**2 + y, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Sum(x**2 + y, (x, -2, 2))**2) == \
r"\left(\sum_{x=-2}^{2} \left(x^{2} + y\right)\right)^{2}"
def test_latex_product():
assert latex(Product(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\prod_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Product(x**2, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} x^{2}"
assert latex(Product(x**2 + y, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Product(x, (x, -2, 2))**2) == \
r"\left(\prod_{x=-2}^{2} x\right)^{2}"
def test_latex_limits():
assert latex(Limit(x, x, oo)) == r"\lim_{x \to \infty} x"
# issue 8175
f = Function('f')
assert latex(Limit(f(x), x, 0)) == r"\lim_{x \to 0^+} f{\left(x \right)}"
assert latex(Limit(f(x), x, 0, "-")) == \
r"\lim_{x \to 0^-} f{\left(x \right)}"
# issue #10806
assert latex(Limit(f(x), x, 0)**2) == \
r"\left(\lim_{x \to 0^+} f{\left(x \right)}\right)^{2}"
# bi-directional limit
assert latex(Limit(f(x), x, 0, dir='+-')) == \
r"\lim_{x \to 0} f{\left(x \right)}"
def test_latex_log():
assert latex(log(x)) == r"\log{\left(x \right)}"
assert latex(ln(x)) == r"\log{\left(x \right)}"
assert latex(log(x), ln_notation=True) == r"\ln{\left(x \right)}"
assert latex(log(x)+log(y)) == \
r"\log{\left(x \right)} + \log{\left(y \right)}"
assert latex(log(x)+log(y), ln_notation=True) == \
r"\ln{\left(x \right)} + \ln{\left(y \right)}"
assert latex(pow(log(x), x)) == r"\log{\left(x \right)}^{x}"
assert latex(pow(log(x), x), ln_notation=True) == \
r"\ln{\left(x \right)}^{x}"
def test_issue_3568():
beta = Symbol(r'\beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
beta = Symbol(r'beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
def test_latex():
assert latex((2*tau)**Rational(7, 2)) == "8 \\sqrt{2} \\tau^{\\frac{7}{2}}"
assert latex((2*mu)**Rational(7, 2), mode='equation*') == \
"\\begin{equation*}8 \\sqrt{2} \\mu^{\\frac{7}{2}}\\end{equation*}"
assert latex((2*mu)**Rational(7, 2), mode='equation', itex=True) == \
"$$8 \\sqrt{2} \\mu^{\\frac{7}{2}}$$"
assert latex([2/x, y]) == r"\left[ \frac{2}{x}, \ y\right]"
def test_latex_dict():
d = {Rational(1): 1, x**2: 2, x: 3, x**3: 4}
assert latex(d) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
D = Dict(d)
assert latex(D) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
def test_latex_list():
ll = [Symbol('omega1'), Symbol('a'), Symbol('alpha')]
assert latex(ll) == r'\left[ \omega_{1}, \ a, \ \alpha\right]'
def test_latex_rational():
# tests issue 3973
assert latex(-Rational(1, 2)) == "- \\frac{1}{2}"
assert latex(Rational(-1, 2)) == "- \\frac{1}{2}"
assert latex(Rational(1, -2)) == "- \\frac{1}{2}"
assert latex(-Rational(-1, 2)) == "\\frac{1}{2}"
assert latex(-Rational(1, 2)*x) == "- \\frac{x}{2}"
assert latex(-Rational(1, 2)*x + Rational(-2, 3)*y) == \
"- \\frac{x}{2} - \\frac{2 y}{3}"
def test_latex_inverse():
# tests issue 4129
assert latex(1/x) == "\\frac{1}{x}"
assert latex(1/(x + y)) == "\\frac{1}{x + y}"
def test_latex_DiracDelta():
assert latex(DiracDelta(x)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x)**2) == r"\left(\delta\left(x\right)\right)^{2}"
assert latex(DiracDelta(x, 0)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x, 5)) == \
r"\delta^{\left( 5 \right)}\left( x \right)"
assert latex(DiracDelta(x, 5)**2) == \
r"\left(\delta^{\left( 5 \right)}\left( x \right)\right)^{2}"
def test_latex_Heaviside():
assert latex(Heaviside(x)) == r"\theta\left(x\right)"
assert latex(Heaviside(x)**2) == r"\left(\theta\left(x\right)\right)^{2}"
def test_latex_KroneckerDelta():
assert latex(KroneckerDelta(x, y)) == r"\delta_{x y}"
assert latex(KroneckerDelta(x, y + 1)) == r"\delta_{x, y + 1}"
# issue 6578
assert latex(KroneckerDelta(x + 1, y)) == r"\delta_{y, x + 1}"
assert latex(Pow(KroneckerDelta(x, y), 2, evaluate=False)) == \
r"\left(\delta_{x y}\right)^{2}"
def test_latex_LeviCivita():
assert latex(LeviCivita(x, y, z)) == r"\varepsilon_{x y z}"
assert latex(LeviCivita(x, y, z)**2) == \
r"\left(\varepsilon_{x y z}\right)^{2}"
assert latex(LeviCivita(x, y, z + 1)) == r"\varepsilon_{x, y, z + 1}"
assert latex(LeviCivita(x, y + 1, z)) == r"\varepsilon_{x, y + 1, z}"
assert latex(LeviCivita(x + 1, y, z)) == r"\varepsilon_{x + 1, y, z}"
def test_mode():
expr = x + y
assert latex(expr) == 'x + y'
assert latex(expr, mode='plain') == 'x + y'
assert latex(expr, mode='inline') == '$x + y$'
assert latex(
expr, mode='equation*') == '\\begin{equation*}x + y\\end{equation*}'
assert latex(
expr, mode='equation') == '\\begin{equation}x + y\\end{equation}'
raises(ValueError, lambda: latex(expr, mode='foo'))
def test_latex_Piecewise():
p = Piecewise((x, x < 1), (x**2, True))
assert latex(p) == "\\begin{cases} x & \\text{for}\\: x < 1 \\\\x^{2} &" \
" \\text{otherwise} \\end{cases}"
assert latex(p, itex=True) == \
"\\begin{cases} x & \\text{for}\\: x \\lt 1 \\\\x^{2} &" \
" \\text{otherwise} \\end{cases}"
p = Piecewise((x, x < 0), (0, x >= 0))
assert latex(p) == '\\begin{cases} x & \\text{for}\\: x < 0 \\\\0 &' \
' \\text{otherwise} \\end{cases}'
A, B = symbols("A B", commutative=False)
p = Piecewise((A**2, Eq(A, B)), (A*B, True))
s = r"\begin{cases} A^{2} & \text{for}\: A = B \\A B & \text{otherwise} \end{cases}"
assert latex(p) == s
assert latex(A*p) == r"A \left(%s\right)" % s
assert latex(p*A) == r"\left(%s\right) A" % s
assert latex(Piecewise((x, x < 1), (x**2, x < 2))) == \
'\\begin{cases} x & ' \
'\\text{for}\\: x < 1 \\\\x^{2} & \\text{for}\\: x < 2 \\end{cases}'
def test_latex_Matrix():
M = Matrix([[1 + x, y], [y, x - 1]])
assert latex(M) == \
r'\left[\begin{matrix}x + 1 & y\\y & x - 1\end{matrix}\right]'
assert latex(M, mode='inline') == \
r'$\left[\begin{smallmatrix}x + 1 & y\\' \
r'y & x - 1\end{smallmatrix}\right]$'
assert latex(M, mat_str='array') == \
r'\left[\begin{array}{cc}x + 1 & y\\y & x - 1\end{array}\right]'
assert latex(M, mat_str='bmatrix') == \
r'\left[\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}\right]'
assert latex(M, mat_delim=None, mat_str='bmatrix') == \
r'\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}'
M2 = Matrix(1, 11, range(11))
assert latex(M2) == \
r'\left[\begin{array}{ccccccccccc}' \
r'0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}\right]'
def test_latex_matrix_with_functions():
t = symbols('t')
theta1 = symbols('theta1', cls=Function)
M = Matrix([[sin(theta1(t)), cos(theta1(t))],
[cos(theta1(t).diff(t)), sin(theta1(t).diff(t))]])
expected = (r'\left[\begin{matrix}\sin{\left('
r'\theta_{1}{\left(t \right)} \right)} & '
r'\cos{\left(\theta_{1}{\left(t \right)} \right)'
r'}\\\cos{\left(\frac{d}{d t} \theta_{1}{\left(t '
r'\right)} \right)} & \sin{\left(\frac{d}{d t} '
r'\theta_{1}{\left(t \right)} \right'
r')}\end{matrix}\right]')
assert latex(M) == expected
def test_latex_NDimArray():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert latex(M) == "x"
M = ArrayType([[1 / x, y], [z, w]])
M1 = ArrayType([1 / x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
assert latex(M) == \
'\\left[\\begin{matrix}\\frac{1}{x} & y\\\\z & w\\end{matrix}\\right]'
assert latex(M1) == \
"\\left[\\begin{matrix}\\frac{1}{x} & y & z\\end{matrix}\\right]"
assert latex(M2) == \
r"\left[\begin{matrix}" \
r"\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right]" \
r"\end{matrix}\right]"
assert latex(M3) == \
r"""\left[\begin{matrix}"""\
r"""\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right]\\"""\
r"""\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{w}{x} & w y\\w z & w^{2}\end{matrix}\right]"""\
r"""\end{matrix}\right]"""
Mrow = ArrayType([[x, y, 1/z]])
Mcolumn = ArrayType([[x], [y], [1/z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
assert latex(Mrow) == \
r"\left[\left[\begin{matrix}x & y & \frac{1}{z}\end{matrix}\right]\right]"
assert latex(Mcolumn) == \
r"\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]"
assert latex(Mcol2) == \
r'\left[\begin{matrix}\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]\end{matrix}\right]'
def test_latex_mul_symbol():
assert latex(4*4**x, mul_symbol='times') == "4 \\times 4^{x}"
assert latex(4*4**x, mul_symbol='dot') == "4 \\cdot 4^{x}"
assert latex(4*4**x, mul_symbol='ldot') == r"4 \,.\, 4^{x}"
assert latex(4*x, mul_symbol='times') == "4 \\times x"
assert latex(4*x, mul_symbol='dot') == "4 \\cdot x"
assert latex(4*x, mul_symbol='ldot') == r"4 \,.\, x"
def test_latex_issue_4381():
y = 4*4**log(2)
assert latex(y) == r'4 \cdot 4^{\log{\left(2 \right)}}'
assert latex(1/y) == r'\frac{1}{4 \cdot 4^{\log{\left(2 \right)}}}'
def test_latex_issue_4576():
assert latex(Symbol("beta_13_2")) == r"\beta_{13 2}"
assert latex(Symbol("beta_132_20")) == r"\beta_{132 20}"
assert latex(Symbol("beta_13")) == r"\beta_{13}"
assert latex(Symbol("x_a_b")) == r"x_{a b}"
assert latex(Symbol("x_1_2_3")) == r"x_{1 2 3}"
assert latex(Symbol("x_a_b1")) == r"x_{a b1}"
assert latex(Symbol("x_a_1")) == r"x_{a 1}"
assert latex(Symbol("x_1_a")) == r"x_{1 a}"
assert latex(Symbol("x_1^aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_1__aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_11^a")) == r"x^{a}_{11}"
assert latex(Symbol("x_11__a")) == r"x^{a}_{11}"
assert latex(Symbol("x_a_a_a_a")) == r"x_{a a a a}"
assert latex(Symbol("x_a_a^a^a")) == r"x^{a a}_{a a}"
assert latex(Symbol("x_a_a__a__a")) == r"x^{a a}_{a a}"
assert latex(Symbol("alpha_11")) == r"\alpha_{11}"
assert latex(Symbol("alpha_11_11")) == r"\alpha_{11 11}"
assert latex(Symbol("alpha_alpha")) == r"\alpha_{\alpha}"
assert latex(Symbol("alpha^aleph")) == r"\alpha^{\aleph}"
assert latex(Symbol("alpha__aleph")) == r"\alpha^{\aleph}"
def test_latex_pow_fraction():
x = Symbol('x')
# Testing exp
assert 'e^{-x}' in latex(exp(-x)/2).replace(' ', '') # Remove Whitespace
# Testing e^{-x} in case future changes alter behavior of muls or fracs
# In particular current output is \frac{1}{2}e^{- x} but perhaps this will
# change to \frac{e^{-x}}{2}
# Testing general, non-exp, power
assert '3^{-x}' in latex(3**-x/2).replace(' ', '')
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
assert latex(A*B*C**-1) == "A B C^{-1}"
assert latex(C**-1*A*B) == "C^{-1} A B"
assert latex(A*C**-1*B) == "A C^{-1} B"
def test_latex_order():
expr = x**3 + x**2*y + y**4 + 3*x*y**3
assert latex(expr, order='lex') == "x^{3} + x^{2} y + 3 x y^{3} + y^{4}"
assert latex(
expr, order='rev-lex') == "y^{4} + 3 x y^{3} + x^{2} y + x^{3}"
assert latex(expr, order='none') == "x^{3} + y^{4} + y x^{2} + 3 x y^{3}"
def test_latex_Lambda():
assert latex(Lambda(x, x + 1)) == \
r"\left( x \mapsto x + 1 \right)"
assert latex(Lambda((x, y), x + 1)) == \
r"\left( \left( x, \ y\right) \mapsto x + 1 \right)"
def test_latex_PolyElement():
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + u + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + u + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x + 1"
assert latex((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == \
r"-\left({u}^{2} - 3 u v + 1\right) {x}^{2} y - \left(u + 1\right) x - 1"
assert latex(-(v**2 + v + 1)*x + 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x + 3 u v + 1"
assert latex(-(v**2 + v + 1)*x - 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x - 3 u v + 1"
def test_latex_FracElement():
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex(x/3) == r"\frac{x}{3}"
assert latex(x/z) == r"\frac{x}{z}"
assert latex(x*y/z) == r"\frac{x y}{z}"
assert latex(x/(z*t)) == r"\frac{x}{z t}"
assert latex(x*y/(z*t)) == r"\frac{x y}{z t}"
assert latex((x - 1)/y) == r"\frac{x - 1}{y}"
assert latex((x + 1)/y) == r"\frac{x + 1}{y}"
assert latex((-x - 1)/y) == r"\frac{-x - 1}{y}"
assert latex((x + 1)/(y*z)) == r"\frac{x + 1}{y z}"
assert latex(-y/(x + 1)) == r"\frac{-y}{x + 1}"
assert latex(y*z/(x + 1)) == r"\frac{y z}{x + 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - u v t - 1}"
def test_latex_Poly():
assert latex(Poly(x**2 + 2 * x, x)) == \
r"\operatorname{Poly}{\left( x^{2} + 2 x, x, domain=\mathbb{Z} \right)}"
assert latex(Poly(x/y, x)) == \
r"\operatorname{Poly}{\left( \frac{1}{y} x, x, domain=\mathbb{Z}\left(y\right) \right)}"
assert latex(Poly(2.0*x + y)) == \
r"\operatorname{Poly}{\left( 2.0 x + 1.0 y, x, y, domain=\mathbb{R} \right)}"
def test_latex_Poly_order():
assert latex(Poly([a, 1, b, 2, c, 3], x)) == \
'\\operatorname{Poly}{\\left( a x^{5} + x^{4} + b x^{3} + 2 x^{2} + c'\
' x + 3, x, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
assert latex(Poly([a, 1, b+c, 2, 3], x)) == \
'\\operatorname{Poly}{\\left( a x^{4} + x^{3} + \\left(b + c\\right) '\
'x^{2} + 2 x + 3, x, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
assert latex(Poly(a*x**3 + x**2*y - x*y - c*y**3 - b*x*y**2 + y - a*x + b,
(x, y))) == \
'\\operatorname{Poly}{\\left( a x^{3} + x^{2}y - b xy^{2} - xy - '\
'a x - c y^{3} + y + b, x, y, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
def test_latex_ComplexRootOf():
assert latex(rootof(x**5 + x + 3, 0)) == \
r"\operatorname{CRootOf} {\left(x^{5} + x + 3, 0\right)}"
def test_latex_RootSum():
assert latex(RootSum(x**5 + x + 3, sin)) == \
r"\operatorname{RootSum} {\left(x^{5} + x + 3, \left( x \mapsto \sin{\left(x \right)} \right)\right)}"
def test_settings():
raises(TypeError, lambda: latex(x*y, method="garbage"))
def test_latex_numbers():
assert latex(catalan(n)) == r"C_{n}"
assert latex(catalan(n)**2) == r"C_{n}^{2}"
assert latex(bernoulli(n)) == r"B_{n}"
assert latex(bernoulli(n)**2) == r"B_{n}^{2}"
assert latex(bell(n)) == r"B_{n}"
assert latex(bell(n)**2) == r"B_{n}^{2}"
assert latex(fibonacci(n)) == r"F_{n}"
assert latex(fibonacci(n)**2) == r"F_{n}^{2}"
assert latex(lucas(n)) == r"L_{n}"
assert latex(lucas(n)**2) == r"L_{n}^{2}"
assert latex(tribonacci(n)) == r"T_{n}"
assert latex(tribonacci(n)**2) == r"T_{n}^{2}"
def test_latex_euler():
assert latex(euler(n)) == r"E_{n}"
assert latex(euler(n, x)) == r"E_{n}\left(x\right)"
assert latex(euler(n, x)**2) == r"E_{n}^{2}\left(x\right)"
def test_lamda():
assert latex(Symbol('lamda')) == r"\lambda"
assert latex(Symbol('Lamda')) == r"\Lambda"
def test_custom_symbol_names():
x = Symbol('x')
y = Symbol('y')
assert latex(x) == "x"
assert latex(x, symbol_names={x: "x_i"}) == "x_i"
assert latex(x + y, symbol_names={x: "x_i"}) == "x_i + y"
assert latex(x**2, symbol_names={x: "x_i"}) == "x_i^{2}"
assert latex(x + y, symbol_names={x: "x_i", y: "y_j"}) == "x_i + y_j"
def test_matAdd():
from sympy import MatrixSymbol
from sympy.printing.latex import LatexPrinter
C = MatrixSymbol('C', 5, 5)
B = MatrixSymbol('B', 5, 5)
l = LatexPrinter()
assert l._print(C - 2*B) in ['- 2 B + C', 'C -2 B']
assert l._print(C + 2*B) in ['2 B + C', 'C + 2 B']
assert l._print(B - 2*C) in ['B - 2 C', '- 2 C + B']
assert l._print(B + 2*C) in ['B + 2 C', '2 C + B']
def test_matMul():
from sympy import MatrixSymbol
from sympy.printing.latex import LatexPrinter
A = MatrixSymbol('A', 5, 5)
B = MatrixSymbol('B', 5, 5)
x = Symbol('x')
lp = LatexPrinter()
assert lp._print_MatMul(2*A) == '2 A'
assert lp._print_MatMul(2*x*A) == '2 x A'
assert lp._print_MatMul(-2*A) == '- 2 A'
assert lp._print_MatMul(1.5*A) == '1.5 A'
assert lp._print_MatMul(sqrt(2)*A) == r'\sqrt{2} A'
assert lp._print_MatMul(-sqrt(2)*A) == r'- \sqrt{2} A'
assert lp._print_MatMul(2*sqrt(2)*x*A) == r'2 \sqrt{2} x A'
assert lp._print_MatMul(-2*A*(A + 2*B)) in [r'- 2 A \left(A + 2 B\right)',
r'- 2 A \left(2 B + A\right)']
def test_latex_MatrixSlice():
from sympy.matrices.expressions import MatrixSymbol
assert latex(MatrixSymbol('X', 10, 10)[:5, 1:9:2]) == \
r'X\left[:5, 1:9:2\right]'
assert latex(MatrixSymbol('X', 10, 10)[5, :5:2]) == \
r'X\left[5, :5:2\right]'
def test_latex_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
from sympy.stats.rv import RandomDomain
X = Normal('x1', 0, 1)
assert latex(where(X > 0)) == r"\text{Domain: }0 < x_{1} \wedge x_{1} < \infty"
D = Die('d1', 6)
assert latex(where(D > 4)) == r"\text{Domain: }d_{1} = 5 \vee d_{1} = 6"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert latex(
pspace(Tuple(A, B)).domain) == \
r"\text{Domain: }0 \leq a \wedge 0 \leq b \wedge a < \infty \wedge b < \infty"
assert latex(RandomDomain(FiniteSet(x), FiniteSet(1, 2))) == \
r'\text{Domain: }\left\{x\right\}\text{ in }\left\{1, 2\right\}'
def test_PrettyPoly():
from sympy.polys.domains import QQ
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert latex(F.convert(x/(x + y))) == latex(x/(x + y))
assert latex(R.convert(x + y)) == latex(x + y)
def test_integral_transforms():
x = Symbol("x")
k = Symbol("k")
f = Function("f")
a = Symbol("a")
b = Symbol("b")
assert latex(MellinTransform(f(x), x, k)) == \
r"\mathcal{M}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseMellinTransform(f(k), k, x, a, b)) == \
r"\mathcal{M}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(LaplaceTransform(f(x), x, k)) == \
r"\mathcal{L}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseLaplaceTransform(f(k), k, x, (a, b))) == \
r"\mathcal{L}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(FourierTransform(f(x), x, k)) == \
r"\mathcal{F}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseFourierTransform(f(k), k, x)) == \
r"\mathcal{F}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(CosineTransform(f(x), x, k)) == \
r"\mathcal{COS}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseCosineTransform(f(k), k, x)) == \
r"\mathcal{COS}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(SineTransform(f(x), x, k)) == \
r"\mathcal{SIN}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseSineTransform(f(k), k, x)) == \
r"\mathcal{SIN}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
def test_PolynomialRingBase():
from sympy.polys.domains import QQ
assert latex(QQ.old_poly_ring(x, y)) == r"\mathbb{Q}\left[x, y\right]"
assert latex(QQ.old_poly_ring(x, y, order="ilex")) == \
r"S_<^{-1}\mathbb{Q}\left[x, y\right]"
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram,
DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert latex(A1) == "A_{1}"
assert latex(f1) == "f_{1}:A_{1}\\rightarrow A_{2}"
assert latex(id_A1) == "id:A_{1}\\rightarrow A_{1}"
assert latex(f2*f1) == "f_{2}\\circ f_{1}:A_{1}\\rightarrow A_{3}"
assert latex(K1) == r"\mathbf{K_{1}}"
d = Diagram()
assert latex(d) == r"\emptyset"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}, " \
r"\ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}"
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}," \
r" \ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}" \
r"\Longrightarrow \left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \left\{unique\right\}\right\}"
# A linear diagram.
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d = Diagram([f, g])
grid = DiagramGrid(d)
assert latex(grid) == "\\begin{array}{cc}\n" \
"A & B \\\\\n" \
" & C \n" \
"\\end{array}\n"
def test_Modules():
from sympy.polys.domains import QQ
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
assert latex(F) == r"{\mathbb{Q}\left[x, y\right]}^{2}"
assert latex(M) == \
r"\left\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle"
I = R.ideal(x**2, y)
assert latex(I) == r"\left\langle {x^{2}},{y} \right\rangle"
Q = F / M
assert latex(Q) == \
r"\frac{{\mathbb{Q}\left[x, y\right]}^{2}}{\left\langle {\left[ {x},"\
r"{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}"
assert latex(Q.submodule([1, x**3/2], [2, y])) == \
r"\left\langle {{\left[ {1},{\frac{x^{3}}{2}} \right]} + {\left"\
r"\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} "\
r"\right\rangle}},{{\left[ {2},{y} \right]} + {\left\langle {\left[ "\
r"{x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}} \right\rangle"
h = homomorphism(QQ.old_poly_ring(x).free_module(2),
QQ.old_poly_ring(x).free_module(2), [0, 0])
assert latex(h) == \
r"{\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]} : "\
r"{{\mathbb{Q}\left[x\right]}^{2}} \to {{\mathbb{Q}\left[x\right]}^{2}}"
def test_QuotientRing():
from sympy.polys.domains import QQ
R = QQ.old_poly_ring(x)/[x**2 + 1]
assert latex(R) == \
r"\frac{\mathbb{Q}\left[x\right]}{\left\langle {x^{2} + 1} \right\rangle}"
assert latex(R.one) == r"{1} + {\left\langle {x^{2} + 1} \right\rangle}"
def test_Tr():
#TODO: Handle indices
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert latex(t) == r'\operatorname{tr}\left(A B\right)'
def test_Adjoint():
from sympy.matrices import MatrixSymbol, Adjoint, Inverse, Transpose
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(Adjoint(X)) == r'X^{\dagger}'
assert latex(Adjoint(X + Y)) == r'\left(X + Y\right)^{\dagger}'
assert latex(Adjoint(X) + Adjoint(Y)) == r'X^{\dagger} + Y^{\dagger}'
assert latex(Adjoint(X*Y)) == r'\left(X Y\right)^{\dagger}'
assert latex(Adjoint(Y)*Adjoint(X)) == r'Y^{\dagger} X^{\dagger}'
assert latex(Adjoint(X**2)) == r'\left(X^{2}\right)^{\dagger}'
assert latex(Adjoint(X)**2) == r'\left(X^{\dagger}\right)^{2}'
assert latex(Adjoint(Inverse(X))) == r'\left(X^{-1}\right)^{\dagger}'
assert latex(Inverse(Adjoint(X))) == r'\left(X^{\dagger}\right)^{-1}'
assert latex(Adjoint(Transpose(X))) == r'\left(X^{T}\right)^{\dagger}'
assert latex(Transpose(Adjoint(X))) == r'\left(X^{\dagger}\right)^{T}'
assert latex(Transpose(Adjoint(X) + Y)) == r'\left(X^{\dagger} + Y\right)^{T}'
assert latex(Transpose(X)) == r'X^{T}'
assert latex(Transpose(X + Y)) == r'\left(X + Y\right)^{T}'
def test_Hadamard():
from sympy.matrices import MatrixSymbol, HadamardProduct
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(HadamardProduct(X, Y*Y)) == r'X \circ Y^{2}'
assert latex(HadamardProduct(X, Y)*Y) == r'\left(X \circ Y\right) Y'
def test_ZeroMatrix():
from sympy import ZeroMatrix
assert latex(ZeroMatrix(1, 1)) == r"\mathbb{0}"
def test_Identity():
from sympy import Identity
assert latex(Identity(1)) == r"\mathbb{I}"
def test_boolean_args_order():
syms = symbols('a:f')
expr = And(*syms)
assert latex(expr) == 'a \\wedge b \\wedge c \\wedge d \\wedge e \\wedge f'
expr = Or(*syms)
assert latex(expr) == 'a \\vee b \\vee c \\vee d \\vee e \\vee f'
expr = Equivalent(*syms)
assert latex(expr) == \
'a \\Leftrightarrow b \\Leftrightarrow c \\Leftrightarrow d \\Leftrightarrow e \\Leftrightarrow f'
expr = Xor(*syms)
assert latex(expr) == \
'a \\veebar b \\veebar c \\veebar d \\veebar e \\veebar f'
def test_imaginary():
i = sqrt(-1)
assert latex(i) == r'i'
def test_builtins_without_args():
assert latex(sin) == r'\sin'
assert latex(cos) == r'\cos'
assert latex(tan) == r'\tan'
assert latex(log) == r'\log'
assert latex(Ei) == r'\operatorname{Ei}'
assert latex(zeta) == r'\zeta'
def test_latex_greek_functions():
# bug because capital greeks that have roman equivalents should not use
# \Alpha, \Beta, \Eta, etc.
s = Function('Alpha')
assert latex(s) == r'A'
assert latex(s(x)) == r'A{\left(x \right)}'
s = Function('Beta')
assert latex(s) == r'B'
s = Function('Eta')
assert latex(s) == r'H'
assert latex(s(x)) == r'H{\left(x \right)}'
# bug because sympy.core.numbers.Pi is special
p = Function('Pi')
# assert latex(p(x)) == r'\Pi{\left(x \right)}'
assert latex(p) == r'\Pi'
# bug because not all greeks are included
c = Function('chi')
assert latex(c(x)) == r'\chi{\left(x \right)}'
assert latex(c) == r'\chi'
def test_translate():
s = 'Alpha'
assert translate(s) == 'A'
s = 'Beta'
assert translate(s) == 'B'
s = 'Eta'
assert translate(s) == 'H'
s = 'omicron'
assert translate(s) == 'o'
s = 'Pi'
assert translate(s) == r'\Pi'
s = 'pi'
assert translate(s) == r'\pi'
s = 'LamdaHatDOT'
assert translate(s) == r'\dot{\hat{\Lambda}}'
def test_other_symbols():
from sympy.printing.latex import other_symbols
for s in other_symbols:
assert latex(symbols(s)) == "\\"+s
def test_modifiers():
# Test each modifier individually in the simplest case
# (with funny capitalizations)
assert latex(symbols("xMathring")) == r"\mathring{x}"
assert latex(symbols("xCheck")) == r"\check{x}"
assert latex(symbols("xBreve")) == r"\breve{x}"
assert latex(symbols("xAcute")) == r"\acute{x}"
assert latex(symbols("xGrave")) == r"\grave{x}"
assert latex(symbols("xTilde")) == r"\tilde{x}"
assert latex(symbols("xPrime")) == r"{x}'"
assert latex(symbols("xddDDot")) == r"\ddddot{x}"
assert latex(symbols("xDdDot")) == r"\dddot{x}"
assert latex(symbols("xDDot")) == r"\ddot{x}"
assert latex(symbols("xBold")) == r"\boldsymbol{x}"
assert latex(symbols("xnOrM")) == r"\left\|{x}\right\|"
assert latex(symbols("xAVG")) == r"\left\langle{x}\right\rangle"
assert latex(symbols("xHat")) == r"\hat{x}"
assert latex(symbols("xDot")) == r"\dot{x}"
assert latex(symbols("xBar")) == r"\bar{x}"
assert latex(symbols("xVec")) == r"\vec{x}"
assert latex(symbols("xAbs")) == r"\left|{x}\right|"
assert latex(symbols("xMag")) == r"\left|{x}\right|"
assert latex(symbols("xPrM")) == r"{x}'"
assert latex(symbols("xBM")) == r"\boldsymbol{x}"
# Test strings that are *only* the names of modifiers
assert latex(symbols("Mathring")) == r"Mathring"
assert latex(symbols("Check")) == r"Check"
assert latex(symbols("Breve")) == r"Breve"
assert latex(symbols("Acute")) == r"Acute"
assert latex(symbols("Grave")) == r"Grave"
assert latex(symbols("Tilde")) == r"Tilde"
assert latex(symbols("Prime")) == r"Prime"
assert latex(symbols("DDot")) == r"\dot{D}"
assert latex(symbols("Bold")) == r"Bold"
assert latex(symbols("NORm")) == r"NORm"
assert latex(symbols("AVG")) == r"AVG"
assert latex(symbols("Hat")) == r"Hat"
assert latex(symbols("Dot")) == r"Dot"
assert latex(symbols("Bar")) == r"Bar"
assert latex(symbols("Vec")) == r"Vec"
assert latex(symbols("Abs")) == r"Abs"
assert latex(symbols("Mag")) == r"Mag"
assert latex(symbols("PrM")) == r"PrM"
assert latex(symbols("BM")) == r"BM"
assert latex(symbols("hbar")) == r"\hbar"
# Check a few combinations
assert latex(symbols("xvecdot")) == r"\dot{\vec{x}}"
assert latex(symbols("xDotVec")) == r"\vec{\dot{x}}"
assert latex(symbols("xHATNorm")) == r"\left\|{\hat{x}}\right\|"
# Check a couple big, ugly combinations
assert latex(symbols('xMathringBm_yCheckPRM__zbreveAbs')) == \
r"\boldsymbol{\mathring{x}}^{\left|{\breve{z}}\right|}_{{\check{y}}'}"
assert latex(symbols('alphadothat_nVECDOT__tTildePrime')) == \
r"\hat{\dot{\alpha}}^{{\tilde{t}}'}_{\dot{\vec{n}}}"
def test_greek_symbols():
assert latex(Symbol('alpha')) == r'\alpha'
assert latex(Symbol('beta')) == r'\beta'
assert latex(Symbol('gamma')) == r'\gamma'
assert latex(Symbol('delta')) == r'\delta'
assert latex(Symbol('epsilon')) == r'\epsilon'
assert latex(Symbol('zeta')) == r'\zeta'
assert latex(Symbol('eta')) == r'\eta'
assert latex(Symbol('theta')) == r'\theta'
assert latex(Symbol('iota')) == r'\iota'
assert latex(Symbol('kappa')) == r'\kappa'
assert latex(Symbol('lambda')) == r'\lambda'
assert latex(Symbol('mu')) == r'\mu'
assert latex(Symbol('nu')) == r'\nu'
assert latex(Symbol('xi')) == r'\xi'
assert latex(Symbol('omicron')) == r'o'
assert latex(Symbol('pi')) == r'\pi'
assert latex(Symbol('rho')) == r'\rho'
assert latex(Symbol('sigma')) == r'\sigma'
assert latex(Symbol('tau')) == r'\tau'
assert latex(Symbol('upsilon')) == r'\upsilon'
assert latex(Symbol('phi')) == r'\phi'
assert latex(Symbol('chi')) == r'\chi'
assert latex(Symbol('psi')) == r'\psi'
assert latex(Symbol('omega')) == r'\omega'
assert latex(Symbol('Alpha')) == r'A'
assert latex(Symbol('Beta')) == r'B'
assert latex(Symbol('Gamma')) == r'\Gamma'
assert latex(Symbol('Delta')) == r'\Delta'
assert latex(Symbol('Epsilon')) == r'E'
assert latex(Symbol('Zeta')) == r'Z'
assert latex(Symbol('Eta')) == r'H'
assert latex(Symbol('Theta')) == r'\Theta'
assert latex(Symbol('Iota')) == r'I'
assert latex(Symbol('Kappa')) == r'K'
assert latex(Symbol('Lambda')) == r'\Lambda'
assert latex(Symbol('Mu')) == r'M'
assert latex(Symbol('Nu')) == r'N'
assert latex(Symbol('Xi')) == r'\Xi'
assert latex(Symbol('Omicron')) == r'O'
assert latex(Symbol('Pi')) == r'\Pi'
assert latex(Symbol('Rho')) == r'P'
assert latex(Symbol('Sigma')) == r'\Sigma'
assert latex(Symbol('Tau')) == r'T'
assert latex(Symbol('Upsilon')) == r'\Upsilon'
assert latex(Symbol('Phi')) == r'\Phi'
assert latex(Symbol('Chi')) == r'X'
assert latex(Symbol('Psi')) == r'\Psi'
assert latex(Symbol('Omega')) == r'\Omega'
assert latex(Symbol('varepsilon')) == r'\varepsilon'
assert latex(Symbol('varkappa')) == r'\varkappa'
assert latex(Symbol('varphi')) == r'\varphi'
assert latex(Symbol('varpi')) == r'\varpi'
assert latex(Symbol('varrho')) == r'\varrho'
assert latex(Symbol('varsigma')) == r'\varsigma'
assert latex(Symbol('vartheta')) == r'\vartheta'
@XFAIL
def test_builtin_without_args_mismatched_names():
assert latex(CosineTransform) == r'\mathcal{COS}'
def test_builtin_no_args():
assert latex(Chi) == r'\operatorname{Chi}'
assert latex(beta) == r'\operatorname{B}'
assert latex(gamma) == r'\Gamma'
assert latex(KroneckerDelta) == r'\delta'
assert latex(DiracDelta) == r'\delta'
assert latex(lowergamma) == r'\gamma'
def test_issue_6853():
p = Function('Pi')
assert latex(p(x)) == r"\Pi{\left(x \right)}"
def test_Mul():
e = Mul(-2, x + 1, evaluate=False)
assert latex(e) == r'- 2 \left(x + 1\right)'
e = Mul(2, x + 1, evaluate=False)
assert latex(e) == r'2 \left(x + 1\right)'
e = Mul(S.One/2, x + 1, evaluate=False)
assert latex(e) == r'\frac{x + 1}{2}'
e = Mul(y, x + 1, evaluate=False)
assert latex(e) == r'y \left(x + 1\right)'
e = Mul(-y, x + 1, evaluate=False)
assert latex(e) == r'- y \left(x + 1\right)'
e = Mul(-2, x + 1)
assert latex(e) == r'- 2 x - 2'
e = Mul(2, x + 1)
assert latex(e) == r'2 x + 2'
def test_Pow():
e = Pow(2, 2, evaluate=False)
assert latex(e) == r'2^{2}'
assert latex(x**(Rational(-1, 3))) == r'\frac{1}{\sqrt[3]{x}}'
x2 = Symbol(r'x^2')
assert latex(x2**2) == r'\left(x^{2}\right)^{2}'
def test_issue_7180():
assert latex(Equivalent(x, y)) == r"x \Leftrightarrow y"
assert latex(Not(Equivalent(x, y))) == r"x \not\Leftrightarrow y"
def test_issue_8409():
assert latex(S.Half**n) == r"\left(\frac{1}{2}\right)^{n}"
def test_issue_8470():
from sympy.parsing.sympy_parser import parse_expr
e = parse_expr("-B*A", evaluate=False)
assert latex(e) == r"A \left(- B\right)"
def test_issue_7117():
# See also issue #5031 (hence the evaluate=False in these).
e = Eq(x + 1, 2*x)
q = Mul(2, e, evaluate=False)
assert latex(q) == r"2 \left(x + 1 = 2 x\right)"
q = Add(6, e, evaluate=False)
assert latex(q) == r"6 + \left(x + 1 = 2 x\right)"
q = Pow(e, 2, evaluate=False)
assert latex(q) == r"\left(x + 1 = 2 x\right)^{2}"
def test_issue_15439():
x = MatrixSymbol('x', 2, 2)
y = MatrixSymbol('y', 2, 2)
assert latex((x * y).subs(y, -y)) == r"x \left(- y\right)"
assert latex((x * y).subs(y, -2*y)) == r"x \left(- 2 y\right)"
assert latex((x * y).subs(x, -x)) == r"- x y"
def test_issue_2934():
assert latex(Symbol(r'\frac{a_1}{b_1}')) == '\\frac{a_1}{b_1}'
def test_issue_10489():
latexSymbolWithBrace = 'C_{x_{0}}'
s = Symbol(latexSymbolWithBrace)
assert latex(s) == latexSymbolWithBrace
assert latex(cos(s)) == r'\cos{\left(C_{x_{0}} \right)}'
def test_issue_12886():
m__1, l__1 = symbols('m__1, l__1')
assert latex(m__1**2 + l__1**2) == \
r'\left(l^{1}\right)^{2} + \left(m^{1}\right)^{2}'
def test_issue_13559():
from sympy.parsing.sympy_parser import parse_expr
expr = parse_expr('5/1', evaluate=False)
assert latex(expr) == r"\frac{5}{1}"
def test_issue_13651():
expr = c + Mul(-1, a + b, evaluate=False)
assert latex(expr) == r"c - \left(a + b\right)"
def test_latex_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
assert latex(he) == latex(1/x) == r"\frac{1}{x}"
assert latex(he**2) == r"\left(\frac{1}{x}\right)^{2}"
assert latex(he + 1) == r"1 + \frac{1}{x}"
assert latex(x*he) == r"x \frac{1}{x}"
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert latex(A[0, 0]) == r"A_{0, 0}"
assert latex(3 * A[0, 0]) == r"3 A_{0, 0}"
F = C[0, 0].subs(C, A - B)
assert latex(F) == r"\left(A - B\right)_{0, 0}"
i, j, k = symbols("i j k")
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
assert latex((M*N)[i, j]) == \
r'\sum_{i_{1}=0}^{k - 1} M_{i, i_{1}} N_{i_{1}, j}'
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A) == r"- A"
assert latex(A - A*B - B) == r"A - A B - B"
assert latex(-A*B - A*B*C - B) == r"- A B - A B C - B"
def test_KroneckerProduct_printing():
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 2, 2)
assert latex(KroneckerProduct(A, B)) == r'A \otimes B'
def test_Quaternion_latex_printing():
q = Quaternion(x, y, z, t)
assert latex(q) == "x + y i + z j + t k"
q = Quaternion(x, y, z, x*t)
assert latex(q) == "x + y i + z j + t x k"
q = Quaternion(x, y, z, x + t)
assert latex(q) == r"x + y i + z j + \left(t + x\right) k"
def test_TensorProduct_printing():
from sympy.tensor.functions import TensorProduct
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert latex(TensorProduct(A, B)) == r"A \otimes B"
def test_WedgeProduct_printing():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert latex(wp) == r"\operatorname{d}x \wedge \operatorname{d}y"
def test_issue_14041():
import sympy.physics.mechanics as me
A_frame = me.ReferenceFrame('A')
thetad, phid = me.dynamicsymbols('theta, phi', 1)
L = Symbol('L')
assert latex(L*(phid + thetad)**2*A_frame.x) == \
r"L \left(\dot{\phi} + \dot{\theta}\right)^{2}\mathbf{\hat{a}_x}"
assert latex((phid + thetad)**2*A_frame.x) == \
r"\left(\dot{\phi} + \dot{\theta}\right)^{2}\mathbf{\hat{a}_x}"
assert latex((phid*thetad)**a*A_frame.x) == \
r"\left(\dot{\phi} \dot{\theta}\right)^{a}\mathbf{\hat{a}_x}"
def test_issue_9216():
expr_1 = Pow(1, -1, evaluate=False)
assert latex(expr_1) == r"1^{-1}"
expr_2 = Pow(1, Pow(1, -1, evaluate=False), evaluate=False)
assert latex(expr_2) == r"1^{1^{-1}}"
expr_3 = Pow(3, -2, evaluate=False)
assert latex(expr_3) == r"\frac{1}{9}"
expr_4 = Pow(1, -2, evaluate=False)
assert latex(expr_4) == r"1^{-2}"
def test_latex_printer_tensor():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
L = TensorIndexType("L")
i, j, k, l = tensor_indices("i j k l", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensorhead("A B C D", [L], [[1]])
H = tensorhead("H", [L, L], [[1], [1]])
K = tensorhead("K", [L, L, L, L], [[1], [1], [1], [1]])
assert latex(i) == "{}^{i}"
assert latex(-i) == "{}_{i}"
expr = A(i)
assert latex(expr) == "A{}^{i}"
expr = A(i0)
assert latex(expr) == "A{}^{i_{0}}"
expr = A(-i)
assert latex(expr) == "A{}_{i}"
expr = -3*A(i)
assert latex(expr) == r"-3A{}^{i}"
expr = K(i, j, -k, -i0)
assert latex(expr) == "K{}^{ij}{}_{ki_{0}}"
expr = K(i, -j, -k, i0)
assert latex(expr) == "K{}^{i}{}_{jk}{}^{i_{0}}"
expr = K(i, -j, k, -i0)
assert latex(expr) == "K{}^{i}{}_{j}{}^{k}{}_{i_{0}}"
expr = H(i, -j)
assert latex(expr) == "H{}^{i}{}_{j}"
expr = H(i, j)
assert latex(expr) == "H{}^{ij}"
expr = H(-i, -j)
assert latex(expr) == "H{}_{ij}"
expr = (1+x)*A(i)
assert latex(expr) == r"\left(x + 1\right)A{}^{i}"
expr = H(i, -i)
assert latex(expr) == "H{}^{L_{0}}{}_{L_{0}}"
expr = H(i, -j)*A(j)*B(k)
assert latex(expr) == "H{}^{i}{}_{L_{0}}A{}^{L_{0}}B{}^{k}"
expr = A(i) + 3*B(i)
assert latex(expr) == "3B{}^{i} + A{}^{i}"
# Test ``TensorElement``:
from sympy.tensor.tensor import TensorElement
expr = TensorElement(K(i, j, k, l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3,j,k=2,l}'
expr = TensorElement(K(i, j, k, l), {i: 3})
assert latex(expr) == 'K{}^{i=3,jkl}'
expr = TensorElement(K(i, -j, k, l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3}{}_{j}{}^{k=2,l}'
expr = TensorElement(K(i, -j, k, -l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3}{}_{j}{}^{k=2}{}_{l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3, -k: 2})
assert latex(expr) == 'K{}^{i=3,j}{}_{k=2,l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3})
assert latex(expr) == 'K{}^{i=3,j}{}_{kl}'
def test_issue_15353():
from sympy import ConditionSet, Tuple, FiniteSet, S, sin, cos
a, x = symbols('a x')
# Obtained from nonlinsolve([(sin(a*x)),cos(a*x)],[x,a])
sol = ConditionSet(Tuple(x, a), FiniteSet(sin(a*x), cos(a*x)), S.Complexes)
assert latex(sol) == \
r'\left\{\left( x, \ a\right) \mid \left( x, \ a\right) \in '\
r'\mathbb{C} \wedge \left\{\sin{\left(a x \right)}, \cos{\left(a x '\
r'\right)}\right\} \right\}'
def test_trace():
# Issue 15303
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A)) == r"\operatorname{tr}\left(A \right)"
assert latex(trace(A**2)) == r"\operatorname{tr}\left(A^{2} \right)"
def test_print_basic():
# Issue 15303
from sympy import Basic, Expr
# dummy class for testing printing where the function is not
# implemented in latex.py
class UnimplementedExpr(Expr):
def __new__(cls, e):
return Basic.__new__(cls, e)
# dummy function for testing
def unimplemented_expr(expr):
return UnimplementedExpr(expr).doit()
# override class name to use superscript / subscript
def unimplemented_expr_sup_sub(expr):
result = UnimplementedExpr(expr)
result.__class__.__name__ = 'UnimplementedExpr_x^1'
return result
assert latex(unimplemented_expr(x)) == r'UnimplementedExpr\left(x\right)'
assert latex(unimplemented_expr(x**2)) == \
r'UnimplementedExpr\left(x^{2}\right)'
assert latex(unimplemented_expr_sup_sub(x)) == \
r'UnimplementedExpr^{1}_{x}\left(x\right)'
def test_MatrixSymbol_bold():
# Issue #15871
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A), mat_symbol_style='bold') == \
r"\operatorname{tr}\left(\mathbf{A} \right)"
assert latex(trace(A), mat_symbol_style='plain') == \
r"\operatorname{tr}\left(A \right)"
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A, mat_symbol_style='bold') == r"- \mathbf{A}"
assert latex(A - A*B - B, mat_symbol_style='bold') == \
r"\mathbf{A} - \mathbf{A} \mathbf{B} - \mathbf{B}"
assert latex(-A*B - A*B*C - B, mat_symbol_style='bold') == \
r"- \mathbf{A} \mathbf{B} - \mathbf{A} \mathbf{B} \mathbf{C} - \mathbf{B}"
A = MatrixSymbol("A_k", 3, 3)
assert latex(A, mat_symbol_style='bold') == r"\mathbf{A_{k}}"
def test_imaginary_unit():
assert latex(1 + I) == '1 + i'
assert latex(1 + I, imaginary_unit='i') == '1 + i'
assert latex(1 + I, imaginary_unit='j') == '1 + j'
assert latex(1 + I, imaginary_unit='foo') == '1 + foo'
assert latex(I, imaginary_unit="ti") == '\\text{i}'
assert latex(I, imaginary_unit="tj") == '\\text{j}'
def test_text_re_im():
assert latex(im(x), gothic_re_im=True) == r'\Im{\left(x\right)}'
assert latex(im(x), gothic_re_im=False) == r'\operatorname{im}{\left(x\right)}'
assert latex(re(x), gothic_re_im=True) == r'\Re{\left(x\right)}'
assert latex(re(x), gothic_re_im=False) == r'\operatorname{re}{\left(x\right)}'
def test_DiffGeomMethods():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
from sympy.diffgeom.rn import R2
m = Manifold('M', 2)
assert latex(m) == r'\text{M}'
p = Patch('P', m)
assert latex(p) == r'\text{P}_{\text{M}}'
rect = CoordSystem('rect', p)
assert latex(rect) == r'\text{rect}^{\text{P}}_{\text{M}}'
b = BaseScalarField(rect, 0)
assert latex(b) == r'\mathbf{rect_{0}}'
g = Function('g')
s_field = g(R2.x, R2.y)
assert latex(Differential(s_field)) == \
r'\operatorname{d}\left(g{\left(\mathbf{x},\mathbf{y} \right)}\right)'
def test_unit_ptinting():
assert latex(5*meter) == r'5 \text{m}'
assert latex(3*gibibyte) == r'3 \text{gibibyte}'
assert latex(4*microgram/second) == r'\frac{4 \mu\text{g}}{\text{s}}'
|
d196d8ca15adcce6d0346ded79f264bef39d712ff1999fde8683155fe1a56695 | from sympy import diff, Integral, Limit, sin, Symbol, Integer, Rational, cos, \
tan, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh, E, I, oo, \
pi, GoldenRatio, EulerGamma, Sum, Eq, Ne, Ge, Lt, Float, Matrix, Basic, \
S, MatrixSymbol, Function, Derivative, log, true, false, Range, Min, Max, \
Lambda, IndexedBase, symbols, zoo, elliptic_f, elliptic_e, elliptic_pi, Ei, \
expint, jacobi, gegenbauer, chebyshevt, chebyshevu, legendre, assoc_legendre, \
laguerre, assoc_laguerre, hermite
from sympy import elliptic_k, totient, reduced_totient, primenu, primeomega, \
fresnelc, fresnels, Heaviside
from sympy.calculus.util import AccumBounds
from sympy.core.containers import Tuple
from sympy.functions.combinatorial.factorials import factorial, factorial2, \
binomial
from sympy.functions.combinatorial.numbers import bernoulli, bell, lucas, \
fibonacci, tribonacci, catalan
from sympy.functions.elementary.complexes import re, im, Abs, conjugate
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.integers import floor, ceiling
from sympy.functions.special.gamma_functions import gamma, lowergamma, uppergamma
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.zeta_functions import polylog, lerchphi, zeta, dirichlet_eta
from sympy.logic.boolalg import And, Or, Implies, Equivalent, Xor, Not
from sympy.matrices.expressions.determinant import Determinant
from sympy.printing.mathml import mathml, MathMLContentPrinter, \
MathMLPresentationPrinter, MathMLPrinter
from sympy.sets.sets import FiniteSet, Union, Intersection, Complement, \
SymmetricDifference, Interval, EmptySet
from sympy.stats.rv import RandomSymbol
from sympy.utilities.pytest import raises
from sympy.vector import CoordSys3D, Cross, Curl, Dot, Divergence, Gradient, Laplacian
x, y, z, a, b, c, d, e, n = symbols('x:z a:e n')
mp = MathMLContentPrinter()
mpp = MathMLPresentationPrinter()
def test_mathml_printer():
m = MathMLPrinter()
assert m.doprint(1+x) == mp.doprint(1+x)
def test_content_printmethod():
assert mp.doprint(1 + x) == '<apply><plus/><ci>x</ci><cn>1</cn></apply>'
def test_content_mathml_core():
mml_1 = mp._print(1 + x)
assert mml_1.nodeName == 'apply'
nodes = mml_1.childNodes
assert len(nodes) == 3
assert nodes[0].nodeName == 'plus'
assert nodes[0].hasChildNodes() is False
assert nodes[0].nodeValue is None
assert nodes[1].nodeName in ['cn', 'ci']
if nodes[1].nodeName == 'cn':
assert nodes[1].childNodes[0].nodeValue == '1'
assert nodes[2].childNodes[0].nodeValue == 'x'
else:
assert nodes[1].childNodes[0].nodeValue == 'x'
assert nodes[2].childNodes[0].nodeValue == '1'
mml_2 = mp._print(x**2)
assert mml_2.nodeName == 'apply'
nodes = mml_2.childNodes
assert nodes[1].childNodes[0].nodeValue == 'x'
assert nodes[2].childNodes[0].nodeValue == '2'
mml_3 = mp._print(2*x)
assert mml_3.nodeName == 'apply'
nodes = mml_3.childNodes
assert nodes[0].nodeName == 'times'
assert nodes[1].childNodes[0].nodeValue == '2'
assert nodes[2].childNodes[0].nodeValue == 'x'
mml = mp._print(Float(1.0, 2)*x)
assert mml.nodeName == 'apply'
nodes = mml.childNodes
assert nodes[0].nodeName == 'times'
assert nodes[1].childNodes[0].nodeValue == '1.0'
assert nodes[2].childNodes[0].nodeValue == 'x'
def test_content_mathml_functions():
mml_1 = mp._print(sin(x))
assert mml_1.nodeName == 'apply'
assert mml_1.childNodes[0].nodeName == 'sin'
assert mml_1.childNodes[1].nodeName == 'ci'
mml_2 = mp._print(diff(sin(x), x, evaluate=False))
assert mml_2.nodeName == 'apply'
assert mml_2.childNodes[0].nodeName == 'diff'
assert mml_2.childNodes[1].nodeName == 'bvar'
assert mml_2.childNodes[1].childNodes[
0].nodeName == 'ci' # below bvar there's <ci>x/ci>
mml_3 = mp._print(diff(cos(x*y), x, evaluate=False))
assert mml_3.nodeName == 'apply'
assert mml_3.childNodes[0].nodeName == 'partialdiff'
assert mml_3.childNodes[1].nodeName == 'bvar'
assert mml_3.childNodes[1].childNodes[
0].nodeName == 'ci' # below bvar there's <ci>x/ci>
def test_content_mathml_limits():
# XXX No unevaluated limits
lim_fun = sin(x)/x
mml_1 = mp._print(Limit(lim_fun, x, 0))
assert mml_1.childNodes[0].nodeName == 'limit'
assert mml_1.childNodes[1].nodeName == 'bvar'
assert mml_1.childNodes[2].nodeName == 'lowlimit'
assert mml_1.childNodes[3].toxml() == mp._print(lim_fun).toxml()
def test_content_mathml_integrals():
integrand = x
mml_1 = mp._print(Integral(integrand, (x, 0, 1)))
assert mml_1.childNodes[0].nodeName == 'int'
assert mml_1.childNodes[1].nodeName == 'bvar'
assert mml_1.childNodes[2].nodeName == 'lowlimit'
assert mml_1.childNodes[3].nodeName == 'uplimit'
assert mml_1.childNodes[4].toxml() == mp._print(integrand).toxml()
def test_content_mathml_matrices():
A = Matrix([1, 2, 3])
B = Matrix([[0, 5, 4], [2, 3, 1], [9, 7, 9]])
mll_1 = mp._print(A)
assert mll_1.childNodes[0].nodeName == 'matrixrow'
assert mll_1.childNodes[0].childNodes[0].nodeName == 'cn'
assert mll_1.childNodes[0].childNodes[0].childNodes[0].nodeValue == '1'
assert mll_1.childNodes[1].nodeName == 'matrixrow'
assert mll_1.childNodes[1].childNodes[0].nodeName == 'cn'
assert mll_1.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mll_1.childNodes[2].nodeName == 'matrixrow'
assert mll_1.childNodes[2].childNodes[0].nodeName == 'cn'
assert mll_1.childNodes[2].childNodes[0].childNodes[0].nodeValue == '3'
mll_2 = mp._print(B)
assert mll_2.childNodes[0].nodeName == 'matrixrow'
assert mll_2.childNodes[0].childNodes[0].nodeName == 'cn'
assert mll_2.childNodes[0].childNodes[0].childNodes[0].nodeValue == '0'
assert mll_2.childNodes[0].childNodes[1].nodeName == 'cn'
assert mll_2.childNodes[0].childNodes[1].childNodes[0].nodeValue == '5'
assert mll_2.childNodes[0].childNodes[2].nodeName == 'cn'
assert mll_2.childNodes[0].childNodes[2].childNodes[0].nodeValue == '4'
assert mll_2.childNodes[1].nodeName == 'matrixrow'
assert mll_2.childNodes[1].childNodes[0].nodeName == 'cn'
assert mll_2.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mll_2.childNodes[1].childNodes[1].nodeName == 'cn'
assert mll_2.childNodes[1].childNodes[1].childNodes[0].nodeValue == '3'
assert mll_2.childNodes[1].childNodes[2].nodeName == 'cn'
assert mll_2.childNodes[1].childNodes[2].childNodes[0].nodeValue == '1'
assert mll_2.childNodes[2].nodeName == 'matrixrow'
assert mll_2.childNodes[2].childNodes[0].nodeName == 'cn'
assert mll_2.childNodes[2].childNodes[0].childNodes[0].nodeValue == '9'
assert mll_2.childNodes[2].childNodes[1].nodeName == 'cn'
assert mll_2.childNodes[2].childNodes[1].childNodes[0].nodeValue == '7'
assert mll_2.childNodes[2].childNodes[2].nodeName == 'cn'
assert mll_2.childNodes[2].childNodes[2].childNodes[0].nodeValue == '9'
def test_content_mathml_sums():
summand = x
mml_1 = mp._print(Sum(summand, (x, 1, 10)))
assert mml_1.childNodes[0].nodeName == 'sum'
assert mml_1.childNodes[1].nodeName == 'bvar'
assert mml_1.childNodes[2].nodeName == 'lowlimit'
assert mml_1.childNodes[3].nodeName == 'uplimit'
assert mml_1.childNodes[4].toxml() == mp._print(summand).toxml()
def test_content_mathml_tuples():
mml_1 = mp._print([2])
assert mml_1.nodeName == 'list'
assert mml_1.childNodes[0].nodeName == 'cn'
assert len(mml_1.childNodes) == 1
mml_2 = mp._print([2, Integer(1)])
assert mml_2.nodeName == 'list'
assert mml_2.childNodes[0].nodeName == 'cn'
assert mml_2.childNodes[1].nodeName == 'cn'
assert len(mml_2.childNodes) == 2
def test_content_mathml_add():
mml = mp._print(x**5 - x**4 + x)
assert mml.childNodes[0].nodeName == 'plus'
assert mml.childNodes[1].childNodes[0].nodeName == 'minus'
assert mml.childNodes[1].childNodes[1].nodeName == 'apply'
def test_content_mathml_Rational():
mml_1 = mp._print(Rational(1, 1))
"""should just return a number"""
assert mml_1.nodeName == 'cn'
mml_2 = mp._print(Rational(2, 5))
assert mml_2.childNodes[0].nodeName == 'divide'
def test_content_mathml_constants():
mml = mp._print(I)
assert mml.nodeName == 'imaginaryi'
mml = mp._print(E)
assert mml.nodeName == 'exponentiale'
mml = mp._print(oo)
assert mml.nodeName == 'infinity'
mml = mp._print(pi)
assert mml.nodeName == 'pi'
assert mathml(GoldenRatio) == '<cn>φ</cn>'
mml = mathml(EulerGamma)
assert mml == '<eulergamma/>'
def test_content_mathml_trig():
mml = mp._print(sin(x))
assert mml.childNodes[0].nodeName == 'sin'
mml = mp._print(cos(x))
assert mml.childNodes[0].nodeName == 'cos'
mml = mp._print(tan(x))
assert mml.childNodes[0].nodeName == 'tan'
mml = mp._print(asin(x))
assert mml.childNodes[0].nodeName == 'arcsin'
mml = mp._print(acos(x))
assert mml.childNodes[0].nodeName == 'arccos'
mml = mp._print(atan(x))
assert mml.childNodes[0].nodeName == 'arctan'
mml = mp._print(sinh(x))
assert mml.childNodes[0].nodeName == 'sinh'
mml = mp._print(cosh(x))
assert mml.childNodes[0].nodeName == 'cosh'
mml = mp._print(tanh(x))
assert mml.childNodes[0].nodeName == 'tanh'
mml = mp._print(asinh(x))
assert mml.childNodes[0].nodeName == 'arcsinh'
mml = mp._print(atanh(x))
assert mml.childNodes[0].nodeName == 'arctanh'
mml = mp._print(acosh(x))
assert mml.childNodes[0].nodeName == 'arccosh'
def test_content_mathml_relational():
mml_1 = mp._print(Eq(x, 1))
assert mml_1.nodeName == 'apply'
assert mml_1.childNodes[0].nodeName == 'eq'
assert mml_1.childNodes[1].nodeName == 'ci'
assert mml_1.childNodes[1].childNodes[0].nodeValue == 'x'
assert mml_1.childNodes[2].nodeName == 'cn'
assert mml_1.childNodes[2].childNodes[0].nodeValue == '1'
mml_2 = mp._print(Ne(1, x))
assert mml_2.nodeName == 'apply'
assert mml_2.childNodes[0].nodeName == 'neq'
assert mml_2.childNodes[1].nodeName == 'cn'
assert mml_2.childNodes[1].childNodes[0].nodeValue == '1'
assert mml_2.childNodes[2].nodeName == 'ci'
assert mml_2.childNodes[2].childNodes[0].nodeValue == 'x'
mml_3 = mp._print(Ge(1, x))
assert mml_3.nodeName == 'apply'
assert mml_3.childNodes[0].nodeName == 'geq'
assert mml_3.childNodes[1].nodeName == 'cn'
assert mml_3.childNodes[1].childNodes[0].nodeValue == '1'
assert mml_3.childNodes[2].nodeName == 'ci'
assert mml_3.childNodes[2].childNodes[0].nodeValue == 'x'
mml_4 = mp._print(Lt(1, x))
assert mml_4.nodeName == 'apply'
assert mml_4.childNodes[0].nodeName == 'lt'
assert mml_4.childNodes[1].nodeName == 'cn'
assert mml_4.childNodes[1].childNodes[0].nodeValue == '1'
assert mml_4.childNodes[2].nodeName == 'ci'
assert mml_4.childNodes[2].childNodes[0].nodeValue == 'x'
def test_content_symbol():
mml = mp._print(x)
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeValue == 'x'
del mml
mml = mp._print(Symbol("x^2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mp._print(Symbol("x__2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mp._print(Symbol("x_2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msub'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mp._print(Symbol("x^3_2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msubsup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
assert mml.childNodes[0].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[2].childNodes[0].nodeValue == '3'
del mml
mml = mp._print(Symbol("x__3_2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msubsup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
assert mml.childNodes[0].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[2].childNodes[0].nodeValue == '3'
del mml
mml = mp._print(Symbol("x_2_a"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msub'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[
0].nodeValue == '2'
assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo'
assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[
0].nodeValue == ' '
assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[
0].nodeValue == 'a'
del mml
mml = mp._print(Symbol("x^2^a"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[
0].nodeValue == '2'
assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo'
assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[
0].nodeValue == ' '
assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[
0].nodeValue == 'a'
del mml
mml = mp._print(Symbol("x__2__a"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[
0].nodeValue == '2'
assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo'
assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[
0].nodeValue == ' '
assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[
0].nodeValue == 'a'
del mml
def test_content_mathml_greek():
mml = mp._print(Symbol('alpha'))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeValue == u'\N{GREEK SMALL LETTER ALPHA}'
assert mp.doprint(Symbol('alpha')) == '<ci>α</ci>'
assert mp.doprint(Symbol('beta')) == '<ci>β</ci>'
assert mp.doprint(Symbol('gamma')) == '<ci>γ</ci>'
assert mp.doprint(Symbol('delta')) == '<ci>δ</ci>'
assert mp.doprint(Symbol('epsilon')) == '<ci>ε</ci>'
assert mp.doprint(Symbol('zeta')) == '<ci>ζ</ci>'
assert mp.doprint(Symbol('eta')) == '<ci>η</ci>'
assert mp.doprint(Symbol('theta')) == '<ci>θ</ci>'
assert mp.doprint(Symbol('iota')) == '<ci>ι</ci>'
assert mp.doprint(Symbol('kappa')) == '<ci>κ</ci>'
assert mp.doprint(Symbol('lambda')) == '<ci>λ</ci>'
assert mp.doprint(Symbol('mu')) == '<ci>μ</ci>'
assert mp.doprint(Symbol('nu')) == '<ci>ν</ci>'
assert mp.doprint(Symbol('xi')) == '<ci>ξ</ci>'
assert mp.doprint(Symbol('omicron')) == '<ci>ο</ci>'
assert mp.doprint(Symbol('pi')) == '<ci>π</ci>'
assert mp.doprint(Symbol('rho')) == '<ci>ρ</ci>'
assert mp.doprint(Symbol('varsigma')) == '<ci>ς</ci>'
assert mp.doprint(Symbol('sigma')) == '<ci>σ</ci>'
assert mp.doprint(Symbol('tau')) == '<ci>τ</ci>'
assert mp.doprint(Symbol('upsilon')) == '<ci>υ</ci>'
assert mp.doprint(Symbol('phi')) == '<ci>φ</ci>'
assert mp.doprint(Symbol('chi')) == '<ci>χ</ci>'
assert mp.doprint(Symbol('psi')) == '<ci>ψ</ci>'
assert mp.doprint(Symbol('omega')) == '<ci>ω</ci>'
assert mp.doprint(Symbol('Alpha')) == '<ci>Α</ci>'
assert mp.doprint(Symbol('Beta')) == '<ci>Β</ci>'
assert mp.doprint(Symbol('Gamma')) == '<ci>Γ</ci>'
assert mp.doprint(Symbol('Delta')) == '<ci>Δ</ci>'
assert mp.doprint(Symbol('Epsilon')) == '<ci>Ε</ci>'
assert mp.doprint(Symbol('Zeta')) == '<ci>Ζ</ci>'
assert mp.doprint(Symbol('Eta')) == '<ci>Η</ci>'
assert mp.doprint(Symbol('Theta')) == '<ci>Θ</ci>'
assert mp.doprint(Symbol('Iota')) == '<ci>Ι</ci>'
assert mp.doprint(Symbol('Kappa')) == '<ci>Κ</ci>'
assert mp.doprint(Symbol('Lambda')) == '<ci>Λ</ci>'
assert mp.doprint(Symbol('Mu')) == '<ci>Μ</ci>'
assert mp.doprint(Symbol('Nu')) == '<ci>Ν</ci>'
assert mp.doprint(Symbol('Xi')) == '<ci>Ξ</ci>'
assert mp.doprint(Symbol('Omicron')) == '<ci>Ο</ci>'
assert mp.doprint(Symbol('Pi')) == '<ci>Π</ci>'
assert mp.doprint(Symbol('Rho')) == '<ci>Ρ</ci>'
assert mp.doprint(Symbol('Sigma')) == '<ci>Σ</ci>'
assert mp.doprint(Symbol('Tau')) == '<ci>Τ</ci>'
assert mp.doprint(Symbol('Upsilon')) == '<ci>Υ</ci>'
assert mp.doprint(Symbol('Phi')) == '<ci>Φ</ci>'
assert mp.doprint(Symbol('Chi')) == '<ci>Χ</ci>'
assert mp.doprint(Symbol('Psi')) == '<ci>Ψ</ci>'
assert mp.doprint(Symbol('Omega')) == '<ci>Ω</ci>'
def test_content_mathml_order():
expr = x**3 + x**2*y + 3*x*y**3 + y**4
mp = MathMLContentPrinter({'order': 'lex'})
mml = mp._print(expr)
assert mml.childNodes[1].childNodes[0].nodeName == 'power'
assert mml.childNodes[1].childNodes[1].childNodes[0].data == 'x'
assert mml.childNodes[1].childNodes[2].childNodes[0].data == '3'
assert mml.childNodes[4].childNodes[0].nodeName == 'power'
assert mml.childNodes[4].childNodes[1].childNodes[0].data == 'y'
assert mml.childNodes[4].childNodes[2].childNodes[0].data == '4'
mp = MathMLContentPrinter({'order': 'rev-lex'})
mml = mp._print(expr)
assert mml.childNodes[1].childNodes[0].nodeName == 'power'
assert mml.childNodes[1].childNodes[1].childNodes[0].data == 'y'
assert mml.childNodes[1].childNodes[2].childNodes[0].data == '4'
assert mml.childNodes[4].childNodes[0].nodeName == 'power'
assert mml.childNodes[4].childNodes[1].childNodes[0].data == 'x'
assert mml.childNodes[4].childNodes[2].childNodes[0].data == '3'
def test_content_settings():
raises(TypeError, lambda: mathml(x, method="garbage"))
def test_presentation_printmethod():
assert mpp.doprint(1 + x) == '<mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow>'
assert mpp.doprint(x**2) == '<msup><mi>x</mi><mn>2</mn></msup>'
assert mpp.doprint(x**-1) == '<mfrac><mn>1</mn><mi>x</mi></mfrac>'
assert mpp.doprint(x**-2) == \
'<mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac>'
assert mpp.doprint(2*x) == \
'<mrow><mn>2</mn><mo>⁢</mo><mi>x</mi></mrow>'
def test_presentation_mathml_core():
mml_1 = mpp._print(1 + x)
assert mml_1.nodeName == 'mrow'
nodes = mml_1.childNodes
assert len(nodes) == 3
assert nodes[0].nodeName in ['mi', 'mn']
assert nodes[1].nodeName == 'mo'
if nodes[0].nodeName == 'mn':
assert nodes[0].childNodes[0].nodeValue == '1'
assert nodes[2].childNodes[0].nodeValue == 'x'
else:
assert nodes[0].childNodes[0].nodeValue == 'x'
assert nodes[2].childNodes[0].nodeValue == '1'
mml_2 = mpp._print(x**2)
assert mml_2.nodeName == 'msup'
nodes = mml_2.childNodes
assert nodes[0].childNodes[0].nodeValue == 'x'
assert nodes[1].childNodes[0].nodeValue == '2'
mml_3 = mpp._print(2*x)
assert mml_3.nodeName == 'mrow'
nodes = mml_3.childNodes
assert nodes[0].childNodes[0].nodeValue == '2'
assert nodes[1].childNodes[0].nodeValue == '⁢'
assert nodes[2].childNodes[0].nodeValue == 'x'
mml = mpp._print(Float(1.0, 2)*x)
assert mml.nodeName == 'mrow'
nodes = mml.childNodes
assert nodes[0].childNodes[0].nodeValue == '1.0'
assert nodes[1].childNodes[0].nodeValue == '⁢'
assert nodes[2].childNodes[0].nodeValue == 'x'
def test_presentation_mathml_functions():
mml_1 = mpp._print(sin(x))
assert mml_1.childNodes[0].childNodes[0
].nodeValue == 'sin'
assert mml_1.childNodes[1].childNodes[0
].childNodes[0].nodeValue == 'x'
mml_2 = mpp._print(diff(sin(x), x, evaluate=False))
assert mml_2.nodeName == 'mrow'
assert mml_2.childNodes[0].childNodes[0
].childNodes[0].childNodes[0].nodeValue == 'ⅆ'
assert mml_2.childNodes[1].childNodes[1
].nodeName == 'mfenced'
assert mml_2.childNodes[0].childNodes[1
].childNodes[0].childNodes[0].nodeValue == 'ⅆ'
mml_3 = mpp._print(diff(cos(x*y), x, evaluate=False))
assert mml_3.childNodes[0].nodeName == 'mfrac'
assert mml_3.childNodes[0].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '∂'
assert mml_3.childNodes[1].childNodes[0
].childNodes[0].nodeValue == 'cos'
def test_print_derivative():
f = Function('f')
d = Derivative(f(x, y, z), x, z, x, z, z, y)
assert mathml(d) == \
'<apply><partialdiff/><bvar><ci>y</ci><ci>z</ci><degree><cn>2</cn></degree><ci>x</ci><ci>z</ci><ci>x</ci></bvar><apply><f/><ci>x</ci><ci>y</ci><ci>z</ci></apply></apply>'
assert mathml(d, printer='presentation') == \
'<mrow><mfrac><mrow><msup><mo>∂</mo><mn>6</mn></msup></mrow><mrow><mo>∂</mo><mi>y</mi><msup><mo>∂</mo><mn>2</mn></msup><mi>z</mi><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>z</mi><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mi>f</mi><mfenced><mi>x</mi><mi>y</mi><mi>z</mi></mfenced></mrow></mrow>'
def test_presentation_mathml_limits():
lim_fun = sin(x)/x
mml_1 = mpp._print(Limit(lim_fun, x, 0))
assert mml_1.childNodes[0].nodeName == 'munder'
assert mml_1.childNodes[0].childNodes[0
].childNodes[0].nodeValue == 'lim'
assert mml_1.childNodes[0].childNodes[1
].childNodes[0].childNodes[0
].nodeValue == 'x'
assert mml_1.childNodes[0].childNodes[1
].childNodes[1].childNodes[0
].nodeValue == '→'
assert mml_1.childNodes[0].childNodes[1
].childNodes[2].childNodes[0
].nodeValue == '0'
def test_presentation_mathml_integrals():
assert mpp.doprint(Integral(x, (x, 0, 1))) == \
'<mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup>'\
'<mi>x</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(log(x), x)) == \
'<mrow><mo>∫</mo><mrow><mi>log</mi><mfenced><mi>x</mi>'\
'</mfenced></mrow><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(x*y, x, y)) == \
'<mrow><mo>∬</mo><mrow><mi>x</mi><mo>⁢</mo>'\
'<mi>y</mi></mrow><mo>ⅆ</mo><mi>y</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
z, w = symbols('z w')
assert mpp.doprint(Integral(x*y*z, x, y, z)) == \
'<mrow><mo>∭</mo><mrow><mi>x</mi><mo>⁢</mo>'\
'<mi>y</mi><mo>⁢</mo><mi>z</mi></mrow><mo>ⅆ</mo>'\
'<mi>z</mi><mo>ⅆ</mo><mi>y</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(x*y*z*w, x, y, z, w)) == \
'<mrow><mo>∫</mo><mo>∫</mo><mo>∫</mo>'\
'<mo>∫</mo><mrow><mi>w</mi><mo>⁢</mo>'\
'<mi>x</mi><mo>⁢</mo><mi>y</mi>'\
'<mo>⁢</mo><mi>z</mi></mrow><mo>ⅆ</mo><mi>w</mi>'\
'<mo>ⅆ</mo><mi>z</mi><mo>ⅆ</mo><mi>y</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(x, x, y, (z, 0, 1))) == \
'<mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup>'\
'<mo>∫</mo><mo>∫</mo><mi>x</mi><mo>ⅆ</mo><mi>z</mi>'\
'<mo>ⅆ</mo><mi>y</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(x, (x, 0))) == \
'<mrow><msup><mo>∫</mo><mn>0</mn></msup><mi>x</mi><mo>ⅆ</mo>'\
'<mi>x</mi></mrow>'
def test_presentation_mathml_matrices():
A = Matrix([1, 2, 3])
B = Matrix([[0, 5, 4], [2, 3, 1], [9, 7, 9]])
mll_1 = mpp._print(A)
assert mll_1.childNodes[0].nodeName == 'mtable'
assert mll_1.childNodes[0].childNodes[0].nodeName == 'mtr'
assert len(mll_1.childNodes[0].childNodes) == 3
assert mll_1.childNodes[0].childNodes[0].childNodes[0].nodeName == 'mtd'
assert len(mll_1.childNodes[0].childNodes[0].childNodes) == 1
assert mll_1.childNodes[0].childNodes[0].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '1'
assert mll_1.childNodes[0].childNodes[1].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '2'
assert mll_1.childNodes[0].childNodes[2].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '3'
mll_2 = mpp._print(B)
assert mll_2.childNodes[0].nodeName == 'mtable'
assert mll_2.childNodes[0].childNodes[0].nodeName == 'mtr'
assert len(mll_2.childNodes[0].childNodes) == 3
assert mll_2.childNodes[0].childNodes[0].childNodes[0].nodeName == 'mtd'
assert len(mll_2.childNodes[0].childNodes[0].childNodes) == 3
assert mll_2.childNodes[0].childNodes[0].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '0'
assert mll_2.childNodes[0].childNodes[0].childNodes[1
].childNodes[0].childNodes[0].nodeValue == '5'
assert mll_2.childNodes[0].childNodes[0].childNodes[2
].childNodes[0].childNodes[0].nodeValue == '4'
assert mll_2.childNodes[0].childNodes[1].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '2'
assert mll_2.childNodes[0].childNodes[1].childNodes[1
].childNodes[0].childNodes[0].nodeValue == '3'
assert mll_2.childNodes[0].childNodes[1].childNodes[2
].childNodes[0].childNodes[0].nodeValue == '1'
assert mll_2.childNodes[0].childNodes[2].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '9'
assert mll_2.childNodes[0].childNodes[2].childNodes[1
].childNodes[0].childNodes[0].nodeValue == '7'
assert mll_2.childNodes[0].childNodes[2].childNodes[2
].childNodes[0].childNodes[0].nodeValue == '9'
def test_presentation_mathml_sums():
summand = x
mml_1 = mpp._print(Sum(summand, (x, 1, 10)))
assert mml_1.childNodes[0].nodeName == 'munderover'
assert len(mml_1.childNodes[0].childNodes) == 3
assert mml_1.childNodes[0].childNodes[0].childNodes[0
].nodeValue == '∑'
assert len(mml_1.childNodes[0].childNodes[1].childNodes) == 3
assert mml_1.childNodes[0].childNodes[2].childNodes[0
].nodeValue == '10'
assert mml_1.childNodes[1].childNodes[0].nodeValue == 'x'
def test_presentation_mathml_add():
mml = mpp._print(x**5 - x**4 + x)
assert len(mml.childNodes) == 5
assert mml.childNodes[0].childNodes[0].childNodes[0
].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].childNodes[0
].nodeValue == '5'
assert mml.childNodes[1].childNodes[0].nodeValue == '-'
assert mml.childNodes[2].childNodes[0].childNodes[0
].nodeValue == 'x'
assert mml.childNodes[2].childNodes[1].childNodes[0
].nodeValue == '4'
assert mml.childNodes[3].childNodes[0].nodeValue == '+'
assert mml.childNodes[4].childNodes[0].nodeValue == 'x'
def test_presentation_mathml_Rational():
mml_1 = mpp._print(Rational(1, 1))
assert mml_1.nodeName == 'mn'
mml_2 = mpp._print(Rational(2, 5))
assert mml_2.nodeName == 'mfrac'
assert mml_2.childNodes[0].childNodes[0].nodeValue == '2'
assert mml_2.childNodes[1].childNodes[0].nodeValue == '5'
def test_presentation_mathml_constants():
mml = mpp._print(I)
assert mml.childNodes[0].nodeValue == 'ⅈ'
mml = mpp._print(E)
assert mml.childNodes[0].nodeValue == 'ⅇ'
mml = mpp._print(oo)
assert mml.childNodes[0].nodeValue == '∞'
mml = mpp._print(pi)
assert mml.childNodes[0].nodeValue == 'π'
assert mathml(GoldenRatio, printer='presentation') == '<mi>Φ</mi>'
assert mathml(zoo, printer='presentation') == \
'<mover><mo>∞</mo><mo>~</mo></mover>'
assert mathml(S.NaN, printer='presentation') == '<mi>NaN</mi>'
def test_presentation_mathml_trig():
mml = mpp._print(sin(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'sin'
mml = mpp._print(cos(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'cos'
mml = mpp._print(tan(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'tan'
mml = mpp._print(asin(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arcsin'
mml = mpp._print(acos(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arccos'
mml = mpp._print(atan(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arctan'
mml = mpp._print(sinh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'sinh'
mml = mpp._print(cosh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'cosh'
mml = mpp._print(tanh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'tanh'
mml = mpp._print(asinh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arcsinh'
mml = mpp._print(atanh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arctanh'
mml = mpp._print(acosh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arccosh'
def test_presentation_mathml_relational():
mml_1 = mpp._print(Eq(x, 1))
assert len(mml_1.childNodes) == 3
assert mml_1.childNodes[0].nodeName == 'mi'
assert mml_1.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml_1.childNodes[1].nodeName == 'mo'
assert mml_1.childNodes[1].childNodes[0].nodeValue == '='
assert mml_1.childNodes[2].nodeName == 'mn'
assert mml_1.childNodes[2].childNodes[0].nodeValue == '1'
mml_2 = mpp._print(Ne(1, x))
assert len(mml_2.childNodes) == 3
assert mml_2.childNodes[0].nodeName == 'mn'
assert mml_2.childNodes[0].childNodes[0].nodeValue == '1'
assert mml_2.childNodes[1].nodeName == 'mo'
assert mml_2.childNodes[1].childNodes[0].nodeValue == '≠'
assert mml_2.childNodes[2].nodeName == 'mi'
assert mml_2.childNodes[2].childNodes[0].nodeValue == 'x'
mml_3 = mpp._print(Ge(1, x))
assert len(mml_3.childNodes) == 3
assert mml_3.childNodes[0].nodeName == 'mn'
assert mml_3.childNodes[0].childNodes[0].nodeValue == '1'
assert mml_3.childNodes[1].nodeName == 'mo'
assert mml_3.childNodes[1].childNodes[0].nodeValue == '≥'
assert mml_3.childNodes[2].nodeName == 'mi'
assert mml_3.childNodes[2].childNodes[0].nodeValue == 'x'
mml_4 = mpp._print(Lt(1, x))
assert len(mml_4.childNodes) == 3
assert mml_4.childNodes[0].nodeName == 'mn'
assert mml_4.childNodes[0].childNodes[0].nodeValue == '1'
assert mml_4.childNodes[1].nodeName == 'mo'
assert mml_4.childNodes[1].childNodes[0].nodeValue == '<'
assert mml_4.childNodes[2].nodeName == 'mi'
assert mml_4.childNodes[2].childNodes[0].nodeValue == 'x'
def test_presentation_symbol():
mml = mpp._print(x)
assert mml.nodeName == 'mi'
assert mml.childNodes[0].nodeValue == 'x'
del mml
mml = mpp._print(Symbol("x^2"))
assert mml.nodeName == 'msup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mpp._print(Symbol("x__2"))
assert mml.nodeName == 'msup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mpp._print(Symbol("x_2"))
assert mml.nodeName == 'msub'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mpp._print(Symbol("x^3_2"))
assert mml.nodeName == 'msubsup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
assert mml.childNodes[2].nodeName == 'mi'
assert mml.childNodes[2].childNodes[0].nodeValue == '3'
del mml
mml = mpp._print(Symbol("x__3_2"))
assert mml.nodeName == 'msubsup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
assert mml.childNodes[2].nodeName == 'mi'
assert mml.childNodes[2].childNodes[0].nodeValue == '3'
del mml
mml = mpp._print(Symbol("x_2_a"))
assert mml.nodeName == 'msub'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mrow'
assert mml.childNodes[1].childNodes[0].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mml.childNodes[1].childNodes[1].nodeName == 'mo'
assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' '
assert mml.childNodes[1].childNodes[2].nodeName == 'mi'
assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a'
del mml
mml = mpp._print(Symbol("x^2^a"))
assert mml.nodeName == 'msup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mrow'
assert mml.childNodes[1].childNodes[0].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mml.childNodes[1].childNodes[1].nodeName == 'mo'
assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' '
assert mml.childNodes[1].childNodes[2].nodeName == 'mi'
assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a'
del mml
mml = mpp._print(Symbol("x__2__a"))
assert mml.nodeName == 'msup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mrow'
assert mml.childNodes[1].childNodes[0].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mml.childNodes[1].childNodes[1].nodeName == 'mo'
assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' '
assert mml.childNodes[1].childNodes[2].nodeName == 'mi'
assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a'
del mml
def test_presentation_mathml_greek():
mml = mpp._print(Symbol('alpha'))
assert mml.nodeName == 'mi'
assert mml.childNodes[0].nodeValue == u'\N{GREEK SMALL LETTER ALPHA}'
assert mpp.doprint(Symbol('alpha')) == '<mi>α</mi>'
assert mpp.doprint(Symbol('beta')) == '<mi>β</mi>'
assert mpp.doprint(Symbol('gamma')) == '<mi>γ</mi>'
assert mpp.doprint(Symbol('delta')) == '<mi>δ</mi>'
assert mpp.doprint(Symbol('epsilon')) == '<mi>ε</mi>'
assert mpp.doprint(Symbol('zeta')) == '<mi>ζ</mi>'
assert mpp.doprint(Symbol('eta')) == '<mi>η</mi>'
assert mpp.doprint(Symbol('theta')) == '<mi>θ</mi>'
assert mpp.doprint(Symbol('iota')) == '<mi>ι</mi>'
assert mpp.doprint(Symbol('kappa')) == '<mi>κ</mi>'
assert mpp.doprint(Symbol('lambda')) == '<mi>λ</mi>'
assert mpp.doprint(Symbol('mu')) == '<mi>μ</mi>'
assert mpp.doprint(Symbol('nu')) == '<mi>ν</mi>'
assert mpp.doprint(Symbol('xi')) == '<mi>ξ</mi>'
assert mpp.doprint(Symbol('omicron')) == '<mi>ο</mi>'
assert mpp.doprint(Symbol('pi')) == '<mi>π</mi>'
assert mpp.doprint(Symbol('rho')) == '<mi>ρ</mi>'
assert mpp.doprint(Symbol('varsigma')) == '<mi>ς</mi>'
assert mpp.doprint(Symbol('sigma')) == '<mi>σ</mi>'
assert mpp.doprint(Symbol('tau')) == '<mi>τ</mi>'
assert mpp.doprint(Symbol('upsilon')) == '<mi>υ</mi>'
assert mpp.doprint(Symbol('phi')) == '<mi>φ</mi>'
assert mpp.doprint(Symbol('chi')) == '<mi>χ</mi>'
assert mpp.doprint(Symbol('psi')) == '<mi>ψ</mi>'
assert mpp.doprint(Symbol('omega')) == '<mi>ω</mi>'
assert mpp.doprint(Symbol('Alpha')) == '<mi>Α</mi>'
assert mpp.doprint(Symbol('Beta')) == '<mi>Β</mi>'
assert mpp.doprint(Symbol('Gamma')) == '<mi>Γ</mi>'
assert mpp.doprint(Symbol('Delta')) == '<mi>Δ</mi>'
assert mpp.doprint(Symbol('Epsilon')) == '<mi>Ε</mi>'
assert mpp.doprint(Symbol('Zeta')) == '<mi>Ζ</mi>'
assert mpp.doprint(Symbol('Eta')) == '<mi>Η</mi>'
assert mpp.doprint(Symbol('Theta')) == '<mi>Θ</mi>'
assert mpp.doprint(Symbol('Iota')) == '<mi>Ι</mi>'
assert mpp.doprint(Symbol('Kappa')) == '<mi>Κ</mi>'
assert mpp.doprint(Symbol('Lambda')) == '<mi>Λ</mi>'
assert mpp.doprint(Symbol('Mu')) == '<mi>Μ</mi>'
assert mpp.doprint(Symbol('Nu')) == '<mi>Ν</mi>'
assert mpp.doprint(Symbol('Xi')) == '<mi>Ξ</mi>'
assert mpp.doprint(Symbol('Omicron')) == '<mi>Ο</mi>'
assert mpp.doprint(Symbol('Pi')) == '<mi>Π</mi>'
assert mpp.doprint(Symbol('Rho')) == '<mi>Ρ</mi>'
assert mpp.doprint(Symbol('Sigma')) == '<mi>Σ</mi>'
assert mpp.doprint(Symbol('Tau')) == '<mi>Τ</mi>'
assert mpp.doprint(Symbol('Upsilon')) == '<mi>Υ</mi>'
assert mpp.doprint(Symbol('Phi')) == '<mi>Φ</mi>'
assert mpp.doprint(Symbol('Chi')) == '<mi>Χ</mi>'
assert mpp.doprint(Symbol('Psi')) == '<mi>Ψ</mi>'
assert mpp.doprint(Symbol('Omega')) == '<mi>Ω</mi>'
def test_presentation_mathml_order():
expr = x**3 + x**2*y + 3*x*y**3 + y**4
mp = MathMLPresentationPrinter({'order': 'lex'})
mml = mp._print(expr)
assert mml.childNodes[0].nodeName == 'msup'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '3'
assert mml.childNodes[6].nodeName == 'msup'
assert mml.childNodes[6].childNodes[0].childNodes[0].nodeValue == 'y'
assert mml.childNodes[6].childNodes[1].childNodes[0].nodeValue == '4'
mp = MathMLPresentationPrinter({'order': 'rev-lex'})
mml = mp._print(expr)
assert mml.childNodes[0].nodeName == 'msup'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'y'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '4'
assert mml.childNodes[6].nodeName == 'msup'
assert mml.childNodes[6].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[6].childNodes[1].childNodes[0].nodeValue == '3'
def test_print_intervals():
a = Symbol('a', real=True)
assert mpp.doprint(Interval(0, a)) == \
'<mrow><mfenced close="]" open="["><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Interval(0, a, False, False)) == \
'<mrow><mfenced close="]" open="["><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Interval(0, a, True, False)) == \
'<mrow><mfenced close="]" open="("><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Interval(0, a, False, True)) == \
'<mrow><mfenced close=")" open="["><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Interval(0, a, True, True)) == \
'<mrow><mfenced close=")" open="("><mn>0</mn><mi>a</mi></mfenced></mrow>'
def test_print_tuples():
assert mpp.doprint(Tuple(0,)) == \
'<mrow><mfenced><mn>0</mn></mfenced></mrow>'
assert mpp.doprint(Tuple(0, a)) == \
'<mrow><mfenced><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Tuple(0, a, a)) == \
'<mrow><mfenced><mn>0</mn><mi>a</mi><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Tuple(0, 1, 2, 3, 4)) == \
'<mrow><mfenced><mn>0</mn><mn>1</mn><mn>2</mn><mn>3</mn><mn>4</mn></mfenced></mrow>'
assert mpp.doprint(Tuple(0, 1, Tuple(2, 3, 4))) == \
'<mrow><mfenced><mn>0</mn><mn>1</mn><mrow><mfenced><mn>2</mn><mn>3'\
'</mn><mn>4</mn></mfenced></mrow></mfenced></mrow>'
def test_print_re_im():
assert mpp.doprint(re(x)) == \
'<mrow><mi mathvariant="fraktur">R</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(im(x)) == \
'<mrow><mi mathvariant="fraktur">I</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(re(x + 1)) == \
'<mrow><mrow><mi mathvariant="fraktur">R</mi><mfenced><mi>x</mi>'\
'</mfenced></mrow><mo>+</mo><mn>1</mn></mrow>'
assert mpp.doprint(im(x + 1)) == \
'<mrow><mi mathvariant="fraktur">I</mi><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_Abs():
assert mpp.doprint(Abs(x)) == \
'<mrow><mfenced close="|" open="|"><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(Abs(x + 1)) == \
'<mrow><mfenced close="|" open="|"><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow>'
def test_print_Determinant():
assert mpp.doprint(Determinant(Matrix([[1, 2], [3, 4]]))) == \
'<mrow><mfenced close="|" open="|"><mfenced close="]" open="["><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></mfenced></mrow>'
def test_presentation_settings():
raises(TypeError, lambda: mathml(x, printer='presentation',
method="garbage"))
def test_toprettyxml_hooking():
# test that the patch doesn't influence the behavior of the standard
# library
import xml.dom.minidom
doc1 = xml.dom.minidom.parseString(
"<apply><plus/><ci>x</ci><cn>1</cn></apply>")
doc2 = xml.dom.minidom.parseString(
"<mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow>")
prettyxml_old1 = doc1.toprettyxml()
prettyxml_old2 = doc2.toprettyxml()
mp.apply_patch()
mp.restore_patch()
assert prettyxml_old1 == doc1.toprettyxml()
assert prettyxml_old2 == doc2.toprettyxml()
def test_print_domains():
from sympy import Complexes, Integers, Naturals, Naturals0, Reals
assert mpp.doprint(Complexes) == '<mi mathvariant="normal">ℂ</mi>'
assert mpp.doprint(Integers) == '<mi mathvariant="normal">ℤ</mi>'
assert mpp.doprint(Naturals) == '<mi mathvariant="normal">ℕ</mi>'
assert mpp.doprint(Naturals0) == \
'<msub><mi mathvariant="normal">ℕ</mi><mn>0</mn></msub>'
assert mpp.doprint(Reals) == '<mi mathvariant="normal">ℝ</mi>'
def test_print_expression_with_minus():
assert mpp.doprint(-x) == '<mrow><mo>-</mo><mi>x</mi></mrow>'
assert mpp.doprint(-x/y) == \
'<mrow><mo>-</mo><mfrac><mi>x</mi><mi>y</mi></mfrac></mrow>'
assert mpp.doprint(-Rational(1, 2)) == \
'<mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow>'
def test_print_AssocOp():
from sympy.core.operations import AssocOp
class TestAssocOp(AssocOp):
identity = 0
expr = TestAssocOp(1, 2)
mpp.doprint(expr) == \
'<mrow><mi>testassocop</mi><mn>2</mn><mn>1</mn></mrow>'
def test_print_basic():
expr = Basic(1, 2)
assert mpp.doprint(expr) == \
'<mrow><mi>basic</mi><mfenced><mn>1</mn><mn>2</mn></mfenced></mrow>'
assert mp.doprint(expr) == '<basic><cn>1</cn><cn>2</cn></basic>'
def test_mat_delim_print():
expr = Matrix([[1, 2], [3, 4]])
assert mathml(expr, printer='presentation', mat_delim='[') == \
'<mfenced close="]" open="["><mtable><mtr><mtd><mn>1</mn></mtd><mtd>'\
'<mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn>'\
'</mtd></mtr></mtable></mfenced>'
assert mathml(expr, printer='presentation', mat_delim='(') == \
'<mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd>'\
'</mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced>'
assert mathml(expr, printer='presentation', mat_delim='') == \
'<mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr>'\
'<mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable>'
def test_ln_notation_print():
expr = log(x)
assert mathml(expr, printer='presentation') == \
'<mrow><mi>log</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mathml(expr, printer='presentation', ln_notation=False) == \
'<mrow><mi>log</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mathml(expr, printer='presentation', ln_notation=True) == \
'<mrow><mi>ln</mi><mfenced><mi>x</mi></mfenced></mrow>'
def test_mul_symbol_print():
expr = x * y
assert mathml(expr, printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mi>y</mi></mrow>'
assert mathml(expr, printer='presentation', mul_symbol=None) == \
'<mrow><mi>x</mi><mo>⁢</mo><mi>y</mi></mrow>'
assert mathml(expr, printer='presentation', mul_symbol='dot') == \
'<mrow><mi>x</mi><mo>·</mo><mi>y</mi></mrow>'
assert mathml(expr, printer='presentation', mul_symbol='ldot') == \
'<mrow><mi>x</mi><mo>․</mo><mi>y</mi></mrow>'
assert mathml(expr, printer='presentation', mul_symbol='times') == \
'<mrow><mi>x</mi><mo>×</mo><mi>y</mi></mrow>'
def test_print_lerchphi():
assert mpp.doprint(lerchphi(1, 2, 3)) == \
'<mrow><mi>Φ</mi><mfenced><mn>1</mn><mn>2</mn><mn>3</mn></mfenced></mrow>'
def test_print_polylog():
assert mp.doprint(polylog(x, y)) == \
'<apply><polylog/><ci>x</ci><ci>y</ci></apply>'
assert mpp.doprint(polylog(x, y)) == \
'<mrow><msub><mi>Li</mi><mi>x</mi></msub><mfenced><mi>y</mi></mfenced></mrow>'
def test_print_set_frozenset():
f = frozenset({1, 5, 3})
assert mpp.doprint(f) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mn>5</mn></mfenced>'
s = set({1, 2, 3})
assert mpp.doprint(s) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>2</mn><mn>3</mn></mfenced>'
def test_print_FiniteSet():
f1 = FiniteSet(x, 1, 3)
assert mpp.doprint(f1) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi></mfenced>'
def test_print_EmptySet():
assert mpp.doprint(EmptySet()) == '<mo>∅</mo>'
def test_print_SetOp():
f1 = FiniteSet(x, 1, 3)
f2 = FiniteSet(y, 2, 4)
assert mpp.doprint(Union(f1, f2, evaluate=False)) == \
'<mrow><mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi>'\
'</mfenced><mo>∪</mo><mfenced close="}" open="{"><mn>2</mn>'\
'<mn>4</mn><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(Intersection(f1, f2, evaluate=False)) == \
'<mrow><mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi>'\
'</mfenced><mo>∩</mo><mfenced close="}" open="{"><mn>2</mn>'\
'<mn>4</mn><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(Complement(f1, f2, evaluate=False)) == \
'<mrow><mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi>'\
'</mfenced><mo>∖</mo><mfenced close="}" open="{"><mn>2</mn>'\
'<mn>4</mn><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(SymmetricDifference(f1, f2, evaluate=False)) == \
'<mrow><mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi>'\
'</mfenced><mo>∆</mo><mfenced close="}" open="{"><mn>2</mn>'\
'<mn>4</mn><mi>y</mi></mfenced></mrow>'
def test_print_logic():
assert mpp.doprint(And(x, y)) == \
'<mrow><mi>x</mi><mo>∧</mo><mi>y</mi></mrow>'
assert mpp.doprint(Or(x, y)) == \
'<mrow><mi>x</mi><mo>∨</mo><mi>y</mi></mrow>'
assert mpp.doprint(Xor(x, y)) == \
'<mrow><mi>x</mi><mo>⊻</mo><mi>y</mi></mrow>'
assert mpp.doprint(Implies(x, y)) == \
'<mrow><mi>x</mi><mo>⇒</mo><mi>y</mi></mrow>'
assert mpp.doprint(Equivalent(x, y)) == \
'<mrow><mi>x</mi><mo>⇔</mo><mi>y</mi></mrow>'
assert mpp.doprint(And(Eq(x, y), x > 4)) == \
'<mrow><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><mo>∧</mo>'\
'<mrow><mi>x</mi><mo>></mo><mn>4</mn></mrow></mrow>'
assert mpp.doprint(And(Eq(x, 3), y < 3, x > y + 1)) == \
'<mrow><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow><mo>∧</mo>'\
'<mrow><mi>x</mi><mo>></mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow>'\
'</mrow><mo>∧</mo><mrow><mi>y</mi><mo><</mo><mn>3</mn></mrow></mrow>'
assert mpp.doprint(Or(Eq(x, y), x > 4)) == \
'<mrow><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><mo>∨</mo>'\
'<mrow><mi>x</mi><mo>></mo><mn>4</mn></mrow></mrow>'
assert mpp.doprint(And(Eq(x, 3), Or(y < 3, x > y + 1))) == \
'<mrow><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow><mo>∧</mo>'\
'<mfenced><mrow><mrow><mi>x</mi><mo>></mo><mrow><mi>y</mi><mo>+</mo>'\
'<mn>1</mn></mrow></mrow><mo>∨</mo><mrow><mi>y</mi><mo><</mo>'\
'<mn>3</mn></mrow></mrow></mfenced></mrow>'
assert mpp.doprint(Not(x)) == '<mrow><mo>¬</mo><mi>x</mi></mrow>'
assert mpp.doprint(Not(And(x, y))) == \
'<mrow><mo>¬</mo><mfenced><mrow><mi>x</mi><mo>∧</mo>'\
'<mi>y</mi></mrow></mfenced></mrow>'
def test_root_notation_print():
assert mathml(x**(S(1)/3), printer='presentation') == \
'<mroot><mi>x</mi><mn>3</mn></mroot>'
assert mathml(x**(S(1)/3), printer='presentation', root_notation=False) ==\
'<msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup>'
assert mathml(x**(S(1)/3), printer='content') == \
'<apply><root/><degree><ci>3</ci></degree><ci>x</ci></apply>'
assert mathml(x**(S(1)/3), printer='content', root_notation=False) == \
'<apply><power/><ci>x</ci><apply><divide/><cn>1</cn><cn>3</cn></apply></apply>'
assert mathml(x**(-S(1)/3), printer='presentation') == \
'<mfrac><mn>1</mn><mroot><mi>x</mi><mn>3</mn></mroot></mfrac>'
assert mathml(x**(-S(1)/3), printer='presentation', root_notation=False) \
== '<mfrac><mn>1</mn><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mfrac>'
def test_fold_frac_powers_print():
expr = x ** Rational(5, 2)
assert mathml(expr, printer='presentation') == \
'<msup><mi>x</mi><mfrac><mn>5</mn><mn>2</mn></mfrac></msup>'
assert mathml(expr, printer='presentation', fold_frac_powers=True) == \
'<msup><mi>x</mi><mfrac bevelled="true"><mn>5</mn><mn>2</mn></mfrac></msup>'
assert mathml(expr, printer='presentation', fold_frac_powers=False) == \
'<msup><mi>x</mi><mfrac><mn>5</mn><mn>2</mn></mfrac></msup>'
def test_fold_short_frac_print():
expr = Rational(2, 5)
assert mathml(expr, printer='presentation') == \
'<mfrac><mn>2</mn><mn>5</mn></mfrac>'
assert mathml(expr, printer='presentation', fold_short_frac=True) == \
'<mfrac bevelled="true"><mn>2</mn><mn>5</mn></mfrac>'
assert mathml(expr, printer='presentation', fold_short_frac=False) == \
'<mfrac><mn>2</mn><mn>5</mn></mfrac>'
def test_print_factorials():
assert mpp.doprint(factorial(x)) == '<mrow><mi>x</mi><mo>!</mo></mrow>'
assert mpp.doprint(factorial(x + 1)) == \
'<mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow>'
assert mpp.doprint(factorial2(x)) == '<mrow><mi>x</mi><mo>!!</mo></mrow>'
assert mpp.doprint(factorial2(x + 1)) == \
'<mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>!!</mo></mrow>'
assert mpp.doprint(binomial(x, y)) == \
'<mfenced><mfrac linethickness="0"><mi>x</mi><mi>y</mi></mfrac></mfenced>'
assert mpp.doprint(binomial(4, x + y)) == \
'<mfenced><mfrac linethickness="0"><mn>4</mn><mrow><mi>x</mi>'\
'<mo>+</mo><mi>y</mi></mrow></mfrac></mfenced>'
def test_print_floor():
expr = floor(x)
assert mathml(expr, printer='presentation') == \
'<mrow><mfenced close="⌋" open="⌊"><mi>x</mi></mfenced></mrow>'
def test_print_ceiling():
expr = ceiling(x)
assert mathml(expr, printer='presentation') == \
'<mrow><mfenced close="⌉" open="⌈"><mi>x</mi></mfenced></mrow>'
def test_print_Lambda():
expr = Lambda(x, x+1)
assert mathml(expr, printer='presentation') == \
'<mfenced><mrow><mi>x</mi><mo>↦</mo><mrow><mi>x</mi><mo>+</mo>'\
'<mn>1</mn></mrow></mrow></mfenced>'
expr = Lambda((x, y), x + y)
assert mathml(expr, printer='presentation') == \
'<mfenced><mrow><mrow><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'\
'<mo>↦</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mrow></mfenced>'
def test_print_conjugate():
assert mpp.doprint(conjugate(x)) == \
'<menclose notation="top"><mi>x</mi></menclose>'
assert mpp.doprint(conjugate(x + 1)) == \
'<mrow><menclose notation="top"><mi>x</mi></menclose><mo>+</mo><mn>1</mn></mrow>'
def test_print_AccumBounds():
a = Symbol('a', real=True)
assert mpp.doprint(AccumBounds(0, 1)) == '<mfenced close="⟩" open="⟨"><mn>0</mn><mn>1</mn></mfenced>'
assert mpp.doprint(AccumBounds(0, a)) == '<mfenced close="⟩" open="⟨"><mn>0</mn><mi>a</mi></mfenced>'
assert mpp.doprint(AccumBounds(a + 1, a + 2)) == '<mfenced close="⟩" open="⟨"><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfenced>'
def test_print_Float():
assert mpp.doprint(Float(1e100)) == '<mrow><mn>1.0</mn><mo>·</mo><msup><mn>10</mn><mn>100</mn></msup></mrow>'
assert mpp.doprint(Float(1e-100)) == '<mrow><mn>1.0</mn><mo>·</mo><msup><mn>10</mn><mn>-100</mn></msup></mrow>'
assert mpp.doprint(Float(-1e100)) == '<mrow><mn>-1.0</mn><mo>·</mo><msup><mn>10</mn><mn>100</mn></msup></mrow>'
assert mpp.doprint(Float(1.0*oo)) == '<mi>∞</mi>'
assert mpp.doprint(Float(-1.0*oo)) == '<mrow><mo>-</mo><mi>∞</mi></mrow>'
def test_print_different_functions():
assert mpp.doprint(gamma(x)) == '<mrow><mi>Γ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(lowergamma(x, y)) == '<mrow><mi>γ</mi><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(uppergamma(x, y)) == '<mrow><mi>Γ</mi><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(zeta(x)) == '<mrow><mi>ζ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(zeta(x, y)) == '<mrow><mi>ζ</mi><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(dirichlet_eta(x)) == '<mrow><mi>η</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(elliptic_k(x)) == '<mrow><mi>Κ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(totient(x)) == '<mrow><mi>ϕ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(reduced_totient(x)) == '<mrow><mi>λ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(primenu(x)) == '<mrow><mi>ν</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(primeomega(x)) == '<mrow><mi>Ω</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(fresnels(x)) == '<mrow><mi>S</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(fresnelc(x)) == '<mrow><mi>C</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(Heaviside(x)) == '<mrow><mi>Θ</mi><mfenced><mi>x</mi></mfenced></mrow>'
def test_mathml_builtins():
assert mpp.doprint(None) == '<mi>None</mi>'
assert mpp.doprint(true) == '<mi>True</mi>'
assert mpp.doprint(false) == '<mi>False</mi>'
def test_mathml_Range():
assert mpp.doprint(Range(1, 51)) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>2</mn><mi>…</mi><mn>50</mn></mfenced>'
assert mpp.doprint(Range(1, 4)) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>2</mn><mn>3</mn></mfenced>'
assert mpp.doprint(Range(0, 3, 1)) == \
'<mfenced close="}" open="{"><mn>0</mn><mn>1</mn><mn>2</mn></mfenced>'
assert mpp.doprint(Range(0, 30, 1)) == \
'<mfenced close="}" open="{"><mn>0</mn><mn>1</mn><mi>…</mi><mn>29</mn></mfenced>'
assert mpp.doprint(Range(30, 1, -1)) == \
'<mfenced close="}" open="{"><mn>30</mn><mn>29</mn><mi>…</mi>'\
'<mn>2</mn></mfenced>'
assert mpp.doprint(Range(0, oo, 2)) == \
'<mfenced close="}" open="{"><mn>0</mn><mn>2</mn><mi>…</mi></mfenced>'
assert mpp.doprint(Range(oo, -2, -2)) == \
'<mfenced close="}" open="{"><mi>…</mi><mn>2</mn><mn>0</mn></mfenced>'
assert mpp.doprint(Range(-2, -oo, -1)) == \
'<mfenced close="}" open="{"><mn>-2</mn><mn>-3</mn><mi>…</mi></mfenced>'
def test_print_exp():
assert mpp.doprint(exp(x)) == \
'<msup><mi>ⅇ</mi><mi>x</mi></msup>'
assert mpp.doprint(exp(1) + exp(2)) == \
'<mrow><mi>ⅇ</mi><mo>+</mo><msup><mi>ⅇ</mi><mn>2</mn></msup></mrow>'
def test_print_MinMax():
assert mpp.doprint(Min(x, y)) == \
'<mrow><mo>min</mo><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(Min(x, 2, x**3)) == \
'<mrow><mo>min</mo><mfenced><mn>2</mn><mi>x</mi><msup><mi>x</mi>'\
'<mn>3</mn></msup></mfenced></mrow>'
assert mpp.doprint(Max(x, y)) == \
'<mrow><mo>max</mo><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(Max(x, 2, x**3)) == \
'<mrow><mo>max</mo><mfenced><mn>2</mn><mi>x</mi><msup><mi>x</mi>'\
'<mn>3</mn></msup></mfenced></mrow>'
def test_mathml_presentation_numbers():
n = Symbol('n')
assert mathml(catalan(n), printer='presentation') == \
'<msub><mi>C</mi><mi>n</mi></msub>'
assert mathml(bernoulli(n), printer='presentation') == \
'<msub><mi>B</mi><mi>n</mi></msub>'
assert mathml(bell(n), printer='presentation') == \
'<msub><mi>B</mi><mi>n</mi></msub>'
assert mathml(fibonacci(n), printer='presentation') == \
'<msub><mi>F</mi><mi>n</mi></msub>'
assert mathml(lucas(n), printer='presentation') == \
'<msub><mi>L</mi><mi>n</mi></msub>'
assert mathml(tribonacci(n), printer='presentation') == \
'<msub><mi>T</mi><mi>n</mi></msub>'
def test_print_matrix_symbol():
A = MatrixSymbol('A', 1, 2)
assert mpp.doprint(A) == '<mi>A</mi>'
assert mp.doprint(A) == '<ci>A</ci>'
assert mathml(A, printer='presentation', mat_symbol_style="bold") == \
'<mi mathvariant="bold">A</mi>'
# No effect in content printer
assert mathml(A, mat_symbol_style="bold") == '<ci>A</ci>'
def test_print_hadamard():
from sympy.matrices.expressions import HadamardProduct
from sympy.matrices.expressions import Transpose
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert mathml(HadamardProduct(X, Y*Y), printer="presentation") == \
'<mrow>' \
'<mi>X</mi>' \
'<mo>∘</mo>' \
'<msup><mi>Y</mi><mn>2</mn></msup>' \
'</mrow>'
assert mathml(HadamardProduct(X, Y)*Y, printer="presentation") == \
'<mrow>' \
'<mfenced>' \
'<mrow><mi>X</mi><mo>∘</mo><mi>Y</mi></mrow>' \
'</mfenced>' \
'<mo>⁢</mo><mi>Y</mi>' \
'</mrow>'
assert mathml(HadamardProduct(X, Y, Y), printer="presentation") == \
'<mrow>' \
'<mi>X</mi><mo>∘</mo>' \
'<mi>Y</mi><mo>∘</mo>' \
'<mi>Y</mi>' \
'</mrow>'
assert mathml(
Transpose(HadamardProduct(X, Y)), printer="presentation") == \
'<msup>' \
'<mfenced>' \
'<mrow><mi>X</mi><mo>∘</mo><mi>Y</mi></mrow>' \
'</mfenced>' \
'<mo>T</mo>' \
'</msup>'
def test_print_random_symbol():
R = RandomSymbol(Symbol('R'))
assert mpp.doprint(R) == '<mi>R</mi>'
assert mp.doprint(R) == '<ci>R</ci>'
def test_print_IndexedBase():
assert mathml(IndexedBase(a)[b], printer='presentation') == \
'<msub><mi>a</mi><mi>b</mi></msub>'
assert mathml(IndexedBase(a)[b, c, d], printer='presentation') == \
'<msub><mi>a</mi><mfenced><mi>b</mi><mi>c</mi><mi>d</mi></mfenced></msub>'
assert mathml(IndexedBase(a)[b]*IndexedBase(c)[d]*IndexedBase(e),
printer='presentation') == \
'<mrow><msub><mi>a</mi><mi>b</mi></msub><mo>⁢'\
'</mo><msub><mi>c</mi><mi>d</mi></msub><mo>⁢</mo><mi>e</mi></mrow>'
def test_print_Indexed():
assert mathml(IndexedBase(a), printer='presentation') == '<mi>a</mi>'
assert mathml(IndexedBase(a/b), printer='presentation') == \
'<mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow>'
assert mathml(IndexedBase((a, b)), printer='presentation') == \
'<mrow><mfenced><mi>a</mi><mi>b</mi></mfenced></mrow>'
def test_print_MatrixElement():
i, j = symbols('i j')
A = MatrixSymbol('A', i, j)
assert mathml(A[0,0],printer = 'presentation') == \
'<msub><mi>A</mi><mfenced close="" open=""><mn>0</mn><mn>0</mn></mfenced></msub>'
assert mathml(A[i,j], printer = 'presentation') == \
'<msub><mi>A</mi><mfenced close="" open=""><mi>i</mi><mi>j</mi></mfenced></msub>'
assert mathml(A[i*j,0], printer = 'presentation') == \
'<msub><mi>A</mi><mfenced close="" open=""><mrow><mi>i</mi><mo>⁢</mo><mi>j</mi></mrow><mn>0</mn></mfenced></msub>'
def test_print_Vector():
ACS = CoordSys3D('A')
assert mathml(Cross(ACS.i, ACS.j*ACS.x*3 + ACS.k), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>×</mo><mfenced><mrow>'\
'<mfenced><mrow><mn>3</mn><mo>⁢</mo><msub>'\
'<mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi></msub>'\
'</mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>+</mo><msub><mover>'\
'<mi mathvariant="bold">k</mi><mo>^</mo></mover><mi mathvariant="bold">'\
'A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Cross(ACS.i, ACS.j), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>×</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow>'
assert mathml(x*Cross(ACS.i, ACS.j), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>×</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mrow>'
assert mathml(Cross(x*ACS.i, ACS.j), printer='presentation') == \
'<mrow><mo>-</mo><mrow><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub>'\
'<mo>×</mo><mfenced><mrow><mfenced><mi>x</mi></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">i</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow>'\
'</mfenced></mrow></mrow>'
assert mathml(Curl(3*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mo>∇</mo><mo>×</mo><mfenced><mrow><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub>'\
'<mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi></msub>'\
'</mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Curl(3*x*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mo>∇</mo><mo>×</mo><mfenced><mrow><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub><mi mathvariant="bold">x'\
'</mi><mi mathvariant="bold">A</mi></msub><mo>⁢</mo>'\
'<mi>x</mi></mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(x*Curl(3*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><mo>∇</mo>'\
'<mo>×</mo><mfenced><mrow><mfenced><mrow><mn>3</mn>'\
'<mo>⁢</mo><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow>'\
'</mfenced></mrow></mrow>'
assert mathml(Curl(3*x*ACS.x*ACS.j + ACS.i), printer='presentation') == \
'<mrow><mo>∇</mo><mo>×</mo><mfenced><mrow><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>+</mo><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub><mi mathvariant="bold">x'\
'</mi><mi mathvariant="bold">A</mi></msub><mo>⁢</mo>'\
'<mi>x</mi></mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Divergence(3*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mo>∇</mo><mo>·</mo><mfenced><mrow><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub><mi mathvariant="bold">x'\
'</mi><mi mathvariant="bold">A</mi></msub></mrow></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(x*Divergence(3*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><mo>∇</mo>'\
'<mo>·</mo><mfenced><mrow><mfenced><mrow><mn>3</mn>'\
'<mo>⁢</mo><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow>'\
'</mfenced></mrow></mrow>'
assert mathml(Divergence(3*x*ACS.x*ACS.j + ACS.i), printer='presentation') == \
'<mrow><mo>∇</mo><mo>·</mo><mfenced><mrow><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>+</mo><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub>'\
'<mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi></msub>'\
'<mo>⁢</mo><mi>x</mi></mrow></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Dot(ACS.i, ACS.j*ACS.x*3+ACS.k), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>·</mo><mfenced><mrow>'\
'<mfenced><mrow><mn>3</mn><mo>⁢</mo><msub>'\
'<mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi></msub>'\
'</mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>+</mo><msub><mover>'\
'<mi mathvariant="bold">k</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Dot(ACS.i, ACS.j), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>·</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow>'
assert mathml(Dot(x*ACS.i, ACS.j), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>·</mo><mfenced><mrow>'\
'<mfenced><mi>x</mi></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(x*Dot(ACS.i, ACS.j), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>·</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mrow>'
assert mathml(Gradient(ACS.x), printer='presentation') == \
'<mrow><mo>∇</mo><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow>'
assert mathml(Gradient(ACS.x + 3*ACS.y), printer='presentation') == \
'<mrow><mo>∇</mo><mfenced><mrow><msub><mi mathvariant="bold">'\
'x</mi><mi mathvariant="bold">A</mi></msub><mo>+</mo><mrow><mn>3</mn>'\
'<mo>⁢</mo><msub><mi mathvariant="bold">y</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mrow></mfenced></mrow>'
assert mathml(x*Gradient(ACS.x), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><mo>∇</mo>'\
'<msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi>'\
'</msub></mrow></mrow>'
assert mathml(Gradient(x*ACS.x), printer='presentation') == \
'<mrow><mo>∇</mo><mfenced><mrow><msub><mi mathvariant="bold">'\
'x</mi><mi mathvariant="bold">A</mi></msub><mo>⁢</mo>'\
'<mi>x</mi></mrow></mfenced></mrow>'
assert mathml(Cross(ACS.x, ACS.z) + Cross(ACS.z, ACS.x), printer='presentation') == \
'<mover><mi mathvariant="bold">0</mi><mo>^</mo></mover>'
assert mathml(Cross(ACS.z, ACS.x), printer='presentation') == \
'<mrow><mo>-</mo><mrow><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub><mo>×</mo><msub>'\
'<mi mathvariant="bold">z</mi><mi mathvariant="bold">A</mi></msub></mrow></mrow>'
assert mathml(Laplacian(ACS.x), printer='presentation') == \
'<mrow><mo>∆</mo><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow>'
assert mathml(Laplacian(ACS.x + 3*ACS.y), printer='presentation') == \
'<mrow><mo>∆</mo><mfenced><mrow><msub><mi mathvariant="bold">'\
'x</mi><mi mathvariant="bold">A</mi></msub><mo>+</mo><mrow><mn>3</mn>'\
'<mo>⁢</mo><msub><mi mathvariant="bold">y</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mrow></mfenced></mrow>'
assert mathml(x*Laplacian(ACS.x), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><mo>∆</mo>'\
'<msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi>'\
'</msub></mrow></mrow>'
assert mathml(Laplacian(x*ACS.x), printer='presentation') == \
'<mrow><mo>∆</mo><mfenced><mrow><msub><mi mathvariant="bold">'\
'x</mi><mi mathvariant="bold">A</mi></msub><mo>⁢</mo>'\
'<mi>x</mi></mrow></mfenced></mrow>'
def test_print_elliptic_f():
assert mathml(elliptic_f(x, y), printer = 'presentation') == \
'<mrow><mi>𝖥</mi><mfenced separators="|"><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mathml(elliptic_f(x/y, y), printer = 'presentation') == \
'<mrow><mi>𝖥</mi><mfenced separators="|"><mrow><mfrac><mi>x</mi><mi>y</mi></mfrac></mrow><mi>y</mi></mfenced></mrow>'
def test_print_elliptic_e():
assert mathml(elliptic_e(x), printer = 'presentation') == \
'<mrow><mi>𝖤</mi><mfenced separators="|"><mi>x</mi></mfenced></mrow>'
assert mathml(elliptic_e(x, y), printer = 'presentation') == \
'<mrow><mi>𝖤</mi><mfenced separators="|"><mi>x</mi><mi>y</mi></mfenced></mrow>'
def test_print_elliptic_pi():
assert mathml(elliptic_pi(x, y), printer = 'presentation') == \
'<mrow><mi>𝛱</mi><mfenced separators="|"><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mathml(elliptic_pi(x, y, z), printer = 'presentation') == \
'<mrow><mi>𝛱</mi><mfenced separators=";|"><mi>x</mi><mi>y</mi><mi>z</mi></mfenced></mrow>'
def test_print_Ei():
assert mathml(Ei(x), printer = 'presentation') == \
'<mrow><mi>Ei</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mathml(Ei(x**y), printer = 'presentation') == \
'<mrow><mi>Ei</mi><mfenced><msup><mi>x</mi><mi>y</mi></msup></mfenced></mrow>'
def test_print_expint():
assert mathml(expint(x, y), printer = 'presentation') == \
'<mrow><msub><mo>E</mo><mi>x</mi></msub><mfenced><mi>y</mi></mfenced></mrow>'
assert mathml(expint(IndexedBase(x)[1], IndexedBase(x)[2]), printer = 'presentation') == \
'<mrow><msub><mo>E</mo><msub><mi>x</mi><mn>1</mn></msub></msub><mfenced><msub><mi>x</mi><mn>2</mn></msub></mfenced></mrow>'
def test_print_jacobi():
assert mathml(jacobi(n, a, b, x), printer = 'presentation') == \
'<mrow><msubsup><mo>P</mo><mi>n</mi><mfenced><mi>a</mi><mi>b</mi></mfenced></msubsup><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_gegenbauer():
assert mathml(gegenbauer(n, a, x), printer = 'presentation') == \
'<mrow><msubsup><mo>C</mo><mi>n</mi><mfenced><mi>a</mi></mfenced></msubsup><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_chebyshevt():
assert mathml(chebyshevt(n, x), printer = 'presentation') == \
'<mrow><msub><mo>T</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_chebyshevu():
assert mathml(chebyshevu(n, x), printer = 'presentation') == \
'<mrow><msub><mo>U</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_legendre():
assert mathml(legendre(n, x), printer = 'presentation') == \
'<mrow><msub><mo>P</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_assoc_legendre():
assert mathml(assoc_legendre(n, a, x), printer = 'presentation') == \
'<mrow><msubsup><mo>P</mo><mi>n</mi><mfenced><mi>a</mi></mfenced></msubsup><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_laguerre():
assert mathml(laguerre(n, x), printer = 'presentation') == \
'<mrow><msub><mo>L</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_assoc_laguerre():
assert mathml(assoc_laguerre(n, a, x), printer = 'presentation') == \
'<mrow><msubsup><mo>L</mo><mi>n</mi><mfenced><mi>a</mi></mfenced></msubsup><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_hermite():
assert mathml(hermite(n, x), printer = 'presentation') == \
'<mrow><msub><mo>H</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_mathml_SingularityFunction():
assert mathml(SingularityFunction(x, 4, 5), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mi>x</mi>' \
'<mo>-</mo><mn>4</mn></mrow></mfenced><mn>5</mn></msup>'
assert mathml(SingularityFunction(x, -3, 4), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mi>x</mi>' \
'<mo>+</mo><mn>3</mn></mrow></mfenced><mn>4</mn></msup>'
assert mathml(SingularityFunction(x, 0, 4), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mi>x</mi></mfenced>' \
'<mn>4</mn></msup>'
assert mathml(SingularityFunction(x, a, n), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mrow>' \
'<mo>-</mo><mi>a</mi></mrow><mo>+</mo><mi>x</mi></mrow></mfenced>' \
'<mi>n</mi></msup>'
assert mathml(SingularityFunction(x, 4, -2), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mi>x</mi>' \
'<mo>-</mo><mn>4</mn></mrow></mfenced><mn>-2</mn></msup>'
assert mathml(SingularityFunction(x, 4, -1), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mi>x</mi>' \
'<mo>-</mo><mn>4</mn></mrow></mfenced><mn>-1</mn></msup>'
def test_mathml_matrix_functions():
from sympy.matrices import MatrixSymbol, Adjoint, Inverse, Transpose
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert mathml(Adjoint(X), printer='presentation') == \
'<msup><mi>X</mi><mo>†</mo></msup>'
assert mathml(Adjoint(X + Y), printer='presentation') == \
'<msup><mfenced><mrow><mi>X</mi><mo>+</mo><mi>Y</mi></mrow></mfenced><mo>†</mo></msup>'
assert mathml(Adjoint(X) + Adjoint(Y), printer='presentation') == \
'<mrow><msup><mi>X</mi><mo>†</mo></msup><mo>+</mo><msup>' \
'<mi>Y</mi><mo>†</mo></msup></mrow>'
assert mathml(Adjoint(X*Y), printer='presentation') == \
'<msup><mfenced><mrow><mi>X</mi><mo>⁢</mo>' \
'<mi>Y</mi></mrow></mfenced><mo>†</mo></msup>'
assert mathml(Adjoint(Y)*Adjoint(X), printer='presentation') == \
'<mrow><msup><mi>Y</mi><mo>†</mo></msup><mo>⁢' \
'</mo><msup><mi>X</mi><mo>†</mo></msup></mrow>'
assert mathml(Adjoint(X**2), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mn>2</mn></msup></mfenced><mo>†</mo></msup>'
assert mathml(Adjoint(X)**2, printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mo>†</mo></msup></mfenced><mn>2</mn></msup>'
assert mathml(Adjoint(Inverse(X)), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mn>-1</mn></msup></mfenced><mo>†</mo></msup>'
assert mathml(Inverse(Adjoint(X)), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mo>†</mo></msup></mfenced><mn>-1</mn></msup>'
assert mathml(Adjoint(Transpose(X)), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mo>T</mo></msup></mfenced><mo>†</mo></msup>'
assert mathml(Transpose(Adjoint(X)), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mo>†</mo></msup></mfenced><mo>T</mo></msup>'
assert mathml(Transpose(Adjoint(X) + Y), printer='presentation') == \
'<msup><mfenced><mrow><msup><mi>X</mi><mo>†</mo></msup>' \
'<mo>+</mo><mi>Y</mi></mrow></mfenced><mo>T</mo></msup>'
assert mathml(Transpose(X), printer='presentation') == \
'<msup><mi>X</mi><mo>T</mo></msup>'
assert mathml(Transpose(X + Y), printer='presentation') == \
'<msup><mfenced><mrow><mi>X</mi><mo>+</mo><mi>Y</mi></mrow></mfenced><mo>T</mo></msup>'
def test_mathml_special_matrices():
from sympy.matrices import Identity, ZeroMatrix
assert mathml(Identity(4), printer='presentation') == '<mi>𝕀</mi>'
assert mathml(ZeroMatrix(2, 2), printer='presentation') == '<mn>𝟘</mn>'
|
79af259b5388f5efb7ea138ac2f6589d8cd4a676db9ef75a20b9a0a5d24359c9 | from sympy.core import (S, pi, oo, symbols, Function, Rational, Integer,
Tuple, Symbol)
from sympy.core import EulerGamma, GoldenRatio, Catalan, Lambda, Mul, Pow
from sympy.functions import (arg, atan2, bernoulli, beta, ceiling, chebyshevu,
chebyshevt, conjugate, DiracDelta, exp, expint,
factorial, floor, harmonic, Heaviside, im,
laguerre, LambertW, log, Max, Min, Piecewise,
polylog, re, RisingFactorial, sign, sinc, sqrt,
zeta)
from sympy.functions import (sin, cos, tan, cot, sec, csc, asin, acos, acot,
atan, asec, acsc, sinh, cosh, tanh, coth, csch,
sech, asinh, acosh, atanh, acoth, asech, acsch)
from sympy.utilities.pytest import raises
from sympy.utilities.lambdify import implemented_function
from sympy.matrices import (eye, Matrix, MatrixSymbol, Identity,
HadamardProduct, SparseMatrix)
from sympy.functions.special.bessel import (jn, yn, besselj, bessely, besseli,
besselk, hankel1, hankel2, airyai,
airybi, airyaiprime, airybiprime)
from sympy.functions.special.gamma_functions import (gamma, lowergamma,
uppergamma, loggamma,
polygamma)
from sympy.functions.special.error_functions import (Chi, Ci, erf, erfc, erfi,
erfcinv, erfinv, fresnelc,
fresnels, li, Shi, Si)
from sympy.utilities.pytest import XFAIL
from sympy.core.compatibility import range
from sympy import octave_code
from sympy import octave_code as mcode
x, y, z = symbols('x,y,z')
def test_Integer():
assert mcode(Integer(67)) == "67"
assert mcode(Integer(-1)) == "-1"
def test_Rational():
assert mcode(Rational(3, 7)) == "3/7"
assert mcode(Rational(18, 9)) == "2"
assert mcode(Rational(3, -7)) == "-3/7"
assert mcode(Rational(-3, -7)) == "3/7"
assert mcode(x + Rational(3, 7)) == "x + 3/7"
assert mcode(Rational(3, 7)*x) == "3*x/7"
def test_Function():
assert mcode(sin(x) ** cos(x)) == "sin(x).^cos(x)"
assert mcode(sign(x)) == "sign(x)"
assert mcode(exp(x)) == "exp(x)"
assert mcode(log(x)) == "log(x)"
assert mcode(factorial(x)) == "factorial(x)"
assert mcode(floor(x)) == "floor(x)"
assert mcode(atan2(y, x)) == "atan2(y, x)"
assert mcode(beta(x, y)) == 'beta(x, y)'
assert mcode(polylog(x, y)) == 'polylog(x, y)'
assert mcode(harmonic(x)) == 'harmonic(x)'
assert mcode(bernoulli(x)) == "bernoulli(x)"
assert mcode(bernoulli(x, y)) == "bernoulli(x, y)"
def test_Function_change_name():
assert mcode(abs(x)) == "abs(x)"
assert mcode(ceiling(x)) == "ceil(x)"
assert mcode(arg(x)) == "angle(x)"
assert mcode(im(x)) == "imag(x)"
assert mcode(re(x)) == "real(x)"
assert mcode(conjugate(x)) == "conj(x)"
assert mcode(chebyshevt(y, x)) == "chebyshevT(y, x)"
assert mcode(chebyshevu(y, x)) == "chebyshevU(y, x)"
assert mcode(laguerre(x, y)) == "laguerreL(x, y)"
assert mcode(Chi(x)) == "coshint(x)"
assert mcode(Shi(x)) == "sinhint(x)"
assert mcode(Ci(x)) == "cosint(x)"
assert mcode(Si(x)) == "sinint(x)"
assert mcode(li(x)) == "logint(x)"
assert mcode(loggamma(x)) == "gammaln(x)"
assert mcode(polygamma(x, y)) == "psi(x, y)"
assert mcode(RisingFactorial(x, y)) == "pochhammer(x, y)"
assert mcode(DiracDelta(x)) == "dirac(x)"
assert mcode(DiracDelta(x, 3)) == "dirac(3, x)"
assert mcode(Heaviside(x)) == "heaviside(x)"
assert mcode(Heaviside(x, y)) == "heaviside(x, y)"
def test_minmax():
assert mcode(Max(x, y) + Min(x, y)) == "max(x, y) + min(x, y)"
assert mcode(Max(x, y, z)) == "max(x, max(y, z))"
assert mcode(Min(x, y, z)) == "min(x, min(y, z))"
def test_Pow():
assert mcode(x**3) == "x.^3"
assert mcode(x**(y**3)) == "x.^(y.^3)"
assert mcode(x**Rational(2, 3)) == 'x.^(2/3)'
g = implemented_function('g', Lambda(x, 2*x))
assert mcode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \
"(3.5*2*x).^(-x + y.^x)./(x.^2 + y)"
# For issue 14160
assert mcode(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False),
evaluate=False)) == '-2*x./(y.*y)'
def test_basic_ops():
assert mcode(x*y) == "x.*y"
assert mcode(x + y) == "x + y"
assert mcode(x - y) == "x - y"
assert mcode(-x) == "-x"
def test_1_over_x_and_sqrt():
# 1.0 and 0.5 would do something different in regular StrPrinter,
# but these are exact in IEEE floating point so no different here.
assert mcode(1/x) == '1./x'
assert mcode(x**-1) == mcode(x**-1.0) == '1./x'
assert mcode(1/sqrt(x)) == '1./sqrt(x)'
assert mcode(x**-S.Half) == mcode(x**-0.5) == '1./sqrt(x)'
assert mcode(sqrt(x)) == 'sqrt(x)'
assert mcode(x**S.Half) == mcode(x**0.5) == 'sqrt(x)'
assert mcode(1/pi) == '1/pi'
assert mcode(pi**-1) == mcode(pi**-1.0) == '1/pi'
assert mcode(pi**-0.5) == '1/sqrt(pi)'
def test_mix_number_mult_symbols():
assert mcode(3*x) == "3*x"
assert mcode(pi*x) == "pi*x"
assert mcode(3/x) == "3./x"
assert mcode(pi/x) == "pi./x"
assert mcode(x/3) == "x/3"
assert mcode(x/pi) == "x/pi"
assert mcode(x*y) == "x.*y"
assert mcode(3*x*y) == "3*x.*y"
assert mcode(3*pi*x*y) == "3*pi*x.*y"
assert mcode(x/y) == "x./y"
assert mcode(3*x/y) == "3*x./y"
assert mcode(x*y/z) == "x.*y./z"
assert mcode(x/y*z) == "x.*z./y"
assert mcode(1/x/y) == "1./(x.*y)"
assert mcode(2*pi*x/y/z) == "2*pi*x./(y.*z)"
assert mcode(3*pi/x) == "3*pi./x"
assert mcode(S(3)/5) == "3/5"
assert mcode(S(3)/5*x) == "3*x/5"
assert mcode(x/y/z) == "x./(y.*z)"
assert mcode((x+y)/z) == "(x + y)./z"
assert mcode((x+y)/(z+x)) == "(x + y)./(x + z)"
assert mcode((x+y)/EulerGamma) == "(x + y)/%s" % EulerGamma.evalf(17)
assert mcode(x/3/pi) == "x/(3*pi)"
assert mcode(S(3)/5*x*y/pi) == "3*x.*y/(5*pi)"
def test_mix_number_pow_symbols():
assert mcode(pi**3) == 'pi^3'
assert mcode(x**2) == 'x.^2'
assert mcode(x**(pi**3)) == 'x.^(pi^3)'
assert mcode(x**y) == 'x.^y'
assert mcode(x**(y**z)) == 'x.^(y.^z)'
assert mcode((x**y)**z) == '(x.^y).^z'
def test_imag():
I = S('I')
assert mcode(I) == "1i"
assert mcode(5*I) == "5i"
assert mcode((S(3)/2)*I) == "3*1i/2"
assert mcode(3+4*I) == "3 + 4i"
assert mcode(sqrt(3)*I) == "sqrt(3)*1i"
def test_constants():
assert mcode(pi) == "pi"
assert mcode(oo) == "inf"
assert mcode(-oo) == "-inf"
assert mcode(S.NegativeInfinity) == "-inf"
assert mcode(S.NaN) == "NaN"
assert mcode(S.Exp1) == "exp(1)"
assert mcode(exp(1)) == "exp(1)"
def test_constants_other():
assert mcode(2*GoldenRatio) == "2*(1+sqrt(5))/2"
assert mcode(2*Catalan) == "2*%s" % Catalan.evalf(17)
assert mcode(2*EulerGamma) == "2*%s" % EulerGamma.evalf(17)
def test_boolean():
assert mcode(x & y) == "x & y"
assert mcode(x | y) == "x | y"
assert mcode(~x) == "~x"
assert mcode(x & y & z) == "x & y & z"
assert mcode(x | y | z) == "x | y | z"
assert mcode((x & y) | z) == "z | x & y"
assert mcode((x | y) & z) == "z & (x | y)"
def test_KroneckerDelta():
from sympy.functions import KroneckerDelta
assert mcode(KroneckerDelta(x, y)) == "double(x == y)"
assert mcode(KroneckerDelta(x, y + 1)) == "double(x == (y + 1))"
assert mcode(KroneckerDelta(2**x, y)) == "double((2.^x) == y)"
def test_Matrices():
assert mcode(Matrix(1, 1, [10])) == "10"
A = Matrix([[1, sin(x/2), abs(x)],
[0, 1, pi],
[0, exp(1), ceiling(x)]]);
expected = "[1 sin(x/2) abs(x); 0 1 pi; 0 exp(1) ceil(x)]"
assert mcode(A) == expected
# row and columns
assert mcode(A[:,0]) == "[1; 0; 0]"
assert mcode(A[0,:]) == "[1 sin(x/2) abs(x)]"
# empty matrices
assert mcode(Matrix(0, 0, [])) == '[]'
assert mcode(Matrix(0, 3, [])) == 'zeros(0, 3)'
# annoying to read but correct
assert mcode(Matrix([[x, x - y, -y]])) == "[x x - y -y]"
def test_vector_entries_hadamard():
# For a row or column, user might to use the other dimension
A = Matrix([[1, sin(2/x), 3*pi/x/5]])
assert mcode(A) == "[1 sin(2./x) 3*pi./(5*x)]"
assert mcode(A.T) == "[1; sin(2./x); 3*pi./(5*x)]"
@XFAIL
def test_Matrices_entries_not_hadamard():
# For Matrix with col >= 2, row >= 2, they need to be scalars
# FIXME: is it worth worrying about this? Its not wrong, just
# leave it user's responsibility to put scalar data for x.
A = Matrix([[1, sin(2/x), 3*pi/x/5], [1, 2, x*y]])
expected = ("[1 sin(2/x) 3*pi/(5*x);\n"
"1 2 x*y]") # <- we give x.*y
assert mcode(A) == expected
def test_MatrixSymbol():
n = Symbol('n', integer=True)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
assert mcode(A*B) == "A*B"
assert mcode(B*A) == "B*A"
assert mcode(2*A*B) == "2*A*B"
assert mcode(B*2*A) == "2*B*A"
assert mcode(A*(B + 3*Identity(n))) == "A*(3*eye(n) + B)"
assert mcode(A**(x**2)) == "A^(x.^2)"
assert mcode(A**3) == "A^3"
assert mcode(A**(S.Half)) == "A^(1/2)"
def test_special_matrices():
assert mcode(6*Identity(3)) == "6*eye(3)"
def test_containers():
assert mcode([1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]) == \
"{1, 2, 3, {4, 5, {6, 7}}, 8, {9, 10}, 11}"
assert mcode((1, 2, (3, 4))) == "{1, 2, {3, 4}}"
assert mcode([1]) == "{1}"
assert mcode((1,)) == "{1}"
assert mcode(Tuple(*[1, 2, 3])) == "{1, 2, 3}"
assert mcode((1, x*y, (3, x**2))) == "{1, x.*y, {3, x.^2}}"
# scalar, matrix, empty matrix and empty list
assert mcode((1, eye(3), Matrix(0, 0, []), [])) == "{1, [1 0 0; 0 1 0; 0 0 1], [], {}}"
def test_octave_noninline():
source = mcode((x+y)/Catalan, assign_to='me', inline=False)
expected = (
"Catalan = %s;\n"
"me = (x + y)/Catalan;"
) % Catalan.evalf(17)
assert source == expected
def test_octave_piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
assert mcode(expr) == "((x < 1).*(x) + (~(x < 1)).*(x.^2))"
assert mcode(expr, assign_to="r") == (
"r = ((x < 1).*(x) + (~(x < 1)).*(x.^2));")
assert mcode(expr, assign_to="r", inline=False) == (
"if (x < 1)\n"
" r = x;\n"
"else\n"
" r = x.^2;\n"
"end")
expr = Piecewise((x**2, x < 1), (x**3, x < 2), (x**4, x < 3), (x**5, True))
expected = ("((x < 1).*(x.^2) + (~(x < 1)).*( ...\n"
"(x < 2).*(x.^3) + (~(x < 2)).*( ...\n"
"(x < 3).*(x.^4) + (~(x < 3)).*(x.^5))))")
assert mcode(expr) == expected
assert mcode(expr, assign_to="r") == "r = " + expected + ";"
assert mcode(expr, assign_to="r", inline=False) == (
"if (x < 1)\n"
" r = x.^2;\n"
"elseif (x < 2)\n"
" r = x.^3;\n"
"elseif (x < 3)\n"
" r = x.^4;\n"
"else\n"
" r = x.^5;\n"
"end")
# Check that Piecewise without a True (default) condition error
expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0))
raises(ValueError, lambda: mcode(expr))
def test_octave_piecewise_times_const():
pw = Piecewise((x, x < 1), (x**2, True))
assert mcode(2*pw) == "2*((x < 1).*(x) + (~(x < 1)).*(x.^2))"
assert mcode(pw/x) == "((x < 1).*(x) + (~(x < 1)).*(x.^2))./x"
assert mcode(pw/(x*y)) == "((x < 1).*(x) + (~(x < 1)).*(x.^2))./(x.*y)"
assert mcode(pw/3) == "((x < 1).*(x) + (~(x < 1)).*(x.^2))/3"
def test_octave_matrix_assign_to():
A = Matrix([[1, 2, 3]])
assert mcode(A, assign_to='a') == "a = [1 2 3];"
A = Matrix([[1, 2], [3, 4]])
assert mcode(A, assign_to='A') == "A = [1 2; 3 4];"
def test_octave_matrix_assign_to_more():
# assigning to Symbol or MatrixSymbol requires lhs/rhs match
A = Matrix([[1, 2, 3]])
B = MatrixSymbol('B', 1, 3)
C = MatrixSymbol('C', 2, 3)
assert mcode(A, assign_to=B) == "B = [1 2 3];"
raises(ValueError, lambda: mcode(A, assign_to=x))
raises(ValueError, lambda: mcode(A, assign_to=C))
def test_octave_matrix_1x1():
A = Matrix([[3]])
B = MatrixSymbol('B', 1, 1)
C = MatrixSymbol('C', 1, 2)
assert mcode(A, assign_to=B) == "B = 3;"
# FIXME?
#assert mcode(A, assign_to=x) == "x = 3;"
raises(ValueError, lambda: mcode(A, assign_to=C))
def test_octave_matrix_elements():
A = Matrix([[x, 2, x*y]])
assert mcode(A[0, 0]**2 + A[0, 1] + A[0, 2]) == "x.^2 + x.*y + 2"
A = MatrixSymbol('AA', 1, 3)
assert mcode(A) == "AA"
assert mcode(A[0, 0]**2 + sin(A[0,1]) + A[0,2]) == \
"sin(AA(1, 2)) + AA(1, 1).^2 + AA(1, 3)"
assert mcode(sum(A)) == "AA(1, 1) + AA(1, 2) + AA(1, 3)"
def test_octave_boolean():
assert mcode(True) == "true"
assert mcode(S.true) == "true"
assert mcode(False) == "false"
assert mcode(S.false) == "false"
def test_octave_not_supported():
assert mcode(S.ComplexInfinity) == (
"% Not supported in Octave:\n"
"% ComplexInfinity\n"
"zoo"
)
f = Function('f')
assert mcode(f(x).diff(x)) == (
"% Not supported in Octave:\n"
"% Derivative\n"
"Derivative(f(x), x)"
)
def test_octave_not_supported_not_on_whitelist():
from sympy import assoc_laguerre
assert mcode(assoc_laguerre(x, y, z)) == (
"% Not supported in Octave:\n"
"% assoc_laguerre\n"
"assoc_laguerre(x, y, z)"
)
def test_octave_expint():
assert mcode(expint(1, x)) == "expint(x)"
assert mcode(expint(2, x)) == (
"% Not supported in Octave:\n"
"% expint\n"
"expint(2, x)"
)
assert mcode(expint(y, x)) == (
"% Not supported in Octave:\n"
"% expint\n"
"expint(y, x)"
)
def test_trick_indent_with_end_else_words():
# words starting with "end" or "else" do not confuse the indenter
t1 = S('endless');
t2 = S('elsewhere');
pw = Piecewise((t1, x < 0), (t2, x <= 1), (1, True))
assert mcode(pw, inline=False) == (
"if (x < 0)\n"
" endless\n"
"elseif (x <= 1)\n"
" elsewhere\n"
"else\n"
" 1\n"
"end")
def test_haramard():
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 3, 3)
v = MatrixSymbol('v', 3, 1)
h = MatrixSymbol('h', 1, 3)
C = HadamardProduct(A, B)
assert mcode(C) == "A.*B"
assert mcode(C*v) == "(A.*B)*v"
assert mcode(h*C*v) == "h*(A.*B)*v"
assert mcode(C*A) == "(A.*B)*A"
# mixing Hadamard and scalar strange b/c we vectorize scalars
assert mcode(C*x*y) == "(x.*y)*(A.*B)"
def test_sparse():
M = SparseMatrix(5, 6, {})
M[2, 2] = 10;
M[1, 2] = 20;
M[1, 3] = 22;
M[0, 3] = 30;
M[3, 0] = x*y;
assert mcode(M) == (
"sparse([4 2 3 1 2], [1 3 3 4 4], [x.*y 20 10 30 22], 5, 6)"
)
def test_sinc():
assert mcode(sinc(x)) == 'sinc(x/pi)'
assert mcode(sinc((x + 3))) == 'sinc((x + 3)/pi)'
assert mcode(sinc(pi*(x + 3))) == 'sinc(x + 3)'
def test_trigfun():
for f in (sin, cos, tan, cot, sec, csc, asin, acos, acot, atan, asec, acsc,
sinh, cosh, tanh, coth, csch, sech, asinh, acosh, atanh, acoth,
asech, acsch):
assert octave_code(f(x) == f.__name__ + '(x)')
def test_specfun():
n = Symbol('n')
for f in [besselj, bessely, besseli, besselk]:
assert octave_code(f(n, x)) == f.__name__ + '(n, x)'
for f in (erfc, erfi, erf, erfinv, erfcinv, fresnelc, fresnels, gamma):
assert octave_code(f(x)) == f.__name__ + '(x)'
assert octave_code(hankel1(n, x)) == 'besselh(n, 1, x)'
assert octave_code(hankel2(n, x)) == 'besselh(n, 2, x)'
assert octave_code(airyai(x)) == 'airy(0, x)'
assert octave_code(airyaiprime(x)) == 'airy(1, x)'
assert octave_code(airybi(x)) == 'airy(2, x)'
assert octave_code(airybiprime(x)) == 'airy(3, x)'
assert octave_code(uppergamma(n, x)) == '(gammainc(x, n, \'upper\').*gamma(n))'
assert octave_code(lowergamma(n, x)) == '(gammainc(x, n).*gamma(n))'
assert octave_code(z**lowergamma(n, x)) == 'z.^(gammainc(x, n).*gamma(n))'
assert octave_code(jn(n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*besselj(n + 1/2, x)/2'
assert octave_code(yn(n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*bessely(n + 1/2, x)/2'
assert octave_code(LambertW(x)) == 'lambertw(x)'
assert octave_code(LambertW(x, n)) == 'lambertw(n, x)'
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert mcode(A[0, 0]) == "A(1, 1)"
assert mcode(3 * A[0, 0]) == "3*A(1, 1)"
F = C[0, 0].subs(C, A - B)
assert mcode(F) == "(A - B)(1, 1)"
def test_zeta_printing_issue_14820():
assert octave_code(zeta(x)) == 'zeta(x)'
assert octave_code(zeta(x, y)) == '% Not supported in Octave:\n% zeta\nzeta(x, y)'
|
07b6ccdd54ed5e77d0f1134b85d7e81974514cab07c4accb1c19ca21932649f7 | # -*- coding: utf-8 -*-
from sympy import (
Add, And, Basic, Derivative, Dict, Eq, Equivalent, FF,
FiniteSet, Function, Ge, Gt, I, Implies, Integral, SingularityFunction,
Lambda, Le, Limit, Lt, Matrix, Mul, Nand, Ne, Nor, Not, O, Or,
Pow, Product, QQ, RR, Rational, Ray, rootof, RootSum, S,
Segment, Subs, Sum, Symbol, Tuple, Trace, Xor, ZZ, conjugate,
groebner, oo, pi, symbols, ilex, grlex, Range, Contains,
SeqPer, SeqFormula, SeqAdd, SeqMul, fourier_series, fps, ITE,
Complement, Interval, Intersection, Union, EulerGamma, GoldenRatio)
from sympy.codegen.ast import (Assignment, AddAugmentedAssignment,
SubAugmentedAssignment, MulAugmentedAssignment, DivAugmentedAssignment, ModAugmentedAssignment)
from sympy.core.compatibility import range, u_decode as u, PY3
from sympy.core.expr import UnevaluatedExpr
from sympy.core.trace import Tr
from sympy.functions import (Abs, Chi, Ci, Ei, KroneckerDelta,
Piecewise, Shi, Si, atan2, beta, binomial, catalan, ceiling, cos,
euler, exp, expint, factorial, factorial2, floor, gamma, hyper, log,
meijerg, sin, sqrt, subfactorial, tan, uppergamma, lerchphi,
elliptic_k, elliptic_f, elliptic_e, elliptic_pi, DiracDelta, bell,
bernoulli, fibonacci, tribonacci, lucas)
from sympy.matrices import Adjoint, Inverse, MatrixSymbol, Transpose, KroneckerProduct
from sympy.physics import mechanics
from sympy.physics.units import joule, degree
from sympy.printing.pretty import pprint, pretty as xpretty
from sympy.printing.pretty.pretty_symbology import center_accent
from sympy.sets import ImageSet
from sympy.sets.setexpr import SetExpr
from sympy.tensor.array import (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray, tensorproduct)
from sympy.tensor.functions import TensorProduct
from sympy.tensor.tensor import (TensorIndexType, tensor_indices, tensorhead,
TensorElement)
from sympy.utilities.pytest import raises, XFAIL
from sympy.vector import CoordSys3D, Gradient, Curl, Divergence, Dot, Cross, Laplacian
import sympy as sym
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
a, b, c, d, x, y, z, k, n = symbols('a,b,c,d,x,y,z,k,n')
f = Function("f")
th = Symbol('theta')
ph = Symbol('phi')
"""
Expressions whose pretty-printing is tested here:
(A '#' to the right of an expression indicates that its various acceptable
orderings are accounted for by the tests.)
BASIC EXPRESSIONS:
oo
(x**2)
1/x
y*x**-2
x**Rational(-5,2)
(-2)**x
Pow(3, 1, evaluate=False)
(x**2 + x + 1) #
1-x #
1-2*x #
x/y
-x/y
(x+2)/y #
(1+x)*y #3
-5*x/(x+10) # correct placement of negative sign
1 - Rational(3,2)*(x+1)
-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5) # issue 5524
ORDERING:
x**2 + x + 1
1 - x
1 - 2*x
2*x**4 + y**2 - x**2 + y**3
RELATIONAL:
Eq(x, y)
Lt(x, y)
Gt(x, y)
Le(x, y)
Ge(x, y)
Ne(x/(y+1), y**2) #
RATIONAL NUMBERS:
y*x**-2
y**Rational(3,2) * x**Rational(-5,2)
sin(x)**3/tan(x)**2
FUNCTIONS (ABS, CONJ, EXP, FUNCTION BRACES, FACTORIAL, FLOOR, CEILING):
(2*x + exp(x)) #
Abs(x)
Abs(x/(x**2+1)) #
Abs(1 / (y - Abs(x)))
factorial(n)
factorial(2*n)
subfactorial(n)
subfactorial(2*n)
factorial(factorial(factorial(n)))
factorial(n+1) #
conjugate(x)
conjugate(f(x+1)) #
f(x)
f(x, y)
f(x/(y+1), y) #
f(x**x**x**x**x**x)
sin(x)**2
conjugate(a+b*I)
conjugate(exp(a+b*I))
conjugate( f(1 + conjugate(f(x))) ) #
f(x/(y+1), y) # denom of first arg
floor(1 / (y - floor(x)))
ceiling(1 / (y - ceiling(x)))
SQRT:
sqrt(2)
2**Rational(1,3)
2**Rational(1,1000)
sqrt(x**2 + 1)
(1 + sqrt(5))**Rational(1,3)
2**(1/x)
sqrt(2+pi)
(2+(1+x**2)/(2+x))**Rational(1,4)+(1+x**Rational(1,1000))/sqrt(3+x**2)
DERIVATIVES:
Derivative(log(x), x, evaluate=False)
Derivative(log(x), x, evaluate=False) + x #
Derivative(log(x) + x**2, x, y, evaluate=False)
Derivative(2*x*y, y, x, evaluate=False) + x**2 #
beta(alpha).diff(alpha)
INTEGRALS:
Integral(log(x), x)
Integral(x**2, x)
Integral((sin(x))**2 / (tan(x))**2)
Integral(x**(2**x), x)
Integral(x**2, (x,1,2))
Integral(x**2, (x,Rational(1,2),10))
Integral(x**2*y**2, x,y)
Integral(x**2, (x, None, 1))
Integral(x**2, (x, 1, None))
Integral(sin(th)/cos(ph), (th,0,pi), (ph, 0, 2*pi))
MATRICES:
Matrix([[x**2+1, 1], [y, x+y]]) #
Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
PIECEWISE:
Piecewise((x,x<1),(x**2,True))
ITE:
ITE(x, y, z)
SEQUENCES (TUPLES, LISTS, DICTIONARIES):
()
[]
{}
(1/x,)
[x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
{x: sin(x)}
{1/x: 1/y, x: sin(x)**2} #
[x**2]
(x**2,)
{x**2: 1}
LIMITS:
Limit(x, x, oo)
Limit(x**2, x, 0)
Limit(1/x, x, 0)
Limit(sin(x)/x, x, 0)
UNITS:
joule => kg*m**2/s
SUBS:
Subs(f(x), x, ph**2)
Subs(f(x).diff(x), x, 0)
Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ORDER:
O(1)
O(1/x)
O(x**2 + y**2)
"""
def pretty(expr, order=None):
"""ASCII pretty-printing"""
return xpretty(expr, order=order, use_unicode=False, wrap_line=False)
def upretty(expr, order=None):
"""Unicode pretty-printing"""
return xpretty(expr, order=order, use_unicode=True, wrap_line=False)
def test_pretty_ascii_str():
assert pretty( 'xxx' ) == 'xxx'
assert pretty( "xxx" ) == 'xxx'
assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx'
assert pretty( 'xxx"xxx' ) == 'xxx\"xxx'
assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx'
assert pretty( "xxx'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\"xxx" ) == 'xxx\"xxx'
assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx'
assert pretty( "xxx\nxxx" ) == 'xxx\nxxx'
def test_pretty_unicode_str():
assert pretty( u'xxx' ) == u'xxx'
assert pretty( u'xxx' ) == u'xxx'
assert pretty( u'xxx\'xxx' ) == u'xxx\'xxx'
assert pretty( u'xxx"xxx' ) == u'xxx\"xxx'
assert pretty( u'xxx\"xxx' ) == u'xxx\"xxx'
assert pretty( u"xxx'xxx" ) == u'xxx\'xxx'
assert pretty( u"xxx\'xxx" ) == u'xxx\'xxx'
assert pretty( u"xxx\"xxx" ) == u'xxx\"xxx'
assert pretty( u"xxx\"xxx\'xxx" ) == u'xxx"xxx\'xxx'
assert pretty( u"xxx\nxxx" ) == u'xxx\nxxx'
def test_upretty_greek():
assert upretty( oo ) == u'∞'
assert upretty( Symbol('alpha^+_1') ) == u'α⁺₁'
assert upretty( Symbol('beta') ) == u'β'
assert upretty(Symbol('lambda')) == u'λ'
def test_upretty_multiindex():
assert upretty( Symbol('beta12') ) == u'β₁₂'
assert upretty( Symbol('Y00') ) == u'Y₀₀'
assert upretty( Symbol('Y_00') ) == u'Y₀₀'
assert upretty( Symbol('F^+-') ) == u'F⁺⁻'
def test_upretty_sub_super():
assert upretty( Symbol('beta_1_2') ) == u'β₁ ₂'
assert upretty( Symbol('beta^1^2') ) == u'β¹ ²'
assert upretty( Symbol('beta_1^2') ) == u'β²₁'
assert upretty( Symbol('beta_10_20') ) == u'β₁₀ ₂₀'
assert upretty( Symbol('beta_ax_gamma^i') ) == u'βⁱₐₓ ᵧ'
assert upretty( Symbol("F^1^2_3_4") ) == u'F¹ ²₃ ₄'
assert upretty( Symbol("F_1_2^3^4") ) == u'F³ ⁴₁ ₂'
assert upretty( Symbol("F_1_2_3_4") ) == u'F₁ ₂ ₃ ₄'
assert upretty( Symbol("F^1^2^3^4") ) == u'F¹ ² ³ ⁴'
def test_upretty_subs_missing_in_24():
assert upretty( Symbol('F_beta') ) == u'Fᵦ'
assert upretty( Symbol('F_gamma') ) == u'Fᵧ'
assert upretty( Symbol('F_rho') ) == u'Fᵨ'
assert upretty( Symbol('F_phi') ) == u'Fᵩ'
assert upretty( Symbol('F_chi') ) == u'Fᵪ'
assert upretty( Symbol('F_a') ) == u'Fₐ'
assert upretty( Symbol('F_e') ) == u'Fₑ'
assert upretty( Symbol('F_i') ) == u'Fᵢ'
assert upretty( Symbol('F_o') ) == u'Fₒ'
assert upretty( Symbol('F_u') ) == u'Fᵤ'
assert upretty( Symbol('F_r') ) == u'Fᵣ'
assert upretty( Symbol('F_v') ) == u'Fᵥ'
assert upretty( Symbol('F_x') ) == u'Fₓ'
def test_missing_in_2X_issue_9047():
if PY3:
assert upretty( Symbol('F_h') ) == u'Fₕ'
assert upretty( Symbol('F_k') ) == u'Fₖ'
assert upretty( Symbol('F_l') ) == u'Fₗ'
assert upretty( Symbol('F_m') ) == u'Fₘ'
assert upretty( Symbol('F_n') ) == u'Fₙ'
assert upretty( Symbol('F_p') ) == u'Fₚ'
assert upretty( Symbol('F_s') ) == u'Fₛ'
assert upretty( Symbol('F_t') ) == u'Fₜ'
def test_upretty_modifiers():
# Accents
assert upretty( Symbol('Fmathring') ) == u'F̊'
assert upretty( Symbol('Fddddot') ) == u'F⃜'
assert upretty( Symbol('Fdddot') ) == u'F⃛'
assert upretty( Symbol('Fddot') ) == u'F̈'
assert upretty( Symbol('Fdot') ) == u'Ḟ'
assert upretty( Symbol('Fcheck') ) == u'F̌'
assert upretty( Symbol('Fbreve') ) == u'F̆'
assert upretty( Symbol('Facute') ) == u'F́'
assert upretty( Symbol('Fgrave') ) == u'F̀'
assert upretty( Symbol('Ftilde') ) == u'F̃'
assert upretty( Symbol('Fhat') ) == u'F̂'
assert upretty( Symbol('Fbar') ) == u'F̅'
assert upretty( Symbol('Fvec') ) == u'F⃗'
assert upretty( Symbol('Fprime') ) == u'F′'
assert upretty( Symbol('Fprm') ) == u'F′'
# No faces are actually implemented, but test to make sure the modifiers are stripped
assert upretty( Symbol('Fbold') ) == u'Fbold'
assert upretty( Symbol('Fbm') ) == u'Fbm'
assert upretty( Symbol('Fcal') ) == u'Fcal'
assert upretty( Symbol('Fscr') ) == u'Fscr'
assert upretty( Symbol('Ffrak') ) == u'Ffrak'
# Brackets
assert upretty( Symbol('Fnorm') ) == u'‖F‖'
assert upretty( Symbol('Favg') ) == u'⟨F⟩'
assert upretty( Symbol('Fabs') ) == u'|F|'
assert upretty( Symbol('Fmag') ) == u'|F|'
# Combinations
assert upretty( Symbol('xvecdot') ) == u'x⃗̇'
assert upretty( Symbol('xDotVec') ) == u'ẋ⃗'
assert upretty( Symbol('xHATNorm') ) == u'‖x̂‖'
assert upretty( Symbol('xMathring_yCheckPRM__zbreveAbs') ) == u'x̊_y̌′__|z̆|'
assert upretty( Symbol('alphadothat_nVECDOT__tTildePrime') ) == u'α̇̂_n⃗̇__t̃′'
assert upretty( Symbol('x_dot') ) == u'x_dot'
assert upretty( Symbol('x__dot') ) == u'x__dot'
def test_pretty_Cycle():
from sympy.combinatorics.permutations import Cycle
assert pretty(Cycle(1, 2)) == '(1 2)'
assert pretty(Cycle(2)) == '(2)'
assert pretty(Cycle(1, 3)(4, 5)) == '(1 3)(4 5)'
assert pretty(Cycle()) == '()'
def test_pretty_basic():
assert pretty( -Rational(1)/2 ) == '-1/2'
assert pretty( -Rational(13)/22 ) == \
"""\
-13 \n\
----\n\
22 \
"""
expr = oo
ascii_str = \
"""\
oo\
"""
ucode_str = \
u("""\
∞\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2)
ascii_str = \
"""\
2\n\
x \
"""
ucode_str = \
u("""\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 1/x
ascii_str = \
"""\
1\n\
-\n\
x\
"""
ucode_str = \
u("""\
1\n\
─\n\
x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# not the same as 1/x
expr = x**-1.0
ascii_str = \
"""\
-1.0\n\
x \
"""
ucode_str = \
("""\
-1.0\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# see issue #2860
expr = Pow(S(2), -1.0, evaluate=False)
ascii_str = \
"""\
-1.0\n\
2 \
"""
ucode_str = \
("""\
-1.0\n\
2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
u("""\
y \n\
──\n\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
#see issue #14033
expr = x**Rational(1, 3)
ascii_str = \
"""\
1/3\n\
x \
"""
ucode_str = \
u("""\
1/3\n\
x \
""")
assert xpretty(expr, use_unicode=False, wrap_line=False,\
root_notation = False) == ascii_str
assert xpretty(expr, use_unicode=True, wrap_line=False,\
root_notation = False) == ucode_str
expr = x**Rational(-5, 2)
ascii_str = \
"""\
1 \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
u("""\
1 \n\
────\n\
5/2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (-2)**x
ascii_str = \
"""\
x\n\
(-2) \
"""
ucode_str = \
u("""\
x\n\
(-2) \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# See issue 4923
expr = Pow(3, 1, evaluate=False)
ascii_str = \
"""\
1\n\
3 \
"""
ucode_str = \
u("""\
1\n\
3 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2 + x + 1)
ascii_str_1 = \
"""\
2\n\
1 + x + x \
"""
ascii_str_2 = \
"""\
2 \n\
x + x + 1\
"""
ascii_str_3 = \
"""\
2 \n\
x + 1 + x\
"""
ucode_str_1 = \
u("""\
2\n\
1 + x + x \
""")
ucode_str_2 = \
u("""\
2 \n\
x + x + 1\
""")
ucode_str_3 = \
u("""\
2 \n\
x + 1 + x\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
expr = 1 - x
ascii_str_1 = \
"""\
1 - x\
"""
ascii_str_2 = \
"""\
-x + 1\
"""
ucode_str_1 = \
u("""\
1 - x\
""")
ucode_str_2 = \
u("""\
-x + 1\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 1 - 2*x
ascii_str_1 = \
"""\
1 - 2*x\
"""
ascii_str_2 = \
"""\
-2*x + 1\
"""
ucode_str_1 = \
u("""\
1 - 2⋅x\
""")
ucode_str_2 = \
u("""\
-2⋅x + 1\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = x/y
ascii_str = \
"""\
x\n\
-\n\
y\
"""
ucode_str = \
u("""\
x\n\
─\n\
y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/y
ascii_str = \
"""\
-x \n\
---\n\
y \
"""
ucode_str = \
u("""\
-x \n\
───\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x + 2)/y
ascii_str_1 = \
"""\
2 + x\n\
-----\n\
y \
"""
ascii_str_2 = \
"""\
x + 2\n\
-----\n\
y \
"""
ucode_str_1 = \
u("""\
2 + x\n\
─────\n\
y \
""")
ucode_str_2 = \
u("""\
x + 2\n\
─────\n\
y \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = (1 + x)*y
ascii_str_1 = \
"""\
y*(1 + x)\
"""
ascii_str_2 = \
"""\
(1 + x)*y\
"""
ascii_str_3 = \
"""\
y*(x + 1)\
"""
ucode_str_1 = \
u("""\
y⋅(1 + x)\
""")
ucode_str_2 = \
u("""\
(1 + x)⋅y\
""")
ucode_str_3 = \
u("""\
y⋅(x + 1)\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
# Test for correct placement of the negative sign
expr = -5*x/(x + 10)
ascii_str_1 = \
"""\
-5*x \n\
------\n\
10 + x\
"""
ascii_str_2 = \
"""\
-5*x \n\
------\n\
x + 10\
"""
ucode_str_1 = \
u("""\
-5⋅x \n\
──────\n\
10 + x\
""")
ucode_str_2 = \
u("""\
-5⋅x \n\
──────\n\
x + 10\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = -S(1)/2 - 3*x
ascii_str = \
"""\
-3*x - 1/2\
"""
ucode_str = \
u("""\
-3⋅x - 1/2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S(1)/2 - 3*x
ascii_str = \
"""\
1/2 - 3*x\
"""
ucode_str = \
u("""\
1/2 - 3⋅x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -S(1)/2 - 3*x/2
ascii_str = \
"""\
3*x 1\n\
- --- - -\n\
2 2\
"""
ucode_str = \
u("""\
3⋅x 1\n\
- ─── - ─\n\
2 2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S(1)/2 - 3*x/2
ascii_str = \
"""\
1 3*x\n\
- - ---\n\
2 2 \
"""
ucode_str = \
u("""\
1 3⋅x\n\
─ - ───\n\
2 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_negative_fractions():
expr = -x/y
ascii_str =\
"""\
-x \n\
---\n\
y \
"""
ucode_str =\
u("""\
-x \n\
───\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x*z/y
ascii_str =\
"""\
-x*z \n\
-----\n\
y \
"""
ucode_str =\
u("""\
-x⋅z \n\
─────\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x**2/y
ascii_str =\
"""\
2\n\
x \n\
--\n\
y \
"""
ucode_str =\
u("""\
2\n\
x \n\
──\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x**2/y
ascii_str =\
"""\
2 \n\
-x \n\
----\n\
y \
"""
ucode_str =\
u("""\
2 \n\
-x \n\
────\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/(y*z)
ascii_str =\
"""\
-x \n\
---\n\
y*z\
"""
ucode_str =\
u("""\
-x \n\
───\n\
y⋅z\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -a/y**2
ascii_str =\
"""\
-a \n\
---\n\
2\n\
y \
"""
ucode_str =\
u("""\
-a \n\
───\n\
2\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**(-a/b)
ascii_str =\
"""\
-a \n\
---\n\
b \n\
y \
"""
ucode_str =\
u("""\
-a \n\
───\n\
b \n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -1/y**2
ascii_str =\
"""\
-1 \n\
---\n\
2\n\
y \
"""
ucode_str =\
u("""\
-1 \n\
───\n\
2\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -10/b**2
ascii_str =\
"""\
-10 \n\
----\n\
2 \n\
b \
"""
ucode_str =\
u("""\
-10 \n\
────\n\
2 \n\
b \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Rational(-200, 37)
ascii_str =\
"""\
-200 \n\
-----\n\
37 \
"""
ucode_str =\
u("""\
-200 \n\
─────\n\
37 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_5524():
assert pretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
"""\
2 / ___ \\\n\
- (5 - y) + (x - 5)*\\-x - 2*\\/ 2 + 5/\
"""
assert upretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
u("""\
2 \n\
- (5 - y) + (x - 5)⋅(-x - 2⋅√2 + 5)\
""")
def test_pretty_ordering():
assert pretty(x**2 + x + 1, order='lex') == \
"""\
2 \n\
x + x + 1\
"""
assert pretty(x**2 + x + 1, order='rev-lex') == \
"""\
2\n\
1 + x + x \
"""
assert pretty(1 - x, order='lex') == '-x + 1'
assert pretty(1 - x, order='rev-lex') == '1 - x'
assert pretty(1 - 2*x, order='lex') == '-2*x + 1'
assert pretty(1 - 2*x, order='rev-lex') == '1 - 2*x'
f = 2*x**4 + y**2 - x**2 + y**3
assert pretty(f, order=None) == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='lex') == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='rev-lex') == \
"""\
2 3 2 4\n\
y + y - x + 2*x \
"""
expr = x - x**3/6 + x**5/120 + O(x**6)
ascii_str = \
"""\
3 5 \n\
x x / 6\\\n\
x - -- + --- + O\\x /\n\
6 120 \
"""
ucode_str = \
u("""\
3 5 \n\
x x ⎛ 6⎞\n\
x - ── + ─── + O⎝x ⎠\n\
6 120 \
""")
assert pretty(expr, order=None) == ascii_str
assert upretty(expr, order=None) == ucode_str
assert pretty(expr, order='lex') == ascii_str
assert upretty(expr, order='lex') == ucode_str
assert pretty(expr, order='rev-lex') == ascii_str
assert upretty(expr, order='rev-lex') == ucode_str
def test_EulerGamma():
assert pretty(EulerGamma) == str(EulerGamma) == "EulerGamma"
assert upretty(EulerGamma) == u"γ"
def test_GoldenRatio():
assert pretty(GoldenRatio) == str(GoldenRatio) == "GoldenRatio"
assert upretty(GoldenRatio) == u"φ"
def test_pretty_relational():
expr = Eq(x, y)
ascii_str = \
"""\
x = y\
"""
ucode_str = \
u("""\
x = y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lt(x, y)
ascii_str = \
"""\
x < y\
"""
ucode_str = \
u("""\
x < y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Gt(x, y)
ascii_str = \
"""\
x > y\
"""
ucode_str = \
u("""\
x > y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Le(x, y)
ascii_str = \
"""\
x <= y\
"""
ucode_str = \
u("""\
x ≤ y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ge(x, y)
ascii_str = \
"""\
x >= y\
"""
ucode_str = \
u("""\
x ≥ y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ne(x/(y + 1), y**2)
ascii_str_1 = \
"""\
x 2\n\
----- != y \n\
1 + y \
"""
ascii_str_2 = \
"""\
x 2\n\
----- != y \n\
y + 1 \
"""
ucode_str_1 = \
u("""\
x 2\n\
───── ≠ y \n\
1 + y \
""")
ucode_str_2 = \
u("""\
x 2\n\
───── ≠ y \n\
y + 1 \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
def test_Assignment():
expr = Assignment(x, y)
ascii_str = \
"""\
x := y\
"""
ucode_str = \
u("""\
x := y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_AugmentedAssignment():
expr = AddAugmentedAssignment(x, y)
ascii_str = \
"""\
x += y\
"""
ucode_str = \
u("""\
x += y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = SubAugmentedAssignment(x, y)
ascii_str = \
"""\
x -= y\
"""
ucode_str = \
u("""\
x -= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = MulAugmentedAssignment(x, y)
ascii_str = \
"""\
x *= y\
"""
ucode_str = \
u("""\
x *= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = DivAugmentedAssignment(x, y)
ascii_str = \
"""\
x /= y\
"""
ucode_str = \
u("""\
x /= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ModAugmentedAssignment(x, y)
ascii_str = \
"""\
x %= y\
"""
ucode_str = \
u("""\
x %= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_7117():
# See also issue #5031 (hence the evaluate=False in these).
e = Eq(x + 1, x/2)
q = Mul(2, e, evaluate=False)
assert upretty(q) == u("""\
⎛ x⎞\n\
2⋅⎜x + 1 = ─⎟\n\
⎝ 2⎠\
""")
q = Add(e, 6, evaluate=False)
assert upretty(q) == u("""\
⎛ x⎞\n\
6 + ⎜x + 1 = ─⎟\n\
⎝ 2⎠\
""")
q = Pow(e, 2, evaluate=False)
assert upretty(q) == u("""\
2\n\
⎛ x⎞ \n\
⎜x + 1 = ─⎟ \n\
⎝ 2⎠ \
""")
e2 = Eq(x, 2)
q = Mul(e, e2, evaluate=False)
assert upretty(q) == u("""\
⎛ x⎞ \n\
⎜x + 1 = ─⎟⋅(x = 2)\n\
⎝ 2⎠ \
""")
def test_pretty_rational():
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
u("""\
y \n\
──\n\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**Rational(3, 2) * x**Rational(-5, 2)
ascii_str = \
"""\
3/2\n\
y \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
u("""\
3/2\n\
y \n\
────\n\
5/2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**3/tan(x)**2
ascii_str = \
"""\
3 \n\
sin (x)\n\
-------\n\
2 \n\
tan (x)\
"""
ucode_str = \
u("""\
3 \n\
sin (x)\n\
───────\n\
2 \n\
tan (x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_functions():
"""Tests for Abs, conjugate, exp, function braces, and factorial."""
expr = (2*x + exp(x))
ascii_str_1 = \
"""\
x\n\
2*x + e \
"""
ascii_str_2 = \
"""\
x \n\
e + 2*x\
"""
ucode_str_1 = \
u("""\
x\n\
2⋅x + ℯ \
""")
ucode_str_2 = \
u("""\
x \n\
ℯ + 2⋅x\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Abs(x)
ascii_str = \
"""\
|x|\
"""
ucode_str = \
u("""\
│x│\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Abs(x/(x**2 + 1))
ascii_str_1 = \
"""\
| x |\n\
|------|\n\
| 2|\n\
|1 + x |\
"""
ascii_str_2 = \
"""\
| x |\n\
|------|\n\
| 2 |\n\
|x + 1|\
"""
ucode_str_1 = \
u("""\
│ x │\n\
│──────│\n\
│ 2│\n\
│1 + x │\
""")
ucode_str_2 = \
u("""\
│ x │\n\
│──────│\n\
│ 2 │\n\
│x + 1│\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Abs(1 / (y - Abs(x)))
ascii_str = \
"""\
| 1 |\n\
|-------|\n\
|y - |x||\
"""
ucode_str = \
u("""\
│ 1 │\n\
│───────│\n\
│y - │x││\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial(n)
ascii_str = \
"""\
n!\
"""
ucode_str = \
u("""\
n!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(2*n)
ascii_str = \
"""\
(2*n)!\
"""
ucode_str = \
u("""\
(2⋅n)!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(factorial(factorial(n)))
ascii_str = \
"""\
((n!)!)!\
"""
ucode_str = \
u("""\
((n!)!)!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(n + 1)
ascii_str_1 = \
"""\
(1 + n)!\
"""
ascii_str_2 = \
"""\
(n + 1)!\
"""
ucode_str_1 = \
u("""\
(1 + n)!\
""")
ucode_str_2 = \
u("""\
(n + 1)!\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = subfactorial(n)
ascii_str = \
"""\
!n\
"""
ucode_str = \
u("""\
!n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = subfactorial(2*n)
ascii_str = \
"""\
!(2*n)\
"""
ucode_str = \
u("""\
!(2⋅n)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial2(n)
ascii_str = \
"""\
n!!\
"""
ucode_str = \
u("""\
n!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(2*n)
ascii_str = \
"""\
(2*n)!!\
"""
ucode_str = \
u("""\
(2⋅n)!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(factorial2(factorial2(n)))
ascii_str = \
"""\
((n!!)!!)!!\
"""
ucode_str = \
u("""\
((n!!)!!)!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(n + 1)
ascii_str_1 = \
"""\
(1 + n)!!\
"""
ascii_str_2 = \
"""\
(n + 1)!!\
"""
ucode_str_1 = \
u("""\
(1 + n)!!\
""")
ucode_str_2 = \
u("""\
(n + 1)!!\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 2*binomial(n, k)
ascii_str = \
"""\
/n\\\n\
2*| |\n\
\\k/\
"""
ucode_str = \
u("""\
⎛n⎞\n\
2⋅⎜ ⎟\n\
⎝k⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(2*n, k)
ascii_str = \
"""\
/2*n\\\n\
2*| |\n\
\\ k /\
"""
ucode_str = \
u("""\
⎛2⋅n⎞\n\
2⋅⎜ ⎟\n\
⎝ k ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(n**2, k)
ascii_str = \
"""\
/ 2\\\n\
|n |\n\
2*| |\n\
\\k /\
"""
ucode_str = \
u("""\
⎛ 2⎞\n\
⎜n ⎟\n\
2⋅⎜ ⎟\n\
⎝k ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
u("""\
C \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
u("""\
C \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bell(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
u("""\
B \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bernoulli(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
u("""\
B \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = fibonacci(n)
ascii_str = \
"""\
F \n\
n\
"""
ucode_str = \
u("""\
F \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = lucas(n)
ascii_str = \
"""\
L \n\
n\
"""
ucode_str = \
u("""\
L \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = tribonacci(n)
ascii_str = \
"""\
T \n\
n\
"""
ucode_str = \
u("""\
T \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(x)
ascii_str = \
"""\
_\n\
x\
"""
ucode_str = \
u("""\
_\n\
x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
f = Function('f')
expr = conjugate(f(x + 1))
ascii_str_1 = \
"""\
________\n\
f(1 + x)\
"""
ascii_str_2 = \
"""\
________\n\
f(x + 1)\
"""
ucode_str_1 = \
u("""\
________\n\
f(1 + x)\
""")
ucode_str_2 = \
u("""\
________\n\
f(x + 1)\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x)
ascii_str = \
"""\
f(x)\
"""
ucode_str = \
u("""\
f(x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x, y)
ascii_str = \
"""\
f(x, y)\
"""
ucode_str = \
u("""\
f(x, y)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
""")
ucode_str_2 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x**x**x**x**x**x)
ascii_str = \
"""\
/ / / / / x\\\\\\\\\\
| | | | \\x /||||
| | | \\x /|||
| | \\x /||
| \\x /|
f\\x /\
"""
ucode_str = \
u("""\
⎛ ⎛ ⎛ ⎛ ⎛ x⎞⎞⎞⎞⎞
⎜ ⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟⎟
⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟
⎜ ⎜ ⎝x ⎠⎟⎟
⎜ ⎝x ⎠⎟
f⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**2
ascii_str = \
"""\
2 \n\
sin (x)\
"""
ucode_str = \
u("""\
2 \n\
sin (x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(a + b*I)
ascii_str = \
"""\
_ _\n\
a - I*b\
"""
ucode_str = \
u("""\
_ _\n\
a - ⅈ⋅b\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(exp(a + b*I))
ascii_str = \
"""\
_ _\n\
a - I*b\n\
e \
"""
ucode_str = \
u("""\
_ _\n\
a - ⅈ⋅b\n\
ℯ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate( f(1 + conjugate(f(x))) )
ascii_str_1 = \
"""\
___________\n\
/ ____\\\n\
f\\1 + f(x)/\
"""
ascii_str_2 = \
"""\
___________\n\
/____ \\\n\
f\\f(x) + 1/\
"""
ucode_str_1 = \
u("""\
___________\n\
⎛ ____⎞\n\
f⎝1 + f(x)⎠\
""")
ucode_str_2 = \
u("""\
___________\n\
⎛____ ⎞\n\
f⎝f(x) + 1⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
""")
ucode_str_2 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = floor(1 / (y - floor(x)))
ascii_str = \
"""\
/ 1 \\\n\
floor|------------|\n\
\\y - floor(x)/\
"""
ucode_str = \
u("""\
⎢ 1 ⎥\n\
⎢───────⎥\n\
⎣y - ⌊x⌋⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ceiling(1 / (y - ceiling(x)))
ascii_str = \
"""\
/ 1 \\\n\
ceiling|--------------|\n\
\\y - ceiling(x)/\
"""
ucode_str = \
u("""\
⎡ 1 ⎤\n\
⎢───────⎥\n\
⎢y - ⌈x⌉⎥\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n)
ascii_str = \
"""\
E \n\
n\
"""
ucode_str = \
u("""\
E \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(1/(1 + 1/(1 + 1/n)))
ascii_str = \
"""\
E \n\
1 \n\
---------\n\
1 \n\
1 + -----\n\
1\n\
1 + -\n\
n\
"""
ucode_str = \
u("""\
E \n\
1 \n\
─────────\n\
1 \n\
1 + ─────\n\
1\n\
1 + ─\n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x)
ascii_str = \
"""\
E (x)\n\
n \
"""
ucode_str = \
u("""\
E (x)\n\
n \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x/2)
ascii_str = \
"""\
/x\\\n\
E |-|\n\
n\\2/\
"""
ucode_str = \
u("""\
⎛x⎞\n\
E ⎜─⎟\n\
n⎝2⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt():
expr = sqrt(2)
ascii_str = \
"""\
___\n\
\\/ 2 \
"""
ucode_str = \
u"√2"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 3)
ascii_str = \
"""\
3 ___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
3 ___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 1000)
ascii_str = \
"""\
1000___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
1000___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(x**2 + 1)
ascii_str = \
"""\
________\n\
/ 2 \n\
\\/ x + 1 \
"""
ucode_str = \
u("""\
________\n\
╱ 2 \n\
╲╱ x + 1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1 + sqrt(5))**Rational(1, 3)
ascii_str = \
"""\
___________\n\
3 / ___ \n\
\\/ 1 + \\/ 5 \
"""
ucode_str = \
u("""\
3 ________\n\
╲╱ 1 + √5 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**(1/x)
ascii_str = \
"""\
x ___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
x ___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(2 + pi)
ascii_str = \
"""\
________\n\
\\/ 2 + pi \
"""
ucode_str = \
u("""\
_______\n\
╲╱ 2 + π \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (2 + (
1 + x**2)/(2 + x))**Rational(1, 4) + (1 + x**Rational(1, 1000))/sqrt(3 + x**2)
ascii_str = \
"""\
____________ \n\
/ 2 1000___ \n\
/ x + 1 \\/ x + 1\n\
4 / 2 + ------ + -----------\n\
\\/ x + 2 ________\n\
/ 2 \n\
\\/ x + 3 \
"""
ucode_str = \
u("""\
____________ \n\
╱ 2 1000___ \n\
╱ x + 1 ╲╱ x + 1\n\
4 ╱ 2 + ────── + ───────────\n\
╲╱ x + 2 ________\n\
╱ 2 \n\
╲╱ x + 3 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt_char_knob():
# See PR #9234.
expr = sqrt(2)
ucode_str1 = \
u("""\
___\n\
╲╱ 2 \
""")
ucode_str2 = \
u"√2"
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=False) == ucode_str1
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=True) == ucode_str2
def test_pretty_sqrt_longsymbol_no_sqrt_char():
# Do not use unicode sqrt char for long symbols (see PR #9234).
expr = sqrt(Symbol('C1'))
ucode_str = \
u("""\
____\n\
╲╱ C₁ \
""")
assert upretty(expr) == ucode_str
def test_pretty_KroneckerDelta():
x, y = symbols("x, y")
expr = KroneckerDelta(x, y)
ascii_str = \
"""\
d \n\
x,y\
"""
ucode_str = \
u("""\
δ \n\
x,y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_product():
n, m, k, l = symbols('n m k l')
f = symbols('f', cls=Function)
expr = Product(f((n/3)**2), (n, k**2, l))
unicode_str = \
u("""\
l \n\
─┬──────┬─ \n\
│ │ ⎛ 2⎞\n\
│ │ ⎜n ⎟\n\
│ │ f⎜──⎟\n\
│ │ ⎝9 ⎠\n\
│ │ \n\
2 \n\
n = k """)
ascii_str = \
"""\
l \n\
__________ \n\
| | / 2\\\n\
| | |n |\n\
| | f|--|\n\
| | \\9 /\n\
| | \n\
2 \n\
n = k """
expr = Product(f((n/3)**2), (n, k**2, l), (l, 1, m))
unicode_str = \
u("""\
m l \n\
─┬──────┬─ ─┬──────┬─ \n\
│ │ │ │ ⎛ 2⎞\n\
│ │ │ │ ⎜n ⎟\n\
│ │ │ │ f⎜──⎟\n\
│ │ │ │ ⎝9 ⎠\n\
│ │ │ │ \n\
l = 1 2 \n\
n = k """)
ascii_str = \
"""\
m l \n\
__________ __________ \n\
| | | | / 2\\\n\
| | | | |n |\n\
| | | | f|--|\n\
| | | | \\9 /\n\
| | | | \n\
l = 1 2 \n\
n = k """
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_pretty_lambda():
# S.IdentityFunction is a special case
expr = Lambda(y, y)
assert pretty(expr) == "x -> x"
assert upretty(expr) == u"x ↦ x"
expr = Lambda(x, x+1)
assert pretty(expr) == "x -> x + 1"
assert upretty(expr) == u"x ↦ x + 1"
expr = Lambda(x, x**2)
ascii_str = \
"""\
2\n\
x -> x \
"""
ucode_str = \
u("""\
2\n\
x ↦ x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda(x, x**2)**2
ascii_str = \
"""\
2
/ 2\\ \n\
\\x -> x / \
"""
ucode_str = \
u("""\
2
⎛ 2⎞ \n\
⎝x ↦ x ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x)
ascii_str = "(x, y) -> x"
ucode_str = u"(x, y) ↦ x"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x**2)
ascii_str = \
"""\
2\n\
(x, y) -> x \
"""
ucode_str = \
u("""\
2\n\
(x, y) ↦ x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_order():
expr = O(1)
ascii_str = \
"""\
O(1)\
"""
ucode_str = \
u("""\
O(1)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x)
ascii_str = \
"""\
/1\\\n\
O|-|\n\
\\x/\
"""
ucode_str = \
u("""\
⎛1⎞\n\
O⎜─⎟\n\
⎝x⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2)
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (0, 0)/\
"""
ucode_str = \
u("""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (0, 0)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1, (x, oo))
ascii_str = \
"""\
O(1; x -> oo)\
"""
ucode_str = \
u("""\
O(1; x → ∞)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x, (x, oo))
ascii_str = \
"""\
/1 \\\n\
O|-; x -> oo|\n\
\\x /\
"""
ucode_str = \
u("""\
⎛1 ⎞\n\
O⎜─; x → ∞⎟\n\
⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2, (x, oo), (y, oo))
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (oo, oo)/\
"""
ucode_str = \
u("""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (∞, ∞)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_derivatives():
# Simple
expr = Derivative(log(x), x, evaluate=False)
ascii_str = \
"""\
d \n\
--(log(x))\n\
dx \
"""
ucode_str = \
u("""\
d \n\
──(log(x))\n\
dx \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(log(x), x, evaluate=False) + x
ascii_str_1 = \
"""\
d \n\
x + --(log(x))\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(log(x)) + x\n\
dx \
"""
ucode_str_1 = \
u("""\
d \n\
x + ──(log(x))\n\
dx \
""")
ucode_str_2 = \
u("""\
d \n\
──(log(x)) + x\n\
dx \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
# basic partial derivatives
expr = Derivative(log(x + y) + x, x)
ascii_str_1 = \
"""\
d \n\
--(log(x + y) + x)\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(x + log(x + y))\n\
dx \
"""
ucode_str_1 = \
u("""\
∂ \n\
──(log(x + y) + x)\n\
∂x \
""")
ucode_str_2 = \
u("""\
∂ \n\
──(x + log(x + y))\n\
∂x \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2], upretty(expr)
# Multiple symbols
expr = Derivative(log(x) + x**2, x, y)
ascii_str_1 = \
"""\
2 \n\
d / 2\\\n\
-----\\log(x) + x /\n\
dy dx \
"""
ascii_str_2 = \
"""\
2 \n\
d / 2 \\\n\
-----\\x + log(x)/\n\
dy dx \
"""
ucode_str_1 = \
u("""\
2 \n\
d ⎛ 2⎞\n\
─────⎝log(x) + x ⎠\n\
dy dx \
""")
ucode_str_2 = \
u("""\
2 \n\
d ⎛ 2 ⎞\n\
─────⎝x + log(x)⎠\n\
dy dx \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, y, x) + x**2
ascii_str_1 = \
"""\
2 \n\
d 2\n\
-----(2*x*y) + x \n\
dx dy \
"""
ascii_str_2 = \
"""\
2 \n\
2 d \n\
x + -----(2*x*y)\n\
dx dy \
"""
ucode_str_1 = \
u("""\
2 \n\
∂ 2\n\
─────(2⋅x⋅y) + x \n\
∂x ∂y \
""")
ucode_str_2 = \
u("""\
2 \n\
2 ∂ \n\
x + ─────(2⋅x⋅y)\n\
∂x ∂y \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, x, x)
ascii_str = \
"""\
2 \n\
d \n\
---(2*x*y)\n\
2 \n\
dx \
"""
ucode_str = \
u("""\
2 \n\
∂ \n\
───(2⋅x⋅y)\n\
2 \n\
∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, 17)
ascii_str = \
"""\
17 \n\
d \n\
----(2*x*y)\n\
17 \n\
dx \
"""
ucode_str = \
u("""\
17 \n\
∂ \n\
────(2⋅x⋅y)\n\
17 \n\
∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, x, y)
ascii_str = \
"""\
3 \n\
d \n\
------(2*x*y)\n\
2 \n\
dy dx \
"""
ucode_str = \
u("""\
3 \n\
∂ \n\
──────(2⋅x⋅y)\n\
2 \n\
∂y ∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# Greek letters
alpha = Symbol('alpha')
beta = Function('beta')
expr = beta(alpha).diff(alpha)
ascii_str = \
"""\
d \n\
------(beta(alpha))\n\
dalpha \
"""
ucode_str = \
u("""\
d \n\
──(β(α))\n\
dα \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(f(x), (x, n))
ascii_str = \
"""\
n \n\
d \n\
---(f(x))\n\
n \n\
dx \
"""
ucode_str = \
u("""\
n \n\
d \n\
───(f(x))\n\
n \n\
dx \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_integrals():
expr = Integral(log(x), x)
ascii_str = \
"""\
/ \n\
| \n\
| log(x) dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ log(x) dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, x)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral((sin(x))**2 / (tan(x))**2)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| sin (x) \n\
| ------- dx\n\
| 2 \n\
| tan (x) \n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ 2 \n\
⎮ sin (x) \n\
⎮ ─────── dx\n\
⎮ 2 \n\
⎮ tan (x) \n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**(2**x), x)
ascii_str = \
"""\
/ \n\
| \n\
| / x\\ \n\
| \\2 / \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ ⎛ x⎞ \n\
⎮ ⎝2 ⎠ \n\
⎮ x dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, 1, 2))
ascii_str = \
"""\
2 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1 \
"""
ucode_str = \
u("""\
2 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, Rational(1, 2), 10))
ascii_str = \
"""\
10 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1/2 \
"""
ucode_str = \
u("""\
10 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1/2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2*y**2, x, y)
ascii_str = \
"""\
/ / \n\
| | \n\
| | 2 2 \n\
| | x *y dx dy\n\
| | \n\
/ / \
"""
ucode_str = \
u("""\
⌠ ⌠ \n\
⎮ ⎮ 2 2 \n\
⎮ ⎮ x ⋅y dx dy\n\
⌡ ⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(sin(th)/cos(ph), (th, 0, pi), (ph, 0, 2*pi))
ascii_str = \
"""\
2*pi pi \n\
/ / \n\
| | \n\
| | sin(theta) \n\
| | ---------- d(theta) d(phi)\n\
| | cos(phi) \n\
| | \n\
/ / \n\
0 0 \
"""
ucode_str = \
u("""\
2⋅π π \n\
⌠ ⌠ \n\
⎮ ⎮ sin(θ) \n\
⎮ ⎮ ────── dθ dφ\n\
⎮ ⎮ cos(φ) \n\
⌡ ⌡ \n\
0 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_matrix():
# Empty Matrix
expr = Matrix()
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(2, 0, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(0, 2, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix([[x**2 + 1, 1], [y, x + y]])
ascii_str_1 = \
"""\
[ 2 ]
[1 + x 1 ]
[ ]
[ y x + y]\
"""
ascii_str_2 = \
"""\
[ 2 ]
[x + 1 1 ]
[ ]
[ y x + y]\
"""
ucode_str_1 = \
u("""\
⎡ 2 ⎤
⎢1 + x 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
""")
ucode_str_2 = \
u("""\
⎡ 2 ⎤
⎢x + 1 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
ascii_str = \
"""\
[x ]
[- y theta]
[y ]
[ ]
[ I*k*phi ]
[0 e 1 ]\
"""
ucode_str = \
u("""\
⎡x ⎤
⎢─ y θ⎥
⎢y ⎥
⎢ ⎥
⎢ ⅈ⋅k⋅φ ⎥
⎣0 ℯ 1⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ndim_arrays():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray, MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert pretty(M) == "x"
assert upretty(M) == "x"
M = ArrayType([[1/x, y], [z, w]])
M1 = ArrayType([1/x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
ascii_str = \
"""\
[1 ]\n\
[- y]\n\
[x ]\n\
[ ]\n\
[z w]\
"""
ucode_str = \
u("""\
⎡1 ⎤\n\
⎢─ y⎥\n\
⎢x ⎥\n\
⎢ ⎥\n\
⎣z w⎦\
""")
assert pretty(M) == ascii_str
assert upretty(M) == ucode_str
ascii_str = \
"""\
[1 ]\n\
[- y z]\n\
[x ]\
"""
ucode_str = \
u("""\
⎡1 ⎤\n\
⎢─ y z⎥\n\
⎣x ⎦\
""")
assert pretty(M1) == ascii_str
assert upretty(M1) == ucode_str
ascii_str = \
"""\
[[1 y] ]\n\
[[-- -] [z ]]\n\
[[ 2 x] [ y 2 ] [- y*z]]\n\
[[x ] [ - y ] [x ]]\n\
[[ ] [ x ] [ ]]\n\
[[z w] [ ] [ 2 ]]\n\
[[- -] [y*z w*y] [z w*z]]\n\
[[x x] ]\
"""
ucode_str = \
u("""\
⎡⎡1 y⎤ ⎤\n\
⎢⎢── ─⎥ ⎡z ⎤⎥\n\
⎢⎢ 2 x⎥ ⎡ y 2 ⎤ ⎢─ y⋅z⎥⎥\n\
⎢⎢x ⎥ ⎢ ─ y ⎥ ⎢x ⎥⎥\n\
⎢⎢ ⎥ ⎢ x ⎥ ⎢ ⎥⎥\n\
⎢⎢z w⎥ ⎢ ⎥ ⎢ 2 ⎥⎥\n\
⎢⎢─ ─⎥ ⎣y⋅z w⋅y⎦ ⎣z w⋅z⎦⎥\n\
⎣⎣x x⎦ ⎦\
""")
assert pretty(M2) == ascii_str
assert upretty(M2) == ucode_str
ascii_str = \
"""\
[ [1 y] ]\n\
[ [-- -] ]\n\
[ [ 2 x] [ y 2 ]]\n\
[ [x ] [ - y ]]\n\
[ [ ] [ x ]]\n\
[ [z w] [ ]]\n\
[ [- -] [y*z w*y]]\n\
[ [x x] ]\n\
[ ]\n\
[[z ] [ w ]]\n\
[[- y*z] [ - w*y]]\n\
[[x ] [ x ]]\n\
[[ ] [ ]]\n\
[[ 2 ] [ 2 ]]\n\
[[z w*z] [w*z w ]]\
"""
ucode_str = \
u("""\
⎡ ⎡1 y⎤ ⎤\n\
⎢ ⎢── ─⎥ ⎥\n\
⎢ ⎢ 2 x⎥ ⎡ y 2 ⎤⎥\n\
⎢ ⎢x ⎥ ⎢ ─ y ⎥⎥\n\
⎢ ⎢ ⎥ ⎢ x ⎥⎥\n\
⎢ ⎢z w⎥ ⎢ ⎥⎥\n\
⎢ ⎢─ ─⎥ ⎣y⋅z w⋅y⎦⎥\n\
⎢ ⎣x x⎦ ⎥\n\
⎢ ⎥\n\
⎢⎡z ⎤ ⎡ w ⎤⎥\n\
⎢⎢─ y⋅z⎥ ⎢ ─ w⋅y⎥⎥\n\
⎢⎢x ⎥ ⎢ x ⎥⎥\n\
⎢⎢ ⎥ ⎢ ⎥⎥\n\
⎢⎢ 2 ⎥ ⎢ 2 ⎥⎥\n\
⎣⎣z w⋅z⎦ ⎣w⋅z w ⎦⎦\
""")
assert pretty(M3) == ascii_str
assert upretty(M3) == ucode_str
Mrow = ArrayType([[x, y, 1 / z]])
Mcolumn = ArrayType([[x], [y], [1 / z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
ascii_str = \
"""\
[[ 1]]\n\
[[x y -]]\n\
[[ z]]\
"""
ucode_str = \
u("""\
⎡⎡ 1⎤⎤\n\
⎢⎢x y ─⎥⎥\n\
⎣⎣ z⎦⎦\
""")
assert pretty(Mrow) == ascii_str
assert upretty(Mrow) == ucode_str
ascii_str = \
"""\
[x]\n\
[ ]\n\
[y]\n\
[ ]\n\
[1]\n\
[-]\n\
[z]\
"""
ucode_str = \
u("""\
⎡x⎤\n\
⎢ ⎥\n\
⎢y⎥\n\
⎢ ⎥\n\
⎢1⎥\n\
⎢─⎥\n\
⎣z⎦\
""")
assert pretty(Mcolumn) == ascii_str
assert upretty(Mcolumn) == ucode_str
ascii_str = \
"""\
[[x]]\n\
[[ ]]\n\
[[y]]\n\
[[ ]]\n\
[[1]]\n\
[[-]]\n\
[[z]]\
"""
ucode_str = \
u("""\
⎡⎡x⎤⎤\n\
⎢⎢ ⎥⎥\n\
⎢⎢y⎥⎥\n\
⎢⎢ ⎥⎥\n\
⎢⎢1⎥⎥\n\
⎢⎢─⎥⎥\n\
⎣⎣z⎦⎦\
""")
assert pretty(Mcol2) == ascii_str
assert upretty(Mcol2) == ucode_str
def test_tensor_TensorProduct():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert upretty(TensorProduct(A, B)) == "A\u2297B"
assert upretty(TensorProduct(A, B, A)) == "A\u2297B\u2297A"
def test_diffgeom_print_WedgeProduct():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert upretty(wp) == u("ⅆ x∧ⅆ y")
def test_Adjoint():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert pretty(Adjoint(X)) == " +\nX "
assert pretty(Adjoint(X + Y)) == " +\n(X + Y) "
assert pretty(Adjoint(X) + Adjoint(Y)) == " + +\nX + Y "
assert pretty(Adjoint(X*Y)) == " +\n(X*Y) "
assert pretty(Adjoint(Y)*Adjoint(X)) == " + +\nY *X "
assert pretty(Adjoint(X**2)) == " +\n/ 2\\ \n\\X / "
assert pretty(Adjoint(X)**2) == " 2\n/ +\\ \n\\X / "
assert pretty(Adjoint(Inverse(X))) == " +\n/ -1\\ \n\\X / "
assert pretty(Inverse(Adjoint(X))) == " -1\n/ +\\ \n\\X / "
assert pretty(Adjoint(Transpose(X))) == " +\n/ T\\ \n\\X / "
assert pretty(Transpose(Adjoint(X))) == " T\n/ +\\ \n\\X / "
assert upretty(Adjoint(X)) == u" †\nX "
assert upretty(Adjoint(X + Y)) == u" †\n(X + Y) "
assert upretty(Adjoint(X) + Adjoint(Y)) == u" † †\nX + Y "
assert upretty(Adjoint(X*Y)) == u" †\n(X⋅Y) "
assert upretty(Adjoint(Y)*Adjoint(X)) == u" † †\nY ⋅X "
assert upretty(Adjoint(X**2)) == \
u" †\n⎛ 2⎞ \n⎝X ⎠ "
assert upretty(Adjoint(X)**2) == \
u" 2\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Inverse(X))) == \
u" †\n⎛ -1⎞ \n⎝X ⎠ "
assert upretty(Inverse(Adjoint(X))) == \
u" -1\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Transpose(X))) == \
u" †\n⎛ T⎞ \n⎝X ⎠ "
assert upretty(Transpose(Adjoint(X))) == \
u" T\n⎛ †⎞ \n⎝X ⎠ "
def test_pretty_Trace_issue_9044():
X = Matrix([[1, 2], [3, 4]])
Y = Matrix([[2, 4], [6, 8]])
ascii_str_1 = \
"""\
/[1 2]\\
tr|[ ]|
\\[3 4]/\
"""
ucode_str_1 = \
u("""\
⎛⎡1 2⎤⎞
tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠\
""")
ascii_str_2 = \
"""\
/[1 2]\\ /[2 4]\\
tr|[ ]| + tr|[ ]|
\\[3 4]/ \\[6 8]/\
"""
ucode_str_2 = \
u("""\
⎛⎡1 2⎤⎞ ⎛⎡2 4⎤⎞
tr⎜⎢ ⎥⎟ + tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠ ⎝⎣6 8⎦⎠\
""")
assert pretty(Trace(X)) == ascii_str_1
assert upretty(Trace(X)) == ucode_str_1
assert pretty(Trace(X) + Trace(Y)) == ascii_str_2
assert upretty(Trace(X) + Trace(Y)) == ucode_str_2
def test_MatrixExpressions():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
assert pretty(X) == upretty(X) == "X"
Y = X[1:2:3, 4:5:6]
ascii_str = ucode_str = "X[1:3, 4:6]"
assert pretty(Y) == ascii_str
assert upretty(Y) == ucode_str
Z = X[1:10:2]
ascii_str = ucode_str = "X[1:10:2, :n]"
assert pretty(Z) == ascii_str
assert upretty(Z) == ucode_str
def test_pretty_dotproduct():
from sympy.matrices import Matrix, MatrixSymbol
from sympy.matrices.expressions.dotproduct import DotProduct
n = symbols("n", integer=True)
A = MatrixSymbol('A', n, 1)
B = MatrixSymbol('B', n, 1)
C = Matrix(1, 3, [1, 2, 3])
D = Matrix(1, 3, [1, 3, 4])
assert pretty(DotProduct(A, B)) == u"A*B"
assert pretty(DotProduct(C, D)) == u"[1 2 3]*[1 3 4]"
assert upretty(DotProduct(A, B)) == u"A⋅B"
assert upretty(DotProduct(C, D)) == u"[1 2 3]⋅[1 3 4]"
def test_pretty_piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
/x for x < 1\n\
| \n\
< 2 \n\
|x otherwise\n\
\\ \
"""
ucode_str = \
u("""\
⎧x for x < 1\n\
⎪ \n\
⎨ 2 \n\
⎪x otherwise\n\
⎩ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
//x for x < 1\\\n\
|| |\n\
-|< 2 |\n\
||x otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x for x < 1⎞\n\
⎜⎪ ⎟\n\
-⎜⎨ 2 ⎟\n\
⎜⎪x otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x + Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x + |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
u("""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x + ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x - Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x - |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
u("""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x - ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x*Piecewise((x, x > 0), (y, True))
ascii_str = \
"""\
//x for x > 0\\\n\
x*|< |\n\
\\\\y otherwise/\
"""
ucode_str = \
u("""\
⎛⎧x for x > 0⎞\n\
x⋅⎜⎨ ⎟\n\
⎝⎩y otherwise⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x >
2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x
> 2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
-|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
-⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((0, Abs(1/y) < 1), (1, Abs(y) < 1), (y*meijerg(((2, 1),
()), ((), (1, 0)), 1/y), True))
ascii_str = \
"""\
/ |1| \n\
| 0 for |-| < 1\n\
| |y| \n\
| \n\
< 1 for |y| < 1\n\
| \n\
| __0, 2 /2, 1 | 1\\ \n\
|y*/__ | | -| otherwise \n\
\\ \\_|2, 2 \\ 1, 0 | y/ \
"""
ucode_str = \
u("""\
⎧ │1│ \n\
⎪ 0 for │─│ < 1\n\
⎪ │y│ \n\
⎪ \n\
⎨ 1 for │y│ < 1\n\
⎪ \n\
⎪ ╭─╮0, 2 ⎛2, 1 │ 1⎞ \n\
⎪y⋅│╶┐ ⎜ │ ─⎟ otherwise \n\
⎩ ╰─╯2, 2 ⎝ 1, 0 │ y⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# XXX: We have to use evaluate=False here because Piecewise._eval_power
# denests the power.
expr = Pow(Piecewise((x, x > 0), (y, True)), 2, evaluate=False)
ascii_str = \
"""\
2\n\
//x for x > 0\\ \n\
|< | \n\
\\\\y otherwise/ \
"""
ucode_str = \
u("""\
2\n\
⎛⎧x for x > 0⎞ \n\
⎜⎨ ⎟ \n\
⎝⎩y otherwise⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ITE():
expr = ITE(x, y, z)
assert pretty(expr) == (
'/y for x \n'
'< \n'
'\\z otherwise'
)
assert upretty(expr) == u("""\
⎧y for x \n\
⎨ \n\
⎩z otherwise\
""")
def test_pretty_seq():
expr = ()
ascii_str = \
"""\
()\
"""
ucode_str = \
u("""\
()\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = []
ascii_str = \
"""\
[]\
"""
ucode_str = \
u("""\
[]\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {}
expr_2 = {}
ascii_str = \
"""\
{}\
"""
ucode_str = \
u("""\
{}\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = (1/x,)
ascii_str = \
"""\
1 \n\
(-,)\n\
x \
"""
ucode_str = \
u("""\
⎛1 ⎞\n\
⎜─,⎟\n\
⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = [x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
[x , -, x, y, -----------]\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎡ 2 ⎤\n\
⎢ 2 1 sin (θ)⎥\n\
⎢x , ─, x, y, ───────⎥\n\
⎢ x 2 ⎥\n\
⎣ cos (φ)⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x: sin(x)}
expr_2 = Dict({x: sin(x)})
ascii_str = \
"""\
{x: sin(x)}\
"""
ucode_str = \
u("""\
{x: sin(x)}\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = {1/x: 1/y, x: sin(x)**2}
expr_2 = Dict({1/x: 1/y, x: sin(x)**2})
ascii_str = \
"""\
1 1 2 \n\
{-: -, x: sin (x)}\n\
x y \
"""
ucode_str = \
u("""\
⎧1 1 2 ⎫\n\
⎨─: ─, x: sin (x)⎬\n\
⎩x y ⎭\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
# There used to be a bug with pretty-printing sequences of even height.
expr = [x**2]
ascii_str = \
"""\
2 \n\
[x ]\
"""
ucode_str = \
u("""\
⎡ 2⎤\n\
⎣x ⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2,)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎝x ,⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎝x ,⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x**2: 1}
expr_2 = Dict({x**2: 1})
ascii_str = \
"""\
2 \n\
{x : 1}\
"""
ucode_str = \
u("""\
⎧ 2 ⎫\n\
⎨x : 1⎬\n\
⎩ ⎭\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
def test_any_object_in_sequence():
# Cf. issue 5306
b1 = Basic()
b2 = Basic(Basic())
expr = [b2, b1]
assert pretty(expr) == "[Basic(Basic()), Basic()]"
assert upretty(expr) == u"[Basic(Basic()), Basic()]"
expr = {b2, b1}
assert pretty(expr) == "{Basic(), Basic(Basic())}"
assert upretty(expr) == u"{Basic(), Basic(Basic())}"
expr = {b2: b1, b1: b2}
expr2 = Dict({b2: b1, b1: b2})
assert pretty(expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert pretty(
expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr) == u"{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr2) == u"{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
def test_print_builtin_set():
assert pretty(set()) == 'set()'
assert upretty(set()) == u'set()'
assert pretty(frozenset()) == 'frozenset()'
assert upretty(frozenset()) == u'frozenset()'
s1 = {1/x, x}
s2 = frozenset(s1)
assert pretty(s1) == \
"""\
1 \n\
{-, x}
x \
"""
assert upretty(s1) == \
u"""\
⎧1 ⎫
⎨─, x⎬
⎩x ⎭\
"""
assert pretty(s2) == \
"""\
1 \n\
frozenset({-, x})
x \
"""
assert upretty(s2) == \
u"""\
⎛⎧1 ⎫⎞
frozenset⎜⎨─, x⎬⎟
⎝⎩x ⎭⎠\
"""
def test_pretty_sets():
s = FiniteSet
assert pretty(s(*[x*y, x**2])) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(s(*range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(s(*range(1, 13))) == "{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty(set([x*y, x**2])) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(set(range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(set(range(1, 13))) == \
"{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty(frozenset([x*y, x**2])) == \
"""\
2 \n\
frozenset({x , x*y})\
"""
assert pretty(frozenset(range(1, 6))) == "frozenset({1, 2, 3, 4, 5})"
assert pretty(frozenset(range(1, 13))) == \
"frozenset({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12})"
assert pretty(Range(0, 3, 1)) == '{0, 1, 2}'
ascii_str = '{0, 1, ..., 29}'
ucode_str = u'{0, 1, …, 29}'
assert pretty(Range(0, 30, 1)) == ascii_str
assert upretty(Range(0, 30, 1)) == ucode_str
ascii_str = '{30, 29, ..., 2}'
ucode_str = u('{30, 29, …, 2}')
assert pretty(Range(30, 1, -1)) == ascii_str
assert upretty(Range(30, 1, -1)) == ucode_str
ascii_str = '{0, 2, ...}'
ucode_str = u'{0, 2, …}'
assert pretty(Range(0, oo, 2)) == ascii_str
assert upretty(Range(0, oo, 2)) == ucode_str
ascii_str = '{..., 2, 0}'
ucode_str = u('{…, 2, 0}')
assert pretty(Range(oo, -2, -2)) == ascii_str
assert upretty(Range(oo, -2, -2)) == ucode_str
ascii_str = '{-2, -3, ...}'
ucode_str = u('{-2, -3, …}')
assert pretty(Range(-2, -oo, -1)) == ascii_str
assert upretty(Range(-2, -oo, -1)) == ucode_str
def test_pretty_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
ascii_str = "SetExpr([1, 3])"
ucode_str = u("SetExpr([1, 3])")
assert pretty(se) == ascii_str
assert upretty(se) == ucode_str
def test_pretty_ImageSet():
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
ascii_str = '{x + y | x in {1, 2, 3} , y in {3, 4}}'
ucode_str = u('{x + y | x ∊ {1, 2, 3} , y ∊ {3, 4}}')
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda(x, x**2), S.Naturals)
ascii_str = \
' 2 \n'\
'{x | x in Naturals}'
ucode_str = u('''\
⎧ 2 ⎫\n\
⎨x | x ∊ ℕ⎬\n\
⎩ ⎭''')
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
def test_pretty_ConditionSet():
from sympy import ConditionSet
ascii_str = '{x | x in (-oo, oo) and sin(x) = 0}'
ucode_str = u'{x | x ∊ ℝ ∧ sin(x) = 0}'
assert pretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ascii_str
assert upretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ucode_str
assert pretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}'
assert upretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == u'{1}'
assert pretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "EmptySet()"
assert upretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == u"∅"
assert pretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}'
assert upretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == u'{2}'
def test_pretty_ComplexRegion():
from sympy import ComplexRegion
ucode_str = u'{x + y⋅ⅈ | x, y ∊ [3, 5] × [4, 6]}'
assert upretty(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == ucode_str
ucode_str = u'{r⋅(ⅈ⋅sin(θ) + cos(θ)) | r, θ ∊ [0, 1] × [0, 2⋅π)}'
assert upretty(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == ucode_str
def test_pretty_Union_issue_10414():
a, b = Interval(2, 3), Interval(4, 7)
ucode_str = u'[2, 3] ∪ [4, 7]'
ascii_str = '[2, 3] U [4, 7]'
assert upretty(Union(a, b)) == ucode_str
assert pretty(Union(a, b)) == ascii_str
def test_pretty_Intersection_issue_10414():
x, y, z, w = symbols('x, y, z, w')
a, b = Interval(x, y), Interval(z, w)
ucode_str = u'[x, y] ∩ [z, w]'
ascii_str = '[x, y] n [z, w]'
assert upretty(Intersection(a, b)) == ucode_str
assert pretty(Intersection(a, b)) == ascii_str
def test_ProductSet_paranthesis():
ucode_str = u'([4, 7] × {1, 2}) ∪ ([2, 3] × [4, 7])'
a, b, c = Interval(2, 3), Interval(4, 7), Interval(1, 9)
assert upretty(Union(a*b, b*FiniteSet(1, 2))) == ucode_str
def test_ProductSet_prod_char_issue_10413():
ascii_str = '[2, 3] x [4, 7]'
ucode_str = u'[2, 3] × [4, 7]'
a, b = Interval(2, 3), Interval(4, 7)
assert pretty(a*b) == ascii_str
assert upretty(a*b) == ucode_str
def test_pretty_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
ascii_str = '[0, 1, 4, 9, ...]'
ucode_str = u'[0, 1, 4, 9, …]'
assert pretty(s1) == ascii_str
assert upretty(s1) == ucode_str
ascii_str = '[1, 2, 1, 2, ...]'
ucode_str = u'[1, 2, 1, 2, …]'
assert pretty(s2) == ascii_str
assert upretty(s2) == ucode_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
ascii_str = '[0, 1, 4]'
ucode_str = u'[0, 1, 4]'
assert pretty(s3) == ascii_str
assert upretty(s3) == ucode_str
ascii_str = '[1, 2, 1]'
ucode_str = u'[1, 2, 1]'
assert pretty(s4) == ascii_str
assert upretty(s4) == ucode_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
ascii_str = '[..., 9, 4, 1, 0]'
ucode_str = u'[…, 9, 4, 1, 0]'
assert pretty(s5) == ascii_str
assert upretty(s5) == ucode_str
ascii_str = '[..., 2, 1, 2, 1]'
ucode_str = u'[…, 2, 1, 2, 1]'
assert pretty(s6) == ascii_str
assert upretty(s6) == ucode_str
ascii_str = '[1, 3, 5, 11, ...]'
ucode_str = u'[1, 3, 5, 11, …]'
assert pretty(SeqAdd(s1, s2)) == ascii_str
assert upretty(SeqAdd(s1, s2)) == ucode_str
ascii_str = '[1, 3, 5]'
ucode_str = u'[1, 3, 5]'
assert pretty(SeqAdd(s3, s4)) == ascii_str
assert upretty(SeqAdd(s3, s4)) == ucode_str
ascii_str = '[..., 11, 5, 3, 1]'
ucode_str = u'[…, 11, 5, 3, 1]'
assert pretty(SeqAdd(s5, s6)) == ascii_str
assert upretty(SeqAdd(s5, s6)) == ucode_str
ascii_str = '[0, 2, 4, 18, ...]'
ucode_str = u'[0, 2, 4, 18, …]'
assert pretty(SeqMul(s1, s2)) == ascii_str
assert upretty(SeqMul(s1, s2)) == ucode_str
ascii_str = '[0, 2, 4]'
ucode_str = u'[0, 2, 4]'
assert pretty(SeqMul(s3, s4)) == ascii_str
assert upretty(SeqMul(s3, s4)) == ucode_str
ascii_str = '[..., 18, 4, 2, 0]'
ucode_str = u'[…, 18, 4, 2, 0]'
assert pretty(SeqMul(s5, s6)) == ascii_str
assert upretty(SeqMul(s5, s6)) == ucode_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
raises(NotImplementedError, lambda: pretty(s7))
raises(NotImplementedError, lambda: upretty(s7))
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
ascii_str = u'[0, b, 4*b]'
ucode_str = u'[0, b, 4⋅b]'
assert pretty(s8) == ascii_str
assert upretty(s8) == ucode_str
def test_pretty_FourierSeries():
f = fourier_series(x, (x, -pi, pi))
ascii_str = \
"""\
2*sin(3*x) \n\
2*sin(x) - sin(2*x) + ---------- + ...\n\
3 \
"""
ucode_str = \
u("""\
2⋅sin(3⋅x) \n\
2⋅sin(x) - sin(2⋅x) + ────────── + …\n\
3 \
""")
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_FormalPowerSeries():
f = fps(log(1 + x))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -k k \n\
\\ -(-1) *x \n\
/ -----------\n\
/ k \n\
/___, \n\
k = 1 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ -k k \n\
╲ -(-1) ⋅x \n\
╱ ───────────\n\
╱ k \n\
╱ \n\
‾‾‾‾ \n\
k = 1 \
""")
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_limits():
expr = Limit(x, x, oo)
ascii_str = \
"""\
lim x\n\
x->oo \
"""
ucode_str = \
u("""\
lim x\n\
x─→∞ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x**2, x, 0)
ascii_str = \
"""\
2\n\
lim x \n\
x->0+ \
"""
ucode_str = \
u("""\
2\n\
lim x \n\
x─→0⁺ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(1/x, x, 0)
ascii_str = \
"""\
1\n\
lim -\n\
x->0+x\
"""
ucode_str = \
u("""\
1\n\
lim ─\n\
x─→0⁺x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0)
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0+\\ x /\
"""
ucode_str = \
u("""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁺⎝ x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0, "-")
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0-\\ x /\
"""
ucode_str = \
u("""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁻⎝ x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x + sin(x), x, 0)
ascii_str = \
"""\
lim (x + sin(x))\n\
x->0+ \
"""
ucode_str = \
u("""\
lim (x + sin(x))\n\
x─→0⁺ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x, x, 0)**2
ascii_str = \
"""\
2\n\
/ lim x\\ \n\
\\x->0+ / \
"""
ucode_str = \
u("""\
2\n\
⎛ lim x⎞ \n\
⎝x─→0⁺ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
u("""\
⎛ ⎛y⎞⎞\n\
lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
2* lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
u("""\
⎛ ⎛y⎞⎞\n\
2⋅ lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x), x, 0, dir='+-')
ascii_str = \
"""\
lim sin(x)\n\
x->0 \
"""
ucode_str = \
u("""\
lim sin(x)\n\
x─→0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ComplexRootOf():
expr = rootof(x**5 + 11*x - 2, 0)
ascii_str = \
"""\
/ 5 \\\n\
CRootOf\\x + 11*x - 2, 0/\
"""
ucode_str = \
u("""\
⎛ 5 ⎞\n\
CRootOf⎝x + 11⋅x - 2, 0⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_RootSum():
expr = RootSum(x**5 + 11*x - 2, auto=False)
ascii_str = \
"""\
/ 5 \\\n\
RootSum\\x + 11*x - 2/\
"""
ucode_str = \
u("""\
⎛ 5 ⎞\n\
RootSum⎝x + 11⋅x - 2⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = RootSum(x**5 + 11*x - 2, Lambda(z, exp(z)))
ascii_str = \
"""\
/ 5 z\\\n\
RootSum\\x + 11*x - 2, z -> e /\
"""
ucode_str = \
u("""\
⎛ 5 z⎞\n\
RootSum⎝x + 11⋅x - 2, z ↦ ℯ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_GroebnerBasis():
expr = groebner([], x, y)
ascii_str = \
"""\
GroebnerBasis([], x, y, domain=ZZ, order=lex)\
"""
ucode_str = \
u("""\
GroebnerBasis([], x, y, domain=ℤ, order=lex)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
expr = groebner(F, x, y, order='grlex')
ascii_str = \
"""\
/[ 2 2 ] \\\n\
GroebnerBasis\\[x - x - 3*y + 1, y - 2*x + y - 1], x, y, domain=ZZ, order=grlex/\
"""
ucode_str = \
u("""\
⎛⎡ 2 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣x - x - 3⋅y + 1, y - 2⋅x + y - 1⎦, x, y, domain=ℤ, order=grlex⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = expr.fglm('lex')
ascii_str = \
"""\
/[ 2 4 3 2 ] \\\n\
GroebnerBasis\\[2*x - y - y + 1, y + 2*y - 3*y - 16*y + 7], x, y, domain=ZZ, order=lex/\
"""
ucode_str = \
u("""\
⎛⎡ 2 4 3 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣2⋅x - y - y + 1, y + 2⋅y - 3⋅y - 16⋅y + 7⎦, x, y, domain=ℤ, order=lex⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_Boolean():
expr = Not(x, evaluate=False)
assert pretty(expr) == "Not(x)"
assert upretty(expr) == u"¬x"
expr = And(x, y)
assert pretty(expr) == "And(x, y)"
assert upretty(expr) == u"x ∧ y"
expr = Or(x, y)
assert pretty(expr) == "Or(x, y)"
assert upretty(expr) == u"x ∨ y"
syms = symbols('a:f')
expr = And(*syms)
assert pretty(expr) == "And(a, b, c, d, e, f)"
assert upretty(expr) == u"a ∧ b ∧ c ∧ d ∧ e ∧ f"
expr = Or(*syms)
assert pretty(expr) == "Or(a, b, c, d, e, f)"
assert upretty(expr) == u"a ∨ b ∨ c ∨ d ∨ e ∨ f"
expr = Xor(x, y, evaluate=False)
assert pretty(expr) == "Xor(x, y)"
assert upretty(expr) == u"x ⊻ y"
expr = Nand(x, y, evaluate=False)
assert pretty(expr) == "Nand(x, y)"
assert upretty(expr) == u"x ⊼ y"
expr = Nor(x, y, evaluate=False)
assert pretty(expr) == "Nor(x, y)"
assert upretty(expr) == u"x ⊽ y"
expr = Implies(x, y, evaluate=False)
assert pretty(expr) == "Implies(x, y)"
assert upretty(expr) == u"x → y"
# don't sort args
expr = Implies(y, x, evaluate=False)
assert pretty(expr) == "Implies(y, x)"
assert upretty(expr) == u"y → x"
expr = Equivalent(x, y, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == u"x ⇔ y"
expr = Equivalent(y, x, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == u"x ⇔ y"
def test_pretty_Domain():
expr = FF(23)
assert pretty(expr) == "GF(23)"
assert upretty(expr) == u"ℤ₂₃"
expr = ZZ
assert pretty(expr) == "ZZ"
assert upretty(expr) == u"ℤ"
expr = QQ
assert pretty(expr) == "QQ"
assert upretty(expr) == u"ℚ"
expr = RR
assert pretty(expr) == "RR"
assert upretty(expr) == u"ℝ"
expr = QQ[x]
assert pretty(expr) == "QQ[x]"
assert upretty(expr) == u"ℚ[x]"
expr = QQ[x, y]
assert pretty(expr) == "QQ[x, y]"
assert upretty(expr) == u"ℚ[x, y]"
expr = ZZ.frac_field(x)
assert pretty(expr) == "ZZ(x)"
assert upretty(expr) == u"ℤ(x)"
expr = ZZ.frac_field(x, y)
assert pretty(expr) == "ZZ(x, y)"
assert upretty(expr) == u"ℤ(x, y)"
expr = QQ.poly_ring(x, y, order=grlex)
assert pretty(expr) == "QQ[x, y, order=grlex]"
assert upretty(expr) == u"ℚ[x, y, order=grlex]"
expr = QQ.poly_ring(x, y, order=ilex)
assert pretty(expr) == "QQ[x, y, order=ilex]"
assert upretty(expr) == u"ℚ[x, y, order=ilex]"
def test_pretty_prec():
assert xpretty(S("0.3"), full_prec=True, wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec="auto", wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec=False, wrap_line=False) == "0.3"
assert xpretty(S("0.3")*x, full_prec=True, use_unicode=False, wrap_line=False) in [
"0.300000000000000*x",
"x*0.300000000000000"
]
assert xpretty(S("0.3")*x, full_prec="auto", use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
assert xpretty(S("0.3")*x, full_prec=False, use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
def test_pprint():
import sys
from sympy.core.compatibility import StringIO
fd = StringIO()
sso = sys.stdout
sys.stdout = fd
try:
pprint(pi, use_unicode=False, wrap_line=False)
finally:
sys.stdout = sso
assert fd.getvalue() == 'pi\n'
def test_pretty_class():
"""Test that the printer dispatcher correctly handles classes."""
class C:
pass # C has no .__class__ and this was causing problems
class D(object):
pass
assert pretty( C ) == str( C )
assert pretty( D ) == str( D )
def test_pretty_no_wrap_line():
huge_expr = 0
for i in range(20):
huge_expr += i*sin(i + x)
assert xpretty(huge_expr ).find('\n') != -1
assert xpretty(huge_expr, wrap_line=False).find('\n') == -1
def test_settings():
raises(TypeError, lambda: pretty(S(4), method="garbage"))
def test_pretty_sum():
from sympy.abc import x, a, b, k, m, n
expr = Sum(k**k, (k, 0, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = 0 \
"""
ucode_str = \
u("""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**k, (k, oo, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = oo \
"""
ucode_str = \
u("""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = ∞ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (k, 0, n**n))
ascii_str = \
"""\
n \n\
n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
n \n\
n \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (
k, x + n + x**2 + n**2 + (x/n) + (1/x), Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
2 2 1 x \n\
k = n + n + x + x + - + - \n\
x n \
"""
ucode_str = \
u("""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
2 2 1 x \n\
k = n + n + x + x + ─ + ─ \n\
x n \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, x + n + x**2 + n**2 + (x/n) + (1/x)))
ascii_str = \
"""\
2 2 1 x \n\
n + n + x + x + - + - \n\
x n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
2 2 1 x \n\
n + n + x + x + ─ + ─ \n\
x n \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x, (x, 0, oo))
ascii_str = \
"""\
oo \n\
__ \n\
\\ ` \n\
) x\n\
/_, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
___ \n\
╲ \n\
╲ \n\
╱ x\n\
╱ \n\
‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**2, (x, 0, oo))
ascii_str = \
u("""\
oo \n\
___ \n\
\\ ` \n\
\\ 2\n\
/ x \n\
/__, \n\
x = 0 \
""")
ucode_str = \
u("""\
∞ \n\
___ \n\
╲ \n\
╲ 2\n\
╱ x \n\
╱ \n\
‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
___ \n\
\\ ` \n\
\\ x\n\
) -\n\
/ 2\n\
/__, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ \n\
╲ x\n\
╱ ─\n\
╱ 2\n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**3/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 3\n\
\\ x \n\
/ --\n\
/ 2 \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ 3\n\
╲ x \n\
╱ ──\n\
╱ 2 \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum((x**3*y**(x/2))**n, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ n\n\
\\ / x\\ \n\
) | -| \n\
/ | 3 2| \n\
/ \\x *y / \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
_____ \n\
╲ \n\
╲ \n\
╲ n\n\
╲ ⎛ x⎞ \n\
╱ ⎜ ─⎟ \n\
╱ ⎜ 3 2⎟ \n\
╱ ⎝x ⋅y ⎠ \n\
╱ \n\
‾‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/x**2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 1 \n\
\\ --\n\
/ 2\n\
/ x \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ 1 \n\
╲ ──\n\
╱ 2\n\
╱ x \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -a \n\
\\ ---\n\
/ b \n\
/ y \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ -a \n\
╲ ───\n\
╱ b \n\
╱ y \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo), (y, 1, 2))
ascii_str = \
"""\
2 oo \n\
____ ____ \n\
\\ ` \\ ` \n\
\\ \\ -a\n\
\\ \\ --\n\
/ / b \n\
/ / y \n\
/___, /___, \n\
y = 1 x = 0 \
"""
ucode_str = \
u("""\
2 ∞ \n\
____ ____ \n\
╲ ╲ \n\
╲ ╲ -a\n\
╲ ╲ ──\n\
╱ ╱ b \n\
╱ ╱ y \n\
╱ ╱ \n\
‾‾‾‾ ‾‾‾‾ \n\
y = 1 x = 0 \
""")
expr = Sum(1/(1 + 1/(
1 + 1/k)) + 1, (k, 111, 1 + 1/n), (k, 1/(1 + m), oo)) + 1/(1 + 1/k)
ascii_str = \
"""\
1 \n\
1 + - \n\
oo n \n\
_____ _____ \n\
\\ ` \\ ` \n\
\\ \\ / 1 \\ \n\
\\ \\ |1 + ---------| \n\
\\ \\ | 1 | 1 \n\
) ) | 1 + -----| + -----\n\
/ / | 1| 1\n\
/ / | 1 + -| 1 + -\n\
/ / \\ k/ k\n\
/____, /____, \n\
1 k = 111 \n\
k = ----- \n\
m + 1 \
"""
ucode_str = \
u("""\
1 \n\
1 + ─ \n\
∞ n \n\
______ ______ \n\
╲ ╲ \n\
╲ ╲ \n\
╲ ╲ ⎛ 1 ⎞ \n\
╲ ╲ ⎜1 + ─────────⎟ \n\
╲ ╲ ⎜ 1 ⎟ 1 \n\
╱ ╱ ⎜ 1 + ─────⎟ + ─────\n\
╱ ╱ ⎜ 1⎟ 1\n\
╱ ╱ ⎜ 1 + ─⎟ 1 + ─\n\
╱ ╱ ⎝ k⎠ k\n\
╱ ╱ \n\
‾‾‾‾‾‾ ‾‾‾‾‾‾ \n\
1 k = 111 \n\
k = ───── \n\
m + 1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_units():
expr = joule
ascii_str1 = \
"""\
2\n\
kilogram*meter \n\
---------------\n\
2 \n\
second \
"""
unicode_str1 = \
u("""\
2\n\
kilogram⋅meter \n\
───────────────\n\
2 \n\
second \
""")
ascii_str2 = \
"""\
2\n\
3*x*y*kilogram*meter \n\
---------------------\n\
2 \n\
second \
"""
unicode_str2 = \
u("""\
2\n\
3⋅x⋅y⋅kilogram⋅meter \n\
─────────────────────\n\
2 \n\
second \
""")
from sympy.physics.units import kg, m, s
assert upretty(expr) == u("joule")
assert pretty(expr) == "joule"
assert upretty(expr.convert_to(kg*m**2/s**2)) == unicode_str1
assert pretty(expr.convert_to(kg*m**2/s**2)) == ascii_str1
assert upretty(3*kg*x*m**2*y/s**2) == unicode_str2
assert pretty(3*kg*x*m**2*y/s**2) == ascii_str2
def test_pretty_Subs():
f = Function('f')
expr = Subs(f(x), x, ph**2)
ascii_str = \
"""\
(f(x))| 2\n\
|x=phi \
"""
unicode_str = \
u("""\
(f(x))│ 2\n\
│x=φ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x), x, 0)
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
\\dx /|x=0\
"""
unicode_str = \
u("""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎝dx ⎠│x=0\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
|dx || \n\
|--------|| \n\
\\ y /|x=0, y=1/2\
"""
unicode_str = \
u("""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎜dx ⎟│ \n\
⎜────────⎟│ \n\
⎝ y ⎠│x=0, y=1/2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_gammas():
assert upretty(lowergamma(x, y)) == u"γ(x, y)"
assert upretty(uppergamma(x, y)) == u"Γ(x, y)"
assert xpretty(gamma(x), use_unicode=True) == u'Γ(x)'
assert xpretty(gamma, use_unicode=True) == u'Γ'
assert xpretty(symbols('gamma', cls=Function)(x), use_unicode=True) == u'γ(x)'
assert xpretty(symbols('gamma', cls=Function), use_unicode=True) == u'γ'
def test_beta():
assert xpretty(beta(x,y), use_unicode=True) == u'Β(x, y)'
assert xpretty(beta(x,y), use_unicode=False) == u'B(x, y)'
assert xpretty(beta, use_unicode=True) == u'Β'
assert xpretty(beta, use_unicode=False) == u'B'
mybeta = Function('beta')
assert xpretty(mybeta(x), use_unicode=True) == u'β(x)'
assert xpretty(mybeta(x, y, z), use_unicode=False) == u'beta(x, y, z)'
assert xpretty(mybeta, use_unicode=True) == u'β'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert xpretty(mygamma, use_unicode=True) == r"mygamma"
assert xpretty(mygamma(x), use_unicode=True) == r"mygamma(x)"
def test_SingularityFunction():
assert xpretty(SingularityFunction(x, 0, n), use_unicode=True) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=True) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=True) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=True) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=True) == (
"""\
n\n\
<x - y> \
""")
assert xpretty(SingularityFunction(x, 0, n), use_unicode=False) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=False) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=False) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=False) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=False) == (
"""\
n\n\
<x - y> \
""")
def test_deltas():
assert xpretty(DiracDelta(x), use_unicode=True) == u'δ(x)'
assert xpretty(DiracDelta(x, 1), use_unicode=True) == \
u("""\
(1) \n\
δ (x)\
""")
assert xpretty(x*DiracDelta(x, 1), use_unicode=True) == \
u("""\
(1) \n\
x⋅δ (x)\
""")
def test_hyper():
expr = hyper((), (), z)
ucode_str = \
u("""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ z⎟\n\
0╵ 0 ⎝ │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | z|\n\
0 0 \\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((), (1,), x)
ucode_str = \
u("""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ x⎟\n\
0╵ 1 ⎝1 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | x|\n\
0 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([2], [1], x)
ucode_str = \
u("""\
┌─ ⎛2 │ ⎞\n\
├─ ⎜ │ x⎟\n\
1╵ 1 ⎝1 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ /2 | \\\n\
| | | x|\n\
1 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi/3, -2*k), (3, 4, 5, -3), x)
ucode_str = \
u("""\
⎛ π │ ⎞\n\
┌─ ⎜ ─, -2⋅k │ ⎟\n\
├─ ⎜ 3 │ x⎟\n\
2╵ 4 ⎜ │ ⎟\n\
⎝3, 4, 5, -3 │ ⎠\
""")
ascii_str = \
"""\
\n\
_ / pi | \\\n\
|_ | --, -2*k | |\n\
| | 3 | x|\n\
2 4 | | |\n\
\\3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi, S('2/3'), -2*k), (3, 4, 5, -3), x**2)
ucode_str = \
u("""\
┌─ ⎛π, 2/3, -2⋅k │ 2⎞\n\
├─ ⎜ │ x ⎟\n\
3╵ 4 ⎝3, 4, 5, -3 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ /pi, 2/3, -2*k | 2\\\n\
| | | x |\n\
3 4 \\ 3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([1, 2], [3, 4], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
┌─ ⎜1, 2 │ 1 + ─────────⎟\n\
├─ ⎜ │ 1 ⎟\n\
2╵ 2 ⎜3, 4 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
""")
ascii_str = \
"""\
\n\
/ | 1 \\\n\
| | -------------|\n\
_ | | 1 |\n\
|_ |1, 2 | 1 + ---------|\n\
| | | 1 |\n\
2 2 |3, 4 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_meijerg():
expr = meijerg([pi, pi, x], [1], [0, 1], [1, 2, 3], z)
ucode_str = \
u("""\
╭─╮2, 3 ⎛π, π, x 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯4, 5 ⎝ 0, 1 1, 2, 3 │ ⎠\
""")
ascii_str = \
"""\
__2, 3 /pi, pi, x 1 | \\\n\
/__ | | z|\n\
\\_|4, 5 \\ 0, 1 1, 2, 3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, pi/7], [2, pi, 5], [], [], z**2)
ucode_str = \
u("""\
⎛ π │ ⎞\n\
╭─╮0, 2 ⎜1, ─ 2, π, 5 │ 2⎟\n\
│╶┐ ⎜ 7 │ z ⎟\n\
╰─╯5, 0 ⎜ │ ⎟\n\
⎝ │ ⎠\
""")
ascii_str = \
"""\
/ pi | \\\n\
__0, 2 |1, -- 2, pi, 5 | 2|\n\
/__ | 7 | z |\n\
\\_|5, 0 | | |\n\
\\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ucode_str = \
u("""\
╭─╮ 1, 10 ⎛1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯11, 2 ⎝ 1 1 │ ⎠\
""")
ascii_str = \
"""\
__ 1, 10 /1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 | \\\n\
/__ | | z|\n\
\\_|11, 2 \\ 1 1 | /\
"""
expr = meijerg([1]*10, [1], [1], [1], z)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, 2, ], [4, 3], [3], [4, 5], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟\n\
│╶┐ ⎜ │ 1 ⎟\n\
╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
""")
ascii_str = \
"""\
/ | 1 \\\n\
| | -------------|\n\
| | 1 |\n\
__1, 2 |1, 2 4, 3 | 1 + ---------|\n\
/__ | | 1 |\n\
\\_|4, 3 | 3 4, 5 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(expr, x)
ucode_str = \
u("""\
⌠ \n\
⎮ ⎛ │ 1 ⎞ \n\
⎮ ⎜ │ ─────────────⎟ \n\
⎮ ⎜ │ 1 ⎟ \n\
⎮ ╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟ \n\
⎮ │╶┐ ⎜ │ 1 ⎟ dx\n\
⎮ ╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟ \n\
⎮ ⎜ │ 1⎟ \n\
⎮ ⎜ │ 1 + ─⎟ \n\
⎮ ⎝ │ x⎠ \n\
⌡ \
""")
ascii_str = \
"""\
/ \n\
| \n\
| / | 1 \\ \n\
| | | -------------| \n\
| | | 1 | \n\
| __1, 2 |1, 2 4, 3 | 1 + ---------| \n\
| /__ | | 1 | dx\n\
| \\_|4, 3 | 3 4, 5 | 1 + -----| \n\
| | | 1| \n\
| | | 1 + -| \n\
| \\ | x/ \n\
| \n\
/ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
expr = A*B*C**-1
ascii_str = \
"""\
-1\n\
A*B*C \
"""
ucode_str = \
u("""\
-1\n\
A⋅B⋅C \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = C**-1*A*B
ascii_str = \
"""\
-1 \n\
C *A*B\
"""
ucode_str = \
u("""\
-1 \n\
C ⋅A⋅B\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B
ascii_str = \
"""\
-1 \n\
A*C *B\
"""
ucode_str = \
u("""\
-1 \n\
A⋅C ⋅B\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B/x
ascii_str = \
"""\
-1 \n\
A*C *B\n\
-------\n\
x \
"""
ucode_str = \
u("""\
-1 \n\
A⋅C ⋅B\n\
───────\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_special_functions():
x, y = symbols("x y")
# atan2
expr = atan2(y/sqrt(200), sqrt(x))
ascii_str = \
"""\
/ ___ \\\n\
|\\/ 2 *y ___|\n\
atan2|-------, \\/ x |\n\
\\ 20 /\
"""
ucode_str = \
u("""\
⎛√2⋅y ⎞\n\
atan2⎜────, √x⎟\n\
⎝ 20 ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_geometry():
e = Segment((0, 1), (0, 2))
assert pretty(e) == 'Segment2D(Point2D(0, 1), Point2D(0, 2))'
e = Ray((1, 1), angle=4.02*pi)
assert pretty(e) == 'Ray2D(Point2D(1, 1), Point2D(2, tan(pi/50) + 1))'
def test_expint():
expr = Ei(x)
string = 'Ei(x)'
assert pretty(expr) == string
assert upretty(expr) == string
expr = expint(1, z)
ucode_str = u"E₁(z)"
ascii_str = "expint(1, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
assert pretty(Shi(x)) == 'Shi(x)'
assert pretty(Si(x)) == 'Si(x)'
assert pretty(Ci(x)) == 'Ci(x)'
assert pretty(Chi(x)) == 'Chi(x)'
assert upretty(Shi(x)) == 'Shi(x)'
assert upretty(Si(x)) == 'Si(x)'
assert upretty(Ci(x)) == 'Ci(x)'
assert upretty(Chi(x)) == 'Chi(x)'
def test_elliptic_functions():
ascii_str = \
"""\
/ 1 \\\n\
K|-----|\n\
\\z + 1/\
"""
ucode_str = \
u("""\
⎛ 1 ⎞\n\
K⎜─────⎟\n\
⎝z + 1⎠\
""")
expr = elliptic_k(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
F|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
F⎜1│─────⎟\n\
⎝ │z + 1⎠\
""")
expr = elliptic_f(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 1 \\\n\
E|-----|\n\
\\z + 1/\
"""
ucode_str = \
u("""\
⎛ 1 ⎞\n\
E⎜─────⎟\n\
⎝z + 1⎠\
""")
expr = elliptic_e(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
E|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
E⎜1│─────⎟\n\
⎝ │z + 1⎠\
""")
expr = elliptic_e(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ |4\\\n\
Pi|3|-|\n\
\\ |x/\
"""
ucode_str = \
u("""\
⎛ │4⎞\n\
Π⎜3│─⎟\n\
⎝ │x⎠\
""")
expr = elliptic_pi(3, 4/x)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 4| \\\n\
Pi|3; -|6|\n\
\\ x| /\
"""
ucode_str = \
u("""\
⎛ 4│ ⎞\n\
Π⎜3; ─│6⎟\n\
⎝ x│ ⎠\
""")
expr = elliptic_pi(3, 4/x, 6)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert upretty(where(X > 0)) == u"Domain: 0 < x₁ ∧ x₁ < ∞"
D = Die('d1', 6)
assert upretty(where(D > 4)) == u'Domain: d₁ = 5 ∨ d₁ = 6'
A = Exponential('a', 1)
B = Exponential('b', 1)
assert upretty(pspace(Tuple(A, B)).domain) == \
u'Domain: 0 ≤ a ∧ 0 ≤ b ∧ a < ∞ ∧ b < ∞'
def test_PrettyPoly():
F = QQ.frac_field(x, y)
R = QQ.poly_ring(x, y)
expr = F.convert(x/(x + y))
assert pretty(expr) == "x/(x + y)"
assert upretty(expr) == u"x/(x + y)"
expr = R.convert(x + y)
assert pretty(expr) == "x + y"
assert upretty(expr) == u"x + y"
def test_issue_6285():
assert pretty(Pow(2, -5, evaluate=False)) == '1 \n--\n 5\n2 '
assert pretty(Pow(x, (1/pi))) == 'pi___\n\\/ x '
def test_issue_6359():
assert pretty(Integral(x**2, x)**2) == \
"""\
2
/ / \\ \n\
| | | \n\
| | 2 | \n\
| | x dx| \n\
| | | \n\
\\/ / \
"""
assert upretty(Integral(x**2, x)**2) == \
u("""\
2
⎛⌠ ⎞ \n\
⎜⎮ 2 ⎟ \n\
⎜⎮ x dx⎟ \n\
⎝⌡ ⎠ \
""")
assert pretty(Sum(x**2, (x, 0, 1))**2) == \
"""\
2
/ 1 \\ \n\
| ___ | \n\
| \\ ` | \n\
| \\ 2| \n\
| / x | \n\
| /__, | \n\
\\x = 0 / \
"""
assert upretty(Sum(x**2, (x, 0, 1))**2) == \
u("""\
2
⎛ 1 ⎞ \n\
⎜ ___ ⎟ \n\
⎜ ╲ ⎟ \n\
⎜ ╲ 2⎟ \n\
⎜ ╱ x ⎟ \n\
⎜ ╱ ⎟ \n\
⎜ ‾‾‾ ⎟ \n\
⎝x = 0 ⎠ \
""")
assert pretty(Product(x**2, (x, 1, 2))**2) == \
"""\
2
/ 2 \\ \n\
|______ | \n\
| | | 2| \n\
| | | x | \n\
| | | | \n\
\\x = 1 / \
"""
assert upretty(Product(x**2, (x, 1, 2))**2) == \
u("""\
2
⎛ 2 ⎞ \n\
⎜─┬──┬─ ⎟ \n\
⎜ │ │ 2⎟ \n\
⎜ │ │ x ⎟ \n\
⎜ │ │ ⎟ \n\
⎝x = 1 ⎠ \
""")
f = Function('f')
assert pretty(Derivative(f(x), x)**2) == \
"""\
2
/d \\ \n\
|--(f(x))| \n\
\\dx / \
"""
assert upretty(Derivative(f(x), x)**2) == \
u("""\
2
⎛d ⎞ \n\
⎜──(f(x))⎟ \n\
⎝dx ⎠ \
""")
def test_issue_6739():
ascii_str = \
"""\
1 \n\
-----\n\
___\n\
\\/ x \
"""
ucode_str = \
u("""\
1 \n\
──\n\
√x\
""")
assert pretty(1/sqrt(x)) == ascii_str
assert upretty(1/sqrt(x)) == ucode_str
def test_complicated_symbol_unchanged():
for symb_name in ["dexpr2_d1tau", "dexpr2^d1tau"]:
assert pretty(Symbol(symb_name)) == symb_name
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram, DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert pretty(A1) == "A1"
assert upretty(A1) == u"A₁"
assert pretty(f1) == "f1:A1-->A2"
assert upretty(f1) == u"f₁:A₁——▶A₂"
assert pretty(id_A1) == "id:A1-->A1"
assert upretty(id_A1) == u"id:A₁——▶A₁"
assert pretty(f2*f1) == "f2*f1:A1-->A3"
assert upretty(f2*f1) == u"f₂∘f₁:A₁——▶A₃"
assert pretty(K1) == "K1"
assert upretty(K1) == u"K₁"
# Test how diagrams are printed.
d = Diagram()
assert pretty(d) == "EmptySet()"
assert upretty(d) == u"∅"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet(), id:A1-->A1: " \
"EmptySet(), id:A2-->A2: EmptySet(), id:A3-->A3: " \
"EmptySet(), f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet()}"
assert upretty(d) == u("{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, " \
"id:A₂——▶A₂: ∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}")
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet(), id:A1-->A1: " \
"EmptySet(), id:A2-->A2: EmptySet(), id:A3-->A3: " \
"EmptySet(), f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet()}" \
" ==> {f2*f1:A1-->A3: {unique}}"
assert upretty(d) == u("{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, id:A₂——▶A₂: " \
"∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}" \
" ══▶ {f₂∘f₁:A₁——▶A₃: {unique}}")
grid = DiagramGrid(d)
assert pretty(grid) == "A1 A2\n \nA3 "
assert upretty(grid) == u"A₁ A₂\n \nA₃ "
def test_PrettyModules():
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
ucode_str = \
u("""\
2\n\
ℚ[x, y] \
""")
ascii_str = \
"""\
2\n\
QQ[x, y] \
"""
assert upretty(F) == ucode_str
assert pretty(F) == ascii_str
ucode_str = \
u("""\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
""")
ascii_str = \
"""\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(M) == ucode_str
assert pretty(M) == ascii_str
I = R.ideal(x**2, y)
ucode_str = \
u("""\
╱ 2 ╲\n\
╲x , y╱\
""")
ascii_str = \
"""\
2 \n\
<x , y>\
"""
assert upretty(I) == ucode_str
assert pretty(I) == ascii_str
Q = F / M
ucode_str = \
u("""\
2 \n\
ℚ[x, y] \n\
─────────────────\n\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
""")
ascii_str = \
"""\
2 \n\
QQ[x, y] \n\
-----------------\n\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(Q) == ucode_str
assert pretty(Q) == ascii_str
ucode_str = \
u("""\
╱⎡ 3⎤ ╲\n\
│⎢ x ⎥ ╱ ⎡ 2⎤╲ ╱ ⎡ 2⎤╲│\n\
│⎢1, ──⎥ + ╲[x, y], ⎣1, x ⎦╱, [2, y] + ╲[x, y], ⎣1, x ⎦╱│\n\
╲⎣ 2 ⎦ ╱\
""")
ascii_str = \
"""\
3 \n\
x 2 2 \n\
<[1, --] + <[x, y], [1, x ]>, [2, y] + <[x, y], [1, x ]>>\n\
2 \
"""
def test_QuotientRing():
R = QQ.old_poly_ring(x)/[x**2 + 1]
ucode_str = \
u("""\
ℚ[x] \n\
────────\n\
╱ 2 ╲\n\
╲x + 1╱\
""")
ascii_str = \
"""\
QQ[x] \n\
--------\n\
2 \n\
<x + 1>\
"""
assert upretty(R) == ucode_str
assert pretty(R) == ascii_str
ucode_str = \
u("""\
╱ 2 ╲\n\
1 + ╲x + 1╱\
""")
ascii_str = \
"""\
2 \n\
1 + <x + 1>\
"""
assert upretty(R.one) == ucode_str
assert pretty(R.one) == ascii_str
def test_Homomorphism():
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x)
expr = homomorphism(R.free_module(1), R.free_module(1), [0])
ucode_str = \
u("""\
1 1\n\
[0] : ℚ[x] ──> ℚ[x] \
""")
ascii_str = \
"""\
1 1\n\
[0] : QQ[x] --> QQ[x] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(2), R.free_module(2), [0, 0])
ucode_str = \
u("""\
⎡0 0⎤ 2 2\n\
⎢ ⎥ : ℚ[x] ──> ℚ[x] \n\
⎣0 0⎦ \
""")
ascii_str = \
"""\
[0 0] 2 2\n\
[ ] : QQ[x] --> QQ[x] \n\
[0 0] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])
ucode_str = \
u("""\
1\n\
1 ℚ[x] \n\
[0] : ℚ[x] ──> ─────\n\
<[x]>\
""")
ascii_str = \
"""\
1\n\
1 QQ[x] \n\
[0] : QQ[x] --> ------\n\
<[x]> \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert pretty(t) == r'Tr(A*B)'
assert upretty(t) == u'Tr(A⋅B)'
def test_pretty_Add():
eq = Mul(-2, x - 2, evaluate=False) + 5
assert pretty(eq) == '5 - 2*(x - 2)'
def test_issue_7179():
assert upretty(Not(Equivalent(x, y))) == u'x ⇎ y'
assert upretty(Not(Implies(x, y))) == u'x ↛ y'
def test_issue_7180():
assert upretty(Equivalent(x, y)) == u'x ⇔ y'
def test_pretty_Complement():
assert pretty(S.Reals - S.Naturals) == '(-oo, oo) \\ Naturals'
assert upretty(S.Reals - S.Naturals) == u'ℝ \\ ℕ'
assert pretty(S.Reals - S.Naturals0) == '(-oo, oo) \\ Naturals0'
assert upretty(S.Reals - S.Naturals0) == u'ℝ \\ ℕ₀'
def test_pretty_SymmetricDifference():
from sympy import SymmetricDifference, Interval
from sympy.utilities.pytest import raises
assert upretty(SymmetricDifference(Interval(2,3), Interval(3,5), \
evaluate = False)) == u'[2, 3] ∆ [3, 5]'
with raises(NotImplementedError):
pretty(SymmetricDifference(Interval(2,3), Interval(3,5), evaluate = False))
def test_pretty_Contains():
assert pretty(Contains(x, S.Integers)) == 'Contains(x, Integers)'
assert upretty(Contains(x, S.Integers)) == u'x ∈ ℤ'
def test_issue_8292():
from sympy.core import sympify
e = sympify('((x+x**4)/(x-1))-(2*(x-1)**4/(x-1)**4)', evaluate=False)
ucode_str = \
u("""\
4 4 \n\
2⋅(x - 1) x + x\n\
- ────────── + ──────\n\
4 x - 1 \n\
(x - 1) \
""")
ascii_str = \
"""\
4 4 \n\
2*(x - 1) x + x\n\
- ---------- + ------\n\
4 x - 1 \n\
(x - 1) \
"""
assert pretty(e) == ascii_str
assert upretty(e) == ucode_str
def test_issue_4335():
y = Function('y')
expr = -y(x).diff(x)
ucode_str = \
u("""\
d \n\
-──(y(x))\n\
dx \
""")
ascii_str = \
"""\
d \n\
- --(y(x))\n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_8344():
from sympy.core import sympify
e = sympify('2*x*y**2/1**2 + 1', evaluate=False)
ucode_str = \
u("""\
2 \n\
2⋅x⋅y \n\
────── + 1\n\
2 \n\
1 \
""")
assert upretty(e) == ucode_str
def test_issue_6324():
x = Pow(2, 3, evaluate=False)
y = Pow(10, -2, evaluate=False)
e = Mul(x, y, evaluate=False)
ucode_str = \
u("""\
3\n\
2 \n\
───\n\
2\n\
10 \
""")
assert upretty(e) == ucode_str
def test_issue_7927():
e = sin(x/2)**cos(x/2)
ucode_str = \
u("""\
⎛x⎞\n\
cos⎜─⎟\n\
⎝2⎠\n\
⎛ ⎛x⎞⎞ \n\
⎜sin⎜─⎟⎟ \n\
⎝ ⎝2⎠⎠ \
""")
assert upretty(e) == ucode_str
e = sin(x)**(S(11)/13)
ucode_str = \
u("""\
11\n\
──\n\
13\n\
(sin(x)) \
""")
assert upretty(e) == ucode_str
def test_issue_6134():
from sympy.abc import lamda, t
phi = Function('phi')
e = lamda*x*Integral(phi(t)*pi*sin(pi*t), (t, 0, 1)) + lamda*x**2*Integral(phi(t)*2*pi*sin(2*pi*t), (t, 0, 1))
ucode_str = \
u("""\
1 1 \n\
2 ⌠ ⌠ \n\
λ⋅x ⋅⎮ 2⋅π⋅φ(t)⋅sin(2⋅π⋅t) dt + λ⋅x⋅⎮ π⋅φ(t)⋅sin(π⋅t) dt\n\
⌡ ⌡ \n\
0 0 \
""")
assert upretty(e) == ucode_str
def test_issue_9877():
ucode_str1 = u'(2, 3) ∪ ([1, 2] \\ {x})'
a, b, c = Interval(2, 3, True, True), Interval(1, 2), FiniteSet(x)
assert upretty(Union(a, Complement(b, c))) == ucode_str1
ucode_str2 = u'{x} ∩ {y} ∩ ({z} \\ [1, 2])'
d, e, f, g = FiniteSet(x), FiniteSet(y), FiniteSet(z), Interval(1, 2)
assert upretty(Intersection(d, e, Complement(f, g))) == ucode_str2
def test_issue_13651():
expr1 = c + Mul(-1, a + b, evaluate=False)
assert pretty(expr1) == 'c - (a + b)'
expr2 = c + Mul(-1, a - b + d, evaluate=False)
assert pretty(expr2) == 'c - (a - b + d)'
def test_pretty_primenu():
from sympy.ntheory.factor_ import primenu
ascii_str1 = "nu(n)"
ucode_str1 = u("ν(n)")
n = symbols('n', integer=True)
assert pretty(primenu(n)) == ascii_str1
assert upretty(primenu(n)) == ucode_str1
def test_pretty_primeomega():
from sympy.ntheory.factor_ import primeomega
ascii_str1 = "Omega(n)"
ucode_str1 = u("Ω(n)")
n = symbols('n', integer=True)
assert pretty(primeomega(n)) == ascii_str1
assert upretty(primeomega(n)) == ucode_str1
def test_pretty_Mod():
from sympy.core import Mod
ascii_str1 = "x mod 7"
ucode_str1 = u("x mod 7")
ascii_str2 = "(x + 1) mod 7"
ucode_str2 = u("(x + 1) mod 7")
ascii_str3 = "2*x mod 7"
ucode_str3 = u("2⋅x mod 7")
ascii_str4 = "(x mod 7) + 1"
ucode_str4 = u("(x mod 7) + 1")
ascii_str5 = "2*(x mod 7)"
ucode_str5 = u("2⋅(x mod 7)")
x = symbols('x', integer=True)
assert pretty(Mod(x, 7)) == ascii_str1
assert upretty(Mod(x, 7)) == ucode_str1
assert pretty(Mod(x + 1, 7)) == ascii_str2
assert upretty(Mod(x + 1, 7)) == ucode_str2
assert pretty(Mod(2 * x, 7)) == ascii_str3
assert upretty(Mod(2 * x, 7)) == ucode_str3
assert pretty(Mod(x, 7) + 1) == ascii_str4
assert upretty(Mod(x, 7) + 1) == ucode_str4
assert pretty(2 * Mod(x, 7)) == ascii_str5
assert upretty(2 * Mod(x, 7)) == ucode_str5
def test_issue_11801():
assert pretty(Symbol("")) == ""
assert upretty(Symbol("")) == ""
def test_pretty_UnevaluatedExpr():
x = symbols('x')
he = UnevaluatedExpr(1/x)
ucode_str = \
u("""\
1\n\
─\n\
x\
""")
assert upretty(he) == ucode_str
ucode_str = \
u("""\
2\n\
⎛1⎞ \n\
⎜─⎟ \n\
⎝x⎠ \
""")
assert upretty(he**2) == ucode_str
ucode_str = \
u("""\
1\n\
1 + ─\n\
x\
""")
assert upretty(he + 1) == ucode_str
ucode_str = \
u('''\
1\n\
x⋅─\n\
x\
''')
assert upretty(x*he) == ucode_str
def test_issue_10472():
M = (Matrix([[0, 0], [0, 0]]), Matrix([0, 0]))
ucode_str = \
u("""\
⎛⎡0 0⎤ ⎡0⎤⎞
⎜⎢ ⎥, ⎢ ⎥⎟
⎝⎣0 0⎦ ⎣0⎦⎠\
""")
assert upretty(M) == ucode_str
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
ascii_str1 = "A_00"
ucode_str1 = u("A₀₀")
assert pretty(A[0, 0]) == ascii_str1
assert upretty(A[0, 0]) == ucode_str1
ascii_str1 = "3*A_00"
ucode_str1 = u("3⋅A₀₀")
assert pretty(3*A[0, 0]) == ascii_str1
assert upretty(3*A[0, 0]) == ucode_str1
ascii_str1 = "(-B + A)[0, 0]"
ucode_str1 = u("(-B + A)[0, 0]")
F = C[0, 0].subs(C, A - B)
assert pretty(F) == ascii_str1
assert upretty(F) == ucode_str1
def test_issue_12675():
from sympy.vector import CoordSys3D
x, y, t, j = symbols('x y t j')
e = CoordSys3D('e')
ucode_str = \
u("""\
⎛ t⎞ \n\
⎜⎛x⎞ ⎟ j_e\n\
⎜⎜─⎟ ⎟ \n\
⎝⎝y⎠ ⎠ \
""")
assert upretty((x/y)**t*e.j) == ucode_str
ucode_str = \
u("""\
⎛1⎞ \n\
⎜─⎟ j_e\n\
⎝y⎠ \
""")
assert upretty((1/y)*e.j) == ucode_str
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert pretty(-A*B*C) == "-A*B*C"
assert pretty(A - B) == "-B + A"
assert pretty(A*B*C - A*B - B*C) == "-A*B -B*C + A*B*C"
# issue #14814
x = MatrixSymbol('x', n, n)
y = MatrixSymbol('y*', n, n)
assert pretty(x + y) == "x + y*"
ascii_str = \
"""\
2 \n\
-2*y* -a*x\
"""
assert pretty(-a*x + -2*y*y) == ascii_str
def test_degree_printing():
expr1 = 90*degree
assert pretty(expr1) == u'90°'
expr2 = x*degree
assert pretty(expr2) == u'x°'
expr3 = cos(x*degree + 90*degree)
assert pretty(expr3) == u'cos(x° + 90°)'
def test_vector_expr_pretty_printing():
A = CoordSys3D('A')
assert upretty(Cross(A.i, A.x*A.i+3*A.y*A.j)) == u("(i_A)×((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(x*Cross(A.i, A.j)) == u('x⋅(i_A)×(j_A)')
assert upretty(Curl(A.x*A.i + 3*A.y*A.j)) == u("∇×((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Divergence(A.x*A.i + 3*A.y*A.j)) == u("∇⋅((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Dot(A.i, A.x*A.i+3*A.y*A.j)) == u("(i_A)⋅((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Gradient(A.x+3*A.y)) == u("∇(x_A + 3⋅y_A)")
assert upretty(Laplacian(A.x+3*A.y)) == u("∆(x_A + 3⋅y_A)")
# TODO: add support for ASCII pretty.
def test_pretty_print_tensor_expr():
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensorhead("A B C D", [L], [[1]])
H = tensorhead("H", [L, L], [[1], [1]])
expr = -i
ascii_str = \
"""\
-i\
"""
ucode_str = \
u("""\
-i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)
ascii_str = \
"""\
i\n\
A \n\
\
"""
ucode_str = \
u("""\
i\n\
A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i0)
ascii_str = \
"""\
i_0\n\
A \n\
\
"""
ucode_str = \
u("""\
i₀\n\
A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(-i)
ascii_str = \
"""\
\n\
A \n\
i\
"""
ucode_str = \
u("""\
\n\
A \n\
i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -3*A(-i)
ascii_str = \
"""\
\n\
-3*A \n\
i\
"""
ucode_str = \
u("""\
\n\
-3⋅A \n\
i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)
ascii_str = \
"""\
i \n\
H \n\
j\
"""
ucode_str = \
u("""\
i \n\
H \n\
j\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -i)
ascii_str = \
"""\
L_0 \n\
H \n\
L_0\
"""
ucode_str = \
u("""\
L₀ \n\
H \n\
L₀\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)*A(j)*B(k)
ascii_str = \
"""\
i L_0 k\n\
H *A *B \n\
L_0 \
"""
ucode_str = \
u("""\
i L₀ k\n\
H ⋅A ⋅B \n\
L₀ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1+x)*A(i)
ascii_str = \
"""\
i\n\
(x + 1)*A \n\
\
"""
ucode_str = \
u("""\
i\n\
(x + 1)⋅A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i) + 3*B(i)
ascii_str = \
"""\
i i\n\
A + 3*B \n\
\
"""
ucode_str = \
u("""\
i i\n\
A + 3⋅B \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_print_tensor_partial_deriv():
from sympy.tensor.toperators import PartialDerivative
from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
i0 = tensor_indices("i0", L)
A, B, C, D = tensorhead("A B C D", [L], [[1]])
H = tensorhead("H", [L, L], [[1], [1]])
expr = PartialDerivative(A(i), A(j))
ascii_str = \
"""\
d / i\\\n\
---|A |\n\
j\\ /\n\
dA \n\
\
"""
ucode_str = \
u("""\
∂ ⎛ i⎞\n\
───⎜A ⎟\n\
j⎝ ⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k \\\n\
A *---|H |\n\
j\\ L_0/\n\
dA \n\
\
"""
ucode_str = \
u("""\
L₀ ∂ ⎛ k ⎞\n\
A ⋅───⎜H ⎟\n\
j⎝ L₀⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(B(k)*C(-i) + 3*H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k k \\\n\
A *---|B *C + 3*H |\n\
j\\ L_0 L_0/\n\
dA \n\
\
"""
ucode_str = \
u("""\
L₀ ∂ ⎛ k k ⎞\n\
A ⋅───⎜B ⋅C + 3⋅H ⎟\n\
j⎝ L₀ L₀⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(-j), D(j))
ascii_str = \
"""\
/ i i\\ d / \\\n\
|A + B |*-----|C |\n\
\\ / L_0\\ L_0/\n\
dD \n\
\
"""
ucode_str = \
u("""\
⎛ i i⎞ ∂ ⎛ ⎞\n\
⎜A + B ⎟⋅────⎜C ⎟\n\
⎝ ⎠ L₀⎝ L₀⎠\n\
∂D \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(-i), D(j))
ascii_str = \
"""\
/ L_0 L_0\\ d / \\\n\
|A + B |*---|C |\n\
\\ / j\\ L_0/\n\
dD \n\
\
"""
ucode_str = \
u("""\
⎛ L₀ L₀⎞ ∂ ⎛ ⎞\n\
⎜A + B ⎟⋅───⎜C ⎟\n\
⎝ ⎠ j⎝ L₀⎠\n\
∂D \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i:1})
ascii_str = \
"""\
i=1,j\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i:1, j:1})
ascii_str = \
"""\
i=1,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {j:1})
ascii_str = \
"""\
i,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
expr = TensorElement(H(-i, j), {-i:1})
ascii_str = \
"""\
j\n\
H \n\
i=1 \
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_15560():
a = MatrixSymbol('a', 1, 1)
e = pretty(a*(KroneckerProduct(a, a)))
result = 'a*(a x a)'
assert e == result
def test_print_lerchphi():
# Part of issue 6013
a = Symbol('a')
pretty(lerchphi(a, 1, 2))
uresult = u'Φ(a, 1, 2)'
aresult = 'lerchphi(a, 1, 2)'
assert pretty(lerchphi(a, 1, 2)) == aresult
assert upretty(lerchphi(a, 1, 2)) == uresult
def test_issue_15583():
N = mechanics.ReferenceFrame('N')
result = '(n_x, n_y, n_z)'
e = pretty((N.x, N.y, N.z))
assert e == result
def test_matrixSymbolBold():
# Issue 15871
def boldpretty(expr):
return xpretty(expr, use_unicode=True, wrap_line=False, mat_symbol_style="bold")
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert boldpretty(trace(A)) == u'tr(𝐀)'
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert boldpretty(-A) == u'-𝐀'
assert boldpretty(A - A*B - B) == u'-𝐁 -𝐀⋅𝐁 + 𝐀'
assert boldpretty(-A*B - A*B*C - B) == u'-𝐁 -𝐀⋅𝐁 -𝐀⋅𝐁⋅𝐂'
A = MatrixSymbol("Addot", 3, 3)
assert boldpretty(A) == u'𝐀̈'
omega = MatrixSymbol("omega", 3, 3)
assert boldpretty(omega) == u'ω'
omega = MatrixSymbol("omeganorm", 3, 3)
assert boldpretty(omega) == u'‖ω‖'
a = Symbol('alpha')
b = Symbol('b')
c = MatrixSymbol("c", 3, 1)
d = MatrixSymbol("d", 3, 1)
assert boldpretty(a*B*c+b*d) == u'b⋅𝐝 + α⋅𝐁⋅𝐜'
d = MatrixSymbol("delta", 3, 1)
B = MatrixSymbol("Beta", 3, 3)
assert boldpretty(a*B*c+b*d) == u'b⋅δ + α⋅Β⋅𝐜'
A = MatrixSymbol("A_2", 3, 3)
assert boldpretty(A) == u'𝐀₂'
def test_center_accent():
assert center_accent('a', u'\N{COMBINING TILDE}') == u'ã'
assert center_accent('aa', u'\N{COMBINING TILDE}') == u'aã'
assert center_accent('aaa', u'\N{COMBINING TILDE}') == u'aãa'
assert center_accent('aaaa', u'\N{COMBINING TILDE}') == u'aaãa'
assert center_accent('aaaaa', u'\N{COMBINING TILDE}') == u'aaãaa'
assert center_accent('abcdefg', u'\N{COMBINING FOUR DOTS ABOVE}') == u'abcd⃜efg'
def test_imaginary_unit():
from sympy import pretty # As it is redefined above
assert pretty(1 + I, use_unicode=False) == '1 + I'
assert pretty(1 + I, use_unicode=True) == u'1 + ⅈ'
assert pretty(1 + I, use_unicode=False, imaginary_unit='j') == '1 + I'
assert pretty(1 + I, use_unicode=True, imaginary_unit='j') == u'1 + ⅉ'
raises(TypeError, lambda: pretty(I, imaginary_unit=I))
raises(ValueError, lambda: pretty(I, imaginary_unit="kkk"))
|
a6f24f64c88dee4e2f813de75df4dd4b322e82cadebc935de599298fe8f3e7da | from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
from sympy.core import Integer, Float
import inspect, re
from sympy import powsimp
if matchpy:
from matchpy import (Operation, CommutativeOperation, AssociativeOperation,
ManyToOneReplacer, OneIdentityOperation, CustomConstraint)
from sympy import Pow, Add, Integral, Basic, Mul, S, Function, E
from sympy.functions import (log, sin, cos, tan, cot, csc, sec, sqrt, erf,
exp as sym_exp, gamma, acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh,
tanh, coth, sech, csch, atan, acsc, asin, acot, acos, asec, fresnels,
fresnelc, erfc, erfi, Ei, uppergamma, polylog, zeta, factorial, polygamma, digamma, li,
expint, LambertW, loggamma)
from sympy.integrals.rubi.utility_function import (Gamma, rubi_exp, rubi_log, ProductLog, PolyGamma,
rubi_unevaluated_expr, process_trig)
from sympy.utilities.matchpy_connector import op_iter, op_len
@doctest_depends_on(modules=('matchpy',))
def rubi_object():
'''
Returns rubi ManyToOneReplacer by adding all rules from different modules.
Uncomment the lines to add integration capabilities of that module.
Currently, there are parsing issues with special_function,
derivative and miscellaneous_integration. Hence they are commented.
'''
from sympy.integrals.rubi.rules.integrand_simplification import integrand_simplification
from sympy.integrals.rubi.rules.linear_products import linear_products
from sympy.integrals.rubi.rules.quadratic_products import quadratic_products
from sympy.integrals.rubi.rules.binomial_products import binomial_products
from sympy.integrals.rubi.rules.trinomial_products import trinomial_products
from sympy.integrals.rubi.rules.miscellaneous_algebraic import miscellaneous_algebraic
from sympy.integrals.rubi.rules.exponential import exponential
from sympy.integrals.rubi.rules.logarithms import logarithms
from sympy.integrals.rubi.rules.sine import sine
from sympy.integrals.rubi.rules.tangent import tangent
from sympy.integrals.rubi.rules.secant import secant
from sympy.integrals.rubi.rules.miscellaneous_trig import miscellaneous_trig
from sympy.integrals.rubi.rules.inverse_trig import inverse_trig
from sympy.integrals.rubi.rules.hyperbolic import hyperbolic
from sympy.integrals.rubi.rules.inverse_hyperbolic import inverse_hyperbolic
from sympy.integrals.rubi.rules.special_functions import special_functions
#from sympy.integrals.rubi.rules.derivative import derivative
#from sympy.integrals.rubi.rules.piecewise_linear import piecewise_linear
from sympy.integrals.rubi.rules.miscellaneous_integration import miscellaneous_integration
rules = []
rules_applied = []
rules += integrand_simplification(rules_applied)
rules += linear_products(rules_applied)
rules += quadratic_products(rules_applied)
rules += binomial_products(rules_applied)
rules += trinomial_products(rules_applied)
rules += miscellaneous_algebraic(rules_applied)
rules += exponential(rules_applied)
rules += logarithms(rules_applied)
rules += special_functions(rules_applied)
rules += sine(rules_applied)
rules += tangent(rules_applied)
rules += secant(rules_applied)
rules += miscellaneous_trig(rules_applied)
rules += inverse_trig(rules_applied)
rules += hyperbolic(rules_applied)
rules += inverse_hyperbolic(rules_applied)
#rubi = piecewise_linear(rubi)
rules += miscellaneous_integration(rules_applied)
rubi = ManyToOneReplacer(*rules)
return rubi, rules_applied, rules
_E = rubi_unevaluated_expr(E)
Integrate = Function('Integrate')
rubi, rules_applied, rules = rubi_object()
def _has_cycle():
if rules_applied.count(rules_applied[-1]) == 1:
return False
if rules_applied[-1] == rules_applied[-2] == rules_applied[-3] == rules_applied[-4] == rules_applied[-5]:
return True
def process_final_integral(expr):
'''
When there is recursion for more than 10 rules or in total 20 rules have been applied
rubi returns `Integrate` in order to stop any further matching. After complete integration,
Integrate needs to be replaced back to Integral. Also rubi's `exp` need to be replaced back
to sympy's general `exp`.
Examples
========
>>> from sympy import Function, E
>>> from sympy.integrals.rubi.rubi import process_final_integral
>>> from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr
>>> Integrate = Function("Integrate")
>>> from sympy.abc import a, x
>>> _E = rubi_unevaluated_expr(E)
>>> process_final_integral(Integrate(a, x))
Integral(a, x)
>>> process_final_integral(_E**5)
exp(5)
'''
if expr.has(Integrate):
expr = expr.replace(Integrate, Integral)
if expr.has(_E):
expr = expr.replace(_E, E)
return expr
def rubi_powsimp(expr):
'''
This function is needed to preprocess an expression as done in matchpy
`x^a*x^b` in matchpy auotmatically transforms to `x^(a+b)`
Examples
========
>>> from sympy.integrals.rubi.rubi import rubi_powsimp
>>> from sympy.abc import a, b, x
>>> rubi_powsimp(x**a*x**b)
x**(a+b)
'''
lst_pow =[]
lst_non_pow = []
if isinstance(expr, Mul):
for i in expr.args:
if isinstance(i, (Pow, exp, sym_exp)):
lst_pow.append(i)
else:
lst_non_pow.append(i)
return powsimp(Mul(*lst_pow))*Mul(*lst_non_pow)
return expr
@doctest_depends_on(modules=('matchpy',))
def rubi_integrate(expr, var, showsteps=False):
'''
Rule based algorithm for integration. Integrates the expression by applying
transformation rules to the expression.
Returns `Integrate` if an expression cannot be integrated.
Parameters
==========
expr : integrand expression
var : variable of integration
Returns Integral object if unable to integrate.
'''
expr = expr.replace(sym_exp, exp)
rules_applied[:] = []
expr = process_trig(expr)
expr = rubi_powsimp(expr)
if isinstance(expr, (int, Integer)) or isinstance(expr, (float, Float)):
return S(expr)*var
if isinstance(expr, Add):
results = 0
for ex in expr.args:
rules_applied[:] = []
results += rubi.replace(Integral(ex, var))
rules_applied[:] = []
return process_final_integral(results)
results = rubi.replace(Integral(expr, var), max_count = 10)
return process_final_integral(results)
@doctest_depends_on(modules=('matchpy',))
def util_rubi_integrate(expr, var, showsteps=False):
expr = process_trig(expr)
expr = expr.replace(sym_exp, exp)
if isinstance(expr, (int, Integer)) or isinstance(expr, (float, Float)):
return S(expr)*var
if isinstance(expr, Add):
return rubi_integrate(expr, var)
if len(rules_applied) > 10:
if _has_cycle() or len(rules_applied) > 20:
return Integrate(expr, var)
results = rubi.replace(Integral(expr, var), max_count = 10)
rules_applied[:] = []
return results
@doctest_depends_on(modules=('matchpy',))
def get_matching_rule_definition(expr, var):
'''
Prints the list or rules which match to `expr`.
Parameters
==========
expr : integrand expression
var : variable of integration
'''
matcher = rubi.matcher
miter = matcher.match(Integral(expr, var))
for fun, e in miter:
print("Rule matching: ")
print(inspect.getsourcefile(fun))
code, lineno = inspect.getsourcelines(fun)
print("On line: ", lineno)
print("\n".join(code))
print("Pattern matching: ")
pattno = int(re.match(r"^\s*rule(\d+)", code[0]).group(1))
print(matcher.patterns[pattno-1])
print(e)
print()
|
d14fbd2bc4771f9c092029be447a341be9cc1db0aa269246464561614bf90b17 | '''
Utility functions for Rubi integration.
See: http://www.apmaths.uwo.ca/~arich/IntegrationRules/PortableDocumentFiles/Integration%20utility%20functions.pdf
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
from sympy.functions.elementary.integers import floor, frac
from sympy.functions import (log as sym_log , sin, cos, tan, cot, csc, sec, sqrt, erf, gamma, uppergamma, polygamma, digamma,
loggamma, factorial, zeta, LambertW)
from sympy.functions.elementary.hyperbolic import acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch
from sympy.functions.elementary.trigonometric import atan, acsc, asin, acot, acos, asec, atan2
from sympy.polys.polytools import Poly, quo, rem, total_degree, degree
from sympy.simplify.simplify import fraction, simplify, cancel, powsimp
from sympy.core.sympify import sympify
from sympy.utilities.iterables import postorder_traversal
from sympy.functions.special.error_functions import fresnelc, fresnels, erfc, erfi, Ei, expint, li, Si, Ci, Shi, Chi
from sympy.functions.elementary.complexes import im, re, Abs
from sympy.core.exprtools import factor_terms
from sympy import (Basic, E, polylog, N, Wild, WildFunction, factor, gcd, Sum, S, I, Mul, Integer, Float, Dict, Symbol, Rational,
Add, hyper, symbols, sqf_list, sqf, Max, factorint, factorrat, Min, sign, E, Function, collect, FiniteSet, nsimplify,
expand_trig, expand, poly, apart, lcm, And, Pow, pi, zoo, oo, Integral, UnevaluatedExpr, PolynomialError, Dummy, exp as sym_exp,
powdenest, PolynomialDivisionFailed, discriminant, UnificationFailed, appellf1)
from sympy.functions.special.hyper import TupleArg
from sympy.functions.special.elliptic_integrals import elliptic_f, elliptic_e, elliptic_pi
from sympy.utilities.iterables import flatten
from random import randint
from sympy.logic.boolalg import Or
class rubi_unevaluated_expr(UnevaluatedExpr):
'''
This is needed to convert `exp` as `Pow`.
sympy's UnevaluatedExpr has an issue with `is_commutative`.
'''
@property
def is_commutative(self):
from sympy.core.logic import fuzzy_and
return fuzzy_and(a.is_commutative for a in self.args)
_E = rubi_unevaluated_expr(E)
class rubi_exp(Function):
'''
sympy's exp is not identified as `Pow`. So it is not matched with `Pow`.
Like `a = exp(2)` is not identified as `Pow(E, 2)`. Rubi rules need it.
So, another exp has been created only for rubi module.
Examples
========
>>> from sympy import Pow, exp as sym_exp
>>> isinstance(sym_exp(2), Pow)
False
>>> from sympy.integrals.rubi.utility_function import rubi_exp
>>> isinstance(rubi_exp(2), Pow)
True
'''
@classmethod
def eval(cls, *args):
return Pow(_E, args[0])
class rubi_log(Function):
'''
For rule matching different `exp` has been used. So for proper results,
`log` is modified little only for case when it encounters rubi's `exp`.
For other cases it is same.
Examples
========
>>> from sympy.integrals.rubi.utility_function import rubi_exp, rubi_log
>>> a = rubi_exp(2)
>>> rubi_log(a)
2
'''
@classmethod
def eval(cls, *args):
if args[0].has(_E):
return sym_log(args[0]).doit()
else:
return sym_log(args[0])
if matchpy:
from matchpy import Arity, Operation, CommutativeOperation, AssociativeOperation, OneIdentityOperation, CustomConstraint, Pattern, ReplacementRule, ManyToOneReplacer
from matchpy.expressions.functions import op_iter, create_operation_expression, op_len
from sympy.integrals.rubi.symbol import WC
from matchpy import is_match, replace_all
from sympy.utilities.matchpy_connector import Operation
class UtilityOperator(Operation):
name = 'UtilityOperator'
arity = Arity.variadic
commutative=False
associative=True
Operation.register(rubi_log)
Operation.register(rubi_exp)
A_, B_, C_, F_, G_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, z_ = [WC(i) for i in 'ABCFGabcdefghijklmnpqrtuvswxz']
a, b, c, d, e = symbols('a b c d e')
class Int(Function):
'''
Integrates given `expr` by matching rubi rules.
'''
@classmethod
def eval(cls, expr, var):
if isinstance(expr, (int, Integer, float, Float)):
return S(expr)*var
from sympy.integrals.rubi.rubi import util_rubi_integrate
return util_rubi_integrate(expr, var)
def replace_pow_exp(z):
'''
This function converts back rubi's `exp` to general sympy's `exp`.
Examples
========
>>> from sympy.integrals.rubi.utility_function import rubi_exp, replace_pow_exp
>>> expr = rubi_exp(5)
>>> expr
E**5
>>> replace_pow_exp(expr)
exp(5)
'''
z = S(z)
if z.has(_E):
z = z.replace(_E, E)
return z
def Simplify(expr):
expr = simplify(expr)
return expr
def Set(expr, value):
return {expr: value}
def With(subs, expr):
if isinstance(subs, dict):
k = list(subs.keys())[0]
expr = expr.xreplace({k: subs[k]})
else:
for i in subs:
k = list(i.keys())[0]
expr = expr.xreplace({k: i[k]})
return expr
def Module(subs, expr):
return With(subs, expr)
def Scan(f, expr):
# evaluates f applied to each element of expr in turn.
for i in expr:
yield f(i)
def MapAnd(f, l, x=None):
# MapAnd[f,l] applies f to the elements of list l until False is returned; else returns True
if x:
for i in l:
if f(i, x) == False:
return False
return True
else:
for i in l:
if f(i) == False:
return False
return True
def FalseQ(u):
if isinstance(u, (Dict, dict)):
return FalseQ(*list(u.values()))
return u == False
def ZeroQ(*expr):
if len(expr) == 1:
if isinstance(expr[0], list):
return list(ZeroQ(i) for i in expr[0])
else:
return Simplify(expr[0]) == 0
else:
return all(ZeroQ(i) for i in expr)
def OneQ(a):
if a == S(1):
return True
return False
def NegativeQ(u):
u = Simplify(u)
if u in (zoo, oo):
return False
if u.is_comparable:
res = u < 0
if not res.is_Relational:
return res
return False
def NonzeroQ(expr):
return Simplify(expr) != 0
def FreeQ(nodes, var):
if var == Int:
return FreeQ(nodes, Integral)
if isinstance(nodes, list):
return not any(S(expr).has(var) for expr in nodes)
else:
nodes = S(nodes)
return not nodes.has(var)
def NFreeQ(nodes, var):
''' Note that in rubi 4.10.8 this function was not defined in `Integration Utility Functions.m`,
but was used in rules. So explicitly its returning `False`
'''
return False
# return not FreeQ(nodes, var)
def List(*var):
return list(var)
def PositiveQ(var):
var = Simplify(var)
if var in (zoo, oo):
return False
if var.is_comparable:
res = var > 0
if not res.is_Relational:
return res
return False
def PositiveIntegerQ(*args):
return all(var.is_Integer and PositiveQ(var) for var in args)
def NegativeIntegerQ(*args):
return all(var.is_Integer and NegativeQ(var) for var in args)
def IntegerQ(var):
var = Simplify(var)
if isinstance(var, (int, Integer)):
return True
else:
return var.is_Integer
def IntegersQ(*var):
return all(IntegerQ(i) for i in var)
def _ComplexNumberQ(var):
i = S(im(var))
if isinstance(i, (Integer, Float)):
return i != 0
else:
return False
def ComplexNumberQ(*var):
"""
ComplexNumberQ(m, n,...) returns True if m, n, ... are all explicit complex numbers, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import ComplexNumberQ
>>> from sympy import I
>>> ComplexNumberQ(1 + I*2, I)
True
>>> ComplexNumberQ(2, I)
False
"""
return all(_ComplexNumberQ(i) for i in var)
def PureComplexNumberQ(*var):
return all((_ComplexNumberQ(i) and re(i)==0) for i in var)
def RealNumericQ(u):
return u.is_real
def PositiveOrZeroQ(u):
return u.is_real and u >= 0
def NegativeOrZeroQ(u):
return u.is_real and u <= 0
def FractionOrNegativeQ(u):
return FractionQ(u) or NegativeQ(u)
def NegQ(var):
return Not(PosQ(var)) and NonzeroQ(var)
def Equal(a, b):
return a == b
def Unequal(a, b):
return a != b
def IntPart(u):
# IntPart[u] returns the sum of the integer terms of u.
if ProductQ(u):
if IntegerQ(First(u)):
return First(u)*IntPart(Rest(u))
elif IntegerQ(u):
return u
elif FractionQ(u):
return IntegerPart(u)
elif SumQ(u):
res = 0
for i in u.args:
res += IntPart(i)
return res
return 0
def FracPart(u):
# FracPart[u] returns the sum of the non-integer terms of u.
if ProductQ(u):
if IntegerQ(First(u)):
return First(u)*FracPart(Rest(u))
if IntegerQ(u):
return 0
elif FractionQ(u):
return FractionalPart(u)
elif SumQ(u):
res = 0
for i in u.args:
res += FracPart(i)
return res
else:
return u
def RationalQ(*nodes):
return all(var.is_Rational for var in nodes)
def ProductQ(expr):
return S(expr).is_Mul
def SumQ(expr):
return expr.is_Add
def NonsumQ(expr):
return not SumQ(expr)
def Subst(a, x, y):
if None in [a, x, y]:
return None
if a.has(Function('Integrate')):
# substituting in `Function(Integrate)` won't take care of properties of Integral
a = a.replace(Function('Integrate'), Integral)
return a.subs(x, y)
# return a.xreplace({x: y})
def First(expr, d=None):
"""
Gives the first element if it exists, or d otherwise.
Examples
========
>>> from sympy.integrals.rubi.utility_function import First
>>> from sympy.abc import a, b, c
>>> First(a + b + c)
a
>>> First(a*b*c)
a
"""
if isinstance(expr, list):
return expr[0]
if isinstance(expr, Symbol):
return expr
else:
if SumQ(expr) or ProductQ(expr):
l = Sort(expr.args)
return l[0]
else:
return expr.args[0]
def Rest(expr):
"""
Gives rest of the elements if it exists
Examples
========
>>> from sympy.integrals.rubi.utility_function import Rest
>>> from sympy.abc import a, b, c
>>> Rest(a + b + c)
b + c
>>> Rest(a*b*c)
b*c
"""
if isinstance(expr, list):
return expr[1:]
else:
if SumQ(expr) or ProductQ(expr):
l = Sort(expr.args)
return expr.func(*l[1:])
else:
return expr.args[1]
def SqrtNumberQ(expr):
# SqrtNumberQ[u] returns True if u^2 is a rational number; else it returns False.
if PowerQ(expr):
m = expr.base
n = expr.exp
return (IntegerQ(n) and SqrtNumberQ(m)) or (IntegerQ(n-S(1)/2) and RationalQ(m))
elif expr.is_Mul:
return all(SqrtNumberQ(i) for i in expr.args)
else:
return RationalQ(expr) or expr == I
def SqrtNumberSumQ(u):
return SumQ(u) and SqrtNumberQ(First(u)) and SqrtNumberQ(Rest(u)) or ProductQ(u) and SqrtNumberQ(First(u)) and SqrtNumberSumQ(Rest(u))
def LinearQ(expr, x):
"""
LinearQ(expr, x) returns True iff u is a polynomial of degree 1.
Examples
========
>>> from sympy.integrals.rubi.utility_function import LinearQ
>>> from sympy.abc import x, y, a
>>> LinearQ(a, x)
False
>>> LinearQ(3*x + y**2, x)
True
>>> LinearQ(3*x + y**2, y)
False
"""
if isinstance(expr, list):
return all(LinearQ(i, x) for i in expr)
elif expr.is_polynomial(x):
if degree(Poly(expr, x), gen=x) == 1:
return True
return False
def Sqrt(a):
return sqrt(a)
def ArcCosh(a):
return acosh(a)
class Util_Coefficient(Function):
def doit(self):
if len(self.args) == 2:
n = 1
else:
n = Simplify(self.args[2])
if NumericQ(n):
expr = expand(self.args[0])
if isinstance(n, (int, Integer)):
return expr.coeff(self.args[1], n)
else:
return expr.coeff(self.args[1]**n)
else:
return self
def Coefficient(expr, var, n=1):
"""
Coefficient(expr, var) gives the coefficient of form in the polynomial expr.
Coefficient(expr, var, n) gives the coefficient of var**n in expr.
Examples
========
>>> from sympy.integrals.rubi.utility_function import Coefficient
>>> from sympy.abc import x, a, b, c
>>> Coefficient(7 + 2*x + 4*x**3, x, 1)
2
>>> Coefficient(a + b*x + c*x**3, x, 0)
a
>>> Coefficient(a + b*x + c*x**3, x, 4)
0
>>> Coefficient(b*x + c*x**3, x, 3)
c
"""
if NumericQ(n):
if expr == 0 or n in (zoo, oo):
return 0
expr = expand(expr)
if isinstance(n, (int, Integer)):
return expr.coeff(var, n)
else:
return expr.coeff(var**n)
return Util_Coefficient(expr, var, n)
def Denominator(var):
var = Simplify(var)
if isinstance(var, Pow):
if isinstance(var.exp, Integer):
if var.exp > 0:
return Pow(Denominator(var.base), var.exp)
elif var.exp < 0:
return Pow(Numerator(var.base), -1*var.exp)
elif isinstance(var, Add):
var = factor(var)
return fraction(var)[1]
def Hypergeometric2F1(a, b, c, z):
return hyper([a, b], [c], z)
def Not(var):
if isinstance(var, bool):
return not var
elif var.is_Relational:
var = False
return not var
def FractionalPart(a):
return frac(a)
def IntegerPart(a):
return floor(a)
def AppellF1(a, b1, b2, c, x, y):
return appellf1(a, b1, b2, c, x, y)
def EllipticPi(*args):
return elliptic_pi(*args)
def EllipticE(*args):
return elliptic_e(*args)
def EllipticF(Phi, m):
return elliptic_f(Phi, m)
def ArcTan(a, b = None):
if b is None:
return atan(a)
else:
return atan2(a, b)
def ArcCot(a):
return acot(a)
def ArcCoth(a):
return acoth(a)
def ArcTanh(a):
return atanh(a)
def ArcSin(a):
return asin(a)
def ArcSinh(a):
return asinh(a)
def ArcCos(a):
return acos(a)
def ArcCsc(a):
return acsc(a)
def ArcSec(a):
return asec(a)
def ArcCsch(a):
return acsch(a)
def ArcSech(a):
return asech(a)
def Sinh(u):
return sinh(u)
def Tanh(u):
return tanh(u)
def Cosh(u):
return cosh(u)
def Sech(u):
return sech(u)
def Csch(u):
return csch(u)
def Coth(u):
return coth(u)
def LessEqual(*args):
for i in range(0, len(args) - 1):
try:
if args[i] > args[i + 1]:
return False
except:
return False
return True
def Less(*args):
for i in range(0, len(args) - 1):
try:
if args[i] >= args[i + 1]:
return False
except:
return False
return True
def Greater(*args):
for i in range(0, len(args) - 1):
try:
if args[i] <= args[i + 1]:
return False
except:
return False
return True
def GreaterEqual(*args):
for i in range(0, len(args) - 1):
try:
if args[i] < args[i + 1]:
return False
except:
return False
return True
def FractionQ(*args):
"""
FractionQ(m, n,...) returns True if m, n, ... are all explicit fractions, else it returns False.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.rubi.utility_function import FractionQ
>>> FractionQ(S('3'))
False
>>> FractionQ(S('3')/S('2'))
True
"""
return all(i.is_Rational for i in args) and all(Denominator(i)!= S(1) for i in args)
def IntLinearcQ(a, b, c, d, m, n, x):
# returns True iff (a+b*x)^m*(c+d*x)^n is integrable wrt x in terms of non-hypergeometric functions.
return IntegerQ(m) or IntegerQ(n) or IntegersQ(S(3)*m, S(3)*n) or IntegersQ(S(4)*m, S(4)*n) or IntegersQ(S(2)*m, S(6)*n) or IntegersQ(S(6)*m, S(2)*n) or IntegerQ(m + n)
Defer = UnevaluatedExpr
def Expand(expr):
return expr.expand()
def IndependentQ(u, x):
"""
If u is free from x IndependentQ(u, x) returns True else False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import IndependentQ
>>> from sympy.abc import x, a, b
>>> IndependentQ(a + b*x, x)
False
>>> IndependentQ(a + b, x)
True
"""
return FreeQ(u, x)
def PowerQ(expr):
return expr.is_Pow or ExpQ(expr)
def IntegerPowerQ(u):
if isinstance(u, sym_exp): #special case for exp
return IntegerQ(u.args[0])
return PowerQ(u) and IntegerQ(u.args[1])
def PositiveIntegerPowerQ(u):
if isinstance(u, sym_exp):
return IntegerQ(u.args[0]) and PositiveQ(u.args[0])
return PowerQ(u) and IntegerQ(u.args[1]) and PositiveQ(u.args[1])
def FractionalPowerQ(u):
if isinstance(u, sym_exp):
return FractionQ(u.args[0])
return PowerQ(u) and FractionQ(u.args[1])
def AtomQ(expr):
expr = sympify(expr)
if isinstance(expr, list):
return False
if expr in [None, True, False, _E]: # [None, True, False] are atoms in mathematica and _E is also an atom
return True
# elif isinstance(expr, list):
# return all(AtomQ(i) for i in expr)
else:
return expr.is_Atom
def ExpQ(u):
u = replace_pow_exp(u)
return Head(u) in (sym_exp, rubi_exp)
def LogQ(u):
return u.func in (sym_log, Log)
def Head(u):
return u.func
def MemberQ(l, u):
if isinstance(l, list):
return u in l
else:
return u in l.args
def TrigQ(u):
if AtomQ(u):
x = u
else:
x = Head(u)
return MemberQ([sin, cos, tan, cot, sec, csc], x)
def SinQ(u):
return Head(u) == sin
def CosQ(u):
return Head(u) == cos
def TanQ(u):
return Head(u) == tan
def CotQ(u):
return Head(u) == cot
def SecQ(u):
return Head(u) == sec
def CscQ(u):
return Head(u) == csc
def Sin(u):
return sin(u)
def Cos(u):
return cos(u)
def Tan(u):
return tan(u)
def Cot(u):
return cot(u)
def Sec(u):
return sec(u)
def Csc(u):
return csc(u)
def HyperbolicQ(u):
if AtomQ(u):
x = u
else:
x = Head(u)
return MemberQ([sinh, cosh, tanh, coth, sech, csch], x)
def SinhQ(u):
return Head(u) == sinh
def CoshQ(u):
return Head(u) == cosh
def TanhQ(u):
return Head(u) == tanh
def CothQ(u):
return Head(u) == coth
def SechQ(u):
return Head(u) == sech
def CschQ(u):
return Head(u) == csch
def InverseTrigQ(u):
if AtomQ(u):
x = u
else:
x = Head(u)
return MemberQ([asin, acos, atan, acot, asec, acsc], x)
def SinCosQ(f):
return MemberQ([sin, cos, sec, csc], Head(f))
def SinhCoshQ(f):
return MemberQ([sinh, cosh, sech, csch], Head(f))
def LeafCount(expr):
return len(list(postorder_traversal(expr)))
def Numerator(u):
u = Simplify(u)
if isinstance(u, Pow):
if isinstance(u.exp, Integer):
if u.exp > 0:
return Pow(Numerator(u.base), u.exp)
elif u.exp < 0:
return Pow(Denominator(u.base), -1*u.exp)
elif isinstance(u, Add):
u = factor(u)
return fraction(u)[0]
def NumberQ(u):
if isinstance(u, (int, float)):
return True
return u.is_number
def NumericQ(u):
return N(u).is_number
def Length(expr):
"""
Returns number of elements in the experssion just as sympy's len.
Examples
========
>>> from sympy.integrals.rubi.utility_function import Length
>>> from sympy.abc import x, a, b
>>> from sympy import cos, sin
>>> Length(a + b)
2
>>> Length(sin(a)*cos(a))
2
"""
if isinstance(expr, list):
return len(expr)
return len(expr.args)
def ListQ(u):
return isinstance(u, list)
def Im(u):
u = S(u)
return im(u.doit())
def Re(u):
u = S(u)
return re(u.doit())
def InverseHyperbolicQ(u):
if not u.is_Atom:
u = Head(u)
return u in [acosh, asinh, atanh, acoth, acsch, acsch]
def InverseFunctionQ(u):
# returns True if u is a call on an inverse function; else returns False.
return LogQ(u) or InverseTrigQ(u) and Length(u) <= 1 or InverseHyperbolicQ(u) or u.func == polylog
def TrigHyperbolicFreeQ(u, x):
# If u is free of trig, hyperbolic and calculus functions involving x, TrigHyperbolicFreeQ[u,x] returns true; else it returns False.
if AtomQ(u):
return True
else:
if TrigQ(u) | HyperbolicQ(u) | CalculusQ(u):
return FreeQ(u, x)
else:
for i in u.args:
if not TrigHyperbolicFreeQ(i, x):
return False
return True
def InverseFunctionFreeQ(u, x):
# If u is free of inverse, calculus and hypergeometric functions involving x, InverseFunctionFreeQ[u,x] returns true; else it returns False.
if AtomQ(u):
return True
else:
if InverseFunctionQ(u) or CalculusQ(u) or u.func == hyper or u.func == appellf1:
return FreeQ(u, x)
else:
for i in u.args:
if not ElementaryFunctionQ(i):
return False
return True
def RealQ(u):
if ListQ(u):
return MapAnd(RealQ, u)
elif NumericQ(u):
return ZeroQ(Im(N(u)))
elif PowerQ(u):
u = u.base
v = u.exp
return RealQ(u) & RealQ(v) & (IntegerQ(v) | PositiveOrZeroQ(u))
elif u.is_Mul:
return all(RealQ(i) for i in u.args)
elif u.is_Add:
return all(RealQ(i) for i in u.args)
elif u.is_Function:
f = u.func
u = u.args[0]
if f in [sin, cos, tan, cot, sec, csc, atan, acot, erf]:
return RealQ(u)
else:
if f in [asin, acos]:
return LE(-1, u, 1)
else:
if f == sym_log:
return PositiveOrZeroQ(u)
else:
return False
else:
return False
def EqQ(u, v):
return ZeroQ(u - v)
def FractionalPowerFreeQ(u):
if AtomQ(u):
return True
elif FractionalPowerQ(u):
return False
def ComplexFreeQ(u):
if AtomQ(u) and Not(ComplexNumberQ(u)):
return True
else:
return False
def PolynomialQ(u, x = None):
if x is None :
return u.is_polynomial()
if isinstance(x, Pow):
if isinstance(x.exp, Integer):
deg = degree(u, x.base)
if u.is_polynomial(x):
if deg % x.exp !=0 :
return False
try:
p = Poly(u, x.base)
except PolynomialError:
return False
c_list = p.all_coeffs()
coeff_list = c_list[:-1:x.exp]
coeff_list += [c_list[-1]]
for i in coeff_list:
if not i == 0:
index = c_list.index(i)
c_list[index] = 0
if all(i == 0 for i in c_list):
return True
else:
return False
return u.is_polynomial(x)
else:
return False
elif isinstance(x.exp, (Float, Rational)): #not full - proof
if FreeQ(simplify(u), x.base) and Exponent(u, x.base) == 0:
if not all(FreeQ(u, i) for i in x.base.free_symbols):
return False
if isinstance(x, Mul):
return all(PolynomialQ(u, i) for i in x.args)
return u.is_polynomial(x)
def FactorSquareFree(u):
return sqf(u)
def PowerOfLinearQ(expr, x):
u = Wild('u')
w = Wild('w')
m = Wild('m')
n = Wild('n')
Match = expr.match(u**m)
if PolynomialQ(Match[u], x) and FreeQ(Match[m], x):
if IntegerQ(Match[m]):
e = FactorSquareFree(Match[u]).match(w**n)
if FreeQ(e[n], x) and LinearQ(e[w], x):
return True
else:
return False
else:
return LinearQ(Match[u], x)
else:
return False
def Exponent(expr, x, h = None):
expr = Expand(S(expr))
if h is None:
if S(expr).is_number or (not expr.has(x)):
return 0
if PolynomialQ(expr, x):
if isinstance(x, Rational):
return degree(Poly(expr, x), x)
return degree(expr, gen = x)
else:
return 0
else:
if S(expr).is_number or (not expr.has(x)):
res = [0]
if expr.is_Add:
expr = collect(expr, x)
lst = []
k = 1
for t in expr.args:
if t.has(x):
if isinstance(x, Rational):
lst += [degree(Poly(t, x), x)]
else:
lst += [degree(t, gen = x)]
else:
if k == 1:
lst += [0]
k += 1
lst.sort()
res = lst
else:
if isinstance(x, Rational):
res = [degree(Poly(expr, x), x)]
else:
res = [degree(expr, gen = x)]
return h(*res)
def QuadraticQ(u, x):
# QuadraticQ(u, x) returns True iff u is a polynomial of degree 2 and not a monomial of the form a x^2
if ListQ(u):
for expr in u:
if Not(PolyQ(expr, x, 2) and Not(Coefficient(expr, x, 0) == 0 and Coefficient(expr, x, 1) == 0)):
return False
return True
else:
return PolyQ(u, x, 2) and Not(Coefficient(u, x, 0) == 0 and Coefficient(u, x, 1) == 0)
def LinearPairQ(u, v, x):
# LinearPairQ(u, v, x) returns True iff u and v are linear not equal x but u/v is a constant wrt x
return LinearQ(u, x) and LinearQ(v, x) and NonzeroQ(u-x) and ZeroQ(Coefficient(u, x, 0)*Coefficient(v, x, 1)-Coefficient(u, x, 1)*Coefficient(v, x, 0))
def BinomialParts(u, x):
if PolynomialQ(u, x):
if Exponent(u, x) > 0:
lst = Exponent(u, x, List)
if len(lst)==1:
return [0, Coefficient(u, x, Exponent(u, x)), Exponent(u,x)]
elif len(lst) == 2 and lst[0] == 0:
return [Coefficient(u, x, 0), Coefficient(u, x, Exponent(u, x)), Exponent(u, x)]
else:
return False
else:
return False
elif PowerQ(u):
if u.base == x and FreeQ(u.exp, x):
return [0, 1, u.exp]
else:
return False
elif ProductQ(u):
if FreeQ(First(u), x):
lst2 = BinomialParts(Rest(u), x)
if AtomQ(lst2):
return False
else:
return [First(u)*lst2[0], First(u)*lst2[1], lst2[2]]
elif FreeQ(Rest(u), x):
lst1 = BinomialParts(First(u), x)
if AtomQ(lst1):
return False
else:
return [Rest(u)*lst1[0], Rest(u)*lst1[1], lst1[2]]
lst1 = BinomialParts(First(u), x)
if AtomQ(lst1):
return False
lst2 = BinomialParts(Rest(u), x)
if AtomQ(lst2):
return False
a = lst1[0]
b = lst1[1]
m = lst1[2]
c = lst2[0]
d = lst2[1]
n = lst2[2]
if ZeroQ(a):
if ZeroQ(c):
return [0, b*d, m + n]
elif ZeroQ(m + n):
return [b*d, b*c, m]
else:
return False
if ZeroQ(c):
if ZeroQ(m + n):
return [b*d, a*d, n]
else:
return False
if EqQ(m, n) and ZeroQ(a*d + b*c):
return [a*c, b*d, 2*m]
else:
return False
elif SumQ(u):
if FreeQ(First(u),x):
lst2 = BinomialParts(Rest(u), x)
if AtomQ(lst2):
return False
else:
return [First(u) + lst2[0], lst2[1], lst2[2]]
elif FreeQ(Rest(u), x):
lst1 = BinomialParts(First(u), x)
if AtomQ(lst1):
return False
else:
return[Rest(u) + lst1[0], lst1[1], lst1[2]]
lst1 = BinomialParts(First(u), x)
if AtomQ(lst1):
return False
lst2 = BinomialParts(Rest(u),x)
if AtomQ(lst2):
return False
if EqQ(lst1[2], lst2[2]):
return [lst1[0] + lst2[0], lst1[1] + lst2[1], lst1[2]]
else:
return False
else:
return False
def TrinomialParts(u, x):
# If u is equivalent to a trinomial of the form a + b*x^n + c*x^(2*n) where n!=0, b!=0 and c!=0, TrinomialParts[u,x] returns the list {a,b,c,n}; else it returns False.
u = sympify(u)
if PolynomialQ(u, x):
lst = CoefficientList(u, x)
if len(lst)<3 or EvenQ(sympify(len(lst))) or ZeroQ((len(lst)+1)/2):
return False
#Catch(
# Scan(Function(if ZeroQ(lst), Null, Throw(False), Drop(Drop(Drop(lst, [(len(lst)+1)/2]), 1), -1];
# [First(lst), lst[(len(lst)+1)/2], Last(lst), (len(lst)-1)/2]):
if PowerQ(u):
if EqQ(u.exp, 2):
lst = BinomialParts(u.base, x)
if not lst or ZeroQ(lst[0]):
return False
else:
return [lst[0]**2, 2*lst[0]*lst[1], lst[1]**2, lst[2]]
else:
return False
if ProductQ(u):
if FreeQ(First(u), x):
lst2 = TrinomialParts(Rest(u), x)
if not lst2:
return False
else:
return [First(u)*lst2[0], First(u)*lst2[1], First(u)*lst2[2], lst2[3]]
if FreeQ(Rest(u), x):
lst1 = TrinomialParts(First(u), x)
if not lst1:
return False
else:
return [Rest(u)*lst1[0], Rest(u)*lst1[1], Rest(u)*lst1[2], lst1[3]]
lst1 = BinomialParts(First(u), x)
if not lst1:
return False
lst2 = BinomialParts(Rest(u), x)
if not lst2:
return False
a = lst1[0]
b = lst1[1]
m = lst1[2]
c = lst2[0]
d = lst2[1]
n = lst2[2]
if EqQ(m, n) and NonzeroQ(a*d+b*c):
return [a*c, a*d + b*c, b*d, m]
else:
return False
if SumQ(u):
if FreeQ(First(u), x):
lst2 = TrinomialParts(Rest(u), x)
if not lst2:
return False
else:
return [First(u)+lst2[0], lst2[1], lst2[2], lst2[3]]
if FreeQ(Rest(u), x):
lst1 = TrinomialParts(First(u), x)
if not lst1:
return False
else:
return [Rest(u)+lst1[0], lst1[1], lst1[2], lst1[3]]
lst1 = TrinomialParts(First(u), x)
if not lst1:
lst3 = BinomialParts(First(u), x)
if not lst3:
return False
lst2 = TrinomialParts(Rest(u), x)
if not lst2:
lst4 = BinomialParts(Rest(u), x)
if not lst4:
return False
if EqQ(lst3[2], 2*lst4[2]):
return [lst3[0]+lst4[0], lst4[1], lst3[1], lst4[2]]
if EqQ(lst4[2], 2*lst3[2]):
return [lst3[0]+lst4[0], lst3[1], lst4[1], lst3[2]]
else:
return False
if EqQ(lst3[2], lst2[3]) and NonzeroQ(lst3[1]+lst2[1]):
return [lst3[0]+lst2[0], lst3[1]+lst2[1], lst2[2], lst2[3]]
if EqQ(lst3[2], 2*lst2[3]) and NonzeroQ(lst3[1]+lst2[2]):
return [lst3[0]+lst2[0], lst2[1], lst3[1]+lst2[2], lst2[3]]
else:
return False
lst2 = TrinomialParts(Rest(u), x)
if AtomQ(lst2):
lst4 = BinomialParts(Rest(u), x)
if not lst4:
return False
if EqQ(lst4[2], lst1[3]) and NonzeroQ(lst1[1]+lst4[0]):
return [lst1[0]+lst4[0], lst1[1]+lst4[1], lst1[2], lst1[3]]
if EqQ(lst4[2], 2*lst1[3]) and NonzeroQ(lst1[2]+lst4[1]):
return [lst1[0]+lst4[0], lst1[1], lst1[2]+lst4[1], lst1[3]]
else:
return False
if EqQ(lst1[3], lst2[3]) and NonzeroQ(lst1[1]+lst2[1]) and NonzeroQ(lst1[2]+lst2[2]):
return [lst1[0]+lst2[0], lst1[1]+lst2[1], lst1[2]+lst2[2], lst1[3]]
else:
return False
else:
return False
def PolyQ(u, x, n=None):
# returns True iff u is a polynomial of degree n.
if ListQ(u):
return all(PolyQ(i, x) for i in u)
if n==None:
if u == x:
return False
elif isinstance(x, Pow):
n = x.exp
x_base = x.base
if FreeQ(n, x_base):
if PositiveIntegerQ(n):
return PolyQ(u, x_base) and (PolynomialQ(u, x) or PolynomialQ(Together(u), x))
elif AtomQ(n):
return PolynomialQ(u, x) and FreeQ(CoefficientList(u, x), x_base)
else:
return False
return PolynomialQ(u, x) or PolynomialQ(u, Together(x))
else:
return PolynomialQ(u, x) and Coefficient(u, x, n) != 0 and Exponent(u, x) == n
def EvenQ(u):
# gives True if expr is an even integer, and False otherwise.
return isinstance(u, (Integer, int)) and u%2 == 0
def OddQ(u):
# gives True if expr is an odd integer, and False otherwise.
return isinstance(u, (Integer, int)) and u%2 == 1
def PerfectSquareQ(u):
# (* If u is a rational number whose squareroot is rational or if u is of the form u1^n1 u2^n2 ...
# and n1, n2, ... are even, PerfectSquareQ[u] returns True; else it returns False. *)
if RationalQ(u):
return Greater(u, 0) and RationalQ(Sqrt(u))
elif PowerQ(u):
return EvenQ(u.exp)
elif ProductQ(u):
return PerfectSquareQ(First(u)) and PerfectSquareQ(Rest(u))
elif SumQ(u):
s = Simplify(u)
if NonsumQ(s):
return PerfectSquareQ(s)
return False
else:
return False
def NiceSqrtAuxQ(u):
if RationalQ(u):
return u > 0
elif PowerQ(u):
return EvenQ(u.exp)
elif ProductQ(u):
return NiceSqrtAuxQ(First(u)) and NiceSqrtAuxQ(Rest(u))
elif SumQ(u):
s = Simplify(u)
return NonsumQ(s) and NiceSqrtAuxQ(s)
else:
return False
def NiceSqrtQ(u):
return Not(NegativeQ(u)) and NiceSqrtAuxQ(u)
def Together(u):
return factor(u)
def PosAux(u):
if RationalQ(u):
return u>0
elif NumberQ(u):
if ZeroQ(Re(u)):
return Im(u) > 0
else:
return Re(u) > 0
elif NumericQ(u):
v = N(u)
if ZeroQ(Re(v)):
return Im(v) > 0
else:
return Re(v) > 0
elif PowerQ(u):
if OddQ(u.exp):
return PosAux(u.base)
else:
return True
elif ProductQ(u):
if PosAux(First(u)):
return PosAux(Rest(u))
else:
return not PosAux(Rest(u))
elif SumQ(u):
return PosAux(First(u))
else:
res = u > 0
if res in(True, False):
return res
return True
def PosQ(u):
# If u is not 0 and has a positive form, PosQ[u] returns True, else it returns False.
return PosAux(TogetherSimplify(u))
def CoefficientList(u, x):
if PolynomialQ(u, x):
return list(reversed(Poly(u, x).all_coeffs()))
else:
return []
def ReplaceAll(expr, args):
if isinstance(args, list):
n_args = {}
for i in args:
n_args.update(i)
return expr.subs(n_args)
return expr.subs(args)
def ExpandLinearProduct(v, u, a, b, x):
# If u is a polynomial in x, ExpandLinearProduct[v,u,a,b,x] expands v*u into a sum of terms of the form c*v*(a+b*x)^n.
if FreeQ([a, b], x) and PolynomialQ(u, x):
lst = CoefficientList(ReplaceAll(u, {x: (x - a)/b}), x)
lst = [SimplifyTerm(i, x) for i in lst]
res = 0
for k in range(1, len(lst)+1):
res = res + Simplify(v*lst[k-1]*(a + b*x)**(k - 1))
return res
return u*v
def GCD(*args):
args = S(args)
if len(args) == 1:
if isinstance(args[0], (int, Integer)):
return args[0]
else:
return S(1)
return gcd(*args)
def ContentFactor(expn):
return factor_terms(expn)
def NumericFactor(u):
# returns the real numeric factor of u.
if NumberQ(u):
if ZeroQ(Im(u)):
return u
elif ZeroQ(Re(u)):
return Im(u)
else:
return S(1)
elif PowerQ(u):
if RationalQ(u.base) and RationalQ(u.exp):
if u.exp > 0:
return 1/Denominator(u.base)
else:
return 1/(1/Denominator(u.base))
else:
return S(1)
elif ProductQ(u):
return Mul(*[NumericFactor(i) for i in u.args])
elif SumQ(u):
if LeafCount(u) < 50:
c = ContentFactor(u)
if SumQ(c):
return S(1)
else:
return NumericFactor(c)
else:
m = NumericFactor(First(u))
n = NumericFactor(Rest(u))
if m < 0 and n < 0:
return -GCD(-m, -n)
else:
return GCD(m, n)
return S(1)
def NonnumericFactors(u):
if NumberQ(u):
if ZeroQ(Im(u)):
return S(1)
elif ZeroQ(Re(u)):
return I
return u
elif PowerQ(u):
if RationalQ(u.base) and FractionQ(u.exp):
return u/NumericFactor(u)
return u
elif ProductQ(u):
result = 1
for i in u.args:
result *= NonnumericFactors(i)
return result
elif SumQ(u):
if LeafCount(u) < 50:
i = ContentFactor(u)
if SumQ(i):
return u
else:
return NonnumericFactors(i)
n = NumericFactor(u)
result = 0
for i in u.args:
result += i/n
return result
return u
def MakeAssocList(u, x, alst=None):
# (* MakeAssocList[u,x,alst] returns an association list of gensymed symbols with the nonatomic
# parameters of a u that are not integer powers, products or sums. *)
if alst is None:
alst = []
u = replace_pow_exp(u)
x = replace_pow_exp(x)
if AtomQ(u):
return alst
elif IntegerPowerQ(u):
return MakeAssocList(u.base, x, alst)
elif ProductQ(u) or SumQ(u):
return MakeAssocList(Rest(u), x, MakeAssocList(First(u), x, alst))
elif FreeQ(u, x):
tmp = []
for i in alst:
if PowerQ(i):
if i.exp == u:
tmp.append(i)
break
elif len(i.args) > 1: # make sure args has length > 1, else causes index error some times
if i.args[1] == u:
tmp.append(i)
break
if tmp == []:
alst.append(u)
return alst
return alst
def GensymSubst(u, x, alst=None):
# (* GensymSubst[u,x,alst] returns u with the kernels in alst free of x replaced by gensymed names. *)
if alst is None:
alst =[]
u = replace_pow_exp(u)
x = replace_pow_exp(x)
if AtomQ(u):
return u
elif IntegerPowerQ(u):
return GensymSubst(u.base, x, alst)**u.exp
elif ProductQ(u) or SumQ(u):
return u.func(*[GensymSubst(i, x, alst) for i in u.args])
elif FreeQ(u, x):
tmp = []
for i in alst:
if PowerQ(i):
if i.exp == u:
tmp.append(i)
break
elif len(i.args) > 1: # make sure args has length > 1, else causes index error some times
if i.args[1] == u:
tmp.append(i)
break
if tmp == []:
return u
return tmp[0][0]
return u
def KernelSubst(u, x, alst):
# (* KernelSubst[u,x,alst] returns u with the gensymed names in alst replaced by kernels free of x. *)
if AtomQ(u):
tmp = []
for i in alst:
if i.args[0] == u:
tmp.append(i)
break
if tmp == []:
return u
elif len(tmp[0].args) > 1: # make sure args has length > 1, else causes index error some times
return tmp[0].args[1]
elif IntegerPowerQ(u):
tmp = KernelSubst(u.base, x, alst)
if u.exp < 0 and ZeroQ(tmp):
return 'Indeterminate'
return tmp**u.exp
elif ProductQ(u) or SumQ(u):
return u.func(*[KernelSubst(i, x, alst) for i in u.args])
return u
def ExpandExpression(u, x):
if AlgebraicFunctionQ(u, x) and Not(RationalFunctionQ(u, x)):
v = ExpandAlgebraicFunction(u, x)
else:
v = S(0)
if SumQ(v):
return ExpandCleanup(v, x)
v = SmartApart(u, x)
if SumQ(v):
return ExpandCleanup(v, x)
v = SmartApart(RationalFunctionFactors(u, x), x, x)
if SumQ(v):
w = NonrationalFunctionFactors(u, x)
return ExpandCleanup(v.func(*[i*w for i in v.args]), x)
v = Expand(u)
if SumQ(v):
return ExpandCleanup(v, x)
v = Expand(u)
if SumQ(v):
return ExpandCleanup(v, x)
return SimplifyTerm(u, x)
def Apart(u, x):
if RationalFunctionQ(u, x):
return apart(u, x)
return u
def SmartApart(*args):
if len(args) == 2:
u, x = args
alst = MakeAssocList(u, x)
tmp = KernelSubst(Apart(GensymSubst(u, x, alst), x), x, alst)
if tmp == 'Indeterminate':
return u
return tmp
u, v, x = args
alst = MakeAssocList(u, x)
tmp = KernelSubst(Apart(GensymSubst(u, x, alst), x), x, alst)
if tmp == 'Indeterminate':
return u
return tmp
def MatchQ(expr, pattern, *var):
# returns the matched arguments after matching pattern with expression
match = expr.match(pattern)
if match:
return tuple(match[i] for i in var)
else:
return None
def PolynomialQuotientRemainder(p, q, x):
return [PolynomialQuotient(p, q, x), PolynomialRemainder(p, q, x)]
def FreeFactors(u, x):
# returns the product of the factors of u free of x.
if ProductQ(u):
result = 1
for i in u.args:
if FreeQ(i, x):
result *= i
return result
elif FreeQ(u, x):
return u
else:
return S(1)
def NonfreeFactors(u, x):
"""
Returns the product of the factors of u not free of x.
Examples
========
>>> from sympy.integrals.rubi.utility_function import NonfreeFactors
>>> from sympy.abc import x, a, b
>>> NonfreeFactors(a, x)
1
>>> NonfreeFactors(x + a, x)
a + x
>>> NonfreeFactors(a*b*x, x)
x
"""
if ProductQ(u):
result = 1
for i in u.args:
if not FreeQ(i, x):
result *= i
return result
elif FreeQ(u, x):
return 1
else:
return u
def RemoveContentAux(expr, x):
return RemoveContentAux_replacer.replace(UtilityOperator(expr, x))
def RemoveContent(u, x):
v = NonfreeFactors(u, x)
w = Together(v)
if EqQ(FreeFactors(w, x), 1):
return RemoveContentAux(v, x)
else:
return RemoveContentAux(NonfreeFactors(w, x), x)
def FreeTerms(u, x):
"""
Returns the sum of the terms of u free of x.
Examples
========
>>> from sympy.integrals.rubi.utility_function import FreeTerms
>>> from sympy.abc import x, a, b
>>> FreeTerms(a, x)
a
>>> FreeTerms(x*a, x)
0
>>> FreeTerms(a*x + b, x)
b
"""
if SumQ(u):
result = 0
for i in u.args:
if FreeQ(i, x):
result += i
return result
elif FreeQ(u, x):
return u
else:
return 0
def NonfreeTerms(u, x):
# returns the sum of the terms of u free of x.
if SumQ(u):
result = S(0)
for i in u.args:
if not FreeQ(i, x):
result += i
return result
elif not FreeQ(u, x):
return u
else:
return S(0)
def ExpandAlgebraicFunction(expr, x):
if ProductQ(expr):
u_ = Wild('u', exclude=[x])
n_ = Wild('n', exclude=[x])
v_ = Wild('v')
pattern = u_*v_
match = expr.match(pattern)
if match:
keys = [u_, v_]
if len(keys) == len(match):
u, v = tuple([match[i] for i in keys])
if SumQ(v):
u, v = v, u
if not FreeQ(u, x) and SumQ(u):
result = 0
for i in u.args:
result += i*v
return result
pattern = u_**n_*v_
match = expr.match(pattern)
if match:
keys = [u_, n_, v_]
if len(keys) == len(match):
u, n, v = tuple([match[i] for i in keys])
if PositiveIntegerQ(n) and SumQ(u):
w = Expand(u**n)
result = 0
for i in w.args:
result += i*v
return result
return expr
def CollectReciprocals(expr, x):
# Basis: e/(a+b x)+f/(c+d x)==(c e+a f+(d e+b f) x)/(a c+(b c+a d) x+b d x^2)
if SumQ(expr):
u_ = Wild('u')
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x])
c_ = Wild('c', exclude=[x])
d_ = Wild('d', exclude=[x])
e_ = Wild('e', exclude=[x])
f_ = Wild('f', exclude=[x])
pattern = u_ + e_/(a_ + b_*x) + f_/(c_+d_*x)
match = expr.match(pattern)
if match:
try: # .match() does not work peoperly always
keys = [u_, a_, b_, c_, d_, e_, f_]
u, a, b, c, d, e, f = tuple([match[i] for i in keys])
if ZeroQ(b*c + a*d) & ZeroQ(d*e + b*f):
return CollectReciprocals(u + (c*e + a*f)/(a*c + b*d*x**2),x)
elif ZeroQ(b*c + a*d) & ZeroQ(c*e + a*f):
return CollectReciprocals(u + (d*e + b*f)*x/(a*c + b*d*x**2),x)
except:
pass
return expr
def ExpandCleanup(u, x):
v = CollectReciprocals(u, x)
if SumQ(v):
res = 0
for i in v.args:
res += SimplifyTerm(i, x)
v = res
if SumQ(v):
return UnifySum(v, x)
else:
return v
else:
return v
def AlgebraicFunctionQ(u, x, flag=False):
if ListQ(u):
if u == []:
return True
elif AlgebraicFunctionQ(First(u), x, flag):
return AlgebraicFunctionQ(Rest(u), x, flag)
else:
return False
elif AtomQ(u) or FreeQ(u, x):
return True
elif PowerQ(u):
if RationalQ(u.exp) | flag & FreeQ(u.exp, x):
return AlgebraicFunctionQ(u.base, x, flag)
elif ProductQ(u) | SumQ(u):
for i in u.args:
if not AlgebraicFunctionQ(i, x, flag):
return False
return True
return False
def Coeff(expr, form, n=1):
if n == 1:
return Coefficient(Together(expr), form, n)
else:
coef1 = Coefficient(expr, form, n)
coef2 = Coefficient(Together(expr), form, n)
if Simplify(coef1 - coef2) == 0:
return coef1
else:
return coef2
def LeadTerm(u):
if SumQ(u):
return First(u)
return u
def RemainingTerms(u):
if SumQ(u):
return Rest(u)
return u
def LeadFactor(u):
# returns the leading factor of u.
if ComplexNumberQ(u) and Re(u) == 0:
if Im(u) == S(1):
return u
else:
return LeadFactor(Im(u))
elif ProductQ(u):
return LeadFactor(First(u))
return u
def RemainingFactors(u):
# returns the remaining factors of u.
if ComplexNumberQ(u) and Re(u) == 0:
if Im(u) == 1:
return S(1)
else:
return I*RemainingFactors(Im(u))
elif ProductQ(u):
return RemainingFactors(First(u))*Rest(u)
return S(1)
def LeadBase(u):
"""
returns the base of the leading factor of u.
Examples
========
>>> from sympy.integrals.rubi.utility_function import LeadBase
>>> from sympy.abc import a, b, c
>>> LeadBase(a**b)
a
>>> LeadBase(a**b*c)
a
"""
v = LeadFactor(u)
if PowerQ(v):
return v.base
return v
def LeadDegree(u):
# returns the degree of the leading factor of u.
v = LeadFactor(u)
if PowerQ(v):
return v.exp
return v
def Numer(expr):
# returns the numerator of u.
if PowerQ(expr):
if expr.exp < 0:
return 1
if ProductQ(expr):
return Mul(*[Numer(i) for i in expr.args])
return Numerator(expr)
def Denom(u):
# returns the denominator of u
if PowerQ(u):
if u.exp < 0:
return u.args[0]**(-u.args[1])
elif ProductQ(u):
return Mul(*[Denom(i) for i in u.args])
return Denominator(u)
def hypergeom(n, d, z):
return hyper(n, d, z)
def Expon(expr, form, h=None):
if h:
return Exponent(Together(expr), form, h)
else:
return Exponent(Together(expr), form)
def MergeMonomials(expr, x):
u_ = Wild('u')
p_ = Wild('p', exclude=[x, 1, 0])
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x, 0])
c_ = Wild('c', exclude=[x])
d_ = Wild('d', exclude=[x, 0])
n_ = Wild('n', exclude=[x])
m_ = Wild('m', exclude=[x])
# Basis: If m/n\[Element]\[DoubleStruckCapitalZ], then z^m (c z^n)^p==(c z^n)^(m/n+p)/c^(m/n)
pattern = u_*(a_ + b_*x)**m_*(c_*(a_ + b_*x)**n_)**p_
match = expr.match(pattern)
if match:
keys = [u_, a_, b_, m_, c_, n_, p_]
if len(keys) == len(match):
u, a, b, m, c, n, p = tuple([match[i] for i in keys])
if IntegerQ(m/n):
if u*(c*(a + b*x)**n)**(m/n + p)/c**(m/n) == S.NaN:
return expr
else:
return u*(c*(a + b*x)**n)**(m/n + p)/c**(m/n)
# Basis: If m\[Element]\[DoubleStruckCapitalZ] \[And] b c-a d==0, then (a+b z)^m==b^m/d^m (c+d z)^m
pattern = u_*(a_ + b_*x)**m_*(c_ + d_*x)**n_
match = expr.match(pattern)
if match:
keys = [u_, a_, b_, m_, c_, d_, n_]
if len(keys) == len(match):
u, a, b, m, c, d, n = tuple([match[i] for i in keys])
if IntegerQ(m) and ZeroQ(b*c - a*d):
if u*b**m/d**m*(c + d*x)**(m + n) == S.NaN:
return expr
else:
return u*b**m/d**m*(c + d*x)**(m + n)
return expr
def PolynomialDivide(u, v, x):
quo = PolynomialQuotient(u, v, x)
rem = PolynomialRemainder(u, v, x)
s = 0
for i in Exponent(quo, x, List):
s += Simp(Together(Coefficient(quo, x, i)*x**i), x)
quo = s
rem = Together(rem)
free = FreeFactors(rem, x)
rem = NonfreeFactors(rem, x)
monomial = x**Exponent(rem, x, Min)
if NegQ(Coefficient(rem, x, 0)):
monomial = -monomial
s = 0
for i in Exponent(rem, x, List):
s += Simp(Together(Coefficient(rem, x, i)*x**i/monomial), x)
rem = s
if BinomialQ(v, x):
return quo + free*monomial*rem/ExpandToSum(v, x)
else:
return quo + free*monomial*rem/v
def BinomialQ(u, x, n=None):
"""
If u is equivalent to an expression of the form a + b*x**n, BinomialQ(u, x, n) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import BinomialQ
>>> from sympy.abc import x
>>> BinomialQ(x**9, x)
True
>>> BinomialQ((1 + x)**3, x)
False
"""
if ListQ(u):
for i in u:
if Not(BinomialQ(i, x, n)):
return False
return True
elif NumberQ(x):
return False
return ListQ(BinomialParts(u, x))
def TrinomialQ(u, x):
"""
If u is equivalent to an expression of the form a + b*x**n + c*x**(2*n) where n, b and c are not 0,
TrinomialQ(u, x) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import TrinomialQ
>>> from sympy.abc import x
>>> TrinomialQ((7 + 2*x**6 + 3*x**12), x)
True
>>> TrinomialQ(x**2, x)
False
"""
if ListQ(u):
for i in u.args:
if Not(TrinomialQ(i, x)):
return False
return True
check = False
u = replace_pow_exp(u)
if PowerQ(u):
if u.exp == 2 and BinomialQ(u.base, x):
check = True
return ListQ(TrinomialParts(u,x)) and Not(QuadraticQ(u, x)) and Not(check)
def GeneralizedBinomialQ(u, x):
"""
If u is equivalent to an expression of the form a*x**q+b*x**n where n, q and b are not 0,
GeneralizedBinomialQ(u, x) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import GeneralizedBinomialQ
>>> from sympy.abc import a, x, q, b, n
>>> GeneralizedBinomialQ(a*x**q, x)
False
"""
if ListQ(u):
return all(GeneralizedBinomialQ(i, x) for i in u)
return ListQ(GeneralizedBinomialParts(u, x))
def GeneralizedTrinomialQ(u, x):
"""
If u is equivalent to an expression of the form a*x**q+b*x**n+c*x**(2*n-q) where n, q, b and c are not 0,
GeneralizedTrinomialQ(u, x) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import GeneralizedTrinomialQ
>>> from sympy.abc import x
>>> GeneralizedTrinomialQ(7 + 2*x**6 + 3*x**12, x)
False
"""
if ListQ(u):
return all(GeneralizedTrinomialQ(i, x) for i in u)
return ListQ(GeneralizedTrinomialParts(u, x))
def FactorSquareFreeList(poly):
r = sqf_list(poly)
result = [[1, 1]]
for i in r[1]:
result.append(list(i))
return result
def PerfectPowerTest(u, x):
# If u (x) is equivalent to a polynomial raised to an integer power greater than 1,
# PerfectPowerTest[u,x] returns u (x) as an expanded polynomial raised to the power;
# else it returns False.
if PolynomialQ(u, x):
lst = FactorSquareFreeList(u)
gcd = 0
v = 1
if lst[0] == [1, 1]:
lst = Rest(lst)
for i in lst:
gcd = GCD(gcd, i[1])
if gcd > 1:
for i in lst:
v = v*i[0]**(i[1]/gcd)
return Expand(v)**gcd
else:
return False
return False
def SquareFreeFactorTest(u, x):
# If u (x) can be square free factored, SquareFreeFactorTest[u,x] returns u (x) in
# factored form; else it returns False.
if PolynomialQ(u, x):
v = FactorSquareFree(u)
if PowerQ(v) or ProductQ(v):
return v
return False
return False
def RationalFunctionQ(u, x):
# If u is a rational function of x, RationalFunctionQ[u,x] returns True; else it returns False.
if AtomQ(u) or FreeQ(u, x):
return True
elif IntegerPowerQ(u):
return RationalFunctionQ(u.base, x)
elif ProductQ(u) or SumQ(u):
for i in u.args:
if Not(RationalFunctionQ(i, x)):
return False
return True
return False
def RationalFunctionFactors(u, x):
# RationalFunctionFactors[u,x] returns the product of the factors of u that are rational functions of x.
if ProductQ(u):
res = 1
for i in u.args:
if RationalFunctionQ(i, x):
res *= i
return res
elif RationalFunctionQ(u, x):
return u
return S(1)
def NonrationalFunctionFactors(u, x):
if ProductQ(u):
res = 1
for i in u.args:
if not RationalFunctionQ(i, x):
res *= i
return res
elif RationalFunctionQ(u, x):
return S(1)
return u
def Reverse(u):
if isinstance(u, list):
return list(reversed(u))
else:
l = list(u.args)
return u.func(*list(reversed(l)))
def RationalFunctionExponents(u, x):
"""
u is a polynomial or rational function of x.
RationalFunctionExponents(u, x) returns a list of the exponent of the
numerator of u and the exponent of the denominator of u.
Examples
========
>>> from sympy.integrals.rubi.utility_function import RationalFunctionExponents
>>> from sympy.abc import x, a
>>> RationalFunctionExponents(x, x)
[1, 0]
>>> RationalFunctionExponents(x**(-1), x)
[0, 1]
>>> RationalFunctionExponents(x**(-1)*a, x)
[0, 1]
"""
if PolynomialQ(u, x):
return [Exponent(u, x), 0]
elif IntegerPowerQ(u):
if PositiveQ(u.exp):
return u.exp*RationalFunctionExponents(u.base, x)
return (-u.exp)*Reverse(RationalFunctionExponents(u.base, x))
elif ProductQ(u):
lst1 = RationalFunctionExponents(First(u), x)
lst2 = RationalFunctionExponents(Rest(u), x)
return [lst1[0] + lst2[0], lst1[1] + lst2[1]]
elif SumQ(u):
v = Together(u)
if SumQ(v):
lst1 = RationalFunctionExponents(First(u), x)
lst2 = RationalFunctionExponents(Rest(u), x)
return [Max(lst1[0] + lst2[1], lst2[0] + lst1[1]), lst1[1] + lst2[1]]
else:
return RationalFunctionExponents(v, x)
return [0, 0]
def RationalFunctionExpand(expr, x):
# expr is a polynomial or rational function of x.
# RationalFunctionExpand[u,x] returns the expansion of the factors of u that are rational functions times the other factors.
def cons_f1(n):
return FractionQ(n)
cons1 = CustomConstraint(cons_f1)
def cons_f2(x, v):
if not isinstance(x, Symbol):
return False
return UnsameQ(v, x)
cons2 = CustomConstraint(cons_f2)
def With1(n, u, x, v):
w = RationalFunctionExpand(u, x)
return If(SumQ(w), Add(*[i*v**n for i in w.args]), v**n*w)
pattern1 = Pattern(UtilityOperator(u_*v_**n_, x_), cons1, cons2)
rule1 = ReplacementRule(pattern1, With1)
def With2(u, x):
v = ExpandIntegrand(u, x)
def _consf_u(a, b, c, d, p, m, n, x):
return And(FreeQ(List(a, b, c, d, p), x), IntegersQ(m, n), Equal(m, Add(n, S(-1))))
cons_u = CustomConstraint(_consf_u)
pat = Pattern(UtilityOperator(x_**WC('m', S(1))*(x_*WC('d', S(1)) + c_)**p_/(x_**n_*WC('b', S(1)) + a_), x_), cons_u)
result_matchq = is_match(UtilityOperator(u, x), pat)
if UnsameQ(v, u) and not result_matchq:
return v
else:
v = ExpandIntegrand(RationalFunctionFactors(u, x), x)
w = NonrationalFunctionFactors(u, x)
if SumQ(v):
return Add(*[i*w for i in v.args])
else:
return v*w
pattern2 = Pattern(UtilityOperator(u_, x_))
rule2 = ReplacementRule(pattern2, With2)
expr = expr.replace(sym_exp, rubi_exp)
res = replace_all(UtilityOperator(expr, x), [rule1, rule2])
return replace_pow_exp(res)
def ExpandIntegrand(expr, x, extra=None):
expr = replace_pow_exp(expr)
if not extra is None:
extra, x = x, extra
w = ExpandIntegrand(extra, x)
r = NonfreeTerms(w, x)
if SumQ(r):
result = [expr*FreeTerms(w, x)]
for i in r.args:
result.append(MergeMonomials(expr*i, x))
return r.func(*result)
else:
return expr*FreeTerms(w, x) + MergeMonomials(expr*r, x)
else:
u_ = Wild('u', exclude=[0, 1])
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x, 0])
F_ = Wild('F', exclude=[0])
c_ = Wild('c', exclude=[x])
d_ = Wild('d', exclude=[x, 0])
n_ = Wild('n', exclude=[0, 1])
pattern = u_*(a_ + b_*F_)**n_
match = expr.match(pattern)
if match:
if MemberQ([asin, acos, asinh, acosh], match[F_].func):
keys = [u_, a_, b_, F_, n_]
if len(match) == len(keys):
u, a, b, F, n = tuple([match[i] for i in keys])
match = F.args[0].match(c_ + d_*x)
if match:
keys = c_, d_
if len(keys) == len(match):
c, d = tuple([match[i] for i in keys])
if PolynomialQ(u, x):
F = F.func
return ExpandLinearProduct((a + b*F(c + d*x))**n, u, c, d, x)
expr = expr.replace(sym_exp, rubi_exp)
res = replace_all(UtilityOperator(expr, x), ExpandIntegrand_rules, max_count = 1)
return replace_pow_exp(res)
def SimplerQ(u, v):
# If u is simpler than v, SimplerQ(u, v) returns True, else it returns False. SimplerQ(u, u) returns False
if IntegerQ(u):
if IntegerQ(v):
if Abs(u)==Abs(v):
return v<0
else:
return Abs(u)<Abs(v)
else:
return True
elif IntegerQ(v):
return False
elif FractionQ(u):
if FractionQ(v):
if Denominator(u) == Denominator(v):
return SimplerQ(Numerator(u), Numerator(v))
else:
return Denominator(u)<Denominator(v)
else:
return True
elif FractionQ(v):
return False
elif (Re(u)==0 or Re(u) == 0) and (Re(v)==0 or Re(v) == 0):
return SimplerQ(Im(u), Im(v))
elif ComplexNumberQ(u):
if ComplexNumberQ(v):
if Re(u) == Re(v):
return SimplerQ(Im(u), Im(v))
else:
return SimplerQ(Re(u),Re(v))
else:
return False
elif NumberQ(u):
if NumberQ(v):
return OrderedQ([u,v])
else:
return True
elif NumberQ(v):
return False
elif AtomQ(u) or (Head(u) == re) or (Head(u) == im):
if AtomQ(v) or (Head(u) == re) or (Head(u) == im):
return OrderedQ([u,v])
else:
return True
elif AtomQ(v) or (Head(u) == re) or (Head(u) == im):
return False
elif Head(u) == Head(v):
if Length(u) == Length(v):
for i in range(len(u.args)):
if not u.args[i] == v.args[i]:
return SimplerQ(u.args[i], v.args[i])
return False
return Length(u) < Length(v)
elif LeafCount(u) < LeafCount(v):
return True
elif LeafCount(v) < LeafCount(u):
return False
return Not(OrderedQ([v,u]))
def SimplerSqrtQ(u, v):
# If Rt(u, 2) is simpler than Rt(v, 2), SimplerSqrtQ(u, v) returns True, else it returns False. SimplerSqrtQ(u, u) returns False
if NegativeQ(v) and Not(NegativeQ(u)):
return True
if NegativeQ(u) and Not(NegativeQ(v)):
return False
sqrtu = Rt(u, S(2))
sqrtv = Rt(v, S(2))
if IntegerQ(sqrtu):
if IntegerQ(sqrtv):
return sqrtu<sqrtv
else:
return True
if IntegerQ(sqrtv):
return False
if RationalQ(sqrtu):
if RationalQ(sqrtv):
return sqrtu<sqrtv
else:
return True
if RationalQ(sqrtv):
return False
if PosQ(u):
if PosQ(v):
return LeafCount(sqrtu)<LeafCount(sqrtv)
else:
return True
if PosQ(v):
return False
if LeafCount(sqrtu)<LeafCount(sqrtv):
return True
if LeafCount(sqrtv)<LeafCount(sqrtu):
return False
else:
return Not(OrderedQ([v, u]))
def SumSimplerQ(u, v):
"""
If u + v is simpler than u, SumSimplerQ(u, v) returns True, else it returns False.
If for every term w of v there is a term of u equal to n*w where n<-1/2, u + v will be simpler than u.
Examples
========
>>> from sympy.integrals.rubi.utility_function import SumSimplerQ
>>> from sympy.abc import x
>>> from sympy import S
>>> SumSimplerQ(S(4 + x),S(3 + x**3))
False
"""
if RationalQ(u, v):
if v == S(0):
return False
elif v > S(0):
return u < -S(1)
else:
return u >= -v
else:
return SumSimplerAuxQ(Expand(u), Expand(v))
def BinomialDegree(u, x):
# if u is a binomial. BinomialDegree[u,x] returns the degree of x in u.
bp = BinomialParts(u, x)
if bp == False:
return bp
return bp[2]
def TrinomialDegree(u, x):
# If u is equivalent to a trinomial of the form a + b*x^n + c*x^(2*n) where n!=0, b!=0 and c!=0, TrinomialDegree[u,x] returns n
t = TrinomialParts(u, x)
if t:
return t[3]
return t
def CancelCommonFactors(u, v):
def _delete_cases(a, b):
# only for CancelCommonFactors
lst = []
deleted = False
for i in a.args:
if i == b and not deleted:
deleted = True
continue
lst.append(i)
return a.func(*lst)
# CancelCommonFactors[u,v] returns {u',v'} are the noncommon factors of u and v respectively.
if ProductQ(u):
if ProductQ(v):
if MemberQ(v, First(u)):
return CancelCommonFactors(Rest(u), _delete_cases(v, First(u)))
else:
lst = CancelCommonFactors(Rest(u), v)
return [First(u)*lst[0], lst[1]]
else:
if MemberQ(u, v):
return [_delete_cases(u, v), 1]
else:
return[u, v]
elif ProductQ(v):
if MemberQ(v, u):
return [1, _delete_cases(v, u)]
else:
return [u, v]
return[u, v]
def SimplerIntegrandQ(u, v, x):
lst = CancelCommonFactors(u, v)
u1 = lst[0]
v1 = lst[1]
if Head(u1) == Head(v1) and Length(u1) == 1 and Length(v1) == 1:
return SimplerIntegrandQ(u1.args[0], v1.args[0], x)
if 4*LeafCount(u1) < 3*LeafCount(v1):
return True
if RationalFunctionQ(u1, x):
if RationalFunctionQ(v1, x):
t1 = 0
t2 = 0
for i in RationalFunctionExponents(u1, x):
t1 += i
for i in RationalFunctionExponents(v1, x):
t2 += i
return t1 < t2
else:
return True
else:
return False
def GeneralizedBinomialDegree(u, x):
b = GeneralizedBinomialParts(u, x)
if b:
return b[2] - b[3]
def GeneralizedBinomialParts(expr, x):
expr = Expand(expr)
if GeneralizedBinomialMatchQ(expr, x):
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
n = Wild('n', exclude=[x])
q = Wild('q', exclude=[x])
Match = expr.match(a*x**q + b*x**n)
if Match and PosQ(Match[q] - Match[n]):
return [Match[b], Match[a], Match[q], Match[n]]
else:
return False
def GeneralizedTrinomialDegree(u, x):
t = GeneralizedTrinomialParts(u, x)
if t:
return t[3] - t[4]
def GeneralizedTrinomialParts(expr, x):
expr = Expand(expr)
if GeneralizedTrinomialMatchQ(expr, x):
a = Wild('a', exclude=[x, 0])
b = Wild('b', exclude=[x, 0])
c = Wild('c', exclude=[x])
n = Wild('n', exclude=[x, 0])
q = Wild('q', exclude=[x])
Match = expr.match(a*x**q + b*x**n+c*x**(2*n-q))
if Match and expr.is_Add:
return [Match[c], Match[b], Match[a], Match[n], 2*Match[n]-Match[q]]
else:
return False
def MonomialQ(u, x):
# If u is of the form a*x^n where n!=0 and a!=0, MonomialQ[u,x] returns True; else False
if isinstance(u, list):
return all(MonomialQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
re = u.match(a*x**b)
if re:
return True
return False
def MonomialSumQ(u, x):
# if u(x) is a sum and each term is free of x or an expression of the form a*x^n, MonomialSumQ(u, x) returns True; else it returns False
if SumQ(u):
for i in u.args:
if Not(FreeQ(i, x) or MonomialQ(i, x)):
return False
return True
@doctest_depends_on(modules=('matchpy',))
def MinimumMonomialExponent(u, x):
"""
u is sum whose terms are monomials. MinimumMonomialExponent(u, x) returns the exponent of the term having the smallest exponent
Examples
========
>>> from sympy.integrals.rubi.utility_function import MinimumMonomialExponent
>>> from sympy.abc import x
>>> MinimumMonomialExponent(x**2 + 5*x**2 + 3*x**5, x)
2
>>> MinimumMonomialExponent(x**2 + 5*x**2 + 1, x)
0
"""
n =MonomialExponent(First(u), x)
for i in u.args:
if PosQ(n - MonomialExponent(i, x)):
n = MonomialExponent(i, x)
return n
def MonomialExponent(u, x):
# u is a monomial. MonomialExponent(u, x) returns the exponent of x in u
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
re = u.match(a*x**b)
if re:
return re[b]
def LinearMatchQ(u, x):
# LinearMatchQ(u, x) returns True iff u matches patterns of the form a+b*x where a and b are free of x
if isinstance(u, list):
return all(LinearMatchQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
re = u.match(a + b*x)
if re:
return True
return False
def PowerOfLinearMatchQ(u, x):
if isinstance(u, list):
for i in u:
if not PowerOfLinearMatchQ(i, x):
return False
return True
else:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x, 0])
m = Wild('m', exclude=[x, 0])
Match = u.match((a + b*x)**m)
if Match:
return True
else:
return False
def QuadraticMatchQ(u, x):
if ListQ(u):
return all(QuadraticMatchQ(i, x) for i in u)
pattern1 = Pattern(UtilityOperator(x_**2*WC('c', 1) + x_*WC('b', 1) + WC('a', 0), x_), CustomConstraint(lambda a, b, c, x: FreeQ([a, b, c], x)))
pattern2 = Pattern(UtilityOperator(x_**2*WC('c', 1) + WC('a', 0), x_), CustomConstraint(lambda a, c, x: FreeQ([a, c], x)))
u1 = UtilityOperator(u, x)
return is_match(u1, pattern1) or is_match(u1, pattern2)
def CubicMatchQ(u, x):
if isinstance(u, list):
return all(CubicMatchQ(i, x) for i in u)
else:
pattern1 = Pattern(UtilityOperator(x_**3*WC('d', 1) + x_**2*WC('c', 1) + x_*WC('b', 1) + WC('a', 0), x_), CustomConstraint(lambda a, b, c, d, x: FreeQ([a, b, c, d], x)))
pattern2 = Pattern(UtilityOperator(x_**3*WC('d', 1) + x_*WC('b', 1) + WC('a', 0), x_), CustomConstraint(lambda a, b, d, x: FreeQ([a, b, d], x)))
pattern3 = Pattern(UtilityOperator(x_**3*WC('d', 1) + x_**2*WC('c', 1) + WC('a', 0), x_), CustomConstraint(lambda a, c, d, x: FreeQ([a, c, d], x)))
pattern4 = Pattern(UtilityOperator(x_**3*WC('d', 1) + WC('a', 0), x_), CustomConstraint(lambda a, d, x: FreeQ([a, d], x)))
u1 = UtilityOperator(u, x)
if is_match(u1, pattern1) or is_match(u1, pattern2) or is_match(u1, pattern3) or is_match(u1, pattern4):
return True
else:
return False
def BinomialMatchQ(u, x):
if isinstance(u, list):
return all(BinomialMatchQ(i, x) for i in u)
else:
pattern = Pattern(UtilityOperator(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)), x_) , CustomConstraint(lambda a, b, n, x: FreeQ([a,b,n],x)))
u = UtilityOperator(u, x)
return is_match(u, pattern)
def TrinomialMatchQ(u, x):
if isinstance(u, list):
return all(TrinomialMatchQ(i, x) for i in u)
else:
pattern = Pattern(UtilityOperator(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)), x_) , CustomConstraint(lambda a, b, c, n, x: FreeQ([a, b, c, n], x)), CustomConstraint(lambda j, n: ZeroQ(j-2*n) ))
u = UtilityOperator(u, x)
return is_match(u, pattern)
def GeneralizedBinomialMatchQ(u, x):
if isinstance(u, list):
return all(GeneralizedBinomialMatchQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x, 0])
b = Wild('b', exclude=[x, 0])
n = Wild('n', exclude=[x, 0])
q = Wild('q', exclude=[x, 0])
Match = u.match(a*x**q + b*x**n)
if Match and len(Match) == 4 and Match[q] != 0 and Match[n] != 0:
return True
else:
return False
def GeneralizedTrinomialMatchQ(u, x):
if isinstance(u, list):
return all(GeneralizedTrinomialMatchQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x, 0])
b = Wild('b', exclude=[x, 0])
n = Wild('n', exclude=[x, 0])
c = Wild('c', exclude=[x, 0])
q = Wild('q', exclude=[x, 0])
Match = u.match(a*x**q + b*x**n + c*x**(2*n - q))
if Match and len(Match) == 5 and 2*Match[n] - Match[q] != 0 and Match[n] != 0:
return True
else:
return False
def QuotientOfLinearsMatchQ(u, x):
if isinstance(u, list):
return all(QuotientOfLinearsMatchQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
d = Wild('d', exclude=[x])
c = Wild('c', exclude=[x])
e = Wild('e')
Match = u.match(e*(a + b*x)/(c + d*x))
if Match and len(Match) == 5:
return True
else:
return False
def PolynomialTermQ(u, x):
a = Wild('a', exclude=[x])
n = Wild('n', exclude=[x])
Match = u.match(a*x**n)
if Match and IntegerQ(Match[n]) and Greater(Match[n], S(0)):
return True
else:
return False
def PolynomialTerms(u, x):
s = 0
for i in u.args:
if PolynomialTermQ(i, x):
s = s + i
return s
def NonpolynomialTerms(u, x):
s = 0
for i in u.args:
if not PolynomialTermQ(i, x):
s = s + i
return s
def PseudoBinomialParts(u, x):
if PolynomialQ(u, x) and Greater(Expon(u, x), S(2)):
n = Expon(u, x)
d = Rt(Coefficient(u, x, n), n)
c = d**(-n + S(1))*Coefficient(u, x, n + S(-1))/n
a = Simplify(u - (c + d*x)**n)
if NonzeroQ(a) and FreeQ(a, x):
return [a, S(1), c, d, n]
else:
return False
else:
return False
def NormalizePseudoBinomial(u, x):
lst = PseudoBinomialParts(u, x)
if lst:
return (lst[0] + lst[1]*(lst[2] + lst[3]*x)**lst[4])
def PseudoBinomialPairQ(u, v, x):
lst1 = PseudoBinomialParts(u, x)
if AtomQ(lst1):
return False
else:
lst2 = PseudoBinomialParts(v, x)
if AtomQ(lst2):
return False
else:
return Drop(lst1, 2) == Drop(lst2, 2)
def PseudoBinomialQ(u, x):
lst = PseudoBinomialParts(u, x)
if lst:
return True
else:
return False
def PolynomialGCD(f, g):
return gcd(f, g)
def PolyGCD(u, v, x):
# (* u and v are polynomials in x. *)
# (* PolyGCD[u,v,x] returns the factors of the gcd of u and v dependent on x. *)
return NonfreeFactors(PolynomialGCD(u, v), x)
def AlgebraicFunctionFactors(u, x, flag=False):
# (* AlgebraicFunctionFactors[u,x] returns the product of the factors of u that are algebraic functions of x. *)
if ProductQ(u):
result = 1
for i in u.args:
if AlgebraicFunctionQ(i, x, flag):
result *= i
return result
if AlgebraicFunctionQ(u, x, flag):
return u
return 1
def NonalgebraicFunctionFactors(u, x):
"""
NonalgebraicFunctionFactors[u,x] returns the product of the factors of u that are not algebraic functions of x.
Examples
========
>>> from sympy.integrals.rubi.utility_function import NonalgebraicFunctionFactors
>>> from sympy.abc import x
>>> from sympy import sin
>>> NonalgebraicFunctionFactors(sin(x), x)
sin(x)
>>> NonalgebraicFunctionFactors(x, x)
1
"""
if ProductQ(u):
result = 1
for i in u.args:
if not AlgebraicFunctionQ(i, x):
result *= i
return result
if AlgebraicFunctionQ(u, x):
return 1
return u
def QuotientOfLinearsP(u, x):
if LinearQ(u, x):
return True
elif SumQ(u):
if FreeQ(u.args[0], x):
return QuotientOfLinearsP(Rest(u), x)
elif LinearQ(Numerator(u), x) and LinearQ(Denominator(u), x):
return True
elif ProductQ(u):
if FreeQ(First(u), x):
return QuotientOfLinearsP(Rest(u), x)
elif Numerator(u) == 1 and PowerQ(u):
return QuotientOfLinearsP(Denominator(u), x)
return u == x or FreeQ(u, x)
def QuotientOfLinearsParts(u, x):
# If u is equivalent to an expression of the form (a+b*x)/(c+d*x), QuotientOfLinearsParts[u,x]
# returns the list {a, b, c, d}.
if LinearQ(u, x):
return [Coefficient(u, x, 0), Coefficient(u, x, 1), 1, 0]
elif PowerQ(u):
if Numerator(u) == 1:
u = Denominator(u)
r = QuotientOfLinearsParts(u, x)
return [r[2], r[3], r[0], r[1]]
elif SumQ(u):
a = First(u)
if FreeQ(a, x):
u = Rest(u)
r = QuotientOfLinearsParts(u, x)
return [r[0] + a*r[2], r[1] + a*r[3], r[2], r[3]]
elif ProductQ(u):
a = First(u)
if FreeQ(a, x):
r = QuotientOfLinearsParts(Rest(u), x)
return [a*r[0], a*r[1], r[2], r[3]]
a = Numerator(u)
d = Denominator(u)
if LinearQ(a, x) and LinearQ(d, x):
return [Coefficient(a, x, 0), Coefficient(a, x, 1), Coefficient(d, x, 0), Coefficient(d, x, 1)]
elif u == x:
return [0, 1, 1, 0]
elif FreeQ(u, x):
return [u, 0, 1, 0]
return [u, 0, 1, 0]
def QuotientOfLinearsQ(u, x):
# (*QuotientOfLinearsQ[u,x] returns True iff u is equivalent to an expression of the form (a+b x)/(c+d x) where b!=0 and d!=0.*)
if ListQ(u):
for i in u:
if not QuotientOfLinearsQ(i, x):
return False
return True
q = QuotientOfLinearsParts(u, x)
return QuotientOfLinearsP(u, x) and NonzeroQ(q[1]) and NonzeroQ(q[3])
def Flatten(l):
return flatten(l)
def Sort(u, r=False):
return sorted(u, key=lambda x: x.sort_key(), reverse=r)
# (*Definition: A number is absurd if it is a rational number, a positive rational number raised to a fractional power, or a product of absurd numbers.*)
def AbsurdNumberQ(u):
# (* AbsurdNumberQ[u] returns True if u is an absurd number, else it returns False. *)
if PowerQ(u):
v = u.exp
u = u.base
return RationalQ(u) and u > 0 and FractionQ(v)
elif ProductQ(u):
return all(AbsurdNumberQ(i) for i in u.args)
return RationalQ(u)
def AbsurdNumberFactors(u):
# (* AbsurdNumberFactors[u] returns the product of the factors of u that are absurd numbers. *)
if AbsurdNumberQ(u):
return u
elif ProductQ(u):
result = S(1)
for i in u.args:
if AbsurdNumberQ(i):
result *= i
return result
return NumericFactor(u)
def NonabsurdNumberFactors(u):
# (* NonabsurdNumberFactors[u] returns the product of the factors of u that are not absurd numbers. *)
if AbsurdNumberQ(u):
return S(1)
elif ProductQ(u):
result = 1
for i in u.args:
result *= NonabsurdNumberFactors(i)
return result
return NonnumericFactors(u)
def SumSimplerAuxQ(u, v):
if SumQ(v):
return (RationalQ(First(v)) or SumSimplerAuxQ(u,First(v))) and (RationalQ(Rest(v)) or SumSimplerAuxQ(u,Rest(v)))
elif SumQ(u):
return SumSimplerAuxQ(First(u), v) or SumSimplerAuxQ(Rest(u), v)
else:
return v!=0 and NonnumericFactors(u)==NonnumericFactors(v) and (NumericFactor(u)/NumericFactor(v)<-1/2 or NumericFactor(u)/NumericFactor(v)==-1/2 and NumericFactor(u)<0)
def Prepend(l1, l2):
if not isinstance(l2, list):
return [l2] + l1
return l2 + l1
def Drop(lst, n):
if isinstance(lst, list):
if isinstance(n, list):
lst = lst[:(n[0]-1)] + lst[n[1]:]
elif n > 0:
lst = lst[n:]
elif n < 0:
lst = lst[:-n]
else:
return lst
return lst
return lst.func(*[i for i in Drop(list(lst.args), n)])
def CombineExponents(lst):
if Length(lst) < 2:
return lst
elif lst[0][0] == lst[1][0]:
return CombineExponents(Prepend(Drop(lst,2),[lst[0][0], lst[0][1] + lst[1][1]]))
return Prepend(CombineExponents(Rest(lst)), First(lst))
def FactorInteger(n, l=None):
if isinstance(n, (int, Integer)):
return sorted(factorint(n, limit=l).items())
else:
return sorted(factorrat(n, limit=l).items())
def FactorAbsurdNumber(m):
# (* m must be an absurd number. FactorAbsurdNumber[m] returns the prime factorization of m *)
# (* as list of base-degree pairs where the bases are prime numbers and the degrees are rational. *)
if RationalQ(m):
return FactorInteger(m)
elif PowerQ(m):
r = FactorInteger(m.base)
return [r[0], r[1]*m.exp]
# CombineExponents[Sort[Flatten[Map[FactorAbsurdNumber,Apply[List,m]],1], Function[i1[[1]]<i2[[1]]]]]
return list((m.as_base_exp(),))
def SubstForInverseFunction(*args):
"""
SubstForInverseFunction(u, v, w, x) returns u with subexpressions equal to v replaced by x and x replaced by w.
Examples
========
>>> from sympy.integrals.rubi.utility_function import SubstForInverseFunction
>>> from sympy.abc import x, a, b
>>> SubstForInverseFunction(a, a, b, x)
a
>>> SubstForInverseFunction(x**a, x**a, b, x)
x
>>> SubstForInverseFunction(a*x**a, a, b, x)
a*b**a
"""
if len(args) == 3:
u, v, x = args[0], args[1], args[2]
return SubstForInverseFunction(u, v, (-Coefficient(v.args[0], x, 0) + InverseFunction(Head(v))(x))/Coefficient(v.args[0], x, 1), x)
elif len(args) == 4:
u, v, w, x = args[0], args[1], args[2], args[3]
if AtomQ(u):
if u == x:
return w
return u
elif Head(u) == Head(v) and ZeroQ(u.args[0] - v.args[0]):
return x
res = [SubstForInverseFunction(i, v, w, x) for i in u.args]
return u.func(*res)
def SubstForFractionalPower(u, v, n, w, x):
# (* SubstForFractionalPower[u,v,n,w,x] returns u with subexpressions equal to v^(m/n) replaced
# by x^m and x replaced by w. *)
if AtomQ(u):
if u == x:
return w
return u
elif FractionalPowerQ(u):
if ZeroQ(u.base - v):
return x**(n*u.exp)
res = [SubstForFractionalPower(i, v, n, w, x) for i in u.args]
return u.func(*res)
def SubstForFractionalPowerOfQuotientOfLinears(u, x):
# (* If u has a subexpression of the form ((a+b*x)/(c+d*x))^(m/n) where m and n>1 are integers,
# SubstForFractionalPowerOfQuotientOfLinears[u,x] returns the list {v,n,(a+b*x)/(c+d*x),b*c-a*d} where v is u
# with subexpressions of the form ((a+b*x)/(c+d*x))^(m/n) replaced by x^m and x replaced
lst = FractionalPowerOfQuotientOfLinears(u, 1, False, x)
if AtomQ(lst) or AtomQ(lst[1]):
return False
n = lst[0]
tmp = lst[1]
lst = QuotientOfLinearsParts(tmp, x)
a, b, c, d = lst[0], lst[1], lst[2], lst[3]
if ZeroQ(d):
return False
lst = Simplify(x**(n - 1)*SubstForFractionalPower(u, tmp, n, (-a + c*x**n)/(b - d*x**n), x)/(b - d*x**n)**2)
return [NonfreeFactors(lst, x), n, tmp, FreeFactors(lst, x)*(b*c - a*d)]
def FractionalPowerOfQuotientOfLinears(u, n, v, x):
# (* If u has a subexpression of the form ((a+b*x)/(c+d*x))^(m/n),
# FractionalPowerOfQuotientOfLinears[u,1,False,x] returns {n,(a+b*x)/(c+d*x)}; else it returns False. *)
if AtomQ(u) or FreeQ(u, x):
return [n, v]
elif CalculusQ(u):
return False
elif FractionalPowerQ(u):
if QuotientOfLinearsQ(u.base, x) and Not(LinearQ(u.base, x)) and (FalseQ(v) or ZeroQ(u.base - v)):
return [LCM(Denominator(u.exp), n), u.base]
lst = [n, v]
for i in u.args:
lst = FractionalPowerOfQuotientOfLinears(i, lst[0], lst[1],x)
if AtomQ(lst):
return False
return lst
def SubstForFractionalPowerQ(u, v, x):
# (* If the substitution x=v^(1/n) will not complicate algebraic subexpressions of u,
# SubstForFractionalPowerQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u) or FreeQ(u, x):
return True
elif FractionalPowerQ(u):
return SubstForFractionalPowerAuxQ(u, v, x)
return all(SubstForFractionalPowerQ(i, v, x) for i in u.args)
def SubstForFractionalPowerAuxQ(u, v, x):
if AtomQ(u):
return False
elif FractionalPowerQ(u):
if ZeroQ(u.base - v):
return True
return any(SubstForFractionalPowerAuxQ(i, v, x) for i in u.args)
def FractionalPowerOfSquareQ(u):
# (* If a subexpression of u is of the form ((v+w)^2)^n where n is a fraction, *)
# (* FractionalPowerOfSquareQ[u] returns (v+w)^2; else it returns False. *)
if AtomQ(u):
return False
elif FractionalPowerQ(u):
a_ = Wild('a', exclude=[0])
b_ = Wild('b', exclude=[0])
c_ = Wild('c', exclude=[0])
match = u.base.match(a_*(b_ + c_)**(S(2)))
if match:
keys = [a_, b_, c_]
if len(keys) == len(match):
a, b, c = tuple(match[i] for i in keys)
if NonsumQ(a):
return (b + c)**S(2)
for i in u.args:
tmp = FractionalPowerOfSquareQ(i)
if Not(FalseQ(tmp)):
return tmp
return False
def FractionalPowerSubexpressionQ(u, v, w):
# (* If a subexpression of u is of the form w^n where n is a fraction but not equal to v, *)
# (* FractionalPowerSubexpressionQ[u,v,w] returns True; else it returns False. *)
if AtomQ(u):
return False
elif FractionalPowerQ(u):
if PositiveQ(u.base/w):
return Not(u.base == v) and LeafCount(w) < 3*LeafCount(v)
for i in u.args:
if FractionalPowerSubexpressionQ(i, v, w):
return True
return False
def Apply(f, lst):
return f(*lst)
def FactorNumericGcd(u):
# (* FactorNumericGcd[u] returns u with the gcd of the numeric coefficients of terms of sums factored out. *)
if PowerQ(u):
if RationalQ(u.exp):
return FactorNumericGcd(u.base)**u.exp
elif ProductQ(u):
res = [FactorNumericGcd(i) for i in u.args]
return Mul(*res)
elif SumQ(u):
g = GCD([NumericFactor(i) for i in u.args])
r = Add(*[i/g for i in u.args])
return g*r
return u
def MergeableFactorQ(bas, deg, v):
# (* MergeableFactorQ[bas,deg,v] returns True iff bas equals the base of a factor of v or bas is a factor of every term of v. *)
if bas == v:
return RationalQ(deg + S(1)) and (deg + 1>=0 or RationalQ(deg) and deg>0)
elif PowerQ(v):
if bas == v.base:
return RationalQ(deg+v.exp) and (deg+v.exp>=0 or RationalQ(deg) and deg>0)
return SumQ(v.base) and IntegerQ(v.exp) and (Not(IntegerQ(deg) or IntegerQ(deg/v.exp))) and MergeableFactorQ(bas, deg/v.exp, v.base)
elif ProductQ(v):
return MergeableFactorQ(bas, deg, First(v)) or MergeableFactorQ(bas, deg, Rest(v))
return SumQ(v) and MergeableFactorQ(bas, deg, First(v)) and MergeableFactorQ(bas, deg, Rest(v))
def MergeFactor(bas, deg, v):
# (* If MergeableFactorQ[bas,deg,v], MergeFactor[bas,deg,v] return the product of bas^deg and v,
# but with bas^deg merged into the factor of v whose base equals bas. *)
if bas == v:
return bas**(deg + 1)
elif PowerQ(v):
if bas == v.base:
return bas**(deg + v.exp)
return MergeFactor(bas, deg/v.exp, v.base**v.exp)
elif ProductQ(v):
if MergeableFactorQ(bas, deg, First(v)):
return MergeFactor(bas, deg, First(v))*Rest(v)
return First(v)*MergeFactor(bas, deg, Rest(v))
return MergeFactor(bas, deg, First(v)) + MergeFactor(bas, deg, Rest(v))
def MergeFactors(u, v):
# (* MergeFactors[u,v] returns the product of u and v, but with the mergeable factors of u merged into v. *)
if ProductQ(u):
return MergeFactors(Rest(u), MergeFactors(First(u), v))
elif PowerQ(u):
if MergeableFactorQ(u.base, u.exp, v):
return MergeFactor(u.base, u.exp, v)
elif RationalQ(u.exp) and u.exp < -1 and MergeableFactorQ(u.base, -S(1), v):
return MergeFactors(u.base**(u.exp + 1), MergeFactor(u.base, -S(1), v))
return u*v
elif MergeableFactorQ(u, S(1), v):
return MergeFactor(u, S(1), v)
return u*v
def TrigSimplifyQ(u):
# (* TrigSimplifyQ[u] returns True if TrigSimplify[u] actually simplifies u; else False. *)
return ActivateTrig(u) != TrigSimplify(u)
def TrigSimplify(u):
# (* TrigSimplify[u] returns a bottom-up trig simplification of u. *)
return ActivateTrig(TrigSimplifyRecur(u))
def TrigSimplifyRecur(u):
if AtomQ(u):
return u
return TrigSimplifyAux(u.func(*[TrigSimplifyRecur(i) for i in u.args]))
def Order(expr1, expr2):
if expr1 == expr2:
return 0
elif expr1.sort_key() > expr2.sort_key():
return -1
return 1
def FactorOrder(u, v):
if u == 1:
if v == 1:
return 0
return -1
elif v == 1:
return 1
return Order(u, v)
def Smallest(num1, num2=None):
if num2 is None:
lst = num1
num = lst[0]
for i in Rest(lst):
num = Smallest(num, i)
return num
return Min(num1, num2)
def OrderedQ(l):
return l == Sort(l)
def MinimumDegree(deg1, deg2):
if RationalQ(deg1):
if RationalQ(deg2):
return Min(deg1, deg2)
return deg1
elif RationalQ(deg2):
return deg2
deg = Simplify(deg1- deg2)
if RationalQ(deg):
if deg > 0:
return deg2
return deg1
elif OrderedQ([deg1, deg2]):
return deg1
return deg2
def PositiveFactors(u):
# (* PositiveFactors[u] returns the positive factors of u *)
if ZeroQ(u):
return S(1)
elif RationalQ(u):
return Abs(u)
elif PositiveQ(u):
return u
elif ProductQ(u):
res = 1
for i in u.args:
res *= PositiveFactors(i)
return res
return 1
def Sign(u):
return sign(u)
def NonpositiveFactors(u):
# (* NonpositiveFactors[u] returns the nonpositive factors of u *)
if ZeroQ(u):
return u
elif RationalQ(u):
return Sign(u)
elif PositiveQ(u):
return S(1)
elif ProductQ(u):
res = S(1)
for i in u.args:
res *= NonpositiveFactors(i)
return res
return u
def PolynomialInAuxQ(u, v, x):
if u == v:
return True
elif AtomQ(u):
return u != x
elif PowerQ(u):
if PowerQ(v):
if u.base == v.base:
return PositiveIntegerQ(u.exp/v.exp)
return PositiveIntegerQ(u.exp) and PolynomialInAuxQ(u.base, v, x)
elif SumQ(u) or ProductQ(u):
for i in u.args:
if Not(PolynomialInAuxQ(i, v, x)):
return False
return True
return False
def PolynomialInQ(u, v, x):
"""
If u is a polynomial in v(x), PolynomialInQ(u, v, x) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import PolynomialInQ
>>> from sympy.abc import x
>>> from sympy import log, S
>>> PolynomialInQ(S(1), log(x), x)
True
>>> PolynomialInQ(log(x), log(x), x)
True
>>> PolynomialInQ(1 + log(x)**2, log(x), x)
True
"""
return PolynomialInAuxQ(u, NonfreeFactors(NonfreeTerms(v, x), x), x)
def ExponentInAux(u, v, x):
if u == v:
return S(1)
elif AtomQ(u):
return S(0)
elif PowerQ(u):
if PowerQ(v):
if u.base == v.base:
return u.exp/v.exp
return u.exp*ExponentInAux(u.base, v, x)
elif ProductQ(u):
return Add(*[ExponentInAux(i, v, x) for i in u.args])
return Max(*[ExponentInAux(i, v, x) for i in u.args])
def ExponentIn(u, v, x):
return ExponentInAux(u, NonfreeFactors(NonfreeTerms(v, x), x), x)
def PolynomialInSubstAux(u, v, x):
if u == v:
return x
elif AtomQ(u):
return u
elif PowerQ(u):
if PowerQ(v):
if u.base == v.base:
return x**(u.exp/v.exp)
return PolynomialInSubstAux(u.base, v, x)**u.exp
return u.func(*[PolynomialInSubstAux(i, v, x) for i in u.args])
def PolynomialInSubst(u, v, x):
# If u is a polynomial in v[x], PolynomialInSubst[u,v,x] returns the polynomial u in x.
w = NonfreeTerms(v, x)
return ReplaceAll(PolynomialInSubstAux(u, NonfreeFactors(w, x), x), {x: x - FreeTerms(v, x)/FreeFactors(w, x)})
def Distrib(u, v):
# Distrib[u,v] returns the sum of u times each term of v.
if SumQ(v):
return Add(*[u*i for i in v.args])
return u*v
def DistributeDegree(u, m):
# DistributeDegree[u,m] returns the product of the factors of u each raised to the mth degree.
if AtomQ(u):
return u**m
elif PowerQ(u):
return u.base**(u.exp*m)
elif ProductQ(u):
return Mul(*[DistributeDegree(i, m) for i in u.args])
return u**m
def FunctionOfPower(*args):
"""
FunctionOfPower[u,x] returns the gcd of the integer degrees of x in u.
Examples
========
>>> from sympy.integrals.rubi.utility_function import FunctionOfPower
>>> from sympy.abc import x
>>> FunctionOfPower(x, x)
1
>>> FunctionOfPower(x**3, x)
3
"""
if len(args) == 2:
return FunctionOfPower(args[0], None, args[1])
u, n, x = args
if FreeQ(u, x):
return n
elif u == x:
return S(1)
elif PowerQ(u):
if u.base == x and IntegerQ(u.exp):
if n is None:
return u.exp
return GCD(n, u.exp)
tmp = n
for i in u.args:
tmp = FunctionOfPower(i, tmp, x)
return tmp
def DivideDegreesOfFactors(u, n):
"""
DivideDegreesOfFactors[u,n] returns the product of the base of the factors of u raised to the degree of the factors divided by n.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.rubi.utility_function import DivideDegreesOfFactors
>>> from sympy.abc import a, b
>>> DivideDegreesOfFactors(a**b, S(3))
a**(b/3)
"""
if ProductQ(u):
return Mul(*[LeadBase(i)**(LeadDegree(i)/n) for i in u.args])
return LeadBase(u)**(LeadDegree(u)/n)
def MonomialFactor(u, x):
# MonomialFactor[u,x] returns the list {n,v} where x^n*v==u and n is free of x.
if AtomQ(u):
if u == x:
return [S(1), S(1)]
return [S(0), u]
elif PowerQ(u):
if IntegerQ(u.exp):
lst = MonomialFactor(u.base, x)
return [lst[0]*u.exp, lst[1]**u.exp]
elif u.base == x and FreeQ(u.exp, x):
return [u.exp, S(1)]
return [S(0), u]
elif ProductQ(u):
lst1 = MonomialFactor(First(u), x)
lst2 = MonomialFactor(Rest(u), x)
return [lst1[0] + lst2[0], lst1[1]*lst2[1]]
elif SumQ(u):
lst = [MonomialFactor(i, x) for i in u.args]
deg = lst[0][0]
for i in Rest(lst):
deg = MinimumDegree(deg, i[0])
if ZeroQ(deg) or RationalQ(deg) and deg < 0:
return [S(0), u]
return [deg, Add(*[x**(i[0] - deg)*i[1] for i in lst])]
return [S(0), u]
def FullSimplify(expr):
return Simplify(expr)
def FunctionOfLinearSubst(u, a, b, x):
if FreeQ(u, x):
return u
elif LinearQ(u, x):
tmp = Coefficient(u, x, 1)
if tmp == b:
tmp = S(1)
else:
tmp = tmp/b
return Coefficient(u, x, S(0)) - a*tmp + tmp*x
elif PowerQ(u):
if FreeQ(u.base, x):
return E**(FullSimplify(FunctionOfLinearSubst(Log(u.base)*u.exp, a, b, x)))
lst = MonomialFactor(u, x)
if ProductQ(u) and NonzeroQ(lst[0]):
if RationalQ(LeadFactor(lst[1])) and LeadFactor(lst[1]) < 0:
return -FunctionOfLinearSubst(DivideDegreesOfFactors(-lst[1], lst[0])*x, a, b, x)**lst[0]
return FunctionOfLinearSubst(DivideDegreesOfFactors(lst[1], lst[0])*x, a, b, x)**lst[0]
return u.func(*[FunctionOfLinearSubst(i, a, b, x) for i in u.args])
def FunctionOfLinear(*args):
# (* If u (x) is equivalent to an expression of the form f (a+b*x) and not the case that a==0 and
# b==1, FunctionOfLinear[u,x] returns the list {f (x),a,b}; else it returns False. *)
if len(args) == 2:
u, x = args
lst = FunctionOfLinear(u, False, False, x, False)
if AtomQ(lst) or FalseQ(lst[0]) or (lst[0] == 0 and lst[1] == 1):
return False
return [FunctionOfLinearSubst(u, lst[0], lst[1], x), lst[0], lst[1]]
u, a, b, x, flag = args
if FreeQ(u, x):
return [a, b]
elif CalculusQ(u):
return False
elif LinearQ(u, x):
if FalseQ(a):
return [Coefficient(u, x, 0), Coefficient(u, x, 1)]
lst = CommonFactors([b, Coefficient(u, x, 1)])
if ZeroQ(Coefficient(u, x, 0)) and Not(flag):
return [0, lst[0]]
elif ZeroQ(b*Coefficient(u, x, 0) - a*Coefficient(u, x, 1)):
return [a/lst[1], lst[0]]
return [0, 1]
elif PowerQ(u):
if FreeQ(u.base, x):
return FunctionOfLinear(Log(u.base)*u.exp, a, b, x, False)
lst = MonomialFactor(u, x)
if ProductQ(u) and NonzeroQ(lst[0]):
if False and IntegerQ(lst[0]) and lst[0] != -1 and FreeQ(lst[1], x):
if RationalQ(LeadFactor(lst[1])) and LeadFactor(lst[1]) < 0:
return FunctionOfLinear(DivideDegreesOfFactors(-lst[1], lst[0])*x, a, b, x, False)
return FunctionOfLinear(DivideDegreesOfFactors(lst[1], lst[0])*x, a, b, x, False)
return False
lst = [a, b]
for i in u.args:
lst = FunctionOfLinear(i, lst[0], lst[1], x, SumQ(u))
if AtomQ(lst):
return False
return lst
def NormalizeIntegrand(u, x):
v = NormalizeLeadTermSigns(NormalizeIntegrandAux(u, x))
if v == NormalizeLeadTermSigns(u):
return u
else:
return v
def NormalizeIntegrandAux(u, x):
if SumQ(u):
l = 0
for i in u.args:
l += NormalizeIntegrandAux(i, x)
return l
if ProductQ(MergeMonomials(u, x)):
l = 1
for i in MergeMonomials(u, x).args:
l *= NormalizeIntegrandFactor(i, x)
return l
else:
return NormalizeIntegrandFactor(MergeMonomials(u, x), x)
def NormalizeIntegrandFactor(u, x):
if PowerQ(u):
if FreeQ(u.exp, x):
bas = NormalizeIntegrandFactorBase(u.base, x)
deg = u.exp
if IntegerQ(deg) and SumQ(bas):
if all(MonomialQ(i, x) for i in bas.args):
mi = MinimumMonomialExponent(bas, x)
q = 0
for i in bas.args:
q += Simplify(i/x**mi)
return x**(mi*deg)*q**deg
else:
return bas**deg
else:
return bas**deg
if PowerQ(u):
if FreeQ(u.base, x):
return u.base**NormalizeIntegrandFactorBase(u.exp, x)
bas = NormalizeIntegrandFactorBase(u, x)
if SumQ(bas):
if all(MonomialQ(i, x) for i in bas.args):
mi = MinimumMonomialExponent(bas, x)
z = 0
for j in bas.args:
z += j/x**mi
return x**mi*z
else:
return bas
else:
return bas
def NormalizeIntegrandFactorBase(expr, x):
m = Wild('m', exclude=[x])
u = Wild('u')
match = expr.match(x**m*u)
if match and SumQ(u):
l = 0
for i in u.args:
l += NormalizeIntegrandFactorBase((x**m*i), x)
return l
if BinomialQ(expr, x):
if BinomialMatchQ(expr, x):
return expr
else:
return ExpandToSum(expr, x)
elif TrinomialQ(expr, x):
if TrinomialMatchQ(expr, x):
return expr
else:
return ExpandToSum(expr, x)
elif ProductQ(expr):
l = 1
for i in expr.args:
l *= NormalizeIntegrandFactor(i, x)
return l
elif PolynomialQ(expr, x) and Exponent(expr, x)<=4:
return ExpandToSum(expr, x)
elif SumQ(expr):
w = Wild('w')
m = Wild('m', exclude=[x])
v = TogetherSimplify(expr)
if SumQ(v) or v.match(x**m*w) and SumQ(w) or LeafCount(v)>LeafCount(expr)+2:
return UnifySum(expr, x)
else:
return NormalizeIntegrandFactorBase(v, x)
else:
return expr
def NormalizeTogether(u):
return NormalizeLeadTermSigns(Together(u))
def NormalizeLeadTermSigns(u):
if ProductQ(u):
t = 1
for i in u.args:
lst = SignOfFactor(i)
if lst[0] == 1:
t *= lst[1]
else:
t *= AbsorbMinusSign(lst[1])
return t
else:
lst = SignOfFactor(u)
if lst[0] == 1:
return lst[1]
else:
return AbsorbMinusSign(lst[1])
def AbsorbMinusSign(expr, *x):
m = Wild('m', exclude=[x])
u = Wild('u')
v = Wild('v')
match = expr.match(u*v**m)
if match:
if len(match) == 3:
if SumQ(match[v]) and OddQ(match[m]):
return match[u]*(-match[v])**match[m]
return -expr
def NormalizeSumFactors(u):
if AtomQ(u):
return u
elif ProductQ(u):
k = 1
for i in u.args:
k *= NormalizeSumFactors(i)
return SignOfFactor(k)[0]*SignOfFactor(k)[1]
elif SumQ(u):
k = 0
for i in u.args:
k += NormalizeSumFactors(i)
return k
else:
return u
def SignOfFactor(u):
if RationalQ(u) and u < 0 or SumQ(u) and NumericFactor(First(u)) < 0:
return [-1, -u]
elif IntegerPowerQ(u):
if SumQ(u.base) and NumericFactor(First(u.base)) < 0:
return [(-1)**u.exp, (-u.base)**u.exp]
elif ProductQ(u):
k = 1
h = 1
for i in u.args:
k *= SignOfFactor(i)[0]
h *= SignOfFactor(i)[1]
return [k, h]
return [1, u]
def NormalizePowerOfLinear(u, x):
v = FactorSquareFree(u)
if PowerQ(v):
if LinearQ(v.base, x) and FreeQ(v.exp, x):
return ExpandToSum(v.base, x)**v.exp
return ExpandToSum(v, x)
def SimplifyIntegrand(u, x):
v = NormalizeLeadTermSigns(NormalizeIntegrandAux(Simplify(u), x))
if 5*LeafCount(v) < 4*LeafCount(u):
return v
if v != NormalizeLeadTermSigns(u):
return v
else:
return u
def SimplifyTerm(u, x):
v = Simplify(u)
w = Together(v)
if LeafCount(v) < LeafCount(w):
return NormalizeIntegrand(v, x)
else:
return NormalizeIntegrand(w, x)
def TogetherSimplify(u):
v = Together(Simplify(Together(u)))
return FixSimplify(v)
def SmartSimplify(u):
v = Simplify(u)
w = factor(v)
if LeafCount(w) < LeafCount(v):
v = w
if Not(FalseQ(w == FractionalPowerOfSquareQ(v))) and FractionalPowerSubexpressionQ(u, w, Expand(w)):
v = SubstForExpn(v, w, Expand(w))
else:
v = FactorNumericGcd(v)
return FixSimplify(v)
def SubstForExpn(u, v, w):
if u == v:
return w
if AtomQ(u):
return u
else:
k = 0
for i in u.args:
k += SubstForExpn(i, v, w)
return k
def ExpandToSum(u, *x):
if len(x) == 1:
x = x[0]
expr = 0
if PolyQ(S(u), x):
for t in Exponent(u, x, List):
expr += Coeff(u, x, t)*x**t
return expr
if BinomialQ(u, x):
i = BinomialParts(u, x)
expr += i[0] + i[1]*x**i[2]
return expr
if TrinomialQ(u, x):
i = TrinomialParts(u, x)
expr += i[0] + i[1]*x**i[3] + i[2]*x**(2*i[3])
return expr
if GeneralizedBinomialMatchQ(u, x):
i = GeneralizedBinomialParts(u, x)
expr += i[0]*x**i[3] + i[1]*x**i[2]
return expr
if GeneralizedTrinomialMatchQ(u, x):
i = GeneralizedTrinomialParts(u, x)
expr += i[0]*x**i[4] + i[1]*x**i[3] + i[2]*x**(2*i[3]-i[4])
return expr
else:
return Expand(u)
else:
v = x[0]
x = x[1]
w = ExpandToSum(v, x)
r = NonfreeTerms(w, x)
if SumQ(r):
k = u*FreeTerms(w, x)
for i in r.args:
k += MergeMonomials(u*i, x)
return k
else:
return u*FreeTerms(w, x) + MergeMonomials(u*r, x)
def UnifySum(u, x):
if SumQ(u):
t = 0
lst = []
for i in u.args:
lst += [i]
for j in UnifyTerms(lst, x):
t += j
return t
else:
return SimplifyTerm(u, x)
def UnifyTerms(lst, x):
if lst==[]:
return lst
else:
return UnifyTerm(First(lst), UnifyTerms(Rest(lst), x), x)
def UnifyTerm(term, lst, x):
if lst==[]:
return [term]
tmp = Simplify(First(lst)/term)
if FreeQ(tmp, x):
return Prepend(Rest(lst), [(1+tmp)*term])
else:
return Prepend(UnifyTerm(term, Rest(lst), x), [First(lst)])
def CalculusQ(u):
return False
def FunctionOfInverseLinear(*args):
# (* If u is a function of an inverse linear binomial of the form 1/(a+b*x),
# FunctionOfInverseLinear[u,x] returns the list {a,b}; else it returns False. *)
if len(args) == 2:
u, x = args
return FunctionOfInverseLinear(u, None, x)
u, lst, x = args
if FreeQ(u, x):
return lst
elif u == x:
return False
elif QuotientOfLinearsQ(u, x):
tmp = Drop(QuotientOfLinearsParts(u, x), 2)
if tmp[1] == 0:
return False
elif lst is None:
return tmp
elif ZeroQ(lst[0]*tmp[1] - lst[1]*tmp[0]):
return lst
return False
elif CalculusQ(u):
return False
tmp = lst
for i in u.args:
tmp = FunctionOfInverseLinear(i, tmp, x)
if AtomQ(tmp):
return False
return tmp
def PureFunctionOfSinhQ(u, v, x):
# (* If u is a pure function of Sinh[v] and/or Csch[v], PureFunctionOfSinhQ[u,v,x] returns True;
# else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and ZeroQ(u.args[0] - v):
return SinhQ(u) or CschQ(u)
for i in u.args:
if Not(PureFunctionOfSinhQ(i, v, x)):
return False
return True
def PureFunctionOfTanhQ(u, v , x):
# (* If u is a pure function of Tanh[v] and/or Coth[v], PureFunctionOfTanhQ[u,v,x] returns True;
# else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and ZeroQ(u.args[0] - v):
return TanhQ(u) or CothQ(u)
for i in u.args:
if Not(PureFunctionOfTanhQ(i, v, x)):
return False
return True
def PureFunctionOfCoshQ(u, v, x):
# (* If u is a pure function of Cosh[v] and/or Sech[v], PureFunctionOfCoshQ[u,v,x] returns True;
# else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and ZeroQ(u.args[0] - v):
return CoshQ(u) or SechQ(u)
for i in u.args:
if Not(PureFunctionOfCoshQ(i, v, x)):
return False
return True
def IntegerQuotientQ(u, v):
# (* If u/v is an integer, IntegerQuotientQ[u,v] returns True; else it returns False. *)
return IntegerQ(Simplify(u/v))
def OddQuotientQ(u, v):
# (* If u/v is odd, OddQuotientQ[u,v] returns True; else it returns False. *)
return OddQ(Simplify(u/v))
def EvenQuotientQ(u, v):
# (* If u/v is even, EvenQuotientQ[u,v] returns True; else it returns False. *)
return EvenQ(Simplify(u/v))
def FindTrigFactor(func1, func2, u, v, flag):
# (* If func[w]^m is a factor of u where m is odd and w is an integer multiple of v,
# FindTrigFactor[func1,func2,u,v,True] returns the list {w,u/func[w]^n}; else it returns False. *)
# (* If func[w]^m is a factor of u where m is odd and w is an integer multiple of v not equal to v,
# FindTrigFactor[func1,func2,u,v,False] returns the list {w,u/func[w]^n}; else it returns False. *)
if u == 1:
return False
elif (Head(LeadBase(u)) == func1 or Head(LeadBase(u)) == func2) and OddQ(LeadDegree(u)) and IntegerQuotientQ(LeadBase(u).args[0], v) and (flag or NonzeroQ(LeadBase(u).args[0] - v)):
return [LeadBase[u].args[0], RemainingFactors(u)]
lst = FindTrigFactor(func1, func2, RemainingFactors(u), v, flag)
if AtomQ(lst):
return False
return [lst[0], LeadFactor(u)*lst[1]]
def FunctionOfSinhQ(u, v, x):
# (* If u is a function of Sinh[v], FunctionOfSinhQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
if OddQuotientQ(u.args[0], v):
# (* Basis: If m odd, Sinh[m*v]^n is a function of Sinh[v]. *)
return SinhQ(u) or CschQ(u)
# (* Basis: If m even, Cos[m*v]^n is a function of Sinh[v]. *)
return CoshQ(u) or SechQ(u)
elif IntegerPowerQ(u):
if HyperbolicQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if EvenQ(u.exp):
# (* Basis: If m integer and n even, Hyper[m*v]^n is a function of Sinh[v]. *)
return True
return FunctionOfSinhQ(u.base, v, x)
elif ProductQ(u):
if CoshQ(u.args[0]) and SinhQ(u.args[1]) and ZeroQ(u.args[0].args[0] - v/2) and ZeroQ(u.args[1].args[0] - v/2):
return FunctionOfSinhQ(Drop(u, 2), v, x)
lst = FindTrigFactor(Sinh, Csch, u, v, False)
if ListQ(lst) and EvenQuotientQ(lst[0], v):
# (* Basis: If m even and n odd, Sinh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *)
return FunctionOfSinhQ(Cosh(v)*lst[1], v, x)
lst = FindTrigFactor(Cosh, Sech, u, v, False)
if ListQ(lst) and OddQuotientQ(lst[0], v):
# (* Basis: If m odd and n odd, Cosh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *)
return FunctionOfSinhQ(Cosh(v)*lst[1], v, x)
lst = FindTrigFactor(Tanh, Coth, u, v, True)
if ListQ(lst):
# (* Basis: If m integer and n odd, Tanh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *)
return FunctionOfSinhQ(Cosh(v)*lst[1], v, x)
return all(FunctionOfSinhQ(i, v, x) for i in u.args)
return all(FunctionOfSinhQ(i, v, x) for i in u.args)
def FunctionOfCoshQ(u, v, x):
#(* If u is a function of Cosh[v], FunctionOfCoshQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
# (* Basis: If m integer, Cosh[m*v]^n is a function of Cosh[v]. *)
return CoshQ(u) or SechQ(u)
elif IntegerPowerQ(u):
if HyperbolicQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if EvenQ(u.exp):
# (* Basis: If m integer and n even, Hyper[m*v]^n is a function of Cosh[v]. *)
return True
return FunctionOfCoshQ(u.base, v, x)
elif ProductQ(u):
lst = FindTrigFactor(Sinh, Csch, u, v, False)
if ListQ(lst):
# (* Basis: If m integer and n odd, Sinh[m*v]^n == Sinh[v]*u where u is a function of Cosh[v]. *)
return FunctionOfCoshQ(Sinh(v)*lst[1], v, x)
lst = FindTrigFactor(Tanh, Coth, u, v, True)
if ListQ(lst):
# (* Basis: If m integer and n odd, Tanh[m*v]^n == Sinh[v]*u where u is a function of Cosh[v]. *)
return FunctionOfCoshQ(Sinh(v)*lst[1], v, x)
return all(FunctionOfCoshQ(i, v, x) for i in u.args)
return all(FunctionOfCoshQ(i, v, x) for i in u.args)
def OddHyperbolicPowerQ(u, v, x):
if SinhQ(u) or CoshQ(u) or SechQ(u) or CschQ(u):
return OddQuotientQ(u.args[0], v)
if PowerQ(u):
return OddQ(u.exp) and OddHyperbolicPowerQ(u.base, v, x)
if ProductQ(u):
if Not(EqQ(FreeFactors(u, x), 1)):
return OddHyperbolicPowerQ(NonfreeFactors(u, x), v, x)
lst = []
for i in u.args:
if Not(FunctionOfTanhQ(i, v, x)):
lst.append(i)
if lst == []:
return True
return Length(lst)==1 and OddHyperbolicPowerQ(lst[0], v, x)
if SumQ(u):
return all(OddHyperbolicPowerQ(i, v, x) for i in u.args)
return False
def FunctionOfTanhQ(u, v, x):
#(* If u is a function of the form f[Tanh[v],Coth[v]] where f is independent of x,
# FunctionOfTanhQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
return TanhQ(u) or CothQ(u) or EvenQuotientQ(u.args[0], v)
elif PowerQ(u):
if EvenQ(u.exp) and HyperbolicQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
return True
elif EvenQ(u.args[1]) and SumQ(u.args[0]):
return FunctionOfTanhQ(Expand(u.args[0]**2, v, x))
if ProductQ(u):
lst = []
for i in u.args:
if Not(FunctionOfTanhQ(i, v, x)):
lst.append(i)
if lst == []:
return True
return Length(lst)==2 and OddHyperbolicPowerQ(lst[0], v, x) and OddHyperbolicPowerQ(lst[1], v, x)
return all(FunctionOfTanhQ(i, v, x) for i in u.args)
def FunctionOfTanhWeight(u, v, x):
"""
u is a function of the form f(tanh(v), coth(v)) where f is independent of x.
FunctionOfTanhWeight(u, v, x) returns a nonnegative number if u is best considered a function of tanh(v), else it returns a negative number.
Examples
========
>>> from sympy import sinh, log, tanh
>>> from sympy.abc import x
>>> from sympy.integrals.rubi.utility_function import FunctionOfTanhWeight
>>> FunctionOfTanhWeight(x, log(x), x)
0
>>> FunctionOfTanhWeight(sinh(log(x)), log(x), x)
0
>>> FunctionOfTanhWeight(tanh(log(x)), log(x), x)
1
"""
if AtomQ(u):
return S(0)
elif CalculusQ(u):
return S(0)
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
if TanhQ(u) and ZeroQ(u.args[0] - v):
return S(1)
elif CothQ(u) and ZeroQ(u.args[0] - v):
return S(-1)
return S(0)
elif PowerQ(u):
if EvenQ(u.exp) and HyperbolicQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if TanhQ(u.base) or CoshQ(u.base) or SechQ(u.base):
return S(1)
return S(-1)
if ProductQ(u):
if all(FunctionOfTanhQ(i, v, x) for i in u.args):
return Add(*[FunctionOfTanhWeight(i, v, x) for i in u.args])
return S(0)
return Add(*[FunctionOfTanhWeight(i, v, x) for i in u.args])
def FunctionOfHyperbolicQ(u, v, x):
# (* If u (x) is equivalent to a function of the form f (Sinh[v],Cosh[v],Tanh[v],Coth[v],Sech[v],Csch[v])
# where f is independent of x, FunctionOfHyperbolicQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
return True
return all(FunctionOfHyperbolicQ(i, v, x) for i in u.args)
def SmartNumerator(expr):
if PowerQ(expr):
n = expr.exp
u = expr.base
if RationalQ(n) and n < 0:
return SmartDenominator(u**(-n))
elif ProductQ(expr):
return Mul(*[SmartNumerator(i) for i in expr.args])
return Numerator(expr)
def SmartDenominator(expr):
if PowerQ(expr):
u = expr.base
n = expr.exp
if RationalQ(n) and n < 0:
return SmartNumerator(u**(-n))
elif ProductQ(expr):
return Mul(*[SmartDenominator(i) for i in expr.args])
return Denominator(expr)
def ActivateTrig(u):
return u
def ExpandTrig(*args):
if len(args) == 2:
u, x = args
return ActivateTrig(ExpandIntegrand(u, x))
u, v, x = args
w = ExpandTrig(v, x)
z = ActivateTrig(u)
if SumQ(w):
return w.func(*[z*i for i in w.args])
return z*w
def TrigExpand(u):
return expand_trig(u)
# SubstForTrig[u_,sin_,cos_,v_,x_] :=
# If[AtomQ[u],
# u,
# If[TrigQ[u] && IntegerQuotientQ[u[[1]],v],
# If[u[[1]]===v || ZeroQ[u[[1]]-v],
# If[SinQ[u],
# sin,
# If[CosQ[u],
# cos,
# If[TanQ[u],
# sin/cos,
# If[CotQ[u],
# cos/sin,
# If[SecQ[u],
# 1/cos,
# 1/sin]]]]],
# Map[Function[SubstForTrig[#,sin,cos,v,x]],
# ReplaceAll[TrigExpand[Head[u][Simplify[u[[1]]/v]*x]],x->v]]],
# If[ProductQ[u] && CosQ[u[[1]]] && SinQ[u[[2]]] && ZeroQ[u[[1,1]]-v/2] && ZeroQ[u[[2,1]]-v/2],
# sin/2*SubstForTrig[Drop[u,2],sin,cos,v,x],
# Map[Function[SubstForTrig[#,sin,cos,v,x]],u]]]]
def SubstForTrig(u, sin_ , cos_, v, x):
# (* u (v) is an expression of the form f (Sin[v],Cos[v],Tan[v],Cot[v],Sec[v],Csc[v]). *)
# (* SubstForTrig[u,sin,cos,v,x] returns the expression f (sin,cos,sin/cos,cos/sin,1/cos,1/sin). *)
if AtomQ(u):
return u
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
if u.args[0] == v or ZeroQ(u.args[0] - v):
if SinQ(u):
return sin_
elif CosQ(u):
return cos_
elif TanQ(u):
return sin_/cos_
elif CotQ(u):
return cos_/sin_
elif SecQ(u):
return 1/cos_
return 1/sin_
r = ReplaceAll(TrigExpand(Head(u)(Simplify(u.args[0]/v*x))), {x: v})
return r.func(*[SubstForTrig(i, sin_, cos_, v, x) for i in r.args])
if ProductQ(u) and CosQ(u.args[0]) and SinQ(u.args[1]) and ZeroQ(u.args[0].args[0] - v/2) and ZeroQ(u.args[1].args[0] - v/2):
return sin(x)/2*SubstForTrig(Drop(u, 2), sin_, cos_, v, x)
return u.func(*[SubstForTrig(i, sin_, cos_, v, x) for i in u.args])
def SubstForHyperbolic(u, sinh_, cosh_, v, x):
# (* u (v) is an expression of the form f (Sinh[v],Cosh[v],Tanh[v],Coth[v],Sech[v],Csch[v]). *)
# (* SubstForHyperbolic[u,sinh,cosh,v,x] returns the expression
# f (sinh,cosh,sinh/cosh,cosh/sinh,1/cosh,1/sinh). *)
if AtomQ(u):
return u
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
if u.args[0] == v or ZeroQ(u.args[0] - v):
if SinhQ(u):
return sinh_
elif CoshQ(u):
return cosh_
elif TanhQ(u):
return sinh_/cosh_
elif CothQ(u):
return cosh_/sinh_
if SechQ(u):
return 1/cosh_
return 1/sinh_
r = ReplaceAll(TrigExpand(Head(u)(Simplify(u.args[0]/v)*x)), {x: v})
return r.func(*[SubstForHyperbolic(i, sinh_, cosh_, v, x) for i in r.args])
elif ProductQ(u) and CoshQ(u.args[0]) and SinhQ(u.args[1]) and ZeroQ(u.args[0].args[0] - v/2) and ZeroQ(u.args[1].args[0] - v/2):
return sinh(x)/2*SubstForHyperbolic(Drop(u, 2), sinh_, cosh_, v, x)
return u.func(*[SubstForHyperbolic(i, sinh_, cosh_, v, x) for i in u.args])
def InertTrigFreeQ(u):
return FreeQ(u, sin) and FreeQ(u, cos) and FreeQ(u, tan) and FreeQ(u, cot) and FreeQ(u, sec) and FreeQ(u, csc)
def LCM(a, b):
return lcm(a, b)
def SubstForFractionalPowerOfLinear(u, x):
# (* If u has a subexpression of the form (a+b*x)^(m/n) where m and n>1 are integers,
# SubstForFractionalPowerOfLinear[u,x] returns the list {v,n,a+b*x,1/b} where v is u
# with subexpressions of the form (a+b*x)^(m/n) replaced by x^m and x replaced
# by -a/b+x^n/b, and all times x^(n-1); else it returns False. *)
lst = FractionalPowerOfLinear(u, S(1), False, x)
if AtomQ(lst) or FalseQ(lst[1]):
return False
n = lst[0]
a = Coefficient(lst[1], x, 0)
b = Coefficient(lst[1], x, 1)
tmp = Simplify(x**(n-1)*SubstForFractionalPower(u, lst[1], n, -a/b + x**n/b, x))
return [NonfreeFactors(tmp, x), n, lst[1], FreeFactors(tmp, x)/b]
def FractionalPowerOfLinear(u, n, v, x):
# If u has a subexpression of the form (a + b*x)**(m/n), FractionalPowerOfLinear(u, 1, False, x) returns [n, a + b*x], else it returns False.
if AtomQ(u) or FreeQ(u, x):
return [n, v]
elif CalculusQ(u):
return False
elif FractionalPowerQ(u):
if LinearQ(u.base, x) and (FalseQ(v) or ZeroQ(u.base - v)):
return [LCM(Denominator(u.exp), n), u.base]
lst = [n, v]
for i in u.args:
lst = FractionalPowerOfLinear(i, lst[0], lst[1], x)
if AtomQ(lst):
return False
return lst
def InverseFunctionOfLinear(u, x):
# (* If u has a subexpression of the form g[a+b*x] where g is an inverse function,
# InverseFunctionOfLinear[u,x] returns g[a+b*x]; else it returns False. *)
if AtomQ(u) or CalculusQ(u) or FreeQ(u, x):
return False
elif InverseFunctionQ(u) and LinearQ(u.args[0], x):
return u
for i in u.args:
tmp = InverseFunctionOfLinear(i, x)
if Not(AtomQ(tmp)):
return tmp
return False
def InertTrigQ(*args):
if len(args) == 1:
f = args[0]
l = [sin,cos,tan,cot,sec,csc]
return any(Head(f) == i for i in l)
elif len(args) == 2:
f, g = args
if f == g:
return InertTrigQ(f)
return InertReciprocalQ(f, g) or InertReciprocalQ(g, f)
else:
f, g, h = args
return InertTrigQ(g, f) and InertTrigQ(g, h)
def InertReciprocalQ(f, g):
return (f.func == sin and g.func == csc) or (f.func == cos and g.func == sec) or (f.func == tan and g.func == cot)
def DeactivateTrig(u, x):
# (* u is a function of trig functions of a linear function of x. *)
# (* DeactivateTrig[u,x] returns u with the trig functions replaced with inert trig functions. *)
return FixInertTrigFunction(DeactivateTrigAux(u, x), x)
def FixInertTrigFunction(u, x):
return u
def DeactivateTrigAux(u, x):
if AtomQ(u):
return u
elif TrigQ(u) and LinearQ(u.args[0], x):
v = ExpandToSum(u.args[0], x)
if SinQ(u):
return sin(v)
elif CosQ(u):
return cos(v)
elif TanQ(u):
return tan(u)
elif CotQ(u):
return cot(v)
elif SecQ(u):
return sec(v)
return csc(v)
elif HyperbolicQ(u) and LinearQ(u.args[0], x):
v = ExpandToSum(I*u.args[0], x)
if SinhQ(u):
return -I*sin(v)
elif CoshQ(u):
return cos(v)
elif TanhQ(u):
return -I*tan(v)
elif CothQ(u):
I*cot(v)
elif SechQ(u):
return sec(v)
return I*csc(v)
return u.func(*[DeactivateTrigAux(i, x) for i in u.args])
def PowerOfInertTrigSumQ(u, func, x):
p_ = Wild('p', exclude=[x])
q_ = Wild('q', exclude=[x])
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x])
c_ = Wild('c', exclude=[x])
d_ = Wild('d', exclude=[x])
n_ = Wild('n', exclude=[x])
w_ = Wild('w')
pattern = (a_ + b_*(c_*func(w_))**p_)**n_
match = u.match(pattern)
if match:
keys = [a_, b_, c_, n_, p_, w_]
if len(keys) == len(match):
return True
pattern = (a_ + b_*(d_*func(w_))**p_ + c_*(d_*func(w_))**q_)**n_
match = u.match(pattern)
if match:
keys = [a_, b_, c_, d_, n_, p_, q_, w_]
if len(keys) == len(match):
return True
return False
def PiecewiseLinearQ(*args):
# (* If the derivative of u wrt x is a constant wrt x, PiecewiseLinearQ[u,x] returns True;
# else it returns False. *)
if len(args) == 3:
u, v, x = args
return PiecewiseLinearQ(u, x) and PiecewiseLinearQ(v, x)
u, x = args
if LinearQ(u, x):
return True
c_ = Wild('c', exclude=[x])
F_ = Wild('F', exclude=[x])
v_ = Wild('v')
match = u.match(Log(c_*F_**v_))
if match:
if len(match) == 3:
if LinearQ(match[v_], x):
return True
try:
F = type(u)
G = type(u.args[0])
v = u.args[0].args[0]
if LinearQ(v, x):
if MemberQ([[atanh, tanh], [atanh, coth], [acoth, coth], [acoth, tanh], [atan, tan], [atan, cot], [acot, cot], [acot, tan]], [F, G]):
return True
except:
pass
return False
def KnownTrigIntegrandQ(lst, u, x):
if u == 1:
return True
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x, 0])
func_ = WildFunction('func')
m_ = Wild('m', exclude=[x])
A_ = Wild('A', exclude=[x])
B_ = Wild('B', exclude=[x, 0])
C_ = Wild('C', exclude=[x, 0])
match = u.match((a_ + b_*func_)**m_)
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match((a_ + b_*func_)**m_*(A_ + B_*func_))
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match(A_ + C_*func_**2)
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match(A_ + B_*func_ + C_*func_**2)
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match((a_ + b_*func_)**m_*(A_ + C_*func_**2))
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match((a_ + b_*func_)**m_*(A_ + B_*func_ + C_*func_**2))
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
return False
def KnownSineIntegrandQ(u, x):
return KnownTrigIntegrandQ([sin, cos], u, x)
def KnownTangentIntegrandQ(u, x):
return KnownTrigIntegrandQ([tan], u, x)
def KnownCotangentIntegrandQ(u, x):
return KnownTrigIntegrandQ([cot], u, x)
def KnownSecantIntegrandQ(u, x):
return KnownTrigIntegrandQ([sec, csc], u, x)
def TryPureTanSubst(u, x):
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x])
c_ = Wild('c', exclude=[x])
G_ = Wild('G')
F = u.func
try:
if MemberQ([atan, acot, atanh, acoth], F):
match = u.args[0].match(c_*(a_ + b_*G_))
if match:
if len(match) == 4:
G = match[G_]
if MemberQ([tan, cot, tanh, coth], G.func):
if LinearQ(G.args[0], x):
return True
except:
pass
return False
def TryTanhSubst(u, x):
if LogQ(u):
return False
elif not FalseQ(FunctionOfLinear(u, x)):
return False
a_ = Wild('a', exclude=[x])
m_ = Wild('m', exclude=[x])
p_ = Wild('p', exclude=[x])
r_, s_, t_, n_, b_, f_, g_ = map(Wild, 'rstnbfg')
match = u.match(r_*(s_ + t_)**n_)
if match:
if len(match) == 4:
r, s, t, n = [match[i] for i in [r_, s_, t_, n_]]
if IntegerQ(n) and PositiveQ(n):
return False
match = u.match(1/(a_ + b_*f_**n_))
if match:
if len(match) == 4:
a, b, f, n = [match[i] for i in [a_, b_, f_, n_]]
if SinhCoshQ(f) and IntegerQ(n) and n > 2:
return False
match = u.match(f_*g_)
if match:
if len(match) == 2:
f, g = match[f_], match[g_]
if SinhCoshQ(f) and SinhCoshQ(g):
if IntegersQ(f.args[0]/x, g.args[0]/x):
return False
match = u.match(r_*(a_*s_**m_)**p_)
if match:
if len(match) == 5:
r, a, s, m, p = [match[i] for i in [r_, a_, s_, m_, p_]]
if Not(m==2 and (s == Sech(x) or s == Csch(x))):
return False
if u != ExpandIntegrand(u, x):
return False
return True
def TryPureTanhSubst(u, x):
F = u.func
a_ = Wild('a', exclude=[x])
G_ = Wild('G')
if F == sym_log:
return False
match = u.args[0].match(a_*G_)
if match and len(match) == 2:
G = match[G_].func
if MemberQ([atanh, acoth], F) and MemberQ([tanh, coth], G):
return False
if u != ExpandIntegrand(u, x):
return False
return True
def AbsurdNumberGCD(*seq):
# (* m, n, ... must be absurd numbers. AbsurdNumberGCD[m,n,...] returns the gcd of m, n, ... *)
lst = list(seq)
if Length(lst) == 1:
return First(lst)
return AbsurdNumberGCDList(FactorAbsurdNumber(First(lst)), FactorAbsurdNumber(AbsurdNumberGCD(*Rest(lst))))
def AbsurdNumberGCDList(lst1, lst2):
# (* lst1 and lst2 must be absurd number prime factorization lists. *)
# (* AbsurdNumberGCDList[lst1,lst2] returns the gcd of the absurd numbers represented by lst1 and lst2. *)
if lst1 == []:
return Mul(*[i[0]**Min(i[1],0) for i in lst2])
elif lst2 == []:
return Mul(*[i[0]**Min(i[1],0) for i in lst1])
elif lst1[0][0] == lst2[0][0]:
if lst1[0][1] <= lst2[0][1]:
return lst1[0][0]**lst1[0][1]*AbsurdNumberGCDList(Rest(lst1), Rest(lst2))
return lst1[0][0]**lst2[0][1]*AbsurdNumberGCDList(Rest(lst1), Rest(lst2))
elif lst1[0][0] < lst2[0][0]:
if lst1[0][1] < 0:
return lst1[0][0]**lst1[0][1]*AbsurdNumberGCDList(Rest(lst1), lst2)
return AbsurdNumberGCDList(Rest(lst1), lst2)
elif lst2[0][1] < 0:
return lst2[0][0]**lst2[0][1]*AbsurdNumberGCDList(lst1, Rest(lst2))
return AbsurdNumberGCDList(lst1, Rest(lst2))
def ExpandTrigExpand(u, F, v, m, n, x):
w = Expand(TrigExpand(F.xreplace({x: n*x}))**m).xreplace({x: v})
if SumQ(w):
t = 0
for i in w.args:
t += u*i
return t
else:
return u*w
def ExpandTrigReduce(*args):
if len(args) == 3:
u = args[0]
v = args[1]
x = args[2]
w = ExpandTrigReduce(v, x)
if SumQ(w):
t = 0
for i in w.args:
t += u*i
return t
else:
return u*w
else:
u = args[0]
x = args[1]
return ExpandTrigReduceAux(u, x)
def ExpandTrigReduceAux(u, x):
v = TrigReduce(u).expand()
if SumQ(v):
t = 0
for i in v.args:
t += NormalizeTrig(i, x)
return t
return NormalizeTrig(v, x)
def NormalizeTrig(v, x):
a = Wild('a', exclude=[x])
n = Wild('n', exclude=[x, 0])
F = Wild('F')
expr = a*F**n
M = v.match(expr)
if M and len(M[F].args) == 1 and PolynomialQ(M[F].args[0], x) and Exponent(M[F].args[0], x)>0:
u = M[F].args[0]
return M[a]*M[F].xreplace({u: ExpandToSum(u, x)})**M[n]
else:
return v
#=================================
def TrigToExp(expr):
ex = expr.rewrite(sin, sym_exp).rewrite(cos, sym_exp).rewrite(tan, sym_exp).rewrite(sec, sym_exp).rewrite(csc, sym_exp).rewrite(cot, sym_exp)
return ex.replace(sym_exp, rubi_exp)
def ExpandTrigToExp(u, *args):
if len(args) == 1:
x = args[0]
return ExpandTrigToExp(1, u, x)
else:
v = args[0]
x = args[1]
w = TrigToExp(v)
k = 0
if SumQ(w):
for i in w.args:
k += SimplifyIntegrand(u*i, x)
w = k
else:
w = SimplifyIntegrand(u*w, x)
return ExpandIntegrand(FreeFactors(w, x), NonfreeFactors(w, x),x)
#======================================
def TrigReduce(i):
"""
TrigReduce(expr) rewrites products and powers of trigonometric functions in expr in terms of trigonometric functions with combined arguments.
Examples
========
>>> from sympy import sin, cos
>>> from sympy.integrals.rubi.utility_function import TrigReduce
>>> from sympy.abc import x
>>> TrigReduce(cos(x)**2)
cos(2*x)/2 + 1/2
>>> TrigReduce(cos(x)**2*sin(x))
sin(x)/4 + sin(3*x)/4
>>> TrigReduce(cos(x)**2+sin(x))
sin(x) + cos(2*x)/2 + 1/2
"""
if SumQ(i):
t = 0
for k in i.args:
t += TrigReduce(k)
return t
if ProductQ(i):
if any(PowerQ(k) for k in i.args):
if (i.rewrite((sin, sinh), sym_exp).rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, sin)).has(I, cosh, sinh):
return i.rewrite((sin, sinh), sym_exp).rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, sin).simplify()
else:
return i.rewrite((sin, sinh), sym_exp).rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, sin)
else:
a = Wild('a')
b = Wild('b')
v = Wild('v')
Match = i.match(v*sin(a)*cos(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*sin(a)*cos(b), v*S(1)/2*(sin(a + b) + sin(a - b)))
Match = i.match(v*sin(a)*sin(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*sin(a)*sin(b), v*S(1)/2*cos(a - b) - cos(a + b))
Match = i.match(v*cos(a)*cos(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*cos(a)*cos(b), v*S(1)/2*cos(a + b) + cos(a - b))
Match = i.match(v*sinh(a)*cosh(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*sinh(a)*cosh(b), v*S(1)/2*(sinh(a + b) + sinh(a - b)))
Match = i.match(v*sinh(a)*sinh(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*sinh(a)*sinh(b), v*S(1)/2*cosh(a - b) - cosh(a + b))
Match = i.match(v*cosh(a)*cosh(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*cosh(a)*cosh(b), v*S(1)/2*cosh(a + b) + cosh(a - b))
if PowerQ(i):
if i.has(sin, sinh):
if (i.rewrite((sin, sinh), sym_exp).expand().rewrite(sym_exp, sin)).has(I, cosh, sinh):
return i.rewrite((sin, sinh), sym_exp).expand().rewrite(sym_exp, sin).simplify()
else:
return i.rewrite((sin, sinh), sym_exp).expand().rewrite(sym_exp, sin)
if i.has(cos, cosh):
if (i.rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, cos)).has(I, cosh, sinh):
return i.rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, cos).simplify()
else:
return i.rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, cos)
return i
def FunctionOfTrig(u, *args):
# If u is a function of trig functions of v where v is a linear function of x,
# FunctionOfTrig[u,x] returns v; else it returns False.
if len(args) == 1:
x = args[0]
v = FunctionOfTrig(u, None, x)
if v:
return v
else:
return False
else:
v, x = args
if AtomQ(u):
if u == x:
return False
else:
return v
if TrigQ(u) and LinearQ(u.args[0], x):
if v is None:
return u.args[0]
else:
a = Coefficient(v, x, 0)
b = Coefficient(v, x, 1)
c = Coefficient(u.args[0], x, 0)
d = Coefficient(u.args[0], x, 1)
if ZeroQ(a*d - b*c) and RationalQ(b/d):
return a/Numerator(b/d) + b*x/Numerator(b/d)
else:
return False
if HyperbolicQ(u) and LinearQ(u.args[0], x):
if v is None:
return I*u.args[0]
a = Coefficient(v, x, 0)
b = Coefficient(v, x, 1)
c = I*Coefficient(u.args[0], x, 0)
d = I*Coefficient(u.args[0], x, 1)
if ZeroQ(a*d - b*c) and RationalQ(b/d):
return a/Numerator(b/d) + b*x/Numerator(b/d)
else:
return False
if CalculusQ(u):
return False
else:
w = v
for i in u.args:
w = FunctionOfTrig(i, w, x)
if FalseQ(w):
return False
return w
def AlgebraicTrigFunctionQ(u, x):
# If u is algebraic function of trig functions, AlgebraicTrigFunctionQ(u,x) returns True; else it returns False.
if AtomQ(u):
return True
elif TrigQ(u) and LinearQ(u.args[0], x):
return True
elif HyperbolicQ(u) and LinearQ(u.args[0], x):
return True
elif PowerQ(u):
if FreeQ(u.exp, x):
return AlgebraicTrigFunctionQ(u.base, x)
elif ProductQ(u) or SumQ(u):
for i in u.args:
if not AlgebraicTrigFunctionQ(i, x):
return False
return True
return False
def FunctionOfHyperbolic(u, *x):
# If u is a function of hyperbolic trig functions of v where v is linear in x,
# FunctionOfHyperbolic(u,x) returns v; else it returns False.
if len(x) == 1:
x = x[0]
v = FunctionOfHyperbolic(u, None, x)
if v==None:
return False
else:
return v
else:
v = x[0]
x = x[1]
if AtomQ(u):
if u == x:
return False
return v
if HyperbolicQ(u) and LinearQ(u.args[0], x):
if v is None:
return u.args[0]
a = Coefficient(v, x, 0)
b = Coefficient(v, x, 1)
c = Coefficient(u.args[0], x, 0)
d = Coefficient(u.args[0], x, 1)
if ZeroQ(a*d - b*c) and RationalQ(b/d):
return a/Numerator(b/d) + b*x/Numerator(b/d)
else:
return False
if CalculusQ(u):
return False
w = v
for i in u.args:
if w == FunctionOfHyperbolic(i, w, x):
return False
return w
def FunctionOfQ(v, u, x, PureFlag=False):
# v is a function of x. If u is a function of v, FunctionOfQ(v, u, x) returns True; else it returns False. *)
if FreeQ(u, x):
return False
elif AtomQ(v):
return True
elif ProductQ(v) and Not(EqQ(FreeFactors(v, x), 1)):
return FunctionOfQ(NonfreeFactors(v, x), u, x, PureFlag)
elif PureFlag:
if SinQ(v) or CscQ(v):
return PureFunctionOfSinQ(u, v.args[0], x)
elif CosQ(v) or SecQ(v):
return PureFunctionOfCosQ(u, v.args[0], x)
elif TanQ(v):
return PureFunctionOfTanQ(u, v.args[0], x)
elif CotQ(v):
return PureFunctionOfCotQ(u, v.args[0], x)
elif SinhQ(v) or CschQ(v):
return PureFunctionOfSinhQ(u, v.args[0], x)
elif CoshQ(v) or SechQ(v):
return PureFunctionOfCoshQ(u, v.args[0], x)
elif TanhQ(v):
return PureFunctionOfTanhQ(u, v.args[0], x)
elif CothQ(v):
return PureFunctionOfCothQ(u, v.args[0], x)
else:
return FunctionOfExpnQ(u, v, x) != False
elif SinQ(v) or CscQ(v):
return FunctionOfSinQ(u, v.args[0], x)
elif CosQ(v) or SecQ(v):
return FunctionOfCosQ(u, v.args[0], x)
elif TanQ(v) or CotQ(v):
FunctionOfTanQ(u, v.args[0], x)
elif SinhQ(v) or CschQ(v):
return FunctionOfSinhQ(u, v.args[0], x)
elif CoshQ(v) or SechQ(v):
return FunctionOfCoshQ(u, v.args[0], x)
elif TanhQ(v) or CothQ(v):
return FunctionOfTanhQ(u, v.args[0], x)
return FunctionOfExpnQ(u, v, x) != False
def FunctionOfExpnQ(u, v, x):
if u == v:
return 1
if AtomQ(u):
if u == x:
return False
else:
return 0
if CalculusQ(u):
return False
if PowerQ(u):
if FreeQ(u.exp, x):
if ZeroQ(u.base - v):
if IntegerQ(u.exp):
return u.exp
else:
return 1
if PowerQ(v):
if FreeQ(v.exp, x) and ZeroQ(u.base-v.base):
if RationalQ(v.exp):
if RationalQ(u.exp) and IntegerQ(u.exp/v.exp) and (v.exp>0 or u.exp<0):
return u.exp/v.exp
else:
return False
if IntegerQ(Simplify(u.exp/v.exp)):
return Simplify(u.exp/v.exp)
else:
return False
return FunctionOfExpnQ(u.base, v, x)
if ProductQ(u) and Not(EqQ(FreeFactors(u, x), 1)):
return FunctionOfExpnQ(NonfreeFactors(u, x), v, x)
if ProductQ(u) and ProductQ(v):
deg1 = FunctionOfExpnQ(First(u), First(v), x)
if deg1==False:
return False
deg2 = FunctionOfExpnQ(Rest(u), Rest(v), x);
if deg1==deg2 and FreeQ(Simplify(u/v^deg1), x):
return deg1
else:
return False
lst = []
for i in u.args:
if FunctionOfExpnQ(i, v, x) is False:
return False
lst.append(FunctionOfExpnQ(i, v, x))
return Apply(GCD, lst)
def PureFunctionOfSinQ(u, v, x):
# If u is a pure function of Sin(v) and/or Csc(v), PureFunctionOfSinQ(u, v, x) returns True; else it returns False.
if AtomQ(u):
return u!=x
if CalculusQ(u):
return False
if TrigQ(u) and ZeroQ(u.args[0]-v):
return SinQ(u) or CscQ(u)
for i in u.args:
if Not(PureFunctionOfSinQ(i, v, x)):
return False
return True
def PureFunctionOfCosQ(u, v, x):
# If u is a pure function of Cos(v) and/or Sec(v), PureFunctionOfCosQ(u, v, x) returns True; else it returns False.
if AtomQ(u):
return u!=x
if CalculusQ(u):
return False
if TrigQ(u) and ZeroQ(u.args[0]-v):
return CosQ(u) or SecQ(u)
for i in u.args:
if Not(PureFunctionOfCosQ(i, v, x)):
return False
return True
def PureFunctionOfTanQ(u, v, x):
# If u is a pure function of Tan(v) and/or Cot(v), PureFunctionOfTanQ(u, v, x) returns True; else it returns False.
if AtomQ(u):
return u!=x
if CalculusQ(u):
return False
if TrigQ(u) and ZeroQ(u.args[0]-v):
return TanQ(u) or CotQ(u)
for i in u.args:
if Not(PureFunctionOfTanQ(i, v, x)):
return False
return True
def PureFunctionOfCotQ(u, v, x):
# If u is a pure function of Cot(v), PureFunctionOfCotQ(u, v, x) returns True; else it returns False.
if AtomQ(u):
return u!=x
if CalculusQ(u):
return False
if TrigQ(u) and ZeroQ(u.args[0]-v):
return CotQ(u)
for i in u.args:
if Not(PureFunctionOfCotQ(i, v, x)):
return False
return True
def FunctionOfCosQ(u, v, x):
# If u is a function of Cos[v], FunctionOfCosQ[u,v,x] returns True; else it returns False.
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
# Basis: If m integer, Cos[m*v]^n is a function of Cos[v]. *)
return CosQ(u) or SecQ(u)
elif IntegerPowerQ(u):
if TrigQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if EvenQ(u.exp):
# Basis: If m integer and n even, Trig[m*v]^n is a function of Cos[v]. *)
return True
return FunctionOfCosQ(u.base, v, x)
elif ProductQ(u):
lst = FindTrigFactor(sin, csc, u, v, False)
if ListQ(lst):
# (* Basis: If m integer and n odd, Sin[m*v]^n == Sin[v]*u where u is a function of Cos[v]. *)
return FunctionOfCosQ(Sin(v)*lst[1], v, x)
lst = FindTrigFactor(tan, cot, u, v, True)
if ListQ(lst):
# (* Basis: If m integer and n odd, Tan[m*v]^n == Sin[v]*u where u is a function of Cos[v]. *)
return FunctionOfCosQ(Sin(v)*lst[1], v, x)
return all(FunctionOfCosQ(i, v, x) for i in u.args)
return all(FunctionOfCosQ(i, v, x) for i in u.args)
def FunctionOfSinQ(u, v, x):
# If u is a function of Sin[v], FunctionOfSinQ[u,v,x] returns True; else it returns False.
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
if OddQuotientQ(u.args[0], v):
# Basis: If m odd, Sin[m*v]^n is a function of Sin[v].
return SinQ(u) or CscQ(u)
# Basis: If m even, Cos[m*v]^n is a function of Sin[v].
return CosQ(u) or SecQ(u)
elif IntegerPowerQ(u):
if TrigQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if EvenQ(u.exp):
# Basis: If m integer and n even, Hyper[m*v]^n is a function of Sin[v].
return True
return FunctionOfSinQ(u.base, v, x)
elif ProductQ(u):
if CosQ(u.args[0]) and SinQ(u.args[1]) and ZeroQ(u.args[0].args[0] - v/2) and ZeroQ(u.args[1].args[0] - v/2):
return FunctionOfSinQ(Drop(u, 2), v, x)
lst = FindTrigFactor(sin, csch, u, v, False)
if ListQ(lst) and EvenQuotientQ(lst[0], v):
# Basis: If m even and n odd, Sin[m*v]^n == Cos[v]*u where u is a function of Sin[v].
return FunctionOfSinQ(Cos(v)*lst[1], v, x)
lst = FindTrigFactor(cos, sec, u, v, False)
if ListQ(lst) and OddQuotientQ(lst[0], v):
# Basis: If m odd and n odd, Cos[m*v]^n == Cos[v]*u where u is a function of Sin[v].
return FunctionOfSinQ(Cos(v)*lst[1], v, x)
lst = FindTrigFactor(tan, cot, u, v, True)
if ListQ(lst):
# Basis: If m integer and n odd, Tan[m*v]^n == Cos[v]*u where u is a function of Sin[v].
return FunctionOfSinQ(Cos(v)*lst[1], v, x)
return all(FunctionOfSinQ(i, v, x) for i in u.args)
return all(FunctionOfSinQ(i, v, x) for i in u.args)
def OddTrigPowerQ(u, v, x):
if SinQ(u) or CosQ(u) or SecQ(u) or CscQ(u):
return OddQuotientQ(u.args[0], v)
if PowerQ(u):
return OddQ(u.exp) and OddTrigPowerQ(u.base, v, x)
if ProductQ(u):
if not FreeFactors(u, x) == 1:
return OddTrigPowerQ(NonfreeFactors(u, x), v, x)
lst = []
for i in u.args:
if Not(FunctionOfTanQ(i, v, x)):
lst.append(i)
if lst == []:
return True
return Length(lst)==1 and OddTrigPowerQ(lst[0], v, x)
if SumQ(u):
return all(OddTrigPowerQ(i, v, x) for i in u.args)
return False
def FunctionOfTanQ(u, v, x):
# If u is a function of the form f[Tan[v],Cot[v]] where f is independent of x,
# FunctionOfTanQ[u,v,x] returns True; else it returns False.
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
return TanQ(u) or CotQ(u) or EvenQuotientQ(u.args[0], v)
elif PowerQ(u):
if EvenQ(u.exp) and TrigQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
return True
elif EvenQ(u.exp) and SumQ(u.base):
return FunctionOfTanQ(Expand(u.base**2, v, x))
if ProductQ(u):
lst = []
for i in u.args:
if Not(FunctionOfTanQ(i, v, x)):
lst.append(i)
if lst == []:
return True
return Length(lst)==2 and OddTrigPowerQ(lst[0], v, x) and OddTrigPowerQ(lst[1], v, x)
return all(FunctionOfTanQ(i, v, x) for i in u.args)
def FunctionOfTanWeight(u, v, x):
# (* u is a function of the form f[Tan[v],Cot[v]] where f is independent of x.
# FunctionOfTanWeight[u,v,x] returns a nonnegative number if u is best considered a function
# of Tan[v]; else it returns a negative number. *)
if AtomQ(u):
return S(0)
elif CalculusQ(u):
return S(0)
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
if TanQ(u) and ZeroQ(u.args[0] - v):
return S(1)
elif CotQ(u) and ZeroQ(u.args[0] - v):
return S(-1)
return S(0)
elif PowerQ(u):
if EvenQ(u.exp) and TrigQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if TanQ(u.base) or CosQ(u.base) or SecQ(u.base):
return S(1)
return S(-1)
if ProductQ(u):
if all(FunctionOfTanQ(i, v, x) for i in u.args):
return Add(*[FunctionOfTanWeight(i, v, x) for i in u.args])
return S(0)
return Add(*[FunctionOfTanWeight(i, v, x) for i in u.args])
def FunctionOfTrigQ(u, v, x):
# If u (x) is equivalent to a function of the form f (Sin[v],Cos[v],Tan[v],Cot[v],Sec[v],Csc[v]) where f is independent of x, FunctionOfTrigQ[u,v,x] returns True; else it returns False.
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
return True
return all(FunctionOfTrigQ(i, v, x) for i in u.args)
def FunctionOfDensePolynomialsQ(u, x):
# If all occurrences of x in u (x) are in dense polynomials, FunctionOfDensePolynomialsQ[u,x] returns True; else it returns False.
if FreeQ(u, x):
return True
if PolynomialQ(u, x):
return Length(Exponent(u,x,List))>1
return all(FunctionOfDensePolynomialsQ(i, x) for i in u.args)
def FunctionOfLog(u, *args):
# If u (x) is equivalent to an expression of the form f (Log[a*x^n]), FunctionOfLog[u,x] returns
# the list {f (x),a*x^n,n}; else it returns False.
if len(args) == 1:
x = args[0]
lst = FunctionOfLog(u, False, False, x)
if AtomQ(lst) or FalseQ(lst[1]) or not isinstance(x, Symbol):
return False
else:
return lst
else:
v = args[0]
n = args[1]
x = args[2]
if AtomQ(u):
if u==x:
return False
else:
return [u, v, n]
if CalculusQ(u):
return False
lst = BinomialParts(u.args[0], x)
if LogQ(u) and ListQ(lst) and ZeroQ(lst[0]):
if FalseQ(v) or u.args[0] == v:
return [x, u.args[0], lst[2]]
else:
return False
lst = [0, v, n]
l = []
for i in u.args:
lst = FunctionOfLog(i, lst[1], lst[2], x)
if AtomQ(lst):
return False
else:
l.append(lst[0])
return [u.func(*l), lst[1], lst[2]]
def PowerVariableExpn(u, m, x):
# If m is an integer, u is an expression of the form f((c*x)**n) and g=GCD(m,n)>1,
# PowerVariableExpn(u,m,x) returns the list {x**(m/g)*f((c*x)**(n/g)),g,c}; else it returns False.
if IntegerQ(m):
lst = PowerVariableDegree(u, m, 1, x)
if not lst:
return False
else:
return [x**(m/lst[0])*PowerVariableSubst(u, lst[0], x), lst[0], lst[1]]
else:
return False
def PowerVariableDegree(u, m, c, x):
if FreeQ(u, x):
return [m, c]
if AtomQ(u) or CalculusQ(u):
return False
if PowerQ(u):
if FreeQ(u.base/x, x):
if ZeroQ(m) or m == u.exp and c == u.base/x:
return [u.exp, u.base/x]
if IntegerQ(u.exp) and IntegerQ(m) and GCD(m, u.exp)>1 and c==u.base/x:
return [GCD(m, u.exp), c]
else:
return False
lst = [m, c]
for i in u.args:
if PowerVariableDegree(i, lst[0], lst[1], x) == False:
return False
lst1 = PowerVariableDegree(i, lst[0], lst[1], x)
if not lst1:
return False
else:
return lst1
def PowerVariableSubst(u, m, x):
if FreeQ(u, x) or AtomQ(u) or CalculusQ(u):
return u
if PowerQ(u):
if FreeQ(u.base/x, x):
return x**(u.exp/m)
if ProductQ(u):
l = 1
for i in u.args:
l *= (PowerVariableSubst(i, m, x))
return l
if SumQ(u):
l = 0
for i in u.args:
l += (PowerVariableSubst(i, m, x))
return l
return u
def EulerIntegrandQ(expr, x):
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
n = Wild('n', exclude=[x, 0])
m = Wild('m', exclude=[x, 0])
p = Wild('p', exclude=[x, 0])
u = Wild('u')
v = Wild('v')
# Pattern 1
M = expr.match((a*x + b*u**n)**p)
if M:
if len(M) == 5 and FreeQ([M[a], M[b]], x) and IntegerQ(M[n] + 1/2) and QuadraticQ(M[u], x) and Not(RationalQ(M[p])) or NegativeIntegerQ(M[p]) and Not(BinomialQ(M[u], x)):
return True
# Pattern 2
M = expr.match(v**m*(a*x + b*u**n)**p)
if M:
if len(M) == 6 and FreeQ([M[a], M[b]], x) and ZeroQ(M[u] - M[v]) and IntegersQ(2*M[m], M[n] + 1/2) and QuadraticQ(M[u], x) and Not(RationalQ(M[p])) or NegativeIntegerQ(M[p]) and Not(BinomialQ(M[u], x)):
return True
# Pattern 3
M = expr.match(u**n*v**p)
if M:
if len(M) == 3 and NegativeIntegerQ(M[p]) and IntegerQ(M[n] + 1/2) and QuadraticQ(M[u], x) and QuadraticQ(M[v], x) and Not(BinomialQ(M[v], x)):
return True
else:
return False
def FunctionOfSquareRootOfQuadratic(u, *args):
if len(args) == 1:
x = args[0]
pattern = Pattern(UtilityOperator(x_**WC('m', 1)*(a_ + x**WC('n', 1)*WC('b', 1))**p_, x), CustomConstraint(lambda a, b, m, n, p, x: FreeQ([a, b, m, n, p], x)))
M = is_match(UtilityOperator(u, args[0]), pattern)
if M:
return False
tmp = FunctionOfSquareRootOfQuadratic(u, False, x)
if AtomQ(tmp) or FalseQ(tmp[0]):
return False
tmp = tmp[0]
a = Coefficient(tmp, x, 0)
b = Coefficient(tmp, x, 1)
c = Coefficient(tmp, x, 2)
if ZeroQ(a) and ZeroQ(b) or ZeroQ(b**2-4*a*c):
return False
if PosQ(c):
sqrt = Rt(c, S(2));
q = a*sqrt + b*x + sqrt*x**2
r = b + 2*sqrt*x
return [Simplify(SquareRootOfQuadraticSubst(u, q/r, (-a+x**2)/r, x)*q/r**2), Simplify(sqrt*x + Sqrt(tmp)), 2]
if PosQ(a):
sqrt = Rt(a, S(2))
q = c*sqrt - b*x + sqrt*x**2
r = c - x**2
return [Simplify(SquareRootOfQuadraticSubst(u, q/r, (-b+2*sqrt*x)/r, x)*q/r**2), Simplify((-sqrt+Sqrt(tmp))/x), 1]
sqrt = Rt(b**2 - 4*a*c, S(2))
r = c - x**2
return[Simplify(-sqrt*SquareRootOfQuadraticSubst(u, -sqrt*x/r, -(b*c+c*sqrt+(-b+sqrt)*x**2)/(2*c*r), x)*x/r**2), FullSimplify(2*c*Sqrt(tmp)/(b-sqrt+2*c*x)), 3]
else:
v = args[0]
x = args[1]
if AtomQ(u) or FreeQ(u, x):
return [v]
if PowerQ(u):
if FreeQ(u.exp, x):
if FractionQ(u.exp) and Denominator(u.exp)==2 and PolynomialQ(u.base, x) and Exponent(u.base, x)==2:
if FalseQ(v) or u.base == v:
return [u.base]
else:
return False
return FunctionOfSquareRootOfQuadratic(u.base, v, x)
if ProductQ(u) or SumQ(u):
lst = [v]
lst1 = []
for i in u.args:
if FunctionOfSquareRootOfQuadratic(i, lst[0], x) == False:
return False
lst1 = FunctionOfSquareRootOfQuadratic(i, lst[0], x)
return lst1
else:
return False
def SquareRootOfQuadraticSubst(u, vv, xx, x):
# SquareRootOfQuadraticSubst(u, vv, xx, x) returns u with fractional powers replaced by vv raised to the power and x replaced by xx.
if AtomQ(u) or FreeQ(u, x):
if u==x:
return xx
return u
if PowerQ(u):
if FreeQ(u.exp, x):
if FractionQ(u.exp) and Denominator(u.exp)==2 and PolynomialQ(u.base, x) and Exponent(u.base, x)==2:
return vv**Numerator(u.exp)
return SquareRootOfQuadraticSubst(u.base, vv, xx, x)**u.exp
elif SumQ(u):
t = 0
for i in u.args:
t += SquareRootOfQuadraticSubst(i, vv, xx, x)
return t
elif ProductQ(u):
t = 1
for i in u.args:
t *= SquareRootOfQuadraticSubst(i, vv, xx, x)
return t
def Divides(y, u, x):
# If u divided by y is free of x, Divides[y,u,x] returns the quotient; else it returns False.
v = Simplify(u/y)
if FreeQ(v, x):
return v
else:
return False
def DerivativeDivides(y, u, x):
'''
If y not equal to x, y is easy to differentiate wrt x, and u divided by the derivative of y
is free of x, DerivativeDivides[y,u,x] returns the quotient; else it returns False.
'''
from matchpy import is_match
pattern0 = Pattern(Mul(a , b_), CustomConstraint(lambda a, b : FreeQ(a, b)))
def f1(y, u, x):
if PolynomialQ(y, x):
return PolynomialQ(u, x) and Exponent(u, x)==Exponent(y, x)-1
else:
return EasyDQ(y, x)
if is_match(y, pattern0):
return False
elif f1(y, u, x):
v = D(y ,x)
if EqQ(v, 0):
return False
else:
v = Simplify(u/v)
if FreeQ(v, x):
return v
else:
return False
else:
return False
def EasyDQ(expr, x):
# If u is easy to differentiate wrt x, EasyDQ(u, x) returns True; else it returns False *)
u = Wild('u',exclude=[1])
m = Wild('m',exclude=[x, 0])
M = expr.match(u*x**m)
if M:
return EasyDQ(M[u], x)
if AtomQ(expr) or FreeQ(expr, x) or Length(expr)==0:
return True
elif CalculusQ(expr):
return False
elif Length(expr)==1:
return EasyDQ(expr.args[0], x)
elif BinomialQ(expr, x) or ProductOfLinearPowersQ(expr, x):
return True
elif RationalFunctionQ(expr, x) and RationalFunctionExponents(expr, x)==[1, 1]:
return True
elif ProductQ(expr):
if FreeQ(First(expr), x):
return EasyDQ(Rest(expr), x)
elif FreeQ(Rest(expr), x):
return EasyDQ(First(expr), x)
else:
return False
elif SumQ(expr):
return EasyDQ(First(expr), x) and EasyDQ(Rest(expr), x)
elif Length(expr)==2:
if FreeQ(expr.args[0], x):
EasyDQ(expr.args[1], x)
elif FreeQ(expr.args[1], x):
return EasyDQ(expr.args[0], x)
else:
return False
return False
def ProductOfLinearPowersQ(u, x):
# ProductOfLinearPowersQ(u, x) returns True iff u is a product of factors of the form v^n where v is linear in x
v = Wild('v')
n = Wild('n', exclude=[x])
M = u.match(v**n)
return FreeQ(u, x) or M and LinearQ(M[v], x) or ProductQ(u) and ProductOfLinearPowersQ(First(u), x) and ProductOfLinearPowersQ(Rest(u), x)
def Rt(u, n):
return RtAux(TogetherSimplify(u), n)
def NthRoot(u, n):
return nsimplify(u**(S(1)/n))
def AtomBaseQ(u):
# If u is an atom or an atom raised to an odd degree, AtomBaseQ(u) returns True; else it returns False
return AtomQ(u) or PowerQ(u) and OddQ(u.args[1]) and AtomBaseQ(u.args[0])
def SumBaseQ(u):
# If u is a sum or a sum raised to an odd degree, SumBaseQ(u) returns True; else it returns False
return SumQ(u) or PowerQ(u) and OddQ(u.args[1]) and SumBaseQ(u.args[0])
def NegSumBaseQ(u):
# If u is a sum or a sum raised to an odd degree whose lead term has a negative form, NegSumBaseQ(u) returns True; else it returns False
return SumQ(u) and NegQ(First(u)) or PowerQ(u) and OddQ(u.args[1]) and NegSumBaseQ(u.args[0])
def AllNegTermQ(u):
# If all terms of u have a negative form, AllNegTermQ(u) returns True; else it returns False
if PowerQ(u):
if OddQ(u.exp):
return AllNegTermQ(u.base)
if SumQ(u):
return NegQ(First(u)) and AllNegTermQ(Rest(u))
return NegQ(u)
def SomeNegTermQ(u):
# If some term of u has a negative form, SomeNegTermQ(u) returns True; else it returns False
if PowerQ(u):
if OddQ(u.exp):
return SomeNegTermQ(u.base)
if SumQ(u):
return NegQ(First(u)) or SomeNegTermQ(Rest(u))
return NegQ(u)
def TrigSquareQ(u):
# If u is an expression of the form Sin(z)^2 or Cos(z)^2, TrigSquareQ(u) returns True, else it returns False
return PowerQ(u) and EqQ(u.args[1], 2) and MemberQ([sin, cos], Head(u.args[0]))
def RtAux(u, n):
if PowerQ(u):
return u.base**(u.exp/n)
if ComplexNumberQ(u):
a = Re(u)
b = Im(u)
if Not(IntegerQ(a) and IntegerQ(b)) and IntegerQ(a/(a**2 + b**2)) and IntegerQ(b/(a**2 + b**2)):
# Basis: a+b*I==1/(a/(a^2+b^2)-b/(a^2+b^2)*I)
return S(1)/RtAux(a/(a**2 + b**2) - b/(a**2 + b**2)*I, n)
else:
return NthRoot(u, n)
if ProductQ(u):
lst = SplitProduct(PositiveQ, u)
if ListQ(lst):
return RtAux(lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(NegativeQ, u)
if ListQ(lst):
if EqQ(lst[0], -1):
v = lst[1]
if PowerQ(v):
if NegativeQ(v.exp):
return 1/RtAux(-v.base**(-v.exp), n)
if ProductQ(v):
if ListQ(SplitProduct(SumBaseQ, v)):
lst = SplitProduct(AllNegTermQ, v)
if ListQ(lst):
return RtAux(-lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(NegSumBaseQ, v)
if ListQ(lst):
return RtAux(-lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(SomeNegTermQ, v)
if ListQ(lst):
return RtAux(-lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(SumBaseQ, v)
return RtAux(-lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(AtomBaseQ, v)
if ListQ(lst):
return RtAux(-lst[0], n)*RtAux(lst[1], n)
else:
return RtAux(-First(v), n)*RtAux(Rest(v), n)
if OddQ(n):
return -RtAux(v, n)
else:
return NthRoot(u, n)
else:
return RtAux(-lst[0], n)*RtAux(-lst[1], n)
lst = SplitProduct(AllNegTermQ, u)
if ListQ(lst) and ListQ(SplitProduct(SumBaseQ, lst[1])):
return RtAux(-lst[0], n)*RtAux(-lst[1], n)
lst = SplitProduct(NegSumBaseQ, u)
if ListQ(lst) and ListQ(SplitProduct(NegSumBaseQ, lst[1])):
return RtAux(-lst[0], n)*RtAux(-lst[1], n)
return u.func(*[RtAux(i, n) for i in u.args])
v = TrigSquare(u)
if Not(AtomQ(v)):
return RtAux(v, n)
if OddQ(n) and NegativeQ(u):
return -RtAux(-u, n)
if OddQ(n) and NegQ(u) and PosQ(-u):
return -RtAux(-u, n)
else:
return NthRoot(u, n)
def TrigSquare(u):
# If u is an expression of the form a-a*Sin(z)^2 or a-a*Cos(z)^2, TrigSquare(u) returns Cos(z)^2 or Sin(z)^2 respectively,
# else it returns False.
if SumQ(u):
for i in u.args:
v = SplitProduct(TrigSquareQ, i)
if v == False or SplitSum(v, u) == False:
return False
lst = SplitSum(SplitProduct(TrigSquareQ, i))
if lst and ZeroQ(lst[1][2] + lst[1]):
if Head(lst[0][0].args[0]) == sin:
return lst[1]*cos(lst[1][1][1][1])**2
return lst[1]*sin(lst[1][1][1][1])**2
else:
return False
else:
return False
def IntSum(u, x):
# If u is free of x or of the form c*(a+b*x)^m, IntSum[u,x] returns the antiderivative of u wrt x;
# else it returns d*Int[v,x] where d*v=u and d is free of x.
return Add(*[Integral(i, x) for i in u.args])
return Simp(FreeTerms(u, x)*x, x) + IntTerm(NonfreeTerms(u, x), x)
def IntTerm(expr, x):
# If u is of the form c*(a+b*x)**m, IntTerm(u,x) returns the antiderivative of u wrt x;
# else it returns d*Int(v,x) where d*v=u and d is free of x.
c = Wild('c', exclude=[x])
m = Wild('m', exclude=[x, 0])
v = Wild('v')
M = expr.match(c/v)
if M and len(M) == 2 and FreeQ(M[c], x) and LinearQ(M[v], x):
return Simp(M[c]*Log(RemoveContent(M[v], x))/Coefficient(M[v], x, 1), x)
M = expr.match(c*v**m)
if M and len(M) == 3 and NonzeroQ(M[m] + 1) and LinearQ(M[v], x):
return Simp(M[c]*M[v]**(M[m] + 1)/(Coefficient(M[v], x, 1)*(M[m] + 1)), x)
if SumQ(expr):
t = 0
for i in expr.args:
t += IntTerm(i, x)
return t
else:
u = expr
return Dist(FreeFactors(u,x), Integral(NonfreeFactors(u, x), x), x)
def Map2(f, lst1, lst2):
result = []
for i in range(0, len(lst1)):
result.append(f(lst1[i], lst2[i]))
return result
def ConstantFactor(u, x):
# (* ConstantFactor[u,x] returns a 2-element list of the factors of u[x] free of x and the
# factors not free of u[x]. Common constant factors of the terms of sums are also collected. *)
if FreeQ(u, x):
return [u, S(1)]
elif AtomQ(u):
return [S(1), u]
elif PowerQ(u):
if FreeQ(u.exp, x):
lst = ConstantFactor(u.base, x)
if IntegerQ(u.exp):
return [lst[0]**u.exp, lst[1]**u.exp]
tmp = PositiveFactors(lst[0])
if tmp == 1:
return [S(1), u]
return [tmp**u.exp, (NonpositiveFactors(lst[0])*lst[1])**u.exp]
elif ProductQ(u):
lst = [ConstantFactor(i, x) for i in u.args]
return [Mul(*[First(i) for i in lst]), Mul(*[i[1] for i in lst])]
elif SumQ(u):
lst1 = [ConstantFactor(i, x) for i in u.args]
if SameQ(*[i[1] for i in lst1]):
return [Add(*[i[0] for i in lst]), lst1[0][1]]
lst2 = CommonFactors([First(i) for i in lst1])
return [First(lst2), Add(*Map2(Mul, Rest(lst2), [i[1] for i in lst1]))]
return [S(1), u]
def SameQ(*args):
for i in range(0, len(args) - 1):
if args[i] != args[i+1]:
return False
return True
def ReplacePart(lst, a, b):
lst[b] = a
return lst
def CommonFactors(lst):
# (* If lst is a list of n terms, CommonFactors[lst] returns a n+1-element list whose first
# element is the product of the factors common to all terms of lst, and whose remaining
# elements are quotients of each term divided by the common factor. *)
lst1 = [NonabsurdNumberFactors(i) for i in lst]
lst2 = [AbsurdNumberFactors(i) for i in lst]
num = AbsurdNumberGCD(*lst2)
common = num
lst2 = [i/num for i in lst2]
while (True):
lst3 = [LeadFactor(i) for i in lst1]
if SameQ(*lst3):
common = common*lst3[0]
lst1 = [RemainingFactors(i) for i in lst1]
elif (all((LogQ(i) and IntegerQ(First(i)) and First(i) > 0) for i in lst3) and
all(RationalQ(i) for i in [FullSimplify(j/First(lst3)) for j in lst3])):
lst4 = [FullSimplify(j/First(lst3)) for j in lst3]
num = GCD(*lst4)
common = common*Log((First(lst3)[0])**num)
lst2 = [lst2[i]*lst4[i]/num for i in range(0, len(lst2))]
lst1 = [RemainingFactors(i) for i in lst1]
lst4 = [LeadDegree(i) for i in lst1]
if SameQ(*[LeadBase(i) for i in lst1]) and RationalQ(*lst4):
num = Smallest(lst4)
base = LeadBase(lst1[0])
if num != 0:
common = common*base**num
lst2 = [lst2[i]*base**(lst4[i] - num) for i in range(0, len(lst2))]
lst1 = [RemainingFactors(i) for i in lst1]
elif (Length(lst1) == 2 and ZeroQ(LeadBase(lst1[0]) + LeadBase(lst1[1])) and
NonzeroQ(lst1[0] - 1) and IntegerQ(lst4[0]) and FractionQ(lst4[1])):
num = Min(lst4)
base = LeadBase(lst1[1])
if num != 0:
common = common*base**num
lst2 = [lst2[0]*(-1)**lst4[0], lst2[1]]
lst2 = [lst2[i]*base**(lst4[i] - num) for i in range(0, len(lst2))]
lst1 = [RemainingFactors(i) for i in lst1]
elif (Length(lst1) == 2 and ZeroQ(lst1[0] + LeadBase(lst1[1])) and
NonzeroQ(lst1[1] - 1) and IntegerQ(lst1[1]) and FractionQ(lst4[0])):
num = Min(lst4)
base = LeadBase(lst1[0])
if num != 0:
common = common*base**num
lst2 = [lst2[0], lst2[1]*(-1)**lst4[1]]
lst2 = [lst2[i]*base**(lst4[i] - num) for i in range(0, len(lst2))]
lst1 = [RemainingFactors(i) for i in lst1]
else:
num = MostMainFactorPosition(lst3)
lst2 = ReplacePart(lst2, lst3[num]*lst2[num], num)
lst1 = ReplacePart(lst1, RemainingFactors(lst1[num]), num)
if all(i==1 for i in lst1):
return Prepend(lst2, common)
def MostMainFactorPosition(lst):
factor = S(1)
num = 0
for i in range(0, Length(lst)):
if FactorOrder(lst[i], factor) > 0:
factor = lst[i]
num = i
return num
SbaseS, SexponS = None, None
SexponFlagS = False
def FunctionOfExponentialQ(u, x):
# (* FunctionOfExponentialQ[u,x] returns True iff u is a function of F^v where F is a constant and v is linear in x, *)
# (* and such an exponential explicitly occurs in u (i.e. not just implicitly in hyperbolic functions). *)
global SbaseS, SexponS, SexponFlagS
SbaseS, SexponS = None, None
SexponFlagS = False
res = FunctionOfExponentialTest(u, x)
return res and SexponFlagS
def FunctionOfExponential(u, x):
global SbaseS, SexponS, SexponFlagS
# (* u is a function of F^v where v is linear in x. FunctionOfExponential[u,x] returns F^v. *)
SbaseS, SexponS = None, None
SexponFlagS = False
FunctionOfExponentialTest(u, x)
return SbaseS**SexponS
def FunctionOfExponentialFunction(u, x):
global SbaseS, SexponS, SexponFlagS
# (* u is a function of F^v where v is linear in x. FunctionOfExponentialFunction[u,x] returns u with F^v replaced by x. *)
SbaseS, SexponS = None, None
SexponFlagS = False
FunctionOfExponentialTest(u, x)
return SimplifyIntegrand(FunctionOfExponentialFunctionAux(u, x), x)
def FunctionOfExponentialFunctionAux(u, x):
# (* u is a function of F^v where v is linear in x, and the fluid variables $base$=F and $expon$=v. *)
# (* FunctionOfExponentialFunctionAux[u,x] returns u with F^v replaced by x. *)
global SbaseS, SexponS, SexponFlagS
if AtomQ(u):
return u
elif PowerQ(u):
if FreeQ(u.base, x) and LinearQ(u.exp, x):
if ZeroQ(Coefficient(SexponS, x, 0)):
return u.base**Coefficient(u.exp, x, 0)*x**FullSimplify(Log(u.base)*Coefficient(u.exp, x, 1)/(Log(SbaseS)*Coefficient(SexponS, x, 1)))
return x**FullSimplify(Log(u.base)*Coefficient(u.exp, x, 1)/(Log(SbaseS)*Coefficient(SexponS, x, 1)))
elif HyperbolicQ(u) and LinearQ(u.args[0], x):
tmp = x**FullSimplify(Coefficient(u.args[0], x, 1)/(Log(SbaseS)*Coefficient(SexponS, x, 1)))
if SinhQ(u):
return tmp/2 - 1/(2*tmp)
elif CoshQ(u):
return tmp/2 + 1/(2*tmp)
elif TanhQ(u):
return (tmp - 1/tmp)/(tmp + 1/tmp)
elif CothQ(u):
return (tmp + 1/tmp)/(tmp - 1/tmp)
elif SechQ(u):
return 2/(tmp + 1/tmp)
return 2/(tmp - 1/tmp)
if PowerQ(u):
if FreeQ(u.base, x) and SumQ(u.exp):
return FunctionOfExponentialFunctionAux(u.base**First(u.exp), x)*FunctionOfExponentialFunctionAux(u.base**Rest(u.exp), x)
return u.func(*[FunctionOfExponentialFunctionAux(i, x) for i in u.args])
def FunctionOfExponentialTest(u, x):
# (* FunctionOfExponentialTest[u,x] returns True iff u is a function of F^v where F is a constant and v is linear in x. *)
# (* Before it is called, the fluid variables $base$ and $expon$ should be set to Null and $exponFlag$ to False. *)
# (* If u is a function of F^v, $base$ and $expon$ are set to F and v, respectively. *)
# (* If an explicit exponential occurs in u, $exponFlag$ is set to True. *)
global SbaseS, SexponS, SexponFlagS
if FreeQ(u, x):
return True
elif u == x or CalculusQ(u):
return False
elif PowerQ(u):
if FreeQ(u.base, x) and LinearQ(u.exp, x):
SexponFlagS = True
return FunctionOfExponentialTestAux(u.base, u.exp, x)
elif HyperbolicQ(u) and LinearQ(u.args[0], x):
return FunctionOfExponentialTestAux(E, u.args[0], x)
if PowerQ(u):
if FreeQ(u.base, x) and SumQ(u.exp):
return FunctionOfExponentialTest(u.base**First(u.exp), x) and FunctionOfExponentialTest(u.base**Rest(u.exp), x)
return all(FunctionOfExponentialTest(i, x) for i in u.args)
def FunctionOfExponentialTestAux(base, expon, x):
global SbaseS, SexponS, SexponFlagS
if SbaseS is None:
SbaseS = base
SexponS = expon
return True
tmp = FullSimplify(Log(base)*Coefficient(expon, x, 1)/(Log(SbaseS)*Coefficient(SexponS, x, 1)))
if Not(RationalQ(tmp)):
return False
elif ZeroQ(Coefficient(SexponS, x, 0)) or NonzeroQ(tmp - FullSimplify(Log(base)*Coefficient(expon, x, 0)/(Log(SbaseS)*Coefficient(SexponS, x, 0)))):
if PositiveIntegerQ(base, SbaseS) and base<SbaseS:
SbaseS = base
SexponS = expon
tmp = 1/tmp
SexponS = Coefficient(SexponS, x, 1)*x/Denominator(tmp)
if tmp < 0 and NegQ(Coefficient(SexponS, x, 1)):
SexponS = -SexponS
return True
else:
return True
if PositiveIntegerQ(base, SbaseS) and base < SbaseS:
SbaseS = base
SexponS = expon
tmp = 1/tmp
SexponS = SexponS/Denominator(tmp)
if tmp < 0 and NegQ(Coefficient(SexponS, x, 1)):
SexponS = -SexponS
return True
return True
def stdev(lst):
"""Calculates the standard deviation for a list of numbers."""
num_items = len(lst)
mean = sum(lst) / num_items
differences = [x - mean for x in lst]
sq_differences = [d ** 2 for d in differences]
ssd = sum(sq_differences)
variance = ssd / num_items
sd = sqrt(variance)
return sd
def rubi_test(expr, x, optimal_output, expand=False, _hyper_check=False, _diff=False, _numerical=False):
#Returns True if (expr - optimal_output) is equal to 0 or a constant
#expr: integrated expression
#x: integration variable
#expand=True equates `expr` with `optimal_output` in expanded form
#_hyper_check=True evaluates numerically
#_diff=True differentiates the expressions before equating
#_numerical=True equates the expressions at random `x`. Normally used for large expressions.
from sympy import nsimplify
if not expr.has(csc, sec, cot, csch, sech, coth):
optimal_output = process_trig(optimal_output)
if expr == optimal_output:
return True
if simplify(expr) == simplify(optimal_output):
return True
if nsimplify(expr) == nsimplify(optimal_output):
return True
if expr.has(sym_exp):
expr = powsimp(powdenest(expr), force=True)
if simplify(expr) == simplify(powsimp(optimal_output, force=True)):
return True
res = expr - optimal_output
if _numerical:
args = res.free_symbols
rand_val = []
try:
for i in range(0, 5): # check at 5 random points
rand_x = randint(1, 40)
substitutions = dict((s, rand_x) for s in args)
rand_val.append(float(abs(res.subs(substitutions).n())))
if stdev(rand_val) < Pow(10, -3):
return True
except:
pass
# return False
dres = res.diff(x)
if _numerical:
args = dres.free_symbols
rand_val = []
try:
for i in range(0, 5): # check at 5 random points
rand_x = randint(1, 40)
substitutions = dict((s, rand_x) for s in args)
rand_val.append(float(abs(dres.subs(substitutions).n())))
if stdev(rand_val) < Pow(10, -3):
return True
# return False
except:
pass
# return False
r = Simplify(nsimplify(res))
if r == 0 or (not r.has(x)):
return True
if _diff:
if dres == 0:
return True
elif Simplify(dres) == 0:
return True
if expand: # expands the expression and equates
e = res.expand()
if Simplify(e) == 0 or (not e.has(x)):
return True
return False
def If(cond, t, f):
# returns t if condition is true else f
if cond:
return t
return f
def IntQuadraticQ(a, b, c, d, e, m, p, x):
# (* IntQuadraticQ[a,b,c,d,e,m,p,x] returns True iff (d+e*x)^m*(a+b*x+c*x^2)^p is integrable wrt x in terms of non-Appell functions. *)
return IntegerQ(p) or PositiveIntegerQ(m) or IntegersQ(2*m, 2*p) or IntegersQ(m, 4*p) or IntegersQ(m, p + S(1)/3) and (ZeroQ(c**2*d**2 - b*c*d*e + b**2*e**2 - 3*a*c*e**2) or ZeroQ(c**2*d**2 - b*c*d*e - 2*b**2*e**2 + 9*a*c*e**2))
def IntBinomialQ(*args):
#(* IntBinomialQ(a,b,c,n,m,p,x) returns True iff (c*x)^m*(a+b*x^n)^p is integrable wrt x in terms of non-hypergeometric functions. *)
if len(args) == 8:
a, b, c, d, n, p, q, x = args
return IntegersQ(p,q) or PositiveIntegerQ(p) or PositiveIntegerQ(q) or (ZeroQ(n-2) or ZeroQ(n-4)) and (IntegersQ(p,4*q) or IntegersQ(4*p,q)) or ZeroQ(n-2) and (IntegersQ(2*p,2*q) or IntegersQ(3*p,q) and ZeroQ(b*c+3*a*d) or IntegersQ(p,3*q) and ZeroQ(3*b*c+a*d))
elif len(args) == 7:
a, b, c, n, m, p, x = args
return IntegerQ(2*p) or IntegerQ((m+1)/n + p) or (ZeroQ(n - 2) or ZeroQ(n - 4)) and IntegersQ(2*m, 4*p) or ZeroQ(n - 2) and IntegerQ(6*p) and (IntegerQ(m) or IntegerQ(m - p))
elif len(args) == 10:
a, b, c, d, e, m, n, p, q, x = args
return IntegersQ(p,q) or PositiveIntegerQ(p) or PositiveIntegerQ(q) or ZeroQ(n-2) and IntegerQ(m) and IntegersQ(2*p,2*q) or ZeroQ(n-4) and (IntegersQ(m,p,2*q) or IntegersQ(m,2*p,q))
def RectifyTangent(*args):
# (* RectifyTangent(u,a,b,r,x) returns an expression whose derivative equals the derivative of r*ArcTan(a+b*Tan(u)) wrt x. *)
if len(args) == 5:
u, a, b, r, x = args
t1 = Together(a)
t2 = Together(b)
if (PureComplexNumberQ(t1) or (ProductQ(t1) and any(PureComplexNumberQ(i) for i in t1.args))) and (PureComplexNumberQ(t2) or ProductQ(t2) and any(PureComplexNumberQ(i) for i in t2.args)):
c = a/I
d = b/I
if NegativeQ(d):
return RectifyTangent(u, -a, -b, -r, x)
e = SmartDenominator(Together(c + d*x))
c = c*e
d = d*e
if EvenQ(Denominator(NumericFactor(Together(u)))):
return I*r*Log(RemoveContent(Simplify((c+e)**2+d**2)+Simplify((c+e)**2-d**2)*Cos(2*u)+Simplify(2*(c+e)*d)*Sin(2*u),x))/4 - I*r*Log(RemoveContent(Simplify((c-e)**2+d**2)+Simplify((c-e)**2-d**2)*Cos(2*u)+Simplify(2*(c-e)*d)*Sin(2*u),x))/4
return I*r*Log(RemoveContent(Simplify((c+e)**2)+Simplify(2*(c+e)*d)*Cos(u)*Sin(u)-Simplify((c+e)**2-d**2)*Sin(u)**2,x))/4 - I*r*Log(RemoveContent(Simplify((c-e)**2)+Simplify(2*(c-e)*d)*Cos(u)*Sin(u)-Simplify((c-e)**2-d**2)*Sin(u)**2,x))/4
elif NegativeQ(b):
return RectifyTangent(u, -a, -b, -r, x)
elif EvenQ(Denominator(NumericFactor(Together(u)))):
return r*SimplifyAntiderivative(u,x) + r*ArcTan(Simplify((2*a*b*Cos(2*u)-(1+a**2-b**2)*Sin(2*u))/(a**2+(1+b)**2+(1+a**2-b**2)*Cos(2*u)+2*a*b*Sin(2*u))))
return r*SimplifyAntiderivative(u,x) - r*ArcTan(ActivateTrig(Simplify((a*b-2*a*b*cos(u)**2+(1+a**2-b**2)*cos(u)*sin(u))/(b*(1+b)+(1+a**2-b**2)*cos(u)**2+2*a*b*cos(u)*sin(u)))))
u, a, b, x = args
t = Together(a)
if PureComplexNumberQ(t) or (ProductQ(t) and any(PureComplexNumberQ(i) for i in t.args)):
c = a/I
if NegativeQ(c):
return RectifyTangent(u, -a, -b, x)
if ZeroQ(c - 1):
if EvenQ(Denominator(NumericFactor(Together(u)))):
return I*b*ArcTanh(Sin(2*u))/2
return I*b*ArcTanh(2*cos(u)*sin(u))/2
e = SmartDenominator(c)
c = c*e
return I*b*Log(RemoveContent(e*Cos(u)+c*Sin(u),x))/2 - I*b*Log(RemoveContent(e*Cos(u)-c*Sin(u),x))/2
elif NegativeQ(a):
return RectifyTangent(u, -a, -b, x)
elif ZeroQ(a - 1):
return b*SimplifyAntiderivative(u, x)
elif EvenQ(Denominator(NumericFactor(Together(u)))):
c = Simplify((1 + a)/(1 - a))
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) - b*ArcTan(NormalizeLeadTermSigns(denr*Sin(2*u)/(numr+denr*Cos(2*u)))),
elif PositiveQ(a - 1):
c = Simplify(1/(a - 1))
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) + b*ArcTan(NormalizeLeadTermSigns(denr*Cos(u)*Sin(u)/(numr+denr*Sin(u)**2))),
c = Simplify(a/(1 - a))
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) - b*ArcTan(NormalizeLeadTermSigns(denr*Cos(u)*Sin(u)/(numr+denr*Cos(u)**2)))
def RectifyCotangent(*args):
#(* RectifyCotangent[u,a,b,r,x] returns an expression whose derivative equals the derivative of r*ArcTan[a+b*Cot[u]] wrt x. *)
if len(args) == 5:
u, a, b, r, x = args
t1 = Together(a)
t2 = Together(b)
if (PureComplexNumberQ(t1) or (ProductQ(t1) and any(PureComplexNumberQ(i) for i in t1.args))) and (PureComplexNumberQ(t2) or ProductQ(t2) and any(PureComplexNumberQ(i) for i in t2.args)):
c = a/I
d = b/I
if NegativeQ(d):
return RectifyTangent(u,-a,-b,-r,x)
e = SmartDenominator(Together(c + d*x))
c = c*e
d = d*e
if EvenQ(Denominator(NumericFactor(Together(u)))):
return I*r*Log(RemoveContent(Simplify((c+e)**2+d**2)-Simplify((c+e)**2-d**2)*Cos(2*u)+Simplify(2*(c+e)*d)*Sin(2*u),x))/4 - I*r*Log(RemoveContent(Simplify((c-e)**2+d**2)-Simplify((c-e)**2-d**2)*Cos(2*u)+Simplify(2*(c-e)*d)*Sin(2*u),x))/4
return I*r*Log(RemoveContent(Simplify((c+e)**2)-Simplify((c+e)**2-d**2)*Cos(u)**2+Simplify(2*(c+e)*d)*Cos(u)*Sin(u),x))/4 - I*r*Log(RemoveContent(Simplify((c-e)**2)-Simplify((c-e)**2-d**2)*Cos(u)**2+Simplify(2*(c-e)*d)*Cos(u)*Sin(u),x))/4
elif NegativeQ(b):
return RectifyCotangent(u,-a,-b,-r,x)
elif EvenQ(Denominator(NumericFactor(Together(u)))):
return -r*SimplifyAntiderivative(u,x) - r*ArcTan(Simplify((2*a*b*Cos(2*u)+(1+a**2-b**2)*Sin(2*u))/(a**2+(1+b)**2-(1+a**2-b**2)*Cos(2*u)+2*a*b*Sin(2*u))))
return -r*SimplifyAntiderivative(u,x) - r*ArcTan(ActivateTrig(Simplify((a*b-2*a*b*sin(u)**2+(1+a**2-b**2)*cos(u)*sin(u))/(b*(1+b)+(1+a**2-b**2)*sin(u)**2+2*a*b*cos(u)*sin(u)))))
u, a, b, x = args
t = Together(a)
if PureComplexNumberQ(t) or (ProductQ(t) and any(PureComplexNumberQ(i) for i in t.args)):
c = a/I
if NegativeQ(c):
return RectifyCotangent(u,-a,-b,x)
elif ZeroQ(c - 1):
if EvenQ(Denominator(NumericFactor(Together(u)))):
return -I*b*ArcTanh(Sin(2*u))/2
return -I*b*ArcTanh(2*Cos(u)*Sin(u))/2
e = SmartDenominator(c)
c = c*e
return -I*b*Log(RemoveContent(c*Cos(u)+e*Sin(u),x))/2 + I*b*Log(RemoveContent(c*Cos(u)-e*Sin(u),x))/2
elif NegativeQ(a):
return RectifyCotangent(u,-a,-b,x)
elif ZeroQ(a-1):
return b*SimplifyAntiderivative(u,x)
elif EvenQ(Denominator(NumericFactor(Together(u)))):
c = Simplify(a - 1)
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) - b*ArcTan(NormalizeLeadTermSigns(denr*Cos(u)*Sin(u)/(numr+denr*Cos(u)**2)))
c = Simplify(a/(1-a))
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) + b*ArcTan(NormalizeLeadTermSigns(denr*Cos(u)*Sin(u)/(numr+denr*Sin(u)**2)))
def Inequality(*args):
f = args[1::2]
e = args[0::2]
r = []
for i in range(0, len(f)):
r.append(f[i](e[i], e[i + 1]))
return all(r)
def Condition(r, c):
# returns r if c is True
if c:
return r
else:
raise NotImplementedError('In Condition()')
def Simp(u, x):
u = replace_pow_exp(u)
return NormalizeSumFactors(SimpHelp(u, x))
def SimpHelp(u, x):
if AtomQ(u):
return u
elif FreeQ(u, x):
v = SmartSimplify(u)
if LeafCount(v) <= LeafCount(u):
return v
return u
elif ProductQ(u):
#m = MatchQ[Rest[u],a_.+n_*Pi+b_.*v_ /; FreeQ[{a,b},x] && Not[FreeQ[v,x]] && EqQ[n^2,1/4]]
#if EqQ(First(u), S(1)/2) and m:
# if
#If[EqQ[First[u],1/2] && MatchQ[Rest[u],a_.+n_*Pi+b_.*v_ /; FreeQ[{a,b},x] && Not[FreeQ[v,x]] && EqQ[n^2,1/4]],
# If[MatchQ[Rest[u],n_*Pi+b_.*v_ /; FreeQ[b,x] && Not[FreeQ[v,x]] && EqQ[n^2,1/4]],
# Map[Function[1/2*#],Rest[u]],
# If[MatchQ[Rest[u],m_*a_.+n_*Pi+p_*b_.*v_ /; FreeQ[{a,b},x] && Not[FreeQ[v,x]] && IntegersQ[m/2,p/2]],
# Map[Function[1/2*#],Rest[u]],
# u]],
v = FreeFactors(u, x)
w = NonfreeFactors(u, x)
v = NumericFactor(v)*SmartSimplify(NonnumericFactors(v)*x**2)/x**2
if ProductQ(w):
w = Mul(*[SimpHelp(i,x) for i in w.args])
else:
w = SimpHelp(w, x)
w = FactorNumericGcd(w)
v = MergeFactors(v, w)
if ProductQ(v):
return Mul(*[SimpFixFactor(i, x) for i in v.args])
return v
elif SumQ(u):
Pi = pi
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x, 0])
n_ = Wild('n', exclude=[x, 0, 0])
pattern = a_ + n_*Pi + b_*x
match = u.match(pattern)
m = False
if match:
if EqQ(match[n_]**3, S(1)/16):
m = True
if m:
return u
elif PolynomialQ(u, x) and Exponent(u, x)<=0:
return SimpHelp(Coefficient(u, x, 0), x)
elif PolynomialQ(u, x) and Exponent(u, x) == 1 and Coefficient(u, x, 0) == 0:
return SimpHelp(Coefficient(u, x, 1), x)*x
v = 0
w = 0
for i in u.args:
if FreeQ(i, x):
v = i + v
else:
w = i + w
v = SmartSimplify(v)
if SumQ(w):
w = Add(*[SimpHelp(i, x) for i in w.args])
else:
w = SimpHelp(w, x)
return v + w
return u.func(*[SimpHelp(i, x) for i in u.args])
def SplitProduct(func, u):
#(* If func[v] is True for a factor v of u, SplitProduct[func,u] returns {v, u/v} where v is the first such factor; else it returns False. *)
if ProductQ(u):
if func(First(u)):
return [First(u), Rest(u)]
lst = SplitProduct(func, Rest(u))
if AtomQ(lst):
return False
return [lst[0], First(u)*lst[1]]
if func(u):
return [u, 1]
return False
def SplitSum(func, u):
# (* If func[v] is nonatomic for a term v of u, SplitSum[func,u] returns {func[v], u-v} where v is the first such term; else it returns False. *)
if SumQ(u):
if Not(AtomQ(func(First(u)))):
return [func(First(u)), Rest(u)]
lst = SplitSum(func, Rest(u))
if AtomQ(lst):
return False
return [lst[0], First(u) + lst[1]]
elif Not(AtomQ(func(u))):
return [func(u), 0]
return False
def SubstFor(*args):
if len(args) == 4:
w, v, u, x = args
# u is a function of v. SubstFor(w,v,u,x) returns w times u with v replaced by x.
return SimplifyIntegrand(w*SubstFor(v, u, x), x)
v, u, x = args
# u is a function of v. SubstFor(v, u, x) returns u with v replaced by x.
if AtomQ(v):
return Subst(u, v, x)
elif Not(EqQ(FreeFactors(v, x), 1)):
return SubstFor(NonfreeFactors(v, x), u, x/FreeFactors(v, x))
elif SinQ(v):
return SubstForTrig(u, x, Sqrt(1 - x**2), v.args[0], x)
elif CosQ(v):
return SubstForTrig(u, Sqrt(1 - x**2), x, v.args[0], x)
elif TanQ(v):
return SubstForTrig(u, x/Sqrt(1 + x**2), 1/Sqrt(1 + x**2), v.args[0], x)
elif CotQ(v):
return SubstForTrig(u, 1/Sqrt(1 + x**2), x/Sqrt(1 + x**2), v.args[0], x)
elif SecQ(v):
return SubstForTrig(u, 1/Sqrt(1 - x**2), 1/x, v.args[0], x)
elif CscQ(v):
return SubstForTrig(u, 1/x, 1/Sqrt(1 - x**2), v.args[0], x)
elif SinhQ(v):
return SubstForHyperbolic(u, x, Sqrt(1 + x**2), v.args[0], x)
elif CoshQ(v):
return SubstForHyperbolic(u, Sqrt( - 1 + x**2), x, v.args[0], x)
elif TanhQ(v):
return SubstForHyperbolic(u, x/Sqrt(1 - x**2), 1/Sqrt(1 - x**2), v.args[0], x)
elif CothQ(v):
return SubstForHyperbolic(u, 1/Sqrt( - 1 + x**2), x/Sqrt( - 1 + x**2), v.args[0], x)
elif SechQ(v):
return SubstForHyperbolic(u, 1/Sqrt( - 1 + x**2), 1/x, v.args[0], x)
elif CschQ(v):
return SubstForHyperbolic(u, 1/x, 1/Sqrt(1 + x**2), v.args[0], x)
else:
return SubstForAux(u, v, x)
def SubstForAux(u, v, x):
# u is a function of v. SubstForAux(u, v, x) returns u with v replaced by x.
if u==v:
return x
elif AtomQ(u):
if PowerQ(v):
if FreeQ(v.exp, x) and ZeroQ(u - v.base):
return x**Simplify(1/v.exp)
return u
elif PowerQ(u):
if FreeQ(u.exp, x):
if ZeroQ(u.base - v):
return x**u.exp
if PowerQ(v):
if FreeQ(v.exp, x) and ZeroQ(u.base - v.base):
return x**Simplify(u.exp/v.exp)
return SubstForAux(u.base, v, x)**u.exp
elif ProductQ(u) and Not(EqQ(FreeFactors(u, x), 1)):
return FreeFactors(u, x)*SubstForAux(NonfreeFactors(u, x), v, x)
elif ProductQ(u) and ProductQ(v):
return SubstForAux(First(u), First(v), x)
return u.func(*[SubstForAux(i, v, x) for i in u.args])
def FresnelS(x):
return fresnels(x)
def FresnelC(x):
return fresnelc(x)
def Erf(x):
return erf(x)
def Erfc(x):
return erfc(x)
def Erfi(x):
return erfi(x)
class Gamma(Function):
@classmethod
def eval(cls,*args):
a = args[0]
if len(args) == 1:
return gamma(a)
else:
b = args[1]
if (NumericQ(a) and NumericQ(b)) or a == 1:
return uppergamma(a, b)
def FunctionOfTrigOfLinearQ(u, x):
# If u is an algebraic function of trig functions of a linear function of x,
# FunctionOfTrigOfLinearQ[u,x] returns True; else it returns False.
if FunctionOfTrig(u, None, x) and AlgebraicTrigFunctionQ(u, x) and FunctionOfLinear(FunctionOfTrig(u, None, x), x):
return True
else:
return False
def ElementaryFunctionQ(u):
# ElementaryExpressionQ[u] returns True if u is a sum, product, or power and all the operands
# are elementary expressions; or if u is a call on a trig, hyperbolic, or inverse function
# and all the arguments are elementary expressions; else it returns False.
if AtomQ(u):
return True
elif SumQ(u) or ProductQ(u) or PowerQ(u) or TrigQ(u) or HyperbolicQ(u) or InverseFunctionQ(u):
for i in u.args:
if not ElementaryFunctionQ(i):
return False
return True
return False
def Complex(a, b):
return a + I*b
def UnsameQ(a, b):
return a != b
@doctest_depends_on(modules=('matchpy',))
def _SimpFixFactor():
replacer = ManyToOneReplacer()
pattern1 = Pattern(UtilityOperator(Pow(Add(Mul(Complex(S(0), c_), WC('a', S(1))), Mul(Complex(S(0), d_), WC('b', S(1)))), WC('p', S(1))), x_), CustomConstraint(lambda p: IntegerQ(p)))
rule1 = ReplacementRule(pattern1, lambda b, c, x, a, p, d : Mul(Pow(I, p), SimpFixFactor(Pow(Add(Mul(a, c), Mul(b, d)), p), x)))
replacer.add(rule1)
pattern2 = Pattern(UtilityOperator(Pow(Add(Mul(Complex(S(0), d_), WC('a', S(1))), Mul(Complex(S(0), e_), WC('b', S(1))), Mul(Complex(S(0), f_), WC('c', S(1)))), WC('p', S(1))), x_), CustomConstraint(lambda p: IntegerQ(p)))
rule2 = ReplacementRule(pattern2, lambda b, c, x, f, a, p, e, d : Mul(Pow(I, p), SimpFixFactor(Pow(Add(Mul(a, d), Mul(b, e), Mul(c, f)), p), x)))
replacer.add(rule2)
pattern3 = Pattern(UtilityOperator(Pow(Add(Mul(WC('a', S(1)), Pow(c_, r_)), Mul(WC('b', S(1)), Pow(x_, WC('n', S(1))))), WC('p', S(1))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, p: IntegersQ(n, p)), CustomConstraint(lambda c: AtomQ(c)), CustomConstraint(lambda r: RationalQ(r)), CustomConstraint(lambda r: Less(r, S(0))))
rule3 = ReplacementRule(pattern3, lambda b, c, r, n, x, a, p : Mul(Pow(c, Mul(r, p)), SimpFixFactor(Pow(Add(a, Mul(Mul(b, Pow(Pow(c, r), S(-1))), Pow(x, n))), p), x)))
replacer.add(rule3)
pattern4 = Pattern(UtilityOperator(Pow(Add(WC('a', S(0)), Mul(WC('b', S(1)), Pow(c_, r_), Pow(x_, WC('n', S(1))))), WC('p', S(1))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, p: IntegersQ(n, p)), CustomConstraint(lambda c: AtomQ(c)), CustomConstraint(lambda r: RationalQ(r)), CustomConstraint(lambda r: Less(r, S(0))))
rule4 = ReplacementRule(pattern4, lambda b, c, r, n, x, a, p : Mul(Pow(c, Mul(r, p)), SimpFixFactor(Pow(Add(Mul(a, Pow(Pow(c, r), S(-1))), Mul(b, Pow(x, n))), p), x)))
replacer.add(rule4)
pattern5 = Pattern(UtilityOperator(Pow(Add(Mul(WC('a', S(1)), Pow(c_, WC('s', S(1)))), Mul(WC('b', S(1)), Pow(c_, WC('r', S(1))), Pow(x_, WC('n', S(1))))), WC('p', S(1))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, p: IntegersQ(n, p)), CustomConstraint(lambda r, s: RationalQ(s, r)), CustomConstraint(lambda r, s: Inequality(S(0), Less, s, LessEqual, r)), CustomConstraint(lambda p, c, s: UnsameQ(Pow(c, Mul(s, p)), S(-1))))
rule5 = ReplacementRule(pattern5, lambda b, c, r, n, x, a, p, s : Mul(Pow(c, Mul(s, p)), SimpFixFactor(Pow(Add(a, Mul(b, Pow(c, Add(r, Mul(S(-1), s))), Pow(x, n))), p), x)))
replacer.add(rule5)
pattern6 = Pattern(UtilityOperator(Pow(Add(Mul(WC('a', S(1)), Pow(c_, WC('s', S(1)))), Mul(WC('b', S(1)), Pow(c_, WC('r', S(1))), Pow(x_, WC('n', S(1))))), WC('p', S(1))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, p: IntegersQ(n, p)), CustomConstraint(lambda r, s: RationalQ(s, r)), CustomConstraint(lambda s, r: Less(S(0), r, s)), CustomConstraint(lambda p, c, r: UnsameQ(Pow(c, Mul(r, p)), S(-1))))
rule6 = ReplacementRule(pattern6, lambda b, c, r, n, x, a, p, s : Mul(Pow(c, Mul(r, p)), SimpFixFactor(Pow(Add(Mul(a, Pow(c, Add(s, Mul(S(-1), r)))), Mul(b, Pow(x, n))), p), x)))
replacer.add(rule6)
return replacer
@doctest_depends_on(modules=('matchpy',))
def SimpFixFactor(expr, x):
r = SimpFixFactor_replacer.replace(UtilityOperator(expr, x))
if isinstance(r, UtilityOperator):
return expr
return r
@doctest_depends_on(modules=('matchpy',))
def _FixSimplify():
Plus = Add
def cons_f1(n):
return OddQ(n)
cons1 = CustomConstraint(cons_f1)
def cons_f2(m):
return RationalQ(m)
cons2 = CustomConstraint(cons_f2)
def cons_f3(n):
return FractionQ(n)
cons3 = CustomConstraint(cons_f3)
def cons_f4(u):
return SqrtNumberSumQ(u)
cons4 = CustomConstraint(cons_f4)
def cons_f5(v):
return SqrtNumberSumQ(v)
cons5 = CustomConstraint(cons_f5)
def cons_f6(u):
return PositiveQ(u)
cons6 = CustomConstraint(cons_f6)
def cons_f7(v):
return PositiveQ(v)
cons7 = CustomConstraint(cons_f7)
def cons_f8(v):
return SqrtNumberSumQ(S(1)/v)
cons8 = CustomConstraint(cons_f8)
def cons_f9(m):
return IntegerQ(m)
cons9 = CustomConstraint(cons_f9)
def cons_f10(u):
return NegativeQ(u)
cons10 = CustomConstraint(cons_f10)
def cons_f11(n, m, a, b):
return RationalQ(a, b, m, n)
cons11 = CustomConstraint(cons_f11)
def cons_f12(a):
return Greater(a, S(0))
cons12 = CustomConstraint(cons_f12)
def cons_f13(b):
return Greater(b, S(0))
cons13 = CustomConstraint(cons_f13)
def cons_f14(p):
return PositiveIntegerQ(p)
cons14 = CustomConstraint(cons_f14)
def cons_f15(p):
return IntegerQ(p)
cons15 = CustomConstraint(cons_f15)
def cons_f16(p, n):
return Greater(-n + p, S(0))
cons16 = CustomConstraint(cons_f16)
def cons_f17(a, b):
return SameQ(a + b, S(0))
cons17 = CustomConstraint(cons_f17)
def cons_f18(n):
return Not(IntegerQ(n))
cons18 = CustomConstraint(cons_f18)
def cons_f19(c, a, b, d):
return ZeroQ(-a*d + b*c)
cons19 = CustomConstraint(cons_f19)
def cons_f20(a):
return Not(RationalQ(a))
cons20 = CustomConstraint(cons_f20)
def cons_f21(t):
return IntegerQ(t)
cons21 = CustomConstraint(cons_f21)
def cons_f22(n, m):
return RationalQ(m, n)
cons22 = CustomConstraint(cons_f22)
def cons_f23(n, m):
return Inequality(S(0), Less, m, LessEqual, n)
cons23 = CustomConstraint(cons_f23)
def cons_f24(p, n, m):
return RationalQ(m, n, p)
cons24 = CustomConstraint(cons_f24)
def cons_f25(p, n, m):
return Inequality(S(0), Less, m, LessEqual, n, LessEqual, p)
cons25 = CustomConstraint(cons_f25)
def cons_f26(p, n, m, q):
return Inequality(S(0), Less, m, LessEqual, n, LessEqual, p, LessEqual, q)
cons26 = CustomConstraint(cons_f26)
def cons_f27(w):
return Not(RationalQ(w))
cons27 = CustomConstraint(cons_f27)
def cons_f28(n):
return Less(n, S(0))
cons28 = CustomConstraint(cons_f28)
def cons_f29(n, w, v):
return ZeroQ(v + w**(-n))
cons29 = CustomConstraint(cons_f29)
def cons_f30(n):
return IntegerQ(n)
cons30 = CustomConstraint(cons_f30)
def cons_f31(w, v):
return ZeroQ(v + w)
cons31 = CustomConstraint(cons_f31)
def cons_f32(p, n):
return IntegerQ(n/p)
cons32 = CustomConstraint(cons_f32)
def cons_f33(w, v):
return ZeroQ(v - w)
cons33 = CustomConstraint(cons_f33)
def cons_f34(p, n):
return IntegersQ(n, n/p)
cons34 = CustomConstraint(cons_f34)
def cons_f35(a):
return AtomQ(a)
cons35 = CustomConstraint(cons_f35)
def cons_f36(b):
return AtomQ(b)
cons36 = CustomConstraint(cons_f36)
pattern1 = Pattern(UtilityOperator((w_ + Complex(S(0), b_)*WC('v', S(1)))**WC('n', S(1))*Complex(S(0), a_)*WC('u', S(1))), cons1)
def replacement1(n, u, w, v, a, b):
return (S(-1))**(n/S(2) + S(1)/2)*a*u*FixSimplify((b*v - w*Complex(S(0), S(1)))**n)
rule1 = ReplacementRule(pattern1, replacement1)
def With2(m, n, u, w, v):
z = u**(m/GCD(m, n))*v**(n/GCD(m, n))
if Or(AbsurdNumberQ(z), SqrtNumberSumQ(z)):
return True
return False
pattern2 = Pattern(UtilityOperator(u_**WC('m', S(1))*v_**n_*WC('w', S(1))), cons2, cons3, cons4, cons5, cons6, cons7, CustomConstraint(With2))
def replacement2(m, n, u, w, v):
z = u**(m/GCD(m, n))*v**(n/GCD(m, n))
return FixSimplify(w*z**GCD(m, n))
rule2 = ReplacementRule(pattern2, replacement2)
def With3(m, n, u, w, v):
z = u**(m/GCD(m, -n))*v**(n/GCD(m, -n))
if Or(AbsurdNumberQ(z), SqrtNumberSumQ(z)):
return True
return False
pattern3 = Pattern(UtilityOperator(u_**WC('m', S(1))*v_**n_*WC('w', S(1))), cons2, cons3, cons4, cons8, cons6, cons7, CustomConstraint(With3))
def replacement3(m, n, u, w, v):
z = u**(m/GCD(m, -n))*v**(n/GCD(m, -n))
return FixSimplify(w*z**GCD(m, -n))
rule3 = ReplacementRule(pattern3, replacement3)
def With4(m, n, u, w, v):
z = v**(n/GCD(m, n))*(-u)**(m/GCD(m, n))
if Or(AbsurdNumberQ(z), SqrtNumberSumQ(z)):
return True
return False
pattern4 = Pattern(UtilityOperator(u_**WC('m', S(1))*v_**n_*WC('w', S(1))), cons9, cons3, cons4, cons5, cons10, cons7, CustomConstraint(With4))
def replacement4(m, n, u, w, v):
z = v**(n/GCD(m, n))*(-u)**(m/GCD(m, n))
return FixSimplify(-w*z**GCD(m, n))
rule4 = ReplacementRule(pattern4, replacement4)
def With5(m, n, u, w, v):
z = v**(n/GCD(m, -n))*(-u)**(m/GCD(m, -n))
if Or(AbsurdNumberQ(z), SqrtNumberSumQ(z)):
return True
return False
pattern5 = Pattern(UtilityOperator(u_**WC('m', S(1))*v_**n_*WC('w', S(1))), cons9, cons3, cons4, cons8, cons10, cons7, CustomConstraint(With5))
def replacement5(m, n, u, w, v):
z = v**(n/GCD(m, -n))*(-u)**(m/GCD(m, -n))
return FixSimplify(-w*z**GCD(m, -n))
rule5 = ReplacementRule(pattern5, replacement5)
def With6(p, m, n, u, w, v, a, b):
c = a**(m/p)*b**n
if RationalQ(c):
return True
return False
pattern6 = Pattern(UtilityOperator(a_**m_*(b_**n_*WC('v', S(1)) + u_)**WC('p', S(1))*WC('w', S(1))), cons11, cons12, cons13, cons14, CustomConstraint(With6))
def replacement6(p, m, n, u, w, v, a, b):
c = a**(m/p)*b**n
return FixSimplify(w*(a**(m/p)*u + c*v)**p)
rule6 = ReplacementRule(pattern6, replacement6)
pattern7 = Pattern(UtilityOperator(a_**WC('m', S(1))*(a_**n_*WC('u', S(1)) + b_**WC('p', S(1))*WC('v', S(1)))*WC('w', S(1))), cons2, cons3, cons15, cons16, cons17)
def replacement7(p, m, n, u, w, v, a, b):
return FixSimplify(a**(m + n)*w*((S(-1))**p*a**(-n + p)*v + u))
rule7 = ReplacementRule(pattern7, replacement7)
def With8(m, d, n, w, c, a, b):
q = b/d
if FreeQ(q, Plus):
return True
return False
pattern8 = Pattern(UtilityOperator((a_ + b_)**WC('m', S(1))*(c_ + d_)**n_*WC('w', S(1))), cons9, cons18, cons19, CustomConstraint(With8))
def replacement8(m, d, n, w, c, a, b):
q = b/d
return FixSimplify(q**m*w*(c + d)**(m + n))
rule8 = ReplacementRule(pattern8, replacement8)
pattern9 = Pattern(UtilityOperator((a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)))**WC('t', S(1))*WC('w', S(1))), cons20, cons21, cons22, cons23)
def replacement9(m, n, u, w, v, a, t):
return FixSimplify(a**(m*t)*w*(a**(-m + n)*v + u)**t)
rule9 = ReplacementRule(pattern9, replacement9)
pattern10 = Pattern(UtilityOperator((a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)) + a_**WC('p', S(1))*WC('z', S(1)))**WC('t', S(1))*WC('w', S(1))), cons20, cons21, cons24, cons25)
def replacement10(p, m, n, u, w, v, a, z, t):
return FixSimplify(a**(m*t)*w*(a**(-m + n)*v + a**(-m + p)*z + u)**t)
rule10 = ReplacementRule(pattern10, replacement10)
pattern11 = Pattern(UtilityOperator((a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)) + a_**WC('p', S(1))*WC('z', S(1)) + a_**WC('q', S(1))*WC('y', S(1)))**WC('t', S(1))*WC('w', S(1))), cons20, cons21, cons24, cons26)
def replacement11(p, m, n, u, q, w, v, a, z, y, t):
return FixSimplify(a**(m*t)*w*(a**(-m + n)*v + a**(-m + p)*z + a**(-m + q)*y + u)**t)
rule11 = ReplacementRule(pattern11, replacement11)
pattern12 = Pattern(UtilityOperator((sqrt(v_)*WC('b', S(1)) + sqrt(v_)*WC('c', S(1)) + sqrt(v_)*WC('d', S(1)) + sqrt(v_)*WC('a', S(1)) + WC('u', S(0)))*WC('w', S(1))))
def replacement12(d, u, w, v, c, a, b):
return FixSimplify(w*(u + sqrt(v)*FixSimplify(a + b + c + d)))
rule12 = ReplacementRule(pattern12, replacement12)
pattern13 = Pattern(UtilityOperator((sqrt(v_)*WC('b', S(1)) + sqrt(v_)*WC('c', S(1)) + sqrt(v_)*WC('a', S(1)) + WC('u', S(0)))*WC('w', S(1))))
def replacement13(u, w, v, c, a, b):
return FixSimplify(w*(u + sqrt(v)*FixSimplify(a + b + c)))
rule13 = ReplacementRule(pattern13, replacement13)
pattern14 = Pattern(UtilityOperator((sqrt(v_)*WC('b', S(1)) + sqrt(v_)*WC('a', S(1)) + WC('u', S(0)))*WC('w', S(1))))
def replacement14(u, w, v, a, b):
return FixSimplify(w*(u + sqrt(v)*FixSimplify(a + b)))
rule14 = ReplacementRule(pattern14, replacement14)
pattern15 = Pattern(UtilityOperator(v_**m_*w_**n_*WC('u', S(1))), cons2, cons27, cons3, cons28, cons29)
def replacement15(m, n, u, w, v):
return -FixSimplify(u*v**(m + S(-1)))
rule15 = ReplacementRule(pattern15, replacement15)
pattern16 = Pattern(UtilityOperator(v_**m_*w_**WC('n', S(1))*WC('u', S(1))), cons2, cons27, cons30, cons31)
def replacement16(m, n, u, w, v):
return (S(-1))**n*FixSimplify(u*v**(m + n))
rule16 = ReplacementRule(pattern16, replacement16)
pattern17 = Pattern(UtilityOperator(w_**WC('n', S(1))*(-v_**WC('p', S(1)))**m_*WC('u', S(1))), cons2, cons27, cons32, cons33)
def replacement17(p, m, n, u, w, v):
return (S(-1))**(n/p)*FixSimplify(u*(-v**p)**(m + n/p))
rule17 = ReplacementRule(pattern17, replacement17)
pattern18 = Pattern(UtilityOperator(w_**WC('n', S(1))*(-v_**WC('p', S(1)))**m_*WC('u', S(1))), cons2, cons27, cons34, cons31)
def replacement18(p, m, n, u, w, v):
return (S(-1))**(n + n/p)*FixSimplify(u*(-v**p)**(m + n/p))
rule18 = ReplacementRule(pattern18, replacement18)
pattern19 = Pattern(UtilityOperator((a_ - b_)**WC('m', S(1))*(a_ + b_)**WC('m', S(1))*WC('u', S(1))), cons9, cons35, cons36)
def replacement19(m, u, a, b):
return u*(a**S(2) - b**S(2))**m
rule19 = ReplacementRule(pattern19, replacement19)
pattern20 = Pattern(UtilityOperator((S(729)*c - e*(-S(20)*e + S(540)))**WC('m', S(1))*WC('u', S(1))), cons2)
def replacement20(m, u):
return u*(a*e**S(2) - b*d*e + c*d**S(2))**m
rule20 = ReplacementRule(pattern20, replacement20)
pattern21 = Pattern(UtilityOperator((S(729)*c + e*(S(20)*e + S(-540)))**WC('m', S(1))*WC('u', S(1))), cons2)
def replacement21(m, u):
return u*(a*e**S(2) - b*d*e + c*d**S(2))**m
rule21 = ReplacementRule(pattern21, replacement21)
pattern22 = Pattern(UtilityOperator(u_))
def replacement22(u):
return u
rule22 = ReplacementRule(pattern22, replacement22)
return [rule1, rule2, rule3, rule4, rule5, rule6, rule7, rule8, rule9, rule10, rule11, rule12, rule13, rule14, rule15, rule16, rule17, rule18, rule19, rule20, rule21, rule22, ]
@doctest_depends_on(modules=('matchpy',))
def FixSimplify(expr):
if isinstance(expr, (list, tuple, TupleArg)):
return [replace_all(UtilityOperator(i), FixSimplify_rules) for i in expr]
return replace_all(UtilityOperator(expr), FixSimplify_rules)
@doctest_depends_on(modules=('matchpy',))
def _SimplifyAntiderivativeSum():
replacer = ManyToOneReplacer()
pattern1 = Pattern(UtilityOperator(Add(Mul(Log(Add(a_, Mul(WC('b', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('A', S(1))), Mul(Log(Cos(u_)), WC('B', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A, n: ZeroQ(Add(Mul(n, A), Mul(S(1), B)))))
rule1 = ReplacementRule(pattern1, lambda n, x, v, b, B, A, u, a : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(a, Pow(Cos(u), n)), Mul(b, Pow(Sin(u), n))), x)))))
replacer.add(rule1)
pattern2 = Pattern(UtilityOperator(Add(Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('b', S(1))), a_)), WC('A', S(1))), Mul(Log(Sin(u_)), WC('B', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A, n: ZeroQ(Add(Mul(n, A), Mul(S(1), B)))))
rule2 = ReplacementRule(pattern2, lambda n, x, v, b, B, A, a, u : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(a, Pow(Sin(u), n)), Mul(b, Pow(Cos(u), n))), x)))))
replacer.add(rule2)
pattern3 = Pattern(UtilityOperator(Add(Mul(Log(Add(a_, Mul(WC('b', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('A', S(1))), Mul(Log(Add(c_, Mul(WC('d', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('B', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda d, x: FreeQ(d, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A: ZeroQ(Add(A, B))))
rule3 = ReplacementRule(pattern3, lambda n, x, v, b, A, B, u, c, d, a : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(a, Pow(Cos(u), n)), Mul(b, Pow(Sin(u), n))), x))), Mul(B, Log(RemoveContent(Add(Mul(c, Pow(Cos(u), n)), Mul(d, Pow(Sin(u), n))), x)))))
replacer.add(rule3)
pattern4 = Pattern(UtilityOperator(Add(Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('b', S(1))), a_)), WC('A', S(1))), Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('d', S(1))), c_)), WC('B', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda d, x: FreeQ(d, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A: ZeroQ(Add(A, B))))
rule4 = ReplacementRule(pattern4, lambda n, x, v, b, A, B, c, a, d, u : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(b, Pow(Cos(u), n)), Mul(a, Pow(Sin(u), n))), x))), Mul(B, Log(RemoveContent(Add(Mul(d, Pow(Cos(u), n)), Mul(c, Pow(Sin(u), n))), x)))))
replacer.add(rule4)
pattern5 = Pattern(UtilityOperator(Add(Mul(Log(Add(a_, Mul(WC('b', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('A', S(1))), Mul(Log(Add(c_, Mul(WC('d', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('B', S(1))), Mul(Log(Add(e_, Mul(WC('f', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('C', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda d, x: FreeQ(d, x)), CustomConstraint(lambda e, x: FreeQ(e, x)), CustomConstraint(lambda f, x: FreeQ(f, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda C, x: FreeQ(C, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A, C: ZeroQ(Add(A, B, C))))
rule5 = ReplacementRule(pattern5, lambda n, e, x, v, b, A, B, u, c, f, d, a, C : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(a, Pow(Cos(u), n)), Mul(b, Pow(Sin(u), n))), x))), Mul(B, Log(RemoveContent(Add(Mul(c, Pow(Cos(u), n)), Mul(d, Pow(Sin(u), n))), x))), Mul(C, Log(RemoveContent(Add(Mul(e, Pow(Cos(u), n)), Mul(f, Pow(Sin(u), n))), x)))))
replacer.add(rule5)
pattern6 = Pattern(UtilityOperator(Add(Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('b', S(1))), a_)), WC('A', S(1))), Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('d', S(1))), c_)), WC('B', S(1))), Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('f', S(1))), e_)), WC('C', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda d, x: FreeQ(d, x)), CustomConstraint(lambda e, x: FreeQ(e, x)), CustomConstraint(lambda f, x: FreeQ(f, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda C, x: FreeQ(C, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A, C: ZeroQ(Add(A, B, C))))
rule6 = ReplacementRule(pattern6, lambda n, e, x, v, b, A, B, c, a, f, d, u, C : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(b, Pow(Cos(u), n)), Mul(a, Pow(Sin(u), n))), x))), Mul(B, Log(RemoveContent(Add(Mul(d, Pow(Cos(u), n)), Mul(c, Pow(Sin(u), n))), x))), Mul(C, Log(RemoveContent(Add(Mul(f, Pow(Cos(u), n)), Mul(e, Pow(Sin(u), n))), x)))))
replacer.add(rule6)
return replacer
@doctest_depends_on(modules=('matchpy',))
def SimplifyAntiderivativeSum(expr, x):
r = SimplifyAntiderivativeSum_replacer.replace(UtilityOperator(expr, x))
if isinstance(r, UtilityOperator):
return expr
return r
@doctest_depends_on(modules=('matchpy',))
def _SimplifyAntiderivative():
replacer = ManyToOneReplacer()
pattern2 = Pattern(UtilityOperator(Log(Mul(c_, u_)), x_), CustomConstraint(lambda c, x: FreeQ(c, x)))
rule2 = ReplacementRule(pattern2, lambda x, c, u : SimplifyAntiderivative(Log(u), x))
replacer.add(rule2)
pattern3 = Pattern(UtilityOperator(Log(Pow(u_, n_)), x_), CustomConstraint(lambda n, x: FreeQ(n, x)))
rule3 = ReplacementRule(pattern3, lambda x, n, u : Mul(n, SimplifyAntiderivative(Log(u), x)))
replacer.add(rule3)
pattern7 = Pattern(UtilityOperator(Log(Pow(f_, u_)), x_), CustomConstraint(lambda f, x: FreeQ(f, x)))
rule7 = ReplacementRule(pattern7, lambda x, f, u : Mul(Log(f), SimplifyAntiderivative(u, x)))
replacer.add(rule7)
pattern8 = Pattern(UtilityOperator(Log(Add(a_, Mul(WC('b', S(1)), Tan(u_)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda b, a: ZeroQ(Add(Pow(a, S(2)), Pow(b, S(2))))))
rule8 = ReplacementRule(pattern8, lambda x, b, u, a : Add(Mul(Mul(b, Pow(a, S(1))), SimplifyAntiderivative(u, x)), Mul(S(1), SimplifyAntiderivative(Log(Cos(u)), x))))
replacer.add(rule8)
pattern9 = Pattern(UtilityOperator(Log(Add(Mul(Cot(u_), WC('b', S(1))), a_)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda b, a: ZeroQ(Add(Pow(a, S(2)), Pow(b, S(2))))))
rule9 = ReplacementRule(pattern9, lambda x, b, u, a : Add(Mul(Mul(Mul(S(1), b), Pow(a, S(1))), SimplifyAntiderivative(u, x)), Mul(S(1), SimplifyAntiderivative(Log(Sin(u)), x))))
replacer.add(rule9)
pattern10 = Pattern(UtilityOperator(ArcTan(Mul(WC('a', S(1)), Tan(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule10 = ReplacementRule(pattern10, lambda x, u, a : RectifyTangent(u, a, S(1), x))
replacer.add(rule10)
pattern11 = Pattern(UtilityOperator(ArcCot(Mul(WC('a', S(1)), Tan(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule11 = ReplacementRule(pattern11, lambda x, u, a : RectifyTangent(u, a, S(1), x))
replacer.add(rule11)
pattern12 = Pattern(UtilityOperator(ArcCot(Mul(WC('a', S(1)), Tanh(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule12 = ReplacementRule(pattern12, lambda x, u, a : Mul(S(1), SimplifyAntiderivative(ArcTan(Mul(a, Tanh(u))), x)))
replacer.add(rule12)
pattern13 = Pattern(UtilityOperator(ArcTanh(Mul(WC('a', S(1)), Tan(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule13 = ReplacementRule(pattern13, lambda x, u, a : RectifyTangent(u, Mul(I, a), Mul(S(1), I), x))
replacer.add(rule13)
pattern14 = Pattern(UtilityOperator(ArcCoth(Mul(WC('a', S(1)), Tan(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule14 = ReplacementRule(pattern14, lambda x, u, a : RectifyTangent(u, Mul(I, a), Mul(S(1), I), x))
replacer.add(rule14)
pattern15 = Pattern(UtilityOperator(ArcTanh(Tanh(u_)), x_))
rule15 = ReplacementRule(pattern15, lambda x, u : SimplifyAntiderivative(u, x))
replacer.add(rule15)
pattern16 = Pattern(UtilityOperator(ArcCoth(Tanh(u_)), x_))
rule16 = ReplacementRule(pattern16, lambda x, u : SimplifyAntiderivative(u, x))
replacer.add(rule16)
pattern17 = Pattern(UtilityOperator(ArcCot(Mul(Cot(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule17 = ReplacementRule(pattern17, lambda x, u, a : RectifyCotangent(u, a, S(1), x))
replacer.add(rule17)
pattern18 = Pattern(UtilityOperator(ArcTan(Mul(Cot(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule18 = ReplacementRule(pattern18, lambda x, u, a : RectifyCotangent(u, a, S(1), x))
replacer.add(rule18)
pattern19 = Pattern(UtilityOperator(ArcTan(Mul(Coth(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule19 = ReplacementRule(pattern19, lambda x, u, a : Mul(S(1), SimplifyAntiderivative(ArcTan(Mul(Tanh(u), Pow(a, S(1)))), x)))
replacer.add(rule19)
pattern20 = Pattern(UtilityOperator(ArcCoth(Mul(Cot(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule20 = ReplacementRule(pattern20, lambda x, u, a : RectifyCotangent(u, Mul(I, a), I, x))
replacer.add(rule20)
pattern21 = Pattern(UtilityOperator(ArcTanh(Mul(Cot(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule21 = ReplacementRule(pattern21, lambda x, u, a : RectifyCotangent(u, Mul(I, a), I, x))
replacer.add(rule21)
pattern22 = Pattern(UtilityOperator(ArcCoth(Coth(u_)), x_))
rule22 = ReplacementRule(pattern22, lambda x, u : SimplifyAntiderivative(u, x))
replacer.add(rule22)
pattern23 = Pattern(UtilityOperator(ArcTanh(Mul(Coth(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule23 = ReplacementRule(pattern23, lambda x, u, a : SimplifyAntiderivative(ArcTanh(Mul(Tanh(u), Pow(a, S(1)))), x))
replacer.add(rule23)
pattern24 = Pattern(UtilityOperator(ArcTanh(Coth(u_)), x_))
rule24 = ReplacementRule(pattern24, lambda x, u : SimplifyAntiderivative(u, x))
replacer.add(rule24)
pattern25 = Pattern(UtilityOperator(ArcTan(Mul(WC('c', S(1)), Add(a_, Mul(WC('b', S(1)), Tan(u_))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda c, a: PositiveQ(Mul(Pow(a, S(2)), Pow(c, S(2))))), CustomConstraint(lambda c, b: PositiveQ(Mul(Pow(b, S(2)), Pow(c, S(2))))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule25 = ReplacementRule(pattern25, lambda x, a, b, u, c : RectifyTangent(u, Mul(a, c), Mul(b, c), S(1), x))
replacer.add(rule25)
pattern26 = Pattern(UtilityOperator(ArcTanh(Mul(WC('c', S(1)), Add(a_, Mul(WC('b', S(1)), Tan(u_))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda c, a: PositiveQ(Mul(Pow(a, S(2)), Pow(c, S(2))))), CustomConstraint(lambda c, b: PositiveQ(Mul(Pow(b, S(2)), Pow(c, S(2))))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule26 = ReplacementRule(pattern26, lambda x, a, b, u, c : RectifyTangent(u, Mul(I, a, c), Mul(I, b, c), Mul(S(1), I), x))
replacer.add(rule26)
pattern27 = Pattern(UtilityOperator(ArcTan(Mul(WC('c', S(1)), Add(Mul(Cot(u_), WC('b', S(1))), a_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda c, a: PositiveQ(Mul(Pow(a, S(2)), Pow(c, S(2))))), CustomConstraint(lambda c, b: PositiveQ(Mul(Pow(b, S(2)), Pow(c, S(2))))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule27 = ReplacementRule(pattern27, lambda x, a, b, u, c : RectifyCotangent(u, Mul(a, c), Mul(b, c), S(1), x))
replacer.add(rule27)
pattern28 = Pattern(UtilityOperator(ArcTanh(Mul(WC('c', S(1)), Add(Mul(Cot(u_), WC('b', S(1))), a_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda c, a: PositiveQ(Mul(Pow(a, S(2)), Pow(c, S(2))))), CustomConstraint(lambda c, b: PositiveQ(Mul(Pow(b, S(2)), Pow(c, S(2))))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule28 = ReplacementRule(pattern28, lambda x, a, b, u, c : RectifyCotangent(u, Mul(I, a, c), Mul(I, b, c), Mul(S(1), I), x))
replacer.add(rule28)
pattern29 = Pattern(UtilityOperator(ArcTan(Add(WC('a', S(0)), Mul(WC('b', S(1)), Tan(u_)), Mul(WC('c', S(1)), Pow(Tan(u_), S(2))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule29 = ReplacementRule(pattern29, lambda x, a, b, u, c : If(EvenQ(Denominator(NumericFactor(Together(u)))), ArcTan(NormalizeTogether(Mul(Add(a, c, S(1), Mul(Add(a, Mul(S(1), c), S(1)), Cos(Mul(S(2), u))), Mul(b, Sin(Mul(S(2), u)))), Pow(Add(a, c, S(1), Mul(Add(a, Mul(S(1), c), S(1)), Cos(Mul(S(2), u))), Mul(b, Sin(Mul(S(2), u)))), S(1))))), ArcTan(NormalizeTogether(Mul(Add(c, Mul(Add(a, Mul(S(1), c), S(1)), Pow(Cos(u), S(2))), Mul(b, Cos(u), Sin(u))), Pow(Add(c, Mul(Add(a, Mul(S(1), c), S(1)), Pow(Cos(u), S(2))), Mul(b, Cos(u), Sin(u))), S(1)))))))
replacer.add(rule29)
pattern30 = Pattern(UtilityOperator(ArcTan(Add(WC('a', S(0)), Mul(WC('b', S(1)), Add(WC('d', S(0)), Mul(WC('e', S(1)), Tan(u_)))), Mul(WC('c', S(1)), Pow(Add(WC('f', S(0)), Mul(WC('g', S(1)), Tan(u_))), S(2))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule30 = ReplacementRule(pattern30, lambda x, d, a, e, f, b, u, c, g : SimplifyAntiderivative(ArcTan(Add(a, Mul(b, d), Mul(c, Pow(f, S(2))), Mul(Add(Mul(b, e), Mul(S(2), c, f, g)), Tan(u)), Mul(c, Pow(g, S(2)), Pow(Tan(u), S(2))))), x))
replacer.add(rule30)
pattern31 = Pattern(UtilityOperator(ArcTan(Add(WC('a', S(0)), Mul(WC('c', S(1)), Pow(Tan(u_), S(2))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule31 = ReplacementRule(pattern31, lambda x, c, u, a : If(EvenQ(Denominator(NumericFactor(Together(u)))), ArcTan(NormalizeTogether(Mul(Add(a, c, S(1), Mul(Add(a, Mul(S(1), c), S(1)), Cos(Mul(S(2), u)))), Pow(Add(a, c, S(1), Mul(Add(a, Mul(S(1), c), S(1)), Cos(Mul(S(2), u)))), S(1))))), ArcTan(NormalizeTogether(Mul(Add(c, Mul(Add(a, Mul(S(1), c), S(1)), Pow(Cos(u), S(2)))), Pow(Add(c, Mul(Add(a, Mul(S(1), c), S(1)), Pow(Cos(u), S(2)))), S(1)))))))
replacer.add(rule31)
pattern32 = Pattern(UtilityOperator(ArcTan(Add(WC('a', S(0)), Mul(WC('c', S(1)), Pow(Add(WC('f', S(0)), Mul(WC('g', S(1)), Tan(u_))), S(2))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule32 = ReplacementRule(pattern32, lambda x, a, f, u, c, g : SimplifyAntiderivative(ArcTan(Add(a, Mul(c, Pow(f, S(2))), Mul(Mul(S(2), c, f, g), Tan(u)), Mul(c, Pow(g, S(2)), Pow(Tan(u), S(2))))), x))
replacer.add(rule32)
return replacer
@doctest_depends_on(modules=('matchpy',))
def SimplifyAntiderivative(expr, x):
r = SimplifyAntiderivative_replacer.replace(UtilityOperator(expr, x))
if isinstance(r, UtilityOperator):
if ProductQ(expr):
u, c = S(1), S(1)
for i in expr.args:
if FreeQ(i, x):
c *= i
else:
u *= i
if FreeQ(c, x) and c != S(1):
v = SimplifyAntiderivative(u, x)
if SumQ(v) and NonsumQ(u):
return Add(*[c*i for i in v.args])
return c*v
elif LogQ(expr):
F = expr.args[0]
if MemberQ([cot, sec, csc, coth, sech, csch], Head(F)):
return -SimplifyAntiderivative(Log(1/F), x)
if MemberQ([Log, atan, acot], Head(expr)):
F = Head(expr)
G = expr.args[0]
if MemberQ([cot, sec, csc, coth, sech, csch], Head(G)):
return -SimplifyAntiderivative(F(1/G), x)
if MemberQ([atanh, acoth], Head(expr)):
F = Head(expr)
G = expr.args[0]
if MemberQ([cot, sec, csc, coth, sech, csch], Head(G)):
return SimplifyAntiderivative(F(1/G), x)
u = expr
if FreeQ(u, x):
return S(0)
elif LogQ(u):
return Log(RemoveContent(u.args[0], x))
elif SumQ(u):
return SimplifyAntiderivativeSum(Add(*[SimplifyAntiderivative(i, x) for i in u.args]), x)
return u
else:
return r
@doctest_depends_on(modules=('matchpy',))
def _TrigSimplifyAux():
replacer = ManyToOneReplacer()
pattern1 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(WC('a', S(1)), Pow(v_, WC('m', S(1)))), Mul(WC('b', S(1)), Pow(v_, WC('n', S(1))))), p_))), CustomConstraint(lambda v: InertTrigQ(v)), CustomConstraint(lambda p: IntegerQ(p)), CustomConstraint(lambda n, m: RationalQ(m, n)), CustomConstraint(lambda n, m: Less(m, n)))
rule1 = ReplacementRule(pattern1, lambda n, a, p, m, u, v, b : Mul(u, Pow(v, Mul(m, p)), Pow(TrigSimplifyAux(Add(a, Mul(b, Pow(v, Add(n, Mul(S(-1), m)))))), p)))
replacer.add(rule1)
pattern2 = Pattern(UtilityOperator(Add(Mul(Pow(cos(u_), S('2')), WC('a', S(1))), WC('v', S(0)), Mul(WC('b', S(1)), Pow(sin(u_), S('2'))))), CustomConstraint(lambda b, a: SameQ(a, b)))
rule2 = ReplacementRule(pattern2, lambda u, v, b, a : Add(a, v))
replacer.add(rule2)
pattern3 = Pattern(UtilityOperator(Add(WC('v', S(0)), Mul(WC('a', S(1)), Pow(sec(u_), S('2'))), Mul(WC('b', S(1)), Pow(tan(u_), S('2'))))), CustomConstraint(lambda b, a: SameQ(a, Mul(S(-1), b))))
rule3 = ReplacementRule(pattern3, lambda u, v, b, a : Add(a, v))
replacer.add(rule3)
pattern4 = Pattern(UtilityOperator(Add(Mul(Pow(csc(u_), S('2')), WC('a', S(1))), Mul(Pow(cot(u_), S('2')), WC('b', S(1))), WC('v', S(0)))), CustomConstraint(lambda b, a: SameQ(a, Mul(S(-1), b))))
rule4 = ReplacementRule(pattern4, lambda u, v, b, a : Add(a, v))
replacer.add(rule4)
pattern5 = Pattern(UtilityOperator(Pow(Add(Mul(Pow(cos(u_), S('2')), WC('a', S(1))), WC('v', S(0)), Mul(WC('b', S(1)), Pow(sin(u_), S('2')))), n_)))
rule5 = ReplacementRule(pattern5, lambda n, a, u, v, b : Pow(Add(Mul(Add(b, Mul(S(-1), a)), Pow(Sin(u), S('2'))), a, v), n))
replacer.add(rule5)
pattern6 = Pattern(UtilityOperator(Add(WC('w', S(0)), u_, Mul(WC('v', S(1)), Pow(sin(z_), S('2'))))), CustomConstraint(lambda u, v: SameQ(u, Mul(S(-1), v))))
rule6 = ReplacementRule(pattern6, lambda u, w, z, v : Add(Mul(u, Pow(Cos(z), S('2'))), w))
replacer.add(rule6)
pattern7 = Pattern(UtilityOperator(Add(Mul(Pow(cos(z_), S('2')), WC('v', S(1))), WC('w', S(0)), u_)), CustomConstraint(lambda u, v: SameQ(u, Mul(S(-1), v))))
rule7 = ReplacementRule(pattern7, lambda z, w, v, u : Add(Mul(u, Pow(Sin(z), S('2'))), w))
replacer.add(rule7)
pattern8 = Pattern(UtilityOperator(Add(WC('w', S(0)), u_, Mul(WC('v', S(1)), Pow(tan(z_), S('2'))))), CustomConstraint(lambda u, v: SameQ(u, v)))
rule8 = ReplacementRule(pattern8, lambda u, w, z, v : Add(Mul(u, Pow(Sec(z), S('2'))), w))
replacer.add(rule8)
pattern9 = Pattern(UtilityOperator(Add(Mul(Pow(cot(z_), S('2')), WC('v', S(1))), WC('w', S(0)), u_)), CustomConstraint(lambda u, v: SameQ(u, v)))
rule9 = ReplacementRule(pattern9, lambda z, w, v, u : Add(Mul(u, Pow(Csc(z), S('2'))), w))
replacer.add(rule9)
pattern10 = Pattern(UtilityOperator(Add(WC('w', S(0)), u_, Mul(WC('v', S(1)), Pow(sec(z_), S('2'))))), CustomConstraint(lambda u, v: SameQ(u, Mul(S(-1), v))))
rule10 = ReplacementRule(pattern10, lambda u, w, z, v : Add(Mul(v, Pow(Tan(z), S('2'))), w))
replacer.add(rule10)
pattern11 = Pattern(UtilityOperator(Add(Mul(Pow(csc(z_), S('2')), WC('v', S(1))), WC('w', S(0)), u_)), CustomConstraint(lambda u, v: SameQ(u, Mul(S(-1), v))))
rule11 = ReplacementRule(pattern11, lambda z, w, v, u : Add(Mul(v, Pow(Cot(z), S('2'))), w))
replacer.add(rule11)
pattern12 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(cos(v_), WC('b', S(1))), a_), S(-1)), Pow(sin(v_), S('2')))), CustomConstraint(lambda b, a: ZeroQ(Add(Pow(a, S('2')), Mul(S(-1), Pow(b, S('2')))))))
rule12 = ReplacementRule(pattern12, lambda u, v, b, a : Mul(u, Add(Mul(S(1), Pow(a, S(-1))), Mul(S(-1), Mul(Cos(v), Pow(b, S(-1)))))))
replacer.add(rule12)
pattern13 = Pattern(UtilityOperator(Mul(Pow(cos(v_), S('2')), WC('u', S(1)), Pow(Add(a_, Mul(WC('b', S(1)), sin(v_))), S(-1)))), CustomConstraint(lambda b, a: ZeroQ(Add(Pow(a, S('2')), Mul(S(-1), Pow(b, S('2')))))))
rule13 = ReplacementRule(pattern13, lambda u, v, b, a : Mul(u, Add(Mul(S(1), Pow(a, S(-1))), Mul(S(-1), Mul(Sin(v), Pow(b, S(-1)))))))
replacer.add(rule13)
pattern14 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(tan(v_), WC('n', S(1))), Pow(Add(a_, Mul(WC('b', S(1)), Pow(tan(v_), WC('n', S(1))))), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule14 = ReplacementRule(pattern14, lambda n, a, u, v, b : Mul(u, Pow(Add(b, Mul(a, Pow(Cot(v), n))), S(-1))))
replacer.add(rule14)
pattern15 = Pattern(UtilityOperator(Mul(Pow(cot(v_), WC('n', S(1))), WC('u', S(1)), Pow(Add(Mul(Pow(cot(v_), WC('n', S(1))), WC('b', S(1))), a_), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule15 = ReplacementRule(pattern15, lambda n, a, u, v, b : Mul(u, Pow(Add(b, Mul(a, Pow(Tan(v), n))), S(-1))))
replacer.add(rule15)
pattern16 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(sec(v_), WC('n', S(1))), Pow(Add(a_, Mul(WC('b', S(1)), Pow(sec(v_), WC('n', S(1))))), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule16 = ReplacementRule(pattern16, lambda n, a, u, v, b : Mul(u, Pow(Add(b, Mul(a, Pow(Cos(v), n))), S(-1))))
replacer.add(rule16)
pattern17 = Pattern(UtilityOperator(Mul(Pow(csc(v_), WC('n', S(1))), WC('u', S(1)), Pow(Add(Mul(Pow(csc(v_), WC('n', S(1))), WC('b', S(1))), a_), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule17 = ReplacementRule(pattern17, lambda n, a, u, v, b : Mul(u, Pow(Add(b, Mul(a, Pow(Sin(v), n))), S(-1))))
replacer.add(rule17)
pattern18 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(a_, Mul(WC('b', S(1)), Pow(sec(v_), WC('n', S(1))))), S(-1)), Pow(tan(v_), WC('n', S(1))))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule18 = ReplacementRule(pattern18, lambda n, a, u, v, b : Mul(u, Mul(Pow(Sin(v), n), Pow(Add(b, Mul(a, Pow(Cos(v), n))), S(-1)))))
replacer.add(rule18)
pattern19 = Pattern(UtilityOperator(Mul(Pow(cot(v_), WC('n', S(1))), WC('u', S(1)), Pow(Add(Mul(Pow(csc(v_), WC('n', S(1))), WC('b', S(1))), a_), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule19 = ReplacementRule(pattern19, lambda n, a, u, v, b : Mul(u, Mul(Pow(Cos(v), n), Pow(Add(b, Mul(a, Pow(Sin(v), n))), S(-1)))))
replacer.add(rule19)
pattern20 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(WC('a', S(1)), Pow(sec(v_), WC('n', S(1)))), Mul(WC('b', S(1)), Pow(tan(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, p: IntegersQ(n, p)))
rule20 = ReplacementRule(pattern20, lambda n, a, p, u, v, b : Mul(u, Pow(Sec(v), Mul(n, p)), Pow(Add(a, Mul(b, Pow(Sin(v), n))), p)))
replacer.add(rule20)
pattern21 = Pattern(UtilityOperator(Mul(Pow(Add(Mul(Pow(csc(v_), WC('n', S(1))), WC('a', S(1))), Mul(Pow(cot(v_), WC('n', S(1))), WC('b', S(1)))), WC('p', S(1))), WC('u', S(1)))), CustomConstraint(lambda n, p: IntegersQ(n, p)))
rule21 = ReplacementRule(pattern21, lambda n, a, p, u, v, b : Mul(u, Pow(Csc(v), Mul(n, p)), Pow(Add(a, Mul(b, Pow(Cos(v), n))), p)))
replacer.add(rule21)
pattern22 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(WC('b', S(1)), Pow(sin(v_), WC('n', S(1)))), Mul(WC('a', S(1)), Pow(tan(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, p: IntegersQ(n, p)))
rule22 = ReplacementRule(pattern22, lambda n, a, p, u, v, b : Mul(u, Pow(Tan(v), Mul(n, p)), Pow(Add(a, Mul(b, Pow(Cos(v), n))), p)))
replacer.add(rule22)
pattern23 = Pattern(UtilityOperator(Mul(Pow(Add(Mul(Pow(cot(v_), WC('n', S(1))), WC('a', S(1))), Mul(Pow(cos(v_), WC('n', S(1))), WC('b', S(1)))), WC('p', S(1))), WC('u', S(1)))), CustomConstraint(lambda n, p: IntegersQ(n, p)))
rule23 = ReplacementRule(pattern23, lambda n, a, p, u, v, b : Mul(u, Pow(Cot(v), Mul(n, p)), Pow(Add(a, Mul(b, Pow(Sin(v), n))), p)))
replacer.add(rule23)
pattern24 = Pattern(UtilityOperator(Mul(Pow(cos(v_), WC('m', S(1))), WC('u', S(1)), Pow(Add(WC('a', S(0)), Mul(WC('c', S(1)), Pow(sec(v_), WC('n', S(1)))), Mul(WC('b', S(1)), Pow(tan(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, p, m: IntegersQ(m, n, p)))
rule24 = ReplacementRule(pattern24, lambda n, a, c, p, m, u, v, b : Mul(u, Pow(Cos(v), Add(m, Mul(S(-1), Mul(n, p)))), Pow(Add(c, Mul(b, Pow(Sin(v), n)), Mul(a, Pow(Cos(v), n))), p)))
replacer.add(rule24)
pattern25 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(sec(v_), WC('m', S(1))), Pow(Add(WC('a', S(0)), Mul(WC('c', S(1)), Pow(sec(v_), WC('n', S(1)))), Mul(WC('b', S(1)), Pow(tan(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, p, m: IntegersQ(m, n, p)))
rule25 = ReplacementRule(pattern25, lambda n, a, c, p, m, u, v, b : Mul(u, Pow(Sec(v), Add(m, Mul(n, p))), Pow(Add(c, Mul(b, Pow(Sin(v), n)), Mul(a, Pow(Cos(v), n))), p)))
replacer.add(rule25)
pattern26 = Pattern(UtilityOperator(Mul(Pow(Add(WC('a', S(0)), Mul(Pow(cot(v_), WC('n', S(1))), WC('b', S(1))), Mul(Pow(csc(v_), WC('n', S(1))), WC('c', S(1)))), WC('p', S(1))), WC('u', S(1)), Pow(sin(v_), WC('m', S(1))))), CustomConstraint(lambda n, p, m: IntegersQ(m, n, p)))
rule26 = ReplacementRule(pattern26, lambda n, a, c, p, m, u, v, b : Mul(u, Pow(Sin(v), Add(m, Mul(S(-1), Mul(n, p)))), Pow(Add(c, Mul(b, Pow(Cos(v), n)), Mul(a, Pow(Sin(v), n))), p)))
replacer.add(rule26)
pattern27 = Pattern(UtilityOperator(Mul(Pow(csc(v_), WC('m', S(1))), Pow(Add(WC('a', S(0)), Mul(Pow(cot(v_), WC('n', S(1))), WC('b', S(1))), Mul(Pow(csc(v_), WC('n', S(1))), WC('c', S(1)))), WC('p', S(1))), WC('u', S(1)))), CustomConstraint(lambda n, p, m: IntegersQ(m, n, p)))
rule27 = ReplacementRule(pattern27, lambda n, a, c, p, m, u, v, b : Mul(u, Pow(Csc(v), Add(m, Mul(n, p))), Pow(Add(c, Mul(b, Pow(Cos(v), n)), Mul(a, Pow(Sin(v), n))), p)))
replacer.add(rule27)
pattern28 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(Pow(csc(v_), WC('m', S(1))), WC('a', S(1))), Mul(WC('b', S(1)), Pow(sin(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, m: IntegersQ(m, n)))
rule28 = ReplacementRule(pattern28, lambda n, a, p, m, u, v, b : If(And(ZeroQ(Add(m, n, S(-2))), ZeroQ(Add(a, b))), Mul(u, Pow(Mul(a, Mul(Pow(Cos(v), S('2')), Pow(Pow(Sin(v), m), S(-1)))), p)), Mul(u, Pow(Mul(Add(a, Mul(b, Pow(Sin(v), Add(m, n)))), Pow(Pow(Sin(v), m), S(-1))), p))))
replacer.add(rule28)
pattern29 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(Pow(cos(v_), WC('n', S(1))), WC('b', S(1))), Mul(WC('a', S(1)), Pow(sec(v_), WC('m', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, m: IntegersQ(m, n)))
rule29 = ReplacementRule(pattern29, lambda n, a, p, m, u, v, b : If(And(ZeroQ(Add(m, n, S(-2))), ZeroQ(Add(a, b))), Mul(u, Pow(Mul(a, Mul(Pow(Sin(v), S('2')), Pow(Pow(Cos(v), m), S(-1)))), p)), Mul(u, Pow(Mul(Add(a, Mul(b, Pow(Cos(v), Add(m, n)))), Pow(Pow(Cos(v), m), S(-1))), p))))
replacer.add(rule29)
pattern30 = Pattern(UtilityOperator(u_))
rule30 = ReplacementRule(pattern30, lambda u : u)
replacer.add(rule30)
return replacer
@doctest_depends_on(modules=('matchpy',))
def TrigSimplifyAux(expr):
return TrigSimplifyAux_replacer.replace(UtilityOperator(expr))
def Cancel(expr):
return cancel(expr)
class Util_Part(Function):
def doit(self):
i = Simplify(self.args[0])
if len(self.args) > 2 :
lst = list(self.args[1:])
else:
lst = self.args[1]
if isinstance(i, (int, Integer)):
if isinstance(lst, list):
return lst[i - 1]
elif AtomQ(lst):
return lst
return lst.args[i - 1]
else:
return self
def Part(lst, i): #see i = -1
if isinstance(lst, list):
return Util_Part(i, *lst).doit()
return Util_Part(i, lst).doit()
def PolyLog(n, p, z=None):
return polylog(n, p)
def D(f, x):
try:
return f.diff(x)
except ValueError:
return Function('D')(f, x)
def IntegralFreeQ(u):
return FreeQ(u, Integral)
def Dist(u, v, x):
#Dist(u,v) returns the sum of u times each term of v, provided v is free of Int
u = replace_pow_exp(u) # to replace back to sympy's exp
v = replace_pow_exp(v)
w = Simp(u*x**2, x)/x**2
if u == 1:
return v
elif u == 0:
return 0
elif NumericFactor(u) < 0 and NumericFactor(-u) > 0:
return -Dist(-u, v, x)
elif SumQ(v):
return Add(*[Dist(u, i, x) for i in v.args])
elif IntegralFreeQ(v):
return Simp(u*v, x)
elif w != u and FreeQ(w, x) and w == Simp(w, x) and w == Simp(w*x**2, x)/x**2:
return Dist(w, v, x)
else:
return Simp(u*v, x)
def PureFunctionOfCothQ(u, v, x):
# If u is a pure function of Coth[v], PureFunctionOfCothQ[u,v,x] returns True;
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and ZeroQ(u.args[0] - v):
return CothQ(u)
return all(PureFunctionOfCothQ(i, v, x) for i in u.args)
def LogIntegral(z):
return li(z)
def ExpIntegralEi(z):
return Ei(z)
def ExpIntegralE(a, b):
return expint(a, b).evalf()
def SinIntegral(z):
return Si(z)
def CosIntegral(z):
return Ci(z)
def SinhIntegral(z):
return Shi(z)
def CoshIntegral(z):
return Chi(z)
class PolyGamma(Function):
@classmethod
def eval(cls, *args):
if len(args) == 2:
return polygamma(args[0], args[1])
return digamma(args[0])
def LogGamma(z):
return loggamma(z)
class ProductLog(Function):
@classmethod
def eval(cls, *args):
if len(args) == 2:
return LambertW(args[1], args[0]).evalf()
return LambertW(args[0]).evalf()
def Factorial(a):
return factorial(a)
def Zeta(*args):
return zeta(*args)
def HypergeometricPFQ(a, b, c):
return hyper(a, b, c)
def Sum_doit(exp, args):
'''
This function perform summation using sympy's `Sum`.
Examples
========
>>> from sympy.integrals.rubi.utility_function import Sum_doit
>>> from sympy.abc import x
>>> Sum_doit(2*x + 2, [x, 0, 1.7])
6
'''
exp = replace_pow_exp(exp)
if not isinstance(args[2], (int, Integer)):
new_args = [args[0], args[1], Floor(args[2])]
return Sum(exp, new_args).doit()
return Sum(exp, args).doit()
def PolynomialQuotient(p, q, x):
try:
p = poly(p, x)
q = poly(q, x)
except:
p = poly(p)
q = poly(q)
try:
return quo(p, q).as_expr()
except (PolynomialDivisionFailed, UnificationFailed):
return p/q
def PolynomialRemainder(p, q, x):
try:
p = poly(p, x)
q = poly(q, x)
except:
p = poly(p)
q = poly(q)
try:
return rem(p, q).as_expr()
except (PolynomialDivisionFailed, UnificationFailed):
return S(0)
def Floor(x, a = None):
if a is None:
return floor(x)
return a*floor(x/a)
def Factor(var):
return factor(var)
def Rule(a, b):
return {a: b}
def Distribute(expr, *args):
if len(args) == 1:
if isinstance(expr, args[0]):
return expr
else:
return expr.expand()
if len(args) == 2:
if isinstance(expr, args[1]):
return expr.expand()
else:
return expr
return expr.expand()
def CoprimeQ(*args):
args = S(args)
g = gcd(*args)
if g == 1:
return True
return False
def Discriminant(a, b):
try:
return discriminant(a, b)
except PolynomialError:
return Function('Discriminant')(a, b)
def Negative(x):
return x < S(0)
def Quotient(m, n):
return Floor(m/n)
def process_trig(expr):
'''
This function processes trigonometric expressions such that all `cot` is
rewritten in terms of `tan`, `sec` in terms of `cos`, `csc` in terms of `sin` and
similarly for `coth`, `sech` and `csch`.
Examples
========
>>> from sympy.integrals.rubi.utility_function import process_trig
>>> from sympy.abc import x
>>> from sympy import coth, cot, csc
>>> process_trig(x*cot(x))
x/tan(x)
>>> process_trig(coth(x)*csc(x))
1/(sin(x)*tanh(x))
'''
expr = expr.replace(lambda x: isinstance(x, cot), lambda x: 1/tan(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, sec), lambda x: 1/cos(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, csc), lambda x: 1/sin(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, coth), lambda x: 1/tanh(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, sech), lambda x: 1/cosh(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, csch), lambda x: 1/sinh(x.args[0]))
return expr
def _ExpandIntegrand():
Plus = Add
Times = Mul
def cons_f1(m):
return PositiveIntegerQ(m)
cons1 = CustomConstraint(cons_f1)
def cons_f2(d, c, b, a):
return ZeroQ(-a*d + b*c)
cons2 = CustomConstraint(cons_f2)
def cons_f3(a, x):
return FreeQ(a, x)
cons3 = CustomConstraint(cons_f3)
def cons_f4(b, x):
return FreeQ(b, x)
cons4 = CustomConstraint(cons_f4)
def cons_f5(c, x):
return FreeQ(c, x)
cons5 = CustomConstraint(cons_f5)
def cons_f6(d, x):
return FreeQ(d, x)
cons6 = CustomConstraint(cons_f6)
def cons_f7(e, x):
return FreeQ(e, x)
cons7 = CustomConstraint(cons_f7)
def cons_f8(f, x):
return FreeQ(f, x)
cons8 = CustomConstraint(cons_f8)
def cons_f9(g, x):
return FreeQ(g, x)
cons9 = CustomConstraint(cons_f9)
def cons_f10(h, x):
return FreeQ(h, x)
cons10 = CustomConstraint(cons_f10)
def cons_f11(e, b, c, f, n, p, F, x, d, m):
if not isinstance(x, Symbol):
return False
return FreeQ(List(F, b, c, d, e, f, m, n, p), x)
cons11 = CustomConstraint(cons_f11)
def cons_f12(F, x):
return FreeQ(F, x)
cons12 = CustomConstraint(cons_f12)
def cons_f13(m, x):
return FreeQ(m, x)
cons13 = CustomConstraint(cons_f13)
def cons_f14(n, x):
return FreeQ(n, x)
cons14 = CustomConstraint(cons_f14)
def cons_f15(p, x):
return FreeQ(p, x)
cons15 = CustomConstraint(cons_f15)
def cons_f16(e, b, c, f, n, a, p, F, x, d, m):
if not isinstance(x, Symbol):
return False
return FreeQ(List(F, a, b, c, d, e, f, m, n, p), x)
cons16 = CustomConstraint(cons_f16)
def cons_f17(n, m):
return IntegersQ(m, n)
cons17 = CustomConstraint(cons_f17)
def cons_f18(n):
return Less(n, S(0))
cons18 = CustomConstraint(cons_f18)
def cons_f19(x, u):
if not isinstance(x, Symbol):
return False
return PolynomialQ(u, x)
cons19 = CustomConstraint(cons_f19)
def cons_f20(G, F, u):
return SameQ(F(u)*G(u), S(1))
cons20 = CustomConstraint(cons_f20)
def cons_f21(q, x):
return FreeQ(q, x)
cons21 = CustomConstraint(cons_f21)
def cons_f22(F):
return MemberQ(List(ArcSin, ArcCos, ArcSinh, ArcCosh), F)
cons22 = CustomConstraint(cons_f22)
def cons_f23(j, n):
return ZeroQ(j - S(2)*n)
cons23 = CustomConstraint(cons_f23)
def cons_f24(A, x):
return FreeQ(A, x)
cons24 = CustomConstraint(cons_f24)
def cons_f25(B, x):
return FreeQ(B, x)
cons25 = CustomConstraint(cons_f25)
def cons_f26(m, u, x):
if not isinstance(x, Symbol):
return False
def _cons_f_u(d, w, c, p, x):
return And(FreeQ(List(c, d), x), IntegerQ(p), Greater(p, m))
cons_u = CustomConstraint(_cons_f_u)
pat = Pattern(UtilityOperator((c_ + x_*WC('d', S(1)))**p_*WC('w', S(1)), x_), cons_u)
result_matchq = is_match(UtilityOperator(u, x), pat)
return Not(And(PositiveIntegerQ(m), result_matchq))
cons26 = CustomConstraint(cons_f26)
def cons_f27(b, v, n, a, x, u, m):
if not isinstance(x, Symbol):
return False
return And(FreeQ(List(a, b, m), x), NegativeIntegerQ(n), Not(IntegerQ(m)), PolynomialQ(u, x), PolynomialQ(v, x),\
RationalQ(m), Less(m, -1), GreaterEqual(Exponent(u, x), (-n - IntegerPart(m))*Exponent(v, x)))
cons27 = CustomConstraint(cons_f27)
def cons_f28(v, n, x, u, m):
if not isinstance(x, Symbol):
return False
return And(FreeQ(List(a, b, m), x), NegativeIntegerQ(n), Not(IntegerQ(m)), PolynomialQ(u, x),\
PolynomialQ(v, x), GreaterEqual(Exponent(u, x), -n*Exponent(v, x)))
cons28 = CustomConstraint(cons_f28)
def cons_f29(n):
return PositiveIntegerQ(n/S(4))
cons29 = CustomConstraint(cons_f29)
def cons_f30(n):
return IntegerQ(n)
cons30 = CustomConstraint(cons_f30)
def cons_f31(n):
return Greater(n, S(1))
cons31 = CustomConstraint(cons_f31)
def cons_f32(n, m):
return Less(S(0), m, n)
cons32 = CustomConstraint(cons_f32)
def cons_f33(n, m):
return OddQ(n/GCD(m, n))
cons33 = CustomConstraint(cons_f33)
def cons_f34(a, b):
return PosQ(a/b)
cons34 = CustomConstraint(cons_f34)
def cons_f35(n, m, p):
return IntegersQ(m, n, p)
cons35 = CustomConstraint(cons_f35)
def cons_f36(n, m, p):
return Less(S(0), m, p, n)
cons36 = CustomConstraint(cons_f36)
def cons_f37(q, n, m, p):
return IntegersQ(m, n, p, q)
cons37 = CustomConstraint(cons_f37)
def cons_f38(n, q, m, p):
return Less(S(0), m, p, q, n)
cons38 = CustomConstraint(cons_f38)
def cons_f39(n):
return IntegerQ(n/S(2))
cons39 = CustomConstraint(cons_f39)
def cons_f40(p):
return NegativeIntegerQ(p)
cons40 = CustomConstraint(cons_f40)
def cons_f41(n, m):
return IntegersQ(m, n/S(2))
cons41 = CustomConstraint(cons_f41)
def cons_f42(n, m):
return Unequal(m, n/S(2))
cons42 = CustomConstraint(cons_f42)
def cons_f43(c, b, a):
return NonzeroQ(-S(4)*a*c + b**S(2))
cons43 = CustomConstraint(cons_f43)
def cons_f44(j, n, m):
return IntegersQ(m, n, j)
cons44 = CustomConstraint(cons_f44)
def cons_f45(n, m):
return Less(S(0), m, S(2)*n)
cons45 = CustomConstraint(cons_f45)
def cons_f46(n, m, p):
return Not(And(Equal(m, n), Equal(p, S(-1))))
cons46 = CustomConstraint(cons_f46)
def cons_f47(v, x):
if not isinstance(x, Symbol):
return False
return PolynomialQ(v, x)
cons47 = CustomConstraint(cons_f47)
def cons_f48(v, x):
if not isinstance(x, Symbol):
return False
return BinomialQ(v, x)
cons48 = CustomConstraint(cons_f48)
def cons_f49(v, x, u):
if not isinstance(x, Symbol):
return False
return Inequality(Exponent(u, x), Equal, Exponent(v, x) + S(-1), GreaterEqual, S(2))
cons49 = CustomConstraint(cons_f49)
def cons_f50(v, x, u):
if not isinstance(x, Symbol):
return False
return GreaterEqual(Exponent(u, x), Exponent(v, x))
cons50 = CustomConstraint(cons_f50)
def cons_f51(p):
return Not(IntegerQ(p))
cons51 = CustomConstraint(cons_f51)
def With2(e, b, c, f, n, a, g, h, x, d, m):
tmp = a*h - b*g
k = Symbol('k')
return f**(e*(c + d*x)**n)*SimplifyTerm(h**(-m)*tmp**m, x)/(g + h*x) + Sum_doit(f**(e*(c + d*x)**n)*(a + b*x)**(-k + m)*SimplifyTerm(b*h**(-k)*tmp**(k - 1), x), List(k, 1, m))
pattern2 = Pattern(UtilityOperator(f_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('e', S(1)))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons3, cons4, cons5, cons6, cons7, cons8, cons9, cons10, cons1, cons2)
rule2 = ReplacementRule(pattern2, With2)
pattern3 = Pattern(UtilityOperator(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('b', S(1)))*x_**WC('m', S(1))*(e_ + x_*WC('f', S(1)))**WC('p', S(1)), x_), cons12, cons4, cons5, cons6, cons7, cons8, cons13, cons14, cons15, cons11)
def replacement3(e, b, c, f, n, p, F, x, d, m):
return If(And(PositiveIntegerQ(m, p), LessEqual(m, p), Or(EqQ(n, S(1)), ZeroQ(-c*f + d*e))), ExpandLinearProduct(F**(b*(c + d*x)**n)*(e + f*x)**p, x**m, e, f, x), If(PositiveIntegerQ(p), Distribute(F**(b*(c + d*x)**n)*x**m*(e + f*x)**p, Plus, Times), ExpandIntegrand(F**(b*(c + d*x)**n), x**m*(e + f*x)**p, x)))
rule3 = ReplacementRule(pattern3, replacement3)
pattern4 = Pattern(UtilityOperator(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*x_**WC('m', S(1))*(e_ + x_*WC('f', S(1)))**WC('p', S(1)), x_), cons12, cons3, cons4, cons5, cons6, cons7, cons8, cons13, cons14, cons15, cons16)
def replacement4(e, b, c, f, n, a, p, F, x, d, m):
return If(And(PositiveIntegerQ(m, p), LessEqual(m, p), Or(EqQ(n, S(1)), ZeroQ(-c*f + d*e))), ExpandLinearProduct(F**(a + b*(c + d*x)**n)*(e + f*x)**p, x**m, e, f, x), If(PositiveIntegerQ(p), Distribute(F**(a + b*(c + d*x)**n)*x**m*(e + f*x)**p, Plus, Times), ExpandIntegrand(F**(a + b*(c + d*x)**n), x**m*(e + f*x)**p, x)))
rule4 = ReplacementRule(pattern4, replacement4)
def With5(b, v, c, n, a, F, u, x, d, m):
if not isinstance(x, Symbol) or not (FreeQ([F, a, b, c, d], x) and IntegersQ(m, n) and n < 0):
return False
w = ExpandIntegrand((a + b*x)**m*(c + d*x)**n, x)
w = ReplaceAll(w, Rule(x, F**v))
if SumQ(w):
return True
return False
pattern5 = Pattern(UtilityOperator((F_**v_*WC('b', S(1)) + a_)**WC('m', S(1))*(F_**v_*WC('d', S(1)) + c_)**n_*WC('u', S(1)), x_), cons12, cons3, cons4, cons5, cons6, cons17, cons18, CustomConstraint(With5))
def replacement5(b, v, c, n, a, F, u, x, d, m):
w = ReplaceAll(ExpandIntegrand((a + b*x)**m*(c + d*x)**n, x), Rule(x, F**v))
return w.func(*[u*i for i in w.args])
rule5 = ReplacementRule(pattern5, replacement5)
def With6(e, b, c, f, n, a, x, u, d, m):
if not isinstance(x, Symbol) or not (FreeQ([a, b, c, d, e, f, m, n], x) and PolynomialQ(u,x)):
return False
v = ExpandIntegrand(u*(a + b*x)**m, x)
if SumQ(v):
return True
return False
pattern6 = Pattern(UtilityOperator(f_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('e', S(1)))*u_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons3, cons4, cons5, cons6, cons7, cons8, cons13, cons14, cons19, CustomConstraint(With6))
def replacement6(e, b, c, f, n, a, x, u, d, m):
v = ExpandIntegrand(u*(a + b*x)**m, x)
return Distribute(f**(e*(c + d*x)**n)*v, Plus, Times)
rule6 = ReplacementRule(pattern6, replacement6)
pattern7 = Pattern(UtilityOperator(u_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*Log((x_**WC('n', S(1))*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons3, cons4, cons5, cons6, cons7, cons13, cons14, cons15, cons19)
def replacement7(e, b, c, n, a, p, x, u, d, m):
return ExpandIntegrand(Log(c*(d + e*x**n)**p), u*(a + b*x)**m, x)
rule7 = ReplacementRule(pattern7, replacement7)
pattern8 = Pattern(UtilityOperator(f_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('e', S(1)))*u_, x_), cons5, cons6, cons7, cons8, cons14, cons19)
def replacement8(e, c, f, n, x, u, d):
return If(EqQ(n, S(1)), ExpandIntegrand(f**(e*(c + d*x)**n), u, x), ExpandLinearProduct(f**(e*(c + d*x)**n), u, c, d, x))
rule8 = ReplacementRule(pattern8, replacement8)
# pattern9 = Pattern(UtilityOperator(F_**u_*(G_*u_*WC('b', S(1)) + a_)**WC('n', S(1)), x_), cons3, cons4, cons17, cons20)
# def replacement9(b, G, n, a, F, u, x, m):
# return ReplaceAll(ExpandIntegrand(x**(-m)*(a + b*x)**n, x), Rule(x, G(u)))
# rule9 = ReplacementRule(pattern9, replacement9)
pattern10 = Pattern(UtilityOperator(u_*(WC('a', S(0)) + WC('b', S(1))*Log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons3, cons4, cons5, cons6, cons7, cons8, cons14, cons15, cons21, cons19)
def replacement10(e, b, c, f, n, a, p, x, u, d, q):
return ExpandLinearProduct((a + b*Log(c*(d*(e + f*x)**p)**q))**n, u, e, f, x)
rule10 = ReplacementRule(pattern10, replacement10)
# pattern11 = Pattern(UtilityOperator(u_*(F_*(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons3, cons4, cons5, cons6, cons14, cons19, cons22)
# def replacement11(b, c, n, a, F, u, x, d):
# return ExpandLinearProduct((a + b*F(c + d*x))**n, u, c, d, x)
# rule11 = ReplacementRule(pattern11, replacement11)
pattern12 = Pattern(UtilityOperator(WC('u', S(1))/(x_**n_*WC('a', S(1)) + sqrt(c_ + x_**j_*WC('d', S(1)))*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons14, cons23)
def replacement12(b, c, n, a, x, u, d, j):
return ExpandIntegrand(u*(a*x**n - b*sqrt(c + d*x**(S(2)*n)))/(-b**S(2)*c + x**(S(2)*n)*(a**S(2) - b**S(2)*d)), x)
rule12 = ReplacementRule(pattern12, replacement12)
pattern13 = Pattern(UtilityOperator((a_ + x_*WC('b', S(1)))**m_/(c_ + x_*WC('d', S(1))), x_), cons3, cons4, cons5, cons6, cons1)
def replacement13(b, c, a, x, d, m):
if RationalQ(a, b, c, d):
return ExpandExpression((a + b*x)**m/(c + d*x), x)
else:
tmp = a*d - b*c
k = Symbol("k")
return Sum_doit((a + b*x)**(-k + m)*SimplifyTerm(b*d**(-k)*tmp**(k + S(-1)), x), List(k, S(1), m)) + SimplifyTerm(d**(-m)*tmp**m, x)/(c + d*x)
rule13 = ReplacementRule(pattern13, replacement13)
pattern14 = Pattern(UtilityOperator((A_ + x_*WC('B', S(1)))*(a_ + x_*WC('b', S(1)))**WC('m', S(1))/(c_ + x_*WC('d', S(1))), x_), cons3, cons4, cons5, cons6, cons24, cons25, cons1)
def replacement14(b, B, A, c, a, x, d, m):
if RationalQ(a, b, c, d, A, B):
return ExpandExpression((A + B*x)*(a + b*x)**m/(c + d*x), x)
else:
tmp1 = (A*d - B*c)/d
tmp2 = ExpandIntegrand((a + b*x)**m/(c + d*x), x)
tmp2 = If(SumQ(tmp2), tmp2.func(*[SimplifyTerm(tmp1*i, x) for i in tmp2.args]), SimplifyTerm(tmp1*tmp2, x))
return SimplifyTerm(B/d, x)*(a + b*x)**m + tmp2
rule14 = ReplacementRule(pattern14, replacement14)
def With15(b, a, x, u, m):
tmp1 = Symbol('tmp1')
tmp2 = Symbol('tmp2')
tmp1 = ExpandLinearProduct((a + b*x)**m, u, a, b, x)
if not IntegerQ(m):
return tmp1
else:
tmp2 = ExpandExpression(u*(a + b*x)**m, x)
if SumQ(tmp2) and LessEqual(LeafCount(tmp2), LeafCount(tmp1) + S(2)):
return tmp2
else:
return tmp1
pattern15 = Pattern(UtilityOperator(u_*(a_ + x_*WC('b', S(1)))**m_, x_), cons3, cons4, cons13, cons19, cons26)
rule15 = ReplacementRule(pattern15, With15)
pattern16 = Pattern(UtilityOperator(u_*v_**n_*(a_ + x_*WC('b', S(1)))**m_, x_), cons27)
def replacement16(b, v, n, a, x, u, m):
s = PolynomialQuotientRemainder(u, v**(-n)*(a+b*x)**(-IntegerPart(m)), x)
return ExpandIntegrand((a + b*x)**FractionalPart(m)*s[0], x) + ExpandIntegrand(v**n*(a + b*x)**m*s[1], x)
rule16 = ReplacementRule(pattern16, replacement16)
pattern17 = Pattern(UtilityOperator(u_*v_**n_*(a_ + x_*WC('b', S(1)))**m_, x_), cons28)
def replacement17(b, v, n, a, x, u, m):
s = PolynomialQuotientRemainder(u, v**(-n),x)
return ExpandIntegrand((a + b*x)**(m)*s[0], x) + ExpandIntegrand(v**n*(a + b*x)**m*s[1], x)
rule17 = ReplacementRule(pattern17, replacement17)
def With18(b, n, a, x, u):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
return r/(S(2)*a*(r + s*u**(n/S(2)))) + r/(S(2)*a*(r - s*u**(n/S(2))))
pattern18 = Pattern(UtilityOperator(S(1)/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons29)
rule18 = ReplacementRule(pattern18, With18)
def With19(b, n, a, x, u):
k = Symbol("k")
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
return Sum_doit(r/(a*n*(-(-1)**(2*k/n)*s*u + r)), List(k, 1, n))
pattern19 = Pattern(UtilityOperator(S(1)/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons30, cons31)
rule19 = ReplacementRule(pattern19, With19)
def With20(b, n, a, x, u, m):
k = Symbol("k")
g = GCD(m, n)
r = Numerator(Rt(a/b, n/GCD(m, n)))
s = Denominator(Rt(a/b, n/GCD(m, n)))
return If(CoprimeQ(g + m, n), Sum_doit((-1)**(-2*k*m/n)*r*(-r/s)**(m/g)/(a*n*((-1)**(2*g*k/n)*s*u**g + r)), List(k, 1, n/g)), Sum_doit((-1)**(2*k*(g + m)/n)*r*(-r/s)**(m/g)/(a*n*((-1)**(2*g*k/n)*r + s*u**g)), List(k, 1, n/g)))
pattern20 = Pattern(UtilityOperator(u_**WC('m', S(1))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons17, cons32, cons33, cons34)
rule20 = ReplacementRule(pattern20, With20)
def With21(b, n, a, x, u, m):
k = Symbol("k")
g = GCD(m, n)
r = Numerator(Rt(-a/b, n/GCD(m, n)))
s = Denominator(Rt(-a/b, n/GCD(m, n)))
return If(Equal(n/g, S(2)), s/(S(2)*b*(r + s*u**g)) - s/(S(2)*b*(r - s*u**g)), If(CoprimeQ(g + m, n), Sum_doit((S(-1))**(-S(2)*k*m/n)*r*(r/s)**(m/g)/(a*n*(-(S(-1))**(S(2)*g*k/n)*s*u**g + r)), List(k, S(1), n/g)), Sum_doit((S(-1))**(S(2)*k*(g + m)/n)*r*(r/s)**(m/g)/(a*n*((S(-1))**(S(2)*g*k/n)*r - s*u**g)), List(k, S(1), n/g))))
pattern21 = Pattern(UtilityOperator(u_**WC('m', S(1))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons17, cons32)
rule21 = ReplacementRule(pattern21, With21)
def With22(b, c, n, a, x, u, d, m):
k = Symbol("k")
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
return Sum_doit((c*r + (-1)**(-2*k*m/n)*d*r*(r/s)**m)/(a*n*(-(-1)**(2*k/n)*s*u + r)), List(k, 1, n))
pattern22 = Pattern(UtilityOperator((c_ + u_**WC('m', S(1))*WC('d', S(1)))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons17, cons32)
rule22 = ReplacementRule(pattern22, With22)
def With23(e, b, c, n, a, p, x, u, d, m):
k = Symbol("k")
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
return Sum_doit((c*r + (-1)**(-2*k*p/n)*e*r*(r/s)**p + (-1)**(-2*k*m/n)*d*r*(r/s)**m)/(a*n*(-(-1)**(2*k/n)*s*u + r)), List(k, 1, n))
pattern23 = Pattern(UtilityOperator((u_**p_*WC('e', S(1)) + u_**WC('m', S(1))*WC('d', S(1)) + WC('c', S(0)))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons7, cons35, cons36)
rule23 = ReplacementRule(pattern23, With23)
def With24(e, b, c, f, n, a, p, x, u, d, q, m):
k = Symbol("k")
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
return Sum_doit((c*r + (-1)**(-2*k*q/n)*f*r*(r/s)**q + (-1)**(-2*k*p/n)*e*r*(r/s)**p + (-1)**(-2*k*m/n)*d*r*(r/s)**m)/(a*n*(-(-1)**(2*k/n)*s*u + r)), List(k, 1, n))
pattern24 = Pattern(UtilityOperator((u_**p_*WC('e', S(1)) + u_**q_*WC('f', S(1)) + u_**WC('m', S(1))*WC('d', S(1)) + WC('c', S(0)))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons7, cons8, cons37, cons38)
rule24 = ReplacementRule(pattern24, With24)
def With25(c, n, a, p, x, u):
q = Symbol('q')
return ReplaceAll(ExpandIntegrand(c**(-p), (c*x - q)**p*(c*x + q)**p, x), List(Rule(q, Rt(-a*c, S(2))), Rule(x, u**(n/S(2)))))
pattern25 = Pattern(UtilityOperator((a_ + u_**WC('n', S(1))*WC('c', S(1)))**p_, x_), cons3, cons5, cons39, cons40)
rule25 = ReplacementRule(pattern25, With25)
def With26(c, n, a, p, x, u, m):
q = Symbol('q')
return ReplaceAll(ExpandIntegrand(c**(-p), x**m*(c*x**(n/S(2)) - q)**p*(c*x**(n/S(2)) + q)**p, x), List(Rule(q, Rt(-a*c, S(2))), Rule(x, u)))
pattern26 = Pattern(UtilityOperator(u_**WC('m', S(1))*(u_**WC('n', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons3, cons5, cons41, cons40, cons32, cons42)
rule26 = ReplacementRule(pattern26, With26)
def With27(b, c, n, a, p, x, u, j):
q = Symbol('q')
return ReplaceAll(ExpandIntegrand(S(4)**(-p)*c**(-p), (b + S(2)*c*x - q)**p*(b + S(2)*c*x + q)**p, x), List(Rule(q, Rt(-S(4)*a*c + b**S(2), S(2))), Rule(x, u**n)))
pattern27 = Pattern(UtilityOperator((u_**WC('j', S(1))*WC('c', S(1)) + u_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons3, cons4, cons5, cons30, cons23, cons40, cons43)
rule27 = ReplacementRule(pattern27, With27)
def With28(b, c, n, a, p, x, u, j, m):
q = Symbol('q')
return ReplaceAll(ExpandIntegrand(S(4)**(-p)*c**(-p), x**m*(b + S(2)*c*x**n - q)**p*(b + S(2)*c*x**n + q)**p, x), List(Rule(q, Rt(-S(4)*a*c + b**S(2), S(2))), Rule(x, u)))
pattern28 = Pattern(UtilityOperator(u_**WC('m', S(1))*(u_**WC('j', S(1))*WC('c', S(1)) + u_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons3, cons4, cons5, cons44, cons23, cons40, cons45, cons46, cons43)
rule28 = ReplacementRule(pattern28, With28)
def With29(b, c, n, a, x, u, d, j):
q = Rt(-a/b, S(2))
return -(c - d*q)/(S(2)*b*q*(q + u**n)) - (c + d*q)/(S(2)*b*q*(q - u**n))
pattern29 = Pattern(UtilityOperator((u_**WC('n', S(1))*WC('d', S(1)) + WC('c', S(0)))/(a_ + u_**WC('j', S(1))*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons14, cons23)
rule29 = ReplacementRule(pattern29, With29)
def With30(e, b, c, f, n, a, g, x, u, d, j):
q = Rt(-S(4)*a*c + b**S(2), S(2))
r = TogetherSimplify((-b*e*g + S(2)*c*(d + e*f))/q)
return (e*g - r)/(b + 2*c*u**n + q) + (e*g + r)/(b + 2*c*u**n - q)
pattern30 = Pattern(UtilityOperator(((u_**WC('n', S(1))*WC('g', S(1)) + WC('f', S(0)))*WC('e', S(1)) + WC('d', S(0)))/(u_**WC('j', S(1))*WC('c', S(1)) + u_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons3, cons4, cons5, cons6, cons7, cons8, cons9, cons14, cons23, cons43)
rule30 = ReplacementRule(pattern30, With30)
def With31(v, x, u):
lst = CoefficientList(u, x)
i = Symbol('i')
return x**Exponent(u, x)*lst[-1]/v + Sum_doit(x**(i - 1)*Part(lst, i), List(i, 1, Exponent(u, x)))/v
pattern31 = Pattern(UtilityOperator(u_/v_, x_), cons19, cons47, cons48, cons49)
rule31 = ReplacementRule(pattern31, With31)
pattern32 = Pattern(UtilityOperator(u_/v_, x_), cons19, cons47, cons50)
def replacement32(v, x, u):
return PolynomialDivide(u, v, x)
rule32 = ReplacementRule(pattern32, replacement32)
pattern33 = Pattern(UtilityOperator(u_*(x_*WC('a', S(1)))**p_, x_), cons51, cons19)
def replacement33(x, a, u, p):
return ExpandToSum((a*x)**p, u, x)
rule33 = ReplacementRule(pattern33, replacement33)
pattern34 = Pattern(UtilityOperator(v_**p_*WC('u', S(1)), x_), cons51)
def replacement34(v, x, u, p):
return ExpandIntegrand(NormalizeIntegrand(v**p, x), u, x)
rule34 = ReplacementRule(pattern34, replacement34)
pattern35 = Pattern(UtilityOperator(u_, x_))
def replacement35(x, u):
return ExpandExpression(u, x)
rule35 = ReplacementRule(pattern35, replacement35)
return [ rule2,rule3, rule4, rule5, rule6, rule7, rule8, rule10, rule12, rule13, rule14, rule15, rule16, rule17, rule18, rule19, rule20, rule21, rule22, rule23, rule24, rule25, rule26, rule27, rule28, rule29, rule30, rule31, rule32, rule33, rule34, rule35]
def _RemoveContentAux():
def cons_f1(b, a):
return IntegersQ(a, b)
cons1 = CustomConstraint(cons_f1)
def cons_f2(b, a):
return Equal(a + b, S(0))
cons2 = CustomConstraint(cons_f2)
def cons_f3(m):
return RationalQ(m)
cons3 = CustomConstraint(cons_f3)
def cons_f4(m, n):
return RationalQ(m, n)
cons4 = CustomConstraint(cons_f4)
def cons_f5(m, n):
return GreaterEqual(-m + n, S(0))
cons5 = CustomConstraint(cons_f5)
def cons_f6(a, x):
return FreeQ(a, x)
cons6 = CustomConstraint(cons_f6)
def cons_f7(m, n, p):
return RationalQ(m, n, p)
cons7 = CustomConstraint(cons_f7)
def cons_f8(m, p):
return GreaterEqual(-m + p, S(0))
cons8 = CustomConstraint(cons_f8)
pattern1 = Pattern(UtilityOperator(a_**m_*WC('u', S(1)) + b_*WC('v', S(1)), x_), cons1, cons2, cons3)
def replacement1(v, x, a, u, m, b):
return If(Greater(m, S(1)), RemoveContentAux(a**(m + S(-1))*u - v, x), RemoveContentAux(-a**(-m + S(1))*v + u, x))
rule1 = ReplacementRule(pattern1, replacement1)
pattern2 = Pattern(UtilityOperator(a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)), x_), cons6, cons4, cons5)
def replacement2(n, v, x, u, m, a):
return RemoveContentAux(a**(-m + n)*v + u, x)
rule2 = ReplacementRule(pattern2, replacement2)
pattern3 = Pattern(UtilityOperator(a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)) + a_**WC('p', S(1))*WC('w', S(1)), x_), cons6, cons7, cons5, cons8)
def replacement3(n, v, x, p, u, w, m, a):
return RemoveContentAux(a**(-m + n)*v + a**(-m + p)*w + u, x)
rule3 = ReplacementRule(pattern3, replacement3)
pattern4 = Pattern(UtilityOperator(u_, x_))
def replacement4(u, x):
return If(And(SumQ(u), NegQ(First(u))), -u, u)
rule4 = ReplacementRule(pattern4, replacement4)
return [rule1, rule2, rule3, rule4, ]
IntHide = Int
Log = rubi_log
Null = None
if matchpy:
RemoveContentAux_replacer = ManyToOneReplacer(* _RemoveContentAux())
ExpandIntegrand_rules = _ExpandIntegrand()
TrigSimplifyAux_replacer = _TrigSimplifyAux()
SimplifyAntiderivative_replacer = _SimplifyAntiderivative()
SimplifyAntiderivativeSum_replacer = _SimplifyAntiderivativeSum()
FixSimplify_rules = _FixSimplify()
SimpFixFactor_replacer = _SimpFixFactor()
|
5b89447f63d8ab0d5102c48f2d5ba16b99a6cee37182178dfff0b4cc77664110 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def cons_f1(a):
return ZeroQ(a)
cons1 = CustomConstraint(cons_f1)
def cons_f2(a, x):
return FreeQ(a, x)
cons2 = CustomConstraint(cons_f2)
def cons_f3(b, x):
return FreeQ(b, x)
cons3 = CustomConstraint(cons_f3)
def cons_f4(n, x):
return FreeQ(n, x)
cons4 = CustomConstraint(cons_f4)
def cons_f5(p, x):
return FreeQ(p, x)
cons5 = CustomConstraint(cons_f5)
def cons_f6(n, j):
return ZeroQ(j - S(2)*n)
cons6 = CustomConstraint(cons_f6)
def cons_f7(c, x):
return FreeQ(c, x)
cons7 = CustomConstraint(cons_f7)
def cons_f8(b):
return ZeroQ(b)
cons8 = CustomConstraint(cons_f8)
def cons_f9(c):
return ZeroQ(c)
cons9 = CustomConstraint(cons_f9)
def cons_f10(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return NFreeQ(v, x)
cons10 = CustomConstraint(cons_f10)
def cons_f11(x, Pm):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolyQ(Pm, x)
cons11 = CustomConstraint(cons_f11)
def cons_f12(p):
return Not(RationalQ(p))
cons12 = CustomConstraint(cons_f12)
def cons_f13(p):
return RationalQ(p)
cons13 = CustomConstraint(cons_f13)
def cons_f14(x, c, a, b):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c), x)
cons14 = CustomConstraint(cons_f14)
def cons_f15(a):
return EqQ(a**S(2), S(1))
cons15 = CustomConstraint(cons_f15)
def cons_f16(u):
return SumQ(u)
cons16 = CustomConstraint(cons_f16)
def cons_f17(m):
return IntegerQ(m)
cons17 = CustomConstraint(cons_f17)
def cons_f18(m):
return Not(IntegerQ(m))
cons18 = CustomConstraint(cons_f18)
def cons_f19(n):
return PositiveIntegerQ(n + S(1)/2)
cons19 = CustomConstraint(cons_f19)
def cons_f20(m, n):
return IntegerQ(m + n)
cons20 = CustomConstraint(cons_f20)
def cons_f21(m, x):
return FreeQ(m, x)
cons21 = CustomConstraint(cons_f21)
def cons_f22(n):
return NegativeIntegerQ(n + S(-1)/2)
cons22 = CustomConstraint(cons_f22)
def cons_f23(n):
return Not(IntegerQ(n))
cons23 = CustomConstraint(cons_f23)
def cons_f24(m, n):
return Not(IntegerQ(m + n))
cons24 = CustomConstraint(cons_f24)
def cons_f25(d, c, b, a):
return ZeroQ(-a*d + b*c)
cons25 = CustomConstraint(cons_f25)
def cons_f26(b, d, c, n, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(Not(IntegerQ(n)), SimplerQ(c + d*x, a + b*x))
cons26 = CustomConstraint(cons_f26)
def cons_f27(d, x):
return FreeQ(d, x)
cons27 = CustomConstraint(cons_f27)
def cons_f28(d, b):
return PositiveQ(b/d)
cons28 = CustomConstraint(cons_f28)
def cons_f29(m, n):
return Not(Or(IntegerQ(m), IntegerQ(n)))
cons29 = CustomConstraint(cons_f29)
def cons_f30(d, m, n, b):
return Not(Or(IntegerQ(m), IntegerQ(n), PositiveQ(b/d)))
cons30 = CustomConstraint(cons_f30)
def cons_f31(m):
return RationalQ(m)
cons31 = CustomConstraint(cons_f31)
def cons_f32(m):
return LessEqual(m, S(-1))
cons32 = CustomConstraint(cons_f32)
def cons_f33(B, C, b, a, A):
return ZeroQ(A*b**S(2) - B*a*b + C*a**S(2))
cons33 = CustomConstraint(cons_f33)
def cons_f34(A, x):
return FreeQ(A, x)
cons34 = CustomConstraint(cons_f34)
def cons_f35(B, x):
return FreeQ(B, x)
cons35 = CustomConstraint(cons_f35)
def cons_f36(C, x):
return FreeQ(C, x)
cons36 = CustomConstraint(cons_f36)
def cons_f37(n, q):
return ZeroQ(n + q)
cons37 = CustomConstraint(cons_f37)
def cons_f38(p):
return IntegerQ(p)
cons38 = CustomConstraint(cons_f38)
def cons_f39(d, a, c, b):
return ZeroQ(a*c - b*d)
cons39 = CustomConstraint(cons_f39)
def cons_f40(m, n):
return Not(And(IntegerQ(m), NegQ(n)))
cons40 = CustomConstraint(cons_f40)
def cons_f41(m, p):
return ZeroQ(m + p)
cons41 = CustomConstraint(cons_f41)
def cons_f42(d, c, b, a):
return ZeroQ(a**S(2)*d + b**S(2)*c)
cons42 = CustomConstraint(cons_f42)
def cons_f43(a):
return PositiveQ(a)
cons43 = CustomConstraint(cons_f43)
def cons_f44(d):
return NegativeQ(d)
cons44 = CustomConstraint(cons_f44)
def cons_f45(c, a, b):
return ZeroQ(-S(4)*a*c + b**S(2))
cons45 = CustomConstraint(cons_f45)
def cons_f46(n, n2):
return ZeroQ(-S(2)*n + n2)
cons46 = CustomConstraint(cons_f46)
def cons_f47(d, c, b, e):
return ZeroQ(-b*e + S(2)*c*d)
cons47 = CustomConstraint(cons_f47)
def cons_f48(e, x):
return FreeQ(e, x)
cons48 = CustomConstraint(cons_f48)
def cons_f49(p, q):
return PosQ(-p + q)
cons49 = CustomConstraint(cons_f49)
def cons_f50(q, x):
return FreeQ(q, x)
cons50 = CustomConstraint(cons_f50)
def cons_f51(p, r):
return PosQ(-p + r)
cons51 = CustomConstraint(cons_f51)
def cons_f52(r, x):
return FreeQ(r, x)
cons52 = CustomConstraint(cons_f52)
def cons_f53(m, n):
return ZeroQ(m - n + S(1))
cons53 = CustomConstraint(cons_f53)
def cons_f54(p):
return NonzeroQ(p + S(1))
cons54 = CustomConstraint(cons_f54)
def cons_f55(b2, a2, a1, b1):
return ZeroQ(a1*b2 + a2*b1)
cons55 = CustomConstraint(cons_f55)
def cons_f56(m, n):
return ZeroQ(m - S(2)*n + S(1))
cons56 = CustomConstraint(cons_f56)
def cons_f57(a1, x):
return FreeQ(a1, x)
cons57 = CustomConstraint(cons_f57)
def cons_f58(b1, x):
return FreeQ(b1, x)
cons58 = CustomConstraint(cons_f58)
def cons_f59(a2, x):
return FreeQ(a2, x)
cons59 = CustomConstraint(cons_f59)
def cons_f60(b2, x):
return FreeQ(b2, x)
cons60 = CustomConstraint(cons_f60)
def cons_f61(x, Qm):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolyQ(Qm, x)
cons61 = CustomConstraint(cons_f61)
def cons_f62(m):
return PositiveIntegerQ(m)
cons62 = CustomConstraint(cons_f62)
def cons_f63(p):
return NegativeIntegerQ(p)
cons63 = CustomConstraint(cons_f63)
def cons_f64(x, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolyQ(Pq, x)
cons64 = CustomConstraint(cons_f64)
def cons_f65(x, Qr):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolyQ(Qr, x)
cons65 = CustomConstraint(cons_f65)
def cons_f66(m):
return NonzeroQ(m + S(1))
cons66 = CustomConstraint(cons_f66)
def cons_f67(x, a, b):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b), x)
cons67 = CustomConstraint(cons_f67)
def cons_f68(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearQ(u, x)
cons68 = CustomConstraint(cons_f68)
def cons_f69(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return NonzeroQ(u - x)
cons69 = CustomConstraint(cons_f69)
def cons_f70(d, c, b, a):
return ZeroQ(a*d + b*c)
cons70 = CustomConstraint(cons_f70)
def cons_f71(d, c, b, a):
return NonzeroQ(-a*d + b*c)
cons71 = CustomConstraint(cons_f71)
def cons_f72(m, n):
return ZeroQ(m + n + S(2))
cons72 = CustomConstraint(cons_f72)
def cons_f73(m):
return PositiveIntegerQ(m + S(1)/2)
cons73 = CustomConstraint(cons_f73)
def cons_f74(m):
return NegativeIntegerQ(m + S(3)/2)
cons74 = CustomConstraint(cons_f74)
def cons_f75(m, c, a):
return Or(IntegerQ(m), And(PositiveQ(a), PositiveQ(c)))
cons75 = CustomConstraint(cons_f75)
def cons_f76(a, c):
return ZeroQ(a + c)
cons76 = CustomConstraint(cons_f76)
def cons_f77(m):
return Not(IntegerQ(S(2)*m))
cons77 = CustomConstraint(cons_f77)
def cons_f78(d, c, a, b):
return PosQ(b*d/(a*c))
cons78 = CustomConstraint(cons_f78)
def cons_f79(m):
return IntegerQ(m + S(1)/2)
cons79 = CustomConstraint(cons_f79)
def cons_f80(n):
return IntegerQ(n + S(1)/2)
cons80 = CustomConstraint(cons_f80)
def cons_f81(m, n):
return Less(S(0), m, n)
cons81 = CustomConstraint(cons_f81)
def cons_f82(m, n):
return Less(m, n, S(0))
cons82 = CustomConstraint(cons_f82)
def cons_f83(m, c, n):
return Or(Not(IntegerQ(n)), And(ZeroQ(c), LessEqual(S(7)*m + S(4)*n, S(0))), Less(S(9)*m + S(5)*n + S(5), S(0)), Greater(m + n + S(2), S(0)))
cons83 = CustomConstraint(cons_f83)
def cons_f84(m):
return NegativeIntegerQ(m)
cons84 = CustomConstraint(cons_f84)
def cons_f85(n):
return IntegerQ(n)
cons85 = CustomConstraint(cons_f85)
def cons_f86(m, n):
return Not(And(PositiveIntegerQ(n), Less(m + n + S(2), S(0))))
cons86 = CustomConstraint(cons_f86)
def cons_f87(n):
return RationalQ(n)
cons87 = CustomConstraint(cons_f87)
def cons_f88(n):
return Greater(n, S(0))
cons88 = CustomConstraint(cons_f88)
def cons_f89(n):
return Less(n, S(-1))
cons89 = CustomConstraint(cons_f89)
def cons_f90(d, c, b, a):
return PosQ((-a*d + b*c)/b)
cons90 = CustomConstraint(cons_f90)
def cons_f91(d, c, b, a):
return NegQ((-a*d + b*c)/b)
cons91 = CustomConstraint(cons_f91)
def cons_f92(n):
return Less(S(-1), n, S(0))
cons92 = CustomConstraint(cons_f92)
def cons_f93(m, n):
return RationalQ(m, n)
cons93 = CustomConstraint(cons_f93)
def cons_f94(m):
return Less(m, S(-1))
cons94 = CustomConstraint(cons_f94)
def cons_f95(m, n):
return Not(And(IntegerQ(n), Not(IntegerQ(m))))
cons95 = CustomConstraint(cons_f95)
def cons_f96(m, n):
return Not(And(IntegerQ(m + n), LessEqual(m + n + S(2), S(0)), Or(FractionQ(m), GreaterEqual(m + S(2)*n + S(1), S(0)))))
cons96 = CustomConstraint(cons_f96)
def cons_f97(m, b, d, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return IntLinearcQ(a, b, c, d, m, n, x)
cons97 = CustomConstraint(cons_f97)
def cons_f98(m, a, n, c):
return Not(And(Less(n, S(-1)), Or(ZeroQ(a), And(NonzeroQ(c), Less(m, n), IntegerQ(n)))))
cons98 = CustomConstraint(cons_f98)
def cons_f99(m, n):
return Unequal(m + n + S(1), S(0))
cons99 = CustomConstraint(cons_f99)
def cons_f100(m, n):
return Not(And(PositiveIntegerQ(m), Or(Not(IntegerQ(n)), Less(S(0), m, n))))
cons100 = CustomConstraint(cons_f100)
def cons_f101(m, n):
return Not(And(IntegerQ(m + n), Less(m + n + S(2), S(0))))
cons101 = CustomConstraint(cons_f101)
def cons_f102(d, b):
return ZeroQ(b + d)
cons102 = CustomConstraint(cons_f102)
def cons_f103(a, c):
return PositiveQ(a + c)
cons103 = CustomConstraint(cons_f103)
def cons_f104(d, c, b, a):
return PositiveQ(-a*d + b*c)
cons104 = CustomConstraint(cons_f104)
def cons_f105(b):
return PositiveQ(b)
cons105 = CustomConstraint(cons_f105)
def cons_f106(d, b):
return ZeroQ(b - d)
cons106 = CustomConstraint(cons_f106)
def cons_f107(m):
return Less(S(-1), m, S(0))
cons107 = CustomConstraint(cons_f107)
def cons_f108(m):
return LessEqual(S(3), Denominator(m), S(4))
cons108 = CustomConstraint(cons_f108)
def cons_f109(d, b):
return PosQ(d/b)
cons109 = CustomConstraint(cons_f109)
def cons_f110(d, b):
return NegQ(d/b)
cons110 = CustomConstraint(cons_f110)
def cons_f111(m, n):
return Equal(m + n + S(1), S(0))
cons111 = CustomConstraint(cons_f111)
def cons_f112(m, n):
return LessEqual(Denominator(n), Denominator(m))
cons112 = CustomConstraint(cons_f112)
def cons_f113(m, n):
return NegativeIntegerQ(m + n + S(2))
cons113 = CustomConstraint(cons_f113)
def cons_f114(m, n):
return Or(SumSimplerQ(m, S(1)), Not(SumSimplerQ(n, S(1))))
cons114 = CustomConstraint(cons_f114)
def cons_f115(d, c, n, b):
return Or(IntegerQ(n), And(PositiveQ(c), Not(And(ZeroQ(n + S(1)/2), ZeroQ(c**S(2) - d**S(2)), PositiveQ(-d/(b*c))))))
cons115 = CustomConstraint(cons_f115)
def cons_f116(d, c, m, b):
return Or(IntegerQ(m), PositiveQ(-d/(b*c)))
cons116 = CustomConstraint(cons_f116)
def cons_f117(c):
return Not(PositiveQ(c))
cons117 = CustomConstraint(cons_f117)
def cons_f118(d, c, b):
return Not(PositiveQ(-d/(b*c)))
cons118 = CustomConstraint(cons_f118)
def cons_f119(d, m, n, c):
return Or(And(RationalQ(m), Not(And(ZeroQ(n + S(1)/2), ZeroQ(c**S(2) - d**S(2))))), Not(RationalQ(n)))
cons119 = CustomConstraint(cons_f119)
def cons_f120(d, c, b, a):
return PositiveQ(b/(-a*d + b*c))
cons120 = CustomConstraint(cons_f120)
def cons_f121(m, b, d, c, n, a):
return Or(RationalQ(m), Not(And(RationalQ(n), PositiveQ(-d/(-a*d + b*c)))))
cons121 = CustomConstraint(cons_f121)
def cons_f122(m, n):
return Or(RationalQ(m), Not(SimplerQ(n + S(1), m + S(1))))
cons122 = CustomConstraint(cons_f122)
def cons_f123(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return NonzeroQ(Coefficient(u, x, S(0)))
cons123 = CustomConstraint(cons_f123)
def cons_f124(m, n):
return ZeroQ(m - n)
cons124 = CustomConstraint(cons_f124)
def cons_f125(f, x):
return FreeQ(f, x)
cons125 = CustomConstraint(cons_f125)
def cons_f126(p, n):
return NonzeroQ(n + p + S(2))
cons126 = CustomConstraint(cons_f126)
def cons_f127(p, f, b, d, a, n, c, e):
return ZeroQ(a*d*f*(n + p + S(2)) - b*(c*f*(p + S(1)) + d*e*(n + S(1))))
cons127 = CustomConstraint(cons_f127)
def cons_f128(p):
return PositiveIntegerQ(p)
cons128 = CustomConstraint(cons_f128)
def cons_f129(a, f, b, e):
return ZeroQ(a*f + b*e)
cons129 = CustomConstraint(cons_f129)
def cons_f130(p, n):
return Not(And(NegativeIntegerQ(n + p + S(2)), Greater(n + S(2)*p, S(0))))
cons130 = CustomConstraint(cons_f130)
def cons_f131(p, n):
return Or(NonzeroQ(n + S(1)), Equal(p, S(1)))
cons131 = CustomConstraint(cons_f131)
def cons_f132(a, f, b, e):
return NonzeroQ(a*f + b*e)
cons132 = CustomConstraint(cons_f132)
def cons_f133(p, f, b, d, a, n, e):
return Or(Not(IntegerQ(n)), Less(S(5)*n + S(9)*p, S(0)), GreaterEqual(n + p + S(1), S(0)), And(GreaterEqual(n + p + S(2), S(0)), RationalQ(a, b, d, e, f)))
cons133 = CustomConstraint(cons_f133)
def cons_f134(p, f, b, d, c, n, a, e):
return Or(NegativeIntegerQ(n, p), ZeroQ(p + S(-1)), And(PositiveIntegerQ(p), Or(Not(IntegerQ(n)), LessEqual(S(5)*n + S(9)*p + S(10), S(0)), GreaterEqual(n + p + S(1), S(0)), And(GreaterEqual(n + p + S(2), S(0)), RationalQ(a, b, c, d, e, f)))))
cons134 = CustomConstraint(cons_f134)
def cons_f135(p, n):
return ZeroQ(n + p + S(2))
cons135 = CustomConstraint(cons_f135)
def cons_f136(p, n):
return Not(And(SumSimplerQ(n, S(1)), Not(SumSimplerQ(p, S(1)))))
cons136 = CustomConstraint(cons_f136)
def cons_f137(p):
return Less(p, S(-1))
cons137 = CustomConstraint(cons_f137)
def cons_f138(p, n, c, e):
return Or(Not(And(RationalQ(n), Less(n, S(-1)))), IntegerQ(p), Not(Or(IntegerQ(n), Not(Or(ZeroQ(e), Not(Or(ZeroQ(c), Less(p, n))))))))
cons138 = CustomConstraint(cons_f138)
def cons_f139(p):
return SumSimplerQ(p, S(1))
cons139 = CustomConstraint(cons_f139)
def cons_f140(p, n):
return NonzeroQ(n + p + S(3))
cons140 = CustomConstraint(cons_f140)
def cons_f141(p, f, b, d, a, n, c, e):
return ZeroQ(-b*(c*f*(p + S(1)) + d*e*(n + S(1)))*(a*d*f*(n + p + S(4)) - b*(c*f*(p + S(2)) + d*e*(n + S(2)))) + d*f*(a**S(2)*d*f*(n + p + S(3)) - b*(a*(c*f*(p + S(1)) + d*e*(n + S(1))) + b*c*e))*(n + p + S(2)))
cons141 = CustomConstraint(cons_f141)
def cons_f142(m, n):
return ZeroQ(m - n + S(-1))
cons142 = CustomConstraint(cons_f142)
def cons_f143(m):
return Not(PositiveIntegerQ(m))
cons143 = CustomConstraint(cons_f143)
def cons_f144(m, n, p):
return NonzeroQ(m + n + p + S(2))
cons144 = CustomConstraint(cons_f144)
def cons_f145(p):
return Less(S(0), p, S(1))
cons145 = CustomConstraint(cons_f145)
def cons_f146(p):
return Greater(p, S(1))
cons146 = CustomConstraint(cons_f146)
def cons_f147(p):
return Not(IntegerQ(p))
cons147 = CustomConstraint(cons_f147)
def cons_f148(n):
return PositiveIntegerQ(n)
cons148 = CustomConstraint(cons_f148)
def cons_f149(p):
return FractionQ(p)
cons149 = CustomConstraint(cons_f149)
def cons_f150(m, n):
return IntegersQ(m, n)
cons150 = CustomConstraint(cons_f150)
def cons_f151(m, p, n):
return Or(IntegerQ(p), And(Greater(m, S(0)), GreaterEqual(n, S(-1))))
cons151 = CustomConstraint(cons_f151)
def cons_f152(p, n):
return Or(And(RationalQ(n), Less(n, S(-1))), And(ZeroQ(n + p + S(3)), NonzeroQ(n + S(1)), Or(SumSimplerQ(n, S(1)), Not(SumSimplerQ(p, S(1))))))
cons152 = CustomConstraint(cons_f152)
def cons_f153(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f), x)
cons153 = CustomConstraint(cons_f153)
def cons_f154(f, b, d, c, a, e):
return ZeroQ(S(2)*b*d*e - f*(a*d + b*c))
cons154 = CustomConstraint(cons_f154)
def cons_f155(m, n):
return ZeroQ(m + n + S(1))
cons155 = CustomConstraint(cons_f155)
def cons_f156(b, d, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return SimplerQ(a + b*x, c + d*x)
cons156 = CustomConstraint(cons_f156)
def cons_f157(m, n, p):
return ZeroQ(m + n + p + S(2))
cons157 = CustomConstraint(cons_f157)
def cons_f158(m, p):
return Not(And(SumSimplerQ(p, S(1)), Not(SumSimplerQ(m, S(1)))))
cons158 = CustomConstraint(cons_f158)
def cons_f159(m, n, p):
return ZeroQ(m + n + p + S(3))
cons159 = CustomConstraint(cons_f159)
def cons_f160(p, m, f, b, d, a, c, n, e):
return ZeroQ(a*d*f*(m + S(1)) + b*c*f*(n + S(1)) + b*d*e*(p + S(1)))
cons160 = CustomConstraint(cons_f160)
def cons_f161(m):
return Or(And(RationalQ(m), Less(m, S(-1))), SumSimplerQ(m, S(1)))
cons161 = CustomConstraint(cons_f161)
def cons_f162(m, n, p):
return RationalQ(m, n, p)
cons162 = CustomConstraint(cons_f162)
def cons_f163(p):
return Greater(p, S(0))
cons163 = CustomConstraint(cons_f163)
def cons_f164(m, n, p):
return Or(IntegersQ(S(2)*m, S(2)*n, S(2)*p), IntegersQ(m, n + p), IntegersQ(p, m + n))
cons164 = CustomConstraint(cons_f164)
def cons_f165(n):
return Greater(n, S(1))
cons165 = CustomConstraint(cons_f165)
def cons_f166(m):
return Greater(m, S(1))
cons166 = CustomConstraint(cons_f166)
def cons_f167(m, n, p):
return NonzeroQ(m + n + p + S(1))
cons167 = CustomConstraint(cons_f167)
def cons_f168(m):
return Greater(m, S(0))
cons168 = CustomConstraint(cons_f168)
def cons_f169(m, n, p):
return Or(IntegersQ(S(2)*m, S(2)*n, S(2)*p), Or(IntegersQ(m, n + p), IntegersQ(p, m + n)))
cons169 = CustomConstraint(cons_f169)
def cons_f170(m, n, p):
return IntegersQ(S(2)*m, S(2)*n, S(2)*p)
cons170 = CustomConstraint(cons_f170)
def cons_f171(p, n):
return Or(IntegerQ(n), IntegersQ(S(2)*n, S(2)*p))
cons171 = CustomConstraint(cons_f171)
def cons_f172(m, n):
return PositiveIntegerQ(m + n + S(1))
cons172 = CustomConstraint(cons_f172)
def cons_f173(m, n):
return Or(And(RationalQ(m), Greater(m, S(0))), And(Not(RationalQ(m)), Or(SumSimplerQ(m, S(-1)), Not(SumSimplerQ(n, S(-1))))))
cons173 = CustomConstraint(cons_f173)
def cons_f174(d, c, f, e):
return PositiveQ(-f/(-c*f + d*e))
cons174 = CustomConstraint(cons_f174)
def cons_f175(d, c, f, e):
return Not(PositiveQ(-f/(-c*f + d*e)))
cons175 = CustomConstraint(cons_f175)
def cons_f176(d, c, f, e):
return NonzeroQ(-c*f + d*e)
cons176 = CustomConstraint(cons_f176)
def cons_f177(c):
return PositiveQ(c)
cons177 = CustomConstraint(cons_f177)
def cons_f178(e):
return PositiveQ(e)
cons178 = CustomConstraint(cons_f178)
def cons_f179(d, b):
return Not(NegativeQ(-b/d))
cons179 = CustomConstraint(cons_f179)
def cons_f180(d, b):
return NegativeQ(-b/d)
cons180 = CustomConstraint(cons_f180)
def cons_f181(c, e):
return Not(And(PositiveQ(c), PositiveQ(e)))
cons181 = CustomConstraint(cons_f181)
def cons_f182(a, f, b, e):
return PositiveQ(b/(-a*f + b*e))
cons182 = CustomConstraint(cons_f182)
def cons_f183(d, c, b, a):
return Not(NegativeQ(-(-a*d + b*c)/d))
cons183 = CustomConstraint(cons_f183)
def cons_f184(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(SimplerQ(c + d*x, a + b*x), PositiveQ(-d/(-a*d + b*c)), PositiveQ(d/(-c*f + d*e)), Not(NegativeQ((-a*d + b*c)/b))))
cons184 = CustomConstraint(cons_f184)
def cons_f185(f, b, d, c, a, e):
return Not(And(PositiveQ(b/(-a*d + b*c)), PositiveQ(b/(-a*f + b*e))))
cons185 = CustomConstraint(cons_f185)
def cons_f186(d, f, b):
return Or(PositiveQ(-b/d), NegativeQ(-b/f))
cons186 = CustomConstraint(cons_f186)
def cons_f187(d, f, b):
return Or(PosQ(-b/d), NegQ(-b/f))
cons187 = CustomConstraint(cons_f187)
def cons_f188(f, b, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return SimplerQ(a + b*x, e + f*x)
cons188 = CustomConstraint(cons_f188)
def cons_f189(f, b, d, c, a, e):
return Or(PositiveQ(-(-a*d + b*c)/d), NegativeQ(-(-a*f + b*e)/f))
cons189 = CustomConstraint(cons_f189)
def cons_f190(f, b, d, c, a, e):
return Or(PosQ(-(-a*d + b*c)/d), NegQ(-(-a*f + b*e)/f))
cons190 = CustomConstraint(cons_f190)
def cons_f191(f, b, d, c, a, e):
return ZeroQ(-a*d*f - b*c*f + S(2)*b*d*e)
cons191 = CustomConstraint(cons_f191)
def cons_f192(m, n):
return PositiveIntegerQ(m - n)
cons192 = CustomConstraint(cons_f192)
def cons_f193(m, n):
return Or(PositiveIntegerQ(m), NegativeIntegerQ(m, n))
cons193 = CustomConstraint(cons_f193)
def cons_f194(m, n, p):
return NegativeIntegerQ(m + n + p + S(2))
cons194 = CustomConstraint(cons_f194)
def cons_f195(m, n, p):
return Or(SumSimplerQ(m, S(1)), And(Not(And(NonzeroQ(n + S(1)), SumSimplerQ(n, S(1)))), Not(And(NonzeroQ(p + S(1)), SumSimplerQ(p, S(1))))))
cons195 = CustomConstraint(cons_f195)
def cons_f196(n):
return NegativeIntegerQ(n)
cons196 = CustomConstraint(cons_f196)
def cons_f197(p, e):
return Or(IntegerQ(p), PositiveQ(e))
cons197 = CustomConstraint(cons_f197)
def cons_f198(d, c, b):
return PositiveQ(-d/(b*c))
cons198 = CustomConstraint(cons_f198)
def cons_f199(p, f, d, c, e):
return Or(IntegerQ(p), PositiveQ(d/(-c*f + d*e)))
cons199 = CustomConstraint(cons_f199)
def cons_f200(b, d, a, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(PositiveQ(d/(a*d - b*c)), SimplerQ(c + d*x, a + b*x)))
cons200 = CustomConstraint(cons_f200)
def cons_f201(d, c, b, a):
return Not(PositiveQ(b/(-a*d + b*c)))
cons201 = CustomConstraint(cons_f201)
def cons_f202(b, d, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(SimplerQ(c + d*x, a + b*x))
cons202 = CustomConstraint(cons_f202)
def cons_f203(f, b, d, a, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(PositiveQ(d/(a*d - b*c)), PositiveQ(d/(-c*f + d*e)), SimplerQ(c + d*x, a + b*x)))
cons203 = CustomConstraint(cons_f203)
def cons_f204(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(PositiveQ(f/(a*f - b*e)), PositiveQ(f/(c*f - d*e)), SimplerQ(e + f*x, a + b*x)))
cons204 = CustomConstraint(cons_f204)
def cons_f205(a, f, b, e):
return Not(PositiveQ(b/(-a*f + b*e)))
cons205 = CustomConstraint(cons_f205)
def cons_f206(f, b, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(SimplerQ(e + f*x, a + b*x))
cons206 = CustomConstraint(cons_f206)
def cons_f207(m, n):
return Or(PositiveIntegerQ(m), IntegersQ(m, n))
cons207 = CustomConstraint(cons_f207)
def cons_f208(g, x):
return FreeQ(g, x)
cons208 = CustomConstraint(cons_f208)
def cons_f209(h, x):
return FreeQ(h, x)
cons209 = CustomConstraint(cons_f209)
def cons_f210(m, n):
return Not(And(SumSimplerQ(n, S(1)), Not(SumSimplerQ(m, S(1)))))
cons210 = CustomConstraint(cons_f210)
def cons_f211(m, n):
return Or(And(RationalQ(m), Less(m, S(-2))), And(ZeroQ(m + n + S(3)), Not(And(RationalQ(n), Less(n, S(-2))))))
cons211 = CustomConstraint(cons_f211)
def cons_f212(m):
return Or(And(RationalQ(m), Inequality(S(-2), LessEqual, m, Less, S(-1))), SumSimplerQ(m, S(1)))
cons212 = CustomConstraint(cons_f212)
def cons_f213(m, n):
return NonzeroQ(m + n + S(3))
cons213 = CustomConstraint(cons_f213)
def cons_f214(m, n):
return NonzeroQ(m + n + S(2))
cons214 = CustomConstraint(cons_f214)
def cons_f215(m, n, p):
return Or(IntegersQ(m, n, p), PositiveIntegerQ(n, p))
cons215 = CustomConstraint(cons_f215)
def cons_f216(f, b, g, d, c, a, x, h, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, h), x)
cons216 = CustomConstraint(cons_f216)
def cons_f217(p, f, b, g, d, c, a, n, x, h, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, h, n, p), x)
cons217 = CustomConstraint(cons_f217)
def cons_f218(f, d, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return SimplerQ(c + d*x, e + f*x)
cons218 = CustomConstraint(cons_f218)
def cons_f219(m, n, p):
return Or(SumSimplerQ(m, S(1)), And(Not(SumSimplerQ(n, S(1))), Not(SumSimplerQ(p, S(1)))))
cons219 = CustomConstraint(cons_f219)
def cons_f220(p, q):
return IntegersQ(p, q)
cons220 = CustomConstraint(cons_f220)
def cons_f221(q):
return PositiveIntegerQ(q)
cons221 = CustomConstraint(cons_f221)
def cons_f222(p, m, f, b, g, d, c, a, n, x, h, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, h, m, n, p, q), x)
cons222 = CustomConstraint(cons_f222)
def cons_f223(p, m, f, b, g, r, i, d, c, a, n, x, h, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, h, i, m, n, p, q, r), x)
cons223 = CustomConstraint(cons_f223)
def cons_f224(i, x):
return FreeQ(i, x)
cons224 = CustomConstraint(cons_f224)
def cons_f225(p):
return NonzeroQ(S(2)*p + S(1))
cons225 = CustomConstraint(cons_f225)
def cons_f226(c, a, b):
return NonzeroQ(-S(4)*a*c + b**S(2))
cons226 = CustomConstraint(cons_f226)
def cons_f227(c, a, b):
return PerfectSquareQ(-S(4)*a*c + b**S(2))
cons227 = CustomConstraint(cons_f227)
def cons_f228(c, a, b):
return Not(PerfectSquareQ(-S(4)*a*c + b**S(2)))
cons228 = CustomConstraint(cons_f228)
def cons_f229(p):
return IntegerQ(S(4)*p)
cons229 = CustomConstraint(cons_f229)
def cons_f230(p):
return Unequal(p, S(-3)/2)
cons230 = CustomConstraint(cons_f230)
def cons_f231(c, a, b):
return PosQ(-S(4)*a*c + b**S(2))
cons231 = CustomConstraint(cons_f231)
def cons_f232(c, a, b):
return PositiveQ(S(4)*a - b**S(2)/c)
cons232 = CustomConstraint(cons_f232)
def cons_f233(x, c, b):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, c), x)
cons233 = CustomConstraint(cons_f233)
def cons_f234(p):
return LessEqual(S(3), Denominator(p), S(4))
cons234 = CustomConstraint(cons_f234)
def cons_f235(p):
return Not(IntegerQ(S(4)*p))
cons235 = CustomConstraint(cons_f235)
def cons_f236(m):
return IntegerQ(m/S(2) + S(1)/2)
cons236 = CustomConstraint(cons_f236)
def cons_f237(m, p):
return ZeroQ(m + S(2)*p + S(1))
cons237 = CustomConstraint(cons_f237)
def cons_f238(m, p):
return NonzeroQ(m + S(2)*p + S(1))
cons238 = CustomConstraint(cons_f238)
def cons_f239(d, c, b, e):
return NonzeroQ(-b*e + S(2)*c*d)
cons239 = CustomConstraint(cons_f239)
def cons_f240(m, p):
return ZeroQ(m + S(2)*p + S(2))
cons240 = CustomConstraint(cons_f240)
def cons_f241(m):
return NonzeroQ(m + S(2))
cons241 = CustomConstraint(cons_f241)
def cons_f242(m, p):
return ZeroQ(m + S(2)*p + S(3))
cons242 = CustomConstraint(cons_f242)
def cons_f243(p):
return NonzeroQ(p + S(3)/2)
cons243 = CustomConstraint(cons_f243)
def cons_f244(m, p):
return RationalQ(m, p)
cons244 = CustomConstraint(cons_f244)
def cons_f245(m):
return Inequality(S(-2), LessEqual, m, Less, S(-1))
cons245 = CustomConstraint(cons_f245)
def cons_f246(p):
return IntegerQ(S(2)*p)
cons246 = CustomConstraint(cons_f246)
def cons_f247(m):
return Less(m, S(-2))
cons247 = CustomConstraint(cons_f247)
def cons_f248(m, p):
return Not(And(NegativeIntegerQ(m + S(2)*p + S(3)), Greater(m + S(3)*p + S(3), S(0))))
cons248 = CustomConstraint(cons_f248)
def cons_f249(m, p):
return NonzeroQ(m + S(2)*p)
cons249 = CustomConstraint(cons_f249)
def cons_f250(m):
return Not(And(RationalQ(m), Less(m, S(-2))))
cons250 = CustomConstraint(cons_f250)
def cons_f251(m, p):
return Not(And(IntegerQ(m), Less(S(0), m, S(2)*p)))
cons251 = CustomConstraint(cons_f251)
def cons_f252(m):
return Inequality(S(0), Less, m, LessEqual, S(1))
cons252 = CustomConstraint(cons_f252)
def cons_f253(m, p):
return NonzeroQ(m + p + S(1))
cons253 = CustomConstraint(cons_f253)
def cons_f254(m, p):
return Or(Not(RationalQ(p)), Inequality(S(-1), LessEqual, p, Less, S(0)), And(IntegerQ(m), Less(S(0), m, S(2)*p)), And(Equal(m, S(1)/2), Less(p, S(0))))
cons254 = CustomConstraint(cons_f254)
def cons_f255(m, p):
return Or(IntegerQ(m), IntegerQ(S(2)*p))
cons255 = CustomConstraint(cons_f255)
def cons_f256(b, d, c, a, e):
return ZeroQ(a*e**S(2) - b*d*e + c*d**S(2))
cons256 = CustomConstraint(cons_f256)
def cons_f257(d, c, e, a):
return ZeroQ(a*e**S(2) + c*d**S(2))
cons257 = CustomConstraint(cons_f257)
def cons_f258(d, m, a, p):
return Or(IntegerQ(p), And(PositiveQ(a), PositiveQ(d), IntegerQ(m + p)))
cons258 = CustomConstraint(cons_f258)
def cons_f259(m, p):
return Or(Less(S(0), -m, p), Less(p, -m, S(0)))
cons259 = CustomConstraint(cons_f259)
def cons_f260(m):
return Unequal(m, S(2))
cons260 = CustomConstraint(cons_f260)
def cons_f261(m):
return Unequal(m, S(-1))
cons261 = CustomConstraint(cons_f261)
def cons_f262(m, p):
return PositiveIntegerQ(m + p)
cons262 = CustomConstraint(cons_f262)
def cons_f263(m, p):
return NegativeIntegerQ(m + S(2)*p + S(2))
cons263 = CustomConstraint(cons_f263)
def cons_f264(m, p):
return Or(Less(m, S(-2)), ZeroQ(m + S(2)*p + S(1)))
cons264 = CustomConstraint(cons_f264)
def cons_f265(m, p):
return Or(Inequality(S(-2), LessEqual, m, Less, S(0)), Equal(m + p + S(1), S(0)))
cons265 = CustomConstraint(cons_f265)
def cons_f266(m):
return GreaterEqual(m, S(1))
cons266 = CustomConstraint(cons_f266)
def cons_f267(m):
return Less(m, S(0))
cons267 = CustomConstraint(cons_f267)
def cons_f268(d):
return PositiveQ(d)
cons268 = CustomConstraint(cons_f268)
def cons_f269(m, p):
return Not(And(ZeroQ(m + S(-3)), Unequal(p, S(1))))
cons269 = CustomConstraint(cons_f269)
def cons_f270(m, p):
return NonzeroQ(m + S(2)*p + S(3))
cons270 = CustomConstraint(cons_f270)
def cons_f271(m, p):
return Not(And(EvenQ(m), Less(m + S(2)*p + S(3), S(0))))
cons271 = CustomConstraint(cons_f271)
def cons_f272(m):
return Not(And(RationalQ(m), Less(m, S(-1))))
cons272 = CustomConstraint(cons_f272)
def cons_f273(m, p):
return Not(And(PositiveIntegerQ(m/S(2) + S(-1)/2), Or(Not(IntegerQ(p)), Less(m, S(2)*p))))
cons273 = CustomConstraint(cons_f273)
def cons_f274(m):
return Not(And(RationalQ(m), Greater(m, S(1))))
cons274 = CustomConstraint(cons_f274)
def cons_f275(c, b, a):
return NegativeQ(c/(-S(4)*a*c + b**S(2)))
cons275 = CustomConstraint(cons_f275)
def cons_f276(m):
return EqQ(m**S(2), S(1)/4)
cons276 = CustomConstraint(cons_f276)
def cons_f277(m, p):
return Or(IntegerQ(S(2)*p), And(IntegerQ(m), RationalQ(p)), OddQ(m))
cons277 = CustomConstraint(cons_f277)
def cons_f278(m, p):
return Or(IntegerQ(S(2)*p), And(IntegerQ(m), RationalQ(p)), IntegerQ(m/S(2) + p + S(3)/2))
cons278 = CustomConstraint(cons_f278)
def cons_f279(b, d, c, a, e):
return NonzeroQ(a*e**S(2) - b*d*e + c*d**S(2))
cons279 = CustomConstraint(cons_f279)
def cons_f280(d, c, e, a):
return NonzeroQ(a*e**S(2) + c*d**S(2))
cons280 = CustomConstraint(cons_f280)
def cons_f281(m, p):
return Not(And(ZeroQ(m + S(-1)), Greater(p, S(1))))
cons281 = CustomConstraint(cons_f281)
def cons_f282(c, a, b):
return NiceSqrtQ(-S(4)*a*c + b**S(2))
cons282 = CustomConstraint(cons_f282)
def cons_f283(a, c):
return NiceSqrtQ(-a*c)
cons283 = CustomConstraint(cons_f283)
def cons_f284(c, a, b):
return Not(NiceSqrtQ(-S(4)*a*c + b**S(2)))
cons284 = CustomConstraint(cons_f284)
def cons_f285(a, c):
return Not(NiceSqrtQ(-a*c))
cons285 = CustomConstraint(cons_f285)
def cons_f286(d, m):
return Or(NonzeroQ(d), Greater(m, S(2)))
cons286 = CustomConstraint(cons_f286)
def cons_f287(p):
return Not(And(RationalQ(p), LessEqual(p, S(-1))))
cons287 = CustomConstraint(cons_f287)
def cons_f288(d, a, b, e):
return ZeroQ(a*e + b*d)
cons288 = CustomConstraint(cons_f288)
def cons_f289(d, c, b, e):
return ZeroQ(b*e + c*d)
cons289 = CustomConstraint(cons_f289)
def cons_f290(m, p):
return PositiveIntegerQ(m - p + S(1))
cons290 = CustomConstraint(cons_f290)
def cons_f291(d, c, b, e):
return NonzeroQ(-b*e + c*d)
cons291 = CustomConstraint(cons_f291)
def cons_f292(m):
return Equal(m**S(2), S(1)/4)
cons292 = CustomConstraint(cons_f292)
def cons_f293(c):
return NegativeQ(c)
cons293 = CustomConstraint(cons_f293)
def cons_f294(b):
return RationalQ(b)
cons294 = CustomConstraint(cons_f294)
def cons_f295(m):
return ZeroQ(m**S(2) + S(-1)/4)
cons295 = CustomConstraint(cons_f295)
def cons_f296(m, p):
return Equal(m + S(2)*p + S(2), S(0))
cons296 = CustomConstraint(cons_f296)
def cons_f297(d, a, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, d, e), x)
cons297 = CustomConstraint(cons_f297)
def cons_f298(m, p):
return Or(IntegerQ(p), And(RationalQ(m), Less(m, S(-1))))
cons298 = CustomConstraint(cons_f298)
def cons_f299(m, p):
return Not(NegativeIntegerQ(m + S(2)*p + S(1)))
cons299 = CustomConstraint(cons_f299)
def cons_f300(p, m, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return IntQuadraticQ(a, b, c, d, e, m, p, x)
cons300 = CustomConstraint(cons_f300)
def cons_f301(p, m, d, a, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return IntQuadraticQ(a, S(0), c, d, e, m, p, x)
cons301 = CustomConstraint(cons_f301)
def cons_f302(m):
return Or(Not(RationalQ(m)), Less(m, S(1)))
cons302 = CustomConstraint(cons_f302)
def cons_f303(m, p):
return Not(NegativeIntegerQ(m + S(2)*p))
cons303 = CustomConstraint(cons_f303)
def cons_f304(m, p):
return Or(Less(m, S(1)), And(NegativeIntegerQ(m + S(2)*p + S(3)), Unequal(m, S(2))))
cons304 = CustomConstraint(cons_f304)
def cons_f305(m):
return If(RationalQ(m), Greater(m, S(1)), SumSimplerQ(m, S(-2)))
cons305 = CustomConstraint(cons_f305)
def cons_f306(p, m, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(And(RationalQ(m), Less(m, S(-1)), IntQuadraticQ(a, b, c, d, e, m, p, x)), And(SumSimplerQ(m, S(1)), IntegerQ(p), NonzeroQ(m + S(1))), And(NegativeIntegerQ(m + S(2)*p + S(3)), NonzeroQ(m + S(1))))
cons306 = CustomConstraint(cons_f306)
def cons_f307(p, m, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(And(RationalQ(m), Less(m, S(-1)), IntQuadraticQ(a, S(0), c, d, e, m, p, x)), And(SumSimplerQ(m, S(1)), IntegerQ(p), NonzeroQ(m + S(1))), And(NegativeIntegerQ(m + S(2)*p + S(3)), NonzeroQ(m + S(1))))
cons307 = CustomConstraint(cons_f307)
def cons_f308(b, d, c, a, e):
return ZeroQ(-S(3)*a*c*e**S(2) + b**S(2)*e**S(2) - b*c*d*e + c**S(2)*d**S(2))
cons308 = CustomConstraint(cons_f308)
def cons_f309(d, c, b, e):
return PosQ(c*e**S(2)*(-b*e + S(2)*c*d))
cons309 = CustomConstraint(cons_f309)
def cons_f310(d, c, e, a):
return ZeroQ(-S(3)*a*e**S(2) + c*d**S(2))
cons310 = CustomConstraint(cons_f310)
def cons_f311(d, c, b, e):
return NegQ(c*e**S(2)*(-b*e + S(2)*c*d))
cons311 = CustomConstraint(cons_f311)
def cons_f312(b, d, c, a, e):
return ZeroQ(S(9)*a*c*e**S(2) - S(2)*b**S(2)*e**S(2) - b*c*d*e + c**S(2)*d**S(2))
cons312 = CustomConstraint(cons_f312)
def cons_f313(c, a, b):
return Not(PositiveQ(S(4)*a - b**S(2)/c))
cons313 = CustomConstraint(cons_f313)
def cons_f314(p):
return Not(IntegerQ(S(2)*p))
cons314 = CustomConstraint(cons_f314)
def cons_f315(d, f, e, g):
return NonzeroQ(-d*g + e*f)
cons315 = CustomConstraint(cons_f315)
def cons_f316(c, f, b, g):
return ZeroQ(-b*g + S(2)*c*f)
cons316 = CustomConstraint(cons_f316)
def cons_f317(m):
return Not(And(RationalQ(m), Greater(m, S(0))))
cons317 = CustomConstraint(cons_f317)
def cons_f318(m, p):
return Or(Not(RationalQ(p)), And(Greater(p, S(0)), Or(Not(IntegerQ(m)), GreaterEqual(m, -S(2)*p + S(-2)), Less(m, -S(4)*p + S(-4)))))
cons318 = CustomConstraint(cons_f318)
def cons_f319(m, p):
return NonzeroQ(m + S(2)*p + S(2))
cons319 = CustomConstraint(cons_f319)
def cons_f320(m, p):
return Or(Not(RationalQ(p)), Less(m, S(2)*p + S(2)))
cons320 = CustomConstraint(cons_f320)
def cons_f321(c, f, b, g):
return NonzeroQ(-b*g + S(2)*c*f)
cons321 = CustomConstraint(cons_f321)
def cons_f322(p):
return Less(p, S(0))
cons322 = CustomConstraint(cons_f322)
def cons_f323(p, m, b, d, c, e):
return Or(And(ZeroQ(m + S(2)*p + S(2)), NonzeroQ(m + S(1))), And(ZeroQ(-b*e + S(2)*c*d), NonzeroQ(m + S(-1))))
cons323 = CustomConstraint(cons_f323)
def cons_f324(m, f, g, d, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(ZeroQ(m + S(-1)), SimplerQ(f + g*x, d + e*x)))
cons324 = CustomConstraint(cons_f324)
def cons_f325(d, m, a, p):
return Or(IntegerQ(p), And(PositiveQ(a), PositiveQ(d), ZeroQ(m + p)))
cons325 = CustomConstraint(cons_f325)
def cons_f326(p, m, g, b, f, d, c, e):
return ZeroQ(e*(p + S(1))*(-b*g + S(2)*c*f) + m*(c*e*f + g*(-b*e + c*d)))
cons326 = CustomConstraint(cons_f326)
def cons_f327(p, m, f, g, d, e):
return ZeroQ(S(2)*e*f*(p + S(1)) + m*(d*g + e*f))
cons327 = CustomConstraint(cons_f327)
def cons_f328(m):
return SumSimplerQ(m, S(-1))
cons328 = CustomConstraint(cons_f328)
def cons_f329(m, p):
return Or(And(RationalQ(m), Less(m, S(-1)), Not(PositiveIntegerQ(m + p + S(1)))), And(RationalQ(m, p), Less(m, S(0)), Less(p, S(-1))), ZeroQ(m + S(2)*p + S(2)))
cons329 = CustomConstraint(cons_f329)
def cons_f330(f, a, g, c):
return ZeroQ(a*g**S(2) + c*f**S(2))
cons330 = CustomConstraint(cons_f330)
def cons_f331(p):
return Less(p, S(-2))
cons331 = CustomConstraint(cons_f331)
def cons_f332(m, p):
return Or(Less(S(0), -m, p + S(1)), Less(p, -m, S(0)))
cons332 = CustomConstraint(cons_f332)
def cons_f333(p, n):
return NegativeIntegerQ(n + S(2)*p)
cons333 = CustomConstraint(cons_f333)
def cons_f334(f, g, b, d, c, e):
return ZeroQ(-b*e*g + c*d*g + c*e*f)
cons334 = CustomConstraint(cons_f334)
def cons_f335(m, n):
return NonzeroQ(m - n + S(-1))
cons335 = CustomConstraint(cons_f335)
def cons_f336(d, f, e, g):
return ZeroQ(d*g + e*f)
cons336 = CustomConstraint(cons_f336)
def cons_f337(m, n):
return ZeroQ(m - n + S(-2))
cons337 = CustomConstraint(cons_f337)
def cons_f338(p, n):
return RationalQ(n, p)
cons338 = CustomConstraint(cons_f338)
def cons_f339(p, n):
return Not(And(IntegerQ(n + p), LessEqual(n + p + S(2), S(0))))
cons339 = CustomConstraint(cons_f339)
def cons_f340(n):
return Not(PositiveIntegerQ(n))
cons340 = CustomConstraint(cons_f340)
def cons_f341(p, n):
return Not(And(IntegerQ(n + p), Less(n + p + S(2), S(0))))
cons341 = CustomConstraint(cons_f341)
def cons_f342(p, n):
return Or(IntegerQ(S(2)*p), IntegerQ(n))
cons342 = CustomConstraint(cons_f342)
def cons_f343(m, p):
return ZeroQ(m + p + S(-1))
cons343 = CustomConstraint(cons_f343)
def cons_f344(p, g, b, f, d, c, n, e):
return ZeroQ(b*e*g*(n + S(1)) - c*d*g*(S(2)*n + p + S(3)) + c*e*f*(p + S(1)))
cons344 = CustomConstraint(cons_f344)
def cons_f345(p, f, g, d, n, e):
return ZeroQ(-d*g*(S(2)*n + p + S(3)) + e*f*(p + S(1)))
cons345 = CustomConstraint(cons_f345)
def cons_f346(n):
return Not(And(RationalQ(n), Less(n, S(-1))))
cons346 = CustomConstraint(cons_f346)
def cons_f347(p):
return IntegerQ(p + S(-1)/2)
cons347 = CustomConstraint(cons_f347)
def cons_f348(m, p):
return Not(And(Less(m, S(0)), Less(p, S(0))))
cons348 = CustomConstraint(cons_f348)
def cons_f349(p):
return Unequal(p, S(1)/2)
cons349 = CustomConstraint(cons_f349)
def cons_f350(f, b, g, d, c, a, e):
return ZeroQ(-S(2)*a*e*g + b*(d*g + e*f) - S(2)*c*d*f)
cons350 = CustomConstraint(cons_f350)
def cons_f351(f, g, d, c, a, e):
return ZeroQ(a*e*g + c*d*f)
cons351 = CustomConstraint(cons_f351)
def cons_f352(m, b, d, c, e):
return Not(And(Equal(m, S(1)), Or(ZeroQ(d), ZeroQ(-b*e + S(2)*c*d))))
cons352 = CustomConstraint(cons_f352)
def cons_f353(d, m):
return Not(And(Equal(m, S(1)), ZeroQ(d)))
cons353 = CustomConstraint(cons_f353)
def cons_f354(p, g, b, f, d, a, c, e):
return ZeroQ(-S(2)*a*c*e*g + b**S(2)*e*g*(p + S(2)) + c*(S(2)*p + S(3))*(-b*(d*g + e*f) + S(2)*c*d*f))
cons354 = CustomConstraint(cons_f354)
def cons_f355(p, g, f, d, a, c, e):
return ZeroQ(a*e*g - c*d*f*(S(2)*p + S(3)))
cons355 = CustomConstraint(cons_f355)
def cons_f356(m):
return Not(RationalQ(m))
cons356 = CustomConstraint(cons_f356)
def cons_f357(p):
return Not(PositiveIntegerQ(p))
cons357 = CustomConstraint(cons_f357)
def cons_f358(m, p):
return ZeroQ(m - p)
cons358 = CustomConstraint(cons_f358)
def cons_f359(m, p):
return Less(m + S(2)*p, S(0))
cons359 = CustomConstraint(cons_f359)
def cons_f360(m, p):
return Not(NegativeIntegerQ(m + S(2)*p + S(3)))
cons360 = CustomConstraint(cons_f360)
def cons_f361(m, p):
return Or(And(RationalQ(m), Less(m, S(-1))), Equal(p, S(1)), And(IntegerQ(p), Not(RationalQ(m))))
cons361 = CustomConstraint(cons_f361)
def cons_f362(m, p):
return Or(IntegerQ(m), IntegerQ(p), IntegersQ(S(2)*m, S(2)*p))
cons362 = CustomConstraint(cons_f362)
def cons_f363(m, p):
return Or(IntegerQ(p), Not(RationalQ(m)), Inequality(S(-1), LessEqual, m, Less, S(0)))
cons363 = CustomConstraint(cons_f363)
def cons_f364(p, m, f, b, g, d, c, a, e):
return Or(And(Equal(m, S(2)), Equal(p, S(-3)), RationalQ(a, b, c, d, e, f, g)), Not(NegativeIntegerQ(m + S(2)*p + S(3))))
cons364 = CustomConstraint(cons_f364)
def cons_f365(p, m, f, g, d, c, a, e):
return Or(And(Equal(m, S(2)), Equal(p, S(-3)), RationalQ(a, c, d, e, f, g)), Not(NegativeIntegerQ(m + S(2)*p + S(3))))
cons365 = CustomConstraint(cons_f365)
def cons_f366(m, f, g, d, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(Equal(m, S(1)), SimplerQ(d + e*x, f + g*x)))
cons366 = CustomConstraint(cons_f366)
def cons_f367(m):
return FractionQ(m)
cons367 = CustomConstraint(cons_f367)
def cons_f368(m, f, g, d, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(Equal(m, S(1)), SimplerQ(f + g*x, d + e*x)))
cons368 = CustomConstraint(cons_f368)
def cons_f369(m, p):
return NegativeIntegerQ(m + S(2)*p + S(3))
cons369 = CustomConstraint(cons_f369)
def cons_f370(b, d, c, a, e):
return ZeroQ(S(4)*c*(a - d) - (b - e)**S(2))
cons370 = CustomConstraint(cons_f370)
def cons_f371(f, b, g, d, a, e):
return ZeroQ(e*f*(b - e) - S(2)*g*(-a*e + b*d))
cons371 = CustomConstraint(cons_f371)
def cons_f372(d, a, b, e):
return NonzeroQ(-a*e + b*d)
cons372 = CustomConstraint(cons_f372)
def cons_f373(f, g, a, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, f, g), x)
cons373 = CustomConstraint(cons_f373)
def cons_f374(f, g, a, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, e, f, g), x)
cons374 = CustomConstraint(cons_f374)
def cons_f375(m, n, p):
return IntegersQ(m, n, p)
cons375 = CustomConstraint(cons_f375)
def cons_f376(p, n):
return IntegersQ(n, p)
cons376 = CustomConstraint(cons_f376)
def cons_f377(d, m, f):
return Or(IntegerQ(m), And(PositiveQ(d), PositiveQ(f)))
cons377 = CustomConstraint(cons_f377)
def cons_f378(m, p, n):
return Or(IntegerQ(p), IntegersQ(m, n))
cons378 = CustomConstraint(cons_f378)
def cons_f379(p, m, f, g, a, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, e, f, g, m, p), x)
cons379 = CustomConstraint(cons_f379)
def cons_f380(p, m, f, b, g, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, m, n, p), x)
cons380 = CustomConstraint(cons_f380)
def cons_f381(p, m, f, g, d, a, c, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, d, e, f, g, m, n, p), x)
cons381 = CustomConstraint(cons_f381)
def cons_f382(d, c, f, a):
return ZeroQ(-a*f + c*d)
cons382 = CustomConstraint(cons_f382)
def cons_f383(d, a, b, e):
return ZeroQ(-a*e + b*d)
cons383 = CustomConstraint(cons_f383)
def cons_f384(f, c, p):
return Or(IntegerQ(p), PositiveQ(c/f))
cons384 = CustomConstraint(cons_f384)
def cons_f385(f, b, d, a, c, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(Not(IntegerQ(q)), LessEqual(LeafCount(d + e*x + f*x**S(2)), LeafCount(a + b*x + c*x**S(2))))
cons385 = CustomConstraint(cons_f385)
def cons_f386(q):
return Not(IntegerQ(q))
cons386 = CustomConstraint(cons_f386)
def cons_f387(c, f):
return Not(PositiveQ(c/f))
cons387 = CustomConstraint(cons_f387)
def cons_f388(f, b, d, a, c, q, e):
return ZeroQ(c*(-S(2)*d*f + e**S(2)*(q + S(2))) + f*(S(2)*q + S(3))*(S(2)*a*f - b*e))
cons388 = CustomConstraint(cons_f388)
def cons_f389(q):
return NonzeroQ(q + S(1))
cons389 = CustomConstraint(cons_f389)
def cons_f390(q):
return NonzeroQ(S(2)*q + S(3))
cons390 = CustomConstraint(cons_f390)
def cons_f391(f, d, a, c, q, e):
return ZeroQ(S(2)*a*f**S(2)*(S(2)*q + S(3)) + c*(-S(2)*d*f + e**S(2)*(q + S(2))))
cons391 = CustomConstraint(cons_f391)
def cons_f392(f, d, a, c, q):
return ZeroQ(S(2)*a*f*q + S(3)*a*f - c*d)
cons392 = CustomConstraint(cons_f392)
def cons_f393(q):
return PositiveIntegerQ(q + S(2))
cons393 = CustomConstraint(cons_f393)
def cons_f394(d, f, e):
return NonzeroQ(-S(4)*d*f + e**S(2))
cons394 = CustomConstraint(cons_f394)
def cons_f395(q):
return RationalQ(q)
cons395 = CustomConstraint(cons_f395)
def cons_f396(q):
return Less(q, S(-1))
cons396 = CustomConstraint(cons_f396)
def cons_f397(f, b, d, a, c, q, e):
return NonzeroQ(c*(-S(2)*d*f + e**S(2)*(q + S(2))) + f*(S(2)*q + S(3))*(S(2)*a*f - b*e))
cons397 = CustomConstraint(cons_f397)
def cons_f398(f, d, a, c, q, e):
return NonzeroQ(S(2)*a*f**S(2)*(S(2)*q + S(3)) + c*(-S(2)*d*f + e**S(2)*(q + S(2))))
cons398 = CustomConstraint(cons_f398)
def cons_f399(f, d, a, c, q):
return NonzeroQ(S(2)*a*f*q + S(3)*a*f - c*d)
cons399 = CustomConstraint(cons_f399)
def cons_f400(q):
return Not(PositiveIntegerQ(q))
cons400 = CustomConstraint(cons_f400)
def cons_f401(q):
return Not(And(RationalQ(q), LessEqual(q, S(-1))))
cons401 = CustomConstraint(cons_f401)
def cons_f402(p, q):
return RationalQ(p, q)
cons402 = CustomConstraint(cons_f402)
def cons_f403(q):
return Greater(q, S(0))
cons403 = CustomConstraint(cons_f403)
def cons_f404(f, b, d, c, a, e):
return NonzeroQ(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))
cons404 = CustomConstraint(cons_f404)
def cons_f405(p, q):
return Not(And(Not(IntegerQ(p)), IntegerQ(q), Less(q, S(-1))))
cons405 = CustomConstraint(cons_f405)
def cons_f406(f, b, d, c, a):
return NonzeroQ(b**S(2)*d*f + (-a*f + c*d)**S(2))
cons406 = CustomConstraint(cons_f406)
def cons_f407(f, d, a, c, e):
return NonzeroQ(a*c*e**S(2) + (-a*f + c*d)**S(2))
cons407 = CustomConstraint(cons_f407)
def cons_f408(p, q):
return NonzeroQ(p + q)
cons408 = CustomConstraint(cons_f408)
def cons_f409(p, q):
return NonzeroQ(S(2)*p + S(2)*q + S(1))
cons409 = CustomConstraint(cons_f409)
def cons_f410(c, f, b, e):
return ZeroQ(-b*f + c*e)
cons410 = CustomConstraint(cons_f410)
def cons_f411(c, f, b, e):
return NonzeroQ(-b*f + c*e)
cons411 = CustomConstraint(cons_f411)
def cons_f412(a, c):
return PosQ(-a*c)
cons412 = CustomConstraint(cons_f412)
def cons_f413(c, a, b):
return NegQ(-S(4)*a*c + b**S(2))
cons413 = CustomConstraint(cons_f413)
def cons_f414(a, c):
return NegQ(-a*c)
cons414 = CustomConstraint(cons_f414)
def cons_f415(p, f, b, d, c, a, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, p, q), x)
cons415 = CustomConstraint(cons_f415)
def cons_f416(p, f, d, a, c, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, d, e, f, p, q), x)
cons416 = CustomConstraint(cons_f416)
def cons_f417(g, b, c, a, h):
return ZeroQ(a*h**S(2) - b*g*h + c*g**S(2))
cons417 = CustomConstraint(cons_f417)
def cons_f418(g, f, d, c, a, h, e):
return ZeroQ(a**S(2)*f*h**S(2) - a*c*e*g*h + c**S(2)*d*g**S(2))
cons418 = CustomConstraint(cons_f418)
def cons_f419(h, c, g, a):
return ZeroQ(a*h**S(2) + c*g**S(2))
cons419 = CustomConstraint(cons_f419)
def cons_f420(f, g, d, c, a, h):
return ZeroQ(a**S(2)*f*h**S(2) + c**S(2)*d*g**S(2))
cons420 = CustomConstraint(cons_f420)
def cons_f421(f, b, c, a, e):
return ZeroQ(a*f**S(2) - b*e*f + c*e**S(2))
cons421 = CustomConstraint(cons_f421)
def cons_f422(c, f, e, a):
return ZeroQ(a*f**S(2) + c*e**S(2))
cons422 = CustomConstraint(cons_f422)
def cons_f423(p, m, f, b, g, c, h, e):
return ZeroQ(b*f*h*(m + p + S(2)) + c*(-e*h*(m + S(2)*p + S(3)) + S(2)*f*g*(p + S(1))))
cons423 = CustomConstraint(cons_f423)
def cons_f424(p, m, f, b, g, d, a, c, h):
return ZeroQ(b*f*g*(p + S(1)) + h*(a*f*(m + S(1)) - c*d*(m + S(2)*p + S(3))))
cons424 = CustomConstraint(cons_f424)
def cons_f425(p, m, f, g, c, h, e):
return ZeroQ(c*(-e*h*(m + S(2)*p + S(3)) + S(2)*f*g*(p + S(1))))
cons425 = CustomConstraint(cons_f425)
def cons_f426(p, m, f, d, a, c, h):
return ZeroQ(h*(a*f*(m + S(1)) - c*d*(m + S(2)*p + S(3))))
cons426 = CustomConstraint(cons_f426)
def cons_f427(p, m, f, b, g, c, h):
return ZeroQ(b*f*h*(m + p + S(2)) + S(2)*c*f*g*(p + S(1)))
cons427 = CustomConstraint(cons_f427)
def cons_f428(m, p):
return Or(IntegersQ(m, p), PositiveIntegerQ(p))
cons428 = CustomConstraint(cons_f428)
def cons_f429(g, b, c, a, h):
return NonzeroQ(a*h**S(2) - b*g*h + c*g**S(2))
cons429 = CustomConstraint(cons_f429)
def cons_f430(h, c, g, a):
return NonzeroQ(a*h**S(2) + c*g**S(2))
cons430 = CustomConstraint(cons_f430)
def cons_f431(g, b, c, a, h):
return NonzeroQ(c*g**S(2) - h*(-a*h + b*g))
cons431 = CustomConstraint(cons_f431)
def cons_f432(p, q):
return Or(Greater(p, S(0)), Greater(q, S(0)))
cons432 = CustomConstraint(cons_f432)
def cons_f433(p, q):
return NonzeroQ(p + q + S(1))
cons433 = CustomConstraint(cons_f433)
def cons_f434(a, c):
return PositiveQ(a*c)
cons434 = CustomConstraint(cons_f434)
def cons_f435(a, c):
return Not(PositiveQ(a*c))
cons435 = CustomConstraint(cons_f435)
def cons_f436(f, h, e, g):
return ZeroQ(e*h - S(2)*f*g)
cons436 = CustomConstraint(cons_f436)
def cons_f437(f, h, e, g):
return NonzeroQ(e*h - S(2)*f*g)
cons437 = CustomConstraint(cons_f437)
def cons_f438(d, h, e, g):
return ZeroQ(S(2)*d*h - e*g)
cons438 = CustomConstraint(cons_f438)
def cons_f439(d, h, e, g):
return NonzeroQ(S(2)*d*h - e*g)
cons439 = CustomConstraint(cons_f439)
def cons_f440(g, b, f, d, a, c, h, e):
return ZeroQ(g**S(2)*(-b*f + c*e) - S(2)*g*h*(-a*f + c*d) + h**S(2)*(-a*e + b*d))
cons440 = CustomConstraint(cons_f440)
def cons_f441(g, f, d, a, c, h, e):
return ZeroQ(a*e*h**S(2) - c*e*g**S(2) + S(2)*g*h*(-a*f + c*d))
cons441 = CustomConstraint(cons_f441)
def cons_f442(g, b, f, d, c, a, h):
return ZeroQ(b*d*h**S(2) - b*f*g**S(2) - S(2)*g*h*(-a*f + c*d))
cons442 = CustomConstraint(cons_f442)
def cons_f443(f, b, d, c, a):
return ZeroQ(c**S(2)*d - f*(-S(3)*a*c + b**S(2)))
cons443 = CustomConstraint(cons_f443)
def cons_f444(g, b, c, a, h):
return ZeroQ(S(9)*a*c*h**S(2) - S(2)*b**S(2)*h**S(2) - b*c*g*h + c**S(2)*g**S(2))
cons444 = CustomConstraint(cons_f444)
def cons_f445(c, h, b, g):
return PositiveQ(-S(9)*c*h**S(2)/(-b*h + S(2)*c*g)**S(2))
cons445 = CustomConstraint(cons_f445)
def cons_f446(d, c, f, a):
return ZeroQ(S(3)*a*f + c*d)
cons446 = CustomConstraint(cons_f446)
def cons_f447(h, c, g, a):
return ZeroQ(S(9)*a*h**S(2) + c*g**S(2))
cons447 = CustomConstraint(cons_f447)
def cons_f448(a):
return Not(PositiveQ(a))
cons448 = CustomConstraint(cons_f448)
def cons_f449(p, f, b, g, d, c, a, x, h, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, h, p, q), x)
cons449 = CustomConstraint(cons_f449)
def cons_f450(p, f, g, d, a, c, x, h, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, d, e, f, g, h, p, q), x)
cons450 = CustomConstraint(cons_f450)
def cons_f451(x, z):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearQ(z, x)
cons451 = CustomConstraint(cons_f451)
def cons_f452(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuadraticQ(List(u, v), x)
cons452 = CustomConstraint(cons_f452)
def cons_f453(x, z, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(LinearMatchQ(z, x), QuadraticMatchQ(List(u, v), x)))
cons453 = CustomConstraint(cons_f453)
def cons_f454(p, q):
return NonzeroQ(S(2)*p + S(2)*q + S(3))
cons454 = CustomConstraint(cons_f454)
def cons_f455(B, C, p, f, b, d, c, a, A, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, A, B, C, p, q), x)
cons455 = CustomConstraint(cons_f455)
def cons_f456(C, p, f, b, d, c, a, A, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, A, C, p, q), x)
cons456 = CustomConstraint(cons_f456)
def cons_f457(B, C, p, f, d, a, c, A, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, d, e, f, A, B, C, p, q), x)
cons457 = CustomConstraint(cons_f457)
def cons_f458(C, p, f, d, a, c, A, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, d, e, f, A, C, p, q), x)
cons458 = CustomConstraint(cons_f458)
def cons_f459(x, p, n, b):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, n, p), x)
cons459 = CustomConstraint(cons_f459)
def cons_f460(p, n):
return ZeroQ(p + S(1) + S(1)/n)
cons460 = CustomConstraint(cons_f460)
def cons_f461(p, n):
return NegativeIntegerQ(p + S(1) + S(1)/n)
cons461 = CustomConstraint(cons_f461)
def cons_f462(n):
return NonzeroQ(S(3)*n + S(1))
cons462 = CustomConstraint(cons_f462)
def cons_f463(n):
return Less(n, S(0))
cons463 = CustomConstraint(cons_f463)
def cons_f464(p, n):
return PositiveIntegerQ(n, p)
cons464 = CustomConstraint(cons_f464)
def cons_f465(p, n):
return Or(IntegerQ(S(2)*p), And(Equal(n, S(2)), IntegerQ(S(4)*p)), And(Equal(n, S(2)), IntegerQ(S(3)*p)), Less(Denominator(p + S(1)/n), Denominator(p)))
cons465 = CustomConstraint(cons_f465)
def cons_f466(a, b):
return PosQ(b/a)
cons466 = CustomConstraint(cons_f466)
def cons_f467(n):
return PositiveIntegerQ(n/S(2) + S(-3)/2)
cons467 = CustomConstraint(cons_f467)
def cons_f468(a, b):
return PosQ(a/b)
cons468 = CustomConstraint(cons_f468)
def cons_f469(a, b):
return NegQ(a/b)
cons469 = CustomConstraint(cons_f469)
def cons_f470(a, b):
return Or(PositiveQ(a), PositiveQ(b))
cons470 = CustomConstraint(cons_f470)
def cons_f471(a, b):
return Or(NegativeQ(a), NegativeQ(b))
cons471 = CustomConstraint(cons_f471)
def cons_f472(a, b):
return Or(PositiveQ(a), NegativeQ(b))
cons472 = CustomConstraint(cons_f472)
def cons_f473(a, b):
return Or(NegativeQ(a), PositiveQ(b))
cons473 = CustomConstraint(cons_f473)
def cons_f474(n):
return PositiveIntegerQ(n/S(4) + S(-1)/2)
cons474 = CustomConstraint(cons_f474)
def cons_f475(a, b):
try:
return Or(PositiveQ(a/b), And(PosQ(a/b), AtomQ(SplitProduct(SumBaseQ, a)), AtomQ(SplitProduct(SumBaseQ, b))))
except (TypeError, AttributeError):
return False
cons475 = CustomConstraint(cons_f475)
def cons_f476(a, b):
return Not(PositiveQ(a/b))
cons476 = CustomConstraint(cons_f476)
def cons_f477(n):
return PositiveIntegerQ(n/S(4) + S(-1))
cons477 = CustomConstraint(cons_f477)
def cons_f478(a, b):
return PositiveQ(a/b)
cons478 = CustomConstraint(cons_f478)
def cons_f479(b):
return PosQ(b)
cons479 = CustomConstraint(cons_f479)
def cons_f480(b):
return NegQ(b)
cons480 = CustomConstraint(cons_f480)
def cons_f481(a):
return PosQ(a)
cons481 = CustomConstraint(cons_f481)
def cons_f482(a):
return NegQ(a)
cons482 = CustomConstraint(cons_f482)
def cons_f483(a, b):
return NegQ(b/a)
cons483 = CustomConstraint(cons_f483)
def cons_f484(a):
return NegativeQ(a)
cons484 = CustomConstraint(cons_f484)
def cons_f485(p):
return Less(S(-1), p, S(0))
cons485 = CustomConstraint(cons_f485)
def cons_f486(p):
return Unequal(p, S(-1)/2)
cons486 = CustomConstraint(cons_f486)
def cons_f487(p, n):
return IntegerQ(p + S(1)/n)
cons487 = CustomConstraint(cons_f487)
def cons_f488(p, n):
return Less(Denominator(p + S(1)/n), Denominator(p))
cons488 = CustomConstraint(cons_f488)
def cons_f489(n):
return FractionQ(n)
cons489 = CustomConstraint(cons_f489)
def cons_f490(n):
return Not(IntegerQ(S(1)/n))
cons490 = CustomConstraint(cons_f490)
def cons_f491(p, n):
return Not(NegativeIntegerQ(p + S(1)/n))
cons491 = CustomConstraint(cons_f491)
def cons_f492(a, p):
return Or(IntegerQ(p), PositiveQ(a))
cons492 = CustomConstraint(cons_f492)
def cons_f493(a, p):
return Not(Or(IntegerQ(p), PositiveQ(a)))
cons493 = CustomConstraint(cons_f493)
def cons_f494(a2, p, a1):
return Or(IntegerQ(p), And(PositiveQ(a1), PositiveQ(a2)))
cons494 = CustomConstraint(cons_f494)
def cons_f495(n):
return PositiveIntegerQ(S(2)*n)
cons495 = CustomConstraint(cons_f495)
def cons_f496(p, n):
return Or(IntegerQ(S(2)*p), Less(Denominator(p + S(1)/n), Denominator(p)))
cons496 = CustomConstraint(cons_f496)
def cons_f497(n):
return NegativeIntegerQ(S(2)*n)
cons497 = CustomConstraint(cons_f497)
def cons_f498(n):
return FractionQ(S(2)*n)
cons498 = CustomConstraint(cons_f498)
def cons_f499(m, c):
return Or(IntegerQ(m), PositiveQ(c))
cons499 = CustomConstraint(cons_f499)
def cons_f500(m, n):
return IntegerQ((m + S(1))/n)
cons500 = CustomConstraint(cons_f500)
def cons_f501(m, n):
return Not(IntegerQ((m + S(1))/n))
cons501 = CustomConstraint(cons_f501)
def cons_f502(n):
return NegQ(n)
cons502 = CustomConstraint(cons_f502)
def cons_f503(m, n, p):
return ZeroQ(p + S(1) + (m + S(1))/n)
cons503 = CustomConstraint(cons_f503)
def cons_f504(m, n, p):
return ZeroQ(p + S(1) + (m + S(1))/(S(2)*n))
cons504 = CustomConstraint(cons_f504)
def cons_f505(m, n):
return IntegerQ((m + S(1))/(S(2)*n))
cons505 = CustomConstraint(cons_f505)
def cons_f506(m, n, p):
return NegativeIntegerQ((m + n*(p + S(1)) + S(1))/n)
cons506 = CustomConstraint(cons_f506)
def cons_f507(m, n, p):
return NegativeIntegerQ((m + S(2)*n*(p + S(1)) + S(1))/(S(2)*n))
cons507 = CustomConstraint(cons_f507)
def cons_f508(m, n, p):
return Not(NegativeIntegerQ((m + n*p + n + S(1))/n))
cons508 = CustomConstraint(cons_f508)
def cons_f509(p, m, b, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return IntBinomialQ(a, b, c, n, m, p, x)
cons509 = CustomConstraint(cons_f509)
def cons_f510(m, n, p):
return NonzeroQ(m + S(2)*n*p + S(1))
cons510 = CustomConstraint(cons_f510)
def cons_f511(a2, p, b2, m, b1, c, n, x, a1):
if isinstance(x, (int, Integer, float, Float)):
return False
return IntBinomialQ(a1*a2, b1*b2, c, n, m, p, x)
cons511 = CustomConstraint(cons_f511)
def cons_f512(m, n, p):
return NonzeroQ(m + n*p + S(1))
cons512 = CustomConstraint(cons_f512)
def cons_f513(m):
return PositiveIntegerQ(m/S(4) + S(-1)/2)
cons513 = CustomConstraint(cons_f513)
def cons_f514(m):
return NegativeIntegerQ(m/S(4) + S(-1)/2)
cons514 = CustomConstraint(cons_f514)
def cons_f515(m):
return IntegerQ(S(2)*m)
cons515 = CustomConstraint(cons_f515)
def cons_f516(m):
return Greater(m, S(3)/2)
cons516 = CustomConstraint(cons_f516)
def cons_f517(m, n):
return Greater(m + S(1), n)
cons517 = CustomConstraint(cons_f517)
def cons_f518(m, n, p):
return Not(NegativeIntegerQ((m + n*(p + S(1)) + S(1))/n))
cons518 = CustomConstraint(cons_f518)
def cons_f519(m, n):
return Greater(m + S(1), S(2)*n)
cons519 = CustomConstraint(cons_f519)
def cons_f520(m, n, p):
return Not(NegativeIntegerQ((m + S(2)*n*(p + S(1)) + S(1))/(S(2)*n)))
cons520 = CustomConstraint(cons_f520)
def cons_f521(n):
return PositiveIntegerQ(n/S(2) + S(-1)/2)
cons521 = CustomConstraint(cons_f521)
def cons_f522(m, n):
return Less(m, n + S(-1))
cons522 = CustomConstraint(cons_f522)
def cons_f523(m, n):
return PositiveIntegerQ(m, n/S(2) + S(-1)/2)
cons523 = CustomConstraint(cons_f523)
def cons_f524(m, n):
return PositiveIntegerQ(m, n/S(4) + S(-1)/2)
cons524 = CustomConstraint(cons_f524)
def cons_f525(m, n):
return PositiveIntegerQ(m, n/S(4))
cons525 = CustomConstraint(cons_f525)
def cons_f526(m, n):
return Less(m, n/S(2))
cons526 = CustomConstraint(cons_f526)
def cons_f527(m, n):
return Inequality(n/S(2), LessEqual, m, Less, n)
cons527 = CustomConstraint(cons_f527)
def cons_f528(m, n):
return PositiveIntegerQ(m, n)
cons528 = CustomConstraint(cons_f528)
def cons_f529(m, n):
return Greater(m, S(2)*n + S(-1))
cons529 = CustomConstraint(cons_f529)
def cons_f530(m, n):
return Greater(m, n + S(-1))
cons530 = CustomConstraint(cons_f530)
def cons_f531(m, n):
return SumSimplerQ(m, -n)
cons531 = CustomConstraint(cons_f531)
def cons_f532(m, n, p):
return NegativeIntegerQ((m + n*p + S(1))/n)
cons532 = CustomConstraint(cons_f532)
def cons_f533(m, n):
return SumSimplerQ(m, -S(2)*n)
cons533 = CustomConstraint(cons_f533)
def cons_f534(m, n, p):
return NegativeIntegerQ((m + S(2)*n*p + S(1))/(S(2)*n))
cons534 = CustomConstraint(cons_f534)
def cons_f535(m, n):
return SumSimplerQ(m, n)
cons535 = CustomConstraint(cons_f535)
def cons_f536(m, n):
return SumSimplerQ(m, S(2)*n)
cons536 = CustomConstraint(cons_f536)
def cons_f537(m, p, n):
return IntegersQ(m, p + (m + S(1))/n)
cons537 = CustomConstraint(cons_f537)
def cons_f538(m, p, n):
return IntegersQ(m, p + (m + S(1))/(S(2)*n))
cons538 = CustomConstraint(cons_f538)
def cons_f539(m, p, n):
return Less(Denominator(p + (m + S(1))/n), Denominator(p))
cons539 = CustomConstraint(cons_f539)
def cons_f540(m, p, n):
return Less(Denominator(p + (m + S(1))/(S(2)*n)), Denominator(p))
cons540 = CustomConstraint(cons_f540)
def cons_f541(m, n):
return IntegerQ(n/(m + S(1)))
cons541 = CustomConstraint(cons_f541)
def cons_f542(m, n):
return IntegerQ(S(2)*n/(m + S(1)))
cons542 = CustomConstraint(cons_f542)
def cons_f543(n):
return Not(IntegerQ(S(2)*n))
cons543 = CustomConstraint(cons_f543)
def cons_f544(m, n, p):
return ZeroQ(p + (m + S(1))/n)
cons544 = CustomConstraint(cons_f544)
def cons_f545(m, n, p):
return ZeroQ(p + (m + S(1))/(S(2)*n))
cons545 = CustomConstraint(cons_f545)
def cons_f546(m, p, n):
return IntegerQ(p + (m + S(1))/n)
cons546 = CustomConstraint(cons_f546)
def cons_f547(m, p, n):
return IntegerQ(p + (m + S(1))/(S(2)*n))
cons547 = CustomConstraint(cons_f547)
def cons_f548(m, n):
return FractionQ((m + S(1))/n)
cons548 = CustomConstraint(cons_f548)
def cons_f549(m, n):
return Or(SumSimplerQ(m, n), SumSimplerQ(m, -n))
cons549 = CustomConstraint(cons_f549)
def cons_f550(a, p):
return Or(NegativeIntegerQ(p), PositiveQ(a))
cons550 = CustomConstraint(cons_f550)
def cons_f551(a, p):
return Not(Or(NegativeIntegerQ(p), PositiveQ(a)))
cons551 = CustomConstraint(cons_f551)
def cons_f552(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearQ(v, x)
cons552 = CustomConstraint(cons_f552)
def cons_f553(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return NonzeroQ(v - x)
cons553 = CustomConstraint(cons_f553)
def cons_f554(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearPairQ(u, v, x)
cons554 = CustomConstraint(cons_f554)
def cons_f555(p, q):
return PositiveIntegerQ(p, q)
cons555 = CustomConstraint(cons_f555)
def cons_f556(p, n):
return ZeroQ(n*p + S(1))
cons556 = CustomConstraint(cons_f556)
def cons_f557(p, n, q):
return ZeroQ(n*(p + q + S(1)) + S(1))
cons557 = CustomConstraint(cons_f557)
def cons_f558(p, n, q):
return ZeroQ(n*(p + q + S(2)) + S(1))
cons558 = CustomConstraint(cons_f558)
def cons_f559(p, b, d, c, a, q):
return ZeroQ(a*d*(p + S(1)) + b*c*(q + S(1)))
cons559 = CustomConstraint(cons_f559)
def cons_f560(p, q):
return Or(And(RationalQ(p), Less(p, S(-1))), Not(And(RationalQ(q), Less(q, S(-1)))))
cons560 = CustomConstraint(cons_f560)
def cons_f561(p, b, d, c, a, n):
return ZeroQ(a*d - b*c*(n*(p + S(1)) + S(1)))
cons561 = CustomConstraint(cons_f561)
def cons_f562(p, n):
return Or(And(RationalQ(p), Less(p, S(-1))), NegativeIntegerQ(p + S(1)/n))
cons562 = CustomConstraint(cons_f562)
def cons_f563(p, n):
return NonzeroQ(n*(p + S(1)) + S(1))
cons563 = CustomConstraint(cons_f563)
def cons_f564(q):
return NegativeIntegerQ(q)
cons564 = CustomConstraint(cons_f564)
def cons_f565(p, q):
return GreaterEqual(p, -q)
cons565 = CustomConstraint(cons_f565)
def cons_f566(d, c, b, a):
return ZeroQ(S(3)*a*d + b*c)
cons566 = CustomConstraint(cons_f566)
def cons_f567(p):
return Or(Equal(p, S(1)/2), Equal(Denominator(p), S(4)))
cons567 = CustomConstraint(cons_f567)
def cons_f568(p):
return Equal(Denominator(p), S(4))
cons568 = CustomConstraint(cons_f568)
def cons_f569(p):
return Or(Equal(p, S(-5)/4), Equal(p, S(-7)/4))
cons569 = CustomConstraint(cons_f569)
def cons_f570(a, b):
return PosQ(a*b)
cons570 = CustomConstraint(cons_f570)
def cons_f571(a, b):
return NegQ(a*b)
cons571 = CustomConstraint(cons_f571)
def cons_f572(p):
return Or(Equal(p, S(3)/4), Equal(p, S(5)/4))
cons572 = CustomConstraint(cons_f572)
def cons_f573(d, c):
return PosQ(d/c)
cons573 = CustomConstraint(cons_f573)
def cons_f574(q):
return Less(S(0), q, S(1))
cons574 = CustomConstraint(cons_f574)
def cons_f575(p, b, d, c, a, n, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
return IntBinomialQ(a, b, c, d, n, p, q, x)
cons575 = CustomConstraint(cons_f575)
def cons_f576(q):
return Greater(q, S(1))
cons576 = CustomConstraint(cons_f576)
def cons_f577(p, q):
return Greater(p + q, S(0))
cons577 = CustomConstraint(cons_f577)
def cons_f578(p, n, q):
return NonzeroQ(n*(p + q) + S(1))
cons578 = CustomConstraint(cons_f578)
def cons_f579(p):
return Not(And(IntegerQ(p), Greater(p, S(1))))
cons579 = CustomConstraint(cons_f579)
def cons_f580(d, c, a, b):
return Not(SimplerSqrtQ(b/a, d/c))
cons580 = CustomConstraint(cons_f580)
def cons_f581(d, c):
return NegQ(d/c)
cons581 = CustomConstraint(cons_f581)
def cons_f582(d, c, a, b):
return Not(And(NegQ(b/a), SimplerSqrtQ(-b/a, -d/c)))
cons582 = CustomConstraint(cons_f582)
def cons_f583(d, c, a, b):
return PositiveQ(a - b*c/d)
cons583 = CustomConstraint(cons_f583)
def cons_f584(n):
return NonzeroQ(n + S(1))
cons584 = CustomConstraint(cons_f584)
def cons_f585(mn, n):
return EqQ(mn, -n)
cons585 = CustomConstraint(cons_f585)
def cons_f586(q):
return IntegerQ(q)
cons586 = CustomConstraint(cons_f586)
def cons_f587(p, n):
return Or(PosQ(n), Not(IntegerQ(p)))
cons587 = CustomConstraint(cons_f587)
def cons_f588(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return PseudoBinomialPairQ(u, v, x)
cons588 = CustomConstraint(cons_f588)
def cons_f589(m, p):
return IntegersQ(p, m/p)
cons589 = CustomConstraint(cons_f589)
def cons_f590(v, p, u, m, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return PseudoBinomialPairQ(u*x**(m/p), v, x)
cons590 = CustomConstraint(cons_f590)
def cons_f591(m, e):
return Or(IntegerQ(m), PositiveQ(e))
cons591 = CustomConstraint(cons_f591)
def cons_f592(p, m, b, d, a, c, n):
return ZeroQ(a*d*(m + S(1)) - b*c*(m + n*(p + S(1)) + S(1)))
cons592 = CustomConstraint(cons_f592)
def cons_f593(n, non2):
return ZeroQ(-n/S(2) + non2)
cons593 = CustomConstraint(cons_f593)
def cons_f594(a2, p, m, b2, b1, d, c, n, a1):
return ZeroQ(a1*a2*d*(m + S(1)) - b1*b2*c*(m + n*(p + S(1)) + S(1)))
cons594 = CustomConstraint(cons_f594)
def cons_f595(m, n, p):
return ZeroQ(m + n*(p + S(1)) + S(1))
cons595 = CustomConstraint(cons_f595)
def cons_f596(n, e):
return Or(IntegerQ(n), PositiveQ(e))
cons596 = CustomConstraint(cons_f596)
def cons_f597(m, n):
return Or(And(Greater(n, S(0)), Less(m, S(-1))), And(Less(n, S(0)), Greater(m + n, S(-1))))
cons597 = CustomConstraint(cons_f597)
def cons_f598(p):
return Not(And(IntegerQ(p), Less(p, S(-1))))
cons598 = CustomConstraint(cons_f598)
def cons_f599(m):
return PositiveIntegerQ(m/S(2))
cons599 = CustomConstraint(cons_f599)
def cons_f600(m, p):
return Or(IntegerQ(p), Equal(m + S(2)*p + S(1), S(0)))
cons600 = CustomConstraint(cons_f600)
def cons_f601(m):
return NegativeIntegerQ(m/S(2))
cons601 = CustomConstraint(cons_f601)
def cons_f602(m, p, n):
return Or(IntegerQ(p), Not(RationalQ(m)), And(PositiveIntegerQ(n), NegativeIntegerQ(p + S(1)/2), LessEqual(S(-1), m, -n*(p + S(1)))))
cons602 = CustomConstraint(cons_f602)
def cons_f603(m, n, p):
return NonzeroQ(m + n*(p + S(1)) + S(1))
cons603 = CustomConstraint(cons_f603)
def cons_f604(m):
return Or(IntegerQ(m), PositiveIntegerQ(S(2)*m + S(2)), Not(RationalQ(m)))
cons604 = CustomConstraint(cons_f604)
def cons_f605(m, n, p):
return NonzeroQ(m + n*(p + S(2)) + S(1))
cons605 = CustomConstraint(cons_f605)
def cons_f606(m, p, q):
return RationalQ(m, p, q)
cons606 = CustomConstraint(cons_f606)
def cons_f607(m, n):
return Greater(m - n + S(1), S(0))
cons607 = CustomConstraint(cons_f607)
def cons_f608(p, m, b, d, c, a, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return IntBinomialQ(a, b, c, d, e, m, n, p, q, x)
cons608 = CustomConstraint(cons_f608)
def cons_f609(m, n):
return Greater(m - n + S(1), n)
cons609 = CustomConstraint(cons_f609)
def cons_f610(m, n):
return Inequality(n, GreaterEqual, m - n + S(1), Greater, S(0))
cons610 = CustomConstraint(cons_f610)
def cons_f611(m, q):
return RationalQ(m, q)
cons611 = CustomConstraint(cons_f611)
def cons_f612(m, n):
return LessEqual(n, m, S(2)*n + S(-1))
cons612 = CustomConstraint(cons_f612)
def cons_f613(m, n):
return IntegersQ(m/S(2), n/S(2))
cons613 = CustomConstraint(cons_f613)
def cons_f614(m, n):
return Less(S(0), m - n + S(1), n)
cons614 = CustomConstraint(cons_f614)
def cons_f615(n):
return LessEqual(n, S(4))
cons615 = CustomConstraint(cons_f615)
def cons_f616(d, c, b, a):
return ZeroQ(-a*d + S(4)*b*c)
cons616 = CustomConstraint(cons_f616)
def cons_f617(m):
return PositiveIntegerQ(m/S(3) + S(-1)/3)
cons617 = CustomConstraint(cons_f617)
def cons_f618(m):
return NegativeIntegerQ(m/S(3) + S(-1)/3)
cons618 = CustomConstraint(cons_f618)
def cons_f619(m):
return IntegerQ(m/S(3) + S(-1)/3)
cons619 = CustomConstraint(cons_f619)
def cons_f620(n):
return Or(EqQ(n, S(2)), EqQ(n, S(4)))
cons620 = CustomConstraint(cons_f620)
def cons_f621(b, d, c, a, n):
return Not(And(EqQ(n, S(2)), SimplerSqrtQ(-b/a, -d/c)))
cons621 = CustomConstraint(cons_f621)
def cons_f622(m, p, q, n):
return IntegersQ(p + (m + S(1))/n, q)
cons622 = CustomConstraint(cons_f622)
def cons_f623(m, n):
return Or(ZeroQ(m - n), ZeroQ(m - S(2)*n + S(1)))
cons623 = CustomConstraint(cons_f623)
def cons_f624(m, p, q):
return IntegersQ(m, p, q)
cons624 = CustomConstraint(cons_f624)
def cons_f625(p):
return GreaterEqual(p, S(-2))
cons625 = CustomConstraint(cons_f625)
def cons_f626(m, q):
return Or(GreaterEqual(q, S(-2)), And(Equal(q, S(-3)), IntegerQ(m/S(2) + S(-1)/2)))
cons626 = CustomConstraint(cons_f626)
def cons_f627(m, n):
return NonzeroQ(m - n + S(1))
cons627 = CustomConstraint(cons_f627)
def cons_f628(r, p, q):
return PositiveIntegerQ(p, q, r)
cons628 = CustomConstraint(cons_f628)
def cons_f629(f, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, n), x)
cons629 = CustomConstraint(cons_f629)
def cons_f630(b, d, c, a, n):
return Not(And(ZeroQ(n + S(-2)), Or(And(PosQ(b/a), PosQ(d/c)), And(NegQ(b/a), Or(PosQ(d/c), And(PositiveQ(a), Or(Not(PositiveQ(c)), SimplerSqrtQ(-b/a, -d/c))))))))
cons630 = CustomConstraint(cons_f630)
def cons_f631(p, n, q):
return NonzeroQ(n*(p + q + S(1)) + S(1))
cons631 = CustomConstraint(cons_f631)
def cons_f632(p, f, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, p, n), x)
cons632 = CustomConstraint(cons_f632)
def cons_f633(p, f, b, d, c, a, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, n, p, q), x)
cons633 = CustomConstraint(cons_f633)
def cons_f634(d, c):
return PositiveQ(d/c)
cons634 = CustomConstraint(cons_f634)
def cons_f635(f, e):
return PositiveQ(f/e)
cons635 = CustomConstraint(cons_f635)
def cons_f636(d, c, f, e):
return Not(SimplerSqrtQ(d/c, f/e))
cons636 = CustomConstraint(cons_f636)
def cons_f637(d, c, f, e):
return Not(SimplerSqrtQ(-f/e, -d/c))
cons637 = CustomConstraint(cons_f637)
def cons_f638(f, e):
return PosQ(f/e)
cons638 = CustomConstraint(cons_f638)
def cons_f639(d, c, f, e):
return Not(And(NegQ(f/e), SimplerSqrtQ(-f/e, -d/c)))
cons639 = CustomConstraint(cons_f639)
def cons_f640(r, q):
return RationalQ(q, r)
cons640 = CustomConstraint(cons_f640)
def cons_f641(r):
return Greater(r, S(1))
cons641 = CustomConstraint(cons_f641)
def cons_f642(q):
return LessEqual(q, S(-1))
cons642 = CustomConstraint(cons_f642)
def cons_f643(d, c, f, e):
return PosQ((-c*f + d*e)/c)
cons643 = CustomConstraint(cons_f643)
def cons_f644(d, c, f, e):
return NegQ((-c*f + d*e)/c)
cons644 = CustomConstraint(cons_f644)
def cons_f645(p, f, b, r, d, c, a, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, n, p, q, r), x)
cons645 = CustomConstraint(cons_f645)
def cons_f646(v, u):
return ZeroQ(u - v)
cons646 = CustomConstraint(cons_f646)
def cons_f647(w, u):
return ZeroQ(u - w)
cons647 = CustomConstraint(cons_f647)
def cons_f648(r):
return IntegerQ(r)
cons648 = CustomConstraint(cons_f648)
def cons_f649(n, n2):
return ZeroQ(-n/S(2) + n2)
cons649 = CustomConstraint(cons_f649)
def cons_f650(e2, f1, f2, e1):
return ZeroQ(e1*f2 + e2*f1)
cons650 = CustomConstraint(cons_f650)
def cons_f651(e2, r, e1):
return Or(IntegerQ(r), And(PositiveQ(e1), PositiveQ(e2)))
cons651 = CustomConstraint(cons_f651)
def cons_f652(e1, x):
return FreeQ(e1, x)
cons652 = CustomConstraint(cons_f652)
def cons_f653(f1, x):
return FreeQ(f1, x)
cons653 = CustomConstraint(cons_f653)
def cons_f654(e2, x):
return FreeQ(e2, x)
cons654 = CustomConstraint(cons_f654)
def cons_f655(f2, x):
return FreeQ(f2, x)
cons655 = CustomConstraint(cons_f655)
def cons_f656(m, g):
return Or(IntegerQ(m), PositiveQ(g))
cons656 = CustomConstraint(cons_f656)
def cons_f657(r, p, q):
return PositiveIntegerQ(p + S(2), q, r)
cons657 = CustomConstraint(cons_f657)
def cons_f658(r, p, q):
return IntegersQ(p, q, r)
cons658 = CustomConstraint(cons_f658)
def cons_f659(f, b, d, c, a, q, e):
return Not(And(Equal(q, S(1)), SimplerQ(-a*d + b*c, -a*f + b*e)))
cons659 = CustomConstraint(cons_f659)
def cons_f660(f, d, c, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(Equal(q, S(1)), SimplerQ(e + f*x**n, c + d*x**n)))
cons660 = CustomConstraint(cons_f660)
def cons_f661(r):
return PositiveIntegerQ(r)
cons661 = CustomConstraint(cons_f661)
def cons_f662(p, m, f, b, g, d, c, a, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, m, n, p, q), x)
cons662 = CustomConstraint(cons_f662)
def cons_f663(p, m, f, b, g, r, d, c, a, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, m, n, p, q, r), x)
cons663 = CustomConstraint(cons_f663)
def cons_f664(p, n):
return ZeroQ(n*(S(2)*p + S(1)) + S(1))
cons664 = CustomConstraint(cons_f664)
def cons_f665(p, n):
return ZeroQ(S(2)*n*(p + S(1)) + S(1))
cons665 = CustomConstraint(cons_f665)
def cons_f666(p, n):
return Or(ZeroQ(S(2)*n*p + S(1)), ZeroQ(n*(S(2)*p + S(-1)) + S(1)))
cons666 = CustomConstraint(cons_f666)
def cons_f667(p):
return IntegerQ(p + S(1)/2)
cons667 = CustomConstraint(cons_f667)
def cons_f668(n):
return NonzeroQ(S(2)*n + S(1))
cons668 = CustomConstraint(cons_f668)
def cons_f669(p, n):
return NonzeroQ(S(2)*n*p + S(1))
cons669 = CustomConstraint(cons_f669)
def cons_f670(p, n):
return NonzeroQ(n*(S(2)*p + S(-1)) + S(1))
cons670 = CustomConstraint(cons_f670)
def cons_f671(p, n):
return NonzeroQ(n*(S(2)*p + S(1)) + S(1))
cons671 = CustomConstraint(cons_f671)
def cons_f672(p, n):
return NonzeroQ(S(2)*n*(p + S(1)) + S(1))
cons672 = CustomConstraint(cons_f672)
def cons_f673(p, n):
return Or(IntegerQ(p), ZeroQ(n + S(-2)))
cons673 = CustomConstraint(cons_f673)
def cons_f674(n):
return PositiveIntegerQ(n/S(2))
cons674 = CustomConstraint(cons_f674)
def cons_f675(c, a, b):
return PositiveQ(-S(4)*a*c + b**S(2))
cons675 = CustomConstraint(cons_f675)
def cons_f676(c, a):
return PositiveQ(c/a)
cons676 = CustomConstraint(cons_f676)
def cons_f677(a, b):
return NegativeQ(b/a)
cons677 = CustomConstraint(cons_f677)
def cons_f678(c, a):
return PosQ(c/a)
cons678 = CustomConstraint(cons_f678)
def cons_f679(c, a):
return NegQ(c/a)
cons679 = CustomConstraint(cons_f679)
def cons_f680(n, n2):
return EqQ(n2, S(2)*n)
cons680 = CustomConstraint(cons_f680)
def cons_f681(n):
return PosQ(n)
cons681 = CustomConstraint(cons_f681)
def cons_f682(m, n, p):
return ZeroQ(m + n*(S(2)*p + S(1)) + S(1))
cons682 = CustomConstraint(cons_f682)
def cons_f683(m, n):
return NonzeroQ(m + n + S(1))
cons683 = CustomConstraint(cons_f683)
def cons_f684(m, n, p):
return ZeroQ(m + S(2)*n*(p + S(1)) + S(1))
cons684 = CustomConstraint(cons_f684)
def cons_f685(m, n):
return Inequality(S(-1), LessEqual, m + n, Less, S(0))
cons685 = CustomConstraint(cons_f685)
def cons_f686(m, n):
return Less(m + n, S(-1))
cons686 = CustomConstraint(cons_f686)
def cons_f687(m, n, p):
return Not(And(NegativeIntegerQ((m + S(2)*n*(p + S(1)) + S(1))/n), Greater(p + (m + S(2)*n*(p + S(1)) + S(1))/n, S(0))))
cons687 = CustomConstraint(cons_f687)
def cons_f688(m, n, p):
return NonzeroQ(m + n*(S(2)*p + S(-1)) + S(1))
cons688 = CustomConstraint(cons_f688)
def cons_f689(m, n, p):
return Not(And(PositiveIntegerQ(m), IntegerQ((m + S(1))/n), Less(S(-1) + (m + S(1))/n, S(2)*p)))
cons689 = CustomConstraint(cons_f689)
def cons_f690(m, n):
return Inequality(n + S(-1), Less, m, LessEqual, S(2)*n + S(-1))
cons690 = CustomConstraint(cons_f690)
def cons_f691(m, p, n):
return Or(IntegerQ(S(2)*p), PositiveIntegerQ((m + S(1))/n))
cons691 = CustomConstraint(cons_f691)
def cons_f692(m, n, p):
return Unequal(m + S(2)*n*p + S(1), S(0))
cons692 = CustomConstraint(cons_f692)
def cons_f693(m, n, p):
return Unequal(m + n*(S(2)*p + S(-1)) + S(1), S(0))
cons693 = CustomConstraint(cons_f693)
def cons_f694(m, p, n):
return Or(IntegerQ(p), And(IntegerQ(S(2)*p), IntegerQ(m), Equal(n, S(2))))
cons694 = CustomConstraint(cons_f694)
def cons_f695(m, n):
return Greater(m, S(3)*n + S(-1))
cons695 = CustomConstraint(cons_f695)
def cons_f696(c, a, b):
return NegativeQ(-S(4)*a*c + b**S(2))
cons696 = CustomConstraint(cons_f696)
def cons_f697(a, c):
return PosQ(a*c)
cons697 = CustomConstraint(cons_f697)
def cons_f698(m, n):
return PositiveIntegerQ(n/S(2), m)
cons698 = CustomConstraint(cons_f698)
def cons_f699(m, n):
return Inequality(S(3)*n/S(2), LessEqual, m, Less, S(2)*n)
cons699 = CustomConstraint(cons_f699)
def cons_f700(m, n):
return Inequality(n/S(2), LessEqual, m, Less, S(3)*n/S(2))
cons700 = CustomConstraint(cons_f700)
def cons_f701(m, n):
return GreaterEqual(m, n)
cons701 = CustomConstraint(cons_f701)
def cons_f702(p):
return NegativeIntegerQ(p + S(1))
cons702 = CustomConstraint(cons_f702)
def cons_f703(p, b, d, c, n, e):
return ZeroQ(b*e*(n*p + S(1)) - c*d*(n*(S(2)*p + S(1)) + S(1)))
cons703 = CustomConstraint(cons_f703)
def cons_f704(p, b, d, c, n, e):
return NonzeroQ(b*e*(n*p + S(1)) - c*d*(n*(S(2)*p + S(1)) + S(1)))
cons704 = CustomConstraint(cons_f704)
def cons_f705(d, c, e, a):
return ZeroQ(-a*e**S(2) + c*d**S(2))
cons705 = CustomConstraint(cons_f705)
def cons_f706(d, e):
return PosQ(d*e)
cons706 = CustomConstraint(cons_f706)
def cons_f707(d, e):
return NegQ(d*e)
cons707 = CustomConstraint(cons_f707)
def cons_f708(d, c, e, a):
return NonzeroQ(-a*e**S(2) + c*d**S(2))
cons708 = CustomConstraint(cons_f708)
def cons_f709(a, c):
return NegQ(a*c)
cons709 = CustomConstraint(cons_f709)
def cons_f710(a, n, c):
return Or(PosQ(a*c), Not(IntegerQ(n)))
cons710 = CustomConstraint(cons_f710)
def cons_f711(b, d, c, a, e):
return Or(PositiveQ(-b/c + S(2)*d/e), And(Not(NegativeQ(-b/c + S(2)*d/e)), ZeroQ(d - e*Rt(a/c, S(2)))))
cons711 = CustomConstraint(cons_f711)
def cons_f712(c, a, b):
return Not(PositiveQ(-S(4)*a*c + b**S(2)))
cons712 = CustomConstraint(cons_f712)
def cons_f713(c, a, n, b):
return Or(PosQ(-S(4)*a*c + b**S(2)), Not(PositiveIntegerQ(n/S(2))))
cons713 = CustomConstraint(cons_f713)
def cons_f714(p, n):
return NonzeroQ(S(2)*n*p + n + S(1))
cons714 = CustomConstraint(cons_f714)
def cons_f715(a, c):
return PositiveQ(-a*c)
cons715 = CustomConstraint(cons_f715)
def cons_f716(p, n, q):
return NonzeroQ(S(2)*n*p + n*q + S(1))
cons716 = CustomConstraint(cons_f716)
def cons_f717(p):
return PositiveIntegerQ(p + S(-1)/2)
cons717 = CustomConstraint(cons_f717)
def cons_f718(c):
return Not(NegativeQ(c))
cons718 = CustomConstraint(cons_f718)
def cons_f719(p):
return NegativeIntegerQ(p + S(1)/2)
cons719 = CustomConstraint(cons_f719)
def cons_f720(d, c, b, e):
return ZeroQ(-b*e + c*d)
cons720 = CustomConstraint(cons_f720)
def cons_f721(d, a):
return Not(And(PositiveQ(a), PositiveQ(d)))
cons721 = CustomConstraint(cons_f721)
def cons_f722(p, q, n):
return Or(And(IntegersQ(p, q), Not(IntegerQ(n))), PositiveIntegerQ(p), And(PositiveIntegerQ(q), Not(IntegerQ(n))))
cons722 = CustomConstraint(cons_f722)
def cons_f723(p, n):
return Not(IntegersQ(n, S(2)*p))
cons723 = CustomConstraint(cons_f723)
def cons_f724(n, q):
return Not(IntegersQ(n, q))
cons724 = CustomConstraint(cons_f724)
def cons_f725(mn, n2):
return EqQ(n2, -S(2)*mn)
cons725 = CustomConstraint(cons_f725)
def cons_f726(mn, x):
return FreeQ(mn, x)
cons726 = CustomConstraint(cons_f726)
def cons_f727(n2):
return PosQ(n2)
cons727 = CustomConstraint(cons_f727)
def cons_f728(n2):
return NegQ(n2)
cons728 = CustomConstraint(cons_f728)
def cons_f729(e2, d2, d1, e1):
return ZeroQ(d1*e2 + d2*e1)
cons729 = CustomConstraint(cons_f729)
def cons_f730(d2, q, d1):
return Or(IntegerQ(q), And(PositiveQ(d1), PositiveQ(d2)))
cons730 = CustomConstraint(cons_f730)
def cons_f731(d1, x):
return FreeQ(d1, x)
cons731 = CustomConstraint(cons_f731)
def cons_f732(d2, x):
return FreeQ(d2, x)
cons732 = CustomConstraint(cons_f732)
def cons_f733(m, f):
return Or(IntegerQ(m), PositiveQ(f))
cons733 = CustomConstraint(cons_f733)
def cons_f734(m, n):
return PositiveIntegerQ(m, n, (m + S(1))/n)
cons734 = CustomConstraint(cons_f734)
def cons_f735(m, q):
return IntegersQ(m, q)
cons735 = CustomConstraint(cons_f735)
def cons_f736(p, n):
return Greater(S(2)*n*p, n + S(-1))
cons736 = CustomConstraint(cons_f736)
def cons_f737(m, n, p, q):
return NonzeroQ(m + S(2)*n*p + n*q + S(1))
cons737 = CustomConstraint(cons_f737)
def cons_f738(m, n, p):
return Unequal(m + n*(S(2)*p + S(1)) + S(1), S(0))
cons738 = CustomConstraint(cons_f738)
def cons_f739(m, n, p):
return NonzeroQ(m + n*(S(2)*p + S(1)) + S(1))
cons739 = CustomConstraint(cons_f739)
def cons_f740(m, n):
return IntegersQ(m, n/S(2))
cons740 = CustomConstraint(cons_f740)
def cons_f741(d, e):
return PositiveQ(d/e)
cons741 = CustomConstraint(cons_f741)
def cons_f742(d, c, b, e):
return PosQ(c*(-b*e + S(2)*c*d)/e)
cons742 = CustomConstraint(cons_f742)
def cons_f743(n):
return IntegerQ(n/S(2))
cons743 = CustomConstraint(cons_f743)
def cons_f744(n):
return Greater(n, S(2))
cons744 = CustomConstraint(cons_f744)
def cons_f745(m, n):
return Less(m, -n)
cons745 = CustomConstraint(cons_f745)
def cons_f746(m, n):
return Greater(m, n)
cons746 = CustomConstraint(cons_f746)
def cons_f747(m, q):
return Or(PositiveIntegerQ(q), IntegersQ(m, q))
cons747 = CustomConstraint(cons_f747)
def cons_f748(p, q):
return Or(PositiveIntegerQ(p), PositiveIntegerQ(q))
cons748 = CustomConstraint(cons_f748)
def cons_f749(m, f):
return Not(Or(IntegerQ(m), PositiveQ(f)))
cons749 = CustomConstraint(cons_f749)
def cons_f750(n, q):
return ZeroQ(n - q)
cons750 = CustomConstraint(cons_f750)
def cons_f751(n, r):
return ZeroQ(-n + r)
cons751 = CustomConstraint(cons_f751)
def cons_f752(n, r, q):
return ZeroQ(-S(2)*n + q + r)
cons752 = CustomConstraint(cons_f752)
def cons_f753(n, q):
return PosQ(n - q)
cons753 = CustomConstraint(cons_f753)
def cons_f754(p, q, n):
return NonzeroQ(p*(S(2)*n - q) + S(1))
cons754 = CustomConstraint(cons_f754)
def cons_f755(n, q):
return ZeroQ(-n + q)
cons755 = CustomConstraint(cons_f755)
def cons_f756(m, n, q):
return Or(And(ZeroQ(m + S(-1)), ZeroQ(n + S(-3)), ZeroQ(q + S(-2))), And(Or(ZeroQ(m + S(1)/2), ZeroQ(m + S(-3)/2), ZeroQ(m + S(-1)/2), ZeroQ(m + S(-5)/2)), ZeroQ(n + S(-3)), ZeroQ(q + S(-1))))
cons756 = CustomConstraint(cons_f756)
def cons_f757(m, n):
return ZeroQ(m - S(3)*n/S(2) + S(3)/2)
cons757 = CustomConstraint(cons_f757)
def cons_f758(n, q):
return ZeroQ(-n + q + S(1))
cons758 = CustomConstraint(cons_f758)
def cons_f759(n, r):
return ZeroQ(-n + r + S(-1))
cons759 = CustomConstraint(cons_f759)
def cons_f760(m, n):
return ZeroQ(m - S(3)*n/S(2) + S(1)/2)
cons760 = CustomConstraint(cons_f760)
def cons_f761(m, p, n):
return Equal(m + p*(n + S(-1)) + S(-1), S(0))
cons761 = CustomConstraint(cons_f761)
def cons_f762(m, p, q, n):
return Equal(m + p*q + S(1), n - q)
cons762 = CustomConstraint(cons_f762)
def cons_f763(m, p, q, n):
return Greater(m + p*q + S(1), n - q)
cons763 = CustomConstraint(cons_f763)
def cons_f764(m, p, q, n):
return Unequal(m + p*(S(2)*n - q) + S(1), S(0))
cons764 = CustomConstraint(cons_f764)
def cons_f765(m, p, q, n):
return Unequal(m + p*q + (n - q)*(S(2)*p + S(-1)) + S(1), S(0))
cons765 = CustomConstraint(cons_f765)
def cons_f766(m, p, q, n):
return LessEqual(m + p*q + S(1), -n + q + S(1))
cons766 = CustomConstraint(cons_f766)
def cons_f767(m, p, q):
return NonzeroQ(m + p*q + S(1))
cons767 = CustomConstraint(cons_f767)
def cons_f768(m, p, q, n):
return Greater(m + p*q + S(1), -n + q)
cons768 = CustomConstraint(cons_f768)
def cons_f769(m, p, q, n):
return Equal(m + p*q + S(1), -(n - q)*(S(2)*p + S(3)))
cons769 = CustomConstraint(cons_f769)
def cons_f770(m, p, q, n):
return Greater(m + p*q + S(1), S(2)*n - S(2)*q)
cons770 = CustomConstraint(cons_f770)
def cons_f771(m, p, q, n):
return Less(m + p*q + S(1), n - q)
cons771 = CustomConstraint(cons_f771)
def cons_f772(m, n, p, q):
return Less(n - q, m + p*q + S(1), S(2)*n - S(2)*q)
cons772 = CustomConstraint(cons_f772)
def cons_f773(p):
return Inequality(S(-1), LessEqual, p, Less, S(0))
cons773 = CustomConstraint(cons_f773)
def cons_f774(m, p, q, n):
return Equal(m + p*q + S(1), S(2)*n - S(2)*q)
cons774 = CustomConstraint(cons_f774)
def cons_f775(m, p, q, n):
return Equal(m + p*q + S(1), -S(2)*(n - q)*(p + S(1)))
cons775 = CustomConstraint(cons_f775)
def cons_f776(m, p, q):
return Less(m + p*q + S(1), S(0))
cons776 = CustomConstraint(cons_f776)
def cons_f777(n, r, q):
return ZeroQ(-n + q + r)
cons777 = CustomConstraint(cons_f777)
def cons_f778(n, q, j):
return ZeroQ(j - S(2)*n + q)
cons778 = CustomConstraint(cons_f778)
def cons_f779(n, q, j):
return ZeroQ(j - n + q)
cons779 = CustomConstraint(cons_f779)
def cons_f780(n):
return ZeroQ(n + S(-3))
cons780 = CustomConstraint(cons_f780)
def cons_f781(q):
return ZeroQ(q + S(-2))
cons781 = CustomConstraint(cons_f781)
def cons_f782(p, q, n):
return NonzeroQ(p*q + (n - q)*(S(2)*p + S(1)) + S(1))
cons782 = CustomConstraint(cons_f782)
def cons_f783(m, p, q, n):
return LessEqual(m + p*q, -n + q)
cons783 = CustomConstraint(cons_f783)
def cons_f784(m, p, q):
return Unequal(m + p*q + S(1), S(0))
cons784 = CustomConstraint(cons_f784)
def cons_f785(m, p, q, n):
return Unequal(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1), S(0))
cons785 = CustomConstraint(cons_f785)
def cons_f786(m, p, q, n):
return Greater(m + p*q, n - q + S(-1))
cons786 = CustomConstraint(cons_f786)
def cons_f787(m, p, q, n):
return Greater(m + p*q, -n + q + S(-1))
cons787 = CustomConstraint(cons_f787)
def cons_f788(m, p, q, n):
return Less(m + p*q, n - q + S(-1))
cons788 = CustomConstraint(cons_f788)
def cons_f789(m, p, q, n):
return GreaterEqual(m + p*q, n - q + S(-1))
cons789 = CustomConstraint(cons_f789)
def cons_f790(m, p, q, n):
return Or(Inequality(S(-1), LessEqual, p, Less, S(0)), Equal(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1), S(0)))
cons790 = CustomConstraint(cons_f790)
def cons_f791(m):
return Or(ZeroQ(m + S(-1)/2), ZeroQ(m + S(1)/2))
cons791 = CustomConstraint(cons_f791)
def cons_f792(q):
return ZeroQ(q + S(-1))
cons792 = CustomConstraint(cons_f792)
def cons_f793(q, j, k):
return ZeroQ(j - k + q)
cons793 = CustomConstraint(cons_f793)
def cons_f794(n, j, k):
return ZeroQ(j - S(2)*k + n)
cons794 = CustomConstraint(cons_f794)
def cons_f795(j, k):
return PosQ(-j + k)
cons795 = CustomConstraint(cons_f795)
def cons_f796(j, x):
return FreeQ(j, x)
cons796 = CustomConstraint(cons_f796)
def cons_f797(k, x):
return FreeQ(k, x)
cons797 = CustomConstraint(cons_f797)
def cons_f798(n, q):
return IntegerQ(n*q)
cons798 = CustomConstraint(cons_f798)
def cons_f799(p, m, f, b, r, d, c, a, n, x, s, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, m, n, p, q, r, s), x)
cons799 = CustomConstraint(cons_f799)
def cons_f800(s, x):
return FreeQ(s, x)
cons800 = CustomConstraint(cons_f800)
def cons_f801(d, b, e):
return PositiveQ(b*d*e)
cons801 = CustomConstraint(cons_f801)
def cons_f802(d, c, b, a):
return PositiveQ(-a*d/b + c)
cons802 = CustomConstraint(cons_f802)
def cons_f803(n):
return IntegerQ(S(1)/n)
cons803 = CustomConstraint(cons_f803)
def cons_f804(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolynomialQ(u, x)
cons804 = CustomConstraint(cons_f804)
def cons_f805(m, r):
return IntegersQ(m, r)
cons805 = CustomConstraint(cons_f805)
def cons_f806(p, b, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, n, p), x)
cons806 = CustomConstraint(cons_f806)
def cons_f807(n, n2):
return ZeroQ(S(2)*n + n2)
cons807 = CustomConstraint(cons_f807)
def cons_f808(n):
return IntegerQ(S(2)*n)
cons808 = CustomConstraint(cons_f808)
def cons_f809(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(LinearMatchQ(u, x))
cons809 = CustomConstraint(cons_f809)
def cons_f810(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearQ(List(u, v), x)
cons810 = CustomConstraint(cons_f810)
def cons_f811(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(LinearMatchQ(List(u, v), x))
cons811 = CustomConstraint(cons_f811)
def cons_f812(v, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearQ(List(u, v, w), x)
cons812 = CustomConstraint(cons_f812)
def cons_f813(v, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(LinearMatchQ(List(u, v, w), x))
cons813 = CustomConstraint(cons_f813)
def cons_f814(v, z, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearQ(List(u, v, w, z), x)
cons814 = CustomConstraint(cons_f814)
def cons_f815(v, z, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(LinearMatchQ(List(u, v, w, z), x))
cons815 = CustomConstraint(cons_f815)
def cons_f816(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuadraticQ(u, x)
cons816 = CustomConstraint(cons_f816)
def cons_f817(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(QuadraticMatchQ(u, x))
cons817 = CustomConstraint(cons_f817)
def cons_f818(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuadraticQ(v, x)
cons818 = CustomConstraint(cons_f818)
def cons_f819(x, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(LinearMatchQ(u, x), QuadraticMatchQ(v, x)))
cons819 = CustomConstraint(cons_f819)
def cons_f820(x, w):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuadraticQ(w, x)
cons820 = CustomConstraint(cons_f820)
def cons_f821(v, x, w, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(LinearMatchQ(List(u, v), x), QuadraticMatchQ(w, x)))
cons821 = CustomConstraint(cons_f821)
def cons_f822(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(QuadraticMatchQ(List(u, v), x))
cons822 = CustomConstraint(cons_f822)
def cons_f823(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return BinomialQ(u, x)
cons823 = CustomConstraint(cons_f823)
def cons_f824(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(BinomialMatchQ(u, x))
cons824 = CustomConstraint(cons_f824)
def cons_f825(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return BinomialQ(List(u, v), x)
cons825 = CustomConstraint(cons_f825)
def cons_f826(x, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return ZeroQ(BinomialDegree(u, x) - BinomialDegree(v, x))
except (TypeError, AttributeError):
return False
cons826 = CustomConstraint(cons_f826)
def cons_f827(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(BinomialMatchQ(List(u, v), x))
cons827 = CustomConstraint(cons_f827)
def cons_f828(v, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return BinomialQ(List(u, v, w), x)
cons828 = CustomConstraint(cons_f828)
def cons_f829(x, w, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return ZeroQ(BinomialDegree(u, x) - BinomialDegree(w, x))
except (TypeError, AttributeError):
return False
cons829 = CustomConstraint(cons_f829)
def cons_f830(v, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(BinomialMatchQ(List(u, v, w), x))
cons830 = CustomConstraint(cons_f830)
def cons_f831(v, z, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return BinomialQ(List(u, v, z), x)
cons831 = CustomConstraint(cons_f831)
def cons_f832(x, z, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return ZeroQ(BinomialDegree(u, x) - BinomialDegree(z, x))
except (TypeError, AttributeError):
return False
cons832 = CustomConstraint(cons_f832)
def cons_f833(v, z, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(BinomialMatchQ(List(u, v, z), x))
cons833 = CustomConstraint(cons_f833)
def cons_f834(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return GeneralizedBinomialQ(u, x)
cons834 = CustomConstraint(cons_f834)
def cons_f835(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(GeneralizedBinomialMatchQ(u, x))
cons835 = CustomConstraint(cons_f835)
def cons_f836(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return TrinomialQ(u, x)
cons836 = CustomConstraint(cons_f836)
def cons_f837(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(TrinomialMatchQ(u, x))
cons837 = CustomConstraint(cons_f837)
def cons_f838(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return TrinomialQ(v, x)
cons838 = CustomConstraint(cons_f838)
def cons_f839(x, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(BinomialMatchQ(u, x), TrinomialMatchQ(v, x)))
cons839 = CustomConstraint(cons_f839)
def cons_f840(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return BinomialQ(v, x)
cons840 = CustomConstraint(cons_f840)
def cons_f841(x, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(BinomialMatchQ(u, x), BinomialMatchQ(v, x)))
cons841 = CustomConstraint(cons_f841)
def cons_f842(x, z):
if isinstance(x, (int, Integer, float, Float)):
return False
return BinomialQ(z, x)
cons842 = CustomConstraint(cons_f842)
def cons_f843(x, z, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(BinomialMatchQ(z, x), TrinomialMatchQ(u, x)))
cons843 = CustomConstraint(cons_f843)
def cons_f844(x, z, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(BinomialMatchQ(z, x), BinomialMatchQ(u, x)))
cons844 = CustomConstraint(cons_f844)
def cons_f845(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return GeneralizedTrinomialQ(u, x)
cons845 = CustomConstraint(cons_f845)
def cons_f846(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(GeneralizedTrinomialMatchQ(u, x))
cons846 = CustomConstraint(cons_f846)
def cons_f847(x, z, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return ZeroQ(BinomialDegree(z, x) - GeneralizedTrinomialDegree(u, x))
except (TypeError, AttributeError):
return False
cons847 = CustomConstraint(cons_f847)
def cons_f848(x, z, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(BinomialMatchQ(z, x), GeneralizedTrinomialMatchQ(u, x)))
cons848 = CustomConstraint(cons_f848)
def cons_f849(n, q):
return ZeroQ(-n/S(4) + q)
cons849 = CustomConstraint(cons_f849)
def cons_f850(n, r):
return ZeroQ(-S(3)*n/S(4) + r)
cons850 = CustomConstraint(cons_f850)
def cons_f851(m, n):
return ZeroQ(S(4)*m - n + S(4))
cons851 = CustomConstraint(cons_f851)
def cons_f852(c, h, e, a):
return ZeroQ(a*h + c*e)
cons852 = CustomConstraint(cons_f852)
def cons_f853(m):
return NegativeIntegerQ(m + S(1))
cons853 = CustomConstraint(cons_f853)
def cons_f854(m, n):
return PositiveIntegerQ(n/(m + S(1)))
cons854 = CustomConstraint(cons_f854)
def cons_f855(x, m, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolyQ(Pq, x**(m + S(1)))
cons855 = CustomConstraint(cons_f855)
def cons_f856(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return NonzeroQ(Coeff(Pq, x, n + S(-1)))
cons856 = CustomConstraint(cons_f856)
def cons_f857(p, n):
return Or(PositiveIntegerQ(p), ZeroQ(n + S(-1)))
cons857 = CustomConstraint(cons_f857)
def cons_f858(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolyQ(Pq, x**n)
cons858 = CustomConstraint(cons_f858)
def cons_f859(x, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(Coeff(Pq, x, S(0)))
cons859 = CustomConstraint(cons_f859)
def cons_f860(Pq):
return SumQ(Pq)
cons860 = CustomConstraint(cons_f860)
def cons_f861(x, m, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return Less(m + Expon(Pq, x) + S(1), S(0))
cons861 = CustomConstraint(cons_f861)
def cons_f862(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return Less(Expon(Pq, x), n + S(-1))
cons862 = CustomConstraint(cons_f862)
def cons_f863(d, a, g, b):
return ZeroQ(a*g + b*d)
cons863 = CustomConstraint(cons_f863)
def cons_f864(a, h, b, e):
return ZeroQ(-S(3)*a*h + b*e)
cons864 = CustomConstraint(cons_f864)
def cons_f865(B, a, b, A):
return ZeroQ(-A**S(3)*b + B**S(3)*a)
cons865 = CustomConstraint(cons_f865)
def cons_f866(B, a, b, A):
return NonzeroQ(-A**S(3)*b + B**S(3)*a)
cons866 = CustomConstraint(cons_f866)
def cons_f867(B, C, A):
return ZeroQ(-A*C + B**S(2))
cons867 = CustomConstraint(cons_f867)
def cons_f868(B, a, C, b):
return ZeroQ(B**S(3)*b + C**S(3)*a)
cons868 = CustomConstraint(cons_f868)
def cons_f869(B, C, b, a, A):
return ZeroQ(A*b**(S(2)/3) - B*a**(S(1)/3)*b**(S(1)/3) - S(2)*C*a**(S(2)/3))
cons869 = CustomConstraint(cons_f869)
def cons_f870(B, a, C, b):
return ZeroQ(B*a**(S(1)/3)*b**(S(1)/3) + S(2)*C*a**(S(2)/3))
cons870 = CustomConstraint(cons_f870)
def cons_f871(a, C, b, A):
return ZeroQ(A*b**(S(2)/3) - S(2)*C*a**(S(2)/3))
cons871 = CustomConstraint(cons_f871)
def cons_f872(B, C, b, a, A):
return ZeroQ(A*(-b)**(S(2)/3) - B*(-a)**(S(1)/3)*(-b)**(S(1)/3) - S(2)*C*(-a)**(S(2)/3))
cons872 = CustomConstraint(cons_f872)
def cons_f873(B, a, C, b):
return ZeroQ(B*(-a)**(S(1)/3)*(-b)**(S(1)/3) + S(2)*C*(-a)**(S(2)/3))
cons873 = CustomConstraint(cons_f873)
def cons_f874(a, C, b, A):
return ZeroQ(A*(-b)**(S(2)/3) - S(2)*C*(-a)**(S(2)/3))
cons874 = CustomConstraint(cons_f874)
def cons_f875(B, C, b, a, A):
return ZeroQ(A*b**(S(2)/3) + B*b**(S(1)/3)*(-a)**(S(1)/3) - S(2)*C*(-a)**(S(2)/3))
cons875 = CustomConstraint(cons_f875)
def cons_f876(B, a, C, b):
return ZeroQ(B*b**(S(1)/3)*(-a)**(S(1)/3) - S(2)*C*(-a)**(S(2)/3))
cons876 = CustomConstraint(cons_f876)
def cons_f877(a, C, b, A):
return ZeroQ(A*b**(S(2)/3) - S(2)*C*(-a)**(S(2)/3))
cons877 = CustomConstraint(cons_f877)
def cons_f878(B, C, b, a, A):
return ZeroQ(A*(-b)**(S(2)/3) + B*a**(S(1)/3)*(-b)**(S(1)/3) - S(2)*C*a**(S(2)/3))
cons878 = CustomConstraint(cons_f878)
def cons_f879(B, a, C, b):
return ZeroQ(B*a**(S(1)/3)*(-b)**(S(1)/3) - S(2)*C*a**(S(2)/3))
cons879 = CustomConstraint(cons_f879)
def cons_f880(a, C, b, A):
return ZeroQ(A*(-b)**(S(2)/3) - S(2)*C*a**(S(2)/3))
cons880 = CustomConstraint(cons_f880)
def cons_f881(B, C, b, a, A):
return ZeroQ(A - B*(a/b)**(S(1)/3) - S(2)*C*(a/b)**(S(2)/3))
cons881 = CustomConstraint(cons_f881)
def cons_f882(B, a, C, b):
return ZeroQ(B*(a/b)**(S(1)/3) + S(2)*C*(a/b)**(S(2)/3))
cons882 = CustomConstraint(cons_f882)
def cons_f883(a, C, b, A):
return ZeroQ(A - S(2)*C*(a/b)**(S(2)/3))
cons883 = CustomConstraint(cons_f883)
def cons_f884(B, C, b, a, A):
return ZeroQ(A - B*Rt(a/b, S(3)) - S(2)*C*Rt(a/b, S(3))**S(2))
cons884 = CustomConstraint(cons_f884)
def cons_f885(B, a, C, b):
return ZeroQ(B*Rt(a/b, S(3)) + S(2)*C*Rt(a/b, S(3))**S(2))
cons885 = CustomConstraint(cons_f885)
def cons_f886(a, C, b, A):
return ZeroQ(A - S(2)*C*Rt(a/b, S(3))**S(2))
cons886 = CustomConstraint(cons_f886)
def cons_f887(B, C, b, a, A):
return ZeroQ(A + B*(-a/b)**(S(1)/3) - S(2)*C*(-a/b)**(S(2)/3))
cons887 = CustomConstraint(cons_f887)
def cons_f888(B, a, C, b):
return ZeroQ(B*(-a/b)**(S(1)/3) - S(2)*C*(-a/b)**(S(2)/3))
cons888 = CustomConstraint(cons_f888)
def cons_f889(a, C, b, A):
return ZeroQ(A - S(2)*C*(-a/b)**(S(2)/3))
cons889 = CustomConstraint(cons_f889)
def cons_f890(B, C, b, a, A):
return ZeroQ(A + B*Rt(-a/b, S(3)) - S(2)*C*Rt(-a/b, S(3))**S(2))
cons890 = CustomConstraint(cons_f890)
def cons_f891(B, a, C, b):
return ZeroQ(B*Rt(-a/b, S(3)) - S(2)*C*Rt(-a/b, S(3))**S(2))
cons891 = CustomConstraint(cons_f891)
def cons_f892(a, C, b, A):
return ZeroQ(A - S(2)*C*Rt(-a/b, S(3))**S(2))
cons892 = CustomConstraint(cons_f892)
def cons_f893(B, a, b, A):
return Or(ZeroQ(-A**S(3)*b + B**S(3)*a), Not(RationalQ(a/b)))
cons893 = CustomConstraint(cons_f893)
def cons_f894(a, b):
return Not(RationalQ(a/b))
cons894 = CustomConstraint(cons_f894)
def cons_f895(a, C, b, A):
return Not(RationalQ(a, b, A, C))
cons895 = CustomConstraint(cons_f895)
def cons_f896(B, C, b, a, A):
return ZeroQ(A - B*(a/b)**(S(1)/3) + C*(a/b)**(S(2)/3))
cons896 = CustomConstraint(cons_f896)
def cons_f897(B, a, C, b):
return ZeroQ(B*(a/b)**(S(1)/3) - C*(a/b)**(S(2)/3))
cons897 = CustomConstraint(cons_f897)
def cons_f898(a, C, b, A):
return ZeroQ(A + C*(a/b)**(S(2)/3))
cons898 = CustomConstraint(cons_f898)
def cons_f899(B, C, b, a, A):
return ZeroQ(A + B*(-a/b)**(S(1)/3) + C*(-a/b)**(S(2)/3))
cons899 = CustomConstraint(cons_f899)
def cons_f900(B, a, C, b):
return ZeroQ(B*(-a/b)**(S(1)/3) + C*(-a/b)**(S(2)/3))
cons900 = CustomConstraint(cons_f900)
def cons_f901(a, C, b, A):
return ZeroQ(A + C*(-a/b)**(S(2)/3))
cons901 = CustomConstraint(cons_f901)
def cons_f902(a, b):
return RationalQ(a/b)
cons902 = CustomConstraint(cons_f902)
def cons_f903(a, b):
return Greater(a/b, S(0))
cons903 = CustomConstraint(cons_f903)
def cons_f904(a, b):
return Less(a/b, S(0))
cons904 = CustomConstraint(cons_f904)
def cons_f905(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return Less(Expon(Pq, x), n)
cons905 = CustomConstraint(cons_f905)
def cons_f906(d, c, b, a):
return ZeroQ(c*Rt(b/a, S(3)) - d*(-sqrt(S(3)) + S(1)))
cons906 = CustomConstraint(cons_f906)
def cons_f907(d, c, b, a):
return NonzeroQ(c*Rt(b/a, S(3)) - d*(-sqrt(S(3)) + S(1)))
cons907 = CustomConstraint(cons_f907)
def cons_f908(d, c, b, a):
return ZeroQ(c*Rt(b/a, S(3)) - d*(S(1) + sqrt(S(3))))
cons908 = CustomConstraint(cons_f908)
def cons_f909(d, c, b, a):
return NonzeroQ(c*Rt(b/a, S(3)) - d*(S(1) + sqrt(S(3))))
cons909 = CustomConstraint(cons_f909)
def cons_f910(d, c, a, b):
return ZeroQ(S(2)*c*Rt(b/a, S(3))**S(2) - d*(-sqrt(S(3)) + S(1)))
cons910 = CustomConstraint(cons_f910)
def cons_f911(d, c, a, b):
return NonzeroQ(S(2)*c*Rt(b/a, S(3))**S(2) - d*(-sqrt(S(3)) + S(1)))
cons911 = CustomConstraint(cons_f911)
def cons_f912(d, c, b, a):
return ZeroQ(-a*d**S(4) + b*c**S(4))
cons912 = CustomConstraint(cons_f912)
def cons_f913(d, c, b, a):
return NonzeroQ(-a*d**S(4) + b*c**S(4))
cons913 = CustomConstraint(cons_f913)
def cons_f914(x, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return NonzeroQ(Coeff(Pq, x, S(0)))
cons914 = CustomConstraint(cons_f914)
def cons_f915(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(PolyQ(Pq, x**(n/S(2))))
cons915 = CustomConstraint(cons_f915)
def cons_f916(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return Equal(Expon(Pq, x), n + S(-1))
cons916 = CustomConstraint(cons_f916)
def cons_f917(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return LessEqual(n + S(-1), Expon(Pq, x))
cons917 = CustomConstraint(cons_f917)
def cons_f918(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(PolyQ(Pq, x), PolyQ(Pq, x**n))
cons918 = CustomConstraint(cons_f918)
def cons_f919(v, n, Pq):
return PolyQ(Pq, v**n)
cons919 = CustomConstraint(cons_f919)
def cons_f920(p, f, b, d, a, c, n, e):
return ZeroQ(a*c*f - e*(a*d + b*c)*(n*(p + S(1)) + S(1)))
cons920 = CustomConstraint(cons_f920)
def cons_f921(p, g, b, d, a, c, n, e):
return ZeroQ(a*c*g - b*d*e*(S(2)*n*(p + S(1)) + S(1)))
cons921 = CustomConstraint(cons_f921)
def cons_f922(p, n):
return ZeroQ(n*(p + S(1)) + S(1))
cons922 = CustomConstraint(cons_f922)
def cons_f923(p, m, f, b, d, a, c, n, e):
return ZeroQ(a*c*f*(m + S(1)) - e*(a*d + b*c)*(m + n*(p + S(1)) + S(1)))
cons923 = CustomConstraint(cons_f923)
def cons_f924(p, m, g, b, d, a, c, n, e):
return ZeroQ(a*c*g*(m + S(1)) - b*d*e*(m + S(2)*n*(p + S(1)) + S(1)))
cons924 = CustomConstraint(cons_f924)
def cons_f925(x, Px):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolynomialQ(Px, x)
cons925 = CustomConstraint(cons_f925)
def cons_f926(p, b, d, a, n, e):
return ZeroQ(a*e - b*d*(n*(p + S(1)) + S(1)))
cons926 = CustomConstraint(cons_f926)
def cons_f927(p, f, d, a, c, n):
return ZeroQ(a*f - c*d*(S(2)*n*(p + S(1)) + S(1)))
cons927 = CustomConstraint(cons_f927)
def cons_f928(d, c, f, a):
return ZeroQ(a*f + c*d)
cons928 = CustomConstraint(cons_f928)
def cons_f929(p, m, b, d, a, n, e):
return ZeroQ(a*e*(m + S(1)) - b*d*(m + n*(p + S(1)) + S(1)))
cons929 = CustomConstraint(cons_f929)
def cons_f930(p, m, f, d, a, c, n):
return ZeroQ(a*f*(m + S(1)) - c*d*(m + S(2)*n*(p + S(1)) + S(1)))
cons930 = CustomConstraint(cons_f930)
def cons_f931(n3, n):
return ZeroQ(-S(3)*n + n3)
cons931 = CustomConstraint(cons_f931)
def cons_f932(p, g, b, d, a, c, n, e):
return ZeroQ(a**S(2)*g*(n + S(1)) - c*(a*e - b*d*(n*(p + S(1)) + S(1)))*(n*(S(2)*p + S(3)) + S(1)))
cons932 = CustomConstraint(cons_f932)
def cons_f933(p, f, b, d, a, c, n, e):
return ZeroQ(a**S(2)*f*(n + S(1)) - a*c*d*(n + S(1))*(S(2)*n*(p + S(1)) + S(1)) - b*(a*e - b*d*(n*(p + S(1)) + S(1)))*(n*(p + S(2)) + S(1)))
cons933 = CustomConstraint(cons_f933)
def cons_f934(p, g, b, d, a, c, n):
return ZeroQ(a**S(2)*g*(n + S(1)) + b*c*d*(n*(p + S(1)) + S(1))*(n*(S(2)*p + S(3)) + S(1)))
cons934 = CustomConstraint(cons_f934)
def cons_f935(p, f, b, d, a, c, n):
return ZeroQ(a**S(2)*f*(n + S(1)) - a*c*d*(n + S(1))*(S(2)*n*(p + S(1)) + S(1)) + b**S(2)*d*(n*(p + S(1)) + S(1))*(n*(p + S(2)) + S(1)))
cons935 = CustomConstraint(cons_f935)
def cons_f936(p, b, d, a, n, c, e):
return ZeroQ(a*c*d*(n + S(1))*(S(2)*n*(p + S(1)) + S(1)) + b*(a*e - b*d*(n*(p + S(1)) + S(1)))*(n*(p + S(2)) + S(1)))
cons936 = CustomConstraint(cons_f936)
def cons_f937(p, b, d, a, n, c):
return ZeroQ(a*c*d*(n + S(1))*(S(2)*n*(p + S(1)) + S(1)) - b**S(2)*d*(n*(p + S(1)) + S(1))*(n*(p + S(2)) + S(1)))
cons937 = CustomConstraint(cons_f937)
def cons_f938(n, q):
return ZeroQ(-n/S(2) + q)
cons938 = CustomConstraint(cons_f938)
def cons_f939(n, r):
return ZeroQ(-S(3)*n/S(2) + r)
cons939 = CustomConstraint(cons_f939)
def cons_f940(n, s):
return ZeroQ(-S(2)*n + s)
cons940 = CustomConstraint(cons_f940)
def cons_f941(m, n):
return ZeroQ(S(2)*m - n + S(2))
cons941 = CustomConstraint(cons_f941)
def cons_f942(d, c, g, a):
return ZeroQ(a*g + c*d)
cons942 = CustomConstraint(cons_f942)
def cons_f943(c, h, e, a):
return ZeroQ(-S(3)*a*h + c*e)
cons943 = CustomConstraint(cons_f943)
def cons_f944(h, c, g, b):
return ZeroQ(-S(2)*b*h + c*g)
cons944 = CustomConstraint(cons_f944)
def cons_f945(g, b, d, c, a, e):
return ZeroQ(S(3)*a*g - S(2)*b*e + S(3)*c*d)
cons945 = CustomConstraint(cons_f945)
def cons_f946(d, c, b, e):
return ZeroQ(-S(2)*b*e + S(3)*c*d)
cons946 = CustomConstraint(cons_f946)
def cons_f947(b, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(NiceSqrtQ(-S(4)*a*c + b**S(2)), Less(Expon(Pq, x), n))
cons947 = CustomConstraint(cons_f947)
def cons_f948(c):
return PosQ(c)
cons948 = CustomConstraint(cons_f948)
def cons_f949(c):
return NegQ(c)
cons949 = CustomConstraint(cons_f949)
def cons_f950(x, n, Pq):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(PolyQ(Pq, x**n))
cons950 = CustomConstraint(cons_f950)
def cons_f951(m):
return NegativeIntegerQ(m + S(-1)/2)
cons951 = CustomConstraint(cons_f951)
def cons_f952(n, j):
return NonzeroQ(-j + n)
cons952 = CustomConstraint(cons_f952)
def cons_f953(p, j, n):
return ZeroQ(j*p + j - n + S(1))
cons953 = CustomConstraint(cons_f953)
def cons_f954(p, n, j):
return NegativeIntegerQ((j - n*p - n + S(-1))/(j - n))
cons954 = CustomConstraint(cons_f954)
def cons_f955(p, j):
return NonzeroQ(j*p + S(1))
cons955 = CustomConstraint(cons_f955)
def cons_f956(p, n, j):
return RationalQ(j, n, p)
cons956 = CustomConstraint(cons_f956)
def cons_f957(n, j):
return Less(S(0), j, n)
cons957 = CustomConstraint(cons_f957)
def cons_f958(p, j):
return Less(j*p + S(1), S(0))
cons958 = CustomConstraint(cons_f958)
def cons_f959(p, n):
return NonzeroQ(n*p + S(1))
cons959 = CustomConstraint(cons_f959)
def cons_f960(p, j, n):
return Greater(j*p + S(1), -j + n)
cons960 = CustomConstraint(cons_f960)
def cons_f961(p):
return PositiveIntegerQ(p + S(1)/2)
cons961 = CustomConstraint(cons_f961)
def cons_f962(p, j):
return ZeroQ(j*p + S(1))
cons962 = CustomConstraint(cons_f962)
def cons_f963(n):
return NonzeroQ(n + S(-2))
cons963 = CustomConstraint(cons_f963)
def cons_f964(n, j):
return RationalQ(j, n)
cons964 = CustomConstraint(cons_f964)
def cons_f965(n, j):
return Less(S(2)*n + S(-2), j, n)
cons965 = CustomConstraint(cons_f965)
def cons_f966(n, j):
return PosQ(-j + n)
cons966 = CustomConstraint(cons_f966)
def cons_f967(n, j):
return IntegerQ(j/n)
cons967 = CustomConstraint(cons_f967)
def cons_f968(m, n, p, j):
return ZeroQ(-j + m + n*p + n + S(1))
cons968 = CustomConstraint(cons_f968)
def cons_f969(c, j):
return Or(IntegerQ(j), PositiveQ(c))
cons969 = CustomConstraint(cons_f969)
def cons_f970(m, n, p, j):
return NegativeIntegerQ((j - m - n*p - n + S(-1))/(j - n))
cons970 = CustomConstraint(cons_f970)
def cons_f971(m, p, j):
return NonzeroQ(j*p + m + S(1))
cons971 = CustomConstraint(cons_f971)
def cons_f972(c, n, j):
return Or(IntegersQ(j, n), PositiveQ(c))
cons972 = CustomConstraint(cons_f972)
def cons_f973(n):
return NonzeroQ(n**S(2) + S(-1))
cons973 = CustomConstraint(cons_f973)
def cons_f974(m, n, p, j):
return RationalQ(j, m, n, p)
cons974 = CustomConstraint(cons_f974)
def cons_f975(m, p, j):
return Less(j*p + m + S(1), S(0))
cons975 = CustomConstraint(cons_f975)
def cons_f976(m, p, j, n):
return Greater(j*p + m + S(1), -j + n)
cons976 = CustomConstraint(cons_f976)
def cons_f977(m, p, j, n):
return PositiveQ(j*p + j + m - n + S(1))
cons977 = CustomConstraint(cons_f977)
def cons_f978(m, p, j):
return NegativeQ(j*p + m + S(1))
cons978 = CustomConstraint(cons_f978)
def cons_f979(m, p, j):
return ZeroQ(j*p + m + S(1))
cons979 = CustomConstraint(cons_f979)
def cons_f980(m, j):
return ZeroQ(-j/S(2) + m + S(1))
cons980 = CustomConstraint(cons_f980)
def cons_f981(j, k):
return NonzeroQ(-j + k)
cons981 = CustomConstraint(cons_f981)
def cons_f982(n, k):
return IntegerQ(k/n)
cons982 = CustomConstraint(cons_f982)
def cons_f983(n, jn, j):
return ZeroQ(jn - j - n)
cons983 = CustomConstraint(cons_f983)
def cons_f984(p, j, m, b, d, a, c, n):
return ZeroQ(a*d*(j*p + m + S(1)) - b*c*(m + n + p*(j + n) + S(1)))
cons984 = CustomConstraint(cons_f984)
def cons_f985(j, e):
return Or(PositiveQ(e), IntegersQ(j))
cons985 = CustomConstraint(cons_f985)
def cons_f986(m, p, j):
return RationalQ(j, m, p)
cons986 = CustomConstraint(cons_f986)
def cons_f987(m, j):
return Inequality(S(0), Less, j, LessEqual, m)
cons987 = CustomConstraint(cons_f987)
def cons_f988(j, e):
return Or(PositiveQ(e), IntegerQ(j))
cons988 = CustomConstraint(cons_f988)
def cons_f989(m, p, j, n):
return Or(Less(j*p + m, S(-1)), And(IntegersQ(m + S(-1)/2, p + S(-1)/2), Less(p, S(0)), Less(m, -n*p + S(-1))))
cons989 = CustomConstraint(cons_f989)
def cons_f990(n, j, e):
return Or(PositiveQ(e), IntegersQ(j, n))
cons990 = CustomConstraint(cons_f990)
def cons_f991(m, n, p, j):
return NonzeroQ(j*p + m - n + S(1))
cons991 = CustomConstraint(cons_f991)
def cons_f992(m, n, p, j):
return NonzeroQ(m + n + p*(j + n) + S(1))
cons992 = CustomConstraint(cons_f992)
def cons_f993(n, j):
return Not(And(ZeroQ(n + S(-1)), ZeroQ(j + S(-1))))
cons993 = CustomConstraint(cons_f993)
def cons_f994(n):
return Less(S(-1), n, S(1))
cons994 = CustomConstraint(cons_f994)
def cons_f995(m):
return Greater(m**S(2), S(1))
cons995 = CustomConstraint(cons_f995)
def cons_f996(n, j):
return PositiveIntegerQ(j, n, j/n)
cons996 = CustomConstraint(cons_f996)
def cons_f997(n, j):
return PositiveIntegerQ(j, n)
cons997 = CustomConstraint(cons_f997)
def cons_f998(n, j):
return Less(j, n)
cons998 = CustomConstraint(cons_f998)
def cons_f999(d, a, b):
return ZeroQ(S(27)*a**S(2)*d + S(4)*b**S(3))
cons999 = CustomConstraint(cons_f999)
def cons_f1000(d, a, b):
return NonzeroQ(S(27)*a**S(2)*d + S(4)*b**S(3))
cons1000 = CustomConstraint(cons_f1000)
def cons_f1001(d, c, a):
return ZeroQ(S(27)*a*d**S(2) + S(4)*c**S(3))
cons1001 = CustomConstraint(cons_f1001)
def cons_f1002(d, c, a):
return NonzeroQ(S(27)*a*d**S(2) + S(4)*c**S(3))
cons1002 = CustomConstraint(cons_f1002)
def cons_f1003(d, c, b):
return ZeroQ(-S(3)*b*d + c**S(2))
cons1003 = CustomConstraint(cons_f1003)
def cons_f1004(c, a, b):
return ZeroQ(-S(3)*a*c + b**S(2))
cons1004 = CustomConstraint(cons_f1004)
def cons_f1005(c, a, b):
return NonzeroQ(-S(3)*a*c + b**S(2))
cons1005 = CustomConstraint(cons_f1005)
def cons_f1006(d, c, b):
return NonzeroQ(-S(3)*b*d + c**S(2))
cons1006 = CustomConstraint(cons_f1006)
def cons_f1007(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolyQ(u, x, S(3))
cons1007 = CustomConstraint(cons_f1007)
def cons_f1008(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(CubicMatchQ(u, x))
cons1008 = CustomConstraint(cons_f1008)
def cons_f1009(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolyQ(v, x, S(3))
cons1009 = CustomConstraint(cons_f1009)
def cons_f1010(x, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(LinearMatchQ(u, x), CubicMatchQ(v, x)))
cons1010 = CustomConstraint(cons_f1010)
def cons_f1011(f, g):
return ZeroQ(f + g)
cons1011 = CustomConstraint(cons_f1011)
def cons_f1012(a, c):
return PosQ(a**S(2)*(S(2)*a - c))
cons1012 = CustomConstraint(cons_f1012)
def cons_f1013(a, c):
return NegQ(a**S(2)*(S(2)*a - c))
cons1013 = CustomConstraint(cons_f1013)
def cons_f1014(d, c, b, e):
return ZeroQ(S(8)*b*e**S(2) - S(4)*c*d*e + d**S(3))
cons1014 = CustomConstraint(cons_f1014)
def cons_f1015(p):
return UnsameQ(p, S(2))
cons1015 = CustomConstraint(cons_f1015)
def cons_f1016(p):
return UnsameQ(p, S(3))
cons1016 = CustomConstraint(cons_f1016)
def cons_f1017(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolynomialQ(v, x)
cons1017 = CustomConstraint(cons_f1017)
def cons_f1018(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Equal(Exponent(v, x), S(4))
cons1018 = CustomConstraint(cons_f1018)
def cons_f1019(d, c, a, b):
return ZeroQ(S(8)*a**S(2)*d - S(4)*a*b*c + b**S(3))
cons1019 = CustomConstraint(cons_f1019)
def cons_f1020(d, b):
return ZeroQ(-b + d)
cons1020 = CustomConstraint(cons_f1020)
def cons_f1021(a, e):
return ZeroQ(-a + e)
cons1021 = CustomConstraint(cons_f1021)
def cons_f1022(x, c, a, b):
if isinstance(x, (int, Integer, float, Float)):
return False
return SumQ(Factor(a*x**S(4) + a + b*x**S(3) + b*x + c*x**S(2)))
cons1022 = CustomConstraint(cons_f1022)
def cons_f1023(D, x):
return FreeQ(D, x)
cons1023 = CustomConstraint(cons_f1023)
def cons_f1024(B, C, e, b, d, c, A):
return ZeroQ(B**S(2)*d - S(2)*B*(S(2)*A*e + C*c) + S(2)*C*(A*d + C*b))
cons1024 = CustomConstraint(cons_f1024)
def cons_f1025(B, C, e, d, c, a, A):
return ZeroQ(-S(4)*A*B*C*d + S(4)*A*e*(S(2)*A*C + B**S(2)) - B**S(3)*d + S(2)*B**S(2)*C*c - S(8)*C**S(3)*a)
cons1025 = CustomConstraint(cons_f1025)
def cons_f1026(B, C, e, d, c, A):
return PosQ(C*(C*(-S(4)*c*e + d**S(2)) + S(2)*e*(-S(4)*A*e + B*d)))
cons1026 = CustomConstraint(cons_f1026)
def cons_f1027(d, C, b, A):
return ZeroQ(A*d + C*b)
cons1027 = CustomConstraint(cons_f1027)
def cons_f1028(a, C, e, A):
return ZeroQ(-A**S(2)*e + C**S(2)*a)
cons1028 = CustomConstraint(cons_f1028)
def cons_f1029(C, d, c, A, e):
return PosQ(C*(-S(8)*A*e**S(2) + C*(-S(4)*c*e + d**S(2))))
cons1029 = CustomConstraint(cons_f1029)
def cons_f1030(B, C, e, d, c, A):
return NegQ(C*(C*(-S(4)*c*e + d**S(2)) + S(2)*e*(-S(4)*A*e + B*d)))
cons1030 = CustomConstraint(cons_f1030)
def cons_f1031(C, d, c, A, e):
return NegQ(C*(-S(8)*A*e**S(2) + C*(-S(4)*c*e + d**S(2))))
cons1031 = CustomConstraint(cons_f1031)
def cons_f1032(B, C, b, D, d, c, A, e):
return ZeroQ(S(4)*d*(-S(2)*B*e + D*c)**S(2) - S(4)*(-S(2)*B*e + D*c)*(-S(8)*A*e**S(2) - S(4)*C*c*e + S(2)*D*b*e + S(3)*D*c*d) + S(8)*(-S(4)*C*e + S(3)*D*d)*(-A*d*e - C*b*e + D*b*d))
cons1032 = CustomConstraint(cons_f1032)
def cons_f1033(B, C, b, D, d, c, a, A, e):
return ZeroQ(S(8)*a*(-S(4)*C*e + S(3)*D*d)**S(3) - S(8)*c*(-S(2)*B*e + D*c)**S(2)*(-S(4)*C*e + S(3)*D*d) + S(8)*d*(-S(4)*A*e + D*b)*(-S(2)*B*e + D*c)*(-S(4)*C*e + S(3)*D*d) + S(8)*d*(-S(2)*B*e + D*c)**S(3) - S(4)*e*(-S(4)*A*e + D*b)*(S(2)*(-S(4)*A*e + D*b)*(-S(4)*C*e + S(3)*D*d) + S(4)*(-S(2)*B*e + D*c)**S(2)))
cons1033 = CustomConstraint(cons_f1033)
def cons_f1034(b, D, d, c, A, e):
return ZeroQ(D**S(2)*c**S(2)*d - D*c*(-S(8)*A*e**S(2) - S(4)*C*c*e + S(2)*D*b*e + S(3)*D*c*d) + S(2)*(-S(4)*C*e + S(3)*D*d)*(-A*d*e - C*b*e + D*b*d))
cons1034 = CustomConstraint(cons_f1034)
def cons_f1035(B, e, b, D, d, a, c, A):
return ZeroQ(S(54)*D**S(3)*a*d**S(3) - S(6)*D*c*d*(-S(2)*B*e + D*c)**S(2) + S(6)*D*d**S(2)*(-S(4)*A*e + D*b)*(-S(2)*B*e + D*c) + S(2)*d*(-S(2)*B*e + D*c)**S(3) - e*(-S(4)*A*e + D*b)*(S(6)*D*d*(-S(4)*A*e + D*b) + S(4)*(-S(2)*B*e + D*c)**S(2)))
cons1035 = CustomConstraint(cons_f1035)
def cons_f1036(a, f, c, e):
return ZeroQ(a*e**S(2) - c*f**S(2))
cons1036 = CustomConstraint(cons_f1036)
def cons_f1037(d, f, b, e):
return ZeroQ(b*e**S(2) - d*f**S(2))
cons1037 = CustomConstraint(cons_f1037)
def cons_f1038(a, f, c, e):
return NonzeroQ(a*e**S(2) - c*f**S(2))
cons1038 = CustomConstraint(cons_f1038)
def cons_f1039(d, f, b, e):
return NonzeroQ(b*e**S(2) - d*f**S(2))
cons1039 = CustomConstraint(cons_f1039)
def cons_f1040(p, n):
return ZeroQ(-S(2)*n + p)
cons1040 = CustomConstraint(cons_f1040)
def cons_f1041(d, c, b):
return ZeroQ(b*c**S(2) - d**S(2))
cons1041 = CustomConstraint(cons_f1041)
def cons_f1042(d, c, b):
return NonzeroQ(b*c**S(2) - d**S(2))
cons1042 = CustomConstraint(cons_f1042)
def cons_f1043(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e), x)
cons1043 = CustomConstraint(cons_f1043)
def cons_f1044(b, d, c, a, e):
return NonzeroQ(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4))
cons1044 = CustomConstraint(cons_f1044)
def cons_f1045(d, c, b, e):
return ZeroQ(b*d*e**S(2) + S(2)*c*d**S(3))
cons1045 = CustomConstraint(cons_f1045)
def cons_f1046(d, c, b, e):
return NonzeroQ(b*d*e**S(2) + S(2)*c*d**S(3))
cons1046 = CustomConstraint(cons_f1046)
def cons_f1047(d, c, e, a):
return NonzeroQ(a*e**S(4) + c*d**S(4))
cons1047 = CustomConstraint(cons_f1047)
def cons_f1048(B, d, e, A):
return ZeroQ(A*e + B*d)
cons1048 = CustomConstraint(cons_f1048)
def cons_f1049(B, a, c, A):
return ZeroQ(A*c + B*a)
cons1049 = CustomConstraint(cons_f1049)
def cons_f1050(d, c, e, a):
return ZeroQ(a*e + c*d)
cons1050 = CustomConstraint(cons_f1050)
def cons_f1051(g, f, b, d, c, a, h, e):
return ZeroQ(-f**S(2)*(a*h**S(2) - b*g*h + c*g**S(2)) + (-d*h + e*g)**S(2))
cons1051 = CustomConstraint(cons_f1051)
def cons_f1052(g, f, b, d, c, h, e):
return ZeroQ(-S(2)*d*e*h + S(2)*e**S(2)*g - f**S(2)*(-b*h + S(2)*c*g))
cons1052 = CustomConstraint(cons_f1052)
def cons_f1053(v, u, j, f, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(LinearMatchQ(u, x), QuadraticMatchQ(v, x), Or(ZeroQ(j), ZeroQ(f + S(-1)))))
cons1053 = CustomConstraint(cons_f1053)
def cons_f1054(v, u, j, k, g, f, x, h):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(-f**S(2)*k**S(2)*(g**S(2)*Coefficient(v, x, S(2)) - g*h*Coefficient(v, x, S(1)) + h**S(2)*Coefficient(v, x, S(0))) + (g*Coefficient(u, x, S(1)) - h*(f*j + Coefficient(u, x, S(0))))**S(2))
cons1054 = CustomConstraint(cons_f1054)
def cons_f1055(c, f, e):
return ZeroQ(-c*f**S(2) + e**S(2))
cons1055 = CustomConstraint(cons_f1055)
def cons_f1056(x, f, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(-f**S(2)*Coefficient(v, x, S(2)) + Coefficient(u, x, S(1))**S(2))
cons1056 = CustomConstraint(cons_f1056)
def cons_f1057(c, g, a, i):
return ZeroQ(-a*i + c*g)
cons1057 = CustomConstraint(cons_f1057)
def cons_f1058(m, p):
return IntegersQ(p, S(2)*m)
cons1058 = CustomConstraint(cons_f1058)
def cons_f1059(m, c, i):
return Or(IntegerQ(m), PositiveQ(i/c))
cons1059 = CustomConstraint(cons_f1059)
def cons_f1060(c, h, b, i):
return ZeroQ(-b*i + c*h)
cons1060 = CustomConstraint(cons_f1060)
def cons_f1061(c, i):
return Not(PositiveQ(i/c))
cons1061 = CustomConstraint(cons_f1061)
def cons_f1062(v, w, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuadraticQ(List(v, w), x)
cons1062 = CustomConstraint(cons_f1062)
def cons_f1063(v, w, u, j, f, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(LinearMatchQ(u, x), QuadraticMatchQ(List(v, w), x), Or(ZeroQ(j), ZeroQ(f + S(-1)))))
cons1063 = CustomConstraint(cons_f1063)
def cons_f1064(v, u, k, f, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(-f**S(2)*k**S(2)*Coefficient(v, x, S(2)) + Coefficient(u, x, S(1))**S(2))
cons1064 = CustomConstraint(cons_f1064)
def cons_f1065(p, n):
return ZeroQ(p - S(2)/n)
cons1065 = CustomConstraint(cons_f1065)
def cons_f1066(c, a, b):
return ZeroQ(a**S(2) - b**S(2)*c)
cons1066 = CustomConstraint(cons_f1066)
def cons_f1067(d, a, b):
return ZeroQ(a**S(2) - b**S(2)*d)
cons1067 = CustomConstraint(cons_f1067)
def cons_f1068(c, b, a):
return ZeroQ(a + b**S(2)*c)
cons1068 = CustomConstraint(cons_f1068)
def cons_f1069(c, b, e, a):
return ZeroQ(a + b**S(2)*c*e)
cons1069 = CustomConstraint(cons_f1069)
def cons_f1070(d, c, b):
return ZeroQ(-b*d**S(2) + c**S(2))
cons1070 = CustomConstraint(cons_f1070)
def cons_f1071(b, e):
return ZeroQ(-b**S(2) + e)
cons1071 = CustomConstraint(cons_f1071)
def cons_f1072(d, c, b, a):
return ZeroQ(-a*d + b*c, S(0))
cons1072 = CustomConstraint(cons_f1072)
def cons_f1073(B, d, a, n, A):
return ZeroQ(-A**S(2)*d*(n + S(-1))**S(2) + B**S(2)*a)
cons1073 = CustomConstraint(cons_f1073)
def cons_f1074(B, d, c, n, A):
return ZeroQ(S(2)*A*d*(n + S(-1)) + B*c)
cons1074 = CustomConstraint(cons_f1074)
def cons_f1075(m, k):
return ZeroQ(k - S(2)*m + S(-2))
cons1075 = CustomConstraint(cons_f1075)
def cons_f1076(B, m, d, a, n, A):
return ZeroQ(-A**S(2)*d*(m - n + S(1))**S(2) + B**S(2)*a*(m + S(1))**S(2))
cons1076 = CustomConstraint(cons_f1076)
def cons_f1077(B, m, d, c, n, A):
return ZeroQ(-S(2)*A*d*(m - n + S(1)) + B*c*(m + S(1)))
cons1077 = CustomConstraint(cons_f1077)
def cons_f1078(f, b, g, d, c, a):
return ZeroQ(-S(12)*a**S(3)*g**S(2) + a**S(2)*c*f**S(2) + S(2)*a*b*g*(a*f + S(3)*c*d) + S(9)*c**S(3)*d**S(2) - c*d*f*(S(6)*a*c + b**S(2)))
cons1078 = CustomConstraint(cons_f1078)
def cons_f1079(f, g, b, d, c, a, e):
return ZeroQ(a**S(3)*c*f**S(2)*g + S(2)*a**S(3)*g**S(2)*(-S(6)*a*g + b*f) - S(3)*a**S(2)*c**S(2)*d*f*g + S(3)*c**S(4)*d**S(2)*e - c**S(3)*d*(-S(12)*a*d*g + a*e*f + S(2)*b*d*f))
cons1079 = CustomConstraint(cons_f1079)
def cons_f1080(d, c, f, a):
return NonzeroQ(-a*f + S(3)*c*d)
cons1080 = CustomConstraint(cons_f1080)
def cons_f1081(g, b, d, c, a):
return NonzeroQ(-S(2)*a**S(2)*g + b*c*d)
cons1081 = CustomConstraint(cons_f1081)
def cons_f1082(f, b, g, d, c, a):
return NonzeroQ(S(4)*a**S(2)*g - a*b*f + b*c*d)
cons1082 = CustomConstraint(cons_f1082)
def cons_f1083(f, g, b, d, a, c):
return PosQ((S(12)*a**S(2)*g**S(2) - a*c*f**S(2) + f*(-S(2)*a*b*g + S(3)*c**S(2)*d))/(c*g*(-a*f + S(3)*c*d)))
cons1083 = CustomConstraint(cons_f1083)
def cons_f1084(f, g, d, c, a):
return ZeroQ(-S(12)*a**S(3)*g**S(2) + a**S(2)*c*f**S(2) - S(6)*a*c**S(2)*d*f + S(9)*c**S(3)*d**S(2))
cons1084 = CustomConstraint(cons_f1084)
def cons_f1085(f, g, d, c, a, e):
return ZeroQ(-S(12)*a**S(4)*g**S(3) + a**S(3)*c*f**S(2)*g - S(3)*a**S(2)*c**S(2)*d*f*g - a*c**S(3)*d*(-S(12)*d*g + e*f) + S(3)*c**S(4)*d**S(2)*e)
cons1085 = CustomConstraint(cons_f1085)
def cons_f1086(f, g, d, a, c):
return PosQ((S(12)*a**S(2)*g**S(2) - a*c*f**S(2) + S(3)*c**S(2)*d*f)/(c*g*(-a*f + S(3)*c*d)))
cons1086 = CustomConstraint(cons_f1086)
def cons_f1087(v):
return SumQ(v)
cons1087 = CustomConstraint(cons_f1087)
def cons_f1088(x, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(MonomialQ(u, x), BinomialQ(v, x)))
cons1088 = CustomConstraint(cons_f1088)
def cons_f1089(x, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(ZeroQ(Coefficient(u, x, S(0))), ZeroQ(Coefficient(v, x, S(0)))))
cons1089 = CustomConstraint(cons_f1089)
def cons_f1090(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return PiecewiseLinearQ(u, x)
cons1090 = CustomConstraint(cons_f1090)
def cons_f1091(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return PiecewiseLinearQ(u, v, x)
cons1091 = CustomConstraint(cons_f1091)
def cons_f1092(n):
return Unequal(n, S(1))
cons1092 = CustomConstraint(cons_f1092)
def cons_f1093(m, n):
return Or(And(RationalQ(m, n), Less(m, S(-1)), Greater(n, S(0)), Not(And(IntegerQ(m + n), Less(m + n + S(2), S(0)), Or(FractionQ(m), GreaterEqual(m + S(2)*n + S(1), S(0)))))), And(PositiveIntegerQ(n, m), LessEqual(n, m)), And(PositiveIntegerQ(n), Not(IntegerQ(m))), And(NegativeIntegerQ(m), Not(IntegerQ(n))))
cons1093 = CustomConstraint(cons_f1093)
def cons_f1094(n):
return Not(RationalQ(n))
cons1094 = CustomConstraint(cons_f1094)
def cons_f1095(n):
return SumSimplerQ(n, S(-1))
cons1095 = CustomConstraint(cons_f1095)
def cons_f1096(m):
return SumSimplerQ(m, S(1))
cons1096 = CustomConstraint(cons_f1096)
def cons_f1097(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(LinearQ(u, x))
cons1097 = CustomConstraint(cons_f1097)
def cons_f1098():
return Not(SameQ(_UseGamma, True))
cons1098 = CustomConstraint(cons_f1098)
def cons_f1099(F, x):
return FreeQ(F, x)
cons1099 = CustomConstraint(cons_f1099)
def cons_f1100(m, f, b, g, d, c, n, x, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, b, c, d, e, f, g, m, n), x)
cons1100 = CustomConstraint(cons_f1100)
def cons_f1101(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return PowerOfLinearQ(u, x)
cons1101 = CustomConstraint(cons_f1101)
def cons_f1102(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(LinearMatchQ(v, x), PowerOfLinearMatchQ(u, x)))
cons1102 = CustomConstraint(cons_f1102)
def cons_f1103(p, m, f, b, g, d, c, a, n, x, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, b, c, d, e, f, g, m, n, p), x)
cons1103 = CustomConstraint(cons_f1103)
def cons_f1104(j, f, g, i, G, n, F, q):
return ZeroQ(f*g*n*log(F) - i*j*q*log(G))
cons1104 = CustomConstraint(cons_f1104)
def cons_f1105(e, j, k, g, f, i, G, n, x, h, q, F):
if isinstance(x, (int, Integer, float, Float)):
return False
return NonzeroQ((G**(j*(h + i*x))*k)**q - (F**(g*(e + f*x)))**n)
cons1105 = CustomConstraint(cons_f1105)
def cons_f1106(b, c, a, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, b, c, n), x)
cons1106 = CustomConstraint(cons_f1106)
def cons_f1107():
return SameQ(_UseGamma, True)
cons1107 = CustomConstraint(cons_f1107)
def cons_f1108(v, w, u, m, c, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(-c*(-Coefficient(u, x, S(0))*Coefficient(w, x, S(1)) + Coefficient(u, x, S(1))*Coefficient(w, x, S(0)))*Coefficient(v, x, S(1))*log(F) + (m + S(1))*Coefficient(u, x, S(1))*Coefficient(w, x, S(1)))
cons1108 = CustomConstraint(cons_f1108)
def cons_f1109(x, w):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolynomialQ(w, x)
cons1109 = CustomConstraint(cons_f1109)
def cons_f1110(h, f, e, n):
return ZeroQ(e - f*h*(n + S(1)))
cons1110 = CustomConstraint(cons_f1110)
def cons_f1111(g, b, c, n, h, F, e):
return ZeroQ(-b*c*e*log(F) + g*h*(n + S(1)))
cons1111 = CustomConstraint(cons_f1111)
def cons_f1112(m, f, n, h, e):
return ZeroQ(e*(m + S(1)) - f*h*(n + S(1)))
cons1112 = CustomConstraint(cons_f1112)
def cons_f1113(b, d, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, b, c, d), x)
cons1113 = CustomConstraint(cons_f1113)
def cons_f1114(n):
return IntegerQ(S(2)/n)
cons1114 = CustomConstraint(cons_f1114)
def cons_f1115(n):
return Not(IntegerQ(S(2)/n))
cons1115 = CustomConstraint(cons_f1115)
def cons_f1116(d, c, f, e):
return ZeroQ(-c*f + d*e)
cons1116 = CustomConstraint(cons_f1116)
def cons_f1117(m, n):
return ZeroQ(-S(2)*m + n + S(-2))
cons1117 = CustomConstraint(cons_f1117)
def cons_f1118(m, n):
return IntegerQ(S(2)*(m + S(1))/n)
cons1118 = CustomConstraint(cons_f1118)
def cons_f1119(m, n):
return Less(S(0), (m + S(1))/n, S(5))
cons1119 = CustomConstraint(cons_f1119)
def cons_f1120(m, n):
return Or(Less(S(0), n, m + S(1)), Less(m, n, S(0)))
cons1120 = CustomConstraint(cons_f1120)
def cons_f1121(m, n):
return Less(S(-4), (m + S(1))/n, S(5))
cons1121 = CustomConstraint(cons_f1121)
def cons_f1122(m, n):
return Or(And(Greater(n, S(0)), Less(m, S(-1))), Inequality(S(0), Less, -n, LessEqual, m + S(1)))
cons1122 = CustomConstraint(cons_f1122)
def cons_f1123(d, f):
return NonzeroQ(-d + f)
cons1123 = CustomConstraint(cons_f1123)
def cons_f1124(c, e):
return NonzeroQ(c*e)
cons1124 = CustomConstraint(cons_f1124)
def cons_f1125(x, u, v):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(LinearMatchQ(u, x), BinomialMatchQ(v, x)))
cons1125 = CustomConstraint(cons_f1125)
def cons_f1126(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return PowerOfLinearQ(v, x)
cons1126 = CustomConstraint(cons_f1126)
def cons_f1127(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(PowerOfLinearMatchQ(v, x))
cons1127 = CustomConstraint(cons_f1127)
def cons_f1128(d, h, c, g):
return ZeroQ(-c*h + d*g)
cons1128 = CustomConstraint(cons_f1128)
def cons_f1129(d, h, c, g):
return NonzeroQ(-c*h + d*g)
cons1129 = CustomConstraint(cons_f1129)
def cons_f1130(b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, b, c), x)
cons1130 = CustomConstraint(cons_f1130)
def cons_f1131(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(QuadraticMatchQ(v, x))
cons1131 = CustomConstraint(cons_f1131)
def cons_f1132(d, c, b, e):
return ZeroQ(b*e - S(2)*c*d)
cons1132 = CustomConstraint(cons_f1132)
def cons_f1133(d, c, b, e):
return NonzeroQ(b*e - S(2)*c*d)
cons1133 = CustomConstraint(cons_f1133)
def cons_f1134(m, b, d, c, a, x, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, b, c, d, e, m), x)
cons1134 = CustomConstraint(cons_f1134)
def cons_f1135(v, d, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(S(2)*e*(c + d*x) - v)
cons1135 = CustomConstraint(cons_f1135)
def cons_f1136(f, b, g, G, d, c, a, n, x, h, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, G, a, b, c, d, e, f, g, h, n), x)
cons1136 = CustomConstraint(cons_f1136)
def cons_f1137(G, x):
return FreeQ(G, x)
cons1137 = CustomConstraint(cons_f1137)
def cons_f1138(g, G, d, h, F, e):
return Not(RationalQ(FullSimplify(g*h*log(G)/(d*e*log(F)))))
cons1138 = CustomConstraint(cons_f1138)
def cons_f1139(s, t, f, b, g, r, G, d, H, a, c, n, x, h, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, G, H, a, b, c, d, e, f, g, h, r, s, t, n), x)
cons1139 = CustomConstraint(cons_f1139)
def cons_f1140(H, x):
return FreeQ(H, x)
cons1140 = CustomConstraint(cons_f1140)
def cons_f1141(t, x):
return FreeQ(t, x)
cons1141 = CustomConstraint(cons_f1141)
def cons_f1142(g, G, d, n, h, F, e):
return ZeroQ(d*e*n*log(F) + g*h*log(G))
cons1142 = CustomConstraint(cons_f1142)
def cons_f1143(t, g, G, d, H, h, s, e, F):
return Not(RationalQ(FullSimplify((g*h*log(G) + s*t*log(H))/(d*e*log(F)))))
cons1143 = CustomConstraint(cons_f1143)
def cons_f1144(v, u):
return ZeroQ(-S(2)*u + v)
cons1144 = CustomConstraint(cons_f1144)
def cons_f1145(d, c, v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(c + d*x + v)
cons1145 = CustomConstraint(cons_f1145)
def cons_f1146(x, w):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearQ(w, x)
cons1146 = CustomConstraint(cons_f1146)
def cons_f1147(v, w):
return ZeroQ(v + w)
cons1147 = CustomConstraint(cons_f1147)
def cons_f1148(v, x, w):
if isinstance(x, (int, Integer, float, Float)):
return False
return If(RationalQ(Coefficient(v, x, S(1))), Greater(Coefficient(v, x, S(1)), S(0)), Less(LeafCount(v), LeafCount(w)))
cons1148 = CustomConstraint(cons_f1148)
def cons_f1149(g, b, d, c, a, n, x, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, b, c, d, e, g, n), x)
cons1149 = CustomConstraint(cons_f1149)
def cons_f1150(g, d, a, c, n, x, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, c, d, e, g, n), x)
cons1150 = CustomConstraint(cons_f1150)
def cons_f1151(x, a, b, F):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, b), x)
cons1151 = CustomConstraint(cons_f1151)
def cons_f1152(n):
return Unequal(n, S(-1))
cons1152 = CustomConstraint(cons_f1152)
def cons_f1153(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return FunctionOfExponentialQ(u, x)
cons1153 = CustomConstraint(cons_f1153)
def cons_f1154(v, w, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return LinearQ(List(v, w), x)
cons1154 = CustomConstraint(cons_f1154)
def cons_f1155(v, w, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(BinomialQ(v + w, x), And(PolynomialQ(v + w, x), LessEqual(Exponent(v + w, x), S(2))))
cons1155 = CustomConstraint(cons_f1155)
def cons_f1156(p, f, d, c, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(c, d, e, f, p, q), x)
cons1156 = CustomConstraint(cons_f1156)
def cons_f1157(d, x, f, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(d, e, f), x)
cons1157 = CustomConstraint(cons_f1157)
def cons_f1158(p, b, q):
return PosQ(b*p*q)
cons1158 = CustomConstraint(cons_f1158)
def cons_f1159(p, b, q):
return NegQ(b*p*q)
cons1159 = CustomConstraint(cons_f1159)
def cons_f1160(h, f, e, g):
return ZeroQ(-e*h + f*g)
cons1160 = CustomConstraint(cons_f1160)
def cons_f1161(m, p):
return ZeroQ(m - p + S(1))
cons1161 = CustomConstraint(cons_f1161)
def cons_f1162(h, p, f):
return Or(IntegerQ(p), PositiveQ(h/f))
cons1162 = CustomConstraint(cons_f1162)
def cons_f1163(h, p, f):
return Not(Or(IntegerQ(p), PositiveQ(h/f)))
cons1163 = CustomConstraint(cons_f1163)
def cons_f1164(m, p, b, q):
return PosQ((m + S(1))/(b*p*q))
cons1164 = CustomConstraint(cons_f1164)
def cons_f1165(m, p, b, q):
return NegQ((m + S(1))/(b*p*q))
cons1165 = CustomConstraint(cons_f1165)
def cons_f1166(f, g, c, h, e):
return ZeroQ(c*(-e*h + f*g) + h)
cons1166 = CustomConstraint(cons_f1166)
def cons_f1167(f, g, c, h, e):
return NonzeroQ(c*(-e*h + f*g) + h)
cons1167 = CustomConstraint(cons_f1167)
def cons_f1168(f, g, c, h, e):
return PositiveQ(c*(e - f*g/h))
cons1168 = CustomConstraint(cons_f1168)
def cons_f1169(h, f, e, g):
return NonzeroQ(-e*h + f*g)
cons1169 = CustomConstraint(cons_f1169)
def cons_f1170(m, n):
return IntegersQ(S(2)*m, S(2)*n)
cons1170 = CustomConstraint(cons_f1170)
def cons_f1171(m, n):
return Or(Equal(n, S(1)), Not(PositiveIntegerQ(m)), And(Equal(n, S(2)), NonzeroQ(m + S(-1))))
cons1171 = CustomConstraint(cons_f1171)
def cons_f1172(j, f, i, c, e):
return ZeroQ(f*i + j*(c - e))
cons1172 = CustomConstraint(cons_f1172)
def cons_f1173(m, n):
return Or(IntegerQ(n), Greater(m, S(0)))
cons1173 = CustomConstraint(cons_f1173)
def cons_f1174(p, j, m, f, b, g, i, d, c, a, n, x, h, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, h, i, j, m, n, p, q), x)
cons1174 = CustomConstraint(cons_f1174)
def cons_f1175(h, f, e, g):
return ZeroQ(e**S(2)*h + f**S(2)*g)
cons1175 = CustomConstraint(cons_f1175)
def cons_f1176(c, e):
return ZeroQ(c - S(2)*e)
cons1176 = CustomConstraint(cons_f1176)
def cons_f1177(c, e):
return PositiveQ(c/(S(2)*e))
cons1177 = CustomConstraint(cons_f1177)
def cons_f1178(c, e, a):
return Or(NonzeroQ(c - S(2)*e), NonzeroQ(a))
cons1178 = CustomConstraint(cons_f1178)
def cons_f1179(f, g, i, h, e):
return ZeroQ(e**S(2)*i - e*f*h + f**S(2)*g)
cons1179 = CustomConstraint(cons_f1179)
def cons_f1180(i, f, e, g):
return ZeroQ(e**S(2)*i + f**S(2)*g)
cons1180 = CustomConstraint(cons_f1180)
def cons_f1181(g):
return PositiveQ(g)
cons1181 = CustomConstraint(cons_f1181)
def cons_f1182(h2, h1, g1, g2):
return ZeroQ(g1*h2 + g2*h1)
cons1182 = CustomConstraint(cons_f1182)
def cons_f1183(g1):
return PositiveQ(g1)
cons1183 = CustomConstraint(cons_f1183)
def cons_f1184(g2):
return PositiveQ(g2)
cons1184 = CustomConstraint(cons_f1184)
def cons_f1185(g1, x):
return FreeQ(g1, x)
cons1185 = CustomConstraint(cons_f1185)
def cons_f1186(h1, x):
return FreeQ(h1, x)
cons1186 = CustomConstraint(cons_f1186)
def cons_f1187(g2, x):
return FreeQ(g2, x)
cons1187 = CustomConstraint(cons_f1187)
def cons_f1188(h2, x):
return FreeQ(h2, x)
cons1188 = CustomConstraint(cons_f1188)
def cons_f1189(g):
return Not(PositiveQ(g))
cons1189 = CustomConstraint(cons_f1189)
def cons_f1190(j, k, g, i, h):
return ZeroQ(h - i*(-g*k + h*j))
cons1190 = CustomConstraint(cons_f1190)
def cons_f1191(h, k, j, g):
return ZeroQ(-g*k + h*j)
cons1191 = CustomConstraint(cons_f1191)
def cons_f1192(F):
return MemberQ(List(Log, ArcSin, ArcCos, ArcTan, ArcCot, ArcSinh, ArcCosh, ArcTanh, ArcCoth), F)
cons1192 = CustomConstraint(cons_f1192)
def cons_f1193(m, r):
return ZeroQ(m + r)
cons1193 = CustomConstraint(cons_f1193)
def cons_f1194(r1, r):
return ZeroQ(-r + r1 + S(1))
cons1194 = CustomConstraint(cons_f1194)
def cons_f1195(b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, n), x)
cons1195 = CustomConstraint(cons_f1195)
def cons_f1196(mn, n):
return ZeroQ(mn + n)
cons1196 = CustomConstraint(cons_f1196)
def cons_f1197(b, d, a, c, e):
return ZeroQ(-a*c*d + b*c*e + d)
cons1197 = CustomConstraint(cons_f1197)
def cons_f1198(x, RFx):
if isinstance(x, (int, Integer, float, Float)):
return False
return RationalFunctionQ(RFx, x)
cons1198 = CustomConstraint(cons_f1198)
def cons_f1199(f, e, g):
return ZeroQ(-S(4)*e*g + f**S(2))
cons1199 = CustomConstraint(cons_f1199)
def cons_f1200(v, p, d, c, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1199(pp, qq, f, cc, dd, e):
return FreeQ(List(cc, dd, e, f, pp, qq), x)
_cons_1199 = CustomConstraint(_cons_f_1199)
pat = Pattern(UtilityOperator(((x*WC('f', S(1)) + WC('e', S(0)))**WC('pp', S(1))*WC('dd', S(1)))**WC('qq', S(1))*WC('cc', S(1)), x), _cons_1199)
result_matchq = is_match(UtilityOperator(c*(d*v**p)**q, x), pat)
return Not(result_matchq)
cons1200 = CustomConstraint(cons_f1200)
def cons_f1201(p, b, r, c, a, n, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, n, p, q, r), x)
cons1201 = CustomConstraint(cons_f1201)
def cons_f1202(p, b, a, n, c, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(SameQ(x**(n*p*q), a*(b*(c*x**n)**p)**q))
cons1202 = CustomConstraint(cons_f1202)
def cons_f1203(n1, n2):
return ZeroQ(n1 + n2)
cons1203 = CustomConstraint(cons_f1203)
def cons_f1204(n1, x):
return FreeQ(n1, x)
cons1204 = CustomConstraint(cons_f1204)
def cons_f1205(d, c, f, g):
return ZeroQ(-c*g + d*f)
cons1205 = CustomConstraint(cons_f1205)
def cons_f1206(d, b, e):
return ZeroQ(-b*e + d)
cons1206 = CustomConstraint(cons_f1206)
def cons_f1207(a, f, b, g):
return ZeroQ(-a*g + b*f)
cons1207 = CustomConstraint(cons_f1207)
def cons_f1208(d, c, f, g):
return NonzeroQ(-c*g + d*f)
cons1208 = CustomConstraint(cons_f1208)
def cons_f1209(a, f, b, g):
return NonzeroQ(-a*g + b*f)
cons1209 = CustomConstraint(cons_f1209)
def cons_f1210(m, m2):
return ZeroQ(m + m2 + S(2))
cons1210 = CustomConstraint(cons_f1210)
def cons_f1211(u, b, d, c, a, x):
return FreeQ(simplify(Mul(u, Add(c, Mul(d, x)), Pow(Add(a, Mul(b, x)), S(-1)))), x)
cons1211 = CustomConstraint(cons_f1211)
def cons_f1212(f, g, b, d, c, a, e):
return ZeroQ(-c*g + d*f - e*(-a*g + b*f))
cons1212 = CustomConstraint(cons_f1212)
def cons_f1213(d, c, f, g):
return ZeroQ(c**S(2)*g + d**S(2)*f)
cons1213 = CustomConstraint(cons_f1213)
def cons_f1214(b, d, c, a, e):
return ZeroQ(-a*d*e - b*c*e + S(2)*c*d)
cons1214 = CustomConstraint(cons_f1214)
def cons_f1215(f, g, d, c, h):
return ZeroQ(c**S(2)*h - c*d*g + d**S(2)*f)
cons1215 = CustomConstraint(cons_f1215)
def cons_f1216(d, h, c, f):
return ZeroQ(c**S(2)*h + d**S(2)*f)
cons1216 = CustomConstraint(cons_f1216)
def cons_f1217(v, u, b, d, c, a, x):
return FreeQ(simplify(Mul(u, Pow(Add(S(1), Mul(S(-1), v)), S(-1)))), x)
cons1217 = CustomConstraint(cons_f1217)
def cons_f1218(v, u, b, d, c, a, x):
return FreeQ(simplify(Mul(u, Add(S(1), Mul(S(-1), v)))), x)
cons1218 = CustomConstraint(cons_f1218)
def cons_f1219(v, u, b, d, c, a, x):
return FreeQ(simplify(Mul(u, Pow(v, S(-1)))), x)
cons1219 = CustomConstraint(cons_f1219)
def cons_f1220(v, u, b, d, c, a, x):
return FreeQ(simplify(Mul(u, v)), x)
cons1220 = CustomConstraint(cons_f1220)
def cons_f1221(f, b, d, c, a, h):
return ZeroQ(-a*c*h + b*d*f)
cons1221 = CustomConstraint(cons_f1221)
def cons_f1222(g, b, d, c, a, h):
return ZeroQ(-a*d*h - b*c*h + b*d*g)
cons1222 = CustomConstraint(cons_f1222)
def cons_f1223(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuotientOfLinearsQ(v, x)
cons1223 = CustomConstraint(cons_f1223)
def cons_f1224(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(QuotientOfLinearsMatchQ(v, x))
cons1224 = CustomConstraint(cons_f1224)
def cons_f1225(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(BinomialMatchQ(v, x))
cons1225 = CustomConstraint(cons_f1225)
def cons_f1226(p, f, b, g, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, n, p), x)
cons1226 = CustomConstraint(cons_f1226)
def cons_f1227(p, f, b, g, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g, p), x)
cons1227 = CustomConstraint(cons_f1227)
def cons_f1228(m):
return IntegerQ(m/S(2) + S(-1)/2)
cons1228 = CustomConstraint(cons_f1228)
def cons_f1229(m):
return Not(IntegerQ(m/S(2) + S(-1)/2))
cons1229 = CustomConstraint(cons_f1229)
def cons_f1230(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return InverseFunctionFreeQ(u, x)
cons1230 = CustomConstraint(cons_f1230)
def cons_f1231(n):
return Not(And(RationalQ(n), Less(n, S(0))))
cons1231 = CustomConstraint(cons_f1231)
def cons_f1232(m, n):
return Or(Equal(n, S(1)), IntegerQ(m))
cons1232 = CustomConstraint(cons_f1232)
def cons_f1233(x, RFx):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(PolynomialQ(RFx, x))
cons1233 = CustomConstraint(cons_f1233)
def cons_f1234(x, Qx, Px):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuadraticQ(List(Qx, Px), x)
cons1234 = CustomConstraint(cons_f1234)
def cons_f1235(x, Qx, Px):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(D(Px/Qx, x))
cons1235 = CustomConstraint(cons_f1235)
def cons_f1236(x, RGx):
if isinstance(x, (int, Integer, float, Float)):
return False
return RationalFunctionQ(RGx, x)
cons1236 = CustomConstraint(cons_f1236)
def cons_f1237(d):
return NonzeroQ(d + S(-1))
cons1237 = CustomConstraint(cons_f1237)
def cons_f1238(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1237(m, g):
return FreeQ(List(g, m), x)
_cons_1237 = CustomConstraint(_cons_f_1237)
pat = Pattern(UtilityOperator((x*WC('g', S(1)))**WC('m', S(1)), x), _cons_1237)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Or(ZeroQ(v + S(-1)), result_matchq)
cons1238 = CustomConstraint(cons_f1238)
def cons_f1239(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return RationalFunctionQ(D(u, x)/u, x)
cons1239 = CustomConstraint(cons_f1239)
def cons_f1240(x, a, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return Or(NonzeroQ(a), Not(And(BinomialQ(u, x), ZeroQ(BinomialDegree(u, x)**S(2) + S(-1)))))
except (TypeError, AttributeError):
return False
cons1240 = CustomConstraint(cons_f1240)
def cons_f1241(x, Qx):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuadraticQ(Qx, x)
cons1241 = CustomConstraint(cons_f1241)
def cons_f1242(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return InverseFunctionFreeQ(v, x)
cons1242 = CustomConstraint(cons_f1242)
def cons_f1243(x, w):
if isinstance(x, (int, Integer, float, Float)):
return False
return InverseFunctionFreeQ(w, x)
cons1243 = CustomConstraint(cons_f1243)
def cons_f1244(p, b, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, n, p), x)
cons1244 = CustomConstraint(cons_f1244)
def cons_f1245(B, a, b, A):
return NonzeroQ(A*b - B*a)
cons1245 = CustomConstraint(cons_f1245)
def cons_f1246(x, a, f):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, f), x)
cons1246 = CustomConstraint(cons_f1246)
def cons_f1247(u):
return NonsumQ(u)
cons1247 = CustomConstraint(cons_f1247)
def cons_f1248(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return AlgebraicFunctionQ(u, x)
cons1248 = CustomConstraint(cons_f1248)
def cons_f1249(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return FunctionOfTrigOfLinearQ(u, x)
cons1249 = CustomConstraint(cons_f1249)
def cons_f1250(n):
return IntegerQ(n/S(2) + S(-1)/2)
cons1250 = CustomConstraint(cons_f1250)
def cons_f1251(m, n):
return Not(And(IntegerQ(m/S(2) + S(-1)/2), Less(S(0), m, n)))
cons1251 = CustomConstraint(cons_f1251)
def cons_f1252(m, n):
return Not(And(IntegerQ(m/S(2) + S(-1)/2), Inequality(S(0), Less, m, LessEqual, n)))
cons1252 = CustomConstraint(cons_f1252)
def cons_f1253(m, n):
return Or(IntegersQ(S(2)*m, S(2)*n), ZeroQ(m + n))
cons1253 = CustomConstraint(cons_f1253)
def cons_f1254(f, b, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, e, f), x)
cons1254 = CustomConstraint(cons_f1254)
def cons_f1255(m, n):
return ZeroQ(m + n)
cons1255 = CustomConstraint(cons_f1255)
def cons_f1256(m):
return Less(S(0), m, S(1))
cons1256 = CustomConstraint(cons_f1256)
def cons_f1257(m, n, b, a):
return Or(RationalQ(n), And(Not(RationalQ(m)), Or(ZeroQ(b + S(-1)), NonzeroQ(a + S(-1)))))
cons1257 = CustomConstraint(cons_f1257)
def cons_f1258(m, f, b, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, e, f, m, n), x)
cons1258 = CustomConstraint(cons_f1258)
def cons_f1259(m, n):
return ZeroQ(m - n + S(2))
cons1259 = CustomConstraint(cons_f1259)
def cons_f1260(m, n):
return NonzeroQ(m - n)
cons1260 = CustomConstraint(cons_f1260)
def cons_f1261(d, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(c, d), x)
cons1261 = CustomConstraint(cons_f1261)
def cons_f1262(x, c):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(c, c), x)
cons1262 = CustomConstraint(cons_f1262)
def cons_f1263(d, c, b, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, c, d), x)
cons1263 = CustomConstraint(cons_f1263)
def cons_f1264(b, d, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d), x)
cons1264 = CustomConstraint(cons_f1264)
def cons_f1265(a, b):
return ZeroQ(a**S(2) - b**S(2))
cons1265 = CustomConstraint(cons_f1265)
def cons_f1266(n):
return PositiveIntegerQ(n + S(-1)/2)
cons1266 = CustomConstraint(cons_f1266)
def cons_f1267(a, b):
return NonzeroQ(a**S(2) - b**S(2))
cons1267 = CustomConstraint(cons_f1267)
def cons_f1268(a, b):
return PositiveQ(a + b)
cons1268 = CustomConstraint(cons_f1268)
def cons_f1269(a, b):
return PositiveQ(a - b)
cons1269 = CustomConstraint(cons_f1269)
def cons_f1270(a, b):
return Not(PositiveQ(a + b))
cons1270 = CustomConstraint(cons_f1270)
def cons_f1271(a, b):
return PositiveQ(a**S(2) - b**S(2))
cons1271 = CustomConstraint(cons_f1271)
def cons_f1272(c):
return SimplerQ(-Pi/S(2) + c, c)
cons1272 = CustomConstraint(cons_f1272)
def cons_f1273(b, d, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, n), x)
cons1273 = CustomConstraint(cons_f1273)
def cons_f1274(p):
return IntegerQ(p/S(2) + S(-1)/2)
cons1274 = CustomConstraint(cons_f1274)
def cons_f1275(m, p, b, a):
return Or(GreaterEqual(p, S(-1)), Not(And(IntegerQ(m + S(1)/2), ZeroQ(a**S(2) - b**S(2)))))
cons1275 = CustomConstraint(cons_f1275)
def cons_f1276(a, p, b):
return Or(IntegerQ(S(2)*p), NonzeroQ(a**S(2) - b**S(2)))
cons1276 = CustomConstraint(cons_f1276)
def cons_f1277(m, p):
return GreaterEqual(S(2)*m + p, S(0))
cons1277 = CustomConstraint(cons_f1277)
def cons_f1278(m, p):
return ZeroQ(m + p + S(1))
cons1278 = CustomConstraint(cons_f1278)
def cons_f1279(p):
return Not(NegativeIntegerQ(p))
cons1279 = CustomConstraint(cons_f1279)
def cons_f1280(m, p):
return NegativeIntegerQ(m + p + S(1))
cons1280 = CustomConstraint(cons_f1280)
def cons_f1281(m, p):
return NonzeroQ(S(2)*m + p + S(1))
cons1281 = CustomConstraint(cons_f1281)
def cons_f1282(m, p):
return ZeroQ(S(2)*m + p + S(-1))
cons1282 = CustomConstraint(cons_f1282)
def cons_f1283(m):
return NonzeroQ(m + S(-1))
cons1283 = CustomConstraint(cons_f1283)
def cons_f1284(m, p):
return PositiveIntegerQ(m + p/S(2) + S(-1)/2)
cons1284 = CustomConstraint(cons_f1284)
def cons_f1285(m, p):
return NonzeroQ(m + p)
cons1285 = CustomConstraint(cons_f1285)
def cons_f1286(m, p):
return LessEqual(p, -S(2)*m)
cons1286 = CustomConstraint(cons_f1286)
def cons_f1287(m, p):
return IntegersQ(m + S(1)/2, S(2)*p)
cons1287 = CustomConstraint(cons_f1287)
def cons_f1288(m, p):
return IntegersQ(S(2)*m, S(2)*p)
cons1288 = CustomConstraint(cons_f1288)
def cons_f1289(m, p):
return Or(Greater(m, S(-2)), ZeroQ(S(2)*m + p + S(1)), And(Equal(m, S(-2)), IntegerQ(p)))
cons1289 = CustomConstraint(cons_f1289)
def cons_f1290(m):
return LessEqual(m, S(-2))
cons1290 = CustomConstraint(cons_f1290)
def cons_f1291(m, p):
return Not(NegativeIntegerQ(m + p + S(1)))
cons1291 = CustomConstraint(cons_f1291)
def cons_f1292(p):
return Not(And(RationalQ(p), GreaterEqual(p, S(1))))
cons1292 = CustomConstraint(cons_f1292)
def cons_f1293(p):
return Greater(p, S(2))
cons1293 = CustomConstraint(cons_f1293)
def cons_f1294(m, p):
return Or(IntegersQ(S(2)*m, S(2)*p), IntegerQ(m))
cons1294 = CustomConstraint(cons_f1294)
def cons_f1295(m, p):
return ZeroQ(m + p + S(2))
cons1295 = CustomConstraint(cons_f1295)
def cons_f1296(m, p):
return NegativeIntegerQ(m + p + S(2))
cons1296 = CustomConstraint(cons_f1296)
def cons_f1297(m, p):
return Not(PositiveIntegerQ(m + p + S(1)))
cons1297 = CustomConstraint(cons_f1297)
def cons_f1298(p):
return IntegerQ(p/S(2) + S(1)/2)
cons1298 = CustomConstraint(cons_f1298)
def cons_f1299(m, p):
return IntegersQ(m, p)
cons1299 = CustomConstraint(cons_f1299)
def cons_f1300(m, p):
return Equal(p, S(2)*m)
cons1300 = CustomConstraint(cons_f1300)
def cons_f1301(m, p):
return IntegersQ(m, p/S(2))
cons1301 = CustomConstraint(cons_f1301)
def cons_f1302(m, p):
return Or(Less(p, S(0)), Greater(m - p/S(2), S(0)))
cons1302 = CustomConstraint(cons_f1302)
def cons_f1303(m):
return Not(And(RationalQ(m), Less(m, S(0))))
cons1303 = CustomConstraint(cons_f1303)
def cons_f1304(m):
return IntegerQ(m + S(-1)/2)
cons1304 = CustomConstraint(cons_f1304)
def cons_f1305(m):
return Not(Less(m, S(-1)))
cons1305 = CustomConstraint(cons_f1305)
def cons_f1306(p):
return IntegerQ(p/S(2))
cons1306 = CustomConstraint(cons_f1306)
def cons_f1307(p):
return IntegersQ(S(2)*p)
cons1307 = CustomConstraint(cons_f1307)
def cons_f1308(p, m, f, b, g, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, e, f, g, m, p), x)
cons1308 = CustomConstraint(cons_f1308)
def cons_f1309(m, n):
return Not(And(IntegerQ(n), Or(And(Less(m, S(0)), Greater(n, S(0))), Less(S(0), n, m), Less(m, n, S(0)))))
cons1309 = CustomConstraint(cons_f1309)
def cons_f1310(n):
return NonzeroQ(n + S(1)/2)
cons1310 = CustomConstraint(cons_f1310)
def cons_f1311(m):
return PositiveIntegerQ(m + S(-1)/2)
cons1311 = CustomConstraint(cons_f1311)
def cons_f1312(m, n):
return Not(And(NegativeIntegerQ(m + n), Greater(S(2)*m + n + S(1), S(0))))
cons1312 = CustomConstraint(cons_f1312)
def cons_f1313(m, n):
return Not(And(PositiveIntegerQ(n + S(-1)/2), Less(n, m)))
cons1313 = CustomConstraint(cons_f1313)
def cons_f1314(m):
return NonzeroQ(m + S(1)/2)
cons1314 = CustomConstraint(cons_f1314)
def cons_f1315(m, n):
return NegativeIntegerQ(m + n + S(1))
cons1315 = CustomConstraint(cons_f1315)
def cons_f1316(m, n):
return Not(And(RationalQ(n), Less(m, n, S(-1))))
cons1316 = CustomConstraint(cons_f1316)
def cons_f1317(m, n):
return Or(FractionQ(m), Not(FractionQ(n)))
cons1317 = CustomConstraint(cons_f1317)
def cons_f1318(m, f, b, d, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, c, d, e, f, m), x)
cons1318 = CustomConstraint(cons_f1318)
def cons_f1319(m, b, d, a, c):
return ZeroQ(a*d*m + b*c*(m + S(1)))
cons1319 = CustomConstraint(cons_f1319)
def cons_f1320(m):
return Less(m, S(-1)/2)
cons1320 = CustomConstraint(cons_f1320)
def cons_f1321(m):
return Not(And(RationalQ(m), Less(m, S(-1)/2)))
cons1321 = CustomConstraint(cons_f1321)
def cons_f1322(d, c):
return ZeroQ(c**S(2) - d**S(2))
cons1322 = CustomConstraint(cons_f1322)
def cons_f1323(d, c):
return NonzeroQ(c**S(2) - d**S(2))
cons1323 = CustomConstraint(cons_f1323)
def cons_f1324(m, n, c):
return Or(IntegersQ(S(2)*m, S(2)*n), IntegerQ(m + S(1)/2), And(IntegerQ(m), ZeroQ(c)))
cons1324 = CustomConstraint(cons_f1324)
def cons_f1325(n):
return Less(S(0), n, S(1))
cons1325 = CustomConstraint(cons_f1325)
def cons_f1326(m, n, c):
return Or(IntegersQ(S(2)*m, S(2)*n), And(IntegerQ(m), ZeroQ(c)))
cons1326 = CustomConstraint(cons_f1326)
def cons_f1327(n):
return Not(And(RationalQ(n), Greater(n, S(0))))
cons1327 = CustomConstraint(cons_f1327)
def cons_f1328(c, n):
return Or(IntegerQ(S(2)*n), ZeroQ(c))
cons1328 = CustomConstraint(cons_f1328)
def cons_f1329(n):
return NonzeroQ(S(2)*n + S(3))
cons1329 = CustomConstraint(cons_f1329)
def cons_f1330(d, a, b):
return ZeroQ(-a/b + d)
cons1330 = CustomConstraint(cons_f1330)
def cons_f1331(d, b):
return PositiveQ(d/b)
cons1331 = CustomConstraint(cons_f1331)
def cons_f1332(d, b):
return Not(PositiveQ(d/b))
cons1332 = CustomConstraint(cons_f1332)
def cons_f1333(m):
return Greater(m, S(2))
cons1333 = CustomConstraint(cons_f1333)
def cons_f1334(m, n):
return Or(IntegerQ(m), IntegersQ(S(2)*m, S(2)*n))
cons1334 = CustomConstraint(cons_f1334)
def cons_f1335(m, n, c, a):
return Not(And(IntegerQ(n), Greater(n, S(2)), Or(Not(IntegerQ(m)), And(ZeroQ(a), NonzeroQ(c)))))
cons1335 = CustomConstraint(cons_f1335)
def cons_f1336(n):
return Less(S(1), n, S(2))
cons1336 = CustomConstraint(cons_f1336)
def cons_f1337(m, a, n):
return Or(And(ZeroQ(a), IntegerQ(m), Not(IntegerQ(n))), Not(And(IntegerQ(S(2)*n), Less(n, S(-1)), Or(And(IntegerQ(n), Not(IntegerQ(m))), ZeroQ(a)))))
cons1337 = CustomConstraint(cons_f1337)
def cons_f1338(d, c):
return PositiveQ(c + d)
cons1338 = CustomConstraint(cons_f1338)
def cons_f1339(d, c):
return PositiveQ(c - d)
cons1339 = CustomConstraint(cons_f1339)
def cons_f1340(d, c):
return Not(PositiveQ(c + d))
cons1340 = CustomConstraint(cons_f1340)
def cons_f1341(d, c):
return PositiveQ(c**S(2) - d**S(2))
cons1341 = CustomConstraint(cons_f1341)
def cons_f1342(d, c, b):
return PosQ((c + d)/b)
cons1342 = CustomConstraint(cons_f1342)
def cons_f1343(c):
return PositiveQ(c**S(2))
cons1343 = CustomConstraint(cons_f1343)
def cons_f1344(d, c, b):
return NegQ((c + d)/b)
cons1344 = CustomConstraint(cons_f1344)
def cons_f1345(d, c, a, b):
return PosQ((a + b)/(c + d))
cons1345 = CustomConstraint(cons_f1345)
def cons_f1346(d, c, a, b):
return NegQ((a + b)/(c + d))
cons1346 = CustomConstraint(cons_f1346)
def cons_f1347(a, b):
return NegativeQ(a**S(2) - b**S(2))
cons1347 = CustomConstraint(cons_f1347)
def cons_f1348(d):
return ZeroQ(d**S(2) + S(-1))
cons1348 = CustomConstraint(cons_f1348)
def cons_f1349(d, b):
return PositiveQ(b*d)
cons1349 = CustomConstraint(cons_f1349)
def cons_f1350(b):
return PositiveQ(b**S(2))
cons1350 = CustomConstraint(cons_f1350)
def cons_f1351(d, b):
return Not(And(ZeroQ(d**S(2) + S(-1)), PositiveQ(b*d)))
cons1351 = CustomConstraint(cons_f1351)
def cons_f1352(d, a, b):
return PosQ((a + b)/d)
cons1352 = CustomConstraint(cons_f1352)
def cons_f1353(a):
return PositiveQ(a**S(2))
cons1353 = CustomConstraint(cons_f1353)
def cons_f1354(d, a, b):
return NegQ((a + b)/d)
cons1354 = CustomConstraint(cons_f1354)
def cons_f1355(d, c, b, a):
return PosQ((c + d)/(a + b))
cons1355 = CustomConstraint(cons_f1355)
def cons_f1356(d, c, b, a):
return NegQ((c + d)/(a + b))
cons1356 = CustomConstraint(cons_f1356)
def cons_f1357(m):
return Less(S(0), m, S(2))
cons1357 = CustomConstraint(cons_f1357)
def cons_f1358(n):
return Less(S(-1), n, S(2))
cons1358 = CustomConstraint(cons_f1358)
def cons_f1359(m, n):
return NonzeroQ(m + n)
cons1359 = CustomConstraint(cons_f1359)
def cons_f1360(m, f, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, m, n), x)
cons1360 = CustomConstraint(cons_f1360)
def cons_f1361(a, p, b, n):
return Or(And(Less(p, S(0)), NonzeroQ(a**S(2) - b**S(2))), Less(S(0), n, p + S(-1)), Less(p + S(1), -n, S(2)*p + S(1)))
cons1361 = CustomConstraint(cons_f1361)
def cons_f1362(p, n):
return Or(Less(S(0), n, p/S(2) + S(1)/2), Inequality(p, LessEqual, -n, Less, S(2)*p + S(-3)), Inequality(S(0), Less, n, LessEqual, -p))
cons1362 = CustomConstraint(cons_f1362)
def cons_f1363(p, f, b, g, d, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, d, e, f, g, n, p), x)
cons1363 = CustomConstraint(cons_f1363)
def cons_f1364(m, n):
return Not(And(IntegerQ(n), Less(n**S(2), m**S(2))))
cons1364 = CustomConstraint(cons_f1364)
def cons_f1365(p, n):
return NonzeroQ(S(2)*n + p + S(1))
cons1365 = CustomConstraint(cons_f1365)
def cons_f1366(m, n, p):
return Not(And(NegativeIntegerQ(m + n + p), Greater(S(2)*m + n + S(3)*p/S(2) + S(1), S(0))))
cons1366 = CustomConstraint(cons_f1366)
def cons_f1367(m, p, n):
return Not(And(PositiveIntegerQ(n + p/S(2) + S(-1)/2), Greater(m - n, S(0))))
cons1367 = CustomConstraint(cons_f1367)
def cons_f1368(m, p):
return ZeroQ(S(2)*m + p + S(1))
cons1368 = CustomConstraint(cons_f1368)
def cons_f1369(m, n, p):
return ZeroQ(m + n + p + S(1))
cons1369 = CustomConstraint(cons_f1369)
def cons_f1370(m, n, p):
return NegativeIntegerQ(m + n + p + S(1))
cons1370 = CustomConstraint(cons_f1370)
def cons_f1371(m, n, p):
return NonzeroQ(m + n + p)
cons1371 = CustomConstraint(cons_f1371)
def cons_f1372(m, n):
return Not(And(RationalQ(n), Less(S(0), n, m)))
cons1372 = CustomConstraint(cons_f1372)
def cons_f1373(p, m, b, d, a, c):
return ZeroQ(a*d*m + b*c*(m + p + S(1)))
cons1373 = CustomConstraint(cons_f1373)
def cons_f1374(m):
return Greater(m, S(-1))
cons1374 = CustomConstraint(cons_f1374)
def cons_f1375(m, p):
return PositiveIntegerQ(m + p/S(2) + S(1)/2)
cons1375 = CustomConstraint(cons_f1375)
def cons_f1376(m):
return Less(m, S(-3)/2)
cons1376 = CustomConstraint(cons_f1376)
def cons_f1377(m):
return Inequality(S(-3)/2, LessEqual, m, Less, S(0))
cons1377 = CustomConstraint(cons_f1377)
def cons_f1378(m, p):
return Or(And(RationalQ(m), Less(m, S(-1))), NegativeIntegerQ(m + p))
cons1378 = CustomConstraint(cons_f1378)
def cons_f1379(p):
return Not(And(RationalQ(p), Less(p, S(-1))))
cons1379 = CustomConstraint(cons_f1379)
def cons_f1380(m, p):
return Equal(S(2)*m + p, S(0))
cons1380 = CustomConstraint(cons_f1380)
def cons_f1381(m, n, p):
return IntegersQ(m, n, p/S(2))
cons1381 = CustomConstraint(cons_f1381)
def cons_f1382(m, p, n):
return Or(And(Greater(m, S(0)), Greater(p, S(0)), Less(-m - p, n, S(-1))), And(Greater(m, S(2)), Less(p, S(0)), Greater(m + p/S(2), S(0))))
cons1382 = CustomConstraint(cons_f1382)
def cons_f1383(m, n):
return Or(NegativeIntegerQ(m), Not(PositiveIntegerQ(n)))
cons1383 = CustomConstraint(cons_f1383)
def cons_f1384(m, p):
return Or(Equal(S(2)*m + p, S(0)), And(Greater(S(2)*m + p, S(0)), Less(p, S(-1))))
cons1384 = CustomConstraint(cons_f1384)
def cons_f1385(m):
return LessEqual(m, S(-1)/2)
cons1385 = CustomConstraint(cons_f1385)
def cons_f1386(m, p):
return NonzeroQ(m + p + S(2))
cons1386 = CustomConstraint(cons_f1386)
def cons_f1387(p, n):
return Or(IntegerQ(p), PositiveIntegerQ(n))
cons1387 = CustomConstraint(cons_f1387)
def cons_f1388(m, p):
return ZeroQ(m + p + S(1)/2)
cons1388 = CustomConstraint(cons_f1388)
def cons_f1389(m, p):
return ZeroQ(m + p + S(3)/2)
cons1389 = CustomConstraint(cons_f1389)
def cons_f1390(m, n):
return Or(PositiveIntegerQ(m), IntegersQ(S(2)*m, S(2)*n))
cons1390 = CustomConstraint(cons_f1390)
def cons_f1391(n):
return Not(Less(n, S(-1)))
cons1391 = CustomConstraint(cons_f1391)
def cons_f1392(m, n):
return Or(Less(m, S(-2)), ZeroQ(m + n + S(4)))
cons1392 = CustomConstraint(cons_f1392)
def cons_f1393(m, n):
return NonzeroQ(m + n + S(4))
cons1393 = CustomConstraint(cons_f1393)
def cons_f1394(m, n):
return Or(Less(n, S(-2)), ZeroQ(m + n + S(4)))
cons1394 = CustomConstraint(cons_f1394)
def cons_f1395(n):
return NonzeroQ(n + S(2))
cons1395 = CustomConstraint(cons_f1395)
def cons_f1396(m, n):
return NonzeroQ(m + n + S(5))
cons1396 = CustomConstraint(cons_f1396)
def cons_f1397(m, n):
return NonzeroQ(m + n + S(6))
cons1397 = CustomConstraint(cons_f1397)
def cons_f1398(m, n, p):
return IntegersQ(m, S(2)*n, p/S(2))
cons1398 = CustomConstraint(cons_f1398)
def cons_f1399(m, p):
return Or(Less(m, S(-1)), And(Equal(m, S(-1)), Greater(p, S(0))))
cons1399 = CustomConstraint(cons_f1399)
def cons_f1400(p, n):
return Or(Less(n, S(0)), PositiveIntegerQ(p + S(1)/2))
cons1400 = CustomConstraint(cons_f1400)
def cons_f1401(p, n):
return IntegersQ(S(2)*n, S(2)*p)
cons1401 = CustomConstraint(cons_f1401)
def cons_f1402(p, n):
return Or(LessEqual(n, S(-2)), And(Equal(n, S(-3)/2), Equal(p, S(3)/2)))
cons1402 = CustomConstraint(cons_f1402)
def cons_f1403(p, n):
return Or(Less(n, S(-1)), And(Equal(p, S(3)/2), Equal(n, S(-1)/2)))
cons1403 = CustomConstraint(cons_f1403)
def cons_f1404(p):
return Less(S(-1), p, S(1))
cons1404 = CustomConstraint(cons_f1404)
def cons_f1405(m, n):
return Or(Greater(m, S(0)), IntegerQ(n))
cons1405 = CustomConstraint(cons_f1405)
def cons_f1406(m, n, p):
return IntegersQ(m, S(2)*n, S(2)*p)
cons1406 = CustomConstraint(cons_f1406)
def cons_f1407(m, n, p):
return Or(LessEqual(n, S(-2)), And(Equal(m, S(-1)), Equal(n, S(-3)/2), Equal(p, S(3)/2)))
cons1407 = CustomConstraint(cons_f1407)
def cons_f1408(p):
return PositiveIntegerQ(p/S(2))
cons1408 = CustomConstraint(cons_f1408)
def cons_f1409(d, c, a, b):
return Or(ZeroQ(a**S(2) - b**S(2)), ZeroQ(c**S(2) - d**S(2)))
cons1409 = CustomConstraint(cons_f1409)
def cons_f1410(d, c):
return ZeroQ(-c + d)
cons1410 = CustomConstraint(cons_f1410)
def cons_f1411(a, b):
return PositiveQ(-a**S(2) + b**S(2))
cons1411 = CustomConstraint(cons_f1411)
def cons_f1412(d, c, b, a):
return NonzeroQ(a*d + b*c)
cons1412 = CustomConstraint(cons_f1412)
def cons_f1413(d, c, a, b):
return Or(NonzeroQ(a**S(2) - b**S(2)), NonzeroQ(c**S(2) - d**S(2)))
cons1413 = CustomConstraint(cons_f1413)
def cons_f1414(p, n):
return ZeroQ(S(2)*n + p)
cons1414 = CustomConstraint(cons_f1414)
def cons_f1415(m, n, p):
return Or(IntegersQ(m, n), IntegersQ(m, p), IntegersQ(n, p))
cons1415 = CustomConstraint(cons_f1415)
def cons_f1416(p):
return NonzeroQ(p + S(-2))
cons1416 = CustomConstraint(cons_f1416)
def cons_f1417(m, n):
return Not(And(IntegerQ(m), IntegerQ(n)))
cons1417 = CustomConstraint(cons_f1417)
def cons_f1418(B, a, b, A):
return ZeroQ(A*b + B*a)
cons1418 = CustomConstraint(cons_f1418)
def cons_f1419(B, m, b, a, n, A):
return ZeroQ(A*b*(m + n + S(1)) + B*a*(m - n))
cons1419 = CustomConstraint(cons_f1419)
def cons_f1420(m, n):
return Or(And(RationalQ(m), Less(m, S(-1)/2)), And(NegativeIntegerQ(m + n), Not(SumSimplerQ(n, S(1)))))
cons1420 = CustomConstraint(cons_f1420)
def cons_f1421(m):
return NonzeroQ(S(2)*m + S(1))
cons1421 = CustomConstraint(cons_f1421)
def cons_f1422(B, m, b, d, a, c, n, A):
return ZeroQ(A*(a*d*m + b*c*(n + S(1))) - B*(a*c*m + b*d*(n + S(1))))
cons1422 = CustomConstraint(cons_f1422)
def cons_f1423(m):
return Greater(m, S(1)/2)
cons1423 = CustomConstraint(cons_f1423)
def cons_f1424(B, b, d, c, n, a, A):
return ZeroQ(A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))
cons1424 = CustomConstraint(cons_f1424)
def cons_f1425(m, n):
return Or(IntegerQ(n), ZeroQ(m + S(1)/2))
cons1425 = CustomConstraint(cons_f1425)
def cons_f1426(B, a, b, A):
return NonzeroQ(A*b + B*a)
cons1426 = CustomConstraint(cons_f1426)
def cons_f1427(m, n, c, a):
return Not(And(IntegerQ(n), Greater(n, S(1)), Or(Not(IntegerQ(m)), And(ZeroQ(a), NonzeroQ(c)))))
cons1427 = CustomConstraint(cons_f1427)
def cons_f1428(B, A):
return ZeroQ(A - B)
cons1428 = CustomConstraint(cons_f1428)
def cons_f1429(B, A):
return NonzeroQ(A - B)
cons1429 = CustomConstraint(cons_f1429)
def cons_f1430(n):
return Equal(n**S(2), S(1)/4)
cons1430 = CustomConstraint(cons_f1430)
def cons_f1431(B, C, m, f, b, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, e, f, B, C, m), x)
cons1431 = CustomConstraint(cons_f1431)
def cons_f1432(m, C, A):
return ZeroQ(A*(m + S(2)) + C*(m + S(1)))
cons1432 = CustomConstraint(cons_f1432)
def cons_f1433(a, C, b, A):
return ZeroQ(A*b**S(2) + C*a**S(2))
cons1433 = CustomConstraint(cons_f1433)
def cons_f1434(B, C, A):
return ZeroQ(A - B + C)
cons1434 = CustomConstraint(cons_f1434)
def cons_f1435(C, A):
return ZeroQ(A + C)
cons1435 = CustomConstraint(cons_f1435)
def cons_f1436(m, n):
return Or(And(RationalQ(m), Less(m, S(-1)/2)), And(ZeroQ(m + n + S(2)), NonzeroQ(S(2)*m + S(1))))
cons1436 = CustomConstraint(cons_f1436)
def cons_f1437(m, n):
return Or(And(RationalQ(n), Less(n, S(-1))), ZeroQ(m + n + S(2)))
cons1437 = CustomConstraint(cons_f1437)
def cons_f1438(m, n, c, a):
return Not(And(IntegerQ(n), Greater(n, S(0)), Or(Not(IntegerQ(m)), And(ZeroQ(a), NonzeroQ(c)))))
cons1438 = CustomConstraint(cons_f1438)
def cons_f1439(a, b):
return ZeroQ(a**S(2) + b**S(2))
cons1439 = CustomConstraint(cons_f1439)
def cons_f1440(a, b):
return NonzeroQ(a**S(2) + b**S(2))
cons1440 = CustomConstraint(cons_f1440)
def cons_f1441(n):
return Not(OddQ(n))
cons1441 = CustomConstraint(cons_f1441)
def cons_f1442(n):
return Unequal(n, S(-2))
cons1442 = CustomConstraint(cons_f1442)
def cons_f1443(n):
return Not(And(RationalQ(n), Or(GreaterEqual(n, S(1)), LessEqual(n, S(-1)))))
cons1443 = CustomConstraint(cons_f1443)
def cons_f1444(a, b):
return PositiveQ(a**S(2) + b**S(2))
cons1444 = CustomConstraint(cons_f1444)
def cons_f1445(a, b):
return Not(Or(PositiveQ(a**S(2) + b**S(2)), ZeroQ(a**S(2) + b**S(2))))
cons1445 = CustomConstraint(cons_f1445)
def cons_f1446(m, n):
return IntegerQ(m/S(2) + n/S(2))
cons1446 = CustomConstraint(cons_f1446)
def cons_f1447(m, n):
return Not(And(Greater(n, S(0)), Greater(m, S(1))))
cons1447 = CustomConstraint(cons_f1447)
def cons_f1448(c, a, b):
return ZeroQ(a**S(2) - b**S(2) - c**S(2))
cons1448 = CustomConstraint(cons_f1448)
def cons_f1449(c, b):
return ZeroQ(b**S(2) + c**S(2))
cons1449 = CustomConstraint(cons_f1449)
def cons_f1450(c, b):
return NonzeroQ(b**S(2) + c**S(2))
cons1450 = CustomConstraint(cons_f1450)
def cons_f1451(c, a, b):
return PositiveQ(a + sqrt(b**S(2) + c**S(2)))
cons1451 = CustomConstraint(cons_f1451)
def cons_f1452(c, a, b):
return NonzeroQ(a**S(2) - b**S(2) - c**S(2))
cons1452 = CustomConstraint(cons_f1452)
def cons_f1453(c, a, b):
return Not(PositiveQ(a + sqrt(b**S(2) + c**S(2))))
cons1453 = CustomConstraint(cons_f1453)
def cons_f1454(a, b):
return ZeroQ(a + b)
cons1454 = CustomConstraint(cons_f1454)
def cons_f1455(a, c):
return ZeroQ(a - c)
cons1455 = CustomConstraint(cons_f1455)
def cons_f1456(a, b):
return NonzeroQ(a - b)
cons1456 = CustomConstraint(cons_f1456)
def cons_f1457(n):
return Unequal(n, S(-3)/2)
cons1457 = CustomConstraint(cons_f1457)
def cons_f1458(B, C, b, c, a, A):
return ZeroQ(A*(b**S(2) + c**S(2)) - a*(B*b + C*c))
cons1458 = CustomConstraint(cons_f1458)
def cons_f1459(C, b, c, a, A):
return ZeroQ(A*(b**S(2) + c**S(2)) - C*a*c)
cons1459 = CustomConstraint(cons_f1459)
def cons_f1460(B, b, c, a, A):
return ZeroQ(A*(b**S(2) + c**S(2)) - B*a*b)
cons1460 = CustomConstraint(cons_f1460)
def cons_f1461(B, C, b, c, a, A):
return NonzeroQ(A*(b**S(2) + c**S(2)) - a*(B*b + C*c))
cons1461 = CustomConstraint(cons_f1461)
def cons_f1462(C, b, c, a, A):
return NonzeroQ(A*(b**S(2) + c**S(2)) - C*a*c)
cons1462 = CustomConstraint(cons_f1462)
def cons_f1463(B, b, c, a, A):
return NonzeroQ(A*(b**S(2) + c**S(2)) - B*a*b)
cons1463 = CustomConstraint(cons_f1463)
def cons_f1464(B, C, b, c, n, a, A):
return ZeroQ(A*a*(n + S(1)) + n*(B*b + C*c))
cons1464 = CustomConstraint(cons_f1464)
def cons_f1465(C, c, a, n, A):
return ZeroQ(A*a*(n + S(1)) + C*c*n)
cons1465 = CustomConstraint(cons_f1465)
def cons_f1466(B, b, a, n, A):
return ZeroQ(A*a*(n + S(1)) + B*b*n)
cons1466 = CustomConstraint(cons_f1466)
def cons_f1467(B, C, b, c, n, a, A):
return NonzeroQ(A*a*(n + S(1)) + n*(B*b + C*c))
cons1467 = CustomConstraint(cons_f1467)
def cons_f1468(C, c, a, n, A):
return NonzeroQ(A*a*(n + S(1)) + C*c*n)
cons1468 = CustomConstraint(cons_f1468)
def cons_f1469(B, b, a, n, A):
return NonzeroQ(A*a*(n + S(1)) + B*b*n)
cons1469 = CustomConstraint(cons_f1469)
def cons_f1470(B, c, C, b):
return ZeroQ(B*b + C*c)
cons1470 = CustomConstraint(cons_f1470)
def cons_f1471(B, c, C, b):
return ZeroQ(B*c - C*b)
cons1471 = CustomConstraint(cons_f1471)
def cons_f1472(B, C, b, c, a, A):
return ZeroQ(A*a - B*b - C*c)
cons1472 = CustomConstraint(cons_f1472)
def cons_f1473(a, C, c, A):
return ZeroQ(A*a - C*c)
cons1473 = CustomConstraint(cons_f1473)
def cons_f1474(B, a, b, A):
return ZeroQ(A*a - B*b)
cons1474 = CustomConstraint(cons_f1474)
def cons_f1475(B, C, b, c, a, A):
return NonzeroQ(A*a - B*b - C*c)
cons1475 = CustomConstraint(cons_f1475)
def cons_f1476(a, C, c, A):
return NonzeroQ(A*a - C*c)
cons1476 = CustomConstraint(cons_f1476)
def cons_f1477(B, a, b, A):
return NonzeroQ(A*a - B*b)
cons1477 = CustomConstraint(cons_f1477)
def cons_f1478(a, b):
return NonzeroQ(a + b)
cons1478 = CustomConstraint(cons_f1478)
def cons_f1479(n):
return EvenQ(n)
cons1479 = CustomConstraint(cons_f1479)
def cons_f1480(m):
return EvenQ(m)
cons1480 = CustomConstraint(cons_f1480)
def cons_f1481(m):
return OddQ(m)
cons1481 = CustomConstraint(cons_f1481)
def cons_f1482(n):
return OddQ(n)
cons1482 = CustomConstraint(cons_f1482)
def cons_f1483(m):
return Not(OddQ(m))
cons1483 = CustomConstraint(cons_f1483)
def cons_f1484(p):
return EvenQ(p)
cons1484 = CustomConstraint(cons_f1484)
def cons_f1485(q):
return EvenQ(q)
cons1485 = CustomConstraint(cons_f1485)
def cons_f1486(p, q):
return Inequality(S(0), Less, p, LessEqual, q)
cons1486 = CustomConstraint(cons_f1486)
def cons_f1487(p, q):
return Less(S(0), q, p)
cons1487 = CustomConstraint(cons_f1487)
def cons_f1488(m, f, d, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(c, d, e, f, m), x)
cons1488 = CustomConstraint(cons_f1488)
def cons_f1489(m):
return Or(Not(RationalQ(m)), Inequality(S(-1), LessEqual, m, Less, S(1)))
cons1489 = CustomConstraint(cons_f1489)
def cons_f1490(m, n, b, a):
return Or(Equal(n, S(1)), PositiveIntegerQ(m), NonzeroQ(a**S(2) - b**S(2)))
cons1490 = CustomConstraint(cons_f1490)
def cons_f1491(m, n):
return Or(Greater(n, S(0)), PositiveIntegerQ(m))
cons1491 = CustomConstraint(cons_f1491)
def cons_f1492(a, b):
return ZeroQ(a - b)
cons1492 = CustomConstraint(cons_f1492)
def cons_f1493(n):
return NegativeIntegerQ(n + S(2))
cons1493 = CustomConstraint(cons_f1493)
def cons_f1494(p, n):
return Or(Equal(n, S(2)), Equal(p, S(-1)))
cons1494 = CustomConstraint(cons_f1494)
def cons_f1495(p, b, d, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, n, p), x)
cons1495 = CustomConstraint(cons_f1495)
def cons_f1496(m, n):
return Or(Greater(m - n + S(1), S(0)), Greater(n, S(2)))
cons1496 = CustomConstraint(cons_f1496)
def cons_f1497(p, m, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, m, n, p), x)
cons1497 = CustomConstraint(cons_f1497)
def cons_f1498(d, c, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(c, d, n), x)
cons1498 = CustomConstraint(cons_f1498)
def cons_f1499(d, x, n):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(d, n), x)
cons1499 = CustomConstraint(cons_f1499)
def cons_f1500(m, n):
return ZeroQ(m - n/S(2) + S(1))
cons1500 = CustomConstraint(cons_f1500)
def cons_f1501(m, n):
return Less(S(0), n, m + S(1))
cons1501 = CustomConstraint(cons_f1501)
def cons_f1502(n):
return NonzeroQ(n + S(-1))
cons1502 = CustomConstraint(cons_f1502)
def cons_f1503(m, n):
return Less(S(0), S(2)*n, m + S(1))
cons1503 = CustomConstraint(cons_f1503)
def cons_f1504(m, n):
return Less(S(0), S(2)*n, -m + S(1))
cons1504 = CustomConstraint(cons_f1504)
def cons_f1505(p):
return Unequal(p, S(-2))
cons1505 = CustomConstraint(cons_f1505)
def cons_f1506(m, d, c, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(c, d, e, m, n), x)
cons1506 = CustomConstraint(cons_f1506)
def cons_f1507(m, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, m), x)
cons1507 = CustomConstraint(cons_f1507)
def cons_f1508(m, n):
return ZeroQ(m + n + S(-1))
cons1508 = CustomConstraint(cons_f1508)
def cons_f1509(m, n):
return IntegersQ(m, n, m/S(2) + n/S(2) + S(-1)/2)
cons1509 = CustomConstraint(cons_f1509)
def cons_f1510(m):
return Unequal(m, S(-2))
cons1510 = CustomConstraint(cons_f1510)
def cons_f1511(m):
return Not(And(RationalQ(m), Greater(m, S(1)), Not(IntegerQ(m/S(2) + S(-1)/2))))
cons1511 = CustomConstraint(cons_f1511)
def cons_f1512(n):
return IntegerQ(n/S(2) + S(1)/2)
cons1512 = CustomConstraint(cons_f1512)
def cons_f1513(m, n):
return Not(IntegersQ(S(2)*m, S(2)*n))
cons1513 = CustomConstraint(cons_f1513)
def cons_f1514(m, n):
return Not(And(IntegerQ(m/S(2)), Less(S(0), m, n + S(1))))
cons1514 = CustomConstraint(cons_f1514)
def cons_f1515(m):
return IntegerQ(m/S(2))
cons1515 = CustomConstraint(cons_f1515)
def cons_f1516(m, n):
return Not(And(IntegerQ(n/S(2) + S(-1)/2), Less(S(0), n, m + S(-1))))
cons1516 = CustomConstraint(cons_f1516)
def cons_f1517(m, n):
return Or(Greater(m, S(1)), And(Equal(m, S(1)), Equal(n, S(-3)/2)))
cons1517 = CustomConstraint(cons_f1517)
def cons_f1518(m, n):
return Or(Less(m, S(-1)), And(Equal(m, S(-1)), Equal(n, S(3)/2)))
cons1518 = CustomConstraint(cons_f1518)
def cons_f1519(m, n):
return NonzeroQ(m + n + S(-1))
cons1519 = CustomConstraint(cons_f1519)
def cons_f1520(m, n):
return Or(Less(m, S(-1)), And(Equal(m, S(-1)), RationalQ(n), Equal(n, S(-1)/2)))
cons1520 = CustomConstraint(cons_f1520)
def cons_f1521(m, n):
return Or(Greater(m, S(1)), And(Equal(m, S(1)), RationalQ(n), Equal(n, S(1)/2)))
cons1521 = CustomConstraint(cons_f1521)
def cons_f1522(x, f, b, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, e, f), x)
cons1522 = CustomConstraint(cons_f1522)
def cons_f1523(n):
return Not(IntegerQ(n/S(2) + S(-1)/2))
cons1523 = CustomConstraint(cons_f1523)
def cons_f1524(m):
return Not(IntegerQ(m/S(2)))
cons1524 = CustomConstraint(cons_f1524)
def cons_f1525(m, p, n):
return NonzeroQ(m*p + n + S(-1))
cons1525 = CustomConstraint(cons_f1525)
def cons_f1526(m, p, n):
return IntegersQ(S(2)*m*p, S(2)*n)
cons1526 = CustomConstraint(cons_f1526)
def cons_f1527(m, p, n):
return NonzeroQ(m*p + n + S(1))
cons1527 = CustomConstraint(cons_f1527)
def cons_f1528(m, b, a):
return Or(IntegerQ(S(2)*m), NonzeroQ(a**S(2) + b**S(2)))
cons1528 = CustomConstraint(cons_f1528)
def cons_f1529(m, n):
return ZeroQ(m/S(2) + n)
cons1529 = CustomConstraint(cons_f1529)
def cons_f1530(m, n):
return ZeroQ(m/S(2) + n + S(-1))
cons1530 = CustomConstraint(cons_f1530)
def cons_f1531(m, n):
return PositiveIntegerQ(m/S(2) + n + S(-1))
cons1531 = CustomConstraint(cons_f1531)
def cons_f1532(m, n):
return Or(And(PositiveIntegerQ(n/S(2)), NegativeIntegerQ(m + S(-1)/2)), And(Equal(n, S(2)), Less(m, S(0))), And(LessEqual(m, S(-1)), Greater(m + n, S(0))), And(NegativeIntegerQ(m), Less(m/S(2) + n + S(-1), S(0)), IntegerQ(n)), And(Equal(n, S(3)/2), Equal(m, S(-1)/2)))
cons1532 = CustomConstraint(cons_f1532)
def cons_f1533(m, n):
return Or(And(PositiveIntegerQ(n/S(2)), NegativeIntegerQ(m + S(-1)/2)), And(Equal(n, S(2)), Less(m, S(0))), And(LessEqual(m, S(-1)), Greater(m + n, S(0))), And(NegativeIntegerQ(m), Less(m/S(2) + n + S(-1), S(0))), And(Equal(n, S(3)/2), Equal(m, S(-1)/2)))
cons1533 = CustomConstraint(cons_f1533)
def cons_f1534(m, n):
return Or(And(NegativeIntegerQ(n/S(2)), PositiveIntegerQ(m + S(-1)/2)), Equal(n, S(-2)), PositiveIntegerQ(m + n), And(IntegersQ(n, m + S(1)/2), Greater(S(2)*m + n + S(1), S(0))))
cons1534 = CustomConstraint(cons_f1534)
def cons_f1535(m, n):
return Not(NegativeIntegerQ(m + n))
cons1535 = CustomConstraint(cons_f1535)
def cons_f1536(m, n):
return NonzeroQ(m + S(2)*n)
cons1536 = CustomConstraint(cons_f1536)
def cons_f1537(m, n):
return PositiveIntegerQ(m + n + S(-1))
cons1537 = CustomConstraint(cons_f1537)
def cons_f1538(m, n):
return NegativeIntegerQ(m + n)
cons1538 = CustomConstraint(cons_f1538)
def cons_f1539(m):
return PositiveIntegerQ(m + S(-1))
cons1539 = CustomConstraint(cons_f1539)
def cons_f1540(m, n):
return Or(And(Less(m, S(5)), Greater(n, S(-4))), And(Equal(m, S(5)), Equal(n, S(-1))))
cons1540 = CustomConstraint(cons_f1540)
def cons_f1541(m, n):
return Not(And(IntegerQ(n), Greater(n, S(0)), Or(Less(m, S(0)), Less(n, m))))
cons1541 = CustomConstraint(cons_f1541)
def cons_f1542(d, a, c, b):
return ZeroQ(a*c + b*d)
cons1542 = CustomConstraint(cons_f1542)
def cons_f1543(d, a, c, b):
return NonzeroQ(a*c + b*d)
cons1543 = CustomConstraint(cons_f1543)
def cons_f1544(d, c):
return ZeroQ(c**S(2) + d**S(2))
cons1544 = CustomConstraint(cons_f1544)
def cons_f1545(d, c):
return NonzeroQ(c**S(2) + d**S(2))
cons1545 = CustomConstraint(cons_f1545)
def cons_f1546(d, a, c, b):
return ZeroQ(S(2)*a*c*d - b*(c**S(2) - d**S(2)))
cons1546 = CustomConstraint(cons_f1546)
def cons_f1547(d, a, c, b):
return NonzeroQ(S(2)*a*c*d - b*(c**S(2) - d**S(2)))
cons1547 = CustomConstraint(cons_f1547)
def cons_f1548(d, c, a, b):
return Or(PerfectSquareQ(a**S(2) + b**S(2)), RationalQ(a, b, c, d))
cons1548 = CustomConstraint(cons_f1548)
def cons_f1549(m):
return Not(And(RationalQ(m), LessEqual(m, S(-1))))
cons1549 = CustomConstraint(cons_f1549)
def cons_f1550(m, a):
return Not(And(ZeroQ(m + S(-2)), ZeroQ(a)))
cons1550 = CustomConstraint(cons_f1550)
def cons_f1551(m, n):
return Equal(m + n, S(0))
cons1551 = CustomConstraint(cons_f1551)
def cons_f1552(m):
return IntegersQ(S(2)*m)
cons1552 = CustomConstraint(cons_f1552)
def cons_f1553(m, n):
return Or(IntegerQ(n), IntegersQ(S(2)*m, S(2)*n))
cons1553 = CustomConstraint(cons_f1553)
def cons_f1554(m, n):
return Or(And(RationalQ(n), GreaterEqual(n, S(-1))), IntegerQ(m))
cons1554 = CustomConstraint(cons_f1554)
def cons_f1555(m, n, c, a):
return Not(And(IntegerQ(n), Greater(n, S(2)), Or(Not(IntegerQ(m)), And(ZeroQ(c), NonzeroQ(a)))))
cons1555 = CustomConstraint(cons_f1555)
def cons_f1556(m, n):
return Or(And(RationalQ(n), Less(n, S(0))), IntegerQ(m))
cons1556 = CustomConstraint(cons_f1556)
def cons_f1557(m, n, c, a):
return Not(And(IntegerQ(n), Less(n, S(-1)), Or(Not(IntegerQ(m)), And(ZeroQ(c), NonzeroQ(a)))))
cons1557 = CustomConstraint(cons_f1557)
def cons_f1558(B, A):
return ZeroQ(A**S(2) + B**S(2))
cons1558 = CustomConstraint(cons_f1558)
def cons_f1559(B, A):
return NonzeroQ(A**S(2) + B**S(2))
cons1559 = CustomConstraint(cons_f1559)
def cons_f1560(m, n, c, a):
return Not(And(IntegerQ(n), Greater(n, S(1)), Or(Not(IntegerQ(m)), And(ZeroQ(c), NonzeroQ(a)))))
cons1560 = CustomConstraint(cons_f1560)
def cons_f1561(C, A):
return ZeroQ(A - C)
cons1561 = CustomConstraint(cons_f1561)
def cons_f1562(B, C, b, a, A):
return NonzeroQ(A*b**S(2) - B*a*b + C*a**S(2))
cons1562 = CustomConstraint(cons_f1562)
def cons_f1563(a, C, b, A):
return NonzeroQ(A*b**S(2) + C*a**S(2))
cons1563 = CustomConstraint(cons_f1563)
def cons_f1564(B, C, b, a, A):
return ZeroQ(A*b - B*a - C*b)
cons1564 = CustomConstraint(cons_f1564)
def cons_f1565(C, A):
return NonzeroQ(A - C)
cons1565 = CustomConstraint(cons_f1565)
def cons_f1566(B, C, b, a, A):
return NonzeroQ(A*b - B*a - C*b)
cons1566 = CustomConstraint(cons_f1566)
def cons_f1567(m, n):
return Or(And(RationalQ(m), Less(m, S(0))), ZeroQ(m + n + S(1)))
cons1567 = CustomConstraint(cons_f1567)
def cons_f1568(m, n, c, a):
return Not(And(IntegerQ(n), Greater(n, S(0)), Or(Not(IntegerQ(m)), And(ZeroQ(c), NonzeroQ(a)))))
cons1568 = CustomConstraint(cons_f1568)
def cons_f1569(n):
return Not(And(RationalQ(n), LessEqual(n, S(-1))))
cons1569 = CustomConstraint(cons_f1569)
def cons_f1570(p, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, n, p), x)
cons1570 = CustomConstraint(cons_f1570)
def cons_f1571(m, n):
return NegativeIntegerQ(m, n)
cons1571 = CustomConstraint(cons_f1571)
def cons_f1572(n):
return NegativeIntegerQ(n + S(1))
cons1572 = CustomConstraint(cons_f1572)
def cons_f1573(n):
return PositiveIntegerQ(S(1)/n)
cons1573 = CustomConstraint(cons_f1573)
def cons_f1574(m, n):
return PositiveIntegerQ((m + S(1))/n)
cons1574 = CustomConstraint(cons_f1574)
def cons_f1575(m, d, c, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(c, d, m, n), x)
cons1575 = CustomConstraint(cons_f1575)
def cons_f1576(p, m, b, d, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, m, n, p), x)
cons1576 = CustomConstraint(cons_f1576)
def cons_f1577(m, n):
return GreaterEqual(m - n, S(0))
cons1577 = CustomConstraint(cons_f1577)
def cons_f1578(q):
return SameQ(q, S(1))
cons1578 = CustomConstraint(cons_f1578)
def cons_f1579(b, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, n), x)
cons1579 = CustomConstraint(cons_f1579)
def cons_f1580(m, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, m, n), x)
cons1580 = CustomConstraint(cons_f1580)
def cons_f1581(m, n):
return ZeroQ(m + n + S(-2))
cons1581 = CustomConstraint(cons_f1581)
def cons_f1582(m, n):
return IntegersQ(m, n, m/S(2) + n/S(2))
cons1582 = CustomConstraint(cons_f1582)
def cons_f1583(m, n):
return Not(And(IntegerQ(m/S(2) + S(1)/2), Less(S(0), m, n)))
cons1583 = CustomConstraint(cons_f1583)
def cons_f1584(m, n):
return Not(PositiveIntegerQ(m/S(2), n/S(2) + S(-1)/2))
cons1584 = CustomConstraint(cons_f1584)
def cons_f1585(n):
return ZeroQ(n**S(2) + S(-1)/4)
cons1585 = CustomConstraint(cons_f1585)
def cons_f1586(n):
return LessEqual(n, S(-1))
cons1586 = CustomConstraint(cons_f1586)
def cons_f1587(f, b, d, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, d, e, f, n), x)
cons1587 = CustomConstraint(cons_f1587)
def cons_f1588(d, a, b):
return PositiveQ(a*d/b)
cons1588 = CustomConstraint(cons_f1588)
def cons_f1589(d, a, b):
return Not(PositiveQ(a*d/b))
cons1589 = CustomConstraint(cons_f1589)
def cons_f1590(n):
return Less(n, S(-1)/2)
cons1590 = CustomConstraint(cons_f1590)
def cons_f1591(m, n):
return Or(Less(n, S(-1)), And(Equal(m, S(3)/2), Equal(n, S(-1)/2)))
cons1591 = CustomConstraint(cons_f1591)
def cons_f1592(m, n):
return Or(IntegersQ(S(2)*m, S(2)*n), IntegerQ(m))
cons1592 = CustomConstraint(cons_f1592)
def cons_f1593(d, a, b):
return NegativeQ(a*d/b)
cons1593 = CustomConstraint(cons_f1593)
def cons_f1594(m, n):
return Or(And(IntegerQ(m), Less(n, S(-1))), And(IntegersQ(m + S(1)/2, S(2)*n), LessEqual(n, S(-1))))
cons1594 = CustomConstraint(cons_f1594)
def cons_f1595(m, n):
return Not(And(IntegerQ(n), Greater(n, S(2)), Not(IntegerQ(m))))
cons1595 = CustomConstraint(cons_f1595)
def cons_f1596(m, n):
return Or(And(IntegerQ(n), Greater(n, S(3))), And(IntegersQ(n + S(1)/2, S(2)*m), Greater(n, S(2))))
cons1596 = CustomConstraint(cons_f1596)
def cons_f1597(m, n):
return NegativeIntegerQ(m + S(1)/2, n)
cons1597 = CustomConstraint(cons_f1597)
def cons_f1598(n):
return Greater(n, S(3))
cons1598 = CustomConstraint(cons_f1598)
def cons_f1599(n):
return IntegersQ(S(2)*n)
cons1599 = CustomConstraint(cons_f1599)
def cons_f1600(m):
return Not(And(IntegerQ(m), Greater(m, S(2))))
cons1600 = CustomConstraint(cons_f1600)
def cons_f1601(n):
return Less(S(0), n, S(3))
cons1601 = CustomConstraint(cons_f1601)
def cons_f1602(m):
return Less(S(-1), m, S(2))
cons1602 = CustomConstraint(cons_f1602)
def cons_f1603(n):
return Less(S(1), n, S(3))
cons1603 = CustomConstraint(cons_f1603)
def cons_f1604(m, f, b, d, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, d, e, f, m, n), x)
cons1604 = CustomConstraint(cons_f1604)
def cons_f1605(m, f, b, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, e, f, m), x)
cons1605 = CustomConstraint(cons_f1605)
def cons_f1606(m, a, p, b):
return Or(ZeroQ(a**S(2) - b**S(2)), IntegersQ(S(2)*m, p))
cons1606 = CustomConstraint(cons_f1606)
def cons_f1607(n):
return IntegerQ(n + S(-1)/2)
cons1607 = CustomConstraint(cons_f1607)
def cons_f1608(m):
return NegativeIntegerQ(m + S(1)/2)
cons1608 = CustomConstraint(cons_f1608)
def cons_f1609(m):
return Or(IntegerQ(m/S(2)), LessEqual(m, S(1)))
cons1609 = CustomConstraint(cons_f1609)
def cons_f1610(m, n):
return Less(m + n, S(2))
cons1610 = CustomConstraint(cons_f1610)
def cons_f1611(m, n):
return Not(And(IntegerQ(n), Greater(m - n, S(0))))
cons1611 = CustomConstraint(cons_f1611)
def cons_f1612(n):
return Greater(n, S(1)/2)
cons1612 = CustomConstraint(cons_f1612)
def cons_f1613(n):
return Not(And(RationalQ(n), LessEqual(n, S(-1)/2)))
cons1613 = CustomConstraint(cons_f1613)
def cons_f1614(m, n):
return MemberQ(List(S(0), S(-1), S(-2)), m + n)
cons1614 = CustomConstraint(cons_f1614)
def cons_f1615(m, n):
return Not(And(PositiveIntegerQ(n + S(1)/2), Less(n + S(1)/2, -m - n)))
cons1615 = CustomConstraint(cons_f1615)
def cons_f1616(m, n):
return Not(And(PositiveIntegerQ(m + S(-1)/2), Less(m, n)))
cons1616 = CustomConstraint(cons_f1616)
def cons_f1617(m, n):
return GreaterEqual(-m + n, S(0))
cons1617 = CustomConstraint(cons_f1617)
def cons_f1618(m, n):
return Greater(m*n, S(0))
cons1618 = CustomConstraint(cons_f1618)
def cons_f1619(m, n):
return Or(NegativeIntegerQ(m, n + S(-1)/2), And(NegativeIntegerQ(m + S(-1)/2, n + S(-1)/2), Less(m, n)))
cons1619 = CustomConstraint(cons_f1619)
def cons_f1620(m, p):
return Or(ZeroQ(p + S(-1)), IntegerQ(m + S(-1)/2))
cons1620 = CustomConstraint(cons_f1620)
def cons_f1621(m, n, p):
return ZeroQ(m + n + p)
cons1621 = CustomConstraint(cons_f1621)
def cons_f1622(m, n, p):
return MemberQ(List(S(-1), S(-2)), m + n + p)
cons1622 = CustomConstraint(cons_f1622)
def cons_f1623(B, m, b, a, A):
return ZeroQ(A*b*(m + S(1)) + B*a*m)
cons1623 = CustomConstraint(cons_f1623)
def cons_f1624(B, m, b, a, A):
return NonzeroQ(A*b*(m + S(1)) + B*a*m)
cons1624 = CustomConstraint(cons_f1624)
def cons_f1625(B, A):
return ZeroQ(A**S(2) - B**S(2))
cons1625 = CustomConstraint(cons_f1625)
def cons_f1626(B, A):
return NonzeroQ(A**S(2) - B**S(2))
cons1626 = CustomConstraint(cons_f1626)
def cons_f1627(B, m, b, a, n, A):
return ZeroQ(A*a*m - B*b*n)
cons1627 = CustomConstraint(cons_f1627)
def cons_f1628(B, b, a, n, A):
return ZeroQ(A*b*(S(2)*n + S(1)) + S(2)*B*a*n)
cons1628 = CustomConstraint(cons_f1628)
def cons_f1629(B, b, a, n, A):
return NonzeroQ(A*b*(S(2)*n + S(1)) + S(2)*B*a*n)
cons1629 = CustomConstraint(cons_f1629)
def cons_f1630(m, n):
return Not(And(IntegerQ(n), Greater(n, S(1)), Not(IntegerQ(m))))
cons1630 = CustomConstraint(cons_f1630)
def cons_f1631(m, n):
return Not(NegativeIntegerQ(m + S(1)/2, n))
cons1631 = CustomConstraint(cons_f1631)
def cons_f1632(m):
return Not(And(IntegerQ(m), Greater(m, S(1))))
cons1632 = CustomConstraint(cons_f1632)
def cons_f1633(m, C, A):
return ZeroQ(A*(m + S(1)) + C*m)
cons1633 = CustomConstraint(cons_f1633)
def cons_f1634(m, C, A):
return NonzeroQ(A*(m + S(1)) + C*m)
cons1634 = CustomConstraint(cons_f1634)
def cons_f1635(B, C, e, m, f, b, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, e, f, A, B, C, m), x)
cons1635 = CustomConstraint(cons_f1635)
def cons_f1636(B, C, e, f, b, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, e, f, A, B, C), x)
cons1636 = CustomConstraint(cons_f1636)
def cons_f1637(C, e, f, b, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, e, f, A, C), x)
cons1637 = CustomConstraint(cons_f1637)
def cons_f1638(m):
return PositiveIntegerQ(S(2)*m)
cons1638 = CustomConstraint(cons_f1638)
def cons_f1639(m, n):
return Or(And(RationalQ(n), Less(n, S(-1)/2)), ZeroQ(m + n + S(1)))
cons1639 = CustomConstraint(cons_f1639)
def cons_f1640(n):
return Not(And(RationalQ(n), Less(n, S(-1)/2)))
cons1640 = CustomConstraint(cons_f1640)
def cons_f1641(B, C, m, f, b, d, a, n, A, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, d, e, f, A, B, C, m, n), x)
cons1641 = CustomConstraint(cons_f1641)
def cons_f1642(C, m, f, b, d, a, n, A, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, d, e, f, A, C, m, n), x)
cons1642 = CustomConstraint(cons_f1642)
def cons_f1643(b, d, c, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, c, d, n), x)
cons1643 = CustomConstraint(cons_f1643)
def cons_f1644(n):
return Unequal(n, S(2))
cons1644 = CustomConstraint(cons_f1644)
def cons_f1645(p):
return NonzeroQ(p + S(-1))
cons1645 = CustomConstraint(cons_f1645)
def cons_f1646(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return KnownSineIntegrandQ(u, x)
cons1646 = CustomConstraint(cons_f1646)
def cons_f1647(B, C, b, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, A, B, C), x)
cons1647 = CustomConstraint(cons_f1647)
def cons_f1648(n1, n):
return ZeroQ(-n + n1 + S(-1))
cons1648 = CustomConstraint(cons_f1648)
def cons_f1649(n, n2):
return ZeroQ(-n + n2 + S(-2))
cons1649 = CustomConstraint(cons_f1649)
def cons_f1650(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return KnownTangentIntegrandQ(u, x)
cons1650 = CustomConstraint(cons_f1650)
def cons_f1651(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return KnownCotangentIntegrandQ(u, x)
cons1651 = CustomConstraint(cons_f1651)
def cons_f1652(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return KnownSecantIntegrandQ(u, x)
cons1652 = CustomConstraint(cons_f1652)
def cons_f1653(d, b):
return NonzeroQ(b**S(2) - d**S(2))
cons1653 = CustomConstraint(cons_f1653)
def cons_f1654(d, b):
return ZeroQ(S(-2) + d/b)
cons1654 = CustomConstraint(cons_f1654)
def cons_f1655(m, p):
return Or(Greater(m, S(3)), Equal(p, S(-3)/2))
cons1655 = CustomConstraint(cons_f1655)
def cons_f1656(m, p):
return Or(Less(p, S(-2)), Equal(m, S(2)))
cons1656 = CustomConstraint(cons_f1656)
def cons_f1657(m, n, p):
return ZeroQ(m + n + S(2)*p + S(2))
cons1657 = CustomConstraint(cons_f1657)
def cons_f1658(m):
return Greater(m, S(3))
cons1658 = CustomConstraint(cons_f1658)
def cons_f1659(p, n):
return NonzeroQ(n + p + S(1))
cons1659 = CustomConstraint(cons_f1659)
def cons_f1660(m, n, p):
return NonzeroQ(m + n + S(2)*p + S(2))
cons1660 = CustomConstraint(cons_f1660)
def cons_f1661(m, p):
return Or(Less(p, S(-2)), Equal(m, S(2)), Equal(m, S(3)))
cons1661 = CustomConstraint(cons_f1661)
def cons_f1662(m, n, p):
return NonzeroQ(m + n + S(2)*p)
cons1662 = CustomConstraint(cons_f1662)
def cons_f1663(d, m, b):
return ZeroQ(-Abs(m + S(2)) + d/b)
cons1663 = CustomConstraint(cons_f1663)
def cons_f1664(F):
return InertTrigQ(F)
cons1664 = CustomConstraint(cons_f1664)
def cons_f1665(F, G):
return InertTrigQ(F, G)
cons1665 = CustomConstraint(cons_f1665)
def cons_f1666(F):
return Or(SameQ(F, Cos), SameQ(F, cos))
cons1666 = CustomConstraint(cons_f1666)
def cons_f1667(F):
return Or(SameQ(F, Sin), SameQ(F, sin))
cons1667 = CustomConstraint(cons_f1667)
def cons_f1668(F):
return Or(SameQ(F, Cot), SameQ(F, cot))
cons1668 = CustomConstraint(cons_f1668)
def cons_f1669(F):
return Or(SameQ(F, Tan), SameQ(F, tan))
cons1669 = CustomConstraint(cons_f1669)
def cons_f1670(F):
return Or(SameQ(F, Sec), SameQ(F, sec))
cons1670 = CustomConstraint(cons_f1670)
def cons_f1671(F):
return Or(SameQ(F, Csc), SameQ(F, csc))
cons1671 = CustomConstraint(cons_f1671)
def cons_f1672(F):
return Or(SameQ(F, sin), SameQ(F, cos))
cons1672 = CustomConstraint(cons_f1672)
def cons_f1673(G):
return Or(SameQ(G, sin), SameQ(G, cos))
cons1673 = CustomConstraint(cons_f1673)
def cons_f1674(H):
return Or(SameQ(H, sin), SameQ(H, cos))
cons1674 = CustomConstraint(cons_f1674)
def cons_f1675(c, b):
return ZeroQ(b - c)
cons1675 = CustomConstraint(cons_f1675)
def cons_f1676(c, b):
return ZeroQ(b + c)
cons1676 = CustomConstraint(cons_f1676)
def cons_f1677(u):
return Not(InertTrigFreeQ(u))
cons1677 = CustomConstraint(cons_f1677)
def cons_f1678(p):
return NegQ(p)
cons1678 = CustomConstraint(cons_f1678)
def cons_f1679(u):
return TrigSimplifyQ(u)
cons1679 = CustomConstraint(cons_f1679)
def cons_f1680(v):
return Not(InertTrigFreeQ(v))
cons1680 = CustomConstraint(cons_f1680)
def cons_f1681(v, w):
return Or(Not(InertTrigFreeQ(v)), Not(InertTrigFreeQ(w)))
cons1681 = CustomConstraint(cons_f1681)
def cons_f1682(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return Not(FalseQ(FunctionOfTrig(u, x)))
except (TypeError, AttributeError):
return False
cons1682 = CustomConstraint(cons_f1682)
def cons_f1683(p):
return SameQ(p, S(1))
cons1683 = CustomConstraint(cons_f1683)
def cons_f1684(p, n):
return Or(EvenQ(n), OddQ(p))
cons1684 = CustomConstraint(cons_f1684)
def cons_f1685(p, n):
return Unequal(n, p)
cons1685 = CustomConstraint(cons_f1685)
def cons_f1686(F):
return TrigQ(F)
cons1686 = CustomConstraint(cons_f1686)
def cons_f1687(G):
return TrigQ(G)
cons1687 = CustomConstraint(cons_f1687)
def cons_f1688(v, w):
return ZeroQ(v - w)
cons1688 = CustomConstraint(cons_f1688)
def cons_f1689(F):
return MemberQ(List(Sin, Cos), F)
cons1689 = CustomConstraint(cons_f1689)
def cons_f1690(G):
return MemberQ(List(Sec, Csc), G)
cons1690 = CustomConstraint(cons_f1690)
def cons_f1691(d, b):
return PositiveIntegerQ(b/d + S(-1))
cons1691 = CustomConstraint(cons_f1691)
def cons_f1692(c, b, F, e):
return NonzeroQ(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2))
cons1692 = CustomConstraint(cons_f1692)
def cons_f1693(b, c, n, F, e):
return NonzeroQ(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2))
cons1693 = CustomConstraint(cons_f1693)
def cons_f1694(m, b, c, F, e):
return NonzeroQ(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*m**S(2))
cons1694 = CustomConstraint(cons_f1694)
def cons_f1695(b, c, n, F, e):
return ZeroQ(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(2))**S(2))
cons1695 = CustomConstraint(cons_f1695)
def cons_f1696(b, c, n, F, e):
return NonzeroQ(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(2))**S(2))
cons1696 = CustomConstraint(cons_f1696)
def cons_f1697(b, c, n, F, e):
return ZeroQ(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(-2))**S(2))
cons1697 = CustomConstraint(cons_f1697)
def cons_f1698(b, c, n, F, e):
return NonzeroQ(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(-2))**S(2))
cons1698 = CustomConstraint(cons_f1698)
def cons_f1699(f, g):
return ZeroQ(f**S(2) - g**S(2))
cons1699 = CustomConstraint(cons_f1699)
def cons_f1700(f, g):
return ZeroQ(f - g)
cons1700 = CustomConstraint(cons_f1700)
def cons_f1701(h, i):
return ZeroQ(h**S(2) - i**S(2))
cons1701 = CustomConstraint(cons_f1701)
def cons_f1702(h, g, f, i):
return ZeroQ(-f*i + g*h)
cons1702 = CustomConstraint(cons_f1702)
def cons_f1703(h, g, f, i):
return ZeroQ(f*i + g*h)
cons1703 = CustomConstraint(cons_f1703)
def cons_f1704(m, n, p):
return PositiveIntegerQ(m, n, p)
cons1704 = CustomConstraint(cons_f1704)
def cons_f1705(H):
return TrigQ(H)
cons1705 = CustomConstraint(cons_f1705)
def cons_f1706(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(LinearQ(u, x), PolyQ(u, x, S(2)))
cons1706 = CustomConstraint(cons_f1706)
def cons_f1707(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(LinearQ(v, x), PolyQ(v, x, S(2)))
cons1707 = CustomConstraint(cons_f1707)
def cons_f1708(p, n, b):
return ZeroQ(b**S(2)*n**S(2)*(p + S(2))**S(2) + S(1))
cons1708 = CustomConstraint(cons_f1708)
def cons_f1709(p, n, b):
return ZeroQ(b**S(2)*n**S(2)*p**S(2) + S(1))
cons1709 = CustomConstraint(cons_f1709)
def cons_f1710(n, b):
return NonzeroQ(b**S(2)*n**S(2) + S(1))
cons1710 = CustomConstraint(cons_f1710)
def cons_f1711(p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*p**S(2) + S(1))
cons1711 = CustomConstraint(cons_f1711)
def cons_f1712(p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*(p + S(2))**S(2) + S(1))
cons1712 = CustomConstraint(cons_f1712)
def cons_f1713(m, p, n, b):
return ZeroQ(b**S(2)*n**S(2)*(p + S(2))**S(2) + (m + S(1))**S(2))
cons1713 = CustomConstraint(cons_f1713)
def cons_f1714(m, p, n, b):
return ZeroQ(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2))
cons1714 = CustomConstraint(cons_f1714)
def cons_f1715(m, n, b):
return NonzeroQ(b**S(2)*n**S(2) + (m + S(1))**S(2))
cons1715 = CustomConstraint(cons_f1715)
def cons_f1716(m, p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2))
cons1716 = CustomConstraint(cons_f1716)
def cons_f1717(m, p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*(p + S(2))**S(2) + (m + S(1))**S(2))
cons1717 = CustomConstraint(cons_f1717)
def cons_f1718(n, b):
return ZeroQ(b**S(2)*n**S(2) + S(1))
cons1718 = CustomConstraint(cons_f1718)
def cons_f1719(p, n, b):
return ZeroQ(b**S(2)*n**S(2)*(p + S(-2))**S(2) + S(1))
cons1719 = CustomConstraint(cons_f1719)
def cons_f1720(p):
return Unequal(p, S(2))
cons1720 = CustomConstraint(cons_f1720)
def cons_f1721(p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*(p + S(-2))**S(2) + S(1))
cons1721 = CustomConstraint(cons_f1721)
def cons_f1722(m, n, b):
return ZeroQ(b**S(2)*n**S(2) + (m + S(1))**S(2))
cons1722 = CustomConstraint(cons_f1722)
def cons_f1723(m, p, n, b):
return ZeroQ(b**S(2)*n**S(2)*(p + S(-2))**S(2) + (m + S(1))**S(2))
cons1723 = CustomConstraint(cons_f1723)
def cons_f1724(m, p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*(p + S(-2))**S(2) + (m + S(1))**S(2))
cons1724 = CustomConstraint(cons_f1724)
def cons_f1725(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return QuotientOfLinearsQ(u, x)
cons1725 = CustomConstraint(cons_f1725)
def cons_f1726(v, x, w):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(And(PolynomialQ(v, x), PolynomialQ(w, x)), And(BinomialQ(List(v, w), x), IndependentQ(v/w, x)))
cons1726 = CustomConstraint(cons_f1726)
def cons_f1727(m, p, q):
return PositiveIntegerQ(m, p, q)
cons1727 = CustomConstraint(cons_f1727)
def cons_f1728(v, w):
return NonzeroQ(v - w)
cons1728 = CustomConstraint(cons_f1728)
def cons_f1729(m, n):
return Or(Equal(n, S(-1)), And(Equal(m, S(1)), Equal(n, S(-2))))
cons1729 = CustomConstraint(cons_f1729)
def cons_f1730(a, c):
return NonzeroQ(a + c)
cons1730 = CustomConstraint(cons_f1730)
def cons_f1731(a, b):
return PosQ(a**S(2) - b**S(2))
cons1731 = CustomConstraint(cons_f1731)
def cons_f1732(a, b):
return NegQ(a**S(2) - b**S(2))
cons1732 = CustomConstraint(cons_f1732)
def cons_f1733(d, b):
return ZeroQ(b**S(2) - d**S(2))
cons1733 = CustomConstraint(cons_f1733)
def cons_f1734(n):
return Inequality(S(-2), LessEqual, n, Less, S(-1))
cons1734 = CustomConstraint(cons_f1734)
def cons_f1735(n):
return Less(n, S(-2))
cons1735 = CustomConstraint(cons_f1735)
def cons_f1736(m, b, d, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, m, n), x)
cons1736 = CustomConstraint(cons_f1736)
def cons_f1737(d, c, e):
return ZeroQ(c**S(2)*d + e)
cons1737 = CustomConstraint(cons_f1737)
def cons_f1738(d):
return Not(PositiveQ(d))
cons1738 = CustomConstraint(cons_f1738)
def cons_f1739(p):
return PositiveIntegerQ(S(2)*p)
cons1739 = CustomConstraint(cons_f1739)
def cons_f1740(d, p):
return Or(IntegerQ(p), PositiveQ(d))
cons1740 = CustomConstraint(cons_f1740)
def cons_f1741(d, p):
return Not(Or(IntegerQ(p), PositiveQ(d)))
cons1741 = CustomConstraint(cons_f1741)
def cons_f1742(d, c, e):
return NonzeroQ(c**S(2)*d + e)
cons1742 = CustomConstraint(cons_f1742)
def cons_f1743(p):
return Or(PositiveIntegerQ(p), NegativeIntegerQ(p + S(1)/2))
cons1743 = CustomConstraint(cons_f1743)
def cons_f1744(p, n):
return Or(Greater(p, S(0)), PositiveIntegerQ(n))
cons1744 = CustomConstraint(cons_f1744)
def cons_f1745(c, f, g):
return ZeroQ(c**S(2)*f**S(2) - g**S(2))
cons1745 = CustomConstraint(cons_f1745)
def cons_f1746(m):
return NegativeIntegerQ(m/S(2) + S(1)/2)
cons1746 = CustomConstraint(cons_f1746)
def cons_f1747(m, p):
return Or(PositiveIntegerQ(m/S(2) + S(1)/2), NegativeIntegerQ(m/S(2) + p + S(3)/2))
cons1747 = CustomConstraint(cons_f1747)
def cons_f1748(m, n):
return Or(RationalQ(m), ZeroQ(n + S(-1)))
cons1748 = CustomConstraint(cons_f1748)
def cons_f1749(m, p, n):
return Or(IntegerQ(m), IntegerQ(p), Equal(n, S(1)))
cons1749 = CustomConstraint(cons_f1749)
def cons_f1750(m, n):
return Or(IntegerQ(m), Equal(n, S(1)))
cons1750 = CustomConstraint(cons_f1750)
def cons_f1751(m):
return Greater(m, S(-3))
cons1751 = CustomConstraint(cons_f1751)
def cons_f1752(p):
return Greater(p, S(-1))
cons1752 = CustomConstraint(cons_f1752)
def cons_f1753(m):
return Not(PositiveIntegerQ(m/S(2) + S(1)/2))
cons1753 = CustomConstraint(cons_f1753)
def cons_f1754(m):
return Less(S(-3), m, S(0))
cons1754 = CustomConstraint(cons_f1754)
def cons_f1755(m, p):
return Or(Greater(p, S(0)), And(PositiveIntegerQ(m/S(2) + S(-1)/2), LessEqual(m + p, S(0))))
cons1755 = CustomConstraint(cons_f1755)
def cons_f1756(p, m, f, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, m, n, p), x)
cons1756 = CustomConstraint(cons_f1756)
def cons_f1757(m, p):
return Less(m + p + S(1), S(0))
cons1757 = CustomConstraint(cons_f1757)
def cons_f1758(d, h, g, e):
return ZeroQ(-S(2)*d*h + e*g)
cons1758 = CustomConstraint(cons_f1758)
def cons_f1759(m, p):
return Or(Less(m, -S(2)*p + S(-1)), Greater(m, S(3)))
cons1759 = CustomConstraint(cons_f1759)
def cons_f1760(m, p, n):
return Or(And(Equal(n, S(1)), Greater(p, S(-1))), Greater(p, S(0)), Equal(m, S(1)), And(Equal(m, S(2)), Less(p, S(-2))))
cons1760 = CustomConstraint(cons_f1760)
def cons_f1761(m, n):
return Or(Greater(m, S(0)), PositiveIntegerQ(n))
cons1761 = CustomConstraint(cons_f1761)
def cons_f1762(B, d, c, A):
return ZeroQ(S(2)*A*c*d + B*(-c**S(2) + S(1)))
cons1762 = CustomConstraint(cons_f1762)
def cons_f1763(B, d, c, C):
return ZeroQ(-B*d + S(2)*C*c)
cons1763 = CustomConstraint(cons_f1763)
def cons_f1764(c):
return ZeroQ(c**S(2) + S(-1))
cons1764 = CustomConstraint(cons_f1764)
def cons_f1765(d, a, b, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, d), x)
cons1765 = CustomConstraint(cons_f1765)
def cons_f1766(m, b, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, n, m), x)
cons1766 = CustomConstraint(cons_f1766)
def cons_f1767(x, n, b):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(b, n), x)
cons1767 = CustomConstraint(cons_f1767)
def cons_f1768(c, b):
return EqQ(b**S(2)*c, S(1))
cons1768 = CustomConstraint(cons_f1768)
def cons_f1769(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(FunctionOfExponentialQ(u, x))
cons1769 = CustomConstraint(cons_f1769)
def cons_f1770(u, m, d, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(FunctionOfQ((c + d*x)**(m + S(1)), u, x))
cons1770 = CustomConstraint(cons_f1770)
def cons_f1771(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1770(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1770 = CustomConstraint(_cons_f_1770)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1770)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1771 = CustomConstraint(cons_f1771)
def cons_f1772(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1771(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1771 = CustomConstraint(_cons_f_1771)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1771)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1772 = CustomConstraint(cons_f1772)
def cons_f1773(d, c, e):
return ZeroQ(c**S(2)*d**S(2) + e**S(2))
cons1773 = CustomConstraint(cons_f1773)
def cons_f1774(d, c, e):
return PositiveQ(I*c*d/e + S(1))
cons1774 = CustomConstraint(cons_f1774)
def cons_f1775(d, c, e):
return NegativeQ(I*c*d/e + S(-1))
cons1775 = CustomConstraint(cons_f1775)
def cons_f1776(d, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(c, d, e), x)
cons1776 = CustomConstraint(cons_f1776)
def cons_f1777(m, a, p):
return Or(Greater(p, S(0)), NonzeroQ(a), IntegerQ(m))
cons1777 = CustomConstraint(cons_f1777)
def cons_f1778(d, c, e):
return ZeroQ(-c**S(2)*d + e)
cons1778 = CustomConstraint(cons_f1778)
def cons_f1779(p):
return NegativeIntegerQ(S(2)*p + S(2))
cons1779 = CustomConstraint(cons_f1779)
def cons_f1780(p):
return Or(IntegerQ(p), NegativeIntegerQ(p + S(1)/2))
cons1780 = CustomConstraint(cons_f1780)
def cons_f1781(m, a):
return Not(And(Equal(m, S(1)), NonzeroQ(a)))
cons1781 = CustomConstraint(cons_f1781)
def cons_f1782(p):
return Unequal(p, S(-5)/2)
cons1782 = CustomConstraint(cons_f1782)
def cons_f1783(m, n, p):
return Or(RationalQ(m), And(EqQ(n, S(1)), IntegerQ(p)))
cons1783 = CustomConstraint(cons_f1783)
def cons_f1784(m, n, p):
return IntegersQ(m, n, S(2)*p)
cons1784 = CustomConstraint(cons_f1784)
def cons_f1785(m, p):
return NegativeIntegerQ(m + S(2)*p + S(1))
cons1785 = CustomConstraint(cons_f1785)
def cons_f1786(m, p):
return Or(And(PositiveIntegerQ(p), Not(And(NegativeIntegerQ(m/S(2) + S(-1)/2), Greater(m + S(2)*p + S(3), S(0))))), And(PositiveIntegerQ(m/S(2) + S(1)/2), Not(And(NegativeIntegerQ(p), Greater(m + S(2)*p + S(3), S(0))))), And(NegativeIntegerQ(m/S(2) + p + S(1)/2), Not(NegativeIntegerQ(m/S(2) + S(-1)/2))))
cons1786 = CustomConstraint(cons_f1786)
def cons_f1787(m, p):
return Or(Greater(p, S(0)), And(Less(p, S(-1)), IntegerQ(m), Unequal(m, S(1))))
cons1787 = CustomConstraint(cons_f1787)
def cons_f1788(p, m, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, m, p), x)
cons1788 = CustomConstraint(cons_f1788)
def cons_f1789(x, c, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(u**S(2) - (S(1) - S(2)*I/(c*x + I))**S(2))
cons1789 = CustomConstraint(cons_f1789)
def cons_f1790(x, c, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(u**S(2) - (S(1) - S(2)*I/(-c*x + I))**S(2))
cons1790 = CustomConstraint(cons_f1790)
def cons_f1791(x, c, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(-(S(1) - S(2)*I/(c*x + I))**S(2) + (-u + S(1))**S(2))
cons1791 = CustomConstraint(cons_f1791)
def cons_f1792(x, c, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(-(S(1) - S(2)*I/(-c*x + I))**S(2) + (-u + S(1))**S(2))
cons1792 = CustomConstraint(cons_f1792)
def cons_f1793(m, n):
return Inequality(S(0), Less, n, LessEqual, m)
cons1793 = CustomConstraint(cons_f1793)
def cons_f1794(m, n):
return Less(S(0), n, m)
cons1794 = CustomConstraint(cons_f1794)
def cons_f1795(d, a, n, c):
return Not(And(Equal(n, S(2)), ZeroQ(-a**S(2)*c + d)))
cons1795 = CustomConstraint(cons_f1795)
def cons_f1796(f, b, g, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, e, f, g), x)
cons1796 = CustomConstraint(cons_f1796)
def cons_f1797(m):
return PositiveIntegerQ(m/S(2) + S(1)/2)
cons1797 = CustomConstraint(cons_f1797)
def cons_f1798(f, c, g):
return ZeroQ(-c**S(2)*f + g)
cons1798 = CustomConstraint(cons_f1798)
def cons_f1799(n):
return OddQ(I*n)
cons1799 = CustomConstraint(cons_f1799)
def cons_f1800(n):
return Not(OddQ(I*n))
cons1800 = CustomConstraint(cons_f1800)
def cons_f1801(d, a, c):
return ZeroQ(a**S(2)*c**S(2) + d**S(2))
cons1801 = CustomConstraint(cons_f1801)
def cons_f1802(c, p):
return Or(IntegerQ(p), PositiveQ(c))
cons1802 = CustomConstraint(cons_f1802)
def cons_f1803(c, p):
return Not(Or(IntegerQ(p), PositiveQ(c)))
cons1803 = CustomConstraint(cons_f1803)
def cons_f1804(d, c, a):
return ZeroQ(a**S(2)*d**S(2) + c**S(2))
cons1804 = CustomConstraint(cons_f1804)
def cons_f1805(n):
return IntegerQ(I*n/S(2))
cons1805 = CustomConstraint(cons_f1805)
def cons_f1806(d, a, c):
return ZeroQ(-a**S(2)*c + d)
cons1806 = CustomConstraint(cons_f1806)
def cons_f1807(n):
return Not(IntegerQ(I*n))
cons1807 = CustomConstraint(cons_f1807)
def cons_f1808(p, n):
return NonzeroQ(n**S(2) + S(4)*(p + S(1))**S(2))
cons1808 = CustomConstraint(cons_f1808)
def cons_f1809(n):
return IntegerQ(I*n/S(2) + S(1)/2)
cons1809 = CustomConstraint(cons_f1809)
def cons_f1810(p, n):
return Not(IntegerQ(-I*n/S(2) + p))
cons1810 = CustomConstraint(cons_f1810)
def cons_f1811(n):
return PositiveIntegerQ(I*n/S(2))
cons1811 = CustomConstraint(cons_f1811)
def cons_f1812(n):
return NegativeIntegerQ(I*n/S(2))
cons1812 = CustomConstraint(cons_f1812)
def cons_f1813(p, n):
return ZeroQ(n**S(2) - S(2)*p + S(-2))
cons1813 = CustomConstraint(cons_f1813)
def cons_f1814(n):
return Not(IntegerQ(I*n/S(2)))
cons1814 = CustomConstraint(cons_f1814)
def cons_f1815(d, c, a):
return ZeroQ(-a**S(2)*d + c)
cons1815 = CustomConstraint(cons_f1815)
def cons_f1816(n):
return RationalQ(I*n)
cons1816 = CustomConstraint(cons_f1816)
def cons_f1817(n):
return Less(S(-1), I*n, S(1))
cons1817 = CustomConstraint(cons_f1817)
def cons_f1818(d, a, b, e):
return ZeroQ(-S(2)*a*e + b*d)
cons1818 = CustomConstraint(cons_f1818)
def cons_f1819(c, b, e, a):
return ZeroQ(b**S(2)*c - e*(a**S(2) + S(1)))
cons1819 = CustomConstraint(cons_f1819)
def cons_f1820(c, p, a):
return Or(IntegerQ(p), PositiveQ(c/(a**S(2) + S(1))))
cons1820 = CustomConstraint(cons_f1820)
def cons_f1821(c, p, a):
return Not(Or(IntegerQ(p), PositiveQ(c/(a**S(2) + S(1)))))
cons1821 = CustomConstraint(cons_f1821)
def cons_f1822(p, n):
return Not(And(IntegerQ(p), EvenQ(I*n)))
cons1822 = CustomConstraint(cons_f1822)
def cons_f1823(p, n):
return Not(And(Not(IntegerQ(p)), OddQ(I*n)))
cons1823 = CustomConstraint(cons_f1823)
def cons_f1824(p):
return LessEqual(p, S(-1))
cons1824 = CustomConstraint(cons_f1824)
def cons_f1825(n):
return NonzeroQ(n**S(2) + S(1))
cons1825 = CustomConstraint(cons_f1825)
def cons_f1826(p, n):
return NonzeroQ(n**S(2) - S(2)*p + S(-2))
cons1826 = CustomConstraint(cons_f1826)
def cons_f1827(m, p):
return LessEqual(S(3), m, -S(2)*p + S(-2))
cons1827 = CustomConstraint(cons_f1827)
def cons_f1828(p, n):
return IntegersQ(S(2)*p, I*n/S(2) + p)
cons1828 = CustomConstraint(cons_f1828)
def cons_f1829(p, n):
return Not(IntegersQ(S(2)*p, I*n/S(2) + p))
cons1829 = CustomConstraint(cons_f1829)
def cons_f1830(B, d, c, A):
return ZeroQ(-S(2)*A*c*d + B*(c**S(2) + S(1)))
cons1830 = CustomConstraint(cons_f1830)
def cons_f1831(x, a, n, b):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, n), x)
cons1831 = CustomConstraint(cons_f1831)
def cons_f1832(m):
return Unequal(m + S(1), S(0))
cons1832 = CustomConstraint(cons_f1832)
def cons_f1833(m, n):
return Unequal(m + S(1), n)
cons1833 = CustomConstraint(cons_f1833)
def cons_f1834(f, b, d, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, f), x)
cons1834 = CustomConstraint(cons_f1834)
def cons_f1835(c, b):
return ZeroQ(b + c**S(2))
cons1835 = CustomConstraint(cons_f1835)
def cons_f1836(s):
return ZeroQ(s**S(2) + S(-1))
cons1836 = CustomConstraint(cons_f1836)
def cons_f1837(v, w):
return ZeroQ(-v**S(2) + w + S(-1))
cons1837 = CustomConstraint(cons_f1837)
def cons_f1838(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return NegQ(Discriminant(v, x))
cons1838 = CustomConstraint(cons_f1838)
def cons_f1839(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1838(w, f, r):
return FreeQ(f, x)
_cons_1838 = CustomConstraint(_cons_f_1838)
pat = Pattern(UtilityOperator(f_**w_*WC('r', S(1)), x), _cons_1838)
result_matchq = is_match(UtilityOperator(u, x), pat)
return result_matchq
cons1839 = CustomConstraint(cons_f1839)
def cons_f1840(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1839(w, f, r):
return FreeQ(f, x)
_cons_1839 = CustomConstraint(_cons_f_1839)
pat = Pattern(UtilityOperator(f_**w_*WC('r', S(1)), x), _cons_1839)
result_matchq = is_match(UtilityOperator(u, x), pat)
return result_matchq
cons1840 = CustomConstraint(cons_f1840)
def cons_f1841(d, c):
return ZeroQ((c + I*d)**S(2) + S(1))
cons1841 = CustomConstraint(cons_f1841)
def cons_f1842(d, c):
return ZeroQ((c - I*d)**S(2) + S(1))
cons1842 = CustomConstraint(cons_f1842)
def cons_f1843(d, c):
return NonzeroQ((c + I*d)**S(2) + S(1))
cons1843 = CustomConstraint(cons_f1843)
def cons_f1844(d, c):
return NonzeroQ((c - I*d)**S(2) + S(1))
cons1844 = CustomConstraint(cons_f1844)
def cons_f1845(d, c):
return ZeroQ((c - d)**S(2) + S(1))
cons1845 = CustomConstraint(cons_f1845)
def cons_f1846(d, c):
return NonzeroQ((c - d)**S(2) + S(1))
cons1846 = CustomConstraint(cons_f1846)
def cons_f1847(x, m, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return FalseQ(PowerVariableExpn(u, m + S(1), x))
except (TypeError, AttributeError):
return False
cons1847 = CustomConstraint(cons_f1847)
def cons_f1848(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1847(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1847 = CustomConstraint(_cons_f_1847)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1847)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1848 = CustomConstraint(cons_f1848)
def cons_f1849(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return FalseQ(FunctionOfLinear(v*(a + b*ArcTan(u)), x))
except (TypeError, AttributeError):
return False
cons1849 = CustomConstraint(cons_f1849)
def cons_f1850(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1849(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1849 = CustomConstraint(_cons_f_1849)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1849)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1850 = CustomConstraint(cons_f1850)
def cons_f1851(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return FalseQ(FunctionOfLinear(v*(a + b*acot(u)), x))
except (TypeError, AttributeError):
return False
cons1851 = CustomConstraint(cons_f1851)
def cons_f1852(v, a, b, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(D(v/(a + b*x), x))
cons1852 = CustomConstraint(cons_f1852)
def cons_f1853(x, w, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(D(w/(a + b*x), x))
cons1853 = CustomConstraint(cons_f1853)
def cons_f1854(m, b, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, m, n), x)
cons1854 = CustomConstraint(cons_f1854)
def cons_f1855(d):
return Negative(d)
cons1855 = CustomConstraint(cons_f1855)
def cons_f1856(d, e):
return Not(And(PositiveQ(e), Negative(d)))
cons1856 = CustomConstraint(cons_f1856)
def cons_f1857(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1856(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1856 = CustomConstraint(_cons_f_1856)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1856)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1857 = CustomConstraint(cons_f1857)
def cons_f1858(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1857(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1857 = CustomConstraint(_cons_f_1857)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1857)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1858 = CustomConstraint(cons_f1858)
def cons_f1859(m, n, b, a):
return Or(Equal(n, S(1)), PositiveIntegerQ(m), NonzeroQ(a**S(2) + b**S(2)))
cons1859 = CustomConstraint(cons_f1859)
def cons_f1860(F):
return HyperbolicQ(F)
cons1860 = CustomConstraint(cons_f1860)
def cons_f1861(G):
return HyperbolicQ(G)
cons1861 = CustomConstraint(cons_f1861)
def cons_f1862(F):
return MemberQ(List(Sinh, Cosh), F)
cons1862 = CustomConstraint(cons_f1862)
def cons_f1863(G):
return MemberQ(List(Sech, Csch), G)
cons1863 = CustomConstraint(cons_f1863)
def cons_f1864(c, b, F, e):
return NonzeroQ(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2))
cons1864 = CustomConstraint(cons_f1864)
def cons_f1865(b, c, n, F, e):
return NonzeroQ(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2))
cons1865 = CustomConstraint(cons_f1865)
def cons_f1866(b, c, n, F, e):
return ZeroQ(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(2))**S(2))
cons1866 = CustomConstraint(cons_f1866)
def cons_f1867(b, c, n, F, e):
return NonzeroQ(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(2))**S(2))
cons1867 = CustomConstraint(cons_f1867)
def cons_f1868(b, c, n, F, e):
return ZeroQ(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(-2))**S(2))
cons1868 = CustomConstraint(cons_f1868)
def cons_f1869(b, c, n, F, e):
return NonzeroQ(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(-2))**S(2))
cons1869 = CustomConstraint(cons_f1869)
def cons_f1870(f, g):
return ZeroQ(f**S(2) + g**S(2))
cons1870 = CustomConstraint(cons_f1870)
def cons_f1871(h, i):
return ZeroQ(h**S(2) + i**S(2))
cons1871 = CustomConstraint(cons_f1871)
def cons_f1872(H):
return HyperbolicQ(H)
cons1872 = CustomConstraint(cons_f1872)
def cons_f1873(p, n, b):
return RationalQ(b, n, p)
cons1873 = CustomConstraint(cons_f1873)
def cons_f1874(p, n, b):
return ZeroQ(b**S(2)*n**S(2)*(p + S(2))**S(2) + S(-1))
cons1874 = CustomConstraint(cons_f1874)
def cons_f1875(n, b):
return ZeroQ(b*n + S(-2))
cons1875 = CustomConstraint(cons_f1875)
def cons_f1876(p, n, b):
return ZeroQ(b**S(2)*n**S(2)*p**S(2) + S(-1))
cons1876 = CustomConstraint(cons_f1876)
def cons_f1877(n, b):
return NonzeroQ(b**S(2)*n**S(2) + S(-1))
cons1877 = CustomConstraint(cons_f1877)
def cons_f1878(p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*p**S(2) + S(-1))
cons1878 = CustomConstraint(cons_f1878)
def cons_f1879(p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*(p + S(2))**S(2) + S(-1))
cons1879 = CustomConstraint(cons_f1879)
def cons_f1880(m, p, n, b):
return ZeroQ(b**S(2)*n**S(2)*(p + S(2))**S(2) - (m + S(1))**S(2))
cons1880 = CustomConstraint(cons_f1880)
def cons_f1881(m, p, n, b):
return ZeroQ(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2))
cons1881 = CustomConstraint(cons_f1881)
def cons_f1882(m, n, b):
return NonzeroQ(b**S(2)*n**S(2) - (m + S(1))**S(2))
cons1882 = CustomConstraint(cons_f1882)
def cons_f1883(m, p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2))
cons1883 = CustomConstraint(cons_f1883)
def cons_f1884(m, p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*(p + S(2))**S(2) - (m + S(1))**S(2))
cons1884 = CustomConstraint(cons_f1884)
def cons_f1885(n, b):
return ZeroQ(b**S(2)*n**S(2) + S(-1))
cons1885 = CustomConstraint(cons_f1885)
def cons_f1886(p, n, b):
return ZeroQ(b**S(2)*n**S(2)*(p + S(-2))**S(2) + S(-1))
cons1886 = CustomConstraint(cons_f1886)
def cons_f1887(p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*(p + S(-2))**S(2) + S(-1))
cons1887 = CustomConstraint(cons_f1887)
def cons_f1888(p, m, n, b):
return RationalQ(b, m, n, p)
cons1888 = CustomConstraint(cons_f1888)
def cons_f1889(m, n, b):
return ZeroQ(b**S(2)*n**S(2) - (m + S(1))**S(2))
cons1889 = CustomConstraint(cons_f1889)
def cons_f1890(m, p, n, b):
return NonzeroQ(b**S(2)*n**S(2)*(p + S(-2))**S(2) - (m + S(1))**S(2))
cons1890 = CustomConstraint(cons_f1890)
def cons_f1891(B, a, b, A):
return ZeroQ(A*a + B*b)
cons1891 = CustomConstraint(cons_f1891)
def cons_f1892(c, d1, e1):
return ZeroQ(-c*d1 + e1)
cons1892 = CustomConstraint(cons_f1892)
def cons_f1893(e2, c, d2):
return ZeroQ(c*d2 + e2)
cons1893 = CustomConstraint(cons_f1893)
def cons_f1894(d1):
return PositiveQ(d1)
cons1894 = CustomConstraint(cons_f1894)
def cons_f1895(d2):
return NegativeQ(d2)
cons1895 = CustomConstraint(cons_f1895)
def cons_f1896(d2, d1):
return Not(And(PositiveQ(d1), NegativeQ(d2)))
cons1896 = CustomConstraint(cons_f1896)
def cons_f1897(d2, d1):
return And(PositiveQ(d1), NegativeQ(d2))
cons1897 = CustomConstraint(cons_f1897)
def cons_f1898(d, c, e):
return NonzeroQ(-c**S(2)*d + e)
cons1898 = CustomConstraint(cons_f1898)
def cons_f1899(d2, p, b, e2, c, a, n, d1, e1, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d1, e1, d2, e2, n, p), x)
cons1899 = CustomConstraint(cons_f1899)
def cons_f1900(c, f, g):
return ZeroQ(c**S(2)*f**S(2) + g**S(2))
cons1900 = CustomConstraint(cons_f1900)
def cons_f1901(p, d1, d2):
return Not(Or(IntegerQ(p), And(PositiveQ(d1), NegativeQ(d2))))
cons1901 = CustomConstraint(cons_f1901)
def cons_f1902(m):
return NonzeroQ(m + S(3))
cons1902 = CustomConstraint(cons_f1902)
def cons_f1903(d2, p, m, f, b, e2, c, a, n, d1, e1, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d1, e1, d2, e2, f, m, n, p), x)
cons1903 = CustomConstraint(cons_f1903)
def cons_f1904(c):
return ZeroQ(c**S(2) + S(1))
cons1904 = CustomConstraint(cons_f1904)
def cons_f1905(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1904(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1904 = CustomConstraint(_cons_f_1904)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1904)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1905 = CustomConstraint(cons_f1905)
def cons_f1906(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1905(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1905 = CustomConstraint(_cons_f_1905)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1905)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1906 = CustomConstraint(cons_f1906)
def cons_f1907(d, c, e):
return ZeroQ(c**S(2)*d**S(2) - e**S(2))
cons1907 = CustomConstraint(cons_f1907)
def cons_f1908(d, c, e):
return PositiveQ(c*d/e + S(1))
cons1908 = CustomConstraint(cons_f1908)
def cons_f1909(d, c, e):
return NegativeQ(c*d/e + S(-1))
cons1909 = CustomConstraint(cons_f1909)
def cons_f1910(x, c, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(u**S(2) - (S(1) - S(2)/(c*x + S(1)))**S(2))
cons1910 = CustomConstraint(cons_f1910)
def cons_f1911(x, c, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(u**S(2) - (S(1) - S(2)/(-c*x + S(1)))**S(2))
cons1911 = CustomConstraint(cons_f1911)
def cons_f1912(x, c, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(-(S(1) - S(2)/(c*x + S(1)))**S(2) + (-u + S(1))**S(2))
cons1912 = CustomConstraint(cons_f1912)
def cons_f1913(x, c, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return ZeroQ(-(S(1) - S(2)/(-c*x + S(1)))**S(2) + (-u + S(1))**S(2))
cons1913 = CustomConstraint(cons_f1913)
def cons_f1914(d, a, n, c):
return Not(And(Equal(n, S(2)), ZeroQ(a**S(2)*c + d)))
cons1914 = CustomConstraint(cons_f1914)
def cons_f1915(c, f, g):
return ZeroQ(c**S(2)*f + g)
cons1915 = CustomConstraint(cons_f1915)
def cons_f1916(d, a, c):
return ZeroQ(a*c + d)
cons1916 = CustomConstraint(cons_f1916)
def cons_f1917(p, n):
return Or(IntegerQ(p), ZeroQ(-n/S(2) + p), ZeroQ(-n/S(2) + p + S(-1)))
cons1917 = CustomConstraint(cons_f1917)
def cons_f1918(d, a, c):
return ZeroQ(a**S(2)*c**S(2) - d**S(2))
cons1918 = CustomConstraint(cons_f1918)
def cons_f1919(d, c, a):
return ZeroQ(-a**S(2)*d**S(2) + c**S(2))
cons1919 = CustomConstraint(cons_f1919)
def cons_f1920(d, a, c):
return ZeroQ(a**S(2)*c + d)
cons1920 = CustomConstraint(cons_f1920)
def cons_f1921(p, n):
return NonzeroQ(n**S(2) - S(4)*(p + S(1))**S(2))
cons1921 = CustomConstraint(cons_f1921)
def cons_f1922(n):
return Not(IntegerQ(n/S(2)))
cons1922 = CustomConstraint(cons_f1922)
def cons_f1923(n):
return PositiveIntegerQ(n/S(2) + S(1)/2)
cons1923 = CustomConstraint(cons_f1923)
def cons_f1924(p, n):
return Not(IntegerQ(-n/S(2) + p))
cons1924 = CustomConstraint(cons_f1924)
def cons_f1925(n):
return NegativeIntegerQ(n/S(2) + S(-1)/2)
cons1925 = CustomConstraint(cons_f1925)
def cons_f1926(n):
return NegativeIntegerQ(n/S(2))
cons1926 = CustomConstraint(cons_f1926)
def cons_f1927(p, n):
return ZeroQ(n**S(2) + S(2)*p + S(2))
cons1927 = CustomConstraint(cons_f1927)
def cons_f1928(d, c, a):
return ZeroQ(a**S(2)*d + c)
cons1928 = CustomConstraint(cons_f1928)
def cons_f1929(c, b, e, a):
return ZeroQ(b**S(2)*c + e*(-a**S(2) + S(1)))
cons1929 = CustomConstraint(cons_f1929)
def cons_f1930(c, p, a):
return Or(IntegerQ(p), PositiveQ(c/(-a**S(2) + S(1))))
cons1930 = CustomConstraint(cons_f1930)
def cons_f1931(c, p, a):
return Not(Or(IntegerQ(p), PositiveQ(c/(-a**S(2) + S(1)))))
cons1931 = CustomConstraint(cons_f1931)
def cons_f1932(p, n):
return ZeroQ(-n/S(2) + p)
cons1932 = CustomConstraint(cons_f1932)
def cons_f1933(d, c, a):
return ZeroQ(a*d + c)
cons1933 = CustomConstraint(cons_f1933)
def cons_f1934(m, p, n):
return Or(IntegerQ(p), ZeroQ(-n/S(2) + p), ZeroQ(-n/S(2) + p + S(-1)), Less(S(-5), m, S(-1)))
cons1934 = CustomConstraint(cons_f1934)
def cons_f1935(p, n):
return Or(IntegerQ(p), Not(IntegerQ(n)))
cons1935 = CustomConstraint(cons_f1935)
def cons_f1936(p, n):
return NonzeroQ(n**S(2) + S(2)*p + S(2))
cons1936 = CustomConstraint(cons_f1936)
def cons_f1937(p, n):
return IntegersQ(S(2)*p, n/S(2) + p)
cons1937 = CustomConstraint(cons_f1937)
def cons_f1938(p, n):
return Not(IntegersQ(S(2)*p, n/S(2) + p))
cons1938 = CustomConstraint(cons_f1938)
def cons_f1939(c, b):
return ZeroQ(b - c**S(2))
cons1939 = CustomConstraint(cons_f1939)
def cons_f1940(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return PosQ(Discriminant(v, x))
cons1940 = CustomConstraint(cons_f1940)
def cons_f1941(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1940(w, f, r):
return FreeQ(f, x)
_cons_1940 = CustomConstraint(_cons_f_1940)
pat = Pattern(UtilityOperator(f_**w_*WC('r', S(1)), x), _cons_1940)
result_matchq = is_match(UtilityOperator(u, x), pat)
return result_matchq
cons1941 = CustomConstraint(cons_f1941)
def cons_f1942(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1941(w, f, r):
return FreeQ(f, x)
_cons_1941 = CustomConstraint(_cons_f_1941)
pat = Pattern(UtilityOperator(f_**w_*WC('r', S(1)), x), _cons_1941)
result_matchq = is_match(UtilityOperator(u, x), pat)
return result_matchq
cons1942 = CustomConstraint(cons_f1942)
def cons_f1943(d, c):
return ZeroQ((c - d)**S(2) + S(-1))
cons1943 = CustomConstraint(cons_f1943)
def cons_f1944(d, c):
return NonzeroQ((c - d)**S(2) + S(-1))
cons1944 = CustomConstraint(cons_f1944)
def cons_f1945(d, c):
return ZeroQ((c + I*d)**S(2) + S(-1))
cons1945 = CustomConstraint(cons_f1945)
def cons_f1946(d, c):
return ZeroQ((c - I*d)**S(2) + S(-1))
cons1946 = CustomConstraint(cons_f1946)
def cons_f1947(d, c):
return NonzeroQ((c + I*d)**S(2) + S(-1))
cons1947 = CustomConstraint(cons_f1947)
def cons_f1948(d, c):
return NonzeroQ((c - I*d)**S(2) + S(-1))
cons1948 = CustomConstraint(cons_f1948)
def cons_f1949(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1948(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1948 = CustomConstraint(_cons_f_1948)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1948)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1949 = CustomConstraint(cons_f1949)
def cons_f1950(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return FalseQ(FunctionOfLinear(v*(a + b*atanh(u)), x))
except (TypeError, AttributeError):
return False
cons1950 = CustomConstraint(cons_f1950)
def cons_f1951(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1950(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1950 = CustomConstraint(_cons_f_1950)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1950)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1951 = CustomConstraint(cons_f1951)
def cons_f1952(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
return FalseQ(FunctionOfLinear(v*(a + b*acoth(u)), x))
except (TypeError, AttributeError):
return False
cons1952 = CustomConstraint(cons_f1952)
def cons_f1953(x, a, p):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, p), x)
cons1953 = CustomConstraint(cons_f1953)
def cons_f1954(x, m, a, p):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, m, p), x)
cons1954 = CustomConstraint(cons_f1954)
def cons_f1955(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1954(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1954 = CustomConstraint(_cons_f_1954)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1954)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1955 = CustomConstraint(cons_f1955)
def cons_f1956(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_1955(d, m, c):
return FreeQ(List(c, d, m), x)
_cons_1955 = CustomConstraint(_cons_f_1955)
pat = Pattern(UtilityOperator((x*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x), _cons_1955)
result_matchq = is_match(UtilityOperator(v, x), pat)
return Not(result_matchq)
cons1956 = CustomConstraint(cons_f1956)
def cons_f1957(d, b):
return ZeroQ(-b**S(2) + d)
cons1957 = CustomConstraint(cons_f1957)
def cons_f1958(d, b):
return ZeroQ(b**S(2) + d)
cons1958 = CustomConstraint(cons_f1958)
def cons_f1959(m):
return Or(Greater(m, S(0)), OddQ(m))
cons1959 = CustomConstraint(cons_f1959)
def cons_f1960(m):
return Or(And(Greater(m, S(0)), EvenQ(m)), Equal(Mod(m, S(4)), S(3)))
cons1960 = CustomConstraint(cons_f1960)
def cons_f1961(c, b):
return ZeroQ(-Pi*b**S(2)/S(2) + c)
cons1961 = CustomConstraint(cons_f1961)
def cons_f1962(m):
return Not(Equal(Mod(m, S(4)), S(2)))
cons1962 = CustomConstraint(cons_f1962)
def cons_f1963(m):
return Equal(Mod(m, S(4)), S(0))
cons1963 = CustomConstraint(cons_f1963)
def cons_f1964(m):
return Not(Equal(Mod(m, S(4)), S(0)))
cons1964 = CustomConstraint(cons_f1964)
def cons_f1965(m):
return Equal(Mod(m, S(4)), S(2))
cons1965 = CustomConstraint(cons_f1965)
def cons_f1966(m, n):
return Or(PositiveIntegerQ(m), NegativeIntegerQ(n), And(RationalQ(m, n), Greater(m, S(0)), Less(n, S(-1))))
cons1966 = CustomConstraint(cons_f1966)
def cons_f1967(m, n):
return Or(PositiveIntegerQ(n), And(RationalQ(m, n), Less(m, S(-1)), Greater(n, S(0))))
cons1967 = CustomConstraint(cons_f1967)
def cons_f1968(n):
return Not(And(IntegerQ(n), LessEqual(n, S(0))))
cons1968 = CustomConstraint(cons_f1968)
def cons_f1969(m, n):
return Or(PositiveIntegerQ(m), PositiveIntegerQ(n), IntegersQ(m, n))
cons1969 = CustomConstraint(cons_f1969)
def cons_f1970(a, c):
return ZeroQ(a - c + S(1))
cons1970 = CustomConstraint(cons_f1970)
def cons_f1971(s):
return NonzeroQ(s + S(-1))
cons1971 = CustomConstraint(cons_f1971)
def cons_f1972(s):
return NonzeroQ(s + S(-2))
cons1972 = CustomConstraint(cons_f1972)
def cons_f1973(p, b, a, n, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, n, p, q), x)
cons1973 = CustomConstraint(cons_f1973)
def cons_f1974(r):
return RationalQ(r)
cons1974 = CustomConstraint(cons_f1974)
def cons_f1975(r):
return Greater(r, S(0))
cons1975 = CustomConstraint(cons_f1975)
def cons_f1976(p, b, d, c, a, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(F, a, b, c, d, n, p), x)
cons1976 = CustomConstraint(cons_f1976)
def cons_f1977(p, n):
return Or(ZeroQ(n*(p + S(-1)) + S(1)), And(IntegerQ(p + S(-1)/2), ZeroQ(n*(p + S(-1)/2) + S(1))))
cons1977 = CustomConstraint(cons_f1977)
def cons_f1978(p, n):
return Or(And(IntegerQ(p), ZeroQ(n*(p + S(1)) + S(1))), And(IntegerQ(p + S(-1)/2), ZeroQ(n*(p + S(1)/2) + S(1))))
cons1978 = CustomConstraint(cons_f1978)
def cons_f1979(m, p, n):
return Or(And(IntegerQ(p + S(-1)/2), IntegerQ(S(2)*(m + n*p + S(1))/n), Greater((m + n*p + S(1))/n, S(0))), And(Not(IntegerQ(p + S(-1)/2)), IntegerQ((m + n*p + S(1))/n), GreaterEqual((m + n*p + S(1))/n, S(0))))
cons1979 = CustomConstraint(cons_f1979)
def cons_f1980(m, p, n):
return Or(ZeroQ(m + S(1)), And(IntegerQ(p + S(-1)/2), IntegerQ(S(-1)/2 + (m + n*p + S(1))/n), Less((m + n*p + S(1))/n, S(0))), And(Not(IntegerQ(p + S(-1)/2)), IntegerQ((m + n*p + S(1))/n), Less((m + n*p + S(1))/n, S(0))))
cons1980 = CustomConstraint(cons_f1980)
def cons_f1981(x, m, a, c):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, m), x)
cons1981 = CustomConstraint(cons_f1981)
def cons_f1982(p, b, d, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, c, d, p), x)
cons1982 = CustomConstraint(cons_f1982)
def cons_f1983(p, n):
return ZeroQ(n*(p + S(-1)) + S(1))
cons1983 = CustomConstraint(cons_f1983)
def cons_f1984(p, n):
return ZeroQ(p + S(1)/n)
cons1984 = CustomConstraint(cons_f1984)
def cons_f1985(p, n):
return ZeroQ(p + S(-1)/2 + S(1)/n)
cons1985 = CustomConstraint(cons_f1985)
def cons_f1986(c, n):
return PosQ(c*n)
cons1986 = CustomConstraint(cons_f1986)
def cons_f1987(c, n):
return NegQ(c*n)
cons1987 = CustomConstraint(cons_f1987)
def cons_f1988(p, n):
return Greater(n*(p + S(-1)) + S(1), S(0))
cons1988 = CustomConstraint(cons_f1988)
def cons_f1989(p, n):
return Less(n*p + S(1), S(0))
cons1989 = CustomConstraint(cons_f1989)
def cons_f1990(d, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, d), x)
cons1990 = CustomConstraint(cons_f1990)
def cons_f1991(d, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, d, n), x)
cons1991 = CustomConstraint(cons_f1991)
def cons_f1992(p, d, a, n, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, c, d, n, p), x)
cons1992 = CustomConstraint(cons_f1992)
def cons_f1993(m, n, p):
return ZeroQ(m + n*(p + S(-1)) + S(1))
cons1993 = CustomConstraint(cons_f1993)
def cons_f1994(m, n, p):
return ZeroQ(m + n*p + S(1))
cons1994 = CustomConstraint(cons_f1994)
def cons_f1995(m, n, p):
return ZeroQ(m + n*(p + S(-1)/2) + S(1))
cons1995 = CustomConstraint(cons_f1995)
def cons_f1996(c, p):
return PosQ(c/(p + S(-1)/2))
cons1996 = CustomConstraint(cons_f1996)
def cons_f1997(c, p):
return NegQ(c/(p + S(-1)/2))
cons1997 = CustomConstraint(cons_f1997)
def cons_f1998(m, p, n):
return RationalQ((m + n*p + S(1))/n)
cons1998 = CustomConstraint(cons_f1998)
def cons_f1999(m, p, n):
return Greater((m + n*p + S(1))/n, S(1))
cons1999 = CustomConstraint(cons_f1999)
def cons_f2000(m, p, n):
return Less((m + n*p + S(1))/n, S(0))
cons2000 = CustomConstraint(cons_f2000)
def cons_f2001(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return FunctionOfQ(ProductLog(x), u, x)
cons2001 = CustomConstraint(cons_f2001)
def cons_f2002(x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
def _cons_f_2001(v, n1):
return ZeroQ(n - n1 - 1)
_cons_2001 = CustomConstraint(_cons_f_2001)
pat = Pattern(UtilityOperator(x**WC('n1', S(1))*WC('v', S(1)), x), _cons_2001)
result_matchq = is_match(UtilityOperator(u, x), pat)
return Not(result_matchq)
cons2002 = CustomConstraint(cons_f2002)
def cons_f2003(g, e):
return ZeroQ(e + g)
cons2003 = CustomConstraint(cons_f2003)
def cons_f2004(d, f):
return ZeroQ(d + f + S(-2))
cons2004 = CustomConstraint(cons_f2004)
def cons_f2005(C, f, d, A, e):
return ZeroQ(A*e**S(2) + C*d*f)
cons2005 = CustomConstraint(cons_f2005)
def cons_f2006(d, B, C, e):
return ZeroQ(-B*e + S(2)*C*(d + S(-1)))
cons2006 = CustomConstraint(cons_f2006)
def cons_f2007(C, e, A):
return ZeroQ(A*e**S(2) + C)
cons2007 = CustomConstraint(cons_f2007)
def cons_f2008(n):
return Not(PositiveQ(n))
cons2008 = CustomConstraint(cons_f2008)
def cons_f2009(v, y):
return ZeroQ(-v + y)
cons2009 = CustomConstraint(cons_f2009)
def cons_f2010(w, y):
return ZeroQ(-w + y)
cons2010 = CustomConstraint(cons_f2010)
def cons_f2011(z, y):
return ZeroQ(y - z)
cons2011 = CustomConstraint(cons_f2011)
def cons_f2012(p, m, b, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return FreeQ(List(a, b, m, n, p), x)
cons2012 = CustomConstraint(cons_f2012)
def cons_f2013(v, w):
return ZeroQ(-v + w)
cons2013 = CustomConstraint(cons_f2013)
def cons_f2014(r, p, q):
return ZeroQ(p - q*(r + S(1)))
cons2014 = CustomConstraint(cons_f2014)
def cons_f2015(r):
return NonzeroQ(r + S(1))
cons2015 = CustomConstraint(cons_f2015)
def cons_f2016(p, r):
return IntegerQ(p/(r + S(1)))
cons2016 = CustomConstraint(cons_f2016)
def cons_f2017(r, p, s, q):
return ZeroQ(p*(s + S(1)) - q*(r + S(1)))
cons2017 = CustomConstraint(cons_f2017)
def cons_f2018(m, p, q):
return ZeroQ(p + q*(m*p + S(1)))
cons2018 = CustomConstraint(cons_f2018)
def cons_f2019(m, p, q, r):
return ZeroQ(p + q*(m*p + r + S(1)))
cons2019 = CustomConstraint(cons_f2019)
def cons_f2020(m, p, s, q):
return ZeroQ(p*(s + S(1)) + q*(m*p + S(1)))
cons2020 = CustomConstraint(cons_f2020)
def cons_f2021(s):
return NonzeroQ(s + S(1))
cons2021 = CustomConstraint(cons_f2021)
def cons_f2022(s, q):
return IntegerQ(q/(s + S(1)))
cons2022 = CustomConstraint(cons_f2022)
def cons_f2023(p, m, r, s, q):
return ZeroQ(p*(s + S(1)) + q*(m*p + r + S(1)))
cons2023 = CustomConstraint(cons_f2023)
def cons_f2024(x, m, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return FunctionOfQ(x**(m + S(1)), u, x)
cons2024 = CustomConstraint(cons_f2024)
def cons_f2025(x, w):
if isinstance(x, (int, Integer, float, Float)):
return False
return NFreeQ(w, x)
cons2025 = CustomConstraint(cons_f2025)
def cons_f2026(x, z):
if isinstance(x, (int, Integer, float, Float)):
return False
return NFreeQ(z, x)
cons2026 = CustomConstraint(cons_f2026)
def cons_f2027(m, a):
return Not(And(EqQ(a, S(1)), EqQ(m, S(1))))
cons2027 = CustomConstraint(cons_f2027)
def cons_f2028(v, m, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(And(EqQ(v, x), EqQ(m, S(1))))
cons2028 = CustomConstraint(cons_f2028)
def cons_f2029(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(RationalFunctionQ(u, x))
cons2029 = CustomConstraint(cons_f2029)
def cons_f2030(v, x):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(LinearQ(v, x))
cons2030 = CustomConstraint(cons_f2030)
def cons_f2031(r, s):
return PosQ(-r + s)
cons2031 = CustomConstraint(cons_f2031)
def cons_f2032(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Not(AlgebraicFunctionQ(u, x))
cons2032 = CustomConstraint(cons_f2032)
def cons_f2033(x, m, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return Or(Greater(m, S(0)), Not(AlgebraicFunctionQ(u, x)))
cons2033 = CustomConstraint(cons_f2033)
def cons_f2034(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return EulerIntegrandQ(u, x)
cons2034 = CustomConstraint(cons_f2034)
def cons_f2035(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
return PolynomialInQ(v, u, x)
cons2035 = CustomConstraint(cons_f2035)
def cons_f2036(d, a):
return ZeroQ(a + d)
cons2036 = CustomConstraint(cons_f2036)
def cons_f2037(p, q):
return ZeroQ(p + q)
cons2037 = CustomConstraint(cons_f2037)
|
ac9e3d33c8572a7426444dedf2bea110eec756afadb2c1164279c9d808324daf | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def quadratic_products(rubi):
from sympy.integrals.rubi.constraints import cons45, cons2, cons3, cons7, cons225, cons5, cons226, cons128, cons227, cons228, cons13, cons163, cons229, cons137, cons230, cons231, cons232, cons233, cons234, cons235, cons68, cons69, cons47, cons236, cons27, cons48, cons21, cons237, cons238, cons239, cons240, cons66, cons241, cons242, cons243, cons244, cons146, cons245, cons246, cons247, cons248, cons249, cons250, cons251, cons252, cons166, cons253, cons31, cons168, cons254, cons255, cons94, cons147, cons256, cons38, cons257, cons258, cons41, cons17, cons259, cons260, cons261, cons262, cons263, cons264, cons265, cons266, cons267, cons43, cons268, cons54, cons269, cons270, cons271, cons272, cons273, cons274, cons275, cons276, cons277, cons278, cons279, cons280, cons281, cons282, cons283, cons284, cons285, cons286, cons18, cons287, cons288, cons289, cons290, cons291, cons292, cons293, cons294, cons295, cons296, cons297, cons298, cons299, cons300, cons301, cons302, cons303, cons304, cons305, cons306, cons307, cons308, cons309, cons310, cons311, cons312, cons313, cons84, cons85, cons314, cons315, cons316, cons125, cons208, cons317, cons318, cons319, cons62, cons320, cons321, cons322, cons323, cons324, cons4, cons325, cons326, cons327, cons139, cons328, cons329, cons330, cons331, cons150, cons332, cons148, cons333, cons196, cons334, cons335, cons336, cons337, cons338, cons89, cons339, cons340, cons341, cons88, cons87, cons342, cons343, cons344, cons126, cons345, cons346, cons207, cons347, cons348, cons349, cons350, cons351, cons352, cons353, cons354, cons355, cons356, cons357, cons358, cons359, cons360, cons361, cons362, cons363, cons364, cons365, cons366, cons367, cons368, cons369, cons370, cons371, cons372, cons373, cons374, cons375, cons149, cons376, cons124, cons377, cons93, cons23, cons165, cons73, cons378, cons80, cons379, cons380, cons381, cons382, cons383, cons384, cons385, cons50, cons386, cons387, cons388, cons389, cons390, cons391, cons392, cons393, cons394, cons395, cons396, cons397, cons398, cons399, cons400, cons401, cons402, cons403, cons404, cons405, cons406, cons407, cons408, cons409, cons410, cons411, cons412, cons413, cons414, cons415, cons416, cons209, cons417, cons418, cons419, cons420, cons421, cons422, cons423, cons424, cons425, cons426, cons427, cons428, cons429, cons430, cons431, cons220, cons432, cons433, cons434, cons435, cons436, cons437, cons438, cons439, cons440, cons441, cons442, cons443, cons444, cons445, cons446, cons447, cons448, cons449, cons450, cons451, cons452, cons453, cons224, cons34, cons35, cons36, cons454, cons455, cons456, cons457, cons458
pattern189 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons45)
def replacement189(x, c, b, a):
rubi.append(189)
return Dist((b/S(2) + c*x)/sqrt(a + b*x + c*x**S(2)), Int(S(1)/(b/S(2) + c*x), x), x)
rule189 = ReplacementRule(pattern189, replacement189)
pattern190 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons45, cons225)
def replacement190(p, b, c, a, x):
rubi.append(190)
return Simp((b + S(2)*c*x)*(a + b*x + c*x**S(2))**p/(S(2)*c*(S(2)*p + S(1))), x)
rule190 = ReplacementRule(pattern190, replacement190)
def With191(p, b, c, a, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(191)
return Dist(c**(-p), Int(Simp(b/S(2) + c*x - q/S(2), x)**p*Simp(b/S(2) + c*x + q/S(2), x)**p, x), x)
pattern191 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons226, cons128, cons227)
rule191 = ReplacementRule(pattern191, With191)
pattern192 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons226, cons128, cons228)
def replacement192(p, b, c, a, x):
rubi.append(192)
return Int(ExpandIntegrand((a + b*x + c*x**S(2))**p, x), x)
rule192 = ReplacementRule(pattern192, replacement192)
pattern193 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons226, cons13, cons163, cons229)
def replacement193(p, b, c, a, x):
rubi.append(193)
return -Dist(p*(-S(4)*a*c + b**S(2))/(S(2)*c*(S(2)*p + S(1))), Int((a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((b + S(2)*c*x)*(a + b*x + c*x**S(2))**p/(S(2)*c*(S(2)*p + S(1))), x)
rule193 = ReplacementRule(pattern193, replacement193)
pattern194 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**(S(-3)/2), x_), cons2, cons3, cons7, cons226)
def replacement194(x, c, a, b):
rubi.append(194)
return Simp(-S(2)*(b + S(2)*c*x)/((-S(4)*a*c + b**S(2))*sqrt(a + b*x + c*x**S(2))), x)
rule194 = ReplacementRule(pattern194, replacement194)
pattern195 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons226, cons13, cons137, cons230, cons229)
def replacement195(p, b, c, a, x):
rubi.append(195)
return -Dist(S(2)*c*(S(2)*p + S(3))/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((b + S(2)*c*x)*(a + b*x + c*x**S(2))**(p + S(1))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule195 = ReplacementRule(pattern195, replacement195)
def With196(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(196)
return Dist(c/q, Int(S(1)/Simp(b/S(2) + c*x - q/S(2), x), x), x) - Dist(c/q, Int(S(1)/Simp(b/S(2) + c*x + q/S(2), x), x), x)
pattern196 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons226, cons231, cons227)
rule196 = ReplacementRule(pattern196, With196)
def With197(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = -S(4)*a*c/b**S(2) + S(1)
if And(RationalQ(q), Or(EqQ(q**S(2), S(1)), Not(RationalQ(-S(4)*a*c + b**S(2))))):
return True
return False
pattern197 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons226, CustomConstraint(With197))
def replacement197(x, c, b, a):
q = -S(4)*a*c/b**S(2) + S(1)
rubi.append(197)
return Dist(-S(2)/b, Subst(Int(S(1)/(q - x**S(2)), x), x, S(1) + S(2)*c*x/b), x)
rule197 = ReplacementRule(pattern197, replacement197)
pattern198 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons226)
def replacement198(x, c, b, a):
rubi.append(198)
return Dist(S(-2), Subst(Int(S(1)/Simp(-S(4)*a*c + b**S(2) - x**S(2), x), x), x, b + S(2)*c*x), x)
rule198 = ReplacementRule(pattern198, replacement198)
pattern199 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons5, cons232)
def replacement199(p, b, c, a, x):
rubi.append(199)
return Dist((-S(4)*c/(-S(4)*a*c + b**S(2)))**(-p)/(S(2)*c), Subst(Int(Simp(-x**S(2)/(-S(4)*a*c + b**S(2)) + S(1), x)**p, x), x, b + S(2)*c*x), x)
rule199 = ReplacementRule(pattern199, replacement199)
pattern200 = Pattern(Integral(S(1)/sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons3, cons7, cons233)
def replacement200(x, c, b):
rubi.append(200)
return Dist(S(2), Subst(Int(S(1)/(-c*x**S(2) + S(1)), x), x, x/sqrt(b*x + c*x**S(2))), x)
rule200 = ReplacementRule(pattern200, replacement200)
pattern201 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons226)
def replacement201(x, c, b, a):
rubi.append(201)
return Dist(S(2), Subst(Int(S(1)/(S(4)*c - x**S(2)), x), x, (b + S(2)*c*x)/sqrt(a + b*x + c*x**S(2))), x)
rule201 = ReplacementRule(pattern201, replacement201)
pattern202 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons3, cons7, cons13, cons234)
def replacement202(x, c, p, b):
rubi.append(202)
return Dist((-c*(b*x + c*x**S(2))/b**S(2))**(-p)*(b*x + c*x**S(2))**p, Int((-c*x/b - c**S(2)*x**S(2)/b**S(2))**p, x), x)
rule202 = ReplacementRule(pattern202, replacement202)
def With203(p, b, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
d = Denominator(p)
if LessEqual(S(3), d, S(4)):
return True
return False
pattern203 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons226, cons13, CustomConstraint(With203))
def replacement203(p, b, c, a, x):
d = Denominator(p)
rubi.append(203)
return Dist(d*sqrt((b + S(2)*c*x)**S(2))/(b + S(2)*c*x), Subst(Int(x**(d*(p + S(1)) + S(-1))/sqrt(-S(4)*a*c + b**S(2) + S(4)*c*x**d), x), x, (a + b*x + c*x**S(2))**(S(1)/d)), x)
rule203 = ReplacementRule(pattern203, replacement203)
def With204(p, b, c, a, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(204)
return -Simp(((-b - S(2)*c*x + q)/(S(2)*q))**(-p + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*Hypergeometric2F1(-p, p + S(1), p + S(2), (b + S(2)*c*x + q)/(S(2)*q))/(q*(p + S(1))), x)
pattern204 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons5, cons226, cons235)
rule204 = ReplacementRule(pattern204, With204)
pattern205 = Pattern(Integral((u_**S(2)*WC('c', S(1)) + u_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons5, cons68, cons69)
def replacement205(p, u, b, c, a, x):
rubi.append(205)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x + c*x**S(2))**p, x), x, u), x)
rule205 = ReplacementRule(pattern205, replacement205)
pattern206 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons45, cons47, cons236)
def replacement206(p, m, b, d, c, a, x, e):
rubi.append(206)
return Simp(c**(-m/S(2) + S(-1)/2)*e**m*(a + b*x + c*x**S(2))**(m/S(2) + p + S(1)/2)/(m + S(2)*p + S(1)), x)
rule206 = ReplacementRule(pattern206, replacement206)
pattern207 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons45, cons47, cons237)
def replacement207(p, m, b, d, c, a, x, e):
rubi.append(207)
return Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*log(RemoveContent(d + e*x, x))/e, x)
rule207 = ReplacementRule(pattern207, replacement207)
pattern208 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons45, cons47, cons238)
def replacement208(p, m, b, d, c, a, x, e):
rubi.append(208)
return Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(2)*p + S(1))), x)
rule208 = ReplacementRule(pattern208, replacement208)
pattern209 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons45, cons239, cons240, cons66)
def replacement209(p, m, b, d, c, a, x, e):
rubi.append(209)
return -Simp((b + S(2)*c*x)*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/((m + S(1))*(-b*e + S(2)*c*d)), x)
rule209 = ReplacementRule(pattern209, replacement209)
pattern210 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons45, cons239)
def replacement210(b, d, c, a, x, e):
rubi.append(210)
return Dist(sqrt(a + b*x + c*x**S(2))/(b + S(2)*c*x), Int((b + S(2)*c*x)/(d + e*x)**S(2), x), x)
rule210 = ReplacementRule(pattern210, replacement210)
pattern211 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons45, cons239, cons241)
def replacement211(m, b, d, c, a, x, e):
rubi.append(211)
return -Dist((-b*e + S(2)*c*d)*sqrt(a + b*x + c*x**S(2))/(e*(b + S(2)*c*x)*(m + S(2))), Int((d + e*x)**m, x), x) + Simp((d + e*x)**(m + S(1))*sqrt(a + b*x + c*x**S(2))/(e*(m + S(2))), x)
rule211 = ReplacementRule(pattern211, replacement211)
pattern212 = Pattern(Integral(S(1)/((x_*WC('e', S(1)) + WC('d', S(0)))**S(2)*sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons45, cons239)
def replacement212(b, d, c, a, x, e):
rubi.append(212)
return Dist(S(2)*c/(-b*e + S(2)*c*d), Int(S(1)/((d + e*x)*sqrt(a + b*x + c*x**S(2))), x), x) + Simp(-S(4)*c*e*sqrt(a + b*x + c*x**S(2))/((d + e*x)*(-b*e + S(2)*c*d)**S(2)), x)
rule212 = ReplacementRule(pattern212, replacement212)
pattern213 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons45, cons239, cons242, cons241)
def replacement213(p, m, b, d, c, a, x, e):
rubi.append(213)
return -Simp((b + S(2)*c*x)*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/((m + S(2))*(-b*e + S(2)*c*d)), x) + Simp(-S(2)*c*e*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/((-b*e + S(2)*c*d)**S(2)*(m*p + S(-1))), x)
rule213 = ReplacementRule(pattern213, replacement213)
pattern214 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons45, cons239, cons243)
def replacement214(p, b, d, c, a, x, e):
rubi.append(214)
return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int((a + b*x + c*x**S(2))**p, x), x) + Simp(e*(a + b*x + c*x**S(2))**(p + S(1))/(S(2)*c*(p + S(1))), x)
rule214 = ReplacementRule(pattern214, replacement214)
pattern215 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons45, cons239, cons244, cons146, cons245, cons246)
def replacement215(p, m, b, d, c, a, x, e):
rubi.append(215)
return Dist(p*(S(2)*p + S(-1))*(-b*e + S(2)*c*d)/(e**S(2)*(m + S(1))*(m + S(2)*p + S(1))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(1))), x) - Simp(p*(b + S(2)*c*x)*(d + e*x)**(m + S(2))*(a + b*x + c*x**S(2))**(p + S(-1))/(e**S(2)*(m + S(1))*(m + S(2)*p + S(1))), x)
rule215 = ReplacementRule(pattern215, replacement215)
pattern216 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons45, cons239, cons244, cons146, cons247, cons246, cons248)
def replacement216(p, m, b, d, c, a, x, e):
rubi.append(216)
return Dist(S(2)*c*p*(S(2)*p + S(-1))/(e**S(2)*(m + S(1))*(m + S(2))), Int((d + e*x)**(m + S(2))*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(1))), x) - Simp(p*(b + S(2)*c*x)*(d + e*x)**(m + S(2))*(a + b*x + c*x**S(2))**(p + S(-1))/(e**S(2)*(m + S(1))*(m + S(2))), x)
rule216 = ReplacementRule(pattern216, replacement216)
pattern217 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons45, cons239, cons13, cons163, cons249, cons238, cons248, cons250, cons251, cons246)
def replacement217(p, m, b, d, c, a, x, e):
rubi.append(217)
return Dist(p*(S(2)*p + S(-1))*(-b*e + S(2)*c*d)**S(2)/(S(2)*c*e**S(2)*(m + S(2)*p)*(m + S(2)*p + S(1))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(2)*p + S(1))), x) - Simp(p*(b + S(2)*c*x)*(d + e*x)**(m + S(1))*(-b*e + S(2)*c*d)*(a + b*x + c*x**S(2))**(p + S(-1))/(S(2)*c*e**S(2)*(m + S(2)*p)*(m + S(2)*p + S(1))), x)
rule217 = ReplacementRule(pattern217, replacement217)
pattern218 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons45, cons239, cons244, cons137, cons252, cons246)
def replacement218(p, m, b, d, c, a, x, e):
rubi.append(218)
return Dist(e**S(2)*m*(m + S(2)*p + S(2))/((p + S(1))*(S(2)*p + S(1))*(-b*e + S(2)*c*d)), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((b + S(2)*c*x)*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/((S(2)*p + S(1))*(-b*e + S(2)*c*d)), x) - Simp(e*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*(m + S(2)*p + S(2))/((p + S(1))*(S(2)*p + S(1))*(-b*e + S(2)*c*d)), x)
rule218 = ReplacementRule(pattern218, replacement218)
pattern219 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons45, cons239, cons244, cons137, cons166, cons246)
def replacement219(p, m, b, d, c, a, x, e):
rubi.append(219)
return Dist(e**S(2)*m*(m + S(-1))/(S(2)*c*(p + S(1))*(S(2)*p + S(1))), Int((d + e*x)**(m + S(-2))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((b + S(2)*c*x)*(d + e*x)**m*(a + b*x + c*x**S(2))**p/(S(2)*c*(S(2)*p + S(1))), x) - Simp(e*m*(d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(S(2)*c*(p + S(1))*(S(2)*p + S(1))), x)
rule219 = ReplacementRule(pattern219, replacement219)
pattern220 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons45, cons239, cons244, cons137, cons253, cons246)
def replacement220(p, m, b, d, c, a, x, e):
rubi.append(220)
return Dist(S(2)*c*e**S(2)*(m + S(2)*p + S(2))*(m + S(2)*p + S(3))/((p + S(1))*(S(2)*p + S(1))*(-b*e + S(2)*c*d)**S(2)), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((b + S(2)*c*x)*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/((S(2)*p + S(1))*(-b*e + S(2)*c*d)), x) + Simp(-S(2)*c*e*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(m + S(2)*p + S(2))/((p + S(1))*(S(2)*p + S(1))*(-b*e + S(2)*c*d)**S(2)), x)
rule220 = ReplacementRule(pattern220, replacement220)
pattern221 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons45, cons239, cons31, cons168, cons238, cons254, cons255)
def replacement221(p, m, b, d, c, a, x, e):
rubi.append(221)
return Dist(m*(-b*e + S(2)*c*d)/(S(2)*c*(m + S(2)*p + S(1))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**p, x), x) + Simp((b + S(2)*c*x)*(d + e*x)**m*(a + b*x + c*x**S(2))**p/(S(2)*c*(m + S(2)*p + S(1))), x)
rule221 = ReplacementRule(pattern221, replacement221)
pattern222 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons45, cons239, cons31, cons94, cons246)
def replacement222(p, m, b, d, c, a, x, e):
rubi.append(222)
return Dist(S(2)*c*(m + S(2)*p + S(2))/((m + S(1))*(-b*e + S(2)*c*d)), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) - Simp((b + S(2)*c*x)*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/((m + S(1))*(-b*e + S(2)*c*d)), x)
rule222 = ReplacementRule(pattern222, replacement222)
pattern223 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons45, cons147, cons239)
def replacement223(p, m, b, d, c, a, x, e):
rubi.append(223)
return Dist(c**(-IntPart(p))*(b/S(2) + c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((b/S(2) + c*x)**(S(2)*p)*(d + e*x)**m, x), x)
rule223 = ReplacementRule(pattern223, replacement223)
pattern224 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons256, cons38)
def replacement224(p, m, b, d, c, a, x, e):
rubi.append(224)
return Int((d + e*x)**(m + p)*(a/d + c*x/e)**p, x)
rule224 = ReplacementRule(pattern224, replacement224)
pattern225 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons257, cons258)
def replacement225(p, m, d, c, a, x, e):
rubi.append(225)
return Int((d + e*x)**(m + p)*(a/d + c*x/e)**p, x)
rule225 = ReplacementRule(pattern225, replacement225)
pattern226 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons256, cons147, cons41)
def replacement226(p, m, b, d, c, a, x, e):
rubi.append(226)
return Simp(e*(d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*(p + S(1))), x)
rule226 = ReplacementRule(pattern226, replacement226)
pattern227 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons5, cons257, cons147, cons41)
def replacement227(p, m, d, c, a, x, e):
rubi.append(227)
return Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))/(c*(p + S(1))), x)
rule227 = ReplacementRule(pattern227, replacement227)
pattern228 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons256, cons147, cons240)
def replacement228(p, m, b, d, a, c, x, e):
rubi.append(228)
return Simp(e*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/((p + S(1))*(-b*e + S(2)*c*d)), x)
rule228 = ReplacementRule(pattern228, replacement228)
pattern229 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons257, cons147, cons240)
def replacement229(p, m, d, c, a, x, e):
rubi.append(229)
return Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(S(2)*c*d*(p + S(1))), x)
rule229 = ReplacementRule(pattern229, replacement229)
pattern230 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**S(2)*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons226, cons256, cons147, cons13, cons137)
def replacement230(p, b, d, c, a, x, e):
rubi.append(230)
return -Dist(e**S(2)*(p + S(2))/(c*(p + S(1))), Int((a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp(e*(d + e*x)*(a + b*x + c*x**S(2))**(p + S(1))/(c*(p + S(1))), x)
rule230 = ReplacementRule(pattern230, replacement230)
pattern231 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**S(2), x_), cons2, cons7, cons27, cons48, cons5, cons257, cons147, cons13, cons137)
def replacement231(p, d, c, a, x, e):
rubi.append(231)
return -Dist(e**S(2)*(p + S(2))/(c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1)), x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)/(c*(p + S(1))), x)
rule231 = ReplacementRule(pattern231, replacement231)
pattern232 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons256, cons147, cons17, cons13, cons259, cons260, cons261)
def replacement232(p, m, b, d, c, a, x, e):
rubi.append(232)
return Int((a/d + c*x/e)**(-m)*(a + b*x + c*x**S(2))**(m + p), x)
rule232 = ReplacementRule(pattern232, replacement232)
pattern233 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons5, cons257, cons147, cons17, cons13, cons259, cons260, cons261)
def replacement233(p, m, d, c, a, x, e):
rubi.append(233)
return Dist(a**(-m)*d**(S(2)*m), Int((a + c*x**S(2))**(m + p)*(d - e*x)**(-m), x), x)
rule233 = ReplacementRule(pattern233, replacement233)
pattern234 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons256, cons147, cons262)
def replacement234(p, m, b, d, a, c, x, e):
rubi.append(234)
return Dist((m + p)*(-b*e + S(2)*c*d)/(c*(m + S(2)*p + S(1))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**p, x), x) + Simp(e*(d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m + S(2)*p + S(1))), x)
rule234 = ReplacementRule(pattern234, replacement234)
pattern235 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons257, cons147, cons262)
def replacement235(p, m, d, c, a, x, e):
rubi.append(235)
return Dist(S(2)*d*(m + p)/(m + S(2)*p + S(1)), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(-1)), x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))/(c*(m + S(2)*p + S(1))), x)
rule235 = ReplacementRule(pattern235, replacement235)
pattern236 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons256, cons147, cons263)
def replacement236(p, m, b, d, c, a, x, e):
rubi.append(236)
return Dist(c*(m + S(2)*p + S(2))/((-b*e + S(2)*c*d)*(m + p + S(1))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) - Simp(e*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/((-b*e + S(2)*c*d)*(m + p + S(1))), x)
rule236 = ReplacementRule(pattern236, replacement236)
pattern237 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons5, cons257, cons147, cons263)
def replacement237(p, m, d, c, a, x, e):
rubi.append(237)
return Dist((m + S(2)*p + S(2))/(S(2)*d*(m + p + S(1))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1)), x), x) - Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(S(2)*c*d*(m + p + S(1))), x)
rule237 = ReplacementRule(pattern237, replacement237)
pattern238 = Pattern(Integral(S(1)/(sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons256)
def replacement238(b, d, c, a, x, e):
rubi.append(238)
return Dist(S(2)*e, Subst(Int(S(1)/(-b*e + S(2)*c*d + e**S(2)*x**S(2)), x), x, sqrt(a + b*x + c*x**S(2))/sqrt(d + e*x)), x)
rule238 = ReplacementRule(pattern238, replacement238)
pattern239 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('c', S(1)))*sqrt(d_ + x_*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons257)
def replacement239(d, c, a, x, e):
rubi.append(239)
return Dist(S(2)*e, Subst(Int(S(1)/(S(2)*c*d + e**S(2)*x**S(2)), x), x, sqrt(a + c*x**S(2))/sqrt(d + e*x)), x)
rule239 = ReplacementRule(pattern239, replacement239)
pattern240 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons256, cons244, cons163, cons264, cons253, cons246)
def replacement240(p, m, b, d, c, a, x, e):
rubi.append(240)
return -Dist(c*p/(e**S(2)*(m + p + S(1))), Int((d + e*x)**(m + S(2))*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + p + S(1))), x)
rule240 = ReplacementRule(pattern240, replacement240)
pattern241 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons257, cons244, cons163, cons264, cons253, cons246)
def replacement241(p, m, d, c, a, x, e):
rubi.append(241)
return -Dist(c*p/(e**S(2)*(m + p + S(1))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**(m + S(2)), x), x) + Simp((a + c*x**S(2))**p*(d + e*x)**(m + S(1))/(e*(m + p + S(1))), x)
rule241 = ReplacementRule(pattern241, replacement241)
pattern242 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons256, cons244, cons163, cons265, cons238, cons246)
def replacement242(p, m, b, d, c, a, x, e):
rubi.append(242)
return -Dist(p*(-b*e + S(2)*c*d)/(e**S(2)*(m + S(2)*p + S(1))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(2)*p + S(1))), x)
rule242 = ReplacementRule(pattern242, replacement242)
pattern243 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons257, cons244, cons163, cons265, cons238, cons246)
def replacement243(p, m, d, c, a, x, e):
rubi.append(243)
return -Dist(S(2)*c*d*p/(e**S(2)*(m + S(2)*p + S(1))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**(m + S(1)), x), x) + Simp((a + c*x**S(2))**p*(d + e*x)**(m + S(1))/(e*(m + S(2)*p + S(1))), x)
rule243 = ReplacementRule(pattern243, replacement243)
pattern244 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons256, cons244, cons137, cons252, cons246)
def replacement244(p, m, b, d, a, c, x, e):
rubi.append(244)
return -Dist((-b*e + S(2)*c*d)*(m + S(2)*p + S(2))/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((d + e*x)**m*(-b*e + S(2)*c*d)*(a + b*x + c*x**S(2))**(p + S(1))/(e*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule244 = ReplacementRule(pattern244, replacement244)
pattern245 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons257, cons244, cons137, cons252, cons246)
def replacement245(p, m, d, c, a, x, e):
rubi.append(245)
return Dist(d*(m + S(2)*p + S(2))/(S(2)*a*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1)), x), x) - Simp(d*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(S(2)*a*e*(p + S(1))), x)
rule245 = ReplacementRule(pattern245, replacement245)
pattern246 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons256, cons244, cons137, cons166, cons246)
def replacement246(p, m, b, d, c, a, x, e):
rubi.append(246)
return -Dist(e**S(2)*(m + p)/(c*(p + S(1))), Int((d + e*x)**(m + S(-2))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*(p + S(1))), x)
rule246 = ReplacementRule(pattern246, replacement246)
pattern247 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons257, cons244, cons137, cons166, cons246)
def replacement247(p, m, d, c, a, x, e):
rubi.append(247)
return -Dist(e**S(2)*(m + p)/(c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-2)), x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))/(c*(p + S(1))), x)
rule247 = ReplacementRule(pattern247, replacement247)
pattern248 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons226, cons256, cons31, cons266, cons238, cons246)
def replacement248(p, m, b, d, a, c, x, e):
rubi.append(248)
return Dist((m + p)*(-b*e + S(2)*c*d)/(c*(m + S(2)*p + S(1))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**p, x), x) + Simp(e*(d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m + S(2)*p + S(1))), x)
rule248 = ReplacementRule(pattern248, replacement248)
pattern249 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons5, cons257, cons31, cons266, cons238, cons246)
def replacement249(p, m, d, c, a, x, e):
rubi.append(249)
return Dist(S(2)*d*(m + p)/(m + S(2)*p + S(1)), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(-1)), x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))/(c*(m + S(2)*p + S(1))), x)
rule249 = ReplacementRule(pattern249, replacement249)
pattern250 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons226, cons256, cons31, cons267, cons253, cons246)
def replacement250(p, m, b, d, c, a, x, e):
rubi.append(250)
return Dist(c*(m + S(2)*p + S(2))/((-b*e + S(2)*c*d)*(m + p + S(1))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) - Simp(e*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/((-b*e + S(2)*c*d)*(m + p + S(1))), x)
rule250 = ReplacementRule(pattern250, replacement250)
pattern251 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons5, cons257, cons31, cons267, cons253, cons246)
def replacement251(p, m, d, c, a, x, e):
rubi.append(251)
return Dist((m + S(2)*p + S(2))/(S(2)*d*(m + p + S(1))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1)), x), x) - Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(S(2)*c*d*(m + p + S(1))), x)
rule251 = ReplacementRule(pattern251, replacement251)
pattern252 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons3, cons7, cons48, cons21, cons147)
def replacement252(p, m, b, c, x, e):
rubi.append(252)
return Dist(x**(-m - p)*(e*x)**m*(b + c*x)**(-p)*(b*x + c*x**S(2))**p, Int(x**(m + p)*(b + c*x)**p, x), x)
rule252 = ReplacementRule(pattern252, replacement252)
pattern253 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons257, cons147, cons43, cons268)
def replacement253(p, m, d, c, a, x, e):
rubi.append(253)
return Int((d + e*x)**(m + p)*(a/d + c*x/e)**p, x)
rule253 = ReplacementRule(pattern253, replacement253)
pattern254 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons256, cons147)
def replacement254(p, m, b, d, c, a, x, e):
rubi.append(254)
return Dist((d + e*x)**(-FracPart(p))*(a/d + c*x/e)**(-FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((d + e*x)**(m + p)*(a/d + c*x/e)**p, x), x)
rule254 = ReplacementRule(pattern254, replacement254)
pattern255 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons257, cons147)
def replacement255(p, m, d, c, a, x, e):
rubi.append(255)
return Dist((a + c*x**S(2))**FracPart(p)*(d + e*x)**(-FracPart(p))*(a/d + c*x/e)**(-FracPart(p)), Int((d + e*x)**(m + p)*(a/d + c*x/e)**p, x), x)
rule255 = ReplacementRule(pattern255, replacement255)
pattern256 = Pattern(Integral(S(1)/((d_ + x_*WC('e', S(1)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons47)
def replacement256(b, d, c, a, x, e):
rubi.append(256)
return Dist(b**S(2)/(d**S(2)*(-S(4)*a*c + b**S(2))), Int((d + e*x)/(a + b*x + c*x**S(2)), x), x) + Dist(-S(4)*b*c/(d*(-S(4)*a*c + b**S(2))), Int(S(1)/(b + S(2)*c*x), x), x)
rule256 = ReplacementRule(pattern256, replacement256)
pattern257 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons47, cons242, cons54)
def replacement257(p, m, b, d, c, a, x, e):
rubi.append(257)
return Simp(S(2)*c*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(e*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule257 = ReplacementRule(pattern257, replacement257)
pattern258 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons47, cons128, cons269)
def replacement258(p, m, b, d, c, a, x, e):
rubi.append(258)
return Int(ExpandIntegrand((d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x)
rule258 = ReplacementRule(pattern258, replacement258)
pattern259 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons47, cons270, cons244, cons163, cons94, cons271, cons246)
def replacement259(p, m, b, d, c, a, x, e):
rubi.append(259)
return -Dist(b*p/(d*e*(m + S(1))), Int((d + e*x)**(m + S(2))*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(1))), x)
rule259 = ReplacementRule(pattern259, replacement259)
pattern260 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons47, cons270, cons13, cons163, cons272, cons273, cons31, cons246)
def replacement260(p, m, b, d, c, a, x, e):
rubi.append(260)
return -Dist(d*p*(-S(4)*a*c + b**S(2))/(b*e*(m + S(2)*p + S(1))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(2)*p + S(1))), x)
rule260 = ReplacementRule(pattern260, replacement260)
pattern261 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons47, cons270, cons244, cons137, cons166, cons246)
def replacement261(p, m, b, d, c, a, x, e):
rubi.append(261)
return -Dist(d*e*(m + S(-1))/(b*(p + S(1))), Int((d + e*x)**(m + S(-2))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp(d*(d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(b*(p + S(1))), x)
rule261 = ReplacementRule(pattern261, replacement261)
pattern262 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons47, cons270, cons13, cons137, cons274, cons31, cons246)
def replacement262(p, m, b, d, c, a, x, e):
rubi.append(262)
return -Dist(S(2)*c*(m + S(2)*p + S(3))/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp(S(2)*c*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(e*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule262 = ReplacementRule(pattern262, replacement262)
pattern263 = Pattern(Integral(S(1)/((d_ + x_*WC('e', S(1)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons47)
def replacement263(b, d, c, a, x, e):
rubi.append(263)
return Dist(S(4)*c, Subst(Int(S(1)/(-S(4)*a*c*e + b**S(2)*e + S(4)*c*e*x**S(2)), x), x, sqrt(a + b*x + c*x**S(2))), x)
rule263 = ReplacementRule(pattern263, replacement263)
pattern264 = Pattern(Integral(S(1)/(sqrt(d_ + x_*WC('e', S(1)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons47, cons275)
def replacement264(b, d, c, a, x, e):
rubi.append(264)
return Dist(S(4)*sqrt(-c/(-S(4)*a*c + b**S(2)))/e, Subst(Int(S(1)/sqrt(Simp(-b**S(2)*x**S(4)/(d**S(2)*(-S(4)*a*c + b**S(2))) + S(1), x)), x), x, sqrt(d + e*x)), x)
rule264 = ReplacementRule(pattern264, replacement264)
pattern265 = Pattern(Integral(sqrt(d_ + x_*WC('e', S(1)))/sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons47, cons275)
def replacement265(b, d, c, a, x, e):
rubi.append(265)
return Dist(S(4)*sqrt(-c/(-S(4)*a*c + b**S(2)))/e, Subst(Int(x**S(2)/sqrt(Simp(-b**S(2)*x**S(4)/(d**S(2)*(-S(4)*a*c + b**S(2))) + S(1), x)), x), x, sqrt(d + e*x)), x)
rule265 = ReplacementRule(pattern265, replacement265)
pattern266 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_/sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons47, cons276)
def replacement266(m, b, d, c, a, x, e):
rubi.append(266)
return Dist(sqrt(-c*(a + b*x + c*x**S(2))/(-S(4)*a*c + b**S(2)))/sqrt(a + b*x + c*x**S(2)), Int((d + e*x)**m/sqrt(-a*c/(-S(4)*a*c + b**S(2)) - b*c*x/(-S(4)*a*c + b**S(2)) - c**S(2)*x**S(2)/(-S(4)*a*c + b**S(2))), x), x)
rule266 = ReplacementRule(pattern266, replacement266)
pattern267 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons226, cons47, cons270, cons31, cons166, cons238, cons277)
def replacement267(p, m, b, d, c, a, x, e):
rubi.append(267)
return Dist(d**S(2)*(m + S(-1))*(-S(4)*a*c + b**S(2))/(b**S(2)*(m + S(2)*p + S(1))), Int((d + e*x)**(m + S(-2))*(a + b*x + c*x**S(2))**p, x), x) + Simp(S(2)*d*(d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(b*(m + S(2)*p + S(1))), x)
rule267 = ReplacementRule(pattern267, replacement267)
pattern268 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons226, cons47, cons270, cons31, cons94, cons278)
def replacement268(p, m, b, d, c, a, x, e):
rubi.append(268)
return Dist(b**S(2)*(m + S(2)*p + S(3))/(d**S(2)*(m + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(2))*(a + b*x + c*x**S(2))**p, x), x) + Simp(-S(2)*b*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(d*(m + S(1))*(-S(4)*a*c + b**S(2))), x)
rule268 = ReplacementRule(pattern268, replacement268)
pattern269 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons47)
def replacement269(p, m, b, d, c, a, x, e):
rubi.append(269)
return Dist(S(1)/e, Subst(Int(x**m*(a - b**S(2)/(S(4)*c) + c*x**S(2)/e**S(2))**p, x), x, d + e*x), x)
rule269 = ReplacementRule(pattern269, replacement269)
pattern270 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons279, cons239, cons128)
def replacement270(p, m, b, d, a, c, x, e):
rubi.append(270)
return Int(ExpandIntegrand((d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x)
rule270 = ReplacementRule(pattern270, replacement270)
pattern271 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons280, cons128, cons281)
def replacement271(p, m, d, c, a, x, e):
rubi.append(271)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(d + e*x)**m, x), x)
rule271 = ReplacementRule(pattern271, replacement271)
def With272(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(272)
return Dist((c*d - e*(b/S(2) - q/S(2)))/q, Int(S(1)/(b/S(2) + c*x - q/S(2)), x), x) - Dist((c*d - e*(b/S(2) + q/S(2)))/q, Int(S(1)/(b/S(2) + c*x + q/S(2)), x), x)
pattern272 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))/(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons282)
rule272 = ReplacementRule(pattern272, With272)
def With273(d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(273)
return Dist(-c*d/(S(2)*q) + e/S(2), Int(S(1)/(c*x + q), x), x) + Dist(c*d/(S(2)*q) + e/S(2), Int(S(1)/(c*x - q), x), x)
pattern273 = Pattern(Integral((d_ + x_*WC('e', S(1)))/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons283)
rule273 = ReplacementRule(pattern273, With273)
pattern274 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))/(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons284)
def replacement274(b, d, c, a, x, e):
rubi.append(274)
return Dist(e/(S(2)*c), Int((b + S(2)*c*x)/(a + b*x + c*x**S(2)), x), x) + Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(S(1)/(a + b*x + c*x**S(2)), x), x)
rule274 = ReplacementRule(pattern274, replacement274)
pattern275 = Pattern(Integral((d_ + x_*WC('e', S(1)))/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons285)
def replacement275(d, c, a, x, e):
rubi.append(275)
return Dist(d, Int(S(1)/(a + c*x**S(2)), x), x) + Dist(e, Int(x/(a + c*x**S(2)), x), x)
rule275 = ReplacementRule(pattern275, replacement275)
pattern276 = Pattern(Integral(sqrt(x_*WC('e', S(1)) + WC('d', S(0)))/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239)
def replacement276(b, d, c, a, x, e):
rubi.append(276)
return Dist(S(2)*e, Subst(Int(x**S(2)/(a*e**S(2) - b*d*e + c*d**S(2) + c*x**S(4) - x**S(2)*(-b*e + S(2)*c*d)), x), x, sqrt(d + e*x)), x)
rule276 = ReplacementRule(pattern276, replacement276)
pattern277 = Pattern(Integral(sqrt(d_ + x_*WC('e', S(1)))/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons280)
def replacement277(d, c, a, x, e):
rubi.append(277)
return Dist(S(2)*e, Subst(Int(x**S(2)/(a*e**S(2) + c*d**S(2) - S(2)*c*d*x**S(2) + c*x**S(4)), x), x, sqrt(d + e*x)), x)
rule277 = ReplacementRule(pattern277, replacement277)
pattern278 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons17, cons166, cons286)
def replacement278(m, b, d, c, a, x, e):
rubi.append(278)
return Int(PolynomialDivide((d + e*x)**m, a + b*x + c*x**S(2), x), x)
rule278 = ReplacementRule(pattern278, replacement278)
pattern279 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons17, cons166, cons286)
def replacement279(m, d, c, a, x, e):
rubi.append(279)
return Int(PolynomialDivide((d + e*x)**m, a + c*x**S(2), x), x)
rule279 = ReplacementRule(pattern279, replacement279)
pattern280 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons31, cons166)
def replacement280(m, b, d, c, a, x, e):
rubi.append(280)
return Dist(S(1)/c, Int((d + e*x)**(m + S(-2))*Simp(-a*e**S(2) + c*d**S(2) + e*x*(-b*e + S(2)*c*d), x)/(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))/(c*(m + S(-1))), x)
rule280 = ReplacementRule(pattern280, replacement280)
pattern281 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons31, cons166)
def replacement281(m, d, c, a, x, e):
rubi.append(281)
return Dist(S(1)/c, Int((d + e*x)**(m + S(-2))*Simp(-a*e**S(2) + c*d**S(2) + S(2)*c*d*e*x, x)/(a + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))/(c*(m + S(-1))), x)
rule281 = ReplacementRule(pattern281, replacement281)
pattern282 = Pattern(Integral(S(1)/((x_*WC('e', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239)
def replacement282(b, d, c, a, x, e):
rubi.append(282)
return Dist(e**S(2)/(a*e**S(2) - b*d*e + c*d**S(2)), Int(S(1)/(d + e*x), x), x) + Dist(S(1)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((-b*e + c*d - c*e*x)/(a + b*x + c*x**S(2)), x), x)
rule282 = ReplacementRule(pattern282, replacement282)
pattern283 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)))*(d_ + x_*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons280)
def replacement283(d, c, a, x, e):
rubi.append(283)
return Dist(e**S(2)/(a*e**S(2) + c*d**S(2)), Int(S(1)/(d + e*x), x), x) + Dist(S(1)/(a*e**S(2) + c*d**S(2)), Int((c*d - c*e*x)/(a + c*x**S(2)), x), x)
rule283 = ReplacementRule(pattern283, replacement283)
pattern284 = Pattern(Integral(S(1)/(sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239)
def replacement284(b, d, c, a, x, e):
rubi.append(284)
return Dist(S(2)*e, Subst(Int(S(1)/(a*e**S(2) - b*d*e + c*d**S(2) + c*x**S(4) - x**S(2)*(-b*e + S(2)*c*d)), x), x, sqrt(d + e*x)), x)
rule284 = ReplacementRule(pattern284, replacement284)
pattern285 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(d_ + x_*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons280)
def replacement285(d, c, a, x, e):
rubi.append(285)
return Dist(S(2)*e, Subst(Int(S(1)/(a*e**S(2) + c*d**S(2) - S(2)*c*d*x**S(2) + c*x**S(4)), x), x, sqrt(d + e*x)), x)
rule285 = ReplacementRule(pattern285, replacement285)
pattern286 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons279, cons239, cons31, cons94)
def replacement286(m, b, d, c, a, x, e):
rubi.append(286)
return Dist(S(1)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((d + e*x)**(m + S(1))*Simp(-b*e + c*d - c*e*x, x)/(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(1))/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule286 = ReplacementRule(pattern286, replacement286)
pattern287 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons21, cons280, cons31, cons94)
def replacement287(m, d, c, a, x, e):
rubi.append(287)
return Dist(c/(a*e**S(2) + c*d**S(2)), Int((d - e*x)*(d + e*x)**(m + S(1))/(a + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(1))/((m + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule287 = ReplacementRule(pattern287, replacement287)
pattern288 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons279, cons239, cons18)
def replacement288(m, b, d, c, a, x, e):
rubi.append(288)
return Int(ExpandIntegrand((d + e*x)**m, S(1)/(a + b*x + c*x**S(2)), x), x)
rule288 = ReplacementRule(pattern288, replacement288)
pattern289 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons21, cons280, cons18)
def replacement289(m, d, c, a, x, e):
rubi.append(289)
return Int(ExpandIntegrand((d + e*x)**m, S(1)/(a + c*x**S(2)), x), x)
rule289 = ReplacementRule(pattern289, replacement289)
pattern290 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239)
def replacement290(b, d, c, a, x, e):
rubi.append(290)
return Simp(-S(2)*(-S(2)*a*e + b*d + x*(-b*e + S(2)*c*d))/((-S(4)*a*c + b**S(2))*sqrt(a + b*x + c*x**S(2))), x)
rule290 = ReplacementRule(pattern290, replacement290)
pattern291 = Pattern(Integral((d_ + x_*WC('e', S(1)))/(a_ + x_**S(2)*WC('c', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons48, cons280)
def replacement291(d, c, a, x, e):
rubi.append(291)
return Simp((-a*e + c*d*x)/(a*c*sqrt(a + c*x**S(2))), x)
rule291 = ReplacementRule(pattern291, replacement291)
pattern292 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons13, cons137, cons230)
def replacement292(p, b, d, c, a, x, e):
rubi.append(292)
return -Dist((S(2)*p + S(3))*(-b*e + S(2)*c*d)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(1))*(-S(2)*a*e + b*d + x*(-b*e + S(2)*c*d))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule292 = ReplacementRule(pattern292, replacement292)
pattern293 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons13, cons137, cons230)
def replacement293(p, d, c, a, x, e):
rubi.append(293)
return Dist(d*(S(2)*p + S(3))/(S(2)*a*(p + S(1))), Int((a + c*x**S(2))**(p + S(1)), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(a*e - c*d*x)/(S(2)*a*c*(p + S(1))), x)
rule293 = ReplacementRule(pattern293, replacement293)
pattern294 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons226, cons279, cons239, cons287)
def replacement294(p, b, d, c, a, x, e):
rubi.append(294)
return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int((a + b*x + c*x**S(2))**p, x), x) + Simp(e*(a + b*x + c*x**S(2))**(p + S(1))/(S(2)*c*(p + S(1))), x)
rule294 = ReplacementRule(pattern294, replacement294)
pattern295 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons5, cons280, cons287)
def replacement295(p, d, c, a, x, e):
rubi.append(295)
return Dist(d, Int((a + c*x**S(2))**p, x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))/(S(2)*c*(p + S(1))), x)
rule295 = ReplacementRule(pattern295, replacement295)
pattern296 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons288, cons289, cons290, cons147)
def replacement296(p, m, b, d, c, a, x, e):
rubi.append(296)
return Dist((d + e*x)**FracPart(p)*(a*d + c*e*x**S(3))**(-FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((d + e*x)**(m - p)*(a*d + c*e*x**S(3))**p, x), x)
rule296 = ReplacementRule(pattern296, replacement296)
pattern297 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons3, cons7, cons27, cons48, cons291, cons239, cons31, cons292, cons293, cons294)
def replacement297(m, b, d, c, x, e):
rubi.append(297)
return Int((d + e*x)**m/(sqrt(b*x)*sqrt(S(1) + c*x/b)), x)
rule297 = ReplacementRule(pattern297, replacement297)
pattern298 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons3, cons7, cons27, cons48, cons291, cons239, cons31, cons292)
def replacement298(m, b, d, c, x, e):
rubi.append(298)
return Dist(sqrt(x)*sqrt(b + c*x)/sqrt(b*x + c*x**S(2)), Int((d + e*x)**m/(sqrt(x)*sqrt(b + c*x)), x), x)
rule298 = ReplacementRule(pattern298, replacement298)
pattern299 = Pattern(Integral(x_**m_/sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons226, cons295)
def replacement299(m, b, c, a, x):
rubi.append(299)
return Dist(S(2), Subst(Int(x**(S(2)*m + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x, sqrt(x)), x)
rule299 = ReplacementRule(pattern299, replacement299)
pattern300 = Pattern(Integral((e_*x_)**m_/sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons48, cons226, cons295)
def replacement300(m, b, c, a, x, e):
rubi.append(300)
return Dist(x**(-m)*(e*x)**m, Int(x**m/sqrt(a + b*x + c*x**S(2)), x), x)
rule300 = ReplacementRule(pattern300, replacement300)
pattern301 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons295)
def replacement301(m, b, d, c, a, x, e):
rubi.append(301)
return Dist(S(2)*sqrt(-c*(a + b*x + c*x**S(2))/(-S(4)*a*c + b**S(2)))*(S(2)*c*(d + e*x)/(-b*e + S(2)*c*d - e*Rt(-S(4)*a*c + b**S(2), S(2))))**(-m)*(d + e*x)**m*Rt(-S(4)*a*c + b**S(2), S(2))/(c*sqrt(a + b*x + c*x**S(2))), Subst(Int((S(2)*e*x**S(2)*Rt(-S(4)*a*c + b**S(2), S(2))/(-b*e + S(2)*c*d - e*Rt(-S(4)*a*c + b**S(2), S(2))) + S(1))**m/sqrt(-x**S(2) + S(1)), x), x, sqrt(S(2))*sqrt((b + S(2)*c*x + Rt(-S(4)*a*c + b**S(2), S(2)))/Rt(-S(4)*a*c + b**S(2), S(2)))/S(2)), x)
rule301 = ReplacementRule(pattern301, replacement301)
pattern302 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_/sqrt(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons295)
def replacement302(m, d, c, a, x, e):
rubi.append(302)
return Dist(S(2)*a*(c*(d + e*x)/(-a*e*Rt(-c/a, S(2)) + c*d))**(-m)*sqrt(S(1) + c*x**S(2)/a)*(d + e*x)**m*Rt(-c/a, S(2))/(c*sqrt(a + c*x**S(2))), Subst(Int((S(2)*a*e*x**S(2)*Rt(-c/a, S(2))/(-a*e*Rt(-c/a, S(2)) + c*d) + S(1))**m/sqrt(-x**S(2) + S(1)), x), x, sqrt(-x*Rt(-c/a, S(2))/S(2) + S(1)/2)), x)
rule302 = ReplacementRule(pattern302, replacement302)
pattern303 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons244, cons296, cons163)
def replacement303(p, m, b, d, c, a, x, e):
rubi.append(303)
return Dist(p*(-S(4)*a*c + b**S(2))/(S(2)*(m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), Int((d + e*x)**(m + S(2))*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) - Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*(-S(2)*a*e + b*d + x*(-b*e + S(2)*c*d))/(S(2)*(m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule303 = ReplacementRule(pattern303, replacement303)
pattern304 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons280, cons244, cons296, cons163)
def replacement304(p, m, d, c, a, x, e):
rubi.append(304)
return -Dist(S(2)*a*c*p/((m + S(1))*(a*e**S(2) + c*d**S(2))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**(m + S(2)), x), x) - Simp((a + c*x**S(2))**p*(d + e*x)**(m + S(1))*(-S(2)*a*e + S(2)*c*d*x)/(S(2)*(m + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule304 = ReplacementRule(pattern304, replacement304)
pattern305 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons244, cons296, cons137)
def replacement305(p, m, b, d, c, a, x, e):
rubi.append(305)
return -Dist(S(2)*(S(2)*p + S(3))*(a*e**S(2) - b*d*e + c*d**S(2))/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(-2))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*(-S(2)*a*e + b*d + x*(-b*e + S(2)*c*d))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule305 = ReplacementRule(pattern305, replacement305)
pattern306 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons280, cons244, cons296, cons137)
def replacement306(p, m, d, c, a, x, e):
rubi.append(306)
return Dist((S(2)*p + S(3))*(a*e**S(2) + c*d**S(2))/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-2)), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(a*e - c*d*x)/(S(2)*a*c*(p + S(1))), x)
rule306 = ReplacementRule(pattern306, replacement306)
pattern307 = Pattern(Integral(S(1)/((x_*WC('e', S(1)) + WC('d', S(0)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons239)
def replacement307(b, d, c, a, x, e):
rubi.append(307)
return Dist(S(-2), Subst(Int(S(1)/(S(4)*a*e**S(2) - S(4)*b*d*e + S(4)*c*d**S(2) - x**S(2)), x), x, (S(2)*a*e - b*d - x*(-b*e + S(2)*c*d))/sqrt(a + b*x + c*x**S(2))), x)
rule307 = ReplacementRule(pattern307, replacement307)
pattern308 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('c', S(1)))*(d_ + x_*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons297)
def replacement308(d, c, a, x, e):
rubi.append(308)
return -Subst(Int(S(1)/(a*e**S(2) + c*d**S(2) - x**S(2)), x), x, (a*e - c*d*x)/sqrt(a + c*x**S(2)))
rule308 = ReplacementRule(pattern308, replacement308)
pattern309 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons279, cons239, cons147, cons240)
def replacement309(p, m, b, d, a, c, x, e):
rubi.append(309)
return -Simp(((b + S(2)*c*x + Rt(-S(4)*a*c + b**S(2), S(2)))*(-b*e + S(2)*c*d + e*Rt(-S(4)*a*c + b**S(2), S(2)))/((b + S(2)*c*x - Rt(-S(4)*a*c + b**S(2), S(2)))*(-b*e + S(2)*c*d - e*Rt(-S(4)*a*c + b**S(2), S(2)))))**(-p)*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*(b + S(2)*c*x - Rt(-S(4)*a*c + b**S(2), S(2)))*Hypergeometric2F1(m + S(1), -p, m + S(2), -S(4)*c*(d + e*x)*Rt(-S(4)*a*c + b**S(2), S(2))/((b + S(2)*c*x - Rt(-S(4)*a*c + b**S(2), S(2)))*(-b*e + S(2)*c*d - e*Rt(-S(4)*a*c + b**S(2), S(2)))))/((m + S(1))*(-b*e + S(2)*c*d + e*Rt(-S(4)*a*c + b**S(2), S(2)))), x)
rule309 = ReplacementRule(pattern309, replacement309)
pattern310 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons280, cons147, cons240)
def replacement310(p, m, d, c, a, x, e):
rubi.append(310)
return Simp(((c*d + e*Rt(-a*c, S(2)))*(c*x + Rt(-a*c, S(2)))/((c*d - e*Rt(-a*c, S(2)))*(c*x - Rt(-a*c, S(2)))))**(-p)*(a + c*x**S(2))**p*(d + e*x)**(m + S(1))*(-c*x + Rt(-a*c, S(2)))*Hypergeometric2F1(m + S(1), -p, m + S(2), S(2)*c*(d + e*x)*Rt(-a*c, S(2))/((c*d - e*Rt(-a*c, S(2)))*(-c*x + Rt(-a*c, S(2)))))/((m + S(1))*(c*d + e*Rt(-a*c, S(2)))), x)
rule310 = ReplacementRule(pattern310, replacement310)
pattern311 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons279, cons239, cons242, cons13, cons137)
def replacement311(p, m, b, d, c, a, x, e):
rubi.append(311)
return Dist(m*(-b*e + S(2)*c*d)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((b + S(2)*c*x)*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule311 = ReplacementRule(pattern311, replacement311)
pattern312 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons5, cons280, cons242, cons13, cons137)
def replacement312(p, m, d, c, a, x, e):
rubi.append(312)
return -Dist(d*m/(S(2)*a*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1)), x), x) - Simp(x*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(S(2)*a*(p + S(1))), x)
rule312 = ReplacementRule(pattern312, replacement312)
pattern313 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons279, cons239, cons242)
def replacement313(p, m, b, d, c, a, x, e):
rubi.append(313)
return Dist((-b*e + S(2)*c*d)/(S(2)*a*e**S(2) - S(2)*b*d*e + S(2)*c*d**S(2)), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) + Simp(e*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule313 = ReplacementRule(pattern313, replacement313)
pattern314 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons5, cons280, cons242)
def replacement314(p, m, d, c, a, x, e):
rubi.append(314)
return Dist(c*d/(a*e**S(2) + c*d**S(2)), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1)), x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(1))/((m + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule314 = ReplacementRule(pattern314, replacement314)
pattern315 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons279, cons239, cons13, cons163, cons298, cons66, cons299, cons300)
def replacement315(p, m, b, d, c, a, x, e):
rubi.append(315)
return -Dist(p/(e*(m + S(1))), Int((b + S(2)*c*x)*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(1))), x)
rule315 = ReplacementRule(pattern315, replacement315)
pattern316 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons280, cons13, cons163, cons298, cons66, cons299, cons301)
def replacement316(p, m, d, c, a, x, e):
rubi.append(316)
return -Dist(S(2)*c*p/(e*(m + S(1))), Int(x*(a + c*x**S(2))**(p + S(-1))*(d + e*x)**(m + S(1)), x), x) + Simp((a + c*x**S(2))**p*(d + e*x)**(m + S(1))/(e*(m + S(1))), x)
rule316 = ReplacementRule(pattern316, replacement316)
pattern317 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons279, cons239, cons13, cons163, cons238, cons302, cons303, cons300)
def replacement317(p, m, b, d, c, a, x, e):
rubi.append(317)
return -Dist(p/(e*(m + S(2)*p + S(1))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(-1))*Simp(-S(2)*a*e + b*d + x*(-b*e + S(2)*c*d), x), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(e*(m + S(2)*p + S(1))), x)
rule317 = ReplacementRule(pattern317, replacement317)
pattern318 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons280, cons13, cons163, cons238, cons302, cons303, cons301)
def replacement318(p, m, d, c, a, x, e):
rubi.append(318)
return Dist(S(2)*p/(e*(m + S(2)*p + S(1))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**m*Simp(a*e - c*d*x, x), x), x) + Simp((a + c*x**S(2))**p*(d + e*x)**(m + S(1))/(e*(m + S(2)*p + S(1))), x)
rule318 = ReplacementRule(pattern318, replacement318)
pattern319 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons244, cons137, cons168, cons304, cons300)
def replacement319(p, m, b, d, c, a, x, e):
rubi.append(319)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*(b*e*m + S(2)*c*d*(S(2)*p + S(3)) + S(2)*c*e*x*(m + S(2)*p + S(3))), x), x) + Simp((b + S(2)*c*x)*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule319 = ReplacementRule(pattern319, replacement319)
pattern320 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons280, cons244, cons137, cons168, cons304, cons301)
def replacement320(p, m, d, c, a, x, e):
rubi.append(320)
return Dist(S(1)/(S(2)*a*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(d*(S(2)*p + S(3)) + e*x*(m + S(2)*p + S(3))), x), x) - Simp(x*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(S(2)*a*(p + S(1))), x)
rule320 = ReplacementRule(pattern320, replacement320)
pattern321 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons239, cons244, cons137, cons166, cons300)
def replacement321(p, m, b, d, c, a, x, e):
rubi.append(321)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(-2))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(-S(2)*c*d**S(2)*(S(2)*p + S(3)) + e*x*(b*e - S(2)*c*d)*(m + S(2)*p + S(2)) + e*(S(2)*a*e*(m + S(-1)) + b*d*(-m + S(2)*p + S(4))), x), x), x) + Simp((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*(-S(2)*a*e + b*d + x*(-b*e + S(2)*c*d))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule321 = ReplacementRule(pattern321, replacement321)
pattern322 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons280, cons244, cons137, cons166, cons301)
def replacement322(p, m, d, c, a, x, e):
rubi.append(322)
return Dist(-S(1)/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-2))*Simp(a*e**S(2)*(m + S(-1)) - c*d**S(2)*(S(2)*p + S(3)) - c*d*e*x*(m + S(2)*p + S(2)), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(a*e - c*d*x)/(S(2)*a*c*(p + S(1))), x)
rule322 = ReplacementRule(pattern322, replacement322)
pattern323 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons226, cons279, cons239, cons13, cons137, cons300)
def replacement323(p, m, b, d, c, a, x, e):
rubi.append(323)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*Simp(-S(2)*a*c*e**S(2)*(m + S(2)*p + S(3)) + b**S(2)*e**S(2)*(m + p + S(2)) + b*c*d*e*(-m + S(2)*p + S(2)) - S(2)*c**S(2)*d**S(2)*(S(2)*p + S(3)) - c*e*x*(-b*e + S(2)*c*d)*(m + S(2)*p + S(4)), x), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(S(2)*a*c*e - b**S(2)*e + b*c*d + c*x*(-b*e + S(2)*c*d))/((p + S(1))*(-S(4)*a*c + b**S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule323 = ReplacementRule(pattern323, replacement323)
pattern324 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons280, cons13, cons137, cons301)
def replacement324(p, m, d, c, a, x, e):
rubi.append(324)
return Dist(S(1)/(S(2)*a*(p + S(1))*(a*e**S(2) + c*d**S(2))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**m*Simp(a*e**S(2)*(m + S(2)*p + S(3)) + c*d**S(2)*(S(2)*p + S(3)) + c*d*e*x*(m + S(2)*p + S(4)), x), x), x) - Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(1))*(a*e + c*d*x)/(S(2)*a*(p + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule324 = ReplacementRule(pattern324, replacement324)
pattern325 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons279, cons239, cons305, cons238, cons300)
def replacement325(p, m, b, d, c, a, x, e):
rubi.append(325)
return Dist(S(1)/(c*(m + S(2)*p + S(1))), Int((d + e*x)**(m + S(-2))*(a + b*x + c*x**S(2))**p*Simp(c*d**S(2)*(m + S(2)*p + S(1)) + e*x*(m + p)*(-b*e + S(2)*c*d) - e*(a*e*(m + S(-1)) + b*d*(p + S(1))), x), x), x) + Simp(e*(d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m + S(2)*p + S(1))), x)
rule325 = ReplacementRule(pattern325, replacement325)
pattern326 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons5, cons280, cons305, cons238, cons301)
def replacement326(p, m, d, c, a, x, e):
rubi.append(326)
return Dist(S(1)/(c*(m + S(2)*p + S(1))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(-2))*Simp(-a*e**S(2)*(m + S(-1)) + c*d**S(2)*(m + S(2)*p + S(1)) + S(2)*c*d*e*x*(m + p), x), x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))/(c*(m + S(2)*p + S(1))), x)
rule326 = ReplacementRule(pattern326, replacement326)
pattern327 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons279, cons239, cons306)
def replacement327(p, m, b, d, c, a, x, e):
rubi.append(327)
return Dist(S(1)/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*Simp(-b*e*(m + p + S(2)) + c*d*(m + S(1)) - c*e*x*(m + S(2)*p + S(3)), x), x), x) + Simp(e*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule327 = ReplacementRule(pattern327, replacement327)
pattern328 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_, x_), cons2, cons7, cons27, cons48, cons21, cons5, cons280, cons307)
def replacement328(p, m, d, c, a, x, e):
rubi.append(328)
return Dist(c/((m + S(1))*(a*e**S(2) + c*d**S(2))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1))*Simp(d*(m + S(1)) - e*x*(m + S(2)*p + S(3)), x), x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(1))/((m + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule328 = ReplacementRule(pattern328, replacement328)
def With329(b, d, c, a, x, e):
q = Rt(S(3)*c*e**S(2)*(-b*e + S(2)*c*d), S(3))
rubi.append(329)
return -Simp(S(3)*c*e*log(d + e*x)/(S(2)*q**S(2)), x) + Simp(S(3)*c*e*log(-b*e + c*d - c*e*x - q*(a + b*x + c*x**S(2))**(S(1)/3))/(S(2)*q**S(2)), x) - Simp(sqrt(S(3))*c*e*ArcTan(sqrt(S(3))/S(3) + S(2)*sqrt(S(3))*(-b*e + c*d - c*e*x)/(S(3)*q*(a + b*x + c*x**S(2))**(S(1)/3)))/q**S(2), x)
pattern329 = Pattern(Integral(S(1)/((x_*WC('e', S(1)) + WC('d', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**(S(1)/3)), x_), cons2, cons3, cons7, cons27, cons48, cons239, cons308, cons309)
rule329 = ReplacementRule(pattern329, With329)
def With330(d, c, a, x, e):
q = Rt(S(6)*c**S(2)*e**S(2)/d**S(2), S(3))
rubi.append(330)
return -Simp(S(3)*c*e*log(d + e*x)/(S(2)*d**S(2)*q**S(2)), x) + Simp(S(3)*c*e*log(c*d - c*e*x - d*q*(a + c*x**S(2))**(S(1)/3))/(S(2)*d**S(2)*q**S(2)), x) - Simp(sqrt(S(3))*c*e*ArcTan(S(2)*sqrt(S(3))*c*(d - e*x)/(S(3)*d*q*(a + c*x**S(2))**(S(1)/3)) + sqrt(S(3))/S(3))/(d**S(2)*q**S(2)), x)
pattern330 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)))**(S(1)/3)*(d_ + x_*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons310)
rule330 = ReplacementRule(pattern330, With330)
def With331(b, d, c, a, x, e):
q = Rt(-S(3)*c*e**S(2)*(-b*e + S(2)*c*d), S(3))
rubi.append(331)
return -Simp(S(3)*c*e*log(d + e*x)/(S(2)*q**S(2)), x) + Simp(S(3)*c*e*log(-b*e + c*d - c*e*x + q*(a + b*x + c*x**S(2))**(S(1)/3))/(S(2)*q**S(2)), x) - Simp(sqrt(S(3))*c*e*ArcTan(sqrt(S(3))/S(3) - S(2)*sqrt(S(3))*(-b*e + c*d - c*e*x)/(S(3)*q*(a + b*x + c*x**S(2))**(S(1)/3)))/q**S(2), x)
pattern331 = Pattern(Integral(S(1)/((x_*WC('e', S(1)) + WC('d', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**(S(1)/3)), x_), cons2, cons3, cons7, cons27, cons48, cons239, cons308, cons311)
rule331 = ReplacementRule(pattern331, With331)
def With332(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(332)
return Dist((b + S(2)*c*x - q)**(S(1)/3)*(b + S(2)*c*x + q)**(S(1)/3)/(a + b*x + c*x**S(2))**(S(1)/3), Int(S(1)/((d + e*x)*(b + S(2)*c*x - q)**(S(1)/3)*(b + S(2)*c*x + q)**(S(1)/3)), x), x)
pattern332 = Pattern(Integral(S(1)/((x_*WC('e', S(1)) + WC('d', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**(S(1)/3)), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons312)
rule332 = ReplacementRule(pattern332, With332)
pattern333 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)))**(S(1)/4)*(d_ + x_*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons280)
def replacement333(d, c, a, x, e):
rubi.append(333)
return Dist(d, Int(S(1)/((a + c*x**S(2))**(S(1)/4)*(d**S(2) - e**S(2)*x**S(2))), x), x) - Dist(e, Int(x/((a + c*x**S(2))**(S(1)/4)*(d**S(2) - e**S(2)*x**S(2))), x), x)
rule333 = ReplacementRule(pattern333, replacement333)
pattern334 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)))**(S(3)/4)*(d_ + x_*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons280)
def replacement334(d, c, a, x, e):
rubi.append(334)
return Dist(d, Int(S(1)/((a + c*x**S(2))**(S(3)/4)*(d**S(2) - e**S(2)*x**S(2))), x), x) - Dist(e, Int(x/((a + c*x**S(2))**(S(3)/4)*(d**S(2) - e**S(2)*x**S(2))), x), x)
rule334 = ReplacementRule(pattern334, replacement334)
pattern335 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons232, cons229)
def replacement335(p, b, d, c, a, x, e):
rubi.append(335)
return Dist((-S(4)*c/(-S(4)*a*c + b**S(2)))**(-p), Subst(Int(Simp(-x**S(2)/(-S(4)*a*c + b**S(2)) + S(1), x)**p/Simp(-b*e + S(2)*c*d + e*x, x), x), x, b + S(2)*c*x), x)
rule335 = ReplacementRule(pattern335, replacement335)
pattern336 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons313, cons229)
def replacement336(p, b, d, c, a, x, e):
rubi.append(336)
return Dist((-c*(a + b*x + c*x**S(2))/(-S(4)*a*c + b**S(2)))**(-p)*(a + b*x + c*x**S(2))**p, Int((-a*c/(-S(4)*a*c + b**S(2)) - b*c*x/(-S(4)*a*c + b**S(2)) - c**S(2)*x**S(2)/(-S(4)*a*c + b**S(2)))**p/(d + e*x), x), x)
rule336 = ReplacementRule(pattern336, replacement336)
pattern337 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons280, cons147, cons43, cons293)
def replacement337(p, m, d, c, a, x, e):
rubi.append(337)
return Int((d + e*x)**m*(-x*Rt(-c, S(2)) + Rt(a, S(2)))**p*(x*Rt(-c, S(2)) + Rt(a, S(2)))**p, x)
rule337 = ReplacementRule(pattern337, replacement337)
def With338(p, m, b, d, a, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(338)
return -Dist((e*(b + S(2)*c*x - q)/(S(2)*c*(d + e*x)))**(-p)*(e*(b + S(2)*c*x + q)/(S(2)*c*(d + e*x)))**(-p)*(a + b*x + c*x**S(2))**p*(S(1)/(d + e*x))**(S(2)*p)/e, Subst(Int(x**(-m - S(2)*p + S(-2))*Simp(-x*(d - e*(b - q)/(S(2)*c)) + S(1), x)**p*Simp(-x*(d - e*(b + q)/(S(2)*c)) + S(1), x)**p, x), x, S(1)/(d + e*x)), x)
pattern338 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons226, cons279, cons239, cons147, cons84)
rule338 = ReplacementRule(pattern338, With338)
def With339(p, m, d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(339)
return -Dist((e*(c*x + q)/(c*(d + e*x)))**(-p)*(-e*(-c*x + q)/(c*(d + e*x)))**(-p)*(a + c*x**S(2))**p*(S(1)/(d + e*x))**(S(2)*p)/e, Subst(Int(x**(-m - S(2)*p + S(-2))*Simp(-x*(d - e*q/c) + S(1), x)**p*Simp(-x*(d + e*q/c) + S(1), x)**p, x), x, S(1)/(d + e*x)), x)
pattern339 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons5, cons280, cons147, cons84)
rule339 = ReplacementRule(pattern339, With339)
def With340(p, m, b, d, a, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(340)
return Dist((-(d + e*x)/(d - e*(b - q)/(S(2)*c)) + S(1))**(-p)*(-(d + e*x)/(d - e*(b + q)/(S(2)*c)) + S(1))**(-p)*(a + b*x + c*x**S(2))**p/e, Subst(Int(x**m*Simp(-x/(d - e*(b - q)/(S(2)*c)) + S(1), x)**p*Simp(-x/(d - e*(b + q)/(S(2)*c)) + S(1), x)**p, x), x, d + e*x), x)
pattern340 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons226, cons279, cons239, cons147)
rule340 = ReplacementRule(pattern340, With340)
def With341(p, m, d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(341)
return Dist((a + c*x**S(2))**p*(-(d + e*x)/(d - e*q/c) + S(1))**(-p)*(-(d + e*x)/(d + e*q/c) + S(1))**(-p)/e, Subst(Int(x**m*Simp(-x/(d - e*q/c) + S(1), x)**p*Simp(-x/(d + e*q/c) + S(1), x)**p, x), x, d + e*x), x)
pattern341 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons280, cons147)
rule341 = ReplacementRule(pattern341, With341)
pattern342 = Pattern(Integral((u_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + u_**S(2)*WC('c', S(1)) + u_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons68, cons69)
def replacement342(p, u, m, b, d, c, a, x, e):
rubi.append(342)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x, u), x)
rule342 = ReplacementRule(pattern342, replacement342)
pattern343 = Pattern(Integral((a_ + u_**S(2)*WC('c', S(1)))**WC('p', S(1))*(u_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons68, cons69)
def replacement343(p, u, m, d, c, a, x, e):
rubi.append(343)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + c*x**S(2))**p*(d + e*x)**m, x), x, u), x)
rule343 = ReplacementRule(pattern343, replacement343)
pattern344 = Pattern(Integral(x_**WC('n', S(1))*(a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons5, cons85, cons314)
def replacement344(p, d, c, n, a, x, e):
rubi.append(344)
return Dist(d, Int(x**n*(a + c*x**S(2))**p, x), x) + Dist(e, Int(x**(n + S(1))*(a + c*x**S(2))**p, x), x)
rule344 = ReplacementRule(pattern344, replacement344)
pattern345 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))/sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons45, cons316)
def replacement345(m, g, b, f, d, c, a, x, e):
rubi.append(345)
return Dist((f + g*x)/sqrt(a + b*x + c*x**S(2)), Int((d + e*x)**m, x), x)
rule345 = ReplacementRule(pattern345, replacement345)
pattern346 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons45, cons316, cons147, cons242)
def replacement346(p, m, g, b, f, d, c, a, x, e):
rubi.append(346)
return -Simp(f*g*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(b*(p + S(1))*(-d*g + e*f)), x)
rule346 = ReplacementRule(pattern346, replacement346)
pattern347 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons45, cons316, cons147, cons244, cons137, cons168)
def replacement347(p, m, g, b, f, d, c, a, x, e):
rubi.append(347)
return -Dist(e*g*m/(S(2)*c*(p + S(1))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp(g*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/(S(2)*c*(p + S(1))), x)
rule347 = ReplacementRule(pattern347, replacement347)
pattern348 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons45, cons316, cons147, cons13, cons137, cons317)
def replacement348(p, m, g, b, f, d, c, a, x, e):
rubi.append(348)
return Dist(e*f*g*(m + S(2)*p + S(3))/(b*(p + S(1))*(-d*g + e*f)), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1)), x), x) - Simp(f*g*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(b*(p + S(1))*(-d*g + e*f)), x)
rule348 = ReplacementRule(pattern348, replacement348)
pattern349 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons45, cons316, cons147, cons31, cons94, cons225, cons318)
def replacement349(p, m, g, b, f, d, c, a, x, e):
rubi.append(349)
return -Dist(g*(S(2)*p + S(1))/(e*(m + S(1))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) + Simp((d + e*x)**(m + S(1))*(f + g*x)*(a + b*x + c*x**S(2))**p/(e*(m + S(1))), x)
rule349 = ReplacementRule(pattern349, replacement349)
pattern350 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons45, cons316, cons147, cons31, cons94, cons319)
def replacement350(p, m, g, b, f, d, c, a, x, e):
rubi.append(350)
return -Dist(g*(m + S(2)*p + S(3))/((m + S(1))*(-d*g + e*f)), Int((d + e*x)**(m + S(1))*(f + g*x)*(a + b*x + c*x**S(2))**p, x), x) + Simp(S(2)*f*g*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(b*(m + S(1))*(-d*g + e*f)), x)
rule350 = ReplacementRule(pattern350, replacement350)
pattern351 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons45, cons316, cons147, cons62, cons319, cons320)
def replacement351(p, m, g, b, f, d, c, a, x, e):
rubi.append(351)
return -Dist(b*m*(-d*g + e*f)/(S(2)*c*f*(m + S(2)*p + S(2))), Int((d + e*x)**(m + S(-1))*(f + g*x)*(a + b*x + c*x**S(2))**p, x), x) + Simp(g*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m + S(2)*p + S(2))), x)
rule351 = ReplacementRule(pattern351, replacement351)
pattern352 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons45, cons316, cons147, cons319)
def replacement352(p, m, g, b, f, d, c, a, x, e):
rubi.append(352)
return Dist((S(2)*p + S(1))*(-d*g + e*f)/(e*(m + S(2)*p + S(2))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x) + Simp((d + e*x)**(m + S(1))*(f + g*x)*(a + b*x + c*x**S(2))**p/(e*(m + S(2)*p + S(2))), x)
rule352 = ReplacementRule(pattern352, replacement352)
pattern353 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons45, cons321, cons147, cons239, cons13, cons322)
def replacement353(p, f, b, g, d, c, a, x, e):
rubi.append(353)
return Dist((-b*g + S(2)*c*f)/(-b*e + S(2)*c*d), Int((a + b*x + c*x**S(2))**p, x), x) - Dist((-d*g + e*f)/(-b*e + S(2)*c*d), Int((b + S(2)*c*x)*(a + b*x + c*x**S(2))**p/(d + e*x), x), x)
rule353 = ReplacementRule(pattern353, replacement353)
pattern354 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons45, cons321, cons147, cons323)
def replacement354(p, m, g, b, f, d, c, a, x, e):
rubi.append(354)
return Dist(g/e, Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) + Dist((-d*g + e*f)/e, Int((d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x)
rule354 = ReplacementRule(pattern354, replacement354)
pattern355 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons45, cons321, cons147, cons239, cons242)
def replacement355(p, m, f, g, b, d, c, a, x, e):
rubi.append(355)
return Dist((-b*g + S(2)*c*f)/(-b*e + S(2)*c*d), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) - Dist((-d*g + e*f)/(-b*e + S(2)*c*d), Int((b + S(2)*c*x)*(d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x)
rule355 = ReplacementRule(pattern355, replacement355)
pattern356 = Pattern(Integral((f_ + x_*WC('g', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons45, cons321, cons147, cons239, cons319, cons270, cons31, cons94)
def replacement356(p, m, g, b, f, d, c, a, x, e):
rubi.append(356)
return Dist((S(2)*c*e*f*(m + S(2)*p + S(2)) - g*(b*e*(m + S(1)) + S(2)*c*d*(S(2)*p + S(1))))/(e*(m + S(1))*(-b*e + S(2)*c*d)), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) - Simp((b + S(2)*c*x)*(d + e*x)**(m + S(1))*(-d*g + e*f)*(a + b*x + c*x**S(2))**p/(e*(m + S(1))*(-b*e + S(2)*c*d)), x)
rule356 = ReplacementRule(pattern356, replacement356)
pattern357 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons45, cons321, cons147, cons239, cons319, cons270, cons272, cons324)
def replacement357(p, m, f, g, b, d, c, a, x, e):
rubi.append(357)
return Dist((S(2)*c*e*f*(m + S(2)*p + S(2)) - g*(b*e*(m + S(1)) + S(2)*c*(S(2)*d*p + d)))/(S(2)*c*e*(m + S(2)*p + S(2))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x) + Simp(g*(b + S(2)*c*x)*(d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p/(S(2)*c*e*(m + S(2)*p + S(2))), x)
rule357 = ReplacementRule(pattern357, replacement357)
pattern358 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons315, cons45, cons147)
def replacement358(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(358)
return Dist(c**(-IntPart(p))*(b/S(2) + c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((b/S(2) + c*x)**(S(2)*p)*(d + e*x)**m*(f + g*x)**n, x), x)
rule358 = ReplacementRule(pattern358, replacement358)
pattern359 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons315, cons226, cons256, cons38)
def replacement359(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(359)
return Int((d + e*x)**(m + p)*(f + g*x)**n*(a/d + c*x/e)**p, x)
rule359 = ReplacementRule(pattern359, replacement359)
pattern360 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons315, cons257, cons325)
def replacement360(p, m, f, g, d, c, n, a, x, e):
rubi.append(360)
return Int((d + e*x)**(m + p)*(f + g*x)**n*(a/d + c*x/e)**p, x)
rule360 = ReplacementRule(pattern360, replacement360)
pattern361 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons256, cons84, cons314)
def replacement361(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(361)
return Dist(d**m*e**m, Int((f + g*x)**n*(a*e + c*d*x)**(-m)*(a + b*x + c*x**S(2))**(m + p), x), x)
rule361 = ReplacementRule(pattern361, replacement361)
pattern362 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons257, cons84, cons314)
def replacement362(p, m, f, g, d, c, n, a, x, e):
rubi.append(362)
return Dist(d**m*e**m, Int((a + c*x**S(2))**(m + p)*(f + g*x)**n*(a*e + c*d*x)**(-m), x), x)
rule362 = ReplacementRule(pattern362, replacement362)
pattern363 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons256, cons326)
def replacement363(p, m, f, g, b, d, a, c, x, e):
rubi.append(363)
return Simp(g*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m + S(2)*p + S(2))), x)
rule363 = ReplacementRule(pattern363, replacement363)
pattern364 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons257, cons327)
def replacement364(p, m, f, g, d, c, a, x, e):
rubi.append(364)
return Simp(g*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(c*(m + S(2)*p + S(2))), x)
rule364 = ReplacementRule(pattern364, replacement364)
pattern365 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons256, cons244, cons137, cons168)
def replacement365(p, m, f, g, b, d, a, c, x, e):
rubi.append(365)
return -Dist(e*(e*(p + S(1))*(-b*g + S(2)*c*f) + m*(c*e*f + g*(-b*e + c*d)))/(c*(p + S(1))*(-b*e + S(2)*c*d)), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((d + e*x)**m*(c*e*f + g*(-b*e + c*d))*(a + b*x + c*x**S(2))**(p + S(1))/(c*(p + S(1))*(-b*e + S(2)*c*d)), x)
rule365 = ReplacementRule(pattern365, replacement365)
pattern366 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons257, cons244, cons137, cons168)
def replacement366(p, m, f, g, d, c, a, x, e):
rubi.append(366)
return -Dist(e*(S(2)*e*f*(p + S(1)) + m*(d*g + e*f))/(S(2)*c*d*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1)), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**m*(d*g + e*f)/(S(2)*c*d*(p + S(1))), x)
rule366 = ReplacementRule(pattern366, replacement366)
pattern367 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons256, cons139, cons328, cons54)
def replacement367(p, m, f, g, b, d, a, c, x, e):
rubi.append(367)
return -Dist(e*(e*(p + S(1))*(-b*g + S(2)*c*f) + m*(c*e*f + g*(-b*e + c*d)))/(c*(p + S(1))*(-b*e + S(2)*c*d)), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((d + e*x)**m*(c*e*f + g*(-b*e + c*d))*(a + b*x + c*x**S(2))**(p + S(1))/(c*(p + S(1))*(-b*e + S(2)*c*d)), x)
rule367 = ReplacementRule(pattern367, replacement367)
pattern368 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons257, cons139, cons328, cons54)
def replacement368(p, m, f, g, d, c, a, x, e):
rubi.append(368)
return -Dist(e*(S(2)*e*f*(p + S(1)) + m*(d*g + e*f))/(S(2)*c*d*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1)), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**m*(d*g + e*f)/(S(2)*c*d*(p + S(1))), x)
rule368 = ReplacementRule(pattern368, replacement368)
pattern369 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons256, cons329, cons253)
def replacement369(p, m, f, g, b, d, a, c, x, e):
rubi.append(369)
return Dist((e*(p + S(1))*(-b*g + S(2)*c*f) + m*(c*e*f + g*(-b*e + c*d)))/(e*(-b*e + S(2)*c*d)*(m + p + S(1))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) + Simp((d + e*x)**m*(d*g - e*f)*(a + b*x + c*x**S(2))**(p + S(1))/((-b*e + S(2)*c*d)*(m + p + S(1))), x)
rule369 = ReplacementRule(pattern369, replacement369)
pattern370 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons257, cons329, cons253)
def replacement370(p, m, f, g, d, c, a, x, e):
rubi.append(370)
return Dist((S(2)*c*e*f*(p + S(1)) + m*(c*d*g + c*e*f))/(S(2)*c*d*e*(m + p + S(1))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1)), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**m*(d*g - e*f)/(S(2)*c*d*(m + p + S(1))), x)
rule370 = ReplacementRule(pattern370, replacement370)
pattern371 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons256, cons319)
def replacement371(p, m, f, g, b, d, a, c, x, e):
rubi.append(371)
return Dist((e*(p + S(1))*(-b*g + S(2)*c*f) + m*(c*e*f + g*(-b*e + c*d)))/(c*e*(m + S(2)*p + S(2))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x) + Simp(g*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m + S(2)*p + S(2))), x)
rule371 = ReplacementRule(pattern371, replacement371)
pattern372 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons257, cons319)
def replacement372(p, m, f, g, d, c, a, x, e):
rubi.append(372)
return Dist((S(2)*e*f*(p + S(1)) + m*(d*g + e*f))/(e*(m + S(2)*p + S(2))), Int((a + c*x**S(2))**p*(d + e*x)**m, x), x) + Simp(g*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(c*(m + S(2)*p + S(2))), x)
rule372 = ReplacementRule(pattern372, replacement372)
pattern373 = Pattern(Integral(x_**S(2)*(a_ + x_**S(2)*WC('c', S(1)))**p_*(f_ + x_*WC('g', S(1))), x_), cons2, cons7, cons125, cons208, cons330, cons13, cons331)
def replacement373(p, f, g, c, a, x):
rubi.append(373)
return -Dist(S(1)/(S(2)*a*c*(p + S(1))), Int(x*(a + c*x**S(2))**(p + S(1))*Simp(S(2)*a*g - c*f*x*(S(2)*p + S(5)), x), x), x) + Simp(x**S(2)*(a + c*x**S(2))**(p + S(1))*(a*g - c*f*x)/(S(2)*a*c*(p + S(1))), x)
rule373 = ReplacementRule(pattern373, replacement373)
pattern374 = Pattern(Integral(x_**S(2)*(a_ + x_**S(2)*WC('c', S(1)))**p_*(f_ + x_*WC('g', S(1))), x_), cons2, cons7, cons125, cons208, cons5, cons330)
def replacement374(p, f, g, c, a, x):
rubi.append(374)
return Dist(S(1)/c, Int((a + c*x**S(2))**(p + S(1))*(f + g*x), x), x) - Dist(f**S(2)/c, Int((a + c*x**S(2))**(p + S(1))/(f - g*x), x), x)
rule374 = ReplacementRule(pattern374, replacement374)
pattern375 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons256, cons147, cons150, cons13, cons332)
def replacement375(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(375)
return Int((f + g*x)**n*(a/d + c*x/e)**(-m)*(a + b*x + c*x**S(2))**(m + p), x)
rule375 = ReplacementRule(pattern375, replacement375)
pattern376 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons257, cons147, cons150, cons13, cons332)
def replacement376(p, m, f, g, d, c, n, a, x, e):
rubi.append(376)
return Dist(a**(-m)*d**(S(2)*m), Int((a + c*x**S(2))**(m + p)*(d - e*x)**(-m)*(f + g*x)**n, x), x)
rule376 = ReplacementRule(pattern376, replacement376)
pattern377 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons256, cons147, cons148, cons333)
def replacement377(p, f, g, b, d, a, n, c, x, e):
rubi.append(377)
return -Dist(S(1)/(d*e*p*(-S(4)*a*c + b**S(2))), Int((f + g*x)**(n + S(-1))*(a + b*x + c*x**S(2))**p*Simp(-S(2)*a*c*(d*g*n - e*f*(S(2)*p + S(1))) + b*(a*e*g*n - c*d*f*(S(2)*p + S(1))) - c*g*x*(-S(2)*a*e + b*d)*(n + S(2)*p + S(1)), x), x), x) - Simp((f + g*x)**n*(a*(-b*e + S(2)*c*d) + c*x*(-S(2)*a*e + b*d))*(a + b*x + c*x**S(2))**p/(d*e*p*(-S(4)*a*c + b**S(2))), x)
rule377 = ReplacementRule(pattern377, replacement377)
pattern378 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons257, cons147, cons148, cons333)
def replacement378(p, f, g, d, c, n, a, x, e):
rubi.append(378)
return -Dist(S(1)/(S(2)*d*e*p), Int((a + c*x**S(2))**p*(f + g*x)**(n + S(-1))*Simp(d*g*n - e*f*(S(2)*p + S(1)) - e*g*x*(n + S(2)*p + S(1)), x), x), x) + Simp((a + c*x**S(2))**p*(d - e*x)*(f + g*x)**n/(S(2)*d*e*p), x)
rule378 = ReplacementRule(pattern378, replacement378)
pattern379 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons256, cons147, cons196, cons333)
def replacement379(p, f, g, b, d, a, n, c, x, e):
rubi.append(379)
return -Dist(S(1)/(d*e*p*(-S(4)*a*c + b**S(2))*(a*g**S(2) - b*f*g + c*f**S(2))), Int((f + g*x)**n*(a + b*x + c*x**S(2))**p*Simp(S(2)*a*c*(a*e*g**S(2)*(n + S(2)*p + S(1)) + c*f*(-d*g*n + S(2)*e*f*p + e*f)) + b**S(2)*g*(-a*e*g*(n + p + S(1)) + c*d*f*p) + b*c*(a*g*(d*g*(n + S(1)) + e*f*(n - S(2)*p)) - c*d*f**S(2)*(S(2)*p + S(1))) + c*g*x*(S(2)*a*c*(d*g + e*f) - b*(a*e*g + c*d*f))*(n + S(2)*p + S(2)), x), x), x) - Simp((f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**p*(a*c*d*(-b*g + S(2)*c*f) - a*e*(S(2)*a*c*g - b**S(2)*g + b*c*f) + c*x*(-a*e*(-b*g + S(2)*c*f) + c*d*(-S(2)*a*g + b*f)))/(d*e*p*(-S(4)*a*c + b**S(2))*(a*g**S(2) - b*f*g + c*f**S(2))), x)
rule379 = ReplacementRule(pattern379, replacement379)
pattern380 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons257, cons147, cons196, cons333)
def replacement380(p, f, g, d, c, n, a, x, e):
rubi.append(380)
return Dist(S(1)/(S(2)*d*e*p*(a*g**S(2) + c*f**S(2))), Int((a + c*x**S(2))**p*(f + g*x)**n*Simp(a*e*g**S(2)*(n + S(2)*p + S(1)) - c*f*(d*g*n - e*(S(2)*f*p + f)) + c*g*x*(d*g + e*f)*(n + S(2)*p + S(2)), x), x), x) + Simp((a + c*x**S(2))**p*(f + g*x)**(n + S(1))*(-a*e*g + c*d*f - c*x*(d*g + e*f))/(S(2)*d*e*p*(a*g**S(2) + c*f**S(2))), x)
rule380 = ReplacementRule(pattern380, replacement380)
pattern381 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons315, cons226, cons256, cons147, cons41, cons334, cons335)
def replacement381(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(381)
return -Simp(e*(d + e*x)**(m + S(-1))*(f + g*x)**n*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m - n + S(-1))), x)
rule381 = ReplacementRule(pattern381, replacement381)
pattern382 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons315, cons257, cons147, cons41, cons336, cons335)
def replacement382(p, m, f, g, d, c, n, a, x, e):
rubi.append(382)
return -Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(f + g*x)**n/(c*(m - n + S(-1))), x)
rule382 = ReplacementRule(pattern382, replacement382)
pattern383 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons315, cons226, cons256, cons147, cons41, cons337)
def replacement383(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(383)
return -Simp(e**S(2)*(d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/((n + S(1))*(-b*e*g + c*d*g + c*e*f)), x)
rule383 = ReplacementRule(pattern383, replacement383)
pattern384 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons315, cons257, cons147, cons41, cons337)
def replacement384(p, m, f, g, d, c, n, a, x, e):
rubi.append(384)
return -Simp(e**S(2)*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))/(c*(n + S(1))*(d*g + e*f)), x)
rule384 = ReplacementRule(pattern384, replacement384)
pattern385 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons256, cons147, cons41, cons338, cons163, cons89, cons339)
def replacement385(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(385)
return Dist(c*m/(e*g*(n + S(1))), Int((d + e*x)**(m + S(1))*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) + Simp((d + e*x)**m*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**p/(g*(n + S(1))), x)
rule385 = ReplacementRule(pattern385, replacement385)
pattern386 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons257, cons147, cons41, cons338, cons163, cons89, cons339)
def replacement386(p, m, f, g, d, c, n, a, x, e):
rubi.append(386)
return Dist(c*m/(e*g*(n + S(1))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**(m + S(1))*(f + g*x)**(n + S(1)), x), x) + Simp((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**(n + S(1))/(g*(n + S(1))), x)
rule386 = ReplacementRule(pattern386, replacement386)
pattern387 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons315, cons226, cons256, cons147, cons41, cons338, cons163, cons335, cons340, cons341)
def replacement387(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(387)
return -Dist(m*(-b*e*g + c*d*g + c*e*f)/(e**S(2)*g*(m - n + S(-1))), Int((d + e*x)**(m + S(1))*(f + g*x)**n*(a + b*x + c*x**S(2))**(p + S(-1)), x), x) - Simp((d + e*x)**m*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**p/(g*(m - n + S(-1))), x)
rule387 = ReplacementRule(pattern387, replacement387)
pattern388 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons4, cons315, cons257, cons147, cons41, cons338, cons163, cons335, cons340, cons341)
def replacement388(p, m, f, g, d, c, n, a, x, e):
rubi.append(388)
return -Dist(c*m*(d*g + e*f)/(e**S(2)*g*(m - n + S(-1))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**(m + S(1))*(f + g*x)**n, x), x) - Simp((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**(n + S(1))/(g*(m - n + S(-1))), x)
rule388 = ReplacementRule(pattern388, replacement388)
pattern389 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons256, cons147, cons41, cons338, cons137, cons88)
def replacement389(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(389)
return -Dist(e*g*n/(c*(p + S(1))), Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-1))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*(f + g*x)**n*(a + b*x + c*x**S(2))**(p + S(1))/(c*(p + S(1))), x)
rule389 = ReplacementRule(pattern389, replacement389)
pattern390 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons257, cons147, cons41, cons338, cons137, cons88)
def replacement390(p, m, f, g, d, c, n, a, x, e):
rubi.append(390)
return -Dist(e*g*n/(c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-1)), x), x) + Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(f + g*x)**n/(c*(p + S(1))), x)
rule390 = ReplacementRule(pattern390, replacement390)
pattern391 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons315, cons226, cons256, cons147, cons41, cons338, cons137)
def replacement391(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(391)
return Dist(e**S(2)*g*(m - n + S(-2))/((p + S(1))*(-b*e*g + c*d*g + c*e*f)), Int((d + e*x)**(m + S(-1))*(f + g*x)**n*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp(e**S(2)*(d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/((p + S(1))*(-b*e*g + c*d*g + c*e*f)), x)
rule391 = ReplacementRule(pattern391, replacement391)
pattern392 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons4, cons315, cons257, cons147, cons41, cons338, cons137)
def replacement392(p, m, f, g, d, c, n, a, x, e):
rubi.append(392)
return Dist(e**S(2)*g*(m - n + S(-2))/(c*(p + S(1))*(d*g + e*f)), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(f + g*x)**n, x), x) + Simp(e**S(2)*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))/(c*(p + S(1))*(d*g + e*f)), x)
rule392 = ReplacementRule(pattern392, replacement392)
pattern393 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons256, cons147, cons41, cons87, cons88, cons335, cons342)
def replacement393(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(393)
return -Dist(n*(-b*e*g + c*d*g + c*e*f)/(c*e*(m - n + S(-1))), Int((d + e*x)**m*(f + g*x)**(n + S(-1))*(a + b*x + c*x**S(2))**p, x), x) - Simp(e*(d + e*x)**(m + S(-1))*(f + g*x)**n*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m - n + S(-1))), x)
rule393 = ReplacementRule(pattern393, replacement393)
pattern394 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons257, cons147, cons41, cons87, cons88, cons335, cons342)
def replacement394(p, m, f, g, d, c, n, a, x, e):
rubi.append(394)
return -Dist(n*(d*g + e*f)/(e*(m - n + S(-1))), Int((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**(n + S(-1)), x), x) - Simp(e*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(f + g*x)**n/(c*(m - n + S(-1))), x)
rule394 = ReplacementRule(pattern394, replacement394)
pattern395 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons256, cons147, cons41, cons87, cons89, cons246)
def replacement395(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(395)
return -Dist(c*e*(m - n + S(-2))/((n + S(1))*(-b*e*g + c*d*g + c*e*f)), Int((d + e*x)**m*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**p, x), x) - Simp(e**S(2)*(d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/((n + S(1))*(-b*e*g + c*d*g + c*e*f)), x)
rule395 = ReplacementRule(pattern395, replacement395)
pattern396 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons257, cons147, cons41, cons87, cons89, cons246)
def replacement396(p, m, f, g, d, c, n, a, x, e):
rubi.append(396)
return -Dist(e*(m - n + S(-2))/((n + S(1))*(d*g + e*f)), Int((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**(n + S(1)), x), x) - Simp(e**S(2)*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))/((n + S(1))*(c*d*g + c*e*f)), x)
rule396 = ReplacementRule(pattern396, replacement396)
pattern397 = Pattern(Integral(sqrt(d_ + x_*WC('e', S(1)))/((x_*WC('g', S(1)) + WC('f', S(0)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons256)
def replacement397(f, b, g, d, a, c, x, e):
rubi.append(397)
return Dist(S(2)*e**S(2), Subst(Int(S(1)/(-b*e*g + c*(d*g + e*f) + e**S(2)*g*x**S(2)), x), x, sqrt(a + b*x + c*x**S(2))/sqrt(d + e*x)), x)
rule397 = ReplacementRule(pattern397, replacement397)
pattern398 = Pattern(Integral(sqrt(d_ + x_*WC('e', S(1)))/(sqrt(a_ + x_**S(2)*WC('c', S(1)))*(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons257)
def replacement398(f, g, d, c, a, x, e):
rubi.append(398)
return Dist(S(2)*e**S(2), Subst(Int(S(1)/(c*(d*g + e*f) + e**S(2)*g*x**S(2)), x), x, sqrt(a + c*x**S(2))/sqrt(d + e*x)), x)
rule398 = ReplacementRule(pattern398, replacement398)
pattern399 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons315, cons226, cons256, cons147, cons343, cons344, cons126)
def replacement399(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(399)
return Simp(e**S(2)*(d + e*x)**(m + S(-2))*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*g*(n + p + S(2))), x)
rule399 = ReplacementRule(pattern399, replacement399)
pattern400 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons315, cons257, cons147, cons343, cons345, cons126)
def replacement400(p, m, f, g, d, c, n, a, x, e):
rubi.append(400)
return Simp(e**S(2)*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-2))*(f + g*x)**(n + S(1))/(c*g*(n + p + S(2))), x)
rule400 = ReplacementRule(pattern400, replacement400)
pattern401 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons256, cons147, cons343, cons87, cons89, cons246)
def replacement401(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(401)
return -Dist(e*(b*e*g*(n + S(1)) - c*d*g*(S(2)*n + p + S(3)) + c*e*f*(p + S(1)))/(g*(n + S(1))*(-b*e*g + c*d*g + c*e*f)), Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**p, x), x) + Simp(e**S(2)*(d + e*x)**(m + S(-2))*(f + g*x)**(n + S(1))*(-d*g + e*f)*(a + b*x + c*x**S(2))**(p + S(1))/(g*(n + S(1))*(-b*e*g + c*d*g + c*e*f)), x)
rule401 = ReplacementRule(pattern401, replacement401)
pattern402 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons257, cons147, cons343, cons87, cons89, cons246)
def replacement402(p, m, f, g, d, c, n, a, x, e):
rubi.append(402)
return -Dist(e*(-d*g*(S(2)*n + p + S(3)) + e*f*(p + S(1)))/(g*(n + S(1))*(d*g + e*f)), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1)), x), x) + Simp(e**S(2)*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-2))*(f + g*x)**(n + S(1))*(-d*g + e*f)/(c*g*(n + S(1))*(d*g + e*f)), x)
rule402 = ReplacementRule(pattern402, replacement402)
pattern403 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons315, cons226, cons256, cons147, cons343, cons346, cons246)
def replacement403(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(403)
return -Dist((b*e*g*(n + S(1)) - c*d*g*(S(2)*n + p + S(3)) + c*e*f*(p + S(1)))/(c*g*(n + p + S(2))), Int((d + e*x)**(m + S(-1))*(f + g*x)**n*(a + b*x + c*x**S(2))**p, x), x) + Simp(e**S(2)*(d + e*x)**(m + S(-2))*(f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*g*(n + p + S(2))), x)
rule403 = ReplacementRule(pattern403, replacement403)
pattern404 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons315, cons257, cons147, cons343, cons346, cons246)
def replacement404(p, m, f, g, d, c, n, a, x, e):
rubi.append(404)
return -Dist((-d*g*(S(2)*n + p + S(3)) + e*f*(p + S(1)))/(g*(n + p + S(2))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(-1))*(f + g*x)**n, x), x) + Simp(e**S(2)*(a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-2))*(f + g*x)**(n + S(1))/(c*g*(n + p + S(2))), x)
rule404 = ReplacementRule(pattern404, replacement404)
pattern405 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons315, cons226, cons256, cons147, cons207)
def replacement405(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(405)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)**n*(a + b*x + c*x**S(2))**p, x), x)
rule405 = ReplacementRule(pattern405, replacement405)
pattern406 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons315, cons257, cons347, cons150, cons348, cons349)
def replacement406(p, m, f, g, d, c, n, a, x, e):
rubi.append(406)
return Int(ExpandIntegrand(S(1)/sqrt(a + c*x**S(2)), (a + c*x**S(2))**(p + S(1)/2)*(d + e*x)**m*(f + g*x)**n, x), x)
rule406 = ReplacementRule(pattern406, replacement406)
pattern407 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons315, cons257, cons147, cons207)
def replacement407(p, m, f, g, d, c, n, a, x, e):
rubi.append(407)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**n, x), x)
rule407 = ReplacementRule(pattern407, replacement407)
pattern408 = Pattern(Integral(x_**S(2)*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons226, cons256)
def replacement408(p, b, d, c, a, x, e):
rubi.append(408)
return -Dist(e**(S(-2)), Int((d - e*x)*(a + b*x + c*x**S(2))**p, x), x) + Dist(d**S(2)/e**S(2), Int((a + b*x + c*x**S(2))**p/(d + e*x), x), x)
rule408 = ReplacementRule(pattern408, replacement408)
pattern409 = Pattern(Integral(x_**S(2)*(a_ + x_**S(2)*WC('c', S(1)))**p_/(d_ + x_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons5, cons257)
def replacement409(p, d, c, a, x, e):
rubi.append(409)
return -Dist(e**(S(-2)), Int((a + c*x**S(2))**p*(d - e*x), x), x) + Dist(d**S(2)/e**S(2), Int((a + c*x**S(2))**p/(d + e*x), x), x)
rule409 = ReplacementRule(pattern409, replacement409)
pattern410 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**S(2)*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons256, cons147, cons270)
def replacement410(p, m, f, b, g, d, a, c, x, e):
rubi.append(410)
return -Dist(S(1)/(c*e**S(2)*(m + S(2)*p + S(3))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**p*Simp(b*e*g*(d*g + e*f*(m + p + S(1))) - c*(d**S(2)*g**S(2) + d*e*f*g*m + e**S(2)*f**S(2)*(m + S(2)*p + S(3))) + e*g*x*(b*e*g*(m + p + S(2)) - c*(d*g*m + e*f*(m + S(2)*p + S(4)))), x), x), x) + Simp(g*(d + e*x)**m*(f + g*x)*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m + S(2)*p + S(3))), x)
rule410 = ReplacementRule(pattern410, replacement410)
pattern411 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**S(2), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons257, cons147, cons270)
def replacement411(p, m, f, g, d, c, a, x, e):
rubi.append(411)
return -Dist(S(1)/(c*e**S(2)*(m + S(2)*p + S(3))), Int((a + c*x**S(2))**p*(d + e*x)**m*Simp(-c*e*g*x*(d*g*m + e*f*(m + S(2)*p + S(4))) - c*(d**S(2)*g**S(2) + d*e*f*g*m + e**S(2)*f**S(2)*(m + S(2)*p + S(3))), x), x), x) + Simp(g*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m*(f + g*x)/(c*(m + S(2)*p + S(3))), x)
rule411 = ReplacementRule(pattern411, replacement411)
pattern412 = Pattern(Integral((x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons3, cons7, cons48, cons125, cons208, cons21, cons4, cons147)
def replacement412(p, m, f, b, g, c, n, x, e):
rubi.append(412)
return Dist(x**(-m - p)*(e*x)**m*(b + c*x)**(-p)*(b*x + c*x**S(2))**p, Int(x**(m + p)*(b + c*x)**p*(f + g*x)**n, x), x)
rule412 = ReplacementRule(pattern412, replacement412)
pattern413 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons315, cons257, cons147, cons43, cons268)
def replacement413(p, m, f, g, d, c, n, a, x, e):
rubi.append(413)
return Int((d + e*x)**(m + p)*(f + g*x)**n*(a/d + c*x/e)**p, x)
rule413 = ReplacementRule(pattern413, replacement413)
pattern414 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons315, cons226, cons256, cons147)
def replacement414(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(414)
return Dist((d + e*x)**(-FracPart(p))*(a/d + c*x/e)**(-FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((d + e*x)**(m + p)*(f + g*x)**n*(a/d + c*x/e)**p, x), x)
rule414 = ReplacementRule(pattern414, replacement414)
pattern415 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons315, cons257, cons147)
def replacement415(p, m, f, g, d, c, n, a, x, e):
rubi.append(415)
return Dist((a + c*x**S(2))**FracPart(p)*(d + e*x)**(-FracPart(p))*(a/d + c*x/e)**(-FracPart(p)), Int((d + e*x)**(m + p)*(f + g*x)**n*(a/d + c*x/e)**p, x), x)
rule415 = ReplacementRule(pattern415, replacement415)
pattern416 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons226, cons279, cons128)
def replacement416(p, m, f, g, b, d, a, c, x, e):
rubi.append(416)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)*(a + b*x + c*x**S(2))**p, x), x)
rule416 = ReplacementRule(pattern416, replacement416)
pattern417 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons280, cons128)
def replacement417(p, m, f, g, d, c, a, x, e):
rubi.append(417)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x), x), x)
rule417 = ReplacementRule(pattern417, replacement417)
pattern418 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))/((x_*WC('e', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279)
def replacement418(f, g, b, d, a, c, x, e):
rubi.append(418)
return Dist(e*(-d*g + e*f)/(a*e**S(2) - b*d*e + c*d**S(2)), Int(S(1)/(d + e*x), x), x) + Dist(S(1)/(a*e**S(2) - b*d*e + c*d**S(2)), Int(Simp(a*e*g - b*e*f + c*d*f - c*x*(-d*g + e*f), x)/(a + b*x + c*x**S(2)), x), x)
rule418 = ReplacementRule(pattern418, replacement418)
pattern419 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))/((a_ + x_**S(2)*WC('c', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280)
def replacement419(f, g, d, c, a, x, e):
rubi.append(419)
return Dist(e*(-d*g + e*f)/(a*e**S(2) + c*d**S(2)), Int(S(1)/(d + e*x), x), x) + Dist(S(1)/(a*e**S(2) + c*d**S(2)), Int(Simp(a*e*g + c*d*f - c*x*(-d*g + e*f), x)/(a + c*x**S(2)), x), x)
rule419 = ReplacementRule(pattern419, replacement419)
pattern420 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons279, cons242, cons350)
def replacement420(p, m, f, g, b, d, a, c, x, e):
rubi.append(420)
return -Simp((d + e*x)**(m + S(1))*(-d*g + e*f)*(a + b*x + c*x**S(2))**(p + S(1))/(S(2)*(p + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule420 = ReplacementRule(pattern420, replacement420)
pattern421 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons280, cons242, cons351)
def replacement421(p, m, f, g, d, c, a, x, e):
rubi.append(421)
return -Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(1))*(-d*g + e*f)/(S(2)*(p + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule421 = ReplacementRule(pattern421, replacement421)
pattern422 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons242, cons13, cons137, cons352)
def replacement422(p, m, f, g, b, d, a, c, x, e):
rubi.append(422)
return -Dist(m*(-S(2)*a*e*g + b*(d*g + e*f) - S(2)*c*d*f)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1)), x), x) + Simp((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*(-S(2)*a*g + b*f + x*(-b*g + S(2)*c*f))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule422 = ReplacementRule(pattern422, replacement422)
pattern423 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons242, cons13, cons137, cons353)
def replacement423(p, m, f, g, d, c, a, x, e):
rubi.append(423)
return -Dist(m*(a*e*g + c*d*f)/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1)), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**m*(a*g - c*f*x)/(S(2)*a*c*(p + S(1))), x)
rule423 = ReplacementRule(pattern423, replacement423)
pattern424 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons279, cons242)
def replacement424(p, m, f, g, b, d, a, c, x, e):
rubi.append(424)
return -Dist((-S(2)*a*e*g + b*(d*g + e*f) - S(2)*c*d*f)/(S(2)*a*e**S(2) - S(2)*b*d*e + S(2)*c*d**S(2)), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) - Simp((d + e*x)**(m + S(1))*(-d*g + e*f)*(a + b*x + c*x**S(2))**(p + S(1))/(S(2)*(p + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule424 = ReplacementRule(pattern424, replacement424)
pattern425 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons280, cons242)
def replacement425(p, m, f, g, d, c, a, x, e):
rubi.append(425)
return Dist((a*e*g + c*d*f)/(a*e**S(2) + c*d**S(2)), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1)), x), x) - Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(1))*(-d*g + e*f)/(S(2)*(p + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule425 = ReplacementRule(pattern425, replacement425)
pattern426 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons226, cons279, cons354)
def replacement426(p, f, g, b, d, a, c, x, e):
rubi.append(426)
return -Simp((a + b*x + c*x**S(2))**(p + S(1))*(b*e*g*(p + S(2)) - S(2)*c*e*g*x*(p + S(1)) - c*(S(2)*p + S(3))*(d*g + e*f))/(S(2)*c**S(2)*(p + S(1))*(S(2)*p + S(3))), x)
rule426 = ReplacementRule(pattern426, replacement426)
pattern427 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('e', S(1)) + WC('d', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons280, cons355)
def replacement427(p, f, g, d, c, a, x, e):
rubi.append(427)
return Simp((a + c*x**S(2))**(p + S(1))*(S(2)*e*g*x*(p + S(1)) + (S(2)*p + S(3))*(d*g + e*f))/(S(2)*c*(p + S(1))*(S(2)*p + S(3))), x)
rule427 = ReplacementRule(pattern427, replacement427)
pattern428 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons13, cons137)
def replacement428(p, f, g, b, d, a, c, x, e):
rubi.append(428)
return -Dist((-S(2)*a*c*e*g + b**S(2)*e*g*(p + S(2)) + c*(S(2)*p + S(3))*(-b*(d*g + e*f) + S(2)*c*d*f))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1)), x), x) - Simp((a + b*x + c*x**S(2))**(p + S(1))*(S(2)*a*c*(d*g + e*f) - b*(a*e*g + c*d*f) - x*(b**S(2)*e*g - b*c*(d*g + e*f) + S(2)*c*(-a*e*g + c*d*f)))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule428 = ReplacementRule(pattern428, replacement428)
pattern429 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('e', S(1)) + WC('d', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons13, cons137)
def replacement429(p, f, g, d, c, a, x, e):
rubi.append(429)
return -Dist((a*e*g - c*d*f*(S(2)*p + S(3)))/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1)), x), x) - Simp((a + c*x**S(2))**(p + S(1))*(-a*(d*g + e*(f + g*x)) + c*d*f*x)/(S(2)*a*c*(p + S(1))), x)
rule429 = ReplacementRule(pattern429, replacement429)
pattern430 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons226, cons279, cons287)
def replacement430(p, f, g, b, d, a, c, x, e):
rubi.append(430)
return Dist((-S(2)*a*c*e*g + b**S(2)*e*g*(p + S(2)) + c*(S(2)*p + S(3))*(-b*(d*g + e*f) + S(2)*c*d*f))/(S(2)*c**S(2)*(S(2)*p + S(3))), Int((a + b*x + c*x**S(2))**p, x), x) - Simp((a + b*x + c*x**S(2))**(p + S(1))*(b*e*g*(p + S(2)) - S(2)*c*e*g*x*(p + S(1)) - c*(S(2)*p + S(3))*(d*g + e*f))/(S(2)*c**S(2)*(p + S(1))*(S(2)*p + S(3))), x)
rule430 = ReplacementRule(pattern430, replacement430)
pattern431 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('e', S(1)) + WC('d', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons280, cons287)
def replacement431(p, f, g, d, c, a, x, e):
rubi.append(431)
return -Dist((a*e*g - c*d*f*(S(2)*p + S(3)))/(c*(S(2)*p + S(3))), Int((a + c*x**S(2))**p, x), x) + Simp((a + c*x**S(2))**(p + S(1))*(S(2)*e*g*x*(p + S(1)) + (S(2)*p + S(3))*(d*g + e*f))/(S(2)*c*(p + S(1))*(S(2)*p + S(3))), x)
rule431 = ReplacementRule(pattern431, replacement431)
pattern432 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**S(2)*WC('c', S(1)))**p_*(f_ + x_*WC('g', S(1))), x_), cons2, cons7, cons48, cons125, cons208, cons5, cons356, cons357)
def replacement432(p, m, g, f, c, a, x, e):
rubi.append(432)
return Dist(f, Int((e*x)**m*(a + c*x**S(2))**p, x), x) + Dist(g/e, Int((e*x)**(m + S(1))*(a + c*x**S(2))**p, x), x)
rule432 = ReplacementRule(pattern432, replacement432)
pattern433 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons358, cons288, cons289)
def replacement433(p, m, f, b, g, d, c, a, x, e):
rubi.append(433)
return Dist((d + e*x)**FracPart(p)*(a*d + c*e*x**S(3))**(-FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((f + g*x)*(a*d + c*e*x**S(3))**p, x), x)
rule433 = ReplacementRule(pattern433, replacement433)
pattern434 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons244, cons163, cons247, cons359, cons360)
def replacement434(p, m, f, g, b, d, a, c, x, e):
rubi.append(434)
return -Dist(p/(e**S(2)*(m + S(1))*(m + S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), Int((d + e*x)**(m + S(2))*(a + b*x + c*x**S(2))**(p + S(-1))*Simp(S(2)*a*c*e*(m + S(2))*(-d*g + e*f) + b**S(2)*e*(d*g*(p + S(1)) - e*f*(m + p + S(2))) + b*(a*e**S(2)*g*(m + S(1)) - c*d*(d*g*(S(2)*p + S(1)) - e*f*(m + S(2)*p + S(2)))) - c*x*(S(2)*c*d*(d*g*(S(2)*p + S(1)) - e*f*(m + S(2)*p + S(2))) - e*(S(2)*a*e*g*(m + S(1)) - b*(d*g*(m - S(2)*p) + e*f*(m + S(2)*p + S(2))))), x), x), x) - Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*(-d*p*(-b*e + S(2)*c*d)*(-d*g + e*f) - e*x*(g*(m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2)) + p*(-b*e + S(2)*c*d)*(-d*g + e*f)) + (d*g - e*f*(m + S(2)))*(a*e**S(2) - b*d*e + c*d**S(2)))/(e**S(2)*(m + S(1))*(m + S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule434 = ReplacementRule(pattern434, replacement434)
pattern435 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons244, cons163, cons247, cons359, cons360)
def replacement435(p, m, f, g, d, c, a, x, e):
rubi.append(435)
return -Dist(p/(e**S(2)*(m + S(1))*(m + S(2))*(a*e**S(2) + c*d**S(2))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**(m + S(2))*Simp(S(2)*a*c*e*(m + S(2))*(-d*g + e*f) - c*x*(-S(2)*a*e**S(2)*g*(m + S(1)) + S(2)*c*d*(d*g*(S(2)*p + S(1)) - e*f*(m + S(2)*p + S(2)))), x), x), x) - Simp((a + c*x**S(2))**p*(d + e*x)**(m + S(1))*(-S(2)*c*d**S(2)*p*(-d*g + e*f) - e*x*(S(2)*c*d*p*(-d*g + e*f) + g*(m + S(1))*(a*e**S(2) + c*d**S(2))) + (a*e**S(2) + c*d**S(2))*(d*g - e*f*(m + S(2))))/(e**S(2)*(m + S(1))*(m + S(2))*(a*e**S(2) + c*d**S(2))), x)
rule435 = ReplacementRule(pattern435, replacement435)
pattern436 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons226, cons279, cons13, cons163, cons361, cons66, cons299, cons362)
def replacement436(p, m, f, g, b, d, a, c, x, e):
rubi.append(436)
return Dist(p/(e**S(2)*(m + S(1))*(m + S(2)*p + S(2))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(-1))*Simp(-b*e*f*(m + S(2)*p + S(2)) + g*(S(2)*a*e*m + S(2)*a*e + S(2)*b*d*p + b*d) + x*(-S(2)*c*e*f*(m + S(2)*p + S(2)) + g*(b*e*m + b*e + S(4)*c*d*p + S(2)*c*d)), x), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*(-d*g*(S(2)*p + S(1)) + e*f*(m + S(2)*p + S(2)) + e*g*x*(m + S(1)))/(e**S(2)*(m + S(1))*(m + S(2)*p + S(2))), x)
rule436 = ReplacementRule(pattern436, replacement436)
pattern437 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons280, cons13, cons163, cons361, cons66, cons299, cons362)
def replacement437(p, m, f, g, d, c, a, x, e):
rubi.append(437)
return Dist(p/(e**S(2)*(m + S(1))*(m + S(2)*p + S(2))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**(m + S(1))*Simp(g*(S(2)*a*e*m + S(2)*a*e) + x*(-S(2)*c*e*f*(m + S(2)*p + S(2)) + g*(S(4)*c*d*p + S(2)*c*d)), x), x), x) + Simp((a + c*x**S(2))**p*(d + e*x)**(m + S(1))*(-d*g*(S(2)*p + S(1)) + e*f*(m + S(2)*p + S(2)) + e*g*x*(m + S(1)))/(e**S(2)*(m + S(1))*(m + S(2)*p + S(2))), x)
rule437 = ReplacementRule(pattern437, replacement437)
pattern438 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons226, cons279, cons13, cons163, cons363, cons303, cons362)
def replacement438(p, m, f, g, b, d, a, c, x, e):
rubi.append(438)
return -Dist(p/(c*e**S(2)*(m + S(2)*p + S(1))*(m + S(2)*p + S(2))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(-1))*Simp(c*e*f*(-S(2)*a*e + b*d)*(m + S(2)*p + S(2)) + g*(a*e*(b*e*m + b*e - S(2)*c*d*m) + b*d*(b*e*p - S(2)*c*d*p - c*d)) + x*(c*e*f*(-b*e + S(2)*c*d)*(m + S(2)*p + S(2)) + g*(b**S(2)*e**S(2)*(m + p + S(1)) - S(2)*c**S(2)*d**S(2)*(S(2)*p + S(1)) - c*e*(S(2)*a*e*(m + S(2)*p + S(1)) + b*d*(m - S(2)*p)))), x), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*(c*e*f*(m + S(2)*p + S(2)) + c*e*g*x*(m + S(2)*p + S(1)) - g*(-b*e*p + S(2)*c*d*p + c*d))/(c*e**S(2)*(m + S(2)*p + S(1))*(m + S(2)*p + S(2))), x)
rule438 = ReplacementRule(pattern438, replacement438)
pattern439 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons280, cons13, cons163, cons363, cons303, cons362)
def replacement439(p, m, f, g, d, c, a, x, e):
rubi.append(439)
return Dist(S(2)*p/(c*e**S(2)*(m + S(2)*p + S(1))*(m + S(2)*p + S(2))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x)**m*Simp(a*c*d*e*g*m + a*c*e**S(2)*f*(m + S(2)*p + S(2)) - x*(c**S(2)*d*e*f*(m + S(2)*p + S(2)) - g*(a*c*e**S(2)*(m + S(2)*p + S(1)) + c**S(2)*d**S(2)*(S(2)*p + S(1)))), x), x), x) + Simp((a + c*x**S(2))**p*(d + e*x)**(m + S(1))*(-c*d*g*(S(2)*p + S(1)) + c*e*f*(m + S(2)*p + S(2)) + c*e*g*x*(m + S(2)*p + S(1)))/(c*e**S(2)*(m + S(2)*p + S(1))*(m + S(2)*p + S(2))), x)
rule439 = ReplacementRule(pattern439, replacement439)
pattern440 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons244, cons137, cons166, cons364)
def replacement440(p, m, f, g, b, d, a, c, x, e):
rubi.append(440)
return -Dist(S(1)/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(-2))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(b*e*g*(a*e*(m + S(-1)) + b*d*(p + S(2))) + S(2)*c**S(2)*d**S(2)*f*(S(2)*p + S(3)) - c*(S(2)*a*e*(d*g*m + e*f*(m + S(-1))) + b*d*(d*g*(S(2)*p + S(3)) - e*f*(m - S(2)*p + S(-4)))) + e*x*(b**S(2)*e*g*(m + p + S(1)) + S(2)*c**S(2)*d*f*(m + S(2)*p + S(2)) - c*(S(2)*a*e*g*m + b*(d*g + e*f)*(m + S(2)*p + S(2)))), x), x), x) - Simp((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*(S(2)*a*c*(d*g + e*f) - b*(a*e*g + c*d*f) - x*(b**S(2)*e*g + S(2)*c**S(2)*d*f - c*(S(2)*a*e*g + b*d*g + b*e*f)))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule440 = ReplacementRule(pattern440, replacement440)
pattern441 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons244, cons137, cons166, cons365)
def replacement441(p, m, f, g, d, c, a, x, e):
rubi.append(441)
return -Dist(S(1)/(S(4)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-2))*Simp(S(2)*a*e*(d*g*m + e*f*(m + S(-1))) - S(2)*c*d**S(2)*f*(S(2)*p + S(3)) + e*x*(S(2)*a*e*g*m - S(2)*c*d*f*(m + S(2)*p + S(2))), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*(S(2)*a*(d*g + e*f) - x*(-S(2)*a*e*g + S(2)*c*d*f))/(S(4)*a*c*(p + S(1))), x)
rule441 = ReplacementRule(pattern441, replacement441)
pattern442 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons244, cons137, cons168, cons366, cons362)
def replacement442(p, m, f, g, b, d, a, c, x, e):
rubi.append(442)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(-e*x*(-b*g + S(2)*c*f)*(m + S(2)*p + S(3)) - f*(b*e*m + S(2)*c*d*(S(2)*p + S(3))) + g*(S(2)*a*e*m + b*d*(S(2)*p + S(3))), x), x), x) + Simp((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*(-S(2)*a*g + b*f + x*(-b*g + S(2)*c*f))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule442 = ReplacementRule(pattern442, replacement442)
pattern443 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons244, cons137, cons168, cons366, cons362)
def replacement443(p, m, f, g, d, c, a, x, e):
rubi.append(443)
return -Dist(S(1)/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(-1))*Simp(a*e*g*m - c*d*f*(S(2)*p + S(3)) - c*e*f*x*(m + S(2)*p + S(3)), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**m*(a*g - c*f*x)/(S(2)*a*c*(p + S(1))), x)
rule443 = ReplacementRule(pattern443, replacement443)
pattern444 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons226, cons279, cons13, cons137, cons362)
def replacement444(p, m, f, g, b, d, a, c, x, e):
rubi.append(444)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), Int((d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*Simp(c*e*x*(-f*(-b*e + S(2)*c*d) + g*(-S(2)*a*e + b*d))*(m + S(2)*p + S(4)) + f*(-S(2)*a*c*e**S(2)*(m + S(2)*p + S(3)) + b**S(2)*e**S(2)*(m + p + S(2)) + b*c*d*e*(-m + S(2)*p + S(2)) - S(2)*c**S(2)*d**S(2)*(S(2)*p + S(3))) - g*(a*e*(b*e*m + b*e - S(2)*c*d*m) - b*d*(-b*e*p - b*e + S(2)*c*d*p + S(3)*c*d)), x), x), x) + Simp((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(-a*g*(-b*e + S(2)*c*d) + c*x*(f*(-b*e + S(2)*c*d) - g*(-S(2)*a*e + b*d)) + f*(S(2)*a*c*e - b**S(2)*e + b*c*d))/((p + S(1))*(-S(4)*a*c + b**S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule444 = ReplacementRule(pattern444, replacement444)
pattern445 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons13, cons137, cons362)
def replacement445(p, m, f, g, d, c, a, x, e):
rubi.append(445)
return Dist(S(1)/(S(2)*a*c*(p + S(1))*(a*e**S(2) + c*d**S(2))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x)**m*Simp(-a*c*d*e*g*m + c*e*x*(a*e*g + c*d*f)*(m + S(2)*p + S(4)) + f*(a*c*e**S(2)*(m + S(2)*p + S(3)) + c**S(2)*d**S(2)*(S(2)*p + S(3))), x), x), x) - Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(1))*(-a*c*d*g + a*c*e*f + c*x*(a*e*g + c*d*f))/(S(2)*a*c*(p + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule445 = ReplacementRule(pattern445, replacement445)
pattern446 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons17)
def replacement446(m, f, g, b, d, a, c, x, e):
rubi.append(446)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)/(a + b*x + c*x**S(2)), x), x)
rule446 = ReplacementRule(pattern446, replacement446)
pattern447 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons17)
def replacement447(m, f, g, d, c, a, x, e):
rubi.append(447)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)/(a + c*x**S(2)), x), x)
rule447 = ReplacementRule(pattern447, replacement447)
pattern448 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons367, cons168)
def replacement448(m, f, g, b, d, a, c, x, e):
rubi.append(448)
return Dist(S(1)/c, Int((d + e*x)**(m + S(-1))*Simp(-a*e*g + c*d*f + x*(-b*e*g + c*d*g + c*e*f), x)/(a + b*x + c*x**S(2)), x), x) + Simp(g*(d + e*x)**m/(c*m), x)
rule448 = ReplacementRule(pattern448, replacement448)
pattern449 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons367, cons168)
def replacement449(m, f, g, d, c, a, x, e):
rubi.append(449)
return Dist(S(1)/c, Int((d + e*x)**(m + S(-1))*Simp(-a*e*g + c*d*f + x*(c*d*g + c*e*f), x)/(a + c*x**S(2)), x), x) + Simp(g*(d + e*x)**m/(c*m), x)
rule449 = ReplacementRule(pattern449, replacement449)
pattern450 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))/(sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279)
def replacement450(f, g, b, d, a, c, x, e):
rubi.append(450)
return Dist(S(2), Subst(Int((-d*g + e*f + g*x**S(2))/(a*e**S(2) - b*d*e + c*d**S(2) + c*x**S(4) - x**S(2)*(-b*e + S(2)*c*d)), x), x, sqrt(d + e*x)), x)
rule450 = ReplacementRule(pattern450, replacement450)
pattern451 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280)
def replacement451(f, g, d, c, a, x, e):
rubi.append(451)
return Dist(S(2), Subst(Int((-d*g + e*f + g*x**S(2))/(a*e**S(2) + c*d**S(2) - S(2)*c*d*x**S(2) + c*x**S(4)), x), x, sqrt(d + e*x)), x)
rule451 = ReplacementRule(pattern451, replacement451)
pattern452 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons226, cons279, cons367, cons94)
def replacement452(m, f, g, b, d, a, c, x, e):
rubi.append(452)
return Dist(S(1)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((d + e*x)**(m + S(1))*Simp(a*e*g - b*e*f + c*d*f - c*x*(-d*g + e*f), x)/(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*(-d*g + e*f)/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule452 = ReplacementRule(pattern452, replacement452)
pattern453 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons315, cons280, cons367, cons94)
def replacement453(m, f, g, d, c, a, x, e):
rubi.append(453)
return Dist(S(1)/(a*e**S(2) + c*d**S(2)), Int((d + e*x)**(m + S(1))*Simp(a*e*g + c*d*f - c*x*(-d*g + e*f), x)/(a + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*(-d*g + e*f)/((m + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule453 = ReplacementRule(pattern453, replacement453)
pattern454 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons356)
def replacement454(m, f, g, b, d, a, c, x, e):
rubi.append(454)
return Int(ExpandIntegrand((d + e*x)**m, (f + g*x)/(a + b*x + c*x**S(2)), x), x)
rule454 = ReplacementRule(pattern454, replacement454)
pattern455 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons356)
def replacement455(m, f, g, d, c, a, x, e):
rubi.append(455)
return Int(ExpandIntegrand((d + e*x)**m, (f + g*x)/(a + c*x**S(2)), x), x)
rule455 = ReplacementRule(pattern455, replacement455)
pattern456 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons226, cons279, cons31, cons168, cons319, cons368, cons362)
def replacement456(p, m, f, g, b, d, a, c, x, e):
rubi.append(456)
return Dist(S(1)/(c*(m + S(2)*p + S(2))), Int((d + e*x)**(m + S(-1))*(a + b*x + c*x**S(2))**p*Simp(d*(p + S(1))*(-b*g + S(2)*c*f) + m*(-a*e*g + c*d*f) + x*(e*(p + S(1))*(-b*g + S(2)*c*f) + m*(-b*e*g + c*d*g + c*e*f)), x), x), x) + Simp(g*(d + e*x)**m*(a + b*x + c*x**S(2))**(p + S(1))/(c*(m + S(2)*p + S(2))), x)
rule456 = ReplacementRule(pattern456, replacement456)
pattern457 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons280, cons31, cons168, cons319, cons368, cons362)
def replacement457(p, m, f, g, d, c, a, x, e):
rubi.append(457)
return Dist(S(1)/(c*(m + S(2)*p + S(2))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(-1))*Simp(-a*e*g*m + c*d*f*(m + S(2)*p + S(2)) + c*x*(d*g*m + e*f*(m + S(2)*p + S(2))), x), x), x) + Simp(g*(a + c*x**S(2))**(p + S(1))*(d + e*x)**m/(c*(m + S(2)*p + S(2))), x)
rule457 = ReplacementRule(pattern457, replacement457)
pattern458 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons226, cons279, cons31, cons94, cons362)
def replacement458(p, m, f, g, b, d, a, c, x, e):
rubi.append(458)
return Dist(S(1)/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*Simp(b*(p + S(1))*(d*g - e*f) - c*x*(-d*g + e*f)*(m + S(2)*p + S(3)) + (m + S(1))*(a*e*g - b*e*f + c*d*f), x), x), x) + Simp((d + e*x)**(m + S(1))*(-d*g + e*f)*(a + b*x + c*x**S(2))**(p + S(1))/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule458 = ReplacementRule(pattern458, replacement458)
pattern459 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons5, cons315, cons280, cons31, cons94, cons362)
def replacement459(p, m, f, g, d, c, a, x, e):
rubi.append(459)
return Dist(S(1)/((m + S(1))*(a*e**S(2) + c*d**S(2))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1))*Simp(-c*x*(-d*g + e*f)*(m + S(2)*p + S(3)) + (m + S(1))*(a*e*g + c*d*f), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(1))*(-d*g + e*f)/((m + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule459 = ReplacementRule(pattern459, replacement459)
pattern460 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons279, cons369, cons66)
def replacement460(p, m, f, g, b, d, a, c, x, e):
rubi.append(460)
return Dist(S(1)/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*Simp(b*(p + S(1))*(d*g - e*f) - c*x*(-d*g + e*f)*(m + S(2)*p + S(3)) + (m + S(1))*(a*e*g - b*e*f + c*d*f), x), x), x) + Simp((d + e*x)**(m + S(1))*(-d*g + e*f)*(a + b*x + c*x**S(2))**(p + S(1))/((m + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule460 = ReplacementRule(pattern460, replacement460)
pattern461 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons280, cons369, cons66)
def replacement461(p, m, f, g, d, c, a, x, e):
rubi.append(461)
return Dist(S(1)/((m + S(1))*(a*e**S(2) + c*d**S(2))), Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1))*Simp(-c*x*(-d*g + e*f)*(m + S(2)*p + S(3)) + (m + S(1))*(a*e*g + c*d*f), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(d + e*x)**(m + S(1))*(-d*g + e*f)/((m + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule461 = ReplacementRule(pattern461, replacement461)
pattern462 = Pattern(Integral((f_ + x_*WC('g', S(1)))/((x_*WC('e', S(1)) + WC('d', S(0)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons370, cons371, cons372)
def replacement462(g, b, f, d, a, c, x, e):
rubi.append(462)
return Dist(S(4)*f*(a - d)/(-a*e + b*d), Subst(Int(S(1)/(S(4)*a - S(4)*d - x**S(2)), x), x, (S(2)*a - S(2)*d + x*(b - e))/sqrt(a + b*x + c*x**S(2))), x)
rule462 = ReplacementRule(pattern462, replacement462)
pattern463 = Pattern(Integral((f_ + x_*WC('g', S(1)))/(sqrt(x_)*sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons125, cons208, cons226)
def replacement463(f, g, b, c, a, x):
rubi.append(463)
return Dist(S(2), Subst(Int((f + g*x**S(2))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x, sqrt(x)), x)
rule463 = ReplacementRule(pattern463, replacement463)
pattern464 = Pattern(Integral((f_ + x_*WC('g', S(1)))/(sqrt(x_)*sqrt(a_ + x_**S(2)*WC('c', S(1)))), x_), cons2, cons7, cons125, cons208, cons373)
def replacement464(f, g, c, a, x):
rubi.append(464)
return Dist(S(2), Subst(Int((f + g*x**S(2))/sqrt(a + c*x**S(4)), x), x, sqrt(x)), x)
rule464 = ReplacementRule(pattern464, replacement464)
pattern465 = Pattern(Integral((f_ + x_*WC('g', S(1)))/(sqrt(e_*x_)*sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons48, cons125, cons208, cons226)
def replacement465(g, b, f, c, a, x, e):
rubi.append(465)
return Dist(sqrt(x)/sqrt(e*x), Int((f + g*x)/(sqrt(x)*sqrt(a + b*x + c*x**S(2))), x), x)
rule465 = ReplacementRule(pattern465, replacement465)
pattern466 = Pattern(Integral((f_ + x_*WC('g', S(1)))/(sqrt(e_*x_)*sqrt(a_ + x_**S(2)*WC('c', S(1)))), x_), cons2, cons7, cons48, cons125, cons208, cons374)
def replacement466(f, g, c, a, x, e):
rubi.append(466)
return Dist(sqrt(x)/sqrt(e*x), Int((f + g*x)/(sqrt(x)*sqrt(a + c*x**S(2))), x), x)
rule466 = ReplacementRule(pattern466, replacement466)
pattern467 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons279)
def replacement467(p, m, f, g, b, d, a, c, x, e):
rubi.append(467)
return Dist(g/e, Int((d + e*x)**(m + S(1))*(a + b*x + c*x**S(2))**p, x), x) + Dist((-d*g + e*f)/e, Int((d + e*x)**m*(a + b*x + c*x**S(2))**p, x), x)
rule467 = ReplacementRule(pattern467, replacement467)
pattern468 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons280)
def replacement468(p, m, f, g, d, c, a, x, e):
rubi.append(468)
return Dist(g/e, Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1)), x), x) + Dist((-d*g + e*f)/e, Int((a + c*x**S(2))**p*(d + e*x)**m, x), x)
rule468 = ReplacementRule(pattern468, replacement468)
pattern469 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons226, cons279, cons375)
def replacement469(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(469)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)**n*(a + b*x + c*x**S(2))**p, x), x)
rule469 = ReplacementRule(pattern469, replacement469)
pattern470 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons280, cons375)
def replacement470(p, m, f, g, d, c, n, a, x, e):
rubi.append(470)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**n, x), x)
rule470 = ReplacementRule(pattern470, replacement470)
pattern471 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_/((x_*WC('e', S(1)) + WC('d', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons149, cons163)
def replacement471(p, f, g, b, d, a, c, x, e):
rubi.append(471)
return -Dist(S(1)/(e*(-d*g + e*f)), Int((a + b*x + c*x**S(2))**(p + S(-1))*Simp(a*e*g - b*e*f + c*d*f - c*x*(-d*g + e*f), x)/(f + g*x), x), x) + Dist((a*e**S(2) - b*d*e + c*d**S(2))/(e*(-d*g + e*f)), Int((a + b*x + c*x**S(2))**(p + S(-1))/(d + e*x), x), x)
rule471 = ReplacementRule(pattern471, replacement471)
pattern472 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_/((x_*WC('e', S(1)) + WC('d', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons149, cons163)
def replacement472(p, f, g, d, c, a, x, e):
rubi.append(472)
return -Dist(S(1)/(e*(-d*g + e*f)), Int((a + c*x**S(2))**(p + S(-1))*Simp(a*e*g + c*d*f - c*x*(-d*g + e*f), x)/(f + g*x), x), x) + Dist((a*e**S(2) + c*d**S(2))/(e*(-d*g + e*f)), Int((a + c*x**S(2))**(p + S(-1))/(d + e*x), x), x)
rule472 = ReplacementRule(pattern472, replacement472)
def With473(p, m, f, g, b, d, a, c, n, x, e):
q = Denominator(m)
rubi.append(473)
return Dist(q/e, Subst(Int(x**(q*(m + S(1)) + S(-1))*(g*x**q/e + (-d*g + e*f)/e)**n*(c*x**(S(2)*q)/e**S(2) - x**q*(-b*e + S(2)*c*d)/e**S(2) + (a*e**S(2) - b*d*e + c*d**S(2))/e**S(2))**p, x), x, (d + e*x)**(S(1)/q)), x)
pattern473 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons376, cons367)
rule473 = ReplacementRule(pattern473, With473)
def With474(p, m, f, g, d, c, n, a, x, e):
q = Denominator(m)
rubi.append(474)
return Dist(q/e, Subst(Int(x**(q*(m + S(1)) + S(-1))*(g*x**q/e + (-d*g + e*f)/e)**n*(-S(2)*c*d*x**q/e**S(2) + c*x**(S(2)*q)/e**S(2) + (a*e**S(2) + c*d**S(2))/e**S(2))**p, x), x, (d + e*x)**(S(1)/q)), x)
pattern474 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons376, cons367)
rule474 = ReplacementRule(pattern474, With474)
pattern475 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(f_ + x_*WC('g', S(1)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons124, cons336, cons377)
def replacement475(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(475)
return Int((d*f + e*g*x**S(2))**m*(a + b*x + c*x**S(2))**p, x)
rule475 = ReplacementRule(pattern475, replacement475)
pattern476 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(f_ + x_*WC('g', S(1)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons124, cons336, cons377)
def replacement476(p, m, g, f, d, a, n, c, x, e):
rubi.append(476)
return Int((a + c*x**S(2))**p*(d*f + e*g*x**S(2))**m, x)
rule476 = ReplacementRule(pattern476, replacement476)
pattern477 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(f_ + x_*WC('g', S(1)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons124, cons336)
def replacement477(p, m, g, b, f, d, c, a, n, x, e):
rubi.append(477)
return Dist((d + e*x)**FracPart(m)*(f + g*x)**FracPart(m)*(d*f + e*g*x**S(2))**(-FracPart(m)), Int((d*f + e*g*x**S(2))**m*(a + b*x + c*x**S(2))**p, x), x)
rule477 = ReplacementRule(pattern477, replacement477)
pattern478 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(f_ + x_*WC('g', S(1)))**n_*(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons124, cons336)
def replacement478(p, m, g, f, d, a, c, n, x, e):
rubi.append(478)
return Dist((d + e*x)**FracPart(m)*(f + g*x)**FracPart(m)*(d*f + e*g*x**S(2))**(-FracPart(m)), Int((a + c*x**S(2))**p*(d*f + e*g*x**S(2))**m, x), x)
rule478 = ReplacementRule(pattern478, replacement478)
pattern479 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons226, cons279, cons93)
def replacement479(m, f, g, b, d, a, c, n, x, e):
rubi.append(479)
return Dist(c, Int(x**S(2)*(d + e*x)**m*(f + g*x)**n, x), x) + Int((a + b*x)*(d + e*x)**m*(f + g*x)**n, x)
rule479 = ReplacementRule(pattern479, replacement479)
pattern480 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons280, cons93)
def replacement480(m, f, g, d, c, n, a, x, e):
rubi.append(480)
return Dist(a, Int((d + e*x)**m*(f + g*x)**n, x), x) + Dist(c, Int(x**S(2)*(d + e*x)**m*(f + g*x)**n, x), x)
rule480 = ReplacementRule(pattern480, replacement480)
pattern481 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons226, cons279, cons18, cons23, cons93, cons168, cons165)
def replacement481(m, f, g, b, d, a, c, n, x, e):
rubi.append(481)
return Dist(c**(S(-2)), Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-2))*Simp(a*b*e*g**S(2) - a*c*d*g**S(2) - S(2)*a*c*e*f*g + c**S(2)*d*f**S(2) + x*(-a*c*e*g**S(2) + b**S(2)*e*g**S(2) - b*c*d*g**S(2) - S(2)*b*c*e*f*g + S(2)*c**S(2)*d*f*g + c**S(2)*e*f**S(2)), x)/(a + b*x + c*x**S(2)), x), x) + Dist(g/c**S(2), Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-2))*Simp(-b*e*g + c*d*g + S(2)*c*e*f + c*e*g*x, x), x), x)
rule481 = ReplacementRule(pattern481, replacement481)
pattern482 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons280, cons18, cons23, cons93, cons168, cons165)
def replacement482(m, f, g, d, c, n, a, x, e):
rubi.append(482)
return Dist(S(1)/c, Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-2))*Simp(-a*d*g**S(2) - S(2)*a*e*f*g + c*d*f**S(2) + x*(-a*e*g**S(2) + S(2)*c*d*f*g + c*e*f**S(2)), x)/(a + c*x**S(2)), x), x) + Dist(g/c, Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-2))*Simp(d*g + S(2)*e*f + e*g*x, x), x), x)
rule482 = ReplacementRule(pattern482, replacement482)
pattern483 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons226, cons279, cons18, cons23, cons93, cons168, cons88)
def replacement483(m, f, g, b, d, a, c, n, x, e):
rubi.append(483)
return Dist(S(1)/c, Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-1))*Simp(-a*e*g + c*d*f + x*(-b*e*g + c*d*g + c*e*f), x)/(a + b*x + c*x**S(2)), x), x) + Dist(e*g/c, Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-1)), x), x)
rule483 = ReplacementRule(pattern483, replacement483)
pattern484 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons280, cons18, cons23, cons93, cons168, cons88)
def replacement484(m, f, g, d, c, n, a, x, e):
rubi.append(484)
return Dist(S(1)/c, Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-1))*Simp(-a*e*g + c*d*f + x*(c*d*g + c*e*f), x)/(a + c*x**S(2)), x), x) + Dist(e*g/c, Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(-1)), x), x)
rule484 = ReplacementRule(pattern484, replacement484)
pattern485 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons226, cons279, cons18, cons23, cons93, cons168, cons89)
def replacement485(m, f, g, b, d, a, c, n, x, e):
rubi.append(485)
return -Dist(g*(-d*g + e*f)/(a*g**S(2) - b*f*g + c*f**S(2)), Int((d + e*x)**(m + S(-1))*(f + g*x)**n, x), x) + Dist(S(1)/(a*g**S(2) - b*f*g + c*f**S(2)), Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))*Simp(a*e*g - b*d*g + c*d*f + c*x*(-d*g + e*f), x)/(a + b*x + c*x**S(2)), x), x)
rule485 = ReplacementRule(pattern485, replacement485)
pattern486 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons280, cons18, cons23, cons93, cons168, cons89)
def replacement486(m, f, g, d, c, n, a, x, e):
rubi.append(486)
return -Dist(g*(-d*g + e*f)/(a*g**S(2) + c*f**S(2)), Int((d + e*x)**(m + S(-1))*(f + g*x)**n, x), x) + Dist(S(1)/(a*g**S(2) + c*f**S(2)), Int((d + e*x)**(m + S(-1))*(f + g*x)**(n + S(1))*Simp(a*e*g + c*d*f + c*x*(-d*g + e*f), x)/(a + c*x**S(2)), x), x)
rule486 = ReplacementRule(pattern486, replacement486)
pattern487 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/(sqrt(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons226, cons279, cons73)
def replacement487(m, f, g, b, d, a, c, x, e):
rubi.append(487)
return Int(ExpandIntegrand(S(1)/(sqrt(d + e*x)*sqrt(f + g*x)), (d + e*x)**(m + S(1)/2)/(a + b*x + c*x**S(2)), x), x)
rule487 = ReplacementRule(pattern487, replacement487)
pattern488 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_/(sqrt(x_*WC('g', S(1)) + WC('f', S(0)))*(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons280, cons73)
def replacement488(m, f, g, d, a, c, x, e):
rubi.append(488)
return Int(ExpandIntegrand(S(1)/(sqrt(d + e*x)*sqrt(f + g*x)), (d + e*x)**(m + S(1)/2)/(a + c*x**S(2)), x), x)
rule488 = ReplacementRule(pattern488, replacement488)
pattern489 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons226, cons279, cons18, cons23)
def replacement489(m, f, g, b, d, a, c, n, x, e):
rubi.append(489)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)**n, S(1)/(a + b*x + c*x**S(2)), x), x)
rule489 = ReplacementRule(pattern489, replacement489)
pattern490 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(a_ + x_**S(2)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons280, cons18, cons23)
def replacement490(m, f, g, d, c, n, a, x, e):
rubi.append(490)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)**n, S(1)/(a + c*x**S(2)), x), x)
rule490 = ReplacementRule(pattern490, replacement490)
pattern491 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons378)
def replacement491(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(491)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)**n*(a + b*x + c*x**S(2))**p, x), x)
rule491 = ReplacementRule(pattern491, replacement491)
pattern492 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons378)
def replacement492(p, m, f, g, d, c, n, a, x, e):
rubi.append(492)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**n, x), x)
rule492 = ReplacementRule(pattern492, replacement492)
pattern493 = Pattern(Integral((x_*WC('g', S(1)))**WC('n', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons21, cons4, cons5, cons358, cons288, cons289)
def replacement493(p, m, g, b, d, c, n, a, x, e):
rubi.append(493)
return Dist((d + e*x)**FracPart(p)*(a*d + c*e*x**S(3))**(-FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((g*x)**n*(a*d + c*e*x**S(3))**p, x), x)
rule493 = ReplacementRule(pattern493, replacement493)
pattern494 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons23, cons147, cons338, cons163, cons89)
def replacement494(p, f, g, b, d, a, c, n, x, e):
rubi.append(494)
return -Dist(S(1)/(e*(-d*g + e*f)), Int((f + g*x)**n*(a + b*x + c*x**S(2))**(p + S(-1))*(a*e*g - b*e*f + c*d*f - c*x*(-d*g + e*f)), x), x) + Dist((a*e**S(2) - b*d*e + c*d**S(2))/(e*(-d*g + e*f)), Int((f + g*x)**(n + S(1))*(a + b*x + c*x**S(2))**(p + S(-1))/(d + e*x), x), x)
rule494 = ReplacementRule(pattern494, replacement494)
pattern495 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons23, cons147, cons338, cons163, cons89)
def replacement495(p, f, g, d, c, n, a, x, e):
rubi.append(495)
return -Dist(S(1)/(e*(-d*g + e*f)), Int((a + c*x**S(2))**(p + S(-1))*(f + g*x)**n*(a*e*g + c*d*f - c*x*(-d*g + e*f)), x), x) + Dist((a*e**S(2) + c*d**S(2))/(e*(-d*g + e*f)), Int((a + c*x**S(2))**(p + S(-1))*(f + g*x)**(n + S(1))/(d + e*x), x), x)
rule495 = ReplacementRule(pattern495, replacement495)
pattern496 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons23, cons147, cons338, cons137, cons88)
def replacement496(p, f, g, b, d, a, c, n, x, e):
rubi.append(496)
return Dist(e*(-d*g + e*f)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f + g*x)**(n + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))/(d + e*x), x), x) + Dist(S(1)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f + g*x)**(n + S(-1))*(a + b*x + c*x**S(2))**p*(a*e*g - b*e*f + c*d*f - c*x*(-d*g + e*f)), x), x)
rule496 = ReplacementRule(pattern496, replacement496)
pattern497 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons23, cons147, cons338, cons137, cons88)
def replacement497(p, f, g, d, c, n, a, x, e):
rubi.append(497)
return Dist(e*(-d*g + e*f)/(a*e**S(2) + c*d**S(2)), Int((a + c*x**S(2))**(p + S(1))*(f + g*x)**(n + S(-1))/(d + e*x), x), x) + Dist(S(1)/(a*e**S(2) + c*d**S(2)), Int((a + c*x**S(2))**p*(f + g*x)**(n + S(-1))*(a*e*g + c*d*f - c*x*(-d*g + e*f)), x), x)
rule497 = ReplacementRule(pattern497, replacement497)
def With498(f, g, b, d, a, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(498)
return Simp(-sqrt(S(2))*sqrt(-g*(b + S(2)*c*x - q)/(-b*g + S(2)*c*f + g*q))*sqrt(-g*(b + S(2)*c*x + q)/(-b*g + S(2)*c*f - g*q))*EllipticPi(e*(-b*g + S(2)*c*f + g*q)/(S(2)*c*(-d*g + e*f)), asin(sqrt(S(2))*sqrt(c/(-b*g + S(2)*c*f + g*q))*sqrt(f + g*x)), (-b*g + S(2)*c*f + g*q)/(-b*g + S(2)*c*f - g*q))/(sqrt(c/(-b*g + S(2)*c*f + g*q))*(-d*g + e*f)*sqrt(a + b*x + c*x**S(2))), x)
pattern498 = Pattern(Integral(S(1)/((x_*WC('e', S(1)) + WC('d', S(0)))*sqrt(x_*WC('g', S(1)) + WC('f', S(0)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279)
rule498 = ReplacementRule(pattern498, With498)
def With499(f, g, d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(499)
return Simp(-S(2)*sqrt(g*(-c*x + q)/(c*f + g*q))*sqrt(-g*(c*x + q)/(c*f - g*q))*EllipticPi(e*(c*f + g*q)/(c*(-d*g + e*f)), asin(sqrt(c/(c*f + g*q))*sqrt(f + g*x)), (c*f + g*q)/(c*f - g*q))/(sqrt(c/(c*f + g*q))*sqrt(a + c*x**S(2))*(-d*g + e*f)), x)
pattern499 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('c', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))*sqrt(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280)
rule499 = ReplacementRule(pattern499, With499)
pattern500 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**n_/((x_*WC('e', S(1)) + WC('d', S(0)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279, cons80)
def replacement500(f, g, b, d, a, c, n, x, e):
rubi.append(500)
return Int(ExpandIntegrand(S(1)/(sqrt(f + g*x)*sqrt(a + b*x + c*x**S(2))), (f + g*x)**(n + S(1)/2)/(d + e*x), x), x)
rule500 = ReplacementRule(pattern500, replacement500)
pattern501 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**n_/(sqrt(a_ + x_**S(2)*WC('c', S(1)))*(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280, cons80)
def replacement501(f, g, d, c, n, a, x, e):
rubi.append(501)
return Int(ExpandIntegrand(S(1)/(sqrt(a + c*x**S(2))*sqrt(f + g*x)), (f + g*x)**(n + S(1)/2)/(d + e*x), x), x)
rule501 = ReplacementRule(pattern501, replacement501)
pattern502 = Pattern(Integral(S(1)/(sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*sqrt(x_*WC('g', S(1)) + WC('f', S(0)))*sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons315, cons226, cons279)
def replacement502(f, g, b, d, a, c, x, e):
rubi.append(502)
return Dist(-S(2)*sqrt((-d*g + e*f)**S(2)*(a + b*x + c*x**S(2))/((d + e*x)**S(2)*(a*g**S(2) - b*f*g + c*f**S(2))))*(d + e*x)/((-d*g + e*f)*sqrt(a + b*x + c*x**S(2))), Subst(Int(S(1)/sqrt(x**S(4)*(a*e**S(2) - b*d*e + c*d**S(2))/(a*g**S(2) - b*f*g + c*f**S(2)) - x**S(2)*(S(2)*a*e*g - b*d*g - b*e*f + S(2)*c*d*f)/(a*g**S(2) - b*f*g + c*f**S(2)) + S(1)), x), x, sqrt(f + g*x)/sqrt(d + e*x)), x)
rule502 = ReplacementRule(pattern502, replacement502)
pattern503 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('c', S(1)))*sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*sqrt(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons315, cons280)
def replacement503(f, g, d, c, a, x, e):
rubi.append(503)
return Dist(-S(2)*sqrt((a + c*x**S(2))*(-d*g + e*f)**S(2)/((d + e*x)**S(2)*(a*g**S(2) + c*f**S(2))))*(d + e*x)/(sqrt(a + c*x**S(2))*(-d*g + e*f)), Subst(Int(S(1)/sqrt(x**S(4)*(a*e**S(2) + c*d**S(2))/(a*g**S(2) + c*f**S(2)) - x**S(2)*(S(2)*a*e*g + S(2)*c*d*f)/(a*g**S(2) + c*f**S(2)) + S(1)), x), x, sqrt(f + g*x)/sqrt(d + e*x)), x)
rule503 = ReplacementRule(pattern503, replacement503)
pattern504 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(f_ + x_*WC('g', S(1)))**S(2), x_), cons2, cons7, cons48, cons125, cons208, cons21, cons5, cons379)
def replacement504(p, m, g, f, c, a, x, e):
rubi.append(504)
return Dist(S(2)*f*g/e, Int((e*x)**(m + S(1))*(a + c*x**S(2))**p, x), x) + Int((e*x)**m*(a + c*x**S(2))**p*(f**S(2) + g**S(2)*x**S(2)), x)
rule504 = ReplacementRule(pattern504, replacement504)
pattern505 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(f_ + x_*WC('g', S(1)))**S(3), x_), cons2, cons7, cons48, cons125, cons208, cons21, cons5, cons379)
def replacement505(p, m, g, f, c, a, x, e):
rubi.append(505)
return Dist(f, Int((e*x)**m*(a + c*x**S(2))**p*(f**S(2) + S(3)*g**S(2)*x**S(2)), x), x) + Dist(g/e, Int((e*x)**(m + S(1))*(a + c*x**S(2))**p*(S(3)*f**S(2) + g**S(2)*x**S(2)), x), x)
rule505 = ReplacementRule(pattern505, replacement505)
pattern506 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons226, cons279, cons148)
def replacement506(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(506)
return Dist(g/e, Int((d + e*x)**(m + S(1))*(f + g*x)**(n + S(-1))*(a + b*x + c*x**S(2))**p, x), x) + Dist((-d*g + e*f)/e, Int((d + e*x)**m*(f + g*x)**(n + S(-1))*(a + b*x + c*x**S(2))**p, x), x)
rule506 = ReplacementRule(pattern506, replacement506)
pattern507 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**n_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons315, cons280, cons148)
def replacement507(p, m, f, g, d, c, n, a, x, e):
rubi.append(507)
return Dist(g/e, Int((a + c*x**S(2))**p*(d + e*x)**(m + S(1))*(f + g*x)**(n + S(-1)), x), x) + Dist((-d*g + e*f)/e, Int((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**(n + S(-1)), x), x)
rule507 = ReplacementRule(pattern507, replacement507)
pattern508 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons380)
def replacement508(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(508)
return Int((d + e*x)**m*(f + g*x)**n*(a + b*x + c*x**S(2))**p, x)
rule508 = ReplacementRule(pattern508, replacement508)
pattern509 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons381)
def replacement509(p, m, f, g, d, c, n, a, x, e):
rubi.append(509)
return Int((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**n, x)
rule509 = ReplacementRule(pattern509, replacement509)
pattern510 = Pattern(Integral((u_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(u_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(a_ + u_**S(2)*WC('c', S(1)) + u_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons68, cons69)
def replacement510(p, u, m, f, g, b, d, c, n, a, x, e):
rubi.append(510)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((d + e*x)**m*(f + g*x)**n*(a + b*x + c*x**S(2))**p, x), x, u), x)
rule510 = ReplacementRule(pattern510, replacement510)
pattern511 = Pattern(Integral((a_ + u_**S(2)*WC('c', S(1)))**WC('p', S(1))*(u_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(u_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons68, cons69)
def replacement511(p, u, m, f, g, d, c, n, a, x, e):
rubi.append(511)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + c*x**S(2))**p*(d + e*x)**m*(f + g*x)**n, x), x, u), x)
rule511 = ReplacementRule(pattern511, replacement511)
pattern512 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons382, cons383, cons384, cons385)
def replacement512(p, f, b, d, c, a, x, q, e):
rubi.append(512)
return Dist((c/f)**p, Int((d + e*x + f*x**S(2))**(p + q), x), x)
rule512 = ReplacementRule(pattern512, replacement512)
pattern513 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons382, cons383, cons147, cons386, cons387)
def replacement513(p, f, b, d, c, a, x, q, e):
rubi.append(513)
return Dist(a**IntPart(p)*d**(-IntPart(p))*(a + b*x + c*x**S(2))**FracPart(p)*(d + e*x + f*x**S(2))**(-FracPart(p)), Int((d + e*x + f*x**S(2))**(p + q), x), x)
rule513 = ReplacementRule(pattern513, replacement513)
pattern514 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons45, cons147)
def replacement514(p, f, b, d, c, a, x, q, e):
rubi.append(514)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((b + S(2)*c*x)**(S(2)*p)*(d + e*x + f*x**S(2))**q, x), x)
rule514 = ReplacementRule(pattern514, replacement514)
pattern515 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons5, cons50, cons45, cons147)
def replacement515(p, f, b, d, c, a, x, q):
rubi.append(515)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((b + S(2)*c*x)**(S(2)*p)*(d + f*x**S(2))**q, x), x)
rule515 = ReplacementRule(pattern515, replacement515)
pattern516 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons50, cons388, cons389, cons390)
def replacement516(f, b, d, c, a, x, q, e):
rubi.append(516)
return Simp((d + e*x + f*x**S(2))**(q + S(1))*(b*f*(S(2)*q + S(3)) - c*e*(q + S(2)) + S(2)*c*f*x*(q + S(1)))/(S(2)*f**S(2)*(q + S(1))*(S(2)*q + S(3))), x)
rule516 = ReplacementRule(pattern516, replacement516)
pattern517 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons50, cons391, cons389, cons390)
def replacement517(f, d, c, a, x, q, e):
rubi.append(517)
return Simp((-c*e*(q + S(2)) + S(2)*c*f*x*(q + S(1)))*(d + e*x + f*x**S(2))**(q + S(1))/(S(2)*f**S(2)*(q + S(1))*(S(2)*q + S(3))), x)
rule517 = ReplacementRule(pattern517, replacement517)
pattern518 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**q_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons125, cons50, cons389, cons392)
def replacement518(f, b, d, c, a, x, q):
rubi.append(518)
return Simp((d + f*x**S(2))**(q + S(1))*(S(2)*a*f*x*(q + S(1)) + b*d)/(S(2)*d*f*(q + S(1))), x)
rule518 = ReplacementRule(pattern518, replacement518)
pattern519 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**q_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons125, cons50, cons393)
def replacement519(f, b, d, c, a, x, q):
rubi.append(519)
return Dist(b, Int(x*(d + f*x**S(2))**q, x), x) + Int((a + c*x**S(2))*(d + f*x**S(2))**q, x)
rule519 = ReplacementRule(pattern519, replacement519)
pattern520 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons394, cons393)
def replacement520(f, b, d, c, a, x, q, e):
rubi.append(520)
return Int(ExpandIntegrand((a + b*x + c*x**S(2))*(d + e*x + f*x**S(2))**q, x), x)
rule520 = ReplacementRule(pattern520, replacement520)
pattern521 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons394, cons393)
def replacement521(f, d, c, a, x, q, e):
rubi.append(521)
return Int(ExpandIntegrand((a + c*x**S(2))*(d + e*x + f*x**S(2))**q, x), x)
rule521 = ReplacementRule(pattern521, replacement521)
pattern522 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons394, cons395, cons396, cons397)
def replacement522(f, b, d, c, a, x, q, e):
rubi.append(522)
return -Dist((c*(-S(2)*d*f + e**S(2)*(q + S(2))) + f*(S(2)*q + S(3))*(S(2)*a*f - b*e))/(f*(q + S(1))*(-S(4)*d*f + e**S(2))), Int((d + e*x + f*x**S(2))**(q + S(1)), x), x) + Simp((d + e*x + f*x**S(2))**(q + S(1))*(a*e*f - S(2)*b*d*f + c*d*e + x*(c*(-S(2)*d*f + e**S(2)) + f*(S(2)*a*f - b*e)))/(f*(q + S(1))*(-S(4)*d*f + e**S(2))), x)
rule522 = ReplacementRule(pattern522, replacement522)
pattern523 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons394, cons395, cons396, cons398)
def replacement523(f, d, c, a, x, q, e):
rubi.append(523)
return -Dist((S(2)*a*f**S(2)*(S(2)*q + S(3)) + c*(-S(2)*d*f + e**S(2)*(q + S(2))))/(f*(q + S(1))*(-S(4)*d*f + e**S(2))), Int((d + e*x + f*x**S(2))**(q + S(1)), x), x) + Simp((d + e*x + f*x**S(2))**(q + S(1))*(a*e*f + c*d*e + x*(S(2)*a*f**S(2) + c*(-S(2)*d*f + e**S(2))))/(f*(q + S(1))*(-S(4)*d*f + e**S(2))), x)
rule523 = ReplacementRule(pattern523, replacement523)
pattern524 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**q_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons125, cons395, cons396, cons399)
def replacement524(f, b, d, c, a, x, q):
rubi.append(524)
return Dist((S(2)*a*f*q + S(3)*a*f - c*d)/(S(2)*d*f*(q + S(1))), Int((d + f*x**S(2))**(q + S(1)), x), x) + Simp((d + f*x**S(2))**(q + S(1))*(b*d - x*(a*f - c*d))/(S(2)*d*f*(q + S(1))), x)
rule524 = ReplacementRule(pattern524, replacement524)
pattern525 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons27, cons48, cons125, cons2, cons3, cons7, cons50, cons394, cons400, cons401, cons397)
def replacement525(f, b, d, c, a, x, q, e):
rubi.append(525)
return Dist((c*(-S(2)*d*f + e**S(2)*(q + S(2))) + f*(S(2)*q + S(3))*(S(2)*a*f - b*e))/(S(2)*f**S(2)*(S(2)*q + S(3))), Int((d + e*x + f*x**S(2))**q, x), x) + Simp((d + e*x + f*x**S(2))**(q + S(1))*(b*f*(S(2)*q + S(3)) - c*e*(q + S(2)) + S(2)*c*f*x*(q + S(1)))/(S(2)*f**S(2)*(q + S(1))*(S(2)*q + S(3))), x)
rule525 = ReplacementRule(pattern525, replacement525)
pattern526 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons27, cons48, cons125, cons2, cons7, cons50, cons394, cons400, cons401, cons398)
def replacement526(f, d, c, a, x, q, e):
rubi.append(526)
return Dist((S(2)*a*f**S(2)*(S(2)*q + S(3)) + c*(-S(2)*d*f + e**S(2)*(q + S(2))))/(S(2)*f**S(2)*(S(2)*q + S(3))), Int((d + e*x + f*x**S(2))**q, x), x) + Simp((-c*e*(q + S(2)) + S(2)*c*f*x*(q + S(1)))*(d + e*x + f*x**S(2))**(q + S(1))/(S(2)*f**S(2)*(q + S(1))*(S(2)*q + S(3))), x)
rule526 = ReplacementRule(pattern526, replacement526)
pattern527 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**q_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons27, cons125, cons2, cons3, cons7, cons50, cons400, cons401, cons399)
def replacement527(f, b, d, c, a, x, q):
rubi.append(527)
return Dist((S(2)*a*f*q + S(3)*a*f - c*d)/(f*(S(2)*q + S(3))), Int((d + f*x**S(2))**q, x), x) + Simp((d + f*x**S(2))**(q + S(1))*(b*f*(S(2)*q + S(3)) + S(2)*c*f*x*(q + S(1)))/(S(2)*f**S(2)*(q + S(1))*(S(2)*q + S(3))), x)
rule527 = ReplacementRule(pattern527, replacement527)
pattern528 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons226, cons394, cons402, cons137, cons403)
def replacement528(p, f, b, d, c, a, x, q, e):
rubi.append(528)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(-1))*Simp(b*e*q + S(2)*c*d*(S(2)*p + S(3)) + S(2)*c*f*x**S(2)*(S(2)*p + S(2)*q + S(3)) + x*(S(2)*b*f*q + S(2)*c*e*(S(2)*p + q + S(3))), x), x), x) + Simp((b + S(2)*c*x)*(a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule528 = ReplacementRule(pattern528, replacement528)
pattern529 = Pattern(Integral((x_**S(2)*WC('f', S(1)) + WC('d', S(0)))**WC('q', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons226, cons402, cons137, cons403)
def replacement529(p, f, b, d, c, a, x, q):
rubi.append(529)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + f*x**S(2))**(q + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(S(2)*b*f*q*x + S(2)*c*d*(S(2)*p + S(3)) + S(2)*c*f*x**S(2)*(S(2)*p + S(2)*q + S(3)), x), x), x) + Simp((b + S(2)*c*x)*(d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(1))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule529 = ReplacementRule(pattern529, replacement529)
pattern530 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons394, cons402, cons137, cons403)
def replacement530(p, f, d, a, c, x, q, e):
rubi.append(530)
return -Dist(-S(1)/(S(4)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(-1))*Simp(S(2)*c*d*(S(2)*p + S(3)) + S(2)*c*e*x*(S(2)*p + q + S(3)) + S(2)*c*f*x**S(2)*(S(2)*p + S(2)*q + S(3)), x), x), x) + Simp(-x*(a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q/(S(2)*a*(p + S(1))), x)
rule530 = ReplacementRule(pattern530, replacement530)
pattern531 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons50, cons226, cons394, cons13, cons137, cons404, cons405)
def replacement531(p, f, b, d, c, a, x, q, e):
rubi.append(531)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*Simp(c*f*x**S(2)*(S(2)*p + S(2)*q + S(5))*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) + S(2)*c*(p + S(1))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2)) - e*(p + q + S(2))*(-S(2)*a*c**S(2)*e - b**S(3)*f + b**S(2)*c*e - b*c*(-S(3)*a*f + c*d)) + x*(S(2)*f*(p + q + S(2))*(S(2)*a*c**S(2)*e + b**S(3)*f - b**S(2)*c*e + b*c*(-S(3)*a*f + c*d)) - (b*f*(p + S(1)) - c*e*(S(2)*p + q + S(4)))*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e))) - (a*f*(p + S(1)) - c*d*(p + S(2)))*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)), x), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(1))*(S(2)*a*c**S(2)*e + b**S(3)*f - b**S(2)*c*e + b*c*(-S(3)*a*f + c*d) + c*x*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)))/((p + S(1))*(-S(4)*a*c + b**S(2))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))), x)
rule531 = ReplacementRule(pattern531, replacement531)
pattern532 = Pattern(Integral((x_**S(2)*WC('f', S(1)) + WC('d', S(0)))**WC('q', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons50, cons226, cons13, cons137, cons406, cons405)
def replacement532(p, f, b, d, c, a, x, q):
rubi.append(532)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(b**S(2)*d*f + (-a*f + c*d)**S(2))), Int((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(1))*Simp(c*f*x**S(2)*(S(2)*p + S(2)*q + S(5))*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d) + S(2)*c*(p + S(1))*(b**S(2)*d*f + (-a*f + c*d)**S(2)) + x*(-b*f*(p + S(1))*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d) + S(2)*f*(b**S(3)*f + b*c*(-S(3)*a*f + c*d))*(p + q + S(2))) - (a*f*(p + S(1)) - c*d*(p + S(2)))*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d), x), x), x) + Simp((d + f*x**S(2))**(q + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(b**S(3)*f + b*c*(-S(3)*a*f + c*d) + c*x*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d))/((p + S(1))*(-S(4)*a*c + b**S(2))*(b**S(2)*d*f + (-a*f + c*d)**S(2))), x)
rule532 = ReplacementRule(pattern532, replacement532)
pattern533 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons50, cons394, cons13, cons137, cons407, cons405)
def replacement533(p, f, d, a, c, x, q, e):
rubi.append(533)
return -Dist(-S(1)/(S(4)*a*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*Simp(S(2)*a*c**S(2)*e**S(2)*(p + q + S(2)) + c*f*x**S(2)*(-S(2)*a*c*f + S(2)*c**S(2)*d)*(S(2)*p + S(2)*q + S(5)) + S(2)*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2)) + x*(S(4)*a*c**S(2)*e*f*(p + q + S(2)) + c*e*(-S(2)*a*c*f + S(2)*c**S(2)*d)*(S(2)*p + q + S(4))) - (-S(2)*a*c*f + S(2)*c**S(2)*d)*(a*f*(p + S(1)) - c*d*(p + S(2))), x), x), x) + Simp(-(a + c*x**S(2))**(p + S(1))*(S(2)*a*c**S(2)*e + c*x*(-S(2)*a*c*f + S(2)*c**S(2)*d))*(d + e*x + f*x**S(2))**(q + S(1))/(S(4)*a*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2))), x)
rule533 = ReplacementRule(pattern533, replacement533)
pattern534 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons50, cons226, cons394, cons13, cons146, cons408, cons409)
def replacement534(p, f, b, d, c, a, x, q, e):
rubi.append(534)
return -Dist(S(1)/(S(2)*f**S(2)*(p + q)*(S(2)*p + S(2)*q + S(1))), Int((a + b*x + c*x**S(2))**(p + S(-2))*(d + e*x + f*x**S(2))**q*Simp(x**S(2)*(c*(p + q)*(-c*(S(2)*d*f*(-S(2)*p + S(1)) + e**S(2)*(S(3)*p + q + S(-1))) + f*(-S(2)*a*f + b*e)*(S(4)*p + S(2)*q + S(-1))) + p*(-p + S(1))*(-b*f + c*e)**S(2)) + x*(S(2)*(-p + S(1))*(S(2)*p + q)*(-a*f + c*d)*(-b*f + c*e) - (p + q)*(b*(c*(S(2)*p + q)*(-S(4)*d*f + e**S(2)) + f*(S(2)*p + S(2)*q + S(1))*(S(2)*a*f - b*e + S(2)*c*d)) + e*f*(-p + S(1))*(-S(4)*a*c + b**S(2)))) + (-p + S(1))*(S(2)*p + q)*(-a*e + b*d)*(-b*f + c*e) - (p + q)*(-a*(c*(S(2)*d*f - e**S(2)*(S(2)*p + q)) + f*(-S(2)*a*f + b*e)*(S(2)*p + S(2)*q + S(1))) + b**S(2)*d*f*(-p + S(1))), x), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(-1))*(d + e*x + f*x**S(2))**(q + S(1))*(b*f*(S(3)*p + S(2)*q) - c*e*(S(2)*p + q) + S(2)*c*f*x*(p + q))/(S(2)*f**S(2)*(p + q)*(S(2)*p + S(2)*q + S(1))), x)
rule534 = ReplacementRule(pattern534, replacement534)
pattern535 = Pattern(Integral((x_**S(2)*WC('f', S(1)) + WC('d', S(0)))**WC('q', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons50, cons226, cons13, cons146, cons408, cons409)
def replacement535(p, f, b, d, c, a, x, q):
rubi.append(535)
return -Dist(S(1)/(S(2)*f*(p + q)*(S(2)*p + S(2)*q + S(1))), Int((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(-2))*Simp(b**S(2)*d*(p + S(-1))*(S(2)*p + q) + x**S(2)*(b**S(2)*f*p*(-p + S(1)) + S(2)*c*(p + q)*(-a*f*(S(4)*p + S(2)*q + S(-1)) + c*d*(S(2)*p + S(-1)))) - x*(S(2)*b*(-p + S(1))*(S(2)*p + q)*(-a*f + c*d) - S(2)*b*(p + q)*(S(2)*c*d*(S(2)*p + q) - (a*f + c*d)*(S(2)*p + S(2)*q + S(1)))) - (p + q)*(-S(2)*a*(-a*f*(S(2)*p + S(2)*q + S(1)) + c*d) + b**S(2)*d*(-p + S(1))), x), x), x) + Simp((d + f*x**S(2))**(q + S(1))*(b*(S(3)*p + S(2)*q) + S(2)*c*x*(p + q))*(a + b*x + c*x**S(2))**(p + S(-1))/(S(2)*f*(p + q)*(S(2)*p + S(2)*q + S(1))), x)
rule535 = ReplacementRule(pattern535, replacement535)
pattern536 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons50, cons394, cons13, cons146, cons408, cons409)
def replacement536(p, f, d, a, c, x, q, e):
rubi.append(536)
return -Dist(S(1)/(S(2)*f**S(2)*(p + q)*(S(2)*p + S(2)*q + S(1))), Int((a + c*x**S(2))**(p + S(-2))*(d + e*x + f*x**S(2))**q*Simp(-a*c*e**S(2)*(-p + S(1))*(S(2)*p + q) + a*(p + q)*(-S(2)*a*f**S(2)*(S(2)*p + S(2)*q + S(1)) + c*(S(2)*d*f - e**S(2)*(S(2)*p + q))) + x**S(2)*(c**S(2)*e**S(2)*p*(-p + S(1)) - c*(p + q)*(S(2)*a*f**S(2)*(S(4)*p + S(2)*q + S(-1)) + c*(S(2)*d*f*(-S(2)*p + S(1)) + e**S(2)*(S(3)*p + q + S(-1))))) + x*(S(4)*a*c*e*f*(-p + S(1))*(p + q) + S(2)*c*e*(-p + S(1))*(S(2)*p + q)*(-a*f + c*d)), x), x), x) - Simp(c*(a + c*x**S(2))**(p + S(-1))*(e*(S(2)*p + q) - S(2)*f*x*(p + q))*(d + e*x + f*x**S(2))**(q + S(1))/(S(2)*f**S(2)*(p + q)*(S(2)*p + S(2)*q + S(1))), x)
rule536 = ReplacementRule(pattern536, replacement536)
def With537(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = a**S(2)*f**S(2) - a*b*e*f - S(2)*a*c*d*f + a*c*e**S(2) + b**S(2)*d*f - b*c*d*e + c**S(2)*d**S(2)
if NonzeroQ(q):
return True
return False
pattern537 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons226, cons394, CustomConstraint(With537))
def replacement537(f, b, d, c, a, x, e):
q = a**S(2)*f**S(2) - a*b*e*f - S(2)*a*c*d*f + a*c*e**S(2) + b**S(2)*d*f - b*c*d*e + c**S(2)*d**S(2)
rubi.append(537)
return Dist(S(1)/q, Int((-a*c*f + b**S(2)*f - b*c*e + c**S(2)*d - x*(-b*c*f + c**S(2)*e))/(a + b*x + c*x**S(2)), x), x) + Dist(S(1)/q, Int((a*f**S(2) - b*e*f - c*d*f + c*e**S(2) + x*(-b*f**S(2) + c*e*f))/(d + e*x + f*x**S(2)), x), x)
rule537 = ReplacementRule(pattern537, replacement537)
def With538(f, b, d, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = a**S(2)*f**S(2) - S(2)*a*c*d*f + b**S(2)*d*f + c**S(2)*d**S(2)
if NonzeroQ(q):
return True
return False
pattern538 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('f', S(1)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons226, CustomConstraint(With538))
def replacement538(f, b, d, c, a, x):
q = a**S(2)*f**S(2) - S(2)*a*c*d*f + b**S(2)*d*f + c**S(2)*d**S(2)
rubi.append(538)
return -Dist(S(1)/q, Int((-a*f**S(2) + b*f**S(2)*x + c*d*f)/(d + f*x**S(2)), x), x) + Dist(S(1)/q, Int((-a*c*f + b**S(2)*f + b*c*f*x + c**S(2)*d)/(a + b*x + c*x**S(2)), x), x)
rule538 = ReplacementRule(pattern538, replacement538)
pattern539 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons226, cons394, cons410)
def replacement539(f, b, d, c, a, x, e):
rubi.append(539)
return Dist(-S(2)*e, Subst(Int(S(1)/(e*(-S(4)*a*f + b*e) - x**S(2)*(-a*e + b*d)), x), x, (e + S(2)*f*x)/sqrt(d + e*x + f*x**S(2))), x)
rule539 = ReplacementRule(pattern539, replacement539)
def With540(f, b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(540)
return Dist(S(2)*c/q, Int(S(1)/((b + S(2)*c*x - q)*sqrt(d + e*x + f*x**S(2))), x), x) - Dist(S(2)*c/q, Int(S(1)/((b + S(2)*c*x + q)*sqrt(d + e*x + f*x**S(2))), x), x)
pattern540 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons226, cons394, cons411, cons231)
rule540 = ReplacementRule(pattern540, With540)
pattern541 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons394, cons412)
def replacement541(f, d, c, a, x, e):
rubi.append(541)
return Dist(S(1)/2, Int(S(1)/((a - x*Rt(-a*c, S(2)))*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(S(1)/2, Int(S(1)/((a + x*Rt(-a*c, S(2)))*sqrt(d + e*x + f*x**S(2))), x), x)
rule541 = ReplacementRule(pattern541, replacement541)
def With542(f, b, d, c, a, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(542)
return Dist(S(2)*c/q, Int(S(1)/(sqrt(d + f*x**S(2))*(b + S(2)*c*x - q)), x), x) - Dist(S(2)*c/q, Int(S(1)/(sqrt(d + f*x**S(2))*(b + S(2)*c*x + q)), x), x)
pattern542 = Pattern(Integral(S(1)/(sqrt(d_ + x_**S(2)*WC('f', S(1)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons226, cons231)
rule542 = ReplacementRule(pattern542, With542)
def With543(f, b, d, c, a, x, e):
q = Rt(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2), S(2))
rubi.append(543)
return -Dist(S(1)/(S(2)*q), Int((-a*f + c*d - q + x*(-b*f + c*e))/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(S(1)/(S(2)*q), Int((-a*f + c*d + q + x*(-b*f + c*e))/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x)
pattern543 = Pattern(Integral(S(1)/((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons226, cons394, cons411, cons413)
rule543 = ReplacementRule(pattern543, With543)
def With544(f, d, a, c, x, e):
q = Rt(a*c*e**S(2) + (-a*f + c*d)**S(2), S(2))
rubi.append(544)
return -Dist(S(1)/(S(2)*q), Int((-a*f + c*d + c*e*x - q)/((a + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(S(1)/(S(2)*q), Int((-a*f + c*d + c*e*x + q)/((a + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x)
pattern544 = Pattern(Integral(S(1)/((x_**S(2)*WC('c', S(1)) + WC('a', S(0)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons394, cons414)
rule544 = ReplacementRule(pattern544, With544)
def With545(f, b, d, c, a, x):
q = Rt(b**S(2)*d*f + (-a*f + c*d)**S(2), S(2))
rubi.append(545)
return -Dist(S(1)/(S(2)*q), Int((-a*f - b*f*x + c*d - q)/(sqrt(d + f*x**S(2))*(a + b*x + c*x**S(2))), x), x) + Dist(S(1)/(S(2)*q), Int((-a*f - b*f*x + c*d + q)/(sqrt(d + f*x**S(2))*(a + b*x + c*x**S(2))), x), x)
pattern545 = Pattern(Integral(S(1)/(sqrt(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons125, cons226, cons413)
rule545 = ReplacementRule(pattern545, With545)
pattern546 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))/(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons226, cons394)
def replacement546(f, b, d, c, a, x, e):
rubi.append(546)
return -Dist(S(1)/f, Int((-a*f + c*d + x*(-b*f + c*e))/(sqrt(a + b*x + c*x**S(2))*(d + e*x + f*x**S(2))), x), x) + Dist(c/f, Int(S(1)/sqrt(a + b*x + c*x**S(2)), x), x)
rule546 = ReplacementRule(pattern546, replacement546)
pattern547 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))/(d_ + x_**S(2)*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons125, cons226)
def replacement547(f, b, d, c, a, x):
rubi.append(547)
return -Dist(S(1)/f, Int((-a*f - b*f*x + c*d)/((d + f*x**S(2))*sqrt(a + b*x + c*x**S(2))), x), x) + Dist(c/f, Int(S(1)/sqrt(a + b*x + c*x**S(2)), x), x)
rule547 = ReplacementRule(pattern547, replacement547)
pattern548 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('c', S(1)))/(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons394)
def replacement548(f, d, c, a, x, e):
rubi.append(548)
return -Dist(S(1)/f, Int((-a*f + c*d + c*e*x)/(sqrt(a + c*x**S(2))*(d + e*x + f*x**S(2))), x), x) + Dist(c/f, Int(S(1)/sqrt(a + c*x**S(2)), x), x)
rule548 = ReplacementRule(pattern548, replacement548)
def With549(f, b, d, c, a, x, e):
r = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(549)
return Dist(sqrt(S(2)*a + x*(b + r))*sqrt(b + S(2)*c*x + r)/sqrt(a + b*x + c*x**S(2)), Int(S(1)/(sqrt(S(2)*a + x*(b + r))*sqrt(b + S(2)*c*x + r)*sqrt(d + e*x + f*x**S(2))), x), x)
pattern549 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons226, cons394)
rule549 = ReplacementRule(pattern549, With549)
def With550(f, b, d, c, a, x):
r = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(550)
return Dist(sqrt(S(2)*a + x*(b + r))*sqrt(b + S(2)*c*x + r)/sqrt(a + b*x + c*x**S(2)), Int(S(1)/(sqrt(S(2)*a + x*(b + r))*sqrt(d + f*x**S(2))*sqrt(b + S(2)*c*x + r)), x), x)
pattern550 = Pattern(Integral(S(1)/(sqrt(d_ + x_**S(2)*WC('f', S(1)))*sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons226)
rule550 = ReplacementRule(pattern550, With550)
pattern551 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons415)
def replacement551(p, f, b, d, c, a, x, q, e):
rubi.append(551)
return Int((a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x)
rule551 = ReplacementRule(pattern551, replacement551)
pattern552 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons5, cons50, cons416)
def replacement552(p, f, d, c, a, x, q, e):
rubi.append(552)
return Int((a + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x)
rule552 = ReplacementRule(pattern552, replacement552)
pattern553 = Pattern(Integral((u_**S(2)*WC('c', S(1)) + u_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons68, cons69)
def replacement553(p, u, f, b, d, c, a, x, q, e):
rubi.append(553)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule553 = ReplacementRule(pattern553, replacement553)
pattern554 = Pattern(Integral((u_**S(2)*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons5, cons50, cons68, cons69)
def replacement554(p, u, f, d, a, c, x, q, e):
rubi.append(554)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule554 = ReplacementRule(pattern554, replacement554)
pattern555 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons382, cons383, cons384, cons385)
def replacement555(p, m, g, b, f, d, c, a, x, h, q, e):
rubi.append(555)
return Dist((c/f)**p, Int((g + h*x)**m*(d + e*x + f*x**S(2))**(p + q), x), x)
rule555 = ReplacementRule(pattern555, replacement555)
pattern556 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons382, cons383, cons147, cons386, cons387)
def replacement556(p, m, g, b, f, d, c, a, x, h, q, e):
rubi.append(556)
return Dist(a**IntPart(p)*d**(-IntPart(p))*(a + b*x + c*x**S(2))**FracPart(p)*(d + e*x + f*x**S(2))**(-FracPart(p)), Int((g + h*x)**m*(d + e*x + f*x**S(2))**(p + q), x), x)
rule556 = ReplacementRule(pattern556, replacement556)
pattern557 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons45)
def replacement557(p, m, g, b, f, d, c, a, x, h, q, e):
rubi.append(557)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((b + S(2)*c*x)**(S(2)*p)*(g + h*x)**m*(d + e*x + f*x**S(2))**q, x), x)
rule557 = ReplacementRule(pattern557, replacement557)
pattern558 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons21, cons5, cons50, cons45)
def replacement558(p, m, g, b, f, d, c, a, x, h, q):
rubi.append(558)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((b + S(2)*c*x)**(S(2)*p)*(d + f*x**S(2))**q*(g + h*x)**m, x), x)
rule558 = ReplacementRule(pattern558, replacement558)
pattern559 = Pattern(Integral((g_ + x_*WC('h', S(1)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons417, cons418, cons17)
def replacement559(p, m, f, b, g, d, c, a, x, h, e):
rubi.append(559)
return Int((f*h*x/c + d*g/a)**m*(a + b*x + c*x**S(2))**(m + p), x)
rule559 = ReplacementRule(pattern559, replacement559)
pattern560 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(g_ + x_*WC('h', S(1)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons419, cons418, cons17)
def replacement560(p, m, f, g, d, c, a, x, h, e):
rubi.append(560)
return Int((a + c*x**S(2))**(m + p)*(f*h*x/c + d*g/a)**m, x)
rule560 = ReplacementRule(pattern560, replacement560)
pattern561 = Pattern(Integral((g_ + x_*WC('h', S(1)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))**WC('m', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons5, cons417, cons420, cons17)
def replacement561(p, m, f, b, g, d, c, a, x, h):
rubi.append(561)
return Int((f*h*x/c + d*g/a)**m*(a + b*x + c*x**S(2))**(m + p), x)
rule561 = ReplacementRule(pattern561, replacement561)
pattern562 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(g_ + x_*WC('h', S(1)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons5, cons419, cons420, cons17)
def replacement562(p, m, f, g, d, c, a, x, h):
rubi.append(562)
return Int((a + c*x**S(2))**(m + p)*(f*h*x/c + d*g/a)**m, x)
rule562 = ReplacementRule(pattern562, replacement562)
pattern563 = Pattern(Integral(x_**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons48, cons125, cons50, cons226, cons421, cons38)
def replacement563(p, f, b, c, a, x, q, e):
rubi.append(563)
return Int((a/e + c*x/f)**p*(e*x + f*x**S(2))**(p + q), x)
rule563 = ReplacementRule(pattern563, replacement563)
pattern564 = Pattern(Integral(x_**WC('p', S(1))*(a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons48, cons125, cons50, cons422, cons38)
def replacement564(p, f, c, a, x, q, e):
rubi.append(564)
return Int((a/e + c*x/f)**p*(e*x + f*x**S(2))**(p + q), x)
rule564 = ReplacementRule(pattern564, replacement564)
pattern565 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons423, cons424, cons270)
def replacement565(p, m, g, b, f, d, a, c, x, h, e):
rubi.append(565)
return Simp(f*(g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*h*(m + S(2)*p + S(3))), x)
rule565 = ReplacementRule(pattern565, replacement565)
pattern566 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons425, cons426, cons270)
def replacement566(p, m, g, f, d, c, a, x, h, e):
rubi.append(566)
return Simp(f*(a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(1))/(c*h*(m + S(2)*p + S(3))), x)
rule566 = ReplacementRule(pattern566, replacement566)
pattern567 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons21, cons5, cons427, cons424, cons270)
def replacement567(p, m, g, b, f, d, a, c, x, h):
rubi.append(567)
return Simp(f*(g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*h*(m + S(2)*p + S(3))), x)
rule567 = ReplacementRule(pattern567, replacement567)
pattern568 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons226, cons394, cons428)
def replacement568(p, m, g, b, f, d, a, c, x, h, e):
rubi.append(568)
return Int(ExpandIntegrand((g + h*x)**m*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2)), x), x)
rule568 = ReplacementRule(pattern568, replacement568)
pattern569 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons394, cons428)
def replacement569(p, m, g, f, d, c, a, x, h, e):
rubi.append(569)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(g + h*x)**m*(d + e*x + f*x**S(2)), x), x)
rule569 = ReplacementRule(pattern569, replacement569)
pattern570 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons21, cons226, cons428)
def replacement570(p, m, g, b, f, d, a, c, x, h):
rubi.append(570)
return Int(ExpandIntegrand((d + f*x**S(2))*(g + h*x)**m*(a + b*x + c*x**S(2))**p, x), x)
rule570 = ReplacementRule(pattern570, replacement570)
pattern571 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(g_ + x_*WC('h', S(1)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons21, cons428)
def replacement571(p, m, f, g, d, c, a, x, h):
rubi.append(571)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(d + f*x**S(2))*(g + h*x)**m, x), x)
rule571 = ReplacementRule(pattern571, replacement571)
pattern572 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons226, cons394, cons31, cons94, cons429)
def replacement572(p, m, g, b, f, d, a, c, x, h, e):
rubi.append(572)
return Dist(S(1)/(h*(m + S(1))*(a*h**S(2) - b*g*h + c*g**S(2))), Int((g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*Simp(-b*(f*g**S(2)*(p + S(1)) - h*(-d*h*(m + p + S(2)) + e*g*(p + S(1)))) + h*(m + S(1))*(a*e*h - a*f*g + c*d*g) - x*(c*(S(2)*f*g**S(2)*(p + S(1)) - h*(-d*h + e*g)*(m + S(2)*p + S(3))) + f*h*(m + S(1))*(-a*h + b*g)), x), x), x) + Simp((g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(d*h**S(2) - e*g*h + f*g**S(2))/(h*(m + S(1))*(a*h**S(2) - b*g*h + c*g**S(2))), x)
rule572 = ReplacementRule(pattern572, replacement572)
pattern573 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**m_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons394, cons31, cons94, cons430)
def replacement573(p, m, g, f, d, c, a, x, h, e):
rubi.append(573)
return Dist(S(1)/(h*(m + S(1))*(a*h**S(2) + c*g**S(2))), Int((a + c*x**S(2))**p*(g + h*x)**(m + S(1))*Simp(h*(m + S(1))*(a*e*h - a*f*g + c*d*g) + x*(a*f*h**S(2)*(m + S(1)) - c*(S(2)*f*g**S(2)*(p + S(1)) - h*(-d*h + e*g)*(m + S(2)*p + S(3)))), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(1))*(d*h**S(2) - e*g*h + f*g**S(2))/(h*(m + S(1))*(a*h**S(2) + c*g**S(2))), x)
rule573 = ReplacementRule(pattern573, replacement573)
pattern574 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**m_*(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons5, cons226, cons31, cons94, cons429)
def replacement574(p, m, g, b, f, d, a, c, x, h):
rubi.append(574)
return Dist(S(1)/(h*(m + S(1))*(a*h**S(2) - b*g*h + c*g**S(2))), Int((g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**p*Simp(-b*(d*h**S(2)*(m + p + S(2)) + f*g**S(2)*(p + S(1))) + h*(m + S(1))*(-a*f*g + c*d*g) - x*(c*(d*h**S(2)*(m + S(2)*p + S(3)) + S(2)*f*g**S(2)*(p + S(1))) + f*h*(m + S(1))*(-a*h + b*g)), x), x), x) + Simp((g + h*x)**(m + S(1))*(d*h**S(2) + f*g**S(2))*(a + b*x + c*x**S(2))**(p + S(1))/(h*(m + S(1))*(a*h**S(2) - b*g*h + c*g**S(2))), x)
rule574 = ReplacementRule(pattern574, replacement574)
pattern575 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(g_ + x_*WC('h', S(1)))**m_*(x_**S(2)*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons5, cons31, cons94, cons430)
def replacement575(p, m, f, g, d, c, a, x, h):
rubi.append(575)
return Dist(S(1)/(h*(m + S(1))*(a*h**S(2) + c*g**S(2))), Int((a + c*x**S(2))**p*(g + h*x)**(m + S(1))*Simp(h*(m + S(1))*(-a*f*g + c*d*g) + x*(a*f*h**S(2)*(m + S(1)) - c*(d*h**S(2)*(m + S(2)*p + S(3)) + S(2)*f*g**S(2)*(p + S(1)))), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(1))*(d*h**S(2) + f*g**S(2))/(h*(m + S(1))*(a*h**S(2) + c*g**S(2))), x)
rule575 = ReplacementRule(pattern575, replacement575)
pattern576 = Pattern(Integral((x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))/((x_*WC('h', S(1)) + WC('g', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**(S(3)/2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons429)
def replacement576(g, b, f, d, a, c, x, h, e):
rubi.append(576)
return Dist((d*h**S(2) - e*g*h + f*g**S(2))/(a*h**S(2) - b*g*h + c*g**S(2)), Int(S(1)/((g + h*x)*sqrt(a + b*x + c*x**S(2))), x), x) + Dist(S(1)/(a*h**S(2) - b*g*h + c*g**S(2)), Int((a*e*h - a*f*g - b*d*h + c*d*g + x*(a*f*h - b*f*g - c*d*h + c*e*g))/(a + b*x + c*x**S(2))**(S(3)/2), x), x)
rule576 = ReplacementRule(pattern576, replacement576)
pattern577 = Pattern(Integral((x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))/((a_ + x_**S(2)*WC('c', S(1)))**(S(3)/2)*(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons394, cons430)
def replacement577(g, f, d, c, a, x, h, e):
rubi.append(577)
return Dist((d*h**S(2) - e*g*h + f*g**S(2))/(a*h**S(2) + c*g**S(2)), Int(S(1)/(sqrt(a + c*x**S(2))*(g + h*x)), x), x) + Dist(S(1)/(a*h**S(2) + c*g**S(2)), Int((a*e*h - a*f*g + c*d*g + x*(a*f*h - c*d*h + c*e*g))/(a + c*x**S(2))**(S(3)/2), x), x)
rule577 = ReplacementRule(pattern577, replacement577)
pattern578 = Pattern(Integral((x_**S(2)*WC('f', S(1)) + WC('d', S(0)))/((x_*WC('h', S(1)) + WC('g', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**(S(3)/2)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons226, cons429)
def replacement578(g, b, f, d, a, c, x, h):
rubi.append(578)
return Dist((d*h**S(2) + f*g**S(2))/(a*h**S(2) - b*g*h + c*g**S(2)), Int(S(1)/((g + h*x)*sqrt(a + b*x + c*x**S(2))), x), x) + Dist(S(1)/(a*h**S(2) - b*g*h + c*g**S(2)), Int((-a*f*g - b*d*h + c*d*g - x*(-a*f*h + b*f*g + c*d*h))/(a + b*x + c*x**S(2))**(S(3)/2), x), x)
rule578 = ReplacementRule(pattern578, replacement578)
pattern579 = Pattern(Integral((x_**S(2)*WC('f', S(1)) + WC('d', S(0)))/((a_ + x_**S(2)*WC('c', S(1)))**(S(3)/2)*(g_ + x_*WC('h', S(1)))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons430)
def replacement579(f, g, d, c, a, x, h):
rubi.append(579)
return Dist((d*h**S(2) + f*g**S(2))/(a*h**S(2) + c*g**S(2)), Int(S(1)/(sqrt(a + c*x**S(2))*(g + h*x)), x), x) + Dist(S(1)/(a*h**S(2) + c*g**S(2)), Int((-a*f*g + c*d*g - x*(-a*f*h + c*d*h))/(a + c*x**S(2))**(S(3)/2), x), x)
rule579 = ReplacementRule(pattern579, replacement579)
pattern580 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**m_*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons244, cons137, cons166)
def replacement580(p, m, g, b, f, d, a, c, x, h, e):
rubi.append(580)
return -Dist(S(1)/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((g + h*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(g*(c*(S(2)*p + S(3))*(-b*e + S(2)*c*d) - f*(S(2)*a*c - b**S(2)*(p + S(2)))) + h*m*(a*b*f - S(2)*a*c*e + b*c*d) + h*x*(c*(-b*e + S(2)*c*d)*(m + S(2)*p + S(3)) - f*(S(2)*a*c*(m + S(1)) - b**S(2)*(m + p + S(2)))), x), x), x) + Simp((g + h*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*(a*b*f - S(2)*a*c*e + b*c*d + x*(c*(-b*e + S(2)*c*d) + f*(-S(2)*a*c + b**S(2))))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule580 = ReplacementRule(pattern580, replacement580)
pattern581 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('h', S(1)) + WC('g', S(0)))**m_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons394, cons244, cons137, cons166)
def replacement581(p, m, g, f, d, c, a, x, h, e):
rubi.append(581)
return -Dist(S(1)/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(-1))*Simp(a*(e*h*m + f*g) - c*d*g*(S(2)*p + S(3)) + h*x*(a*f*(m + S(1)) - c*d*(m + S(2)*p + S(3))), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(g + h*x)**m*(a*e - x*(-a*f + c*d))/(S(2)*a*c*(p + S(1))), x)
rule581 = ReplacementRule(pattern581, replacement581)
pattern582 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**m_*(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons226, cons244, cons137, cons166)
def replacement582(p, m, g, b, f, d, a, c, x, h):
rubi.append(582)
return -Dist(S(1)/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((g + h*x)**(m + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(g*(S(2)*c**S(2)*d*(S(2)*p + S(3)) - f*(S(2)*a*c - b**S(2)*(p + S(2)))) + h*m*(a*b*f + b*c*d) + h*x*(S(2)*c**S(2)*d*(m + S(2)*p + S(3)) - f*(S(2)*a*c*(m + S(1)) - b**S(2)*(m + p + S(2)))), x), x), x) + Simp((g + h*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*(a*b*f + b*c*d + x*(S(2)*c**S(2)*d + f*(-S(2)*a*c + b**S(2))))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule582 = ReplacementRule(pattern582, replacement582)
pattern583 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(g_ + x_*WC('h', S(1)))**m_*(x_**S(2)*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons244, cons137, cons166)
def replacement583(p, m, f, g, d, c, a, x, h):
rubi.append(583)
return -Dist(S(1)/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(-1))*Simp(a*f*g - c*d*g*(S(2)*p + S(3)) + h*x*(a*f*(m + S(1)) - c*d*(m + S(2)*p + S(3))), x), x), x) - Simp(x*(a + c*x**S(2))**(p + S(1))*(g + h*x)**m*(-a*f + c*d)/(S(2)*a*c*(p + S(1))), x)
rule583 = ReplacementRule(pattern583, replacement583)
pattern584 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons226, cons394, cons13, cons137, cons431)
def replacement584(p, m, g, b, f, d, a, c, x, h, e):
rubi.append(584)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(c*g**S(2) - h*(-a*h + b*g))), Int((g + h*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*Simp(g*(p + S(2))*(-S(2)*a*(-c*e*h + c*f*g) + b**S(2)*f*g - b*(a*f*h + c*d*h + c*e*g) + S(2)*c**S(2)*d*g) + h*x*(m + S(2)*p + S(4))*(-S(2)*a*(-c*e*h + c*f*g) + b**S(2)*f*g - b*(a*f*h + c*d*h + c*e*g) + S(2)*c**S(2)*d*g) - h*(-(-a*e + b*d)*(-b*h + S(2)*c*g) + (-a*f + c*d)*(-S(2)*a*h + b*g))*(m + p + S(2)) + (p + S(1))*(c*g**S(2) - h*(-a*h + b*g))*(S(2)*a*f - b*e + S(2)*c*d), x), x), x) - Simp((g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(-x*(b*f*(-a*h + b*g) + S(2)*c**S(2)*d*g - c*(-S(2)*a*e*h + S(2)*a*f*g + b*d*h + b*e*g)) - (-a*e + b*d)*(-b*h + S(2)*c*g) + (-a*f + c*d)*(-S(2)*a*h + b*g))/((p + S(1))*(-S(4)*a*c + b**S(2))*(c*g**S(2) - h*(-a*h + b*g))), x)
rule584 = ReplacementRule(pattern584, replacement584)
pattern585 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons394, cons13, cons137, cons430)
def replacement585(p, m, g, f, d, c, a, x, h, e):
rubi.append(585)
return Dist(S(1)/(S(2)*a*c*(p + S(1))*(a*h**S(2) + c*g**S(2))), Int((a + c*x**S(2))**(p + S(1))*(g + h*x)**m*Simp(g*(p + S(2))*(-a*(-c*e*h + c*f*g) + c**S(2)*d*g) + h*x*(-a*(-c*e*h + c*f*g) + c**S(2)*d*g)*(m + S(2)*p + S(4)) - h*(a*c*e*g - a*h*(-a*f + c*d))*(m + p + S(2)) + (p + S(1))*(a*f + c*d)*(a*h**S(2) + c*g**S(2)), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(1))*(a*c*e*g - a*h*(-a*f + c*d) - c*x*(a*e*h - a*f*g + c*d*g))/(S(2)*a*c*(p + S(1))*(a*h**S(2) + c*g**S(2))), x)
rule585 = ReplacementRule(pattern585, replacement585)
pattern586 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons21, cons226, cons13, cons137, cons431)
def replacement586(p, m, g, b, f, d, a, c, x, h):
rubi.append(586)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(c*g**S(2) - h*(-a*h + b*g))), Int((g + h*x)**m*(a + b*x + c*x**S(2))**(p + S(1))*Simp(g*(p + S(2))*(-S(2)*a*c*f*g + b**S(2)*f*g - b*(a*f*h + c*d*h) + S(2)*c**S(2)*d*g) + h*x*(m + S(2)*p + S(4))*(-S(2)*a*c*f*g + b**S(2)*f*g - b*(a*f*h + c*d*h) + S(2)*c**S(2)*d*g) - h*(-b*d*(-b*h + S(2)*c*g) + (-a*f + c*d)*(-S(2)*a*h + b*g))*(m + p + S(2)) + S(2)*(p + S(1))*(a*f + c*d)*(c*g**S(2) - h*(-a*h + b*g)), x), x), x) - Simp((g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(-b*d*(-b*h + S(2)*c*g) - x*(b*f*(-a*h + b*g) + S(2)*c**S(2)*d*g - c*(S(2)*a*f*g + b*d*h)) + (-a*f + c*d)*(-S(2)*a*h + b*g))/((p + S(1))*(-S(4)*a*c + b**S(2))*(c*g**S(2) - h*(-a*h + b*g))), x)
rule586 = ReplacementRule(pattern586, replacement586)
pattern587 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**p_*(g_ + x_*WC('h', S(1)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons21, cons13, cons137, cons430)
def replacement587(p, m, f, g, d, c, a, x, h):
rubi.append(587)
return Dist(S(1)/(S(2)*a*c*(p + S(1))*(a*h**S(2) + c*g**S(2))), Int((a + c*x**S(2))**(p + S(1))*(g + h*x)**m*Simp(a*h**S(2)*(-a*f + c*d)*(m + p + S(2)) + g*(p + S(2))*(-a*c*f*g + c**S(2)*d*g) + h*x*(-a*c*f*g + c**S(2)*d*g)*(m + S(2)*p + S(4)) + (p + S(1))*(a*f + c*d)*(a*h**S(2) + c*g**S(2)), x), x), x) - Simp((a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(1))*(a*h*(-a*f + c*d) + c*x*(-a*f*g + c*d*g))/(S(2)*a*c*(p + S(1))*(a*h**S(2) + c*g**S(2))), x)
rule587 = ReplacementRule(pattern587, replacement587)
pattern588 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons226, cons394, cons242)
def replacement588(p, m, g, b, f, d, a, c, x, h, e):
rubi.append(588)
return -Dist(h**(S(-2)), Int((g + h*x)**m*(a + b*x + c*x**S(2))**p*(-d*h**S(2) + f*g**S(2) + h*x*(-e*h + S(2)*f*g)), x), x) + Dist(f/h**S(2), Int((g + h*x)**(m + S(2))*(a + b*x + c*x**S(2))**p, x), x)
rule588 = ReplacementRule(pattern588, replacement588)
pattern589 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons394, cons242)
def replacement589(p, m, g, f, d, c, a, x, h, e):
rubi.append(589)
return -Dist(h**(S(-2)), Int((a + c*x**S(2))**p*(g + h*x)**m*(-d*h**S(2) + f*g**S(2) + h*x*(-e*h + S(2)*f*g)), x), x) + Dist(f/h**S(2), Int((a + c*x**S(2))**p*(g + h*x)**(m + S(2)), x), x)
rule589 = ReplacementRule(pattern589, replacement589)
pattern590 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons21, cons5, cons226, cons242)
def replacement590(p, m, g, b, f, d, a, c, x, h):
rubi.append(590)
return -Dist(h**(S(-2)), Int((g + h*x)**m*(a + b*x + c*x**S(2))**p*(-d*h**S(2) + f*g**S(2) + S(2)*f*g*h*x), x), x) + Dist(f/h**S(2), Int((g + h*x)**(m + S(2))*(a + b*x + c*x**S(2))**p, x), x)
rule590 = ReplacementRule(pattern590, replacement590)
pattern591 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(g_ + x_*WC('h', S(1)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons21, cons5, cons242)
def replacement591(p, m, f, g, d, c, a, x, h):
rubi.append(591)
return -Dist(h**(S(-2)), Int((a + c*x**S(2))**p*(g + h*x)**m*(-d*h**S(2) + f*g**S(2) + S(2)*f*g*h*x), x), x) + Dist(f/h**S(2), Int((a + c*x**S(2))**p*(g + h*x)**(m + S(2)), x), x)
rule591 = ReplacementRule(pattern591, replacement591)
pattern592 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons226, cons394, cons270)
def replacement592(p, m, g, b, f, d, a, c, x, h, e):
rubi.append(592)
return -Dist(S(1)/(c*h*(m + S(2)*p + S(3))), Int((g + h*x)**m*(a + b*x + c*x**S(2))**p*Simp(b*f*g*(p + S(1)) + h*(a*f*(m + S(1)) - c*d*(m + S(2)*p + S(3))) + x*(b*f*h*(m + p + S(2)) + c*(-e*h*(m + S(2)*p + S(3)) + S(2)*f*g*(p + S(1)))), x), x), x) + Simp(f*(g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*h*(m + S(2)*p + S(3))), x)
rule592 = ReplacementRule(pattern592, replacement592)
pattern593 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons394, cons270)
def replacement593(p, m, g, f, d, c, a, x, h, e):
rubi.append(593)
return -Dist(S(1)/(c*h*(m + S(2)*p + S(3))), Int((a + c*x**S(2))**p*(g + h*x)**m*Simp(c*x*(-e*h*(m + S(2)*p + S(3)) + S(2)*f*g*(p + S(1))) + h*(a*f*(m + S(1)) - c*d*(m + S(2)*p + S(3))), x), x), x) + Simp(f*(a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(1))/(c*h*(m + S(2)*p + S(3))), x)
rule593 = ReplacementRule(pattern593, replacement593)
pattern594 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons21, cons5, cons226, cons270)
def replacement594(p, m, g, b, f, d, a, c, x, h):
rubi.append(594)
return -Dist(S(1)/(c*h*(m + S(2)*p + S(3))), Int((g + h*x)**m*(a + b*x + c*x**S(2))**p*Simp(b*f*g*(p + S(1)) + f*x*(b*h*(m + p + S(2)) + S(2)*c*g*(p + S(1))) + h*(a*f*(m + S(1)) - c*d*(m + S(2)*p + S(3))), x), x), x) + Simp(f*(g + h*x)**(m + S(1))*(a + b*x + c*x**S(2))**(p + S(1))/(c*h*(m + S(2)*p + S(3))), x)
rule594 = ReplacementRule(pattern594, replacement594)
pattern595 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)))*(g_ + x_*WC('h', S(1)))**WC('m', S(1)), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons21, cons5, cons270)
def replacement595(p, m, f, g, d, c, a, x, h):
rubi.append(595)
return -Dist(S(1)/(c*h*(m + S(2)*p + S(3))), Int((a + c*x**S(2))**p*(g + h*x)**m*Simp(S(2)*c*f*g*x*(p + S(1)) + h*(a*f*(m + S(1)) - c*d*(m + S(2)*p + S(3))), x), x), x) + Simp(f*(a + c*x**S(2))**(p + S(1))*(g + h*x)**(m + S(1))/(c*h*(m + S(2)*p + S(3))), x)
rule595 = ReplacementRule(pattern595, replacement595)
pattern596 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons220, cons163)
def replacement596(p, g, b, f, d, c, a, x, h, q, e):
rubi.append(596)
return Int(ExpandIntegrand((g + h*x)*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x)
rule596 = ReplacementRule(pattern596, replacement596)
pattern597 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons394, cons220, cons432)
def replacement597(p, g, f, d, c, a, x, h, q, e):
rubi.append(597)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(g + h*x)*(d + e*x + f*x**S(2))**q, x), x)
rule597 = ReplacementRule(pattern597, replacement597)
pattern598 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons402, cons137, cons403)
def replacement598(p, g, b, f, d, c, a, x, h, q, e):
rubi.append(598)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(-1))*Simp(-d*(S(2)*p + S(3))*(b*h - S(2)*c*g) + e*q*(-S(2)*a*h + b*g) - f*x**S(2)*(b*h - S(2)*c*g)*(S(2)*p + S(2)*q + S(3)) + x*(-e*(b*h - S(2)*c*g)*(S(2)*p + q + S(3)) + S(2)*f*q*(-S(2)*a*h + b*g)), x), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*(-S(2)*a*h + b*g - x*(b*h - S(2)*c*g))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule598 = ReplacementRule(pattern598, replacement598)
pattern599 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons394, cons402, cons137, cons403)
def replacement599(p, g, f, d, c, a, x, h, q, e):
rubi.append(599)
return Dist(S(1)/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(-1))*Simp(-a*e*h*q + c*d*g*(S(2)*p + S(3)) + c*f*g*x**S(2)*(S(2)*p + S(2)*q + S(3)) + x*(-S(2)*a*f*h*q + c*e*g*(S(2)*p + q + S(3))), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(a*h - c*g*x)*(d + e*x + f*x**S(2))**q/(S(2)*a*c*(p + S(1))), x)
rule599 = ReplacementRule(pattern599, replacement599)
pattern600 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons226, cons402, cons137, cons403)
def replacement600(p, g, b, f, d, c, a, x, h, q):
rubi.append(600)
return -Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + f*x**S(2))**(q + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(-d*(S(2)*p + S(3))*(b*h - S(2)*c*g) + S(2)*f*q*x*(-S(2)*a*h + b*g) - f*x**S(2)*(b*h - S(2)*c*g)*(S(2)*p + S(2)*q + S(3)), x), x), x) + Simp((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(1))*(-S(2)*a*h + b*g - x*(b*h - S(2)*c*g))/((p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule600 = ReplacementRule(pattern600, replacement600)
pattern601 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons50, cons226, cons394, cons13, cons137, cons404, cons405)
def replacement601(p, g, b, f, d, c, a, x, h, q, e):
rubi.append(601)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*Simp(-c*f*x**S(2)*(S(2)*p + S(2)*q + S(5))*(S(2)*a*c*e*h + b**S(2)*f*g - b*(a*f*h + c*d*h + c*e*g) + S(2)*c*g*(-a*f + c*d)) - e*(c*g*(S(2)*a*c*e - b*(a*f + c*d)) + (-a*h + b*g)*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)))*(p + q + S(2)) - x*(S(2)*f*(c*g*(S(2)*a*c*e - b*(a*f + c*d)) + (-a*h + b*g)*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)))*(p + q + S(2)) - (b*f*(p + S(1)) - c*e*(S(2)*p + q + S(4)))*(S(2)*a*c*e*h + b**S(2)*f*g - b*(a*f*h + c*d*h + c*e*g) + S(2)*c*g*(-a*f + c*d))) + (p + S(1))*(b*h - S(2)*c*g)*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2)) + (a*f*(p + S(1)) - c*d*(p + S(2)))*(S(2)*a*c*e*h + b**S(2)*f*g - b*(a*f*h + c*d*h + c*e*g) + S(2)*c*g*(-a*f + c*d)), x), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(1))*(c*g*(S(2)*a*c*e - b*(a*f + c*d)) + c*x*(g*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) - h*(a*b*f - S(2)*a*c*e + b*c*d)) + (-a*h + b*g)*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)))/((p + S(1))*(-S(4)*a*c + b**S(2))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))), x)
rule601 = ReplacementRule(pattern601, replacement601)
pattern602 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons50, cons394, cons13, cons137, cons407, cons405)
def replacement602(p, g, f, d, c, a, x, h, q, e):
rubi.append(602)
return Dist(-S(1)/(S(4)*a*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*Simp(-c*f*x**S(2)*(S(2)*a*c*e*h + S(2)*c*g*(-a*f + c*d))*(S(2)*p + S(2)*q + S(5)) - S(2)*c*g*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2)) - e*(S(2)*a*c**S(2)*e*g - a*h*(-S(2)*a*c*f + S(2)*c**S(2)*d))*(p + q + S(2)) - x*(c*e*(S(2)*a*c*e*h + S(2)*c*g*(-a*f + c*d))*(S(2)*p + q + S(4)) + S(2)*f*(S(2)*a*c**S(2)*e*g - a*h*(-S(2)*a*c*f + S(2)*c**S(2)*d))*(p + q + S(2))) + (a*f*(p + S(1)) - c*d*(p + S(2)))*(S(2)*a*c*e*h + S(2)*c*g*(-a*f + c*d)), x), x), x) + Simp(-(a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(1))*(S(2)*a*c**S(2)*e*g - a*h*(-S(2)*a*c*f + S(2)*c**S(2)*d) + c*x*(S(2)*a*c*e*h + g*(-S(2)*a*c*f + S(2)*c**S(2)*d)))/(S(4)*a*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2))), x)
rule602 = ReplacementRule(pattern602, replacement602)
pattern603 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons50, cons226, cons13, cons137, cons406, cons405)
def replacement603(p, g, b, f, d, c, a, x, h, q):
rubi.append(603)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(b**S(2)*d*f + (-a*f + c*d)**S(2))), Int((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(1))*Simp(-c*f*x**S(2)*(S(2)*p + S(2)*q + S(5))*(b**S(2)*f*g - b*(a*f*h + c*d*h) + S(2)*c*g*(-a*f + c*d)) - x*(-b*f*(p + S(1))*(b**S(2)*f*g - b*(a*f*h + c*d*h) + S(2)*c*g*(-a*f + c*d)) + S(2)*f*(-b*c*g*(a*f + c*d) + (-a*h + b*g)*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d))*(p + q + S(2))) + (p + S(1))*(b*h - S(2)*c*g)*(b**S(2)*d*f + (-a*f + c*d)**S(2)) + (a*f*(p + S(1)) - c*d*(p + S(2)))*(b**S(2)*f*g - b*(a*f*h + c*d*h) + S(2)*c*g*(-a*f + c*d)), x), x), x) + Simp((d + f*x**S(2))**(q + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(-b*c*g*(a*f + c*d) + c*x*(g*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d) - h*(a*b*f + b*c*d)) + (-a*h + b*g)*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d))/((p + S(1))*(-S(4)*a*c + b**S(2))*(b**S(2)*d*f + (-a*f + c*d)**S(2))), x)
rule603 = ReplacementRule(pattern603, replacement603)
pattern604 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons50, cons226, cons394, cons13, cons163, cons433)
def replacement604(p, g, b, f, d, c, a, x, h, q, e):
rubi.append(604)
return -Dist(S(1)/(S(2)*f*(p + q + S(1))), Int((a + b*x + c*x**S(2))**(p + S(-1))*(d + e*x + f*x**S(2))**q*Simp(a*(e*h - S(2)*f*g)*(p + q + S(1)) + h*p*(-a*e + b*d) + x**S(2)*(c*(e*h - S(2)*f*g)*(p + q + S(1)) + h*p*(-b*f + c*e)) + x*(b*(e*h - S(2)*f*g)*(p + q + S(1)) + S(2)*h*p*(-a*f + c*d)), x), x), x) + Simp(h*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**(q + S(1))/(S(2)*f*(p + q + S(1))), x)
rule604 = ReplacementRule(pattern604, replacement604)
pattern605 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons50, cons394, cons13, cons163, cons433)
def replacement605(p, g, f, d, c, a, x, h, q, e):
rubi.append(605)
return Dist(S(1)/(S(2)*f*(p + q + S(1))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x + f*x**S(2))**q*Simp(a*e*h*p - a*(e*h - S(2)*f*g)*(p + q + S(1)) - S(2)*h*p*x*(-a*f + c*d) - x**S(2)*(c*e*h*p + c*(e*h - S(2)*f*g)*(p + q + S(1))), x), x), x) + Simp(h*(a + c*x**S(2))**p*(d + e*x + f*x**S(2))**(q + S(1))/(S(2)*f*(p + q + S(1))), x)
rule605 = ReplacementRule(pattern605, replacement605)
pattern606 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons50, cons226, cons13, cons163, cons433)
def replacement606(p, g, b, f, d, c, a, x, h, q):
rubi.append(606)
return -Dist(S(1)/(S(2)*f*(p + q + S(1))), Int((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(-1))*Simp(-S(2)*a*f*g*(p + q + S(1)) + b*d*h*p + x**S(2)*(-b*f*h*p - S(2)*c*f*g*(p + q + S(1))) + x*(-S(2)*b*f*g*(p + q + S(1)) + S(2)*h*p*(-a*f + c*d)), x), x), x) + Simp(h*(d + f*x**S(2))**(q + S(1))*(a + b*x + c*x**S(2))**p/(S(2)*f*(p + q + S(1))), x)
rule606 = ReplacementRule(pattern606, replacement606)
def With607(g, b, f, d, c, a, x, h, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = a**S(2)*f**S(2) - a*b*e*f - S(2)*a*c*d*f + a*c*e**S(2) + b**S(2)*d*f - b*c*d*e + c**S(2)*d**S(2)
if NonzeroQ(q):
return True
return False
pattern607 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, CustomConstraint(With607))
def replacement607(g, b, f, d, c, a, x, h, e):
q = a**S(2)*f**S(2) - a*b*e*f - S(2)*a*c*d*f + a*c*e**S(2) + b**S(2)*d*f - b*c*d*e + c**S(2)*d**S(2)
rubi.append(607)
return Dist(S(1)/q, Int(Simp(-a*b*f*h + a*c*e*h - a*c*f*g + b**S(2)*f*g - b*c*e*g + c**S(2)*d*g + c*x*(-a*f*h + b*f*g + c*d*h - c*e*g), x)/(a + b*x + c*x**S(2)), x), x) + Dist(S(1)/q, Int(Simp(a*f**S(2)*g + b*d*f*h - b*e*f*g - c*d*e*h - c*d*f*g + c*e**S(2)*g - f*x*(-a*f*h + b*f*g + c*d*h - c*e*g), x)/(d + e*x + f*x**S(2)), x), x)
rule607 = ReplacementRule(pattern607, replacement607)
def With608(g, b, f, d, c, a, x, h):
if isinstance(x, (int, Integer, float, Float)):
return False
q = a**S(2)*f**S(2) - S(2)*a*c*d*f + b**S(2)*d*f + c**S(2)*d**S(2)
if NonzeroQ(q):
return True
return False
pattern608 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((d_ + x_**S(2)*WC('f', S(1)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons226, CustomConstraint(With608))
def replacement608(g, b, f, d, c, a, x, h):
q = a**S(2)*f**S(2) - S(2)*a*c*d*f + b**S(2)*d*f + c**S(2)*d**S(2)
rubi.append(608)
return Dist(S(1)/q, Int(Simp(a*f**S(2)*g + b*d*f*h - c*d*f*g - f*x*(-a*f*h + b*f*g + c*d*h), x)/(d + f*x**S(2)), x), x) + Dist(S(1)/q, Int(Simp(-a*b*f*h - a*c*f*g + b**S(2)*f*g + c**S(2)*d*g + c*x*(-a*f*h + b*f*g + c*d*h), x)/(a + b*x + c*x**S(2)), x), x)
rule608 = ReplacementRule(pattern608, replacement608)
pattern609 = Pattern(Integral((g_ + x_*WC('h', S(1)))/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(d_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons434)
def replacement609(f, g, d, c, a, x, h):
rubi.append(609)
return Dist(g, Int(S(1)/((a + c*x**S(2))*sqrt(d + f*x**S(2))), x), x) + Dist(h, Int(x/((a + c*x**S(2))*sqrt(d + f*x**S(2))), x), x)
rule609 = ReplacementRule(pattern609, replacement609)
def With610(f, g, d, c, a, x, h):
q = Rt(-a*c, S(2))
rubi.append(610)
return -Dist((c*g - h*q)/(S(2)*q), Int(S(1)/(sqrt(d + f*x**S(2))*(c*x + q)), x), x) - Dist((c*g + h*q)/(S(2)*q), Int(S(1)/(sqrt(d + f*x**S(2))*(-c*x + q)), x), x)
pattern610 = Pattern(Integral((g_ + x_*WC('h', S(1)))/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(d_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons435)
rule610 = ReplacementRule(pattern610, With610)
pattern611 = Pattern(Integral((g_ + x_*WC('h', S(1)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons410, cons436)
def replacement611(f, b, g, d, c, a, x, h, e):
rubi.append(611)
return Dist(-S(2)*g, Subst(Int(S(1)/(-a*e + b*d - b*x**S(2)), x), x, sqrt(d + e*x + f*x**S(2))), x)
rule611 = ReplacementRule(pattern611, replacement611)
pattern612 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons410, cons437)
def replacement612(g, b, f, d, c, a, x, h, e):
rubi.append(612)
return Dist(h/(S(2)*f), Int((e + S(2)*f*x)/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) - Dist((e*h - S(2)*f*g)/(S(2)*f), Int(S(1)/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x)
rule612 = ReplacementRule(pattern612, replacement612)
pattern613 = Pattern(Integral(x_/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons226, cons394, cons383)
def replacement613(f, b, d, c, a, x, e):
rubi.append(613)
return Dist(-S(2)*e, Subst(Int((-d*x**S(2) + S(1))/(-b*f + c*e + d**S(2)*x**S(4)*(-b*f + c*e) - e*x**S(2)*(S(2)*a*f - b*e + S(2)*c*d)), x), x, (S(1) + x*(e + sqrt(-S(4)*d*f + e**S(2)))/(S(2)*d))/sqrt(d + e*x + f*x**S(2))), x)
rule613 = ReplacementRule(pattern613, replacement613)
pattern614 = Pattern(Integral((g_ + x_*WC('h', S(1)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons383, cons438)
def replacement614(f, b, g, d, c, a, x, h, e):
rubi.append(614)
return Dist(g, Subst(Int(S(1)/(a + x**S(2)*(-a*f + c*d)), x), x, x/sqrt(d + e*x + f*x**S(2))), x)
rule614 = ReplacementRule(pattern614, replacement614)
pattern615 = Pattern(Integral((g_ + x_*WC('h', S(1)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons383, cons439)
def replacement615(f, b, g, d, c, a, x, h, e):
rubi.append(615)
return Dist(h/e, Int((S(2)*d + e*x)/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) - Dist((S(2)*d*h - e*g)/e, Int(S(1)/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x)
rule615 = ReplacementRule(pattern615, replacement615)
pattern616 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons372, cons440)
def replacement616(g, b, f, d, a, c, x, h, e):
rubi.append(616)
return Dist(-S(2)*g*(-S(2)*a*h + b*g), Subst(Int(S(1)/Simp(g*(-S(4)*a*c + b**S(2))*(-S(2)*a*h + b*g) - x**S(2)*(-a*e + b*d), x), x), x, Simp(-S(2)*a*h + b*g - x*(b*h - S(2)*c*g), x)/sqrt(d + e*x + f*x**S(2))), x)
rule616 = ReplacementRule(pattern616, replacement616)
pattern617 = Pattern(Integral((g_ + x_*WC('h', S(1)))/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons441)
def replacement617(f, g, d, c, a, x, h, e):
rubi.append(617)
return Dist(-S(2)*a*g*h, Subst(Int(S(1)/Simp(S(2)*a**S(2)*c*g*h + a*e*x**S(2), x), x), x, Simp(a*h - c*g*x, x)/sqrt(d + e*x + f*x**S(2))), x)
rule617 = ReplacementRule(pattern617, replacement617)
pattern618 = Pattern(Integral((g_ + x_*WC('h', S(1)))/(sqrt(d_ + x_**S(2)*WC('f', S(1)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons226, cons442)
def replacement618(f, b, g, d, c, a, x, h):
rubi.append(618)
return Dist(-S(2)*g*(-S(2)*a*h + b*g), Subst(Int(S(1)/Simp(-b*d*x**S(2) + g*(-S(4)*a*c + b**S(2))*(-S(2)*a*h + b*g), x), x), x, Simp(-S(2)*a*h + b*g - x*(b*h - S(2)*c*g), x)/sqrt(d + f*x**S(2))), x)
rule618 = ReplacementRule(pattern618, replacement618)
def With619(g, b, f, d, c, a, x, h, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(619)
return Dist((S(2)*c*g - h*(b - q))/q, Int(S(1)/((b + S(2)*c*x - q)*sqrt(d + e*x + f*x**S(2))), x), x) - Dist((S(2)*c*g - h*(b + q))/q, Int(S(1)/((b + S(2)*c*x + q)*sqrt(d + e*x + f*x**S(2))), x), x)
pattern619 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons231)
rule619 = ReplacementRule(pattern619, With619)
def With620(g, f, d, c, a, x, h, e):
q = Rt(-a*c, S(2))
rubi.append(620)
return Dist(-c*g/(S(2)*q) + h/S(2), Int(S(1)/((c*x + q)*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(c*g/(S(2)*q) + h/S(2), Int(S(1)/((c*x - q)*sqrt(d + e*x + f*x**S(2))), x), x)
pattern620 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons394, cons412)
rule620 = ReplacementRule(pattern620, With620)
def With621(g, b, f, d, c, a, x, h):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(621)
return Dist((S(2)*c*g - h*(b - q))/q, Int(S(1)/(sqrt(d + f*x**S(2))*(b + S(2)*c*x - q)), x), x) - Dist((S(2)*c*g - h*(b + q))/q, Int(S(1)/(sqrt(d + f*x**S(2))*(b + S(2)*c*x + q)), x), x)
pattern621 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/(sqrt(d_ + x_**S(2)*WC('f', S(1)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons226, cons231)
rule621 = ReplacementRule(pattern621, With621)
def With622(g, b, f, d, a, c, x, h, e):
q = Rt(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2), S(2))
rubi.append(622)
return Dist(S(1)/(S(2)*q), Int(Simp(-g*(-a*f + c*d - q) + h*(-a*e + b*d) - x*(g*(-b*f + c*e) - h*(-a*f + c*d + q)), x)/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) - Dist(S(1)/(S(2)*q), Int(Simp(-g*(-a*f + c*d + q) + h*(-a*e + b*d) - x*(g*(-b*f + c*e) - h*(-a*f + c*d - q)), x)/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x)
pattern622 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394, cons372, cons413)
rule622 = ReplacementRule(pattern622, With622)
def With623(g, f, d, c, a, x, h, e):
q = Rt(a*c*e**S(2) + (-a*f + c*d)**S(2), S(2))
rubi.append(623)
return Dist(S(1)/(S(2)*q), Int(Simp(-a*e*h - g*(-a*f + c*d - q) + x*(-c*e*g + h*(-a*f + c*d + q)), x)/((a + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) - Dist(S(1)/(S(2)*q), Int(Simp(-a*e*h - g*(-a*f + c*d + q) + x*(-c*e*g + h*(-a*f + c*d - q)), x)/((a + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x)
pattern623 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons394, cons414)
rule623 = ReplacementRule(pattern623, With623)
def With624(g, b, f, d, a, c, x, h):
q = Rt(b**S(2)*d*f + (-a*f + c*d)**S(2), S(2))
rubi.append(624)
return Dist(S(1)/(S(2)*q), Int(Simp(b*d*h - g*(-a*f + c*d - q) + x*(b*f*g + h*(-a*f + c*d + q)), x)/(sqrt(d + f*x**S(2))*(a + b*x + c*x**S(2))), x), x) - Dist(S(1)/(S(2)*q), Int(Simp(b*d*h - g*(-a*f + c*d + q) + x*(b*f*g + h*(-a*f + c*d - q)), x)/(sqrt(d + f*x**S(2))*(a + b*x + c*x**S(2))), x), x)
pattern624 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/(sqrt(d_ + x_**S(2)*WC('f', S(1)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons226, cons413)
rule624 = ReplacementRule(pattern624, With624)
def With625(g, b, f, d, c, a, x, h, e):
s = Rt(-S(4)*a*c + b**S(2), S(2))
t = Rt(-S(4)*d*f + e**S(2), S(2))
rubi.append(625)
return Dist(sqrt(S(2)*a + x*(b + s))*sqrt(S(2)*d + x*(e + t))*sqrt(b + S(2)*c*x + s)*sqrt(e + S(2)*f*x + t)/(sqrt(a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), Int((g + h*x)/(sqrt(S(2)*a + x*(b + s))*sqrt(S(2)*d + x*(e + t))*sqrt(b + S(2)*c*x + s)*sqrt(e + S(2)*f*x + t)), x), x)
pattern625 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/(sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons394)
rule625 = ReplacementRule(pattern625, With625)
def With626(g, b, f, d, c, a, x, h):
s = Rt(-S(4)*a*c + b**S(2), S(2))
t = Rt(-S(4)*d*f, S(2))
rubi.append(626)
return Dist(sqrt(S(2)*a + x*(b + s))*sqrt(S(2)*d + t*x)*sqrt(S(2)*f*x + t)*sqrt(b + S(2)*c*x + s)/(sqrt(d + f*x**S(2))*sqrt(a + b*x + c*x**S(2))), Int((g + h*x)/(sqrt(S(2)*a + x*(b + s))*sqrt(S(2)*d + t*x)*sqrt(S(2)*f*x + t)*sqrt(b + S(2)*c*x + s)), x), x)
pattern626 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/(sqrt(d_ + x_**S(2)*WC('f', S(1)))*sqrt(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons209, cons226)
rule626 = ReplacementRule(pattern626, With626)
def With627(g, b, f, d, a, c, x, h, e):
q = S(3)**(S(2)/3)*(-c*h**S(2)/(-b*h + S(2)*c*g)**S(2))**(S(1)/3)
rubi.append(627)
return -Simp(S(3)*h*q*log((-S(3)*h*(b + S(2)*c*x)/(-b*h + S(2)*c*g) + S(1))**(S(2)/3) + S(2)**(S(1)/3)*(S(3)*h*(b + S(2)*c*x)/(-b*h + S(2)*c*g) + S(1))**(S(1)/3))/(S(2)*f), x) + Simp(h*q*log(d + e*x + f*x**S(2))/(S(2)*f), x) + Simp(sqrt(S(3))*h*q*ArcTan(-S(2)**(S(2)/3)*sqrt(S(3))*(-S(3)*h*(b + S(2)*c*x)/(-b*h + S(2)*c*g) + S(1))**(S(2)/3)/(S(3)*(S(3)*h*(b + S(2)*c*x)/(-b*h + S(2)*c*g) + S(1))**(S(1)/3)) + sqrt(S(3))/S(3))/f, x)
pattern627 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**(S(1)/3)*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons410, cons443, cons444, cons445)
rule627 = ReplacementRule(pattern627, With627)
pattern628 = Pattern(Integral((g_ + x_*WC('h', S(1)))/((a_ + x_**S(2)*WC('c', S(1)))**(S(1)/3)*(d_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons446, cons447, cons43)
def replacement628(f, g, d, c, a, x, h):
rubi.append(628)
return Simp(S(2)**(S(1)/3)*h*log(d + f*x**S(2))/(S(4)*a**(S(1)/3)*f), x) - Simp(S(3)*S(2)**(S(1)/3)*h*log((S(1) - S(3)*h*x/g)**(S(2)/3) + S(2)**(S(1)/3)*(S(1) + S(3)*h*x/g)**(S(1)/3))/(S(4)*a**(S(1)/3)*f), x) + Simp(S(2)**(S(1)/3)*sqrt(S(3))*h*ArcTan(-S(2)**(S(2)/3)*sqrt(S(3))*(S(1) - S(3)*h*x/g)**(S(2)/3)/(S(3)*(S(1) + S(3)*h*x/g)**(S(1)/3)) + sqrt(S(3))/S(3))/(S(2)*a**(S(1)/3)*f), x)
rule628 = ReplacementRule(pattern628, replacement628)
def With629(g, b, f, d, a, c, x, h, e):
q = -c/(-S(4)*a*c + b**S(2))
rubi.append(629)
return Dist((q*(a + b*x + c*x**S(2)))**(S(1)/3)/(a + b*x + c*x**S(2))**(S(1)/3), Int((g + h*x)/((d + e*x + f*x**S(2))*(a*q + b*q*x + c*q*x**S(2))**(S(1)/3)), x), x)
pattern629 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/((x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**(S(1)/3)*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons410, cons443, cons444, cons313)
rule629 = ReplacementRule(pattern629, With629)
pattern630 = Pattern(Integral((g_ + x_*WC('h', S(1)))/((a_ + x_**S(2)*WC('c', S(1)))**(S(1)/3)*(d_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons7, cons27, cons125, cons208, cons209, cons446, cons447, cons448)
def replacement630(f, g, d, c, a, x, h):
rubi.append(630)
return Dist((S(1) + c*x**S(2)/a)**(S(1)/3)/(a + c*x**S(2))**(S(1)/3), Int((g + h*x)/((S(1) + c*x**S(2)/a)**(S(1)/3)*(d + f*x**S(2))), x), x)
rule630 = ReplacementRule(pattern630, replacement630)
pattern631 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons449)
def replacement631(p, g, b, f, d, a, c, x, h, q, e):
rubi.append(631)
return Int((g + h*x)*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x)
rule631 = ReplacementRule(pattern631, replacement631)
pattern632 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))*(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons450)
def replacement632(p, g, f, d, a, c, x, h, q, e):
rubi.append(632)
return Int((a + c*x**S(2))**p*(g + h*x)*(d + e*x + f*x**S(2))**q, x)
rule632 = ReplacementRule(pattern632, replacement632)
pattern633 = Pattern(Integral((u_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(u_**S(2)*WC('c', S(1)) + u_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons68, cons69)
def replacement633(p, u, m, g, b, f, d, a, c, x, h, q, e):
rubi.append(633)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((g + h*x)**m*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule633 = ReplacementRule(pattern633, replacement633)
pattern634 = Pattern(Integral((u_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(u_**S(2)*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons68, cons69)
def replacement634(p, u, m, g, f, d, c, a, x, h, q, e):
rubi.append(634)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + c*x**S(2))**p*(g + h*x)**m*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule634 = ReplacementRule(pattern634, replacement634)
pattern635 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1))*z_**WC('m', S(1)), x_), cons21, cons5, cons50, cons451, cons452, cons453)
def replacement635(v, z, p, u, m, x, q):
rubi.append(635)
return Int(ExpandToSum(u, x)**p*ExpandToSum(v, x)**q*ExpandToSum(z, x)**m, x)
rule635 = ReplacementRule(pattern635, replacement635)
pattern636 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(f_ + x_*WC('g', S(1)))**WC('n', S(1))*(x_*WC('i', S(1)) + WC('h', S(0)))**WC('q', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons21, cons4, cons5, cons50, cons336, cons124, cons377)
def replacement636(p, m, g, b, f, i, d, a, n, c, x, h, q, e):
rubi.append(636)
return Int((h + i*x)**q*(d*f + e*g*x**S(2))**m*(a + b*x + c*x**S(2))**p, x)
rule636 = ReplacementRule(pattern636, replacement636)
pattern637 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1))*(x_*WC('i', S(1)) + WC('h', S(0)))**WC('q', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons21, cons4, cons5, cons50, cons128, cons84)
def replacement637(p, m, f, g, b, i, d, a, n, c, x, h, q, e):
rubi.append(637)
return Int(ExpandIntegrand((d + e*x)**m*(f + g*x)**n*(h + i*x)**q*(a + b*x + c*x**S(2))**p, x), x)
rule637 = ReplacementRule(pattern637, replacement637)
pattern638 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(f_ + x_*WC('g', S(1)))**n_*(x_*WC('i', S(1)) + WC('h', S(0)))**WC('q', S(1))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons21, cons4, cons5, cons50, cons336, cons124)
def replacement638(p, m, g, b, f, i, d, a, c, n, x, h, q, e):
rubi.append(638)
return Dist((d + e*x)**FracPart(m)*(f + g*x)**FracPart(m)*(d*f + e*g*x**S(2))**(-FracPart(m)), Int((h + i*x)**q*(d*f + e*g*x**S(2))**m*(a + b*x + c*x**S(2))**p, x), x)
rule638 = ReplacementRule(pattern638, replacement638)
pattern639 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons382, cons383, cons384, cons385)
def replacement639(B, p, C, e, f, b, d, c, a, x, q, A):
rubi.append(639)
return Dist((c/f)**p, Int((A + B*x + C*x**S(2))*(d + e*x + f*x**S(2))**(p + q), x), x)
rule639 = ReplacementRule(pattern639, replacement639)
pattern640 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons5, cons50, cons382, cons383, cons384, cons385)
def replacement640(C, p, e, f, b, d, c, a, x, q, A):
rubi.append(640)
return Dist((c/f)**p, Int((A + C*x**S(2))*(d + e*x + f*x**S(2))**(p + q), x), x)
rule640 = ReplacementRule(pattern640, replacement640)
pattern641 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons382, cons383, cons147, cons386, cons387)
def replacement641(B, p, C, e, f, b, d, c, a, x, q, A):
rubi.append(641)
return Dist(a**IntPart(p)*d**(-IntPart(p))*(a + b*x + c*x**S(2))**FracPart(p)*(d + e*x + f*x**S(2))**(-FracPart(p)), Int((A + B*x + C*x**S(2))*(d + e*x + f*x**S(2))**(p + q), x), x)
rule641 = ReplacementRule(pattern641, replacement641)
pattern642 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons5, cons50, cons382, cons383, cons147, cons386, cons387)
def replacement642(C, p, e, f, b, d, c, a, x, q, A):
rubi.append(642)
return Dist(a**IntPart(p)*d**(-IntPart(p))*(a + b*x + c*x**S(2))**FracPart(p)*(d + e*x + f*x**S(2))**(-FracPart(p)), Int((A + C*x**S(2))*(d + e*x + f*x**S(2))**(p + q), x), x)
rule642 = ReplacementRule(pattern642, replacement642)
pattern643 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons45)
def replacement643(B, p, C, e, f, b, d, c, a, x, q, A):
rubi.append(643)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((b + S(2)*c*x)**(S(2)*p)*(A + B*x + C*x**S(2))*(d + e*x + f*x**S(2))**q, x), x)
rule643 = ReplacementRule(pattern643, replacement643)
pattern644 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons5, cons50, cons45)
def replacement644(C, p, e, f, b, d, c, a, x, q, A):
rubi.append(644)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((A + C*x**S(2))*(b + S(2)*c*x)**(S(2)*p)*(d + e*x + f*x**S(2))**q, x), x)
rule644 = ReplacementRule(pattern644, replacement644)
pattern645 = Pattern(Integral((x_**S(2)*WC('f', S(1)) + WC('d', S(0)))**WC('q', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons35, cons36, cons5, cons50, cons45)
def replacement645(B, p, C, f, b, d, c, a, x, q, A):
rubi.append(645)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((b + S(2)*c*x)**(S(2)*p)*(d + f*x**S(2))**q*(A + B*x + C*x**S(2)), x), x)
rule645 = ReplacementRule(pattern645, replacement645)
pattern646 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))**WC('q', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons36, cons5, cons50, cons45)
def replacement646(C, p, f, b, d, c, a, x, q, A):
rubi.append(646)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x)**(-S(2)*FracPart(p))*(a + b*x + c*x**S(2))**FracPart(p), Int((A + C*x**S(2))*(b + S(2)*c*x)**(S(2)*p)*(d + f*x**S(2))**q, x), x)
rule646 = ReplacementRule(pattern646, replacement646)
pattern647 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons226, cons394, cons220, cons163)
def replacement647(B, p, C, e, f, b, d, c, a, x, q, A):
rubi.append(647)
return Int(ExpandIntegrand((A + B*x + C*x**S(2))*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x)
rule647 = ReplacementRule(pattern647, replacement647)
pattern648 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons226, cons394, cons220, cons163)
def replacement648(C, p, e, f, b, d, c, a, x, q, A):
rubi.append(648)
return Int(ExpandIntegrand((A + C*x**S(2))*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x)
rule648 = ReplacementRule(pattern648, replacement648)
pattern649 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons394, cons220, cons432)
def replacement649(B, C, p, e, f, d, c, a, x, q, A):
rubi.append(649)
return Int(ExpandIntegrand((a + c*x**S(2))**p*(A + B*x + C*x**S(2))*(d + e*x + f*x**S(2))**q, x), x)
rule649 = ReplacementRule(pattern649, replacement649)
pattern650 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons36, cons394, cons220, cons432)
def replacement650(C, p, f, d, c, a, A, x, q, e):
rubi.append(650)
return Int(ExpandIntegrand((A + C*x**S(2))*(a + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x)
rule650 = ReplacementRule(pattern650, replacement650)
pattern651 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons226, cons394, cons402, cons137, cons403)
def replacement651(B, p, C, e, f, b, d, c, a, x, q, A):
rubi.append(651)
return -Dist(S(1)/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(-1))*Simp(-d*(C*(S(2)*a*c - b**S(2)*(p + S(2))) + c*(S(2)*p + S(3))*(-S(2)*A*c + B*b)) + e*q*(A*b*c - S(2)*B*a*c + C*a*b) - f*x**S(2)*(C*(S(2)*a*c*(S(2)*q + S(1)) - b**S(2)*(p + S(2)*q + S(2))) + c*(-S(2)*A*c + B*b)*(S(2)*p + S(2)*q + S(3))) + x*(-e*(C*(S(2)*a*c*(q + S(1)) - b**S(2)*(p + q + S(2))) + c*(-S(2)*A*c + B*b)*(S(2)*p + q + S(3))) + S(2)*f*q*(A*b*c - S(2)*B*a*c + C*a*b)), x), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*(A*b*c - S(2)*B*a*c + C*a*b - x*(-C*(-S(2)*a*c + b**S(2)) + c*(-S(2)*A*c + B*b)))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule651 = ReplacementRule(pattern651, replacement651)
pattern652 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons226, cons394, cons402, cons137, cons403)
def replacement652(C, p, e, f, b, d, c, a, x, q, A):
rubi.append(652)
return -Dist(S(1)/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(-1))*Simp(A*c*(b*e*q + S(2)*c*d*(S(2)*p + S(3))) - C*(-a*b*e*q + S(2)*a*c*d - b**S(2)*d*(p + S(2))) - f*x**S(2)*(-S(2)*A*c**S(2)*(S(2)*p + S(2)*q + S(3)) + C*(S(2)*a*c*(S(2)*q + S(1)) - b**S(2)*(p + S(2)*q + S(2)))) + x*(S(2)*A*c*(b*f*q + c*e*(S(2)*p + q + S(3))) + C*(S(2)*a*b*f*q - S(2)*a*c*e*(q + S(1)) + b**S(2)*e*(p + q + S(2)))), x), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*(A*b*c + C*a*b + x*(S(2)*A*c**S(2) + C*(-S(2)*a*c + b**S(2))))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule652 = ReplacementRule(pattern652, replacement652)
pattern653 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons394, cons402, cons137, cons403)
def replacement653(B, C, p, e, f, d, c, a, x, q, A):
rubi.append(653)
return -Dist(-S(1)/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(-1))*Simp(A*c*d*(S(2)*p + S(3)) - a*(B*e*q + C*d) - f*x**S(2)*(-A*c*(S(2)*p + S(2)*q + S(3)) + C*a*(S(2)*q + S(1))) + x*(A*c*e*(S(2)*p + q + S(3)) - a*(S(2)*B*f*q + C*e*(q + S(1)))), x), x), x) + Simp((a + c*x**S(2))**(p + S(1))*(B*a - x*(A*c - C*a))*(d + e*x + f*x**S(2))**q/(S(2)*a*c*(p + S(1))), x)
rule653 = ReplacementRule(pattern653, replacement653)
pattern654 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons36, cons394, cons402, cons137, cons403)
def replacement654(C, p, f, d, c, a, A, x, q, e):
rubi.append(654)
return Dist(S(1)/(S(2)*a*c*(p + S(1))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(-1))*Simp(A*c*d*(S(2)*p + S(3)) - C*a*d - f*x**S(2)*(-A*c*(S(2)*p + S(2)*q + S(3)) + C*a*(S(2)*q + S(1))) + x*(A*c*e*(S(2)*p + q + S(3)) - C*a*e*(q + S(1))), x), x), x) - Simp(x*(a + c*x**S(2))**(p + S(1))*(A*c - C*a)*(d + e*x + f*x**S(2))**q/(S(2)*a*c*(p + S(1))), x)
rule654 = ReplacementRule(pattern654, replacement654)
pattern655 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons35, cons36, cons226, cons402, cons137, cons403)
def replacement655(B, p, C, f, b, d, c, a, x, q, A):
rubi.append(655)
return -Dist(S(1)/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + f*x**S(2))**(q + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(-d*(C*(S(2)*a*c - b**S(2)*(p + S(2))) + c*(S(2)*p + S(3))*(-S(2)*A*c + B*b)) + S(2)*f*q*x*(A*b*c - S(2)*B*a*c + C*a*b) - f*x**S(2)*(C*(S(2)*a*c*(S(2)*q + S(1)) - b**S(2)*(p + S(2)*q + S(2))) + c*(-S(2)*A*c + B*b)*(S(2)*p + S(2)*q + S(3))), x), x), x) + Simp((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(1))*(A*b*c - S(2)*B*a*c + C*a*b - x*(-C*(-S(2)*a*c + b**S(2)) + c*(-S(2)*A*c + B*b)))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule655 = ReplacementRule(pattern655, replacement655)
pattern656 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons36, cons226, cons402, cons137, cons403)
def replacement656(C, p, f, b, d, c, a, x, q, A):
rubi.append(656)
return -Dist(S(1)/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((d + f*x**S(2))**(q + S(-1))*(a + b*x + c*x**S(2))**(p + S(1))*Simp(S(2)*A*c**S(2)*d*(S(2)*p + S(3)) - C*(S(2)*a*c*d - b**S(2)*d*(p + S(2))) - f*x**S(2)*(-S(2)*A*c**S(2)*(S(2)*p + S(2)*q + S(3)) + C*(S(2)*a*c*(S(2)*q + S(1)) - b**S(2)*(p + S(2)*q + S(2)))) + x*(S(2)*A*b*c*f*q + S(2)*C*a*b*f*q), x), x), x) + Simp((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(1))*(A*b*c + C*a*b + x*(S(2)*A*c**S(2) + C*(-S(2)*a*c + b**S(2))))/(c*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule656 = ReplacementRule(pattern656, replacement656)
pattern657 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons50, cons226, cons394, cons13, cons137, cons404, cons405)
def replacement657(B, p, C, e, f, b, d, c, a, x, q, A):
rubi.append(657)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*Simp(-c*f*x**S(2)*(S(2)*p + S(2)*q + S(5))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-B*c*e - C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(A*c*e + B*a*f + B*c*d + C*a*e)) - e*((A*b - B*a)*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) + (A*c - C*a)*(S(2)*a*c*e - b*(a*f + c*d)))*(p + q + S(2)) - x*(S(2)*f*((A*b - B*a)*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) + (A*c - C*a)*(S(2)*a*c*e - b*(a*f + c*d)))*(p + q + S(2)) - (b*f*(p + S(1)) - c*e*(S(2)*p + q + S(4)))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-B*c*e - C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(A*c*e + B*a*f + B*c*d + C*a*e))) + (p + S(1))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))*(-S(2)*A*c + B*b - S(2)*C*a) + (a*f*(p + S(1)) - c*d*(p + S(2)))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-B*c*e - C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(A*c*e + B*a*f + B*c*d + C*a*e)), x), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(1))*(c*x*(A*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) - B*(a*b*f - S(2)*a*c*e + b*c*d) + C*(-a*b*e - S(2)*a*(-a*f + c*d) + b**S(2)*d)) + (A*b - B*a)*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) + (A*c - C*a)*(S(2)*a*c*e - b*(a*f + c*d)))/((p + S(1))*(-S(4)*a*c + b**S(2))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))), x)
rule657 = ReplacementRule(pattern657, replacement657)
pattern658 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons50, cons226, cons394, cons13, cons137, cons404, cons405)
def replacement658(C, p, e, f, b, d, c, a, x, q, A):
rubi.append(658)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))), Int((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*Simp(-c*f*x**S(2)*(S(2)*p + S(2)*q + S(5))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(A*c*e + C*a*e)) - e*(A*b*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) + (A*c - C*a)*(S(2)*a*c*e - b*(a*f + c*d)))*(p + q + S(2)) - x*(S(2)*f*(A*b*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) + (A*c - C*a)*(S(2)*a*c*e - b*(a*f + c*d)))*(p + q + S(2)) - (b*f*(p + S(1)) - c*e*(S(2)*p + q + S(4)))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(A*c*e + C*a*e))) + (p + S(1))*(-S(2)*A*c - S(2)*C*a)*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2)) + (a*f*(p + S(1)) - c*d*(p + S(2)))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(A*c*e + C*a*e)), x), x), x) + Simp((a + b*x + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(1))*(A*b*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) + c*x*(A*(b**S(2)*f + S(2)*c**S(2)*d - c*(S(2)*a*f + b*e)) + C*(-a*b*e - S(2)*a*(-a*f + c*d) + b**S(2)*d)) + (A*c - C*a)*(S(2)*a*c*e - b*(a*f + c*d)))/((p + S(1))*(-S(4)*a*c + b**S(2))*(-(-a*e + b*d)*(-b*f + c*e) + (-a*f + c*d)**S(2))), x)
rule658 = ReplacementRule(pattern658, replacement658)
pattern659 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons50, cons394, cons13, cons137, cons407, cons405)
def replacement659(B, C, p, e, f, d, c, a, x, q, A):
rubi.append(659)
return Dist(-S(1)/(S(4)*a*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*Simp(-c*f*x**S(2)*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-B*c*e - C*a*f + C*c*d))*(S(2)*p + S(2)*q + S(5)) - e*(-B*a*(-S(2)*a*c*f + S(2)*c**S(2)*d) + S(2)*a*c*e*(A*c - C*a))*(p + q + S(2)) - x*(c*e*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-B*c*e - C*a*f + C*c*d))*(S(2)*p + q + S(4)) + S(2)*f*(-B*a*(-S(2)*a*c*f + S(2)*c**S(2)*d) + S(2)*a*c*e*(A*c - C*a))*(p + q + S(2))) + (p + S(1))*(-S(2)*A*c - S(2)*C*a)*(a*c*e**S(2) + (-a*f + c*d)**S(2)) + (S(2)*A*c*(-a*f + c*d) - S(2)*a*(-B*c*e - C*a*f + C*c*d))*(a*f*(p + S(1)) - c*d*(p + S(2))), x), x), x) + Simp(-(a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**(q + S(1))*(-B*a*(-S(2)*a*c*f + S(2)*c**S(2)*d) + S(2)*a*c*e*(A*c - C*a) + c*x*(A*(-S(2)*a*c*f + S(2)*c**S(2)*d) + S(2)*B*a*c*e - S(2)*C*a*(-a*f + c*d)))/(S(4)*a*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2))), x)
rule659 = ReplacementRule(pattern659, replacement659)
pattern660 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons36, cons50, cons394, cons13, cons137, cons407, cons405)
def replacement660(C, p, f, d, c, a, A, x, q, e):
rubi.append(660)
return Dist(-S(1)/(S(4)*a*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2))), Int((a + c*x**S(2))**(p + S(1))*(d + e*x + f*x**S(2))**q*Simp(-S(2)*a*c*e**S(2)*(A*c - C*a)*(p + q + S(2)) - c*f*x**S(2)*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d))*(S(2)*p + S(2)*q + S(5)) - x*(S(4)*a*c*e*f*(A*c - C*a)*(p + q + S(2)) + c*e*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d))*(S(2)*p + q + S(4))) + (p + S(1))*(-S(2)*A*c - S(2)*C*a)*(a*c*e**S(2) + (-a*f + c*d)**S(2)) + (S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d))*(a*f*(p + S(1)) - c*d*(p + S(2))), x), x), x) + Simp(-(a + c*x**S(2))**(p + S(1))*(S(2)*a*c*e*(A*c - C*a) + c*x*(A*(-S(2)*a*c*f + S(2)*c**S(2)*d) - S(2)*C*a*(-a*f + c*d)))*(d + e*x + f*x**S(2))**(q + S(1))/(S(4)*a*c*(p + S(1))*(a*c*e**S(2) + (-a*f + c*d)**S(2))), x)
rule660 = ReplacementRule(pattern660, replacement660)
pattern661 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons35, cons36, cons50, cons226, cons13, cons137, cons406, cons405)
def replacement661(B, p, C, f, b, d, c, a, x, q, A):
rubi.append(661)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(b**S(2)*d*f + (-a*f + c*d)**S(2))), Int((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(1))*Simp(-c*f*x**S(2)*(S(2)*p + S(2)*q + S(5))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(B*a*f + B*c*d)) - x*(-b*f*(p + S(1))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(B*a*f + B*c*d)) + S(2)*f*(-b*(A*c - C*a)*(a*f + c*d) + (A*b - B*a)*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d))*(p + q + S(2))) + (p + S(1))*(b**S(2)*d*f + (-a*f + c*d)**S(2))*(-S(2)*A*c + B*b - S(2)*C*a) + (a*f*(p + S(1)) - c*d*(p + S(2)))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d) - b*(B*a*f + B*c*d)), x), x), x) + Simp((d + f*x**S(2))**(q + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(-b*(A*c - C*a)*(a*f + c*d) + c*x*(A*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d) - B*(a*b*f + b*c*d) + C*(-S(2)*a*(-a*f + c*d) + b**S(2)*d)) + (A*b - B*a)*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d))/((p + S(1))*(-S(4)*a*c + b**S(2))*(b**S(2)*d*f + (-a*f + c*d)**S(2))), x)
rule661 = ReplacementRule(pattern661, replacement661)
pattern662 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons36, cons50, cons226, cons13, cons137, cons406, cons405)
def replacement662(C, p, f, b, d, c, a, x, q, A):
rubi.append(662)
return Dist(S(1)/((p + S(1))*(-S(4)*a*c + b**S(2))*(b**S(2)*d*f + (-a*f + c*d)**S(2))), Int((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(1))*Simp(-c*f*x**S(2)*(S(2)*p + S(2)*q + S(5))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d)) - x*(-b*f*(p + S(1))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d)) + S(2)*f*(A*b*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d) - b*(A*c - C*a)*(a*f + c*d))*(p + q + S(2))) + (p + S(1))*(-S(2)*A*c - S(2)*C*a)*(b**S(2)*d*f + (-a*f + c*d)**S(2)) + (a*f*(p + S(1)) - c*d*(p + S(2)))*(S(2)*A*c*(-a*f + c*d) - S(2)*a*(-C*a*f + C*c*d) + b**S(2)*(A*f + C*d)), x), x), x) + Simp((d + f*x**S(2))**(q + S(1))*(a + b*x + c*x**S(2))**(p + S(1))*(A*b*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d) - b*(A*c - C*a)*(a*f + c*d) + c*x*(A*(-S(2)*a*c*f + b**S(2)*f + S(2)*c**S(2)*d) + C*(-S(2)*a*(-a*f + c*d) + b**S(2)*d)))/((p + S(1))*(-S(4)*a*c + b**S(2))*(b**S(2)*d*f + (-a*f + c*d)**S(2))), x)
rule662 = ReplacementRule(pattern662, replacement662)
pattern663 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons50, cons226, cons394, cons13, cons163, cons433, cons454)
def replacement663(B, p, C, e, f, b, d, c, a, x, q, A):
rubi.append(663)
return -Dist(S(1)/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), Int((a + b*x + c*x**S(2))**(p + S(-1))*(d + e*x + f*x**S(2))**q*Simp(p*(-a*e + b*d)*(C*(q + S(1))*(-b*f + c*e) - c*(-B*f + C*e)*(S(2)*p + S(2)*q + S(3))) + x**S(2)*(p*(-b*f + c*e)*(C*(q + S(1))*(-b*f + c*e) - c*(-B*f + C*e)*(S(2)*p + S(2)*q + S(3))) + (C*f**S(2)*p*(-S(4)*a*c + b**S(2)) - c**S(2)*(C*(-S(4)*d*f + e**S(2))*(S(2)*p + q + S(2)) + f*(S(2)*p + S(2)*q + S(3))*(S(2)*A*f - B*e + S(2)*C*d)))*(p + q + S(1))) + x*(S(2)*p*(-a*f + c*d)*(C*(q + S(1))*(-b*f + c*e) - c*(-B*f + C*e)*(S(2)*p + S(2)*q + S(3))) + (C*e*f*p*(-S(4)*a*c + b**S(2)) - b*c*(C*(-S(4)*d*f + e**S(2))*(S(2)*p + q + S(2)) + f*(S(2)*p + S(2)*q + S(3))*(S(2)*A*f - B*e + S(2)*C*d)))*(p + q + S(1))) + (C*b**S(2)*d*f*p + a*c*(C*(S(2)*d*f - e**S(2)*(S(2)*p + q + S(2))) + f*(-S(2)*A*f + B*e)*(S(2)*p + S(2)*q + S(3))))*(p + q + S(1)), x), x), x) + Simp((a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**(q + S(1))*(B*c*f*(S(2)*p + S(2)*q + S(3)) + S(2)*C*c*f*x*(p + q + S(1)) + C*(b*f*p - c*e*(S(2)*p + q + S(2))))/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), x)
rule663 = ReplacementRule(pattern663, replacement663)
pattern664 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons50, cons226, cons394, cons13, cons163, cons433, cons454)
def replacement664(C, p, e, f, b, d, c, a, x, q, A):
rubi.append(664)
return -Dist(S(1)/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), Int((a + b*x + c*x**S(2))**(p + S(-1))*(d + e*x + f*x**S(2))**q*Simp(p*(-a*e + b*d)*(-C*c*e*(S(2)*p + S(2)*q + S(3)) + C*(q + S(1))*(-b*f + c*e)) + x**S(2)*(p*(-b*f + c*e)*(-C*c*e*(S(2)*p + S(2)*q + S(3)) + C*(q + S(1))*(-b*f + c*e)) + (C*f**S(2)*p*(-S(4)*a*c + b**S(2)) - c**S(2)*(C*(-S(4)*d*f + e**S(2))*(S(2)*p + q + S(2)) + f*(S(2)*A*f + S(2)*C*d)*(S(2)*p + S(2)*q + S(3))))*(p + q + S(1))) + x*(S(2)*p*(-a*f + c*d)*(-C*c*e*(S(2)*p + S(2)*q + S(3)) + C*(q + S(1))*(-b*f + c*e)) + (C*e*f*p*(-S(4)*a*c + b**S(2)) - b*c*(C*(-S(4)*d*f + e**S(2))*(S(2)*p + q + S(2)) + f*(S(2)*A*f + S(2)*C*d)*(S(2)*p + S(2)*q + S(3))))*(p + q + S(1))) + (C*b**S(2)*d*f*p + a*c*(-S(2)*A*f**S(2)*(S(2)*p + S(2)*q + S(3)) + C*(S(2)*d*f - e**S(2)*(S(2)*p + q + S(2)))))*(p + q + S(1)), x), x), x) + Simp((S(2)*C*c*f*x*(p + q + S(1)) + C*(b*f*p - c*e*(S(2)*p + q + S(2))))*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**(q + S(1))/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), x)
rule664 = ReplacementRule(pattern664, replacement664)
pattern665 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons50, cons394, cons13, cons163, cons433, cons454)
def replacement665(B, C, p, e, f, d, c, a, x, q, A):
rubi.append(665)
return -Dist(S(1)/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x + f*x**S(2))**q*Simp(a*c*(C*(S(2)*d*f - e**S(2)*(S(2)*p + q + S(2))) + f*(-S(2)*A*f + B*e)*(S(2)*p + S(2)*q + S(3)))*(p + q + S(1)) - a*e*p*(C*c*e*(q + S(1)) - c*(-B*f + C*e)*(S(2)*p + S(2)*q + S(3))) + x**S(2)*(c*e*p*(C*c*e*(q + S(1)) - c*(-B*f + C*e)*(S(2)*p + S(2)*q + S(3))) + (-S(4)*C*a*c*f**S(2)*p - c**S(2)*(C*(-S(4)*d*f + e**S(2))*(S(2)*p + q + S(2)) + f*(S(2)*p + S(2)*q + S(3))*(S(2)*A*f - B*e + S(2)*C*d)))*(p + q + S(1))) + x*(-S(4)*C*a*c*e*f*p*(p + q + S(1)) + S(2)*p*(-a*f + c*d)*(C*c*e*(q + S(1)) - c*(-B*f + C*e)*(S(2)*p + S(2)*q + S(3)))), x), x), x) + Simp((a + c*x**S(2))**p*(d + e*x + f*x**S(2))**(q + S(1))*(B*c*f*(S(2)*p + S(2)*q + S(3)) - C*c*e*(S(2)*p + q + S(2)) + S(2)*C*c*f*x*(p + q + S(1)))/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), x)
rule665 = ReplacementRule(pattern665, replacement665)
pattern666 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons36, cons50, cons394, cons13, cons163, cons433, cons454)
def replacement666(C, p, f, d, c, a, A, x, q, e):
rubi.append(666)
return -Dist(S(1)/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), Int((a + c*x**S(2))**(p + S(-1))*(d + e*x + f*x**S(2))**q*Simp(a*c*(-S(2)*A*f**S(2)*(S(2)*p + S(2)*q + S(3)) + C*(S(2)*d*f - e**S(2)*(S(2)*p + q + S(2))))*(p + q + S(1)) - a*e*p*(C*c*e*(q + S(1)) - C*c*e*(S(2)*p + S(2)*q + S(3))) + x**S(2)*(c*e*p*(C*c*e*(q + S(1)) - C*c*e*(S(2)*p + S(2)*q + S(3))) + (-S(4)*C*a*c*f**S(2)*p - c**S(2)*(C*(-S(4)*d*f + e**S(2))*(S(2)*p + q + S(2)) + f*(S(2)*A*f + S(2)*C*d)*(S(2)*p + S(2)*q + S(3))))*(p + q + S(1))) + x*(-S(4)*C*a*c*e*f*p*(p + q + S(1)) + S(2)*p*(-a*f + c*d)*(C*c*e*(q + S(1)) - C*c*e*(S(2)*p + S(2)*q + S(3)))), x), x), x) + Simp((a + c*x**S(2))**p*(-C*c*e*(S(2)*p + q + S(2)) + S(2)*C*c*f*x*(p + q + S(1)))*(d + e*x + f*x**S(2))**(q + S(1))/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), x)
rule666 = ReplacementRule(pattern666, replacement666)
pattern667 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons35, cons36, cons50, cons226, cons13, cons163, cons433, cons454)
def replacement667(B, p, C, f, b, d, c, a, x, q, A):
rubi.append(667)
return -Dist(S(1)/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), Int((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(-1))*Simp(b*d*p*(B*c*f*(S(2)*p + S(2)*q + S(3)) - C*b*f*(q + S(1))) + x**S(2)*(-b*f*p*(B*c*f*(S(2)*p + S(2)*q + S(3)) - C*b*f*(q + S(1))) + (C*f**S(2)*p*(-S(4)*a*c + b**S(2)) - c**S(2)*(-S(4)*C*d*f*(S(2)*p + q + S(2)) + f*(S(2)*A*f + S(2)*C*d)*(S(2)*p + S(2)*q + S(3))))*(p + q + S(1))) + x*(-b*c*(-S(4)*C*d*f*(S(2)*p + q + S(2)) + f*(S(2)*A*f + S(2)*C*d)*(S(2)*p + S(2)*q + S(3)))*(p + q + S(1)) + S(2)*p*(-a*f + c*d)*(B*c*f*(S(2)*p + S(2)*q + S(3)) - C*b*f*(q + S(1)))) + (C*b**S(2)*d*f*p + a*c*(-S(2)*A*f**S(2)*(S(2)*p + S(2)*q + S(3)) + S(2)*C*d*f))*(p + q + S(1)), x), x), x) + Simp((d + f*x**S(2))**(q + S(1))*(a + b*x + c*x**S(2))**p*(B*c*f*(S(2)*p + S(2)*q + S(3)) + C*b*f*p + S(2)*C*c*f*x*(p + q + S(1)))/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), x)
rule667 = ReplacementRule(pattern667, replacement667)
pattern668 = Pattern(Integral((d_ + x_**S(2)*WC('f', S(1)))**WC('q', S(1))*(x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons36, cons50, cons226, cons13, cons163, cons433, cons454)
def replacement668(C, p, f, b, d, c, a, x, q, A):
rubi.append(668)
return -Dist(S(1)/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), Int((d + f*x**S(2))**q*(a + b*x + c*x**S(2))**(p + S(-1))*Simp(-C*b**S(2)*d*f*p*(q + S(1)) + x**S(2)*(C*b**S(2)*f**S(2)*p*(q + S(1)) + (C*f**S(2)*p*(-S(4)*a*c + b**S(2)) - c**S(2)*(-S(4)*C*d*f*(S(2)*p + q + S(2)) + f*(S(2)*A*f + S(2)*C*d)*(S(2)*p + S(2)*q + S(3))))*(p + q + S(1))) + x*(-S(2)*C*b*f*p*(q + S(1))*(-a*f + c*d) - b*c*(-S(4)*C*d*f*(S(2)*p + q + S(2)) + f*(S(2)*A*f + S(2)*C*d)*(S(2)*p + S(2)*q + S(3)))*(p + q + S(1))) + (C*b**S(2)*d*f*p + a*c*(-S(2)*A*f**S(2)*(S(2)*p + S(2)*q + S(3)) + S(2)*C*d*f))*(p + q + S(1)), x), x), x) + Simp((d + f*x**S(2))**(q + S(1))*(C*b*f*p + S(2)*C*c*f*x*(p + q + S(1)))*(a + b*x + c*x**S(2))**p/(S(2)*c*f**S(2)*(p + q + S(1))*(S(2)*p + S(2)*q + S(3))), x)
rule668 = ReplacementRule(pattern668, replacement668)
def With669(B, C, e, f, b, d, c, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
q = a**S(2)*f**S(2) - a*b*e*f - S(2)*a*c*d*f + a*c*e**S(2) + b**S(2)*d*f - b*c*d*e + c**S(2)*d**S(2)
if NonzeroQ(q):
return True
return False
pattern669 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons226, cons394, CustomConstraint(With669))
def replacement669(B, C, e, f, b, d, c, a, x, A):
q = a**S(2)*f**S(2) - a*b*e*f - S(2)*a*c*d*f + a*c*e**S(2) + b**S(2)*d*f - b*c*d*e + c**S(2)*d**S(2)
rubi.append(669)
return Dist(S(1)/q, Int((-A*a*c*f + A*b**S(2)*f - A*b*c*e + A*c**S(2)*d - B*a*b*f + B*a*c*e + C*a**S(2)*f - C*a*c*d + c*x*(A*b*f - A*c*e - B*a*f + B*c*d + C*a*e - C*b*d))/(a + b*x + c*x**S(2)), x), x) + Dist(S(1)/q, Int((A*a*f**S(2) - A*b*e*f - A*c*d*f + A*c*e**S(2) + B*b*d*f - B*c*d*e - C*a*d*f + C*c*d**S(2) - f*x*(A*b*f - A*c*e - B*a*f + B*c*d + C*a*e - C*b*d))/(d + e*x + f*x**S(2)), x), x)
rule669 = ReplacementRule(pattern669, replacement669)
def With670(C, f, b, d, c, a, A, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = a**S(2)*f**S(2) - a*b*e*f - S(2)*a*c*d*f + a*c*e**S(2) + b**S(2)*d*f - b*c*d*e + c**S(2)*d**S(2)
if NonzeroQ(q):
return True
return False
pattern670 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*(d_ + x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons226, cons394, CustomConstraint(With670))
def replacement670(C, f, b, d, c, a, A, x, e):
q = a**S(2)*f**S(2) - a*b*e*f - S(2)*a*c*d*f + a*c*e**S(2) + b**S(2)*d*f - b*c*d*e + c**S(2)*d**S(2)
rubi.append(670)
return Dist(S(1)/q, Int((-A*a*c*f + A*b**S(2)*f - A*b*c*e + A*c**S(2)*d + C*a**S(2)*f - C*a*c*d + c*x*(A*b*f - A*c*e + C*a*e - C*b*d))/(a + b*x + c*x**S(2)), x), x) + Dist(S(1)/q, Int((A*a*f**S(2) - A*b*e*f - A*c*d*f + A*c*e**S(2) - C*a*d*f + C*c*d**S(2) - f*x*(A*b*f - A*c*e + C*a*e - C*b*d))/(d + e*x + f*x**S(2)), x), x)
rule670 = ReplacementRule(pattern670, replacement670)
def With671(B, C, f, b, d, c, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
q = a**S(2)*f**S(2) - S(2)*a*c*d*f + b**S(2)*d*f + c**S(2)*d**S(2)
if NonzeroQ(q):
return True
return False
pattern671 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/((d_ + x_**S(2)*WC('f', S(1)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons35, cons36, cons226, CustomConstraint(With671))
def replacement671(B, C, f, b, d, c, a, x, A):
q = a**S(2)*f**S(2) - S(2)*a*c*d*f + b**S(2)*d*f + c**S(2)*d**S(2)
rubi.append(671)
return Dist(S(1)/q, Int((A*a*f**S(2) - A*c*d*f + B*b*d*f - C*a*d*f + C*c*d**S(2) - f*x*(A*b*f - B*a*f + B*c*d - C*b*d))/(d + f*x**S(2)), x), x) + Dist(S(1)/q, Int((-A*a*c*f + A*b**S(2)*f + A*c**S(2)*d - B*a*b*f + C*a**S(2)*f - C*a*c*d + c*x*(A*b*f - B*a*f + B*c*d - C*b*d))/(a + b*x + c*x**S(2)), x), x)
rule671 = ReplacementRule(pattern671, replacement671)
def With672(C, f, b, d, c, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
q = a**S(2)*f**S(2) - S(2)*a*c*d*f + b**S(2)*d*f + c**S(2)*d**S(2)
if NonzeroQ(q):
return True
return False
pattern672 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/((d_ + x_**S(2)*WC('f', S(1)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons36, cons226, CustomConstraint(With672))
def replacement672(C, f, b, d, c, a, x, A):
q = a**S(2)*f**S(2) - S(2)*a*c*d*f + b**S(2)*d*f + c**S(2)*d**S(2)
rubi.append(672)
return Dist(S(1)/q, Int((A*a*f**S(2) - A*c*d*f - C*a*d*f + C*c*d**S(2) - f*x*(A*b*f - C*b*d))/(d + f*x**S(2)), x), x) + Dist(S(1)/q, Int((-A*a*c*f + A*b**S(2)*f + A*c**S(2)*d + C*a**S(2)*f - C*a*c*d + c*x*(A*b*f - C*b*d))/(a + b*x + c*x**S(2)), x), x)
rule672 = ReplacementRule(pattern672, replacement672)
pattern673 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons226, cons394)
def replacement673(B, C, e, f, b, d, c, a, x, A):
rubi.append(673)
return Dist(S(1)/c, Int((A*c - C*a + x*(B*c - C*b))/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(C/c, Int(S(1)/sqrt(d + e*x + f*x**S(2)), x), x)
rule673 = ReplacementRule(pattern673, replacement673)
pattern674 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons226, cons394)
def replacement674(C, e, f, b, d, c, a, x, A):
rubi.append(674)
return Dist(S(1)/c, Int((A*c - C*a - C*b*x)/((a + b*x + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(C/c, Int(S(1)/sqrt(d + e*x + f*x**S(2)), x), x)
rule674 = ReplacementRule(pattern674, replacement674)
pattern675 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons394)
def replacement675(B, C, e, f, d, c, a, x, A):
rubi.append(675)
return Dist(S(1)/c, Int((A*c + B*c*x - C*a)/((a + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(C/c, Int(S(1)/sqrt(d + e*x + f*x**S(2)), x), x)
rule675 = ReplacementRule(pattern675, replacement675)
pattern676 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/((a_ + x_**S(2)*WC('c', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons36, cons394)
def replacement676(C, f, d, c, a, A, x, e):
rubi.append(676)
return Dist(C/c, Int(S(1)/sqrt(d + e*x + f*x**S(2)), x), x) + Dist((A*c - C*a)/c, Int(S(1)/((a + c*x**S(2))*sqrt(d + e*x + f*x**S(2))), x), x)
rule676 = ReplacementRule(pattern676, replacement676)
pattern677 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(sqrt(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons35, cons36, cons226)
def replacement677(B, C, f, b, d, c, a, x, A):
rubi.append(677)
return Dist(S(1)/c, Int((A*c - C*a + x*(B*c - C*b))/(sqrt(d + f*x**S(2))*(a + b*x + c*x**S(2))), x), x) + Dist(C/c, Int(S(1)/sqrt(d + f*x**S(2)), x), x)
rule677 = ReplacementRule(pattern677, replacement677)
pattern678 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(sqrt(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons36, cons226)
def replacement678(C, f, b, d, c, a, x, A):
rubi.append(678)
return Dist(S(1)/c, Int((A*c - C*a - C*b*x)/(sqrt(d + f*x**S(2))*(a + b*x + c*x**S(2))), x), x) + Dist(C/c, Int(S(1)/sqrt(d + f*x**S(2)), x), x)
rule678 = ReplacementRule(pattern678, replacement678)
pattern679 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons455)
def replacement679(B, C, e, p, f, b, d, c, a, x, q, A):
rubi.append(679)
return Int((A + B*x + C*x**S(2))*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x)
rule679 = ReplacementRule(pattern679, replacement679)
pattern680 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons5, cons50, cons456)
def replacement680(C, e, p, f, b, d, c, a, x, q, A):
rubi.append(680)
return Int((A + C*x**S(2))*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x)
rule680 = ReplacementRule(pattern680, replacement680)
pattern681 = Pattern(Integral((x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons457)
def replacement681(B, C, e, p, f, d, a, c, x, q, A):
rubi.append(681)
return Int((a + c*x**S(2))**p*(A + B*x + C*x**S(2))*(d + e*x + f*x**S(2))**q, x)
rule681 = ReplacementRule(pattern681, replacement681)
pattern682 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_*(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons34, cons36, cons5, cons50, cons458)
def replacement682(C, e, p, f, d, a, c, x, q, A):
rubi.append(682)
return Int((A + C*x**S(2))*(a + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x)
rule682 = ReplacementRule(pattern682, replacement682)
pattern683 = Pattern(Integral((u_**S(2)*WC('C', S(1)) + u_*WC('B', S(1)) + WC('A', S(0)))*(u_**S(2)*WC('c', S(1)) + u_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons68, cons69)
def replacement683(B, p, C, e, u, f, b, d, c, a, x, q, A):
rubi.append(683)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((A + B*x + C*x**S(2))*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule683 = ReplacementRule(pattern683, replacement683)
pattern684 = Pattern(Integral((u_*WC('B', S(1)) + WC('A', S(0)))*(u_**S(2)*WC('c', S(1)) + u_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons68, cons69)
def replacement684(B, p, e, u, f, b, d, c, a, x, q, A):
rubi.append(684)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((A + B*x)*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule684 = ReplacementRule(pattern684, replacement684)
pattern685 = Pattern(Integral((u_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(u_**S(2)*WC('c', S(1)) + u_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons5, cons50, cons68, cons69)
def replacement685(p, C, e, u, f, b, d, c, a, x, q, A):
rubi.append(685)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((A + C*x**S(2))*(a + b*x + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule685 = ReplacementRule(pattern685, replacement685)
pattern686 = Pattern(Integral((u_**S(2)*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('C', S(1)) + u_*WC('B', S(1)) + WC('A', S(0)))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons68, cons69)
def replacement686(B, p, C, e, u, f, d, a, c, x, q, A):
rubi.append(686)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + c*x**S(2))**p*(A + B*x + C*x**S(2))*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule686 = ReplacementRule(pattern686, replacement686)
pattern687 = Pattern(Integral((u_*WC('B', S(1)) + WC('A', S(0)))*(u_**S(2)*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons5, cons50, cons68, cons69)
def replacement687(B, p, e, u, f, d, a, c, x, q, A):
rubi.append(687)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((A + B*x)*(a + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule687 = ReplacementRule(pattern687, replacement687)
pattern688 = Pattern(Integral((u_**S(2)*WC('C', S(1)) + WC('A', S(0)))*(u_**S(2)*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**S(2)*WC('f', S(1)) + u_*WC('e', S(1)) + WC('d', S(0)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons36, cons5, cons50, cons68, cons69)
def replacement688(C, p, e, u, f, d, a, c, x, q, A):
rubi.append(688)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((A + C*x**S(2))*(a + c*x**S(2))**p*(d + e*x + f*x**S(2))**q, x), x, u), x)
rule688 = ReplacementRule(pattern688, replacement688)
return [rule189, rule190, rule191, rule192, rule193, rule194, rule195, rule196, rule197, rule198, rule199, rule200, rule201, rule202, rule203, rule204, rule205, rule206, rule207, rule208, rule209, rule210, rule211, rule212, rule213, rule214, rule215, rule216, rule217, rule218, rule219, rule220, rule221, rule222, rule223, rule224, rule225, rule226, rule227, rule228, rule229, rule230, rule231, rule232, rule233, rule234, rule235, rule236, rule237, rule238, rule239, rule240, rule241, rule242, rule243, rule244, rule245, rule246, rule247, rule248, rule249, rule250, rule251, rule252, rule253, rule254, rule255, rule256, rule257, rule258, rule259, rule260, rule261, rule262, rule263, rule264, rule265, rule266, rule267, rule268, rule269, rule270, rule271, rule272, rule273, rule274, rule275, rule276, rule277, rule278, rule279, rule280, rule281, rule282, rule283, rule284, rule285, rule286, rule287, rule288, rule289, rule290, rule291, rule292, rule293, rule294, rule295, rule296, rule297, rule298, rule299, rule300, rule301, rule302, rule303, rule304, rule305, rule306, rule307, rule308, rule309, rule310, rule311, rule312, rule313, rule314, rule315, rule316, rule317, rule318, rule319, rule320, rule321, rule322, rule323, rule324, rule325, rule326, rule327, rule328, rule329, rule330, rule331, rule332, rule333, rule334, rule335, rule336, rule337, rule338, rule339, rule340, rule341, rule342, rule343, rule344, rule345, rule346, rule347, rule348, rule349, rule350, rule351, rule352, rule353, rule354, rule355, rule356, rule357, rule358, rule359, rule360, rule361, rule362, rule363, rule364, rule365, rule366, rule367, rule368, rule369, rule370, rule371, rule372, rule373, rule374, rule375, rule376, rule377, rule378, rule379, rule380, rule381, rule382, rule383, rule384, rule385, rule386, rule387, rule388, rule389, rule390, rule391, rule392, rule393, rule394, rule395, rule396, rule397, rule398, rule399, rule400, rule401, rule402, rule403, rule404, rule405, rule406, rule407, rule408, rule409, rule410, rule411, rule412, rule413, rule414, rule415, rule416, rule417, rule418, rule419, rule420, rule421, rule422, rule423, rule424, rule425, rule426, rule427, rule428, rule429, rule430, rule431, rule432, rule433, rule434, rule435, rule436, rule437, rule438, rule439, rule440, rule441, rule442, rule443, rule444, rule445, rule446, rule447, rule448, rule449, rule450, rule451, rule452, rule453, rule454, rule455, rule456, rule457, rule458, rule459, rule460, rule461, rule462, rule463, rule464, rule465, rule466, rule467, rule468, rule469, rule470, rule471, rule472, rule473, rule474, rule475, rule476, rule477, rule478, rule479, rule480, rule481, rule482, rule483, rule484, rule485, rule486, rule487, rule488, rule489, rule490, rule491, rule492, rule493, rule494, rule495, rule496, rule497, rule498, rule499, rule500, rule501, rule502, rule503, rule504, rule505, rule506, rule507, rule508, rule509, rule510, rule511, rule512, rule513, rule514, rule515, rule516, rule517, rule518, rule519, rule520, rule521, rule522, rule523, rule524, rule525, rule526, rule527, rule528, rule529, rule530, rule531, rule532, rule533, rule534, rule535, rule536, rule537, rule538, rule539, rule540, rule541, rule542, rule543, rule544, rule545, rule546, rule547, rule548, rule549, rule550, rule551, rule552, rule553, rule554, rule555, rule556, rule557, rule558, rule559, rule560, rule561, rule562, rule563, rule564, rule565, rule566, rule567, rule568, rule569, rule570, rule571, rule572, rule573, rule574, rule575, rule576, rule577, rule578, rule579, rule580, rule581, rule582, rule583, rule584, rule585, rule586, rule587, rule588, rule589, rule590, rule591, rule592, rule593, rule594, rule595, rule596, rule597, rule598, rule599, rule600, rule601, rule602, rule603, rule604, rule605, rule606, rule607, rule608, rule609, rule610, rule611, rule612, rule613, rule614, rule615, rule616, rule617, rule618, rule619, rule620, rule621, rule622, rule623, rule624, rule625, rule626, rule627, rule628, rule629, rule630, rule631, rule632, rule633, rule634, rule635, rule636, rule637, rule638, rule639, rule640, rule641, rule642, rule643, rule644, rule645, rule646, rule647, rule648, rule649, rule650, rule651, rule652, rule653, rule654, rule655, rule656, rule657, rule658, rule659, rule660, rule661, rule662, rule663, rule664, rule665, rule666, rule667, rule668, rule669, rule670, rule671, rule672, rule673, rule674, rule675, rule676, rule677, rule678, rule679, rule680, rule681, rule682, rule683, rule684, rule685, rule686, rule687, rule688, ]
|
2f0bd455a0d58585368fe3458138860ad0a8a3a787af1e5d88e41c270e18c4b6 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def binomial_products(rubi):
from sympy.integrals.rubi.constraints import cons459, cons3, cons4, cons5, cons460, cons2, cons461, cons54, cons462, cons87, cons463, cons38, cons464, cons148, cons13, cons163, cons465, cons466, cons43, cons448, cons67, cons137, cons467, cons468, cons469, cons470, cons471, cons472, cons473, cons474, cons475, cons476, cons477, cons478, cons479, cons480, cons481, cons482, cons483, cons484, cons105, cons485, cons486, cons487, cons488, cons196, cons489, cons128, cons357, cons490, cons491, cons492, cons493, cons68, cons69, cons55, cons494, cons57, cons58, cons59, cons60, cons495, cons496, cons497, cons498, cons147, cons7, cons21, cons499, cons500, cons501, cons18, cons502, cons503, cons66, cons504, cons505, cons506, cons507, cons17, cons244, cons94, cons508, cons509, cons510, cons511, cons512, cons513, cons514, cons515, cons516, cons517, cons518, cons519, cons520, cons521, cons62, cons522, cons523, cons524, cons525, cons526, cons527, cons528, cons529, cons31, cons530, cons531, cons532, cons533, cons534, cons535, cons536, cons367, cons537, cons538, cons539, cons540, cons356, cons541, cons23, cons542, cons543, cons544, cons545, cons546, cons547, cons548, cons549, cons550, cons551, cons552, cons553, cons554, cons71, cons555, cons27, cons220, cons50, cons556, cons85, cons557, cons395, cons403, cons63, cons558, cons559, cons560, cons561, cons562, cons563, cons564, cons565, cons566, cons567, cons568, cons569, cons70, cons570, cons571, cons572, cons573, cons402, cons574, cons575, cons576, cons405, cons577, cons578, cons579, cons580, cons581, cons177, cons582, cons583, cons117, cons584, cons585, cons586, cons587, cons386, cons588, cons589, cons590, cons591, cons48, cons53, cons592, cons593, cons594, cons595, cons596, cons93, cons597, cons598, cons599, cons600, cons601, cons602, cons603, cons604, cons88, cons605, cons606, cons607, cons608, cons609, cons610, cons611, cons612, cons613, cons614, cons615, cons616, cons617, cons618, cons619, cons620, cons621, cons622, cons623, cons624, cons625, cons626, cons627, cons46, cons628, cons125, cons629, cons630, cons631, cons153, cons632, cons633, cons176, cons634, cons635, cons636, cons637, cons638, cons178, cons639, cons640, cons396, cons641, cons52, cons642, cons643, cons644, cons645, cons646, cons647, cons648, cons649, cons650, cons651, cons652, cons653, cons654, cons655, cons656, cons208, cons657, cons658, cons659, cons660, cons661, cons380, cons662, cons663
pattern689 = Pattern(Integral((x_**n_*WC('b', S(1)))**p_, x_), cons3, cons4, cons5, cons459)
def replacement689(x, p, n, b):
rubi.append(689)
return Dist(b**IntPart(p)*x**(-n*FracPart(p))*(b*x**n)**FracPart(p), Int(x**(n*p), x), x)
rule689 = ReplacementRule(pattern689, replacement689)
pattern690 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons460)
def replacement690(p, b, a, n, x):
rubi.append(690)
return Simp(x*(a + b*x**n)**(p + S(1))/a, x)
rule690 = ReplacementRule(pattern690, replacement690)
pattern691 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons461, cons54)
def replacement691(p, b, a, n, x):
rubi.append(691)
return Dist((n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1)), x), x) - Simp(x*(a + b*x**n)**(p + S(1))/(a*n*(p + S(1))), x)
rule691 = ReplacementRule(pattern691, replacement691)
pattern692 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons4, cons462)
def replacement692(x, a, n, b):
rubi.append(692)
return Int(a**S(2) + S(2)*a*b*x**n + b**S(2)*x**(S(2)*n), x)
rule692 = ReplacementRule(pattern692, replacement692)
pattern693 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons87, cons463, cons38)
def replacement693(p, b, a, n, x):
rubi.append(693)
return Int(x**(n*p)*(a*x**(-n) + b)**p, x)
rule693 = ReplacementRule(pattern693, replacement693)
pattern694 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons464)
def replacement694(p, b, a, n, x):
rubi.append(694)
return Int(ExpandIntegrand((a + b*x**n)**p, x), x)
rule694 = ReplacementRule(pattern694, replacement694)
pattern695 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons148, cons13, cons163, cons465)
def replacement695(p, b, a, n, x):
rubi.append(695)
return Dist(a*n*p/(n*p + S(1)), Int((a + b*x**n)**(p + S(-1)), x), x) + Simp(x*(a + b*x**n)**p/(n*p + S(1)), x)
rule695 = ReplacementRule(pattern695, replacement695)
pattern696 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-5)/4), x_), cons2, cons3, cons466, cons43)
def replacement696(x, a, b):
rubi.append(696)
return Simp(S(2)*EllipticE(ArcTan(x*Rt(b/a, S(2)))/S(2), S(2))/(a**(S(5)/4)*Rt(b/a, S(2))), x)
rule696 = ReplacementRule(pattern696, replacement696)
pattern697 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-5)/4), x_), cons2, cons3, cons466, cons448)
def replacement697(x, a, b):
rubi.append(697)
return Dist((S(1) + b*x**S(2)/a)**(S(1)/4)/(a*(a + b*x**S(2))**(S(1)/4)), Int((S(1) + b*x**S(2)/a)**(S(-5)/4), x), x)
rule697 = ReplacementRule(pattern697, replacement697)
pattern698 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-7)/6), x_), cons2, cons3, cons67)
def replacement698(x, a, b):
rubi.append(698)
return Dist(S(1)/((a/(a + b*x**S(2)))**(S(2)/3)*(a + b*x**S(2))**(S(2)/3)), Subst(Int((-b*x**S(2) + S(1))**(S(-1)/3), x), x, x/sqrt(a + b*x**S(2))), x)
rule698 = ReplacementRule(pattern698, replacement698)
pattern699 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons148, cons13, cons137, cons465)
def replacement699(p, b, a, n, x):
rubi.append(699)
return Dist((n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1)), x), x) - Simp(x*(a + b*x**n)**(p + S(1))/(a*n*(p + S(1))), x)
rule699 = ReplacementRule(pattern699, replacement699)
pattern700 = Pattern(Integral(S(1)/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement700(x, a, b):
rubi.append(700)
return Dist(S(1)/(S(3)*Rt(a, S(3))**S(2)), Int((-x*Rt(b, S(3)) + S(2)*Rt(a, S(3)))/(x**S(2)*Rt(b, S(3))**S(2) - x*Rt(a, S(3))*Rt(b, S(3)) + Rt(a, S(3))**S(2)), x), x) + Dist(S(1)/(S(3)*Rt(a, S(3))**S(2)), Int(S(1)/(x*Rt(b, S(3)) + Rt(a, S(3))), x), x)
rule700 = ReplacementRule(pattern700, replacement700)
def With701(x, a, n, b):
r = Numerator(Rt(a/b, n))
s = Denominator(Rt(a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r - s*x*cos(Pi*(S(2)*k + S(-1))/n))/(r**S(2) - S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r - s*x*cos(Pi*(2*k - 1)/n))/(r**2 - 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(Dist(2*r/(a*n), Sum_doit(u, List(k, 1, n/2 - S.Half)), x) + r*Int(1/(r + s*x), x)/(a*n), x)
pattern701 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons467, cons468)
rule701 = ReplacementRule(pattern701, With701)
def With702(x, a, n, b):
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r + s*x*cos(Pi*(S(2)*k + S(-1))/n))/(r**S(2) + S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r + s*x*cos(Pi*(2*k - 1)/n))/(r**2 + 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(Dist(2*r/(a*n), Sum_doit(u, List(k, 1, n/2 - S.Half)), x) + r*Int(1/(r - s*x), x)/(a*n), x)
pattern702 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons467, cons469)
rule702 = ReplacementRule(pattern702, With702)
pattern703 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons468, cons470)
def replacement703(x, a, b):
rubi.append(703)
return Simp(ArcTan(x*Rt(b, S(2))/Rt(a, S(2)))/(Rt(a, S(2))*Rt(b, S(2))), x)
rule703 = ReplacementRule(pattern703, replacement703)
pattern704 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons468, cons471)
def replacement704(x, a, b):
rubi.append(704)
return -Simp(ArcTan(x*Rt(-b, S(2))/Rt(-a, S(2)))/(Rt(-a, S(2))*Rt(-b, S(2))), x)
rule704 = ReplacementRule(pattern704, replacement704)
pattern705 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons468)
def replacement705(x, a, b):
rubi.append(705)
return Simp(ArcTan(x/Rt(a/b, S(2)))*Rt(a/b, S(2))/a, x)
rule705 = ReplacementRule(pattern705, replacement705)
pattern706 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons469, cons472)
def replacement706(x, a, b):
rubi.append(706)
return Simp(atanh(x*Rt(-b, S(2))/Rt(a, S(2)))/(Rt(a, S(2))*Rt(-b, S(2))), x)
rule706 = ReplacementRule(pattern706, replacement706)
pattern707 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons469, cons473)
def replacement707(x, a, b):
rubi.append(707)
return -Simp(atanh(x*Rt(b, S(2))/Rt(-a, S(2)))/(Rt(-a, S(2))*Rt(b, S(2))), x)
rule707 = ReplacementRule(pattern707, replacement707)
pattern708 = Pattern(Integral(S(1)/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons469)
def replacement708(x, a, b):
rubi.append(708)
return Simp(Rt(-a/b, S(2))*atanh(x/Rt(-a/b, S(2)))/a, x)
rule708 = ReplacementRule(pattern708, replacement708)
def With709(x, a, n, b):
r = Numerator(Rt(a/b, n))
s = Denominator(Rt(a/b, n))
k = Symbol('k')
u = Symbol('u')
v = Symbol('v')
u = Int((r - s*x*cos(Pi*(S(2)*k + S(-1))/n))/(r**S(2) - S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x) + Int((r + s*x*cos(Pi*(S(2)*k + S(-1))/n))/(r**S(2) + S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r - s*x*cos(Pi*(2*k - 1)/n))/(r**2 - 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x) + Int((r + s*x*cos(Pi*(2*k - 1)/n))/(r**2 + 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(Dist(2*r/(a*n), Sum_doit(u, List(k, 1, n/4 - S.Half)), x) + 2*r**2*Int(1/(r**2 + s**2*x**2), x)/(a*n), x)
pattern709 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons474, cons468)
rule709 = ReplacementRule(pattern709, With709)
def With710(x, a, n, b):
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r - s*x*cos(S(2)*Pi*k/n))/(r**S(2) - S(2)*r*s*x*cos(S(2)*Pi*k/n) + s**S(2)*x**S(2)), x) + Int((r + s*x*cos(S(2)*Pi*k/n))/(r**S(2) + S(2)*r*s*x*cos(S(2)*Pi*k/n) + s**S(2)*x**S(2)), x)
u = Int((r - s*x*cos(2*Pi*k/n))/(r**2 - 2*r*s*x*cos(2*Pi*k/n) + s**2*x**2), x) + Int((r + s*x*cos(2*Pi*k/n))/(r**2 + 2*r*s*x*cos(2*Pi*k/n) + s**2*x**2), x)
return Simp(Dist(2*r/(a*n), Sum_doit(u, List(k, 1, n/4 - S.Half)), x) + 2*r**2*Int(1/(r**2 - s**2*x**2), x)/(a*n), x)
pattern710 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons474, cons469)
rule710 = ReplacementRule(pattern710, With710)
def With711(x, a, b):
r = Numerator(Rt(a/b, S(2)))
s = Denominator(Rt(a/b, S(2)))
rubi.append(711)
return Dist(S(1)/(S(2)*r), Int((r - s*x**S(2))/(a + b*x**S(4)), x), x) + Dist(S(1)/(S(2)*r), Int((r + s*x**S(2))/(a + b*x**S(4)), x), x)
pattern711 = Pattern(Integral(S(1)/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons475)
rule711 = ReplacementRule(pattern711, With711)
def With712(x, a, b):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
rubi.append(712)
return Dist(r/(S(2)*a), Int(S(1)/(r - s*x**S(2)), x), x) + Dist(r/(S(2)*a), Int(S(1)/(r + s*x**S(2)), x), x)
pattern712 = Pattern(Integral(S(1)/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons476)
rule712 = ReplacementRule(pattern712, With712)
def With713(x, a, n, b):
r = Numerator(Rt(a/b, S(4)))
s = Denominator(Rt(a/b, S(4)))
rubi.append(713)
return Dist(sqrt(S(2))*r/(S(4)*a), Int((sqrt(S(2))*r - s*x**(n/S(4)))/(r**S(2) - sqrt(S(2))*r*s*x**(n/S(4)) + s**S(2)*x**(n/S(2))), x), x) + Dist(sqrt(S(2))*r/(S(4)*a), Int((sqrt(S(2))*r + s*x**(n/S(4)))/(r**S(2) + sqrt(S(2))*r*s*x**(n/S(4)) + s**S(2)*x**(n/S(2))), x), x)
pattern713 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons477, cons478)
rule713 = ReplacementRule(pattern713, With713)
def With714(x, a, n, b):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
rubi.append(714)
return Dist(r/(S(2)*a), Int(S(1)/(r - s*x**(n/S(2))), x), x) + Dist(r/(S(2)*a), Int(S(1)/(r + s*x**(n/S(2))), x), x)
pattern714 = Pattern(Integral(S(1)/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons477, cons476)
rule714 = ReplacementRule(pattern714, With714)
pattern715 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons43, cons479)
def replacement715(x, a, b):
rubi.append(715)
return Simp(asinh(x*Rt(b, S(2))/sqrt(a))/Rt(b, S(2)), x)
rule715 = ReplacementRule(pattern715, replacement715)
pattern716 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons43, cons480)
def replacement716(x, a, b):
rubi.append(716)
return Simp(asin(x*Rt(-b, S(2))/sqrt(a))/Rt(-b, S(2)), x)
rule716 = ReplacementRule(pattern716, replacement716)
pattern717 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons448)
def replacement717(x, a, b):
rubi.append(717)
return Subst(Int(S(1)/(-b*x**S(2) + S(1)), x), x, x/sqrt(a + b*x**S(2)))
rule717 = ReplacementRule(pattern717, replacement717)
def With718(x, a, b):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(718)
return Simp(S(2)*S(3)**(S(3)/4)*sqrt((r**S(2)*x**S(2) - r*s*x + s**S(2))/(r*x + s*(S(1) + sqrt(S(3))))**S(2))*sqrt(sqrt(S(3)) + S(2))*(r*x + s)*EllipticF(asin((r*x + s*(-sqrt(S(3)) + S(1)))/(r*x + s*(S(1) + sqrt(S(3))))), S(-7) - S(4)*sqrt(S(3)))/(S(3)*r*sqrt(s*(r*x + s)/(r*x + s*(S(1) + sqrt(S(3))))**S(2))*sqrt(a + b*x**S(3))), x)
pattern718 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons481)
rule718 = ReplacementRule(pattern718, With718)
def With719(x, a, b):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(719)
return Simp(S(2)*S(3)**(S(3)/4)*sqrt((r**S(2)*x**S(2) - r*s*x + s**S(2))/(r*x + s*(-sqrt(S(3)) + S(1)))**S(2))*sqrt(-sqrt(S(3)) + S(2))*(r*x + s)*EllipticF(asin((r*x + s*(S(1) + sqrt(S(3))))/(r*x + s*(-sqrt(S(3)) + S(1)))), S(-7) + S(4)*sqrt(S(3)))/(S(3)*r*sqrt(-s*(r*x + s)/(r*x + s*(-sqrt(S(3)) + S(1)))**S(2))*sqrt(a + b*x**S(3))), x)
pattern719 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons482)
rule719 = ReplacementRule(pattern719, With719)
def With720(x, a, b):
q = Rt(b/a, S(4))
rubi.append(720)
return Simp(sqrt((a + b*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(q**S(2)*x**S(2) + S(1))*EllipticF(S(2)*ArcTan(q*x), S(1)/2)/(S(2)*q*sqrt(a + b*x**S(4))), x)
pattern720 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons466)
rule720 = ReplacementRule(pattern720, With720)
pattern721 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons483, cons43)
def replacement721(x, a, b):
rubi.append(721)
return Simp(EllipticF(asin(x*Rt(-b, S(4))/Rt(a, S(4))), S(-1))/(Rt(a, S(4))*Rt(-b, S(4))), x)
rule721 = ReplacementRule(pattern721, replacement721)
def With722(x, a, b):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-a*b, S(2))
if IntegerQ(q):
return True
return False
pattern722 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons484, cons105, CustomConstraint(With722))
def replacement722(x, a, b):
q = Rt(-a*b, S(2))
rubi.append(722)
return Simp(sqrt(S(2))*sqrt((a + q*x**S(2))/q)*sqrt(-a + q*x**S(2))*EllipticF(asin(sqrt(S(2))*x/sqrt((a + q*x**S(2))/q)), S(1)/2)/(S(2)*sqrt(-a)*sqrt(a + b*x**S(4))), x)
rule722 = ReplacementRule(pattern722, replacement722)
def With723(x, a, b):
q = Rt(-a*b, S(2))
rubi.append(723)
return Simp(sqrt(S(2))*sqrt((a + q*x**S(2))/q)*sqrt((a - q*x**S(2))/(a + q*x**S(2)))*EllipticF(asin(sqrt(S(2))*x/sqrt((a + q*x**S(2))/q)), S(1)/2)/(S(2)*sqrt(a/(a + q*x**S(2)))*sqrt(a + b*x**S(4))), x)
pattern723 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons484, cons105)
rule723 = ReplacementRule(pattern723, With723)
pattern724 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons483, cons448)
def replacement724(x, a, b):
rubi.append(724)
return Dist(sqrt(S(1) + b*x**S(4)/a)/sqrt(a + b*x**S(4)), Int(S(1)/sqrt(S(1) + b*x**S(4)/a), x), x)
rule724 = ReplacementRule(pattern724, replacement724)
def With725(x, a, b):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(725)
return Simp(S(3)**(S(3)/4)*x*sqrt((r**S(2)*x**S(4) - r*s*x**S(2) + s**S(2))/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)**S(2))*(r*x**S(2) + s)*EllipticF(acos((r*x**S(2)*(-sqrt(S(3)) + S(1)) + s)/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)), sqrt(S(3))/S(4) + S(1)/2)/(S(6)*s*sqrt(r*x**S(2)*(r*x**S(2) + s)/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)**S(2))*sqrt(a + b*x**S(6))), x)
pattern725 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(6)*WC('b', S(1))), x_), cons2, cons3, cons67)
rule725 = ReplacementRule(pattern725, With725)
pattern726 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(8)*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement726(x, a, b):
rubi.append(726)
return Dist(S(1)/2, Int((-x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x) + Dist(S(1)/2, Int((x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x)
rule726 = ReplacementRule(pattern726, replacement726)
pattern727 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/4), x_), cons2, cons3, cons466)
def replacement727(x, a, b):
rubi.append(727)
return -Dist(a, Int((a + b*x**S(2))**(S(-5)/4), x), x) + Simp(S(2)*x/(a + b*x**S(2))**(S(1)/4), x)
rule727 = ReplacementRule(pattern727, replacement727)
pattern728 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/4), x_), cons2, cons3, cons483, cons43)
def replacement728(x, a, b):
rubi.append(728)
return Simp(S(2)*EllipticE(asin(x*Rt(-b/a, S(2)))/S(2), S(2))/(a**(S(1)/4)*Rt(-b/a, S(2))), x)
rule728 = ReplacementRule(pattern728, replacement728)
pattern729 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/4), x_), cons2, cons3, cons483, cons448)
def replacement729(x, a, b):
rubi.append(729)
return Dist((S(1) + b*x**S(2)/a)**(S(1)/4)/(a + b*x**S(2))**(S(1)/4), Int((S(1) + b*x**S(2)/a)**(S(-1)/4), x), x)
rule729 = ReplacementRule(pattern729, replacement729)
pattern730 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-3)/4), x_), cons2, cons3, cons43, cons466)
def replacement730(x, a, b):
rubi.append(730)
return Simp(S(2)*EllipticF(ArcTan(x*Rt(b/a, S(2)))/S(2), S(2))/(a**(S(3)/4)*Rt(b/a, S(2))), x)
rule730 = ReplacementRule(pattern730, replacement730)
pattern731 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-3)/4), x_), cons2, cons3, cons43, cons483)
def replacement731(x, a, b):
rubi.append(731)
return Simp(S(2)*EllipticF(asin(x*Rt(-b/a, S(2)))/S(2), S(2))/(a**(S(3)/4)*Rt(-b/a, S(2))), x)
rule731 = ReplacementRule(pattern731, replacement731)
pattern732 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-3)/4), x_), cons2, cons3, cons448)
def replacement732(x, a, b):
rubi.append(732)
return Dist((S(1) + b*x**S(2)/a)**(S(3)/4)/(a + b*x**S(2))**(S(3)/4), Int((S(1) + b*x**S(2)/a)**(S(-3)/4), x), x)
rule732 = ReplacementRule(pattern732, replacement732)
pattern733 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/3), x_), cons2, cons3, cons67)
def replacement733(x, a, b):
rubi.append(733)
return Dist(S(3)*sqrt(b*x**S(2))/(S(2)*b*x), Subst(Int(x/sqrt(-a + x**S(3)), x), x, (a + b*x**S(2))**(S(1)/3)), x)
rule733 = ReplacementRule(pattern733, replacement733)
pattern734 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-2)/3), x_), cons2, cons3, cons67)
def replacement734(x, a, b):
rubi.append(734)
return Dist(S(3)*sqrt(b*x**S(2))/(S(2)*b*x), Subst(Int(S(1)/sqrt(-a + x**S(3)), x), x, (a + b*x**S(2))**(S(1)/3)), x)
rule734 = ReplacementRule(pattern734, replacement734)
pattern735 = Pattern(Integral((a_ + x_**S(4)*WC('b', S(1)))**(S(-3)/4), x_), cons2, cons3, cons67)
def replacement735(x, a, b):
rubi.append(735)
return Dist(x**S(3)*(a/(b*x**S(4)) + S(1))**(S(3)/4)/(a + b*x**S(4))**(S(3)/4), Int(S(1)/(x**S(3)*(a/(b*x**S(4)) + S(1))**(S(3)/4)), x), x)
rule735 = ReplacementRule(pattern735, replacement735)
pattern736 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(-1)/6), x_), cons2, cons3, cons67)
def replacement736(x, a, b):
rubi.append(736)
return -Dist(a/S(2), Int((a + b*x**S(2))**(S(-7)/6), x), x) + Simp(S(3)*x/(S(2)*(a + b*x**S(2))**(S(1)/6)), x)
rule736 = ReplacementRule(pattern736, replacement736)
pattern737 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons148, cons13, cons485, cons486, cons487)
def replacement737(p, b, a, n, x):
rubi.append(737)
return Dist(a**(p + S(1)/n), Subst(Int((-b*x**n + S(1))**(-p + S(-1) - S(1)/n), x), x, x*(a + b*x**n)**(-S(1)/n)), x)
rule737 = ReplacementRule(pattern737, replacement737)
pattern738 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons148, cons13, cons485, cons486, cons488)
def replacement738(p, b, a, n, x):
rubi.append(738)
return Dist((a/(a + b*x**n))**(p + S(1)/n)*(a + b*x**n)**(p + S(1)/n), Subst(Int((-b*x**n + S(1))**(-p + S(-1) - S(1)/n), x), x, x*(a + b*x**n)**(-S(1)/n)), x)
rule738 = ReplacementRule(pattern738, replacement738)
pattern739 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons196)
def replacement739(p, b, a, n, x):
rubi.append(739)
return -Subst(Int((a + b*x**(-n))**p/x**S(2), x), x, S(1)/x)
rule739 = ReplacementRule(pattern739, replacement739)
def With740(p, b, a, n, x):
k = Denominator(n)
rubi.append(740)
return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*x**(k*n))**p, x), x, x**(S(1)/k)), x)
pattern740 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons489)
rule740 = ReplacementRule(pattern740, With740)
pattern741 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons128)
def replacement741(p, b, a, n, x):
rubi.append(741)
return Int(ExpandIntegrand((a + b*x**n)**p, x), x)
rule741 = ReplacementRule(pattern741, replacement741)
pattern742 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons357, cons490, cons491, cons492)
def replacement742(p, b, a, n, x):
rubi.append(742)
return Simp(a**p*x*Hypergeometric2F1(-p, S(1)/n, S(1) + S(1)/n, -b*x**n/a), x)
rule742 = ReplacementRule(pattern742, replacement742)
pattern743 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons357, cons490, cons491, cons493)
def replacement743(p, b, a, n, x):
rubi.append(743)
return Dist(a**IntPart(p)*(S(1) + b*x**n/a)**(-FracPart(p))*(a + b*x**n)**FracPart(p), Int((S(1) + b*x**n/a)**p, x), x)
rule743 = ReplacementRule(pattern743, replacement743)
pattern744 = Pattern(Integral((u_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons4, cons5, cons68, cons69)
def replacement744(p, u, b, a, n, x):
rubi.append(744)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x**n)**p, x), x, u), x)
rule744 = ReplacementRule(pattern744, replacement744)
pattern745 = Pattern(Integral((x_**n_*WC('b1', S(1)) + WC('a1', S(0)))**WC('p', S(1))*(x_**n_*WC('b2', S(1)) + WC('a2', S(0)))**WC('p', S(1)), x_), cons57, cons58, cons59, cons60, cons4, cons5, cons55, cons494)
def replacement745(a2, p, b2, b1, n, x, a1):
rubi.append(745)
return Int((a1*a2 + b1*b2*x**(S(2)*n))**p, x)
rule745 = ReplacementRule(pattern745, replacement745)
pattern746 = Pattern(Integral((a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons57, cons58, cons59, cons60, cons55, cons495, cons13, cons163, cons496)
def replacement746(p, a2, b2, b1, n, x, a1):
rubi.append(746)
return Dist(S(2)*a1*a2*n*p/(S(2)*n*p + S(1)), Int((a1 + b1*x**n)**(p + S(-1))*(a2 + b2*x**n)**(p + S(-1)), x), x) + Simp(x*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p/(S(2)*n*p + S(1)), x)
rule746 = ReplacementRule(pattern746, replacement746)
pattern747 = Pattern(Integral((a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**p_*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons55, cons495, cons13, cons137, cons496)
def replacement747(p, a2, b2, b1, n, x, a1):
rubi.append(747)
return Dist((S(2)*n*(p + S(1)) + S(1))/(S(2)*a1*a2*n*(p + S(1))), Int((a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) - Simp(x*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*a1*a2*n*(p + S(1))), x)
rule747 = ReplacementRule(pattern747, replacement747)
pattern748 = Pattern(Integral((a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons5, cons55, cons497)
def replacement748(p, a2, b2, b1, n, x, a1):
rubi.append(748)
return -Subst(Int((a1 + b1*x**(-n))**p*(a2 + b2*x**(-n))**p/x**S(2), x), x, S(1)/x)
rule748 = ReplacementRule(pattern748, replacement748)
def With749(p, a2, b2, b1, n, x, a1):
k = Denominator(S(2)*n)
rubi.append(749)
return Dist(k, Subst(Int(x**(k + S(-1))*(a1 + b1*x**(k*n))**p*(a2 + b2*x**(k*n))**p, x), x, x**(S(1)/k)), x)
pattern749 = Pattern(Integral((a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons5, cons55, cons498)
rule749 = ReplacementRule(pattern749, With749)
pattern750 = Pattern(Integral((x_**n_*WC('b1', S(1)) + WC('a1', S(0)))**p_*(x_**n_*WC('b2', S(1)) + WC('a2', S(0)))**p_, x_), cons57, cons58, cons59, cons60, cons4, cons5, cons55, cons147)
def replacement750(a2, p, b2, b1, n, x, a1):
rubi.append(750)
return Dist((a1 + b1*x**n)**FracPart(p)*(a2 + b2*x**n)**FracPart(p)*(a1*a2 + b1*b2*x**(S(2)*n))**(-FracPart(p)), Int((a1*a2 + b1*b2*x**(S(2)*n))**p, x), x)
rule750 = ReplacementRule(pattern750, replacement750)
pattern751 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons5, cons55, cons494)
def replacement751(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(751)
return Int((c*x)**m*(a1*a2 + b1*b2*x**(S(2)*n))**p, x)
rule751 = ReplacementRule(pattern751, replacement751)
pattern752 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)))**p_, x_), cons3, cons7, cons21, cons4, cons5, cons499, cons500)
def replacement752(p, m, b, c, n, x):
rubi.append(752)
return Dist(b**(S(1) - (m + S(1))/n)*c**m/n, Subst(Int((b*x)**(p + S(-1) + (m + S(1))/n), x), x, x**n), x)
rule752 = ReplacementRule(pattern752, replacement752)
pattern753 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons3, cons7, cons21, cons4, cons5, cons499, cons501)
def replacement753(p, m, b, c, n, x):
rubi.append(753)
return Dist(b**IntPart(p)*c**m*x**(-n*FracPart(p))*(b*x**n)**FracPart(p), Int(x**(m + n*p), x), x)
rule753 = ReplacementRule(pattern753, replacement753)
pattern754 = Pattern(Integral((c_*x_)**m_*(x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons3, cons7, cons21, cons4, cons5, cons18)
def replacement754(p, m, b, c, n, x):
rubi.append(754)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(b*x**n)**p, x), x)
rule754 = ReplacementRule(pattern754, replacement754)
pattern755 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons38, cons502)
def replacement755(p, m, b, a, n, x):
rubi.append(755)
return Int(x**(m + n*p)*(a*x**(-n) + b)**p, x)
rule755 = ReplacementRule(pattern755, replacement755)
pattern756 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons503, cons66)
def replacement756(p, m, b, c, a, n, x):
rubi.append(756)
return Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*(m + S(1))), x)
rule756 = ReplacementRule(pattern756, replacement756)
pattern757 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons5, cons55, cons504, cons66)
def replacement757(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(757)
return Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(a1*a2*c*(m + S(1))), x)
rule757 = ReplacementRule(pattern757, replacement757)
pattern758 = Pattern(Integral(x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons5, cons500)
def replacement758(p, m, b, a, n, x):
rubi.append(758)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x)**p, x), x, x**n), x)
rule758 = ReplacementRule(pattern758, replacement758)
pattern759 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons21, cons4, cons5, cons55, cons505)
def replacement759(p, a2, m, b2, b1, n, x, a1):
rubi.append(759)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a1 + b1*x)**p*(a2 + b2*x)**p, x), x, x**n), x)
rule759 = ReplacementRule(pattern759, replacement759)
pattern760 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons500)
def replacement760(p, m, b, c, a, n, x):
rubi.append(760)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
rule760 = ReplacementRule(pattern760, replacement760)
pattern761 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons5, cons55, cons505)
def replacement761(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(761)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
rule761 = ReplacementRule(pattern761, replacement761)
pattern762 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons128)
def replacement762(p, m, b, c, a, n, x):
rubi.append(762)
return Int(ExpandIntegrand((c*x)**m*(a + b*x**n)**p, x), x)
rule762 = ReplacementRule(pattern762, replacement762)
pattern763 = Pattern(Integral(x_**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons5, cons506, cons66)
def replacement763(p, m, b, a, n, x):
rubi.append(763)
return -Dist(b*(m + n*(p + S(1)) + S(1))/(a*(m + S(1))), Int(x**(m + n)*(a + b*x**n)**p, x), x) + Simp(x**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*(m + S(1))), x)
rule763 = ReplacementRule(pattern763, replacement763)
pattern764 = Pattern(Integral(x_**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons21, cons4, cons5, cons55, cons507, cons66)
def replacement764(p, a2, m, b2, b1, n, x, a1):
rubi.append(764)
return -Dist(b1*b2*(m + S(2)*n*(p + S(1)) + S(1))/(a1*a2*(m + S(1))), Int(x**(m + S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp(x**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(a1*a2*(m + S(1))), x)
rule764 = ReplacementRule(pattern764, replacement764)
pattern765 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons506, cons54)
def replacement765(p, m, b, c, a, n, x):
rubi.append(765)
return Dist((m + n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((c*x)**m*(a + b*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*n*(p + S(1))), x)
rule765 = ReplacementRule(pattern765, replacement765)
pattern766 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons5, cons55, cons507, cons54)
def replacement766(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(766)
return Dist((m + S(2)*n*(p + S(1)) + S(1))/(S(2)*a1*a2*n*(p + S(1))), Int((c*x)**m*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*a1*a2*c*n*(p + S(1))), x)
rule766 = ReplacementRule(pattern766, replacement766)
def With767(p, m, b, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
pattern767 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons148, cons17, CustomConstraint(With767))
def replacement767(p, m, b, a, n, x):
k = GCD(m + S(1), n)
rubi.append(767)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a + b*x**(n/k))**p, x), x, x**k), x)
rule767 = ReplacementRule(pattern767, replacement767)
def With768(p, a2, m, b2, b1, n, x, a1):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), S(2)*n)
if Unequal(k, S(1)):
return True
return False
pattern768 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons5, cons55, cons495, cons17, CustomConstraint(With768))
def replacement768(p, a2, m, b2, b1, n, x, a1):
k = GCD(m + S(1), S(2)*n)
rubi.append(768)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a1 + b1*x**(n/k))**p*(a2 + b2*x**(n/k))**p, x), x, x**k), x)
rule768 = ReplacementRule(pattern768, replacement768)
pattern769 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons148, cons244, cons163, cons94, cons508, cons509)
def replacement769(p, m, b, c, a, n, x):
rubi.append(769)
return -Dist(b*c**(-n)*n*p/(m + S(1)), Int((c*x)**(m + n)*(a + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**p/(c*(m + S(1))), x)
rule769 = ReplacementRule(pattern769, replacement769)
pattern770 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons55, cons495, cons244, cons163, cons510, cons511)
def replacement770(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(770)
return Dist(S(2)*a1*a2*n*p/(m + S(2)*n*p + S(1)), Int((c*x)**m*(a1 + b1*x**n)**(p + S(-1))*(a2 + b2*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p/(c*(m + S(2)*n*p + S(1))), x)
rule770 = ReplacementRule(pattern770, replacement770)
pattern771 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons148, cons244, cons163, cons512, cons509)
def replacement771(p, m, b, c, a, n, x):
rubi.append(771)
return Dist(a*n*p/(m + n*p + S(1)), Int((c*x)**m*(a + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**p/(c*(m + n*p + S(1))), x)
rule771 = ReplacementRule(pattern771, replacement771)
pattern772 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons466)
def replacement772(x, a, b):
rubi.append(772)
return Dist(x*(a/(b*x**S(4)) + S(1))**(S(1)/4)/(b*(a + b*x**S(4))**(S(1)/4)), Int(S(1)/(x**S(3)*(a/(b*x**S(4)) + S(1))**(S(5)/4)), x), x)
rule772 = ReplacementRule(pattern772, replacement772)
pattern773 = Pattern(Integral(x_**m_/(a_ + x_**S(4)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons466, cons513)
def replacement773(x, m, b, a):
rubi.append(773)
return -Dist(a*(m + S(-3))/(b*(m + S(-4))), Int(x**(m + S(-4))/(a + b*x**S(4))**(S(5)/4), x), x) + Simp(x**(m + S(-3))/(b*(a + b*x**S(4))**(S(1)/4)*(m + S(-4))), x)
rule773 = ReplacementRule(pattern773, replacement773)
pattern774 = Pattern(Integral(x_**m_/(a_ + x_**S(4)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons466, cons514)
def replacement774(x, m, b, a):
rubi.append(774)
return -Dist(b*m/(a*(m + S(1))), Int(x**(m + S(4))/(a + b*x**S(4))**(S(5)/4), x), x) + Simp(x**(m + S(1))/(a*(a + b*x**S(4))**(S(1)/4)*(m + S(1))), x)
rule774 = ReplacementRule(pattern774, replacement774)
pattern775 = Pattern(Integral(sqrt(x_*WC('c', S(1)))/(a_ + x_**S(2)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons7, cons466)
def replacement775(x, c, b, a):
rubi.append(775)
return Dist(sqrt(c*x)*(a/(b*x**S(2)) + S(1))**(S(1)/4)/(b*(a + b*x**S(2))**(S(1)/4)), Int(S(1)/(x**S(2)*(a/(b*x**S(2)) + S(1))**(S(5)/4)), x), x)
rule775 = ReplacementRule(pattern775, replacement775)
pattern776 = Pattern(Integral((x_*WC('c', S(1)))**m_/(a_ + x_**S(2)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons7, cons466, cons515, cons516)
def replacement776(m, b, c, a, x):
rubi.append(776)
return -Dist(S(2)*a*c**S(2)*(m + S(-1))/(b*(S(2)*m + S(-3))), Int((c*x)**(m + S(-2))/(a + b*x**S(2))**(S(5)/4), x), x) + Simp(S(2)*c*(c*x)**(m + S(-1))/(b*(a + b*x**S(2))**(S(1)/4)*(S(2)*m + S(-3))), x)
rule776 = ReplacementRule(pattern776, replacement776)
pattern777 = Pattern(Integral((x_*WC('c', S(1)))**m_/(a_ + x_**S(2)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons7, cons466, cons515, cons94)
def replacement777(m, b, c, a, x):
rubi.append(777)
return -Dist(b*(S(2)*m + S(1))/(S(2)*a*c**S(2)*(m + S(1))), Int((c*x)**(m + S(2))/(a + b*x**S(2))**(S(5)/4), x), x) + Simp((c*x)**(m + S(1))/(a*c*(a + b*x**S(2))**(S(1)/4)*(m + S(1))), x)
rule777 = ReplacementRule(pattern777, replacement777)
pattern778 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1)))**(S(5)/4), x_), cons2, cons3, cons483)
def replacement778(x, a, b):
rubi.append(778)
return -Dist(S(1)/b, Int(S(1)/(x**S(2)*(a + b*x**S(4))**(S(1)/4)), x), x) - Simp(S(1)/(b*x*(a + b*x**S(4))**(S(1)/4)), x)
rule778 = ReplacementRule(pattern778, replacement778)
pattern779 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons148, cons244, cons137, cons517, cons518, cons509)
def replacement779(p, m, b, c, a, n, x):
rubi.append(779)
return -Dist(c**n*(m - n + S(1))/(b*n*(p + S(1))), Int((c*x)**(m - n)*(a + b*x**n)**(p + S(1)), x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule779 = ReplacementRule(pattern779, replacement779)
pattern780 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons55, cons495, cons244, cons137, cons519, cons520, cons511)
def replacement780(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(780)
return -Dist(c**(S(2)*n)*(m - S(2)*n + S(1))/(S(2)*b1*b2*n*(p + S(1))), Int((c*x)**(m - S(2)*n)*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) + Simp(c**(S(2)*n + S(-1))*(c*x)**(m - S(2)*n + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*b1*b2*n*(p + S(1))), x)
rule780 = ReplacementRule(pattern780, replacement780)
pattern781 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons148, cons244, cons137, cons509)
def replacement781(p, m, b, c, a, n, x):
rubi.append(781)
return Dist((m + n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((c*x)**m*(a + b*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*n*(p + S(1))), x)
rule781 = ReplacementRule(pattern781, replacement781)
pattern782 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons55, cons495, cons244, cons137, cons511)
def replacement782(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(782)
return Dist((m + S(2)*n*(p + S(1)) + S(1))/(S(2)*a1*a2*n*(p + S(1))), Int((c*x)**m*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*a1*a2*c*n*(p + S(1))), x)
rule782 = ReplacementRule(pattern782, replacement782)
pattern783 = Pattern(Integral(x_/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement783(x, a, b):
rubi.append(783)
return Dist(S(1)/(S(3)*Rt(a, S(3))*Rt(b, S(3))), Int((x*Rt(b, S(3)) + Rt(a, S(3)))/(x**S(2)*Rt(b, S(3))**S(2) - x*Rt(a, S(3))*Rt(b, S(3)) + Rt(a, S(3))**S(2)), x), x) - Dist(S(1)/(S(3)*Rt(a, S(3))*Rt(b, S(3))), Int(S(1)/(x*Rt(b, S(3)) + Rt(a, S(3))), x), x)
rule783 = ReplacementRule(pattern783, replacement783)
def With784(m, b, a, n, x):
r = Numerator(Rt(a/b, n))
s = Denominator(Rt(a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r*cos(Pi*m*(S(2)*k + S(-1))/n) - s*x*cos(Pi*(S(2)*k + S(-1))*(m + S(1))/n))/(r**S(2) - S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r*cos(Pi*m*(2*k - 1)/n) - s*x*cos(Pi*(2*k - 1)*(m + 1)/n))/(r**2 - 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(Dist(2*r**(m + 1)*s**(-m)/(a*n), Sum_doit(u, List(k, 1, n/2 - S.Half)), x) - s**(-m)*(-r)**(m + 1)*Int(1/(r + s*x), x)/(a*n), x)
pattern784 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons521, cons62, cons522, cons468)
rule784 = ReplacementRule(pattern784, With784)
def With785(m, b, a, n, x):
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r*cos(Pi*m*(S(2)*k + S(-1))/n) + s*x*cos(Pi*(S(2)*k + S(-1))*(m + S(1))/n))/(r**S(2) + S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r*cos(Pi*m*(2*k - 1)/n) + s*x*cos(Pi*(2*k - 1)*(m + 1)/n))/(r**2 + 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(-Dist(2*s**(-m)*(-r)**(m + 1)/(a*n), Sum_doit(u, List(k, 1, n/2 - S.Half)), x) + r**(m + 1)*s**(-m)*Int(1/(r - s*x), x)/(a*n), x)
pattern785 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons523, cons62, cons522, cons469)
rule785 = ReplacementRule(pattern785, With785)
def With786(m, b, a, n, x):
r = Numerator(Rt(a/b, n))
s = Denominator(Rt(a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r*cos(Pi*m*(S(2)*k + S(-1))/n) - s*x*cos(Pi*(S(2)*k + S(-1))*(m + S(1))/n))/(r**S(2) - S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x) + Int((r*cos(Pi*m*(S(2)*k + S(-1))/n) + s*x*cos(Pi*(S(2)*k + S(-1))*(m + S(1))/n))/(r**S(2) + S(2)*r*s*x*cos(Pi*(S(2)*k + S(-1))/n) + s**S(2)*x**S(2)), x)
u = Int((r*cos(Pi*m*(2*k - 1)/n) - s*x*cos(Pi*(2*k - 1)*(m + 1)/n))/(r**2 - 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x) + Int((r*cos(Pi*m*(2*k - 1)/n) + s*x*cos(Pi*(2*k - 1)*(m + 1)/n))/(r**2 + 2*r*s*x*cos(Pi*(2*k - 1)/n) + s**2*x**2), x)
return Simp(2*(-1)**(m/2)*r**(m + 2)*s**(-m)*Int(1/(r**2 + s**2*x**2), x)/(a*n) + Dist(2*r**(m + 1)*s**(-m)/(a*n), Sum_doit(u, List(k, 1, n/4 - S.Half)), x), x)
pattern786 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons524, cons62, cons522, cons468)
rule786 = ReplacementRule(pattern786, With786)
def With787(m, b, a, n, x):
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
k = Symbol('k')
u = Symbol('u')
u = Int((r*cos(S(2)*Pi*k*m/n) - s*x*cos(S(2)*Pi*k*(m + S(1))/n))/(r**S(2) - S(2)*r*s*x*cos(S(2)*Pi*k/n) + s**S(2)*x**S(2)), x) + Int((r*cos(S(2)*Pi*k*m/n) + s*x*cos(S(2)*Pi*k*(m + S(1))/n))/(r**S(2) + S(2)*r*s*x*cos(S(2)*Pi*k/n) + s**S(2)*x**S(2)), x)
u = Int((r*cos(2*Pi*k*m/n) - s*x*cos(2*Pi*k*(m + 1)/n))/(r**2 - 2*r*s*x*cos(2*Pi*k/n) + s**2*x**2), x) + Int((r*cos(2*Pi*k*m/n) + s*x*cos(2*Pi*k*(m + 1)/n))/(r**2 + 2*r*s*x*cos(2*Pi*k/n) + s**2*x**2), x)
return Simp(Dist(2*r**(m + 1)*s**(-m)/(a*n), Sum_doit(u, List(k, 1, n/4 - S.Half)), x) + 2*r**(m + 2)*s**(-m)*Int(1/(r**2 - s**2*x**2), x)/(a*n), x)
pattern787 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons524, cons62, cons522, cons469)
rule787 = ReplacementRule(pattern787, With787)
def With788(x, a, b):
r = Numerator(Rt(a/b, S(2)))
s = Denominator(Rt(a/b, S(2)))
rubi.append(788)
return -Dist(S(1)/(S(2)*s), Int((r - s*x**S(2))/(a + b*x**S(4)), x), x) + Dist(S(1)/(S(2)*s), Int((r + s*x**S(2))/(a + b*x**S(4)), x), x)
pattern788 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons475)
rule788 = ReplacementRule(pattern788, With788)
def With789(x, a, b):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
rubi.append(789)
return -Dist(s/(S(2)*b), Int(S(1)/(r - s*x**S(2)), x), x) + Dist(s/(S(2)*b), Int(S(1)/(r + s*x**S(2)), x), x)
pattern789 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons476)
rule789 = ReplacementRule(pattern789, With789)
def With790(m, b, a, n, x):
r = Numerator(Rt(a/b, S(4)))
s = Denominator(Rt(a/b, S(4)))
rubi.append(790)
return Dist(sqrt(S(2))*s**S(3)/(S(4)*b*r), Int(x**(m - n/S(4))/(r**S(2) - sqrt(S(2))*r*s*x**(n/S(4)) + s**S(2)*x**(n/S(2))), x), x) - Dist(sqrt(S(2))*s**S(3)/(S(4)*b*r), Int(x**(m - n/S(4))/(r**S(2) + sqrt(S(2))*r*s*x**(n/S(4)) + s**S(2)*x**(n/S(2))), x), x)
pattern790 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons525, cons62, cons522, cons478)
rule790 = ReplacementRule(pattern790, With790)
def With791(m, b, a, n, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
rubi.append(791)
return Dist(r/(S(2)*a), Int(x**m/(r - s*x**(n/S(2))), x), x) + Dist(r/(S(2)*a), Int(x**m/(r + s*x**(n/S(2))), x), x)
pattern791 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons525, cons62, cons526, cons476)
rule791 = ReplacementRule(pattern791, With791)
def With792(m, b, a, n, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
rubi.append(792)
return -Dist(s/(S(2)*b), Int(x**(m - n/S(2))/(r - s*x**(n/S(2))), x), x) + Dist(s/(S(2)*b), Int(x**(m - n/S(2))/(r + s*x**(n/S(2))), x), x)
pattern792 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons525, cons62, cons527, cons476)
rule792 = ReplacementRule(pattern792, With792)
pattern793 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons528, cons529)
def replacement793(m, b, a, n, x):
rubi.append(793)
return Int(PolynomialDivide(x**m, a + b*x**n, x), x)
rule793 = ReplacementRule(pattern793, replacement793)
def With794(x, a, b):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(794)
return Dist(S(1)/r, Int((r*x + s*(-sqrt(S(3)) + S(1)))/sqrt(a + b*x**S(3)), x), x) + Dist(sqrt(S(2))*s/(r*sqrt(sqrt(S(3)) + S(2))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
pattern794 = Pattern(Integral(x_/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons481)
rule794 = ReplacementRule(pattern794, With794)
def With795(x, a, b):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(795)
return Dist(S(1)/r, Int((r*x + s*(S(1) + sqrt(S(3))))/sqrt(a + b*x**S(3)), x), x) - Dist(sqrt(S(2))*s/(r*sqrt(-sqrt(S(3)) + S(2))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
pattern795 = Pattern(Integral(x_/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons482)
rule795 = ReplacementRule(pattern795, With795)
def With796(x, a, b):
q = Rt(b/a, S(2))
rubi.append(796)
return -Dist(S(1)/q, Int((-q*x**S(2) + S(1))/sqrt(a + b*x**S(4)), x), x) + Dist(S(1)/q, Int(S(1)/sqrt(a + b*x**S(4)), x), x)
pattern796 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons466)
rule796 = ReplacementRule(pattern796, With796)
def With797(x, a, b):
q = Rt(-b/a, S(2))
rubi.append(797)
return -Dist(S(1)/q, Int((-q*x**S(2) + S(1))/sqrt(a + b*x**S(4)), x), x) + Dist(S(1)/q, Int(S(1)/sqrt(a + b*x**S(4)), x), x)
pattern797 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons484, cons105)
rule797 = ReplacementRule(pattern797, With797)
def With798(x, a, b):
q = Rt(-b/a, S(2))
rubi.append(798)
return Dist(S(1)/q, Int((q*x**S(2) + S(1))/sqrt(a + b*x**S(4)), x), x) - Dist(S(1)/q, Int(S(1)/sqrt(a + b*x**S(4)), x), x)
pattern798 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons483)
rule798 = ReplacementRule(pattern798, With798)
def With799(x, a, b):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(799)
return -Dist(S(1)/(S(2)*r**S(2)), Int((-S(2)*r**S(2)*x**S(4) + s**S(2)*(S(-1) + sqrt(S(3))))/sqrt(a + b*x**S(6)), x), x) + Dist(s**S(2)*(S(-1) + sqrt(S(3)))/(S(2)*r**S(2)), Int(S(1)/sqrt(a + b*x**S(6)), x), x)
pattern799 = Pattern(Integral(x_**S(4)/sqrt(a_ + x_**S(6)*WC('b', S(1))), x_), cons2, cons3, cons67)
rule799 = ReplacementRule(pattern799, With799)
pattern800 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(8)*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement800(x, a, b):
rubi.append(800)
return -Dist(S(1)/(S(2)*Rt(b/a, S(4))), Int((-x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x) + Dist(S(1)/(S(2)*Rt(b/a, S(4))), Int((x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x)
rule800 = ReplacementRule(pattern800, replacement800)
pattern801 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4), x_), cons2, cons3, cons466)
def replacement801(x, a, b):
rubi.append(801)
return -Dist(a/S(2), Int(x**S(2)/(a + b*x**S(4))**(S(5)/4), x), x) + Simp(x**S(3)/(S(2)*(a + b*x**S(4))**(S(1)/4)), x)
rule801 = ReplacementRule(pattern801, replacement801)
pattern802 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4), x_), cons2, cons3, cons483)
def replacement802(x, a, b):
rubi.append(802)
return Dist(a/(S(2)*b), Int(S(1)/(x**S(2)*(a + b*x**S(4))**(S(1)/4)), x), x) + Simp((a + b*x**S(4))**(S(3)/4)/(S(2)*b*x), x)
rule802 = ReplacementRule(pattern802, replacement802)
pattern803 = Pattern(Integral(S(1)/(x_**S(2)*(a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4)), x_), cons2, cons3, cons466)
def replacement803(x, a, b):
rubi.append(803)
return -Dist(b, Int(x**S(2)/(a + b*x**S(4))**(S(5)/4), x), x) - Simp(S(1)/(x*(a + b*x**S(4))**(S(1)/4)), x)
rule803 = ReplacementRule(pattern803, replacement803)
pattern804 = Pattern(Integral(S(1)/(x_**S(2)*(a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4)), x_), cons2, cons3, cons483)
def replacement804(x, a, b):
rubi.append(804)
return Dist(x*(a/(b*x**S(4)) + S(1))**(S(1)/4)/(a + b*x**S(4))**(S(1)/4), Int(S(1)/(x**S(3)*(a/(b*x**S(4)) + S(1))**(S(1)/4)), x), x)
rule804 = ReplacementRule(pattern804, replacement804)
pattern805 = Pattern(Integral(sqrt(c_*x_)/(a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4), x_), cons2, cons3, cons7, cons466)
def replacement805(x, c, b, a):
rubi.append(805)
return -Dist(a/S(2), Int(sqrt(c*x)/(a + b*x**S(2))**(S(5)/4), x), x) + Simp(x*sqrt(c*x)/(a + b*x**S(2))**(S(1)/4), x)
rule805 = ReplacementRule(pattern805, replacement805)
pattern806 = Pattern(Integral(sqrt(c_*x_)/(a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4), x_), cons2, cons3, cons7, cons483)
def replacement806(x, c, b, a):
rubi.append(806)
return Dist(a*c**S(2)/(S(2)*b), Int(S(1)/((c*x)**(S(3)/2)*(a + b*x**S(2))**(S(1)/4)), x), x) + Simp(c*(a + b*x**S(2))**(S(3)/4)/(b*sqrt(c*x)), x)
rule806 = ReplacementRule(pattern806, replacement806)
pattern807 = Pattern(Integral(S(1)/((x_*WC('c', S(1)))**(S(3)/2)*(a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4)), x_), cons2, cons3, cons7, cons466)
def replacement807(x, c, b, a):
rubi.append(807)
return -Dist(b/c**S(2), Int(sqrt(c*x)/(a + b*x**S(2))**(S(5)/4), x), x) + Simp(-S(2)/(c*sqrt(c*x)*(a + b*x**S(2))**(S(1)/4)), x)
rule807 = ReplacementRule(pattern807, replacement807)
pattern808 = Pattern(Integral(S(1)/((x_*WC('c', S(1)))**(S(3)/2)*(a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4)), x_), cons2, cons3, cons7, cons483)
def replacement808(x, c, b, a):
rubi.append(808)
return Dist(sqrt(c*x)*(a/(b*x**S(2)) + S(1))**(S(1)/4)/(c**S(2)*(a + b*x**S(2))**(S(1)/4)), Int(S(1)/(x**S(2)*(a/(b*x**S(2)) + S(1))**(S(1)/4)), x), x)
rule808 = ReplacementRule(pattern808, replacement808)
pattern809 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons148, cons31, cons530, cons512, cons509)
def replacement809(p, m, b, c, a, n, x):
rubi.append(809)
return -Dist(a*c**n*(m - n + S(1))/(b*(m + n*p + S(1))), Int((c*x)**(m - n)*(a + b*x**n)**p, x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))/(b*(m + n*p + S(1))), x)
rule809 = ReplacementRule(pattern809, replacement809)
pattern810 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons148, cons531, cons512, cons532)
def replacement810(p, m, b, c, a, n, x):
rubi.append(810)
return -Dist(a*c**n*(m - n + S(1))/(b*(m + n*p + S(1))), Int((c*x)**(m - n)*(a + b*x**n)**p, x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))/(b*(m + n*p + S(1))), x)
rule810 = ReplacementRule(pattern810, replacement810)
pattern811 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons5, cons55, cons495, cons31, cons529, cons510, cons511)
def replacement811(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(811)
return -Dist(a1*a2*c**(S(2)*n)*(m - S(2)*n + S(1))/(b1*b2*(m + S(2)*n*p + S(1))), Int((c*x)**(m - S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp(c**(S(2)*n + S(-1))*(c*x)**(m - S(2)*n + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(b1*b2*(m + S(2)*n*p + S(1))), x)
rule811 = ReplacementRule(pattern811, replacement811)
pattern812 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons5, cons55, cons495, cons533, cons510, cons534)
def replacement812(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(812)
return -Dist(a1*a2*c**(S(2)*n)*(m - S(2)*n + S(1))/(b1*b2*(m + S(2)*n*p + S(1))), Int((c*x)**(m - S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp(c**(S(2)*n + S(-1))*(c*x)**(m - S(2)*n + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(b1*b2*(m + S(2)*n*p + S(1))), x)
rule812 = ReplacementRule(pattern812, replacement812)
pattern813 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons148, cons31, cons94, cons509)
def replacement813(p, m, b, c, a, n, x):
rubi.append(813)
return -Dist(b*c**(-n)*(m + n*(p + S(1)) + S(1))/(a*(m + S(1))), Int((c*x)**(m + n)*(a + b*x**n)**p, x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*(m + S(1))), x)
rule813 = ReplacementRule(pattern813, replacement813)
pattern814 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons148, cons535, cons532)
def replacement814(p, m, b, c, a, n, x):
rubi.append(814)
return -Dist(b*c**(-n)*(m + n*(p + S(1)) + S(1))/(a*(m + S(1))), Int((c*x)**(m + n)*(a + b*x**n)**p, x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*(m + S(1))), x)
rule814 = ReplacementRule(pattern814, replacement814)
pattern815 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons5, cons55, cons495, cons31, cons94, cons511)
def replacement815(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(815)
return -Dist(b1*b2*c**(-S(2)*n)*(m + S(2)*n*(p + S(1)) + S(1))/(a1*a2*(m + S(1))), Int((c*x)**(m + S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(a1*a2*c*(m + S(1))), x)
rule815 = ReplacementRule(pattern815, replacement815)
pattern816 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons5, cons55, cons495, cons536, cons534)
def replacement816(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(816)
return -Dist(b1*b2*c**(-S(2)*n)*(m + S(2)*n*(p + S(1)) + S(1))/(a1*a2*(m + S(1))), Int((c*x)**(m + S(2)*n)*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x) + Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(a1*a2*c*(m + S(1))), x)
rule816 = ReplacementRule(pattern816, replacement816)
def With817(p, m, b, c, a, n, x):
k = Denominator(m)
rubi.append(817)
return Dist(k/c, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*c**(-n)*x**(k*n))**p, x), x, (c*x)**(S(1)/k)), x)
pattern817 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons148, cons367, cons509)
rule817 = ReplacementRule(pattern817, With817)
def With818(p, a2, m, b2, b1, c, n, x, a1):
k = Denominator(m)
rubi.append(818)
return Dist(k/c, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a1 + b1*c**(-n)*x**(k*n))**p*(a2 + b2*c**(-n)*x**(k*n))**p, x), x, (c*x)**(S(1)/k)), x)
pattern818 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons5, cons55, cons495, cons367, cons511)
rule818 = ReplacementRule(pattern818, With818)
pattern819 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons148, cons13, cons485, cons486, cons537)
def replacement819(p, m, b, a, n, x):
rubi.append(819)
return Dist(a**(p + (m + S(1))/n), Subst(Int(x**m*(-b*x**n + S(1))**(-p + S(-1) - (m + S(1))/n), x), x, x*(a + b*x**n)**(-S(1)/n)), x)
rule819 = ReplacementRule(pattern819, replacement819)
pattern820 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons55, cons495, cons13, cons485, cons486, cons538)
def replacement820(p, a2, m, b2, b1, n, x, a1):
rubi.append(820)
return Dist((a1*a2)**(p + (m + S(1))/(S(2)*n)), Subst(Int(x**m*(-b1*x**n + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n))*(-b2*x**n + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n)), x), x, x*(a1 + b1*x**n)**(-S(1)/(S(2)*n))*(a2 + b2*x**n)**(-S(1)/(S(2)*n))), x)
rule820 = ReplacementRule(pattern820, replacement820)
pattern821 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons148, cons13, cons485, cons486, cons17, cons539)
def replacement821(p, m, b, a, n, x):
rubi.append(821)
return Dist((a/(a + b*x**n))**(p + (m + S(1))/n)*(a + b*x**n)**(p + (m + S(1))/n), Subst(Int(x**m*(-b*x**n + S(1))**(-p + S(-1) - (m + S(1))/n), x), x, x*(a + b*x**n)**(-S(1)/n)), x)
rule821 = ReplacementRule(pattern821, replacement821)
pattern822 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons55, cons495, cons13, cons485, cons486, cons17, cons540)
def replacement822(p, a2, m, b2, b1, n, x, a1):
rubi.append(822)
return Dist((a1/(a1 + b1*x**n))**(p + (m + S(1))/(S(2)*n))*(a2/(a2 + b2*x**n))**(p + (m + S(1))/(S(2)*n))*(a1 + b1*x**n)**(p + (m + S(1))/(S(2)*n))*(a2 + b2*x**n)**(p + (m + S(1))/(S(2)*n)), Subst(Int(x**m*(-b1*x**n + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n))*(-b2*x**n + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n)), x), x, x*(a1 + b1*x**n)**(-S(1)/(S(2)*n))*(a2 + b2*x**n)**(-S(1)/(S(2)*n))), x)
rule822 = ReplacementRule(pattern822, replacement822)
pattern823 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons196, cons17)
def replacement823(p, m, b, a, n, x):
rubi.append(823)
return -Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p, x), x, S(1)/x)
rule823 = ReplacementRule(pattern823, replacement823)
pattern824 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons5, cons55, cons497, cons17)
def replacement824(p, a2, m, b2, b1, n, x, a1):
rubi.append(824)
return -Subst(Int(x**(-m + S(-2))*(a1 + b1*x**(-n))**p*(a2 + b2*x**(-n))**p, x), x, S(1)/x)
rule824 = ReplacementRule(pattern824, replacement824)
def With825(p, m, b, c, a, n, x):
k = Denominator(m)
rubi.append(825)
return -Dist(k/c, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*c**(-n)*x**(-k*n))**p, x), x, (c*x)**(-S(1)/k)), x)
pattern825 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons196, cons367)
rule825 = ReplacementRule(pattern825, With825)
def With826(p, a2, m, b2, b1, c, n, x, a1):
k = Denominator(m)
rubi.append(826)
return -Dist(k/c, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a1 + b1*c**(-n)*x**(-k*n))**p*(a2 + b2*c**(-n)*x**(-k*n))**p, x), x, (c*x)**(-S(1)/k)), x)
pattern826 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons5, cons55, cons497, cons367)
rule826 = ReplacementRule(pattern826, With826)
pattern827 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons196, cons356)
def replacement827(p, m, b, c, a, n, x):
rubi.append(827)
return -Dist((c*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p, x), x, S(1)/x), x)
rule827 = ReplacementRule(pattern827, replacement827)
pattern828 = Pattern(Integral((x_*WC('c', S(1)))**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons5, cons55, cons497, cons356)
def replacement828(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(828)
return -Dist((c*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a1 + b1*x**(-n))**p*(a2 + b2*x**(-n))**p, x), x, S(1)/x), x)
rule828 = ReplacementRule(pattern828, replacement828)
def With829(p, m, b, a, n, x):
k = Denominator(n)
rubi.append(829)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*x**(k*n))**p, x), x, x**(S(1)/k)), x)
pattern829 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons5, cons489)
rule829 = ReplacementRule(pattern829, With829)
def With830(p, a2, m, b2, b1, n, x, a1):
k = Denominator(S(2)*n)
rubi.append(830)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a1 + b1*x**(k*n))**p*(a2 + b2*x**(k*n))**p, x), x, x**(S(1)/k)), x)
pattern830 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons21, cons5, cons55, cons498)
rule830 = ReplacementRule(pattern830, With830)
pattern831 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons489)
def replacement831(p, m, b, c, a, n, x):
rubi.append(831)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
rule831 = ReplacementRule(pattern831, replacement831)
pattern832 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons5, cons55, cons498)
def replacement832(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(832)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
rule832 = ReplacementRule(pattern832, replacement832)
pattern833 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons5, cons541, cons23)
def replacement833(p, m, b, a, n, x):
rubi.append(833)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p, x), x, x**(m + S(1))), x)
rule833 = ReplacementRule(pattern833, replacement833)
pattern834 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons21, cons4, cons5, cons55, cons542, cons543)
def replacement834(p, a2, m, b2, b1, n, x, a1):
rubi.append(834)
return Dist(S(1)/(m + S(1)), Subst(Int((a1 + b1*x**(n/(m + S(1))))**p*(a2 + b2*x**(n/(m + S(1))))**p, x), x, x**(m + S(1))), x)
rule834 = ReplacementRule(pattern834, replacement834)
pattern835 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons541, cons23)
def replacement835(p, m, b, c, a, n, x):
rubi.append(835)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
rule835 = ReplacementRule(pattern835, replacement835)
pattern836 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons5, cons55, cons542, cons543)
def replacement836(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(836)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
rule836 = ReplacementRule(pattern836, replacement836)
pattern837 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons544, cons13, cons163)
def replacement837(p, m, b, a, n, x):
rubi.append(837)
return -Dist(b*n*p/(m + S(1)), Int(x**(m + n)*(a + b*x**n)**(p + S(-1)), x), x) + Simp(x**(m + S(1))*(a + b*x**n)**p/(m + S(1)), x)
rule837 = ReplacementRule(pattern837, replacement837)
pattern838 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons21, cons4, cons55, cons545, cons13, cons163)
def replacement838(p, a2, m, b2, b1, n, x, a1):
rubi.append(838)
return -Dist(S(2)*b1*b2*n*p/(m + S(1)), Int(x**(m + n)*(a1 + b1*x**n)**(p + S(-1))*(a2 + b2*x**n)**(p + S(-1)), x), x) + Simp(x**(m + S(1))*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p/(m + S(1)), x)
rule838 = ReplacementRule(pattern838, replacement838)
pattern839 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons544, cons13, cons163)
def replacement839(p, m, b, c, a, n, x):
rubi.append(839)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
rule839 = ReplacementRule(pattern839, replacement839)
pattern840 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons55, cons545, cons13, cons163)
def replacement840(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(840)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
rule840 = ReplacementRule(pattern840, replacement840)
pattern841 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons546, cons13, cons163, cons512)
def replacement841(p, m, b, c, a, n, x):
rubi.append(841)
return Dist(a*n*p/(m + n*p + S(1)), Int((c*x)**m*(a + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a + b*x**n)**p/(c*(m + n*p + S(1))), x)
rule841 = ReplacementRule(pattern841, replacement841)
pattern842 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons55, cons547, cons13, cons163, cons510)
def replacement842(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(842)
return Dist(S(2)*a1*a2*n*p/(m + S(2)*n*p + S(1)), Int((c*x)**m*(a1 + b1*x**n)**(p + S(-1))*(a2 + b2*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p/(c*(m + S(2)*n*p + S(1))), x)
rule842 = ReplacementRule(pattern842, replacement842)
def With843(p, m, b, a, n, x):
k = Denominator(p)
rubi.append(843)
return Dist(a**(p + (m + S(1))/n)*k/n, Subst(Int(x**(k*(m + S(1))/n + S(-1))*(-b*x**k + S(1))**(-p + S(-1) - (m + S(1))/n), x), x, x**(n/k)*(a + b*x**n)**(-S(1)/k)), x)
pattern843 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons546, cons13, cons485)
rule843 = ReplacementRule(pattern843, With843)
def With844(p, a2, m, b2, b1, n, x, a1):
k = Denominator(p)
rubi.append(844)
return Dist(k*(a1*a2)**(p + (m + S(1))/(S(2)*n))/(S(2)*n), Subst(Int(x**(k*(m + S(1))/(S(2)*n) + S(-1))*(-b1*x**k + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n))*(-b2*x**k + S(1))**(-p + S(-1) - (m + S(1))/(S(2)*n)), x), x, x**(S(2)*n/k)*(a1 + b1*x**n)**(-S(1)/k)*(a2 + b2*x**n)**(-S(1)/k)), x)
pattern844 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons21, cons4, cons55, cons547, cons13, cons485)
rule844 = ReplacementRule(pattern844, With844)
pattern845 = Pattern(Integral((c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons546, cons13, cons485)
def replacement845(p, m, b, c, a, n, x):
rubi.append(845)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a + b*x**n)**p, x), x)
rule845 = ReplacementRule(pattern845, replacement845)
pattern846 = Pattern(Integral((c_*x_)**m_*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons55, cons547, cons13, cons485)
def replacement846(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(846)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a1 + b1*x**n)**p*(a2 + b2*x**n)**p, x), x)
rule846 = ReplacementRule(pattern846, replacement846)
pattern847 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons546, cons13, cons137)
def replacement847(p, m, b, c, a, n, x):
rubi.append(847)
return Dist((m + n*(p + S(1)) + S(1))/(a*n*(p + S(1))), Int((c*x)**m*(a + b*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*n*(p + S(1))), x)
rule847 = ReplacementRule(pattern847, replacement847)
pattern848 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons55, cons546, cons13, cons137)
def replacement848(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(848)
return Dist((m + S(2)*n*(p + S(1)) + S(1))/(S(2)*a1*a2*n*(p + S(1))), Int((c*x)**m*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1)), x), x) - Simp((c*x)**(m + S(1))*(a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*a1*a2*c*n*(p + S(1))), x)
rule848 = ReplacementRule(pattern848, replacement848)
def With849(m, b, a, n, x):
mn = m - n
rubi.append(849)
return -Dist(a/b, Int(x**mn/(a + b*x**n), x), x) + Simp(x**(mn + S(1))/(b*(mn + S(1))), x)
pattern849 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons21, cons4, cons548, cons531)
rule849 = ReplacementRule(pattern849, With849)
pattern850 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons21, cons4, cons548, cons535)
def replacement850(m, b, a, n, x):
rubi.append(850)
return -Dist(b/a, Int(x**(m + n)/(a + b*x**n), x), x) + Simp(x**(m + S(1))/(a*(m + S(1))), x)
rule850 = ReplacementRule(pattern850, replacement850)
pattern851 = Pattern(Integral((c_*x_)**m_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons21, cons4, cons548, cons549)
def replacement851(m, b, c, a, n, x):
rubi.append(851)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m/(a + b*x**n), x), x)
rule851 = ReplacementRule(pattern851, replacement851)
pattern852 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons357, cons550)
def replacement852(p, m, b, c, a, n, x):
rubi.append(852)
return Simp(a**p*(c*x)**(m + S(1))*Hypergeometric2F1(-p, (m + S(1))/n, S(1) + (m + S(1))/n, -b*x**n/a)/(c*(m + S(1))), x)
rule852 = ReplacementRule(pattern852, replacement852)
pattern853 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons357, cons551)
def replacement853(p, m, b, c, a, n, x):
rubi.append(853)
return Dist(a**IntPart(p)*(S(1) + b*x**n/a)**(-FracPart(p))*(a + b*x**n)**FracPart(p), Int((c*x)**m*(S(1) + b*x**n/a)**p, x), x)
rule853 = ReplacementRule(pattern853, replacement853)
pattern854 = Pattern(Integral(x_**WC('m', S(1))*(a_ + v_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons4, cons5, cons552, cons17, cons553)
def replacement854(v, p, m, b, a, n, x):
rubi.append(854)
return Dist(Coefficient(v, x, S(1))**(-m + S(-1)), Subst(Int(SimplifyIntegrand((a + b*x**n)**p*(x - Coefficient(v, x, S(0)))**m, x), x), x, v), x)
rule854 = ReplacementRule(pattern854, replacement854)
pattern855 = Pattern(Integral(u_**WC('m', S(1))*(a_ + v_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons21, cons4, cons5, cons554)
def replacement855(v, p, u, m, b, a, n, x):
rubi.append(855)
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n)**p, x), x, v), x)
rule855 = ReplacementRule(pattern855, replacement855)
pattern856 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**n_*WC('b1', S(1)))**p_*(a2_ + x_**n_*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons5, cons55, cons147)
def replacement856(p, a2, m, b2, b1, c, n, x, a1):
rubi.append(856)
return Dist((a1 + b1*x**n)**FracPart(p)*(a2 + b2*x**n)**FracPart(p)*(a1*a2 + b1*b2*x**(S(2)*n))**(-FracPart(p)), Int((c*x)**m*(a1*a2 + b1*b2*x**(S(2)*n))**p, x), x)
rule856 = ReplacementRule(pattern856, replacement856)
pattern857 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons71, cons555)
def replacement857(p, b, d, a, n, c, x, q):
rubi.append(857)
return Int(ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule857 = ReplacementRule(pattern857, replacement857)
pattern858 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons71, cons220, cons502)
def replacement858(p, b, d, a, n, c, x, q):
rubi.append(858)
return Int(x**(n*(p + q))*(a*x**(-n) + b)**p*(c*x**(-n) + d)**q, x)
rule858 = ReplacementRule(pattern858, replacement858)
pattern859 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons50, cons71, cons196)
def replacement859(p, b, d, a, n, c, x, q):
rubi.append(859)
return -Subst(Int((a + b*x**(-n))**p*(c + d*x**(-n))**q/x**S(2), x), x, S(1)/x)
rule859 = ReplacementRule(pattern859, replacement859)
def With860(p, b, d, a, n, c, x, q):
g = Denominator(n)
rubi.append(860)
return Dist(g, Subst(Int(x**(g + S(-1))*(a + b*x**(g*n))**p*(c + d*x**(g*n))**q, x), x, x**(S(1)/g)), x)
pattern860 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons50, cons71, cons489)
rule860 = ReplacementRule(pattern860, With860)
pattern861 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71, cons556, cons85)
def replacement861(p, b, d, a, n, c, x):
rubi.append(861)
return Subst(Int(S(1)/(c - x**n*(-a*d + b*c)), x), x, x*(a + b*x**n)**(-S(1)/n))
rule861 = ReplacementRule(pattern861, replacement861)
pattern862 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons71, cons557, cons395, cons403, cons54)
def replacement862(p, b, d, a, n, c, x, q):
rubi.append(862)
return -Dist(c*q/(a*(p + S(1))), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1)), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*n*(p + S(1))), x)
rule862 = ReplacementRule(pattern862, replacement862)
pattern863 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons50, cons71, cons557, cons63)
def replacement863(p, b, d, a, n, c, x, q):
rubi.append(863)
return Simp(a**p*c**(-p + S(-1))*x*(c + d*x**n)**(-S(1)/n)*Hypergeometric2F1(S(1)/n, -p, S(1) + S(1)/n, -x**n*(-a*d + b*c)/(a*(c + d*x**n))), x)
rule863 = ReplacementRule(pattern863, replacement863)
pattern864 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons71, cons557)
def replacement864(p, b, d, a, n, c, x, q):
rubi.append(864)
return Simp(x*(c*(a + b*x**n)/(a*(c + d*x**n)))**(-p)*(a + b*x**n)**p*(c + d*x**n)**(-p - S(1)/n)*Hypergeometric2F1(S(1)/n, -p, S(1) + S(1)/n, -x**n*(-a*d + b*c)/(a*(c + d*x**n)))/c, x)
rule864 = ReplacementRule(pattern864, replacement864)
pattern865 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons71, cons558, cons559)
def replacement865(p, b, d, a, n, c, x, q):
rubi.append(865)
return Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*c), x)
rule865 = ReplacementRule(pattern865, replacement865)
pattern866 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons50, cons71, cons558, cons560, cons54)
def replacement866(p, b, d, a, n, c, x, q):
rubi.append(866)
return Dist((b*c + n*(p + S(1))*(-a*d + b*c))/(a*n*(p + S(1))*(-a*d + b*c)), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**q, x), x) - Simp(b*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*n*(p + S(1))*(-a*d + b*c)), x)
rule866 = ReplacementRule(pattern866, replacement866)
pattern867 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons71, cons561)
def replacement867(p, b, d, a, n, c, x):
rubi.append(867)
return Simp(c*x*(a + b*x**n)**(p + S(1))/a, x)
rule867 = ReplacementRule(pattern867, replacement867)
pattern868 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons71, cons562)
def replacement868(p, b, d, a, n, c, x):
rubi.append(868)
return -Dist((a*d - b*c*(n*(p + S(1)) + S(1)))/(a*b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1)), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(-a*d + b*c)/(a*b*n*(p + S(1))), x)
rule868 = ReplacementRule(pattern868, replacement868)
pattern869 = Pattern(Integral((c_ + x_**n_*WC('d', S(1)))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons71, cons87, cons463)
def replacement869(b, d, a, n, c, x):
rubi.append(869)
return -Dist((-a*d + b*c)/a, Int(S(1)/(a*x**(-n) + b), x), x) + Simp(c*x/a, x)
rule869 = ReplacementRule(pattern869, replacement869)
pattern870 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons71, cons563)
def replacement870(p, b, d, a, n, c, x):
rubi.append(870)
return -Dist((a*d - b*c*(n*(p + S(1)) + S(1)))/(b*(n*(p + S(1)) + S(1))), Int((a + b*x**n)**p, x), x) + Simp(d*x*(a + b*x**n)**(p + S(1))/(b*(n*(p + S(1)) + S(1))), x)
rule870 = ReplacementRule(pattern870, replacement870)
pattern871 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons71, cons464, cons564, cons565)
def replacement871(p, b, d, a, n, c, x, q):
rubi.append(871)
return Int(PolynomialDivide((a + b*x**n)**p, (c + d*x**n)**(-q), x), x)
rule871 = ReplacementRule(pattern871, replacement871)
pattern872 = Pattern(Integral(S(1)/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons4, cons71)
def replacement872(b, d, a, n, c, x):
rubi.append(872)
return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*x**n), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*x**n), x), x)
rule872 = ReplacementRule(pattern872, replacement872)
pattern873 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**(S(1)/3)*(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71, cons566, cons466)
def replacement873(b, d, c, a, x):
rubi.append(873)
return Dist(sqrt(S(3))/(S(2)*c), Int(S(1)/((a + b*x**S(2))**(S(1)/3)*(-x*Rt(b/a, S(2)) + sqrt(S(3)))), x), x) + Dist(sqrt(S(3))/(S(2)*c), Int(S(1)/((a + b*x**S(2))**(S(1)/3)*(x*Rt(b/a, S(2)) + sqrt(S(3)))), x), x)
rule873 = ReplacementRule(pattern873, replacement873)
pattern874 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**(S(1)/3)*(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71, cons566, cons483)
def replacement874(b, d, c, a, x):
rubi.append(874)
return Dist(S(1)/6, Int((-x*Rt(-b/a, S(2)) + S(3))/((a + b*x**S(2))**(S(1)/3)*(c + d*x**S(2))), x), x) + Dist(S(1)/6, Int((x*Rt(-b/a, S(2)) + S(3))/((a + b*x**S(2))**(S(1)/3)*(c + d*x**S(2))), x), x)
rule874 = ReplacementRule(pattern874, replacement874)
pattern875 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**(S(2)/3)/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71, cons566)
def replacement875(b, d, c, a, x):
rubi.append(875)
return Dist(b/d, Int((a + b*x**S(2))**(S(-1)/3), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/((a + b*x**S(2))**(S(1)/3)*(c + d*x**S(2))), x), x)
rule875 = ReplacementRule(pattern875, replacement875)
pattern876 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**(S(1)/4)*(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement876(b, d, c, a, x):
rubi.append(876)
return Dist(sqrt(-b*x**S(2)/a)/(S(2)*x), Subst(Int(S(1)/(sqrt(-b*x/a)*(a + b*x)**(S(1)/4)*(c + d*x)), x), x, x**S(2)), x)
rule876 = ReplacementRule(pattern876, replacement876)
pattern877 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**(S(3)/4)*(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement877(b, d, c, a, x):
rubi.append(877)
return Dist(sqrt(-b*x**S(2)/a)/(S(2)*x), Subst(Int(S(1)/(sqrt(-b*x/a)*(a + b*x)**(S(3)/4)*(c + d*x)), x), x, x**S(2)), x)
rule877 = ReplacementRule(pattern877, replacement877)
pattern878 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**WC('p', S(1))/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71, cons13, cons163, cons567)
def replacement878(p, b, d, a, c, x):
rubi.append(878)
return Dist(b/d, Int((a + b*x**S(2))**(p + S(-1)), x), x) - Dist((-a*d + b*c)/d, Int((a + b*x**S(2))**(p + S(-1))/(c + d*x**S(2)), x), x)
rule878 = ReplacementRule(pattern878, replacement878)
pattern879 = Pattern(Integral((a_ + x_**S(2)*WC('b', S(1)))**p_/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71, cons13, cons137, cons568, cons569)
def replacement879(p, b, d, a, c, x):
rubi.append(879)
return Dist(b/(-a*d + b*c), Int((a + b*x**S(2))**p, x), x) - Dist(d/(-a*d + b*c), Int((a + b*x**S(2))**(p + S(1))/(c + d*x**S(2)), x), x)
rule879 = ReplacementRule(pattern879, replacement879)
pattern880 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('b', S(1)))/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons70, cons570)
def replacement880(b, d, c, a, x):
rubi.append(880)
return Dist(a/c, Subst(Int(S(1)/(-S(4)*a*b*x**S(4) + S(1)), x), x, x/sqrt(a + b*x**S(4))), x)
rule880 = ReplacementRule(pattern880, replacement880)
def With881(b, d, c, a, x):
q = Rt(-a*b, S(4))
rubi.append(881)
return Simp(a*ArcTan(q*x*(a + q**S(2)*x**S(2))/(a*sqrt(a + b*x**S(4))))/(S(2)*c*q), x) + Simp(a*atanh(q*x*(a - q**S(2)*x**S(2))/(a*sqrt(a + b*x**S(4))))/(S(2)*c*q), x)
pattern881 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('b', S(1)))/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons70, cons571)
rule881 = ReplacementRule(pattern881, With881)
pattern882 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('b', S(1)))/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement882(b, d, c, a, x):
rubi.append(882)
return Dist(b/d, Int(S(1)/sqrt(a + b*x**S(4)), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/(sqrt(a + b*x**S(4))*(c + d*x**S(4))), x), x)
rule882 = ReplacementRule(pattern882, replacement882)
pattern883 = Pattern(Integral((a_ + x_**S(4)*WC('b', S(1)))**(S(1)/4)/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement883(b, d, c, a, x):
rubi.append(883)
return Dist(sqrt(a/(a + b*x**S(4)))*sqrt(a + b*x**S(4)), Subst(Int(S(1)/((c - x**S(4)*(-a*d + b*c))*sqrt(-b*x**S(4) + S(1))), x), x, x/(a + b*x**S(4))**(S(1)/4)), x)
rule883 = ReplacementRule(pattern883, replacement883)
pattern884 = Pattern(Integral((a_ + x_**S(4)*WC('b', S(1)))**p_/(c_ + x_**S(4)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71, cons13, cons572)
def replacement884(p, b, d, a, c, x):
rubi.append(884)
return Dist(b/d, Int((a + b*x**S(4))**(p + S(-1)), x), x) - Dist((-a*d + b*c)/d, Int((a + b*x**S(4))**(p + S(-1))/(c + d*x**S(4)), x), x)
rule884 = ReplacementRule(pattern884, replacement884)
pattern885 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('b', S(1)))*(c_ + x_**S(4)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement885(b, d, c, a, x):
rubi.append(885)
return Dist(S(1)/(S(2)*c), Int(S(1)/(sqrt(a + b*x**S(4))*(-x**S(2)*Rt(-d/c, S(2)) + S(1))), x), x) + Dist(S(1)/(S(2)*c), Int(S(1)/(sqrt(a + b*x**S(4))*(x**S(2)*Rt(-d/c, S(2)) + S(1))), x), x)
rule885 = ReplacementRule(pattern885, replacement885)
pattern886 = Pattern(Integral(S(1)/((a_ + x_**S(4)*WC('b', S(1)))**(S(3)/4)*(c_ + x_**S(4)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement886(b, d, c, a, x):
rubi.append(886)
return Dist(b/(-a*d + b*c), Int((a + b*x**S(4))**(S(-3)/4), x), x) - Dist(d/(-a*d + b*c), Int((a + b*x**S(4))**(S(1)/4)/(c + d*x**S(4)), x), x)
rule886 = ReplacementRule(pattern886, replacement886)
pattern887 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons466, cons573)
def replacement887(b, d, c, a, x):
rubi.append(887)
return Simp(sqrt(a + b*x**S(2))*EllipticE(ArcTan(x*Rt(d/c, S(2))), S(1) - b*c/(a*d))/(c*sqrt(c*(a + b*x**S(2))/(a*(c + d*x**S(2))))*sqrt(c + d*x**S(2))*Rt(d/c, S(2))), x)
rule887 = ReplacementRule(pattern887, replacement887)
pattern888 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons71, cons402, cons137, cons574, cons575)
def replacement888(p, b, d, a, n, c, x, q):
rubi.append(888)
return Dist(S(1)/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(n*(p + S(1)) + S(1)) + d*x**n*(n*(p + q + S(1)) + S(1)), x), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*n*(p + S(1))), x)
rule888 = ReplacementRule(pattern888, replacement888)
pattern889 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons71, cons402, cons137, cons576, cons575)
def replacement889(p, b, d, a, n, c, x, q):
rubi.append(889)
return -Dist(S(1)/(a*b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-2))*Simp(c*(a*d - b*c*(n*(p + S(1)) + S(1))) + d*x**n*(a*d*(n*(q + S(-1)) + S(1)) - b*c*(n*(p + q) + S(1))), x), x), x) + Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*(a*d - b*c)/(a*b*n*(p + S(1))), x)
rule889 = ReplacementRule(pattern889, replacement889)
pattern890 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons50, cons71, cons13, cons137, cons405, cons575)
def replacement890(p, b, d, a, n, c, x, q):
rubi.append(890)
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(b*c + b*d*x**n*(n*(p + q + S(2)) + S(1)) + n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp(b*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*n*(p + S(1))*(-a*d + b*c)), x)
rule890 = ReplacementRule(pattern890, replacement890)
pattern891 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons71, cons148, cons220, cons577)
def replacement891(p, b, d, a, n, c, x, q):
rubi.append(891)
return Int(ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule891 = ReplacementRule(pattern891, replacement891)
pattern892 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons71, cons395, cons576, cons578, cons579, cons575)
def replacement892(p, b, d, a, n, c, x, q):
rubi.append(892)
return Dist(S(1)/(b*(n*(p + q) + S(1))), Int((a + b*x**n)**p*(c + d*x**n)**(q + S(-2))*Simp(c*(-a*d + b*c*(n*(p + q) + S(1))) + d*x**n*(-a*d*(n*(q + S(-1)) + S(1)) + b*c*(n*(p + S(2)*q + S(-1)) + S(1))), x), x), x) + Simp(d*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))/(b*(n*(p + q) + S(1))), x)
rule892 = ReplacementRule(pattern892, replacement892)
pattern893 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons71, cons402, cons403, cons163, cons575)
def replacement893(p, b, d, a, n, c, x, q):
rubi.append(893)
return Dist(n/(n*(p + q) + S(1)), Int((a + b*x**n)**(p + S(-1))*(c + d*x**n)**(q + S(-1))*Simp(a*c*(p + q) + x**n*(a*d*(p + q) + q*(-a*d + b*c)), x), x), x) + Simp(x*(a + b*x**n)**p*(c + d*x**n)**q/(n*(p + q) + S(1)), x)
rule893 = ReplacementRule(pattern893, replacement893)
pattern894 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons573, cons466, cons580)
def replacement894(b, d, c, a, x):
rubi.append(894)
return Simp(sqrt(a + b*x**S(2))*EllipticF(ArcTan(x*Rt(d/c, S(2))), S(1) - b*c/(a*d))/(a*sqrt(c*(a + b*x**S(2))/(a*(c + d*x**S(2))))*sqrt(c + d*x**S(2))*Rt(d/c, S(2))), x)
rule894 = ReplacementRule(pattern894, replacement894)
pattern895 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons581, cons177, cons43, cons582)
def replacement895(b, d, c, a, x):
rubi.append(895)
return Simp(EllipticF(asin(x*Rt(-d/c, S(2))), b*c/(a*d))/(sqrt(a)*sqrt(c)*Rt(-d/c, S(2))), x)
rule895 = ReplacementRule(pattern895, replacement895)
pattern896 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons581, cons177, cons583)
def replacement896(b, d, c, a, x):
rubi.append(896)
return -Simp(EllipticF(acos(x*Rt(-d/c, S(2))), b*c/(-a*d + b*c))/(sqrt(c)*sqrt(a - b*c/d)*Rt(-d/c, S(2))), x)
rule896 = ReplacementRule(pattern896, replacement896)
pattern897 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons117)
def replacement897(b, d, c, a, x):
rubi.append(897)
return Dist(sqrt(S(1) + d*x**S(2)/c)/sqrt(c + d*x**S(2)), Int(S(1)/(sqrt(S(1) + d*x**S(2)/c)*sqrt(a + b*x**S(2))), x), x)
rule897 = ReplacementRule(pattern897, replacement897)
pattern898 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons573, cons466)
def replacement898(b, d, c, a, x):
rubi.append(898)
return Dist(a, Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) + Dist(b, Int(x**S(2)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
rule898 = ReplacementRule(pattern898, replacement898)
pattern899 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons573, cons483)
def replacement899(b, d, c, a, x):
rubi.append(899)
return Dist(b/d, Int(sqrt(c + d*x**S(2))/sqrt(a + b*x**S(2)), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
rule899 = ReplacementRule(pattern899, replacement899)
pattern900 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons581, cons177, cons43)
def replacement900(b, d, c, a, x):
rubi.append(900)
return Simp(sqrt(a)*EllipticE(asin(x*Rt(-d/c, S(2))), b*c/(a*d))/(sqrt(c)*Rt(-d/c, S(2))), x)
rule900 = ReplacementRule(pattern900, replacement900)
pattern901 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons581, cons177, cons583)
def replacement901(b, d, c, a, x):
rubi.append(901)
return -Simp(sqrt(a - b*c/d)*EllipticE(acos(x*Rt(-d/c, S(2))), b*c/(-a*d + b*c))/(sqrt(c)*Rt(-d/c, S(2))), x)
rule901 = ReplacementRule(pattern901, replacement901)
pattern902 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons581, cons177, cons448)
def replacement902(b, d, c, a, x):
rubi.append(902)
return Dist(sqrt(a + b*x**S(2))/sqrt(S(1) + b*x**S(2)/a), Int(sqrt(S(1) + b*x**S(2)/a)/sqrt(c + d*x**S(2)), x), x)
rule902 = ReplacementRule(pattern902, replacement902)
pattern903 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/sqrt(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons581, cons117)
def replacement903(b, d, c, a, x):
rubi.append(903)
return Dist(sqrt(S(1) + d*x**S(2)/c)/sqrt(c + d*x**S(2)), Int(sqrt(a + b*x**S(2))/sqrt(S(1) + d*x**S(2)/c), x), x)
rule903 = ReplacementRule(pattern903, replacement903)
pattern904 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons50, cons71, cons128)
def replacement904(p, b, d, a, n, c, x, q):
rubi.append(904)
return Int(ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule904 = ReplacementRule(pattern904, replacement904)
pattern905 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons71, cons584, cons43, cons177)
def replacement905(p, b, d, a, n, c, x, q):
rubi.append(905)
return Simp(a**p*c**q*x*AppellF1(S(1)/n, -p, -q, S(1) + S(1)/n, -b*x**n/a, -d*x**n/c), x)
rule905 = ReplacementRule(pattern905, replacement905)
pattern906 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons71, cons584, cons448)
def replacement906(p, b, d, a, n, c, x, q):
rubi.append(906)
return Dist(a**IntPart(p)*(S(1) + b*x**n/a)**(-FracPart(p))*(a + b*x**n)**FracPart(p), Int((S(1) + b*x**n/a)**p*(c + d*x**n)**q, x), x)
rule906 = ReplacementRule(pattern906, replacement906)
pattern907 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons585, cons586, cons587)
def replacement907(mn, p, b, d, c, n, a, x, q):
rubi.append(907)
return Int(x**(-n*q)*(a + b*x**n)**p*(c*x**n + d)**q, x)
rule907 = ReplacementRule(pattern907, replacement907)
pattern908 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons585, cons386, cons147)
def replacement908(mn, p, b, d, c, n, a, x, q):
rubi.append(908)
return Dist(x**(n*FracPart(q))*(c + d*x**(-n))**FracPart(q)*(c*x**n + d)**(-FracPart(q)), Int(x**(-n*q)*(a + b*x**n)**p*(c*x**n + d)**q, x), x)
rule908 = ReplacementRule(pattern908, replacement908)
pattern909 = Pattern(Integral((u_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(u_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons68, cons69)
def replacement909(p, u, b, d, c, a, n, x, q):
rubi.append(909)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x**n)**p*(c + d*x**n)**q, x), x, u), x)
rule909 = ReplacementRule(pattern909, replacement909)
pattern910 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1)), x_), cons5, cons50, cons588)
def replacement910(v, p, u, x, q):
rubi.append(910)
return Int(NormalizePseudoBinomial(u, x)**p*NormalizePseudoBinomial(v, x)**q, x)
rule910 = ReplacementRule(pattern910, replacement910)
pattern911 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1))*x_**WC('m', S(1)), x_), cons5, cons50, cons589, cons590)
def replacement911(v, p, u, m, x, q):
rubi.append(911)
return Int(NormalizePseudoBinomial(v, x)**q*NormalizePseudoBinomial(u*x**(m/p), x)**p, x)
rule911 = ReplacementRule(pattern911, replacement911)
pattern912 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons591, cons500)
def replacement912(p, m, b, d, c, n, x, q, e):
rubi.append(912)
return Dist(b**(S(1) - (m + S(1))/n)*e**m/n, Subst(Int((b*x)**(p + S(-1) + (m + S(1))/n)*(c + d*x)**q, x), x, x**n), x)
rule912 = ReplacementRule(pattern912, replacement912)
pattern913 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons591, cons501)
def replacement913(p, m, b, d, c, n, x, q, e):
rubi.append(913)
return Dist(b**IntPart(p)*e**m*x**(-n*FracPart(p))*(b*x**n)**FracPart(p), Int(x**(m + n*p)*(c + d*x**n)**q, x), x)
rule913 = ReplacementRule(pattern913, replacement913)
pattern914 = Pattern(Integral((e_*x_)**m_*(x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons18)
def replacement914(p, m, b, d, c, n, x, q, e):
rubi.append(914)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(b*x**n)**p*(c + d*x**n)**q, x), x)
rule914 = ReplacementRule(pattern914, replacement914)
pattern915 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons50, cons71, cons53)
def replacement915(p, m, b, d, a, n, c, x, q):
rubi.append(915)
return Dist(S(1)/n, Subst(Int((a + b*x)**p*(c + d*x)**q, x), x, x**n), x)
rule915 = ReplacementRule(pattern915, replacement915)
pattern916 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons71, cons220, cons502)
def replacement916(p, m, b, d, a, n, c, x, q):
rubi.append(916)
return Int(x**(m + n*(p + q))*(a*x**(-n) + b)**p*(c*x**(-n) + d)**q, x)
rule916 = ReplacementRule(pattern916, replacement916)
pattern917 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons50, cons71, cons500)
def replacement917(p, m, b, d, a, n, c, x, q):
rubi.append(917)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x)**p*(c + d*x)**q, x), x, x**n), x)
rule917 = ReplacementRule(pattern917, replacement917)
pattern918 = Pattern(Integral((e_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons71, cons500)
def replacement918(p, m, b, d, a, n, c, x, q, e):
rubi.append(918)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule918 = ReplacementRule(pattern918, replacement918)
pattern919 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons71, cons555)
def replacement919(p, m, b, d, a, n, c, x, q, e):
rubi.append(919)
return Int(ExpandIntegrand((e*x)**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule919 = ReplacementRule(pattern919, replacement919)
pattern920 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons71, cons592, cons66)
def replacement920(p, m, b, d, a, n, c, x, e):
rubi.append(920)
return Simp(c*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*e*(m + S(1))), x)
rule920 = ReplacementRule(pattern920, replacement920)
pattern921 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons57, cons58, cons59, cons60, cons7, cons27, cons48, cons21, cons4, cons5, cons593, cons55, cons594, cons66)
def replacement921(p, a2, m, b2, b1, d, c, n, non2, x, a1, e):
rubi.append(921)
return Simp(c*(e*x)**(m + S(1))*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1))/(a1*a2*e*(m + S(1))), x)
rule921 = ReplacementRule(pattern921, replacement921)
pattern922 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons71, cons595, cons596, cons93, cons597)
def replacement922(p, m, b, d, a, n, c, x, e):
rubi.append(922)
return Dist(d*e**(-n), Int((e*x)**(m + n)*(a + b*x**n)**p, x), x) + Simp(c*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*e*(m + S(1))), x)
rule922 = ReplacementRule(pattern922, replacement922)
pattern923 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons71, cons595, cons66)
def replacement923(p, m, b, d, a, n, c, x, e):
rubi.append(923)
return Dist(d/b, Int((e*x)**m*(a + b*x**n)**(p + S(1)), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(-a*d + b*c)/(a*b*e*(m + S(1))), x)
rule923 = ReplacementRule(pattern923, replacement923)
pattern924 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons71, cons596, cons93, cons597, cons598)
def replacement924(p, m, b, d, a, n, c, x, e):
rubi.append(924)
return Dist(e**(-n)*(a*d*(m + S(1)) - b*c*(m + n*(p + S(1)) + S(1)))/(a*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p, x), x) + Simp(c*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*e*(m + S(1))), x)
rule924 = ReplacementRule(pattern924, replacement924)
pattern925 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons57, cons58, cons59, cons60, cons7, cons27, cons48, cons5, cons593, cons55, cons596, cons93, cons597, cons598)
def replacement925(p, a2, m, b2, b1, d, c, n, non2, x, a1, e):
rubi.append(925)
return Dist(e**(-n)*(a1*a2*d*(m + S(1)) - b1*b2*c*(m + n*(p + S(1)) + S(1)))/(a1*a2*(m + S(1))), Int((e*x)**(m + n)*(a1 + b1*x**(n/S(2)))**p*(a2 + b2*x**(n/S(2)))**p, x), x) + Simp(c*(e*x)**(m + S(1))*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1))/(a1*a2*e*(m + S(1))), x)
rule925 = ReplacementRule(pattern925, replacement925)
pattern926 = Pattern(Integral(x_**m_*(a_ + x_**S(2)*WC('b', S(1)))**p_*(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71, cons13, cons137, cons599, cons600)
def replacement926(p, m, b, d, a, c, x):
rubi.append(926)
return Dist(b**(-m/S(2) + S(-1))/(S(2)*(p + S(1))), Int((a + b*x**S(2))**(p + S(1))*ExpandToSum(S(2)*b*x**S(2)*(p + S(1))*Together((b**(m/S(2))*x**(m + S(-2))*(c + d*x**S(2)) - (-a)**(m/S(2) + S(-1))*(-a*d + b*c))/(a + b*x**S(2))) - (-a)**(m/S(2) + S(-1))*(-a*d + b*c), x), x), x) + Simp(b**(-m/S(2) + S(-1))*x*(-a)**(m/S(2) + S(-1))*(a + b*x**S(2))**(p + S(1))*(-a*d + b*c)/(S(2)*(p + S(1))), x)
rule926 = ReplacementRule(pattern926, replacement926)
pattern927 = Pattern(Integral(x_**m_*(a_ + x_**S(2)*WC('b', S(1)))**p_*(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons71, cons13, cons137, cons601, cons600)
def replacement927(p, m, b, d, a, c, x):
rubi.append(927)
return Dist(b**(-m/S(2) + S(-1))/(S(2)*(p + S(1))), Int(x**m*(a + b*x**S(2))**(p + S(1))*ExpandToSum(S(2)*b*(p + S(1))*Together((b**(m/S(2))*(c + d*x**S(2)) - x**(-m + S(2))*(-a)**(m/S(2) + S(-1))*(-a*d + b*c))/(a + b*x**S(2))) - x**(-m)*(-a)**(m/S(2) + S(-1))*(-a*d + b*c), x), x), x) + Simp(b**(-m/S(2) + S(-1))*x*(-a)**(m/S(2) + S(-1))*(a + b*x**S(2))**(p + S(1))*(-a*d + b*c)/(S(2)*(p + S(1))), x)
rule927 = ReplacementRule(pattern927, replacement927)
pattern928 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons71, cons13, cons137, cons602)
def replacement928(p, m, b, d, a, n, c, x, e):
rubi.append(928)
return -Dist((a*d*(m + S(1)) - b*c*(m + n*(p + S(1)) + S(1)))/(a*b*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1)), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(-a*d + b*c)/(a*b*e*n*(p + S(1))), x)
rule928 = ReplacementRule(pattern928, replacement928)
pattern929 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons57, cons58, cons59, cons60, cons7, cons27, cons48, cons21, cons4, cons593, cons55, cons13, cons137, cons602)
def replacement929(p, a2, m, b2, b1, d, c, n, non2, x, a1, e):
rubi.append(929)
return -Dist((a1*a2*d*(m + S(1)) - b1*b2*c*(m + n*(p + S(1)) + S(1)))/(a1*a2*b1*b2*n*(p + S(1))), Int((e*x)**m*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1)), x), x) - Simp((e*x)**(m + S(1))*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1))*(-a1*a2*d + b1*b2*c)/(a1*a2*b1*b2*e*n*(p + S(1))), x)
rule929 = ReplacementRule(pattern929, replacement929)
pattern930 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons71, cons603)
def replacement930(p, m, b, d, a, n, c, x, e):
rubi.append(930)
return -Dist((a*d*(m + S(1)) - b*c*(m + n*(p + S(1)) + S(1)))/(b*(m + n*(p + S(1)) + S(1))), Int((e*x)**m*(a + b*x**n)**p, x), x) + Simp(d*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(b*e*(m + n*(p + S(1)) + S(1))), x)
rule930 = ReplacementRule(pattern930, replacement930)
pattern931 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1))), x_), cons57, cons58, cons59, cons60, cons7, cons27, cons48, cons21, cons4, cons5, cons593, cons55, cons603)
def replacement931(p, a2, m, b2, b1, d, c, n, non2, x, a1, e):
rubi.append(931)
return -Dist((a1*a2*d*(m + S(1)) - b1*b2*c*(m + n*(p + S(1)) + S(1)))/(b1*b2*(m + n*(p + S(1)) + S(1))), Int((e*x)**m*(a1 + b1*x**(n/S(2)))**p*(a2 + b2*x**(n/S(2)))**p, x), x) + Simp(d*(e*x)**(m + S(1))*(a1 + b1*x**(n/S(2)))**(p + S(1))*(a2 + b2*x**(n/S(2)))**(p + S(1))/(b1*b2*e*(m + n*(p + S(1)) + S(1))), x)
rule931 = ReplacementRule(pattern931, replacement931)
pattern932 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons71, cons148, cons128, cons604)
def replacement932(p, m, b, d, a, n, c, x, e):
rubi.append(932)
return Int(ExpandIntegrand((e*x)**m*(a + b*x**n)**p/(c + d*x**n), x), x)
rule932 = ReplacementRule(pattern932, replacement932)
pattern933 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons71, cons148, cons93, cons94, cons88)
def replacement933(p, m, b, d, a, n, c, x, e):
rubi.append(933)
return -Dist(e**(-n)/(a*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p*Simp(-a*d**S(2)*x**n*(m + S(1)) + b*c**S(2)*n*(p + S(1)) + c*(m + S(1))*(-S(2)*a*d + b*c), x), x), x) + Simp(c**S(2)*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*e*(m + S(1))), x)
rule933 = ReplacementRule(pattern933, replacement933)
pattern934 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons71, cons148, cons13, cons137)
def replacement934(p, m, b, d, a, n, c, x, e):
rubi.append(934)
return Dist(S(1)/(a*b**S(2)*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*Simp(a*b*d**S(2)*n*x**n*(p + S(1)) + b**S(2)*c**S(2)*n*(p + S(1)) + (m + S(1))*(-a*d + b*c)**S(2), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(-a*d + b*c)**S(2)/(a*b**S(2)*e*n*(p + S(1))), x)
rule934 = ReplacementRule(pattern934, replacement934)
pattern935 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons71, cons148, cons605)
def replacement935(p, m, b, d, a, n, c, x, e):
rubi.append(935)
return Dist(S(1)/(b*(m + n*(p + S(2)) + S(1))), Int((e*x)**m*(a + b*x**n)**p*Simp(b*c**S(2)*(m + n*(p + S(2)) + S(1)) + d*x**n*(S(2)*b*c*n*(p + S(1)) + (-a*d + S(2)*b*c)*(m + n + S(1))), x), x), x) + Simp(d**S(2)*e**(-n + S(-1))*(e*x)**(m + n + S(1))*(a + b*x**n)**(p + S(1))/(b*(m + n*(p + S(2)) + S(1))), x)
rule935 = ReplacementRule(pattern935, replacement935)
def With936(p, m, b, d, a, n, c, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
pattern936 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons5, cons50, cons71, cons148, cons17, CustomConstraint(With936))
def replacement936(p, m, b, d, a, n, c, x, q):
k = GCD(m + S(1), n)
rubi.append(936)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a + b*x**(n/k))**p*(c + d*x**(n/k))**q, x), x, x**k), x)
rule936 = ReplacementRule(pattern936, replacement936)
def With937(p, m, b, d, a, n, c, x, q, e):
k = Denominator(m)
rubi.append(937)
return Dist(k/e, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*e**(-n)*x**(k*n))**p*(c + d*e**(-n)*x**(k*n))**q, x), x, (e*x)**(S(1)/k)), x)
pattern937 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons71, cons148, cons367, cons38)
rule937 = ReplacementRule(pattern937, With937)
pattern938 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons71, cons148, cons606, cons137, cons403, cons607, cons608)
def replacement938(p, m, b, d, a, n, c, x, q, e):
rubi.append(938)
return -Dist(e**n/(b*n*(p + S(1))), Int((e*x)**(m - n)*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(m - n + S(1)) + d*x**n*(m + n*(q + S(-1)) + S(1)), x), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*n*(p + S(1))), x)
rule938 = ReplacementRule(pattern938, replacement938)
pattern939 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons71, cons148, cons402, cons137, cons576, cons608)
def replacement939(p, m, b, d, a, n, c, x, q, e):
rubi.append(939)
return Dist(S(1)/(a*b*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-2))*Simp(c*(b*c*n*(p + S(1)) + (m + S(1))*(-a*d + b*c)) + d*x**n*(b*c*n*(p + S(1)) + (-a*d + b*c)*(m + n*(q + S(-1)) + S(1))), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*(-a*d + b*c)/(a*b*e*n*(p + S(1))), x)
rule939 = ReplacementRule(pattern939, replacement939)
pattern940 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons71, cons148, cons402, cons137, cons574, cons608)
def replacement940(p, m, b, d, a, n, c, x, q, e):
rubi.append(940)
return Dist(S(1)/(a*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(m + n*(p + S(1)) + S(1)) + d*x**n*(m + n*(p + q + S(1)) + S(1)), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*e*n*(p + S(1))), x)
rule940 = ReplacementRule(pattern940, replacement940)
pattern941 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons50, cons71, cons148, cons244, cons137, cons609, cons608)
def replacement941(p, m, b, d, a, n, c, x, q, e):
rubi.append(941)
return Dist(e**(S(2)*n)/(b*n*(p + S(1))*(-a*d + b*c)), Int((e*x)**(m - S(2)*n)*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(a*c*(m - S(2)*n + S(1)) + x**n*(a*d*(m + n*q - n + S(1)) + b*c*n*(p + S(1))), x), x), x) - Simp(a*e**(S(2)*n + S(-1))*(e*x)**(m - S(2)*n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(b*n*(p + S(1))*(-a*d + b*c)), x)
rule941 = ReplacementRule(pattern941, replacement941)
pattern942 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons50, cons71, cons148, cons244, cons137, cons610, cons608)
def replacement942(p, m, b, d, a, n, c, x, q, e):
rubi.append(942)
return -Dist(e**n/(n*(p + S(1))*(-a*d + b*c)), Int((e*x)**(m - n)*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(m - n + S(1)) + d*x**n*(m + n*(p + q + S(1)) + S(1)), x), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(n*(p + S(1))*(-a*d + b*c)), x)
rule942 = ReplacementRule(pattern942, replacement942)
pattern943 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons50, cons71, cons148, cons13, cons137, cons608)
def replacement943(p, m, b, d, a, n, c, x, q, e):
rubi.append(943)
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(b*c*(m + S(1)) + b*d*x**n*(m + n*(p + q + S(2)) + S(1)) + n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp(b*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*e*n*(p + S(1))*(-a*d + b*c)), x)
rule943 = ReplacementRule(pattern943, replacement943)
pattern944 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons71, cons148, cons606, cons403, cons94, cons163, cons608)
def replacement944(p, m, b, d, a, n, c, x, q, e):
rubi.append(944)
return -Dist(e**(-n)*n/(m + S(1)), Int((e*x)**(m + n)*(a + b*x**n)**(p + S(-1))*(c + d*x**n)**(q + S(-1))*Simp(a*d*q + b*c*p + b*d*x**n*(p + q), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**p*(c + d*x**n)**q/(e*(m + S(1))), x)
rule944 = ReplacementRule(pattern944, replacement944)
pattern945 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons71, cons148, cons611, cons576, cons94, cons608)
def replacement945(p, m, b, d, a, n, c, x, q, e):
rubi.append(945)
return -Dist(e**(-n)/(a*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**(q + S(-2))*Simp(c*n*(a*d*(q + S(-1)) + b*c*(p + S(1))) + c*(m + S(1))*(-a*d + b*c) + d*x**n*(b*c*n*(p + q) + (m + S(1))*(-a*d + b*c)), x), x), x) + Simp(c*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))/(a*e*(m + S(1))), x)
rule945 = ReplacementRule(pattern945, replacement945)
pattern946 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons71, cons148, cons611, cons574, cons94, cons608)
def replacement946(p, m, b, d, a, n, c, x, q, e):
rubi.append(946)
return -Dist(e**(-n)/(a*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(b*c*(m + S(1)) + d*x**n*(b*n*(p + q + S(1)) + b*(m + S(1))) + n*(a*d*q + b*c*(p + S(1))), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*e*(m + S(1))), x)
rule946 = ReplacementRule(pattern946, replacement946)
pattern947 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons71, cons148, cons402, cons403, cons163, cons608)
def replacement947(p, m, b, d, a, n, c, x, q, e):
rubi.append(947)
return Dist(n/(m + n*(p + q) + S(1)), Int((e*x)**m*(a + b*x**n)**(p + S(-1))*(c + d*x**n)**(q + S(-1))*Simp(a*c*(p + q) + x**n*(a*d*(p + q) + q*(-a*d + b*c)), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**p*(c + d*x**n)**q/(e*(m + n*(p + q) + S(1))), x)
rule947 = ReplacementRule(pattern947, replacement947)
pattern948 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons71, cons148, cons395, cons576, cons608)
def replacement948(p, m, b, d, a, n, c, x, q, e):
rubi.append(948)
return Dist(S(1)/(b*(m + n*(p + q) + S(1))), Int((e*x)**m*(a + b*x**n)**p*(c + d*x**n)**(q + S(-2))*Simp(c*(b*c*n*(p + q) + (m + S(1))*(-a*d + b*c)) + x**n*(b*c*d*n*(p + q) + d*n*(q + S(-1))*(-a*d + b*c) + d*(m + S(1))*(-a*d + b*c)), x), x), x) + Simp(d*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))/(b*e*(m + n*(p + q) + S(1))), x)
rule948 = ReplacementRule(pattern948, replacement948)
pattern949 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons71, cons148, cons611, cons403, cons607, cons608)
def replacement949(p, m, b, d, a, n, c, x, q, e):
rubi.append(949)
return -Dist(e**n/(b*(m + n*(p + q) + S(1))), Int((e*x)**(m - n)*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(a*c*(m - n + S(1)) + x**n*(a*d*(m - n + S(1)) - n*q*(-a*d + b*c)), x), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*(m + n*(p + q) + S(1))), x)
rule949 = ReplacementRule(pattern949, replacement949)
pattern950 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons71, cons148, cons31, cons609, cons608)
def replacement950(p, m, b, d, a, n, c, x, q, e):
rubi.append(950)
return -Dist(e**(S(2)*n)/(b*d*(m + n*(p + q) + S(1))), Int((e*x)**(m - S(2)*n)*(a + b*x**n)**p*(c + d*x**n)**q*Simp(a*c*(m - S(2)*n + S(1)) + x**n*(a*d*(m + n*(q + S(-1)) + S(1)) + b*c*(m + n*(p + S(-1)) + S(1))), x), x), x) + Simp(e**(S(2)*n + S(-1))*(e*x)**(m - S(2)*n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(b*d*(m + n*(p + q) + S(1))), x)
rule950 = ReplacementRule(pattern950, replacement950)
pattern951 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons71, cons148, cons31, cons94, cons608)
def replacement951(p, m, b, d, a, n, c, x, q, e):
rubi.append(951)
return -Dist(e**(-n)/(a*c*(m + S(1))), Int((e*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q*Simp(b*d*x**n*(m + n*(p + q + S(2)) + S(1)) + n*(a*d*q + b*c*p) + (a*d + b*c)*(m + n + S(1)), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*c*e*(m + S(1))), x)
rule951 = ReplacementRule(pattern951, replacement951)
pattern952 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons71, cons148, cons31, cons612)
def replacement952(m, b, d, a, n, c, x, e):
rubi.append(952)
return -Dist(a*e**n/(-a*d + b*c), Int((e*x)**(m - n)/(a + b*x**n), x), x) + Dist(c*e**n/(-a*d + b*c), Int((e*x)**(m - n)/(c + d*x**n), x), x)
rule952 = ReplacementRule(pattern952, replacement952)
pattern953 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons71, cons148)
def replacement953(m, b, d, a, n, c, x, e):
rubi.append(953)
return Dist(b/(-a*d + b*c), Int((e*x)**m/(a + b*x**n), x), x) - Dist(d/(-a*d + b*c), Int((e*x)**m/(c + d*x**n), x), x)
rule953 = ReplacementRule(pattern953, replacement953)
pattern954 = Pattern(Integral(x_**m_/((a_ + x_**n_*WC('b', S(1)))*sqrt(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71, cons613, cons614, cons615)
def replacement954(m, b, d, a, n, c, x):
rubi.append(954)
return Dist(S(1)/b, Int(x**(m - n)/sqrt(c + d*x**n), x), x) - Dist(a/b, Int(x**(m - n)/((a + b*x**n)*sqrt(c + d*x**n)), x), x)
rule954 = ReplacementRule(pattern954, replacement954)
def With955(b, d, c, a, x):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
rubi.append(955)
return -Dist(s/(S(2)*b), Int(S(1)/(sqrt(c + d*x**S(4))*(r - s*x**S(2))), x), x) + Dist(s/(S(2)*b), Int(S(1)/(sqrt(c + d*x**S(4))*(r + s*x**S(2))), x), x)
pattern955 = Pattern(Integral(x_**S(2)/((a_ + x_**S(4)*WC('b', S(1)))*sqrt(c_ + x_**S(4)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71)
rule955 = ReplacementRule(pattern955, With955)
def With956(b, d, c, a, x):
q = Rt(d/c, S(3))
rubi.append(956)
return Simp(S(2)**(S(1)/3)*q*log(-S(2)**(S(1)/3)*q*x + S(1) - sqrt(c + d*x**S(3))/sqrt(c))/(S(12)*b*sqrt(c)), x) - Simp(S(2)**(S(1)/3)*q*log(-S(2)**(S(1)/3)*q*x + S(1) + sqrt(c + d*x**S(3))/sqrt(c))/(S(12)*b*sqrt(c)), x) + Simp(S(2)**(S(1)/3)*q*atanh(sqrt(c + d*x**S(3))/sqrt(c))/(S(18)*b*sqrt(c)), x) - Simp(S(2)**(S(1)/3)*sqrt(S(3))*q*ArcTan(sqrt(S(3))/S(3) + S(2)**(S(2)/3)*sqrt(S(3))*(sqrt(c) - sqrt(c + d*x**S(3)))/(S(3)*sqrt(c)*q*x))/(S(18)*b*sqrt(c)), x) + Simp(S(2)**(S(1)/3)*sqrt(S(3))*q*ArcTan(sqrt(S(3))/S(3) + S(2)**(S(2)/3)*sqrt(S(3))*(sqrt(c) + sqrt(c + d*x**S(3)))/(S(3)*sqrt(c)*q*x))/(S(18)*b*sqrt(c)), x)
pattern956 = Pattern(Integral(x_/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(c_ + x_**S(3)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71, cons616)
rule956 = ReplacementRule(pattern956, With956)
pattern957 = Pattern(Integral(x_**m_/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(c_ + x_**S(3)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71, cons616, cons617)
def replacement957(m, b, d, a, c, x):
rubi.append(957)
return Dist(S(1)/b, Int(x**(m + S(-3))/sqrt(c + d*x**S(3)), x), x) - Dist(a/b, Int(x**(m + S(-3))/((a + b*x**S(3))*sqrt(c + d*x**S(3))), x), x)
rule957 = ReplacementRule(pattern957, replacement957)
pattern958 = Pattern(Integral(x_**m_/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(c_ + x_**S(3)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71, cons616, cons618)
def replacement958(m, b, d, a, c, x):
rubi.append(958)
return Dist(S(1)/a, Int(x**m/sqrt(c + d*x**S(3)), x), x) - Dist(b/a, Int(x**(m + S(3))/((a + b*x**S(3))*sqrt(c + d*x**S(3))), x), x)
rule958 = ReplacementRule(pattern958, replacement958)
pattern959 = Pattern(Integral(x_**S(2)*sqrt(c_ + x_**S(4)*WC('d', S(1)))/(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement959(b, d, c, a, x):
rubi.append(959)
return Dist(d/b, Int(x**S(2)/sqrt(c + d*x**S(4)), x), x) + Dist((-a*d + b*c)/b, Int(x**S(2)/((a + b*x**S(4))*sqrt(c + d*x**S(4))), x), x)
rule959 = ReplacementRule(pattern959, replacement959)
pattern960 = Pattern(Integral(x_**WC('m', S(1))*sqrt(c_ + x_**S(3)*WC('d', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons71, cons616, cons619)
def replacement960(m, b, d, a, c, x):
rubi.append(960)
return Dist(d/b, Int(x**m/sqrt(c + d*x**S(3)), x), x) + Dist((-a*d + b*c)/b, Int(x**m/((a + b*x**S(3))*sqrt(c + d*x**S(3))), x), x)
rule960 = ReplacementRule(pattern960, replacement960)
pattern961 = Pattern(Integral(x_**S(2)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71, cons466, cons573, cons580)
def replacement961(b, d, c, a, x):
rubi.append(961)
return -Dist(c/b, Int(sqrt(a + b*x**S(2))/(c + d*x**S(2))**(S(3)/2), x), x) + Simp(x*sqrt(a + b*x**S(2))/(b*sqrt(c + d*x**S(2))), x)
rule961 = ReplacementRule(pattern961, replacement961)
pattern962 = Pattern(Integral(x_**n_/(sqrt(a_ + x_**n_*WC('b', S(1)))*sqrt(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons71, cons620, cons621)
def replacement962(b, d, a, n, c, x):
rubi.append(962)
return Dist(S(1)/b, Int(sqrt(a + b*x**n)/sqrt(c + d*x**n), x), x) - Dist(a/b, Int(S(1)/(sqrt(a + b*x**n)*sqrt(c + d*x**n)), x), x)
rule962 = ReplacementRule(pattern962, replacement962)
def With963(p, m, b, d, a, n, c, x, q):
k = Denominator(p)
rubi.append(963)
return Dist(a**(p + (m + S(1))/n)*k/n, Subst(Int(x**(k*(m + S(1))/n + S(-1))*(c - x**k*(-a*d + b*c))**q*(-b*x**k + S(1))**(-p - q + S(-1) - (m + S(1))/n), x), x, x**(n/k)*(a + b*x**n)**(-S(1)/k)), x)
pattern963 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons148, cons244, cons622, cons485)
rule963 = ReplacementRule(pattern963, With963)
pattern964 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons5, cons50, cons71, cons196, cons17)
def replacement964(p, m, b, d, a, n, c, x, q):
rubi.append(964)
return -Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p*(c + d*x**(-n))**q, x), x, S(1)/x)
rule964 = ReplacementRule(pattern964, replacement964)
def With965(p, m, b, d, a, n, c, x, q, e):
g = Denominator(m)
rubi.append(965)
return -Dist(g/e, Subst(Int(x**(-g*(m + S(1)) + S(-1))*(a + b*e**(-n)*x**(-g*n))**p*(c + d*e**(-n)*x**(-g*n))**q, x), x, (e*x)**(-S(1)/g)), x)
pattern965 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons196, cons367)
rule965 = ReplacementRule(pattern965, With965)
pattern966 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons50, cons71, cons196, cons356)
def replacement966(p, m, b, d, a, n, c, x, q, e):
rubi.append(966)
return -Dist((e*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p*(c + d*x**(-n))**q, x), x, S(1)/x), x)
rule966 = ReplacementRule(pattern966, replacement966)
def With967(p, m, b, d, a, n, c, x, q):
g = Denominator(n)
rubi.append(967)
return Dist(g, Subst(Int(x**(g*(m + S(1)) + S(-1))*(a + b*x**(g*n))**p*(c + d*x**(g*n))**q, x), x, x**(S(1)/g)), x)
pattern967 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons50, cons71, cons489)
rule967 = ReplacementRule(pattern967, With967)
pattern968 = Pattern(Integral((e_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons50, cons71, cons489)
def replacement968(p, m, b, d, a, n, c, x, q, e):
rubi.append(968)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule968 = ReplacementRule(pattern968, replacement968)
pattern969 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons50, cons71, cons541, cons23)
def replacement969(p, m, b, d, a, n, c, x, q):
rubi.append(969)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p*(c + d*x**(n/(m + S(1))))**q, x), x, x**(m + S(1))), x)
rule969 = ReplacementRule(pattern969, replacement969)
pattern970 = Pattern(Integral((e_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons71, cons541, cons23)
def replacement970(p, m, b, d, a, n, c, x, q, e):
rubi.append(970)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule970 = ReplacementRule(pattern970, replacement970)
pattern971 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons71, cons402, cons137, cons576, cons608)
def replacement971(p, m, b, d, a, n, c, x, q, e):
rubi.append(971)
return Dist(S(1)/(a*b*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-2))*Simp(c*(b*c*n*(p + S(1)) + (m + S(1))*(-a*d + b*c)) + d*x**n*(b*c*n*(p + S(1)) + (-a*d + b*c)*(m + n*(q + S(-1)) + S(1))), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*(-a*d + b*c)/(a*b*e*n*(p + S(1))), x)
rule971 = ReplacementRule(pattern971, replacement971)
pattern972 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons71, cons402, cons137, cons574, cons608)
def replacement972(p, m, b, d, a, n, c, x, q, e):
rubi.append(972)
return Dist(S(1)/(a*n*(p + S(1))), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(m + n*(p + S(1)) + S(1)) + d*x**n*(m + n*(p + q + S(1)) + S(1)), x), x), x) - Simp((e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*e*n*(p + S(1))), x)
rule972 = ReplacementRule(pattern972, replacement972)
pattern973 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons50, cons71, cons13, cons137, cons608)
def replacement973(p, m, b, d, a, n, c, x, q, e):
rubi.append(973)
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((e*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(b*c*(m + S(1)) + b*d*x**n*(m + n*(p + q + S(2)) + S(1)) + n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp(b*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*e*n*(p + S(1))*(-a*d + b*c)), x)
rule973 = ReplacementRule(pattern973, replacement973)
pattern974 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons71, cons402, cons403, cons163, cons608)
def replacement974(p, m, b, d, a, n, c, x, q, e):
rubi.append(974)
return Dist(n/(m + n*(p + q) + S(1)), Int((e*x)**m*(a + b*x**n)**(p + S(-1))*(c + d*x**n)**(q + S(-1))*Simp(a*c*(p + q) + x**n*(a*d*(p + q) + q*(-a*d + b*c)), x), x), x) + Simp((e*x)**(m + S(1))*(a + b*x**n)**p*(c + d*x**n)**q/(e*(m + n*(p + q) + S(1))), x)
rule974 = ReplacementRule(pattern974, replacement974)
pattern975 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons71, cons395, cons576, cons608)
def replacement975(p, m, b, d, a, n, c, x, q, e):
rubi.append(975)
return Dist(S(1)/(b*(m + n*(p + q) + S(1))), Int((e*x)**m*(a + b*x**n)**p*(c + d*x**n)**(q + S(-2))*Simp(c*(b*c*n*(p + q) + (m + S(1))*(-a*d + b*c)) + x**n*(b*c*d*n*(p + q) + d*n*(q + S(-1))*(-a*d + b*c) + d*(m + S(1))*(-a*d + b*c)), x), x), x) + Simp(d*(e*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))/(b*e*(m + n*(p + q) + S(1))), x)
rule975 = ReplacementRule(pattern975, replacement975)
pattern976 = Pattern(Integral(x_**m_/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons71, cons623)
def replacement976(m, b, d, a, n, c, x):
rubi.append(976)
return -Dist(a/(-a*d + b*c), Int(x**(m - n)/(a + b*x**n), x), x) + Dist(c/(-a*d + b*c), Int(x**(m - n)/(c + d*x**n), x), x)
rule976 = ReplacementRule(pattern976, replacement976)
pattern977 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons21, cons71)
def replacement977(m, b, d, a, n, c, x, e):
rubi.append(977)
return Dist(b/(-a*d + b*c), Int((e*x)**m/(a + b*x**n), x), x) - Dist(d/(-a*d + b*c), Int((e*x)**m/(c + d*x**n), x), x)
rule977 = ReplacementRule(pattern977, replacement977)
pattern978 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons71, cons624, cons625, cons626)
def replacement978(p, m, b, d, a, n, c, x, q, e):
rubi.append(978)
return Int(ExpandIntegrand((e*x)**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule978 = ReplacementRule(pattern978, replacement978)
pattern979 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons585, cons586, cons587)
def replacement979(mn, p, m, b, d, c, n, a, x, q):
rubi.append(979)
return Int(x**(m - n*q)*(a + b*x**n)**p*(c*x**n + d)**q, x)
rule979 = ReplacementRule(pattern979, replacement979)
pattern980 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons50, cons585, cons386, cons147)
def replacement980(mn, p, m, b, d, c, n, a, x, q):
rubi.append(980)
return Dist(x**(n*FracPart(q))*(c + d*x**(-n))**FracPart(q)*(c*x**n + d)**(-FracPart(q)), Int(x**(m - n*q)*(a + b*x**n)**p*(c*x**n + d)**q, x), x)
rule980 = ReplacementRule(pattern980, replacement980)
pattern981 = Pattern(Integral((e_*x_)**m_*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons585)
def replacement981(mn, p, m, b, d, a, n, c, x, q, e):
rubi.append(981)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**(-n))**q, x), x)
rule981 = ReplacementRule(pattern981, replacement981)
pattern982 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons71, cons66, cons627, cons43, cons177)
def replacement982(p, m, b, d, a, n, c, x, q, e):
rubi.append(982)
return Simp(a**p*c**q*(e*x)**(m + S(1))*AppellF1((m + S(1))/n, -p, -q, S(1) + (m + S(1))/n, -b*x**n/a, -d*x**n/c)/(e*(m + S(1))), x)
rule982 = ReplacementRule(pattern982, replacement982)
pattern983 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons71, cons66, cons627, cons448)
def replacement983(p, m, b, d, a, n, c, x, q, e):
rubi.append(983)
return Dist(a**IntPart(p)*(S(1) + b*x**n/a)**(-FracPart(p))*(a + b*x**n)**FracPart(p), Int((e*x)**m*(S(1) + b*x**n/a)**p*(c + d*x**n)**q, x), x)
rule983 = ReplacementRule(pattern983, replacement983)
pattern984 = Pattern(Integral(x_**WC('m', S(1))*(v_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons552, cons17, cons553)
def replacement984(v, p, m, b, d, c, a, n, x, q):
rubi.append(984)
return Dist(Coefficient(v, x, S(1))**(-m + S(-1)), Subst(Int(SimplifyIntegrand((a + b*x**n)**p*(c + d*x**n)**q*(x - Coefficient(v, x, S(0)))**m, x), x), x, v), x)
rule984 = ReplacementRule(pattern984, replacement984)
pattern985 = Pattern(Integral(u_**WC('m', S(1))*(v_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons50, cons554)
def replacement985(v, p, u, m, b, d, c, a, n, x, q):
rubi.append(985)
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x, v), x)
rule985 = ReplacementRule(pattern985, replacement985)
pattern986 = Pattern(Integral((a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('u', S(1)), x_), cons57, cons58, cons59, cons60, cons7, cons27, cons4, cons5, cons50, cons593, cons55, cons494)
def replacement986(p, a2, u, b2, b1, d, c, n, non2, x, a1, q):
rubi.append(986)
return Int(u*(c + d*x**n)**q*(a1*a2 + b1*b2*x**n)**p, x)
rule986 = ReplacementRule(pattern986, replacement986)
pattern987 = Pattern(Integral((a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)) + x_**WC('n2', S(1))*WC('e', S(1)))**WC('q', S(1))*WC('u', S(1)), x_), cons57, cons58, cons59, cons60, cons7, cons27, cons48, cons4, cons5, cons50, cons593, cons46, cons55, cons494)
def replacement987(p, a2, u, b2, n2, b1, d, c, n, non2, x, a1, q, e):
rubi.append(987)
return Int(u*(a1*a2 + b1*b2*x**n)**p*(c + d*x**n + e*x**(S(2)*n))**q, x)
rule987 = ReplacementRule(pattern987, replacement987)
pattern988 = Pattern(Integral((a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**p_*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**p_*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('u', S(1)), x_), cons57, cons58, cons59, cons60, cons7, cons27, cons4, cons5, cons50, cons593, cons55)
def replacement988(p, a2, u, b2, b1, d, c, n, non2, x, a1, q):
rubi.append(988)
return Dist((a1 + b1*x**(n/S(2)))**FracPart(p)*(a2 + b2*x**(n/S(2)))**FracPart(p)*(a1*a2 + b1*b2*x**n)**(-FracPart(p)), Int(u*(c + d*x**n)**q*(a1*a2 + b1*b2*x**n)**p, x), x)
rule988 = ReplacementRule(pattern988, replacement988)
pattern989 = Pattern(Integral((a1_ + x_**WC('non2', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('non2', S(1))*WC('b2', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)) + x_**WC('n2', S(1))*WC('e', S(1)))**WC('q', S(1))*WC('u', S(1)), x_), cons57, cons58, cons59, cons60, cons7, cons27, cons48, cons4, cons5, cons50, cons593, cons46, cons55)
def replacement989(p, a2, u, b2, n2, b1, d, c, n, non2, x, a1, q, e):
rubi.append(989)
return Dist((a1 + b1*x**(n/S(2)))**FracPart(p)*(a2 + b2*x**(n/S(2)))**FracPart(p)*(a1*a2 + b1*b2*x**n)**(-FracPart(p)), Int(u*(a1*a2 + b1*b2*x**n)**p*(c + d*x**n + e*x**(S(2)*n))**q, x), x)
rule989 = ReplacementRule(pattern989, replacement989)
pattern990 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons628)
def replacement990(p, f, b, r, d, a, n, c, x, q, e):
rubi.append(990)
return Int(ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
rule990 = ReplacementRule(pattern990, replacement990)
pattern991 = Pattern(Integral((e_ + x_**n_*WC('f', S(1)))/((a_ + x_**n_*WC('b', S(1)))*(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons629)
def replacement991(f, b, d, a, n, c, x, e):
rubi.append(991)
return Dist((-a*f + b*e)/(-a*d + b*c), Int(S(1)/(a + b*x**n), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int(S(1)/(c + d*x**n), x), x)
rule991 = ReplacementRule(pattern991, replacement991)
pattern992 = Pattern(Integral((e_ + x_**n_*WC('f', S(1)))/((a_ + x_**n_*WC('b', S(1)))*sqrt(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons629)
def replacement992(f, b, d, a, n, c, x, e):
rubi.append(992)
return Dist(f/b, Int(S(1)/sqrt(c + d*x**n), x), x) + Dist((-a*f + b*e)/b, Int(S(1)/((a + b*x**n)*sqrt(c + d*x**n)), x), x)
rule992 = ReplacementRule(pattern992, replacement992)
pattern993 = Pattern(Integral((e_ + x_**n_*WC('f', S(1)))/(sqrt(a_ + x_**n_*WC('b', S(1)))*sqrt(c_ + x_**n_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons630)
def replacement993(f, b, d, a, n, c, x, e):
rubi.append(993)
return Dist(f/b, Int(sqrt(a + b*x**n)/sqrt(c + d*x**n), x), x) + Dist((-a*f + b*e)/b, Int(S(1)/(sqrt(a + b*x**n)*sqrt(c + d*x**n)), x), x)
rule993 = ReplacementRule(pattern993, replacement993)
pattern994 = Pattern(Integral((e_ + x_**S(2)*WC('f', S(1)))/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons466, cons573)
def replacement994(f, b, d, a, c, x, e):
rubi.append(994)
return Dist((-a*f + b*e)/(-a*d + b*c), Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int(sqrt(a + b*x**S(2))/(c + d*x**S(2))**(S(3)/2), x), x)
rule994 = ReplacementRule(pattern994, replacement994)
pattern995 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons402, cons137, cons403)
def replacement995(p, f, b, d, a, n, c, x, q, e):
rubi.append(995)
return Dist(S(1)/(a*b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(-a*f + b*e*n*(p + S(1)) + b*e) + d*x**n*(b*e*n*(p + S(1)) + (-a*f + b*e)*(n*q + S(1))), x), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*(-a*f + b*e)/(a*b*n*(p + S(1))), x)
rule995 = ReplacementRule(pattern995, replacement995)
pattern996 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons50, cons13, cons137)
def replacement996(p, f, b, d, a, n, c, x, q, e):
rubi.append(996)
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(-a*f + b*e) + d*x**n*(-a*f + b*e)*(n*(p + q + S(2)) + S(1)) + e*n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp(x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))*(-a*f + b*e)/(a*n*(p + S(1))*(-a*d + b*c)), x)
rule996 = ReplacementRule(pattern996, replacement996)
pattern997 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons395, cons403, cons631)
def replacement997(p, f, b, d, a, n, c, x, q, e):
rubi.append(997)
return Dist(S(1)/(b*(n*(p + q + S(1)) + S(1))), Int((a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(c*(-a*f + b*e*n*(p + q + S(1)) + b*e) + x**n*(b*d*e*n*(p + q + S(1)) + d*(-a*f + b*e) + f*n*q*(-a*d + b*c)), x), x), x) + Simp(f*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*(n*(p + q + S(1)) + S(1))), x)
rule997 = ReplacementRule(pattern997, replacement997)
pattern998 = Pattern(Integral((e_ + x_**S(4)*WC('f', S(1)))/((a_ + x_**S(4)*WC('b', S(1)))**(S(3)/4)*(c_ + x_**S(4)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement998(f, b, d, a, c, x, e):
rubi.append(998)
return Dist((-a*f + b*e)/(-a*d + b*c), Int((a + b*x**S(4))**(S(-3)/4), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int((a + b*x**S(4))**(S(1)/4)/(c + d*x**S(4)), x), x)
rule998 = ReplacementRule(pattern998, replacement998)
pattern999 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(e_ + x_**n_*WC('f', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons4, cons632)
def replacement999(p, f, b, d, a, n, c, x, e):
rubi.append(999)
return Dist(f/d, Int((a + b*x**n)**p, x), x) + Dist((-c*f + d*e)/d, Int((a + b*x**n)**p/(c + d*x**n), x), x)
rule999 = ReplacementRule(pattern999, replacement999)
pattern1000 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons50, cons633)
def replacement1000(p, f, b, d, a, n, c, x, q, e):
rubi.append(1000)
return Dist(e, Int((a + b*x**n)**p*(c + d*x**n)**q, x), x) + Dist(f, Int(x**n*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule1000 = ReplacementRule(pattern1000, replacement1000)
pattern1001 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))*(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement1001(f, b, d, a, c, x, e):
rubi.append(1001)
return Dist(b/(-a*d + b*c), Int(S(1)/((a + b*x**S(2))*sqrt(e + f*x**S(2))), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/((c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
rule1001 = ReplacementRule(pattern1001, replacement1001)
pattern1002 = Pattern(Integral(S(1)/(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons7, cons27, cons48, cons125, cons176)
def replacement1002(f, d, c, x, e):
rubi.append(1002)
return Dist(S(1)/c, Int(S(1)/(x**S(2)*sqrt(e + f*x**S(2))), x), x) - Dist(d/c, Int(S(1)/((c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
rule1002 = ReplacementRule(pattern1002, replacement1002)
pattern1003 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons634, cons635, cons636)
def replacement1003(f, b, d, a, c, x, e):
rubi.append(1003)
return Dist(d/b, Int(sqrt(e + f*x**S(2))/sqrt(c + d*x**S(2)), x), x) + Dist((-a*d + b*c)/b, Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
rule1003 = ReplacementRule(pattern1003, replacement1003)
pattern1004 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons637)
def replacement1004(f, b, d, a, c, x, e):
rubi.append(1004)
return Dist(d/b, Int(sqrt(e + f*x**S(2))/sqrt(c + d*x**S(2)), x), x) + Dist((-a*d + b*c)/b, Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
rule1004 = ReplacementRule(pattern1004, replacement1004)
pattern1005 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons573, cons638, cons636)
def replacement1005(f, b, d, a, c, x, e):
rubi.append(1005)
return Dist(b/(-a*f + b*e), Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) - Dist(f/(-a*f + b*e), Int(S(1)/(sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
rule1005 = ReplacementRule(pattern1005, replacement1005)
pattern1006 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons581, cons177, cons178, cons639)
def replacement1006(f, b, d, a, c, x, e):
rubi.append(1006)
return Simp(EllipticPi(b*c/(a*d), asin(x*Rt(-d/c, S(2))), c*f/(d*e))/(a*sqrt(c)*sqrt(e)*Rt(-d/c, S(2))), x)
rule1006 = ReplacementRule(pattern1006, replacement1006)
pattern1007 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons117)
def replacement1007(f, b, d, a, c, x, e):
rubi.append(1007)
return Dist(sqrt(S(1) + d*x**S(2)/c)/sqrt(c + d*x**S(2)), Int(S(1)/(sqrt(S(1) + d*x**S(2)/c)*(a + b*x**S(2))*sqrt(e + f*x**S(2))), x), x)
rule1007 = ReplacementRule(pattern1007, replacement1007)
pattern1008 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons573)
def replacement1008(f, b, d, a, c, x, e):
rubi.append(1008)
return Simp(c*sqrt(e + f*x**S(2))*EllipticPi(S(1) - b*c/(a*d), ArcTan(x*Rt(d/c, S(2))), -c*f/(d*e) + S(1))/(a*e*sqrt(c*(e + f*x**S(2))/(e*(c + d*x**S(2))))*sqrt(c + d*x**S(2))*Rt(d/c, S(2))), x)
rule1008 = ReplacementRule(pattern1008, replacement1008)
pattern1009 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))/((a_ + x_**S(2)*WC('b', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons581)
def replacement1009(f, b, d, a, c, x, e):
rubi.append(1009)
return Dist(d/b, Int(S(1)/(sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/((a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
rule1009 = ReplacementRule(pattern1009, replacement1009)
pattern1010 = Pattern(Integral(sqrt(e_ + x_**S(2)*WC('f', S(1)))/((a_ + x_**S(2)*WC('b', S(1)))*(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons573, cons638)
def replacement1010(f, b, d, a, c, x, e):
rubi.append(1010)
return Dist(b/(-a*d + b*c), Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(e + f*x**S(2))/(c + d*x**S(2))**(S(3)/2), x), x)
rule1010 = ReplacementRule(pattern1010, replacement1010)
pattern1011 = Pattern(Integral((e_ + x_**S(2)*WC('f', S(1)))**(S(3)/2)/((a_ + x_**S(2)*WC('b', S(1)))*(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons573, cons638)
def replacement1011(f, b, d, a, c, x, e):
rubi.append(1011)
return Dist((-a*f + b*e)/(-a*d + b*c), Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int(sqrt(e + f*x**S(2))/(c + d*x**S(2))**(S(3)/2), x), x)
rule1011 = ReplacementRule(pattern1011, replacement1011)
pattern1012 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2)*sqrt(e_ + x_**S(2)*WC('f', S(1)))/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons573, cons638)
def replacement1012(f, b, d, a, c, x, e):
rubi.append(1012)
return Dist(d/b**S(2), Int(sqrt(e + f*x**S(2))*(-a*d + S(2)*b*c + b*d*x**S(2))/sqrt(c + d*x**S(2)), x), x) + Dist((-a*d + b*c)**S(2)/b**S(2), Int(sqrt(e + f*x**S(2))/((a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x)
rule1012 = ReplacementRule(pattern1012, replacement1012)
pattern1013 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**q_*(e_ + x_**S(2)*WC('f', S(1)))**r_/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons640, cons396, cons641)
def replacement1013(f, b, r, d, a, c, x, q, e):
rubi.append(1013)
return Dist(b*(-a*f + b*e)/(-a*d + b*c)**S(2), Int((c + d*x**S(2))**(q + S(2))*(e + f*x**S(2))**(r + S(-1))/(a + b*x**S(2)), x), x) - Dist((-a*d + b*c)**(S(-2)), Int((c + d*x**S(2))**q*(e + f*x**S(2))**(r + S(-1))*(-a*d**S(2)*e - b*c**S(2)*f + S(2)*b*c*d*e + d**S(2)*x**S(2)*(-a*f + b*e)), x), x)
rule1013 = ReplacementRule(pattern1013, replacement1013)
pattern1014 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**q_*(e_ + x_**S(2)*WC('f', S(1)))**r_/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons52, cons395, cons576)
def replacement1014(f, b, r, d, a, c, x, q, e):
rubi.append(1014)
return Dist(d/b, Int((c + d*x**S(2))**(q + S(-1))*(e + f*x**S(2))**r, x), x) + Dist((-a*d + b*c)/b, Int((c + d*x**S(2))**(q + S(-1))*(e + f*x**S(2))**r/(a + b*x**S(2)), x), x)
rule1014 = ReplacementRule(pattern1014, replacement1014)
pattern1015 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**q_*(e_ + x_**S(2)*WC('f', S(1)))**r_/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons52, cons395, cons396)
def replacement1015(f, b, r, d, a, c, x, q, e):
rubi.append(1015)
return Dist(b**S(2)/(-a*d + b*c)**S(2), Int((c + d*x**S(2))**(q + S(2))*(e + f*x**S(2))**r/(a + b*x**S(2)), x), x) - Dist(d/(-a*d + b*c)**S(2), Int((c + d*x**S(2))**q*(e + f*x**S(2))**r*(-a*d + S(2)*b*c + b*d*x**S(2)), x), x)
rule1015 = ReplacementRule(pattern1015, replacement1015)
pattern1016 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**q_*(e_ + x_**S(2)*WC('f', S(1)))**r_/(a_ + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons52, cons395, cons642)
def replacement1016(f, b, r, d, a, c, x, q, e):
rubi.append(1016)
return Dist(b/(-a*d + b*c), Int((c + d*x**S(2))**(q + S(1))*(e + f*x**S(2))**r/(a + b*x**S(2)), x), x) - Dist(d/(-a*d + b*c), Int((c + d*x**S(2))**q*(e + f*x**S(2))**r, x), x)
rule1016 = ReplacementRule(pattern1016, replacement1016)
pattern1017 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))/(a_ + x_**S(2)*WC('b', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement1017(f, b, d, a, c, x, e):
rubi.append(1017)
return Dist((-a**S(2)*d*f + b**S(2)*c*e)/(S(2)*a*b**S(2)), Int(S(1)/((a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Dist(d*f/(S(2)*a*b**S(2)), Int((a - b*x**S(2))/(sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Simp(x*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))/(S(2)*a*(a + b*x**S(2))), x)
rule1017 = ReplacementRule(pattern1017, replacement1017)
pattern1018 = Pattern(Integral(S(1)/((a_ + x_**S(2)*WC('b', S(1)))**S(2)*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement1018(f, b, d, a, c, x, e):
rubi.append(1018)
return Dist((S(3)*a**S(2)*d*f - S(2)*a*b*(c*f + d*e) + b**S(2)*c*e)/(S(2)*a*(-a*d + b*c)*(-a*f + b*e)), Int(S(1)/((a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) - Dist(d*f/(S(2)*a*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x**S(2))/(sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Simp(b**S(2)*x*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))/(S(2)*a*(a + b*x**S(2))*(-a*d + b*c)*(-a*f + b*e)), x)
rule1018 = ReplacementRule(pattern1018, replacement1018)
pattern1019 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons52, cons63, cons395, cons403)
def replacement1019(p, f, b, r, d, a, n, c, x, q, e):
rubi.append(1019)
return Dist(d/b, Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*(e + f*x**n)**r, x), x) + Dist((-a*d + b*c)/b, Int((a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*(e + f*x**n)**r, x), x)
rule1019 = ReplacementRule(pattern1019, replacement1019)
pattern1020 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons50, cons63, cons395, cons642)
def replacement1020(p, f, b, r, d, a, n, c, x, q, e):
rubi.append(1020)
return Dist(b/(-a*d + b*c), Int((a + b*x**n)**p*(c + d*x**n)**(q + S(1))*(e + f*x**n)**r, x), x) - Dist(d/(-a*d + b*c), Int((a + b*x**n)**(p + S(1))*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
rule1020 = ReplacementRule(pattern1020, replacement1020)
pattern1021 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement1021(f, b, d, a, c, x, e):
rubi.append(1021)
return Dist(sqrt(a*(e + f*x**S(2))/(e*(a + b*x**S(2))))*sqrt(c + d*x**S(2))/(c*sqrt(a*(c + d*x**S(2))/(c*(a + b*x**S(2))))*sqrt(e + f*x**S(2))), Subst(Int(S(1)/(sqrt(S(1) - x**S(2)*(-a*d + b*c)/c)*sqrt(S(1) - x**S(2)*(-a*f + b*e)/e)), x), x, x/sqrt(a + b*x**S(2))), x)
rule1021 = ReplacementRule(pattern1021, replacement1021)
pattern1022 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))/(sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement1022(f, b, d, a, c, x, e):
rubi.append(1022)
return Dist(a*sqrt(a*(e + f*x**S(2))/(e*(a + b*x**S(2))))*sqrt(c + d*x**S(2))/(c*sqrt(a*(c + d*x**S(2))/(c*(a + b*x**S(2))))*sqrt(e + f*x**S(2))), Subst(Int(S(1)/(sqrt(S(1) - x**S(2)*(-a*d + b*c)/c)*sqrt(S(1) - x**S(2)*(-a*f + b*e)/e)*(-b*x**S(2) + S(1))), x), x, x/sqrt(a + b*x**S(2))), x)
rule1022 = ReplacementRule(pattern1022, replacement1022)
pattern1023 = Pattern(Integral(sqrt(c_ + x_**S(2)*WC('d', S(1)))/((a_ + x_**S(2)*WC('b', S(1)))**(S(3)/2)*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement1023(f, b, d, a, c, x, e):
rubi.append(1023)
return Dist(sqrt(a*(e + f*x**S(2))/(e*(a + b*x**S(2))))*sqrt(c + d*x**S(2))/(a*sqrt(a*(c + d*x**S(2))/(c*(a + b*x**S(2))))*sqrt(e + f*x**S(2))), Subst(Int(sqrt(S(1) - x**S(2)*(-a*d + b*c)/c)/sqrt(S(1) - x**S(2)*(-a*f + b*e)/e), x), x, x/sqrt(a + b*x**S(2))), x)
rule1023 = ReplacementRule(pattern1023, replacement1023)
pattern1024 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))/sqrt(e_ + x_**S(2)*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons643)
def replacement1024(f, b, d, a, c, x, e):
rubi.append(1024)
return -Dist(c*(-c*f + d*e)/(S(2)*f), Int(sqrt(a + b*x**S(2))/((c + d*x**S(2))**(S(3)/2)*sqrt(e + f*x**S(2))), x), x) - Dist((-a*d*f - b*c*f + b*d*e)/(S(2)*d*f), Int(sqrt(c + d*x**S(2))/(sqrt(a + b*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Dist(b*c*(-c*f + d*e)/(S(2)*d*f), Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Simp(d*x*sqrt(a + b*x**S(2))*sqrt(e + f*x**S(2))/(S(2)*f*sqrt(c + d*x**S(2))), x)
rule1024 = ReplacementRule(pattern1024, replacement1024)
pattern1025 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))/sqrt(e_ + x_**S(2)*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons644)
def replacement1025(f, b, d, a, c, x, e):
rubi.append(1025)
return -Dist((-a*d*f - b*c*f + b*d*e)/(S(2)*f**S(2)), Int(sqrt(e + f*x**S(2))/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))), x), x) + Dist(e*(-a*f + b*e)/(S(2)*f), Int(sqrt(c + d*x**S(2))/(sqrt(a + b*x**S(2))*(e + f*x**S(2))**(S(3)/2)), x), x) + Dist((-a*f + b*e)*(-S(2)*c*f + d*e)/(S(2)*f**S(2)), Int(S(1)/(sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) + Simp(x*sqrt(a + b*x**S(2))*sqrt(c + d*x**S(2))/(S(2)*sqrt(e + f*x**S(2))), x)
rule1025 = ReplacementRule(pattern1025, replacement1025)
pattern1026 = Pattern(Integral(sqrt(a_ + x_**S(2)*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))/(e_ + x_**S(2)*WC('f', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement1026(f, b, d, a, c, x, e):
rubi.append(1026)
return Dist(b/f, Int(sqrt(c + d*x**S(2))/(sqrt(a + b*x**S(2))*sqrt(e + f*x**S(2))), x), x) - Dist((-a*f + b*e)/f, Int(sqrt(c + d*x**S(2))/(sqrt(a + b*x**S(2))*(e + f*x**S(2))**(S(3)/2)), x), x)
rule1026 = ReplacementRule(pattern1026, replacement1026)
def With1027(p, f, b, r, d, a, n, c, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x)
if SumQ(u):
return True
return False
pattern1027 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons52, cons148, CustomConstraint(With1027))
def replacement1027(p, f, b, r, d, a, n, c, x, q, e):
u = ExpandIntegrand((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x)
rubi.append(1027)
return Int(u, x)
rule1027 = ReplacementRule(pattern1027, replacement1027)
pattern1028 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons52, cons196)
def replacement1028(p, f, b, r, d, a, n, c, x, q, e):
rubi.append(1028)
return -Subst(Int((a + b*x**(-n))**p*(c + d*x**(-n))**q*(e + f*x**(-n))**r/x**S(2), x), x, S(1)/x)
rule1028 = ReplacementRule(pattern1028, replacement1028)
pattern1029 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons50, cons52, cons645)
def replacement1029(p, f, b, r, d, a, n, c, x, q, e):
rubi.append(1029)
return Int((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x)
rule1029 = ReplacementRule(pattern1029, replacement1029)
pattern1030 = Pattern(Integral((u_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1))*(w_**n_*WC('f', S(1)) + WC('e', S(0)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons4, cons50, cons52, cons646, cons647, cons68, cons69)
def replacement1030(v, w, p, u, f, b, r, d, c, a, n, x, q, e):
rubi.append(1030)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x, u), x)
rule1030 = ReplacementRule(pattern1030, replacement1030)
pattern1031 = Pattern(Integral((c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons52, cons585, cons586)
def replacement1031(mn, p, f, b, r, d, a, n, c, x, q, e):
rubi.append(1031)
return Int(x**(-n*q)*(a + b*x**n)**p*(e + f*x**n)**r*(c*x**n + d)**q, x)
rule1031 = ReplacementRule(pattern1031, replacement1031)
pattern1032 = Pattern(Integral((c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons50, cons585, cons38, cons648)
def replacement1032(mn, p, f, b, r, d, a, n, c, x, q, e):
rubi.append(1032)
return Int(x**(n*(p + r))*(c + d*x**(-n))**q*(a*x**(-n) + b)**p*(e*x**(-n) + f)**r, x)
rule1032 = ReplacementRule(pattern1032, replacement1032)
pattern1033 = Pattern(Integral((c_ + x_**WC('mn', S(1))*WC('d', S(1)))**q_*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons50, cons52, cons585, cons386)
def replacement1033(mn, p, f, b, r, d, a, n, c, x, q, e):
rubi.append(1033)
return Dist(x**(n*FracPart(q))*(c + d*x**(-n))**FracPart(q)*(c*x**n + d)**(-FracPart(q)), Int(x**(-n*q)*(a + b*x**n)**p*(e + f*x**n)**r*(c*x**n + d)**q, x), x)
rule1033 = ReplacementRule(pattern1033, replacement1033)
pattern1034 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e1_ + x_**WC('n2', S(1))*WC('f1', S(1)))**WC('r', S(1))*(e2_ + x_**WC('n2', S(1))*WC('f2', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons652, cons653, cons654, cons655, cons4, cons5, cons50, cons52, cons649, cons650, cons651)
def replacement1034(p, f2, b, n2, r, d, e2, f1, a, n, c, e1, x, q):
rubi.append(1034)
return Int((a + b*x**n)**p*(c + d*x**n)**q*(e1*e2 + f1*f2*x**n)**r, x)
rule1034 = ReplacementRule(pattern1034, replacement1034)
pattern1035 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e1_ + x_**WC('n2', S(1))*WC('f1', S(1)))**WC('r', S(1))*(e2_ + x_**WC('n2', S(1))*WC('f2', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons652, cons653, cons654, cons655, cons4, cons5, cons50, cons52, cons649, cons650)
def replacement1035(p, f2, b, n2, r, d, e2, f1, a, n, c, e1, x, q):
rubi.append(1035)
return Dist((e1 + f1*x**(n/S(2)))**FracPart(r)*(e2 + f2*x**(n/S(2)))**FracPart(r)*(e1*e2 + f1*f2*x**n)**(-FracPart(r)), Int((a + b*x**n)**p*(c + d*x**n)**q*(e1*e2 + f1*f2*x**n)**r, x), x)
rule1035 = ReplacementRule(pattern1035, replacement1035)
pattern1036 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons52, cons656, cons500)
def replacement1036(p, m, g, b, f, r, d, c, n, x, q, e):
rubi.append(1036)
return Dist(b**(S(1) - (m + S(1))/n)*g**m/n, Subst(Int((b*x)**(p + S(-1) + (m + S(1))/n)*(c + d*x)**q*(e + f*x)**r, x), x, x**n), x)
rule1036 = ReplacementRule(pattern1036, replacement1036)
pattern1037 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons52, cons656, cons501)
def replacement1037(p, m, g, b, f, r, d, c, n, x, q, e):
rubi.append(1037)
return Dist(b**IntPart(p)*g**m*x**(-n*FracPart(p))*(b*x**n)**FracPart(p), Int(x**(m + n*p)*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
rule1037 = ReplacementRule(pattern1037, replacement1037)
pattern1038 = Pattern(Integral((g_*x_)**m_*(x_**WC('n', S(1))*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons52, cons18)
def replacement1038(p, m, f, b, r, g, d, c, n, x, q, e):
rubi.append(1038)
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
rule1038 = ReplacementRule(pattern1038, replacement1038)
pattern1039 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons657)
def replacement1039(p, m, g, b, f, r, d, a, n, c, x, q, e):
rubi.append(1039)
return Int(ExpandIntegrand((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
rule1039 = ReplacementRule(pattern1039, replacement1039)
pattern1040 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons52, cons53)
def replacement1040(p, m, f, b, r, d, a, n, c, x, q, e):
rubi.append(1040)
return Dist(S(1)/n, Subst(Int((a + b*x)**p*(c + d*x)**q*(e + f*x)**r, x), x, x**n), x)
rule1040 = ReplacementRule(pattern1040, replacement1040)
pattern1041 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons658, cons502)
def replacement1041(p, m, f, b, r, d, a, n, c, x, q, e):
rubi.append(1041)
return Int(x**(m + n*(p + q + r))*(a*x**(-n) + b)**p*(c*x**(-n) + d)**q*(e*x**(-n) + f)**r, x)
rule1041 = ReplacementRule(pattern1041, replacement1041)
pattern1042 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons52, cons500)
def replacement1042(p, m, f, b, r, d, a, n, c, x, q, e):
rubi.append(1042)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x)**p*(c + d*x)**q*(e + f*x)**r, x), x, x**n), x)
rule1042 = ReplacementRule(pattern1042, replacement1042)
pattern1043 = Pattern(Integral((g_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons52, cons500)
def replacement1043(p, m, f, b, r, g, d, a, n, c, x, q, e):
rubi.append(1043)
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
rule1043 = ReplacementRule(pattern1043, replacement1043)
def With1044(p, m, f, b, r, d, a, n, c, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
pattern1044 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons52, cons148, cons17, CustomConstraint(With1044))
def replacement1044(p, m, f, b, r, d, a, n, c, x, q, e):
k = GCD(m + S(1), n)
rubi.append(1044)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a + b*x**(n/k))**p*(c + d*x**(n/k))**q*(e + f*x**(n/k))**r, x), x, x**k), x)
rule1044 = ReplacementRule(pattern1044, replacement1044)
def With1045(p, m, f, g, b, r, d, a, n, c, x, q, e):
k = Denominator(m)
rubi.append(1045)
return Dist(k/g, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*g**(-n)*x**(k*n))**p*(c + d*g**(-n)*x**(k*n))**q*(e + f*g**(-n)*x**(k*n))**r, x), x, (g*x)**(S(1)/k)), x)
pattern1045 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1)))**r_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons50, cons52, cons148, cons367)
rule1045 = ReplacementRule(pattern1045, With1045)
pattern1046 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons148, cons402, cons137, cons403, cons659)
def replacement1046(p, m, g, b, f, d, a, n, c, x, q, e):
rubi.append(1046)
return Dist(S(1)/(a*b*n*(p + S(1))), Int((g*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(b*e*n*(p + S(1)) + (m + S(1))*(-a*f + b*e)) + d*x**n*(b*e*n*(p + S(1)) + (-a*f + b*e)*(m + n*q + S(1))), x), x), x) - Simp((g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*(-a*f + b*e)/(a*b*g*n*(p + S(1))), x)
rule1046 = ReplacementRule(pattern1046, replacement1046)
pattern1047 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons50, cons148, cons244, cons137, cons607)
def replacement1047(p, m, f, g, b, d, a, n, c, x, q, e):
rubi.append(1047)
return -Dist(g**n/(b*n*(p + S(1))*(-a*d + b*c)), Int((g*x)**(m - n)*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(-a*f + b*e)*(m - n + S(1)) + x**n*(-b*n*(p + S(1))*(c*f - d*e) + d*(-a*f + b*e)*(m + n*q + S(1))), x), x), x) + Simp(g**(n + S(-1))*(g*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))*(-a*f + b*e)/(b*n*(p + S(1))*(-a*d + b*c)), x)
rule1047 = ReplacementRule(pattern1047, replacement1047)
pattern1048 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons50, cons148, cons13, cons137)
def replacement1048(p, m, f, g, b, d, a, n, c, x, q, e):
rubi.append(1048)
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((g*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(m + S(1))*(-a*f + b*e) + d*x**n*(-a*f + b*e)*(m + n*(p + q + S(2)) + S(1)) + e*n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp((g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))*(-a*f + b*e)/(a*g*n*(p + S(1))*(-a*d + b*c)), x)
rule1048 = ReplacementRule(pattern1048, replacement1048)
pattern1049 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons148, cons611, cons403, cons94, cons660)
def replacement1049(p, m, g, b, f, d, a, n, c, x, q, e):
rubi.append(1049)
return -Dist(g**(-n)/(a*(m + S(1))), Int((g*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(c*(m + S(1))*(-a*f + b*e) + d*x**n*(b*e*n*(p + q + S(1)) + (m + S(1))*(-a*f + b*e)) + e*n*(a*d*q + b*c*(p + S(1))), x), x), x) + Simp(e*(g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(a*g*(m + S(1))), x)
rule1049 = ReplacementRule(pattern1049, replacement1049)
pattern1050 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons148, cons395, cons403, cons660)
def replacement1050(p, m, g, b, f, d, a, n, c, x, q, e):
rubi.append(1050)
return Dist(S(1)/(b*(m + n*(p + q + S(1)) + S(1))), Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(c*(b*e*n*(p + q + S(1)) + (m + S(1))*(-a*f + b*e)) + x**n*(b*d*e*n*(p + q + S(1)) + d*(m + S(1))*(-a*f + b*e) + f*n*q*(-a*d + b*c)), x), x), x) + Simp(f*(g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*g*(m + n*(p + q + S(1)) + S(1))), x)
rule1050 = ReplacementRule(pattern1050, replacement1050)
pattern1051 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons50, cons148, cons31, cons530)
def replacement1051(p, m, g, b, f, d, a, n, c, x, q, e):
rubi.append(1051)
return -Dist(g**n/(b*d*(m + n*(p + q + S(1)) + S(1))), Int((g*x)**(m - n)*(a + b*x**n)**p*(c + d*x**n)**q*Simp(a*c*f*(m - n + S(1)) + x**n*(a*d*f*(m + n*q + S(1)) + b*(c*f*(m + n*p + S(1)) - d*e*(m + n*(p + q + S(1)) + S(1)))), x), x), x) + Simp(f*g**(n + S(-1))*(g*x)**(m - n + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(b*d*(m + n*(p + q + S(1)) + S(1))), x)
rule1051 = ReplacementRule(pattern1051, replacement1051)
pattern1052 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons50, cons148, cons31, cons94)
def replacement1052(p, m, g, b, f, d, a, n, c, x, q, e):
rubi.append(1052)
return Dist(g**(-n)/(a*c*(m + S(1))), Int((g*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q*Simp(a*c*f*(m + S(1)) - b*d*e*x**n*(m + n*(p + q + S(2)) + S(1)) - e*n*(a*d*q + b*c*p) - e*(a*d + b*c)*(m + n + S(1)), x), x), x) + Simp(e*(g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))/(a*c*g*(m + S(1))), x)
rule1052 = ReplacementRule(pattern1052, replacement1052)
pattern1053 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(e_ + x_**n_*WC('f', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons148)
def replacement1053(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(1053)
return Int(ExpandIntegrand((g*x)**m*(a + b*x**n)**p*(e + f*x**n)/(c + d*x**n), x), x)
rule1053 = ReplacementRule(pattern1053, replacement1053)
pattern1054 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons50, cons148)
def replacement1054(p, m, g, b, f, d, a, n, c, x, q, e):
rubi.append(1054)
return Dist(e, Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x) + Dist(e**(-n)*f, Int((g*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule1054 = ReplacementRule(pattern1054, replacement1054)
pattern1055 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons50, cons148, cons661)
def replacement1055(p, m, g, b, f, r, d, a, n, c, x, q, e):
rubi.append(1055)
return Dist(e, Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**(r + S(-1)), x), x) + Dist(e**(-n)*f, Int((g*x)**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**(r + S(-1)), x), x)
rule1055 = ReplacementRule(pattern1055, replacement1055)
pattern1056 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons52, cons196, cons17)
def replacement1056(p, m, f, b, r, d, a, n, c, x, q, e):
rubi.append(1056)
return -Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p*(c + d*x**(-n))**q*(e + f*x**(-n))**r, x), x, S(1)/x)
rule1056 = ReplacementRule(pattern1056, replacement1056)
def With1057(p, m, g, b, f, r, d, a, n, c, x, q, e):
k = Denominator(m)
rubi.append(1057)
return -Dist(k/g, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*g**(-n)*x**(-k*n))**p*(c + d*g**(-n)*x**(-k*n))**q*(e + f*g**(-n)*x**(-k*n))**r, x), x, (g*x)**(-S(1)/k)), x)
pattern1057 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons50, cons52, cons196, cons367)
rule1057 = ReplacementRule(pattern1057, With1057)
pattern1058 = Pattern(Integral((x_*WC('g', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons50, cons52, cons196, cons356)
def replacement1058(p, m, g, b, f, r, d, a, n, c, x, q, e):
rubi.append(1058)
return -Dist((g*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**(-n))**p*(c + d*x**(-n))**q*(e + f*x**(-n))**r, x), x, S(1)/x), x)
rule1058 = ReplacementRule(pattern1058, replacement1058)
def With1059(p, m, f, b, r, d, a, n, c, x, q, e):
k = Denominator(n)
rubi.append(1059)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*x**(k*n))**p*(c + d*x**(k*n))**q*(e + f*x**(k*n))**r, x), x, x**(S(1)/k)), x)
pattern1059 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons50, cons52, cons489)
rule1059 = ReplacementRule(pattern1059, With1059)
pattern1060 = Pattern(Integral((g_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons50, cons52, cons489)
def replacement1060(p, m, f, b, r, g, d, a, n, c, x, q, e):
rubi.append(1060)
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
rule1060 = ReplacementRule(pattern1060, replacement1060)
pattern1061 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons52, cons541)
def replacement1061(p, m, f, b, r, d, a, n, c, x, q, e):
rubi.append(1061)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p*(c + d*x**(n/(m + S(1))))**q*(e + f*x**(n/(m + S(1))))**r, x), x, x**(m + S(1))), x)
rule1061 = ReplacementRule(pattern1061, replacement1061)
pattern1062 = Pattern(Integral((g_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons52, cons541)
def replacement1062(p, m, f, b, r, g, d, a, n, c, x, q, e):
rubi.append(1062)
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x)
rule1062 = ReplacementRule(pattern1062, replacement1062)
pattern1063 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons402, cons137, cons403, cons659)
def replacement1063(p, m, g, b, f, d, a, n, c, x, q, e):
rubi.append(1063)
return Dist(S(1)/(a*b*n*(p + S(1))), Int((g*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(-1))*Simp(c*(b*e*n*(p + S(1)) + (m + S(1))*(-a*f + b*e)) + d*x**n*(b*e*n*(p + S(1)) + (-a*f + b*e)*(m + n*q + S(1))), x), x), x) - Simp((g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*(-a*f + b*e)/(a*b*g*n*(p + S(1))), x)
rule1063 = ReplacementRule(pattern1063, replacement1063)
pattern1064 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons50, cons13, cons137)
def replacement1064(p, m, f, g, b, d, a, n, c, x, q, e):
rubi.append(1064)
return Dist(S(1)/(a*n*(p + S(1))*(-a*d + b*c)), Int((g*x)**m*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q*Simp(c*(m + S(1))*(-a*f + b*e) + d*x**n*(-a*f + b*e)*(m + n*(p + q + S(2)) + S(1)) + e*n*(p + S(1))*(-a*d + b*c), x), x), x) - Simp((g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(q + S(1))*(-a*f + b*e)/(a*g*n*(p + S(1))*(-a*d + b*c)), x)
rule1064 = ReplacementRule(pattern1064, replacement1064)
pattern1065 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons395, cons403, cons660)
def replacement1065(p, m, g, b, f, d, a, n, c, x, q, e):
rubi.append(1065)
return Dist(S(1)/(b*(m + n*(p + q + S(1)) + S(1))), Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**(q + S(-1))*Simp(c*(b*e*n*(p + q + S(1)) + (m + S(1))*(-a*f + b*e)) + x**n*(b*d*e*n*(p + q + S(1)) + d*(m + S(1))*(-a*f + b*e) + f*n*q*(-a*d + b*c)), x), x), x) + Simp(f*(g*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**q/(b*g*(m + n*(p + q + S(1)) + S(1))), x)
rule1065 = ReplacementRule(pattern1065, replacement1065)
pattern1066 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(e_ + x_**n_*WC('f', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons380)
def replacement1066(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(1066)
return Int(ExpandIntegrand((g*x)**m*(a + b*x**n)**p*(e + f*x**n)/(c + d*x**n), x), x)
rule1066 = ReplacementRule(pattern1066, replacement1066)
pattern1067 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*(c_ + x_**n_*WC('d', S(1)))**q_*(e_ + x_**n_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons662)
def replacement1067(p, m, f, g, b, d, a, n, c, x, q, e):
rubi.append(1067)
return Dist(e, Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x) + Dist(f*x**(-m)*(g*x)**m, Int(x**(m + n)*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule1067 = ReplacementRule(pattern1067, replacement1067)
pattern1068 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons52, cons585, cons586)
def replacement1068(mn, p, m, f, b, r, d, c, n, a, x, q, e):
rubi.append(1068)
return Int(x**(m - n*q)*(a + b*x**n)**p*(e + f*x**n)**r*(c*x**n + d)**q, x)
rule1068 = ReplacementRule(pattern1068, replacement1068)
pattern1069 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons50, cons585, cons38, cons648)
def replacement1069(mn, p, m, f, b, r, d, a, n, c, x, q, e):
rubi.append(1069)
return Int(x**(m + n*(p + r))*(c + d*x**(-n))**q*(a*x**(-n) + b)**p*(e*x**(-n) + f)**r, x)
rule1069 = ReplacementRule(pattern1069, replacement1069)
pattern1070 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**q_*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons52, cons585, cons386)
def replacement1070(mn, p, m, f, b, r, d, a, n, c, x, q, e):
rubi.append(1070)
return Dist(x**(n*FracPart(q))*(c + d*x**(-n))**FracPart(q)*(c*x**n + d)**(-FracPart(q)), Int(x**(m - n*q)*(a + b*x**n)**p*(e + f*x**n)**r*(c*x**n + d)**q, x), x)
rule1070 = ReplacementRule(pattern1070, replacement1070)
pattern1071 = Pattern(Integral((g_*x_)**m_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('mn', S(1))*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons52, cons585)
def replacement1071(mn, p, m, f, b, r, g, d, c, n, a, x, q, e):
rubi.append(1071)
return Dist(g**IntPart(m)*x**(-FracPart(m))*(g*x)**FracPart(m), Int(x**m*(a + b*x**n)**p*(c + d*x**(-n))**q*(e + f*x**n)**r, x), x)
rule1071 = ReplacementRule(pattern1071, replacement1071)
pattern1072 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e_ + x_**n_*WC('f', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons52, cons663)
def replacement1072(p, m, g, b, f, r, d, a, n, c, x, q, e):
rubi.append(1072)
return Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x)
rule1072 = ReplacementRule(pattern1072, replacement1072)
pattern1073 = Pattern(Integral(u_**WC('m', S(1))*(e_ + v_**n_*WC('f', S(1)))**WC('r', S(1))*(v_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons52, cons554)
def replacement1073(v, p, u, m, f, b, r, d, c, a, n, x, q, e):
rubi.append(1073)
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q*(e + f*x**n)**r, x), x, v), x)
rule1073 = ReplacementRule(pattern1073, replacement1073)
pattern1074 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e1_ + x_**WC('n2', S(1))*WC('f1', S(1)))**WC('r', S(1))*(e2_ + x_**WC('n2', S(1))*WC('f2', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons652, cons653, cons654, cons655, cons208, cons21, cons4, cons5, cons50, cons52, cons649, cons650, cons651)
def replacement1074(p, m, g, f2, n2, r, b, d, e2, f1, a, n, c, e1, x, q):
rubi.append(1074)
return Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e1*e2 + f1*f2*x**n)**r, x)
rule1074 = ReplacementRule(pattern1074, replacement1074)
pattern1075 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(e1_ + x_**WC('n2', S(1))*WC('f1', S(1)))**WC('r', S(1))*(e2_ + x_**WC('n2', S(1))*WC('f2', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons27, cons652, cons653, cons654, cons655, cons208, cons21, cons4, cons5, cons50, cons52, cons649, cons650)
def replacement1075(p, m, g, f2, n2, r, b, d, e2, f1, a, n, c, e1, x, q):
rubi.append(1075)
return Dist((e1 + f1*x**(n/S(2)))**FracPart(r)*(e2 + f2*x**(n/S(2)))**FracPart(r)*(e1*e2 + f1*f2*x**n)**(-FracPart(r)), Int((g*x)**m*(a + b*x**n)**p*(c + d*x**n)**q*(e1*e2 + f1*f2*x**n)**r, x), x)
rule1075 = ReplacementRule(pattern1075, replacement1075)
return [rule689, rule690, rule691, rule692, rule693, rule694, rule695, rule696, rule697, rule698, rule699, rule700, rule701, rule702, rule703, rule704, rule705, rule706, rule707, rule708, rule709, rule710, rule711, rule712, rule713, rule714, rule715, rule716, rule717, rule718, rule719, rule720, rule721, rule722, rule723, rule724, rule725, rule726, rule727, rule728, rule729, rule730, rule731, rule732, rule733, rule734, rule735, rule736, rule737, rule738, rule739, rule740, rule741, rule742, rule743, rule744, rule745, rule746, rule747, rule748, rule749, rule750, rule751, rule752, rule753, rule754, rule755, rule756, rule757, rule758, rule759, rule760, rule761, rule762, rule763, rule764, rule765, rule766, rule767, rule768, rule769, rule770, rule771, rule772, rule773, rule774, rule775, rule776, rule777, rule778, rule779, rule780, rule781, rule782, rule783, rule784, rule785, rule786, rule787, rule788, rule789, rule790, rule791, rule792, rule793, rule794, rule795, rule796, rule797, rule798, rule799, rule800, rule801, rule802, rule803, rule804, rule805, rule806, rule807, rule808, rule809, rule810, rule811, rule812, rule813, rule814, rule815, rule816, rule817, rule818, rule819, rule820, rule821, rule822, rule823, rule824, rule825, rule826, rule827, rule828, rule829, rule830, rule831, rule832, rule833, rule834, rule835, rule836, rule837, rule838, rule839, rule840, rule841, rule842, rule843, rule844, rule845, rule846, rule847, rule848, rule849, rule850, rule851, rule852, rule853, rule854, rule855, rule856, rule857, rule858, rule859, rule860, rule861, rule862, rule863, rule864, rule865, rule866, rule867, rule868, rule869, rule870, rule871, rule872, rule873, rule874, rule875, rule876, rule877, rule878, rule879, rule880, rule881, rule882, rule883, rule884, rule885, rule886, rule887, rule888, rule889, rule890, rule891, rule892, rule893, rule894, rule895, rule896, rule897, rule898, rule899, rule900, rule901, rule902, rule903, rule904, rule905, rule906, rule907, rule908, rule909, rule910, rule911, rule912, rule913, rule914, rule915, rule916, rule917, rule918, rule919, rule920, rule921, rule922, rule923, rule924, rule925, rule926, rule927, rule928, rule929, rule930, rule931, rule932, rule933, rule934, rule935, rule936, rule937, rule938, rule939, rule940, rule941, rule942, rule943, rule944, rule945, rule946, rule947, rule948, rule949, rule950, rule951, rule952, rule953, rule954, rule955, rule956, rule957, rule958, rule959, rule960, rule961, rule962, rule963, rule964, rule965, rule966, rule967, rule968, rule969, rule970, rule971, rule972, rule973, rule974, rule975, rule976, rule977, rule978, rule979, rule980, rule981, rule982, rule983, rule984, rule985, rule986, rule987, rule988, rule989, rule990, rule991, rule992, rule993, rule994, rule995, rule996, rule997, rule998, rule999, rule1000, rule1001, rule1002, rule1003, rule1004, rule1005, rule1006, rule1007, rule1008, rule1009, rule1010, rule1011, rule1012, rule1013, rule1014, rule1015, rule1016, rule1017, rule1018, rule1019, rule1020, rule1021, rule1022, rule1023, rule1024, rule1025, rule1026, rule1027, rule1028, rule1029, rule1030, rule1031, rule1032, rule1033, rule1034, rule1035, rule1036, rule1037, rule1038, rule1039, rule1040, rule1041, rule1042, rule1043, rule1044, rule1045, rule1046, rule1047, rule1048, rule1049, rule1050, rule1051, rule1052, rule1053, rule1054, rule1055, rule1056, rule1057, rule1058, rule1059, rule1060, rule1061, rule1062, rule1063, rule1064, rule1065, rule1066, rule1067, rule1068, rule1069, rule1070, rule1071, rule1072, rule1073, rule1074, rule1075, ]
|
aea41bd101c18a89d2efb038934b8d2bb7301f6054efb49e9f4882f71b6da0ec | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def inverse_hyperbolic(rubi):
from sympy.integrals.rubi.constraints import cons87, cons88, cons2, cons3, cons7, cons89, cons1579, cons4, cons148, cons66, cons27, cons21, cons62, cons1734, cons1735, cons1736, cons1778, cons268, cons48, cons1892, cons1893, cons1894, cons1895, cons731, cons652, cons732, cons654, cons584, cons1738, cons1896, cons128, cons1737, cons338, cons163, cons38, cons347, cons137, cons230, cons667, cons5, cons1739, cons1740, cons961, cons1897, cons1741, cons1898, cons1743, cons1742, cons1744, cons1570, cons1899, cons336, cons1900, cons147, cons125, cons208, cons54, cons242, cons1746, cons1747, cons486, cons162, cons94, cons93, cons272, cons1748, cons17, cons166, cons274, cons1749, cons1750, cons18, cons238, cons237, cons1751, cons246, cons1752, cons1901, cons1753, cons1754, cons1902, cons1755, cons1756, cons1903, cons209, cons925, cons464, cons84, cons1757, cons1758, cons719, cons168, cons1759, cons1760, cons267, cons717, cons1761, cons1608, cons14, cons150, cons1198, cons1273, cons1360, cons1830, cons1763, cons34, cons35, cons36, cons1762, cons1904, cons165, cons1442, cons1765, cons1764, cons1766, cons1767, cons528, cons1230, cons1769, cons1770, cons1905, cons1906, cons85, cons804, cons31, cons340, cons1907, cons1908, cons1909, cons1776, cons1043, cons1777, cons1497, cons13, cons1779, cons1780, cons1781, cons1782, cons240, cons241, cons146, cons1783, cons1510, cons1784, cons1152, cons319, cons1785, cons1786, cons1787, cons1788, cons1910, cons1911, cons1912, cons1913, cons1793, cons1794, cons1914, cons1796, cons601, cons1797, cons261, cons1915, cons1482, cons1441, cons1916, cons1250, cons1917, cons1918, cons1802, cons1803, cons1919, cons743, cons177, cons117, cons1920, cons23, cons1921, cons1922, cons1923, cons1924, cons1925, cons674, cons1926, cons1927, cons1928, cons994, cons1580, cons1818, cons1929, cons1930, cons1931, cons1932, cons1933, cons1934, cons1935, cons1824, cons973, cons1936, cons1827, cons1937, cons1938, cons1094, cons1831, cons1832, cons1833, cons1834, cons1939, cons383, cons808, cons1586, cons818, cons463, cons1940, cons1941, cons1942, cons1943, cons1944, cons67, cons1945, cons1946, cons1947, cons1948, cons1847, cons1949, cons1950, cons1951, cons1952, cons1854, cons178, cons1855, cons1856, cons1299, cons1953, cons1954, cons1955, cons1956
pattern6084 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons87, cons88)
def replacement6084(b, a, n, c, x):
rubi.append(6084)
return -Dist(b*c*n, Int(x*(a + b*asinh(c*x))**(n + S(-1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*asinh(c*x))**n, x)
rule6084 = ReplacementRule(pattern6084, replacement6084)
pattern6085 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons87, cons88)
def replacement6085(b, a, n, c, x):
rubi.append(6085)
return -Dist(b*c*n, Int(x*(a + b*acosh(c*x))**(n + S(-1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) + Simp(x*(a + b*acosh(c*x))**n, x)
rule6085 = ReplacementRule(pattern6085, replacement6085)
pattern6086 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons87, cons89)
def replacement6086(b, a, n, c, x):
rubi.append(6086)
return -Dist(c/(b*(n + S(1))), Int(x*(a + b*asinh(c*x))**(n + S(1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*asinh(c*x))**(n + S(1))*sqrt(c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule6086 = ReplacementRule(pattern6086, replacement6086)
pattern6087 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons87, cons89)
def replacement6087(b, a, n, c, x):
rubi.append(6087)
return -Dist(c/(b*(n + S(1))), Int(x*(a + b*acosh(c*x))**(n + S(1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) + Simp((a + b*acosh(c*x))**(n + S(1))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6087 = ReplacementRule(pattern6087, replacement6087)
pattern6088 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6088(b, a, n, c, x):
rubi.append(6088)
return Dist(S(1)/(b*c), Subst(Int(x**n*cosh(a/b - x/b), x), x, a + b*asinh(c*x)), x)
rule6088 = ReplacementRule(pattern6088, replacement6088)
pattern6089 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6089(b, a, n, c, x):
rubi.append(6089)
return -Dist(S(1)/(b*c), Subst(Int(x**n*sinh(a/b - x/b), x), x, a + b*acosh(c*x)), x)
rule6089 = ReplacementRule(pattern6089, replacement6089)
pattern6090 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/x_, x_), cons2, cons3, cons7, cons148)
def replacement6090(b, a, n, c, x):
rubi.append(6090)
return Subst(Int((a + b*x)**n/tanh(x), x), x, asinh(c*x))
rule6090 = ReplacementRule(pattern6090, replacement6090)
pattern6091 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/x_, x_), cons2, cons3, cons7, cons148)
def replacement6091(b, a, n, c, x):
rubi.append(6091)
return Subst(Int((a + b*x)**n*tanh(x), x), x, acosh(c*x))
rule6091 = ReplacementRule(pattern6091, replacement6091)
pattern6092 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons148, cons66)
def replacement6092(m, b, d, a, n, c, x):
rubi.append(6092)
return -Dist(b*c*n/(d*(m + S(1))), Int((d*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x) + Simp((d*x)**(m + S(1))*(a + b*asinh(c*x))**n/(d*(m + S(1))), x)
rule6092 = ReplacementRule(pattern6092, replacement6092)
pattern6093 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons148, cons66)
def replacement6093(m, b, d, a, n, c, x):
rubi.append(6093)
return -Dist(b*c*n/(d*(m + S(1))), Int((d*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) + Simp((d*x)**(m + S(1))*(a + b*acosh(c*x))**n/(d*(m + S(1))), x)
rule6093 = ReplacementRule(pattern6093, replacement6093)
pattern6094 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons88)
def replacement6094(m, b, a, c, n, x):
rubi.append(6094)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**(m + S(1))*(a + b*asinh(c*x))**n/(m + S(1)), x)
rule6094 = ReplacementRule(pattern6094, replacement6094)
pattern6095 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons88)
def replacement6095(m, b, a, c, n, x):
rubi.append(6095)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) + Simp(x**(m + S(1))*(a + b*acosh(c*x))**n/(m + S(1)), x)
rule6095 = ReplacementRule(pattern6095, replacement6095)
pattern6096 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons1734)
def replacement6096(m, b, a, c, n, x):
rubi.append(6096)
return -Dist(c**(-m + S(-1))/(b*(n + S(1))), Subst(Int(ExpandTrigReduce((a + b*x)**(n + S(1)), (m + (m + S(1))*sinh(x)**S(2))*sinh(x)**(m + S(-1)), x), x), x, asinh(c*x)), x) + Simp(x**m*(a + b*asinh(c*x))**(n + S(1))*sqrt(c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule6096 = ReplacementRule(pattern6096, replacement6096)
pattern6097 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons1734)
def replacement6097(m, b, a, c, n, x):
rubi.append(6097)
return Dist(c**(-m + S(-1))/(b*(n + S(1))), Subst(Int(ExpandTrigReduce((a + b*x)**(n + S(1))*(m - (m + S(1))*cosh(x)**S(2))*cosh(x)**(m + S(-1)), x), x), x, acosh(c*x)), x) + Simp(x**m*(a + b*acosh(c*x))**(n + S(1))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6097 = ReplacementRule(pattern6097, replacement6097)
pattern6098 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons1735)
def replacement6098(m, b, a, c, n, x):
rubi.append(6098)
return -Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*asinh(c*x))**(n + S(1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x) - Dist(c*(m + S(1))/(b*(n + S(1))), Int(x**(m + S(1))*(a + b*asinh(c*x))**(n + S(1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**m*(a + b*asinh(c*x))**(n + S(1))*sqrt(c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule6098 = ReplacementRule(pattern6098, replacement6098)
pattern6099 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons1735)
def replacement6099(m, b, a, c, n, x):
rubi.append(6099)
return Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*acosh(c*x))**(n + S(1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) - Dist(c*(m + S(1))/(b*(n + S(1))), Int(x**(m + S(1))*(a + b*acosh(c*x))**(n + S(1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) + Simp(x**m*(a + b*acosh(c*x))**(n + S(1))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6099 = ReplacementRule(pattern6099, replacement6099)
pattern6100 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons62)
def replacement6100(m, b, a, c, n, x):
rubi.append(6100)
return Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*sinh(x)**m*cosh(x), x), x, asinh(c*x)), x)
rule6100 = ReplacementRule(pattern6100, replacement6100)
pattern6101 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons62)
def replacement6101(m, b, a, c, n, x):
rubi.append(6101)
return Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*sinh(x)*cosh(x)**m, x), x, acosh(c*x)), x)
rule6101 = ReplacementRule(pattern6101, replacement6101)
pattern6102 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1736)
def replacement6102(m, b, d, a, n, c, x):
rubi.append(6102)
return Int((d*x)**m*(a + b*asinh(c*x))**n, x)
rule6102 = ReplacementRule(pattern6102, replacement6102)
pattern6103 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1736)
def replacement6103(m, b, d, a, n, c, x):
rubi.append(6103)
return Int((d*x)**m*(a + b*acosh(c*x))**n, x)
rule6103 = ReplacementRule(pattern6103, replacement6103)
pattern6104 = Pattern(Integral(S(1)/(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons268)
def replacement6104(b, d, a, c, x, e):
rubi.append(6104)
return Simp(log(a + b*asinh(c*x))/(b*c*sqrt(d)), x)
rule6104 = ReplacementRule(pattern6104, replacement6104)
pattern6105 = Pattern(Integral(S(1)/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons1894, cons1895)
def replacement6105(d2, b, e2, a, c, d1, e1, x):
rubi.append(6105)
return Simp(log(a + b*acosh(c*x))/(b*c*sqrt(-d1*d2)), x)
rule6105 = ReplacementRule(pattern6105, replacement6105)
pattern6106 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons268, cons584)
def replacement6106(b, d, a, n, c, x, e):
rubi.append(6106)
return Simp((a + b*asinh(c*x))**(n + S(1))/(b*c*sqrt(d)*(n + S(1))), x)
rule6106 = ReplacementRule(pattern6106, replacement6106)
pattern6107 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons1892, cons1893, cons1894, cons1895, cons584)
def replacement6107(d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6107)
return Simp((a + b*acosh(c*x))**(n + S(1))/(b*c*sqrt(-d1*d2)*(n + S(1))), x)
rule6107 = ReplacementRule(pattern6107, replacement6107)
pattern6108 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons1738)
def replacement6108(b, d, a, n, c, x, e):
rubi.append(6108)
return Dist(sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*asinh(c*x))**n/sqrt(c**S(2)*x**S(2) + S(1)), x), x)
rule6108 = ReplacementRule(pattern6108, replacement6108)
pattern6109 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons1892, cons1893, cons1896)
def replacement6109(d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6109)
return Dist(sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(sqrt(d1 + e1*x)*sqrt(d2 + e2*x)), Int((a + b*acosh(c*x))**n/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x)
rule6109 = ReplacementRule(pattern6109, replacement6109)
def With6110(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(6110)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(c*x), u, x)
pattern6110 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons128)
rule6110 = ReplacementRule(pattern6110, With6110)
def With6111(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(6111)
return -Dist(b*c, Int(SimplifyIntegrand(u/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x), x) + Dist(a + b*acosh(c*x), u, x)
pattern6111 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons128)
rule6111 = ReplacementRule(pattern6111, With6111)
pattern6112 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons88, cons163, cons38)
def replacement6112(p, b, d, a, n, c, x, e):
rubi.append(6112)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*n*(-d)**p/(S(2)*p + S(1)), Int(x*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp(x*(a + b*acosh(c*x))**n*(d + e*x**S(2))**p/(S(2)*p + S(1)), x)
rule6112 = ReplacementRule(pattern6112, replacement6112)
pattern6113 = Pattern(Integral(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement6113(b, d, a, n, c, x, e):
rubi.append(6113)
return Dist(sqrt(d + e*x**S(2))/(S(2)*sqrt(c**S(2)*x**S(2) + S(1))), Int((a + b*asinh(c*x))**n/sqrt(c**S(2)*x**S(2) + S(1)), x), x) - Dist(b*c*n*sqrt(d + e*x**S(2))/(S(2)*sqrt(c**S(2)*x**S(2) + S(1))), Int(x*(a + b*asinh(c*x))**(n + S(-1)), x), x) + Simp(x*(a + b*asinh(c*x))**n*sqrt(d + e*x**S(2))/S(2), x)
rule6113 = ReplacementRule(pattern6113, replacement6113)
pattern6114 = Pattern(Integral(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons87, cons88)
def replacement6114(d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6114)
return -Dist(sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(S(2)*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int((a + b*acosh(c*x))**n/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) - Dist(b*c*n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(S(2)*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int(x*(a + b*acosh(c*x))**(n + S(-1)), x), x) + Simp(x*(a + b*acosh(c*x))**n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/S(2), x)
rule6114 = ReplacementRule(pattern6114, replacement6114)
pattern6115 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons338, cons88, cons163)
def replacement6115(p, b, d, a, n, c, x, e):
rubi.append(6115)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*p + S(1)), Int(x*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp(x*(a + b*asinh(c*x))**n*(d + e*x**S(2))**p/(S(2)*p + S(1)), x)
rule6115 = ReplacementRule(pattern6115, replacement6115)
pattern6116 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons338, cons88, cons163, cons347)
def replacement6116(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6116)
return Dist(S(2)*d1*d2*p/(S(2)*p + S(1)), Int((a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(-1))*(d2 + e2*x)**(p + S(-1)), x), x) - Dist(b*c*n*(-d1*d2)**(p + S(-1)/2)*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/((S(2)*p + S(1))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int(x*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(-1)/2), x), x) + Simp(x*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p/(S(2)*p + S(1)), x)
rule6116 = ReplacementRule(pattern6116, replacement6116)
pattern6117 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons338, cons88, cons163)
def replacement6117(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6117)
return Dist(S(2)*d1*d2*p/(S(2)*p + S(1)), Int((a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(-1))*(d2 + e2*x)**(p + S(-1)), x), x) - Dist(b*c*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(S(2)*p + S(1)), Int(x*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp(x*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p/(S(2)*p + S(1)), x)
rule6117 = ReplacementRule(pattern6117, replacement6117)
pattern6118 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement6118(b, d, a, n, c, x, e):
rubi.append(6118)
return -Dist(b*c*n*sqrt(c**S(2)*x**S(2) + S(1))/(d*sqrt(d + e*x**S(2))), Int(x*(a + b*asinh(c*x))**(n + S(-1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*asinh(c*x))**n/(d*sqrt(d + e*x**S(2))), x)
rule6118 = ReplacementRule(pattern6118, replacement6118)
pattern6119 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/((d1_ + x_*WC('e1', S(1)))**(S(3)/2)*(d2_ + x_*WC('e2', S(1)))**(S(3)/2)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons87, cons88)
def replacement6119(d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6119)
return Dist(b*c*n*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(d1*d2*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)), Int(x*(a + b*acosh(c*x))**(n + S(-1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*acosh(c*x))**n/(d1*d2*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)), x)
rule6119 = ReplacementRule(pattern6119, replacement6119)
pattern6120 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons88, cons137, cons38)
def replacement6120(p, b, d, a, n, c, x, e):
rubi.append(6120)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b*c*n*(-d)**p/(S(2)*p + S(2)), Int(x*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) - Simp(x*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x)
rule6120 = ReplacementRule(pattern6120, replacement6120)
pattern6121 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons338, cons88, cons137, cons230)
def replacement6121(p, b, d, a, n, c, x, e):
rubi.append(6121)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*(p + S(1))), Int(x*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) - Simp(x*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x)
rule6121 = ReplacementRule(pattern6121, replacement6121)
pattern6122 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons338, cons88, cons137, cons230, cons667)
def replacement6122(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6122)
return Dist((S(2)*p + S(3))/(S(2)*d1*d2*(p + S(1))), Int((a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1)), x), x) - Dist(b*c*n*(-d1*d2)**(p + S(1)/2)*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(S(2)*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)*(p + S(1))), Int(x*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(1)/2), x), x) - Simp(x*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(S(2)*d1*d2*(p + S(1))), x)
rule6122 = ReplacementRule(pattern6122, replacement6122)
pattern6123 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons338, cons88, cons137, cons230)
def replacement6123(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6123)
return Dist((S(2)*p + S(3))/(S(2)*d1*d2*(p + S(1))), Int((a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1)), x), x) - Dist(b*c*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(S(2)*(p + S(1))), Int(x*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) - Simp(x*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(S(2)*d1*d2*(p + S(1))), x)
rule6123 = ReplacementRule(pattern6123, replacement6123)
pattern6124 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148)
def replacement6124(b, d, a, n, c, x, e):
rubi.append(6124)
return Dist(S(1)/(c*d), Subst(Int((a + b*x)**n/cosh(x), x), x, asinh(c*x)), x)
rule6124 = ReplacementRule(pattern6124, replacement6124)
pattern6125 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement6125(b, d, a, n, c, x, e):
rubi.append(6125)
return -Dist(S(1)/(c*d), Subst(Int((a + b*x)**n/sinh(x), x), x, acosh(c*x)), x)
rule6125 = ReplacementRule(pattern6125, replacement6125)
pattern6126 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons89, cons38)
def replacement6126(p, b, d, a, c, n, x, e):
rubi.append(6126)
return -Dist(c*(-d)**p*(S(2)*p + S(1))/(b*(n + S(1))), Int(x*(a + b*acosh(c*x))**(n + S(1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((-d)**p*(a + b*acosh(c*x))**(n + S(1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2)/(b*c*(n + S(1))), x)
rule6126 = ReplacementRule(pattern6126, replacement6126)
pattern6127 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1778, cons87, cons89)
def replacement6127(p, b, d, a, c, n, x, e):
rubi.append(6127)
return -Dist(c*d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(S(2)*p + S(1))*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*(n + S(1))), Int(x*(a + b*asinh(c*x))**(n + S(1))*(c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((a + b*asinh(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule6127 = ReplacementRule(pattern6127, replacement6127)
pattern6128 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons5, cons1892, cons1893, cons87, cons89, cons347)
def replacement6128(p, d2, b, e2, a, c, n, d1, e1, x):
rubi.append(6128)
return -Dist(c*(-d1*d2)**(p + S(-1)/2)*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)*(S(2)*p + S(1))/(b*(n + S(1))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int(x*(a + b*acosh(c*x))**(n + S(1))*(c**S(2)*x**S(2) + S(-1))**(p + S(-1)/2), x), x) + Simp((a + b*acosh(c*x))**(n + S(1))*(d1 + e1*x)**p*(d2 + e2*x)**p*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6128 = ReplacementRule(pattern6128, replacement6128)
pattern6129 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons5, cons1892, cons1893, cons87, cons89)
def replacement6129(p, d2, b, e2, a, c, n, d1, e1, x):
rubi.append(6129)
return -Dist(c*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(S(2)*p + S(1))*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(b*(n + S(1))), Int(x*(a + b*acosh(c*x))**(n + S(1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((a + b*acosh(c*x))**(n + S(1))*(d1 + e1*x)**p*(d2 + e2*x)**p*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6129 = ReplacementRule(pattern6129, replacement6129)
pattern6130 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons1739, cons1740)
def replacement6130(p, b, d, a, n, c, x, e):
rubi.append(6130)
return Dist(d**p/c, Subst(Int((a + b*x)**n*cosh(x)**(S(2)*p + S(1)), x), x, asinh(c*x)), x)
rule6130 = ReplacementRule(pattern6130, replacement6130)
pattern6131 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons128)
def replacement6131(p, b, d, a, n, c, x, e):
rubi.append(6131)
return Dist((-d)**p/c, Subst(Int((a + b*x)**n*sinh(x)**(S(2)*p + S(1)), x), x, acosh(c*x)), x)
rule6131 = ReplacementRule(pattern6131, replacement6131)
pattern6132 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons1892, cons1893, cons961, cons1897)
def replacement6132(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6132)
return Dist((-d1*d2)**p/c, Subst(Int((a + b*x)**n*sinh(x)**(S(2)*p + S(1)), x), x, acosh(c*x)), x)
rule6132 = ReplacementRule(pattern6132, replacement6132)
pattern6133 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons1739, cons1741)
def replacement6133(p, b, d, a, n, c, x, e):
rubi.append(6133)
return Dist(d**(p + S(-1)/2)*sqrt(d + e*x**S(2))/sqrt(c**S(2)*x**S(2) + S(1)), Int((a + b*asinh(c*x))**n*(c**S(2)*x**S(2) + S(1))**p, x), x)
rule6133 = ReplacementRule(pattern6133, replacement6133)
pattern6134 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons1892, cons1893, cons1739, cons1896)
def replacement6134(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6134)
return Dist((-d1*d2)**(p + S(-1)/2)*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int((a + b*acosh(c*x))**n*(c*x + S(-1))**p*(c*x + S(1))**p, x), x)
rule6134 = ReplacementRule(pattern6134, replacement6134)
def With6135(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(6135)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(c*x), u, x)
pattern6135 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1898, cons1743)
rule6135 = ReplacementRule(pattern6135, With6135)
def With6136(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(6136)
return -Dist(b*c, Int(SimplifyIntegrand(u/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x), x) + Dist(a + b*acosh(c*x), u, x)
pattern6136 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1742, cons1743)
rule6136 = ReplacementRule(pattern6136, With6136)
pattern6137 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1898, cons38, cons1744)
def replacement6137(p, b, d, a, n, c, x, e):
rubi.append(6137)
return Int(ExpandIntegrand((a + b*asinh(c*x))**n, (d + e*x**S(2))**p, x), x)
rule6137 = ReplacementRule(pattern6137, replacement6137)
pattern6138 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1742, cons38, cons1744)
def replacement6138(p, b, d, a, n, c, x, e):
rubi.append(6138)
return Int(ExpandIntegrand((a + b*acosh(c*x))**n, (d + e*x**S(2))**p, x), x)
rule6138 = ReplacementRule(pattern6138, replacement6138)
pattern6139 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement6139(p, b, d, a, n, c, x, e):
rubi.append(6139)
return Int((a + b*asinh(c*x))**n*(d + e*x**S(2))**p, x)
rule6139 = ReplacementRule(pattern6139, replacement6139)
pattern6140 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons38)
def replacement6140(p, b, d, a, n, c, x, e):
rubi.append(6140)
return Int((a + b*acosh(c*x))**n*(d + e*x**S(2))**p, x)
rule6140 = ReplacementRule(pattern6140, replacement6140)
pattern6141 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons5, cons1899)
def replacement6141(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6141)
return Int((a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, x)
rule6141 = ReplacementRule(pattern6141, replacement6141)
pattern6142 = Pattern(Integral((d_ + x_*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons336, cons1900, cons147)
def replacement6142(p, g, b, f, d, a, n, c, x, e):
rubi.append(6142)
return Dist((d + e*x)**FracPart(p)*(f + g*x)**FracPart(p)*(d*f + e*g*x**S(2))**(-FracPart(p)), Int((a + b*asinh(c*x))**n*(d*f + e*g*x**S(2))**p, x), x)
rule6142 = ReplacementRule(pattern6142, replacement6142)
pattern6143 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1737, cons147)
def replacement6143(p, b, d, a, n, c, x, e):
rubi.append(6143)
return Dist((-d)**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p)), Int((a + b*acosh(c*x))**n*(c*x + S(-1))**p*(c*x + S(1))**p, x), x)
rule6143 = ReplacementRule(pattern6143, replacement6143)
pattern6144 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148)
def replacement6144(b, d, a, n, c, x, e):
rubi.append(6144)
return Dist(S(1)/e, Subst(Int((a + b*x)**n*tanh(x), x), x, asinh(c*x)), x)
rule6144 = ReplacementRule(pattern6144, replacement6144)
pattern6145 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement6145(b, d, a, n, c, x, e):
rubi.append(6145)
return Dist(S(1)/e, Subst(Int((a + b*x)**n/tanh(x), x), x, acosh(c*x)), x)
rule6145 = ReplacementRule(pattern6145, replacement6145)
pattern6146 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons54, cons38)
def replacement6146(p, b, d, a, n, c, x, e):
rubi.append(6146)
return -Dist(b*n*(-d)**p/(S(2)*c*(p + S(1))), Int((a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp((a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6146 = ReplacementRule(pattern6146, replacement6146)
pattern6147 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1778, cons87, cons88, cons54)
def replacement6147(p, b, d, a, n, c, x, e):
rubi.append(6147)
return -Dist(b*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6147 = ReplacementRule(pattern6147, replacement6147)
pattern6148 = Pattern(Integral(x_*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons5, cons1892, cons1893, cons87, cons88, cons54, cons667)
def replacement6148(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6148)
return -Dist(b*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(1)/2), x), x) + Simp((a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(S(2)*e1*e2*(p + S(1))), x)
rule6148 = ReplacementRule(pattern6148, replacement6148)
pattern6149 = Pattern(Integral(x_*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons5, cons1892, cons1893, cons87, cons88, cons54)
def replacement6149(p, d2, b, e2, a, n, c, d1, e1, x):
rubi.append(6149)
return -Dist(b*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp((a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(S(2)*e1*e2*(p + S(1))), x)
rule6149 = ReplacementRule(pattern6149, replacement6149)
pattern6150 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148)
def replacement6150(b, d, a, n, c, x, e):
rubi.append(6150)
return Dist(S(1)/d, Subst(Int((a + b*x)**n/(sinh(x)*cosh(x)), x), x, asinh(c*x)), x)
rule6150 = ReplacementRule(pattern6150, replacement6150)
pattern6151 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement6151(b, d, a, n, c, x, e):
rubi.append(6151)
return -Dist(S(1)/d, Subst(Int((a + b*x)**n/(sinh(x)*cosh(x)), x), x, acosh(c*x)), x)
rule6151 = ReplacementRule(pattern6151, replacement6151)
pattern6152 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1737, cons87, cons88, cons242, cons66, cons38)
def replacement6152(p, m, f, b, d, a, n, c, x, e):
rubi.append(6152)
return Dist(b*c*n*(-d)**p/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*f*(m + S(1))), x)
rule6152 = ReplacementRule(pattern6152, replacement6152)
pattern6153 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1778, cons87, cons88, cons242, cons66)
def replacement6153(p, m, f, b, d, a, n, c, x, e):
rubi.append(6153)
return -Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*f*(m + S(1))), x)
rule6153 = ReplacementRule(pattern6153, replacement6153)
pattern6154 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons5, cons1892, cons1893, cons87, cons88, cons242, cons66, cons667)
def replacement6154(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6154)
return Dist(b*c*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(d1*d2*f*(m + S(1))), x)
rule6154 = ReplacementRule(pattern6154, replacement6154)
pattern6155 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons5, cons1892, cons1893, cons87, cons88, cons242, cons66)
def replacement6155(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6155)
return Dist(b*c*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(d1*d2*f*(m + S(1))), x)
rule6155 = ReplacementRule(pattern6155, replacement6155)
pattern6156 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))/x_, x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons128)
def replacement6156(p, b, d, a, c, x, e):
rubi.append(6156)
return Dist(d, Int((a + b*asinh(c*x))*(d + e*x**S(2))**(p + S(-1))/x, x), x) - Dist(b*c*d**p/(S(2)*p), Int((c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((a + b*asinh(c*x))*(d + e*x**S(2))**p/(S(2)*p), x)
rule6156 = ReplacementRule(pattern6156, replacement6156)
pattern6157 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))/x_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons128)
def replacement6157(p, b, d, a, c, x, e):
rubi.append(6157)
return Dist(d, Int((a + b*acosh(c*x))*(d + e*x**S(2))**(p + S(-1))/x, x), x) - Dist(b*c*(-d)**p/(S(2)*p), Int((c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((a + b*acosh(c*x))*(d + e*x**S(2))**p/(S(2)*p), x)
rule6157 = ReplacementRule(pattern6157, replacement6157)
pattern6158 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1778, cons128, cons1746)
def replacement6158(p, m, f, b, d, a, c, x, e):
rubi.append(6158)
return -Dist(S(2)*e*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*asinh(c*x))*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*d**p/(f*(m + S(1))), Int((f*x)**(m + S(1))*(c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))*(d + e*x**S(2))**p/(f*(m + S(1))), x)
rule6158 = ReplacementRule(pattern6158, replacement6158)
pattern6159 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons128, cons1746)
def replacement6159(p, m, f, b, d, a, c, x, e):
rubi.append(6159)
return -Dist(S(2)*e*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acosh(c*x))*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*(-d)**p/(f*(m + S(1))), Int((f*x)**(m + S(1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))*(d + e*x**S(2))**p/(f*(m + S(1))), x)
rule6159 = ReplacementRule(pattern6159, replacement6159)
def With6160(p, m, f, b, d, a, c, x, e):
u = IntHide((f*x)**m*(d + e*x**S(2))**p, x)
rubi.append(6160)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(c*x), u, x)
pattern6160 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1778, cons128)
rule6160 = ReplacementRule(pattern6160, With6160)
def With6161(p, m, f, b, d, a, c, x, e):
u = IntHide((f*x)**m*(d + e*x**S(2))**p, x)
rubi.append(6161)
return -Dist(b*c, Int(SimplifyIntegrand(u/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x), x) + Dist(a + b*acosh(c*x), u, x)
pattern6161 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons128)
rule6161 = ReplacementRule(pattern6161, With6161)
def With6162(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(c**S(2)*x**S(2) + S(1))**p, x)
rubi.append(6162)
return Dist(d**p*(a + b*asinh(c*x)), u, x) - Dist(b*c*d**p, Int(SimplifyIntegrand(u/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x)
pattern6162 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons347, cons1747, cons486, cons268)
rule6162 = ReplacementRule(pattern6162, With6162)
def With6163(p, d2, m, b, e2, a, c, d1, e1, x):
u = IntHide(x**m*(c*x + S(-1))**p*(c*x + S(1))**p, x)
rubi.append(6163)
return Dist((-d1*d2)**p*(a + b*acosh(c*x)), u, x) - Dist(b*c*(-d1*d2)**p, Int(SimplifyIntegrand(u/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x), x)
pattern6163 = Pattern(Integral(x_**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons347, cons1747, cons486, cons1894, cons1895)
rule6163 = ReplacementRule(pattern6163, With6163)
def With6164(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(c**S(2)*x**S(2) + S(1))**p, x)
rubi.append(6164)
return -Dist(b*c*d**(p + S(-1)/2)*sqrt(d + e*x**S(2))/sqrt(c**S(2)*x**S(2) + S(1)), Int(SimplifyIntegrand(u/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(c*x), Int(x**m*(d + e*x**S(2))**p, x), x)
pattern6164 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons961, cons1747)
rule6164 = ReplacementRule(pattern6164, With6164)
def With6165(p, d2, m, b, e2, a, c, d1, e1, x):
u = IntHide(x**m*(c*x + S(-1))**p*(c*x + S(1))**p, x)
rubi.append(6165)
return -Dist(b*c*(-d1*d2)**(p + S(-1)/2)*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int(SimplifyIntegrand(u/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x), x) + Dist(a + b*acosh(c*x), Int(x**m*(d1 + e1*x)**p*(d2 + e2*x)**p, x), x)
pattern6165 = Pattern(Integral(x_**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons961, cons1747)
rule6165 = ReplacementRule(pattern6165, With6165)
pattern6166 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons162, cons88, cons163, cons94, cons38)
def replacement6166(p, m, f, b, d, a, n, c, x, e):
rubi.append(6166)
return -Dist(S(2)*e*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*n*(-d)**p/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**p/(f*(m + S(1))), x)
rule6166 = ReplacementRule(pattern6166, replacement6166)
pattern6167 = Pattern(Integral((x_*WC('f', S(1)))**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1778, cons93, cons88, cons94)
def replacement6167(m, f, b, d, a, n, c, x, e):
rubi.append(6167)
return -Dist(c**S(2)*sqrt(d + e*x**S(2))/(f**S(2)*(m + S(1))*sqrt(c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(2))*(a + b*asinh(c*x))**n/sqrt(c**S(2)*x**S(2) + S(1)), x), x) - Dist(b*c*n*sqrt(d + e*x**S(2))/(f*(m + S(1))*sqrt(c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1)), x), x) + Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))**n*sqrt(d + e*x**S(2))/(f*(m + S(1))), x)
rule6167 = ReplacementRule(pattern6167, replacement6167)
pattern6168 = Pattern(Integral((x_*WC('f', S(1)))**m_*sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons1892, cons1893, cons93, cons88, cons94)
def replacement6168(d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6168)
return -Dist(c**S(2)*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(f**S(2)*(m + S(1))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int((f*x)**(m + S(2))*(a + b*acosh(c*x))**n/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) - Dist(b*c*n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(f*(m + S(1))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1)), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(f*(m + S(1))), x)
rule6168 = ReplacementRule(pattern6168, replacement6168)
pattern6169 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1778, cons162, cons88, cons163, cons94)
def replacement6169(p, m, f, b, d, a, n, c, x, e):
rubi.append(6169)
return -Dist(S(2)*e*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**p/(f*(m + S(1))), x)
rule6169 = ReplacementRule(pattern6169, replacement6169)
pattern6170 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons1892, cons1893, cons162, cons88, cons163, cons94, cons347)
def replacement6170(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6170)
return -Dist(S(2)*e1*e2*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(-1))*(d2 + e2*x)**(p + S(-1)), x), x) - Dist(b*c*n*(-d1*d2)**(p + S(-1)/2)*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(f*(m + S(1))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p/(f*(m + S(1))), x)
rule6170 = ReplacementRule(pattern6170, replacement6170)
pattern6171 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons338, cons88, cons163, cons272, cons38, cons1748)
def replacement6171(p, m, f, b, d, a, n, c, x, e):
rubi.append(6171)
return Dist(S(2)*d*p/(m + S(2)*p + S(1)), Int((f*x)**m*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*n*(-d)**p/(f*(m + S(2)*p + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**p/(f*(m + S(2)*p + S(1))), x)
rule6171 = ReplacementRule(pattern6171, replacement6171)
pattern6172 = Pattern(Integral((x_*WC('f', S(1)))**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1778, cons87, cons88, cons272, cons1748)
def replacement6172(m, f, b, d, a, n, c, x, e):
rubi.append(6172)
return Dist(sqrt(d + e*x**S(2))/((m + S(2))*sqrt(c**S(2)*x**S(2) + S(1))), Int((f*x)**m*(a + b*asinh(c*x))**n/sqrt(c**S(2)*x**S(2) + S(1)), x), x) - Dist(b*c*n*sqrt(d + e*x**S(2))/(f*(m + S(2))*sqrt(c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1)), x), x) + Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))**n*sqrt(d + e*x**S(2))/(f*(m + S(2))), x)
rule6172 = ReplacementRule(pattern6172, replacement6172)
pattern6173 = Pattern(Integral((x_*WC('f', S(1)))**m_*sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons1892, cons1893, cons87, cons88, cons272, cons1748)
def replacement6173(d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6173)
return -Dist(sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/((m + S(2))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int((f*x)**m*(a + b*acosh(c*x))**n/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) - Dist(b*c*n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(f*(m + S(2))*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1)), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(f*(m + S(2))), x)
rule6173 = ReplacementRule(pattern6173, replacement6173)
pattern6174 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1778, cons338, cons88, cons163, cons272, cons1748)
def replacement6174(p, m, f, b, d, a, n, c, x, e):
rubi.append(6174)
return Dist(S(2)*d*p/(m + S(2)*p + S(1)), Int((f*x)**m*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(2)*p + S(1))), Int((f*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**p/(f*(m + S(2)*p + S(1))), x)
rule6174 = ReplacementRule(pattern6174, replacement6174)
pattern6175 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons1892, cons1893, cons338, cons88, cons163, cons272, cons347, cons1748)
def replacement6175(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6175)
return Dist(S(2)*d1*d2*p/(m + S(2)*p + S(1)), Int((f*x)**m*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(-1))*(d2 + e2*x)**(p + S(-1)), x), x) - Dist(b*c*n*(-d1*d2)**(p + S(-1)/2)*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(f*sqrt(c*x + S(-1))*sqrt(c*x + S(1))*(m + S(2)*p + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p/(f*(m + S(2)*p + S(1))), x)
rule6175 = ReplacementRule(pattern6175, replacement6175)
pattern6176 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons1737, cons93, cons88, cons94, cons17, cons38)
def replacement6176(p, m, f, b, d, a, n, c, x, e):
rubi.append(6176)
return Dist(c**S(2)*(m + S(2)*p + S(3))/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**p, x), x) + Dist(b*c*n*(-d)**p/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*f*(m + S(1))), x)
rule6176 = ReplacementRule(pattern6176, replacement6176)
pattern6177 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons1778, cons93, cons88, cons94, cons17)
def replacement6177(p, m, f, b, d, a, n, c, x, e):
rubi.append(6177)
return -Dist(c**S(2)*(m + S(2)*p + S(3))/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**p, x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*f*(m + S(1))), x)
rule6177 = ReplacementRule(pattern6177, replacement6177)
pattern6178 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons5, cons1892, cons1893, cons93, cons88, cons94, cons17, cons667)
def replacement6178(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6178)
return Dist(c**S(2)*(m + S(2)*p + S(3))/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, x), x) + Dist(b*c*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(d1*d2*f*(m + S(1))), x)
rule6178 = ReplacementRule(pattern6178, replacement6178)
pattern6179 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons5, cons1892, cons1893, cons93, cons88, cons94, cons17)
def replacement6179(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6179)
return Dist(c**S(2)*(m + S(2)*p + S(3))/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, x), x) + Dist(b*c*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(d1*d2*f*(m + S(1))), x)
rule6179 = ReplacementRule(pattern6179, replacement6179)
pattern6180 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons93, cons88, cons137, cons166, cons38)
def replacement6180(p, m, f, b, d, a, n, c, x, e):
rubi.append(6180)
return -Dist(f**S(2)*(m + S(-1))/(S(2)*e*(p + S(1))), Int((f*x)**(m + S(-2))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b*f*n*(-d)**p/(S(2)*c*(p + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6180 = ReplacementRule(pattern6180, replacement6180)
pattern6181 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1778, cons162, cons88, cons137, cons166)
def replacement6181(p, m, f, b, d, a, n, c, x, e):
rubi.append(6181)
return -Dist(f**S(2)*(m + S(-1))/(S(2)*e*(p + S(1))), Int((f*x)**(m + S(-2))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b*d**IntPart(p)*f*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((f*x)**(m + S(-1))*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6181 = ReplacementRule(pattern6181, replacement6181)
pattern6182 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons1892, cons1893, cons162, cons88, cons137, cons166, cons667)
def replacement6182(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6182)
return -Dist(f**S(2)*(m + S(-1))/(S(2)*e1*e2*(p + S(1))), Int((f*x)**(m + S(-2))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1)), x), x) - Dist(b*f*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(S(2)*e1*e2*(p + S(1))), x)
rule6182 = ReplacementRule(pattern6182, replacement6182)
pattern6183 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons1892, cons1893, cons162, cons88, cons137, cons147, cons166)
def replacement6183(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6183)
return -Dist(f**S(2)*(m + S(-1))/(S(2)*e1*e2*(p + S(1))), Int((f*x)**(m + S(-2))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1)), x), x) - Dist(b*f*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(S(2)*e1*e2*(p + S(1))), x)
rule6183 = ReplacementRule(pattern6183, replacement6183)
pattern6184 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons338, cons88, cons137, cons274, cons38)
def replacement6184(p, m, f, b, d, a, n, c, x, e):
rubi.append(6184)
return Dist((m + S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((f*x)**m*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b*c*n*(-d)**p/(S(2)*f*(p + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) - Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*f*(p + S(1))), x)
rule6184 = ReplacementRule(pattern6184, replacement6184)
pattern6185 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1778, cons338, cons88, cons137, cons274, cons1749)
def replacement6185(p, m, f, b, d, a, n, c, x, e):
rubi.append(6185)
return Dist((m + S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((f*x)**m*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*f*(p + S(1))), Int((f*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) - Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*f*(p + S(1))), x)
rule6185 = ReplacementRule(pattern6185, replacement6185)
pattern6186 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons1892, cons1893, cons338, cons88, cons137, cons274, cons1750, cons667)
def replacement6186(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6186)
return Dist((m + S(2)*p + S(3))/(S(2)*d1*d2*(p + S(1))), Int((f*x)**m*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1)), x), x) - Dist(b*c*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(S(2)*f*(p + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(1)/2), x), x) - Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(S(2)*d1*d2*f*(p + S(1))), x)
rule6186 = ReplacementRule(pattern6186, replacement6186)
pattern6187 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons1892, cons1893, cons338, cons88, cons137, cons274, cons1749)
def replacement6187(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6187)
return Dist((m + S(2)*p + S(3))/(S(2)*d1*d2*(p + S(1))), Int((f*x)**m*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1)), x), x) - Dist(b*c*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(S(2)*f*(p + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) - Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(S(2)*d1*d2*f*(p + S(1))), x)
rule6187 = ReplacementRule(pattern6187, replacement6187)
pattern6188 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1778, cons93, cons88, cons166, cons17)
def replacement6188(m, f, b, d, a, n, c, x, e):
rubi.append(6188)
return -Dist(f**S(2)*(m + S(-1))/(c**S(2)*m), Int((f*x)**(m + S(-2))*(a + b*asinh(c*x))**n/sqrt(d + e*x**S(2)), x), x) - Dist(b*f*n*sqrt(c**S(2)*x**S(2) + S(1))/(c*m*sqrt(d + e*x**S(2))), Int((f*x)**(m + S(-1))*(a + b*asinh(c*x))**(n + S(-1)), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*asinh(c*x))**n*sqrt(d + e*x**S(2))/(e*m), x)
rule6188 = ReplacementRule(pattern6188, replacement6188)
pattern6189 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons1892, cons1893, cons93, cons88, cons166, cons17)
def replacement6189(d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6189)
return Dist(f**S(2)*(m + S(-1))/(c**S(2)*m), Int((f*x)**(m + S(-2))*(a + b*acosh(c*x))**n/(sqrt(d1 + e1*x)*sqrt(d2 + e2*x)), x), x) + Dist(b*f*n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(c*d1*d2*m*sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(-1)), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acosh(c*x))**n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)/(e1*e2*m), x)
rule6189 = ReplacementRule(pattern6189, replacement6189)
pattern6190 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons268, cons148, cons17)
def replacement6190(m, b, d, a, n, c, x, e):
rubi.append(6190)
return Dist(c**(-m + S(-1))/sqrt(d), Subst(Int((a + b*x)**n*sinh(x)**m, x), x, asinh(c*x)), x)
rule6190 = ReplacementRule(pattern6190, replacement6190)
pattern6191 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1892, cons1893, cons148, cons1894, cons1895, cons17)
def replacement6191(d2, m, b, e2, a, n, c, d1, e1, x):
rubi.append(6191)
return Dist(c**(-m + S(-1))/sqrt(-d1*d2), Subst(Int((a + b*x)**n*cosh(x)**m, x), x, acosh(c*x)), x)
rule6191 = ReplacementRule(pattern6191, replacement6191)
pattern6192 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1778, cons268, cons18)
def replacement6192(m, f, b, d, a, c, x, e):
rubi.append(6192)
return Simp((f*x)**(m + S(1))*(a + b*asinh(c*x))*Hypergeometric2F1(S(1)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, -c**S(2)*x**S(2))/(sqrt(d)*f*(m + S(1))), x) - Simp(b*c*(f*x)**(m + S(2))*HypergeometricPFQ(List(S(1), m/S(2) + S(1), m/S(2) + S(1)), List(m/S(2) + S(3)/2, m/S(2) + S(2)), -c**S(2)*x**S(2))/(sqrt(d)*f**S(2)*(m + S(1))*(m + S(2))), x)
rule6192 = ReplacementRule(pattern6192, replacement6192)
pattern6193 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons1892, cons1893, cons1894, cons1895, cons18)
def replacement6193(d2, m, f, b, e2, a, c, d1, e1, x):
rubi.append(6193)
return Simp(b*c*(f*x)**(m + S(2))*HypergeometricPFQ(List(S(1), m/S(2) + S(1), m/S(2) + S(1)), List(m/S(2) + S(3)/2, m/S(2) + S(2)), c**S(2)*x**S(2))/(f**S(2)*sqrt(-d1*d2)*(m + S(1))*(m + S(2))), x) + Simp((f*x)**(m + S(1))*(a + b*acosh(c*x))*sqrt(-c**S(2)*x**S(2) + S(1))*Hypergeometric2F1(S(1)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, c**S(2)*x**S(2))/(f*sqrt(d1 + e1*x)*sqrt(d2 + e2*x)*(m + S(1))), x)
rule6193 = ReplacementRule(pattern6193, replacement6193)
pattern6194 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1778, cons87, cons88, cons1738, cons1750)
def replacement6194(m, f, b, d, a, n, c, x, e):
rubi.append(6194)
return Dist(sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((f*x)**m*(a + b*asinh(c*x))**n/sqrt(c**S(2)*x**S(2) + S(1)), x), x)
rule6194 = ReplacementRule(pattern6194, replacement6194)
pattern6195 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons1892, cons1893, cons87, cons88, cons1896, cons1750)
def replacement6195(d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6195)
return Dist(sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(sqrt(d1 + e1*x)*sqrt(d2 + e2*x)), Int((f*x)**m*(a + b*acosh(c*x))**n/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x)
rule6195 = ReplacementRule(pattern6195, replacement6195)
pattern6196 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons1737, cons93, cons88, cons166, cons238, cons38, cons17)
def replacement6196(p, m, f, b, d, a, n, c, x, e):
rubi.append(6196)
return Dist(f**S(2)*(m + S(-1))/(c**S(2)*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-2))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**p, x), x) - Dist(b*f*n*(-d)**p/(c*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acosh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(e*(m + S(2)*p + S(1))), x)
rule6196 = ReplacementRule(pattern6196, replacement6196)
pattern6197 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons1778, cons93, cons88, cons166, cons238, cons17)
def replacement6197(p, m, f, b, d, a, n, c, x, e):
rubi.append(6197)
return -Dist(f**S(2)*(m + S(-1))/(c**S(2)*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-2))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**p, x), x) - Dist(b*d**IntPart(p)*f*n*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(c*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-1))*(a + b*asinh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*asinh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(e*(m + S(2)*p + S(1))), x)
rule6197 = ReplacementRule(pattern6197, replacement6197)
pattern6198 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons5, cons1892, cons1893, cons93, cons88, cons166, cons238, cons17, cons667)
def replacement6198(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6198)
return Dist(f**S(2)*(m + S(-1))/(c**S(2)*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-2))*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, x), x) - Dist(b*f*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(c*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(-1))*(c**S(2)*x**S(2) + S(-1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(e1*e2*(m + S(2)*p + S(1))), x)
rule6198 = ReplacementRule(pattern6198, replacement6198)
pattern6199 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons5, cons1892, cons1893, cons93, cons88, cons166, cons238, cons17)
def replacement6199(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6199)
return Dist(f**S(2)*(m + S(-1))/(c**S(2)*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-2))*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, x), x) - Dist(b*f*n*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(c*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(-1))*(c*x + S(-1))**(p + S(1)/2)*(c*x + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acosh(c*x))**n*(d1 + e1*x)**(p + S(1))*(d2 + e2*x)**(p + S(1))/(e1*e2*(m + S(2)*p + S(1))), x)
rule6199 = ReplacementRule(pattern6199, replacement6199)
pattern6200 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1737, cons87, cons89, cons237, cons38)
def replacement6200(p, m, f, b, d, a, c, n, x, e):
rubi.append(6200)
return Dist(f*m*(-d)**p/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*acosh(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6200 = ReplacementRule(pattern6200, replacement6200)
pattern6201 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1778, cons87, cons89, cons237)
def replacement6201(p, m, f, b, d, a, c, n, x, e):
rubi.append(6201)
return -Dist(d**IntPart(p)*f*m*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*asinh(c*x))**(n + S(1))*(c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*asinh(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule6201 = ReplacementRule(pattern6201, replacement6201)
pattern6202 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons5, cons1892, cons1893, cons87, cons89, cons237, cons347)
def replacement6202(p, d2, m, f, b, e2, a, c, n, d1, e1, x):
rubi.append(6202)
return Dist(f*m*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(1))*(c**S(2)*x**S(2) + S(-1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*acosh(c*x))**(n + S(1))*(d1 + e1*x)**p*(d2 + e2*x)**p*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6202 = ReplacementRule(pattern6202, replacement6202)
pattern6203 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons5, cons1892, cons1893, cons87, cons89, cons237)
def replacement6203(p, d2, m, f, b, e2, a, c, n, d1, e1, x):
rubi.append(6203)
return Dist(f*m*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*acosh(c*x))**(n + S(1))*(d1 + e1*x)**p*(d2 + e2*x)**p*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6203 = ReplacementRule(pattern6203, replacement6203)
pattern6204 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1778, cons87, cons89, cons268)
def replacement6204(m, f, b, d, a, c, n, x, e):
rubi.append(6204)
return -Dist(f*m/(b*c*sqrt(d)*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*asinh(c*x))**(n + S(1)), x), x) + Simp((f*x)**m*(a + b*asinh(c*x))**(n + S(1))/(b*c*sqrt(d)*(n + S(1))), x)
rule6204 = ReplacementRule(pattern6204, replacement6204)
pattern6205 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons1892, cons1893, cons87, cons89, cons1894, cons1895)
def replacement6205(d2, m, f, b, e2, a, c, n, d1, e1, x):
rubi.append(6205)
return -Dist(f*m/(b*c*sqrt(-d1*d2)*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(1)), x), x) + Simp((f*x)**m*(a + b*acosh(c*x))**(n + S(1))/(b*c*sqrt(-d1*d2)*(n + S(1))), x)
rule6205 = ReplacementRule(pattern6205, replacement6205)
pattern6206 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1778, cons87, cons89, cons1738)
def replacement6206(m, f, b, d, a, c, n, x, e):
rubi.append(6206)
return Dist(sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((f*x)**m*(a + b*asinh(c*x))**n/sqrt(c**S(2)*x**S(2) + S(1)), x), x)
rule6206 = ReplacementRule(pattern6206, replacement6206)
pattern6207 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons1892, cons1893, cons87, cons89, cons1896)
def replacement6207(d2, m, f, b, e2, a, c, n, d1, e1, x):
rubi.append(6207)
return Dist(sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(sqrt(d1 + e1*x)*sqrt(d2 + e2*x)), Int((f*x)**m*(a + b*acosh(c*x))**n/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x)
rule6207 = ReplacementRule(pattern6207, replacement6207)
pattern6208 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons87, cons89, cons17, cons1751, cons128)
def replacement6208(p, m, f, b, d, a, c, n, x, e):
rubi.append(6208)
return Dist(f*m*(-d)**p/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) - Dist(c*(-d)**p*(m + S(2)*p + S(1))/(b*f*(n + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(1))*(c*x + S(-1))**(p + S(-1)/2)*(c*x + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*acosh(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6208 = ReplacementRule(pattern6208, replacement6208)
pattern6209 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1778, cons87, cons89, cons17, cons1751, cons1739)
def replacement6209(p, m, f, b, d, a, c, n, x, e):
rubi.append(6209)
return -Dist(d**IntPart(p)*f*m*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*asinh(c*x))**(n + S(1))*(c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) - Dist(c*d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p))*(m + S(2)*p + S(1))/(b*f*(n + S(1))), Int((f*x)**(m + S(1))*(a + b*asinh(c*x))**(n + S(1))*(c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*asinh(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule6209 = ReplacementRule(pattern6209, replacement6209)
pattern6210 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons1892, cons1893, cons87, cons89, cons17, cons1751, cons961)
def replacement6210(p, d2, m, f, b, e2, a, c, n, d1, e1, x):
rubi.append(6210)
return Dist(f*m*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acosh(c*x))**(n + S(1))*(c**S(2)*x**S(2) + S(-1))**(p + S(-1)/2), x), x) - Dist(c*(-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p))*(m + S(2)*p + S(1))/(b*f*(n + S(1))), Int((f*x)**(m + S(1))*(a + b*acosh(c*x))**(n + S(1))*(c**S(2)*x**S(2) + S(-1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*acosh(c*x))**(n + S(1))*(d1 + e1*x)**p*(d2 + e2*x)**p*sqrt(c*x + S(-1))*sqrt(c*x + S(1))/(b*c*(n + S(1))), x)
rule6210 = ReplacementRule(pattern6210, replacement6210)
pattern6211 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons246, cons1752, cons62, cons1740)
def replacement6211(p, m, b, d, a, n, c, x, e):
rubi.append(6211)
return Dist(c**(-m + S(-1))*d**p, Subst(Int((a + b*x)**n*sinh(x)**m*cosh(x)**(S(2)*p + S(1)), x), x, asinh(c*x)), x)
rule6211 = ReplacementRule(pattern6211, replacement6211)
pattern6212 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons128, cons62)
def replacement6212(p, m, b, d, a, n, c, x, e):
rubi.append(6212)
return Dist(c**(-m + S(-1))*(-d)**p, Subst(Int((a + b*x)**n*sinh(x)**(S(2)*p + S(1))*cosh(x)**m, x), x, acosh(c*x)), x)
rule6212 = ReplacementRule(pattern6212, replacement6212)
pattern6213 = Pattern(Integral(x_**WC('m', S(1))*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons1892, cons1893, cons667, cons1752, cons62, cons1897)
def replacement6213(p, d2, m, b, e2, a, n, c, d1, e1, x):
rubi.append(6213)
return Dist(c**(-m + S(-1))*(-d1*d2)**p, Subst(Int((a + b*x)**n*sinh(x)**(S(2)*p + S(1))*cosh(x)**m, x), x, acosh(c*x)), x)
rule6213 = ReplacementRule(pattern6213, replacement6213)
pattern6214 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons246, cons1752, cons62, cons1741)
def replacement6214(p, m, b, d, a, c, n, x, e):
rubi.append(6214)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int(x**m*(a + b*asinh(c*x))**n*(c**S(2)*x**S(2) + S(1))**p, x), x)
rule6214 = ReplacementRule(pattern6214, replacement6214)
pattern6215 = Pattern(Integral(x_**WC('m', S(1))*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons1892, cons1893, cons246, cons1752, cons62, cons1901)
def replacement6215(p, d2, m, b, e2, a, n, c, d1, e1, x):
rubi.append(6215)
return Dist((-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p)), Int(x**m*(a + b*acosh(c*x))**n*(c*x + S(-1))**p*(c*x + S(1))**p, x), x)
rule6215 = ReplacementRule(pattern6215, replacement6215)
pattern6216 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1778, cons268, cons961, cons1753, cons17, cons1754)
def replacement6216(p, m, f, b, d, a, n, c, x, e):
rubi.append(6216)
return Int(ExpandIntegrand((a + b*asinh(c*x))**n/sqrt(d + e*x**S(2)), (f*x)**m*(d + e*x**S(2))**(p + S(1)/2), x), x)
rule6216 = ReplacementRule(pattern6216, replacement6216)
pattern6217 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons4, cons1892, cons1893, cons1894, cons1895, cons961, cons1753, cons17, cons1754)
def replacement6217(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6217)
return Int(ExpandIntegrand((a + b*acosh(c*x))**n/(sqrt(d1 + e1*x)*sqrt(d2 + e2*x)), (f*x)**m*(d1 + e1*x)**(p + S(1)/2)*(d2 + e2*x)**(p + S(1)/2), x), x)
rule6217 = ReplacementRule(pattern6217, replacement6217)
pattern6218 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1742, cons66, cons1902)
def replacement6218(m, f, b, d, a, c, x, e):
rubi.append(6218)
return -Dist(b*c/(f*(m + S(1))*(m + S(3))), Int((f*x)**(m + S(1))*(d*(m + S(3)) + e*x**S(2)*(m + S(1)))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) + Simp(d*(f*x)**(m + S(1))*(a + b*acosh(c*x))/(f*(m + S(1))), x) + Simp(e*(f*x)**(m + S(3))*(a + b*acosh(c*x))/(f**S(3)*(m + S(3))), x)
rule6218 = ReplacementRule(pattern6218, replacement6218)
pattern6219 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1898, cons54)
def replacement6219(p, b, d, a, c, x, e):
rubi.append(6219)
return -Dist(b*c/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*asinh(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6219 = ReplacementRule(pattern6219, replacement6219)
pattern6220 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1742, cons54)
def replacement6220(p, b, d, a, c, x, e):
rubi.append(6220)
return -Dist(b*c/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) + Simp((a + b*acosh(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6220 = ReplacementRule(pattern6220, replacement6220)
def With6221(p, m, f, b, d, a, c, x, e):
u = IntHide((f*x)**m*(d + e*x**S(2))**p, x)
rubi.append(6221)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(c*x), u, x)
pattern6221 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1898, cons38, cons1755)
rule6221 = ReplacementRule(pattern6221, With6221)
def With6222(p, m, f, b, d, a, c, x, e):
u = IntHide((f*x)**m*(d + e*x**S(2))**p, x)
rubi.append(6222)
return -Dist(b*c, Int(SimplifyIntegrand(u/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x), x) + Dist(a + b*acosh(c*x), u, x)
pattern6222 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1742, cons38, cons1755)
rule6222 = ReplacementRule(pattern6222, With6222)
pattern6223 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1898, cons148, cons38, cons17)
def replacement6223(p, m, f, b, d, a, n, c, x, e):
rubi.append(6223)
return Int(ExpandIntegrand((a + b*asinh(c*x))**n, (f*x)**m*(d + e*x**S(2))**p, x), x)
rule6223 = ReplacementRule(pattern6223, replacement6223)
pattern6224 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1742, cons148, cons38, cons17)
def replacement6224(p, m, f, b, d, a, n, c, x, e):
rubi.append(6224)
return Int(ExpandIntegrand((a + b*acosh(c*x))**n, (f*x)**m*(d + e*x**S(2))**p, x), x)
rule6224 = ReplacementRule(pattern6224, replacement6224)
pattern6225 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons1756)
def replacement6225(p, m, f, b, d, a, n, c, x, e):
rubi.append(6225)
return Int((f*x)**m*(a + b*asinh(c*x))**n*(d + e*x**S(2))**p, x)
rule6225 = ReplacementRule(pattern6225, replacement6225)
pattern6226 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons38)
def replacement6226(p, m, f, b, d, a, n, c, x, e):
rubi.append(6226)
return Int((f*x)**m*(a + b*acosh(c*x))**n*(d + e*x**S(2))**p, x)
rule6226 = ReplacementRule(pattern6226, replacement6226)
pattern6227 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d1_ + x_*WC('e1', S(1)))**WC('p', S(1))*(d2_ + x_*WC('e2', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons21, cons4, cons5, cons1903)
def replacement6227(p, d2, m, f, b, e2, a, n, c, d1, e1, x):
rubi.append(6227)
return Int((f*x)**m*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, x)
rule6227 = ReplacementRule(pattern6227, replacement6227)
pattern6228 = Pattern(Integral((x_*WC('h', S(1)))**WC('m', S(1))*(d_ + x_*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons336, cons1900, cons147)
def replacement6228(p, m, g, b, f, d, a, n, c, x, h, e):
rubi.append(6228)
return Dist((d + e*x)**FracPart(p)*(f + g*x)**FracPart(p)*(d*f + e*g*x**S(2))**(-FracPart(p)), Int((h*x)**m*(a + b*asinh(c*x))**n*(d*f + e*g*x**S(2))**p, x), x)
rule6228 = ReplacementRule(pattern6228, replacement6228)
pattern6229 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons1737, cons147)
def replacement6229(p, m, f, b, d, a, n, c, x, e):
rubi.append(6229)
return Dist((-d)**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p)), Int((f*x)**m*(a + b*acosh(c*x))**n*(c*x + S(-1))**p*(c*x + S(1))**p, x), x)
rule6229 = ReplacementRule(pattern6229, replacement6229)
pattern6230 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons148)
def replacement6230(b, d, a, n, c, x, e):
rubi.append(6230)
return Subst(Int((a + b*x)**n*cosh(x)/(c*d + e*sinh(x)), x), x, asinh(c*x))
rule6230 = ReplacementRule(pattern6230, replacement6230)
pattern6231 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons148)
def replacement6231(b, d, a, n, c, x, e):
rubi.append(6231)
return Subst(Int((a + b*x)**n*sinh(x)/(c*d + e*cosh(x)), x), x, acosh(c*x))
rule6231 = ReplacementRule(pattern6231, replacement6231)
pattern6232 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons148, cons66)
def replacement6232(m, b, d, a, n, c, x, e):
rubi.append(6232)
return -Dist(b*c*n/(e*(m + S(1))), Int((a + b*asinh(c*x))**(n + S(-1))*(d + e*x)**(m + S(1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*asinh(c*x))**n*(d + e*x)**(m + S(1))/(e*(m + S(1))), x)
rule6232 = ReplacementRule(pattern6232, replacement6232)
pattern6233 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons148, cons66)
def replacement6233(m, b, d, a, n, c, x, e):
rubi.append(6233)
return -Dist(b*c*n/(e*(m + S(1))), Int((a + b*acosh(c*x))**(n + S(-1))*(d + e*x)**(m + S(1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x) + Simp((a + b*acosh(c*x))**n*(d + e*x)**(m + S(1))/(e*(m + S(1))), x)
rule6233 = ReplacementRule(pattern6233, replacement6233)
pattern6234 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons62, cons87, cons89)
def replacement6234(m, b, d, a, c, n, x, e):
rubi.append(6234)
return Int(ExpandIntegrand((a + b*asinh(c*x))**n*(d + e*x)**m, x), x)
rule6234 = ReplacementRule(pattern6234, replacement6234)
pattern6235 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons62, cons87, cons89)
def replacement6235(m, b, d, a, c, n, x, e):
rubi.append(6235)
return Int(ExpandIntegrand((a + b*acosh(c*x))**n*(d + e*x)**m, x), x)
rule6235 = ReplacementRule(pattern6235, replacement6235)
pattern6236 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons62)
def replacement6236(m, b, d, a, n, c, x, e):
rubi.append(6236)
return Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*(c*d + e*sinh(x))**m*cosh(x), x), x, asinh(c*x)), x)
rule6236 = ReplacementRule(pattern6236, replacement6236)
pattern6237 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons62)
def replacement6237(m, b, d, a, n, c, x, e):
rubi.append(6237)
return Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*(c*d + e*cosh(x))**m*sinh(x), x), x, acosh(c*x)), x)
rule6237 = ReplacementRule(pattern6237, replacement6237)
def With6238(b, Px, a, c, x):
u = IntHide(Px, x)
rubi.append(6238)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(c*x), u, x)
pattern6238 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons925)
rule6238 = ReplacementRule(pattern6238, With6238)
def With6239(b, Px, a, c, x):
u = IntHide(Px, x)
rubi.append(6239)
return -Dist(b*c*sqrt(-c**S(2)*x**S(2) + S(1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acosh(c*x), u, x)
pattern6239 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons925)
rule6239 = ReplacementRule(pattern6239, With6239)
pattern6240 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons925)
def replacement6240(b, Px, a, n, c, x):
rubi.append(6240)
return Int(ExpandIntegrand(Px*(a + b*asinh(c*x))**n, x), x)
rule6240 = ReplacementRule(pattern6240, replacement6240)
pattern6241 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons925)
def replacement6241(b, Px, a, n, c, x):
rubi.append(6241)
return Int(ExpandIntegrand(Px*(a + b*acosh(c*x))**n, x), x)
rule6241 = ReplacementRule(pattern6241, replacement6241)
def With6242(m, b, Px, d, a, c, x, e):
u = IntHide(Px*(d + e*x)**m, x)
rubi.append(6242)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(c*x), u, x)
pattern6242 = Pattern(Integral(Px_*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons925)
rule6242 = ReplacementRule(pattern6242, With6242)
def With6243(m, b, Px, d, a, c, x, e):
u = IntHide(Px*(d + e*x)**m, x)
rubi.append(6243)
return -Dist(b*c*sqrt(-c**S(2)*x**S(2) + S(1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acosh(c*x), u, x)
pattern6243 = Pattern(Integral(Px_*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons925)
rule6243 = ReplacementRule(pattern6243, With6243)
def With6244(p, m, f, b, g, d, a, c, n, x, e):
u = IntHide((d + e*x)**m*(f + g*x)**p, x)
rubi.append(6244)
return -Dist(b*c*n, Int(SimplifyIntegrand(u*(a + b*asinh(c*x))**(n + S(-1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist((a + b*asinh(c*x))**n, u, x)
pattern6244 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons464, cons84, cons1757)
rule6244 = ReplacementRule(pattern6244, With6244)
def With6245(p, m, f, b, g, d, a, c, n, x, e):
u = IntHide((d + e*x)**m*(f + g*x)**p, x)
rubi.append(6245)
return -Dist(b*c*n, Int(SimplifyIntegrand(u*(a + b*acosh(c*x))**(n + S(-1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x), x) + Dist((a + b*acosh(c*x))**n, u, x)
pattern6245 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons464, cons84, cons1757)
rule6245 = ReplacementRule(pattern6245, With6245)
def With6246(p, f, b, g, d, a, c, n, x, h, e):
u = IntHide((f + g*x + h*x**S(2))**p/(d + e*x)**S(2), x)
rubi.append(6246)
return -Dist(b*c*n, Int(SimplifyIntegrand(u*(a + b*asinh(c*x))**(n + S(-1))/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist((a + b*asinh(c*x))**n, u, x)
pattern6246 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_*(x_**S(2)*WC('h', S(1)) + x_*WC('g', S(1)) + WC('f', S(0)))**WC('p', S(1))/(d_ + x_*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons464, cons1758)
rule6246 = ReplacementRule(pattern6246, With6246)
def With6247(p, f, b, g, d, a, c, n, x, h, e):
u = IntHide((f + g*x + h*x**S(2))**p/(d + e*x)**S(2), x)
rubi.append(6247)
return -Dist(b*c*n, Int(SimplifyIntegrand(u*(a + b*acosh(c*x))**(n + S(-1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), x), x), x) + Dist((a + b*acosh(c*x))**n, u, x)
pattern6247 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_*(x_**S(2)*WC('h', S(1)) + x_*WC('g', S(1)) + WC('f', S(0)))**WC('p', S(1))/(d_ + x_*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons464, cons1758)
rule6247 = ReplacementRule(pattern6247, With6247)
pattern6248 = Pattern(Integral(Px_*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons925, cons148, cons17)
def replacement6248(m, b, Px, d, a, n, c, x, e):
rubi.append(6248)
return Int(ExpandIntegrand(Px*(a + b*asinh(c*x))**n*(d + e*x)**m, x), x)
rule6248 = ReplacementRule(pattern6248, replacement6248)
pattern6249 = Pattern(Integral(Px_*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons925, cons148, cons17)
def replacement6249(m, b, Px, d, a, n, c, x, e):
rubi.append(6249)
return Int(ExpandIntegrand(Px*(a + b*acosh(c*x))**n*(d + e*x)**m, x), x)
rule6249 = ReplacementRule(pattern6249, replacement6249)
def With6250(p, m, g, b, f, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p*(f + g*x)**m, x)
rubi.append(6250)
return -Dist(b*c, Int(Dist(S(1)/sqrt(c**S(2)*x**S(2) + S(1)), u, x), x), x) + Dist(a + b*asinh(c*x), u, x)
pattern6250 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1778, cons17, cons719, cons268, cons168, cons1759)
rule6250 = ReplacementRule(pattern6250, With6250)
def With6251(p, d2, m, g, b, f, e2, a, c, d1, e1, x):
u = IntHide((d1 + e1*x)**p*(d2 + e2*x)**p*(f + g*x)**m, x)
rubi.append(6251)
return -Dist(b*c, Int(Dist(S(1)/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), u, x), x), x) + Dist(a + b*acosh(c*x), u, x)
pattern6251 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons1892, cons1893, cons17, cons719, cons1894, cons1895, cons168, cons1759)
rule6251 = ReplacementRule(pattern6251, With6251)
pattern6252 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1778, cons17, cons667, cons268, cons148, cons168, cons1760)
def replacement6252(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(6252)
return Int(ExpandIntegrand((a + b*asinh(c*x))**n*(d + e*x**S(2))**p, (f + g*x)**m, x), x)
rule6252 = ReplacementRule(pattern6252, replacement6252)
pattern6253 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons1892, cons1893, cons17, cons667, cons1894, cons1895, cons148, cons168, cons1760)
def replacement6253(p, d2, m, g, b, f, e2, a, n, c, d1, e1, x):
rubi.append(6253)
return Int(ExpandIntegrand((a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, (f + g*x)**m, x), x)
rule6253 = ReplacementRule(pattern6253, replacement6253)
pattern6254 = Pattern(Integral(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(x_*WC('g', S(1)) + WC('f', S(0)))**m_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1778, cons17, cons268, cons148, cons267)
def replacement6254(m, f, b, g, d, a, n, c, x, e):
rubi.append(6254)
return -Dist(S(1)/(b*c*sqrt(d)*(n + S(1))), Int((a + b*asinh(c*x))**(n + S(1))*(f + g*x)**(m + S(-1))*(d*g*m + S(2)*e*f*x + e*g*x**S(2)*(m + S(2))), x), x) + Simp((a + b*asinh(c*x))**(n + S(1))*(d + e*x**S(2))*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule6254 = ReplacementRule(pattern6254, replacement6254)
pattern6255 = Pattern(Integral(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))*(f_ + x_*WC('g', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons1892, cons1893, cons17, cons1894, cons1895, cons148, cons267)
def replacement6255(d2, m, g, b, f, e2, a, n, c, d1, e1, x):
rubi.append(6255)
return -Dist(S(1)/(b*c*sqrt(-d1*d2)*(n + S(1))), Int((a + b*acosh(c*x))**(n + S(1))*(f + g*x)**(m + S(-1))*(d1*d2*g*m + S(2)*e1*e2*f*x + e1*e2*g*x**S(2)*(m + S(2))), x), x) + Simp((a + b*acosh(c*x))**(n + S(1))*(f + g*x)**m*(d1*d2 + e1*e2*x**S(2))/(b*c*sqrt(-d1*d2)*(n + S(1))), x)
rule6255 = ReplacementRule(pattern6255, replacement6255)
pattern6256 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1778, cons17, cons961, cons268, cons148)
def replacement6256(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(6256)
return Int(ExpandIntegrand((a + b*asinh(c*x))**n*sqrt(d + e*x**S(2)), (d + e*x**S(2))**(p + S(-1)/2)*(f + g*x)**m, x), x)
rule6256 = ReplacementRule(pattern6256, replacement6256)
pattern6257 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons1892, cons1893, cons17, cons961, cons1894, cons1895, cons148)
def replacement6257(p, d2, m, g, b, f, e2, a, n, c, d1, e1, x):
rubi.append(6257)
return Int(ExpandIntegrand((a + b*acosh(c*x))**n*sqrt(d1 + e1*x)*sqrt(d2 + e2*x), (d1 + e1*x)**(p + S(-1)/2)*(d2 + e2*x)**(p + S(-1)/2)*(f + g*x)**m, x), x)
rule6257 = ReplacementRule(pattern6257, replacement6257)
pattern6258 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1778, cons17, cons717, cons268, cons148, cons267)
def replacement6258(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(6258)
return -Dist(S(1)/(b*c*sqrt(d)*(n + S(1))), Int(ExpandIntegrand((a + b*asinh(c*x))**(n + S(1))*(f + g*x)**(m + S(-1)), (d + e*x**S(2))**(p + S(-1)/2)*(d*g*m + e*f*x*(S(2)*p + S(1)) + e*g*x**S(2)*(m + S(2)*p + S(1))), x), x), x) + Simp((a + b*asinh(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1)/2)*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule6258 = ReplacementRule(pattern6258, replacement6258)
pattern6259 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons1892, cons1893, cons17, cons717, cons1894, cons1895, cons148, cons267)
def replacement6259(p, d2, m, g, b, f, e2, a, n, c, d1, e1, x):
rubi.append(6259)
return -Dist(S(1)/(b*c*sqrt(-d1*d2)*(n + S(1))), Int(ExpandIntegrand((a + b*acosh(c*x))**(n + S(1))*(f + g*x)**(m + S(-1)), (d1 + e1*x)**(p + S(-1)/2)*(d2 + e2*x)**(p + S(-1)/2)*(d1*d2*g*m + e1*e2*f*x*(S(2)*p + S(1)) + e1*e2*g*x**S(2)*(m + S(2)*p + S(1))), x), x), x) + Simp((a + b*acosh(c*x))**(n + S(1))*(d1 + e1*x)**(p + S(1)/2)*(d2 + e2*x)**(p + S(1)/2)*(f + g*x)**m/(b*c*sqrt(-d1*d2)*(n + S(1))), x)
rule6259 = ReplacementRule(pattern6259, replacement6259)
pattern6260 = Pattern(Integral((f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1778, cons17, cons268, cons168, cons87, cons89)
def replacement6260(m, g, b, f, d, a, c, n, x, e):
rubi.append(6260)
return -Dist(g*m/(b*c*sqrt(d)*(n + S(1))), Int((a + b*asinh(c*x))**(n + S(1))*(f + g*x)**(m + S(-1)), x), x) + Simp((a + b*asinh(c*x))**(n + S(1))*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule6260 = ReplacementRule(pattern6260, replacement6260)
pattern6261 = Pattern(Integral((f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons1892, cons1893, cons17, cons1894, cons1895, cons168, cons87, cons89)
def replacement6261(d2, m, g, b, f, e2, a, c, n, d1, e1, x):
rubi.append(6261)
return -Dist(g*m/(b*c*sqrt(-d1*d2)*(n + S(1))), Int((a + b*acosh(c*x))**(n + S(1))*(f + g*x)**(m + S(-1)), x), x) + Simp((a + b*acosh(c*x))**(n + S(1))*(f + g*x)**m/(b*c*sqrt(-d1*d2)*(n + S(1))), x)
rule6261 = ReplacementRule(pattern6261, replacement6261)
pattern6262 = Pattern(Integral((f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1778, cons17, cons268, cons1761)
def replacement6262(m, g, b, f, d, a, n, c, x, e):
rubi.append(6262)
return Dist(c**(-m + S(-1))/sqrt(d), Subst(Int((a + b*x)**n*(c*f + g*sinh(x))**m, x), x, asinh(c*x)), x)
rule6262 = ReplacementRule(pattern6262, replacement6262)
pattern6263 = Pattern(Integral((f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons4, cons1892, cons1893, cons17, cons1894, cons1895, cons1761)
def replacement6263(d2, m, g, b, f, e2, a, n, c, d1, e1, x):
rubi.append(6263)
return Dist(c**(-m + S(-1))/sqrt(-d1*d2), Subst(Int((a + b*x)**n*(c*f + g*cosh(x))**m, x), x, acosh(c*x)), x)
rule6263 = ReplacementRule(pattern6263, replacement6263)
pattern6264 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1778, cons17, cons719, cons268, cons148)
def replacement6264(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(6264)
return Int(ExpandIntegrand((a + b*asinh(c*x))**n/sqrt(d + e*x**S(2)), (d + e*x**S(2))**(p + S(1)/2)*(f + g*x)**m, x), x)
rule6264 = ReplacementRule(pattern6264, replacement6264)
pattern6265 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons1892, cons1893, cons17, cons719, cons1894, cons1895, cons148)
def replacement6265(p, d2, m, g, b, f, e2, a, n, c, d1, e1, x):
rubi.append(6265)
return Int(ExpandIntegrand((a + b*acosh(c*x))**n/(sqrt(d1 + e1*x)*sqrt(d2 + e2*x)), (d1 + e1*x)**(p + S(1)/2)*(d2 + e2*x)**(p + S(1)/2)*(f + g*x)**m, x), x)
rule6265 = ReplacementRule(pattern6265, replacement6265)
pattern6266 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1778, cons17, cons347, cons1738)
def replacement6266(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(6266)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((a + b*asinh(c*x))**n*(f + g*x)**m*(c**S(2)*x**S(2) + S(1))**p, x), x)
rule6266 = ReplacementRule(pattern6266, replacement6266)
pattern6267 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1737, cons17, cons347)
def replacement6267(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(6267)
return Dist((-d)**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p)), Int((a + b*acosh(c*x))**n*(f + g*x)**m*(c*x + S(-1))**p*(c*x + S(1))**p, x), x)
rule6267 = ReplacementRule(pattern6267, replacement6267)
pattern6268 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons4, cons1892, cons1893, cons17, cons347, cons1896)
def replacement6268(p, d2, m, g, b, f, e2, a, n, c, d1, e1, x):
rubi.append(6268)
return Dist((-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((a + b*acosh(c*x))**n*(f + g*x)**m*(c*x + S(-1))**p*(c*x + S(1))**p, x), x)
rule6268 = ReplacementRule(pattern6268, replacement6268)
pattern6269 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1)))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons1778, cons268, cons148)
def replacement6269(m, f, g, b, d, a, c, n, x, h, e):
rubi.append(6269)
return -Dist(g*m/(b*c*sqrt(d)*(n + S(1))), Int((a + b*asinh(c*x))**(n + S(1))/(f + g*x), x), x) + Simp((a + b*asinh(c*x))**(n + S(1))*log(h*(f + g*x)**m)/(b*c*sqrt(d)*(n + S(1))), x)
rule6269 = ReplacementRule(pattern6269, replacement6269)
pattern6270 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1)))/(sqrt(d1_ + x_*WC('e1', S(1)))*sqrt(d2_ + x_*WC('e2', S(1)))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons209, cons21, cons1892, cons1893, cons1894, cons1895, cons148)
def replacement6270(d2, m, f, g, b, e2, a, c, n, d1, e1, x, h):
rubi.append(6270)
return -Dist(g*m/(b*c*sqrt(-d1*d2)*(n + S(1))), Int((a + b*acosh(c*x))**(n + S(1))/(f + g*x), x), x) + Simp((a + b*acosh(c*x))**(n + S(1))*log(h*(f + g*x)**m)/(b*c*sqrt(-d1*d2)*(n + S(1))), x)
rule6270 = ReplacementRule(pattern6270, replacement6270)
pattern6271 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons1778, cons347, cons1738)
def replacement6271(p, m, f, g, b, d, a, c, n, x, h, e):
rubi.append(6271)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((a + b*asinh(c*x))**n*(c**S(2)*x**S(2) + S(1))**p*log(h*(f + g*x)**m), x), x)
rule6271 = ReplacementRule(pattern6271, replacement6271)
pattern6272 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons1737, cons347)
def replacement6272(p, m, f, g, b, d, a, c, n, x, h, e):
rubi.append(6272)
return Dist((-d)**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p)), Int((a + b*acosh(c*x))**n*(c*x + S(-1))**p*(c*x + S(1))**p*log(h*(f + g*x)**m), x), x)
rule6272 = ReplacementRule(pattern6272, replacement6272)
pattern6273 = Pattern(Integral((d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1))), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons209, cons21, cons4, cons1892, cons1893, cons347, cons1896)
def replacement6273(p, d2, m, f, g, b, e2, a, c, n, d1, e1, x, h):
rubi.append(6273)
return Dist((-d1*d2)**IntPart(p)*(d1 + e1*x)**FracPart(p)*(d2 + e2*x)**FracPart(p)*(c*x + S(-1))**(-FracPart(p))*(c*x + S(1))**(-FracPart(p)), Int((a + b*acosh(c*x))**n*(c*x + S(-1))**p*(c*x + S(1))**p*log(h*(f + g*x)**m), x), x)
rule6273 = ReplacementRule(pattern6273, replacement6273)
def With6274(m, g, b, f, d, a, c, x, e):
u = IntHide((d + e*x)**m*(f + g*x)**m, x)
rubi.append(6274)
return -Dist(b*c, Int(Dist(S(1)/sqrt(c**S(2)*x**S(2) + S(1)), u, x), x), x) + Dist(a + b*asinh(c*x), u, x)
pattern6274 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(f_ + x_*WC('g', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1608)
rule6274 = ReplacementRule(pattern6274, With6274)
def With6275(m, g, b, f, d, a, c, x, e):
u = IntHide((d + e*x)**m*(f + g*x)**m, x)
rubi.append(6275)
return -Dist(b*c, Int(Dist(S(1)/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), u, x), x), x) + Dist(a + b*acosh(c*x), u, x)
pattern6275 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(f_ + x_*WC('g', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1608)
rule6275 = ReplacementRule(pattern6275, With6275)
pattern6276 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons17)
def replacement6276(m, g, b, f, d, a, n, c, x, e):
rubi.append(6276)
return Int(ExpandIntegrand((a + b*asinh(c*x))**n, (d + e*x)**m*(f + g*x)**m, x), x)
rule6276 = ReplacementRule(pattern6276, replacement6276)
pattern6277 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons17)
def replacement6277(m, g, b, f, d, a, n, c, x, e):
rubi.append(6277)
return Int(ExpandIntegrand((a + b*acosh(c*x))**n, (d + e*x)**m*(f + g*x)**m, x), x)
rule6277 = ReplacementRule(pattern6277, replacement6277)
def With6278(u, b, a, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
v = IntHide(u, x)
if InverseFunctionFreeQ(v, x):
return True
return False
pattern6278 = Pattern(Integral(u_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons14, CustomConstraint(With6278))
def replacement6278(u, b, a, c, x):
v = IntHide(u, x)
rubi.append(6278)
return -Dist(b*c, Int(SimplifyIntegrand(v/sqrt(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(c*x), v, x)
rule6278 = ReplacementRule(pattern6278, replacement6278)
def With6279(u, b, a, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
v = IntHide(u, x)
if InverseFunctionFreeQ(v, x):
return True
return False
pattern6279 = Pattern(Integral(u_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons14, CustomConstraint(With6279))
def replacement6279(u, b, a, c, x):
v = IntHide(u, x)
rubi.append(6279)
return -Dist(b*c*sqrt(-c**S(2)*x**S(2) + S(1))/(sqrt(c*x + S(-1))*sqrt(c*x + S(1))), Int(SimplifyIntegrand(v/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acosh(c*x), v, x)
rule6279 = ReplacementRule(pattern6279, replacement6279)
def With6280(p, b, Px, d, a, c, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(Px*(a + b*asinh(c*x))**n*(d + e*x**S(2))**p, x)
if SumQ(u):
return True
return False
pattern6280 = Pattern(Integral(Px_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons925, cons1778, cons347, CustomConstraint(With6280))
def replacement6280(p, b, Px, d, a, c, n, x, e):
u = ExpandIntegrand(Px*(a + b*asinh(c*x))**n*(d + e*x**S(2))**p, x)
rubi.append(6280)
return Int(u, x)
rule6280 = ReplacementRule(pattern6280, replacement6280)
def With6281(p, d2, b, Px, e2, a, c, n, d1, e1, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(Px*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, x)
if SumQ(u):
return True
return False
pattern6281 = Pattern(Integral(Px_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons925, cons1892, cons1893, cons347, CustomConstraint(With6281))
def replacement6281(p, d2, b, Px, e2, a, c, n, d1, e1, x):
u = ExpandIntegrand(Px*(a + b*acosh(c*x))**n*(d1 + e1*x)**p*(d2 + e2*x)**p, x)
rubi.append(6281)
return Int(u, x)
rule6281 = ReplacementRule(pattern6281, replacement6281)
def With6282(p, m, g, b, f, Px, d, a, n, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(Px*(a + b*asinh(c*x))**n*(f + g*(d + e*x**S(2))**p)**m, x)
if SumQ(u):
return True
return False
pattern6282 = Pattern(Integral((f_ + (d_ + x_**S(2)*WC('e', S(1)))**p_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))*WC('Px', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons925, cons1778, cons961, cons150, CustomConstraint(With6282))
def replacement6282(p, m, g, b, f, Px, d, a, n, c, x, e):
u = ExpandIntegrand(Px*(a + b*asinh(c*x))**n*(f + g*(d + e*x**S(2))**p)**m, x)
rubi.append(6282)
return Int(u, x)
rule6282 = ReplacementRule(pattern6282, replacement6282)
def With6283(p, d2, m, g, b, f, Px, e2, a, n, c, d1, e1, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(Px*(a + b*acosh(c*x))**n*(f + g*(d1 + e1*x)**p*(d2 + e2*x)**p)**m, x)
if SumQ(u):
return True
return False
pattern6283 = Pattern(Integral((f_ + (d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))*WC('Px', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons208, cons925, cons1892, cons1893, cons961, cons150, CustomConstraint(With6283))
def replacement6283(p, d2, m, g, b, f, Px, e2, a, n, c, d1, e1, x):
u = ExpandIntegrand(Px*(a + b*acosh(c*x))**n*(f + g*(d1 + e1*x)**p*(d2 + e2*x)**p)**m, x)
rubi.append(6283)
return Int(u, x)
rule6283 = ReplacementRule(pattern6283, replacement6283)
def With6284(x, c, n, RFx):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(asinh(c*x)**n, RFx, x)
if SumQ(u):
return True
return False
pattern6284 = Pattern(Integral(RFx_*asinh(x_*WC('c', S(1)))**WC('n', S(1)), x_), cons7, cons1198, cons148, CustomConstraint(With6284))
def replacement6284(x, c, n, RFx):
u = ExpandIntegrand(asinh(c*x)**n, RFx, x)
rubi.append(6284)
return Int(u, x)
rule6284 = ReplacementRule(pattern6284, replacement6284)
def With6285(x, c, n, RFx):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(acosh(c*x)**n, RFx, x)
if SumQ(u):
return True
return False
pattern6285 = Pattern(Integral(RFx_*acosh(x_*WC('c', S(1)))**WC('n', S(1)), x_), cons7, cons1198, cons148, CustomConstraint(With6285))
def replacement6285(x, c, n, RFx):
u = ExpandIntegrand(acosh(c*x)**n, RFx, x)
rubi.append(6285)
return Int(u, x)
rule6285 = ReplacementRule(pattern6285, replacement6285)
pattern6286 = Pattern(Integral(RFx_*(a_ + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons1198, cons148)
def replacement6286(RFx, b, c, n, a, x):
rubi.append(6286)
return Int(ExpandIntegrand(RFx*(a + b*asinh(c*x))**n, x), x)
rule6286 = ReplacementRule(pattern6286, replacement6286)
pattern6287 = Pattern(Integral(RFx_*(a_ + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons1198, cons148)
def replacement6287(RFx, b, c, n, a, x):
rubi.append(6287)
return Int(ExpandIntegrand(RFx*(a + b*acosh(c*x))**n, x), x)
rule6287 = ReplacementRule(pattern6287, replacement6287)
def With6288(p, RFx, d, c, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((d + e*x**S(2))**p*asinh(c*x)**n, RFx, x)
if SumQ(u):
return True
return False
pattern6288 = Pattern(Integral(RFx_*(d_ + x_**S(2)*WC('e', S(1)))**p_*asinh(x_*WC('c', S(1)))**WC('n', S(1)), x_), cons7, cons27, cons48, cons1198, cons148, cons1778, cons347, CustomConstraint(With6288))
def replacement6288(p, RFx, d, c, n, x, e):
u = ExpandIntegrand((d + e*x**S(2))**p*asinh(c*x)**n, RFx, x)
rubi.append(6288)
return Int(u, x)
rule6288 = ReplacementRule(pattern6288, replacement6288)
def With6289(p, RFx, d2, e2, c, n, d1, e1, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((d1 + e1*x)**p*(d2 + e2*x)**p*acosh(c*x)**n, RFx, x)
if SumQ(u):
return True
return False
pattern6289 = Pattern(Integral(RFx_*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_*acosh(x_*WC('c', S(1)))**WC('n', S(1)), x_), cons7, cons731, cons652, cons732, cons654, cons1198, cons148, cons1892, cons1893, cons347, CustomConstraint(With6289))
def replacement6289(p, RFx, d2, e2, c, n, d1, e1, x):
u = ExpandIntegrand((d1 + e1*x)**p*(d2 + e2*x)**p*acosh(c*x)**n, RFx, x)
rubi.append(6289)
return Int(u, x)
rule6289 = ReplacementRule(pattern6289, replacement6289)
pattern6290 = Pattern(Integral(RFx_*(a_ + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons1198, cons148, cons1778, cons347)
def replacement6290(p, RFx, b, d, c, n, a, x, e):
rubi.append(6290)
return Int(ExpandIntegrand((d + e*x**S(2))**p, RFx*(a + b*asinh(c*x))**n, x), x)
rule6290 = ReplacementRule(pattern6290, replacement6290)
pattern6291 = Pattern(Integral(RFx_*(a_ + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))*(d1_ + x_*WC('e1', S(1)))**p_*(d2_ + x_*WC('e2', S(1)))**p_, x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons1198, cons148, cons1892, cons1893, cons347)
def replacement6291(p, RFx, d2, b, e2, c, n, a, d1, e1, x):
rubi.append(6291)
return Int(ExpandIntegrand((d1 + e1*x)**p*(d2 + e2*x)**p, RFx*(a + b*acosh(c*x))**n, x), x)
rule6291 = ReplacementRule(pattern6291, replacement6291)
pattern6292 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(x_*WC('c', S(1))))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6292(u, b, a, n, c, x):
rubi.append(6292)
return Int(u*(a + b*asinh(c*x))**n, x)
rule6292 = ReplacementRule(pattern6292, replacement6292)
pattern6293 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_*WC('c', S(1))))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6293(u, b, a, n, c, x):
rubi.append(6293)
return Int(u*(a + b*acosh(c*x))**n, x)
rule6293 = ReplacementRule(pattern6293, replacement6293)
pattern6294 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1273)
def replacement6294(b, d, c, a, n, x):
rubi.append(6294)
return Dist(S(1)/d, Subst(Int((a + b*asinh(x))**n, x), x, c + d*x), x)
rule6294 = ReplacementRule(pattern6294, replacement6294)
pattern6295 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1273)
def replacement6295(b, d, c, a, n, x):
rubi.append(6295)
return Dist(S(1)/d, Subst(Int((a + b*acosh(x))**n, x), x, c + d*x), x)
rule6295 = ReplacementRule(pattern6295, replacement6295)
pattern6296 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement6296(m, f, b, d, a, n, c, x, e):
rubi.append(6296)
return Dist(S(1)/d, Subst(Int((a + b*asinh(x))**n*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule6296 = ReplacementRule(pattern6296, replacement6296)
pattern6297 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement6297(m, f, b, d, a, n, c, x, e):
rubi.append(6297)
return Dist(S(1)/d, Subst(Int((a + b*acosh(x))**n*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule6297 = ReplacementRule(pattern6297, replacement6297)
pattern6298 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons36, cons4, cons5, cons1830, cons1763)
def replacement6298(B, C, p, b, d, a, n, c, x, A):
rubi.append(6298)
return Dist(S(1)/d, Subst(Int((a + b*asinh(x))**n*(C*x**S(2)/d**S(2) + C/d**S(2))**p, x), x, c + d*x), x)
rule6298 = ReplacementRule(pattern6298, replacement6298)
pattern6299 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons36, cons4, cons5, cons1762, cons1763)
def replacement6299(B, C, p, b, d, a, n, c, x, A):
rubi.append(6299)
return Dist(S(1)/d, Subst(Int((a + b*acosh(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p, x), x, c + d*x), x)
rule6299 = ReplacementRule(pattern6299, replacement6299)
pattern6300 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons1830, cons1763)
def replacement6300(B, C, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(6300)
return Dist(S(1)/d, Subst(Int((a + b*asinh(x))**n*(C*x**S(2)/d**S(2) + C/d**S(2))**p*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule6300 = ReplacementRule(pattern6300, replacement6300)
pattern6301 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons1762, cons1763)
def replacement6301(B, C, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(6301)
return Dist(S(1)/d, Subst(Int((a + b*acosh(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule6301 = ReplacementRule(pattern6301, replacement6301)
pattern6302 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons1904)
def replacement6302(b, d, c, a, x):
rubi.append(6302)
return Simp(x*sqrt(a + b*asinh(c + d*x**S(2))), x) - Simp(sqrt(Pi)*x*(-c*sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*FresnelC(sqrt(-c/(Pi*b))*sqrt(a + b*asinh(c + d*x**S(2))))/(sqrt(-c/b)*(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2)))), x) + Simp(sqrt(Pi)*x*(c*sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*FresnelS(sqrt(-c/(Pi*b))*sqrt(a + b*asinh(c + d*x**S(2))))/(sqrt(-c/b)*(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2)))), x)
rule6302 = ReplacementRule(pattern6302, replacement6302)
pattern6303 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_**S(2)*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons1904, cons87, cons165)
def replacement6303(b, d, c, a, n, x):
rubi.append(6303)
return Dist(S(4)*b**S(2)*n*(n + S(-1)), Int((a + b*asinh(c + d*x**S(2)))**(n + S(-2)), x), x) + Simp(x*(a + b*asinh(c + d*x**S(2)))**n, x) - Simp(S(2)*b*n*(a + b*asinh(c + d*x**S(2)))**(n + S(-1))*sqrt(S(2)*c*d*x**S(2) + d**S(2)*x**S(4))/(d*x), x)
rule6303 = ReplacementRule(pattern6303, replacement6303)
pattern6304 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons1904)
def replacement6304(b, d, c, a, x):
rubi.append(6304)
return Simp(x*(-c*sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*SinhIntegral((a + b*asinh(c + d*x**S(2)))/(S(2)*b))/(S(2)*b*(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2)))), x) + Simp(x*(c*cosh(a/(S(2)*b)) - sinh(a/(S(2)*b)))*CoshIntegral((a + b*asinh(c + d*x**S(2)))/(S(2)*b))/(S(2)*b*(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2)))), x)
rule6304 = ReplacementRule(pattern6304, replacement6304)
pattern6305 = Pattern(Integral(S(1)/sqrt(WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons1904)
def replacement6305(b, d, c, a, x):
rubi.append(6305)
return Simp(sqrt(S(2))*sqrt(Pi)*x*(c + S(-1))*(sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erf(sqrt(S(2))*sqrt(a + b*asinh(c + d*x**S(2)))/(S(2)*sqrt(b)))/(S(4)*sqrt(b)*(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2)))), x) + Simp(sqrt(S(2))*sqrt(Pi)*x*(c + S(1))*(-sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erfi(sqrt(S(2))*sqrt(a + b*asinh(c + d*x**S(2)))/(S(2)*sqrt(b)))/(S(4)*sqrt(b)*(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2)))), x)
rule6305 = ReplacementRule(pattern6305, replacement6305)
pattern6306 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_**S(2)*WC('d', S(1))))**(S(-3)/2), x_), cons2, cons3, cons7, cons27, cons1904)
def replacement6306(b, d, c, a, x):
rubi.append(6306)
return -Simp(sqrt(S(2)*c*d*x**S(2) + d**S(2)*x**S(4))/(b*d*x*sqrt(a + b*asinh(c + d*x**S(2)))), x) - Simp(sqrt(Pi)*x*(-c/b)**(S(3)/2)*(-c*sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*FresnelC(sqrt(-c/(Pi*b))*sqrt(a + b*asinh(c + d*x**S(2))))/(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2))), x) + Simp(sqrt(Pi)*x*(-c/b)**(S(3)/2)*(c*sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*FresnelS(sqrt(-c/(Pi*b))*sqrt(a + b*asinh(c + d*x**S(2))))/(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2))), x)
rule6306 = ReplacementRule(pattern6306, replacement6306)
pattern6307 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_**S(2)*WC('d', S(1))))**(S(-2)), x_), cons2, cons3, cons7, cons27, cons1904)
def replacement6307(b, d, c, a, x):
rubi.append(6307)
return Simp(x*(-c*sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*CoshIntegral((a + b*asinh(c + d*x**S(2)))/(S(2)*b))/(S(4)*b**S(2)*(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2)))), x) + Simp(x*(c*cosh(a/(S(2)*b)) - sinh(a/(S(2)*b)))*SinhIntegral((a + b*asinh(c + d*x**S(2)))/(S(2)*b))/(S(4)*b**S(2)*(c*sinh(asinh(c + d*x**S(2))/S(2)) + cosh(asinh(c + d*x**S(2))/S(2)))), x) - Simp(sqrt(S(2)*c*d*x**S(2) + d**S(2)*x**S(4))/(S(2)*b*d*x*(a + b*asinh(c + d*x**S(2)))), x)
rule6307 = ReplacementRule(pattern6307, replacement6307)
pattern6308 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asinh(c_ + x_**S(2)*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons1904, cons87, cons89, cons1442)
def replacement6308(b, d, c, a, n, x):
rubi.append(6308)
return Dist(S(1)/(S(4)*b**S(2)*(n + S(1))*(n + S(2))), Int((a + b*asinh(c + d*x**S(2)))**(n + S(2)), x), x) - Simp(x*(a + b*asinh(c + d*x**S(2)))**(n + S(2))/(S(4)*b**S(2)*(n + S(1))*(n + S(2))), x) + Simp((a + b*asinh(c + d*x**S(2)))**(n + S(1))*sqrt(S(2)*c*d*x**S(2) + d**S(2)*x**S(4))/(S(2)*b*d*x*(n + S(1))), x)
rule6308 = ReplacementRule(pattern6308, replacement6308)
pattern6309 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(1))), x_), cons2, cons3, cons27, cons1765)
def replacement6309(d, a, b, x):
rubi.append(6309)
return Simp(S(2)*sqrt(a + b*acosh(d*x**S(2) + S(1)))*sinh(acosh(d*x**S(2) + S(1))/S(2))**S(2)/(d*x), x) - Simp(sqrt(S(2))*sqrt(Pi)*sqrt(b)*(-sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erfi(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(1)))/(S(2)*sqrt(b)))*sinh(acosh(d*x**S(2) + S(1))/S(2))/(S(2)*d*x), x) + Simp(sqrt(S(2))*sqrt(Pi)*sqrt(b)*(sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erf(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(1)))/(S(2)*sqrt(b)))*sinh(acosh(d*x**S(2) + S(1))/S(2))/(S(2)*d*x), x)
rule6309 = ReplacementRule(pattern6309, replacement6309)
pattern6310 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(-1))), x_), cons2, cons3, cons27, cons1765)
def replacement6310(d, a, b, x):
rubi.append(6310)
return Simp(S(2)*sqrt(a + b*acosh(d*x**S(2) + S(-1)))*cosh(acosh(d*x**S(2) + S(-1))/S(2))**S(2)/(d*x), x) - Simp(sqrt(S(2))*sqrt(Pi)*sqrt(b)*(-sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erfi(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*sqrt(b)))*cosh(acosh(d*x**S(2) + S(-1))/S(2))/(S(2)*d*x), x) - Simp(sqrt(S(2))*sqrt(Pi)*sqrt(b)*(sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erf(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*sqrt(b)))*cosh(acosh(d*x**S(2) + S(-1))/S(2))/(S(2)*d*x), x)
rule6310 = ReplacementRule(pattern6310, replacement6310)
pattern6311 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(c_ + x_**S(2)*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons1764, cons87, cons165)
def replacement6311(b, d, c, a, n, x):
rubi.append(6311)
return Dist(S(4)*b**S(2)*n*(n + S(-1)), Int((a + b*acosh(c + d*x**S(2)))**(n + S(-2)), x), x) + Simp(x*(a + b*acosh(c + d*x**S(2)))**n, x) - Simp(S(2)*b*n*(a + b*acosh(c + d*x**S(2)))**(n + S(-1))*(S(2)*c*d*x**S(2) + d**S(2)*x**S(4))/(d*x*sqrt(c + d*x**S(2) + S(-1))*sqrt(c + d*x**S(2) + S(1))), x)
rule6311 = ReplacementRule(pattern6311, replacement6311)
pattern6312 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(1))), x_), cons2, cons3, cons27, cons1765)
def replacement6312(d, a, b, x):
rubi.append(6312)
return Simp(sqrt(S(2))*x*CoshIntegral((a + b*acosh(d*x**S(2) + S(1)))/(S(2)*b))*cosh(a/(S(2)*b))/(S(2)*b*sqrt(d*x**S(2))), x) - Simp(sqrt(S(2))*x*SinhIntegral((a + b*acosh(d*x**S(2) + S(1)))/(S(2)*b))*sinh(a/(S(2)*b))/(S(2)*b*sqrt(d*x**S(2))), x)
rule6312 = ReplacementRule(pattern6312, replacement6312)
pattern6313 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(-1))), x_), cons2, cons3, cons27, cons1765)
def replacement6313(d, a, b, x):
rubi.append(6313)
return -Simp(sqrt(S(2))*x*CoshIntegral((a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*b))*sinh(a/(S(2)*b))/(S(2)*b*sqrt(d*x**S(2))), x) + Simp(sqrt(S(2))*x*SinhIntegral((a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*b))*cosh(a/(S(2)*b))/(S(2)*b*sqrt(d*x**S(2))), x)
rule6313 = ReplacementRule(pattern6313, replacement6313)
pattern6314 = Pattern(Integral(S(1)/sqrt(WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(1))), x_), cons2, cons3, cons27, cons1765)
def replacement6314(d, a, b, x):
rubi.append(6314)
return Simp(sqrt(S(2))*sqrt(Pi)*(-sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erfi(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(1)))/(S(2)*sqrt(b)))*sinh(acosh(d*x**S(2) + S(1))/S(2))/(S(2)*sqrt(b)*d*x), x) + Simp(sqrt(S(2))*sqrt(Pi)*(sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erf(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(1)))/(S(2)*sqrt(b)))*sinh(acosh(d*x**S(2) + S(1))/S(2))/(S(2)*sqrt(b)*d*x), x)
rule6314 = ReplacementRule(pattern6314, replacement6314)
pattern6315 = Pattern(Integral(S(1)/sqrt(WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(-1))), x_), cons2, cons3, cons27, cons1765)
def replacement6315(d, a, b, x):
rubi.append(6315)
return Simp(sqrt(S(2))*sqrt(Pi)*(-sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erfi(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*sqrt(b)))*cosh(acosh(d*x**S(2) + S(-1))/S(2))/(S(2)*sqrt(b)*d*x), x) - Simp(sqrt(S(2))*sqrt(Pi)*(sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erf(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*sqrt(b)))*cosh(acosh(d*x**S(2) + S(-1))/S(2))/(S(2)*sqrt(b)*d*x), x)
rule6315 = ReplacementRule(pattern6315, replacement6315)
pattern6316 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(1)))**(S(-3)/2), x_), cons2, cons3, cons27, cons1765)
def replacement6316(d, a, b, x):
rubi.append(6316)
return -Simp(sqrt(d*x**S(2))*sqrt(d*x**S(2) + S(2))/(b*d*x*sqrt(a + b*acosh(d*x**S(2) + S(1)))), x) + Simp(sqrt(S(2))*sqrt(Pi)*(-sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erfi(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(1)))/(S(2)*sqrt(b)))*sinh(acosh(d*x**S(2) + S(1))/S(2))/(S(2)*b**(S(3)/2)*d*x), x) - Simp(sqrt(S(2))*sqrt(Pi)*(sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erf(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(1)))/(S(2)*sqrt(b)))*sinh(acosh(d*x**S(2) + S(1))/S(2))/(S(2)*b**(S(3)/2)*d*x), x)
rule6316 = ReplacementRule(pattern6316, replacement6316)
pattern6317 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(-1)))**(S(-3)/2), x_), cons2, cons3, cons27, cons1765)
def replacement6317(d, a, b, x):
rubi.append(6317)
return -Simp(sqrt(d*x**S(2))*sqrt(d*x**S(2) + S(-2))/(b*d*x*sqrt(a + b*acosh(d*x**S(2) + S(-1)))), x) + Simp(sqrt(S(2))*sqrt(Pi)*(-sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erfi(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*sqrt(b)))*cosh(acosh(d*x**S(2) + S(-1))/S(2))/(S(2)*b**(S(3)/2)*d*x), x) + Simp(sqrt(S(2))*sqrt(Pi)*(sinh(a/(S(2)*b)) + cosh(a/(S(2)*b)))*Erf(sqrt(S(2))*sqrt(a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*sqrt(b)))*cosh(acosh(d*x**S(2) + S(-1))/S(2))/(S(2)*b**(S(3)/2)*d*x), x)
rule6317 = ReplacementRule(pattern6317, replacement6317)
pattern6318 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(1)))**(S(-2)), x_), cons2, cons3, cons27, cons1765)
def replacement6318(d, a, b, x):
rubi.append(6318)
return -Simp(sqrt(S(2))*x*CoshIntegral((a + b*acosh(d*x**S(2) + S(1)))/(S(2)*b))*sinh(a/(S(2)*b))/(S(4)*b**S(2)*sqrt(d*x**S(2))), x) + Simp(sqrt(S(2))*x*SinhIntegral((a + b*acosh(d*x**S(2) + S(1)))/(S(2)*b))*cosh(a/(S(2)*b))/(S(4)*b**S(2)*sqrt(d*x**S(2))), x) - Simp(sqrt(d*x**S(2))*sqrt(d*x**S(2) + S(2))/(S(2)*b*d*x*(a + b*acosh(d*x**S(2) + S(1)))), x)
rule6318 = ReplacementRule(pattern6318, replacement6318)
pattern6319 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(x_**S(2)*WC('d', S(1)) + S(-1)))**(S(-2)), x_), cons2, cons3, cons27, cons1765)
def replacement6319(d, a, b, x):
rubi.append(6319)
return Simp(sqrt(S(2))*x*CoshIntegral((a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*b))*cosh(a/(S(2)*b))/(S(4)*b**S(2)*sqrt(d*x**S(2))), x) - Simp(sqrt(S(2))*x*SinhIntegral((a + b*acosh(d*x**S(2) + S(-1)))/(S(2)*b))*sinh(a/(S(2)*b))/(S(4)*b**S(2)*sqrt(d*x**S(2))), x) - Simp(sqrt(d*x**S(2))*sqrt(d*x**S(2) + S(-2))/(S(2)*b*d*x*(a + b*acosh(d*x**S(2) + S(-1)))), x)
rule6319 = ReplacementRule(pattern6319, replacement6319)
pattern6320 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acosh(c_ + x_**S(2)*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons1764, cons87, cons89, cons1442)
def replacement6320(b, d, c, a, n, x):
rubi.append(6320)
return Dist(S(1)/(S(4)*b**S(2)*(n + S(1))*(n + S(2))), Int((a + b*acosh(c + d*x**S(2)))**(n + S(2)), x), x) - Simp(x*(a + b*acosh(c + d*x**S(2)))**(n + S(2))/(S(4)*b**S(2)*(n + S(1))*(n + S(2))), x) + Simp((a + b*acosh(c + d*x**S(2)))**(n + S(1))*(S(2)*c*x**S(2) + d*x**S(4))/(S(2)*b*x*(n + S(1))*sqrt(c + d*x**S(2) + S(-1))*sqrt(c + d*x**S(2) + S(1))), x)
rule6320 = ReplacementRule(pattern6320, replacement6320)
pattern6321 = Pattern(Integral(asinh(x_**p_*WC('a', S(1)))**WC('n', S(1))/x_, x_), cons2, cons5, cons148)
def replacement6321(x, a, n, p):
rubi.append(6321)
return Dist(S(1)/p, Subst(Int(x**n/tanh(x), x), x, asinh(a*x**p)), x)
rule6321 = ReplacementRule(pattern6321, replacement6321)
pattern6322 = Pattern(Integral(acosh(x_**p_*WC('a', S(1)))**WC('n', S(1))/x_, x_), cons2, cons5, cons148)
def replacement6322(x, a, n, p):
rubi.append(6322)
return Dist(S(1)/p, Subst(Int(x**n*tanh(x), x), x, acosh(a*x**p)), x)
rule6322 = ReplacementRule(pattern6322, replacement6322)
pattern6323 = Pattern(Integral(WC('u', S(1))*asinh(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement6323(u, m, b, c, n, a, x):
rubi.append(6323)
return Int(u*acsch(a/c + b*x**n/c)**m, x)
rule6323 = ReplacementRule(pattern6323, replacement6323)
pattern6324 = Pattern(Integral(WC('u', S(1))*acosh(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement6324(u, m, b, c, n, a, x):
rubi.append(6324)
return Int(u*asech(a/c + b*x**n/c)**m, x)
rule6324 = ReplacementRule(pattern6324, replacement6324)
pattern6325 = Pattern(Integral(asinh(sqrt(x_**S(2)*WC('b', S(1)) + S(-1)))**WC('n', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + S(-1)), x_), cons3, cons4, cons1767)
def replacement6325(x, n, b):
rubi.append(6325)
return Dist(sqrt(b*x**S(2))/(b*x), Subst(Int(asinh(x)**n/sqrt(x**S(2) + S(1)), x), x, sqrt(b*x**S(2) + S(-1))), x)
rule6325 = ReplacementRule(pattern6325, replacement6325)
pattern6326 = Pattern(Integral(acosh(sqrt(x_**S(2)*WC('b', S(1)) + S(1)))**WC('n', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + S(1)), x_), cons3, cons4, cons1767)
def replacement6326(x, n, b):
rubi.append(6326)
return Dist(sqrt(sqrt(b*x**S(2) + S(1)) + S(-1))*sqrt(sqrt(b*x**S(2) + S(1)) + S(1))/(b*x), Subst(Int(acosh(x)**n/(sqrt(x + S(-1))*sqrt(x + S(1))), x), x, sqrt(b*x**S(2) + S(1))), x)
rule6326 = ReplacementRule(pattern6326, replacement6326)
pattern6327 = Pattern(Integral(f_**(WC('c', S(1))*asinh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))), x_), cons2, cons3, cons7, cons125, cons148)
def replacement6327(f, b, c, a, n, x):
rubi.append(6327)
return Dist(S(1)/b, Subst(Int(f**(c*x**n)*cosh(x), x), x, asinh(a + b*x)), x)
rule6327 = ReplacementRule(pattern6327, replacement6327)
pattern6328 = Pattern(Integral(f_**(WC('c', S(1))*acosh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))), x_), cons2, cons3, cons7, cons125, cons148)
def replacement6328(f, b, c, a, n, x):
rubi.append(6328)
return Dist(S(1)/b, Subst(Int(f**(c*x**n)*sinh(x), x), x, acosh(a + b*x)), x)
rule6328 = ReplacementRule(pattern6328, replacement6328)
pattern6329 = Pattern(Integral(f_**(WC('c', S(1))*asinh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)))*x_**WC('m', S(1)), x_), cons2, cons3, cons7, cons125, cons528)
def replacement6329(m, f, b, c, a, n, x):
rubi.append(6329)
return Dist(S(1)/b, Subst(Int(f**(c*x**n)*(-a/b + sinh(x)/b)**m*cosh(x), x), x, asinh(a + b*x)), x)
rule6329 = ReplacementRule(pattern6329, replacement6329)
pattern6330 = Pattern(Integral(f_**(WC('c', S(1))*acosh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)))*x_**WC('m', S(1)), x_), cons2, cons3, cons7, cons125, cons528)
def replacement6330(m, f, b, c, a, n, x):
rubi.append(6330)
return Dist(S(1)/b, Subst(Int(f**(c*x**n)*(-a/b + cosh(x)/b)**m*sinh(x), x), x, acosh(a + b*x)), x)
rule6330 = ReplacementRule(pattern6330, replacement6330)
pattern6331 = Pattern(Integral(asinh(u_), x_), cons1230, cons1769)
def replacement6331(x, u):
rubi.append(6331)
return -Int(SimplifyIntegrand(x*D(u, x)/sqrt(u**S(2) + S(1)), x), x) + Simp(x*asinh(u), x)
rule6331 = ReplacementRule(pattern6331, replacement6331)
pattern6332 = Pattern(Integral(acosh(u_), x_), cons1230, cons1769)
def replacement6332(x, u):
rubi.append(6332)
return -Int(SimplifyIntegrand(x*D(u, x)/(sqrt(u + S(-1))*sqrt(u + S(1))), x), x) + Simp(x*acosh(u), x)
rule6332 = ReplacementRule(pattern6332, replacement6332)
pattern6333 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asinh(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1769)
def replacement6333(u, m, b, d, c, a, x):
rubi.append(6333)
return -Dist(b/(d*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/sqrt(u**S(2) + S(1)), x), x), x) + Simp((a + b*asinh(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule6333 = ReplacementRule(pattern6333, replacement6333)
pattern6334 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acosh(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1769)
def replacement6334(u, m, b, d, c, a, x):
rubi.append(6334)
return -Dist(b/(d*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(sqrt(u + S(-1))*sqrt(u + S(1))), x), x), x) + Simp((a + b*acosh(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule6334 = ReplacementRule(pattern6334, replacement6334)
def With6335(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern6335 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*asinh(u_)), x_), cons2, cons3, cons1230, cons1905, CustomConstraint(With6335))
def replacement6335(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(6335)
return -Dist(b, Int(SimplifyIntegrand(w*D(u, x)/sqrt(u**S(2) + S(1)), x), x), x) + Dist(a + b*asinh(u), w, x)
rule6335 = ReplacementRule(pattern6335, replacement6335)
def With6336(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern6336 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*acosh(u_)), x_), cons2, cons3, cons1230, cons1906, CustomConstraint(With6336))
def replacement6336(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(6336)
return -Dist(b, Int(SimplifyIntegrand(w*D(u, x)/(sqrt(u + S(-1))*sqrt(u + S(1))), x), x), x) + Dist(a + b*acosh(u), w, x)
rule6336 = ReplacementRule(pattern6336, replacement6336)
pattern6337 = Pattern(Integral(exp(WC('n', S(1))*asinh(u_)), x_), cons85, cons804)
def replacement6337(x, n, u):
rubi.append(6337)
return Int((u + sqrt(u**S(2) + S(1)))**n, x)
rule6337 = ReplacementRule(pattern6337, replacement6337)
pattern6338 = Pattern(Integral(x_**WC('m', S(1))*exp(WC('n', S(1))*asinh(u_)), x_), cons31, cons85, cons804)
def replacement6338(x, m, n, u):
rubi.append(6338)
return Int(x**m*(u + sqrt(u**S(2) + S(1)))**n, x)
rule6338 = ReplacementRule(pattern6338, replacement6338)
pattern6339 = Pattern(Integral(exp(WC('n', S(1))*acosh(u_)), x_), cons85, cons804)
def replacement6339(x, n, u):
rubi.append(6339)
return Int((u + sqrt(u + S(-1))*sqrt(u + S(1)))**n, x)
rule6339 = ReplacementRule(pattern6339, replacement6339)
pattern6340 = Pattern(Integral(x_**WC('m', S(1))*exp(WC('n', S(1))*acosh(u_)), x_), cons31, cons85, cons804)
def replacement6340(x, m, n, u):
rubi.append(6340)
return Int(x**m*(u + sqrt(u + S(-1))*sqrt(u + S(1)))**n, x)
rule6340 = ReplacementRule(pattern6340, replacement6340)
pattern6341 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons148)
def replacement6341(b, a, n, c, x):
rubi.append(6341)
return -Dist(b*c*n, Int(x*(a + b*atanh(c*x))**(n + S(-1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*atanh(c*x))**n, x)
rule6341 = ReplacementRule(pattern6341, replacement6341)
pattern6342 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons148)
def replacement6342(b, a, n, c, x):
rubi.append(6342)
return -Dist(b*c*n, Int(x*(a + b*acoth(c*x))**(n + S(-1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*acoth(c*x))**n, x)
rule6342 = ReplacementRule(pattern6342, replacement6342)
pattern6343 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons340)
def replacement6343(b, a, n, c, x):
rubi.append(6343)
return Int((a + b*atanh(c*x))**n, x)
rule6343 = ReplacementRule(pattern6343, replacement6343)
pattern6344 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons340)
def replacement6344(b, a, n, c, x):
rubi.append(6344)
return Int((a + b*acoth(c*x))**n, x)
rule6344 = ReplacementRule(pattern6344, replacement6344)
pattern6345 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1907, cons148)
def replacement6345(b, d, a, n, c, x, e):
rubi.append(6345)
return Dist(b*c*n/e, Int((a + b*atanh(c*x))**(n + S(-1))*log(S(2)*d/(d + e*x))/(-c**S(2)*x**S(2) + S(1)), x), x) - Simp((a + b*atanh(c*x))**n*log(S(2)*d/(d + e*x))/e, x)
rule6345 = ReplacementRule(pattern6345, replacement6345)
pattern6346 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1907, cons148)
def replacement6346(b, d, a, n, c, x, e):
rubi.append(6346)
return Dist(b*c*n/e, Int((a + b*acoth(c*x))**(n + S(-1))*log(S(2)*d/(d + e*x))/(-c**S(2)*x**S(2) + S(1)), x), x) - Simp((a + b*acoth(c*x))**n*log(S(2)*d/(d + e*x))/e, x)
rule6346 = ReplacementRule(pattern6346, replacement6346)
pattern6347 = Pattern(Integral(atanh(x_*WC('c', S(1)))/(d_ + x_*WC('e', S(1))), x_), cons7, cons27, cons48, cons1908, cons1909)
def replacement6347(x, d, c, e):
rubi.append(6347)
return -Simp(PolyLog(S(2), Simp(c*(d + e*x)/(c*d - e), x))/(S(2)*e), x) + Simp(PolyLog(S(2), Simp(c*(d + e*x)/(c*d + e), x))/(S(2)*e), x) - Simp(log(d + e*x)*atanh(c*d/e)/e, x)
rule6347 = ReplacementRule(pattern6347, replacement6347)
pattern6348 = Pattern(Integral(atanh(x_*WC('c', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons7, cons27, cons48, cons1776)
def replacement6348(d, c, x, e):
rubi.append(6348)
return -Dist(S(1)/2, Int(log(-c*x + S(1))/(d + e*x), x), x) + Dist(S(1)/2, Int(log(c*x + S(1))/(d + e*x), x), x)
rule6348 = ReplacementRule(pattern6348, replacement6348)
pattern6349 = Pattern(Integral(acoth(x_*WC('c', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons7, cons27, cons48, cons1776)
def replacement6349(d, c, x, e):
rubi.append(6349)
return -Dist(S(1)/2, Int(log(S(1) - S(1)/(c*x))/(d + e*x), x), x) + Dist(S(1)/2, Int(log(S(1) + S(1)/(c*x))/(d + e*x), x), x)
rule6349 = ReplacementRule(pattern6349, replacement6349)
pattern6350 = Pattern(Integral((a_ + WC('b', S(1))*atanh(x_*WC('c', S(1))))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement6350(b, d, c, a, x, e):
rubi.append(6350)
return Dist(b, Int(atanh(c*x)/(d + e*x), x), x) + Simp(a*log(RemoveContent(d + e*x, x))/e, x)
rule6350 = ReplacementRule(pattern6350, replacement6350)
pattern6351 = Pattern(Integral((a_ + WC('b', S(1))*acoth(x_*WC('c', S(1))))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement6351(b, d, c, a, x, e):
rubi.append(6351)
return Dist(b, Int(acoth(c*x)/(d + e*x), x), x) + Simp(a*log(RemoveContent(d + e*x, x))/e, x)
rule6351 = ReplacementRule(pattern6351, replacement6351)
pattern6352 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement6352(p, b, d, a, c, x, e):
rubi.append(6352)
return -Dist(b*c/(e*(p + S(1))), Int((d + e*x)**(p + S(1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*atanh(c*x))*(d + e*x)**(p + S(1))/(e*(p + S(1))), x)
rule6352 = ReplacementRule(pattern6352, replacement6352)
pattern6353 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement6353(p, b, d, a, c, x, e):
rubi.append(6353)
return -Dist(b*c/(e*(p + S(1))), Int((d + e*x)**(p + S(1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*acoth(c*x))*(d + e*x)**(p + S(1))/(e*(p + S(1))), x)
rule6353 = ReplacementRule(pattern6353, replacement6353)
pattern6354 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_/x_, x_), cons2, cons3, cons7, cons85, cons165)
def replacement6354(b, a, n, c, x):
rubi.append(6354)
return -Dist(S(2)*b*c*n, Int((a + b*atanh(c*x))**(n + S(-1))*atanh(S(1) - S(2)/(-c*x + S(1)))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(S(2)*(a + b*atanh(c*x))**n*atanh(S(1) - S(2)/(-c*x + S(1))), x)
rule6354 = ReplacementRule(pattern6354, replacement6354)
pattern6355 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_/x_, x_), cons2, cons3, cons7, cons85, cons165)
def replacement6355(b, a, n, c, x):
rubi.append(6355)
return -Dist(S(2)*b*c*n, Int((a + b*acoth(c*x))**(n + S(-1))*acoth(S(1) - S(2)/(-c*x + S(1)))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(S(2)*(a + b*acoth(c*x))**n*acoth(S(1) - S(2)/(-c*x + S(1))), x)
rule6355 = ReplacementRule(pattern6355, replacement6355)
pattern6356 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons21, cons85, cons165, cons66)
def replacement6356(m, b, a, c, n, x):
rubi.append(6356)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*atanh(c*x))**(n + S(-1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**(m + S(1))*(a + b*atanh(c*x))**n/(m + S(1)), x)
rule6356 = ReplacementRule(pattern6356, replacement6356)
pattern6357 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons21, cons85, cons165, cons66)
def replacement6357(m, b, a, c, n, x):
rubi.append(6357)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*acoth(c*x))**(n + S(-1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**(m + S(1))*(a + b*acoth(c*x))**n/(m + S(1)), x)
rule6357 = ReplacementRule(pattern6357, replacement6357)
pattern6358 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons464)
def replacement6358(p, b, d, a, n, c, x, e):
rubi.append(6358)
return Int(ExpandIntegrand((a + b*atanh(c*x))**n*(d + e*x)**p, x), x)
rule6358 = ReplacementRule(pattern6358, replacement6358)
pattern6359 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons464)
def replacement6359(p, b, d, a, n, c, x, e):
rubi.append(6359)
return Int(ExpandIntegrand((a + b*acoth(c*x))**n*(d + e*x)**p, x), x)
rule6359 = ReplacementRule(pattern6359, replacement6359)
pattern6360 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons340)
def replacement6360(p, b, d, a, c, n, x, e):
rubi.append(6360)
return Int((a + b*atanh(c*x))**n*(d + e*x)**p, x)
rule6360 = ReplacementRule(pattern6360, replacement6360)
pattern6361 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons340)
def replacement6361(p, b, d, a, c, n, x, e):
rubi.append(6361)
return Int((a + b*acoth(c*x))**n*(d + e*x)**p, x)
rule6361 = ReplacementRule(pattern6361, replacement6361)
pattern6362 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1907, cons148, cons31, cons168)
def replacement6362(m, b, d, a, n, c, x, e):
rubi.append(6362)
return Dist(S(1)/e, Int(x**(m + S(-1))*(a + b*atanh(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-1))*(a + b*atanh(c*x))**n/(d + e*x), x), x)
rule6362 = ReplacementRule(pattern6362, replacement6362)
pattern6363 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1907, cons148, cons31, cons168)
def replacement6363(m, b, d, a, n, c, x, e):
rubi.append(6363)
return Dist(S(1)/e, Int(x**(m + S(-1))*(a + b*acoth(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-1))*(a + b*acoth(c*x))**n/(d + e*x), x), x)
rule6363 = ReplacementRule(pattern6363, replacement6363)
pattern6364 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1907, cons148)
def replacement6364(b, d, a, n, c, x, e):
rubi.append(6364)
return -Dist(b*c*n/d, Int((a + b*atanh(c*x))**(n + S(-1))*log(S(2)*e*x/(d + e*x))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*atanh(c*x))**n*log(S(2)*e*x/(d + e*x))/d, x)
rule6364 = ReplacementRule(pattern6364, replacement6364)
pattern6365 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1907, cons148)
def replacement6365(b, d, a, n, c, x, e):
rubi.append(6365)
return -Dist(b*c*n/d, Int((a + b*acoth(c*x))**(n + S(-1))*log(S(2)*e*x/(d + e*x))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*acoth(c*x))**n*log(S(2)*e*x/(d + e*x))/d, x)
rule6365 = ReplacementRule(pattern6365, replacement6365)
pattern6366 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1907, cons148, cons31, cons94)
def replacement6366(m, b, d, a, n, c, x, e):
rubi.append(6366)
return Dist(S(1)/d, Int(x**m*(a + b*atanh(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(1))*(a + b*atanh(c*x))**n/(d + e*x), x), x)
rule6366 = ReplacementRule(pattern6366, replacement6366)
pattern6367 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1907, cons148, cons31, cons94)
def replacement6367(m, b, d, a, n, c, x, e):
rubi.append(6367)
return Dist(S(1)/d, Int(x**m*(a + b*acoth(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(1))*(a + b*acoth(c*x))**n/(d + e*x), x), x)
rule6367 = ReplacementRule(pattern6367, replacement6367)
pattern6368 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons1777)
def replacement6368(p, m, b, d, a, n, c, x, e):
rubi.append(6368)
return Int(ExpandIntegrand(x**m*(a + b*atanh(c*x))**n*(d + e*x)**p, x), x)
rule6368 = ReplacementRule(pattern6368, replacement6368)
pattern6369 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons1777)
def replacement6369(p, m, b, d, a, n, c, x, e):
rubi.append(6369)
return Int(ExpandIntegrand(x**m*(a + b*acoth(c*x))**n*(d + e*x)**p, x), x)
rule6369 = ReplacementRule(pattern6369, replacement6369)
pattern6370 = Pattern(Integral(x_**WC('m', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement6370(p, m, b, d, a, n, c, x, e):
rubi.append(6370)
return Int(x**m*(a + b*atanh(c*x))**n*(d + e*x)**p, x)
rule6370 = ReplacementRule(pattern6370, replacement6370)
pattern6371 = Pattern(Integral(x_**WC('m', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement6371(p, m, b, d, a, n, c, x, e):
rubi.append(6371)
return Int(x**m*(a + b*acoth(c*x))**n*(d + e*x)**p, x)
rule6371 = ReplacementRule(pattern6371, replacement6371)
pattern6372 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons13, cons163)
def replacement6372(p, b, d, a, c, x, e):
rubi.append(6372)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*atanh(c*x))*(d + e*x**S(2))**(p + S(-1)), x), x) + Simp(x*(a + b*atanh(c*x))*(d + e*x**S(2))**p/(S(2)*p + S(1)), x) + Simp(b*(d + e*x**S(2))**p/(S(2)*c*p*(S(2)*p + S(1))), x)
rule6372 = ReplacementRule(pattern6372, replacement6372)
pattern6373 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons13, cons163)
def replacement6373(p, b, d, a, c, x, e):
rubi.append(6373)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*acoth(c*x))*(d + e*x**S(2))**(p + S(-1)), x), x) + Simp(x*(a + b*acoth(c*x))*(d + e*x**S(2))**p/(S(2)*p + S(1)), x) + Simp(b*(d + e*x**S(2))**p/(S(2)*c*p*(S(2)*p + S(1))), x)
rule6373 = ReplacementRule(pattern6373, replacement6373)
pattern6374 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons163, cons165)
def replacement6374(p, b, d, a, c, n, x, e):
rubi.append(6374)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b**S(2)*d*n*(n + S(-1))/(S(2)*p*(S(2)*p + S(1))), Int((a + b*atanh(c*x))**(n + S(-2))*(d + e*x**S(2))**(p + S(-1)), x), x) + Simp(x*(a + b*atanh(c*x))**n*(d + e*x**S(2))**p/(S(2)*p + S(1)), x) + Simp(b*n*(a + b*atanh(c*x))**(n + S(-1))*(d + e*x**S(2))**p/(S(2)*c*p*(S(2)*p + S(1))), x)
rule6374 = ReplacementRule(pattern6374, replacement6374)
pattern6375 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons163, cons165)
def replacement6375(p, b, d, a, c, n, x, e):
rubi.append(6375)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b**S(2)*d*n*(n + S(-1))/(S(2)*p*(S(2)*p + S(1))), Int((a + b*acoth(c*x))**(n + S(-2))*(d + e*x**S(2))**(p + S(-1)), x), x) + Simp(x*(a + b*acoth(c*x))**n*(d + e*x**S(2))**p/(S(2)*p + S(1)), x) + Simp(b*n*(a + b*acoth(c*x))**(n + S(-1))*(d + e*x**S(2))**p/(S(2)*c*p*(S(2)*p + S(1))), x)
rule6375 = ReplacementRule(pattern6375, replacement6375)
pattern6376 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons1737)
def replacement6376(b, d, a, c, x, e):
rubi.append(6376)
return Simp(log(RemoveContent(a + b*atanh(c*x), x))/(b*c*d), x)
rule6376 = ReplacementRule(pattern6376, replacement6376)
pattern6377 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons1737)
def replacement6377(b, d, a, c, x, e):
rubi.append(6377)
return Simp(log(RemoveContent(a + b*acoth(c*x), x))/(b*c*d), x)
rule6377 = ReplacementRule(pattern6377, replacement6377)
pattern6378 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons584)
def replacement6378(b, d, a, n, c, x, e):
rubi.append(6378)
return Simp((a + b*atanh(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule6378 = ReplacementRule(pattern6378, replacement6378)
pattern6379 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons584)
def replacement6379(b, d, a, n, c, x, e):
rubi.append(6379)
return Simp((a + b*acoth(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule6379 = ReplacementRule(pattern6379, replacement6379)
pattern6380 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons268)
def replacement6380(b, d, a, c, x, e):
rubi.append(6380)
return Simp(-S(2)*(a + b*atanh(c*x))*ArcTan(sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/(c*sqrt(d)), x) - Simp(I*b*PolyLog(S(2), -I*sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/(c*sqrt(d)), x) + Simp(I*b*PolyLog(S(2), I*sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/(c*sqrt(d)), x)
rule6380 = ReplacementRule(pattern6380, replacement6380)
pattern6381 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons268)
def replacement6381(b, d, a, c, x, e):
rubi.append(6381)
return Simp(-S(2)*(a + b*acoth(c*x))*ArcTan(sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/(c*sqrt(d)), x) - Simp(I*b*PolyLog(S(2), -I*sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/(c*sqrt(d)), x) + Simp(I*b*PolyLog(S(2), I*sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/(c*sqrt(d)), x)
rule6381 = ReplacementRule(pattern6381, replacement6381)
pattern6382 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148, cons268)
def replacement6382(b, d, a, n, c, x, e):
rubi.append(6382)
return Dist(S(1)/(c*sqrt(d)), Subst(Int((a + b*x)**n/cosh(x), x), x, atanh(c*x)), x)
rule6382 = ReplacementRule(pattern6382, replacement6382)
pattern6383 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148, cons268)
def replacement6383(b, d, a, n, c, x, e):
rubi.append(6383)
return -Dist(x*sqrt(S(1) - S(1)/(c**S(2)*x**S(2)))/sqrt(d + e*x**S(2)), Subst(Int((a + b*x)**n/sinh(x), x), x, acoth(c*x)), x)
rule6383 = ReplacementRule(pattern6383, replacement6383)
pattern6384 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148, cons1738)
def replacement6384(b, d, a, n, c, x, e):
rubi.append(6384)
return Dist(sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*atanh(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x)
rule6384 = ReplacementRule(pattern6384, replacement6384)
pattern6385 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148, cons1738)
def replacement6385(b, d, a, n, c, x, e):
rubi.append(6385)
return Dist(sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*acoth(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x)
rule6385 = ReplacementRule(pattern6385, replacement6385)
pattern6386 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement6386(b, d, a, n, c, x, e):
rubi.append(6386)
return -Dist(b*c*n/S(2), Int(x*(a + b*atanh(c*x))**(n + S(-1))/(d + e*x**S(2))**S(2), x), x) + Simp(x*(a + b*atanh(c*x))**n/(S(2)*d*(d + e*x**S(2))), x) + Simp((a + b*atanh(c*x))**(n + S(1))/(S(2)*b*c*d**S(2)*(n + S(1))), x)
rule6386 = ReplacementRule(pattern6386, replacement6386)
pattern6387 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement6387(b, d, a, n, c, x, e):
rubi.append(6387)
return -Dist(b*c*n/S(2), Int(x*(a + b*acoth(c*x))**(n + S(-1))/(d + e*x**S(2))**S(2), x), x) + Simp(x*(a + b*acoth(c*x))**n/(S(2)*d*(d + e*x**S(2))), x) + Simp((a + b*acoth(c*x))**(n + S(1))/(S(2)*b*c*d**S(2)*(n + S(1))), x)
rule6387 = ReplacementRule(pattern6387, replacement6387)
pattern6388 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1737)
def replacement6388(b, d, a, c, x, e):
rubi.append(6388)
return -Simp(b/(c*d*sqrt(d + e*x**S(2))), x) + Simp(x*(a + b*atanh(c*x))/(d*sqrt(d + e*x**S(2))), x)
rule6388 = ReplacementRule(pattern6388, replacement6388)
pattern6389 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1737)
def replacement6389(b, d, a, c, x, e):
rubi.append(6389)
return -Simp(b/(c*d*sqrt(d + e*x**S(2))), x) + Simp(x*(a + b*acoth(c*x))/(d*sqrt(d + e*x**S(2))), x)
rule6389 = ReplacementRule(pattern6389, replacement6389)
pattern6390 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons13, cons137, cons230)
def replacement6390(p, b, d, a, c, x, e):
rubi.append(6390)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*atanh(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*(d + e*x**S(2))**(p + S(1))/(S(4)*c*d*(p + S(1))**S(2)), x) - Simp(x*(a + b*atanh(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x)
rule6390 = ReplacementRule(pattern6390, replacement6390)
pattern6391 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons13, cons137, cons230)
def replacement6391(p, b, d, a, c, x, e):
rubi.append(6391)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*acoth(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*(d + e*x**S(2))**(p + S(1))/(S(4)*c*d*(p + S(1))**S(2)), x) - Simp(x*(a + b*acoth(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x)
rule6391 = ReplacementRule(pattern6391, replacement6391)
pattern6392 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons165)
def replacement6392(b, d, a, c, n, x, e):
rubi.append(6392)
return Dist(b**S(2)*n*(n + S(-1)), Int((a + b*atanh(c*x))**(n + S(-2))/(d + e*x**S(2))**(S(3)/2), x), x) + Simp(x*(a + b*atanh(c*x))**n/(d*sqrt(d + e*x**S(2))), x) - Simp(b*n*(a + b*atanh(c*x))**(n + S(-1))/(c*d*sqrt(d + e*x**S(2))), x)
rule6392 = ReplacementRule(pattern6392, replacement6392)
pattern6393 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons165)
def replacement6393(b, d, a, c, n, x, e):
rubi.append(6393)
return Dist(b**S(2)*n*(n + S(-1)), Int((a + b*acoth(c*x))**(n + S(-2))/(d + e*x**S(2))**(S(3)/2), x), x) + Simp(x*(a + b*acoth(c*x))**n/(d*sqrt(d + e*x**S(2))), x) - Simp(b*n*(a + b*acoth(c*x))**(n + S(-1))/(c*d*sqrt(d + e*x**S(2))), x)
rule6393 = ReplacementRule(pattern6393, replacement6393)
pattern6394 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons137, cons165, cons230)
def replacement6394(p, b, d, a, c, n, x, e):
rubi.append(6394)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Dist(b**S(2)*n*(n + S(-1))/(S(4)*(p + S(1))**S(2)), Int((a + b*atanh(c*x))**(n + S(-2))*(d + e*x**S(2))**p, x), x) - Simp(x*(a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x) - Simp(b*n*(a + b*atanh(c*x))**(n + S(-1))*(d + e*x**S(2))**(p + S(1))/(S(4)*c*d*(p + S(1))**S(2)), x)
rule6394 = ReplacementRule(pattern6394, replacement6394)
pattern6395 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons137, cons165, cons230)
def replacement6395(p, b, d, a, c, n, x, e):
rubi.append(6395)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Dist(b**S(2)*n*(n + S(-1))/(S(4)*(p + S(1))**S(2)), Int((a + b*acoth(c*x))**(n + S(-2))*(d + e*x**S(2))**p, x), x) - Simp(x*(a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x) - Simp(b*n*(a + b*acoth(c*x))**(n + S(-1))*(d + e*x**S(2))**(p + S(1))/(S(4)*c*d*(p + S(1))**S(2)), x)
rule6395 = ReplacementRule(pattern6395, replacement6395)
pattern6396 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons137, cons89)
def replacement6396(p, b, d, a, c, n, x, e):
rubi.append(6396)
return Dist(S(2)*c*(p + S(1))/(b*(n + S(1))), Int(x*(a + b*atanh(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp((a + b*atanh(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule6396 = ReplacementRule(pattern6396, replacement6396)
pattern6397 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons137, cons89)
def replacement6397(p, b, d, a, c, n, x, e):
rubi.append(6397)
return Dist(S(2)*c*(p + S(1))/(b*(n + S(1))), Int(x*(a + b*acoth(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp((a + b*acoth(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule6397 = ReplacementRule(pattern6397, replacement6397)
pattern6398 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1779, cons1740)
def replacement6398(p, b, d, a, n, c, x, e):
rubi.append(6398)
return Dist(d**p/c, Subst(Int((a + b*x)**n*cosh(x)**(-S(2)*p + S(-2)), x), x, atanh(c*x)), x)
rule6398 = ReplacementRule(pattern6398, replacement6398)
pattern6399 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1779, cons1741)
def replacement6399(p, b, d, a, n, c, x, e):
rubi.append(6399)
return Dist(d**(p + S(1)/2)*sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*atanh(c*x))**n*(-c**S(2)*x**S(2) + S(1))**p, x), x)
rule6399 = ReplacementRule(pattern6399, replacement6399)
pattern6400 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1779, cons38)
def replacement6400(p, b, d, a, n, c, x, e):
rubi.append(6400)
return -Dist((-d)**p/c, Subst(Int((a + b*x)**n*sinh(x)**(-S(2)*p + S(-2)), x), x, acoth(c*x)), x)
rule6400 = ReplacementRule(pattern6400, replacement6400)
pattern6401 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1779, cons147)
def replacement6401(p, b, d, a, n, c, x, e):
rubi.append(6401)
return -Dist(x*(-d)**(p + S(1)/2)*sqrt((c**S(2)*x**S(2) + S(-1))/(c**S(2)*x**S(2)))/sqrt(d + e*x**S(2)), Subst(Int((a + b*x)**n*sinh(x)**(-S(2)*p + S(-2)), x), x, acoth(c*x)), x)
rule6401 = ReplacementRule(pattern6401, replacement6401)
pattern6402 = Pattern(Integral(atanh(x_*WC('c', S(1)))/(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons7, cons27, cons48, cons1776)
def replacement6402(d, c, x, e):
rubi.append(6402)
return -Dist(S(1)/2, Int(log(-c*x + S(1))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int(log(c*x + S(1))/(d + e*x**S(2)), x), x)
rule6402 = ReplacementRule(pattern6402, replacement6402)
pattern6403 = Pattern(Integral(acoth(x_*WC('c', S(1)))/(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons7, cons27, cons48, cons1776)
def replacement6403(d, c, x, e):
rubi.append(6403)
return -Dist(S(1)/2, Int(log(S(1) - S(1)/(c*x))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int(log(S(1) + S(1)/(c*x))/(d + e*x**S(2)), x), x)
rule6403 = ReplacementRule(pattern6403, replacement6403)
pattern6404 = Pattern(Integral((a_ + WC('b', S(1))*atanh(x_*WC('c', S(1))))/(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement6404(b, d, c, a, x, e):
rubi.append(6404)
return Dist(a, Int(S(1)/(d + e*x**S(2)), x), x) + Dist(b, Int(atanh(c*x)/(d + e*x**S(2)), x), x)
rule6404 = ReplacementRule(pattern6404, replacement6404)
pattern6405 = Pattern(Integral((a_ + WC('b', S(1))*acoth(x_*WC('c', S(1))))/(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement6405(b, d, c, a, x, e):
rubi.append(6405)
return Dist(a, Int(S(1)/(d + e*x**S(2)), x), x) + Dist(b, Int(acoth(c*x)/(d + e*x**S(2)), x), x)
rule6405 = ReplacementRule(pattern6405, replacement6405)
def With6406(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(6406)
return -Dist(b*c, Int(ExpandIntegrand(u/(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*atanh(c*x), u, x)
pattern6406 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1780)
rule6406 = ReplacementRule(pattern6406, With6406)
def With6407(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(6407)
return -Dist(b*c, Int(ExpandIntegrand(u/(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acoth(c*x), u, x)
pattern6407 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1780)
rule6407 = ReplacementRule(pattern6407, With6407)
pattern6408 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons148)
def replacement6408(p, b, d, a, n, c, x, e):
rubi.append(6408)
return Int(ExpandIntegrand((a + b*atanh(c*x))**n*(d + e*x**S(2))**p, x), x)
rule6408 = ReplacementRule(pattern6408, replacement6408)
pattern6409 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons148)
def replacement6409(p, b, d, a, n, c, x, e):
rubi.append(6409)
return Int(ExpandIntegrand((a + b*acoth(c*x))**n*(d + e*x**S(2))**p, x), x)
rule6409 = ReplacementRule(pattern6409, replacement6409)
pattern6410 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement6410(p, b, d, a, n, c, x, e):
rubi.append(6410)
return Int((a + b*atanh(c*x))**n*(d + e*x**S(2))**p, x)
rule6410 = ReplacementRule(pattern6410, replacement6410)
pattern6411 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement6411(p, b, d, a, n, c, x, e):
rubi.append(6411)
return Int((a + b*acoth(c*x))**n*(d + e*x**S(2))**p, x)
rule6411 = ReplacementRule(pattern6411, replacement6411)
pattern6412 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons93, cons88, cons166)
def replacement6412(m, b, d, a, n, c, x, e):
rubi.append(6412)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*atanh(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*atanh(c*x))**n/(d + e*x**S(2)), x), x)
rule6412 = ReplacementRule(pattern6412, replacement6412)
pattern6413 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons93, cons88, cons166)
def replacement6413(m, b, d, a, n, c, x, e):
rubi.append(6413)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*acoth(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*acoth(c*x))**n/(d + e*x**S(2)), x), x)
rule6413 = ReplacementRule(pattern6413, replacement6413)
pattern6414 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons93, cons88, cons94)
def replacement6414(m, b, d, a, n, c, x, e):
rubi.append(6414)
return Dist(S(1)/d, Int(x**m*(a + b*atanh(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*atanh(c*x))**n/(d + e*x**S(2)), x), x)
rule6414 = ReplacementRule(pattern6414, replacement6414)
pattern6415 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons93, cons88, cons94)
def replacement6415(m, b, d, a, n, c, x, e):
rubi.append(6415)
return Dist(S(1)/d, Int(x**m*(a + b*acoth(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*acoth(c*x))**n/(d + e*x**S(2)), x), x)
rule6415 = ReplacementRule(pattern6415, replacement6415)
pattern6416 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement6416(b, d, a, n, c, x, e):
rubi.append(6416)
return Dist(S(1)/(c*d), Int((a + b*atanh(c*x))**n/(-c*x + S(1)), x), x) + Simp((a + b*atanh(c*x))**(n + S(1))/(b*e*(n + S(1))), x)
rule6416 = ReplacementRule(pattern6416, replacement6416)
pattern6417 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement6417(b, d, a, n, c, x, e):
rubi.append(6417)
return Dist(S(1)/(c*d), Int((a + b*acoth(c*x))**n/(-c*x + S(1)), x), x) + Simp((a + b*acoth(c*x))**(n + S(1))/(b*e*(n + S(1))), x)
rule6417 = ReplacementRule(pattern6417, replacement6417)
pattern6418 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons340, cons584)
def replacement6418(b, d, a, c, n, x, e):
rubi.append(6418)
return -Dist(S(1)/(b*c*d*(n + S(1))), Int((a + b*atanh(c*x))**(n + S(1)), x), x) + Simp(x*(a + b*atanh(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule6418 = ReplacementRule(pattern6418, replacement6418)
pattern6419 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons340, cons584)
def replacement6419(b, d, a, c, n, x, e):
rubi.append(6419)
return -Dist(S(1)/(b*c*d*(n + S(1))), Int((a + b*acoth(c*x))**(n + S(1)), x), x) - Simp(x*(a + b*acoth(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule6419 = ReplacementRule(pattern6419, replacement6419)
pattern6420 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons93, cons88, cons166)
def replacement6420(m, b, d, a, n, c, x, e):
rubi.append(6420)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*atanh(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*atanh(c*x))**n/(d + e*x**S(2)), x), x)
rule6420 = ReplacementRule(pattern6420, replacement6420)
pattern6421 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons93, cons88, cons166)
def replacement6421(m, b, d, a, n, c, x, e):
rubi.append(6421)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*acoth(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*acoth(c*x))**n/(d + e*x**S(2)), x), x)
rule6421 = ReplacementRule(pattern6421, replacement6421)
pattern6422 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement6422(b, d, a, n, c, x, e):
rubi.append(6422)
return Dist(S(1)/d, Int((a + b*atanh(c*x))**n/(x*(c*x + S(1))), x), x) + Simp((a + b*atanh(c*x))**(n + S(1))/(b*d*(n + S(1))), x)
rule6422 = ReplacementRule(pattern6422, replacement6422)
pattern6423 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement6423(b, d, a, n, c, x, e):
rubi.append(6423)
return Dist(S(1)/d, Int((a + b*acoth(c*x))**n/(x*(c*x + S(1))), x), x) + Simp((a + b*acoth(c*x))**(n + S(1))/(b*d*(n + S(1))), x)
rule6423 = ReplacementRule(pattern6423, replacement6423)
pattern6424 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons93, cons88, cons94)
def replacement6424(m, b, d, a, n, c, x, e):
rubi.append(6424)
return Dist(S(1)/d, Int(x**m*(a + b*atanh(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*atanh(c*x))**n/(d + e*x**S(2)), x), x)
rule6424 = ReplacementRule(pattern6424, replacement6424)
pattern6425 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons93, cons88, cons94)
def replacement6425(m, b, d, a, n, c, x, e):
rubi.append(6425)
return Dist(S(1)/d, Int(x**m*(a + b*acoth(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*acoth(c*x))**n/(d + e*x**S(2)), x), x)
rule6425 = ReplacementRule(pattern6425, replacement6425)
pattern6426 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons87, cons89)
def replacement6426(m, b, d, a, c, n, x, e):
rubi.append(6426)
return -Dist(m/(b*c*d*(n + S(1))), Int(x**(m + S(-1))*(a + b*atanh(c*x))**(n + S(1)), x), x) + Simp(x**m*(a + b*atanh(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule6426 = ReplacementRule(pattern6426, replacement6426)
pattern6427 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons87, cons89)
def replacement6427(m, b, d, a, c, n, x, e):
rubi.append(6427)
return -Dist(m/(b*c*d*(n + S(1))), Int(x**(m + S(-1))*(a + b*acoth(c*x))**(n + S(1)), x), x) + Simp(x**m*(a + b*acoth(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule6427 = ReplacementRule(pattern6427, replacement6427)
pattern6428 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons17, cons1781)
def replacement6428(m, b, d, a, c, x, e):
rubi.append(6428)
return Int(ExpandIntegrand(a + b*atanh(c*x), x**m/(d + e*x**S(2)), x), x)
rule6428 = ReplacementRule(pattern6428, replacement6428)
pattern6429 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons17, cons1781)
def replacement6429(m, b, d, a, c, x, e):
rubi.append(6429)
return Int(ExpandIntegrand(a + b*acoth(c*x), x**m/(d + e*x**S(2)), x), x)
rule6429 = ReplacementRule(pattern6429, replacement6429)
pattern6430 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons54)
def replacement6430(p, b, d, a, n, c, x, e):
rubi.append(6430)
return Dist(b*n/(S(2)*c*(p + S(1))), Int((a + b*atanh(c*x))**(n + S(-1))*(d + e*x**S(2))**p, x), x) + Simp((a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6430 = ReplacementRule(pattern6430, replacement6430)
pattern6431 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons54)
def replacement6431(p, b, d, a, n, c, x, e):
rubi.append(6431)
return Dist(b*n/(S(2)*c*(p + S(1))), Int((a + b*acoth(c*x))**(n + S(-1))*(d + e*x**S(2))**p, x), x) + Simp((a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6431 = ReplacementRule(pattern6431, replacement6431)
pattern6432 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons89, cons1442)
def replacement6432(b, d, a, c, n, x, e):
rubi.append(6432)
return Dist(S(4)/(b**S(2)*(n + S(1))*(n + S(2))), Int(x*(a + b*atanh(c*x))**(n + S(2))/(d + e*x**S(2))**S(2), x), x) + Simp((a + b*atanh(c*x))**(n + S(2))*(c**S(2)*x**S(2) + S(1))/(b**S(2)*e*(d + e*x**S(2))*(n + S(1))*(n + S(2))), x) + Simp(x*(a + b*atanh(c*x))**(n + S(1))/(b*c*d*(d + e*x**S(2))*(n + S(1))), x)
rule6432 = ReplacementRule(pattern6432, replacement6432)
pattern6433 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons89, cons1442)
def replacement6433(b, d, a, c, n, x, e):
rubi.append(6433)
return Dist(S(4)/(b**S(2)*(n + S(1))*(n + S(2))), Int(x*(a + b*acoth(c*x))**(n + S(2))/(d + e*x**S(2))**S(2), x), x) + Simp((a + b*acoth(c*x))**(n + S(2))*(c**S(2)*x**S(2) + S(1))/(b**S(2)*e*(d + e*x**S(2))*(n + S(1))*(n + S(2))), x) + Simp(x*(a + b*acoth(c*x))**(n + S(1))/(b*c*d*(d + e*x**S(2))*(n + S(1))), x)
rule6433 = ReplacementRule(pattern6433, replacement6433)
pattern6434 = Pattern(Integral(x_**S(2)*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons13, cons137, cons1782)
def replacement6434(p, b, d, a, c, x, e):
rubi.append(6434)
return Dist(S(1)/(S(2)*c**S(2)*d*(p + S(1))), Int((a + b*atanh(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*(d + e*x**S(2))**(p + S(1))/(S(4)*c**S(3)*d*(p + S(1))**S(2)), x) - Simp(x*(a + b*atanh(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*c**S(2)*d*(p + S(1))), x)
rule6434 = ReplacementRule(pattern6434, replacement6434)
pattern6435 = Pattern(Integral(x_**S(2)*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons13, cons137, cons1782)
def replacement6435(p, b, d, a, c, x, e):
rubi.append(6435)
return Dist(S(1)/(S(2)*c**S(2)*d*(p + S(1))), Int((a + b*acoth(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*(d + e*x**S(2))**(p + S(1))/(S(4)*c**S(3)*d*(p + S(1))**S(2)), x) - Simp(x*(a + b*acoth(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*c**S(2)*d*(p + S(1))), x)
rule6435 = ReplacementRule(pattern6435, replacement6435)
pattern6436 = Pattern(Integral(x_**S(2)*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement6436(b, d, a, n, c, x, e):
rubi.append(6436)
return -Dist(b*n/(S(2)*c), Int(x*(a + b*atanh(c*x))**(n + S(-1))/(d + e*x**S(2))**S(2), x), x) - Simp((a + b*atanh(c*x))**(n + S(1))/(S(2)*b*c**S(3)*d**S(2)*(n + S(1))), x) + Simp(x*(a + b*atanh(c*x))**n/(S(2)*c**S(2)*d*(d + e*x**S(2))), x)
rule6436 = ReplacementRule(pattern6436, replacement6436)
pattern6437 = Pattern(Integral(x_**S(2)*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement6437(b, d, a, n, c, x, e):
rubi.append(6437)
return -Dist(b*n/(S(2)*c), Int(x*(a + b*acoth(c*x))**(n + S(-1))/(d + e*x**S(2))**S(2), x), x) - Simp((a + b*acoth(c*x))**(n + S(1))/(S(2)*b*c**S(3)*d**S(2)*(n + S(1))), x) + Simp(x*(a + b*acoth(c*x))**n/(S(2)*c**S(2)*d*(d + e*x**S(2))), x)
rule6437 = ReplacementRule(pattern6437, replacement6437)
pattern6438 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons240, cons13, cons137)
def replacement6438(p, m, b, d, a, c, x, e):
rubi.append(6438)
return -Dist((m + S(-1))/(c**S(2)*d*m), Int(x**(m + S(-2))*(a + b*atanh(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*x**m*(d + e*x**S(2))**(p + S(1))/(c*d*m**S(2)), x) + Simp(x**(m + S(-1))*(a + b*atanh(c*x))*(d + e*x**S(2))**(p + S(1))/(c**S(2)*d*m), x)
rule6438 = ReplacementRule(pattern6438, replacement6438)
pattern6439 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons240, cons13, cons137)
def replacement6439(p, m, b, d, a, c, x, e):
rubi.append(6439)
return -Dist((m + S(-1))/(c**S(2)*d*m), Int(x**(m + S(-2))*(a + b*acoth(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*x**m*(d + e*x**S(2))**(p + S(1))/(c*d*m**S(2)), x) + Simp(x**(m + S(-1))*(a + b*acoth(c*x))*(d + e*x**S(2))**(p + S(1))/(c**S(2)*d*m), x)
rule6439 = ReplacementRule(pattern6439, replacement6439)
pattern6440 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons240, cons338, cons137, cons165)
def replacement6440(p, m, b, d, a, c, n, x, e):
rubi.append(6440)
return Dist(b**S(2)*n*(n + S(-1))/m**S(2), Int(x**m*(a + b*atanh(c*x))**(n + S(-2))*(d + e*x**S(2))**p, x), x) - Dist((m + S(-1))/(c**S(2)*d*m), Int(x**(m + S(-2))*(a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Simp(x**(m + S(-1))*(a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(c**S(2)*d*m), x) - Simp(b*n*x**m*(a + b*atanh(c*x))**(n + S(-1))*(d + e*x**S(2))**(p + S(1))/(c*d*m**S(2)), x)
rule6440 = ReplacementRule(pattern6440, replacement6440)
pattern6441 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons240, cons338, cons137, cons165)
def replacement6441(p, m, b, d, a, c, n, x, e):
rubi.append(6441)
return Dist(b**S(2)*n*(n + S(-1))/m**S(2), Int(x**m*(a + b*acoth(c*x))**(n + S(-2))*(d + e*x**S(2))**p, x), x) - Dist((m + S(-1))/(c**S(2)*d*m), Int(x**(m + S(-2))*(a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Simp(x**(m + S(-1))*(a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(1))/(c**S(2)*d*m), x) - Simp(b*n*x**m*(a + b*acoth(c*x))**(n + S(-1))*(d + e*x**S(2))**(p + S(1))/(c*d*m**S(2)), x)
rule6441 = ReplacementRule(pattern6441, replacement6441)
pattern6442 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1737, cons240, cons87, cons89)
def replacement6442(p, m, b, d, a, c, n, x, e):
rubi.append(6442)
return -Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*atanh(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp(x**m*(a + b*atanh(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule6442 = ReplacementRule(pattern6442, replacement6442)
pattern6443 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1737, cons240, cons87, cons89)
def replacement6443(p, m, b, d, a, c, n, x, e):
rubi.append(6443)
return -Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*acoth(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp(x**m*(a + b*acoth(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule6443 = ReplacementRule(pattern6443, replacement6443)
pattern6444 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1737, cons242, cons87, cons88, cons66)
def replacement6444(p, m, b, d, a, n, c, x, e):
rubi.append(6444)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*atanh(c*x))**(n + S(-1))*(d + e*x**S(2))**p, x), x) + Simp(x**(m + S(1))*(a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*(m + S(1))), x)
rule6444 = ReplacementRule(pattern6444, replacement6444)
pattern6445 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1737, cons242, cons87, cons88, cons66)
def replacement6445(p, m, b, d, a, n, c, x, e):
rubi.append(6445)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*acoth(c*x))**(n + S(-1))*(d + e*x**S(2))**p, x), x) + Simp(x**(m + S(1))*(a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*(m + S(1))), x)
rule6445 = ReplacementRule(pattern6445, replacement6445)
pattern6446 = Pattern(Integral(x_**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons241)
def replacement6446(m, b, d, a, c, x, e):
rubi.append(6446)
return Dist(d/(m + S(2)), Int(x**m*(a + b*atanh(c*x))/sqrt(d + e*x**S(2)), x), x) - Dist(b*c*d/(m + S(2)), Int(x**(m + S(1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*atanh(c*x))*sqrt(d + e*x**S(2))/(m + S(2)), x)
rule6446 = ReplacementRule(pattern6446, replacement6446)
pattern6447 = Pattern(Integral(x_**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons241)
def replacement6447(m, b, d, a, c, x, e):
rubi.append(6447)
return Dist(d/(m + S(2)), Int(x**m*(a + b*acoth(c*x))/sqrt(d + e*x**S(2)), x), x) - Dist(b*c*d/(m + S(2)), Int(x**(m + S(1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*acoth(c*x))*sqrt(d + e*x**S(2))/(m + S(2)), x)
rule6447 = ReplacementRule(pattern6447, replacement6447)
pattern6448 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons148, cons38, cons146)
def replacement6448(p, m, b, d, a, n, c, x, e):
rubi.append(6448)
return Int(ExpandIntegrand(x**m*(a + b*atanh(c*x))**n*(d + e*x**S(2))**p, x), x)
rule6448 = ReplacementRule(pattern6448, replacement6448)
pattern6449 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons148, cons38, cons146)
def replacement6449(p, m, b, d, a, n, c, x, e):
rubi.append(6449)
return Int(ExpandIntegrand(x**m*(a + b*acoth(c*x))**n*(d + e*x**S(2))**p, x), x)
rule6449 = ReplacementRule(pattern6449, replacement6449)
pattern6450 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons13, cons163, cons148, cons1783)
def replacement6450(p, m, b, d, a, n, c, x, e):
rubi.append(6450)
return Dist(d, Int(x**m*(a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(c**S(2)*d, Int(x**(m + S(2))*(a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x)
rule6450 = ReplacementRule(pattern6450, replacement6450)
pattern6451 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1737, cons13, cons163, cons148, cons1783)
def replacement6451(p, m, b, d, a, n, c, x, e):
rubi.append(6451)
return Dist(d, Int(x**m*(a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(c**S(2)*d, Int(x**(m + S(2))*(a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x)
rule6451 = ReplacementRule(pattern6451, replacement6451)
pattern6452 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons93, cons88, cons166)
def replacement6452(m, b, d, a, n, c, x, e):
rubi.append(6452)
return Dist((m + S(-1))/(c**S(2)*m), Int(x**(m + S(-2))*(a + b*atanh(c*x))**n/sqrt(d + e*x**S(2)), x), x) + Dist(b*n/(c*m), Int(x**(m + S(-1))*(a + b*atanh(c*x))**(n + S(-1))/sqrt(d + e*x**S(2)), x), x) - Simp(x**(m + S(-1))*(a + b*atanh(c*x))**n*sqrt(d + e*x**S(2))/(c**S(2)*d*m), x)
rule6452 = ReplacementRule(pattern6452, replacement6452)
pattern6453 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons93, cons88, cons166)
def replacement6453(m, b, d, a, n, c, x, e):
rubi.append(6453)
return Dist((m + S(-1))/(c**S(2)*m), Int(x**(m + S(-2))*(a + b*acoth(c*x))**n/sqrt(d + e*x**S(2)), x), x) + Dist(b*n/(c*m), Int(x**(m + S(-1))*(a + b*acoth(c*x))**(n + S(-1))/sqrt(d + e*x**S(2)), x), x) - Simp(x**(m + S(-1))*(a + b*acoth(c*x))**n*sqrt(d + e*x**S(2))/(c**S(2)*d*m), x)
rule6453 = ReplacementRule(pattern6453, replacement6453)
pattern6454 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons268)
def replacement6454(b, d, a, c, x, e):
rubi.append(6454)
return Simp(b*PolyLog(S(2), -sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/sqrt(d), x) - Simp(b*PolyLog(S(2), sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/sqrt(d), x) + Simp(-S(2)*(a + b*atanh(c*x))*atanh(sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/sqrt(d), x)
rule6454 = ReplacementRule(pattern6454, replacement6454)
pattern6455 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons268)
def replacement6455(b, d, a, c, x, e):
rubi.append(6455)
return Simp(b*PolyLog(S(2), -sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/sqrt(d), x) - Simp(b*PolyLog(S(2), sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/sqrt(d), x) + Simp(-S(2)*(a + b*acoth(c*x))*atanh(sqrt(-c*x + S(1))/sqrt(c*x + S(1)))/sqrt(d), x)
rule6455 = ReplacementRule(pattern6455, replacement6455)
pattern6456 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**n_/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148, cons268)
def replacement6456(b, d, a, c, n, x, e):
rubi.append(6456)
return Dist(S(1)/sqrt(d), Subst(Int((a + b*x)**n/sinh(x), x), x, atanh(c*x)), x)
rule6456 = ReplacementRule(pattern6456, replacement6456)
pattern6457 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**n_/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148, cons268)
def replacement6457(b, d, a, c, n, x, e):
rubi.append(6457)
return -Dist(c*x*sqrt(S(1) - S(1)/(c**S(2)*x**S(2)))/sqrt(d + e*x**S(2)), Subst(Int((a + b*x)**n/cosh(x), x), x, acoth(c*x)), x)
rule6457 = ReplacementRule(pattern6457, replacement6457)
pattern6458 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148, cons1738)
def replacement6458(b, d, a, n, c, x, e):
rubi.append(6458)
return Dist(sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*atanh(c*x))**n/(x*sqrt(-c**S(2)*x**S(2) + S(1))), x), x)
rule6458 = ReplacementRule(pattern6458, replacement6458)
pattern6459 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148, cons1738)
def replacement6459(b, d, a, n, c, x, e):
rubi.append(6459)
return Dist(sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*acoth(c*x))**n/(x*sqrt(-c**S(2)*x**S(2) + S(1))), x), x)
rule6459 = ReplacementRule(pattern6459, replacement6459)
pattern6460 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(x_**S(2)*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement6460(b, d, a, n, c, x, e):
rubi.append(6460)
return Dist(b*c*n, Int((a + b*atanh(c*x))**(n + S(-1))/(x*sqrt(d + e*x**S(2))), x), x) - Simp((a + b*atanh(c*x))**n*sqrt(d + e*x**S(2))/(d*x), x)
rule6460 = ReplacementRule(pattern6460, replacement6460)
pattern6461 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/(x_**S(2)*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement6461(b, d, a, n, c, x, e):
rubi.append(6461)
return Dist(b*c*n, Int((a + b*acoth(c*x))**(n + S(-1))/(x*sqrt(d + e*x**S(2))), x), x) - Simp((a + b*acoth(c*x))**n*sqrt(d + e*x**S(2))/(d*x), x)
rule6461 = ReplacementRule(pattern6461, replacement6461)
pattern6462 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons93, cons88, cons94, cons1510)
def replacement6462(m, b, d, a, n, c, x, e):
rubi.append(6462)
return Dist(c**S(2)*(m + S(2))/(m + S(1)), Int(x**(m + S(2))*(a + b*atanh(c*x))**n/sqrt(d + e*x**S(2)), x), x) - Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*atanh(c*x))**(n + S(-1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*atanh(c*x))**n*sqrt(d + e*x**S(2))/(d*(m + S(1))), x)
rule6462 = ReplacementRule(pattern6462, replacement6462)
pattern6463 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons93, cons88, cons94, cons1510)
def replacement6463(m, b, d, a, n, c, x, e):
rubi.append(6463)
return Dist(c**S(2)*(m + S(2))/(m + S(1)), Int(x**(m + S(2))*(a + b*acoth(c*x))**n/sqrt(d + e*x**S(2)), x), x) - Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*acoth(c*x))**(n + S(-1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*acoth(c*x))**n*sqrt(d + e*x**S(2))/(d*(m + S(1))), x)
rule6463 = ReplacementRule(pattern6463, replacement6463)
pattern6464 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons1784, cons137, cons166, cons1152)
def replacement6464(p, m, b, d, a, n, c, x, e):
rubi.append(6464)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*atanh(c*x))**n*(d + e*x**S(2))**p, x), x)
rule6464 = ReplacementRule(pattern6464, replacement6464)
pattern6465 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons1784, cons137, cons166, cons1152)
def replacement6465(p, m, b, d, a, n, c, x, e):
rubi.append(6465)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*acoth(c*x))**n*(d + e*x**S(2))**p, x), x)
rule6465 = ReplacementRule(pattern6465, replacement6465)
pattern6466 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons1784, cons137, cons267, cons1152)
def replacement6466(p, m, b, d, a, n, c, x, e):
rubi.append(6466)
return Dist(S(1)/d, Int(x**m*(a + b*atanh(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*atanh(c*x))**n*(d + e*x**S(2))**p, x), x)
rule6466 = ReplacementRule(pattern6466, replacement6466)
pattern6467 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons1784, cons137, cons267, cons1152)
def replacement6467(p, m, b, d, a, n, c, x, e):
rubi.append(6467)
return Dist(S(1)/d, Int(x**m*(a + b*acoth(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*acoth(c*x))**n*(d + e*x**S(2))**p, x), x)
rule6467 = ReplacementRule(pattern6467, replacement6467)
pattern6468 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons162, cons137, cons89, cons319)
def replacement6468(p, m, b, d, a, n, c, x, e):
rubi.append(6468)
return -Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*atanh(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Dist(c*(m + S(2)*p + S(2))/(b*(n + S(1))), Int(x**(m + S(1))*(a + b*atanh(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp(x**m*(a + b*atanh(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule6468 = ReplacementRule(pattern6468, replacement6468)
pattern6469 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons162, cons137, cons89, cons319)
def replacement6469(p, m, b, d, a, n, c, x, e):
rubi.append(6469)
return -Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*acoth(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Dist(c*(m + S(2)*p + S(2))/(b*(n + S(1))), Int(x**(m + S(1))*(a + b*acoth(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp(x**m*(a + b*acoth(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule6469 = ReplacementRule(pattern6469, replacement6469)
pattern6470 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons62, cons1785, cons1740)
def replacement6470(p, m, b, d, a, n, c, x, e):
rubi.append(6470)
return Dist(c**(-m + S(-1))*d**p, Subst(Int((a + b*x)**n*sinh(x)**m*cosh(x)**(-m - S(2)*p + S(-2)), x), x, atanh(c*x)), x)
rule6470 = ReplacementRule(pattern6470, replacement6470)
pattern6471 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons62, cons1785, cons1741)
def replacement6471(p, m, b, d, a, n, c, x, e):
rubi.append(6471)
return Dist(d**(p + S(1)/2)*sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int(x**m*(a + b*atanh(c*x))**n*(-c**S(2)*x**S(2) + S(1))**p, x), x)
rule6471 = ReplacementRule(pattern6471, replacement6471)
pattern6472 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons62, cons1785, cons38)
def replacement6472(p, m, b, d, a, n, c, x, e):
rubi.append(6472)
return -Dist(c**(-m + S(-1))*(-d)**p, Subst(Int((a + b*x)**n*sinh(x)**(-m - S(2)*p + S(-2))*cosh(x)**m, x), x, acoth(c*x)), x)
rule6472 = ReplacementRule(pattern6472, replacement6472)
pattern6473 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons62, cons1785, cons147)
def replacement6473(p, m, b, d, a, n, c, x, e):
rubi.append(6473)
return -Dist(c**(-m)*x*(-d)**(p + S(1)/2)*sqrt((c**S(2)*x**S(2) + S(-1))/(c**S(2)*x**S(2)))/sqrt(d + e*x**S(2)), Subst(Int((a + b*x)**n*sinh(x)**(-m - S(2)*p + S(-2))*cosh(x)**m, x), x, acoth(c*x)), x)
rule6473 = ReplacementRule(pattern6473, replacement6473)
pattern6474 = Pattern(Integral(x_*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement6474(p, b, d, a, c, x, e):
rubi.append(6474)
return -Dist(b*c/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*atanh(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6474 = ReplacementRule(pattern6474, replacement6474)
pattern6475 = Pattern(Integral(x_*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement6475(p, b, d, a, c, x, e):
rubi.append(6475)
return -Dist(b*c/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*acoth(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6475 = ReplacementRule(pattern6475, replacement6475)
def With6476(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(d + e*x**S(2))**p, x)
rubi.append(6476)
return -Dist(b*c, Int(SimplifyIntegrand(u/(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*atanh(c*x), u, x)
pattern6476 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1786)
rule6476 = ReplacementRule(pattern6476, With6476)
def With6477(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(d + e*x**S(2))**p, x)
rubi.append(6477)
return -Dist(b*c, Int(SimplifyIntegrand(u/(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acoth(c*x), u, x)
pattern6477 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1786)
rule6477 = ReplacementRule(pattern6477, With6477)
pattern6478 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons1787)
def replacement6478(p, m, b, d, a, n, c, x, e):
rubi.append(6478)
return Int(ExpandIntegrand((a + b*atanh(c*x))**n, x**m*(d + e*x**S(2))**p, x), x)
rule6478 = ReplacementRule(pattern6478, replacement6478)
pattern6479 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons1787)
def replacement6479(p, m, b, d, a, n, c, x, e):
rubi.append(6479)
return Int(ExpandIntegrand((a + b*acoth(c*x))**n, x**m*(d + e*x**S(2))**p, x), x)
rule6479 = ReplacementRule(pattern6479, replacement6479)
pattern6480 = Pattern(Integral(x_**WC('m', S(1))*(a_ + WC('b', S(1))*atanh(x_*WC('c', S(1))))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1788)
def replacement6480(p, m, b, d, c, a, x, e):
rubi.append(6480)
return Dist(a, Int(x**m*(d + e*x**S(2))**p, x), x) + Dist(b, Int(x**m*(d + e*x**S(2))**p*atanh(c*x), x), x)
rule6480 = ReplacementRule(pattern6480, replacement6480)
pattern6481 = Pattern(Integral(x_**WC('m', S(1))*(a_ + WC('b', S(1))*acoth(x_*WC('c', S(1))))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1788)
def replacement6481(p, m, b, d, c, a, x, e):
rubi.append(6481)
return Dist(a, Int(x**m*(d + e*x**S(2))**p, x), x) + Dist(b, Int(x**m*(d + e*x**S(2))**p*acoth(c*x), x), x)
rule6481 = ReplacementRule(pattern6481, replacement6481)
pattern6482 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement6482(p, m, b, d, a, n, c, x, e):
rubi.append(6482)
return Int(x**m*(a + b*atanh(c*x))**n*(d + e*x**S(2))**p, x)
rule6482 = ReplacementRule(pattern6482, replacement6482)
pattern6483 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement6483(p, m, b, d, a, n, c, x, e):
rubi.append(6483)
return Int(x**m*(a + b*acoth(c*x))**n*(d + e*x**S(2))**p, x)
rule6483 = ReplacementRule(pattern6483, replacement6483)
pattern6484 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))*atanh(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons1910)
def replacement6484(u, b, d, a, n, c, x, e):
rubi.append(6484)
return -Dist(S(1)/2, Int((a + b*atanh(c*x))**n*log(-u + S(1))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int((a + b*atanh(c*x))**n*log(u + S(1))/(d + e*x**S(2)), x), x)
rule6484 = ReplacementRule(pattern6484, replacement6484)
pattern6485 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))*acoth(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons1910)
def replacement6485(u, b, d, a, n, c, x, e):
rubi.append(6485)
return -Dist(S(1)/2, Int((a + b*acoth(c*x))**n*log(SimplifyIntegrand(S(1) - S(1)/u, x))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int((a + b*acoth(c*x))**n*log(SimplifyIntegrand(S(1) + S(1)/u, x))/(d + e*x**S(2)), x), x)
rule6485 = ReplacementRule(pattern6485, replacement6485)
pattern6486 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))*atanh(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons1911)
def replacement6486(u, b, d, a, n, c, x, e):
rubi.append(6486)
return -Dist(S(1)/2, Int((a + b*atanh(c*x))**n*log(-u + S(1))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int((a + b*atanh(c*x))**n*log(u + S(1))/(d + e*x**S(2)), x), x)
rule6486 = ReplacementRule(pattern6486, replacement6486)
pattern6487 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))*acoth(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons1911)
def replacement6487(u, b, d, a, n, c, x, e):
rubi.append(6487)
return -Dist(S(1)/2, Int((a + b*acoth(c*x))**n*log(SimplifyIntegrand(S(1) - S(1)/u, x))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int((a + b*acoth(c*x))**n*log(SimplifyIntegrand(S(1) + S(1)/u, x))/(d + e*x**S(2)), x), x)
rule6487 = ReplacementRule(pattern6487, replacement6487)
pattern6488 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))*log(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons1912)
def replacement6488(u, b, d, a, n, c, x, e):
rubi.append(6488)
return -Dist(b*n/S(2), Int((a + b*atanh(c*x))**(n + S(-1))*PolyLog(S(2), Together(-u + S(1)))/(d + e*x**S(2)), x), x) + Simp((a + b*atanh(c*x))**n*PolyLog(S(2), Together(-u + S(1)))/(S(2)*c*d), x)
rule6488 = ReplacementRule(pattern6488, replacement6488)
pattern6489 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))*log(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons1912)
def replacement6489(u, b, d, a, n, c, x, e):
rubi.append(6489)
return -Dist(b*n/S(2), Int((a + b*acoth(c*x))**(n + S(-1))*PolyLog(S(2), Together(-u + S(1)))/(d + e*x**S(2)), x), x) + Simp((a + b*acoth(c*x))**n*PolyLog(S(2), Together(-u + S(1)))/(S(2)*c*d), x)
rule6489 = ReplacementRule(pattern6489, replacement6489)
pattern6490 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))*log(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons1913)
def replacement6490(u, b, d, a, n, c, x, e):
rubi.append(6490)
return Dist(b*n/S(2), Int((a + b*atanh(c*x))**(n + S(-1))*PolyLog(S(2), Together(-u + S(1)))/(d + e*x**S(2)), x), x) - Simp((a + b*atanh(c*x))**n*PolyLog(S(2), Together(-u + S(1)))/(S(2)*c*d), x)
rule6490 = ReplacementRule(pattern6490, replacement6490)
pattern6491 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))*log(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons1913)
def replacement6491(u, b, d, a, n, c, x, e):
rubi.append(6491)
return Dist(b*n/S(2), Int((a + b*acoth(c*x))**(n + S(-1))*PolyLog(S(2), Together(-u + S(1)))/(d + e*x**S(2)), x), x) - Simp((a + b*acoth(c*x))**n*PolyLog(S(2), Together(-u + S(1)))/(S(2)*c*d), x)
rule6491 = ReplacementRule(pattern6491, replacement6491)
pattern6492 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))*PolyLog(p_, u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons1910)
def replacement6492(p, u, b, d, a, n, c, x, e):
rubi.append(6492)
return Dist(b*n/S(2), Int((a + b*atanh(c*x))**(n + S(-1))*PolyLog(p + S(1), u)/(d + e*x**S(2)), x), x) - Simp((a + b*atanh(c*x))**n*PolyLog(p + S(1), u)/(S(2)*c*d), x)
rule6492 = ReplacementRule(pattern6492, replacement6492)
pattern6493 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))*PolyLog(p_, u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons1910)
def replacement6493(p, u, b, d, a, n, c, x, e):
rubi.append(6493)
return Dist(b*n/S(2), Int((a + b*acoth(c*x))**(n + S(-1))*PolyLog(p + S(1), u)/(d + e*x**S(2)), x), x) - Simp((a + b*acoth(c*x))**n*PolyLog(p + S(1), u)/(S(2)*c*d), x)
rule6493 = ReplacementRule(pattern6493, replacement6493)
pattern6494 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))*PolyLog(p_, u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons1911)
def replacement6494(p, u, b, d, a, n, c, x, e):
rubi.append(6494)
return -Dist(b*n/S(2), Int((a + b*atanh(c*x))**(n + S(-1))*PolyLog(p + S(1), u)/(d + e*x**S(2)), x), x) + Simp((a + b*atanh(c*x))**n*PolyLog(p + S(1), u)/(S(2)*c*d), x)
rule6494 = ReplacementRule(pattern6494, replacement6494)
pattern6495 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))*PolyLog(p_, u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons1911)
def replacement6495(p, u, b, d, a, n, c, x, e):
rubi.append(6495)
return -Dist(b*n/S(2), Int((a + b*acoth(c*x))**(n + S(-1))*PolyLog(p + S(1), u)/(d + e*x**S(2)), x), x) + Simp((a + b*acoth(c*x))**n*PolyLog(p + S(1), u)/(S(2)*c*d), x)
rule6495 = ReplacementRule(pattern6495, replacement6495)
pattern6496 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons1737)
def replacement6496(b, d, a, c, x, e):
rubi.append(6496)
return Simp((-log(a + b*acoth(c*x)) + log(a + b*atanh(c*x)))/(b**S(2)*c*d*(acoth(c*x) - atanh(c*x))), x)
rule6496 = ReplacementRule(pattern6496, replacement6496)
pattern6497 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons150, cons1793)
def replacement6497(m, b, d, a, n, c, x, e):
rubi.append(6497)
return -Dist(n/(m + S(1)), Int((a + b*acoth(c*x))**(m + S(1))*(a + b*atanh(c*x))**(n + S(-1))/(d + e*x**S(2)), x), x) + Simp((a + b*acoth(c*x))**(m + S(1))*(a + b*atanh(c*x))**n/(b*c*d*(m + S(1))), x)
rule6497 = ReplacementRule(pattern6497, replacement6497)
pattern6498 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**WC('m', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons150, cons1794)
def replacement6498(m, b, d, a, n, c, x, e):
rubi.append(6498)
return -Dist(n/(m + S(1)), Int((a + b*acoth(c*x))**(n + S(-1))*(a + b*atanh(c*x))**(m + S(1))/(d + e*x**S(2)), x), x) + Simp((a + b*acoth(c*x))**n*(a + b*atanh(c*x))**(m + S(1))/(b*c*d*(m + S(1))), x)
rule6498 = ReplacementRule(pattern6498, replacement6498)
pattern6499 = Pattern(Integral(atanh(x_*WC('a', S(1)))/(c_ + x_**WC('n', S(1))*WC('d', S(1))), x_), cons2, cons7, cons27, cons85, cons1914)
def replacement6499(d, a, n, c, x):
rubi.append(6499)
return -Dist(S(1)/2, Int(log(-a*x + S(1))/(c + d*x**n), x), x) + Dist(S(1)/2, Int(log(a*x + S(1))/(c + d*x**n), x), x)
rule6499 = ReplacementRule(pattern6499, replacement6499)
pattern6500 = Pattern(Integral(acoth(x_*WC('a', S(1)))/(c_ + x_**WC('n', S(1))*WC('d', S(1))), x_), cons2, cons7, cons27, cons85, cons1914)
def replacement6500(d, a, n, c, x):
rubi.append(6500)
return -Dist(S(1)/2, Int(log(S(1) - S(1)/(a*x))/(c + d*x**n), x), x) + Dist(S(1)/2, Int(log(S(1) + S(1)/(a*x))/(c + d*x**n), x), x)
rule6500 = ReplacementRule(pattern6500, replacement6500)
pattern6501 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1796)
def replacement6501(f, b, g, d, a, c, x, e):
rubi.append(6501)
return -Dist(b*c, Int(x*(d + e*log(f + g*x**S(2)))/(-c**S(2)*x**S(2) + S(1)), x), x) - Dist(S(2)*e*g, Int(x**S(2)*(a + b*atanh(c*x))/(f + g*x**S(2)), x), x) + Simp(x*(a + b*atanh(c*x))*(d + e*log(f + g*x**S(2))), x)
rule6501 = ReplacementRule(pattern6501, replacement6501)
pattern6502 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1796)
def replacement6502(f, b, g, d, a, c, x, e):
rubi.append(6502)
return -Dist(b*c, Int(x*(d + e*log(f + g*x**S(2)))/(-c**S(2)*x**S(2) + S(1)), x), x) - Dist(S(2)*e*g, Int(x**S(2)*(a + b*acoth(c*x))/(f + g*x**S(2)), x), x) + Simp(x*(a + b*acoth(c*x))*(d + e*log(f + g*x**S(2))), x)
rule6502 = ReplacementRule(pattern6502, replacement6502)
pattern6503 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons601)
def replacement6503(m, f, b, g, d, a, c, x, e):
rubi.append(6503)
return -Dist(b*c/(m + S(1)), Int(x**(m + S(1))*(d + e*log(f + g*x**S(2)))/(-c**S(2)*x**S(2) + S(1)), x), x) - Dist(S(2)*e*g/(m + S(1)), Int(x**(m + S(2))*(a + b*atanh(c*x))/(f + g*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*atanh(c*x))*(d + e*log(f + g*x**S(2)))/(m + S(1)), x)
rule6503 = ReplacementRule(pattern6503, replacement6503)
pattern6504 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons601)
def replacement6504(m, f, b, g, d, a, c, x, e):
rubi.append(6504)
return -Dist(b*c/(m + S(1)), Int(x**(m + S(1))*(d + e*log(f + g*x**S(2)))/(-c**S(2)*x**S(2) + S(1)), x), x) - Dist(S(2)*e*g/(m + S(1)), Int(x**(m + S(2))*(a + b*acoth(c*x))/(f + g*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*acoth(c*x))*(d + e*log(f + g*x**S(2)))/(m + S(1)), x)
rule6504 = ReplacementRule(pattern6504, replacement6504)
def With6505(m, f, b, g, d, a, c, x, e):
u = IntHide(x**m*(d + e*log(f + g*x**S(2))), x)
rubi.append(6505)
return -Dist(b*c, Int(ExpandIntegrand(u/(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*atanh(c*x), u, x)
pattern6505 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1797)
rule6505 = ReplacementRule(pattern6505, With6505)
def With6506(m, f, b, g, d, a, c, x, e):
u = IntHide(x**m*(d + e*log(f + g*x**S(2))), x)
rubi.append(6506)
return -Dist(b*c, Int(ExpandIntegrand(u/(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acoth(c*x), u, x)
pattern6506 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1797)
rule6506 = ReplacementRule(pattern6506, With6506)
def With6507(m, f, b, g, d, a, c, x, e):
u = IntHide(x**m*(a + b*atanh(c*x)), x)
rubi.append(6507)
return -Dist(S(2)*e*g, Int(ExpandIntegrand(u*x/(f + g*x**S(2)), x), x), x) + Dist(d + e*log(f + g*x**S(2)), u, x)
pattern6507 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons17, cons261)
rule6507 = ReplacementRule(pattern6507, With6507)
def With6508(m, f, b, g, d, a, c, x, e):
u = IntHide(x**m*(a + b*acoth(c*x)), x)
rubi.append(6508)
return -Dist(S(2)*e*g, Int(ExpandIntegrand(u*x/(f + g*x**S(2)), x), x), x) + Dist(d + e*log(f + g*x**S(2)), u, x)
pattern6508 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons17, cons261)
rule6508 = ReplacementRule(pattern6508, With6508)
pattern6509 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*atanh(x_*WC('c', S(1))))**S(2)*(WC('d', S(0)) + WC('e', S(1))*log(f_ + x_**S(2)*WC('g', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1915)
def replacement6509(g, b, f, d, a, c, x, e):
rubi.append(6509)
return Dist(b/c, Int((a + b*atanh(c*x))*(d + e*log(f + g*x**S(2))), x), x) + Dist(b*c*e, Int(x**S(2)*(a + b*atanh(c*x))/(-c**S(2)*x**S(2) + S(1)), x), x) - Simp(e*x**S(2)*(a + b*atanh(c*x))**S(2)/S(2), x) + Simp((a + b*atanh(c*x))**S(2)*(d + e*log(f + g*x**S(2)))*(f + g*x**S(2))/(S(2)*g), x)
rule6509 = ReplacementRule(pattern6509, replacement6509)
pattern6510 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acoth(x_*WC('c', S(1))))**S(2)*(WC('d', S(0)) + WC('e', S(1))*log(f_ + x_**S(2)*WC('g', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1915)
def replacement6510(g, b, f, d, a, c, x, e):
rubi.append(6510)
return Dist(b/c, Int((a + b*acoth(c*x))*(d + e*log(f + g*x**S(2))), x), x) + Dist(b*c*e, Int(x**S(2)*(a + b*acoth(c*x))/(-c**S(2)*x**S(2) + S(1)), x), x) - Simp(e*x**S(2)*(a + b*acoth(c*x))**S(2)/S(2), x) + Simp((a + b*acoth(c*x))**S(2)*(d + e*log(f + g*x**S(2)))*(f + g*x**S(2))/(S(2)*g), x)
rule6510 = ReplacementRule(pattern6510, replacement6510)
pattern6511 = Pattern(Integral(exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons1482)
def replacement6511(x, a, n):
rubi.append(6511)
return Int((-a*x + S(1))**(-n/S(2) + S(1)/2)*(a*x + S(1))**(n/S(2) + S(1)/2)/sqrt(-a**S(2)*x**S(2) + S(1)), x)
rule6511 = ReplacementRule(pattern6511, replacement6511)
pattern6512 = Pattern(Integral(x_**WC('m', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons21, cons1482)
def replacement6512(x, m, a, n):
rubi.append(6512)
return Int(x**m*(-a*x + S(1))**(-n/S(2) + S(1)/2)*(a*x + S(1))**(n/S(2) + S(1)/2)/sqrt(-a**S(2)*x**S(2) + S(1)), x)
rule6512 = ReplacementRule(pattern6512, replacement6512)
pattern6513 = Pattern(Integral(exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons4, cons1441)
def replacement6513(x, a, n):
rubi.append(6513)
return Int((-a*x + S(1))**(-n/S(2))*(a*x + S(1))**(n/S(2)), x)
rule6513 = ReplacementRule(pattern6513, replacement6513)
pattern6514 = Pattern(Integral(x_**WC('m', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons21, cons4, cons1441)
def replacement6514(x, m, a, n):
rubi.append(6514)
return Int(x**m*(-a*x + S(1))**(-n/S(2))*(a*x + S(1))**(n/S(2)), x)
rule6514 = ReplacementRule(pattern6514, replacement6514)
pattern6515 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1916, cons1250, cons246)
def replacement6515(p, d, a, n, c, x):
rubi.append(6515)
return Dist(c**n, Int((c + d*x)**(-n + p)*(-a**S(2)*x**S(2) + S(1))**(n/S(2)), x), x)
rule6515 = ReplacementRule(pattern6515, replacement6515)
pattern6516 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons5, cons1916, cons1250, cons1917, cons246)
def replacement6516(p, m, f, d, a, n, c, x, e):
rubi.append(6516)
return Dist(c**n, Int((c + d*x)**(-n + p)*(e + f*x)**m*(-a**S(2)*x**S(2) + S(1))**(n/S(2)), x), x)
rule6516 = ReplacementRule(pattern6516, replacement6516)
pattern6517 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1918, cons1802)
def replacement6517(p, u, d, a, n, c, x):
rubi.append(6517)
return Dist(c**p, Int(u*(S(1) + d*x/c)**p*(-a*x + S(1))**(-n/S(2))*(a*x + S(1))**(n/S(2)), x), x)
rule6517 = ReplacementRule(pattern6517, replacement6517)
pattern6518 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1918, cons1803)
def replacement6518(p, u, d, a, n, c, x):
rubi.append(6518)
return Int(u*(c + d*x)**p*(-a*x + S(1))**(-n/S(2))*(a*x + S(1))**(n/S(2)), x)
rule6518 = ReplacementRule(pattern6518, replacement6518)
pattern6519 = Pattern(Integral((c_ + WC('d', S(1))/x_)**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1919, cons38)
def replacement6519(p, u, d, a, n, c, x):
rubi.append(6519)
return Dist(d**p, Int(u*x**(-p)*(c*x/d + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6519 = ReplacementRule(pattern6519, replacement6519)
pattern6520 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*WC('u', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1919, cons147, cons743, cons177)
def replacement6520(p, u, d, a, n, c, x):
rubi.append(6520)
return Dist((S(-1))**(n/S(2))*c**p, Int(u*(S(1) - S(1)/(a*x))**(-n/S(2))*(S(1) + S(1)/(a*x))**(n/S(2))*(S(1) + d/(c*x))**p, x), x)
rule6520 = ReplacementRule(pattern6520, replacement6520)
pattern6521 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*WC('u', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1919, cons147, cons743, cons117)
def replacement6521(p, u, d, a, n, c, x):
rubi.append(6521)
return Int(u*(c + d/x)**p*(-a*x + S(1))**(-n/S(2))*(a*x + S(1))**(n/S(2)), x)
rule6521 = ReplacementRule(pattern6521, replacement6521)
pattern6522 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*WC('u', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1919, cons147)
def replacement6522(p, u, d, a, n, c, x):
rubi.append(6522)
return Dist(x**p*(c + d/x)**p*(c*x/d + S(1))**(-p), Int(u*x**(-p)*(c*x/d + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6522 = ReplacementRule(pattern6522, replacement6522)
pattern6523 = Pattern(Integral(exp(n_*atanh(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons4, cons1920, cons23)
def replacement6523(d, a, n, c, x):
rubi.append(6523)
return Simp((-a*x + n)*exp(n*atanh(a*x))/(a*c*sqrt(c + d*x**S(2))*(n**S(2) + S(-1))), x)
rule6523 = ReplacementRule(pattern6523, replacement6523)
pattern6524 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons13, cons137, cons23, cons1921, cons246)
def replacement6524(p, d, a, n, c, x):
rubi.append(6524)
return -Dist(S(2)*(p + S(1))*(S(2)*p + S(3))/(c*(n**S(2) - S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*atanh(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*(S(2)*a*x*(p + S(1)) + n)*exp(n*atanh(a*x))/(a*c*(n**S(2) - S(4)*(p + S(1))**S(2))), x)
rule6524 = ReplacementRule(pattern6524, replacement6524)
pattern6525 = Pattern(Integral(exp(WC('n', S(1))*atanh(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons7, cons27, cons4, cons1920, cons1922)
def replacement6525(d, a, n, c, x):
rubi.append(6525)
return Simp(exp(n*atanh(a*x))/(a*c*n), x)
rule6525 = ReplacementRule(pattern6525, replacement6525)
pattern6526 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1920, cons38, cons1923, cons1924)
def replacement6526(p, d, a, n, c, x):
rubi.append(6526)
return Dist(c**p, Int((a*x + S(1))**n*(-a**S(2)*x**S(2) + S(1))**(-n/S(2) + p), x), x)
rule6526 = ReplacementRule(pattern6526, replacement6526)
pattern6527 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1920, cons38, cons1925, cons1924)
def replacement6527(p, d, a, n, c, x):
rubi.append(6527)
return Dist(c**p, Int((-a*x + S(1))**(-n)*(-a**S(2)*x**S(2) + S(1))**(n/S(2) + p), x), x)
rule6527 = ReplacementRule(pattern6527, replacement6527)
pattern6528 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1920, cons1802)
def replacement6528(p, d, a, n, c, x):
rubi.append(6528)
return Dist(c**p, Int((-a*x + S(1))**(-n/S(2) + p)*(a*x + S(1))**(n/S(2) + p), x), x)
rule6528 = ReplacementRule(pattern6528, replacement6528)
pattern6529 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1920, cons1803, cons674)
def replacement6529(p, d, a, n, c, x):
rubi.append(6529)
return Dist(c**(n/S(2)), Int((c + d*x**S(2))**(-n/S(2) + p)*(a*x + S(1))**n, x), x)
rule6529 = ReplacementRule(pattern6529, replacement6529)
pattern6530 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1920, cons1803, cons1926)
def replacement6530(p, d, a, n, c, x):
rubi.append(6530)
return Dist(c**(-n/S(2)), Int((c + d*x**S(2))**(n/S(2) + p)*(-a*x + S(1))**(-n), x), x)
rule6530 = ReplacementRule(pattern6530, replacement6530)
pattern6531 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1920, cons1803)
def replacement6531(p, d, a, n, c, x):
rubi.append(6531)
return Dist(c**IntPart(p)*(c + d*x**S(2))**FracPart(p)*(-a**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((-a**S(2)*x**S(2) + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6531 = ReplacementRule(pattern6531, replacement6531)
pattern6532 = Pattern(Integral(x_*exp(n_*atanh(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons4, cons1920, cons23)
def replacement6532(d, a, n, c, x):
rubi.append(6532)
return Simp((-a*n*x + S(1))*exp(n*atanh(a*x))/(d*sqrt(c + d*x**S(2))*(n**S(2) + S(-1))), x)
rule6532 = ReplacementRule(pattern6532, replacement6532)
pattern6533 = Pattern(Integral(x_*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons13, cons137, cons23, cons246)
def replacement6533(p, d, a, n, c, x):
rubi.append(6533)
return -Dist(a*c*n/(S(2)*d*(p + S(1))), Int((c + d*x**S(2))**p*exp(n*atanh(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*exp(n*atanh(a*x))/(S(2)*d*(p + S(1))), x)
rule6533 = ReplacementRule(pattern6533, replacement6533)
pattern6534 = Pattern(Integral(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons1927, cons23)
def replacement6534(p, d, a, n, c, x):
rubi.append(6534)
return Simp((c + d*x**S(2))**(p + S(1))*(-a*n*x + S(1))*exp(n*atanh(a*x))/(a*d*n*(n**S(2) + S(-1))), x)
rule6534 = ReplacementRule(pattern6534, replacement6534)
pattern6535 = Pattern(Integral(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons13, cons137, cons23, cons1921, cons246)
def replacement6535(p, d, a, n, c, x):
rubi.append(6535)
return Dist((n**S(2) + S(2)*p + S(2))/(d*(n**S(2) - S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*atanh(a*x)), x), x) - Simp((c + d*x**S(2))**(p + S(1))*(S(2)*a*x*(p + S(1)) + n)*exp(n*atanh(a*x))/(a*d*(n**S(2) - S(4)*(p + S(1))**S(2))), x)
rule6535 = ReplacementRule(pattern6535, replacement6535)
pattern6536 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons5, cons1920, cons1802, cons1923, cons1924)
def replacement6536(p, m, d, a, n, c, x):
rubi.append(6536)
return Dist(c**p, Int(x**m*(a*x + S(1))**n*(-a**S(2)*x**S(2) + S(1))**(-n/S(2) + p), x), x)
rule6536 = ReplacementRule(pattern6536, replacement6536)
pattern6537 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons5, cons1920, cons1802, cons1925, cons1924)
def replacement6537(p, m, d, a, n, c, x):
rubi.append(6537)
return Dist(c**p, Int(x**m*(-a*x + S(1))**(-n)*(-a**S(2)*x**S(2) + S(1))**(n/S(2) + p), x), x)
rule6537 = ReplacementRule(pattern6537, replacement6537)
pattern6538 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1920, cons1802)
def replacement6538(p, m, d, a, n, c, x):
rubi.append(6538)
return Dist(c**p, Int(x**m*(-a*x + S(1))**(-n/S(2) + p)*(a*x + S(1))**(n/S(2) + p), x), x)
rule6538 = ReplacementRule(pattern6538, replacement6538)
pattern6539 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons5, cons1920, cons1803, cons674)
def replacement6539(p, m, d, a, n, c, x):
rubi.append(6539)
return Dist(c**(n/S(2)), Int(x**m*(c + d*x**S(2))**(-n/S(2) + p)*(a*x + S(1))**n, x), x)
rule6539 = ReplacementRule(pattern6539, replacement6539)
pattern6540 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons5, cons1920, cons1803, cons1926)
def replacement6540(p, m, d, a, n, c, x):
rubi.append(6540)
return Dist(c**(-n/S(2)), Int(x**m*(c + d*x**S(2))**(n/S(2) + p)*(-a*x + S(1))**(-n), x), x)
rule6540 = ReplacementRule(pattern6540, replacement6540)
pattern6541 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1920, cons1803, cons1922)
def replacement6541(p, m, d, a, n, c, x):
rubi.append(6541)
return Dist(c**IntPart(p)*(c + d*x**S(2))**FracPart(p)*(-a**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int(x**m*(-a**S(2)*x**S(2) + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6541 = ReplacementRule(pattern6541, replacement6541)
pattern6542 = Pattern(Integral(u_*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1920, cons1802)
def replacement6542(p, u, d, a, n, c, x):
rubi.append(6542)
return Dist(c**p, Int(u*(-a*x + S(1))**(-n/S(2) + p)*(a*x + S(1))**(n/S(2) + p), x), x)
rule6542 = ReplacementRule(pattern6542, replacement6542)
pattern6543 = Pattern(Integral(u_*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1920, cons1803, cons743)
def replacement6543(p, u, d, a, n, c, x):
rubi.append(6543)
return Dist(c**IntPart(p)*(c + d*x**S(2))**FracPart(p)*(-a*x + S(1))**(-FracPart(p))*(a*x + S(1))**(-FracPart(p)), Int(u*(-a*x + S(1))**(-n/S(2) + p)*(a*x + S(1))**(n/S(2) + p), x), x)
rule6543 = ReplacementRule(pattern6543, replacement6543)
pattern6544 = Pattern(Integral(u_*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1920, cons1803, cons1922)
def replacement6544(p, u, d, a, n, c, x):
rubi.append(6544)
return Dist(c**IntPart(p)*(c + d*x**S(2))**FracPart(p)*(-a**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int(u*(-a**S(2)*x**S(2) + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6544 = ReplacementRule(pattern6544, replacement6544)
pattern6545 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1928, cons38)
def replacement6545(p, u, d, a, n, c, x):
rubi.append(6545)
return Dist(d**p, Int(u*x**(-S(2)*p)*(-a**S(2)*x**S(2) + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6545 = ReplacementRule(pattern6545, replacement6545)
pattern6546 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**p_*WC('u', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1928, cons147, cons743, cons177)
def replacement6546(p, u, d, a, n, c, x):
rubi.append(6546)
return Dist(c**p, Int(u*(S(1) - S(1)/(a*x))**p*(S(1) + S(1)/(a*x))**p*exp(n*atanh(a*x)), x), x)
rule6546 = ReplacementRule(pattern6546, replacement6546)
pattern6547 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**p_*WC('u', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1928, cons147, cons743, cons117)
def replacement6547(p, u, d, a, n, c, x):
rubi.append(6547)
return Dist(x**(S(2)*p)*(c + d/x**S(2))**p*(-a*x + S(1))**(-p)*(a*x + S(1))**(-p), Int(u*x**(-S(2)*p)*(-a*x + S(1))**p*(a*x + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6547 = ReplacementRule(pattern6547, replacement6547)
pattern6548 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**p_*WC('u', S(1))*exp(WC('n', S(1))*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1928, cons147, cons1922)
def replacement6548(p, u, d, a, n, c, x):
rubi.append(6548)
return Dist(x**(S(2)*p)*(c + d/x**S(2))**p*(c*x**S(2)/d + S(1))**(-p), Int(u*x**(-S(2)*p)*(c*x**S(2)/d + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6548 = ReplacementRule(pattern6548, replacement6548)
pattern6549 = Pattern(Integral(exp(WC('n', S(1))*atanh((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6549(b, c, n, a, x):
rubi.append(6549)
return Int((-a*c - b*c*x + S(1))**(-n/S(2))*(a*c + b*c*x + S(1))**(n/S(2)), x)
rule6549 = ReplacementRule(pattern6549, replacement6549)
pattern6550 = Pattern(Integral(x_**m_*exp(n_*atanh((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons84, cons87, cons994)
def replacement6550(m, b, c, n, a, x):
rubi.append(6550)
return Dist(S(4)*b**(-m + S(-1))*c**(-m + S(-1))/n, Subst(Int(x**(S(2)/n)*(x**(S(2)/n) + S(1))**(-m + S(-2))*(-a*c + x**(S(2)/n)*(-a*c + S(1)) + S(-1))**m, x), x, (-c*(a + b*x) + S(1))**(-n/S(2))*(c*(a + b*x) + S(1))**(n/S(2))), x)
rule6550 = ReplacementRule(pattern6550, replacement6550)
pattern6551 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*exp(WC('n', S(1))*atanh((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1580)
def replacement6551(m, b, d, c, n, a, x, e):
rubi.append(6551)
return Int((d + e*x)**m*(-a*c - b*c*x + S(1))**(-n/S(2))*(a*c + b*c*x + S(1))**(n/S(2)), x)
rule6551 = ReplacementRule(pattern6551, replacement6551)
pattern6552 = Pattern(Integral((c_ + x_**S(2)*WC('e', S(1)) + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*atanh(a_ + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1818, cons1929, cons1930)
def replacement6552(p, u, b, d, a, n, c, x, e):
rubi.append(6552)
return Dist((c/(-a**S(2) + S(1)))**p, Int(u*(-a - b*x + S(1))**(-n/S(2) + p)*(a + b*x + S(1))**(n/S(2) + p), x), x)
rule6552 = ReplacementRule(pattern6552, replacement6552)
pattern6553 = Pattern(Integral((c_ + x_**S(2)*WC('e', S(1)) + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*atanh(a_ + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1818, cons1929, cons1931)
def replacement6553(p, u, b, d, a, n, c, x, e):
rubi.append(6553)
return Dist((c + d*x + e*x**S(2))**p*(-a**S(2) - S(2)*a*b*x - b**S(2)*x**S(2) + S(1))**(-p), Int(u*(-a**S(2) - S(2)*a*b*x - b**S(2)*x**S(2) + S(1))**p*exp(n*atanh(a*x)), x), x)
rule6553 = ReplacementRule(pattern6553, replacement6553)
pattern6554 = Pattern(Integral(WC('u', S(1))*exp(WC('n', S(1))*atanh(WC('c', S(1))/(x_*WC('b', S(1)) + WC('a', S(0))))), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6554(u, b, c, n, a, x):
rubi.append(6554)
return Int(u*exp(n*acoth(a/c + b*x/c)), x)
rule6554 = ReplacementRule(pattern6554, replacement6554)
pattern6555 = Pattern(Integral(WC('u', S(1))*exp(n_*acoth(x_*WC('a', S(1)))), x_), cons2, cons743)
def replacement6555(x, a, n, u):
rubi.append(6555)
return Dist((S(-1))**(n/S(2)), Int(u*exp(n*atanh(a*x)), x), x)
rule6555 = ReplacementRule(pattern6555, replacement6555)
pattern6556 = Pattern(Integral(exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons1482)
def replacement6556(x, a, n):
rubi.append(6556)
return -Subst(Int((S(1) - x/a)**(-n/S(2) + S(1)/2)*(S(1) + x/a)**(n/S(2) + S(1)/2)/(x**S(2)*sqrt(S(1) - x**S(2)/a**S(2))), x), x, S(1)/x)
rule6556 = ReplacementRule(pattern6556, replacement6556)
pattern6557 = Pattern(Integral(x_**WC('m', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons1482, cons17)
def replacement6557(x, m, a, n):
rubi.append(6557)
return -Subst(Int(x**(-m + S(-2))*(S(1) - x/a)**(-n/S(2) + S(1)/2)*(S(1) + x/a)**(n/S(2) + S(1)/2)/sqrt(S(1) - x**S(2)/a**S(2)), x), x, S(1)/x)
rule6557 = ReplacementRule(pattern6557, replacement6557)
pattern6558 = Pattern(Integral(exp(n_*acoth(x_*WC('a', S(1)))), x_), cons2, cons4, cons23)
def replacement6558(x, a, n):
rubi.append(6558)
return -Subst(Int((S(1) - x/a)**(-n/S(2))*(S(1) + x/a)**(n/S(2))/x**S(2), x), x, S(1)/x)
rule6558 = ReplacementRule(pattern6558, replacement6558)
pattern6559 = Pattern(Integral(x_**WC('m', S(1))*exp(n_*acoth(x_*WC('a', S(1)))), x_), cons2, cons4, cons23, cons17)
def replacement6559(x, m, a, n):
rubi.append(6559)
return -Subst(Int(x**(-m + S(-2))*(S(1) - x/a)**(-n/S(2))*(S(1) + x/a)**(n/S(2)), x), x, S(1)/x)
rule6559 = ReplacementRule(pattern6559, replacement6559)
pattern6560 = Pattern(Integral(x_**m_*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons21, cons1482, cons18)
def replacement6560(x, m, a, n):
rubi.append(6560)
return -Dist(x**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(S(1) - x/a)**(-n/S(2) + S(1)/2)*(S(1) + x/a)**(n/S(2) + S(1)/2)/sqrt(S(1) - x**S(2)/a**S(2)), x), x, S(1)/x), x)
rule6560 = ReplacementRule(pattern6560, replacement6560)
pattern6561 = Pattern(Integral(x_**m_*exp(n_*acoth(x_*WC('a', S(1)))), x_), cons2, cons21, cons4, cons23, cons18)
def replacement6561(x, m, a, n):
rubi.append(6561)
return -Dist(x**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(S(1) - x/a)**(-n/S(2))*(S(1) + x/a)**(n/S(2)), x), x, S(1)/x), x)
rule6561 = ReplacementRule(pattern6561, replacement6561)
pattern6562 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1916, cons1932, cons1922)
def replacement6562(p, d, a, n, c, x):
rubi.append(6562)
return Simp((c + d*x)**p*(a*x + S(1))*exp(n*acoth(a*x))/(a*(p + S(1))), x)
rule6562 = ReplacementRule(pattern6562, replacement6562)
pattern6563 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1918, cons1922, cons38)
def replacement6563(p, u, d, a, n, c, x):
rubi.append(6563)
return Dist(d**p, Int(u*x**p*(c/(d*x) + S(1))**p*exp(n*acoth(a*x)), x), x)
rule6563 = ReplacementRule(pattern6563, replacement6563)
pattern6564 = Pattern(Integral((c_ + x_*WC('d', S(1)))**p_*WC('u', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1918, cons1922, cons147)
def replacement6564(p, u, d, a, n, c, x):
rubi.append(6564)
return Dist(x**(-p)*(c + d*x)**p*(c/(d*x) + S(1))**(-p), Int(u*x**p*(c/(d*x) + S(1))**p*exp(n*acoth(a*x)), x), x)
rule6564 = ReplacementRule(pattern6564, replacement6564)
pattern6565 = Pattern(Integral((c_ + WC('d', S(1))/x_)**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1933, cons1250, cons1917, cons246)
def replacement6565(p, d, a, n, c, x):
rubi.append(6565)
return -Dist(c**n, Subst(Int((S(1) - x**S(2)/a**S(2))**(n/S(2))*(c + d*x)**(-n + p)/x**S(2), x), x, S(1)/x), x)
rule6565 = ReplacementRule(pattern6565, replacement6565)
pattern6566 = Pattern(Integral(x_**WC('m', S(1))*(c_ + WC('d', S(1))/x_)**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1933, cons1250, cons17, cons1934, cons246)
def replacement6566(p, m, d, a, n, c, x):
rubi.append(6566)
return -Dist(c**n, Subst(Int(x**(-m + S(-2))*(S(1) - x**S(2)/a**S(2))**(n/S(2))*(c + d*x)**(-n + p), x), x, S(1)/x), x)
rule6566 = ReplacementRule(pattern6566, replacement6566)
pattern6567 = Pattern(Integral((c_ + WC('d', S(1))/x_)**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1919, cons1922, cons1802)
def replacement6567(p, d, a, n, c, x):
rubi.append(6567)
return -Dist(c**p, Subst(Int((S(1) - x/a)**(-n/S(2))*(S(1) + x/a)**(n/S(2))*(S(1) + d*x/c)**p/x**S(2), x), x, S(1)/x), x)
rule6567 = ReplacementRule(pattern6567, replacement6567)
pattern6568 = Pattern(Integral(x_**WC('m', S(1))*(c_ + WC('d', S(1))/x_)**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1919, cons1922, cons1802, cons17)
def replacement6568(p, m, d, a, n, c, x):
rubi.append(6568)
return -Dist(c**p, Subst(Int(x**(-m + S(-2))*(S(1) - x/a)**(-n/S(2))*(S(1) + x/a)**(n/S(2))*(S(1) + d*x/c)**p, x), x, S(1)/x), x)
rule6568 = ReplacementRule(pattern6568, replacement6568)
pattern6569 = Pattern(Integral(x_**m_*(c_ + WC('d', S(1))/x_)**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1919, cons1922, cons1802, cons18)
def replacement6569(p, m, d, a, n, c, x):
rubi.append(6569)
return -Dist(c**p*x**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(S(1) - x/a)**(-n/S(2))*(S(1) + x/a)**(n/S(2))*(S(1) + d*x/c)**p, x), x, S(1)/x), x)
rule6569 = ReplacementRule(pattern6569, replacement6569)
pattern6570 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*WC('u', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1919, cons1922, cons1803)
def replacement6570(p, u, d, a, n, c, x):
rubi.append(6570)
return Dist((S(1) + d/(c*x))**(-p)*(c + d/x)**p, Int(u*(S(1) + d/(c*x))**p*exp(n*acoth(a*x)), x), x)
rule6570 = ReplacementRule(pattern6570, replacement6570)
pattern6571 = Pattern(Integral(exp(WC('n', S(1))*acoth(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons7, cons27, cons4, cons1920, cons1922)
def replacement6571(d, a, n, c, x):
rubi.append(6571)
return Simp(exp(n*acoth(a*x))/(a*c*n), x)
rule6571 = ReplacementRule(pattern6571, replacement6571)
pattern6572 = Pattern(Integral(exp(n_*acoth(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons4, cons1920, cons23)
def replacement6572(d, a, n, c, x):
rubi.append(6572)
return Simp((-a*x + n)*exp(n*acoth(a*x))/(a*c*sqrt(c + d*x**S(2))*(n**S(2) + S(-1))), x)
rule6572 = ReplacementRule(pattern6572, replacement6572)
pattern6573 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons1922, cons13, cons137, cons230, cons1921, cons1935)
def replacement6573(p, d, a, n, c, x):
rubi.append(6573)
return -Dist(S(2)*(p + S(1))*(S(2)*p + S(3))/(c*(n**S(2) - S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*acoth(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*(S(2)*a*x*(p + S(1)) + n)*exp(n*acoth(a*x))/(a*c*(n**S(2) - S(4)*(p + S(1))**S(2))), x)
rule6573 = ReplacementRule(pattern6573, replacement6573)
pattern6574 = Pattern(Integral(x_*exp(n_*acoth(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons4, cons1920, cons23)
def replacement6574(d, a, n, c, x):
rubi.append(6574)
return -Simp((-a*n*x + S(1))*exp(n*acoth(a*x))/(a**S(2)*c*sqrt(c + d*x**S(2))*(n**S(2) + S(-1))), x)
rule6574 = ReplacementRule(pattern6574, replacement6574)
pattern6575 = Pattern(Integral(x_*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons1922, cons13, cons1824, cons230, cons1921, cons1935)
def replacement6575(p, d, a, n, c, x):
rubi.append(6575)
return -Dist(n*(S(2)*p + S(3))/(a*c*(n**S(2) - S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*acoth(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*(a*n*x + S(2)*p + S(2))*exp(n*acoth(a*x))/(a**S(2)*c*(n**S(2) - S(4)*(p + S(1))**S(2))), x)
rule6575 = ReplacementRule(pattern6575, replacement6575)
pattern6576 = Pattern(Integral(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons1922, cons1927, cons973)
def replacement6576(p, d, a, n, c, x):
rubi.append(6576)
return -Simp((c + d*x**S(2))**(p + S(1))*(S(2)*a*x*(p + S(1)) + n)*exp(n*acoth(a*x))/(a**S(3)*c*n**S(2)*(n**S(2) + S(-1))), x)
rule6576 = ReplacementRule(pattern6576, replacement6576)
pattern6577 = Pattern(Integral(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons1922, cons13, cons1824, cons1936, cons1921, cons1935)
def replacement6577(p, d, a, n, c, x):
rubi.append(6577)
return -Dist((n**S(2) + S(2)*p + S(2))/(a**S(2)*c*(n**S(2) - S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*acoth(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*(S(2)*a*x*(p + S(1)) + n)*exp(n*acoth(a*x))/(a**S(3)*c*(n**S(2) - S(4)*(p + S(1))**S(2))), x)
rule6577 = ReplacementRule(pattern6577, replacement6577)
pattern6578 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons1922, cons17, cons13, cons1827, cons38)
def replacement6578(p, m, d, a, n, c, x):
rubi.append(6578)
return -Dist(a**(-m + S(-1))*(-c)**p, Subst(Int((S(1)/tanh(x))**(m + S(2)*p + S(2))*exp(n*x)*cosh(x)**(-S(2)*p + S(-2)), x), x, acoth(a*x)), x)
rule6578 = ReplacementRule(pattern6578, replacement6578)
pattern6579 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1920, cons1922, cons38)
def replacement6579(p, u, d, a, n, c, x):
rubi.append(6579)
return Dist(d**p, Int(u*x**(S(2)*p)*(S(1) - S(1)/(a**S(2)*x**S(2)))**p*exp(n*acoth(a*x)), x), x)
rule6579 = ReplacementRule(pattern6579, replacement6579)
pattern6580 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*WC('u', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1920, cons1922, cons147)
def replacement6580(p, u, d, a, n, c, x):
rubi.append(6580)
return Dist(x**(-S(2)*p)*(S(1) - S(1)/(a**S(2)*x**S(2)))**(-p)*(c + d*x**S(2))**p, Int(u*x**(S(2)*p)*(S(1) - S(1)/(a**S(2)*x**S(2)))**p*exp(n*acoth(a*x)), x), x)
rule6580 = ReplacementRule(pattern6580, replacement6580)
pattern6581 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1928, cons1922, cons1802, cons1937)
def replacement6581(p, u, d, a, n, c, x):
rubi.append(6581)
return Dist(a**(-S(2)*p)*c**p, Int(u*x**(-S(2)*p)*(a*x + S(-1))**(-n/S(2) + p)*(a*x + S(1))**(n/S(2) + p), x), x)
rule6581 = ReplacementRule(pattern6581, replacement6581)
pattern6582 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1928, cons1922, cons1802, cons1938)
def replacement6582(p, d, a, n, c, x):
rubi.append(6582)
return -Dist(c**p, Subst(Int((S(1) - x/a)**(-n/S(2) + p)*(S(1) + x/a)**(n/S(2) + p)/x**S(2), x), x, S(1)/x), x)
rule6582 = ReplacementRule(pattern6582, replacement6582)
pattern6583 = Pattern(Integral(x_**WC('m', S(1))*(c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1928, cons1922, cons1802, cons1938, cons17)
def replacement6583(p, m, d, a, n, c, x):
rubi.append(6583)
return -Dist(c**p, Subst(Int(x**(-m + S(-2))*(S(1) - x/a)**(-n/S(2) + p)*(S(1) + x/a)**(n/S(2) + p), x), x, S(1)/x), x)
rule6583 = ReplacementRule(pattern6583, replacement6583)
pattern6584 = Pattern(Integral(x_**m_*(c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1928, cons1922, cons1802, cons1938, cons18)
def replacement6584(p, m, d, a, n, c, x):
rubi.append(6584)
return -Dist(c**p*x**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(S(1) - x/a)**(-n/S(2) + p)*(S(1) + x/a)**(n/S(2) + p), x), x, S(1)/x), x)
rule6584 = ReplacementRule(pattern6584, replacement6584)
pattern6585 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**p_*WC('u', S(1))*exp(WC('n', S(1))*acoth(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1928, cons1922, cons1803)
def replacement6585(p, u, d, a, n, c, x):
rubi.append(6585)
return Dist(c**IntPart(p)*(S(1) - S(1)/(a**S(2)*x**S(2)))**(-FracPart(p))*(c + d/x**S(2))**FracPart(p), Int(u*(S(1) - S(1)/(a**S(2)*x**S(2)))**p*exp(n*acoth(a*x)), x), x)
rule6585 = ReplacementRule(pattern6585, replacement6585)
pattern6586 = Pattern(Integral(WC('u', S(1))*exp(n_*acoth((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons743)
def replacement6586(u, b, c, a, n, x):
rubi.append(6586)
return Dist((S(-1))**(n/S(2)), Int(u*exp(n*atanh(c*(a + b*x))), x), x)
rule6586 = ReplacementRule(pattern6586, replacement6586)
pattern6587 = Pattern(Integral(exp(WC('n', S(1))*acoth((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1922)
def replacement6587(b, c, n, a, x):
rubi.append(6587)
return Dist((c*(a + b*x))**(n/S(2))*(S(1) + S(1)/(c*(a + b*x)))**(n/S(2))*(a*c + b*c*x + S(1))**(-n/S(2)), Int((a*c + b*c*x + S(-1))**(-n/S(2))*(a*c + b*c*x + S(1))**(n/S(2)), x), x)
rule6587 = ReplacementRule(pattern6587, replacement6587)
pattern6588 = Pattern(Integral(x_**m_*exp(n_*acoth((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons84, cons87, cons994)
def replacement6588(m, b, c, n, a, x):
rubi.append(6588)
return Dist(-S(4)*b**(-m + S(-1))*c**(-m + S(-1))/n, Subst(Int(x**(S(2)/n)*(x**(S(2)/n) + S(-1))**(-m + S(-2))*(a*c + x**(S(2)/n)*(-a*c + S(1)) + S(1))**m, x), x, (S(1) - S(1)/(c*(a + b*x)))**(-n/S(2))*(S(1) + S(1)/(c*(a + b*x)))**(n/S(2))), x)
rule6588 = ReplacementRule(pattern6588, replacement6588)
pattern6589 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*exp(WC('n', S(1))*acoth((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1922)
def replacement6589(m, b, d, c, n, a, x, e):
rubi.append(6589)
return Dist((c*(a + b*x))**(n/S(2))*(S(1) + S(1)/(c*(a + b*x)))**(n/S(2))*(a*c + b*c*x + S(1))**(-n/S(2)), Int((d + e*x)**m*(a*c + b*c*x + S(-1))**(-n/S(2))*(a*c + b*c*x + S(1))**(n/S(2)), x), x)
rule6589 = ReplacementRule(pattern6589, replacement6589)
pattern6590 = Pattern(Integral((c_ + x_**S(2)*WC('e', S(1)) + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acoth(a_ + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1922, cons1818, cons1929, cons1930)
def replacement6590(p, u, b, d, a, n, c, x, e):
rubi.append(6590)
return Dist((c/(-a**S(2) + S(1)))**p*((a + b*x + S(1))/(a + b*x))**(n/S(2))*((a + b*x)/(a + b*x + S(1)))**(n/S(2))*(-a - b*x + S(1))**(n/S(2))*(a + b*x + S(-1))**(-n/S(2)), Int(u*(-a - b*x + S(1))**(-n/S(2) + p)*(a + b*x + S(1))**(n/S(2) + p), x), x)
rule6590 = ReplacementRule(pattern6590, replacement6590)
pattern6591 = Pattern(Integral((c_ + x_**S(2)*WC('e', S(1)) + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acoth(a_ + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1922, cons1818, cons1929, cons1931)
def replacement6591(p, u, b, d, a, n, c, x, e):
rubi.append(6591)
return Dist((c + d*x + e*x**S(2))**p*(-a**S(2) - S(2)*a*b*x - b**S(2)*x**S(2) + S(1))**(-p), Int(u*(-a**S(2) - S(2)*a*b*x - b**S(2)*x**S(2) + S(1))**p*exp(n*acoth(a*x)), x), x)
rule6591 = ReplacementRule(pattern6591, replacement6591)
pattern6592 = Pattern(Integral(WC('u', S(1))*exp(WC('n', S(1))*acoth(WC('c', S(1))/(x_*WC('b', S(1)) + WC('a', S(0))))), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6592(u, b, c, n, a, x):
rubi.append(6592)
return Int(u*exp(n*atanh(a/c + b*x/c)), x)
rule6592 = ReplacementRule(pattern6592, replacement6592)
pattern6593 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons148)
def replacement6593(b, d, c, a, n, x):
rubi.append(6593)
return Dist(S(1)/d, Subst(Int((a + b*atanh(x))**n, x), x, c + d*x), x)
rule6593 = ReplacementRule(pattern6593, replacement6593)
pattern6594 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons148)
def replacement6594(b, d, c, a, n, x):
rubi.append(6594)
return Dist(S(1)/d, Subst(Int((a + b*acoth(x))**n, x), x, c + d*x), x)
rule6594 = ReplacementRule(pattern6594, replacement6594)
pattern6595 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(c_ + x_*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons340)
def replacement6595(b, d, c, a, n, x):
rubi.append(6595)
return Int((a + b*atanh(c + d*x))**n, x)
rule6595 = ReplacementRule(pattern6595, replacement6595)
pattern6596 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(c_ + x_*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons340)
def replacement6596(b, d, c, a, n, x):
rubi.append(6596)
return Int((a + b*acoth(c + d*x))**n, x)
rule6596 = ReplacementRule(pattern6596, replacement6596)
pattern6597 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons148)
def replacement6597(m, f, b, d, a, n, c, x, e):
rubi.append(6597)
return Dist(S(1)/d, Subst(Int((a + b*atanh(x))**n*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule6597 = ReplacementRule(pattern6597, replacement6597)
pattern6598 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons148)
def replacement6598(m, f, b, d, a, n, c, x, e):
rubi.append(6598)
return Dist(S(1)/d, Subst(Int((a + b*acoth(x))**n*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule6598 = ReplacementRule(pattern6598, replacement6598)
pattern6599 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**m_*(WC('a', S(0)) + WC('b', S(1))*atanh(c_ + x_*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons340)
def replacement6599(m, f, b, d, a, c, n, x, e):
rubi.append(6599)
return Int((a + b*atanh(c + d*x))**n*(e + f*x)**m, x)
rule6599 = ReplacementRule(pattern6599, replacement6599)
pattern6600 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**m_*(WC('a', S(0)) + WC('b', S(1))*acoth(c_ + x_*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons340)
def replacement6600(m, f, b, d, a, c, n, x, e):
rubi.append(6600)
return Int((a + b*acoth(c + d*x))**n*(e + f*x)**m, x)
rule6600 = ReplacementRule(pattern6600, replacement6600)
pattern6601 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*atanh(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons36, cons4, cons5, cons1762, cons1763)
def replacement6601(B, C, p, b, d, a, n, c, x, A):
rubi.append(6601)
return Dist(S(1)/d, Subst(Int((a + b*atanh(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p, x), x, c + d*x), x)
rule6601 = ReplacementRule(pattern6601, replacement6601)
pattern6602 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acoth(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons36, cons4, cons5, cons1762, cons1763)
def replacement6602(B, C, p, b, d, a, n, c, x, A):
rubi.append(6602)
return Dist(S(1)/d, Subst(Int((a + b*acoth(x))**n*(C*x**S(2)/d**S(2) + C/d**S(2))**p, x), x, c + d*x), x)
rule6602 = ReplacementRule(pattern6602, replacement6602)
pattern6603 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons1762, cons1763)
def replacement6603(B, C, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(6603)
return Dist(S(1)/d, Subst(Int((a + b*atanh(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule6603 = ReplacementRule(pattern6603, replacement6603)
pattern6604 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons1762, cons1763)
def replacement6604(B, C, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(6604)
return Dist(S(1)/d, Subst(Int((a + b*acoth(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule6604 = ReplacementRule(pattern6604, replacement6604)
pattern6605 = Pattern(Integral(atanh(a_ + x_*WC('b', S(1)))/(c_ + x_**WC('n', S(1))*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons87)
def replacement6605(b, d, a, n, c, x):
rubi.append(6605)
return -Dist(S(1)/2, Int(log(-a - b*x + S(1))/(c + d*x**n), x), x) + Dist(S(1)/2, Int(log(a + b*x + S(1))/(c + d*x**n), x), x)
rule6605 = ReplacementRule(pattern6605, replacement6605)
pattern6606 = Pattern(Integral(acoth(a_ + x_*WC('b', S(1)))/(c_ + x_**WC('n', S(1))*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons87)
def replacement6606(b, d, a, n, c, x):
rubi.append(6606)
return -Dist(S(1)/2, Int(log((a + b*x + S(-1))/(a + b*x))/(c + d*x**n), x), x) + Dist(S(1)/2, Int(log((a + b*x + S(1))/(a + b*x))/(c + d*x**n), x), x)
rule6606 = ReplacementRule(pattern6606, replacement6606)
pattern6607 = Pattern(Integral(atanh(a_ + x_*WC('b', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons1094)
def replacement6607(b, d, c, a, n, x):
rubi.append(6607)
return Int(atanh(a + b*x)/(c + d*x**n), x)
rule6607 = ReplacementRule(pattern6607, replacement6607)
pattern6608 = Pattern(Integral(acoth(a_ + x_*WC('b', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons1094)
def replacement6608(b, d, c, a, n, x):
rubi.append(6608)
return Int(acoth(a + b*x)/(c + d*x**n), x)
rule6608 = ReplacementRule(pattern6608, replacement6608)
pattern6609 = Pattern(Integral(atanh(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons4, cons1831)
def replacement6609(x, a, n, b):
rubi.append(6609)
return -Dist(b*n, Int(x**n/(-a**S(2) - S(2)*a*b*x**n - b**S(2)*x**(S(2)*n) + S(1)), x), x) + Simp(x*atanh(a + b*x**n), x)
rule6609 = ReplacementRule(pattern6609, replacement6609)
pattern6610 = Pattern(Integral(acoth(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons4, cons1831)
def replacement6610(x, a, n, b):
rubi.append(6610)
return -Dist(b*n, Int(x**n/(-a**S(2) - S(2)*a*b*x**n - b**S(2)*x**(S(2)*n) + S(1)), x), x) + Simp(x*acoth(a + b*x**n), x)
rule6610 = ReplacementRule(pattern6610, replacement6610)
pattern6611 = Pattern(Integral(atanh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))/x_, x_), cons2, cons3, cons4, cons1831)
def replacement6611(x, a, n, b):
rubi.append(6611)
return -Dist(S(1)/2, Int(log(-a - b*x**n + S(1))/x, x), x) + Dist(S(1)/2, Int(log(a + b*x**n + S(1))/x, x), x)
rule6611 = ReplacementRule(pattern6611, replacement6611)
pattern6612 = Pattern(Integral(acoth(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))/x_, x_), cons2, cons3, cons4, cons1831)
def replacement6612(x, a, n, b):
rubi.append(6612)
return -Dist(S(1)/2, Int(log(S(1) - S(1)/(a + b*x**n))/x, x), x) + Dist(S(1)/2, Int(log(S(1) + S(1)/(a + b*x**n))/x, x), x)
rule6612 = ReplacementRule(pattern6612, replacement6612)
pattern6613 = Pattern(Integral(x_**WC('m', S(1))*atanh(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons93, cons1832, cons1833)
def replacement6613(m, b, a, n, x):
rubi.append(6613)
return -Dist(b*n/(m + S(1)), Int(x**(m + n)/(-a**S(2) - S(2)*a*b*x**n - b**S(2)*x**(S(2)*n) + S(1)), x), x) + Simp(x**(m + S(1))*atanh(a + b*x**n)/(m + S(1)), x)
rule6613 = ReplacementRule(pattern6613, replacement6613)
pattern6614 = Pattern(Integral(x_**WC('m', S(1))*acoth(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons93, cons1832, cons1833)
def replacement6614(m, b, a, n, x):
rubi.append(6614)
return -Dist(b*n/(m + S(1)), Int(x**(m + n)/(-a**S(2) - S(2)*a*b*x**n - b**S(2)*x**(S(2)*n) + S(1)), x), x) + Simp(x**(m + S(1))*acoth(a + b*x**n)/(m + S(1)), x)
rule6614 = ReplacementRule(pattern6614, replacement6614)
pattern6615 = Pattern(Integral(atanh(f_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons1834)
def replacement6615(f, b, d, c, a, x):
rubi.append(6615)
return -Dist(S(1)/2, Int(log(-a - b*f**(c + d*x) + S(1)), x), x) + Dist(S(1)/2, Int(log(a + b*f**(c + d*x) + S(1)), x), x)
rule6615 = ReplacementRule(pattern6615, replacement6615)
pattern6616 = Pattern(Integral(acoth(f_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons1834)
def replacement6616(f, b, d, c, a, x):
rubi.append(6616)
return -Dist(S(1)/2, Int(log(S(1) - S(1)/(a + b*f**(c + d*x))), x), x) + Dist(S(1)/2, Int(log(S(1) + S(1)/(a + b*f**(c + d*x))), x), x)
rule6616 = ReplacementRule(pattern6616, replacement6616)
pattern6617 = Pattern(Integral(x_**WC('m', S(1))*atanh(f_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons17, cons168)
def replacement6617(m, f, b, d, c, a, x):
rubi.append(6617)
return -Dist(S(1)/2, Int(x**m*log(-a - b*f**(c + d*x) + S(1)), x), x) + Dist(S(1)/2, Int(x**m*log(a + b*f**(c + d*x) + S(1)), x), x)
rule6617 = ReplacementRule(pattern6617, replacement6617)
pattern6618 = Pattern(Integral(x_**WC('m', S(1))*acoth(f_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons17, cons168)
def replacement6618(m, f, b, d, c, a, x):
rubi.append(6618)
return -Dist(S(1)/2, Int(x**m*log(S(1) - S(1)/(a + b*f**(c + d*x))), x), x) + Dist(S(1)/2, Int(x**m*log(S(1) + S(1)/(a + b*f**(c + d*x))), x), x)
rule6618 = ReplacementRule(pattern6618, replacement6618)
pattern6619 = Pattern(Integral(WC('u', S(1))*atanh(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement6619(u, m, b, c, n, a, x):
rubi.append(6619)
return Int(u*acoth(a/c + b*x**n/c)**m, x)
rule6619 = ReplacementRule(pattern6619, replacement6619)
pattern6620 = Pattern(Integral(WC('u', S(1))*acoth(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement6620(u, m, b, c, n, a, x):
rubi.append(6620)
return Int(u*atanh(a/c + b*x**n/c)**m, x)
rule6620 = ReplacementRule(pattern6620, replacement6620)
pattern6621 = Pattern(Integral(S(1)/(sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0)))*atanh(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))), x_), cons2, cons3, cons7, cons1939)
def replacement6621(x, c, b, a):
rubi.append(6621)
return Simp(log(atanh(c*x/sqrt(a + b*x**S(2))))/c, x)
rule6621 = ReplacementRule(pattern6621, replacement6621)
pattern6622 = Pattern(Integral(S(1)/(sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0)))*acoth(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))), x_), cons2, cons3, cons7, cons1939)
def replacement6622(x, c, b, a):
rubi.append(6622)
return -Simp(log(acoth(c*x/sqrt(a + b*x**S(2))))/c, x)
rule6622 = ReplacementRule(pattern6622, replacement6622)
pattern6623 = Pattern(Integral(atanh(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons21, cons1939, cons66)
def replacement6623(m, b, c, a, x):
rubi.append(6623)
return Simp(atanh(c*x/sqrt(a + b*x**S(2)))**(m + S(1))/(c*(m + S(1))), x)
rule6623 = ReplacementRule(pattern6623, replacement6623)
pattern6624 = Pattern(Integral(acoth(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons21, cons1939, cons66)
def replacement6624(m, b, c, a, x):
rubi.append(6624)
return -Simp(acoth(c*x/sqrt(a + b*x**S(2)))**(m + S(1))/(c*(m + S(1))), x)
rule6624 = ReplacementRule(pattern6624, replacement6624)
pattern6625 = Pattern(Integral(atanh(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))/sqrt(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1939, cons383)
def replacement6625(m, b, d, c, a, x, e):
rubi.append(6625)
return Dist(sqrt(a + b*x**S(2))/sqrt(d + e*x**S(2)), Int(atanh(c*x/sqrt(a + b*x**S(2)))**m/sqrt(a + b*x**S(2)), x), x)
rule6625 = ReplacementRule(pattern6625, replacement6625)
pattern6626 = Pattern(Integral(acoth(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))/sqrt(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1939, cons383)
def replacement6626(m, b, d, c, a, x, e):
rubi.append(6626)
return Dist(sqrt(a + b*x**S(2))/sqrt(d + e*x**S(2)), Int(acoth(c*x/sqrt(a + b*x**S(2)))**m/sqrt(a + b*x**S(2)), x), x)
rule6626 = ReplacementRule(pattern6626, replacement6626)
def With6627(d, a, n, c, x):
u = IntHide((c + d*x**S(2))**n, x)
rubi.append(6627)
return -Dist(a, Int(Dist(S(1)/(-a**S(2)*x**S(2) + S(1)), u, x), x), x) + Dist(atanh(a*x), u, x)
pattern6627 = Pattern(Integral((x_**S(2)*WC('d', S(1)) + WC('c', S(0)))**n_*atanh(x_*WC('a', S(1))), x_), cons2, cons7, cons27, cons808, cons1586)
rule6627 = ReplacementRule(pattern6627, With6627)
def With6628(d, a, n, c, x):
u = IntHide((c + d*x**S(2))**n, x)
rubi.append(6628)
return -Dist(a, Int(Dist(S(1)/(-a**S(2)*x**S(2) + S(1)), u, x), x), x) + Dist(acoth(a*x), u, x)
pattern6628 = Pattern(Integral((x_**S(2)*WC('d', S(1)) + WC('c', S(0)))**n_*acoth(x_*WC('a', S(1))), x_), cons2, cons7, cons27, cons808, cons1586)
rule6628 = ReplacementRule(pattern6628, With6628)
def With6629(v, x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
tmp = InverseFunctionOfLinear(u, x)
res = And(Not(FalseQ(tmp)), SameQ(Head(tmp), ArcTanh), ZeroQ(-D(v, x)**S(2) + Discriminant(v, x)*Part(tmp, S(1))**S(2)))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6629 = Pattern(Integral(u_*v_**WC('n', S(1)), x_), cons818, cons85, cons463, cons1940, cons1941, CustomConstraint(With6629))
def replacement6629(v, x, n, u):
tmp = InverseFunctionOfLinear(u, x)
rubi.append(6629)
return Dist((-Discriminant(v, x)/(S(4)*Coefficient(v, x, S(2))))**n/Coefficient(Part(tmp, S(1)), x, S(1)), Subst(Int(SimplifyIntegrand((S(1)/cosh(x))**(S(2)*n + S(2))*SubstForInverseFunction(u, tmp, x), x), x), x, tmp), x)
rule6629 = ReplacementRule(pattern6629, replacement6629)
def With6630(v, x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
tmp = InverseFunctionOfLinear(u, x)
res = And(Not(FalseQ(tmp)), SameQ(Head(tmp), ArcCoth), ZeroQ(-D(v, x)**S(2) + Discriminant(v, x)*Part(tmp, S(1))**S(2)))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6630 = Pattern(Integral(u_*v_**WC('n', S(1)), x_), cons818, cons85, cons463, cons1940, cons1942, CustomConstraint(With6630))
def replacement6630(v, x, n, u):
tmp = InverseFunctionOfLinear(u, x)
rubi.append(6630)
return Dist((-Discriminant(v, x)/(S(4)*Coefficient(v, x, S(2))))**n/Coefficient(Part(tmp, S(1)), x, S(1)), Subst(Int(SimplifyIntegrand((-S(1)/sinh(x)**S(2))**(n + S(1))*SubstForInverseFunction(u, tmp, x), x), x), x, tmp), x)
rule6630 = ReplacementRule(pattern6630, replacement6630)
pattern6631 = Pattern(Integral(atanh(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1943)
def replacement6631(b, d, c, a, x):
rubi.append(6631)
return Dist(b, Int(x/(c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp(x*atanh(c + d*tanh(a + b*x)), x)
rule6631 = ReplacementRule(pattern6631, replacement6631)
pattern6632 = Pattern(Integral(acoth(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1943)
def replacement6632(b, d, c, a, x):
rubi.append(6632)
return Dist(b, Int(x/(c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp(x*acoth(c + d*tanh(a + b*x)), x)
rule6632 = ReplacementRule(pattern6632, replacement6632)
pattern6633 = Pattern(Integral(atanh(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1943)
def replacement6633(b, d, c, a, x):
rubi.append(6633)
return Dist(b, Int(x/(-c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp(x*atanh(c + d/tanh(a + b*x)), x)
rule6633 = ReplacementRule(pattern6633, replacement6633)
pattern6634 = Pattern(Integral(acoth(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1943)
def replacement6634(b, d, c, a, x):
rubi.append(6634)
return Dist(b, Int(x/(-c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp(x*acoth(c + d/tanh(a + b*x)), x)
rule6634 = ReplacementRule(pattern6634, replacement6634)
pattern6635 = Pattern(Integral(atanh(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1944)
def replacement6635(b, d, c, a, x):
rubi.append(6635)
return Dist(b*(-c - d + S(1)), Int(x*exp(S(2)*a + S(2)*b*x)/(-c + d + (-c - d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) - Dist(b*(c + d + S(1)), Int(x*exp(S(2)*a + S(2)*b*x)/(c - d + (c + d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Simp(x*atanh(c + d*tanh(a + b*x)), x)
rule6635 = ReplacementRule(pattern6635, replacement6635)
pattern6636 = Pattern(Integral(acoth(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1944)
def replacement6636(b, d, c, a, x):
rubi.append(6636)
return Dist(b*(-c - d + S(1)), Int(x*exp(S(2)*a + S(2)*b*x)/(-c + d + (-c - d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) - Dist(b*(c + d + S(1)), Int(x*exp(S(2)*a + S(2)*b*x)/(c - d + (c + d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Simp(x*acoth(c + d*tanh(a + b*x)), x)
rule6636 = ReplacementRule(pattern6636, replacement6636)
pattern6637 = Pattern(Integral(atanh(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1944)
def replacement6637(b, d, c, a, x):
rubi.append(6637)
return -Dist(b*(-c - d + S(1)), Int(x*exp(S(2)*a + S(2)*b*x)/(-c + d - (-c - d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Dist(b*(c + d + S(1)), Int(x*exp(S(2)*a + S(2)*b*x)/(c - d - (c + d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Simp(x*atanh(c + d/tanh(a + b*x)), x)
rule6637 = ReplacementRule(pattern6637, replacement6637)
pattern6638 = Pattern(Integral(acoth(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1944)
def replacement6638(b, d, c, a, x):
rubi.append(6638)
return -Dist(b*(-c - d + S(1)), Int(x*exp(S(2)*a + S(2)*b*x)/(-c + d - (-c - d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Dist(b*(c + d + S(1)), Int(x*exp(S(2)*a + S(2)*b*x)/(c - d - (c + d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Simp(x*acoth(c + d/tanh(a + b*x)), x)
rule6638 = ReplacementRule(pattern6638, replacement6638)
pattern6639 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1943)
def replacement6639(m, f, b, d, c, a, x, e):
rubi.append(6639)
return Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp((e + f*x)**(m + S(1))*atanh(c + d*tanh(a + b*x))/(f*(m + S(1))), x)
rule6639 = ReplacementRule(pattern6639, replacement6639)
pattern6640 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1943)
def replacement6640(m, f, b, d, c, a, x, e):
rubi.append(6640)
return Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp((e + f*x)**(m + S(1))*acoth(c + d*tanh(a + b*x))/(f*(m + S(1))), x)
rule6640 = ReplacementRule(pattern6640, replacement6640)
pattern6641 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1943)
def replacement6641(m, f, b, d, c, a, x, e):
rubi.append(6641)
return Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(-c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp((e + f*x)**(m + S(1))*atanh(c + d/tanh(a + b*x))/(f*(m + S(1))), x)
rule6641 = ReplacementRule(pattern6641, replacement6641)
pattern6642 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1943)
def replacement6642(m, f, b, d, c, a, x, e):
rubi.append(6642)
return Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(-c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp((e + f*x)**(m + S(1))*acoth(c + d/tanh(a + b*x))/(f*(m + S(1))), x)
rule6642 = ReplacementRule(pattern6642, replacement6642)
pattern6643 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1944)
def replacement6643(m, f, b, d, c, a, x, e):
rubi.append(6643)
return Dist(b*(-c - d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(-c + d + (-c - d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) - Dist(b*(c + d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(c - d + (c + d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*atanh(c + d*tanh(a + b*x))/(f*(m + S(1))), x)
rule6643 = ReplacementRule(pattern6643, replacement6643)
pattern6644 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1944)
def replacement6644(m, f, b, d, c, a, x, e):
rubi.append(6644)
return Dist(b*(-c - d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(-c + d + (-c - d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) - Dist(b*(c + d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(c - d + (c + d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*acoth(c + d*tanh(a + b*x))/(f*(m + S(1))), x)
rule6644 = ReplacementRule(pattern6644, replacement6644)
pattern6645 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1944)
def replacement6645(m, f, b, d, c, a, x, e):
rubi.append(6645)
return -Dist(b*(-c - d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(-c + d - (-c - d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Dist(b*(c + d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(c - d - (c + d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*atanh(c + d/tanh(a + b*x))/(f*(m + S(1))), x)
rule6645 = ReplacementRule(pattern6645, replacement6645)
pattern6646 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1944)
def replacement6646(m, f, b, d, c, a, x, e):
rubi.append(6646)
return -Dist(b*(-c - d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(-c + d - (-c - d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Dist(b*(c + d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(c - d - (c + d + S(1))*exp(S(2)*a + S(2)*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*acoth(c + d/tanh(a + b*x))/(f*(m + S(1))), x)
rule6646 = ReplacementRule(pattern6646, replacement6646)
pattern6647 = Pattern(Integral(atanh(tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons67)
def replacement6647(x, a, b):
rubi.append(6647)
return -Dist(b, Int(x/cos(S(2)*a + S(2)*b*x), x), x) + Simp(x*atanh(tan(a + b*x)), x)
rule6647 = ReplacementRule(pattern6647, replacement6647)
pattern6648 = Pattern(Integral(acoth(tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons67)
def replacement6648(x, a, b):
rubi.append(6648)
return -Dist(b, Int(x/cos(S(2)*a + S(2)*b*x), x), x) + Simp(x*acoth(tan(a + b*x)), x)
rule6648 = ReplacementRule(pattern6648, replacement6648)
pattern6649 = Pattern(Integral(atanh(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons67)
def replacement6649(x, a, b):
rubi.append(6649)
return -Dist(b, Int(x/cos(S(2)*a + S(2)*b*x), x), x) + Simp(x*atanh(S(1)/tan(a + b*x)), x)
rule6649 = ReplacementRule(pattern6649, replacement6649)
pattern6650 = Pattern(Integral(acoth(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons67)
def replacement6650(x, a, b):
rubi.append(6650)
return -Dist(b, Int(x/cos(S(2)*a + S(2)*b*x), x), x) + Simp(x*acoth(S(1)/tan(a + b*x)), x)
rule6650 = ReplacementRule(pattern6650, replacement6650)
pattern6651 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement6651(m, f, b, a, x, e):
rubi.append(6651)
return -Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/cos(S(2)*a + S(2)*b*x), x), x) + Simp((e + f*x)**(m + S(1))*atanh(tan(a + b*x))/(f*(m + S(1))), x)
rule6651 = ReplacementRule(pattern6651, replacement6651)
pattern6652 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement6652(m, f, b, a, x, e):
rubi.append(6652)
return -Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/cos(S(2)*a + S(2)*b*x), x), x) + Simp((e + f*x)**(m + S(1))*acoth(tan(a + b*x))/(f*(m + S(1))), x)
rule6652 = ReplacementRule(pattern6652, replacement6652)
pattern6653 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement6653(m, f, b, a, x, e):
rubi.append(6653)
return -Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/cos(S(2)*a + S(2)*b*x), x), x) + Simp((e + f*x)**(m + S(1))*atanh(S(1)/tan(a + b*x))/(f*(m + S(1))), x)
rule6653 = ReplacementRule(pattern6653, replacement6653)
pattern6654 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement6654(m, f, b, a, x, e):
rubi.append(6654)
return -Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/cos(S(2)*a + S(2)*b*x), x), x) + Simp((e + f*x)**(m + S(1))*acoth(S(1)/tan(a + b*x))/(f*(m + S(1))), x)
rule6654 = ReplacementRule(pattern6654, replacement6654)
pattern6655 = Pattern(Integral(atanh(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1945)
def replacement6655(b, d, c, a, x):
rubi.append(6655)
return Dist(I*b, Int(x/(c*exp(S(2)*I*a + S(2)*I*b*x) + c + I*d), x), x) + Simp(x*atanh(c + d*tan(a + b*x)), x)
rule6655 = ReplacementRule(pattern6655, replacement6655)
pattern6656 = Pattern(Integral(acoth(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1945)
def replacement6656(b, d, c, a, x):
rubi.append(6656)
return Dist(I*b, Int(x/(c*exp(S(2)*I*a + S(2)*I*b*x) + c + I*d), x), x) + Simp(x*acoth(c + d*tan(a + b*x)), x)
rule6656 = ReplacementRule(pattern6656, replacement6656)
pattern6657 = Pattern(Integral(atanh(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1946)
def replacement6657(b, d, c, a, x):
rubi.append(6657)
return Dist(I*b, Int(x/(-c*exp(S(2)*I*a + S(2)*I*b*x) + c - I*d), x), x) + Simp(x*atanh(c + d/tan(a + b*x)), x)
rule6657 = ReplacementRule(pattern6657, replacement6657)
pattern6658 = Pattern(Integral(acoth(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1946)
def replacement6658(b, d, c, a, x):
rubi.append(6658)
return Dist(I*b, Int(x/(-c*exp(S(2)*I*a + S(2)*I*b*x) + c - I*d), x), x) + Simp(x*acoth(c + d/tan(a + b*x)), x)
rule6658 = ReplacementRule(pattern6658, replacement6658)
pattern6659 = Pattern(Integral(atanh(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1947)
def replacement6659(b, d, c, a, x):
rubi.append(6659)
return Dist(I*b*(-c + I*d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(-c - I*d + (-c + I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) - Dist(I*b*(c - I*d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(c + I*d + (c - I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp(x*atanh(c + d*tan(a + b*x)), x)
rule6659 = ReplacementRule(pattern6659, replacement6659)
pattern6660 = Pattern(Integral(acoth(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1947)
def replacement6660(b, d, c, a, x):
rubi.append(6660)
return Dist(I*b*(-c + I*d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(-c - I*d + (-c + I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) - Dist(I*b*(c - I*d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(c + I*d + (c - I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp(x*acoth(c + d*tan(a + b*x)), x)
rule6660 = ReplacementRule(pattern6660, replacement6660)
pattern6661 = Pattern(Integral(atanh(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1948)
def replacement6661(b, d, c, a, x):
rubi.append(6661)
return -Dist(I*b*(-c - I*d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(-c + I*d - (-c - I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Dist(I*b*(c + I*d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(c - I*d - (c + I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp(x*atanh(c + d/tan(a + b*x)), x)
rule6661 = ReplacementRule(pattern6661, replacement6661)
pattern6662 = Pattern(Integral(acoth(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1948)
def replacement6662(b, d, c, a, x):
rubi.append(6662)
return -Dist(I*b*(-c - I*d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(-c + I*d - (-c - I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Dist(I*b*(c + I*d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(c - I*d - (c + I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp(x*acoth(c + d/tan(a + b*x)), x)
rule6662 = ReplacementRule(pattern6662, replacement6662)
pattern6663 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1945)
def replacement6663(m, f, b, d, c, a, x, e):
rubi.append(6663)
return Dist(I*b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(c*exp(S(2)*I*a + S(2)*I*b*x) + c + I*d), x), x) + Simp((e + f*x)**(m + S(1))*atanh(c + d*tan(a + b*x))/(f*(m + S(1))), x)
rule6663 = ReplacementRule(pattern6663, replacement6663)
pattern6664 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1945)
def replacement6664(m, f, b, d, c, a, x, e):
rubi.append(6664)
return Dist(I*b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(c*exp(S(2)*I*a + S(2)*I*b*x) + c + I*d), x), x) + Simp((e + f*x)**(m + S(1))*acoth(c + d*tan(a + b*x))/(f*(m + S(1))), x)
rule6664 = ReplacementRule(pattern6664, replacement6664)
pattern6665 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1946)
def replacement6665(m, f, b, d, c, a, x, e):
rubi.append(6665)
return Dist(I*b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(-c*exp(S(2)*I*a + S(2)*I*b*x) + c - I*d), x), x) + Simp((e + f*x)**(m + S(1))*atanh(c + d/tan(a + b*x))/(f*(m + S(1))), x)
rule6665 = ReplacementRule(pattern6665, replacement6665)
pattern6666 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1946)
def replacement6666(m, f, b, d, c, a, x, e):
rubi.append(6666)
return Dist(I*b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(-c*exp(S(2)*I*a + S(2)*I*b*x) + c - I*d), x), x) + Simp((e + f*x)**(m + S(1))*acoth(c + d/tan(a + b*x))/(f*(m + S(1))), x)
rule6666 = ReplacementRule(pattern6666, replacement6666)
pattern6667 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1947)
def replacement6667(m, f, b, d, c, a, x, e):
rubi.append(6667)
return Dist(I*b*(-c + I*d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(-c - I*d + (-c + I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) - Dist(I*b*(c - I*d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(c + I*d + (c - I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*atanh(c + d*tan(a + b*x))/(f*(m + S(1))), x)
rule6667 = ReplacementRule(pattern6667, replacement6667)
pattern6668 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1947)
def replacement6668(m, f, b, d, c, a, x, e):
rubi.append(6668)
return Dist(I*b*(-c + I*d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(-c - I*d + (-c + I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) - Dist(I*b*(c - I*d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(c + I*d + (c - I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*acoth(c + d*tan(a + b*x))/(f*(m + S(1))), x)
rule6668 = ReplacementRule(pattern6668, replacement6668)
pattern6669 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*atanh(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1948)
def replacement6669(m, f, b, d, c, a, x, e):
rubi.append(6669)
return -Dist(I*b*(-c - I*d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(-c + I*d - (-c - I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Dist(I*b*(c + I*d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(c - I*d - (c + I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*atanh(c + d/tan(a + b*x))/(f*(m + S(1))), x)
rule6669 = ReplacementRule(pattern6669, replacement6669)
pattern6670 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acoth(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1948)
def replacement6670(m, f, b, d, c, a, x, e):
rubi.append(6670)
return -Dist(I*b*(-c - I*d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(-c + I*d - (-c - I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Dist(I*b*(c + I*d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(c - I*d - (c + I*d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*acoth(c + d/tan(a + b*x))/(f*(m + S(1))), x)
rule6670 = ReplacementRule(pattern6670, replacement6670)
pattern6671 = Pattern(Integral(atanh(u_), x_), cons1230)
def replacement6671(x, u):
rubi.append(6671)
return -Int(SimplifyIntegrand(x*D(u, x)/(-u**S(2) + S(1)), x), x) + Simp(x*atanh(u), x)
rule6671 = ReplacementRule(pattern6671, replacement6671)
pattern6672 = Pattern(Integral(acoth(u_), x_), cons1230)
def replacement6672(x, u):
rubi.append(6672)
return -Int(SimplifyIntegrand(x*D(u, x)/(-u**S(2) + S(1)), x), x) + Simp(x*acoth(u), x)
rule6672 = ReplacementRule(pattern6672, replacement6672)
pattern6673 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*atanh(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1847)
def replacement6673(u, m, b, d, c, a, x):
rubi.append(6673)
return -Dist(b/(d*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(-u**S(2) + S(1)), x), x), x) + Simp((a + b*atanh(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule6673 = ReplacementRule(pattern6673, replacement6673)
pattern6674 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acoth(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1847)
def replacement6674(u, m, b, d, c, a, x):
rubi.append(6674)
return -Dist(b/(d*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(-u**S(2) + S(1)), x), x), x) + Simp((a + b*acoth(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule6674 = ReplacementRule(pattern6674, replacement6674)
def With6675(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern6675 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*atanh(u_)), x_), cons2, cons3, cons1230, cons1949, cons1950, CustomConstraint(With6675))
def replacement6675(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(6675)
return -Dist(b, Int(SimplifyIntegrand(w*D(u, x)/(-u**S(2) + S(1)), x), x), x) + Dist(a + b*atanh(u), w, x)
rule6675 = ReplacementRule(pattern6675, replacement6675)
def With6676(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern6676 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*acoth(u_)), x_), cons2, cons3, cons1230, cons1951, cons1952, CustomConstraint(With6676))
def replacement6676(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(6676)
return -Dist(b, Int(SimplifyIntegrand(w*D(u, x)/(-u**S(2) + S(1)), x), x), x) + Dist(a + b*acoth(u), w, x)
rule6676 = ReplacementRule(pattern6676, replacement6676)
pattern6677 = Pattern(Integral(asech(x_*WC('c', S(1))), x_), cons7, cons7)
def replacement6677(x, c):
rubi.append(6677)
return Dist(sqrt(c*x + S(1))*sqrt(S(1)/(c*x + S(1))), Int(S(1)/(sqrt(-c*x + S(1))*sqrt(c*x + S(1))), x), x) + Simp(x*asech(c*x), x)
rule6677 = ReplacementRule(pattern6677, replacement6677)
pattern6678 = Pattern(Integral(acsch(x_*WC('c', S(1))), x_), cons7, cons7)
def replacement6678(x, c):
rubi.append(6678)
return Dist(S(1)/c, Int(S(1)/(x*sqrt(S(1) + S(1)/(c**S(2)*x**S(2)))), x), x) + Simp(x*acsch(c*x), x)
rule6678 = ReplacementRule(pattern6678, replacement6678)
pattern6679 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6679(b, a, n, c, x):
rubi.append(6679)
return -Dist(S(1)/c, Subst(Int((a + b*x)**n*tanh(x)/cosh(x), x), x, asech(c*x)), x)
rule6679 = ReplacementRule(pattern6679, replacement6679)
pattern6680 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons1579)
def replacement6680(b, a, n, c, x):
rubi.append(6680)
return -Dist(S(1)/c, Subst(Int((a + b*x)**n/(sinh(x)*tanh(x)), x), x, acsch(c*x)), x)
rule6680 = ReplacementRule(pattern6680, replacement6680)
pattern6681 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))/x_, x_), cons2, cons3, cons7, cons14)
def replacement6681(x, a, c, b):
rubi.append(6681)
return -Subst(Int((a + b*acosh(x/c))/x, x), x, S(1)/x)
rule6681 = ReplacementRule(pattern6681, replacement6681)
pattern6682 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))/x_, x_), cons2, cons3, cons7, cons14)
def replacement6682(x, a, c, b):
rubi.append(6682)
return -Subst(Int((a + b*asinh(x/c))/x, x), x, S(1)/x)
rule6682 = ReplacementRule(pattern6682, replacement6682)
pattern6683 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons66)
def replacement6683(m, b, a, c, x):
rubi.append(6683)
return Dist(b*sqrt(c*x + S(1))*sqrt(S(1)/(c*x + S(1)))/(m + S(1)), Int(x**m/(sqrt(-c*x + S(1))*sqrt(c*x + S(1))), x), x) + Simp(x**(m + S(1))*(a + b*asech(c*x))/(m + S(1)), x)
rule6683 = ReplacementRule(pattern6683, replacement6683)
pattern6684 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons66)
def replacement6684(m, b, a, c, x):
rubi.append(6684)
return Dist(b/(c*(m + S(1))), Int(x**(m + S(-1))/sqrt(S(1) + S(1)/(c**S(2)*x**S(2))), x), x) + Simp(x**(m + S(1))*(a + b*acsch(c*x))/(m + S(1)), x)
rule6684 = ReplacementRule(pattern6684, replacement6684)
pattern6685 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons17)
def replacement6685(m, b, a, c, n, x):
rubi.append(6685)
return -Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*(S(1)/cosh(x))**(m + S(1))*tanh(x), x), x, asech(c*x)), x)
rule6685 = ReplacementRule(pattern6685, replacement6685)
pattern6686 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons17)
def replacement6686(m, b, a, c, n, x):
rubi.append(6686)
return -Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*(S(1)/sinh(x))**(m + S(1))/tanh(x), x), x, acsch(c*x)), x)
rule6686 = ReplacementRule(pattern6686, replacement6686)
pattern6687 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons1854)
def replacement6687(m, b, a, n, c, x):
rubi.append(6687)
return Int(x**m*(a + b*asech(c*x))**n, x)
rule6687 = ReplacementRule(pattern6687, replacement6687)
pattern6688 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons1854)
def replacement6688(m, b, a, n, c, x):
rubi.append(6688)
return Int(x**m*(a + b*acsch(c*x))**n, x)
rule6688 = ReplacementRule(pattern6688, replacement6688)
def With6689(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(6689)
return Dist(b*sqrt(c*x + S(1))*sqrt(S(1)/(c*x + S(1))), Int(SimplifyIntegrand(u/(x*sqrt(-c*x + S(1))*sqrt(c*x + S(1))), x), x), x) + Dist(a + b*asech(c*x), u, x)
pattern6689 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1743)
rule6689 = ReplacementRule(pattern6689, With6689)
def With6690(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(6690)
return -Dist(b*c*x/sqrt(-c**S(2)*x**S(2)), Int(SimplifyIntegrand(u/(x*sqrt(-c**S(2)*x**S(2) + S(-1))), x), x), x) + Dist(a + b*acsch(c*x), u, x)
pattern6690 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1743)
rule6690 = ReplacementRule(pattern6690, With6690)
pattern6691 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons38)
def replacement6691(p, b, d, a, n, c, x, e):
rubi.append(6691)
return -Subst(Int(x**(-S(2)*p + S(-2))*(a + b*acosh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x)
rule6691 = ReplacementRule(pattern6691, replacement6691)
pattern6692 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons38)
def replacement6692(p, b, d, a, n, c, x, e):
rubi.append(6692)
return -Subst(Int(x**(-S(2)*p + S(-2))*(a + b*asinh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x)
rule6692 = ReplacementRule(pattern6692, replacement6692)
pattern6693 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons667, cons178, cons1855)
def replacement6693(p, b, d, a, n, c, x, e):
rubi.append(6693)
return -Dist(sqrt(x**S(2))/x, Subst(Int(x**(-S(2)*p + S(-2))*(a + b*acosh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule6693 = ReplacementRule(pattern6693, replacement6693)
pattern6694 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons667, cons178, cons1855)
def replacement6694(p, b, d, a, n, c, x, e):
rubi.append(6694)
return -Dist(sqrt(x**S(2))/x, Subst(Int(x**(-S(2)*p + S(-2))*(a + b*asinh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule6694 = ReplacementRule(pattern6694, replacement6694)
pattern6695 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons667, cons1856)
def replacement6695(p, b, d, a, n, c, x, e):
rubi.append(6695)
return -Dist(sqrt(d + e*x**S(2))/(x*sqrt(d/x**S(2) + e)), Subst(Int(x**(-S(2)*p + S(-2))*(a + b*acosh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule6695 = ReplacementRule(pattern6695, replacement6695)
pattern6696 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons667, cons1856)
def replacement6696(p, b, d, a, n, c, x, e):
rubi.append(6696)
return -Dist(sqrt(d + e*x**S(2))/(x*sqrt(d/x**S(2) + e)), Subst(Int(x**(-S(2)*p + S(-2))*(a + b*asinh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule6696 = ReplacementRule(pattern6696, replacement6696)
pattern6697 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement6697(p, b, d, a, n, c, x, e):
rubi.append(6697)
return Int((a + b*asech(c*x))**n*(d + e*x**S(2))**p, x)
rule6697 = ReplacementRule(pattern6697, replacement6697)
pattern6698 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement6698(p, b, d, a, n, c, x, e):
rubi.append(6698)
return Int((a + b*acsch(c*x))**n*(d + e*x**S(2))**p, x)
rule6698 = ReplacementRule(pattern6698, replacement6698)
pattern6699 = Pattern(Integral(x_*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement6699(p, b, d, a, c, x, e):
rubi.append(6699)
return Dist(b*sqrt(c*x + S(1))*sqrt(S(1)/(c*x + S(1)))/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(x*sqrt(-c*x + S(1))*sqrt(c*x + S(1))), x), x) + Simp((a + b*asech(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6699 = ReplacementRule(pattern6699, replacement6699)
pattern6700 = Pattern(Integral(x_*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement6700(p, b, d, a, c, x, e):
rubi.append(6700)
return -Dist(b*c*x/(S(2)*e*sqrt(-c**S(2)*x**S(2))*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(x*sqrt(-c**S(2)*x**S(2) + S(-1))), x), x) + Simp((a + b*acsch(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule6700 = ReplacementRule(pattern6700, replacement6700)
def With6701(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(d + e*x**S(2))**p, x)
rubi.append(6701)
return Dist(b*sqrt(c*x + S(1))*sqrt(S(1)/(c*x + S(1))), Int(SimplifyIntegrand(u/(x*sqrt(-c*x + S(1))*sqrt(c*x + S(1))), x), x), x) + Dist(a + b*asech(c*x), u, x)
pattern6701 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1786)
rule6701 = ReplacementRule(pattern6701, With6701)
def With6702(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(d + e*x**S(2))**p, x)
rubi.append(6702)
return -Dist(b*c*x/sqrt(-c**S(2)*x**S(2)), Int(SimplifyIntegrand(u/(x*sqrt(-c**S(2)*x**S(2) + S(-1))), x), x), x) + Dist(a + b*acsch(c*x), u, x)
pattern6702 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1786)
rule6702 = ReplacementRule(pattern6702, With6702)
pattern6703 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1299)
def replacement6703(p, m, b, d, a, n, c, x, e):
rubi.append(6703)
return -Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*acosh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x)
rule6703 = ReplacementRule(pattern6703, replacement6703)
pattern6704 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1299)
def replacement6704(p, m, b, d, a, n, c, x, e):
rubi.append(6704)
return -Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*asinh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x)
rule6704 = ReplacementRule(pattern6704, replacement6704)
pattern6705 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons17, cons667, cons178, cons1855)
def replacement6705(p, m, b, d, a, n, c, x, e):
rubi.append(6705)
return -Dist(sqrt(x**S(2))/x, Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*acosh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule6705 = ReplacementRule(pattern6705, replacement6705)
pattern6706 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons17, cons667, cons178, cons1855)
def replacement6706(p, m, b, d, a, n, c, x, e):
rubi.append(6706)
return -Dist(sqrt(x**S(2))/x, Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*asinh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule6706 = ReplacementRule(pattern6706, replacement6706)
pattern6707 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons17, cons667, cons1856)
def replacement6707(p, m, b, d, a, n, c, x, e):
rubi.append(6707)
return -Dist(sqrt(d + e*x**S(2))/(x*sqrt(d/x**S(2) + e)), Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*acosh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule6707 = ReplacementRule(pattern6707, replacement6707)
pattern6708 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons17, cons667, cons1856)
def replacement6708(p, m, b, d, a, n, c, x, e):
rubi.append(6708)
return -Dist(sqrt(d + e*x**S(2))/(x*sqrt(d/x**S(2) + e)), Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*asinh(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule6708 = ReplacementRule(pattern6708, replacement6708)
pattern6709 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement6709(p, m, b, d, a, n, c, x, e):
rubi.append(6709)
return Int(x**m*(a + b*asech(c*x))**n*(d + e*x**S(2))**p, x)
rule6709 = ReplacementRule(pattern6709, replacement6709)
pattern6710 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement6710(p, m, b, d, a, n, c, x, e):
rubi.append(6710)
return Int(x**m*(a + b*acsch(c*x))**n*(d + e*x**S(2))**p, x)
rule6710 = ReplacementRule(pattern6710, replacement6710)
pattern6711 = Pattern(Integral(asech(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement6711(x, a, b):
rubi.append(6711)
return Int(sqrt((-a - b*x + S(1))/(a + b*x + S(1)))/(-a - b*x + S(1)), x) + Simp((a + b*x)*asech(a + b*x)/b, x)
rule6711 = ReplacementRule(pattern6711, replacement6711)
pattern6712 = Pattern(Integral(acsch(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement6712(x, a, b):
rubi.append(6712)
return Int(S(1)/(sqrt(S(1) + (a + b*x)**(S(-2)))*(a + b*x)), x) + Simp((a + b*x)*acsch(a + b*x)/b, x)
rule6712 = ReplacementRule(pattern6712, replacement6712)
pattern6713 = Pattern(Integral(asech(a_ + x_*WC('b', S(1)))**n_, x_), cons2, cons3, cons4, cons1831)
def replacement6713(x, a, n, b):
rubi.append(6713)
return -Dist(S(1)/b, Subst(Int(x**n*tanh(x)/cosh(x), x), x, asech(a + b*x)), x)
rule6713 = ReplacementRule(pattern6713, replacement6713)
pattern6714 = Pattern(Integral(acsch(a_ + x_*WC('b', S(1)))**n_, x_), cons2, cons3, cons4, cons1831)
def replacement6714(x, a, n, b):
rubi.append(6714)
return -Dist(S(1)/b, Subst(Int(x**n/(sinh(x)*tanh(x)), x), x, acsch(a + b*x)), x)
rule6714 = ReplacementRule(pattern6714, replacement6714)
pattern6715 = Pattern(Integral(asech(a_ + x_*WC('b', S(1)))/x_, x_), cons2, cons3, cons67)
def replacement6715(x, a, b):
rubi.append(6715)
return Simp(log(S(1) - (-sqrt(-a**S(2) + S(1)) + S(1))*exp(-asech(a + b*x))/a)*asech(a + b*x), x) + Simp(log(S(1) - (sqrt(-a**S(2) + S(1)) + S(1))*exp(-asech(a + b*x))/a)*asech(a + b*x), x) - Simp(log(S(1) + exp(-S(2)*asech(a + b*x)))*asech(a + b*x), x) - Simp(PolyLog(S(2), (-sqrt(-a**S(2) + S(1)) + S(1))*exp(-asech(a + b*x))/a), x) - Simp(PolyLog(S(2), (sqrt(-a**S(2) + S(1)) + S(1))*exp(-asech(a + b*x))/a), x) + Simp(PolyLog(S(2), -exp(-S(2)*asech(a + b*x)))/S(2), x)
rule6715 = ReplacementRule(pattern6715, replacement6715)
pattern6716 = Pattern(Integral(acsch(a_ + x_*WC('b', S(1)))/x_, x_), cons2, cons3, cons67)
def replacement6716(x, a, b):
rubi.append(6716)
return Simp(log(S(1) + (-sqrt(a**S(2) + S(1)) + S(1))*exp(acsch(a + b*x))/a)*acsch(a + b*x), x) + Simp(log(S(1) + (sqrt(a**S(2) + S(1)) + S(1))*exp(acsch(a + b*x))/a)*acsch(a + b*x), x) - Simp(log(S(1) - exp(-S(2)*acsch(a + b*x)))*acsch(a + b*x), x) + Simp(PolyLog(S(2), -(-sqrt(a**S(2) + S(1)) + S(1))*exp(acsch(a + b*x))/a), x) + Simp(PolyLog(S(2), -(sqrt(a**S(2) + S(1)) + S(1))*exp(acsch(a + b*x))/a), x) + Simp(PolyLog(S(2), exp(-S(2)*acsch(a + b*x)))/S(2), x) - Simp(acsch(a + b*x)**S(2), x)
rule6716 = ReplacementRule(pattern6716, replacement6716)
pattern6717 = Pattern(Integral(x_**WC('m', S(1))*asech(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons21, cons17, cons66)
def replacement6717(x, m, b, a):
rubi.append(6717)
return Dist(b**(-m + S(-1))/(m + S(1)), Subst(Int(x**(-m + S(-1))*((-a*x)**(m + S(1)) - (-a*x + S(1))**(m + S(1)))/(sqrt(x + S(-1))*sqrt(x + S(1))), x), x, S(1)/(a + b*x)), x) - Simp(b**(-m + S(-1))*(-b**(m + S(1))*x**(m + S(1)) + (-a)**(m + S(1)))*asech(a + b*x)/(m + S(1)), x)
rule6717 = ReplacementRule(pattern6717, replacement6717)
pattern6718 = Pattern(Integral(x_**WC('m', S(1))*acsch(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons21, cons17, cons66)
def replacement6718(x, m, b, a):
rubi.append(6718)
return Dist(b**(-m + S(-1))/(m + S(1)), Subst(Int(x**(-m + S(-1))*((-a*x)**(m + S(1)) - (-a*x + S(1))**(m + S(1)))/sqrt(x**S(2) + S(1)), x), x, S(1)/(a + b*x)), x) - Simp(b**(-m + S(-1))*(-b**(m + S(1))*x**(m + S(1)) + (-a)**(m + S(1)))*acsch(a + b*x)/(m + S(1)), x)
rule6718 = ReplacementRule(pattern6718, replacement6718)
pattern6719 = Pattern(Integral(x_**WC('m', S(1))*asech(a_ + x_*WC('b', S(1)))**n_, x_), cons2, cons3, cons4, cons62)
def replacement6719(m, b, a, n, x):
rubi.append(6719)
return -Dist(b**(-m + S(-1)), Subst(Int(x**n*(-a + S(1)/cosh(x))**m*tanh(x)/cosh(x), x), x, asech(a + b*x)), x)
rule6719 = ReplacementRule(pattern6719, replacement6719)
pattern6720 = Pattern(Integral(x_**WC('m', S(1))*acsch(a_ + x_*WC('b', S(1)))**n_, x_), cons2, cons3, cons4, cons62)
def replacement6720(m, b, a, n, x):
rubi.append(6720)
return -Dist(b**(-m + S(-1)), Subst(Int(x**n*(-a + S(1)/sinh(x))**m/(sinh(x)*tanh(x)), x), x, acsch(a + b*x)), x)
rule6720 = ReplacementRule(pattern6720, replacement6720)
pattern6721 = Pattern(Integral(WC('u', S(1))*asech(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement6721(u, m, b, c, n, a, x):
rubi.append(6721)
return Int(u*acosh(a/c + b*x**n/c)**m, x)
rule6721 = ReplacementRule(pattern6721, replacement6721)
pattern6722 = Pattern(Integral(WC('u', S(1))*acsch(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement6722(u, m, b, c, n, a, x):
rubi.append(6722)
return Int(u*asinh(a/c + b*x**n/c)**m, x)
rule6722 = ReplacementRule(pattern6722, replacement6722)
pattern6723 = Pattern(Integral(exp(asech(x_*WC('a', S(1)))), x_), cons2, cons2)
def replacement6723(x, a):
rubi.append(6723)
return Dist(S(1)/a, Int(sqrt((-a*x + S(1))/(a*x + S(1)))/(x*(-a*x + S(1))), x), x) + Simp(log(x)/a, x) + Simp(x*exp(asech(a*x)), x)
rule6723 = ReplacementRule(pattern6723, replacement6723)
pattern6724 = Pattern(Integral(exp(asech(x_**p_*WC('a', S(1)))), x_), cons2, cons5, cons1953)
def replacement6724(x, a, p):
rubi.append(6724)
return Dist(p/a, Int(x**(-p), x), x) + Dist(p*sqrt(a*x**p + S(1))*sqrt(S(1)/(a*x**p + S(1)))/a, Int(x**(-p)/(sqrt(-a*x**p + S(1))*sqrt(a*x**p + S(1))), x), x) + Simp(x*exp(asech(a*x**p)), x)
rule6724 = ReplacementRule(pattern6724, replacement6724)
pattern6725 = Pattern(Integral(exp(acsch(x_**WC('p', S(1))*WC('a', S(1)))), x_), cons2, cons5, cons1953)
def replacement6725(x, a, p):
rubi.append(6725)
return Dist(S(1)/a, Int(x**(-p), x), x) + Int(sqrt(S(1) + x**(-S(2)*p)/a**S(2)), x)
rule6725 = ReplacementRule(pattern6725, replacement6725)
pattern6726 = Pattern(Integral(exp(WC('n', S(1))*asech(u_)), x_), cons85)
def replacement6726(x, n, u):
rubi.append(6726)
return Int((sqrt((-u + S(1))/(u + S(1))) + sqrt((-u + S(1))/(u + S(1)))/u + S(1)/u)**n, x)
rule6726 = ReplacementRule(pattern6726, replacement6726)
pattern6727 = Pattern(Integral(exp(WC('n', S(1))*acsch(u_)), x_), cons85)
def replacement6727(x, n, u):
rubi.append(6727)
return Int((sqrt(S(1) + u**(S(-2))) + S(1)/u)**n, x)
rule6727 = ReplacementRule(pattern6727, replacement6727)
pattern6728 = Pattern(Integral(exp(asech(x_**WC('p', S(1))*WC('a', S(1))))/x_, x_), cons2, cons5, cons1953)
def replacement6728(x, a, p):
rubi.append(6728)
return Dist(sqrt(a*x**p + S(1))*sqrt(S(1)/(a*x**p + S(1)))/a, Int(x**(-p + S(-1))*sqrt(-a*x**p + S(1))*sqrt(a*x**p + S(1)), x), x) - Simp(x**(-p)/(a*p), x)
rule6728 = ReplacementRule(pattern6728, replacement6728)
pattern6729 = Pattern(Integral(x_**WC('m', S(1))*exp(asech(x_**WC('p', S(1))*WC('a', S(1)))), x_), cons2, cons21, cons5, cons66)
def replacement6729(x, m, a, p):
rubi.append(6729)
return Dist(p/(a*(m + S(1))), Int(x**(m - p), x), x) + Dist(p*sqrt(a*x**p + S(1))*sqrt(S(1)/(a*x**p + S(1)))/(a*(m + S(1))), Int(x**(m - p)/(sqrt(-a*x**p + S(1))*sqrt(a*x**p + S(1))), x), x) + Simp(x**(m + S(1))*exp(asech(a*x**p))/(m + S(1)), x)
rule6729 = ReplacementRule(pattern6729, replacement6729)
pattern6730 = Pattern(Integral(x_**WC('m', S(1))*exp(acsch(x_**WC('p', S(1))*WC('a', S(1)))), x_), cons2, cons21, cons5, cons1954)
def replacement6730(x, m, a, p):
rubi.append(6730)
return Dist(S(1)/a, Int(x**(m - p), x), x) + Int(x**m*sqrt(S(1) + x**(-S(2)*p)/a**S(2)), x)
rule6730 = ReplacementRule(pattern6730, replacement6730)
pattern6731 = Pattern(Integral(x_**WC('m', S(1))*exp(WC('n', S(1))*asech(u_)), x_), cons21, cons85)
def replacement6731(x, m, n, u):
rubi.append(6731)
return Int(x**m*(sqrt((-u + S(1))/(u + S(1))) + sqrt((-u + S(1))/(u + S(1)))/u + S(1)/u)**n, x)
rule6731 = ReplacementRule(pattern6731, replacement6731)
pattern6732 = Pattern(Integral(x_**WC('m', S(1))*exp(WC('n', S(1))*acsch(u_)), x_), cons21, cons85)
def replacement6732(x, m, n, u):
rubi.append(6732)
return Int(x**m*(sqrt(S(1) + u**(S(-2))) + S(1)/u)**n, x)
rule6732 = ReplacementRule(pattern6732, replacement6732)
pattern6733 = Pattern(Integral(asech(u_), x_), cons1230, cons1769)
def replacement6733(x, u):
rubi.append(6733)
return Dist(sqrt(-u**S(2) + S(1))/(u*sqrt(S(-1) + S(1)/u)*sqrt(S(1) + S(1)/u)), Int(SimplifyIntegrand(x*D(u, x)/(u*sqrt(-u**S(2) + S(1))), x), x), x) + Simp(x*asech(u), x)
rule6733 = ReplacementRule(pattern6733, replacement6733)
pattern6734 = Pattern(Integral(acsch(u_), x_), cons1230, cons1769)
def replacement6734(x, u):
rubi.append(6734)
return -Dist(u/sqrt(-u**S(2)), Int(SimplifyIntegrand(x*D(u, x)/(u*sqrt(-u**S(2) + S(-1))), x), x), x) + Simp(x*acsch(u), x)
rule6734 = ReplacementRule(pattern6734, replacement6734)
pattern6735 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asech(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1769)
def replacement6735(u, m, b, d, c, a, x):
rubi.append(6735)
return Dist(b*sqrt(-u**S(2) + S(1))/(d*u*sqrt(S(-1) + S(1)/u)*sqrt(S(1) + S(1)/u)*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(u*sqrt(-u**S(2) + S(1))), x), x), x) + Simp((a + b*asech(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule6735 = ReplacementRule(pattern6735, replacement6735)
pattern6736 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsch(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1769)
def replacement6736(u, m, b, d, c, a, x):
rubi.append(6736)
return -Dist(b*u/(d*sqrt(-u**S(2))*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(u*sqrt(-u**S(2) + S(-1))), x), x), x) + Simp((a + b*acsch(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule6736 = ReplacementRule(pattern6736, replacement6736)
def With6737(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern6737 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*asech(u_)), x_), cons2, cons3, cons1230, cons1955, CustomConstraint(With6737))
def replacement6737(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(6737)
return Dist(b*sqrt(-u**S(2) + S(1))/(u*sqrt(S(-1) + S(1)/u)*sqrt(S(1) + S(1)/u)), Int(SimplifyIntegrand(w*D(u, x)/(u*sqrt(-u**S(2) + S(1))), x), x), x) + Dist(a + b*asech(u), w, x)
rule6737 = ReplacementRule(pattern6737, replacement6737)
def With6738(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern6738 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*acsch(u_)), x_), cons2, cons3, cons1230, cons1956, CustomConstraint(With6738))
def replacement6738(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(6738)
return -Dist(b*u/sqrt(-u**S(2)), Int(SimplifyIntegrand(w*D(u, x)/(u*sqrt(-u**S(2) + S(-1))), x), x), x) + Dist(a + b*acsch(u), w, x)
rule6738 = ReplacementRule(pattern6738, replacement6738)
return [rule6084, rule6085, rule6086, rule6087, rule6088, rule6089, rule6090, rule6091, rule6092, rule6093, rule6094, rule6095, rule6096, rule6097, rule6098, rule6099, rule6100, rule6101, rule6102, rule6103, rule6104, rule6105, rule6106, rule6107, rule6108, rule6109, rule6110, rule6111, rule6112, rule6113, rule6114, rule6115, rule6116, rule6117, rule6118, rule6119, rule6120, rule6121, rule6122, rule6123, rule6124, rule6125, rule6126, rule6127, rule6128, rule6129, rule6130, rule6131, rule6132, rule6133, rule6134, rule6135, rule6136, rule6137, rule6138, rule6139, rule6140, rule6141, rule6142, rule6143, rule6144, rule6145, rule6146, rule6147, rule6148, rule6149, rule6150, rule6151, rule6152, rule6153, rule6154, rule6155, rule6156, rule6157, rule6158, rule6159, rule6160, rule6161, rule6162, rule6163, rule6164, rule6165, rule6166, rule6167, rule6168, rule6169, rule6170, rule6171, rule6172, rule6173, rule6174, rule6175, rule6176, rule6177, rule6178, rule6179, rule6180, rule6181, rule6182, rule6183, rule6184, rule6185, rule6186, rule6187, rule6188, rule6189, rule6190, rule6191, rule6192, rule6193, rule6194, rule6195, rule6196, rule6197, rule6198, rule6199, rule6200, rule6201, rule6202, rule6203, rule6204, rule6205, rule6206, rule6207, rule6208, rule6209, rule6210, rule6211, rule6212, rule6213, rule6214, rule6215, rule6216, rule6217, rule6218, rule6219, rule6220, rule6221, rule6222, rule6223, rule6224, rule6225, rule6226, rule6227, rule6228, rule6229, rule6230, rule6231, rule6232, rule6233, rule6234, rule6235, rule6236, rule6237, rule6238, rule6239, rule6240, rule6241, rule6242, rule6243, rule6244, rule6245, rule6246, rule6247, rule6248, rule6249, rule6250, rule6251, rule6252, rule6253, rule6254, rule6255, rule6256, rule6257, rule6258, rule6259, rule6260, rule6261, rule6262, rule6263, rule6264, rule6265, rule6266, rule6267, rule6268, rule6269, rule6270, rule6271, rule6272, rule6273, rule6274, rule6275, rule6276, rule6277, rule6278, rule6279, rule6280, rule6281, rule6282, rule6283, rule6284, rule6285, rule6286, rule6287, rule6288, rule6289, rule6290, rule6291, rule6292, rule6293, rule6294, rule6295, rule6296, rule6297, rule6298, rule6299, rule6300, rule6301, rule6302, rule6303, rule6304, rule6305, rule6306, rule6307, rule6308, rule6309, rule6310, rule6311, rule6312, rule6313, rule6314, rule6315, rule6316, rule6317, rule6318, rule6319, rule6320, rule6321, rule6322, rule6323, rule6324, rule6325, rule6326, rule6327, rule6328, rule6329, rule6330, rule6331, rule6332, rule6333, rule6334, rule6335, rule6336, rule6337, rule6338, rule6339, rule6340, rule6341, rule6342, rule6343, rule6344, rule6345, rule6346, rule6347, rule6348, rule6349, rule6350, rule6351, rule6352, rule6353, rule6354, rule6355, rule6356, rule6357, rule6358, rule6359, rule6360, rule6361, rule6362, rule6363, rule6364, rule6365, rule6366, rule6367, rule6368, rule6369, rule6370, rule6371, rule6372, rule6373, rule6374, rule6375, rule6376, rule6377, rule6378, rule6379, rule6380, rule6381, rule6382, rule6383, rule6384, rule6385, rule6386, rule6387, rule6388, rule6389, rule6390, rule6391, rule6392, rule6393, rule6394, rule6395, rule6396, rule6397, rule6398, rule6399, rule6400, rule6401, rule6402, rule6403, rule6404, rule6405, rule6406, rule6407, rule6408, rule6409, rule6410, rule6411, rule6412, rule6413, rule6414, rule6415, rule6416, rule6417, rule6418, rule6419, rule6420, rule6421, rule6422, rule6423, rule6424, rule6425, rule6426, rule6427, rule6428, rule6429, rule6430, rule6431, rule6432, rule6433, rule6434, rule6435, rule6436, rule6437, rule6438, rule6439, rule6440, rule6441, rule6442, rule6443, rule6444, rule6445, rule6446, rule6447, rule6448, rule6449, rule6450, rule6451, rule6452, rule6453, rule6454, rule6455, rule6456, rule6457, rule6458, rule6459, rule6460, rule6461, rule6462, rule6463, rule6464, rule6465, rule6466, rule6467, rule6468, rule6469, rule6470, rule6471, rule6472, rule6473, rule6474, rule6475, rule6476, rule6477, rule6478, rule6479, rule6480, rule6481, rule6482, rule6483, rule6484, rule6485, rule6486, rule6487, rule6488, rule6489, rule6490, rule6491, rule6492, rule6493, rule6494, rule6495, rule6496, rule6497, rule6498, rule6499, rule6500, rule6501, rule6502, rule6503, rule6504, rule6505, rule6506, rule6507, rule6508, rule6509, rule6510, rule6511, rule6512, rule6513, rule6514, rule6515, rule6516, rule6517, rule6518, rule6519, rule6520, rule6521, rule6522, rule6523, rule6524, rule6525, rule6526, rule6527, rule6528, rule6529, rule6530, rule6531, rule6532, rule6533, rule6534, rule6535, rule6536, rule6537, rule6538, rule6539, rule6540, rule6541, rule6542, rule6543, rule6544, rule6545, rule6546, rule6547, rule6548, rule6549, rule6550, rule6551, rule6552, rule6553, rule6554, rule6555, rule6556, rule6557, rule6558, rule6559, rule6560, rule6561, rule6562, rule6563, rule6564, rule6565, rule6566, rule6567, rule6568, rule6569, rule6570, rule6571, rule6572, rule6573, rule6574, rule6575, rule6576, rule6577, rule6578, rule6579, rule6580, rule6581, rule6582, rule6583, rule6584, rule6585, rule6586, rule6587, rule6588, rule6589, rule6590, rule6591, rule6592, rule6593, rule6594, rule6595, rule6596, rule6597, rule6598, rule6599, rule6600, rule6601, rule6602, rule6603, rule6604, rule6605, rule6606, rule6607, rule6608, rule6609, rule6610, rule6611, rule6612, rule6613, rule6614, rule6615, rule6616, rule6617, rule6618, rule6619, rule6620, rule6621, rule6622, rule6623, rule6624, rule6625, rule6626, rule6627, rule6628, rule6629, rule6630, rule6631, rule6632, rule6633, rule6634, rule6635, rule6636, rule6637, rule6638, rule6639, rule6640, rule6641, rule6642, rule6643, rule6644, rule6645, rule6646, rule6647, rule6648, rule6649, rule6650, rule6651, rule6652, rule6653, rule6654, rule6655, rule6656, rule6657, rule6658, rule6659, rule6660, rule6661, rule6662, rule6663, rule6664, rule6665, rule6666, rule6667, rule6668, rule6669, rule6670, rule6671, rule6672, rule6673, rule6674, rule6675, rule6676, rule6677, rule6678, rule6679, rule6680, rule6681, rule6682, rule6683, rule6684, rule6685, rule6686, rule6687, rule6688, rule6689, rule6690, rule6691, rule6692, rule6693, rule6694, rule6695, rule6696, rule6697, rule6698, rule6699, rule6700, rule6701, rule6702, rule6703, rule6704, rule6705, rule6706, rule6707, rule6708, rule6709, rule6710, rule6711, rule6712, rule6713, rule6714, rule6715, rule6716, rule6717, rule6718, rule6719, rule6720, rule6721, rule6722, rule6723, rule6724, rule6725, rule6726, rule6727, rule6728, rule6729, rule6730, rule6731, rule6732, rule6733, rule6734, rule6735, rule6736, rule6737, rule6738, ]
|
46b9245a7cad572c8daa2d28e3566ac2251b1cda10b5b26e782975016f323af1 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def inverse_trig(rubi):
from sympy.integrals.rubi.constraints import cons87, cons88, cons2, cons3, cons7, cons89, cons1579, cons4, cons148, cons66, cons27, cons21, cons62, cons1734, cons1735, cons1736, cons1737, cons268, cons48, cons584, cons1738, cons128, cons338, cons163, cons137, cons230, cons5, cons1739, cons1740, cons1741, cons1742, cons1743, cons38, cons1744, cons1570, cons336, cons1745, cons147, cons125, cons208, cons54, cons242, cons1746, cons347, cons1747, cons486, cons961, cons93, cons94, cons162, cons272, cons1748, cons17, cons166, cons274, cons1749, cons18, cons1750, cons238, cons237, cons1751, cons246, cons1752, cons1753, cons1754, cons1755, cons1756, cons209, cons925, cons464, cons84, cons1757, cons1758, cons719, cons168, cons1759, cons667, cons1760, cons267, cons717, cons1761, cons1608, cons14, cons150, cons1198, cons1273, cons1360, cons1762, cons1763, cons34, cons35, cons36, cons1764, cons1765, cons165, cons1442, cons1766, cons1767, cons1768, cons1230, cons1769, cons1770, cons1771, cons1772, cons340, cons1773, cons1774, cons1775, cons1776, cons1043, cons85, cons31, cons1777, cons1497, cons1778, cons13, cons1779, cons1780, cons1781, cons1782, cons240, cons241, cons146, cons1783, cons1510, cons1784, cons1152, cons319, cons1785, cons1786, cons1787, cons1788, cons1789, cons1790, cons1791, cons1792, cons1793, cons1794, cons1795, cons1796, cons601, cons1797, cons261, cons1798, cons1799, cons1800, cons1801, cons1802, cons1803, cons1804, cons1805, cons177, cons117, cons1806, cons1807, cons1808, cons1809, cons1810, cons1811, cons1812, cons1813, cons1814, cons1815, cons1816, cons1817, cons1580, cons1818, cons1819, cons1820, cons1821, cons1822, cons1823, cons1824, cons1825, cons1826, cons1827, cons1828, cons1829, cons1830, cons1094, cons1831, cons1832, cons1833, cons1834, cons1835, cons383, cons1836, cons1837, cons818, cons463, cons1838, cons1839, cons1840, cons1841, cons1842, cons1843, cons1844, cons67, cons1845, cons1846, cons1847, cons1848, cons1849, cons1850, cons1851, cons552, cons1146, cons1852, cons1853, cons1242, cons1243, cons1854, cons178, cons1855, cons1856, cons1299, cons1857, cons1858
pattern5031 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons87, cons88)
def replacement5031(b, a, n, c, x):
rubi.append(5031)
return -Dist(b*c*n, Int(x*(a + b*asin(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*asin(c*x))**n, x)
rule5031 = ReplacementRule(pattern5031, replacement5031)
pattern5032 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons87, cons88)
def replacement5032(b, a, n, c, x):
rubi.append(5032)
return Dist(b*c*n, Int(x*(a + b*acos(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*acos(c*x))**n, x)
rule5032 = ReplacementRule(pattern5032, replacement5032)
pattern5033 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons87, cons89)
def replacement5033(b, a, n, c, x):
rubi.append(5033)
return Dist(c/(b*(n + S(1))), Int(x*(a + b*asin(c*x))**(n + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*asin(c*x))**(n + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5033 = ReplacementRule(pattern5033, replacement5033)
pattern5034 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons87, cons89)
def replacement5034(b, a, n, c, x):
rubi.append(5034)
return -Dist(c/(b*(n + S(1))), Int(x*(a + b*acos(c*x))**(n + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) - Simp((a + b*acos(c*x))**(n + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5034 = ReplacementRule(pattern5034, replacement5034)
pattern5035 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5035(b, a, n, c, x):
rubi.append(5035)
return Dist(S(1)/(b*c), Subst(Int(x**n*cos(a/b - x/b), x), x, a + b*asin(c*x)), x)
rule5035 = ReplacementRule(pattern5035, replacement5035)
pattern5036 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5036(b, a, n, c, x):
rubi.append(5036)
return Dist(S(1)/(b*c), Subst(Int(x**n*sin(a/b - x/b), x), x, a + b*acos(c*x)), x)
rule5036 = ReplacementRule(pattern5036, replacement5036)
pattern5037 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/x_, x_), cons2, cons3, cons7, cons148)
def replacement5037(b, a, n, c, x):
rubi.append(5037)
return Subst(Int((a + b*x)**n/tan(x), x), x, asin(c*x))
rule5037 = ReplacementRule(pattern5037, replacement5037)
pattern5038 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/x_, x_), cons2, cons3, cons7, cons148)
def replacement5038(b, a, n, c, x):
rubi.append(5038)
return -Subst(Int((a + b*x)**n*tan(x), x), x, acos(c*x))
rule5038 = ReplacementRule(pattern5038, replacement5038)
pattern5039 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons148, cons66)
def replacement5039(m, b, d, a, n, c, x):
rubi.append(5039)
return -Dist(b*c*n/(d*(m + S(1))), Int((d*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((d*x)**(m + S(1))*(a + b*asin(c*x))**n/(d*(m + S(1))), x)
rule5039 = ReplacementRule(pattern5039, replacement5039)
pattern5040 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons148, cons66)
def replacement5040(m, b, d, a, n, c, x):
rubi.append(5040)
return Dist(b*c*n/(d*(m + S(1))), Int((d*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((d*x)**(m + S(1))*(a + b*acos(c*x))**n/(d*(m + S(1))), x)
rule5040 = ReplacementRule(pattern5040, replacement5040)
pattern5041 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons88)
def replacement5041(m, b, a, c, n, x):
rubi.append(5041)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*asin(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**(m + S(1))*(a + b*asin(c*x))**n/(m + S(1)), x)
rule5041 = ReplacementRule(pattern5041, replacement5041)
pattern5042 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons88)
def replacement5042(m, b, a, c, n, x):
rubi.append(5042)
return Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*acos(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**(m + S(1))*(a + b*acos(c*x))**n/(m + S(1)), x)
rule5042 = ReplacementRule(pattern5042, replacement5042)
pattern5043 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons1734)
def replacement5043(m, b, a, c, n, x):
rubi.append(5043)
return -Dist(c**(-m + S(-1))/(b*(n + S(1))), Subst(Int(ExpandTrigReduce((a + b*x)**(n + S(1)), (m - (m + S(1))*sin(x)**S(2))*sin(x)**(m + S(-1)), x), x), x, asin(c*x)), x) + Simp(x**m*(a + b*asin(c*x))**(n + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5043 = ReplacementRule(pattern5043, replacement5043)
pattern5044 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons1734)
def replacement5044(m, b, a, c, n, x):
rubi.append(5044)
return -Dist(c**(-m + S(-1))/(b*(n + S(1))), Subst(Int(ExpandTrigReduce((a + b*x)**(n + S(1)), (m - (m + S(1))*cos(x)**S(2))*cos(x)**(m + S(-1)), x), x), x, acos(c*x)), x) - Simp(x**m*(a + b*acos(c*x))**(n + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5044 = ReplacementRule(pattern5044, replacement5044)
pattern5045 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons1735)
def replacement5045(m, b, a, c, n, x):
rubi.append(5045)
return -Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*asin(c*x))**(n + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Dist(c*(m + S(1))/(b*(n + S(1))), Int(x**(m + S(1))*(a + b*asin(c*x))**(n + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**m*(a + b*asin(c*x))**(n + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5045 = ReplacementRule(pattern5045, replacement5045)
pattern5046 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons62, cons87, cons1735)
def replacement5046(m, b, a, c, n, x):
rubi.append(5046)
return Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*acos(c*x))**(n + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) - Dist(c*(m + S(1))/(b*(n + S(1))), Int(x**(m + S(1))*(a + b*acos(c*x))**(n + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) - Simp(x**m*(a + b*acos(c*x))**(n + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5046 = ReplacementRule(pattern5046, replacement5046)
pattern5047 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons62)
def replacement5047(m, b, a, c, n, x):
rubi.append(5047)
return Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*sin(x)**m*cos(x), x), x, asin(c*x)), x)
rule5047 = ReplacementRule(pattern5047, replacement5047)
pattern5048 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons62)
def replacement5048(m, b, a, c, n, x):
rubi.append(5048)
return -Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*sin(x)*cos(x)**m, x), x, acos(c*x)), x)
rule5048 = ReplacementRule(pattern5048, replacement5048)
pattern5049 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1736)
def replacement5049(m, b, d, a, n, c, x):
rubi.append(5049)
return Int((d*x)**m*(a + b*asin(c*x))**n, x)
rule5049 = ReplacementRule(pattern5049, replacement5049)
pattern5050 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1736)
def replacement5050(m, b, d, a, n, c, x):
rubi.append(5050)
return Int((d*x)**m*(a + b*acos(c*x))**n, x)
rule5050 = ReplacementRule(pattern5050, replacement5050)
pattern5051 = Pattern(Integral(S(1)/(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons268)
def replacement5051(b, d, a, c, x, e):
rubi.append(5051)
return Simp(log(a + b*asin(c*x))/(b*c*sqrt(d)), x)
rule5051 = ReplacementRule(pattern5051, replacement5051)
pattern5052 = Pattern(Integral(S(1)/(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons268)
def replacement5052(b, d, a, c, x, e):
rubi.append(5052)
return -Simp(log(a + b*acos(c*x))/(b*c*sqrt(d)), x)
rule5052 = ReplacementRule(pattern5052, replacement5052)
pattern5053 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons268, cons584)
def replacement5053(b, d, a, n, c, x, e):
rubi.append(5053)
return Simp((a + b*asin(c*x))**(n + S(1))/(b*c*sqrt(d)*(n + S(1))), x)
rule5053 = ReplacementRule(pattern5053, replacement5053)
pattern5054 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons268, cons584)
def replacement5054(b, d, a, n, c, x, e):
rubi.append(5054)
return -Simp((a + b*acos(c*x))**(n + S(1))/(b*c*sqrt(d)*(n + S(1))), x)
rule5054 = ReplacementRule(pattern5054, replacement5054)
pattern5055 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1738)
def replacement5055(b, d, a, n, c, x, e):
rubi.append(5055)
return Dist(sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*asin(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x)
rule5055 = ReplacementRule(pattern5055, replacement5055)
pattern5056 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1738)
def replacement5056(b, d, a, n, c, x, e):
rubi.append(5056)
return Dist(sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*acos(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x)
rule5056 = ReplacementRule(pattern5056, replacement5056)
def With5057(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(5057)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asin(c*x), u, x)
pattern5057 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons128)
rule5057 = ReplacementRule(pattern5057, With5057)
def With5058(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(5058)
return Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acos(c*x), u, x)
pattern5058 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons128)
rule5058 = ReplacementRule(pattern5058, With5058)
pattern5059 = Pattern(Integral(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement5059(b, d, a, n, c, x, e):
rubi.append(5059)
return Dist(sqrt(d + e*x**S(2))/(S(2)*sqrt(-c**S(2)*x**S(2) + S(1))), Int((a + b*asin(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) - Dist(b*c*n*sqrt(d + e*x**S(2))/(S(2)*sqrt(-c**S(2)*x**S(2) + S(1))), Int(x*(a + b*asin(c*x))**(n + S(-1)), x), x) + Simp(x*(a + b*asin(c*x))**n*sqrt(d + e*x**S(2))/S(2), x)
rule5059 = ReplacementRule(pattern5059, replacement5059)
pattern5060 = Pattern(Integral(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement5060(b, d, a, n, c, x, e):
rubi.append(5060)
return Dist(sqrt(d + e*x**S(2))/(S(2)*sqrt(-c**S(2)*x**S(2) + S(1))), Int((a + b*acos(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Dist(b*c*n*sqrt(d + e*x**S(2))/(S(2)*sqrt(-c**S(2)*x**S(2) + S(1))), Int(x*(a + b*acos(c*x))**(n + S(-1)), x), x) + Simp(x*(a + b*acos(c*x))**n*sqrt(d + e*x**S(2))/S(2), x)
rule5060 = ReplacementRule(pattern5060, replacement5060)
pattern5061 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons88, cons163)
def replacement5061(p, b, d, a, n, c, x, e):
rubi.append(5061)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*p + S(1)), Int(x*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp(x*(a + b*asin(c*x))**n*(d + e*x**S(2))**p/(S(2)*p + S(1)), x)
rule5061 = ReplacementRule(pattern5061, replacement5061)
pattern5062 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons88, cons163)
def replacement5062(p, b, d, a, n, c, x, e):
rubi.append(5062)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) + Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*p + S(1)), Int(x*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp(x*(a + b*acos(c*x))**n*(d + e*x**S(2))**p/(S(2)*p + S(1)), x)
rule5062 = ReplacementRule(pattern5062, replacement5062)
pattern5063 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons268)
def replacement5063(b, d, a, n, c, x, e):
rubi.append(5063)
return -Dist(b*c*n/sqrt(d), Int(x*(a + b*asin(c*x))**(n + S(-1))/(d + e*x**S(2)), x), x) + Simp(x*(a + b*asin(c*x))**n/(d*sqrt(d + e*x**S(2))), x)
rule5063 = ReplacementRule(pattern5063, replacement5063)
pattern5064 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88, cons268)
def replacement5064(b, d, a, n, c, x, e):
rubi.append(5064)
return Dist(b*c*n/sqrt(d), Int(x*(a + b*acos(c*x))**(n + S(-1))/(d + e*x**S(2)), x), x) + Simp(x*(a + b*acos(c*x))**n/(d*sqrt(d + e*x**S(2))), x)
rule5064 = ReplacementRule(pattern5064, replacement5064)
pattern5065 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement5065(b, d, a, n, c, x, e):
rubi.append(5065)
return -Dist(b*c*n*sqrt(-c**S(2)*x**S(2) + S(1))/(d*sqrt(d + e*x**S(2))), Int(x*(a + b*asin(c*x))**(n + S(-1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*asin(c*x))**n/(d*sqrt(d + e*x**S(2))), x)
rule5065 = ReplacementRule(pattern5065, replacement5065)
pattern5066 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons87, cons88)
def replacement5066(b, d, a, n, c, x, e):
rubi.append(5066)
return Dist(b*c*n*sqrt(-c**S(2)*x**S(2) + S(1))/(d*sqrt(d + e*x**S(2))), Int(x*(a + b*acos(c*x))**(n + S(-1))/(-c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*acos(c*x))**n/(d*sqrt(d + e*x**S(2))), x)
rule5066 = ReplacementRule(pattern5066, replacement5066)
pattern5067 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons88, cons137, cons230)
def replacement5067(p, b, d, a, n, c, x, e):
rubi.append(5067)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*(p + S(1))), Int(x*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) - Simp(x*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x)
rule5067 = ReplacementRule(pattern5067, replacement5067)
pattern5068 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons338, cons88, cons137, cons230)
def replacement5068(p, b, d, a, n, c, x, e):
rubi.append(5068)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*(p + S(1))), Int(x*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) - Simp(x*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x)
rule5068 = ReplacementRule(pattern5068, replacement5068)
pattern5069 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement5069(b, d, a, n, c, x, e):
rubi.append(5069)
return Dist(S(1)/(c*d), Subst(Int((a + b*x)**n/cos(x), x), x, asin(c*x)), x)
rule5069 = ReplacementRule(pattern5069, replacement5069)
pattern5070 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement5070(b, d, a, n, c, x, e):
rubi.append(5070)
return -Dist(S(1)/(c*d), Subst(Int((a + b*x)**n/sin(x), x), x, acos(c*x)), x)
rule5070 = ReplacementRule(pattern5070, replacement5070)
pattern5071 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons89)
def replacement5071(p, b, d, a, c, n, x, e):
rubi.append(5071)
return Dist(c*d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(S(2)*p + S(1))*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*(n + S(1))), Int(x*(a + b*asin(c*x))**(n + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((a + b*asin(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5071 = ReplacementRule(pattern5071, replacement5071)
pattern5072 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons89)
def replacement5072(p, b, d, a, c, n, x, e):
rubi.append(5072)
return -Dist(c*d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(S(2)*p + S(1))*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*(n + S(1))), Int(x*(a + b*acos(c*x))**(n + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) - Simp((a + b*acos(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5072 = ReplacementRule(pattern5072, replacement5072)
pattern5073 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1739, cons1740)
def replacement5073(p, b, d, a, n, c, x, e):
rubi.append(5073)
return Dist(d**p/c, Subst(Int((a + b*x)**n*cos(x)**(S(2)*p + S(1)), x), x, asin(c*x)), x)
rule5073 = ReplacementRule(pattern5073, replacement5073)
pattern5074 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1739, cons1740)
def replacement5074(p, b, d, a, n, c, x, e):
rubi.append(5074)
return -Dist(d**p/c, Subst(Int((a + b*x)**n*sin(x)**(S(2)*p + S(1)), x), x, acos(c*x)), x)
rule5074 = ReplacementRule(pattern5074, replacement5074)
pattern5075 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1739, cons1741)
def replacement5075(p, b, d, a, n, c, x, e):
rubi.append(5075)
return Dist(d**(p + S(-1)/2)*sqrt(d + e*x**S(2))/sqrt(-c**S(2)*x**S(2) + S(1)), Int((a + b*asin(c*x))**n*(-c**S(2)*x**S(2) + S(1))**p, x), x)
rule5075 = ReplacementRule(pattern5075, replacement5075)
pattern5076 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons1739, cons1741)
def replacement5076(p, b, d, a, n, c, x, e):
rubi.append(5076)
return Dist(d**(p + S(-1)/2)*sqrt(d + e*x**S(2))/sqrt(-c**S(2)*x**S(2) + S(1)), Int((a + b*acos(c*x))**n*(-c**S(2)*x**S(2) + S(1))**p, x), x)
rule5076 = ReplacementRule(pattern5076, replacement5076)
def With5077(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(5077)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asin(c*x), u, x)
pattern5077 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1742, cons1743)
rule5077 = ReplacementRule(pattern5077, With5077)
def With5078(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(5078)
return Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acos(c*x), u, x)
pattern5078 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1742, cons1743)
rule5078 = ReplacementRule(pattern5078, With5078)
pattern5079 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1742, cons38, cons1744)
def replacement5079(p, b, d, a, n, c, x, e):
rubi.append(5079)
return Int(ExpandIntegrand((a + b*asin(c*x))**n, (d + e*x**S(2))**p, x), x)
rule5079 = ReplacementRule(pattern5079, replacement5079)
pattern5080 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1742, cons38, cons1744)
def replacement5080(p, b, d, a, n, c, x, e):
rubi.append(5080)
return Int(ExpandIntegrand((a + b*acos(c*x))**n, (d + e*x**S(2))**p, x), x)
rule5080 = ReplacementRule(pattern5080, replacement5080)
pattern5081 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement5081(p, b, d, a, n, c, x, e):
rubi.append(5081)
return Int((a + b*asin(c*x))**n*(d + e*x**S(2))**p, x)
rule5081 = ReplacementRule(pattern5081, replacement5081)
pattern5082 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement5082(p, b, d, a, n, c, x, e):
rubi.append(5082)
return Int((a + b*acos(c*x))**n*(d + e*x**S(2))**p, x)
rule5082 = ReplacementRule(pattern5082, replacement5082)
pattern5083 = Pattern(Integral((d_ + x_*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons336, cons1745, cons147)
def replacement5083(p, g, b, f, d, a, n, c, x, e):
rubi.append(5083)
return Dist((d + e*x)**FracPart(p)*(f + g*x)**FracPart(p)*(d*f + e*g*x**S(2))**(-FracPart(p)), Int((a + b*asin(c*x))**n*(d*f + e*g*x**S(2))**p, x), x)
rule5083 = ReplacementRule(pattern5083, replacement5083)
pattern5084 = Pattern(Integral((d_ + x_*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons336, cons1745, cons147)
def replacement5084(p, g, b, f, d, a, n, c, x, e):
rubi.append(5084)
return Dist((d + e*x)**FracPart(p)*(f + g*x)**FracPart(p)*(d*f + e*g*x**S(2))**(-FracPart(p)), Int((a + b*acos(c*x))**n*(d*f + e*g*x**S(2))**p, x), x)
rule5084 = ReplacementRule(pattern5084, replacement5084)
pattern5085 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement5085(b, d, a, n, c, x, e):
rubi.append(5085)
return -Dist(S(1)/e, Subst(Int((a + b*x)**n*tan(x), x), x, asin(c*x)), x)
rule5085 = ReplacementRule(pattern5085, replacement5085)
pattern5086 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement5086(b, d, a, n, c, x, e):
rubi.append(5086)
return Dist(S(1)/e, Subst(Int((a + b*x)**n/tan(x), x), x, acos(c*x)), x)
rule5086 = ReplacementRule(pattern5086, replacement5086)
pattern5087 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons54)
def replacement5087(p, b, d, a, n, c, x, e):
rubi.append(5087)
return Dist(b*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5087 = ReplacementRule(pattern5087, replacement5087)
pattern5088 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1737, cons87, cons88, cons54)
def replacement5088(p, b, d, a, n, c, x, e):
rubi.append(5088)
return -Dist(b*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5088 = ReplacementRule(pattern5088, replacement5088)
pattern5089 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement5089(b, d, a, n, c, x, e):
rubi.append(5089)
return Dist(S(1)/d, Subst(Int((a + b*x)**n/(sin(x)*cos(x)), x), x, asin(c*x)), x)
rule5089 = ReplacementRule(pattern5089, replacement5089)
pattern5090 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons148)
def replacement5090(b, d, a, n, c, x, e):
rubi.append(5090)
return -Dist(S(1)/d, Subst(Int((a + b*x)**n/(sin(x)*cos(x)), x), x, acos(c*x)), x)
rule5090 = ReplacementRule(pattern5090, replacement5090)
pattern5091 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1737, cons87, cons88, cons242, cons66)
def replacement5091(p, m, f, b, d, a, n, c, x, e):
rubi.append(5091)
return -Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*f*(m + S(1))), x)
rule5091 = ReplacementRule(pattern5091, replacement5091)
pattern5092 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1737, cons87, cons88, cons242, cons66)
def replacement5092(p, m, f, b, d, a, n, c, x, e):
rubi.append(5092)
return Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*f*(m + S(1))), x)
rule5092 = ReplacementRule(pattern5092, replacement5092)
pattern5093 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))/x_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons128)
def replacement5093(p, b, d, a, c, x, e):
rubi.append(5093)
return Dist(d, Int((a + b*asin(c*x))*(d + e*x**S(2))**(p + S(-1))/x, x), x) - Dist(b*c*d**p/(S(2)*p), Int((-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((a + b*asin(c*x))*(d + e*x**S(2))**p/(S(2)*p), x)
rule5093 = ReplacementRule(pattern5093, replacement5093)
pattern5094 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))/x_, x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons128)
def replacement5094(p, b, d, a, c, x, e):
rubi.append(5094)
return Dist(d, Int((a + b*acos(c*x))*(d + e*x**S(2))**(p + S(-1))/x, x), x) + Dist(b*c*d**p/(S(2)*p), Int((-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((a + b*acos(c*x))*(d + e*x**S(2))**p/(S(2)*p), x)
rule5094 = ReplacementRule(pattern5094, replacement5094)
pattern5095 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons128, cons1746)
def replacement5095(p, m, f, b, d, a, c, x, e):
rubi.append(5095)
return -Dist(S(2)*e*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*asin(c*x))*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*d**p/(f*(m + S(1))), Int((f*x)**(m + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asin(c*x))*(d + e*x**S(2))**p/(f*(m + S(1))), x)
rule5095 = ReplacementRule(pattern5095, replacement5095)
pattern5096 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons128, cons1746)
def replacement5096(p, m, f, b, d, a, c, x, e):
rubi.append(5096)
return -Dist(S(2)*e*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acos(c*x))*(d + e*x**S(2))**(p + S(-1)), x), x) + Dist(b*c*d**p/(f*(m + S(1))), Int((f*x)**(m + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acos(c*x))*(d + e*x**S(2))**p/(f*(m + S(1))), x)
rule5096 = ReplacementRule(pattern5096, replacement5096)
def With5097(p, m, f, b, d, a, c, x, e):
u = IntHide((f*x)**m*(d + e*x**S(2))**p, x)
rubi.append(5097)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asin(c*x), u, x)
pattern5097 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons128)
rule5097 = ReplacementRule(pattern5097, With5097)
def With5098(p, m, f, b, d, a, c, x, e):
u = IntHide((f*x)**m*(d + e*x**S(2))**p, x)
rubi.append(5098)
return Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acos(c*x), u, x)
pattern5098 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons128)
rule5098 = ReplacementRule(pattern5098, With5098)
def With5099(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(-c**S(2)*x**S(2) + S(1))**p, x)
rubi.append(5099)
return Dist(d**p*(a + b*asin(c*x)), u, x) - Dist(b*c*d**p, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x)
pattern5099 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons347, cons1747, cons486, cons268)
rule5099 = ReplacementRule(pattern5099, With5099)
def With5100(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(-c**S(2)*x**S(2) + S(1))**p, x)
rubi.append(5100)
return Dist(d**p*(a + b*acos(c*x)), u, x) + Dist(b*c*d**p, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x)
pattern5100 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons347, cons1747, cons486, cons268)
rule5100 = ReplacementRule(pattern5100, With5100)
def With5101(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(-c**S(2)*x**S(2) + S(1))**p, x)
rubi.append(5101)
return -Dist(b*c*d**(p + S(-1)/2)*sqrt(d + e*x**S(2))/sqrt(-c**S(2)*x**S(2) + S(1)), Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asin(c*x), Int(x**m*(d + e*x**S(2))**p, x), x)
pattern5101 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons961, cons1747)
rule5101 = ReplacementRule(pattern5101, With5101)
def With5102(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(-c**S(2)*x**S(2) + S(1))**p, x)
rubi.append(5102)
return Dist(b*c*d**(p + S(-1)/2)*sqrt(d + e*x**S(2))/sqrt(-c**S(2)*x**S(2) + S(1)), Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acos(c*x), Int(x**m*(d + e*x**S(2))**p, x), x)
pattern5102 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons961, cons1747)
rule5102 = ReplacementRule(pattern5102, With5102)
pattern5103 = Pattern(Integral((x_*WC('f', S(1)))**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons93, cons88, cons94)
def replacement5103(m, f, b, d, a, n, c, x, e):
rubi.append(5103)
return Dist(c**S(2)*sqrt(d + e*x**S(2))/(f**S(2)*(m + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(2))*(a + b*asin(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) - Dist(b*c*n*sqrt(d + e*x**S(2))/(f*(m + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(-1)), x), x) + Simp((f*x)**(m + S(1))*(a + b*asin(c*x))**n*sqrt(d + e*x**S(2))/(f*(m + S(1))), x)
rule5103 = ReplacementRule(pattern5103, replacement5103)
pattern5104 = Pattern(Integral((x_*WC('f', S(1)))**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons93, cons88, cons94)
def replacement5104(m, f, b, d, a, n, c, x, e):
rubi.append(5104)
return Dist(c**S(2)*sqrt(d + e*x**S(2))/(f**S(2)*(m + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(2))*(a + b*acos(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Dist(b*c*n*sqrt(d + e*x**S(2))/(f*(m + S(1))*sqrt(-c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(-1)), x), x) + Simp((f*x)**(m + S(1))*(a + b*acos(c*x))**n*sqrt(d + e*x**S(2))/(f*(m + S(1))), x)
rule5104 = ReplacementRule(pattern5104, replacement5104)
pattern5105 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons162, cons88, cons163, cons94)
def replacement5105(p, m, f, b, d, a, n, c, x, e):
rubi.append(5105)
return -Dist(S(2)*e*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asin(c*x))**n*(d + e*x**S(2))**p/(f*(m + S(1))), x)
rule5105 = ReplacementRule(pattern5105, replacement5105)
pattern5106 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons162, cons88, cons163, cons94)
def replacement5106(p, m, f, b, d, a, n, c, x, e):
rubi.append(5106)
return -Dist(S(2)*e*p/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) + Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acos(c*x))**n*(d + e*x**S(2))**p/(f*(m + S(1))), x)
rule5106 = ReplacementRule(pattern5106, replacement5106)
pattern5107 = Pattern(Integral((x_*WC('f', S(1)))**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons87, cons88, cons272, cons1748)
def replacement5107(m, f, b, d, a, n, c, x, e):
rubi.append(5107)
return Dist(sqrt(d + e*x**S(2))/((m + S(2))*sqrt(-c**S(2)*x**S(2) + S(1))), Int((f*x)**m*(a + b*asin(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) - Dist(b*c*n*sqrt(d + e*x**S(2))/(f*(m + S(2))*sqrt(-c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(-1)), x), x) + Simp((f*x)**(m + S(1))*(a + b*asin(c*x))**n*sqrt(d + e*x**S(2))/(f*(m + S(2))), x)
rule5107 = ReplacementRule(pattern5107, replacement5107)
pattern5108 = Pattern(Integral((x_*WC('f', S(1)))**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons87, cons88, cons272, cons1748)
def replacement5108(m, f, b, d, a, n, c, x, e):
rubi.append(5108)
return Dist(sqrt(d + e*x**S(2))/((m + S(2))*sqrt(-c**S(2)*x**S(2) + S(1))), Int((f*x)**m*(a + b*acos(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Dist(b*c*n*sqrt(d + e*x**S(2))/(f*(m + S(2))*sqrt(-c**S(2)*x**S(2) + S(1))), Int((f*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(-1)), x), x) + Simp((f*x)**(m + S(1))*(a + b*acos(c*x))**n*sqrt(d + e*x**S(2))/(f*(m + S(2))), x)
rule5108 = ReplacementRule(pattern5108, replacement5108)
pattern5109 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons338, cons88, cons163, cons272, cons1748)
def replacement5109(p, m, f, b, d, a, n, c, x, e):
rubi.append(5109)
return Dist(S(2)*d*p/(m + S(2)*p + S(1)), Int((f*x)**m*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(2)*p + S(1))), Int((f*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asin(c*x))**n*(d + e*x**S(2))**p/(f*(m + S(2)*p + S(1))), x)
rule5109 = ReplacementRule(pattern5109, replacement5109)
pattern5110 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons338, cons88, cons163, cons272, cons1748)
def replacement5110(p, m, f, b, d, a, n, c, x, e):
rubi.append(5110)
return Dist(S(2)*d*p/(m + S(2)*p + S(1)), Int((f*x)**m*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) + Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(2)*p + S(1))), Int((f*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acos(c*x))**n*(d + e*x**S(2))**p/(f*(m + S(2)*p + S(1))), x)
rule5110 = ReplacementRule(pattern5110, replacement5110)
pattern5111 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons1737, cons93, cons88, cons94, cons17)
def replacement5111(p, m, f, b, d, a, n, c, x, e):
rubi.append(5111)
return Dist(c**S(2)*(m + S(2)*p + S(3))/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*asin(c*x))**n*(d + e*x**S(2))**p, x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*f*(m + S(1))), x)
rule5111 = ReplacementRule(pattern5111, replacement5111)
pattern5112 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons1737, cons93, cons88, cons94, cons17)
def replacement5112(p, m, f, b, d, a, n, c, x, e):
rubi.append(5112)
return Dist(c**S(2)*(m + S(2)*p + S(3))/(f**S(2)*(m + S(1))), Int((f*x)**(m + S(2))*(a + b*acos(c*x))**n*(d + e*x**S(2))**p, x), x) + Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(f*(m + S(1))), Int((f*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp((f*x)**(m + S(1))*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*f*(m + S(1))), x)
rule5112 = ReplacementRule(pattern5112, replacement5112)
pattern5113 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons162, cons88, cons137, cons166)
def replacement5113(p, m, f, b, d, a, n, c, x, e):
rubi.append(5113)
return -Dist(f**S(2)*(m + S(-1))/(S(2)*e*(p + S(1))), Int((f*x)**(m + S(-2))*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Dist(b*d**IntPart(p)*f*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((f*x)**(m + S(-1))*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5113 = ReplacementRule(pattern5113, replacement5113)
pattern5114 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons162, cons88, cons137, cons166)
def replacement5114(p, m, f, b, d, a, n, c, x, e):
rubi.append(5114)
return -Dist(f**S(2)*(m + S(-1))/(S(2)*e*(p + S(1))), Int((f*x)**(m + S(-2))*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b*d**IntPart(p)*f*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*c*(p + S(1))), Int((f*x)**(m + S(-1))*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5114 = ReplacementRule(pattern5114, replacement5114)
pattern5115 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons338, cons88, cons137, cons274, cons1749)
def replacement5115(p, m, f, b, d, a, n, c, x, e):
rubi.append(5115)
return Dist((m + S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((f*x)**m*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) + Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*f*(p + S(1))), Int((f*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) - Simp((f*x)**(m + S(1))*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*f*(p + S(1))), x)
rule5115 = ReplacementRule(pattern5115, replacement5115)
pattern5116 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons338, cons88, cons137, cons274, cons1749)
def replacement5116(p, m, f, b, d, a, n, c, x, e):
rubi.append(5116)
return Dist((m + S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((f*x)**m*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b*c*d**IntPart(p)*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(S(2)*f*(p + S(1))), Int((f*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) - Simp((f*x)**(m + S(1))*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*f*(p + S(1))), x)
rule5116 = ReplacementRule(pattern5116, replacement5116)
pattern5117 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons93, cons88, cons166, cons17)
def replacement5117(m, f, b, d, a, n, c, x, e):
rubi.append(5117)
return Dist(f**S(2)*(m + S(-1))/(c**S(2)*m), Int((f*x)**(m + S(-2))*(a + b*asin(c*x))**n/sqrt(d + e*x**S(2)), x), x) + Dist(b*f*n*sqrt(-c**S(2)*x**S(2) + S(1))/(c*m*sqrt(d + e*x**S(2))), Int((f*x)**(m + S(-1))*(a + b*asin(c*x))**(n + S(-1)), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*asin(c*x))**n*sqrt(d + e*x**S(2))/(e*m), x)
rule5117 = ReplacementRule(pattern5117, replacement5117)
pattern5118 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons93, cons88, cons166, cons17)
def replacement5118(m, f, b, d, a, n, c, x, e):
rubi.append(5118)
return Dist(f**S(2)*(m + S(-1))/(c**S(2)*m), Int((f*x)**(m + S(-2))*(a + b*acos(c*x))**n/sqrt(d + e*x**S(2)), x), x) - Dist(b*f*n*sqrt(-c**S(2)*x**S(2) + S(1))/(c*m*sqrt(d + e*x**S(2))), Int((f*x)**(m + S(-1))*(a + b*acos(c*x))**(n + S(-1)), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acos(c*x))**n*sqrt(d + e*x**S(2))/(e*m), x)
rule5118 = ReplacementRule(pattern5118, replacement5118)
pattern5119 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons268, cons148, cons17)
def replacement5119(m, b, d, a, n, c, x, e):
rubi.append(5119)
return Dist(c**(-m + S(-1))/sqrt(d), Subst(Int((a + b*x)**n*sin(x)**m, x), x, asin(c*x)), x)
rule5119 = ReplacementRule(pattern5119, replacement5119)
pattern5120 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1737, cons268, cons148, cons17)
def replacement5120(m, b, d, a, n, c, x, e):
rubi.append(5120)
return -Dist(c**(-m + S(-1))/sqrt(d), Subst(Int((a + b*x)**n*cos(x)**m, x), x, acos(c*x)), x)
rule5120 = ReplacementRule(pattern5120, replacement5120)
pattern5121 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons268, cons18)
def replacement5121(m, f, b, d, a, c, x, e):
rubi.append(5121)
return Simp((f*x)**(m + S(1))*(a + b*asin(c*x))*Hypergeometric2F1(S(1)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, c**S(2)*x**S(2))/(sqrt(d)*f*(m + S(1))), x) - Simp(b*c*(f*x)**(m + S(2))*HypergeometricPFQ(List(S(1), m/S(2) + S(1), m/S(2) + S(1)), List(m/S(2) + S(3)/2, m/S(2) + S(2)), c**S(2)*x**S(2))/(sqrt(d)*f**S(2)*(m + S(1))*(m + S(2))), x)
rule5121 = ReplacementRule(pattern5121, replacement5121)
pattern5122 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons268, cons18)
def replacement5122(m, f, b, d, a, c, x, e):
rubi.append(5122)
return Simp((f*x)**(m + S(1))*(a + b*acos(c*x))*Hypergeometric2F1(S(1)/2, m/S(2) + S(1)/2, m/S(2) + S(3)/2, c**S(2)*x**S(2))/(sqrt(d)*f*(m + S(1))), x) + Simp(b*c*(f*x)**(m + S(2))*HypergeometricPFQ(List(S(1), m/S(2) + S(1), m/S(2) + S(1)), List(m/S(2) + S(3)/2, m/S(2) + S(2)), c**S(2)*x**S(2))/(sqrt(d)*f**S(2)*(m + S(1))*(m + S(2))), x)
rule5122 = ReplacementRule(pattern5122, replacement5122)
pattern5123 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons87, cons88, cons1738, cons1750)
def replacement5123(m, f, b, d, a, n, c, x, e):
rubi.append(5123)
return Dist(sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((f*x)**m*(a + b*asin(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x)
rule5123 = ReplacementRule(pattern5123, replacement5123)
pattern5124 = Pattern(Integral((x_*WC('f', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons87, cons88, cons1738, cons1750)
def replacement5124(m, f, b, d, a, n, c, x, e):
rubi.append(5124)
return Dist(sqrt(-c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((f*x)**m*(a + b*acos(c*x))**n/sqrt(-c**S(2)*x**S(2) + S(1)), x), x)
rule5124 = ReplacementRule(pattern5124, replacement5124)
pattern5125 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons1737, cons93, cons88, cons166, cons238, cons17)
def replacement5125(p, m, f, b, d, a, n, c, x, e):
rubi.append(5125)
return Dist(f**S(2)*(m + S(-1))/(c**S(2)*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-2))*(a + b*asin(c*x))**n*(d + e*x**S(2))**p, x), x) + Dist(b*d**IntPart(p)*f*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(c*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-1))*(a + b*asin(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*asin(c*x))**n*(d + e*x**S(2))**(p + S(1))/(e*(m + S(2)*p + S(1))), x)
rule5125 = ReplacementRule(pattern5125, replacement5125)
pattern5126 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons1737, cons93, cons88, cons166, cons238, cons17)
def replacement5126(p, m, f, b, d, a, n, c, x, e):
rubi.append(5126)
return Dist(f**S(2)*(m + S(-1))/(c**S(2)*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-2))*(a + b*acos(c*x))**n*(d + e*x**S(2))**p, x), x) - Dist(b*d**IntPart(p)*f*n*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(c*(m + S(2)*p + S(1))), Int((f*x)**(m + S(-1))*(a + b*acos(c*x))**(n + S(-1))*(-c**S(2)*x**S(2) + S(1))**(p + S(1)/2), x), x) + Simp(f*(f*x)**(m + S(-1))*(a + b*acos(c*x))**n*(d + e*x**S(2))**(p + S(1))/(e*(m + S(2)*p + S(1))), x)
rule5126 = ReplacementRule(pattern5126, replacement5126)
pattern5127 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1737, cons87, cons89, cons237)
def replacement5127(p, m, f, b, d, a, c, n, x, e):
rubi.append(5127)
return -Dist(d**IntPart(p)*f*m*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*asin(c*x))**(n + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*asin(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5127 = ReplacementRule(pattern5127, replacement5127)
pattern5128 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1737, cons87, cons89, cons237)
def replacement5128(p, m, f, b, d, a, c, n, x, e):
rubi.append(5128)
return Dist(d**IntPart(p)*f*m*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acos(c*x))**(n + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) - Simp((f*x)**m*(a + b*acos(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5128 = ReplacementRule(pattern5128, replacement5128)
pattern5129 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons87, cons89, cons268)
def replacement5129(m, f, b, d, a, c, n, x, e):
rubi.append(5129)
return -Dist(f*m/(b*c*sqrt(d)*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*asin(c*x))**(n + S(1)), x), x) + Simp((f*x)**m*(a + b*asin(c*x))**(n + S(1))/(b*c*sqrt(d)*(n + S(1))), x)
rule5129 = ReplacementRule(pattern5129, replacement5129)
pattern5130 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1737, cons87, cons89, cons268)
def replacement5130(m, f, b, d, a, c, n, x, e):
rubi.append(5130)
return Dist(f*m/(b*c*sqrt(d)*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acos(c*x))**(n + S(1)), x), x) - Simp((f*x)**m*(a + b*acos(c*x))**(n + S(1))/(b*c*sqrt(d)*(n + S(1))), x)
rule5130 = ReplacementRule(pattern5130, replacement5130)
pattern5131 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons87, cons89, cons17, cons1751, cons1739)
def replacement5131(p, m, f, b, d, a, c, n, x, e):
rubi.append(5131)
return -Dist(d**IntPart(p)*f*m*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*asin(c*x))**(n + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Dist(c*d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))*(m + S(2)*p + S(1))/(b*f*(n + S(1))), Int((f*x)**(m + S(1))*(a + b*asin(c*x))**(n + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) + Simp((f*x)**m*(a + b*asin(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5131 = ReplacementRule(pattern5131, replacement5131)
pattern5132 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1737, cons87, cons89, cons17, cons1751, cons1739)
def replacement5132(p, m, f, b, d, a, c, n, x, e):
rubi.append(5132)
return Dist(d**IntPart(p)*f*m*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))/(b*c*(n + S(1))), Int((f*x)**(m + S(-1))*(a + b*acos(c*x))**(n + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) - Dist(c*d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p))*(m + S(2)*p + S(1))/(b*f*(n + S(1))), Int((f*x)**(m + S(1))*(a + b*acos(c*x))**(n + S(1))*(-c**S(2)*x**S(2) + S(1))**(p + S(-1)/2), x), x) - Simp((f*x)**m*(a + b*acos(c*x))**(n + S(1))*(d + e*x**S(2))**p*sqrt(-c**S(2)*x**S(2) + S(1))/(b*c*(n + S(1))), x)
rule5132 = ReplacementRule(pattern5132, replacement5132)
pattern5133 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons246, cons1752, cons62, cons1740)
def replacement5133(p, m, b, d, a, n, c, x, e):
rubi.append(5133)
return Dist(c**(-m + S(-1))*d**p, Subst(Int((a + b*x)**n*sin(x)**m*cos(x)**(S(2)*p + S(1)), x), x, asin(c*x)), x)
rule5133 = ReplacementRule(pattern5133, replacement5133)
pattern5134 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons246, cons1752, cons62, cons1740)
def replacement5134(p, m, b, d, a, n, c, x, e):
rubi.append(5134)
return -Dist(c**(-m + S(-1))*d**p, Subst(Int((a + b*x)**n*sin(x)**(S(2)*p + S(1))*cos(x)**m, x), x, acos(c*x)), x)
rule5134 = ReplacementRule(pattern5134, replacement5134)
pattern5135 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons246, cons1752, cons62, cons1741)
def replacement5135(p, m, b, d, a, c, n, x, e):
rubi.append(5135)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int(x**m*(a + b*asin(c*x))**n*(-c**S(2)*x**S(2) + S(1))**p, x), x)
rule5135 = ReplacementRule(pattern5135, replacement5135)
pattern5136 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons246, cons1752, cons62, cons1741)
def replacement5136(p, m, b, d, a, c, n, x, e):
rubi.append(5136)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int(x**m*(a + b*acos(c*x))**n*(-c**S(2)*x**S(2) + S(1))**p, x), x)
rule5136 = ReplacementRule(pattern5136, replacement5136)
pattern5137 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1737, cons268, cons961, cons1753, cons17, cons1754)
def replacement5137(p, m, f, b, d, a, n, c, x, e):
rubi.append(5137)
return Int(ExpandIntegrand((a + b*asin(c*x))**n/sqrt(d + e*x**S(2)), (f*x)**m*(d + e*x**S(2))**(p + S(1)/2), x), x)
rule5137 = ReplacementRule(pattern5137, replacement5137)
pattern5138 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1737, cons268, cons961, cons1753, cons17, cons1754)
def replacement5138(p, m, f, b, d, a, n, c, x, e):
rubi.append(5138)
return Int(ExpandIntegrand((a + b*acos(c*x))**n/sqrt(d + e*x**S(2)), (f*x)**m*(d + e*x**S(2))**(p + S(1)/2), x), x)
rule5138 = ReplacementRule(pattern5138, replacement5138)
pattern5139 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1742, cons54)
def replacement5139(p, b, d, a, c, x, e):
rubi.append(5139)
return -Dist(b*c/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*asin(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5139 = ReplacementRule(pattern5139, replacement5139)
pattern5140 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1742, cons54)
def replacement5140(p, b, d, a, c, x, e):
rubi.append(5140)
return Dist(b*c/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*acos(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5140 = ReplacementRule(pattern5140, replacement5140)
def With5141(p, m, f, b, d, a, c, x, e):
u = IntHide((f*x)**m*(d + e*x**S(2))**p, x)
rubi.append(5141)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asin(c*x), u, x)
pattern5141 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1742, cons38, cons1755)
rule5141 = ReplacementRule(pattern5141, With5141)
def With5142(p, m, f, b, d, a, c, x, e):
u = IntHide((f*x)**m*(d + e*x**S(2))**p, x)
rubi.append(5142)
return Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acos(c*x), u, x)
pattern5142 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1742, cons38, cons1755)
rule5142 = ReplacementRule(pattern5142, With5142)
pattern5143 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1742, cons148, cons38, cons17)
def replacement5143(p, m, f, b, d, a, n, c, x, e):
rubi.append(5143)
return Int(ExpandIntegrand((a + b*asin(c*x))**n, (f*x)**m*(d + e*x**S(2))**p, x), x)
rule5143 = ReplacementRule(pattern5143, replacement5143)
pattern5144 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1742, cons148, cons38, cons17)
def replacement5144(p, m, f, b, d, a, n, c, x, e):
rubi.append(5144)
return Int(ExpandIntegrand((a + b*acos(c*x))**n, (f*x)**m*(d + e*x**S(2))**p, x), x)
rule5144 = ReplacementRule(pattern5144, replacement5144)
pattern5145 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons1756)
def replacement5145(p, m, f, b, d, a, n, c, x, e):
rubi.append(5145)
return Int((f*x)**m*(a + b*asin(c*x))**n*(d + e*x**S(2))**p, x)
rule5145 = ReplacementRule(pattern5145, replacement5145)
pattern5146 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons1756)
def replacement5146(p, m, f, b, d, a, n, c, x, e):
rubi.append(5146)
return Int((f*x)**m*(a + b*acos(c*x))**n*(d + e*x**S(2))**p, x)
rule5146 = ReplacementRule(pattern5146, replacement5146)
pattern5147 = Pattern(Integral((x_*WC('h', S(1)))**WC('m', S(1))*(d_ + x_*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons336, cons1745, cons147)
def replacement5147(p, m, g, b, f, d, a, n, c, x, h, e):
rubi.append(5147)
return Dist((d + e*x)**FracPart(p)*(f + g*x)**FracPart(p)*(d*f + e*g*x**S(2))**(-FracPart(p)), Int((h*x)**m*(a + b*asin(c*x))**n*(d*f + e*g*x**S(2))**p, x), x)
rule5147 = ReplacementRule(pattern5147, replacement5147)
pattern5148 = Pattern(Integral((x_*WC('h', S(1)))**WC('m', S(1))*(d_ + x_*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons336, cons1745, cons147)
def replacement5148(p, m, g, b, f, d, a, n, c, x, h, e):
rubi.append(5148)
return Dist((d + e*x)**FracPart(p)*(f + g*x)**FracPart(p)*(d*f + e*g*x**S(2))**(-FracPart(p)), Int((h*x)**m*(a + b*acos(c*x))**n*(d*f + e*g*x**S(2))**p, x), x)
rule5148 = ReplacementRule(pattern5148, replacement5148)
pattern5149 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons148)
def replacement5149(b, d, a, n, c, x, e):
rubi.append(5149)
return Subst(Int((a + b*x)**n*cos(x)/(c*d + e*sin(x)), x), x, asin(c*x))
rule5149 = ReplacementRule(pattern5149, replacement5149)
pattern5150 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons148)
def replacement5150(b, d, a, n, c, x, e):
rubi.append(5150)
return -Subst(Int((a + b*x)**n*sin(x)/(c*d + e*cos(x)), x), x, acos(c*x))
rule5150 = ReplacementRule(pattern5150, replacement5150)
pattern5151 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons148, cons66)
def replacement5151(m, b, d, a, n, c, x, e):
rubi.append(5151)
return -Dist(b*c*n/(e*(m + S(1))), Int((a + b*asin(c*x))**(n + S(-1))*(d + e*x)**(m + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*asin(c*x))**n*(d + e*x)**(m + S(1))/(e*(m + S(1))), x)
rule5151 = ReplacementRule(pattern5151, replacement5151)
pattern5152 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons148, cons66)
def replacement5152(m, b, d, a, n, c, x, e):
rubi.append(5152)
return Dist(b*c*n/(e*(m + S(1))), Int((a + b*acos(c*x))**(n + S(-1))*(d + e*x)**(m + S(1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*acos(c*x))**n*(d + e*x)**(m + S(1))/(e*(m + S(1))), x)
rule5152 = ReplacementRule(pattern5152, replacement5152)
pattern5153 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons62, cons87, cons89)
def replacement5153(m, b, d, a, c, n, x, e):
rubi.append(5153)
return Int(ExpandIntegrand((a + b*asin(c*x))**n*(d + e*x)**m, x), x)
rule5153 = ReplacementRule(pattern5153, replacement5153)
pattern5154 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons62, cons87, cons89)
def replacement5154(m, b, d, a, c, n, x, e):
rubi.append(5154)
return Int(ExpandIntegrand((a + b*acos(c*x))**n*(d + e*x)**m, x), x)
rule5154 = ReplacementRule(pattern5154, replacement5154)
pattern5155 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons62)
def replacement5155(m, b, d, a, c, n, x, e):
rubi.append(5155)
return Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*(c*d + e*sin(x))**m*cos(x), x), x, asin(c*x)), x)
rule5155 = ReplacementRule(pattern5155, replacement5155)
pattern5156 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons62)
def replacement5156(m, b, d, a, c, n, x, e):
rubi.append(5156)
return -Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*(c*d + e*cos(x))**m*sin(x), x), x, acos(c*x)), x)
rule5156 = ReplacementRule(pattern5156, replacement5156)
def With5157(b, Px, a, c, x):
u = IntHide(Px, x)
rubi.append(5157)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asin(c*x), u, x)
pattern5157 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons925)
rule5157 = ReplacementRule(pattern5157, With5157)
def With5158(b, Px, a, c, x):
u = IntHide(Px, x)
rubi.append(5158)
return Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acos(c*x), u, x)
pattern5158 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons925)
rule5158 = ReplacementRule(pattern5158, With5158)
pattern5159 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons925)
def replacement5159(b, Px, a, n, c, x):
rubi.append(5159)
return Int(ExpandIntegrand(Px*(a + b*asin(c*x))**n, x), x)
rule5159 = ReplacementRule(pattern5159, replacement5159)
pattern5160 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons925)
def replacement5160(b, Px, a, n, c, x):
rubi.append(5160)
return Int(ExpandIntegrand(Px*(a + b*acos(c*x))**n, x), x)
rule5160 = ReplacementRule(pattern5160, replacement5160)
def With5161(m, b, Px, d, a, c, x, e):
u = IntHide(Px*(d + e*x)**m, x)
rubi.append(5161)
return -Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asin(c*x), u, x)
pattern5161 = Pattern(Integral(Px_*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons925)
rule5161 = ReplacementRule(pattern5161, With5161)
def With5162(m, b, Px, d, a, c, x, e):
u = IntHide(Px*(d + e*x)**m, x)
rubi.append(5162)
return Dist(b*c, Int(SimplifyIntegrand(u/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acos(c*x), u, x)
pattern5162 = Pattern(Integral(Px_*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons925)
rule5162 = ReplacementRule(pattern5162, With5162)
def With5163(p, m, f, b, g, d, a, c, n, x, e):
u = IntHide((d + e*x)**m*(f + g*x)**p, x)
rubi.append(5163)
return -Dist(b*c*n, Int(SimplifyIntegrand(u*(a + b*asin(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist((a + b*asin(c*x))**n, u, x)
pattern5163 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons464, cons84, cons1757)
rule5163 = ReplacementRule(pattern5163, With5163)
def With5164(p, m, f, b, g, d, a, c, n, x, e):
u = IntHide((d + e*x)**m*(f + g*x)**p, x)
rubi.append(5164)
return Dist(b*c*n, Int(SimplifyIntegrand(u*(a + b*acos(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist((a + b*acos(c*x))**n, u, x)
pattern5164 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons464, cons84, cons1757)
rule5164 = ReplacementRule(pattern5164, With5164)
def With5165(p, f, b, g, d, a, c, n, x, h, e):
u = IntHide((f + g*x + h*x**S(2))**p/(d + e*x)**S(2), x)
rubi.append(5165)
return -Dist(b*c*n, Int(SimplifyIntegrand(u*(a + b*asin(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist((a + b*asin(c*x))**n, u, x)
pattern5165 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_*(x_**S(2)*WC('h', S(1)) + x_*WC('g', S(1)) + WC('f', S(0)))**WC('p', S(1))/(d_ + x_*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons464, cons1758)
rule5165 = ReplacementRule(pattern5165, With5165)
def With5166(p, f, b, g, d, a, c, n, x, h, e):
u = IntHide((f + g*x + h*x**S(2))**p/(d + e*x)**S(2), x)
rubi.append(5166)
return Dist(b*c*n, Int(SimplifyIntegrand(u*(a + b*acos(c*x))**(n + S(-1))/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist((a + b*acos(c*x))**n, u, x)
pattern5166 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_*(x_**S(2)*WC('h', S(1)) + x_*WC('g', S(1)) + WC('f', S(0)))**WC('p', S(1))/(d_ + x_*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons464, cons1758)
rule5166 = ReplacementRule(pattern5166, With5166)
pattern5167 = Pattern(Integral(Px_*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons925, cons148, cons17)
def replacement5167(m, b, Px, d, a, c, n, x, e):
rubi.append(5167)
return Int(ExpandIntegrand(Px*(a + b*asin(c*x))**n*(d + e*x)**m, x), x)
rule5167 = ReplacementRule(pattern5167, replacement5167)
pattern5168 = Pattern(Integral(Px_*(d_ + x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons925, cons148, cons17)
def replacement5168(m, b, Px, d, a, c, n, x, e):
rubi.append(5168)
return Int(ExpandIntegrand(Px*(a + b*acos(c*x))**n*(d + e*x)**m, x), x)
rule5168 = ReplacementRule(pattern5168, replacement5168)
def With5169(p, m, g, b, f, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p*(f + g*x)**m, x)
rubi.append(5169)
return -Dist(b*c, Int(Dist(S(1)/sqrt(-c**S(2)*x**S(2) + S(1)), u, x), x), x) + Dist(a + b*asin(c*x), u, x)
pattern5169 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons719, cons268, cons168, cons1759)
rule5169 = ReplacementRule(pattern5169, With5169)
def With5170(p, m, g, b, f, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p*(f + g*x)**m, x)
rubi.append(5170)
return Dist(b*c, Int(Dist(S(1)/sqrt(-c**S(2)*x**S(2) + S(1)), u, x), x), x) + Dist(a + b*acos(c*x), u, x)
pattern5170 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons719, cons268, cons168, cons1759)
rule5170 = ReplacementRule(pattern5170, With5170)
pattern5171 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons667, cons268, cons148, cons168, cons1760)
def replacement5171(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5171)
return Int(ExpandIntegrand((a + b*asin(c*x))**n*(d + e*x**S(2))**p, (f + g*x)**m, x), x)
rule5171 = ReplacementRule(pattern5171, replacement5171)
pattern5172 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons667, cons268, cons148, cons168, cons1760)
def replacement5172(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5172)
return Int(ExpandIntegrand((a + b*acos(c*x))**n*(d + e*x**S(2))**p, (f + g*x)**m, x), x)
rule5172 = ReplacementRule(pattern5172, replacement5172)
pattern5173 = Pattern(Integral(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(f_ + x_*WC('g', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons268, cons148, cons267)
def replacement5173(m, g, b, f, d, a, n, c, x, e):
rubi.append(5173)
return -Dist(S(1)/(b*c*sqrt(d)*(n + S(1))), Int((a + b*asin(c*x))**(n + S(1))*(f + g*x)**(m + S(-1))*(d*g*m + S(2)*e*f*x + e*g*x**S(2)*(m + S(2))), x), x) + Simp((a + b*asin(c*x))**(n + S(1))*(d + e*x**S(2))*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule5173 = ReplacementRule(pattern5173, replacement5173)
pattern5174 = Pattern(Integral(sqrt(d_ + x_**S(2)*WC('e', S(1)))*(f_ + x_*WC('g', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons268, cons148, cons267)
def replacement5174(m, g, b, f, d, a, n, c, x, e):
rubi.append(5174)
return Dist(S(1)/(b*c*sqrt(d)*(n + S(1))), Int((a + b*acos(c*x))**(n + S(1))*(f + g*x)**(m + S(-1))*(d*g*m + S(2)*e*f*x + e*g*x**S(2)*(m + S(2))), x), x) - Simp((a + b*acos(c*x))**(n + S(1))*(d + e*x**S(2))*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule5174 = ReplacementRule(pattern5174, replacement5174)
pattern5175 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons961, cons268, cons148)
def replacement5175(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5175)
return Int(ExpandIntegrand((a + b*asin(c*x))**n*sqrt(d + e*x**S(2)), (d + e*x**S(2))**(p + S(-1)/2)*(f + g*x)**m, x), x)
rule5175 = ReplacementRule(pattern5175, replacement5175)
pattern5176 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons961, cons268, cons148)
def replacement5176(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5176)
return Int(ExpandIntegrand((a + b*acos(c*x))**n*sqrt(d + e*x**S(2)), (d + e*x**S(2))**(p + S(-1)/2)*(f + g*x)**m, x), x)
rule5176 = ReplacementRule(pattern5176, replacement5176)
pattern5177 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons717, cons268, cons148, cons267)
def replacement5177(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5177)
return -Dist(S(1)/(b*c*sqrt(d)*(n + S(1))), Int(ExpandIntegrand((a + b*asin(c*x))**(n + S(1))*(f + g*x)**(m + S(-1)), (d + e*x**S(2))**(p + S(-1)/2)*(d*g*m + e*f*x*(S(2)*p + S(1)) + e*g*x**S(2)*(m + S(2)*p + S(1))), x), x), x) + Simp((a + b*asin(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1)/2)*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule5177 = ReplacementRule(pattern5177, replacement5177)
pattern5178 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons717, cons268, cons148, cons267)
def replacement5178(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5178)
return Dist(S(1)/(b*c*sqrt(d)*(n + S(1))), Int(ExpandIntegrand((a + b*acos(c*x))**(n + S(1))*(f + g*x)**(m + S(-1)), (d + e*x**S(2))**(p + S(-1)/2)*(d*g*m + e*f*x*(S(2)*p + S(1)) + e*g*x**S(2)*(m + S(2)*p + S(1))), x), x), x) - Simp((a + b*acos(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1)/2)*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule5178 = ReplacementRule(pattern5178, replacement5178)
pattern5179 = Pattern(Integral((f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**n_/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons268, cons168, cons87, cons89)
def replacement5179(m, g, b, f, d, a, c, n, x, e):
rubi.append(5179)
return -Dist(g*m/(b*c*sqrt(d)*(n + S(1))), Int((a + b*asin(c*x))**(n + S(1))*(f + g*x)**(m + S(-1)), x), x) + Simp((a + b*asin(c*x))**(n + S(1))*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule5179 = ReplacementRule(pattern5179, replacement5179)
pattern5180 = Pattern(Integral((f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**n_/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons268, cons168, cons87, cons89)
def replacement5180(m, g, b, f, d, a, c, n, x, e):
rubi.append(5180)
return Dist(g*m/(b*c*sqrt(d)*(n + S(1))), Int((a + b*acos(c*x))**(n + S(1))*(f + g*x)**(m + S(-1)), x), x) - Simp((a + b*acos(c*x))**(n + S(1))*(f + g*x)**m/(b*c*sqrt(d)*(n + S(1))), x)
rule5180 = ReplacementRule(pattern5180, replacement5180)
pattern5181 = Pattern(Integral((f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1737, cons17, cons268, cons1761)
def replacement5181(m, g, b, f, d, a, n, c, x, e):
rubi.append(5181)
return Dist(c**(-m + S(-1))/sqrt(d), Subst(Int((a + b*x)**n*(c*f + g*sin(x))**m, x), x, asin(c*x)), x)
rule5181 = ReplacementRule(pattern5181, replacement5181)
pattern5182 = Pattern(Integral((f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1737, cons17, cons268, cons1761)
def replacement5182(m, g, b, f, d, a, n, c, x, e):
rubi.append(5182)
return -Dist(c**(-m + S(-1))/sqrt(d), Subst(Int((a + b*x)**n*(c*f + g*cos(x))**m, x), x, acos(c*x)), x)
rule5182 = ReplacementRule(pattern5182, replacement5182)
pattern5183 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons719, cons268, cons148)
def replacement5183(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5183)
return Int(ExpandIntegrand((a + b*asin(c*x))**n/sqrt(d + e*x**S(2)), (d + e*x**S(2))**(p + S(1)/2)*(f + g*x)**m, x), x)
rule5183 = ReplacementRule(pattern5183, replacement5183)
pattern5184 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1737, cons17, cons719, cons268, cons148)
def replacement5184(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5184)
return Int(ExpandIntegrand((a + b*acos(c*x))**n/sqrt(d + e*x**S(2)), (d + e*x**S(2))**(p + S(1)/2)*(f + g*x)**m, x), x)
rule5184 = ReplacementRule(pattern5184, replacement5184)
pattern5185 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1737, cons17, cons347, cons1738)
def replacement5185(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5185)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((a + b*asin(c*x))**n*(f + g*x)**m*(-c**S(2)*x**S(2) + S(1))**p, x), x)
rule5185 = ReplacementRule(pattern5185, replacement5185)
pattern5186 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1737, cons17, cons347, cons1738)
def replacement5186(p, m, g, b, f, d, a, n, c, x, e):
rubi.append(5186)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((a + b*acos(c*x))**n*(f + g*x)**m*(-c**S(2)*x**S(2) + S(1))**p, x), x)
rule5186 = ReplacementRule(pattern5186, replacement5186)
pattern5187 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1)))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons1737, cons268, cons148)
def replacement5187(m, f, g, b, d, a, c, n, x, h, e):
rubi.append(5187)
return -Dist(g*m/(b*c*sqrt(d)*(n + S(1))), Int((a + b*asin(c*x))**(n + S(1))/(f + g*x), x), x) + Simp((a + b*asin(c*x))**(n + S(1))*log(h*(f + g*x)**m)/(b*c*sqrt(d)*(n + S(1))), x)
rule5187 = ReplacementRule(pattern5187, replacement5187)
pattern5188 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1)))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons1737, cons268, cons148)
def replacement5188(m, f, g, b, d, a, c, n, x, h, e):
rubi.append(5188)
return Dist(g*m/(b*c*sqrt(d)*(n + S(1))), Int((a + b*acos(c*x))**(n + S(1))/(f + g*x), x), x) - Simp((a + b*acos(c*x))**(n + S(1))*log(h*(f + g*x)**m)/(b*c*sqrt(d)*(n + S(1))), x)
rule5188 = ReplacementRule(pattern5188, replacement5188)
pattern5189 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons1737, cons347, cons1738)
def replacement5189(p, m, f, g, b, d, a, c, n, x, h, e):
rubi.append(5189)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((a + b*asin(c*x))**n*(-c**S(2)*x**S(2) + S(1))**p*log(h*(f + g*x)**m), x), x)
rule5189 = ReplacementRule(pattern5189, replacement5189)
pattern5190 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*WC('h', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons1737, cons347, cons1738)
def replacement5190(p, m, f, g, b, d, a, c, n, x, h, e):
rubi.append(5190)
return Dist(d**IntPart(p)*(d + e*x**S(2))**FracPart(p)*(-c**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((a + b*acos(c*x))**n*(-c**S(2)*x**S(2) + S(1))**p*log(h*(f + g*x)**m), x), x)
rule5190 = ReplacementRule(pattern5190, replacement5190)
def With5191(m, g, b, f, d, a, c, x, e):
u = IntHide((d + e*x)**m*(f + g*x)**m, x)
rubi.append(5191)
return -Dist(b*c, Int(Dist(S(1)/sqrt(-c**S(2)*x**S(2) + S(1)), u, x), x), x) + Dist(a + b*asin(c*x), u, x)
pattern5191 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(f_ + x_*WC('g', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1608)
rule5191 = ReplacementRule(pattern5191, With5191)
def With5192(m, g, b, f, d, a, c, x, e):
u = IntHide((d + e*x)**m*(f + g*x)**m, x)
rubi.append(5192)
return Dist(b*c, Int(Dist(S(1)/sqrt(-c**S(2)*x**S(2) + S(1)), u, x), x), x) + Dist(a + b*acos(c*x), u, x)
pattern5192 = Pattern(Integral((d_ + x_*WC('e', S(1)))**m_*(f_ + x_*WC('g', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1608)
rule5192 = ReplacementRule(pattern5192, With5192)
pattern5193 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons17)
def replacement5193(m, g, b, f, d, a, n, c, x, e):
rubi.append(5193)
return Int(ExpandIntegrand((a + b*asin(c*x))**n*(d + e*x)**m*(f + g*x)**m, x), x)
rule5193 = ReplacementRule(pattern5193, replacement5193)
pattern5194 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('m', S(1))*(f_ + x_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons17)
def replacement5194(m, g, b, f, d, a, n, c, x, e):
rubi.append(5194)
return Int(ExpandIntegrand((a + b*acos(c*x))**n*(d + e*x)**m*(f + g*x)**m, x), x)
rule5194 = ReplacementRule(pattern5194, replacement5194)
def With5195(u, b, a, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
v = IntHide(u, x)
if InverseFunctionFreeQ(v, x):
return True
return False
pattern5195 = Pattern(Integral(u_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons14, CustomConstraint(With5195))
def replacement5195(u, b, a, c, x):
v = IntHide(u, x)
rubi.append(5195)
return -Dist(b*c, Int(SimplifyIntegrand(v/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*asin(c*x), v, x)
rule5195 = ReplacementRule(pattern5195, replacement5195)
def With5196(u, b, a, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
v = IntHide(u, x)
if InverseFunctionFreeQ(v, x):
return True
return False
pattern5196 = Pattern(Integral(u_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons14, CustomConstraint(With5196))
def replacement5196(u, b, a, c, x):
v = IntHide(u, x)
rubi.append(5196)
return Dist(b*c, Int(SimplifyIntegrand(v/sqrt(-c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acos(c*x), v, x)
rule5196 = ReplacementRule(pattern5196, replacement5196)
def With5197(p, b, Px, d, a, n, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(Px*(a + b*asin(c*x))**n*(d + e*x**S(2))**p, x)
if SumQ(u):
return True
return False
pattern5197 = Pattern(Integral(Px_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons925, cons1737, cons347, CustomConstraint(With5197))
def replacement5197(p, b, Px, d, a, n, c, x, e):
u = ExpandIntegrand(Px*(a + b*asin(c*x))**n*(d + e*x**S(2))**p, x)
rubi.append(5197)
return Int(u, x)
rule5197 = ReplacementRule(pattern5197, replacement5197)
def With5198(p, b, Px, d, a, n, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(Px*(a + b*acos(c*x))**n*(d + e*x**S(2))**p, x)
if SumQ(u):
return True
return False
pattern5198 = Pattern(Integral(Px_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons925, cons1737, cons347, CustomConstraint(With5198))
def replacement5198(p, b, Px, d, a, n, c, x, e):
u = ExpandIntegrand(Px*(a + b*acos(c*x))**n*(d + e*x**S(2))**p, x)
rubi.append(5198)
return Int(u, x)
rule5198 = ReplacementRule(pattern5198, replacement5198)
def With5199(p, m, g, b, f, Px, d, a, n, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(Px*(a + b*asin(c*x))**n*(f + g*(d + e*x**S(2))**p)**m, x)
if SumQ(u):
return True
return False
pattern5199 = Pattern(Integral((f_ + (d_ + x_**S(2)*WC('e', S(1)))**p_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))*WC('Px', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons925, cons1737, cons961, cons150, CustomConstraint(With5199))
def replacement5199(p, m, g, b, f, Px, d, a, n, c, x, e):
u = ExpandIntegrand(Px*(a + b*asin(c*x))**n*(f + g*(d + e*x**S(2))**p)**m, x)
rubi.append(5199)
return Int(u, x)
rule5199 = ReplacementRule(pattern5199, replacement5199)
def With5200(p, m, g, b, f, Px, d, a, n, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(Px*(a + b*acos(c*x))**n*(f + g*(d + e*x**S(2))**p)**m, x)
if SumQ(u):
return True
return False
pattern5200 = Pattern(Integral((f_ + (d_ + x_**S(2)*WC('e', S(1)))**p_*WC('g', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))*WC('Px', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons925, cons1737, cons961, cons150, CustomConstraint(With5200))
def replacement5200(p, m, g, b, f, Px, d, a, n, c, x, e):
u = ExpandIntegrand(Px*(a + b*acos(c*x))**n*(f + g*(d + e*x**S(2))**p)**m, x)
rubi.append(5200)
return Int(u, x)
rule5200 = ReplacementRule(pattern5200, replacement5200)
def With5201(x, c, n, RFx):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(asin(c*x)**n, RFx, x)
if SumQ(u):
return True
return False
pattern5201 = Pattern(Integral(RFx_*asin(x_*WC('c', S(1)))**WC('n', S(1)), x_), cons7, cons1198, cons148, CustomConstraint(With5201))
def replacement5201(x, c, n, RFx):
u = ExpandIntegrand(asin(c*x)**n, RFx, x)
rubi.append(5201)
return Int(u, x)
rule5201 = ReplacementRule(pattern5201, replacement5201)
def With5202(x, c, n, RFx):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(acos(c*x)**n, RFx, x)
if SumQ(u):
return True
return False
pattern5202 = Pattern(Integral(RFx_*acos(x_*WC('c', S(1)))**WC('n', S(1)), x_), cons7, cons1198, cons148, CustomConstraint(With5202))
def replacement5202(x, c, n, RFx):
u = ExpandIntegrand(acos(c*x)**n, RFx, x)
rubi.append(5202)
return Int(u, x)
rule5202 = ReplacementRule(pattern5202, replacement5202)
pattern5203 = Pattern(Integral(RFx_*(a_ + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons1198, cons148)
def replacement5203(RFx, b, c, n, a, x):
rubi.append(5203)
return Int(ExpandIntegrand(RFx*(a + b*asin(c*x))**n, x), x)
rule5203 = ReplacementRule(pattern5203, replacement5203)
pattern5204 = Pattern(Integral(RFx_*(a_ + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons1198, cons148)
def replacement5204(RFx, b, c, n, a, x):
rubi.append(5204)
return Int(ExpandIntegrand(RFx*(a + b*acos(c*x))**n, x), x)
rule5204 = ReplacementRule(pattern5204, replacement5204)
def With5205(p, RFx, d, c, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((d + e*x**S(2))**p*asin(c*x)**n, RFx, x)
if SumQ(u):
return True
return False
pattern5205 = Pattern(Integral(RFx_*(d_ + x_**S(2)*WC('e', S(1)))**p_*asin(x_*WC('c', S(1)))**WC('n', S(1)), x_), cons7, cons27, cons48, cons1198, cons148, cons1737, cons347, CustomConstraint(With5205))
def replacement5205(p, RFx, d, c, n, x, e):
u = ExpandIntegrand((d + e*x**S(2))**p*asin(c*x)**n, RFx, x)
rubi.append(5205)
return Int(u, x)
rule5205 = ReplacementRule(pattern5205, replacement5205)
def With5206(p, RFx, d, c, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((d + e*x**S(2))**p*acos(c*x)**n, RFx, x)
if SumQ(u):
return True
return False
pattern5206 = Pattern(Integral(RFx_*(d_ + x_**S(2)*WC('e', S(1)))**p_*acos(x_*WC('c', S(1)))**WC('n', S(1)), x_), cons7, cons27, cons48, cons1198, cons148, cons1737, cons347, CustomConstraint(With5206))
def replacement5206(p, RFx, d, c, n, x, e):
u = ExpandIntegrand((d + e*x**S(2))**p*acos(c*x)**n, RFx, x)
rubi.append(5206)
return Int(u, x)
rule5206 = ReplacementRule(pattern5206, replacement5206)
pattern5207 = Pattern(Integral(RFx_*(a_ + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons1198, cons148, cons1737, cons347)
def replacement5207(p, RFx, b, d, c, n, a, x, e):
rubi.append(5207)
return Int(ExpandIntegrand((d + e*x**S(2))**p, RFx*(a + b*asin(c*x))**n, x), x)
rule5207 = ReplacementRule(pattern5207, replacement5207)
pattern5208 = Pattern(Integral(RFx_*(a_ + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons1198, cons148, cons1737, cons347)
def replacement5208(p, RFx, b, d, c, n, a, x, e):
rubi.append(5208)
return Int(ExpandIntegrand((d + e*x**S(2))**p, RFx*(a + b*acos(c*x))**n, x), x)
rule5208 = ReplacementRule(pattern5208, replacement5208)
pattern5209 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(x_*WC('c', S(1))))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5209(u, b, a, n, c, x):
rubi.append(5209)
return Int(u*(a + b*asin(c*x))**n, x)
rule5209 = ReplacementRule(pattern5209, replacement5209)
pattern5210 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_*WC('c', S(1))))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5210(u, b, a, n, c, x):
rubi.append(5210)
return Int(u*(a + b*acos(c*x))**n, x)
rule5210 = ReplacementRule(pattern5210, replacement5210)
pattern5211 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1273)
def replacement5211(b, d, c, a, n, x):
rubi.append(5211)
return Dist(S(1)/d, Subst(Int((a + b*asin(x))**n, x), x, c + d*x), x)
rule5211 = ReplacementRule(pattern5211, replacement5211)
pattern5212 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1273)
def replacement5212(b, d, c, a, n, x):
rubi.append(5212)
return Dist(S(1)/d, Subst(Int((a + b*acos(x))**n, x), x, c + d*x), x)
rule5212 = ReplacementRule(pattern5212, replacement5212)
pattern5213 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement5213(m, f, b, d, a, n, c, x, e):
rubi.append(5213)
return Dist(S(1)/d, Subst(Int((a + b*asin(x))**n*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule5213 = ReplacementRule(pattern5213, replacement5213)
pattern5214 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement5214(m, f, b, d, a, n, c, x, e):
rubi.append(5214)
return Dist(S(1)/d, Subst(Int((a + b*acos(x))**n*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule5214 = ReplacementRule(pattern5214, replacement5214)
pattern5215 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons36, cons4, cons5, cons1762, cons1763)
def replacement5215(B, C, p, b, d, a, n, c, x, A):
rubi.append(5215)
return Dist(S(1)/d, Subst(Int((a + b*asin(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p, x), x, c + d*x), x)
rule5215 = ReplacementRule(pattern5215, replacement5215)
pattern5216 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons36, cons4, cons5, cons1762, cons1763)
def replacement5216(B, C, p, b, d, a, n, c, x, A):
rubi.append(5216)
return Dist(S(1)/d, Subst(Int((a + b*acos(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p, x), x, c + d*x), x)
rule5216 = ReplacementRule(pattern5216, replacement5216)
pattern5217 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons1762, cons1763)
def replacement5217(B, C, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(5217)
return Dist(S(1)/d, Subst(Int((a + b*asin(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule5217 = ReplacementRule(pattern5217, replacement5217)
pattern5218 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons1762, cons1763)
def replacement5218(B, C, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(5218)
return Dist(S(1)/d, Subst(Int((a + b*acos(x))**n*(C*x**S(2)/d**S(2) - C/d**S(2))**p*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule5218 = ReplacementRule(pattern5218, replacement5218)
pattern5219 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons1764)
def replacement5219(b, d, c, a, x):
rubi.append(5219)
return Simp(x*sqrt(a + b*asin(c + d*x**S(2))), x) + Simp(sqrt(Pi)*x*(-c*sin(a/(S(2)*b)) + cos(a/(S(2)*b)))*FresnelS(sqrt(c/(Pi*b))*sqrt(a + b*asin(c + d*x**S(2))))/(sqrt(c/b)*(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2)))), x) - Simp(sqrt(Pi)*x*(c*sin(a/(S(2)*b)) + cos(a/(S(2)*b)))*FresnelC(sqrt(c/(Pi*b))*sqrt(a + b*asin(c + d*x**S(2))))/(sqrt(c/b)*(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2)))), x)
rule5219 = ReplacementRule(pattern5219, replacement5219)
pattern5220 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(1))), x_), cons2, cons3, cons27, cons1765)
def replacement5220(d, a, b, x):
rubi.append(5220)
return Simp(-S(2)*sqrt(a + b*acos(d*x**S(2) + S(1)))*sin(acos(d*x**S(2) + S(1))/S(2))**S(2)/(d*x), x) - Simp(S(2)*sqrt(Pi)*FresnelC(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(1))))*sin(a/(S(2)*b))*sin(acos(d*x**S(2) + S(1))/S(2))/(d*x*sqrt(S(1)/b)), x) + Simp(S(2)*sqrt(Pi)*FresnelS(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(1))))*sin(acos(d*x**S(2) + S(1))/S(2))*cos(a/(S(2)*b))/(d*x*sqrt(S(1)/b)), x)
rule5220 = ReplacementRule(pattern5220, replacement5220)
pattern5221 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(-1))), x_), cons2, cons3, cons27, cons1765)
def replacement5221(d, a, b, x):
rubi.append(5221)
return Simp(S(2)*sqrt(a + b*acos(d*x**S(2) + S(-1)))*cos(acos(d*x**S(2) + S(-1))/S(2))**S(2)/(d*x), x) - Simp(S(2)*sqrt(Pi)*FresnelC(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(-1))))*cos(a/(S(2)*b))*cos(acos(d*x**S(2) + S(-1))/S(2))/(d*x*sqrt(S(1)/b)), x) - Simp(S(2)*sqrt(Pi)*FresnelS(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(-1))))*sin(a/(S(2)*b))*cos(acos(d*x**S(2) + S(-1))/S(2))/(d*x*sqrt(S(1)/b)), x)
rule5221 = ReplacementRule(pattern5221, replacement5221)
pattern5222 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_**S(2)*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons1764, cons87, cons165)
def replacement5222(b, d, c, a, n, x):
rubi.append(5222)
return -Dist(S(4)*b**S(2)*n*(n + S(-1)), Int((a + b*asin(c + d*x**S(2)))**(n + S(-2)), x), x) + Simp(x*(a + b*asin(c + d*x**S(2)))**n, x) + Simp(S(2)*b*n*(a + b*asin(c + d*x**S(2)))**(n + S(-1))*sqrt(-S(2)*c*d*x**S(2) - d**S(2)*x**S(4))/(d*x), x)
rule5222 = ReplacementRule(pattern5222, replacement5222)
pattern5223 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(c_ + x_**S(2)*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons1764, cons87, cons165)
def replacement5223(b, d, c, a, n, x):
rubi.append(5223)
return -Dist(S(4)*b**S(2)*n*(n + S(-1)), Int((a + b*acos(c + d*x**S(2)))**(n + S(-2)), x), x) + Simp(x*(a + b*acos(c + d*x**S(2)))**n, x) - Simp(S(2)*b*n*(a + b*acos(c + d*x**S(2)))**(n + S(-1))*sqrt(-S(2)*c*d*x**S(2) - d**S(2)*x**S(4))/(d*x), x)
rule5223 = ReplacementRule(pattern5223, replacement5223)
pattern5224 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons1764)
def replacement5224(b, d, c, a, x):
rubi.append(5224)
return -Simp(x*(c*cos(a/(S(2)*b)) - sin(a/(S(2)*b)))*CosIntegral(c*(a + b*asin(c + d*x**S(2)))/(S(2)*b))/(S(2)*b*(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2)))), x) - Simp(x*(c*cos(a/(S(2)*b)) + sin(a/(S(2)*b)))*SinIntegral(c*(a + b*asin(c + d*x**S(2)))/(S(2)*b))/(S(2)*b*(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2)))), x)
rule5224 = ReplacementRule(pattern5224, replacement5224)
pattern5225 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(1))), x_), cons2, cons3, cons27, cons1765)
def replacement5225(d, a, b, x):
rubi.append(5225)
return Simp(sqrt(S(2))*x*CosIntegral((a + b*acos(d*x**S(2) + S(1)))/(S(2)*b))*cos(a/(S(2)*b))/(S(2)*b*sqrt(-d*x**S(2))), x) + Simp(sqrt(S(2))*x*SinIntegral((a + b*acos(d*x**S(2) + S(1)))/(S(2)*b))*sin(a/(S(2)*b))/(S(2)*b*sqrt(-d*x**S(2))), x)
rule5225 = ReplacementRule(pattern5225, replacement5225)
pattern5226 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(-1))), x_), cons2, cons3, cons27, cons1765)
def replacement5226(d, a, b, x):
rubi.append(5226)
return Simp(sqrt(S(2))*x*CosIntegral((a + b*acos(d*x**S(2) + S(-1)))/(S(2)*b))*sin(a/(S(2)*b))/(S(2)*b*sqrt(d*x**S(2))), x) - Simp(sqrt(S(2))*x*SinIntegral((a + b*acos(d*x**S(2) + S(-1)))/(S(2)*b))*cos(a/(S(2)*b))/(S(2)*b*sqrt(d*x**S(2))), x)
rule5226 = ReplacementRule(pattern5226, replacement5226)
pattern5227 = Pattern(Integral(S(1)/sqrt(WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons1764)
def replacement5227(b, d, c, a, x):
rubi.append(5227)
return -Simp(sqrt(Pi)*x*(-c*sin(a/(S(2)*b)) + cos(a/(S(2)*b)))*FresnelC(sqrt(a + b*asin(c + d*x**S(2)))/(sqrt(Pi)*sqrt(b*c)))/(sqrt(b*c)*(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2)))), x) - Simp(sqrt(Pi)*x*(c*sin(a/(S(2)*b)) + cos(a/(S(2)*b)))*FresnelS(sqrt(a + b*asin(c + d*x**S(2)))/(sqrt(Pi)*sqrt(b*c)))/(sqrt(b*c)*(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2)))), x)
rule5227 = ReplacementRule(pattern5227, replacement5227)
pattern5228 = Pattern(Integral(S(1)/sqrt(WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(1))), x_), cons2, cons3, cons27, cons1765)
def replacement5228(d, a, b, x):
rubi.append(5228)
return Simp(-S(2)*sqrt(Pi/b)*FresnelC(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(1))))*sin(acos(d*x**S(2) + S(1))/S(2))*cos(a/(S(2)*b))/(d*x), x) - Simp(S(2)*sqrt(Pi/b)*FresnelS(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(1))))*sin(a/(S(2)*b))*sin(acos(d*x**S(2) + S(1))/S(2))/(d*x), x)
rule5228 = ReplacementRule(pattern5228, replacement5228)
pattern5229 = Pattern(Integral(S(1)/sqrt(WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(-1))), x_), cons2, cons3, cons27, cons1765)
def replacement5229(d, a, b, x):
rubi.append(5229)
return Simp(S(2)*sqrt(Pi/b)*FresnelC(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(-1))))*sin(a/(S(2)*b))*cos(acos(d*x**S(2) + S(-1))/S(2))/(d*x), x) - Simp(S(2)*sqrt(Pi/b)*FresnelS(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(-1))))*cos(a/(S(2)*b))*cos(acos(d*x**S(2) + S(-1))/S(2))/(d*x), x)
rule5229 = ReplacementRule(pattern5229, replacement5229)
pattern5230 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_**S(2)*WC('d', S(1))))**(S(-3)/2), x_), cons2, cons3, cons7, cons27, cons1764)
def replacement5230(b, d, c, a, x):
rubi.append(5230)
return -Simp(sqrt(-S(2)*c*d*x**S(2) - d**S(2)*x**S(4))/(b*d*x*sqrt(a + b*asin(c + d*x**S(2)))), x) + Simp(sqrt(Pi)*x*(c/b)**(S(3)/2)*(-c*sin(a/(S(2)*b)) + cos(a/(S(2)*b)))*FresnelS(sqrt(c/(Pi*b))*sqrt(a + b*asin(c + d*x**S(2))))/(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2))), x) - Simp(sqrt(Pi)*x*(c/b)**(S(3)/2)*(c*sin(a/(S(2)*b)) + cos(a/(S(2)*b)))*FresnelC(sqrt(c/(Pi*b))*sqrt(a + b*asin(c + d*x**S(2))))/(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2))), x)
rule5230 = ReplacementRule(pattern5230, replacement5230)
pattern5231 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(1)))**(S(-3)/2), x_), cons2, cons3, cons27, cons1765)
def replacement5231(d, a, b, x):
rubi.append(5231)
return Simp(sqrt(-d**S(2)*x**S(4) - S(2)*d*x**S(2))/(b*d*x*sqrt(a + b*acos(d*x**S(2) + S(1)))), x) - Simp(S(2)*sqrt(Pi)*(S(1)/b)**(S(3)/2)*FresnelC(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(1))))*sin(a/(S(2)*b))*sin(acos(d*x**S(2) + S(1))/S(2))/(d*x), x) + Simp(S(2)*sqrt(Pi)*(S(1)/b)**(S(3)/2)*FresnelS(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(1))))*sin(acos(d*x**S(2) + S(1))/S(2))*cos(a/(S(2)*b))/(d*x), x)
rule5231 = ReplacementRule(pattern5231, replacement5231)
pattern5232 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(-1)))**(S(-3)/2), x_), cons2, cons3, cons27, cons1765)
def replacement5232(d, a, b, x):
rubi.append(5232)
return Simp(sqrt(-d**S(2)*x**S(4) + S(2)*d*x**S(2))/(b*d*x*sqrt(a + b*acos(d*x**S(2) + S(-1)))), x) - Simp(S(2)*sqrt(Pi)*(S(1)/b)**(S(3)/2)*FresnelC(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(-1))))*cos(a/(S(2)*b))*cos(acos(d*x**S(2) + S(-1))/S(2))/(d*x), x) - Simp(S(2)*sqrt(Pi)*(S(1)/b)**(S(3)/2)*FresnelS(sqrt(S(1)/(Pi*b))*sqrt(a + b*acos(d*x**S(2) + S(-1))))*sin(a/(S(2)*b))*cos(acos(d*x**S(2) + S(-1))/S(2))/(d*x), x)
rule5232 = ReplacementRule(pattern5232, replacement5232)
pattern5233 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_**S(2)*WC('d', S(1))))**(S(-2)), x_), cons2, cons3, cons7, cons27, cons1764)
def replacement5233(b, d, c, a, x):
rubi.append(5233)
return Simp(x*(-c*sin(a/(S(2)*b)) + cos(a/(S(2)*b)))*SinIntegral(c*(a + b*asin(c + d*x**S(2)))/(S(2)*b))/(S(4)*b**S(2)*(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2)))), x) - Simp(x*(c*sin(a/(S(2)*b)) + cos(a/(S(2)*b)))*CosIntegral(c*(a + b*asin(c + d*x**S(2)))/(S(2)*b))/(S(4)*b**S(2)*(-c*sin(asin(c + d*x**S(2))/S(2)) + cos(asin(c + d*x**S(2))/S(2)))), x) - Simp(sqrt(-S(2)*c*d*x**S(2) - d**S(2)*x**S(4))/(S(2)*b*d*x*(a + b*asin(c + d*x**S(2)))), x)
rule5233 = ReplacementRule(pattern5233, replacement5233)
pattern5234 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(1)))**(S(-2)), x_), cons2, cons3, cons27, cons1765)
def replacement5234(d, a, b, x):
rubi.append(5234)
return Simp(sqrt(-d**S(2)*x**S(4) - S(2)*d*x**S(2))/(S(2)*b*d*x*(a + b*acos(d*x**S(2) + S(1)))), x) + Simp(sqrt(S(2))*x*CosIntegral((a + b*acos(d*x**S(2) + S(1)))/(S(2)*b))*sin(a/(S(2)*b))/(S(4)*b**S(2)*sqrt(-d*x**S(2))), x) - Simp(sqrt(S(2))*x*SinIntegral((a + b*acos(d*x**S(2) + S(1)))/(S(2)*b))*cos(a/(S(2)*b))/(S(4)*b**S(2)*sqrt(-d*x**S(2))), x)
rule5234 = ReplacementRule(pattern5234, replacement5234)
pattern5235 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(x_**S(2)*WC('d', S(1)) + S(-1)))**(S(-2)), x_), cons2, cons3, cons27, cons1765)
def replacement5235(d, a, b, x):
rubi.append(5235)
return Simp(sqrt(-d**S(2)*x**S(4) + S(2)*d*x**S(2))/(S(2)*b*d*x*(a + b*acos(d*x**S(2) + S(-1)))), x) - Simp(sqrt(S(2))*x*CosIntegral((a + b*acos(d*x**S(2) + S(-1)))/(S(2)*b))*cos(a/(S(2)*b))/(S(4)*b**S(2)*sqrt(d*x**S(2))), x) - Simp(sqrt(S(2))*x*SinIntegral((a + b*acos(d*x**S(2) + S(-1)))/(S(2)*b))*sin(a/(S(2)*b))/(S(4)*b**S(2)*sqrt(d*x**S(2))), x)
rule5235 = ReplacementRule(pattern5235, replacement5235)
pattern5236 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asin(c_ + x_**S(2)*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons1764, cons87, cons89, cons1442)
def replacement5236(b, d, c, a, n, x):
rubi.append(5236)
return -Dist(S(1)/(S(4)*b**S(2)*(n + S(1))*(n + S(2))), Int((a + b*asin(c + d*x**S(2)))**(n + S(2)), x), x) + Simp(x*(a + b*asin(c + d*x**S(2)))**(n + S(2))/(S(4)*b**S(2)*(n + S(1))*(n + S(2))), x) + Simp((a + b*asin(c + d*x**S(2)))**(n + S(1))*sqrt(-S(2)*c*d*x**S(2) - d**S(2)*x**S(4))/(S(2)*b*d*x*(n + S(1))), x)
rule5236 = ReplacementRule(pattern5236, replacement5236)
pattern5237 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acos(c_ + x_**S(2)*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons1764, cons87, cons89, cons1442)
def replacement5237(b, d, c, a, n, x):
rubi.append(5237)
return -Dist(S(1)/(S(4)*b**S(2)*(n + S(1))*(n + S(2))), Int((a + b*acos(c + d*x**S(2)))**(n + S(2)), x), x) + Simp(x*(a + b*acos(c + d*x**S(2)))**(n + S(2))/(S(4)*b**S(2)*(n + S(1))*(n + S(2))), x) - Simp((a + b*acos(c + d*x**S(2)))**(n + S(1))*sqrt(-S(2)*c*d*x**S(2) - d**S(2)*x**S(4))/(S(2)*b*d*x*(n + S(1))), x)
rule5237 = ReplacementRule(pattern5237, replacement5237)
pattern5238 = Pattern(Integral(asin(x_**p_*WC('a', S(1)))**WC('n', S(1))/x_, x_), cons2, cons5, cons148)
def replacement5238(x, a, n, p):
rubi.append(5238)
return Dist(S(1)/p, Subst(Int(x**n/tan(x), x), x, asin(a*x**p)), x)
rule5238 = ReplacementRule(pattern5238, replacement5238)
pattern5239 = Pattern(Integral(acos(x_**p_*WC('a', S(1)))**WC('n', S(1))/x_, x_), cons2, cons5, cons148)
def replacement5239(x, a, n, p):
rubi.append(5239)
return -Dist(S(1)/p, Subst(Int(x**n*tan(x), x), x, acos(a*x**p)), x)
rule5239 = ReplacementRule(pattern5239, replacement5239)
pattern5240 = Pattern(Integral(WC('u', S(1))*asin(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement5240(u, m, b, c, n, a, x):
rubi.append(5240)
return Int(u*acsc(a/c + b*x**n/c)**m, x)
rule5240 = ReplacementRule(pattern5240, replacement5240)
pattern5241 = Pattern(Integral(WC('u', S(1))*acos(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement5241(u, m, b, c, n, a, x):
rubi.append(5241)
return Int(u*asec(a/c + b*x**n/c)**m, x)
rule5241 = ReplacementRule(pattern5241, replacement5241)
pattern5242 = Pattern(Integral(asin(sqrt(x_**S(2)*WC('b', S(1)) + S(1)))**WC('n', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + S(1)), x_), cons3, cons4, cons1767)
def replacement5242(x, n, b):
rubi.append(5242)
return Dist(sqrt(-b*x**S(2))/(b*x), Subst(Int(asin(x)**n/sqrt(-x**S(2) + S(1)), x), x, sqrt(b*x**S(2) + S(1))), x)
rule5242 = ReplacementRule(pattern5242, replacement5242)
pattern5243 = Pattern(Integral(acos(sqrt(x_**S(2)*WC('b', S(1)) + S(1)))**WC('n', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + S(1)), x_), cons3, cons4, cons1767)
def replacement5243(x, n, b):
rubi.append(5243)
return Dist(sqrt(-b*x**S(2))/(b*x), Subst(Int(acos(x)**n/sqrt(-x**S(2) + S(1)), x), x, sqrt(b*x**S(2) + S(1))), x)
rule5243 = ReplacementRule(pattern5243, replacement5243)
pattern5244 = Pattern(Integral(f_**(WC('c', S(1))*asin(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)))*WC('u', S(1)), x_), cons2, cons3, cons7, cons125, cons148)
def replacement5244(u, f, b, c, a, n, x):
rubi.append(5244)
return Dist(S(1)/b, Subst(Int(f**(c*x**n)*ReplaceAll(u, Rule(x, -a/b + sin(x)/b))*cos(x), x), x, asin(a + b*x)), x)
rule5244 = ReplacementRule(pattern5244, replacement5244)
pattern5245 = Pattern(Integral(f_**(WC('c', S(1))*acos(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)))*WC('u', S(1)), x_), cons2, cons3, cons7, cons125, cons148)
def replacement5245(u, f, b, c, a, n, x):
rubi.append(5245)
return -Dist(S(1)/b, Subst(Int(f**(c*x**n)*ReplaceAll(u, Rule(x, -a/b + cos(x)/b))*sin(x), x), x, acos(a + b*x)), x)
rule5245 = ReplacementRule(pattern5245, replacement5245)
pattern5246 = Pattern(Integral(asin(x_**S(2)*WC('a', S(1)) + sqrt(c_ + x_**S(2)*WC('d', S(1)))*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons1768)
def replacement5246(b, d, c, a, x):
rubi.append(5246)
return -Dist(x*sqrt(a**S(2)*x**S(2) + S(2)*a*b*sqrt(c + d*x**S(2)) + b**S(2)*d)/sqrt(-x**S(2)*(a**S(2)*x**S(2) + S(2)*a*b*sqrt(c + d*x**S(2)) + b**S(2)*d)), Int(x*(S(2)*a*sqrt(c + d*x**S(2)) + b*d)/(sqrt(c + d*x**S(2))*sqrt(a**S(2)*x**S(2) + S(2)*a*b*sqrt(c + d*x**S(2)) + b**S(2)*d)), x), x) + Simp(x*asin(a*x**S(2) + b*sqrt(c + d*x**S(2))), x)
rule5246 = ReplacementRule(pattern5246, replacement5246)
pattern5247 = Pattern(Integral(acos(x_**S(2)*WC('a', S(1)) + sqrt(c_ + x_**S(2)*WC('d', S(1)))*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons1768)
def replacement5247(b, d, c, a, x):
rubi.append(5247)
return Dist(x*sqrt(a**S(2)*x**S(2) + S(2)*a*b*sqrt(c + d*x**S(2)) + b**S(2)*d)/sqrt(-x**S(2)*(a**S(2)*x**S(2) + S(2)*a*b*sqrt(c + d*x**S(2)) + b**S(2)*d)), Int(x*(S(2)*a*sqrt(c + d*x**S(2)) + b*d)/(sqrt(c + d*x**S(2))*sqrt(a**S(2)*x**S(2) + S(2)*a*b*sqrt(c + d*x**S(2)) + b**S(2)*d)), x), x) + Simp(x*acos(a*x**S(2) + b*sqrt(c + d*x**S(2))), x)
rule5247 = ReplacementRule(pattern5247, replacement5247)
pattern5248 = Pattern(Integral(asin(u_), x_), cons1230, cons1769)
def replacement5248(x, u):
rubi.append(5248)
return -Int(SimplifyIntegrand(x*D(u, x)/sqrt(-u**S(2) + S(1)), x), x) + Simp(x*asin(u), x)
rule5248 = ReplacementRule(pattern5248, replacement5248)
pattern5249 = Pattern(Integral(acos(u_), x_), cons1230, cons1769)
def replacement5249(x, u):
rubi.append(5249)
return Int(SimplifyIntegrand(x*D(u, x)/sqrt(-u**S(2) + S(1)), x), x) + Simp(x*acos(u), x)
rule5249 = ReplacementRule(pattern5249, replacement5249)
pattern5250 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asin(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1769)
def replacement5250(u, m, b, d, c, a, x):
rubi.append(5250)
return -Dist(b/(d*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/sqrt(-u**S(2) + S(1)), x), x), x) + Simp((a + b*asin(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule5250 = ReplacementRule(pattern5250, replacement5250)
pattern5251 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acos(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1769)
def replacement5251(u, m, b, d, c, a, x):
rubi.append(5251)
return Dist(b/(d*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/sqrt(-u**S(2) + S(1)), x), x), x) + Simp((a + b*acos(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule5251 = ReplacementRule(pattern5251, replacement5251)
def With5252(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern5252 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*asin(u_)), x_), cons2, cons3, cons1230, cons1771, CustomConstraint(With5252))
def replacement5252(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(5252)
return -Dist(b, Int(SimplifyIntegrand(w*D(u, x)/sqrt(-u**S(2) + S(1)), x), x), x) + Dist(a + b*asin(u), w, x)
rule5252 = ReplacementRule(pattern5252, replacement5252)
def With5253(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern5253 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*acos(u_)), x_), cons2, cons3, cons1230, cons1772, CustomConstraint(With5253))
def replacement5253(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(5253)
return Dist(b, Int(SimplifyIntegrand(w*D(u, x)/sqrt(-u**S(2) + S(1)), x), x), x) + Dist(a + b*acos(u), w, x)
rule5253 = ReplacementRule(pattern5253, replacement5253)
pattern5254 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons148)
def replacement5254(b, a, n, c, x):
rubi.append(5254)
return -Dist(b*c*n, Int(x*(a + b*ArcTan(c*x))**(n + S(-1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*ArcTan(c*x))**n, x)
rule5254 = ReplacementRule(pattern5254, replacement5254)
pattern5255 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons148)
def replacement5255(b, a, n, c, x):
rubi.append(5255)
return Dist(b*c*n, Int(x*(a + b*acot(c*x))**(n + S(-1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp(x*(a + b*acot(c*x))**n, x)
rule5255 = ReplacementRule(pattern5255, replacement5255)
pattern5256 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons4, cons340)
def replacement5256(b, a, n, c, x):
rubi.append(5256)
return Int((a + b*ArcTan(c*x))**n, x)
rule5256 = ReplacementRule(pattern5256, replacement5256)
pattern5257 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons340)
def replacement5257(b, a, n, c, x):
rubi.append(5257)
return Int((a + b*acot(c*x))**n, x)
rule5257 = ReplacementRule(pattern5257, replacement5257)
pattern5258 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1773, cons148)
def replacement5258(b, d, a, n, c, x, e):
rubi.append(5258)
return Dist(b*c*n/e, Int((a + b*ArcTan(c*x))**(n + S(-1))*log(S(2)*d/(d + e*x))/(c**S(2)*x**S(2) + S(1)), x), x) - Simp((a + b*ArcTan(c*x))**n*log(S(2)*d/(d + e*x))/e, x)
rule5258 = ReplacementRule(pattern5258, replacement5258)
pattern5259 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1773, cons148)
def replacement5259(b, d, a, n, c, x, e):
rubi.append(5259)
return -Dist(b*c*n/e, Int((a + b*acot(c*x))**(n + S(-1))*log(S(2)*d/(d + e*x))/(c**S(2)*x**S(2) + S(1)), x), x) - Simp((a + b*acot(c*x))**n*log(S(2)*d/(d + e*x))/e, x)
rule5259 = ReplacementRule(pattern5259, replacement5259)
pattern5260 = Pattern(Integral(ArcTan(x_*WC('c', S(1)))/(d_ + x_*WC('e', S(1))), x_), cons7, cons27, cons48, cons1774, cons1775)
def replacement5260(x, d, c, e):
rubi.append(5260)
return Simp(I*PolyLog(S(2), Simp(I*c*(d + e*x)/(I*c*d - e), x))/(S(2)*e), x) - Simp(I*PolyLog(S(2), Simp(I*c*(d + e*x)/(I*c*d + e), x))/(S(2)*e), x) - Simp(ArcTan(c*d/e)*log(d + e*x)/e, x)
rule5260 = ReplacementRule(pattern5260, replacement5260)
pattern5261 = Pattern(Integral(ArcTan(x_*WC('c', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons7, cons27, cons48, cons1776)
def replacement5261(d, c, x, e):
rubi.append(5261)
return Dist(I/S(2), Int(log(-I*c*x + S(1))/(d + e*x), x), x) - Dist(I/S(2), Int(log(I*c*x + S(1))/(d + e*x), x), x)
rule5261 = ReplacementRule(pattern5261, replacement5261)
pattern5262 = Pattern(Integral(acot(x_*WC('c', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons7, cons27, cons48, cons1776)
def replacement5262(d, c, x, e):
rubi.append(5262)
return Dist(I/S(2), Int(log(S(1) - I/(c*x))/(d + e*x), x), x) - Dist(I/S(2), Int(log(S(1) + I/(c*x))/(d + e*x), x), x)
rule5262 = ReplacementRule(pattern5262, replacement5262)
pattern5263 = Pattern(Integral((a_ + ArcTan(x_*WC('c', S(1)))*WC('b', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement5263(b, d, c, a, x, e):
rubi.append(5263)
return Dist(b, Int(ArcTan(c*x)/(d + e*x), x), x) + Simp(a*log(RemoveContent(d + e*x, x))/e, x)
rule5263 = ReplacementRule(pattern5263, replacement5263)
pattern5264 = Pattern(Integral((a_ + WC('b', S(1))*acot(x_*WC('c', S(1))))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement5264(b, d, c, a, x, e):
rubi.append(5264)
return Dist(b, Int(acot(c*x)/(d + e*x), x), x) + Simp(a*log(RemoveContent(d + e*x, x))/e, x)
rule5264 = ReplacementRule(pattern5264, replacement5264)
pattern5265 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement5265(p, b, d, a, c, x, e):
rubi.append(5265)
return -Dist(b*c/(e*(p + S(1))), Int((d + e*x)**(p + S(1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*ArcTan(c*x))*(d + e*x)**(p + S(1))/(e*(p + S(1))), x)
rule5265 = ReplacementRule(pattern5265, replacement5265)
pattern5266 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement5266(p, b, d, a, c, x, e):
rubi.append(5266)
return Dist(b*c/(e*(p + S(1))), Int((d + e*x)**(p + S(1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*acot(c*x))*(d + e*x)**(p + S(1))/(e*(p + S(1))), x)
rule5266 = ReplacementRule(pattern5266, replacement5266)
pattern5267 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_/x_, x_), cons2, cons3, cons7, cons85, cons165)
def replacement5267(b, a, n, c, x):
rubi.append(5267)
return -Dist(S(2)*b*c*n, Int((a + b*ArcTan(c*x))**(n + S(-1))*atanh(S(1) - S(2)*I/(-c*x + I))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp(S(2)*(a + b*ArcTan(c*x))**n*atanh(S(1) - S(2)*I/(-c*x + I)), x)
rule5267 = ReplacementRule(pattern5267, replacement5267)
pattern5268 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_/x_, x_), cons2, cons3, cons7, cons85, cons165)
def replacement5268(b, a, n, c, x):
rubi.append(5268)
return Dist(S(2)*b*c*n, Int((a + b*acot(c*x))**(n + S(-1))*acoth(S(1) - S(2)*I/(-c*x + I))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp(S(2)*(a + b*acot(c*x))**n*acoth(S(1) - S(2)*I/(-c*x + I)), x)
rule5268 = ReplacementRule(pattern5268, replacement5268)
pattern5269 = Pattern(Integral(x_**WC('m', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons21, cons85, cons165, cons66)
def replacement5269(m, b, a, c, n, x):
rubi.append(5269)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*ArcTan(c*x))**(n + S(-1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**(m + S(1))*(a + b*ArcTan(c*x))**n/(m + S(1)), x)
rule5269 = ReplacementRule(pattern5269, replacement5269)
pattern5270 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons21, cons85, cons165, cons66)
def replacement5270(m, b, a, c, n, x):
rubi.append(5270)
return Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*acot(c*x))**(n + S(-1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp(x**(m + S(1))*(a + b*acot(c*x))**n/(m + S(1)), x)
rule5270 = ReplacementRule(pattern5270, replacement5270)
pattern5271 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons464)
def replacement5271(p, b, d, a, n, c, x, e):
rubi.append(5271)
return Int(ExpandIntegrand((a + b*ArcTan(c*x))**n*(d + e*x)**p, x), x)
rule5271 = ReplacementRule(pattern5271, replacement5271)
pattern5272 = Pattern(Integral((d_ + x_*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons464)
def replacement5272(p, b, d, a, n, c, x, e):
rubi.append(5272)
return Int(ExpandIntegrand((a + b*acot(c*x))**n*(d + e*x)**p, x), x)
rule5272 = ReplacementRule(pattern5272, replacement5272)
pattern5273 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement5273(p, b, d, a, n, c, x, e):
rubi.append(5273)
return Int((a + b*ArcTan(c*x))**n*(d + e*x)**p, x)
rule5273 = ReplacementRule(pattern5273, replacement5273)
pattern5274 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement5274(p, b, d, a, n, c, x, e):
rubi.append(5274)
return Int((a + b*acot(c*x))**n*(d + e*x)**p, x)
rule5274 = ReplacementRule(pattern5274, replacement5274)
pattern5275 = Pattern(Integral(x_**WC('m', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1773, cons148, cons31, cons168)
def replacement5275(m, b, d, a, n, c, x, e):
rubi.append(5275)
return Dist(S(1)/e, Int(x**(m + S(-1))*(a + b*ArcTan(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-1))*(a + b*ArcTan(c*x))**n/(d + e*x), x), x)
rule5275 = ReplacementRule(pattern5275, replacement5275)
pattern5276 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1773, cons148, cons31, cons168)
def replacement5276(m, b, d, a, n, c, x, e):
rubi.append(5276)
return Dist(S(1)/e, Int(x**(m + S(-1))*(a + b*acot(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-1))*(a + b*acot(c*x))**n/(d + e*x), x), x)
rule5276 = ReplacementRule(pattern5276, replacement5276)
pattern5277 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_*(d_ + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1773, cons148)
def replacement5277(b, d, a, n, c, x, e):
rubi.append(5277)
return -Dist(b*c*n/d, Int((a + b*ArcTan(c*x))**(n + S(-1))*log(S(2)*e*x/(d + e*x))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*ArcTan(c*x))**n*log(S(2)*e*x/(d + e*x))/d, x)
rule5277 = ReplacementRule(pattern5277, replacement5277)
pattern5278 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1773, cons148)
def replacement5278(b, d, a, n, c, x, e):
rubi.append(5278)
return Dist(b*c*n/d, Int((a + b*acot(c*x))**(n + S(-1))*log(S(2)*e*x/(d + e*x))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*acot(c*x))**n*log(S(2)*e*x/(d + e*x))/d, x)
rule5278 = ReplacementRule(pattern5278, replacement5278)
pattern5279 = Pattern(Integral(x_**m_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1773, cons148, cons31, cons94)
def replacement5279(m, b, d, a, n, c, x, e):
rubi.append(5279)
return Dist(S(1)/d, Int(x**m*(a + b*ArcTan(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(1))*(a + b*ArcTan(c*x))**n/(d + e*x), x), x)
rule5279 = ReplacementRule(pattern5279, replacement5279)
pattern5280 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1773, cons148, cons31, cons94)
def replacement5280(m, b, d, a, n, c, x, e):
rubi.append(5280)
return Dist(S(1)/d, Int(x**m*(a + b*acot(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(1))*(a + b*acot(c*x))**n/(d + e*x), x), x)
rule5280 = ReplacementRule(pattern5280, replacement5280)
pattern5281 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons1777)
def replacement5281(p, m, b, d, a, n, c, x, e):
rubi.append(5281)
return Int(ExpandIntegrand(x**m*(a + b*ArcTan(c*x))**n*(d + e*x)**p, x), x)
rule5281 = ReplacementRule(pattern5281, replacement5281)
pattern5282 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons1777)
def replacement5282(p, m, b, d, a, n, c, x, e):
rubi.append(5282)
return Int(ExpandIntegrand(x**m*(a + b*acot(c*x))**n*(d + e*x)**p, x), x)
rule5282 = ReplacementRule(pattern5282, replacement5282)
pattern5283 = Pattern(Integral(x_**WC('m', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5283(p, m, b, d, a, n, c, x, e):
rubi.append(5283)
return Int(x**m*(a + b*ArcTan(c*x))**n*(d + e*x)**p, x)
rule5283 = ReplacementRule(pattern5283, replacement5283)
pattern5284 = Pattern(Integral(x_**WC('m', S(1))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5284(p, m, b, d, a, n, c, x, e):
rubi.append(5284)
return Int(x**m*(a + b*acot(c*x))**n*(d + e*x)**p, x)
rule5284 = ReplacementRule(pattern5284, replacement5284)
pattern5285 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons13, cons163)
def replacement5285(p, b, d, a, c, x, e):
rubi.append(5285)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*ArcTan(c*x))*(d + e*x**S(2))**(p + S(-1)), x), x) + Simp(x*(a + b*ArcTan(c*x))*(d + e*x**S(2))**p/(S(2)*p + S(1)), x) - Simp(b*(d + e*x**S(2))**p/(S(2)*c*p*(S(2)*p + S(1))), x)
rule5285 = ReplacementRule(pattern5285, replacement5285)
pattern5286 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons13, cons163)
def replacement5286(p, b, d, a, c, x, e):
rubi.append(5286)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*acot(c*x))*(d + e*x**S(2))**(p + S(-1)), x), x) + Simp(x*(a + b*acot(c*x))*(d + e*x**S(2))**p/(S(2)*p + S(1)), x) + Simp(b*(d + e*x**S(2))**p/(S(2)*c*p*(S(2)*p + S(1))), x)
rule5286 = ReplacementRule(pattern5286, replacement5286)
pattern5287 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons338, cons163, cons165)
def replacement5287(p, b, d, a, c, n, x, e):
rubi.append(5287)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) + Dist(b**S(2)*d*n*(n + S(-1))/(S(2)*p*(S(2)*p + S(1))), Int((a + b*ArcTan(c*x))**(n + S(-2))*(d + e*x**S(2))**(p + S(-1)), x), x) + Simp(x*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**p/(S(2)*p + S(1)), x) - Simp(b*n*(a + b*ArcTan(c*x))**(n + S(-1))*(d + e*x**S(2))**p/(S(2)*c*p*(S(2)*p + S(1))), x)
rule5287 = ReplacementRule(pattern5287, replacement5287)
pattern5288 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons338, cons163, cons165)
def replacement5288(p, b, d, a, c, n, x, e):
rubi.append(5288)
return Dist(S(2)*d*p/(S(2)*p + S(1)), Int((a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) + Dist(b**S(2)*d*n*(n + S(-1))/(S(2)*p*(S(2)*p + S(1))), Int((a + b*acot(c*x))**(n + S(-2))*(d + e*x**S(2))**(p + S(-1)), x), x) + Simp(x*(a + b*acot(c*x))**n*(d + e*x**S(2))**p/(S(2)*p + S(1)), x) + Simp(b*n*(a + b*acot(c*x))**(n + S(-1))*(d + e*x**S(2))**p/(S(2)*c*p*(S(2)*p + S(1))), x)
rule5288 = ReplacementRule(pattern5288, replacement5288)
pattern5289 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778)
def replacement5289(b, d, a, c, x, e):
rubi.append(5289)
return Simp(log(RemoveContent(a + b*ArcTan(c*x), x))/(b*c*d), x)
rule5289 = ReplacementRule(pattern5289, replacement5289)
pattern5290 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons1778)
def replacement5290(b, d, a, c, x, e):
rubi.append(5290)
return -Simp(log(RemoveContent(a + b*acot(c*x), x))/(b*c*d), x)
rule5290 = ReplacementRule(pattern5290, replacement5290)
pattern5291 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons584)
def replacement5291(b, d, a, n, c, x, e):
rubi.append(5291)
return Simp((a + b*ArcTan(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule5291 = ReplacementRule(pattern5291, replacement5291)
pattern5292 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons584)
def replacement5292(b, d, a, n, c, x, e):
rubi.append(5292)
return -Simp((a + b*acot(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule5292 = ReplacementRule(pattern5292, replacement5292)
pattern5293 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons268)
def replacement5293(b, d, a, c, x, e):
rubi.append(5293)
return Simp(I*b*PolyLog(S(2), -I*sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/(c*sqrt(d)), x) - Simp(I*b*PolyLog(S(2), I*sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/(c*sqrt(d)), x) + Simp(-S(2)*I*(a + b*ArcTan(c*x))*ArcTan(sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/(c*sqrt(d)), x)
rule5293 = ReplacementRule(pattern5293, replacement5293)
pattern5294 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons268)
def replacement5294(b, d, a, c, x, e):
rubi.append(5294)
return -Simp(I*b*PolyLog(S(2), -I*sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/(c*sqrt(d)), x) + Simp(I*b*PolyLog(S(2), I*sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/(c*sqrt(d)), x) + Simp(-S(2)*I*(a + b*acot(c*x))*ArcTan(sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/(c*sqrt(d)), x)
rule5294 = ReplacementRule(pattern5294, replacement5294)
pattern5295 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148, cons268)
def replacement5295(b, d, a, n, c, x, e):
rubi.append(5295)
return Dist(S(1)/(c*sqrt(d)), Subst(Int((a + b*x)**n/cos(x), x), x, ArcTan(c*x)), x)
rule5295 = ReplacementRule(pattern5295, replacement5295)
pattern5296 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148, cons268)
def replacement5296(b, d, a, n, c, x, e):
rubi.append(5296)
return -Dist(x*sqrt(S(1) + S(1)/(c**S(2)*x**S(2)))/sqrt(d + e*x**S(2)), Subst(Int((a + b*x)**n/sin(x), x), x, acot(c*x)), x)
rule5296 = ReplacementRule(pattern5296, replacement5296)
pattern5297 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148, cons1738)
def replacement5297(b, d, a, n, c, x, e):
rubi.append(5297)
return Dist(sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*ArcTan(c*x))**n/sqrt(c**S(2)*x**S(2) + S(1)), x), x)
rule5297 = ReplacementRule(pattern5297, replacement5297)
pattern5298 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148, cons1738)
def replacement5298(b, d, a, n, c, x, e):
rubi.append(5298)
return Dist(sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*acot(c*x))**n/sqrt(c**S(2)*x**S(2) + S(1)), x), x)
rule5298 = ReplacementRule(pattern5298, replacement5298)
pattern5299 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement5299(b, d, a, n, c, x, e):
rubi.append(5299)
return -Dist(b*c*n/S(2), Int(x*(a + b*ArcTan(c*x))**(n + S(-1))/(d + e*x**S(2))**S(2), x), x) + Simp(x*(a + b*ArcTan(c*x))**n/(S(2)*d*(d + e*x**S(2))), x) + Simp((a + b*ArcTan(c*x))**(n + S(1))/(S(2)*b*c*d**S(2)*(n + S(1))), x)
rule5299 = ReplacementRule(pattern5299, replacement5299)
pattern5300 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement5300(b, d, a, n, c, x, e):
rubi.append(5300)
return Dist(b*c*n/S(2), Int(x*(a + b*acot(c*x))**(n + S(-1))/(d + e*x**S(2))**S(2), x), x) + Simp(x*(a + b*acot(c*x))**n/(S(2)*d*(d + e*x**S(2))), x) - Simp((a + b*acot(c*x))**(n + S(1))/(S(2)*b*c*d**S(2)*(n + S(1))), x)
rule5300 = ReplacementRule(pattern5300, replacement5300)
pattern5301 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1778)
def replacement5301(b, d, a, c, x, e):
rubi.append(5301)
return Simp(b/(c*d*sqrt(d + e*x**S(2))), x) + Simp(x*(a + b*ArcTan(c*x))/(d*sqrt(d + e*x**S(2))), x)
rule5301 = ReplacementRule(pattern5301, replacement5301)
pattern5302 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1778)
def replacement5302(b, d, a, c, x, e):
rubi.append(5302)
return -Simp(b/(c*d*sqrt(d + e*x**S(2))), x) + Simp(x*(a + b*acot(c*x))/(d*sqrt(d + e*x**S(2))), x)
rule5302 = ReplacementRule(pattern5302, replacement5302)
pattern5303 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons13, cons137, cons230)
def replacement5303(p, b, d, a, c, x, e):
rubi.append(5303)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*ArcTan(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) + Simp(b*(d + e*x**S(2))**(p + S(1))/(S(4)*c*d*(p + S(1))**S(2)), x) - Simp(x*(a + b*ArcTan(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x)
rule5303 = ReplacementRule(pattern5303, replacement5303)
pattern5304 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons13, cons137, cons230)
def replacement5304(p, b, d, a, c, x, e):
rubi.append(5304)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*acot(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*(d + e*x**S(2))**(p + S(1))/(S(4)*c*d*(p + S(1))**S(2)), x) - Simp(x*(a + b*acot(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x)
rule5304 = ReplacementRule(pattern5304, replacement5304)
pattern5305 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons165)
def replacement5305(b, d, a, c, n, x, e):
rubi.append(5305)
return -Dist(b**S(2)*n*(n + S(-1)), Int((a + b*ArcTan(c*x))**(n + S(-2))/(d + e*x**S(2))**(S(3)/2), x), x) + Simp(x*(a + b*ArcTan(c*x))**n/(d*sqrt(d + e*x**S(2))), x) + Simp(b*n*(a + b*ArcTan(c*x))**(n + S(-1))/(c*d*sqrt(d + e*x**S(2))), x)
rule5305 = ReplacementRule(pattern5305, replacement5305)
pattern5306 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons165)
def replacement5306(b, d, a, c, n, x, e):
rubi.append(5306)
return -Dist(b**S(2)*n*(n + S(-1)), Int((a + b*acot(c*x))**(n + S(-2))/(d + e*x**S(2))**(S(3)/2), x), x) + Simp(x*(a + b*acot(c*x))**n/(d*sqrt(d + e*x**S(2))), x) - Simp(b*n*(a + b*acot(c*x))**(n + S(-1))/(c*d*sqrt(d + e*x**S(2))), x)
rule5306 = ReplacementRule(pattern5306, replacement5306)
pattern5307 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons338, cons137, cons165, cons230)
def replacement5307(p, b, d, a, c, n, x, e):
rubi.append(5307)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b**S(2)*n*(n + S(-1))/(S(4)*(p + S(1))**S(2)), Int((a + b*ArcTan(c*x))**(n + S(-2))*(d + e*x**S(2))**p, x), x) - Simp(x*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x) + Simp(b*n*(a + b*ArcTan(c*x))**(n + S(-1))*(d + e*x**S(2))**(p + S(1))/(S(4)*c*d*(p + S(1))**S(2)), x)
rule5307 = ReplacementRule(pattern5307, replacement5307)
pattern5308 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons338, cons137, cons165, cons230)
def replacement5308(p, b, d, a, c, n, x, e):
rubi.append(5308)
return Dist((S(2)*p + S(3))/(S(2)*d*(p + S(1))), Int((a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(b**S(2)*n*(n + S(-1))/(S(4)*(p + S(1))**S(2)), Int((a + b*acot(c*x))**(n + S(-2))*(d + e*x**S(2))**p, x), x) - Simp(x*(a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*d*(p + S(1))), x) - Simp(b*n*(a + b*acot(c*x))**(n + S(-1))*(d + e*x**S(2))**(p + S(1))/(S(4)*c*d*(p + S(1))**S(2)), x)
rule5308 = ReplacementRule(pattern5308, replacement5308)
pattern5309 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons338, cons137, cons89)
def replacement5309(p, b, d, a, c, n, x, e):
rubi.append(5309)
return -Dist(S(2)*c*(p + S(1))/(b*(n + S(1))), Int(x*(a + b*ArcTan(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp((a + b*ArcTan(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule5309 = ReplacementRule(pattern5309, replacement5309)
pattern5310 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons338, cons137, cons89)
def replacement5310(p, b, d, a, c, n, x, e):
rubi.append(5310)
return Dist(S(2)*c*(p + S(1))/(b*(n + S(1))), Int(x*(a + b*acot(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) - Simp((a + b*acot(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule5310 = ReplacementRule(pattern5310, replacement5310)
pattern5311 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons1779, cons1740)
def replacement5311(p, b, d, a, n, c, x, e):
rubi.append(5311)
return Dist(d**p/c, Subst(Int((a + b*x)**n*cos(x)**(-S(2)*p + S(-2)), x), x, ArcTan(c*x)), x)
rule5311 = ReplacementRule(pattern5311, replacement5311)
pattern5312 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons1779, cons1741)
def replacement5312(p, b, d, a, n, c, x, e):
rubi.append(5312)
return Dist(d**(p + S(1)/2)*sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*ArcTan(c*x))**n*(c**S(2)*x**S(2) + S(1))**p, x), x)
rule5312 = ReplacementRule(pattern5312, replacement5312)
pattern5313 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons1779, cons38)
def replacement5313(p, b, d, a, n, c, x, e):
rubi.append(5313)
return -Dist(d**p/c, Subst(Int((a + b*x)**n*sin(x)**(-S(2)*p + S(-2)), x), x, acot(c*x)), x)
rule5313 = ReplacementRule(pattern5313, replacement5313)
pattern5314 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons1779, cons147)
def replacement5314(p, b, d, a, n, c, x, e):
rubi.append(5314)
return -Dist(d**(p + S(1)/2)*x*sqrt((c**S(2)*x**S(2) + S(1))/(c**S(2)*x**S(2)))/sqrt(d + e*x**S(2)), Subst(Int((a + b*x)**n*sin(x)**(-S(2)*p + S(-2)), x), x, acot(c*x)), x)
rule5314 = ReplacementRule(pattern5314, replacement5314)
pattern5315 = Pattern(Integral(ArcTan(x_*WC('c', S(1)))/(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons7, cons27, cons48, cons1776)
def replacement5315(d, c, x, e):
rubi.append(5315)
return Dist(I/S(2), Int(log(-I*c*x + S(1))/(d + e*x**S(2)), x), x) - Dist(I/S(2), Int(log(I*c*x + S(1))/(d + e*x**S(2)), x), x)
rule5315 = ReplacementRule(pattern5315, replacement5315)
pattern5316 = Pattern(Integral(acot(x_*WC('c', S(1)))/(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons7, cons27, cons48, cons1776)
def replacement5316(d, c, x, e):
rubi.append(5316)
return Dist(I/S(2), Int(log(S(1) - I/(c*x))/(d + e*x**S(2)), x), x) - Dist(I/S(2), Int(log(S(1) + I/(c*x))/(d + e*x**S(2)), x), x)
rule5316 = ReplacementRule(pattern5316, replacement5316)
pattern5317 = Pattern(Integral((a_ + ArcTan(x_*WC('c', S(1)))*WC('b', S(1)))/(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement5317(b, d, c, a, x, e):
rubi.append(5317)
return Dist(a, Int(S(1)/(d + e*x**S(2)), x), x) + Dist(b, Int(ArcTan(c*x)/(d + e*x**S(2)), x), x)
rule5317 = ReplacementRule(pattern5317, replacement5317)
pattern5318 = Pattern(Integral((a_ + WC('b', S(1))*acot(x_*WC('c', S(1))))/(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement5318(b, d, c, a, x, e):
rubi.append(5318)
return Dist(a, Int(S(1)/(d + e*x**S(2)), x), x) + Dist(b, Int(acot(c*x)/(d + e*x**S(2)), x), x)
rule5318 = ReplacementRule(pattern5318, replacement5318)
def With5319(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(5319)
return -Dist(b*c, Int(ExpandIntegrand(u/(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*ArcTan(c*x), u, x)
pattern5319 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1780)
rule5319 = ReplacementRule(pattern5319, With5319)
def With5320(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(5320)
return Dist(b*c, Int(ExpandIntegrand(u/(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acot(c*x), u, x)
pattern5320 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1780)
rule5320 = ReplacementRule(pattern5320, With5320)
pattern5321 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons148)
def replacement5321(p, b, d, a, n, c, x, e):
rubi.append(5321)
return Int(ExpandIntegrand((a + b*ArcTan(c*x))**n*(d + e*x**S(2))**p, x), x)
rule5321 = ReplacementRule(pattern5321, replacement5321)
pattern5322 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons148)
def replacement5322(p, b, d, a, n, c, x, e):
rubi.append(5322)
return Int(ExpandIntegrand((a + b*acot(c*x))**n*(d + e*x**S(2))**p, x), x)
rule5322 = ReplacementRule(pattern5322, replacement5322)
pattern5323 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement5323(p, b, d, a, n, c, x, e):
rubi.append(5323)
return Int((a + b*ArcTan(c*x))**n*(d + e*x**S(2))**p, x)
rule5323 = ReplacementRule(pattern5323, replacement5323)
pattern5324 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement5324(p, b, d, a, n, c, x, e):
rubi.append(5324)
return Int((a + b*acot(c*x))**n*(d + e*x**S(2))**p, x)
rule5324 = ReplacementRule(pattern5324, replacement5324)
pattern5325 = Pattern(Integral(x_**m_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons93, cons88, cons166)
def replacement5325(m, b, d, a, n, c, x, e):
rubi.append(5325)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*ArcTan(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*ArcTan(c*x))**n/(d + e*x**S(2)), x), x)
rule5325 = ReplacementRule(pattern5325, replacement5325)
pattern5326 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons93, cons88, cons166)
def replacement5326(m, b, d, a, n, c, x, e):
rubi.append(5326)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*acot(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*acot(c*x))**n/(d + e*x**S(2)), x), x)
rule5326 = ReplacementRule(pattern5326, replacement5326)
pattern5327 = Pattern(Integral(x_**m_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons93, cons88, cons94)
def replacement5327(m, b, d, a, n, c, x, e):
rubi.append(5327)
return Dist(S(1)/d, Int(x**m*(a + b*ArcTan(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*ArcTan(c*x))**n/(d + e*x**S(2)), x), x)
rule5327 = ReplacementRule(pattern5327, replacement5327)
pattern5328 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons93, cons88, cons94)
def replacement5328(m, b, d, a, n, c, x, e):
rubi.append(5328)
return Dist(S(1)/d, Int(x**m*(a + b*acot(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*acot(c*x))**n/(d + e*x**S(2)), x), x)
rule5328 = ReplacementRule(pattern5328, replacement5328)
pattern5329 = Pattern(Integral(x_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148)
def replacement5329(b, d, a, n, c, x, e):
rubi.append(5329)
return -Dist(S(1)/(c*d), Int((a + b*ArcTan(c*x))**n/(-c*x + I), x), x) - Simp(I*(a + b*ArcTan(c*x))**(n + S(1))/(b*e*(n + S(1))), x)
rule5329 = ReplacementRule(pattern5329, replacement5329)
pattern5330 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148)
def replacement5330(b, d, a, n, c, x, e):
rubi.append(5330)
return -Dist(S(1)/(c*d), Int((a + b*acot(c*x))**n/(-c*x + I), x), x) + Simp(I*(a + b*acot(c*x))**(n + S(1))/(b*e*(n + S(1))), x)
rule5330 = ReplacementRule(pattern5330, replacement5330)
pattern5331 = Pattern(Integral(x_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons340, cons584)
def replacement5331(b, d, a, c, n, x, e):
rubi.append(5331)
return -Dist(S(1)/(b*c*d*(n + S(1))), Int((a + b*ArcTan(c*x))**(n + S(1)), x), x) + Simp(x*(a + b*ArcTan(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule5331 = ReplacementRule(pattern5331, replacement5331)
pattern5332 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons340, cons584)
def replacement5332(b, d, a, c, n, x, e):
rubi.append(5332)
return Dist(S(1)/(b*c*d*(n + S(1))), Int((a + b*acot(c*x))**(n + S(1)), x), x) - Simp(x*(a + b*acot(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule5332 = ReplacementRule(pattern5332, replacement5332)
pattern5333 = Pattern(Integral(x_**m_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons93, cons88, cons166)
def replacement5333(m, b, d, a, n, c, x, e):
rubi.append(5333)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*ArcTan(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*ArcTan(c*x))**n/(d + e*x**S(2)), x), x)
rule5333 = ReplacementRule(pattern5333, replacement5333)
pattern5334 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons93, cons88, cons166)
def replacement5334(m, b, d, a, n, c, x, e):
rubi.append(5334)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*acot(c*x))**n, x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*acot(c*x))**n/(d + e*x**S(2)), x), x)
rule5334 = ReplacementRule(pattern5334, replacement5334)
pattern5335 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement5335(b, d, a, n, c, x, e):
rubi.append(5335)
return Dist(I/d, Int((a + b*ArcTan(c*x))**n/(x*(c*x + I)), x), x) - Simp(I*(a + b*ArcTan(c*x))**(n + S(1))/(b*d*(n + S(1))), x)
rule5335 = ReplacementRule(pattern5335, replacement5335)
pattern5336 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(x_*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement5336(b, d, a, n, c, x, e):
rubi.append(5336)
return Dist(I/d, Int((a + b*acot(c*x))**n/(x*(c*x + I)), x), x) + Simp(I*(a + b*acot(c*x))**(n + S(1))/(b*d*(n + S(1))), x)
rule5336 = ReplacementRule(pattern5336, replacement5336)
pattern5337 = Pattern(Integral(x_**m_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons93, cons88, cons94)
def replacement5337(m, b, d, a, n, c, x, e):
rubi.append(5337)
return Dist(S(1)/d, Int(x**m*(a + b*ArcTan(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*ArcTan(c*x))**n/(d + e*x**S(2)), x), x)
rule5337 = ReplacementRule(pattern5337, replacement5337)
pattern5338 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons93, cons88, cons94)
def replacement5338(m, b, d, a, n, c, x, e):
rubi.append(5338)
return Dist(S(1)/d, Int(x**m*(a + b*acot(c*x))**n, x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*acot(c*x))**n/(d + e*x**S(2)), x), x)
rule5338 = ReplacementRule(pattern5338, replacement5338)
pattern5339 = Pattern(Integral(x_**m_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons87, cons89)
def replacement5339(m, b, d, a, c, n, x, e):
rubi.append(5339)
return -Dist(m/(b*c*d*(n + S(1))), Int(x**(m + S(-1))*(a + b*ArcTan(c*x))**(n + S(1)), x), x) + Simp(x**m*(a + b*ArcTan(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule5339 = ReplacementRule(pattern5339, replacement5339)
pattern5340 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons87, cons89)
def replacement5340(m, b, d, a, c, n, x, e):
rubi.append(5340)
return Dist(m/(b*c*d*(n + S(1))), Int(x**(m + S(-1))*(a + b*acot(c*x))**(n + S(1)), x), x) - Simp(x**m*(a + b*acot(c*x))**(n + S(1))/(b*c*d*(n + S(1))), x)
rule5340 = ReplacementRule(pattern5340, replacement5340)
pattern5341 = Pattern(Integral(x_**WC('m', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons17, cons1781)
def replacement5341(m, b, d, a, c, x, e):
rubi.append(5341)
return Int(ExpandIntegrand(a + b*ArcTan(c*x), x**m/(d + e*x**S(2)), x), x)
rule5341 = ReplacementRule(pattern5341, replacement5341)
pattern5342 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons17, cons1781)
def replacement5342(m, b, d, a, c, x, e):
rubi.append(5342)
return Int(ExpandIntegrand(a + b*acot(c*x), x**m/(d + e*x**S(2)), x), x)
rule5342 = ReplacementRule(pattern5342, replacement5342)
pattern5343 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1778, cons87, cons88, cons54)
def replacement5343(p, b, d, a, n, c, x, e):
rubi.append(5343)
return -Dist(b*n/(S(2)*c*(p + S(1))), Int((a + b*ArcTan(c*x))**(n + S(-1))*(d + e*x**S(2))**p, x), x) + Simp((a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5343 = ReplacementRule(pattern5343, replacement5343)
pattern5344 = Pattern(Integral(x_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1778, cons87, cons88, cons54)
def replacement5344(p, b, d, a, n, c, x, e):
rubi.append(5344)
return Dist(b*n/(S(2)*c*(p + S(1))), Int((a + b*acot(c*x))**(n + S(-1))*(d + e*x**S(2))**p, x), x) + Simp((a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5344 = ReplacementRule(pattern5344, replacement5344)
pattern5345 = Pattern(Integral(x_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons89, cons1442)
def replacement5345(b, d, a, c, n, x, e):
rubi.append(5345)
return -Dist(S(4)/(b**S(2)*(n + S(1))*(n + S(2))), Int(x*(a + b*ArcTan(c*x))**(n + S(2))/(d + e*x**S(2))**S(2), x), x) - Simp((a + b*ArcTan(c*x))**(n + S(2))*(-c**S(2)*x**S(2) + S(1))/(b**S(2)*e*(d + e*x**S(2))*(n + S(1))*(n + S(2))), x) + Simp(x*(a + b*ArcTan(c*x))**(n + S(1))/(b*c*d*(d + e*x**S(2))*(n + S(1))), x)
rule5345 = ReplacementRule(pattern5345, replacement5345)
pattern5346 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons89, cons1442)
def replacement5346(b, d, a, c, n, x, e):
rubi.append(5346)
return -Dist(S(4)/(b**S(2)*(n + S(1))*(n + S(2))), Int(x*(a + b*acot(c*x))**(n + S(2))/(d + e*x**S(2))**S(2), x), x) - Simp((a + b*acot(c*x))**(n + S(2))*(-c**S(2)*x**S(2) + S(1))/(b**S(2)*e*(d + e*x**S(2))*(n + S(1))*(n + S(2))), x) - Simp(x*(a + b*acot(c*x))**(n + S(1))/(b*c*d*(d + e*x**S(2))*(n + S(1))), x)
rule5346 = ReplacementRule(pattern5346, replacement5346)
pattern5347 = Pattern(Integral(x_**S(2)*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons13, cons137, cons1782)
def replacement5347(p, b, d, a, c, x, e):
rubi.append(5347)
return -Dist(S(1)/(S(2)*c**S(2)*d*(p + S(1))), Int((a + b*ArcTan(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*(d + e*x**S(2))**(p + S(1))/(S(4)*c**S(3)*d*(p + S(1))**S(2)), x) + Simp(x*(a + b*ArcTan(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*c**S(2)*d*(p + S(1))), x)
rule5347 = ReplacementRule(pattern5347, replacement5347)
pattern5348 = Pattern(Integral(x_**S(2)*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons13, cons137, cons1782)
def replacement5348(p, b, d, a, c, x, e):
rubi.append(5348)
return -Dist(S(1)/(S(2)*c**S(2)*d*(p + S(1))), Int((a + b*acot(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) + Simp(b*(d + e*x**S(2))**(p + S(1))/(S(4)*c**S(3)*d*(p + S(1))**S(2)), x) + Simp(x*(a + b*acot(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*c**S(2)*d*(p + S(1))), x)
rule5348 = ReplacementRule(pattern5348, replacement5348)
pattern5349 = Pattern(Integral(x_**S(2)*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement5349(b, d, a, n, c, x, e):
rubi.append(5349)
return Dist(b*n/(S(2)*c), Int(x*(a + b*ArcTan(c*x))**(n + S(-1))/(d + e*x**S(2))**S(2), x), x) + Simp((a + b*ArcTan(c*x))**(n + S(1))/(S(2)*b*c**S(3)*d**S(2)*(n + S(1))), x) - Simp(x*(a + b*ArcTan(c*x))**n/(S(2)*c**S(2)*d*(d + e*x**S(2))), x)
rule5349 = ReplacementRule(pattern5349, replacement5349)
pattern5350 = Pattern(Integral(x_**S(2)*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement5350(b, d, a, n, c, x, e):
rubi.append(5350)
return -Dist(b*n/(S(2)*c), Int(x*(a + b*acot(c*x))**(n + S(-1))/(d + e*x**S(2))**S(2), x), x) - Simp((a + b*acot(c*x))**(n + S(1))/(S(2)*b*c**S(3)*d**S(2)*(n + S(1))), x) - Simp(x*(a + b*acot(c*x))**n/(S(2)*c**S(2)*d*(d + e*x**S(2))), x)
rule5350 = ReplacementRule(pattern5350, replacement5350)
pattern5351 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons240, cons13, cons137)
def replacement5351(p, m, b, d, a, c, x, e):
rubi.append(5351)
return Dist((m + S(-1))/(c**S(2)*d*m), Int(x**(m + S(-2))*(a + b*ArcTan(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) + Simp(b*x**m*(d + e*x**S(2))**(p + S(1))/(c*d*m**S(2)), x) - Simp(x**(m + S(-1))*(a + b*ArcTan(c*x))*(d + e*x**S(2))**(p + S(1))/(c**S(2)*d*m), x)
rule5351 = ReplacementRule(pattern5351, replacement5351)
pattern5352 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons240, cons13, cons137)
def replacement5352(p, m, b, d, a, c, x, e):
rubi.append(5352)
return Dist((m + S(-1))/(c**S(2)*d*m), Int(x**(m + S(-2))*(a + b*acot(c*x))*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(b*x**m*(d + e*x**S(2))**(p + S(1))/(c*d*m**S(2)), x) - Simp(x**(m + S(-1))*(a + b*acot(c*x))*(d + e*x**S(2))**(p + S(1))/(c**S(2)*d*m), x)
rule5352 = ReplacementRule(pattern5352, replacement5352)
pattern5353 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons240, cons338, cons137, cons165)
def replacement5353(p, m, b, d, a, c, n, x, e):
rubi.append(5353)
return -Dist(b**S(2)*n*(n + S(-1))/m**S(2), Int(x**m*(a + b*ArcTan(c*x))**(n + S(-2))*(d + e*x**S(2))**p, x), x) + Dist((m + S(-1))/(c**S(2)*d*m), Int(x**(m + S(-2))*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(x**(m + S(-1))*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(1))/(c**S(2)*d*m), x) + Simp(b*n*x**m*(a + b*ArcTan(c*x))**(n + S(-1))*(d + e*x**S(2))**(p + S(1))/(c*d*m**S(2)), x)
rule5353 = ReplacementRule(pattern5353, replacement5353)
pattern5354 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons240, cons338, cons137, cons165)
def replacement5354(p, m, b, d, a, c, n, x, e):
rubi.append(5354)
return -Dist(b**S(2)*n*(n + S(-1))/m**S(2), Int(x**m*(a + b*acot(c*x))**(n + S(-2))*(d + e*x**S(2))**p, x), x) + Dist((m + S(-1))/(c**S(2)*d*m), Int(x**(m + S(-2))*(a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Simp(x**(m + S(-1))*(a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(1))/(c**S(2)*d*m), x) - Simp(b*n*x**m*(a + b*acot(c*x))**(n + S(-1))*(d + e*x**S(2))**(p + S(1))/(c*d*m**S(2)), x)
rule5354 = ReplacementRule(pattern5354, replacement5354)
pattern5355 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1778, cons240, cons87, cons89)
def replacement5355(p, m, b, d, a, c, n, x, e):
rubi.append(5355)
return -Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*ArcTan(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp(x**m*(a + b*ArcTan(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule5355 = ReplacementRule(pattern5355, replacement5355)
pattern5356 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1778, cons240, cons87, cons89)
def replacement5356(p, m, b, d, a, c, n, x, e):
rubi.append(5356)
return Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*acot(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) - Simp(x**m*(a + b*acot(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule5356 = ReplacementRule(pattern5356, replacement5356)
pattern5357 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1778, cons242, cons87, cons88, cons66)
def replacement5357(p, m, b, d, a, n, c, x, e):
rubi.append(5357)
return -Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*ArcTan(c*x))**(n + S(-1))*(d + e*x**S(2))**p, x), x) + Simp(x**(m + S(1))*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*(m + S(1))), x)
rule5357 = ReplacementRule(pattern5357, replacement5357)
pattern5358 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1778, cons242, cons87, cons88, cons66)
def replacement5358(p, m, b, d, a, n, c, x, e):
rubi.append(5358)
return Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*acot(c*x))**(n + S(-1))*(d + e*x**S(2))**p, x), x) + Simp(x**(m + S(1))*(a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(1))/(d*(m + S(1))), x)
rule5358 = ReplacementRule(pattern5358, replacement5358)
pattern5359 = Pattern(Integral(x_**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons241)
def replacement5359(m, b, d, a, c, x, e):
rubi.append(5359)
return Dist(d/(m + S(2)), Int(x**m*(a + b*ArcTan(c*x))/sqrt(d + e*x**S(2)), x), x) - Dist(b*c*d/(m + S(2)), Int(x**(m + S(1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*ArcTan(c*x))*sqrt(d + e*x**S(2))/(m + S(2)), x)
rule5359 = ReplacementRule(pattern5359, replacement5359)
pattern5360 = Pattern(Integral(x_**m_*sqrt(d_ + x_**S(2)*WC('e', S(1)))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons241)
def replacement5360(m, b, d, a, c, x, e):
rubi.append(5360)
return Dist(d/(m + S(2)), Int(x**m*(a + b*acot(c*x))/sqrt(d + e*x**S(2)), x), x) + Dist(b*c*d/(m + S(2)), Int(x**(m + S(1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*acot(c*x))*sqrt(d + e*x**S(2))/(m + S(2)), x)
rule5360 = ReplacementRule(pattern5360, replacement5360)
pattern5361 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons148, cons38, cons146)
def replacement5361(p, m, b, d, a, n, c, x, e):
rubi.append(5361)
return Int(ExpandIntegrand(x**m*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**p, x), x)
rule5361 = ReplacementRule(pattern5361, replacement5361)
pattern5362 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons148, cons38, cons146)
def replacement5362(p, m, b, d, a, n, c, x, e):
rubi.append(5362)
return Int(ExpandIntegrand(x**m*(a + b*acot(c*x))**n*(d + e*x**S(2))**p, x), x)
rule5362 = ReplacementRule(pattern5362, replacement5362)
pattern5363 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons13, cons163, cons148, cons1783)
def replacement5363(p, m, b, d, a, n, c, x, e):
rubi.append(5363)
return Dist(d, Int(x**m*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) + Dist(c**S(2)*d, Int(x**(m + S(2))*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x)
rule5363 = ReplacementRule(pattern5363, replacement5363)
pattern5364 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1778, cons13, cons163, cons148, cons1783)
def replacement5364(p, m, b, d, a, n, c, x, e):
rubi.append(5364)
return Dist(d, Int(x**m*(a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x) + Dist(c**S(2)*d, Int(x**(m + S(2))*(a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(-1)), x), x)
rule5364 = ReplacementRule(pattern5364, replacement5364)
pattern5365 = Pattern(Integral(x_**m_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons93, cons88, cons166)
def replacement5365(m, b, d, a, n, c, x, e):
rubi.append(5365)
return -Dist((m + S(-1))/(c**S(2)*m), Int(x**(m + S(-2))*(a + b*ArcTan(c*x))**n/sqrt(d + e*x**S(2)), x), x) - Dist(b*n/(c*m), Int(x**(m + S(-1))*(a + b*ArcTan(c*x))**(n + S(-1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(-1))*(a + b*ArcTan(c*x))**n*sqrt(d + e*x**S(2))/(c**S(2)*d*m), x)
rule5365 = ReplacementRule(pattern5365, replacement5365)
pattern5366 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons93, cons88, cons166)
def replacement5366(m, b, d, a, n, c, x, e):
rubi.append(5366)
return -Dist((m + S(-1))/(c**S(2)*m), Int(x**(m + S(-2))*(a + b*acot(c*x))**n/sqrt(d + e*x**S(2)), x), x) + Dist(b*n/(c*m), Int(x**(m + S(-1))*(a + b*acot(c*x))**(n + S(-1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(-1))*(a + b*acot(c*x))**n*sqrt(d + e*x**S(2))/(c**S(2)*d*m), x)
rule5366 = ReplacementRule(pattern5366, replacement5366)
pattern5367 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons268)
def replacement5367(b, d, a, c, x, e):
rubi.append(5367)
return Simp(-S(2)*(a + b*ArcTan(c*x))*atanh(sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/sqrt(d), x) + Simp(I*b*PolyLog(S(2), -sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/sqrt(d), x) - Simp(I*b*PolyLog(S(2), sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/sqrt(d), x)
rule5367 = ReplacementRule(pattern5367, replacement5367)
pattern5368 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons268)
def replacement5368(b, d, a, c, x, e):
rubi.append(5368)
return Simp(-S(2)*(a + b*acot(c*x))*atanh(sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/sqrt(d), x) - Simp(I*b*PolyLog(S(2), -sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/sqrt(d), x) + Simp(I*b*PolyLog(S(2), sqrt(I*c*x + S(1))/sqrt(-I*c*x + S(1)))/sqrt(d), x)
rule5368 = ReplacementRule(pattern5368, replacement5368)
pattern5369 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148, cons268)
def replacement5369(b, d, a, c, n, x, e):
rubi.append(5369)
return Dist(S(1)/sqrt(d), Subst(Int((a + b*x)**n/sin(x), x), x, ArcTan(c*x)), x)
rule5369 = ReplacementRule(pattern5369, replacement5369)
pattern5370 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**n_/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148, cons268)
def replacement5370(b, d, a, c, n, x, e):
rubi.append(5370)
return -Dist(c*x*sqrt(S(1) + S(1)/(c**S(2)*x**S(2)))/sqrt(d + e*x**S(2)), Subst(Int((a + b*x)**n/cos(x), x), x, acot(c*x)), x)
rule5370 = ReplacementRule(pattern5370, replacement5370)
pattern5371 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148, cons1738)
def replacement5371(b, d, a, n, c, x, e):
rubi.append(5371)
return Dist(sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*ArcTan(c*x))**n/(x*sqrt(c**S(2)*x**S(2) + S(1))), x), x)
rule5371 = ReplacementRule(pattern5371, replacement5371)
pattern5372 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(x_*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons148, cons1738)
def replacement5372(b, d, a, n, c, x, e):
rubi.append(5372)
return Dist(sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int((a + b*acot(c*x))**n/(x*sqrt(c**S(2)*x**S(2) + S(1))), x), x)
rule5372 = ReplacementRule(pattern5372, replacement5372)
pattern5373 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement5373(b, d, a, n, c, x, e):
rubi.append(5373)
return Dist(b*c*n, Int((a + b*ArcTan(c*x))**(n + S(-1))/(x*sqrt(d + e*x**S(2))), x), x) - Simp((a + b*ArcTan(c*x))**n*sqrt(d + e*x**S(2))/(d*x), x)
rule5373 = ReplacementRule(pattern5373, replacement5373)
pattern5374 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(x_**S(2)*sqrt(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88)
def replacement5374(b, d, a, n, c, x, e):
rubi.append(5374)
return -Dist(b*c*n, Int((a + b*acot(c*x))**(n + S(-1))/(x*sqrt(d + e*x**S(2))), x), x) - Simp((a + b*acot(c*x))**n*sqrt(d + e*x**S(2))/(d*x), x)
rule5374 = ReplacementRule(pattern5374, replacement5374)
pattern5375 = Pattern(Integral(x_**m_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons93, cons88, cons94, cons1510)
def replacement5375(m, b, d, a, n, c, x, e):
rubi.append(5375)
return -Dist(c**S(2)*(m + S(2))/(m + S(1)), Int(x**(m + S(2))*(a + b*ArcTan(c*x))**n/sqrt(d + e*x**S(2)), x), x) - Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*ArcTan(c*x))**(n + S(-1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*ArcTan(c*x))**n*sqrt(d + e*x**S(2))/(d*(m + S(1))), x)
rule5375 = ReplacementRule(pattern5375, replacement5375)
pattern5376 = Pattern(Integral(x_**m_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons93, cons88, cons94, cons1510)
def replacement5376(m, b, d, a, n, c, x, e):
rubi.append(5376)
return -Dist(c**S(2)*(m + S(2))/(m + S(1)), Int(x**(m + S(2))*(a + b*acot(c*x))**n/sqrt(d + e*x**S(2)), x), x) + Dist(b*c*n/(m + S(1)), Int(x**(m + S(1))*(a + b*acot(c*x))**(n + S(-1))/sqrt(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*acot(c*x))**n*sqrt(d + e*x**S(2))/(d*(m + S(1))), x)
rule5376 = ReplacementRule(pattern5376, replacement5376)
pattern5377 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons1784, cons137, cons166, cons1152)
def replacement5377(p, m, b, d, a, n, c, x, e):
rubi.append(5377)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**p, x), x)
rule5377 = ReplacementRule(pattern5377, replacement5377)
pattern5378 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons1784, cons137, cons166, cons1152)
def replacement5378(p, m, b, d, a, n, c, x, e):
rubi.append(5378)
return Dist(S(1)/e, Int(x**(m + S(-2))*(a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(d/e, Int(x**(m + S(-2))*(a + b*acot(c*x))**n*(d + e*x**S(2))**p, x), x)
rule5378 = ReplacementRule(pattern5378, replacement5378)
pattern5379 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons1784, cons137, cons267, cons1152)
def replacement5379(p, m, b, d, a, n, c, x, e):
rubi.append(5379)
return Dist(S(1)/d, Int(x**m*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**p, x), x)
rule5379 = ReplacementRule(pattern5379, replacement5379)
pattern5380 = Pattern(Integral(x_**m_*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons1784, cons137, cons267, cons1152)
def replacement5380(p, m, b, d, a, n, c, x, e):
rubi.append(5380)
return Dist(S(1)/d, Int(x**m*(a + b*acot(c*x))**n*(d + e*x**S(2))**(p + S(1)), x), x) - Dist(e/d, Int(x**(m + S(2))*(a + b*acot(c*x))**n*(d + e*x**S(2))**p, x), x)
rule5380 = ReplacementRule(pattern5380, replacement5380)
pattern5381 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons162, cons137, cons89, cons319)
def replacement5381(p, m, b, d, a, n, c, x, e):
rubi.append(5381)
return -Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*ArcTan(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) - Dist(c*(m + S(2)*p + S(2))/(b*(n + S(1))), Int(x**(m + S(1))*(a + b*ArcTan(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Simp(x**m*(a + b*ArcTan(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule5381 = ReplacementRule(pattern5381, replacement5381)
pattern5382 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons162, cons137, cons89, cons319)
def replacement5382(p, m, b, d, a, n, c, x, e):
rubi.append(5382)
return Dist(m/(b*c*(n + S(1))), Int(x**(m + S(-1))*(a + b*acot(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) + Dist(c*(m + S(2)*p + S(2))/(b*(n + S(1))), Int(x**(m + S(1))*(a + b*acot(c*x))**(n + S(1))*(d + e*x**S(2))**p, x), x) - Simp(x**m*(a + b*acot(c*x))**(n + S(1))*(d + e*x**S(2))**(p + S(1))/(b*c*d*(n + S(1))), x)
rule5382 = ReplacementRule(pattern5382, replacement5382)
pattern5383 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons62, cons1785, cons1740)
def replacement5383(p, m, b, d, a, n, c, x, e):
rubi.append(5383)
return Dist(c**(-m + S(-1))*d**p, Subst(Int((a + b*x)**n*sin(x)**m*cos(x)**(-m - S(2)*p + S(-2)), x), x, ArcTan(c*x)), x)
rule5383 = ReplacementRule(pattern5383, replacement5383)
pattern5384 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons62, cons1785, cons1741)
def replacement5384(p, m, b, d, a, n, c, x, e):
rubi.append(5384)
return Dist(d**(p + S(1)/2)*sqrt(c**S(2)*x**S(2) + S(1))/sqrt(d + e*x**S(2)), Int(x**m*(a + b*ArcTan(c*x))**n*(c**S(2)*x**S(2) + S(1))**p, x), x)
rule5384 = ReplacementRule(pattern5384, replacement5384)
pattern5385 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons62, cons1785, cons38)
def replacement5385(p, m, b, d, a, n, c, x, e):
rubi.append(5385)
return -Dist(c**(-m + S(-1))*d**p, Subst(Int((a + b*x)**n*sin(x)**(-m - S(2)*p + S(-2))*cos(x)**m, x), x, acot(c*x)), x)
rule5385 = ReplacementRule(pattern5385, replacement5385)
pattern5386 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**p_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1778, cons62, cons1785, cons147)
def replacement5386(p, m, b, d, a, n, c, x, e):
rubi.append(5386)
return -Dist(c**(-m)*d**(p + S(1)/2)*x*sqrt((c**S(2)*x**S(2) + S(1))/(c**S(2)*x**S(2)))/sqrt(d + e*x**S(2)), Subst(Int((a + b*x)**n*sin(x)**(-m - S(2)*p + S(-2))*cos(x)**m, x), x, acot(c*x)), x)
rule5386 = ReplacementRule(pattern5386, replacement5386)
pattern5387 = Pattern(Integral(x_*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement5387(p, b, d, a, c, x, e):
rubi.append(5387)
return -Dist(b*c/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*ArcTan(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5387 = ReplacementRule(pattern5387, replacement5387)
pattern5388 = Pattern(Integral(x_*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement5388(p, b, d, a, c, x, e):
rubi.append(5388)
return Dist(b*c/(S(2)*e*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(c**S(2)*x**S(2) + S(1)), x), x) + Simp((a + b*acot(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5388 = ReplacementRule(pattern5388, replacement5388)
def With5389(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(d + e*x**S(2))**p, x)
rubi.append(5389)
return -Dist(b*c, Int(SimplifyIntegrand(u/(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*ArcTan(c*x), u, x)
pattern5389 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1786)
rule5389 = ReplacementRule(pattern5389, With5389)
def With5390(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(d + e*x**S(2))**p, x)
rubi.append(5390)
return Dist(b*c, Int(SimplifyIntegrand(u/(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acot(c*x), u, x)
pattern5390 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1786)
rule5390 = ReplacementRule(pattern5390, With5390)
pattern5391 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons1787)
def replacement5391(p, m, b, d, a, n, c, x, e):
rubi.append(5391)
return Int(ExpandIntegrand((a + b*ArcTan(c*x))**n, x**m*(d + e*x**S(2))**p, x), x)
rule5391 = ReplacementRule(pattern5391, replacement5391)
pattern5392 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons1787)
def replacement5392(p, m, b, d, a, n, c, x, e):
rubi.append(5392)
return Int(ExpandIntegrand((a + b*acot(c*x))**n, x**m*(d + e*x**S(2))**p, x), x)
rule5392 = ReplacementRule(pattern5392, replacement5392)
pattern5393 = Pattern(Integral(x_**WC('m', S(1))*(a_ + ArcTan(x_*WC('c', S(1)))*WC('b', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1788)
def replacement5393(p, m, b, d, c, a, x, e):
rubi.append(5393)
return Dist(a, Int(x**m*(d + e*x**S(2))**p, x), x) + Dist(b, Int(x**m*(d + e*x**S(2))**p*ArcTan(c*x), x), x)
rule5393 = ReplacementRule(pattern5393, replacement5393)
pattern5394 = Pattern(Integral(x_**WC('m', S(1))*(a_ + WC('b', S(1))*acot(x_*WC('c', S(1))))*(d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1788)
def replacement5394(p, m, b, d, c, a, x, e):
rubi.append(5394)
return Dist(a, Int(x**m*(d + e*x**S(2))**p, x), x) + Dist(b, Int(x**m*(d + e*x**S(2))**p*acot(c*x), x), x)
rule5394 = ReplacementRule(pattern5394, replacement5394)
pattern5395 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5395(p, m, b, d, a, n, c, x, e):
rubi.append(5395)
return Int(x**m*(a + b*ArcTan(c*x))**n*(d + e*x**S(2))**p, x)
rule5395 = ReplacementRule(pattern5395, replacement5395)
pattern5396 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5396(p, m, b, d, a, n, c, x, e):
rubi.append(5396)
return Int(x**m*(a + b*acot(c*x))**n*(d + e*x**S(2))**p, x)
rule5396 = ReplacementRule(pattern5396, replacement5396)
pattern5397 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*atanh(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88, cons1789)
def replacement5397(u, b, d, a, n, c, x, e):
rubi.append(5397)
return -Dist(S(1)/2, Int((a + b*ArcTan(c*x))**n*log(-u + S(1))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int((a + b*ArcTan(c*x))**n*log(u + S(1))/(d + e*x**S(2)), x), x)
rule5397 = ReplacementRule(pattern5397, replacement5397)
pattern5398 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))*acoth(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88, cons1789)
def replacement5398(u, b, d, a, n, c, x, e):
rubi.append(5398)
return -Dist(S(1)/2, Int((a + b*acot(c*x))**n*log(SimplifyIntegrand(S(1) - S(1)/u, x))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int((a + b*acot(c*x))**n*log(SimplifyIntegrand(S(1) + S(1)/u, x))/(d + e*x**S(2)), x), x)
rule5398 = ReplacementRule(pattern5398, replacement5398)
pattern5399 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*atanh(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88, cons1790)
def replacement5399(u, b, d, a, n, c, x, e):
rubi.append(5399)
return -Dist(S(1)/2, Int((a + b*ArcTan(c*x))**n*log(-u + S(1))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int((a + b*ArcTan(c*x))**n*log(u + S(1))/(d + e*x**S(2)), x), x)
rule5399 = ReplacementRule(pattern5399, replacement5399)
pattern5400 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))*acoth(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88, cons1790)
def replacement5400(u, b, d, a, n, c, x, e):
rubi.append(5400)
return -Dist(S(1)/2, Int((a + b*acot(c*x))**n*log(SimplifyIntegrand(S(1) - S(1)/u, x))/(d + e*x**S(2)), x), x) + Dist(S(1)/2, Int((a + b*acot(c*x))**n*log(SimplifyIntegrand(S(1) + S(1)/u, x))/(d + e*x**S(2)), x), x)
rule5400 = ReplacementRule(pattern5400, replacement5400)
pattern5401 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*log(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88, cons1791)
def replacement5401(u, b, d, a, n, c, x, e):
rubi.append(5401)
return -Dist(I*b*n/S(2), Int((a + b*ArcTan(c*x))**(n + S(-1))*PolyLog(S(2), Together(-u + S(1)))/(d + e*x**S(2)), x), x) + Simp(I*(a + b*ArcTan(c*x))**n*PolyLog(S(2), Together(-u + S(1)))/(S(2)*c*d), x)
rule5401 = ReplacementRule(pattern5401, replacement5401)
pattern5402 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))*log(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88, cons1791)
def replacement5402(u, b, d, a, n, c, x, e):
rubi.append(5402)
return Dist(I*b*n/S(2), Int((a + b*acot(c*x))**(n + S(-1))*PolyLog(S(2), Together(-u + S(1)))/(d + e*x**S(2)), x), x) + Simp(I*(a + b*acot(c*x))**n*PolyLog(S(2), Together(-u + S(1)))/(S(2)*c*d), x)
rule5402 = ReplacementRule(pattern5402, replacement5402)
pattern5403 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*log(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88, cons1792)
def replacement5403(u, b, d, a, n, c, x, e):
rubi.append(5403)
return Dist(I*b*n/S(2), Int((a + b*ArcTan(c*x))**(n + S(-1))*PolyLog(S(2), Together(-u + S(1)))/(d + e*x**S(2)), x), x) - Simp(I*(a + b*ArcTan(c*x))**n*PolyLog(S(2), Together(-u + S(1)))/(S(2)*c*d), x)
rule5403 = ReplacementRule(pattern5403, replacement5403)
pattern5404 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))*log(u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons87, cons88, cons1792)
def replacement5404(u, b, d, a, n, c, x, e):
rubi.append(5404)
return -Dist(I*b*n/S(2), Int((a + b*acot(c*x))**(n + S(-1))*PolyLog(S(2), Together(-u + S(1)))/(d + e*x**S(2)), x), x) - Simp(I*(a + b*acot(c*x))**n*PolyLog(S(2), Together(-u + S(1)))/(S(2)*c*d), x)
rule5404 = ReplacementRule(pattern5404, replacement5404)
pattern5405 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*PolyLog(p_, u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1778, cons87, cons88, cons1789)
def replacement5405(p, u, b, d, a, n, c, x, e):
rubi.append(5405)
return Dist(I*b*n/S(2), Int((a + b*ArcTan(c*x))**(n + S(-1))*PolyLog(p + S(1), u)/(d + e*x**S(2)), x), x) - Simp(I*(a + b*ArcTan(c*x))**n*PolyLog(p + S(1), u)/(S(2)*c*d), x)
rule5405 = ReplacementRule(pattern5405, replacement5405)
pattern5406 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))*PolyLog(p_, u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1778, cons87, cons88, cons1789)
def replacement5406(p, u, b, d, a, n, c, x, e):
rubi.append(5406)
return -Dist(I*b*n/S(2), Int((a + b*acot(c*x))**(n + S(-1))*PolyLog(p + S(1), u)/(d + e*x**S(2)), x), x) - Simp(I*(a + b*acot(c*x))**n*PolyLog(p + S(1), u)/(S(2)*c*d), x)
rule5406 = ReplacementRule(pattern5406, replacement5406)
pattern5407 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*PolyLog(p_, u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1778, cons87, cons88, cons1790)
def replacement5407(p, u, b, d, a, n, c, x, e):
rubi.append(5407)
return -Dist(I*b*n/S(2), Int((a + b*ArcTan(c*x))**(n + S(-1))*PolyLog(p + S(1), u)/(d + e*x**S(2)), x), x) + Simp(I*(a + b*ArcTan(c*x))**n*PolyLog(p + S(1), u)/(S(2)*c*d), x)
rule5407 = ReplacementRule(pattern5407, replacement5407)
pattern5408 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))*PolyLog(p_, u_)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1778, cons87, cons88, cons1790)
def replacement5408(p, u, b, d, a, n, c, x, e):
rubi.append(5408)
return Dist(I*b*n/S(2), Int((a + b*acot(c*x))**(n + S(-1))*PolyLog(p + S(1), u)/(d + e*x**S(2)), x), x) + Simp(I*(a + b*acot(c*x))**n*PolyLog(p + S(1), u)/(S(2)*c*d), x)
rule5408 = ReplacementRule(pattern5408, replacement5408)
pattern5409 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons1778)
def replacement5409(b, d, a, c, x, e):
rubi.append(5409)
return Simp((log(a + b*ArcTan(c*x)) - log(a + b*acot(c*x)))/(b*c*d*(S(2)*a + b*ArcTan(c*x) + b*acot(c*x))), x)
rule5409 = ReplacementRule(pattern5409, replacement5409)
pattern5410 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('m', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons150, cons1793)
def replacement5410(m, b, d, a, n, c, x, e):
rubi.append(5410)
return Dist(n/(m + S(1)), Int((a + b*ArcTan(c*x))**(n + S(-1))*(a + b*acot(c*x))**(m + S(1))/(d + e*x**S(2)), x), x) - Simp((a + b*ArcTan(c*x))**n*(a + b*acot(c*x))**(m + S(1))/(b*c*d*(m + S(1))), x)
rule5410 = ReplacementRule(pattern5410, replacement5410)
pattern5411 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**WC('n', S(1))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1778, cons150, cons1794)
def replacement5411(m, b, d, a, n, c, x, e):
rubi.append(5411)
return Dist(n/(m + S(1)), Int((a + b*ArcTan(c*x))**(m + S(1))*(a + b*acot(c*x))**(n + S(-1))/(d + e*x**S(2)), x), x) + Simp((a + b*ArcTan(c*x))**(m + S(1))*(a + b*acot(c*x))**n/(b*c*d*(m + S(1))), x)
rule5411 = ReplacementRule(pattern5411, replacement5411)
pattern5412 = Pattern(Integral(ArcTan(x_*WC('a', S(1)))/(c_ + x_**WC('n', S(1))*WC('d', S(1))), x_), cons2, cons7, cons27, cons85, cons1795)
def replacement5412(d, a, n, c, x):
rubi.append(5412)
return Dist(I/S(2), Int(log(-I*a*x + S(1))/(c + d*x**n), x), x) - Dist(I/S(2), Int(log(I*a*x + S(1))/(c + d*x**n), x), x)
rule5412 = ReplacementRule(pattern5412, replacement5412)
pattern5413 = Pattern(Integral(acot(x_*WC('a', S(1)))/(c_ + x_**WC('n', S(1))*WC('d', S(1))), x_), cons2, cons7, cons27, cons85, cons1795)
def replacement5413(d, a, n, c, x):
rubi.append(5413)
return Dist(I/S(2), Int(log(S(1) - I/(a*x))/(c + d*x**n), x), x) - Dist(I/S(2), Int(log(S(1) + I/(a*x))/(c + d*x**n), x), x)
rule5413 = ReplacementRule(pattern5413, replacement5413)
pattern5414 = Pattern(Integral((ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1796)
def replacement5414(f, b, g, d, a, c, x, e):
rubi.append(5414)
return -Dist(b*c, Int(x*(d + e*log(f + g*x**S(2)))/(c**S(2)*x**S(2) + S(1)), x), x) - Dist(S(2)*e*g, Int(x**S(2)*(a + b*ArcTan(c*x))/(f + g*x**S(2)), x), x) + Simp(x*(a + b*ArcTan(c*x))*(d + e*log(f + g*x**S(2))), x)
rule5414 = ReplacementRule(pattern5414, replacement5414)
pattern5415 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1796)
def replacement5415(f, b, g, d, a, c, x, e):
rubi.append(5415)
return Dist(b*c, Int(x*(d + e*log(f + g*x**S(2)))/(c**S(2)*x**S(2) + S(1)), x), x) - Dist(S(2)*e*g, Int(x**S(2)*(a + b*acot(c*x))/(f + g*x**S(2)), x), x) + Simp(x*(a + b*acot(c*x))*(d + e*log(f + g*x**S(2))), x)
rule5415 = ReplacementRule(pattern5415, replacement5415)
pattern5416 = Pattern(Integral(x_**WC('m', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons601)
def replacement5416(m, f, b, g, d, a, c, x, e):
rubi.append(5416)
return -Dist(b*c/(m + S(1)), Int(x**(m + S(1))*(d + e*log(f + g*x**S(2)))/(c**S(2)*x**S(2) + S(1)), x), x) - Dist(S(2)*e*g/(m + S(1)), Int(x**(m + S(2))*(a + b*ArcTan(c*x))/(f + g*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*ArcTan(c*x))*(d + e*log(f + g*x**S(2)))/(m + S(1)), x)
rule5416 = ReplacementRule(pattern5416, replacement5416)
pattern5417 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons601)
def replacement5417(m, f, b, g, d, a, c, x, e):
rubi.append(5417)
return Dist(b*c/(m + S(1)), Int(x**(m + S(1))*(d + e*log(f + g*x**S(2)))/(c**S(2)*x**S(2) + S(1)), x), x) - Dist(S(2)*e*g/(m + S(1)), Int(x**(m + S(2))*(a + b*acot(c*x))/(f + g*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*acot(c*x))*(d + e*log(f + g*x**S(2)))/(m + S(1)), x)
rule5417 = ReplacementRule(pattern5417, replacement5417)
def With5418(m, f, b, g, d, a, c, x, e):
u = IntHide(x**m*(d + e*log(f + g*x**S(2))), x)
rubi.append(5418)
return -Dist(b*c, Int(ExpandIntegrand(u/(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*ArcTan(c*x), u, x)
pattern5418 = Pattern(Integral(x_**WC('m', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1797)
rule5418 = ReplacementRule(pattern5418, With5418)
def With5419(m, f, b, g, d, a, c, x, e):
u = IntHide(x**m*(d + e*log(f + g*x**S(2))), x)
rubi.append(5419)
return Dist(b*c, Int(ExpandIntegrand(u/(c**S(2)*x**S(2) + S(1)), x), x), x) + Dist(a + b*acot(c*x), u, x)
pattern5419 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1797)
rule5419 = ReplacementRule(pattern5419, With5419)
def With5420(m, f, b, g, d, a, c, x, e):
u = IntHide(x**m*(a + b*ArcTan(c*x)), x)
rubi.append(5420)
return -Dist(S(2)*e*g, Int(ExpandIntegrand(u*x/(f + g*x**S(2)), x), x), x) + Dist(d + e*log(f + g*x**S(2)), u, x)
pattern5420 = Pattern(Integral(x_**WC('m', S(1))*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons17, cons261)
rule5420 = ReplacementRule(pattern5420, With5420)
def With5421(m, f, b, g, d, a, c, x, e):
u = IntHide(x**m*(a + b*acot(c*x)), x)
rubi.append(5421)
return -Dist(S(2)*e*g, Int(ExpandIntegrand(u*x/(f + g*x**S(2)), x), x), x) + Dist(d + e*log(f + g*x**S(2)), u, x)
pattern5421 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))*(WC('d', S(0)) + WC('e', S(1))*log(x_**S(2)*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons17, cons261)
rule5421 = ReplacementRule(pattern5421, With5421)
pattern5422 = Pattern(Integral(x_*(ArcTan(x_*WC('c', S(1)))*WC('b', S(1)) + WC('a', S(0)))**S(2)*(WC('d', S(0)) + WC('e', S(1))*log(f_ + x_**S(2)*WC('g', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1798)
def replacement5422(g, b, f, d, a, c, x, e):
rubi.append(5422)
return -Dist(b/c, Int((a + b*ArcTan(c*x))*(d + e*log(f + g*x**S(2))), x), x) + Dist(b*c*e, Int(x**S(2)*(a + b*ArcTan(c*x))/(c**S(2)*x**S(2) + S(1)), x), x) - Simp(e*x**S(2)*(a + b*ArcTan(c*x))**S(2)/S(2), x) + Simp((a + b*ArcTan(c*x))**S(2)*(d + e*log(f + g*x**S(2)))*(f + g*x**S(2))/(S(2)*g), x)
rule5422 = ReplacementRule(pattern5422, replacement5422)
pattern5423 = Pattern(Integral(x_*(WC('a', S(0)) + WC('b', S(1))*acot(x_*WC('c', S(1))))**S(2)*(WC('d', S(0)) + WC('e', S(1))*log(f_ + x_**S(2)*WC('g', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1798)
def replacement5423(g, b, f, d, a, c, x, e):
rubi.append(5423)
return Dist(b/c, Int((a + b*acot(c*x))*(d + e*log(f + g*x**S(2))), x), x) - Dist(b*c*e, Int(x**S(2)*(a + b*acot(c*x))/(c**S(2)*x**S(2) + S(1)), x), x) - Simp(e*x**S(2)*(a + b*acot(c*x))**S(2)/S(2), x) + Simp((a + b*acot(c*x))**S(2)*(d + e*log(f + g*x**S(2)))*(f + g*x**S(2))/(S(2)*g), x)
rule5423 = ReplacementRule(pattern5423, replacement5423)
pattern5424 = Pattern(Integral(exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons1799)
def replacement5424(x, a, n):
rubi.append(5424)
return Int((-I*a*x + S(1))**(I*n/S(2) + S(1)/2)*(I*a*x + S(1))**(-I*n/S(2) + S(1)/2)/sqrt(a**S(2)*x**S(2) + S(1)), x)
rule5424 = ReplacementRule(pattern5424, replacement5424)
pattern5425 = Pattern(Integral(x_**WC('m', S(1))*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons21, cons1799)
def replacement5425(x, m, a, n):
rubi.append(5425)
return Int(x**m*(-I*a*x + S(1))**(I*n/S(2) + S(1)/2)*(I*a*x + S(1))**(-I*n/S(2) + S(1)/2)/sqrt(a**S(2)*x**S(2) + S(1)), x)
rule5425 = ReplacementRule(pattern5425, replacement5425)
pattern5426 = Pattern(Integral(exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons4, cons1800)
def replacement5426(x, a, n):
rubi.append(5426)
return Int((-I*a*x + S(1))**(I*n/S(2))*(I*a*x + S(1))**(-I*n/S(2)), x)
rule5426 = ReplacementRule(pattern5426, replacement5426)
pattern5427 = Pattern(Integral(x_**WC('m', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons21, cons4, cons1800)
def replacement5427(x, m, a, n):
rubi.append(5427)
return Int(x**m*(-I*a*x + S(1))**(I*n/S(2))*(I*a*x + S(1))**(-I*n/S(2)), x)
rule5427 = ReplacementRule(pattern5427, replacement5427)
pattern5428 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1801, cons1802)
def replacement5428(p, u, d, a, n, c, x):
rubi.append(5428)
return Dist(c**p, Int(u*(S(1) + d*x/c)**p*(-I*a*x + S(1))**(I*n/S(2))*(I*a*x + S(1))**(-I*n/S(2)), x), x)
rule5428 = ReplacementRule(pattern5428, replacement5428)
pattern5429 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1801, cons1803)
def replacement5429(p, u, d, a, n, c, x):
rubi.append(5429)
return Int(u*(c + d*x)**p*(-I*a*x + S(1))**(I*n/S(2))*(I*a*x + S(1))**(-I*n/S(2)), x)
rule5429 = ReplacementRule(pattern5429, replacement5429)
pattern5430 = Pattern(Integral((c_ + WC('d', S(1))/x_)**WC('p', S(1))*WC('u', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons1804, cons38)
def replacement5430(p, u, d, a, n, c, x):
rubi.append(5430)
return Dist(d**p, Int(u*x**(-p)*(c*x/d + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5430 = ReplacementRule(pattern5430, replacement5430)
pattern5431 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*WC('u', S(1))*exp(n_*atanh(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1804, cons147, cons1805, cons177)
def replacement5431(p, u, d, a, n, c, x):
rubi.append(5431)
return Dist((S(-1))**(n/S(2))*c**p, Int(u*(S(1) - I/(a*x))**(-I*n/S(2))*(S(1) + I/(a*x))**(I*n/S(2))*(S(1) + d/(c*x))**p, x), x)
rule5431 = ReplacementRule(pattern5431, replacement5431)
pattern5432 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*WC('u', S(1))*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1804, cons147, cons1805, cons117)
def replacement5432(p, u, d, a, n, c, x):
rubi.append(5432)
return Int(u*(c + d/x)**p*(-I*a*x + S(1))**(I*n/S(2))*(I*a*x + S(1))**(-I*n/S(2)), x)
rule5432 = ReplacementRule(pattern5432, replacement5432)
pattern5433 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*WC('u', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1804, cons147)
def replacement5433(p, u, d, a, n, c, x):
rubi.append(5433)
return Dist(x**p*(c + d/x)**p*(c*x/d + S(1))**(-p), Int(u*x**(-p)*(c*x/d + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5433 = ReplacementRule(pattern5433, replacement5433)
pattern5434 = Pattern(Integral(exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1)))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons4, cons1806, cons1807)
def replacement5434(d, a, n, c, x):
rubi.append(5434)
return Simp((a*x + n)*exp(n*ArcTan(a*x))/(a*c*sqrt(c + d*x**S(2))*(n**S(2) + S(1))), x)
rule5434 = ReplacementRule(pattern5434, replacement5434)
pattern5435 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons1806, cons13, cons137, cons1807, cons1808, cons246)
def replacement5435(p, d, a, n, c, x):
rubi.append(5435)
return Dist(S(2)*(p + S(1))*(S(2)*p + S(3))/(c*(n**S(2) + S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*ArcTan(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*(-S(2)*a*x*(p + S(1)) + n)*exp(n*ArcTan(a*x))/(a*c*(n**S(2) + S(4)*(p + S(1))**S(2))), x)
rule5435 = ReplacementRule(pattern5435, replacement5435)
pattern5436 = Pattern(Integral(exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1)))/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons7, cons27, cons4, cons1806)
def replacement5436(d, a, n, c, x):
rubi.append(5436)
return Simp(exp(n*ArcTan(a*x))/(a*c*n), x)
rule5436 = ReplacementRule(pattern5436, replacement5436)
pattern5437 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1806, cons38, cons1809, cons1810)
def replacement5437(p, d, a, n, c, x):
rubi.append(5437)
return Dist(c**p, Int((a**S(2)*x**S(2) + S(1))**(-I*n/S(2) + p)*(-I*a*x + S(1))**(I*n), x), x)
rule5437 = ReplacementRule(pattern5437, replacement5437)
pattern5438 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1806, cons1802)
def replacement5438(p, d, a, n, c, x):
rubi.append(5438)
return Dist(c**p, Int((-I*a*x + S(1))**(I*n/S(2) + p)*(I*a*x + S(1))**(-I*n/S(2) + p), x), x)
rule5438 = ReplacementRule(pattern5438, replacement5438)
pattern5439 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1806, cons1803, cons1811)
def replacement5439(p, d, a, n, c, x):
rubi.append(5439)
return Dist(c**(I*n/S(2)), Int((c + d*x**S(2))**(-I*n/S(2) + p)*(-I*a*x + S(1))**(I*n), x), x)
rule5439 = ReplacementRule(pattern5439, replacement5439)
pattern5440 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1806, cons1803, cons1812)
def replacement5440(p, d, a, n, c, x):
rubi.append(5440)
return Dist(c**(-I*n/S(2)), Int((c + d*x**S(2))**(I*n/S(2) + p)*(I*a*x + S(1))**(-I*n), x), x)
rule5440 = ReplacementRule(pattern5440, replacement5440)
pattern5441 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1806, cons1803)
def replacement5441(p, d, a, n, c, x):
rubi.append(5441)
return Dist(c**IntPart(p)*(c + d*x**S(2))**FracPart(p)*(a**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int((a**S(2)*x**S(2) + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5441 = ReplacementRule(pattern5441, replacement5441)
pattern5442 = Pattern(Integral(x_*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1)))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons4, cons1806, cons1807)
def replacement5442(d, a, n, c, x):
rubi.append(5442)
return -Simp((-a*n*x + S(1))*exp(n*ArcTan(a*x))/(d*sqrt(c + d*x**S(2))*(n**S(2) + S(1))), x)
rule5442 = ReplacementRule(pattern5442, replacement5442)
pattern5443 = Pattern(Integral(x_*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons1806, cons13, cons137, cons1807, cons246)
def replacement5443(p, d, a, n, c, x):
rubi.append(5443)
return -Dist(a*c*n/(S(2)*d*(p + S(1))), Int((c + d*x**S(2))**p*exp(n*ArcTan(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*exp(n*ArcTan(a*x))/(S(2)*d*(p + S(1))), x)
rule5443 = ReplacementRule(pattern5443, replacement5443)
pattern5444 = Pattern(Integral(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons1806, cons1813, cons1807)
def replacement5444(p, d, a, n, c, x):
rubi.append(5444)
return -Simp((c + d*x**S(2))**(p + S(1))*(-a*n*x + S(1))*exp(n*ArcTan(a*x))/(a*d*n*(n**S(2) + S(1))), x)
rule5444 = ReplacementRule(pattern5444, replacement5444)
pattern5445 = Pattern(Integral(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons1806, cons13, cons137, cons1807, cons1808, cons246)
def replacement5445(p, d, a, n, c, x):
rubi.append(5445)
return Dist((n**S(2) - S(2)*p + S(-2))/(d*(n**S(2) + S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*ArcTan(a*x)), x), x) - Simp((c + d*x**S(2))**(p + S(1))*(-S(2)*a*x*(p + S(1)) + n)*exp(n*ArcTan(a*x))/(a*d*(n**S(2) + S(4)*(p + S(1))**S(2))), x)
rule5445 = ReplacementRule(pattern5445, replacement5445)
pattern5446 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons5, cons1806, cons1802, cons1809, cons1810)
def replacement5446(p, m, d, a, n, c, x):
rubi.append(5446)
return Dist(c**p, Int(x**m*(a**S(2)*x**S(2) + S(1))**(-I*n/S(2) + p)*(-I*a*x + S(1))**(I*n), x), x)
rule5446 = ReplacementRule(pattern5446, replacement5446)
pattern5447 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1806, cons1802)
def replacement5447(p, m, d, a, n, c, x):
rubi.append(5447)
return Dist(c**p, Int(x**m*(-I*a*x + S(1))**(I*n/S(2) + p)*(I*a*x + S(1))**(-I*n/S(2) + p), x), x)
rule5447 = ReplacementRule(pattern5447, replacement5447)
pattern5448 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons5, cons1806, cons1803, cons1811)
def replacement5448(p, m, d, a, n, c, x):
rubi.append(5448)
return Dist(c**(I*n/S(2)), Int(x**m*(c + d*x**S(2))**(-I*n/S(2) + p)*(-I*a*x + S(1))**(I*n), x), x)
rule5448 = ReplacementRule(pattern5448, replacement5448)
pattern5449 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons5, cons1806, cons1803, cons1812)
def replacement5449(p, m, d, a, n, c, x):
rubi.append(5449)
return Dist(c**(-I*n/S(2)), Int(x**m*(c + d*x**S(2))**(I*n/S(2) + p)*(I*a*x + S(1))**(-I*n), x), x)
rule5449 = ReplacementRule(pattern5449, replacement5449)
pattern5450 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1806, cons1803)
def replacement5450(p, m, d, a, n, c, x):
rubi.append(5450)
return Dist(c**IntPart(p)*(c + d*x**S(2))**FracPart(p)*(a**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int(x**m*(a**S(2)*x**S(2) + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5450 = ReplacementRule(pattern5450, replacement5450)
pattern5451 = Pattern(Integral(u_*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1806, cons1802)
def replacement5451(p, u, d, a, n, c, x):
rubi.append(5451)
return Dist(c**p, Int(u*(-I*a*x + S(1))**(I*n/S(2) + p)*(I*a*x + S(1))**(-I*n/S(2) + p), x), x)
rule5451 = ReplacementRule(pattern5451, replacement5451)
pattern5452 = Pattern(Integral(u_*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1806, cons1802, cons1805)
def replacement5452(p, u, d, a, n, c, x):
rubi.append(5452)
return Dist(c**IntPart(p)*(c + d*x**S(2))**FracPart(p)*(-I*a*x + S(1))**(-FracPart(p))*(I*a*x + S(1))**(-FracPart(p)), Int(u*(-I*a*x + S(1))**(I*n/S(2) + p)*(I*a*x + S(1))**(-I*n/S(2) + p), x), x)
rule5452 = ReplacementRule(pattern5452, replacement5452)
pattern5453 = Pattern(Integral(u_*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1806, cons1803, cons1814)
def replacement5453(p, u, d, a, n, c, x):
rubi.append(5453)
return Dist(c**IntPart(p)*(c + d*x**S(2))**FracPart(p)*(a**S(2)*x**S(2) + S(1))**(-FracPart(p)), Int(u*(a**S(2)*x**S(2) + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5453 = ReplacementRule(pattern5453, replacement5453)
pattern5454 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*WC('u', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons1815, cons38)
def replacement5454(p, u, d, a, n, c, x):
rubi.append(5454)
return Dist(d**p, Int(u*x**(-S(2)*p)*(a**S(2)*x**S(2) + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5454 = ReplacementRule(pattern5454, replacement5454)
pattern5455 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**p_*WC('u', S(1))*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons5, cons1815, cons147, cons1805, cons177)
def replacement5455(p, u, d, a, n, c, x):
rubi.append(5455)
return Dist(c**p, Int(u*(S(1) - I/(a*x))**p*(S(1) + I/(a*x))**p*exp(n*ArcTan(a*x)), x), x)
rule5455 = ReplacementRule(pattern5455, replacement5455)
pattern5456 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**p_*WC('u', S(1))*exp(n_*ArcTan(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1815, cons147, cons1805, cons117)
def replacement5456(p, u, d, a, n, c, x):
rubi.append(5456)
return Dist(x**(S(2)*p)*(c + d/x**S(2))**p*(-I*a*x + S(1))**(-p)*(I*a*x + S(1))**(-p), Int(u*x**(-S(2)*p)*(-I*a*x + S(1))**p*(I*a*x + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5456 = ReplacementRule(pattern5456, replacement5456)
pattern5457 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**p_*WC('u', S(1))*exp(ArcTan(x_*WC('a', S(1)))*WC('n', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1815, cons147, cons1814)
def replacement5457(p, u, d, a, n, c, x):
rubi.append(5457)
return Dist(x**(S(2)*p)*(c + d/x**S(2))**p*(a**S(2)*x**S(2) + S(1))**(-p), Int(u*x**(-S(2)*p)*(a**S(2)*x**S(2) + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5457 = ReplacementRule(pattern5457, replacement5457)
pattern5458 = Pattern(Integral(exp(ArcTan((a_ + x_*WC('b', S(1)))*WC('c', S(1)))*WC('n', S(1))), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5458(b, c, n, a, x):
rubi.append(5458)
return Int((-I*a*c - I*b*c*x + S(1))**(I*n/S(2))*(I*a*c + I*b*c*x + S(1))**(-I*n/S(2)), x)
rule5458 = ReplacementRule(pattern5458, replacement5458)
pattern5459 = Pattern(Integral(x_**m_*exp(n_*ArcTan((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons84, cons1816, cons1817)
def replacement5459(m, b, c, n, a, x):
rubi.append(5459)
return Dist(S(4)*I**(-m)*b**(-m + S(-1))*c**(-m + S(-1))/n, Subst(Int(x**(-S(2)*I/n)*(S(1) + x**(-S(2)*I/n))**(-m + S(-2))*(-I*a*c + S(1) - x**(-S(2)*I/n)*(I*a*c + S(1)))**m, x), x, (-I*c*(a + b*x) + S(1))**(I*n/S(2))*(I*c*(a + b*x) + S(1))**(-I*n/S(2))), x)
rule5459 = ReplacementRule(pattern5459, replacement5459)
pattern5460 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*exp(ArcTan((a_ + x_*WC('b', S(1)))*WC('c', S(1)))*WC('n', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1580)
def replacement5460(m, b, d, c, n, a, x, e):
rubi.append(5460)
return Int((d + e*x)**m*(-I*a*c - I*b*c*x + S(1))**(I*n/S(2))*(I*a*c + I*b*c*x + S(1))**(-I*n/S(2)), x)
rule5460 = ReplacementRule(pattern5460, replacement5460)
pattern5461 = Pattern(Integral((c_ + x_**S(2)*WC('e', S(1)) + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(ArcTan(a_ + x_*WC('b', S(1)))*WC('n', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1818, cons1819, cons1820)
def replacement5461(p, u, b, d, a, n, c, x, e):
rubi.append(5461)
return Dist((c/(a**S(2) + S(1)))**p, Int(u*(-I*a - I*b*x + S(1))**(I*n/S(2) + p)*(I*a + I*b*x + S(1))**(-I*n/S(2) + p), x), x)
rule5461 = ReplacementRule(pattern5461, replacement5461)
pattern5462 = Pattern(Integral((c_ + x_**S(2)*WC('e', S(1)) + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(ArcTan(a_ + x_*WC('b', S(1)))*WC('n', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1818, cons1819, cons1821)
def replacement5462(p, u, b, d, a, n, c, x, e):
rubi.append(5462)
return Dist((c + d*x + e*x**S(2))**p*(a**S(2) + S(2)*a*b*x + b**S(2)*x**S(2) + S(1))**(-p), Int(u*(a**S(2) + S(2)*a*b*x + b**S(2)*x**S(2) + S(1))**p*exp(n*ArcTan(a*x)), x), x)
rule5462 = ReplacementRule(pattern5462, replacement5462)
pattern5463 = Pattern(Integral(WC('u', S(1))*exp(ArcTan(WC('c', S(1))/(x_*WC('b', S(1)) + WC('a', S(0))))*WC('n', S(1))), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5463(u, b, c, n, a, x):
rubi.append(5463)
return Int(u*exp(n*acot(a/c + b*x/c)), x)
rule5463 = ReplacementRule(pattern5463, replacement5463)
pattern5464 = Pattern(Integral(WC('u', S(1))*exp(n_*acot(x_*WC('a', S(1)))), x_), cons2, cons1805)
def replacement5464(x, a, n, u):
rubi.append(5464)
return Dist((S(-1))**(I*n/S(2)), Int(u*exp(-n*ArcTan(a*x)), x), x)
rule5464 = ReplacementRule(pattern5464, replacement5464)
pattern5465 = Pattern(Integral(exp(n_*acot(x_*WC('a', S(1)))), x_), cons2, cons1799)
def replacement5465(x, a, n):
rubi.append(5465)
return -Subst(Int((S(1) - I*x/a)**(I*n/S(2) + S(1)/2)*(S(1) + I*x/a)**(-I*n/S(2) + S(1)/2)/(x**S(2)*sqrt(S(1) + x**S(2)/a**S(2))), x), x, S(1)/x)
rule5465 = ReplacementRule(pattern5465, replacement5465)
pattern5466 = Pattern(Integral(x_**WC('m', S(1))*exp(n_*acot(x_*WC('a', S(1)))), x_), cons2, cons1799, cons17)
def replacement5466(x, m, a, n):
rubi.append(5466)
return -Subst(Int(x**(-m + S(-2))*(S(1) - I*x/a)**(I*n/S(2) + S(1)/2)*(S(1) + I*x/a)**(-I*n/S(2) + S(1)/2)/sqrt(S(1) + x**S(2)/a**S(2)), x), x, S(1)/x)
rule5466 = ReplacementRule(pattern5466, replacement5466)
pattern5467 = Pattern(Integral(exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons4, cons1807)
def replacement5467(x, a, n):
rubi.append(5467)
return -Subst(Int((S(1) - I*x/a)**(I*n/S(2))*(S(1) + I*x/a)**(-I*n/S(2))/x**S(2), x), x, S(1)/x)
rule5467 = ReplacementRule(pattern5467, replacement5467)
pattern5468 = Pattern(Integral(x_**WC('m', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons4, cons1807, cons17)
def replacement5468(x, m, a, n):
rubi.append(5468)
return -Subst(Int(x**(-m + S(-2))*(S(1) - I*x/a)**(n/S(2))*(S(1) + I*x/a)**(-n/S(2)), x), x, S(1)/x)
rule5468 = ReplacementRule(pattern5468, replacement5468)
pattern5469 = Pattern(Integral(x_**m_*exp(n_*acot(x_*WC('a', S(1)))), x_), cons2, cons21, cons1799, cons18)
def replacement5469(x, m, a, n):
rubi.append(5469)
return -Dist(x**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(S(1) - I*x/a)**(I*n/S(2) + S(1)/2)*(S(1) + I*x/a)**(-I*n/S(2) + S(1)/2)/sqrt(S(1) + x**S(2)/a**S(2)), x), x, S(1)/x), x)
rule5469 = ReplacementRule(pattern5469, replacement5469)
pattern5470 = Pattern(Integral(x_**m_*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons21, cons4, cons1814, cons18)
def replacement5470(x, m, a, n):
rubi.append(5470)
return -Subst(Int(x**(-m + S(-2))*(S(1) - I*x/a)**(n/S(2))*(S(1) + I*x/a)**(-n/S(2)), x), x, S(1)/x)
rule5470 = ReplacementRule(pattern5470, replacement5470)
pattern5471 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1801, cons1814, cons38)
def replacement5471(p, u, d, a, n, c, x):
rubi.append(5471)
return Dist(d**p, Int(u*x**p*(c/(d*x) + S(1))**p*exp(n*acot(a*x)), x), x)
rule5471 = ReplacementRule(pattern5471, replacement5471)
pattern5472 = Pattern(Integral((c_ + x_*WC('d', S(1)))**p_*WC('u', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1801, cons1814, cons147)
def replacement5472(p, u, d, a, n, c, x):
rubi.append(5472)
return Dist(x**(-p)*(c + d*x)**p*(c/(d*x) + S(1))**(-p), Int(u*x**p*(c/(d*x) + S(1))**p*exp(n*acot(a*x)), x), x)
rule5472 = ReplacementRule(pattern5472, replacement5472)
pattern5473 = Pattern(Integral((c_ + WC('d', S(1))/x_)**WC('p', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1804, cons1814, cons1802)
def replacement5473(p, d, a, n, c, x):
rubi.append(5473)
return -Dist(c**p, Subst(Int((S(1) - I*x/a)**(I*n/S(2))*(S(1) + I*x/a)**(-I*n/S(2))*(S(1) + d*x/c)**p/x**S(2), x), x, S(1)/x), x)
rule5473 = ReplacementRule(pattern5473, replacement5473)
pattern5474 = Pattern(Integral(x_**WC('m', S(1))*(c_ + WC('d', S(1))/x_)**WC('p', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1804, cons1814, cons1802, cons17)
def replacement5474(p, m, d, a, n, c, x):
rubi.append(5474)
return -Dist(c**p, Subst(Int(x**(-m + S(-2))*(S(1) - I*x/a)**(I*n/S(2))*(S(1) + I*x/a)**(-I*n/S(2))*(S(1) + d*x/c)**p, x), x, S(1)/x), x)
rule5474 = ReplacementRule(pattern5474, replacement5474)
pattern5475 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1804, cons1814, cons1803)
def replacement5475(p, d, a, n, c, x):
rubi.append(5475)
return Dist((S(1) + d/(c*x))**(-p)*(c + d/x)**p, Int((S(1) + d/(c*x))**p*exp(n*acot(a*x)), x), x)
rule5475 = ReplacementRule(pattern5475, replacement5475)
pattern5476 = Pattern(Integral(x_**m_*(c_ + WC('d', S(1))/x_)**WC('p', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1804, cons1814, cons1802, cons18)
def replacement5476(p, m, d, a, n, c, x):
rubi.append(5476)
return -Dist(c**p*x**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(S(1) - I*x/a)**(I*n/S(2))*(S(1) + I*x/a)**(-I*n/S(2))*(S(1) + d*x/c)**p, x), x, S(1)/x), x)
rule5476 = ReplacementRule(pattern5476, replacement5476)
pattern5477 = Pattern(Integral((c_ + WC('d', S(1))/x_)**p_*WC('u', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1804, cons1814, cons1803)
def replacement5477(p, u, d, a, n, c, x):
rubi.append(5477)
return Dist((S(1) + d/(c*x))**(-p)*(c + d/x)**p, Int(u*(S(1) + d/(c*x))**p*exp(n*acot(a*x)), x), x)
rule5477 = ReplacementRule(pattern5477, replacement5477)
pattern5478 = Pattern(Integral(exp(WC('n', S(1))*acot(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1))), x_), cons2, cons7, cons27, cons4, cons1806)
def replacement5478(d, a, n, c, x):
rubi.append(5478)
return -Simp(exp(n*acot(a*x))/(a*c*n), x)
rule5478 = ReplacementRule(pattern5478, replacement5478)
pattern5479 = Pattern(Integral(exp(WC('n', S(1))*acot(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons4, cons1806, cons1800)
def replacement5479(d, a, n, c, x):
rubi.append(5479)
return -Simp((-a*x + n)*exp(n*acot(a*x))/(a*c*sqrt(c + d*x**S(2))*(n**S(2) + S(1))), x)
rule5479 = ReplacementRule(pattern5479, replacement5479)
pattern5480 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1806, cons13, cons137, cons230, cons1808, cons1822, cons1823)
def replacement5480(p, d, a, n, c, x):
rubi.append(5480)
return Dist(S(2)*(p + S(1))*(S(2)*p + S(3))/(c*(n**S(2) + S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*acot(a*x)), x), x) - Simp((c + d*x**S(2))**(p + S(1))*(S(2)*a*x*(p + S(1)) + n)*exp(n*acot(a*x))/(a*c*(n**S(2) + S(4)*(p + S(1))**S(2))), x)
rule5480 = ReplacementRule(pattern5480, replacement5480)
pattern5481 = Pattern(Integral(x_*exp(WC('n', S(1))*acot(x_*WC('a', S(1))))/(c_ + x_**S(2)*WC('d', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons4, cons1806, cons1800)
def replacement5481(d, a, n, c, x):
rubi.append(5481)
return -Simp((a*n*x + S(1))*exp(n*acot(a*x))/(a**S(2)*c*sqrt(c + d*x**S(2))*(n**S(2) + S(1))), x)
rule5481 = ReplacementRule(pattern5481, replacement5481)
pattern5482 = Pattern(Integral(x_*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1806, cons13, cons1824, cons230, cons1808, cons1822, cons1823)
def replacement5482(p, d, a, n, c, x):
rubi.append(5482)
return Dist(n*(S(2)*p + S(3))/(a*c*(n**S(2) + S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*acot(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*(-a*n*x + S(2)*p + S(2))*exp(n*acot(a*x))/(a**S(2)*c*(n**S(2) + S(4)*(p + S(1))**S(2))), x)
rule5482 = ReplacementRule(pattern5482, replacement5482)
pattern5483 = Pattern(Integral(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1806, cons1813, cons1825)
def replacement5483(p, d, a, n, c, x):
rubi.append(5483)
return Simp((c + d*x**S(2))**(p + S(1))*(S(2)*a*x*(p + S(1)) + n)*exp(n*acot(a*x))/(a**S(3)*c*n**S(2)*(n**S(2) + S(1))), x)
rule5483 = ReplacementRule(pattern5483, replacement5483)
pattern5484 = Pattern(Integral(x_**S(2)*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1806, cons13, cons1824, cons1826, cons1808, cons1822, cons1823)
def replacement5484(p, d, a, n, c, x):
rubi.append(5484)
return Dist((n**S(2) - S(2)*p + S(-2))/(a**S(2)*c*(n**S(2) + S(4)*(p + S(1))**S(2))), Int((c + d*x**S(2))**(p + S(1))*exp(n*acot(a*x)), x), x) + Simp((c + d*x**S(2))**(p + S(1))*(S(2)*a*x*(p + S(1)) + n)*exp(n*acot(a*x))/(a**S(3)*c*(n**S(2) + S(4)*(p + S(1))**S(2))), x)
rule5484 = ReplacementRule(pattern5484, replacement5484)
pattern5485 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**S(2)*WC('d', S(1)))**p_*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1806, cons17, cons1827, cons38)
def replacement5485(p, m, d, a, n, c, x):
rubi.append(5485)
return -Dist(a**(-m + S(-1))*c**p, Subst(Int((S(1)/tan(x))**(m + S(2)*p + S(2))*exp(n*x)*cos(x)**(-S(2)*p + S(-2)), x), x, acot(a*x)), x)
rule5485 = ReplacementRule(pattern5485, replacement5485)
pattern5486 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons1806, cons1814, cons38)
def replacement5486(p, u, d, a, n, c, x):
rubi.append(5486)
return Dist(d**p, Int(u*x**(S(2)*p)*(S(1) + S(1)/(a**S(2)*x**S(2)))**p*exp(n*acot(a*x)), x), x)
rule5486 = ReplacementRule(pattern5486, replacement5486)
pattern5487 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))**p_*WC('u', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1806, cons1814, cons147)
def replacement5487(p, u, d, a, n, c, x):
rubi.append(5487)
return Dist(x**(-S(2)*p)*(S(1) + S(1)/(a**S(2)*x**S(2)))**(-p)*(c + d*x**S(2))**p, Int(u*x**(S(2)*p)*(S(1) + S(1)/(a**S(2)*x**S(2)))**p*exp(n*acot(a*x)), x), x)
rule5487 = ReplacementRule(pattern5487, replacement5487)
pattern5488 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1815, cons1814, cons1802, cons1828)
def replacement5488(p, u, d, a, n, c, x):
rubi.append(5488)
return Dist(c**p*(I*a)**(-S(2)*p), Int(u*x**(-S(2)*p)*(I*a*x + S(-1))**(-I*n/S(2) + p)*(I*a*x + S(1))**(I*n/S(2) + p), x), x)
rule5488 = ReplacementRule(pattern5488, replacement5488)
pattern5489 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1815, cons1814, cons1802, cons1829)
def replacement5489(p, d, a, n, c, x):
rubi.append(5489)
return -Dist(c**p, Subst(Int((S(1) - I*x/a)**(I*n/S(2) + p)*(S(1) + I*x/a)**(-I*n/S(2) + p)/x**S(2), x), x, S(1)/x), x)
rule5489 = ReplacementRule(pattern5489, replacement5489)
pattern5490 = Pattern(Integral(x_**WC('m', S(1))*(c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1815, cons1814, cons1802, cons1829, cons17)
def replacement5490(p, m, d, a, n, c, x):
rubi.append(5490)
return -Dist(c**p, Subst(Int(x**(-m + S(-2))*(S(1) - I*x/a)**(I*n/S(2) + p)*(S(1) + I*x/a)**(-I*n/S(2) + p), x), x, S(1)/x), x)
rule5490 = ReplacementRule(pattern5490, replacement5490)
pattern5491 = Pattern(Integral(x_**m_*(c_ + WC('d', S(1))/x_**S(2))**WC('p', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons1815, cons1814, cons1802, cons1829, cons18)
def replacement5491(p, m, d, a, n, c, x):
rubi.append(5491)
return -Dist(c**p*x**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(S(1) - I*x/a)**(I*n/S(2) + p)*(S(1) + I*x/a)**(-I*n/S(2) + p), x), x, S(1)/x), x)
rule5491 = ReplacementRule(pattern5491, replacement5491)
pattern5492 = Pattern(Integral((c_ + WC('d', S(1))/x_**S(2))**p_*WC('u', S(1))*exp(WC('n', S(1))*acot(x_*WC('a', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1815, cons1814, cons1803)
def replacement5492(p, u, d, a, n, c, x):
rubi.append(5492)
return Dist((S(1) + S(1)/(a**S(2)*x**S(2)))**(-p)*(c + d/x**S(2))**p, Int(u*(S(1) + S(1)/(a**S(2)*x**S(2)))**p*exp(n*acot(a*x)), x), x)
rule5492 = ReplacementRule(pattern5492, replacement5492)
pattern5493 = Pattern(Integral(WC('u', S(1))*exp(n_*acot((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons1805)
def replacement5493(u, b, c, a, n, x):
rubi.append(5493)
return Dist((S(-1))**(I*n/S(2)), Int(u*exp(-n*ArcTan(c*(a + b*x))), x), x)
rule5493 = ReplacementRule(pattern5493, replacement5493)
pattern5494 = Pattern(Integral(exp(WC('n', S(1))*acot((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1814)
def replacement5494(b, c, n, a, x):
rubi.append(5494)
return Dist((I*c*(a + b*x))**(I*n/S(2))*(S(1) - I/(c*(a + b*x)))**(I*n/S(2))*(I*a*c + I*b*c*x + S(1))**(-I*n/S(2)), Int((I*a*c + I*b*c*x + S(-1))**(-I*n/S(2))*(I*a*c + I*b*c*x + S(1))**(I*n/S(2)), x), x)
rule5494 = ReplacementRule(pattern5494, replacement5494)
pattern5495 = Pattern(Integral(x_**m_*exp(n_*acoth((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons84, cons1816, cons1817)
def replacement5495(m, b, c, n, a, x):
rubi.append(5495)
return Dist(S(4)*I**(-m)*b**(-m + S(-1))*c**(-m + S(-1))/n, Subst(Int(x**(-S(2)*I/n)*(S(-1) + x**(-S(2)*I/n))**(-m + S(-2))*(I*a*c + S(1) + x**(-S(2)*I/n)*(-I*a*c + S(1)))**m, x), x, (S(1) - I/(c*(a + b*x)))**(I*n/S(2))*(S(1) + I/(c*(a + b*x)))**(-I*n/S(2))), x)
rule5495 = ReplacementRule(pattern5495, replacement5495)
pattern5496 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*exp(WC('n', S(1))*acoth((a_ + x_*WC('b', S(1)))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1814)
def replacement5496(m, b, d, c, n, a, x, e):
rubi.append(5496)
return Dist((I*c*(a + b*x))**(I*n/S(2))*(S(1) - I/(c*(a + b*x)))**(I*n/S(2))*(I*a*c + I*b*c*x + S(1))**(-I*n/S(2)), Int((d + e*x)**m*(I*a*c + I*b*c*x + S(-1))**(-I*n/S(2))*(I*a*c + I*b*c*x + S(1))**(I*n/S(2)), x), x)
rule5496 = ReplacementRule(pattern5496, replacement5496)
pattern5497 = Pattern(Integral((c_ + x_**S(2)*WC('e', S(1)) + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acot(a_ + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1814, cons1818, cons1819, cons1820)
def replacement5497(p, u, b, d, a, n, c, x, e):
rubi.append(5497)
return Dist((c/(a**S(2) + S(1)))**p*((I*a + I*b*x + S(1))/(I*a + I*b*x))**(I*n/S(2))*((I*a + I*b*x)/(I*a + I*b*x + S(1)))**(I*n/S(2))*(-I*a - I*b*x + S(1))**(I*n/S(2))*(I*a + I*b*x + S(-1))**(-I*n/S(2)), Int(u*(-I*a - I*b*x + S(1))**(-I*n/S(2) + p)*(I*a + I*b*x + S(1))**(I*n/S(2) + p), x), x)
rule5497 = ReplacementRule(pattern5497, replacement5497)
pattern5498 = Pattern(Integral((c_ + x_**S(2)*WC('e', S(1)) + x_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1))*exp(WC('n', S(1))*acot(a_ + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1814, cons1818, cons1819, cons1821)
def replacement5498(p, u, b, d, a, n, c, x, e):
rubi.append(5498)
return Dist((c + d*x + e*x**S(2))**p*(a**S(2) + S(2)*a*b*x + b**S(2)*x**S(2) + S(1))**(-p), Int(u*(a**S(2) + S(2)*a*b*x + b**S(2)*x**S(2) + S(1))**p*exp(n*acot(a*x)), x), x)
rule5498 = ReplacementRule(pattern5498, replacement5498)
pattern5499 = Pattern(Integral(WC('u', S(1))*exp(WC('n', S(1))*acot(WC('c', S(1))/(x_*WC('b', S(1)) + WC('a', S(0))))), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5499(u, b, c, n, a, x):
rubi.append(5499)
return Int(u*exp(n*ArcTan(a/c + b*x/c)), x)
rule5499 = ReplacementRule(pattern5499, replacement5499)
pattern5500 = Pattern(Integral((ArcTan(c_ + x_*WC('d', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons148)
def replacement5500(b, d, c, a, n, x):
rubi.append(5500)
return Dist(S(1)/d, Subst(Int((a + b*ArcTan(x))**n, x), x, c + d*x), x)
rule5500 = ReplacementRule(pattern5500, replacement5500)
pattern5501 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons148)
def replacement5501(b, d, c, a, n, x):
rubi.append(5501)
return Dist(S(1)/d, Subst(Int((a + b*acot(x))**n, x), x, c + d*x), x)
rule5501 = ReplacementRule(pattern5501, replacement5501)
pattern5502 = Pattern(Integral((ArcTan(c_ + x_*WC('d', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons340)
def replacement5502(b, d, c, a, n, x):
rubi.append(5502)
return Int((a + b*ArcTan(c + d*x))**n, x)
rule5502 = ReplacementRule(pattern5502, replacement5502)
pattern5503 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(c_ + x_*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons340)
def replacement5503(b, d, c, a, n, x):
rubi.append(5503)
return Int((a + b*acot(c + d*x))**n, x)
rule5503 = ReplacementRule(pattern5503, replacement5503)
pattern5504 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(ArcTan(c_ + x_*WC('d', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons148)
def replacement5504(m, f, b, d, a, n, c, x, e):
rubi.append(5504)
return Dist(S(1)/d, Subst(Int((a + b*ArcTan(x))**n*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule5504 = ReplacementRule(pattern5504, replacement5504)
pattern5505 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(c_ + x_*WC('d', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons148)
def replacement5505(m, f, b, d, a, n, c, x, e):
rubi.append(5505)
return Dist(S(1)/d, Subst(Int((a + b*acot(x))**n*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule5505 = ReplacementRule(pattern5505, replacement5505)
pattern5506 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**m_*(ArcTan(c_ + x_*WC('d', S(1)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons340)
def replacement5506(m, f, b, d, a, c, n, x, e):
rubi.append(5506)
return Int((a + b*ArcTan(c + d*x))**n*(e + f*x)**m, x)
rule5506 = ReplacementRule(pattern5506, replacement5506)
pattern5507 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**m_*(WC('a', S(0)) + WC('b', S(1))*acot(c_ + x_*WC('d', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons340)
def replacement5507(m, f, b, d, a, c, n, x, e):
rubi.append(5507)
return Int((a + b*acot(c + d*x))**n*(e + f*x)**m, x)
rule5507 = ReplacementRule(pattern5507, replacement5507)
pattern5508 = Pattern(Integral((ArcTan(c_ + x_*WC('d', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons36, cons4, cons5, cons1830, cons1763)
def replacement5508(B, C, p, b, d, a, n, c, x, A):
rubi.append(5508)
return Dist(S(1)/d, Subst(Int((a + b*ArcTan(x))**n*(C*x**S(2)/d**S(2) + C/d**S(2))**p, x), x, c + d*x), x)
rule5508 = ReplacementRule(pattern5508, replacement5508)
pattern5509 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acot(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons36, cons4, cons5, cons1830, cons1763)
def replacement5509(B, C, p, b, d, a, n, c, x, A):
rubi.append(5509)
return Dist(S(1)/d, Subst(Int((a + b*acot(x))**n*(C*x**S(2)/d**S(2) + C/d**S(2))**p, x), x, c + d*x), x)
rule5509 = ReplacementRule(pattern5509, replacement5509)
pattern5510 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(ArcTan(c_ + x_*WC('d', S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons1830, cons1763)
def replacement5510(B, C, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(5510)
return Dist(S(1)/d, Subst(Int((a + b*ArcTan(x))**n*(C*x**S(2)/d**S(2) + C/d**S(2))**p*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule5510 = ReplacementRule(pattern5510, replacement5510)
pattern5511 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(c_ + x_*WC('d', S(1))))**WC('n', S(1))*(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons1830, cons1763)
def replacement5511(B, C, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(5511)
return Dist(S(1)/d, Subst(Int((a + b*acot(x))**n*(C*x**S(2)/d**S(2) + C/d**S(2))**p*(f*x/d + (-c*f + d*e)/d)**m, x), x, c + d*x), x)
rule5511 = ReplacementRule(pattern5511, replacement5511)
pattern5512 = Pattern(Integral(ArcTan(a_ + x_*WC('b', S(1)))/(c_ + x_**WC('n', S(1))*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons87)
def replacement5512(b, d, a, n, c, x):
rubi.append(5512)
return Dist(I/S(2), Int(log(-I*a - I*b*x + S(1))/(c + d*x**n), x), x) - Dist(I/S(2), Int(log(I*a + I*b*x + S(1))/(c + d*x**n), x), x)
rule5512 = ReplacementRule(pattern5512, replacement5512)
pattern5513 = Pattern(Integral(acot(a_ + x_*WC('b', S(1)))/(c_ + x_**WC('n', S(1))*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons87)
def replacement5513(b, d, a, n, c, x):
rubi.append(5513)
return Dist(I/S(2), Int(log((a + b*x - I)/(a + b*x))/(c + d*x**n), x), x) - Dist(I/S(2), Int(log((a + b*x + I)/(a + b*x))/(c + d*x**n), x), x)
rule5513 = ReplacementRule(pattern5513, replacement5513)
pattern5514 = Pattern(Integral(ArcTan(a_ + x_*WC('b', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons1094)
def replacement5514(b, d, c, a, n, x):
rubi.append(5514)
return Int(ArcTan(a + b*x)/(c + d*x**n), x)
rule5514 = ReplacementRule(pattern5514, replacement5514)
pattern5515 = Pattern(Integral(acot(a_ + x_*WC('b', S(1)))/(c_ + x_**n_*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons1094)
def replacement5515(b, d, c, a, n, x):
rubi.append(5515)
return Int(acot(a + b*x)/(c + d*x**n), x)
rule5515 = ReplacementRule(pattern5515, replacement5515)
pattern5516 = Pattern(Integral(ArcTan(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons4, cons1831)
def replacement5516(x, a, n, b):
rubi.append(5516)
return -Dist(b*n, Int(x**n/(a**S(2) + S(2)*a*b*x**n + b**S(2)*x**(S(2)*n) + S(1)), x), x) + Simp(x*ArcTan(a + b*x**n), x)
rule5516 = ReplacementRule(pattern5516, replacement5516)
pattern5517 = Pattern(Integral(acot(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons4, cons1831)
def replacement5517(x, a, n, b):
rubi.append(5517)
return Dist(b*n, Int(x**n/(a**S(2) + S(2)*a*b*x**n + b**S(2)*x**(S(2)*n) + S(1)), x), x) + Simp(x*acot(a + b*x**n), x)
rule5517 = ReplacementRule(pattern5517, replacement5517)
pattern5518 = Pattern(Integral(ArcTan(x_**n_*WC('b', S(1)) + WC('a', S(0)))/x_, x_), cons2, cons3, cons4, cons1831)
def replacement5518(x, a, n, b):
rubi.append(5518)
return Dist(I/S(2), Int(log(-I*a - I*b*x**n + S(1))/x, x), x) - Dist(I/S(2), Int(log(I*a + I*b*x**n + S(1))/x, x), x)
rule5518 = ReplacementRule(pattern5518, replacement5518)
pattern5519 = Pattern(Integral(acot(x_**n_*WC('b', S(1)) + WC('a', S(0)))/x_, x_), cons2, cons3, cons4, cons1831)
def replacement5519(x, a, n, b):
rubi.append(5519)
return Dist(I/S(2), Int(log(S(1) - I/(a + b*x**n))/x, x), x) - Dist(I/S(2), Int(log(S(1) + I/(a + b*x**n))/x, x), x)
rule5519 = ReplacementRule(pattern5519, replacement5519)
pattern5520 = Pattern(Integral(x_**WC('m', S(1))*ArcTan(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons93, cons1832, cons1833)
def replacement5520(m, b, a, n, x):
rubi.append(5520)
return -Dist(b*n/(m + S(1)), Int(x**(m + n)/(a**S(2) + S(2)*a*b*x**n + b**S(2)*x**(S(2)*n) + S(1)), x), x) + Simp(x**(m + S(1))*ArcTan(a + b*x**n)/(m + S(1)), x)
rule5520 = ReplacementRule(pattern5520, replacement5520)
pattern5521 = Pattern(Integral(x_**WC('m', S(1))*acot(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons93, cons1832, cons1833)
def replacement5521(m, b, a, n, x):
rubi.append(5521)
return Dist(b*n/(m + S(1)), Int(x**(m + n)/(a**S(2) + S(2)*a*b*x**n + b**S(2)*x**(S(2)*n) + S(1)), x), x) + Simp(x**(m + S(1))*acot(a + b*x**n)/(m + S(1)), x)
rule5521 = ReplacementRule(pattern5521, replacement5521)
pattern5522 = Pattern(Integral(ArcTan(f_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons1834)
def replacement5522(f, b, d, c, a, x):
rubi.append(5522)
return Dist(I/S(2), Int(log(-I*a - I*b*f**(c + d*x) + S(1)), x), x) - Dist(I/S(2), Int(log(I*a + I*b*f**(c + d*x) + S(1)), x), x)
rule5522 = ReplacementRule(pattern5522, replacement5522)
pattern5523 = Pattern(Integral(acot(f_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons1834)
def replacement5523(f, b, d, c, a, x):
rubi.append(5523)
return Dist(I/S(2), Int(log(S(1) - I/(a + b*f**(c + d*x))), x), x) - Dist(I/S(2), Int(log(S(1) + I/(a + b*f**(c + d*x))), x), x)
rule5523 = ReplacementRule(pattern5523, replacement5523)
pattern5524 = Pattern(Integral(x_**WC('m', S(1))*ArcTan(f_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons17, cons168)
def replacement5524(m, f, b, d, c, a, x):
rubi.append(5524)
return Dist(I/S(2), Int(x**m*log(-I*a - I*b*f**(c + d*x) + S(1)), x), x) - Dist(I/S(2), Int(x**m*log(I*a + I*b*f**(c + d*x) + S(1)), x), x)
rule5524 = ReplacementRule(pattern5524, replacement5524)
pattern5525 = Pattern(Integral(x_**WC('m', S(1))*acot(f_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons125, cons17, cons168)
def replacement5525(m, f, b, d, c, a, x):
rubi.append(5525)
return Dist(I/S(2), Int(x**m*log(S(1) - I/(a + b*f**(c + d*x))), x), x) - Dist(I/S(2), Int(x**m*log(S(1) + I/(a + b*f**(c + d*x))), x), x)
rule5525 = ReplacementRule(pattern5525, replacement5525)
pattern5526 = Pattern(Integral(ArcTan(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement5526(u, m, b, c, n, a, x):
rubi.append(5526)
return Int(u*acot(a/c + b*x**n/c)**m, x)
rule5526 = ReplacementRule(pattern5526, replacement5526)
pattern5527 = Pattern(Integral(WC('u', S(1))*acot(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement5527(u, m, b, c, n, a, x):
rubi.append(5527)
return Int(u*ArcTan(a/c + b*x**n/c)**m, x)
rule5527 = ReplacementRule(pattern5527, replacement5527)
pattern5528 = Pattern(Integral(S(1)/(sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0)))*ArcTan(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))), x_), cons2, cons3, cons7, cons1835)
def replacement5528(x, c, b, a):
rubi.append(5528)
return Simp(log(ArcTan(c*x/sqrt(a + b*x**S(2))))/c, x)
rule5528 = ReplacementRule(pattern5528, replacement5528)
pattern5529 = Pattern(Integral(S(1)/(sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0)))*acot(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))), x_), cons2, cons3, cons7, cons1835)
def replacement5529(x, c, b, a):
rubi.append(5529)
return -Simp(log(acot(c*x/sqrt(a + b*x**S(2))))/c, x)
rule5529 = ReplacementRule(pattern5529, replacement5529)
pattern5530 = Pattern(Integral(ArcTan(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons21, cons1835, cons66)
def replacement5530(m, b, c, a, x):
rubi.append(5530)
return Simp(ArcTan(c*x/sqrt(a + b*x**S(2)))**(m + S(1))/(c*(m + S(1))), x)
rule5530 = ReplacementRule(pattern5530, replacement5530)
pattern5531 = Pattern(Integral(acot(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons21, cons1835, cons66)
def replacement5531(m, b, c, a, x):
rubi.append(5531)
return -Simp(acot(c*x/sqrt(a + b*x**S(2)))**(m + S(1))/(c*(m + S(1))), x)
rule5531 = ReplacementRule(pattern5531, replacement5531)
pattern5532 = Pattern(Integral(ArcTan(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))/sqrt(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1835, cons383)
def replacement5532(m, b, d, c, a, x, e):
rubi.append(5532)
return Dist(sqrt(a + b*x**S(2))/sqrt(d + e*x**S(2)), Int(ArcTan(c*x/sqrt(a + b*x**S(2)))**m/sqrt(a + b*x**S(2)), x), x)
rule5532 = ReplacementRule(pattern5532, replacement5532)
pattern5533 = Pattern(Integral(acot(x_*WC('c', S(1))/sqrt(x_**S(2)*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))/sqrt(x_**S(2)*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1835, cons383)
def replacement5533(m, b, d, c, a, x, e):
rubi.append(5533)
return Dist(sqrt(a + b*x**S(2))/sqrt(d + e*x**S(2)), Int(acot(c*x/sqrt(a + b*x**S(2)))**m/sqrt(a + b*x**S(2)), x), x)
rule5533 = ReplacementRule(pattern5533, replacement5533)
pattern5534 = Pattern(Integral(ArcTan(v_ + sqrt(w_)*WC('s', S(1)))*WC('u', S(1)), x_), cons1836, cons1837)
def replacement5534(v, w, u, x, s):
rubi.append(5534)
return Dist(S(1)/2, Int(u*ArcTan(v), x), x) + Dist(Pi*s/S(4), Int(u, x), x)
rule5534 = ReplacementRule(pattern5534, replacement5534)
pattern5535 = Pattern(Integral(WC('u', S(1))*acot(v_ + sqrt(w_)*WC('s', S(1))), x_), cons1836, cons1837)
def replacement5535(v, w, u, x, s):
rubi.append(5535)
return -Dist(S(1)/2, Int(u*ArcTan(v), x), x) + Dist(Pi*s/S(4), Int(u, x), x)
rule5535 = ReplacementRule(pattern5535, replacement5535)
def With5536(v, x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
tmp = InverseFunctionOfLinear(u, x)
res = And(Not(FalseQ(tmp)), SameQ(Head(tmp), ArcTan), ZeroQ(D(v, x)**S(2) + Discriminant(v, x)*Part(tmp, S(1))**S(2)))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern5536 = Pattern(Integral(u_*v_**WC('n', S(1)), x_), cons818, cons85, cons463, cons1838, cons1839, CustomConstraint(With5536))
def replacement5536(v, x, n, u):
tmp = InverseFunctionOfLinear(u, x)
rubi.append(5536)
return Dist((-Discriminant(v, x)/(S(4)*Coefficient(v, x, S(2))))**n/Coefficient(Part(tmp, S(1)), x, S(1)), Subst(Int(SimplifyIntegrand((S(1)/cos(x))**(S(2)*n + S(2))*SubstForInverseFunction(u, tmp, x), x), x), x, tmp), x)
rule5536 = ReplacementRule(pattern5536, replacement5536)
def With5537(v, x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
tmp = InverseFunctionOfLinear(u, x)
res = And(Not(FalseQ(tmp)), SameQ(Head(tmp), ArcCot), ZeroQ(D(v, x)**S(2) + Discriminant(v, x)*Part(tmp, S(1))**S(2)))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern5537 = Pattern(Integral(u_*v_**WC('n', S(1)), x_), cons818, cons85, cons463, cons1838, cons1840, CustomConstraint(With5537))
def replacement5537(v, x, n, u):
tmp = InverseFunctionOfLinear(u, x)
rubi.append(5537)
return -Dist((-Discriminant(v, x)/(S(4)*Coefficient(v, x, S(2))))**n/Coefficient(Part(tmp, S(1)), x, S(1)), Subst(Int(SimplifyIntegrand((S(1)/sin(x))**(S(2)*n + S(2))*SubstForInverseFunction(u, tmp, x), x), x), x, tmp), x)
rule5537 = ReplacementRule(pattern5537, replacement5537)
pattern5538 = Pattern(Integral(ArcTan(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1841)
def replacement5538(b, d, c, a, x):
rubi.append(5538)
return -Dist(I*b, Int(x/(c*exp(S(2)*I*a + S(2)*I*b*x) + c + I*d), x), x) + Simp(x*ArcTan(c + d*tan(a + b*x)), x)
rule5538 = ReplacementRule(pattern5538, replacement5538)
pattern5539 = Pattern(Integral(acot(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1841)
def replacement5539(b, d, c, a, x):
rubi.append(5539)
return Dist(I*b, Int(x/(c*exp(S(2)*I*a + S(2)*I*b*x) + c + I*d), x), x) + Simp(x*acot(c + d*tan(a + b*x)), x)
rule5539 = ReplacementRule(pattern5539, replacement5539)
pattern5540 = Pattern(Integral(ArcTan(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1842)
def replacement5540(b, d, c, a, x):
rubi.append(5540)
return -Dist(I*b, Int(x/(-c*exp(S(2)*I*a + S(2)*I*b*x) + c - I*d), x), x) + Simp(x*ArcTan(c + d/tan(a + b*x)), x)
rule5540 = ReplacementRule(pattern5540, replacement5540)
pattern5541 = Pattern(Integral(acot(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1842)
def replacement5541(b, d, c, a, x):
rubi.append(5541)
return Dist(I*b, Int(x/(-c*exp(S(2)*I*a + S(2)*I*b*x) + c - I*d), x), x) + Simp(x*acot(c + d/tan(a + b*x)), x)
rule5541 = ReplacementRule(pattern5541, replacement5541)
pattern5542 = Pattern(Integral(ArcTan(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1843)
def replacement5542(b, d, c, a, x):
rubi.append(5542)
return Dist(b*(-I*c - d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(-I*c + d + (-I*c - d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) - Dist(b*(I*c + d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(I*c - d + (I*c + d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp(x*ArcTan(c + d*tan(a + b*x)), x)
rule5542 = ReplacementRule(pattern5542, replacement5542)
pattern5543 = Pattern(Integral(acot(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1843)
def replacement5543(b, d, c, a, x):
rubi.append(5543)
return -Dist(b*(-I*c - d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(-I*c + d + (-I*c - d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Dist(b*(I*c + d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(I*c - d + (I*c + d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp(x*acot(c + d*tan(a + b*x)), x)
rule5543 = ReplacementRule(pattern5543, replacement5543)
pattern5544 = Pattern(Integral(ArcTan(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1843)
def replacement5544(b, d, c, a, x):
rubi.append(5544)
return -Dist(b*(-I*c + d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(-I*c - d - (-I*c + d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Dist(b*(I*c - d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(I*c + d - (I*c - d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp(x*ArcTan(c + d/tan(a + b*x)), x)
rule5544 = ReplacementRule(pattern5544, replacement5544)
pattern5545 = Pattern(Integral(acot(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1844)
def replacement5545(b, d, c, a, x):
rubi.append(5545)
return Dist(b*(-I*c + d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(-I*c - d - (-I*c + d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) - Dist(b*(I*c - d + S(1)), Int(x*exp(S(2)*I*a + S(2)*I*b*x)/(I*c + d - (I*c - d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp(x*acot(c + d/tan(a + b*x)), x)
rule5545 = ReplacementRule(pattern5545, replacement5545)
pattern5546 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1841)
def replacement5546(m, f, b, d, c, a, x, e):
rubi.append(5546)
return -Dist(I*b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(c*exp(S(2)*I*a + S(2)*I*b*x) + c + I*d), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(c + d*tan(a + b*x))/(f*(m + S(1))), x)
rule5546 = ReplacementRule(pattern5546, replacement5546)
pattern5547 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1841)
def replacement5547(m, f, b, d, c, a, x, e):
rubi.append(5547)
return Dist(I*b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(c*exp(S(2)*I*a + S(2)*I*b*x) + c + I*d), x), x) + Simp((e + f*x)**(m + S(1))*acot(c + d*tan(a + b*x))/(f*(m + S(1))), x)
rule5547 = ReplacementRule(pattern5547, replacement5547)
pattern5548 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1842)
def replacement5548(m, f, b, d, c, a, x, e):
rubi.append(5548)
return -Dist(I*b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(-c*exp(S(2)*I*a + S(2)*I*b*x) + c - I*d), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(c + d/tan(a + b*x))/(f*(m + S(1))), x)
rule5548 = ReplacementRule(pattern5548, replacement5548)
pattern5549 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1842)
def replacement5549(m, f, b, d, c, a, x, e):
rubi.append(5549)
return Dist(I*b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(-c*exp(S(2)*I*a + S(2)*I*b*x) + c - I*d), x), x) + Simp((e + f*x)**(m + S(1))*acot(c + d/tan(a + b*x))/(f*(m + S(1))), x)
rule5549 = ReplacementRule(pattern5549, replacement5549)
pattern5550 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1843)
def replacement5550(m, f, b, d, c, a, x, e):
rubi.append(5550)
return Dist(b*(-I*c - d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(-I*c + d + (-I*c - d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) - Dist(b*(I*c + d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(I*c - d + (I*c + d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(c + d*tan(a + b*x))/(f*(m + S(1))), x)
rule5550 = ReplacementRule(pattern5550, replacement5550)
pattern5551 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1843)
def replacement5551(m, f, b, d, c, a, x, e):
rubi.append(5551)
return -Dist(b*(-I*c - d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(-I*c + d + (-I*c - d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Dist(b*(I*c + d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(I*c - d + (I*c + d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*acot(c + d*tan(a + b*x))/(f*(m + S(1))), x)
rule5551 = ReplacementRule(pattern5551, replacement5551)
pattern5552 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1844)
def replacement5552(m, f, b, d, c, a, x, e):
rubi.append(5552)
return -Dist(b*(-I*c + d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(-I*c - d - (-I*c + d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Dist(b*(I*c - d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(I*c + d - (I*c - d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(c + d/tan(a + b*x))/(f*(m + S(1))), x)
rule5552 = ReplacementRule(pattern5552, replacement5552)
pattern5553 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1844)
def replacement5553(m, f, b, d, c, a, x, e):
rubi.append(5553)
return Dist(b*(-I*c + d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(-I*c - d - (-I*c + d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) - Dist(b*(I*c - d + S(1))/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*I*a + S(2)*I*b*x)/(I*c + d - (I*c - d + S(1))*exp(S(2)*I*a + S(2)*I*b*x) + S(1)), x), x) + Simp((e + f*x)**(m + S(1))*acot(c + d/tan(a + b*x))/(f*(m + S(1))), x)
rule5553 = ReplacementRule(pattern5553, replacement5553)
pattern5554 = Pattern(Integral(ArcTan(tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons67)
def replacement5554(x, a, b):
rubi.append(5554)
return -Dist(b, Int(x/cosh(S(2)*a + S(2)*b*x), x), x) + Simp(x*ArcTan(tanh(a + b*x)), x)
rule5554 = ReplacementRule(pattern5554, replacement5554)
pattern5555 = Pattern(Integral(acot(tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons67)
def replacement5555(x, a, b):
rubi.append(5555)
return Dist(b, Int(x/cosh(S(2)*a + S(2)*b*x), x), x) + Simp(x*acot(tanh(a + b*x)), x)
rule5555 = ReplacementRule(pattern5555, replacement5555)
pattern5556 = Pattern(Integral(ArcTan(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons67)
def replacement5556(x, a, b):
rubi.append(5556)
return Dist(b, Int(x/cosh(S(2)*a + S(2)*b*x), x), x) + Simp(x*ArcTan(S(1)/tanh(a + b*x)), x)
rule5556 = ReplacementRule(pattern5556, replacement5556)
pattern5557 = Pattern(Integral(acot(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons67)
def replacement5557(x, a, b):
rubi.append(5557)
return -Dist(b, Int(x/cosh(S(2)*a + S(2)*b*x), x), x) + Simp(x*acot(S(1)/tanh(a + b*x)), x)
rule5557 = ReplacementRule(pattern5557, replacement5557)
pattern5558 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement5558(m, f, b, a, x, e):
rubi.append(5558)
return -Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/cosh(S(2)*a + S(2)*b*x), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(tanh(a + b*x))/(f*(m + S(1))), x)
rule5558 = ReplacementRule(pattern5558, replacement5558)
pattern5559 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement5559(m, f, b, a, x, e):
rubi.append(5559)
return Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/cosh(S(2)*a + S(2)*b*x), x), x) + Simp((e + f*x)**(m + S(1))*acot(tanh(a + b*x))/(f*(m + S(1))), x)
rule5559 = ReplacementRule(pattern5559, replacement5559)
pattern5560 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement5560(m, f, b, a, x, e):
rubi.append(5560)
return Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/cosh(S(2)*a + S(2)*b*x), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(S(1)/tanh(a + b*x))/(f*(m + S(1))), x)
rule5560 = ReplacementRule(pattern5560, replacement5560)
pattern5561 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement5561(m, f, b, a, x, e):
rubi.append(5561)
return -Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/cosh(S(2)*a + S(2)*b*x), x), x) + Simp((e + f*x)**(m + S(1))*acot(S(1)/tanh(a + b*x))/(f*(m + S(1))), x)
rule5561 = ReplacementRule(pattern5561, replacement5561)
pattern5562 = Pattern(Integral(ArcTan(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1845)
def replacement5562(b, d, c, a, x):
rubi.append(5562)
return -Dist(b, Int(x/(c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp(x*ArcTan(c + d*tanh(a + b*x)), x)
rule5562 = ReplacementRule(pattern5562, replacement5562)
pattern5563 = Pattern(Integral(acot(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1845)
def replacement5563(b, d, c, a, x):
rubi.append(5563)
return Dist(b, Int(x/(c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp(x*acot(c + d*tanh(a + b*x)), x)
rule5563 = ReplacementRule(pattern5563, replacement5563)
pattern5564 = Pattern(Integral(ArcTan(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1845)
def replacement5564(b, d, c, a, x):
rubi.append(5564)
return -Dist(b, Int(x/(-c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp(x*ArcTan(c + d/tanh(a + b*x)), x)
rule5564 = ReplacementRule(pattern5564, replacement5564)
pattern5565 = Pattern(Integral(acot(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1845)
def replacement5565(b, d, c, a, x):
rubi.append(5565)
return Dist(b, Int(x/(-c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp(x*acot(c + d/tanh(a + b*x)), x)
rule5565 = ReplacementRule(pattern5565, replacement5565)
pattern5566 = Pattern(Integral(ArcTan(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1846)
def replacement5566(b, d, c, a, x):
rubi.append(5566)
return Dist(I*b*(-c - d + I), Int(x*exp(S(2)*a + S(2)*b*x)/(-c + d + (-c - d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) - Dist(I*b*(c + d + I), Int(x*exp(S(2)*a + S(2)*b*x)/(c - d + (c + d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Simp(x*ArcTan(c + d*tanh(a + b*x)), x)
rule5566 = ReplacementRule(pattern5566, replacement5566)
pattern5567 = Pattern(Integral(acot(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1846)
def replacement5567(b, d, c, a, x):
rubi.append(5567)
return -Dist(I*b*(-c - d + I), Int(x*exp(S(2)*a + S(2)*b*x)/(-c + d + (-c - d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Dist(I*b*(c + d + I), Int(x*exp(S(2)*a + S(2)*b*x)/(c - d + (c + d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Simp(x*acot(c + d*tanh(a + b*x)), x)
rule5567 = ReplacementRule(pattern5567, replacement5567)
pattern5568 = Pattern(Integral(ArcTan(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1846)
def replacement5568(b, d, c, a, x):
rubi.append(5568)
return -Dist(I*b*(-c - d + I), Int(x*exp(S(2)*a + S(2)*b*x)/(-c + d - (-c - d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Dist(I*b*(c + d + I), Int(x*exp(S(2)*a + S(2)*b*x)/(c - d - (c + d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Simp(x*ArcTan(c + d/tanh(a + b*x)), x)
rule5568 = ReplacementRule(pattern5568, replacement5568)
pattern5569 = Pattern(Integral(acot(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1846)
def replacement5569(b, d, c, a, x):
rubi.append(5569)
return Dist(I*b*(-c - d + I), Int(x*exp(S(2)*a + S(2)*b*x)/(-c + d - (-c - d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) - Dist(I*b*(c + d + I), Int(x*exp(S(2)*a + S(2)*b*x)/(c - d - (c + d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Simp(x*acot(c + d/tanh(a + b*x)), x)
rule5569 = ReplacementRule(pattern5569, replacement5569)
pattern5570 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1845)
def replacement5570(m, f, b, d, c, a, x, e):
rubi.append(5570)
return -Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(c + d*tanh(a + b*x))/(f*(m + S(1))), x)
rule5570 = ReplacementRule(pattern5570, replacement5570)
pattern5571 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1845)
def replacement5571(m, f, b, d, c, a, x, e):
rubi.append(5571)
return Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp((e + f*x)**(m + S(1))*acot(c + d*tanh(a + b*x))/(f*(m + S(1))), x)
rule5571 = ReplacementRule(pattern5571, replacement5571)
pattern5572 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1845)
def replacement5572(m, f, b, d, c, a, x, e):
rubi.append(5572)
return -Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(-c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(c + d/tanh(a + b*x))/(f*(m + S(1))), x)
rule5572 = ReplacementRule(pattern5572, replacement5572)
pattern5573 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1845)
def replacement5573(m, f, b, d, c, a, x, e):
rubi.append(5573)
return Dist(b/(f*(m + S(1))), Int((e + f*x)**(m + S(1))/(-c*exp(S(2)*a + S(2)*b*x) + c - d), x), x) + Simp((e + f*x)**(m + S(1))*acot(c + d/tanh(a + b*x))/(f*(m + S(1))), x)
rule5573 = ReplacementRule(pattern5573, replacement5573)
pattern5574 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1846)
def replacement5574(m, f, b, d, c, a, x, e):
rubi.append(5574)
return Dist(I*b*(-c - d + I)/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(-c + d + (-c - d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) - Dist(I*b*(c + d + I)/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(c - d + (c + d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(c + d*tanh(a + b*x))/(f*(m + S(1))), x)
rule5574 = ReplacementRule(pattern5574, replacement5574)
pattern5575 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(WC('c', S(0)) + WC('d', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1846)
def replacement5575(m, f, b, d, c, a, x, e):
rubi.append(5575)
return -Dist(I*b*(-c - d + I)/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(-c + d + (-c - d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Dist(I*b*(c + d + I)/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(c - d + (c + d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Simp((e + f*x)**(m + S(1))*acot(c + d*tanh(a + b*x))/(f*(m + S(1))), x)
rule5575 = ReplacementRule(pattern5575, replacement5575)
pattern5576 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*ArcTan(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1846)
def replacement5576(m, f, b, d, c, a, x, e):
rubi.append(5576)
return -Dist(I*b*(-c - d + I)/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(-c + d - (-c - d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Dist(I*b*(c + d + I)/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(c - d - (c + d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Simp((e + f*x)**(m + S(1))*ArcTan(c + d/tanh(a + b*x))/(f*(m + S(1))), x)
rule5576 = ReplacementRule(pattern5576, replacement5576)
pattern5577 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*acot(WC('c', S(0)) + WC('d', S(1))/tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1846)
def replacement5577(m, f, b, d, c, a, x, e):
rubi.append(5577)
return Dist(I*b*(-c - d + I)/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(-c + d - (-c - d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) - Dist(I*b*(c + d + I)/(f*(m + S(1))), Int((e + f*x)**(m + S(1))*exp(S(2)*a + S(2)*b*x)/(c - d - (c + d + I)*exp(S(2)*a + S(2)*b*x) + I), x), x) + Simp((e + f*x)**(m + S(1))*acot(c + d/tanh(a + b*x))/(f*(m + S(1))), x)
rule5577 = ReplacementRule(pattern5577, replacement5577)
pattern5578 = Pattern(Integral(ArcTan(u_), x_), cons1230)
def replacement5578(x, u):
rubi.append(5578)
return -Int(SimplifyIntegrand(x*D(u, x)/(u**S(2) + S(1)), x), x) + Simp(x*ArcTan(u), x)
rule5578 = ReplacementRule(pattern5578, replacement5578)
pattern5579 = Pattern(Integral(acot(u_), x_), cons1230)
def replacement5579(x, u):
rubi.append(5579)
return Int(SimplifyIntegrand(x*D(u, x)/(u**S(2) + S(1)), x), x) + Simp(x*acot(u), x)
rule5579 = ReplacementRule(pattern5579, replacement5579)
pattern5580 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(ArcTan(u_)*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1847)
def replacement5580(u, m, b, d, c, a, x):
rubi.append(5580)
return -Dist(b/(d*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(u**S(2) + S(1)), x), x), x) + Simp((a + b*ArcTan(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule5580 = ReplacementRule(pattern5580, replacement5580)
pattern5581 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acot(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1847)
def replacement5581(u, m, b, d, c, a, x):
rubi.append(5581)
return Dist(b/(d*(m + S(1))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(u**S(2) + S(1)), x), x), x) + Simp((a + b*acot(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule5581 = ReplacementRule(pattern5581, replacement5581)
def With5582(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern5582 = Pattern(Integral(v_*(ArcTan(u_)*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons1230, cons1848, cons1849, CustomConstraint(With5582))
def replacement5582(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(5582)
return -Dist(b, Int(SimplifyIntegrand(w*D(u, x)/(u**S(2) + S(1)), x), x), x) + Dist(a + b*ArcTan(u), w, x)
rule5582 = ReplacementRule(pattern5582, replacement5582)
def With5583(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern5583 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*acot(u_)), x_), cons2, cons3, cons1230, cons1850, cons1851, CustomConstraint(With5583))
def replacement5583(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(5583)
return Dist(b, Int(SimplifyIntegrand(w*D(u, x)/(u**S(2) + S(1)), x), x), x) + Dist(a + b*acot(u), w, x)
rule5583 = ReplacementRule(pattern5583, replacement5583)
pattern5584 = Pattern(Integral(ArcTan(v_)*log(w_)/(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons552, cons1146, cons1852, cons1853)
def replacement5584(v, w, b, a, x):
rubi.append(5584)
return Dist(I/S(2), Int(log(w)*log(-I*v + S(1))/(a + b*x), x), x) - Dist(I/S(2), Int(log(w)*log(I*v + S(1))/(a + b*x), x), x)
rule5584 = ReplacementRule(pattern5584, replacement5584)
pattern5585 = Pattern(Integral(ArcTan(v_)*log(w_), x_), cons1242, cons1243)
def replacement5585(v, w, x):
rubi.append(5585)
return -Int(SimplifyIntegrand(x*ArcTan(v)*D(w, x)/w, x), x) - Int(SimplifyIntegrand(x*D(v, x)*log(w)/(v**S(2) + S(1)), x), x) + Simp(x*ArcTan(v)*log(w), x)
rule5585 = ReplacementRule(pattern5585, replacement5585)
pattern5586 = Pattern(Integral(log(w_)*acot(v_), x_), cons1242, cons1243)
def replacement5586(v, w, x):
rubi.append(5586)
return -Int(SimplifyIntegrand(x*D(w, x)*acot(v)/w, x), x) + Int(SimplifyIntegrand(x*D(v, x)*log(w)/(v**S(2) + S(1)), x), x) + Simp(x*log(w)*acot(v), x)
rule5586 = ReplacementRule(pattern5586, replacement5586)
def With5587(v, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
z = IntHide(u, x)
if InverseFunctionFreeQ(z, x):
return True
return False
pattern5587 = Pattern(Integral(u_*ArcTan(v_)*log(w_), x_), cons1242, cons1243, CustomConstraint(With5587))
def replacement5587(v, w, u, x):
z = IntHide(u, x)
rubi.append(5587)
return Dist(ArcTan(v)*log(w), z, x) - Int(SimplifyIntegrand(z*ArcTan(v)*D(w, x)/w, x), x) - Int(SimplifyIntegrand(z*D(v, x)*log(w)/(v**S(2) + S(1)), x), x)
rule5587 = ReplacementRule(pattern5587, replacement5587)
def With5588(v, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
z = IntHide(u, x)
if InverseFunctionFreeQ(z, x):
return True
return False
pattern5588 = Pattern(Integral(u_*log(w_)*acot(v_), x_), cons1242, cons1243, CustomConstraint(With5588))
def replacement5588(v, w, u, x):
z = IntHide(u, x)
rubi.append(5588)
return Dist(log(w)*acot(v), z, x) - Int(SimplifyIntegrand(z*D(w, x)*acot(v)/w, x), x) + Int(SimplifyIntegrand(z*D(v, x)*log(w)/(v**S(2) + S(1)), x), x)
rule5588 = ReplacementRule(pattern5588, replacement5588)
pattern5589 = Pattern(Integral(asec(x_*WC('c', S(1))), x_), cons7, cons7)
def replacement5589(x, c):
rubi.append(5589)
return -Dist(S(1)/c, Int(S(1)/(x*sqrt(S(1) - S(1)/(c**S(2)*x**S(2)))), x), x) + Simp(x*asec(c*x), x)
rule5589 = ReplacementRule(pattern5589, replacement5589)
pattern5590 = Pattern(Integral(acsc(x_*WC('c', S(1))), x_), cons7, cons7)
def replacement5590(x, c):
rubi.append(5590)
return Dist(S(1)/c, Int(S(1)/(x*sqrt(S(1) - S(1)/(c**S(2)*x**S(2)))), x), x) + Simp(x*acsc(c*x), x)
rule5590 = ReplacementRule(pattern5590, replacement5590)
pattern5591 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5591(b, a, n, c, x):
rubi.append(5591)
return Dist(S(1)/c, Subst(Int((a + b*x)**n*tan(x)/cos(x), x), x, asec(c*x)), x)
rule5591 = ReplacementRule(pattern5591, replacement5591)
pattern5592 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5592(b, a, n, c, x):
rubi.append(5592)
return -Dist(S(1)/c, Subst(Int((a + b*x)**n/(sin(x)*tan(x)), x), x, acsc(c*x)), x)
rule5592 = ReplacementRule(pattern5592, replacement5592)
pattern5593 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))/x_, x_), cons2, cons3, cons7, cons14)
def replacement5593(x, a, c, b):
rubi.append(5593)
return -Subst(Int((a + b*acos(x/c))/x, x), x, S(1)/x)
rule5593 = ReplacementRule(pattern5593, replacement5593)
pattern5594 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))/x_, x_), cons2, cons3, cons7, cons14)
def replacement5594(x, a, c, b):
rubi.append(5594)
return -Subst(Int((a + b*asin(x/c))/x, x), x, S(1)/x)
rule5594 = ReplacementRule(pattern5594, replacement5594)
pattern5595 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons66)
def replacement5595(m, b, a, c, x):
rubi.append(5595)
return -Dist(b/(c*(m + S(1))), Int(x**(m + S(-1))/sqrt(S(1) - S(1)/(c**S(2)*x**S(2))), x), x) + Simp(x**(m + S(1))*(a + b*asec(c*x))/(m + S(1)), x)
rule5595 = ReplacementRule(pattern5595, replacement5595)
pattern5596 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons66)
def replacement5596(m, b, a, c, x):
rubi.append(5596)
return Dist(b/(c*(m + S(1))), Int(x**(m + S(-1))/sqrt(S(1) - S(1)/(c**S(2)*x**S(2))), x), x) + Simp(x**(m + S(1))*(a + b*acsc(c*x))/(m + S(1)), x)
rule5596 = ReplacementRule(pattern5596, replacement5596)
pattern5597 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons17)
def replacement5597(m, b, a, c, n, x):
rubi.append(5597)
return Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*(S(1)/cos(x))**(m + S(1))*tan(x), x), x, asec(c*x)), x)
rule5597 = ReplacementRule(pattern5597, replacement5597)
pattern5598 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons4, cons17)
def replacement5598(m, b, a, c, n, x):
rubi.append(5598)
return -Dist(c**(-m + S(-1)), Subst(Int((a + b*x)**n*(S(1)/sin(x))**(m + S(1))/tan(x), x), x, acsc(c*x)), x)
rule5598 = ReplacementRule(pattern5598, replacement5598)
pattern5599 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons1854)
def replacement5599(m, b, a, n, c, x):
rubi.append(5599)
return Int(x**m*(a + b*asec(c*x))**n, x)
rule5599 = ReplacementRule(pattern5599, replacement5599)
pattern5600 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons1854)
def replacement5600(m, b, a, n, c, x):
rubi.append(5600)
return Int(x**m*(a + b*acsc(c*x))**n, x)
rule5600 = ReplacementRule(pattern5600, replacement5600)
def With5601(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(5601)
return -Dist(b*c*x/sqrt(c**S(2)*x**S(2)), Int(SimplifyIntegrand(u/(x*sqrt(c**S(2)*x**S(2) + S(-1))), x), x), x) + Dist(a + b*asec(c*x), u, x)
pattern5601 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1743)
rule5601 = ReplacementRule(pattern5601, With5601)
def With5602(p, b, d, a, c, x, e):
u = IntHide((d + e*x**S(2))**p, x)
rubi.append(5602)
return Dist(b*c*x/sqrt(c**S(2)*x**S(2)), Int(SimplifyIntegrand(u/(x*sqrt(c**S(2)*x**S(2) + S(-1))), x), x), x) + Dist(a + b*acsc(c*x), u, x)
pattern5602 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1743)
rule5602 = ReplacementRule(pattern5602, With5602)
pattern5603 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons38)
def replacement5603(p, b, d, a, n, c, x, e):
rubi.append(5603)
return -Subst(Int(x**(-S(2)*p + S(-2))*(a + b*acos(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x)
rule5603 = ReplacementRule(pattern5603, replacement5603)
pattern5604 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons38)
def replacement5604(p, b, d, a, n, c, x, e):
rubi.append(5604)
return -Subst(Int(x**(-S(2)*p + S(-2))*(a + b*asin(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x)
rule5604 = ReplacementRule(pattern5604, replacement5604)
pattern5605 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons667, cons178, cons1855)
def replacement5605(p, b, d, a, n, c, x, e):
rubi.append(5605)
return -Dist(sqrt(x**S(2))/x, Subst(Int(x**(-S(2)*p + S(-2))*(a + b*acos(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule5605 = ReplacementRule(pattern5605, replacement5605)
pattern5606 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons667, cons178, cons1855)
def replacement5606(p, b, d, a, n, c, x, e):
rubi.append(5606)
return -Dist(sqrt(x**S(2))/x, Subst(Int(x**(-S(2)*p + S(-2))*(a + b*asin(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule5606 = ReplacementRule(pattern5606, replacement5606)
pattern5607 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons667, cons1856)
def replacement5607(p, b, d, a, n, c, x, e):
rubi.append(5607)
return -Dist(sqrt(d + e*x**S(2))/(x*sqrt(d/x**S(2) + e)), Subst(Int(x**(-S(2)*p + S(-2))*(a + b*acos(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule5607 = ReplacementRule(pattern5607, replacement5607)
pattern5608 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons667, cons1856)
def replacement5608(p, b, d, a, n, c, x, e):
rubi.append(5608)
return -Dist(sqrt(d + e*x**S(2))/(x*sqrt(d/x**S(2) + e)), Subst(Int(x**(-S(2)*p + S(-2))*(a + b*asin(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule5608 = ReplacementRule(pattern5608, replacement5608)
pattern5609 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement5609(p, b, d, a, n, c, x, e):
rubi.append(5609)
return Int((a + b*asec(c*x))**n*(d + e*x**S(2))**p, x)
rule5609 = ReplacementRule(pattern5609, replacement5609)
pattern5610 = Pattern(Integral((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement5610(p, b, d, a, n, c, x, e):
rubi.append(5610)
return Int((a + b*acsc(c*x))**n*(d + e*x**S(2))**p, x)
rule5610 = ReplacementRule(pattern5610, replacement5610)
pattern5611 = Pattern(Integral(x_*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement5611(p, b, d, a, c, x, e):
rubi.append(5611)
return -Dist(b*c*x/(S(2)*e*sqrt(c**S(2)*x**S(2))*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(x*sqrt(c**S(2)*x**S(2) + S(-1))), x), x) + Simp((a + b*asec(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5611 = ReplacementRule(pattern5611, replacement5611)
pattern5612 = Pattern(Integral(x_*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons54)
def replacement5612(p, b, d, a, c, x, e):
rubi.append(5612)
return Dist(b*c*x/(S(2)*e*sqrt(c**S(2)*x**S(2))*(p + S(1))), Int((d + e*x**S(2))**(p + S(1))/(x*sqrt(c**S(2)*x**S(2) + S(-1))), x), x) + Simp((a + b*acsc(c*x))*(d + e*x**S(2))**(p + S(1))/(S(2)*e*(p + S(1))), x)
rule5612 = ReplacementRule(pattern5612, replacement5612)
def With5613(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(d + e*x**S(2))**p, x)
rubi.append(5613)
return -Dist(b*c*x/sqrt(c**S(2)*x**S(2)), Int(SimplifyIntegrand(u/(x*sqrt(c**S(2)*x**S(2) + S(-1))), x), x), x) + Dist(a + b*asec(c*x), u, x)
pattern5613 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1786)
rule5613 = ReplacementRule(pattern5613, With5613)
def With5614(p, m, b, d, a, c, x, e):
u = IntHide(x**m*(d + e*x**S(2))**p, x)
rubi.append(5614)
return Dist(b*c*x/sqrt(c**S(2)*x**S(2)), Int(SimplifyIntegrand(u/(x*sqrt(c**S(2)*x**S(2) + S(-1))), x), x), x) + Dist(a + b*acsc(c*x), u, x)
pattern5614 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1786)
rule5614 = ReplacementRule(pattern5614, With5614)
pattern5615 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1299)
def replacement5615(p, m, b, d, a, n, c, x, e):
rubi.append(5615)
return -Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*acos(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x)
rule5615 = ReplacementRule(pattern5615, replacement5615)
pattern5616 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1299)
def replacement5616(p, m, b, d, a, n, c, x, e):
rubi.append(5616)
return -Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*asin(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x)
rule5616 = ReplacementRule(pattern5616, replacement5616)
pattern5617 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons17, cons667, cons178, cons1855)
def replacement5617(p, m, b, d, a, n, c, x, e):
rubi.append(5617)
return -Dist(sqrt(x**S(2))/x, Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*acos(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule5617 = ReplacementRule(pattern5617, replacement5617)
pattern5618 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons17, cons667, cons178, cons1855)
def replacement5618(p, m, b, d, a, n, c, x, e):
rubi.append(5618)
return -Dist(sqrt(x**S(2))/x, Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*asin(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule5618 = ReplacementRule(pattern5618, replacement5618)
pattern5619 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons17, cons667, cons1856)
def replacement5619(p, m, b, d, a, n, c, x, e):
rubi.append(5619)
return -Dist(sqrt(d + e*x**S(2))/(x*sqrt(d/x**S(2) + e)), Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*acos(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule5619 = ReplacementRule(pattern5619, replacement5619)
pattern5620 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**p_*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1737, cons17, cons667, cons1856)
def replacement5620(p, m, b, d, a, n, c, x, e):
rubi.append(5620)
return -Dist(sqrt(d + e*x**S(2))/(x*sqrt(d/x**S(2) + e)), Subst(Int(x**(-m - S(2)*p + S(-2))*(a + b*asin(x/c))**n*(d*x**S(2) + e)**p, x), x, S(1)/x), x)
rule5620 = ReplacementRule(pattern5620, replacement5620)
pattern5621 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5621(p, m, b, d, a, n, c, x, e):
rubi.append(5621)
return Int(x**m*(a + b*asec(c*x))**n*(d + e*x**S(2))**p, x)
rule5621 = ReplacementRule(pattern5621, replacement5621)
pattern5622 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(x_*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5622(p, m, b, d, a, n, c, x, e):
rubi.append(5622)
return Int(x**m*(a + b*acsc(c*x))**n*(d + e*x**S(2))**p, x)
rule5622 = ReplacementRule(pattern5622, replacement5622)
pattern5623 = Pattern(Integral(asec(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement5623(x, a, b):
rubi.append(5623)
return -Int(S(1)/(sqrt(S(1) - S(1)/(a + b*x)**S(2))*(a + b*x)), x) + Simp((a + b*x)*asec(a + b*x)/b, x)
rule5623 = ReplacementRule(pattern5623, replacement5623)
pattern5624 = Pattern(Integral(acsc(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement5624(x, a, b):
rubi.append(5624)
return Int(S(1)/(sqrt(S(1) - S(1)/(a + b*x)**S(2))*(a + b*x)), x) + Simp((a + b*x)*acsc(a + b*x)/b, x)
rule5624 = ReplacementRule(pattern5624, replacement5624)
pattern5625 = Pattern(Integral(asec(a_ + x_*WC('b', S(1)))**n_, x_), cons2, cons3, cons4, cons1831)
def replacement5625(x, a, n, b):
rubi.append(5625)
return Dist(S(1)/b, Subst(Int(x**n*tan(x)/cos(x), x), x, asec(a + b*x)), x)
rule5625 = ReplacementRule(pattern5625, replacement5625)
pattern5626 = Pattern(Integral(acsc(a_ + x_*WC('b', S(1)))**n_, x_), cons2, cons3, cons4, cons1831)
def replacement5626(x, a, n, b):
rubi.append(5626)
return -Dist(S(1)/b, Subst(Int(x**n/(sin(x)*tan(x)), x), x, acsc(a + b*x)), x)
rule5626 = ReplacementRule(pattern5626, replacement5626)
pattern5627 = Pattern(Integral(asec(a_ + x_*WC('b', S(1)))/x_, x_), cons2, cons3, cons67)
def replacement5627(x, a, b):
rubi.append(5627)
return -Simp(I*PolyLog(S(2), (-sqrt(-a**S(2) + S(1)) + S(1))*exp(I*asec(a + b*x))/a), x) - Simp(I*PolyLog(S(2), (sqrt(-a**S(2) + S(1)) + S(1))*exp(I*asec(a + b*x))/a), x) + Simp(I*PolyLog(S(2), -exp(S(2)*I*asec(a + b*x)))/S(2), x) + Simp(log(S(1) - (-sqrt(-a**S(2) + S(1)) + S(1))*exp(I*asec(a + b*x))/a)*asec(a + b*x), x) + Simp(log(S(1) - (sqrt(-a**S(2) + S(1)) + S(1))*exp(I*asec(a + b*x))/a)*asec(a + b*x), x) - Simp(log(exp(S(2)*I*asec(a + b*x)) + S(1))*asec(a + b*x), x)
rule5627 = ReplacementRule(pattern5627, replacement5627)
pattern5628 = Pattern(Integral(acsc(a_ + x_*WC('b', S(1)))/x_, x_), cons2, cons3, cons67)
def replacement5628(x, a, b):
rubi.append(5628)
return Simp(I*PolyLog(S(2), I*(-sqrt(-a**S(2) + S(1)) + S(1))*exp(-I*acsc(a + b*x))/a), x) + Simp(I*PolyLog(S(2), I*(sqrt(-a**S(2) + S(1)) + S(1))*exp(-I*acsc(a + b*x))/a), x) + Simp(I*PolyLog(S(2), exp(S(2)*I*acsc(a + b*x)))/S(2), x) + Simp(I*acsc(a + b*x)**S(2), x) + Simp(log(S(1) - I*(-sqrt(-a**S(2) + S(1)) + S(1))*exp(-I*acsc(a + b*x))/a)*acsc(a + b*x), x) + Simp(log(S(1) - I*(sqrt(-a**S(2) + S(1)) + S(1))*exp(-I*acsc(a + b*x))/a)*acsc(a + b*x), x) - Simp(log(-exp(S(2)*I*acsc(a + b*x)) + S(1))*acsc(a + b*x), x)
rule5628 = ReplacementRule(pattern5628, replacement5628)
pattern5629 = Pattern(Integral(x_**WC('m', S(1))*asec(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons21, cons17, cons66)
def replacement5629(x, m, b, a):
rubi.append(5629)
return -Dist(b**(-m + S(-1))/(m + S(1)), Subst(Int(x**(-m + S(-1))*((-a*x)**(m + S(1)) - (-a*x + S(1))**(m + S(1)))/sqrt(-x**S(2) + S(1)), x), x, S(1)/(a + b*x)), x) - Simp(b**(-m + S(-1))*(-b**(m + S(1))*x**(m + S(1)) + (-a)**(m + S(1)))*asec(a + b*x)/(m + S(1)), x)
rule5629 = ReplacementRule(pattern5629, replacement5629)
pattern5630 = Pattern(Integral(x_**WC('m', S(1))*acsc(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons21, cons17, cons66)
def replacement5630(x, m, b, a):
rubi.append(5630)
return Dist(b**(-m + S(-1))/(m + S(1)), Subst(Int(x**(-m + S(-1))*((-a*x)**(m + S(1)) - (-a*x + S(1))**(m + S(1)))/sqrt(-x**S(2) + S(1)), x), x, S(1)/(a + b*x)), x) - Simp(b**(-m + S(-1))*(-b**(m + S(1))*x**(m + S(1)) + (-a)**(m + S(1)))*acsc(a + b*x)/(m + S(1)), x)
rule5630 = ReplacementRule(pattern5630, replacement5630)
pattern5631 = Pattern(Integral(x_**WC('m', S(1))*asec(a_ + x_*WC('b', S(1)))**n_, x_), cons2, cons3, cons4, cons62)
def replacement5631(m, b, a, n, x):
rubi.append(5631)
return Dist(b**(-m + S(-1)), Subst(Int(x**n*(-a + S(1)/cos(x))**m*tan(x)/cos(x), x), x, asec(a + b*x)), x)
rule5631 = ReplacementRule(pattern5631, replacement5631)
pattern5632 = Pattern(Integral(x_**WC('m', S(1))*acsc(a_ + x_*WC('b', S(1)))**n_, x_), cons2, cons3, cons4, cons62)
def replacement5632(m, b, a, n, x):
rubi.append(5632)
return -Dist(b**(-m + S(-1)), Subst(Int(x**n*(-a + S(1)/sin(x))**m/(sin(x)*tan(x)), x), x, acsc(a + b*x)), x)
rule5632 = ReplacementRule(pattern5632, replacement5632)
pattern5633 = Pattern(Integral(WC('u', S(1))*asec(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement5633(u, m, b, c, n, a, x):
rubi.append(5633)
return Int(u*acos(a/c + b*x**n/c)**m, x)
rule5633 = ReplacementRule(pattern5633, replacement5633)
pattern5634 = Pattern(Integral(WC('u', S(1))*acsc(WC('c', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons4, cons21, cons1766)
def replacement5634(u, m, b, c, n, a, x):
rubi.append(5634)
return Int(u*asin(a/c + b*x**n/c)**m, x)
rule5634 = ReplacementRule(pattern5634, replacement5634)
pattern5635 = Pattern(Integral(f_**(WC('c', S(1))*asec(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)))*WC('u', S(1)), x_), cons2, cons3, cons7, cons125, cons148)
def replacement5635(u, f, b, c, a, n, x):
rubi.append(5635)
return Dist(S(1)/b, Subst(Int(f**(c*x**n)*ReplaceAll(u, Rule(x, -a/b + S(1)/(b*cos(x))))*tan(x)/cos(x), x), x, asec(a + b*x)), x)
rule5635 = ReplacementRule(pattern5635, replacement5635)
pattern5636 = Pattern(Integral(f_**(WC('c', S(1))*acsc(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)))*WC('u', S(1)), x_), cons2, cons3, cons7, cons125, cons148)
def replacement5636(u, f, b, c, a, n, x):
rubi.append(5636)
return -Dist(S(1)/b, Subst(Int(f**(c*x**n)*ReplaceAll(u, Rule(x, -a/b + S(1)/(b*sin(x))))/(sin(x)*tan(x)), x), x, acsc(a + b*x)), x)
rule5636 = ReplacementRule(pattern5636, replacement5636)
pattern5637 = Pattern(Integral(asec(u_), x_), cons1230, cons1769)
def replacement5637(x, u):
rubi.append(5637)
return -Dist(u/sqrt(u**S(2)), Int(SimplifyIntegrand(x*D(u, x)/(u*sqrt(u**S(2) + S(-1))), x), x), x) + Simp(x*asec(u), x)
rule5637 = ReplacementRule(pattern5637, replacement5637)
pattern5638 = Pattern(Integral(acsc(u_), x_), cons1230, cons1769)
def replacement5638(x, u):
rubi.append(5638)
return Dist(u/sqrt(u**S(2)), Int(SimplifyIntegrand(x*D(u, x)/(u*sqrt(u**S(2) + S(-1))), x), x), x) + Simp(x*acsc(u), x)
rule5638 = ReplacementRule(pattern5638, replacement5638)
pattern5639 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*asec(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1769)
def replacement5639(u, m, b, d, c, a, x):
rubi.append(5639)
return -Dist(b*u/(d*(m + S(1))*sqrt(u**S(2))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(u*sqrt(u**S(2) + S(-1))), x), x), x) + Simp((a + b*asec(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule5639 = ReplacementRule(pattern5639, replacement5639)
pattern5640 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*acsc(u_)), x_), cons2, cons3, cons7, cons27, cons21, cons66, cons1230, cons1770, cons1769)
def replacement5640(u, m, b, d, c, a, x):
rubi.append(5640)
return Dist(b*u/(d*(m + S(1))*sqrt(u**S(2))), Int(SimplifyIntegrand((c + d*x)**(m + S(1))*D(u, x)/(u*sqrt(u**S(2) + S(-1))), x), x), x) + Simp((a + b*acsc(u))*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule5640 = ReplacementRule(pattern5640, replacement5640)
def With5641(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern5641 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*asec(u_)), x_), cons2, cons3, cons1230, cons1857, CustomConstraint(With5641))
def replacement5641(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(5641)
return -Dist(b*u/sqrt(u**S(2)), Int(SimplifyIntegrand(w*D(u, x)/(u*sqrt(u**S(2) + S(-1))), x), x), x) + Dist(a + b*asec(u), w, x)
rule5641 = ReplacementRule(pattern5641, replacement5641)
def With5642(v, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern5642 = Pattern(Integral(v_*(WC('a', S(0)) + WC('b', S(1))*acsc(u_)), x_), cons2, cons3, cons1230, cons1858, CustomConstraint(With5642))
def replacement5642(v, u, b, a, x):
w = IntHide(v, x)
rubi.append(5642)
return Dist(b*u/sqrt(u**S(2)), Int(SimplifyIntegrand(w*D(u, x)/(u*sqrt(u**S(2) + S(-1))), x), x), x) + Dist(a + b*acsc(u), w, x)
rule5642 = ReplacementRule(pattern5642, replacement5642)
return [rule5031, rule5032, rule5033, rule5034, rule5035, rule5036, rule5037, rule5038, rule5039, rule5040, rule5041, rule5042, rule5043, rule5044, rule5045, rule5046, rule5047, rule5048, rule5049, rule5050, rule5051, rule5052, rule5053, rule5054, rule5055, rule5056, rule5057, rule5058, rule5059, rule5060, rule5061, rule5062, rule5063, rule5064, rule5065, rule5066, rule5067, rule5068, rule5069, rule5070, rule5071, rule5072, rule5073, rule5074, rule5075, rule5076, rule5077, rule5078, rule5079, rule5080, rule5081, rule5082, rule5083, rule5084, rule5085, rule5086, rule5087, rule5088, rule5089, rule5090, rule5091, rule5092, rule5093, rule5094, rule5095, rule5096, rule5097, rule5098, rule5099, rule5100, rule5101, rule5102, rule5103, rule5104, rule5105, rule5106, rule5107, rule5108, rule5109, rule5110, rule5111, rule5112, rule5113, rule5114, rule5115, rule5116, rule5117, rule5118, rule5119, rule5120, rule5121, rule5122, rule5123, rule5124, rule5125, rule5126, rule5127, rule5128, rule5129, rule5130, rule5131, rule5132, rule5133, rule5134, rule5135, rule5136, rule5137, rule5138, rule5139, rule5140, rule5141, rule5142, rule5143, rule5144, rule5145, rule5146, rule5147, rule5148, rule5149, rule5150, rule5151, rule5152, rule5153, rule5154, rule5155, rule5156, rule5157, rule5158, rule5159, rule5160, rule5161, rule5162, rule5163, rule5164, rule5165, rule5166, rule5167, rule5168, rule5169, rule5170, rule5171, rule5172, rule5173, rule5174, rule5175, rule5176, rule5177, rule5178, rule5179, rule5180, rule5181, rule5182, rule5183, rule5184, rule5185, rule5186, rule5187, rule5188, rule5189, rule5190, rule5191, rule5192, rule5193, rule5194, rule5195, rule5196, rule5197, rule5198, rule5199, rule5200, rule5201, rule5202, rule5203, rule5204, rule5205, rule5206, rule5207, rule5208, rule5209, rule5210, rule5211, rule5212, rule5213, rule5214, rule5215, rule5216, rule5217, rule5218, rule5219, rule5220, rule5221, rule5222, rule5223, rule5224, rule5225, rule5226, rule5227, rule5228, rule5229, rule5230, rule5231, rule5232, rule5233, rule5234, rule5235, rule5236, rule5237, rule5238, rule5239, rule5240, rule5241, rule5242, rule5243, rule5244, rule5245, rule5246, rule5247, rule5248, rule5249, rule5250, rule5251, rule5252, rule5253, rule5254, rule5255, rule5256, rule5257, rule5258, rule5259, rule5260, rule5261, rule5262, rule5263, rule5264, rule5265, rule5266, rule5267, rule5268, rule5269, rule5270, rule5271, rule5272, rule5273, rule5274, rule5275, rule5276, rule5277, rule5278, rule5279, rule5280, rule5281, rule5282, rule5283, rule5284, rule5285, rule5286, rule5287, rule5288, rule5289, rule5290, rule5291, rule5292, rule5293, rule5294, rule5295, rule5296, rule5297, rule5298, rule5299, rule5300, rule5301, rule5302, rule5303, rule5304, rule5305, rule5306, rule5307, rule5308, rule5309, rule5310, rule5311, rule5312, rule5313, rule5314, rule5315, rule5316, rule5317, rule5318, rule5319, rule5320, rule5321, rule5322, rule5323, rule5324, rule5325, rule5326, rule5327, rule5328, rule5329, rule5330, rule5331, rule5332, rule5333, rule5334, rule5335, rule5336, rule5337, rule5338, rule5339, rule5340, rule5341, rule5342, rule5343, rule5344, rule5345, rule5346, rule5347, rule5348, rule5349, rule5350, rule5351, rule5352, rule5353, rule5354, rule5355, rule5356, rule5357, rule5358, rule5359, rule5360, rule5361, rule5362, rule5363, rule5364, rule5365, rule5366, rule5367, rule5368, rule5369, rule5370, rule5371, rule5372, rule5373, rule5374, rule5375, rule5376, rule5377, rule5378, rule5379, rule5380, rule5381, rule5382, rule5383, rule5384, rule5385, rule5386, rule5387, rule5388, rule5389, rule5390, rule5391, rule5392, rule5393, rule5394, rule5395, rule5396, rule5397, rule5398, rule5399, rule5400, rule5401, rule5402, rule5403, rule5404, rule5405, rule5406, rule5407, rule5408, rule5409, rule5410, rule5411, rule5412, rule5413, rule5414, rule5415, rule5416, rule5417, rule5418, rule5419, rule5420, rule5421, rule5422, rule5423, rule5424, rule5425, rule5426, rule5427, rule5428, rule5429, rule5430, rule5431, rule5432, rule5433, rule5434, rule5435, rule5436, rule5437, rule5438, rule5439, rule5440, rule5441, rule5442, rule5443, rule5444, rule5445, rule5446, rule5447, rule5448, rule5449, rule5450, rule5451, rule5452, rule5453, rule5454, rule5455, rule5456, rule5457, rule5458, rule5459, rule5460, rule5461, rule5462, rule5463, rule5464, rule5465, rule5466, rule5467, rule5468, rule5469, rule5470, rule5471, rule5472, rule5473, rule5474, rule5475, rule5476, rule5477, rule5478, rule5479, rule5480, rule5481, rule5482, rule5483, rule5484, rule5485, rule5486, rule5487, rule5488, rule5489, rule5490, rule5491, rule5492, rule5493, rule5494, rule5495, rule5496, rule5497, rule5498, rule5499, rule5500, rule5501, rule5502, rule5503, rule5504, rule5505, rule5506, rule5507, rule5508, rule5509, rule5510, rule5511, rule5512, rule5513, rule5514, rule5515, rule5516, rule5517, rule5518, rule5519, rule5520, rule5521, rule5522, rule5523, rule5524, rule5525, rule5526, rule5527, rule5528, rule5529, rule5530, rule5531, rule5532, rule5533, rule5534, rule5535, rule5536, rule5537, rule5538, rule5539, rule5540, rule5541, rule5542, rule5543, rule5544, rule5545, rule5546, rule5547, rule5548, rule5549, rule5550, rule5551, rule5552, rule5553, rule5554, rule5555, rule5556, rule5557, rule5558, rule5559, rule5560, rule5561, rule5562, rule5563, rule5564, rule5565, rule5566, rule5567, rule5568, rule5569, rule5570, rule5571, rule5572, rule5573, rule5574, rule5575, rule5576, rule5577, rule5578, rule5579, rule5580, rule5581, rule5582, rule5583, rule5584, rule5585, rule5586, rule5587, rule5588, rule5589, rule5590, rule5591, rule5592, rule5593, rule5594, rule5595, rule5596, rule5597, rule5598, rule5599, rule5600, rule5601, rule5602, rule5603, rule5604, rule5605, rule5606, rule5607, rule5608, rule5609, rule5610, rule5611, rule5612, rule5613, rule5614, rule5615, rule5616, rule5617, rule5618, rule5619, rule5620, rule5621, rule5622, rule5623, rule5624, rule5625, rule5626, rule5627, rule5628, rule5629, rule5630, rule5631, rule5632, rule5633, rule5634, rule5635, rule5636, rule5637, rule5638, rule5639, rule5640, rule5641, rule5642, ]
|
98903fe9a14b59cf899bb228421332b5f4ace2fafd0587eded9606f7533f543a | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def piecewise_linear(rubi):
from sympy.integrals.rubi.constraints import cons1090, cons21, cons1091, cons87, cons88, cons1092, cons89, cons23, cons72, cons66, cons4, cons1093, cons214, cons683, cons100, cons101, cons1094, cons1095, cons31, cons94, cons356, cons1096, cons18, cons1097, cons2, cons3
def With1882(x, m, u):
c = D(u, x)
rubi.append(1882)
return Dist(S(1)/c, Subst(Int(x**m, x), x, u), x)
pattern1882 = Pattern(Integral(u_**WC('m', S(1)), x_), cons21, cons1090)
rule1882 = ReplacementRule(pattern1882, With1882)
def With1883(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1883 = Pattern(Integral(v_/u_, x_), cons1091, CustomConstraint(With1883))
def replacement1883(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1883)
return -Dist((-a*v + b*u)/a, Int(S(1)/u, x), x) + Simp(b*x/a, x)
rule1883 = ReplacementRule(pattern1883, replacement1883)
def With1884(v, x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1884 = Pattern(Integral(v_**n_/u_, x_), cons1091, cons87, cons88, cons1092, CustomConstraint(With1884))
def replacement1884(v, x, n, u):
a = D(u, x)
b = D(v, x)
rubi.append(1884)
return -Dist((-a*v + b*u)/a, Int(v**(n + S(-1))/u, x), x) + Simp(v**n/(a*n), x)
rule1884 = ReplacementRule(pattern1884, replacement1884)
def With1885(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1885 = Pattern(Integral(S(1)/(u_*v_), x_), cons1091, CustomConstraint(With1885))
def replacement1885(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1885)
return -Dist(a/(-a*v + b*u), Int(S(1)/u, x), x) + Dist(b/(-a*v + b*u), Int(S(1)/v, x), x)
rule1885 = ReplacementRule(pattern1885, replacement1885)
def With1886(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if And(NonzeroQ(-a*v + b*u), PosQ((-a*v + b*u)/a)):
return True
return False
pattern1886 = Pattern(Integral(S(1)/(u_*sqrt(v_)), x_), cons1091, CustomConstraint(With1886))
def replacement1886(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1886)
return Simp(S(2)*ArcTan(sqrt(v)/Rt((-a*v + b*u)/a, S(2)))/(a*Rt((-a*v + b*u)/a, S(2))), x)
rule1886 = ReplacementRule(pattern1886, replacement1886)
def With1887(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if And(NonzeroQ(-a*v + b*u), NegQ((-a*v + b*u)/a)):
return True
return False
pattern1887 = Pattern(Integral(S(1)/(u_*sqrt(v_)), x_), cons1091, CustomConstraint(With1887))
def replacement1887(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1887)
return Simp(-S(2)*atanh(sqrt(v)/Rt(-(-a*v + b*u)/a, S(2)))/(a*Rt(-(-a*v + b*u)/a, S(2))), x)
rule1887 = ReplacementRule(pattern1887, replacement1887)
def With1888(v, x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1888 = Pattern(Integral(v_**n_/u_, x_), cons1091, cons87, cons89, CustomConstraint(With1888))
def replacement1888(v, x, n, u):
a = D(u, x)
b = D(v, x)
rubi.append(1888)
return -Dist(a/(-a*v + b*u), Int(v**(n + S(1))/u, x), x) + Simp(v**(n + S(1))/((n + S(1))*(-a*v + b*u)), x)
rule1888 = ReplacementRule(pattern1888, replacement1888)
def With1889(v, x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1889 = Pattern(Integral(v_**n_/u_, x_), cons1091, cons23, CustomConstraint(With1889))
def replacement1889(v, x, n, u):
a = D(u, x)
b = D(v, x)
rubi.append(1889)
return Simp(v**(n + S(1))*Hypergeometric2F1(S(1), n + S(1), n + S(2), -a*v/(-a*v + b*u))/((n + S(1))*(-a*v + b*u)), x)
rule1889 = ReplacementRule(pattern1889, replacement1889)
def With1890(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if And(NonzeroQ(-a*v + b*u), PosQ(a*b)):
return True
return False
pattern1890 = Pattern(Integral(S(1)/(sqrt(u_)*sqrt(v_)), x_), cons1091, CustomConstraint(With1890))
def replacement1890(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1890)
return Simp(S(2)*atanh(sqrt(u)*Rt(a*b, S(2))/(a*sqrt(v)))/Rt(a*b, S(2)), x)
rule1890 = ReplacementRule(pattern1890, replacement1890)
def With1891(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if And(NonzeroQ(-a*v + b*u), NegQ(a*b)):
return True
return False
pattern1891 = Pattern(Integral(S(1)/(sqrt(u_)*sqrt(v_)), x_), cons1091, CustomConstraint(With1891))
def replacement1891(v, x, u):
a = D(u, x)
b = D(v, x)
rubi.append(1891)
return Simp(S(2)*ArcTan(sqrt(u)*Rt(-a*b, S(2))/(a*sqrt(v)))/Rt(-a*b, S(2)), x)
rule1891 = ReplacementRule(pattern1891, replacement1891)
def With1892(v, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1892 = Pattern(Integral(u_**m_*v_**n_, x_), cons21, cons4, cons1091, cons72, cons66, CustomConstraint(With1892))
def replacement1892(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1892)
return -Simp(u**(m + S(1))*v**(n + S(1))/((m + S(1))*(-a*v + b*u)), x)
rule1892 = ReplacementRule(pattern1892, replacement1892)
def With1893(v, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1893 = Pattern(Integral(u_**m_*v_**WC('n', S(1)), x_), cons21, cons4, cons1091, cons66, cons1093, CustomConstraint(With1893))
def replacement1893(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1893)
return -Dist(b*n/(a*(m + S(1))), Int(u**(m + S(1))*v**(n + S(-1)), x), x) + Simp(u**(m + S(1))*v**n/(a*(m + S(1))), x)
rule1893 = ReplacementRule(pattern1893, replacement1893)
def With1894(v, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1894 = Pattern(Integral(u_**m_*v_**WC('n', S(1)), x_), cons1091, cons214, cons87, cons88, cons683, cons100, cons101, CustomConstraint(With1894))
def replacement1894(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1894)
return -Dist(n*(-a*v + b*u)/(a*(m + n + S(1))), Int(u**m*v**(n + S(-1)), x), x) + Simp(u**(m + S(1))*v**n/(a*(m + n + S(1))), x)
rule1894 = ReplacementRule(pattern1894, replacement1894)
def With1895(v, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1895 = Pattern(Integral(u_**m_*v_**n_, x_), cons1091, cons683, cons1094, cons1095, CustomConstraint(With1895))
def replacement1895(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1895)
return -Dist(n*(-a*v + b*u)/(a*(m + n + S(1))), Int(u**m*v**(n + S(-1)), x), x) + Simp(u**(m + S(1))*v**n/(a*(m + n + S(1))), x)
rule1895 = ReplacementRule(pattern1895, replacement1895)
def With1896(v, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1896 = Pattern(Integral(u_**m_*v_**n_, x_), cons1091, cons214, cons31, cons94, CustomConstraint(With1896))
def replacement1896(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1896)
return Dist(b*(m + n + S(2))/((m + S(1))*(-a*v + b*u)), Int(u**(m + S(1))*v**n, x), x) - Simp(u**(m + S(1))*v**(n + S(1))/((m + S(1))*(-a*v + b*u)), x)
rule1896 = ReplacementRule(pattern1896, replacement1896)
def With1897(v, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1897 = Pattern(Integral(u_**m_*v_**n_, x_), cons1091, cons356, cons1096, CustomConstraint(With1897))
def replacement1897(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1897)
return Dist(b*(m + n + S(2))/((m + S(1))*(-a*v + b*u)), Int(u**(m + S(1))*v**n, x), x) - Simp(u**(m + S(1))*v**(n + S(1))/((m + S(1))*(-a*v + b*u)), x)
rule1897 = ReplacementRule(pattern1897, replacement1897)
def With1898(v, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
a = D(u, x)
b = D(v, x)
if NonzeroQ(-a*v + b*u):
return True
return False
pattern1898 = Pattern(Integral(u_**m_*v_**n_, x_), cons1091, cons18, cons23, CustomConstraint(With1898))
def replacement1898(v, u, m, n, x):
a = D(u, x)
b = D(v, x)
rubi.append(1898)
return Simp(u**m*v**(n + S(1))*(b*u/(-a*v + b*u))**(-m)*Hypergeometric2F1(-m, n + S(1), n + S(2), -a*v/(-a*v + b*u))/(b*(n + S(1))), x)
rule1898 = ReplacementRule(pattern1898, replacement1898)
def With1899(u, b, a, n, x):
c = D(u, x)
rubi.append(1899)
return -Dist(c*n/b, Int(u**(n + S(-1))*(a + b*x)*log(a + b*x), x), x) - Int(u**n, x) + Simp(u**n*(a + b*x)*log(a + b*x)/b, x)
pattern1899 = Pattern(Integral(u_**WC('n', S(1))*log(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons1090, cons1097, cons87, cons88)
rule1899 = ReplacementRule(pattern1899, With1899)
def With1900(u, m, b, a, n, x):
c = D(u, x)
rubi.append(1900)
return -Dist(c*n/(b*(m + S(1))), Int(u**(n + S(-1))*(a + b*x)**(m + S(1))*log(a + b*x), x), x) - Dist(S(1)/(m + S(1)), Int(u**n*(a + b*x)**m, x), x) + Simp(u**n*(a + b*x)**(m + S(1))*log(a + b*x)/(b*(m + S(1))), x)
pattern1900 = Pattern(Integral(u_**WC('n', S(1))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*log(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons1090, cons1097, cons87, cons88, cons66)
rule1900 = ReplacementRule(pattern1900, With1900)
return [rule1882, rule1883, rule1884, rule1885, rule1886, rule1887, rule1888, rule1889, rule1890, rule1891, rule1892, rule1893, rule1894, rule1895, rule1896, rule1897, rule1898, rule1899, rule1900, ]
|
2219edf07e7cefe01595b56eda1f7440d85a9660e015acec1b0e6792619ab3b0 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def miscellaneous_integration(rubi):
from sympy.integrals.rubi.constraints import cons147, cons2002, cons2, cons3, cons7, cons4, cons5, cons386, cons27, cons50, cons2003, cons2004, cons2005, cons2006, cons48, cons125, cons208, cons34, cons35, cons36, cons1099, cons2007, cons66, cons21, cons84, cons1037, cons1036, cons38, cons2008, cons10, cons2009, cons2010, cons2011, cons209, cons1831, cons1244, cons2012, cons46, cons2013, cons2014, cons2015, cons2016, cons52, cons2017, cons800, cons2018, cons17, cons2019, cons586, cons2020, cons2021, cons2022, cons2023, cons2024, cons2025, cons2026, cons2027, cons2028, cons667, cons196, cons2029, cons840, cons2030, cons18, cons2031, cons148, cons45, cons2032, cons1854, cons1247, cons261, cons2033, cons367, cons2034, cons67, cons1479, cons744, cons1482, cons165, cons2035, cons2036, cons1676, cons1255, cons2037, cons347
pattern6931 = Pattern(Integral(u_*((x_*WC('b', S(1)) + WC('a', S(0)))**n_*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons147, cons2002)
def replacement6931(p, u, b, c, a, n, x):
rubi.append(6931)
return Dist(c**IntPart(p)*(c*(a + b*x)**n)**FracPart(p)*(a + b*x)**(-n*FracPart(p)), Int(u*(a + b*x)**(n*p), x), x)
rule6931 = ReplacementRule(pattern6931, replacement6931)
pattern6932 = Pattern(Integral(((d_*(x_*WC('b', S(1)) + WC('a', S(0))))**p_*WC('c', S(1)))**q_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons50, cons147, cons386)
def replacement6932(p, u, b, d, c, a, x, q):
rubi.append(6932)
return Dist((c*(d*(a + b*x))**p)**q*(a + b*x)**(-p*q), Int(u*(a + b*x)**(p*q), x), x)
rule6932 = ReplacementRule(pattern6932, replacement6932)
pattern6933 = Pattern(Integral((((x_*WC('b', S(1)) + WC('a', S(0)))**n_*WC('d', S(1)))**p_*WC('c', S(1)))**q_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons147, cons386)
def replacement6933(p, u, b, d, c, a, n, x, q):
rubi.append(6933)
return Dist((c*(d*(a + b*x)**n)**p)**q*(a + b*x)**(-n*p*q), Int(u*(a + b*x)**(n*p*q), x), x)
rule6933 = ReplacementRule(pattern6933, replacement6933)
pattern6934 = Pattern(Integral((F_*sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*WC('b', S(1))*WC('c', S(1))/sqrt(x_*WC('g', S(1)) + WC('f', S(0))) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons34, cons35, cons36, cons1099, cons2003, cons2004, cons2005, cons2006)
def replacement6934(B, C, e, f, b, g, d, a, c, n, x, F, A):
rubi.append(6934)
return Dist(g/C, Subst(Int((a + b*F(c*x))**n/x, x), x, sqrt(d + e*x)/sqrt(f + g*x)), x)
rule6934 = ReplacementRule(pattern6934, replacement6934)
pattern6935 = Pattern(Integral((F_*sqrt(x_*WC('e', S(1)) + S(1))*WC('b', S(1))*WC('c', S(1))/sqrt(x_*WC('g', S(1)) + S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*WC('C', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons48, cons208, cons34, cons36, cons1099, cons2003, cons2007)
def replacement6935(C, e, g, b, c, a, n, x, F, A):
rubi.append(6935)
return Dist(g/C, Subst(Int((a + b*F(c*x))**n/x, x), x, sqrt(e*x + S(1))/sqrt(g*x + S(1))), x)
rule6935 = ReplacementRule(pattern6935, replacement6935)
pattern6936 = Pattern(Integral((F_**(sqrt(x_*WC('e', S(1)) + WC('d', S(0)))*WC('c', S(1))/sqrt(x_*WC('g', S(1)) + WC('f', S(0))))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons34, cons35, cons36, cons1099, cons2003, cons2004, cons2005, cons2006)
def replacement6936(B, C, f, b, g, d, c, a, n, A, x, F, e):
rubi.append(6936)
return Dist(g/C, Subst(Int((F**(c*x)*b + a)**n/x, x), x, sqrt(d + e*x)/sqrt(f + g*x)), x)
rule6936 = ReplacementRule(pattern6936, replacement6936)
pattern6937 = Pattern(Integral((F_**(sqrt(x_*WC('e', S(1)) + S(1))*WC('c', S(1))/sqrt(x_*WC('g', S(1)) + S(1)))*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/(x_**S(2)*WC('C', S(1)) + WC('A', S(0))), x_), cons2, cons3, cons7, cons48, cons208, cons34, cons36, cons1099, cons2003, cons2007)
def replacement6937(C, g, b, c, a, n, A, x, F, e):
rubi.append(6937)
return Dist(g/C, Subst(Int((F**(c*x)*b + a)**n/x, x), x, sqrt(e*x + S(1))/sqrt(g*x + S(1))), x)
rule6937 = ReplacementRule(pattern6937, replacement6937)
def With6938(y, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6938 = Pattern(Integral(u_/y_, x_), CustomConstraint(With6938))
def replacement6938(y, x, u):
q = DerivativeDivides(y, u, x)
rubi.append(6938)
return Simp(q*log(RemoveContent(y, x)), x)
rule6938 = ReplacementRule(pattern6938, replacement6938)
def With6939(y, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(w*y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6939 = Pattern(Integral(u_/(w_*y_), x_), CustomConstraint(With6939))
def replacement6939(y, w, u, x):
q = DerivativeDivides(w*y, u, x)
rubi.append(6939)
return Simp(q*log(RemoveContent(w*y, x)), x)
rule6939 = ReplacementRule(pattern6939, replacement6939)
def With6940(y, m, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6940 = Pattern(Integral(u_*y_**WC('m', S(1)), x_), cons21, cons66, CustomConstraint(With6940))
def replacement6940(y, m, u, x):
q = DerivativeDivides(y, u, x)
rubi.append(6940)
return Simp(q*y**(m + S(1))/(m + S(1)), x)
rule6940 = ReplacementRule(pattern6940, replacement6940)
def With6941(z, y, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y*z, u*z**(-m + n), x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6941 = Pattern(Integral(u_*y_**WC('m', S(1))*z_**WC('n', S(1)), x_), cons21, cons4, cons66, CustomConstraint(With6941))
def replacement6941(z, y, u, m, n, x):
q = DerivativeDivides(y*z, u*z**(-m + n), x)
rubi.append(6941)
return Simp(q*y**(m + S(1))*z**(m + S(1))/(m + S(1)), x)
rule6941 = ReplacementRule(pattern6941, replacement6941)
def With6942(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
v = SimplifyIntegrand(u, x)
if SimplerIntegrandQ(v, u, x):
return True
return False
pattern6942 = Pattern(Integral(u_, x_), CustomConstraint(With6942))
def replacement6942(x, u):
v = SimplifyIntegrand(u, x)
rubi.append(6942)
return Int(v, x)
rule6942 = ReplacementRule(pattern6942, replacement6942)
pattern6943 = Pattern(Integral((sqrt(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_**WC('n', S(1))*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1)))**m_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons84, cons1037)
def replacement6943(u, m, f, b, d, a, n, c, x, e):
rubi.append(6943)
return Dist((a*e**S(2) - c*f**S(2))**m, Int(ExpandIntegrand(u*(e*sqrt(a + b*x**n) - f*sqrt(c + d*x**n))**(-m), x), x), x)
rule6943 = ReplacementRule(pattern6943, replacement6943)
pattern6944 = Pattern(Integral((sqrt(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_**WC('n', S(1))*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1)))**m_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons84, cons1036)
def replacement6944(u, m, f, b, d, a, n, c, x, e):
rubi.append(6944)
return Dist((b*e**S(2) - d*f**S(2))**m, Int(ExpandIntegrand(u*x**(m*n)*(e*sqrt(a + b*x**n) - f*sqrt(c + d*x**n))**(-m), x), x), x)
rule6944 = ReplacementRule(pattern6944, replacement6944)
pattern6945 = Pattern(Integral(u_**WC('m', S(1))*w_*(u_**n_*WC('a', S(1)) + v_)**WC('p', S(1)), x_), cons2, cons21, cons4, cons38, cons2008, cons10)
def replacement6945(v, w, p, u, m, a, n, x):
rubi.append(6945)
return Int(u**(m + n*p)*w*(a + u**(-n)*v)**p, x)
rule6945 = ReplacementRule(pattern6945, replacement6945)
def With6946(v, u, y, m, b, d, c, n, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6946 = Pattern(Integral(u_*(v_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(y_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons2009, CustomConstraint(With6946))
def replacement6946(v, u, y, m, b, d, c, n, a, x):
q = DerivativeDivides(y, u, x)
rubi.append(6946)
return Dist(q, Subst(Int((a + b*x)**m*(c + d*x)**n, x), x, y), x)
rule6946 = ReplacementRule(pattern6946, replacement6946)
def With6947(v, w, p, u, y, m, f, b, d, c, n, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6947 = Pattern(Integral(u_*(v_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(w_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*(y_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons2009, cons2010, CustomConstraint(With6947))
def replacement6947(v, w, p, u, y, m, f, b, d, c, n, a, x, e):
q = DerivativeDivides(y, u, x)
rubi.append(6947)
return Dist(q, Subst(Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p, x), x, y), x)
rule6947 = ReplacementRule(pattern6947, replacement6947)
def With6948(v, z, w, p, u, y, m, f, b, g, d, c, n, a, x, h, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
r = DerivativeDivides(y, u, x)
res = Not(FalseQ(r))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6948 = Pattern(Integral(u_*(v_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(w_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*(y_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(z_*WC('h', S(1)) + WC('g', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons50, cons2009, cons2010, cons2011, CustomConstraint(With6948))
def replacement6948(v, z, w, p, u, y, m, f, b, g, d, c, n, a, x, h, q, e):
r = DerivativeDivides(y, u, x)
rubi.append(6948)
return Dist(r, Subst(Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p*(g + h*x)**q, x), x, y), x)
rule6948 = ReplacementRule(pattern6948, replacement6948)
def With6949(y, u, b, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6949 = Pattern(Integral((a_ + y_**n_*WC('b', S(1)))*WC('u', S(1)), x_), cons2, cons3, cons4, cons1831, CustomConstraint(With6949))
def replacement6949(y, u, b, a, n, x):
q = DerivativeDivides(y, u, x)
rubi.append(6949)
return Dist(a, Int(u, x), x) + Dist(b*q, Subst(Int(x**n, x), x, y), x)
rule6949 = ReplacementRule(pattern6949, replacement6949)
def With6950(p, y, u, b, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6950 = Pattern(Integral((y_**n_*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons4, cons5, cons1244, CustomConstraint(With6950))
def replacement6950(p, y, u, b, a, n, x):
q = DerivativeDivides(y, u, x)
rubi.append(6950)
return Dist(q, Subst(Int((a + b*x**n)**p, x), x, y), x)
rule6950 = ReplacementRule(pattern6950, replacement6950)
def With6951(v, p, u, y, m, b, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = Symbol('q')
r = Symbol('r')
r = Divides(y**m, v**m, x)
q = DerivativeDivides(y, u, x)
res = And(Not(FalseQ(Set(r, Divides(y**m, v**m, x)))), Not(FalseQ(Set(q, DerivativeDivides(y, u, x)))))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6951 = Pattern(Integral(v_**WC('m', S(1))*(y_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons21, cons4, cons5, cons2012, CustomConstraint(With6951))
def replacement6951(v, p, u, y, m, b, a, n, x):
q = Symbol('q')
r = Symbol('r')
r = Divides(y**m, v**m, x)
q = DerivativeDivides(y, u, x)
rubi.append(6951)
return Dist(q*r, Subst(Int(x**m*(a + b*x**n)**p, x), x, y), x)
rule6951 = ReplacementRule(pattern6951, replacement6951)
def With6952(v, p, u, y, b, n2, a, c, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6952 = Pattern(Integral((v_**WC('n2', S(1))*WC('c', S(1)) + y_**n_*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons46, cons2009, CustomConstraint(With6952))
def replacement6952(v, p, u, y, b, n2, a, c, n, x):
q = DerivativeDivides(y, u, x)
rubi.append(6952)
return Dist(q, Subst(Int((a + b*x**n + c*x**(S(2)*n))**p, x), x, y), x)
rule6952 = ReplacementRule(pattern6952, replacement6952)
def With6953(B, v, w, p, u, y, b, n2, c, a, n, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6953 = Pattern(Integral((A_ + y_**n_*WC('B', S(1)))*(v_**n_*WC('b', S(1)) + w_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons5, cons46, cons2009, cons2010, CustomConstraint(With6953))
def replacement6953(B, v, w, p, u, y, b, n2, c, a, n, x, A):
q = DerivativeDivides(y, u, x)
rubi.append(6953)
return Dist(q, Subst(Int((A + B*x**n)*(a + b*x**n + c*x**(S(2)*n))**p, x), x, y), x)
rule6953 = ReplacementRule(pattern6953, replacement6953)
def With6954(B, w, p, u, y, n2, a, c, n, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6954 = Pattern(Integral((A_ + y_**n_*WC('B', S(1)))*(w_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons7, cons34, cons35, cons4, cons5, cons46, cons2010, CustomConstraint(With6954))
def replacement6954(B, w, p, u, y, n2, a, c, n, x, A):
q = DerivativeDivides(y, u, x)
rubi.append(6954)
return Dist(q, Subst(Int((A + B*x**n)*(a + c*x**(S(2)*n))**p, x), x, y), x)
rule6954 = ReplacementRule(pattern6954, replacement6954)
def With6955(v, w, p, u, y, m, b, n2, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = Symbol('q')
r = Symbol('r')
r = Divides(y**m, v**m, x)
q = DerivativeDivides(y, u, x)
res = And(Not(FalseQ(Set(r, Divides(y**m, v**m, x)))), Not(FalseQ(Set(q, DerivativeDivides(y, u, x)))))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6955 = Pattern(Integral(v_**WC('m', S(1))*(w_**WC('n2', S(1))*WC('c', S(1)) + y_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons46, cons2010, CustomConstraint(With6955))
def replacement6955(v, w, p, u, y, m, b, n2, c, a, n, x):
q = Symbol('q')
r = Symbol('r')
r = Divides(y**m, v**m, x)
q = DerivativeDivides(y, u, x)
rubi.append(6955)
return Dist(q*r, Subst(Int(x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x, y), x)
rule6955 = ReplacementRule(pattern6955, replacement6955)
def With6956(B, v, w, p, z, u, y, m, b, n2, c, a, n, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = Symbol('q')
r = Symbol('r')
r = Divides(y**m, z**m, x)
q = DerivativeDivides(y, u, x)
res = And(Not(FalseQ(Set(r, Divides(y**m, z**m, x)))), Not(FalseQ(Set(q, DerivativeDivides(y, u, x)))))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6956 = Pattern(Integral(z_**WC('m', S(1))*(A_ + y_**n_*WC('B', S(1)))*(v_**n_*WC('b', S(1)) + w_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons21, cons4, cons5, cons46, cons2009, cons2010, CustomConstraint(With6956))
def replacement6956(B, v, w, p, z, u, y, m, b, n2, c, a, n, x, A):
q = Symbol('q')
r = Symbol('r')
r = Divides(y**m, z**m, x)
q = DerivativeDivides(y, u, x)
rubi.append(6956)
return Dist(q*r, Subst(Int(x**m*(A + B*x**n)*(a + b*x**n + c*x**(S(2)*n))**p, x), x, y), x)
rule6956 = ReplacementRule(pattern6956, replacement6956)
def With6957(B, z, w, p, u, y, m, n2, a, c, n, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = Symbol('q')
r = Symbol('r')
r = Divides(y**m, z**m, x)
q = DerivativeDivides(y, u, x)
res = And(Not(FalseQ(Set(r, Divides(y**m, z**m, x)))), Not(FalseQ(Set(q, DerivativeDivides(y, u, x)))))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6957 = Pattern(Integral(z_**WC('m', S(1))*(A_ + y_**n_*WC('B', S(1)))*(w_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons7, cons34, cons35, cons21, cons4, cons5, cons46, cons2010, CustomConstraint(With6957))
def replacement6957(B, z, w, p, u, y, m, n2, a, c, n, x, A):
q = Symbol('q')
r = Symbol('r')
r = Divides(y**m, z**m, x)
q = DerivativeDivides(y, u, x)
rubi.append(6957)
return Dist(q*r, Subst(Int(x**m*(A + B*x**n)*(a + c*x**(S(2)*n))**p, x), x, y), x)
rule6957 = ReplacementRule(pattern6957, replacement6957)
def With6958(v, p, u, y, m, b, d, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(y, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6958 = Pattern(Integral((v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('p', S(1))*(y_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons2009, CustomConstraint(With6958))
def replacement6958(v, p, u, y, m, b, d, c, a, n, x):
q = DerivativeDivides(y, u, x)
rubi.append(6958)
return Dist(q, Subst(Int((a + b*x**n)**m*(c + d*x**n)**p, x), x, y), x)
rule6958 = ReplacementRule(pattern6958, replacement6958)
def With6959(v, w, p, u, y, m, f, b, d, c, a, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
r = DerivativeDivides(y, u, x)
res = Not(FalseQ(r))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6959 = Pattern(Integral((v_**n_*WC('d', S(1)) + WC('c', S(0)))**WC('p', S(1))*(w_**n_*WC('f', S(1)) + WC('e', S(0)))**WC('q', S(1))*(y_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons2009, cons2010, CustomConstraint(With6959))
def replacement6959(v, w, p, u, y, m, f, b, d, c, a, n, x, q, e):
r = DerivativeDivides(y, u, x)
rubi.append(6959)
return Dist(r, Subst(Int((a + b*x**n)**m*(c + d*x**n)**p*(e + f*x**n)**q, x), x, y), x)
rule6959 = ReplacementRule(pattern6959, replacement6959)
def With6960(v, x, u, F):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(v, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6960 = Pattern(Integral(F_**v_*u_, x_), cons1099, cons1099, CustomConstraint(With6960))
def replacement6960(v, x, u, F):
q = DerivativeDivides(v, u, x)
rubi.append(6960)
return Simp(F**v*q/log(F), x)
rule6960 = ReplacementRule(pattern6960, replacement6960)
def With6961(v, w, u, m, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(v, u, x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6961 = Pattern(Integral(F_**v_*u_*w_**WC('m', S(1)), x_), cons1099, cons21, cons2013, CustomConstraint(With6961))
def replacement6961(v, w, u, m, x, F):
q = DerivativeDivides(v, u, x)
rubi.append(6961)
return Dist(q, Subst(Int(F**x*x**m, x), x, v), x)
rule6961 = ReplacementRule(pattern6961, replacement6961)
def With6962(v, w, p, u, m, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
c = u/(v*D(w, x) + w*D(v, x))
if FreeQ(c, x):
return True
return False
pattern6962 = Pattern(Integral(u_*(a_ + v_**WC('p', S(1))*w_**WC('p', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons21, cons5, cons38, CustomConstraint(With6962))
def replacement6962(v, w, p, u, m, b, a, x):
c = u/(v*D(w, x) + w*D(v, x))
rubi.append(6962)
return Dist(c, Subst(Int((a + b*x**p)**m, x), x, v*w), x)
rule6962 = ReplacementRule(pattern6962, replacement6962)
def With6963(v, w, p, u, m, b, r, a, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
c = u/(p*w*D(v, x) + q*v*D(w, x))
if FreeQ(c, x):
return True
return False
pattern6963 = Pattern(Integral(u_*v_**WC('r', S(1))*(a_ + v_**WC('p', S(1))*w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons21, cons5, cons50, cons52, cons2014, cons2015, cons2016, CustomConstraint(With6963))
def replacement6963(v, w, p, u, m, b, r, a, x, q):
c = u/(p*w*D(v, x) + q*v*D(w, x))
rubi.append(6963)
return Dist(c*p/(r + S(1)), Subst(Int((a + b*x**(p/(r + S(1))))**m, x), x, v**(r + S(1))*w), x)
rule6963 = ReplacementRule(pattern6963, replacement6963)
def With6964(v, w, p, u, m, b, r, a, x, s, q):
if isinstance(x, (int, Integer, float, Float)):
return False
c = u/(p*w*D(v, x) + q*v*D(w, x))
if FreeQ(c, x):
return True
return False
pattern6964 = Pattern(Integral(u_*v_**WC('r', S(1))*w_**WC('s', S(1))*(a_ + v_**WC('p', S(1))*w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons21, cons5, cons50, cons52, cons800, cons2017, cons2015, cons2016, CustomConstraint(With6964))
def replacement6964(v, w, p, u, m, b, r, a, x, s, q):
c = u/(p*w*D(v, x) + q*v*D(w, x))
rubi.append(6964)
return Dist(c*p/(r + S(1)), Subst(Int((a + b*x**(p/(r + S(1))))**m, x), x, v**(r + S(1))*w**(s + S(1))), x)
rule6964 = ReplacementRule(pattern6964, replacement6964)
def With6965(v, w, p, u, m, b, a, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
c = u/(p*w*D(v, x) - q*v*D(w, x))
if FreeQ(c, x):
return True
return False
pattern6965 = Pattern(Integral(u_*(v_**WC('p', S(1))*WC('a', S(1)) + w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons21, cons5, cons50, cons2018, cons38, cons17, CustomConstraint(With6965))
def replacement6965(v, w, p, u, m, b, a, x, q):
c = u/(p*w*D(v, x) - q*v*D(w, x))
rubi.append(6965)
return Dist(c*p, Subst(Int((a*x**p + b)**m, x), x, v*w**(m*q + S(1))), x)
rule6965 = ReplacementRule(pattern6965, replacement6965)
def With6966(v, w, p, u, m, b, r, a, x, q):
if isinstance(x, (int, Integer, float, Float)):
return False
c = u/(p*w*D(v, x) - q*v*D(w, x))
if FreeQ(c, x):
return True
return False
pattern6966 = Pattern(Integral(u_*v_**WC('r', S(1))*(v_**WC('p', S(1))*WC('a', S(1)) + w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons21, cons5, cons50, cons52, cons2019, cons586, cons17, CustomConstraint(With6966))
def replacement6966(v, w, p, u, m, b, r, a, x, q):
c = u/(p*w*D(v, x) - q*v*D(w, x))
rubi.append(6966)
return -Dist(c*q, Subst(Int((a + b*x**q)**m, x), x, v**(m*p + r + S(1))*w), x)
rule6966 = ReplacementRule(pattern6966, replacement6966)
def With6967(v, w, p, u, m, b, a, x, s, q):
if isinstance(x, (int, Integer, float, Float)):
return False
c = u/(p*w*D(v, x) - q*v*D(w, x))
if FreeQ(c, x):
return True
return False
pattern6967 = Pattern(Integral(u_*w_**WC('s', S(1))*(v_**WC('p', S(1))*WC('a', S(1)) + w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons21, cons5, cons50, cons800, cons2020, cons2021, cons2022, cons17, CustomConstraint(With6967))
def replacement6967(v, w, p, u, m, b, a, x, s, q):
c = u/(p*w*D(v, x) - q*v*D(w, x))
rubi.append(6967)
return -Dist(c*q/(s + S(1)), Subst(Int((a + b*x**(q/(s + S(1))))**m, x), x, v**(m*p + S(1))*w**(s + S(1))), x)
rule6967 = ReplacementRule(pattern6967, replacement6967)
def With6968(v, w, p, u, m, b, r, a, x, s, q):
if isinstance(x, (int, Integer, float, Float)):
return False
c = u/(p*w*D(v, x) - q*v*D(w, x))
if FreeQ(c, x):
return True
return False
pattern6968 = Pattern(Integral(u_*v_**WC('r', S(1))*w_**WC('s', S(1))*(v_**WC('p', S(1))*WC('a', S(1)) + w_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons21, cons5, cons50, cons52, cons800, cons2023, cons2021, cons2022, cons17, CustomConstraint(With6968))
def replacement6968(v, w, p, u, m, b, r, a, x, s, q):
c = u/(p*w*D(v, x) - q*v*D(w, x))
rubi.append(6968)
return -Dist(c*q/(s + S(1)), Subst(Int((a + b*x**(q/(s + S(1))))**m, x), x, v**(m*p + r + S(1))*w**(s + S(1))), x)
rule6968 = ReplacementRule(pattern6968, replacement6968)
pattern6969 = Pattern(Integral(u_*x_**WC('m', S(1)), x_), cons21, cons66, cons2024)
def replacement6969(x, m, u):
rubi.append(6969)
return Dist(S(1)/(m + S(1)), Subst(Int(SubstFor(x**(m + S(1)), u, x), x), x, x**(m + S(1))), x)
rule6969 = ReplacementRule(pattern6969, replacement6969)
def With6970(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = SubstForFractionalPowerOfLinear(u, x)
res = And(Not(FalseQ(lst)), SubstForFractionalPowerQ(u, Part(lst, S(3)), x))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6970 = Pattern(Integral(u_, x_), CustomConstraint(With6970))
def replacement6970(x, u):
lst = SubstForFractionalPowerOfLinear(u, x)
rubi.append(6970)
return Dist(Part(lst, S(2))*Part(lst, S(4)), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(3))**(S(1)/Part(lst, S(2)))), x)
rule6970 = ReplacementRule(pattern6970, replacement6970)
def With6971(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = SubstForFractionalPowerOfQuotientOfLinears(u, x)
res = Not(FalseQ(lst))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6971 = Pattern(Integral(u_, x_), CustomConstraint(With6971))
def replacement6971(x, u):
lst = SubstForFractionalPowerOfQuotientOfLinears(u, x)
rubi.append(6971)
return Dist(Part(lst, S(2))*Part(lst, S(4)), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(3))**(S(1)/Part(lst, S(2)))), x)
rule6971 = ReplacementRule(pattern6971, replacement6971)
pattern6972 = Pattern(Integral((v_**WC('m', S(1))*w_**WC('n', S(1))*z_**WC('q', S(1))*WC('a', S(1)))**p_*WC('u', S(1)), x_), cons2, cons21, cons4, cons5, cons50, cons147, cons10, cons2025, cons2026)
def replacement6972(v, z, w, p, u, m, a, n, x, q):
rubi.append(6972)
return Dist(a**IntPart(p)*v**(-m*FracPart(p))*w**(-n*FracPart(p))*z**(-q*FracPart(p))*(a*v**m*w**n*z**q)**FracPart(p), Int(u*v**(m*p)*w**(n*p)*z**(p*q), x), x)
rule6972 = ReplacementRule(pattern6972, replacement6972)
pattern6973 = Pattern(Integral((v_**WC('m', S(1))*w_**WC('n', S(1))*WC('a', S(1)))**p_*WC('u', S(1)), x_), cons2, cons21, cons4, cons5, cons147, cons10, cons2025)
def replacement6973(v, w, p, u, m, a, n, x):
rubi.append(6973)
return Dist(a**IntPart(p)*v**(-m*FracPart(p))*w**(-n*FracPart(p))*(a*v**m*w**n)**FracPart(p), Int(u*v**(m*p)*w**(n*p), x), x)
rule6973 = ReplacementRule(pattern6973, replacement6973)
pattern6974 = Pattern(Integral((v_**WC('m', S(1))*WC('a', S(1)))**p_*WC('u', S(1)), x_), cons2, cons21, cons5, cons147, cons10, cons2027, cons2028)
def replacement6974(v, p, u, m, a, x):
rubi.append(6974)
return Dist(a**IntPart(p)*v**(-m*FracPart(p))*(a*v**m)**FracPart(p), Int(u*v**(m*p), x), x)
rule6974 = ReplacementRule(pattern6974, replacement6974)
pattern6975 = Pattern(Integral((x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons5, cons667, cons196, cons2029)
def replacement6975(p, u, b, a, n, x):
rubi.append(6975)
return Dist(FullSimplify(x**(-n/S(2))*sqrt(a + b*x**n)/sqrt(a*x**(-n) + b)), Int(u*x**(n*p)*(a*x**(-n) + b)**p, x), x)
rule6975 = ReplacementRule(pattern6975, replacement6975)
pattern6976 = Pattern(Integral((v_**n_*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons5, cons147, cons196, cons840, cons2030)
def replacement6976(v, p, u, b, a, n, x):
rubi.append(6976)
return Dist(v**(-n*FracPart(p))*(a + b*v**n)**FracPart(p)*(a*v**(-n) + b)**(-FracPart(p)), Int(u*v**(n*p)*(a*v**(-n) + b)**p, x), x)
rule6976 = ReplacementRule(pattern6976, replacement6976)
pattern6977 = Pattern(Integral((v_**n_*x_**WC('m', S(1))*WC('b', S(1)) + WC('a', S(0)))**p_*WC('u', S(1)), x_), cons2, cons3, cons21, cons5, cons147, cons196, cons840)
def replacement6977(v, p, u, m, b, a, n, x):
rubi.append(6977)
return Dist(v**(-n*FracPart(p))*(a + b*v**n*x**m)**FracPart(p)*(a*v**(-n) + b*x**m)**(-FracPart(p)), Int(u*v**(n*p)*(a*v**(-n) + b*x**m)**p, x), x)
rule6977 = ReplacementRule(pattern6977, replacement6977)
def With6978(u, m, b, r, a, x, s):
if isinstance(x, (int, Integer, float, Float)):
return False
v = x**(-r*FracPart(m))*(a + b*x**(-r + s))**(-FracPart(m))*(a*x**r + b*x**s)**FracPart(m)
if Not(EqQ(v, S(1))):
return True
return False
pattern6978 = Pattern(Integral((x_**WC('r', S(1))*WC('a', S(1)) + x_**WC('s', S(1))*WC('b', S(1)))**m_*WC('u', S(1)), x_), cons2, cons3, cons21, cons52, cons800, cons18, cons2031, CustomConstraint(With6978))
def replacement6978(u, m, b, r, a, x, s):
v = x**(-r*FracPart(m))*(a + b*x**(-r + s))**(-FracPart(m))*(a*x**r + b*x**s)**FracPart(m)
rubi.append(6978)
return Dist(v, Int(u*x**(m*r)*(a + b*x**(-r + s))**m, x), x)
rule6978 = ReplacementRule(pattern6978, replacement6978)
def With6979(u, b, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
v = RationalFunctionExpand(u/(a + b*x**n), x)
if SumQ(v):
return True
return False
pattern6979 = Pattern(Integral(u_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons148, CustomConstraint(With6979))
def replacement6979(u, b, a, n, x):
v = RationalFunctionExpand(u/(a + b*x**n), x)
rubi.append(6979)
return Int(v, x)
rule6979 = ReplacementRule(pattern6979, replacement6979)
pattern6980 = Pattern(Integral(u_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons46, cons45, cons38, cons2032)
def replacement6980(p, u, b, n2, c, a, n, x):
rubi.append(6980)
return Dist(S(4)**(-p)*c**(-p), Int(u*(b + S(2)*c*x**n)**(S(2)*p), x), x)
rule6980 = ReplacementRule(pattern6980, replacement6980)
pattern6981 = Pattern(Integral(u_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons46, cons45, cons147, cons2032)
def replacement6981(p, u, b, n2, c, a, n, x):
rubi.append(6981)
return Dist((b + S(2)*c*x**n)**(-S(2)*p)*(a + b*x**n + c*x**(S(2)*n))**p, Int(u*(b + S(2)*c*x**n)**(S(2)*p), x), x)
rule6981 = ReplacementRule(pattern6981, replacement6981)
def With6982(u, b, n2, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
v = RationalFunctionExpand(u/(a + b*x**n + c*x**(S(2)*n)), x)
if SumQ(v):
return True
return False
pattern6982 = Pattern(Integral(u_/(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons46, cons148, CustomConstraint(With6982))
def replacement6982(u, b, n2, c, a, n, x):
v = RationalFunctionExpand(u/(a + b*x**n + c*x**(S(2)*n)), x)
rubi.append(6982)
return Int(v, x)
rule6982 = ReplacementRule(pattern6982, replacement6982)
pattern6983 = Pattern(Integral(WC('u', S(1))/(x_**WC('m', S(1))*WC('a', S(1)) + sqrt(x_**n_*WC('c', S(1)))*WC('b', S(1))), x_), cons2, cons3, cons7, cons21, cons4, cons1854)
def replacement6983(u, m, b, a, c, n, x):
rubi.append(6983)
return Int(u*(a*x**m - b*sqrt(c*x**n))/(a**S(2)*x**(S(2)*m) - b**S(2)*c*x**n), x)
rule6983 = ReplacementRule(pattern6983, replacement6983)
def With6984(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = FunctionOfLinear(u, x)
res = Not(FalseQ(lst))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6984 = Pattern(Integral(u_, x_), CustomConstraint(With6984))
def replacement6984(x, u):
lst = FunctionOfLinear(u, x)
rubi.append(6984)
return Dist(S(1)/Part(lst, S(3)), Subst(Int(Part(lst, S(1)), x), x, x*Part(lst, S(3)) + Part(lst, S(2))), x)
rule6984 = ReplacementRule(pattern6984, replacement6984)
def With6985(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = PowerVariableExpn(u, S(0), x)
res = And(Not(FalseQ(lst)), NonzeroQ(Part(lst, S(2))))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6985 = Pattern(Integral(u_/x_, x_), cons1247, cons2029, CustomConstraint(With6985))
def replacement6985(x, u):
lst = PowerVariableExpn(u, S(0), x)
rubi.append(6985)
return Dist(S(1)/Part(lst, S(2)), Subst(Int(NormalizeIntegrand(Part(lst, S(1))/x, x), x), x, (x*Part(lst, S(3)))**Part(lst, S(2))), x)
rule6985 = ReplacementRule(pattern6985, replacement6985)
def With6986(x, m, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = PowerVariableExpn(u, m + S(1), x)
res = And(Not(FalseQ(lst)), NonzeroQ(-m + Part(lst, S(2)) + S(-1)))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6986 = Pattern(Integral(u_*x_**WC('m', S(1)), x_), cons17, cons261, cons1247, cons2033, CustomConstraint(With6986))
def replacement6986(x, m, u):
lst = PowerVariableExpn(u, m + S(1), x)
rubi.append(6986)
return Dist(S(1)/Part(lst, S(2)), Subst(Int(NormalizeIntegrand(Part(lst, S(1))/x, x), x), x, (x*Part(lst, S(3)))**Part(lst, S(2))), x)
rule6986 = ReplacementRule(pattern6986, replacement6986)
def With6987(x, m, u):
k = Denominator(m)
rubi.append(6987)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*ReplaceAll(u, Rule(x, x**k)), x), x, x**(S(1)/k)), x)
pattern6987 = Pattern(Integral(u_*x_**m_, x_), cons367)
rule6987 = ReplacementRule(pattern6987, With6987)
def With6988(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = FunctionOfSquareRootOfQuadratic(u, x)
res = Not(FalseQ(lst))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6988 = Pattern(Integral(u_, x_), cons2034, CustomConstraint(With6988))
def replacement6988(x, u):
lst = FunctionOfSquareRootOfQuadratic(u, x)
rubi.append(6988)
return Dist(S(2), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(2))), x)
rule6988 = ReplacementRule(pattern6988, replacement6988)
pattern6989 = Pattern(Integral(S(1)/(a_ + v_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement6989(v, a, b, x):
rubi.append(6989)
return Dist(S(1)/(S(2)*a), Int(Together(S(1)/(-v/Rt(-a/b, S(2)) + S(1))), x), x) + Dist(S(1)/(S(2)*a), Int(Together(S(1)/(v/Rt(-a/b, S(2)) + S(1))), x), x)
rule6989 = ReplacementRule(pattern6989, replacement6989)
pattern6990 = Pattern(Integral(S(1)/(a_ + v_**n_*WC('b', S(1))), x_), cons2, cons3, cons1479, cons744)
def replacement6990(v, b, a, n, x):
rubi.append(6990)
return Dist(S(2)/(a*n), Sum_doit(Int(Together(S(1)/(S(1) - (S(-1))**(-S(4)*k/n)*v**S(2)/Rt(-a/b, n/S(2)))), x), List(k, S(1), n/S(2))), x)
rule6990 = ReplacementRule(pattern6990, replacement6990)
pattern6991 = Pattern(Integral(S(1)/(a_ + v_**n_*WC('b', S(1))), x_), cons2, cons3, cons1482, cons165)
def replacement6991(v, b, a, n, x):
rubi.append(6991)
return Dist(S(1)/(a*n), Sum_doit(Int(Together(S(1)/(S(1) - (S(-1))**(-S(2)*k/n)*v/Rt(-a/b, n))), x), List(k, S(1), n)), x)
rule6991 = ReplacementRule(pattern6991, replacement6991)
pattern6992 = Pattern(Integral(v_/(a_ + u_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons148, cons2035)
def replacement6992(v, u, b, a, n, x):
rubi.append(6992)
return Int(ReplaceAll(ExpandIntegrand(PolynomialInSubst(v, u, x)/(a + b*x**n), x), Rule(x, u)), x)
rule6992 = ReplacementRule(pattern6992, replacement6992)
def With6993(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
v = NormalizeIntegrand(u, x)
if UnsameQ(v, u):
return True
return False
pattern6993 = Pattern(Integral(u_, x_), CustomConstraint(With6993))
def replacement6993(x, u):
v = NormalizeIntegrand(u, x)
rubi.append(6993)
return Int(v, x)
rule6993 = ReplacementRule(pattern6993, replacement6993)
def With6994(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
v = ExpandIntegrand(u, x)
if SumQ(v):
return True
return False
pattern6994 = Pattern(Integral(u_, x_), CustomConstraint(With6994))
def replacement6994(x, u):
v = ExpandIntegrand(u, x)
rubi.append(6994)
return Int(v, x)
rule6994 = ReplacementRule(pattern6994, replacement6994)
pattern6995 = Pattern(Integral((x_**WC('m', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(x_**WC('n', S(1))*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons50, cons2036, cons1676, cons1255, cons2037)
def replacement6995(p, u, m, b, d, a, c, n, x, q):
rubi.append(6995)
return Dist(x**(-m*p)*(a + b*x**m)**p*(c + d*x**n)**q, Int(u*x**(m*p), x), x)
rule6995 = ReplacementRule(pattern6995, replacement6995)
pattern6996 = Pattern(Integral(u_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons46, cons45, cons347)
def replacement6996(p, u, b, n2, c, n, a, x):
rubi.append(6996)
return Dist((S(4)*c)**(-p + S(1)/2)*sqrt(a + b*x**n + c*x**(S(2)*n))/(b + S(2)*c*x**n), Int(u*(b + S(2)*c*x**n)**(S(2)*p), x), x)
rule6996 = ReplacementRule(pattern6996, replacement6996)
def With6997(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = SubstForFractionalPowerOfLinear(u, x)
res = Not(FalseQ(lst))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6997 = Pattern(Integral(u_, x_), CustomConstraint(With6997))
def replacement6997(x, u):
lst = SubstForFractionalPowerOfLinear(u, x)
rubi.append(6997)
return Dist(Part(lst, S(2))*Part(lst, S(4)), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(3))**(S(1)/Part(lst, S(2)))), x)
rule6997 = ReplacementRule(pattern6997, replacement6997)
pattern6998 = Pattern(Integral(u_, x_))
def replacement6998(x, u):
rubi.append(6998)
return Int(u, x)
rule6998 = ReplacementRule(pattern6998, replacement6998)
return [rule6931, rule6932, rule6933, rule6934, rule6935, rule6936, rule6937, rule6938, rule6939, rule6940, rule6941, rule6942, rule6943, rule6944, rule6945, rule6946, rule6947, rule6948, rule6949, rule6950, rule6951, rule6952, rule6953, rule6954, rule6955, rule6956, rule6957, rule6958, rule6959, rule6960, rule6961, rule6962, rule6963, rule6964, rule6965, rule6966, rule6967, rule6968, rule6969, rule6970, rule6971, rule6972, rule6973, rule6974, rule6975, rule6976, rule6977, rule6978, rule6979, rule6980, rule6981, rule6982, rule6983, rule6984, rule6985, rule6986, rule6987, rule6988, rule6989, rule6990, rule6991, rule6992, rule6993, rule6994, rule6995, rule6996, rule6997, rule6998, ]
|
861b885a0dadfd35dec443d147d7acbea78f51170dcd2151a391041c93c3633a | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def exponential(rubi):
from sympy.integrals.rubi.constraints import cons31, cons168, cons515, cons1098, cons1099, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons94, cons17, cons18, cons21, cons1100, cons128, cons2, cons244, cons137, cons552, cons1101, cons1102, cons5, cons380, cons54, cons1103, cons1104, cons1105, cons209, cons224, cons796, cons797, cons50, cons1106, cons804, cons1107, cons812, cons1108, cons1109, cons1110, cons1111, cons584, cons1112, cons1113, cons479, cons480, cons1114, cons196, cons23, cons1115, cons53, cons1116, cons1117, cons1118, cons1119, cons85, cons1120, cons356, cons531, cons1121, cons1122, cons535, cons93, cons1123, cons1124, cons176, cons367, cons166, cons744, cons68, cons840, cons1125, cons1126, cons1127, cons25, cons71, cons1128, cons1129, cons1130, cons818, cons1131, cons1132, cons1133, cons1134, cons819, cons1135, cons1136, cons1137, cons1138, cons148, cons810, cons811, cons1139, cons1140, cons52, cons800, cons1141, cons1142, cons1143, cons813, cons1144, cons226, cons62, cons1145, cons1146, cons1147, cons1148, cons1149, cons1150, cons1151, cons463, cons1152, cons43, cons448, cons1153, cons1154, cons1155, cons1017
pattern1901 = Pattern(Integral((F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1)))*WC('b', S(1)))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons31, cons168, cons515, cons1098)
def replacement1901(m, f, g, b, d, c, n, x, F, e):
rubi.append(1901)
return -Dist(d*m/(f*g*n*log(F)), Int((F**(g*(e + f*x))*b)**n*(c + d*x)**(m + S(-1)), x), x) + Simp((F**(g*(e + f*x))*b)**n*(c + d*x)**m/(f*g*n*log(F)), x)
rule1901 = ReplacementRule(pattern1901, replacement1901)
pattern1902 = Pattern(Integral((F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1)))*WC('b', S(1)))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons1099, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons31, cons94, cons515, cons1098)
def replacement1902(m, f, g, b, d, c, n, x, F, e):
rubi.append(1902)
return -Dist(f*g*n*log(F)/(d*(m + S(1))), Int((F**(g*(e + f*x))*b)**n*(c + d*x)**(m + S(1)), x), x) + Simp((F**(g*(e + f*x))*b)**n*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule1902 = ReplacementRule(pattern1902, replacement1902)
pattern1903 = Pattern(Integral(F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons1099, cons7, cons27, cons48, cons125, cons208, cons1098)
def replacement1903(f, g, d, c, x, F, e):
rubi.append(1903)
return Simp(F**(g*(-c*f/d + e))*ExpIntegralEi(f*g*(c + d*x)*log(F)/d)/d, x)
rule1903 = ReplacementRule(pattern1903, replacement1903)
pattern1904 = Pattern(Integral(F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons7, cons27, cons48, cons125, cons208, cons17)
def replacement1904(m, f, g, d, c, x, F, e):
rubi.append(1904)
return Simp(F**(g*(-c*f/d + e))*f**(-m + S(-1))*g**(-m + S(-1))*(-d)**m*Gamma(m + S(1), -f*g*(c + d*x)*log(F)/d)*log(F)**(-m + S(-1)), x)
rule1904 = ReplacementRule(pattern1904, replacement1904)
pattern1905 = Pattern(Integral(F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons1099, cons7, cons27, cons48, cons125, cons208, cons1098)
def replacement1905(f, g, d, c, x, F, e):
rubi.append(1905)
return Dist(S(2)/d, Subst(Int(F**(g*(-c*f/d + e) + f*g*x**S(2)/d), x), x, sqrt(c + d*x)), x)
rule1905 = ReplacementRule(pattern1905, replacement1905)
pattern1906 = Pattern(Integral(F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1)))*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons1099, cons7, cons27, cons48, cons125, cons208, cons21, cons18)
def replacement1906(m, f, g, d, c, x, F, e):
rubi.append(1906)
return -Simp(F**(g*(-c*f/d + e))*(-f*g*log(F)/d)**(-IntPart(m) + S(-1))*(-f*g*(c + d*x)*log(F)/d)**(-FracPart(m))*(c + d*x)**FracPart(m)*Gamma(m + S(1), -f*g*(c + d*x)*log(F)/d)/d, x)
rule1906 = ReplacementRule(pattern1906, replacement1906)
pattern1907 = Pattern(Integral((F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1)))*WC('b', S(1)))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons1100)
def replacement1907(m, f, g, b, d, c, n, x, F, e):
rubi.append(1907)
return Dist(F**(-g*n*(e + f*x))*(F**(g*(e + f*x))*b)**n, Int(F**(g*n*(e + f*x))*(c + d*x)**m, x), x)
rule1907 = ReplacementRule(pattern1907, replacement1907)
pattern1908 = Pattern(Integral((a_ + (F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons128)
def replacement1908(p, m, f, g, b, d, c, n, a, x, F, e):
rubi.append(1908)
return Int(ExpandIntegrand((c + d*x)**m, (a + b*(F**(g*(e + f*x)))**n)**p, x), x)
rule1908 = ReplacementRule(pattern1908, replacement1908)
pattern1909 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + (F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons31, cons168)
def replacement1909(m, f, g, b, d, c, n, a, x, F, e):
rubi.append(1909)
return Dist(d*m/(a*f*g*n*log(F)), Int((c + d*x)**(m + S(-1))*log(a*(F**(g*(e + f*x)))**(-n)/b + S(1)), x), x) - Simp((c + d*x)**m*log(a*(F**(g*(e + f*x)))**(-n)/b + S(1))/(a*f*g*n*log(F)), x)
rule1909 = ReplacementRule(pattern1909, replacement1909)
def With1910(p, m, f, g, b, d, c, n, a, x, F, e):
u = IntHide((a + b*(F**(g*(e + f*x)))**n)**p, x)
rubi.append(1910)
return -Dist(d*m, Int(u*(c + d*x)**(m + S(-1)), x), x) + Dist((c + d*x)**m, u, x)
pattern1910 = Pattern(Integral((a_ + (F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1)))**p_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons244, cons168, cons137)
rule1910 = ReplacementRule(pattern1910, With1910)
pattern1911 = Pattern(Integral(u_**WC('m', S(1))*((F_**(v_*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons1099, cons2, cons3, cons208, cons4, cons5, cons552, cons1101, cons1102, cons17)
def replacement1911(v, p, u, m, g, b, a, n, x, F):
rubi.append(1911)
return Int((a + b*(F**(g*ExpandToSum(v, x)))**n)**p*NormalizePowerOfLinear(u, x)**m, x)
rule1911 = ReplacementRule(pattern1911, replacement1911)
def With1912(v, p, u, m, g, b, a, n, x, F):
uu = NormalizePowerOfLinear(u, x)
z = Symbol('z')
z = If(And(PowerQ(uu), FreeQ(Part(uu, S(2)), x)), Part(uu, S(1))**(m*Part(uu, S(2))), uu**m)
z = If(And(PowerQ(uu), FreeQ(Part(uu, 2), x)), Part(uu, 1)**(m*Part(uu, 2)), uu**m)
return Simp(uu**m*Int(z*(a + b*(F**(g*ExpandToSum(v, x)))**n)**p, x)/z, x)
pattern1912 = Pattern(Integral(u_**WC('m', S(1))*((F_**(v_*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons1099, cons2, cons3, cons208, cons21, cons4, cons5, cons552, cons1101, cons1102, cons18)
rule1912 = ReplacementRule(pattern1912, With1912)
pattern1913 = Pattern(Integral((a_ + (F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons380)
def replacement1913(p, m, f, g, b, d, c, n, a, x, F, e):
rubi.append(1913)
return Int((a + b*(F**(g*(e + f*x)))**n)**p*(c + d*x)**m, x)
rule1913 = ReplacementRule(pattern1913, replacement1913)
pattern1914 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))/(a_ + (F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons31, cons168)
def replacement1914(m, f, g, b, d, c, n, a, x, F, e):
rubi.append(1914)
return -Dist(d*m/(b*f*g*n*log(F)), Int((c + d*x)**(m + S(-1))*log(S(1) + b*(F**(g*(e + f*x)))**n/a), x), x) + Simp((c + d*x)**m*log(S(1) + b*(F**(g*(e + f*x)))**n/a)/(b*f*g*n*log(F)), x)
rule1914 = ReplacementRule(pattern1914, replacement1914)
pattern1915 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*((F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons54)
def replacement1915(p, m, f, g, b, d, a, n, c, x, F, e):
rubi.append(1915)
return -Dist(d*m/(b*f*g*n*(p + S(1))*log(F)), Int((a + b*(F**(g*(e + f*x)))**n)**(p + S(1))*(c + d*x)**(m + S(-1)), x), x) + Simp((a + b*(F**(g*(e + f*x)))**n)**(p + S(1))*(c + d*x)**m/(b*f*g*n*(p + S(1))*log(F)), x)
rule1915 = ReplacementRule(pattern1915, replacement1915)
pattern1916 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*((F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*(F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons1103)
def replacement1916(p, m, f, g, b, d, a, n, c, x, F, e):
rubi.append(1916)
return Int((a + b*(F**(g*(e + f*x)))**n)**p*(c + d*x)**m*(F**(g*(e + f*x)))**n, x)
rule1916 = ReplacementRule(pattern1916, replacement1916)
pattern1917 = Pattern(Integral((G_**((x_*WC('i', S(1)) + WC('h', S(0)))*WC('j', S(1)))*WC('k', S(1)))**WC('q', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*((F_**((x_*WC('f', S(1)) + WC('e', S(0)))*WC('g', S(1))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons796, cons797, cons21, cons4, cons5, cons50, cons1104, cons1105)
def replacement1917(p, j, k, m, f, g, b, i, d, G, a, n, c, x, h, q, e, F):
rubi.append(1917)
return Dist((G**(j*(h + i*x))*k)**q*(F**(g*(e + f*x)))**(-n), Int((a + b*(F**(g*(e + f*x)))**n)**p*(c + d*x)**m*(F**(g*(e + f*x)))**n, x), x)
rule1917 = ReplacementRule(pattern1917, replacement1917)
pattern1918 = Pattern(Integral((F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons4, cons1106)
def replacement1918(b, c, n, a, x, F):
rubi.append(1918)
return Simp((F**(c*(a + b*x)))**n/(b*c*n*log(F)), x)
rule1918 = ReplacementRule(pattern1918, replacement1918)
pattern1919 = Pattern(Integral(F_**(v_*WC('c', S(1)))*u_, x_), cons1099, cons7, cons804, cons552, cons1107)
def replacement1919(v, u, c, x, F):
rubi.append(1919)
return Int(ExpandIntegrand(F**(c*ExpandToSum(v, x))*u, x), x)
rule1919 = ReplacementRule(pattern1919, replacement1919)
pattern1920 = Pattern(Integral(F_**(v_*WC('c', S(1)))*u_, x_), cons1099, cons7, cons804, cons552, cons1098)
def replacement1920(v, u, c, x, F):
rubi.append(1920)
return Int(ExpandIntegrand(F**(c*ExpandToSum(v, x)), u, x), x)
rule1920 = ReplacementRule(pattern1920, replacement1920)
pattern1921 = Pattern(Integral(F_**(v_*WC('c', S(1)))*u_**WC('m', S(1))*w_, x_), cons1099, cons7, cons21, cons812, cons1108)
def replacement1921(v, w, u, m, c, x, F):
rubi.append(1921)
return Simp(F**(c*v)*u**(m + S(1))*Coefficient(w, x, S(1))/(c*Coefficient(u, x, S(1))*Coefficient(v, x, S(1))*log(F)), x)
rule1921 = ReplacementRule(pattern1921, replacement1921)
pattern1922 = Pattern(Integral(F_**(v_*WC('c', S(1)))*u_**WC('m', S(1))*w_, x_), cons1099, cons7, cons1109, cons552, cons1101, cons17, cons1107)
def replacement1922(v, w, u, m, c, x, F):
rubi.append(1922)
return Int(ExpandIntegrand(F**(c*ExpandToSum(v, x))*w*NormalizePowerOfLinear(u, x)**m, x), x)
rule1922 = ReplacementRule(pattern1922, replacement1922)
pattern1923 = Pattern(Integral(F_**(v_*WC('c', S(1)))*u_**WC('m', S(1))*w_, x_), cons1099, cons7, cons1109, cons552, cons1101, cons17, cons1098)
def replacement1923(v, w, u, m, c, x, F):
rubi.append(1923)
return Int(ExpandIntegrand(F**(c*ExpandToSum(v, x)), w*NormalizePowerOfLinear(u, x)**m, x), x)
rule1923 = ReplacementRule(pattern1923, replacement1923)
def With1924(v, w, u, m, c, x, F):
uu = NormalizePowerOfLinear(u, x)
z = Symbol('z')
z = If(And(PowerQ(uu), FreeQ(Part(uu, S(2)), x)), Part(uu, S(1))**(m*Part(uu, S(2))), uu**m)
z = If(And(PowerQ(uu), FreeQ(Part(uu, 2), x)), Part(uu, 1)**(m*Part(uu, 2)), uu**m)
return Simp(uu**m*Int(ExpandIntegrand(F**(c*ExpandToSum(v, x))*w*z, x), x)/z, x)
pattern1924 = Pattern(Integral(F_**(v_*WC('c', S(1)))*u_**WC('m', S(1))*w_, x_), cons1099, cons7, cons21, cons1109, cons552, cons1101, cons18)
rule1924 = ReplacementRule(pattern1924, With1924)
pattern1925 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(e_ + (x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1))*log(x_*WC('d', S(1))))*log(x_*WC('d', S(1)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons1110, cons1111, cons584)
def replacement1925(f, b, g, d, c, n, a, x, h, F, e):
rubi.append(1925)
return Simp(F**(c*(a + b*x))*e*x*log(d*x)**(n + S(1))/(n + S(1)), x)
rule1925 = ReplacementRule(pattern1925, replacement1925)
pattern1926 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*x_**WC('m', S(1))*(e_ + (x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1))*log(x_*WC('d', S(1))))*log(x_*WC('d', S(1)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons1112, cons1111, cons584)
def replacement1926(m, f, b, g, d, c, n, a, x, h, F, e):
rubi.append(1926)
return Simp(F**(c*(a + b*x))*e*x**(m + S(1))*log(d*x)**(n + S(1))/(n + S(1)), x)
rule1926 = ReplacementRule(pattern1926, replacement1926)
pattern1927 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons1113)
def replacement1927(b, d, c, a, x, F):
rubi.append(1927)
return Simp(F**(a + b*(c + d*x))/(b*d*log(F)), x)
rule1927 = ReplacementRule(pattern1927, replacement1927)
pattern1928 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**S(2)*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons479)
def replacement1928(b, d, c, a, x, F):
rubi.append(1928)
return Simp(F**a*sqrt(Pi)*Erfi((c + d*x)*Rt(b*log(F), S(2)))/(S(2)*d*Rt(b*log(F), S(2))), x)
rule1928 = ReplacementRule(pattern1928, replacement1928)
pattern1929 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**S(2)*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons480)
def replacement1929(b, d, c, a, x, F):
rubi.append(1929)
return Simp(F**a*sqrt(Pi)*Erf((c + d*x)*Rt(-b*log(F), S(2)))/(S(2)*d*Rt(-b*log(F), S(2))), x)
rule1929 = ReplacementRule(pattern1929, replacement1929)
pattern1930 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons1114, cons196)
def replacement1930(b, d, c, a, n, x, F):
rubi.append(1930)
return -Dist(b*n*log(F), Int(F**(a + b*(c + d*x)**n)*(c + d*x)**n, x), x) + Simp(F**(a + b*(c + d*x)**n)*(c + d*x)/d, x)
rule1930 = ReplacementRule(pattern1930, replacement1930)
def With1931(b, d, c, a, n, x, F):
k = Denominator(n)
rubi.append(1931)
return Dist(k/d, Subst(Int(F**(a + b*x**(k*n))*x**(k + S(-1)), x), x, (c + d*x)**(S(1)/k)), x)
pattern1931 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons1114, cons23)
rule1931 = ReplacementRule(pattern1931, With1931)
pattern1932 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons4, cons1115)
def replacement1932(b, d, c, a, n, x, F):
rubi.append(1932)
return -Simp(F**a*(-b*(c + d*x)**n*log(F))**(-S(1)/n)*(c + d*x)*Gamma(S(1)/n, -b*(c + d*x)**n*log(F))/(d*n), x)
rule1932 = ReplacementRule(pattern1932, replacement1932)
pattern1933 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons53, cons1116)
def replacement1933(m, f, b, d, c, a, n, x, F, e):
rubi.append(1933)
return Simp(F**(a + b*(c + d*x)**n)*(c + d*x)**(-n)*(e + f*x)**n/(b*f*n*log(F)), x)
rule1933 = ReplacementRule(pattern1933, replacement1933)
pattern1934 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))/(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1116)
def replacement1934(f, b, d, c, a, n, x, F, e):
rubi.append(1934)
return Simp(F**a*ExpIntegralEi(b*(c + d*x)**n*log(F))/(f*n), x)
rule1934 = ReplacementRule(pattern1934, replacement1934)
pattern1935 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons21, cons4, cons1117)
def replacement1935(m, b, d, c, a, n, x, F):
rubi.append(1935)
return Dist(S(1)/(d*(m + S(1))), Subst(Int(F**(a + b*x**S(2)), x), x, (c + d*x)**(m + S(1))), x)
rule1935 = ReplacementRule(pattern1935, replacement1935)
pattern1936 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons31, cons1118, cons1119, cons85, cons1120)
def replacement1936(m, b, d, c, a, n, x, F):
rubi.append(1936)
return -Dist((m - n + S(1))/(b*n*log(F)), Int(F**(a + b*(c + d*x)**n)*(c + d*x)**(m - n), x), x) + Simp(F**(a + b*(c + d*x)**n)*(c + d*x)**(m - n + S(1))/(b*d*n*log(F)), x)
rule1936 = ReplacementRule(pattern1936, replacement1936)
pattern1937 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons21, cons4, cons1118, cons1119, cons356, cons531)
def replacement1937(m, b, d, c, a, n, x, F):
rubi.append(1937)
return -Dist((m - n + S(1))/(b*n*log(F)), Int(F**(a + b*(c + d*x)**n)*(c + d*x)**(m - n), x), x) + Simp(F**(a + b*(c + d*x)**n)*(c + d*x)**(m - n + S(1))/(b*d*n*log(F)), x)
rule1937 = ReplacementRule(pattern1937, replacement1937)
pattern1938 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons31, cons1118, cons1121, cons85, cons1122)
def replacement1938(m, b, d, c, a, n, x, F):
rubi.append(1938)
return -Dist(b*n*log(F)/(m + S(1)), Int(F**(a + b*(c + d*x)**n)*(c + d*x)**(m + n), x), x) + Simp(F**(a + b*(c + d*x)**n)*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule1938 = ReplacementRule(pattern1938, replacement1938)
pattern1939 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons21, cons4, cons1118, cons1121, cons356, cons535)
def replacement1939(m, b, d, c, a, n, x, F):
rubi.append(1939)
return -Dist(b*n*log(F)/(m + S(1)), Int(F**(a + b*(c + d*x)**n)*(c + d*x)**(m + n), x), x) + Simp(F**(a + b*(c + d*x)**n)*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule1939 = ReplacementRule(pattern1939, replacement1939)
def With1940(m, b, d, c, a, n, x, F):
k = Denominator(n)
rubi.append(1940)
return Dist(k/d, Subst(Int(F**(a + b*x**(k*n))*x**(k*(m + S(1)) + S(-1)), x), x, (c + d*x)**(S(1)/k)), x)
pattern1940 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons93, cons1118, cons1119, cons23)
rule1940 = ReplacementRule(pattern1940, With1940)
pattern1941 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1116, cons1118, cons1123, cons18, cons1124)
def replacement1941(m, f, b, d, c, a, n, x, F, e):
rubi.append(1941)
return Dist((c + d*x)**(-m)*(e + f*x)**m, Int(F**(a + b*(c + d*x)**n)*(c + d*x)**m, x), x)
rule1941 = ReplacementRule(pattern1941, replacement1941)
pattern1942 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1116)
def replacement1942(m, f, b, d, c, a, n, x, F, e):
rubi.append(1942)
return -Simp(F**a*(-b*(c + d*x)**n*log(F))**(-(m + S(1))/n)*(e + f*x)**(m + S(1))*Gamma((m + S(1))/n, -b*(c + d*x)**n*log(F))/(f*n), x)
rule1942 = ReplacementRule(pattern1942, replacement1942)
pattern1943 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**S(2)*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons176, cons367, cons166)
def replacement1943(m, f, b, d, c, a, x, F, e):
rubi.append(1943)
return Dist((-c*f + d*e)/d, Int(F**(a + b*(c + d*x)**S(2))*(e + f*x)**(m + S(-1)), x), x) - Dist(f**S(2)*(m + S(-1))/(S(2)*b*d**S(2)*log(F)), Int(F**(a + b*(c + d*x)**S(2))*(e + f*x)**(m + S(-2)), x), x) + Simp(F**(a + b*(c + d*x)**S(2))*f*(e + f*x)**(m + S(-1))/(S(2)*b*d**S(2)*log(F)), x)
rule1943 = ReplacementRule(pattern1943, replacement1943)
pattern1944 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**S(2)*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons176, cons31, cons94)
def replacement1944(m, f, b, d, c, a, x, F, e):
rubi.append(1944)
return -Dist(S(2)*b*d**S(2)*log(F)/(f**S(2)*(m + S(1))), Int(F**(a + b*(c + d*x)**S(2))*(e + f*x)**(m + S(2)), x), x) + Dist(S(2)*b*d*(-c*f + d*e)*log(F)/(f**S(2)*(m + S(1))), Int(F**(a + b*(c + d*x)**S(2))*(e + f*x)**(m + S(1)), x), x) + Simp(F**(a + b*(c + d*x)**S(2))*(e + f*x)**(m + S(1))/(f*(m + S(1))), x)
rule1944 = ReplacementRule(pattern1944, replacement1944)
pattern1945 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons176, cons85, cons744, cons31, cons94)
def replacement1945(m, f, b, d, c, a, n, x, F, e):
rubi.append(1945)
return -Dist(b*d*n*log(F)/(f*(m + S(1))), Int(F**(a + b*(c + d*x)**n)*(c + d*x)**(n + S(-1))*(e + f*x)**(m + S(1)), x), x) + Simp(F**(a + b*(c + d*x)**n)*(e + f*x)**(m + S(1))/(f*(m + S(1))), x)
rule1945 = ReplacementRule(pattern1945, replacement1945)
pattern1946 = Pattern(Integral(F_**(WC('a', S(0)) + WC('b', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))/(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons176)
def replacement1946(f, b, d, c, a, x, F, e):
rubi.append(1946)
return Dist(d/f, Int(F**(a + b/(c + d*x))/(c + d*x), x), x) - Dist((-c*f + d*e)/f, Int(F**(a + b/(c + d*x))/((c + d*x)*(e + f*x)), x), x)
rule1946 = ReplacementRule(pattern1946, replacement1946)
pattern1947 = Pattern(Integral(F_**(WC('a', S(0)) + WC('b', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))*(x_*WC('f', S(1)) + WC('e', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons176, cons17, cons94)
def replacement1947(m, f, b, d, c, a, x, F, e):
rubi.append(1947)
return Dist(b*d*log(F)/(f*(m + S(1))), Int(F**(a + b/(c + d*x))*(e + f*x)**(m + S(1))/(c + d*x)**S(2), x), x) + Simp(F**(a + b/(c + d*x))*(e + f*x)**(m + S(1))/(f*(m + S(1))), x)
rule1947 = ReplacementRule(pattern1947, replacement1947)
pattern1948 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))/(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons176)
def replacement1948(f, b, d, c, a, n, x, F, e):
rubi.append(1948)
return Int(F**(a + b*(c + d*x)**n)/(e + f*x), x)
rule1948 = ReplacementRule(pattern1948, replacement1948)
pattern1949 = Pattern(Integral(F_**v_*u_**WC('m', S(1)), x_), cons1099, cons21, cons68, cons840, cons1125)
def replacement1949(v, u, m, x, F):
rubi.append(1949)
return Int(F**ExpandToSum(v, x)*ExpandToSum(u, x)**m, x)
rule1949 = ReplacementRule(pattern1949, replacement1949)
pattern1950 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**n_*WC('b', S(1)) + WC('a', S(0)))*u_, x_), cons1099, cons2, cons3, cons7, cons27, cons4, cons804)
def replacement1950(u, b, d, c, a, n, x, F):
rubi.append(1950)
return Int(ExpandLinearProduct(F**(a + b*(c + d*x)**n), u, c, d, x), x)
rule1950 = ReplacementRule(pattern1950, replacement1950)
pattern1951 = Pattern(Integral(F_**(v_*WC('b', S(1)) + WC('a', S(0)))*WC('u', S(1)), x_), cons1099, cons2, cons3, cons804, cons1126, cons1127)
def replacement1951(v, u, b, a, x, F):
rubi.append(1951)
return Int(F**(a + b*NormalizePowerOfLinear(v, x))*u, x)
rule1951 = ReplacementRule(pattern1951, replacement1951)
pattern1952 = Pattern(Integral(F_**(WC('a', S(0)) + WC('b', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))/((x_*WC('f', S(1)) + WC('e', S(0)))*(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons1116)
def replacement1952(f, b, g, d, c, a, x, h, F, e):
rubi.append(1952)
return -Dist(d/(f*(-c*h + d*g)), Subst(Int(F**(a + b*d*x/(-c*h + d*g) - b*h/(-c*h + d*g))/x, x), x, (g + h*x)/(c + d*x)), x)
rule1952 = ReplacementRule(pattern1952, replacement1952)
pattern1953 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))) + WC('e', S(0)))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons25)
def replacement1953(m, f, b, g, d, c, a, x, h, F, e):
rubi.append(1953)
return Dist(F**(b*f/d + e), Int((g + h*x)**m, x), x)
rule1953 = ReplacementRule(pattern1953, replacement1953)
pattern1954 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))) + WC('e', S(0)))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons71, cons1128)
def replacement1954(m, f, b, g, d, c, a, x, h, F, e):
rubi.append(1954)
return Int(F**(-f*(-a*d + b*c)/(d*(c + d*x)) + (b*f + d*e)/d)*(g + h*x)**m, x)
rule1954 = ReplacementRule(pattern1954, replacement1954)
pattern1955 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))) + WC('e', S(0)))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons71, cons1129)
def replacement1955(f, b, g, d, c, a, x, h, F, e):
rubi.append(1955)
return Dist(d/h, Int(F**(e + f*(a + b*x)/(c + d*x))/(c + d*x), x), x) - Dist((-c*h + d*g)/h, Int(F**(e + f*(a + b*x)/(c + d*x))/((c + d*x)*(g + h*x)), x), x)
rule1955 = ReplacementRule(pattern1955, replacement1955)
pattern1956 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))) + WC('e', S(0)))*(x_*WC('h', S(1)) + WC('g', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons71, cons1129, cons17, cons94)
def replacement1956(m, f, b, g, d, c, a, x, h, F, e):
rubi.append(1956)
return -Dist(f*(-a*d + b*c)*log(F)/(h*(m + S(1))), Int(F**(e + f*(a + b*x)/(c + d*x))*(g + h*x)**(m + S(1))/(c + d*x)**S(2), x), x) + Simp(F**(e + f*(a + b*x)/(c + d*x))*(g + h*x)**(m + S(1))/(h*(m + S(1))), x)
rule1956 = ReplacementRule(pattern1956, replacement1956)
pattern1957 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))) + WC('e', S(0)))/((x_*WC('h', S(1)) + WC('g', S(0)))*(x_*WC('j', S(1)) + WC('i', S(0)))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons1128)
def replacement1957(j, f, b, g, i, d, c, a, x, h, F, e):
rubi.append(1957)
return -Dist(d/(h*(-c*j + d*i)), Subst(Int(F**(e - f*x*(-a*d + b*c)/(-c*j + d*i) + f*(-a*j + b*i)/(-c*j + d*i))/x, x), x, (i + j*x)/(c + d*x)), x)
rule1957 = ReplacementRule(pattern1957, replacement1957)
pattern1958 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons1130)
def replacement1958(b, c, a, x, F):
rubi.append(1958)
return Dist(F**(a - b**S(2)/(S(4)*c)), Int(F**((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x)
rule1958 = ReplacementRule(pattern1958, replacement1958)
pattern1959 = Pattern(Integral(F_**v_, x_), cons1099, cons818, cons1131)
def replacement1959(v, x, F):
rubi.append(1959)
return Int(F**ExpandToSum(v, x), x)
rule1959 = ReplacementRule(pattern1959, replacement1959)
pattern1960 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1132)
def replacement1960(b, d, c, a, x, F, e):
rubi.append(1960)
return Simp(F**(a + b*x + c*x**S(2))*e/(S(2)*c*log(F)), x)
rule1960 = ReplacementRule(pattern1960, replacement1960)
pattern1961 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1132, cons31, cons166)
def replacement1961(m, b, d, c, a, x, F, e):
rubi.append(1961)
return -Dist(e**S(2)*(m + S(-1))/(S(2)*c*log(F)), Int(F**(a + b*x + c*x**S(2))*(d + e*x)**(m + S(-2)), x), x) + Simp(F**(a + b*x + c*x**S(2))*e*(d + e*x)**(m + S(-1))/(S(2)*c*log(F)), x)
rule1961 = ReplacementRule(pattern1961, replacement1961)
pattern1962 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1132)
def replacement1962(b, d, c, a, x, F, e):
rubi.append(1962)
return Simp(F**(a - b**S(2)/(S(4)*c))*ExpIntegralEi((b + S(2)*c*x)**S(2)*log(F)/(S(4)*c))/(S(2)*e), x)
rule1962 = ReplacementRule(pattern1962, replacement1962)
pattern1963 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1132, cons31, cons94)
def replacement1963(m, b, d, c, a, x, F, e):
rubi.append(1963)
return -Dist(S(2)*c*log(F)/(e**S(2)*(m + S(1))), Int(F**(a + b*x + c*x**S(2))*(d + e*x)**(m + S(2)), x), x) + Simp(F**(a + b*x + c*x**S(2))*(d + e*x)**(m + S(1))/(e*(m + S(1))), x)
rule1963 = ReplacementRule(pattern1963, replacement1963)
pattern1964 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1133)
def replacement1964(b, d, c, a, x, F, e):
rubi.append(1964)
return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int(F**(a + b*x + c*x**S(2)), x), x) + Simp(F**(a + b*x + c*x**S(2))*e/(S(2)*c*log(F)), x)
rule1964 = ReplacementRule(pattern1964, replacement1964)
pattern1965 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1133, cons31, cons166)
def replacement1965(m, b, d, c, a, x, F, e):
rubi.append(1965)
return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int(F**(a + b*x + c*x**S(2))*(d + e*x)**(m + S(-1)), x), x) - Dist(e**S(2)*(m + S(-1))/(S(2)*c*log(F)), Int(F**(a + b*x + c*x**S(2))*(d + e*x)**(m + S(-2)), x), x) + Simp(F**(a + b*x + c*x**S(2))*e*(d + e*x)**(m + S(-1))/(S(2)*c*log(F)), x)
rule1965 = ReplacementRule(pattern1965, replacement1965)
pattern1966 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1133, cons31, cons94)
def replacement1966(m, b, d, c, a, x, F, e):
rubi.append(1966)
return -Dist(S(2)*c*log(F)/(e**S(2)*(m + S(1))), Int(F**(a + b*x + c*x**S(2))*(d + e*x)**(m + S(2)), x), x) - Dist((b*e - S(2)*c*d)*log(F)/(e**S(2)*(m + S(1))), Int(F**(a + b*x + c*x**S(2))*(d + e*x)**(m + S(1)), x), x) + Simp(F**(a + b*x + c*x**S(2))*(d + e*x)**(m + S(1))/(e*(m + S(1))), x)
rule1966 = ReplacementRule(pattern1966, replacement1966)
pattern1967 = Pattern(Integral(F_**(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons21, cons1134)
def replacement1967(m, b, d, c, a, x, F, e):
rubi.append(1967)
return Int(F**(a + b*x + c*x**S(2))*(d + e*x)**m, x)
rule1967 = ReplacementRule(pattern1967, replacement1967)
pattern1968 = Pattern(Integral(F_**v_*u_**WC('m', S(1)), x_), cons1099, cons21, cons68, cons818, cons819)
def replacement1968(v, u, m, x, F):
rubi.append(1968)
return Int(F**ExpandToSum(v, x)*ExpandToSum(u, x)**m, x)
rule1968 = ReplacementRule(pattern1968, replacement1968)
def With1969(v, m, b, d, c, a, n, x, F, e):
u = IntHide(F**(e*(c + d*x))*(F**v*b + a)**n, x)
rubi.append(1969)
return -Dist(m, Int(u*x**(m + S(-1)), x), x) + Dist(x**m, u, x)
pattern1969 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*x_**WC('m', S(1))*(F_**v_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1135, cons31, cons168, cons196)
rule1969 = ReplacementRule(pattern1969, With1969)
def With1970(f, b, g, G, d, c, n, a, x, h, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
m = FullSimplify(g*h*log(G)/(d*e*log(F)))
if And(RationalQ(m), GreaterEqual(Abs(m), S(1))):
return True
return False
pattern1970 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**WC('n', S(1)), x_), cons1099, cons1137, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons1136, CustomConstraint(With1970))
def replacement1970(f, b, g, G, d, c, n, a, x, h, F, e):
m = FullSimplify(g*h*log(G)/(d*e*log(F)))
rubi.append(1970)
return Dist(G**(-c*g*h/d + f*h)*Denominator(m)/(d*e*log(F)), Subst(Int(x**(Numerator(m) + S(-1))*(a + b*x**Denominator(m))**n, x), x, F**(e*(c + d*x)/Denominator(m))), x)
rule1970 = ReplacementRule(pattern1970, replacement1970)
def With1971(f, b, g, G, d, c, n, a, x, h, F, e):
if isinstance(x, (int, Integer, float, Float)):
return False
m = FullSimplify(d*e*log(F)/(g*h*log(G)))
if And(RationalQ(m), Greater(Abs(m), S(1))):
return True
return False
pattern1971 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**WC('n', S(1)), x_), cons1099, cons1137, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons1136, CustomConstraint(With1971))
def replacement1971(f, b, g, G, d, c, n, a, x, h, F, e):
m = FullSimplify(d*e*log(F)/(g*h*log(G)))
rubi.append(1971)
return Dist(Denominator(m)/(g*h*log(G)), Subst(Int(x**(Denominator(m) + S(-1))*(F**(c*e - d*e*f/g)*b*x**Numerator(m) + a)**n, x), x, G**(h*(f + g*x)/Denominator(m))), x)
rule1971 = ReplacementRule(pattern1971, replacement1971)
pattern1972 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**WC('n', S(1)), x_), cons1099, cons1137, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons1138, cons148)
def replacement1972(f, b, g, G, d, c, n, a, x, h, F, e):
rubi.append(1972)
return Int(G**(f*h)*G**(g*h*x)*(F**(c*e)*F**(d*e*x)*b + a)**n, x)
rule1972 = ReplacementRule(pattern1972, replacement1972)
pattern1973 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**n_, x_), cons1099, cons1137, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons1138, cons196)
def replacement1973(f, b, g, G, d, c, a, n, x, h, F, e):
rubi.append(1973)
return Simp(G**(h*(f + g*x))*a**n*Hypergeometric2F1(-n, g*h*log(G)/(d*e*log(F)), S(1) + g*h*log(G)/(d*e*log(F)), -F**(e*(c + d*x))*b/a)/(g*h*log(G)), x)
rule1973 = ReplacementRule(pattern1973, replacement1973)
pattern1974 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**n_, x_), cons1099, cons1137, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons1138, cons23)
def replacement1974(f, b, g, G, d, c, a, n, x, h, F, e):
rubi.append(1974)
return Simp(G**(h*(f + g*x))*(F**(e*(c + d*x))*b + a)**(n + S(1))*Hypergeometric2F1(S(1), n + S(1) + g*h*log(G)/(d*e*log(F)), S(1) + g*h*log(G)/(d*e*log(F)), -F**(e*(c + d*x))*b/a)/(a*g*h*log(G)), x)
rule1974 = ReplacementRule(pattern1974, replacement1974)
pattern1975 = Pattern(Integral(G_**(u_*WC('h', S(1)))*(F_**(v_*WC('e', S(1)))*WC('b', S(1)) + a_)**n_, x_), cons1099, cons1137, cons2, cons3, cons48, cons209, cons4, cons810, cons811)
def replacement1975(v, u, b, G, a, n, x, h, F, e):
rubi.append(1975)
return Int(G**(h*ExpandToSum(u, x))*(F**(e*ExpandToSum(v, x))*b + a)**n, x)
rule1975 = ReplacementRule(pattern1975, replacement1975)
def With1976(t, f, b, g, r, G, d, c, n, a, H, x, h, s, e, F):
if isinstance(x, (int, Integer, float, Float)):
return False
m = FullSimplify((g*h*log(G) + s*t*log(H))/(d*e*log(F)))
if RationalQ(m):
return True
return False
pattern1976 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*H_**((x_*WC('s', S(1)) + WC('r', S(0)))*WC('t', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**WC('n', S(1)), x_), cons1099, cons1137, cons1140, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons52, cons800, cons1141, cons4, cons1139, CustomConstraint(With1976))
def replacement1976(t, f, b, g, r, G, d, c, n, a, H, x, h, s, e, F):
m = FullSimplify((g*h*log(G) + s*t*log(H))/(d*e*log(F)))
rubi.append(1976)
return Dist(G**(-c*g*h/d + f*h)*H**(-c*s*t/d + r*t)*Denominator(m)/(d*e*log(F)), Subst(Int(x**(Numerator(m) + S(-1))*(a + b*x**Denominator(m))**n, x), x, F**(e*(c + d*x)/Denominator(m))), x)
rule1976 = ReplacementRule(pattern1976, replacement1976)
pattern1977 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*H_**((x_*WC('s', S(1)) + WC('r', S(0)))*WC('t', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**WC('n', S(1)), x_), cons1099, cons1137, cons1140, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons52, cons800, cons1141, cons1142, cons85)
def replacement1977(t, f, b, g, r, G, d, c, n, a, H, x, h, s, e, F):
rubi.append(1977)
return Dist(G**(h*(-c*g/d + f)), Int(H**(t*(r + s*x))*(b + F**(-e*(c + d*x))*a)**n, x), x)
rule1977 = ReplacementRule(pattern1977, replacement1977)
pattern1978 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*H_**((x_*WC('s', S(1)) + WC('r', S(0)))*WC('t', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**WC('n', S(1)), x_), cons1099, cons1137, cons1140, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons52, cons800, cons1141, cons1143, cons148)
def replacement1978(t, f, b, g, r, G, d, c, n, a, H, x, h, s, e, F):
rubi.append(1978)
return Int(G**(f*h)*G**(g*h*x)*H**(r*t)*H**(s*t*x)*(F**(c*e)*F**(d*e*x)*b + a)**n, x)
rule1978 = ReplacementRule(pattern1978, replacement1978)
pattern1979 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*H_**((x_*WC('s', S(1)) + WC('r', S(0)))*WC('t', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**n_, x_), cons1099, cons1137, cons1140, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons52, cons800, cons1141, cons1143, cons196)
def replacement1979(t, f, b, g, r, G, d, c, a, n, H, x, h, s, e, F):
rubi.append(1979)
return Simp(G**(h*(f + g*x))*H**(t*(r + s*x))*a**n*Hypergeometric2F1(-n, (g*h*log(G) + s*t*log(H))/(d*e*log(F)), S(1) + (g*h*log(G) + s*t*log(H))/(d*e*log(F)), -F**(e*(c + d*x))*b/a)/(g*h*log(G) + s*t*log(H)), x)
rule1979 = ReplacementRule(pattern1979, replacement1979)
pattern1980 = Pattern(Integral(G_**((x_*WC('g', S(1)) + WC('f', S(0)))*WC('h', S(1)))*H_**((x_*WC('s', S(1)) + WC('r', S(0)))*WC('t', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + a_)**n_, x_), cons1099, cons1137, cons1140, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons52, cons800, cons1141, cons4, cons1143, cons23)
def replacement1980(t, f, b, g, r, G, d, c, a, n, H, x, h, s, e, F):
rubi.append(1980)
return Simp(G**(h*(f + g*x))*H**(t*(r + s*x))*((F**(e*(c + d*x))*b + a)/a)**(-n)*(F**(e*(c + d*x))*b + a)**n*Hypergeometric2F1(-n, (g*h*log(G) + s*t*log(H))/(d*e*log(F)), S(1) + (g*h*log(G) + s*t*log(H))/(d*e*log(F)), -F**(e*(c + d*x))*b/a)/(g*h*log(G) + s*t*log(H)), x)
rule1980 = ReplacementRule(pattern1980, replacement1980)
pattern1981 = Pattern(Integral(G_**(u_*WC('h', S(1)))*H_**(w_*WC('t', S(1)))*(F_**(v_*WC('e', S(1)))*WC('b', S(1)) + a_)**n_, x_), cons1099, cons1137, cons1140, cons2, cons3, cons48, cons209, cons1141, cons4, cons812, cons813)
def replacement1981(v, w, u, t, b, G, a, n, H, x, h, F, e):
rubi.append(1981)
return Int(G**(h*ExpandToSum(u, x))*H**(t*ExpandToSum(w, x))*(F**(e*ExpandToSum(v, x))*b + a)**n, x)
rule1981 = ReplacementRule(pattern1981, replacement1981)
pattern1982 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + x_**WC('n', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons54)
def replacement1982(p, b, d, c, a, n, x, F, e):
rubi.append(1982)
return -Dist(a*n/(b*d*e*log(F)), Int(x**(n + S(-1))*(F**(e*(c + d*x))*b + a*x**n)**p, x), x) + Simp((F**(e*(c + d*x))*b + a*x**n)**(p + S(1))/(b*d*e*(p + S(1))*log(F)), x)
rule1982 = ReplacementRule(pattern1982, replacement1982)
pattern1983 = Pattern(Integral(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*x_**WC('m', S(1))*(F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1)))*WC('b', S(1)) + x_**WC('n', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons54)
def replacement1983(p, m, b, d, c, a, n, x, F, e):
rubi.append(1983)
return -Dist(a*n/(b*d*e*log(F)), Int(x**(m + n + S(-1))*(F**(e*(c + d*x))*b + a*x**n)**p, x), x) - Dist(m/(b*d*e*(p + S(1))*log(F)), Int(x**(m + S(-1))*(F**(e*(c + d*x))*b + a*x**n)**(p + S(1)), x), x) + Simp(x**m*(F**(e*(c + d*x))*b + a*x**n)**(p + S(1))/(b*d*e*(p + S(1))*log(F)), x)
rule1983 = ReplacementRule(pattern1983, replacement1983)
def With1984(v, u, m, f, b, g, c, a, x, F):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1984)
return Dist(S(2)*c/q, Int((f + g*x)**m/(S(2)*F**u*c + b - q), x), x) - Dist(S(2)*c/q, Int((f + g*x)**m/(S(2)*F**u*c + b + q), x), x)
pattern1984 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/(F_**u_*WC('b', S(1)) + F_**v_*WC('c', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons125, cons208, cons1144, cons68, cons226, cons62)
rule1984 = ReplacementRule(pattern1984, With1984)
def With1985(v, u, m, f, b, g, c, a, x, F):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1985)
return Dist(S(2)*c/q, Int(F**u*(f + g*x)**m/(S(2)*F**u*c + b - q), x), x) - Dist(S(2)*c/q, Int(F**u*(f + g*x)**m/(S(2)*F**u*c + b + q), x), x)
pattern1985 = Pattern(Integral(F_**u_*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/(F_**u_*WC('b', S(1)) + F_**v_*WC('c', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons125, cons208, cons1144, cons68, cons226, cons62)
rule1985 = ReplacementRule(pattern1985, With1985)
def With1986(v, u, m, f, b, g, i, c, a, x, h, F):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1986)
return -Dist(-i + (-b*i + S(2)*c*h)/q, Int((f + g*x)**m/(S(2)*F**u*c + b + q), x), x) + Dist(i + (-b*i + S(2)*c*h)/q, Int((f + g*x)**m/(S(2)*F**u*c + b - q), x), x)
pattern1986 = Pattern(Integral((F_**u_*WC('i', S(1)) + h_)*(x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/(F_**u_*WC('b', S(1)) + F_**v_*WC('c', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons125, cons208, cons209, cons224, cons1144, cons68, cons226, cons62)
rule1986 = ReplacementRule(pattern1986, With1986)
def With1987(v, m, b, d, c, a, x, F):
u = IntHide(S(1)/(F**v*b + F**(c + d*x)*a), x)
rubi.append(1987)
return -Dist(m, Int(u*x**(m + S(-1)), x), x) + Simp(u*x**m, x)
pattern1987 = Pattern(Integral(x_**WC('m', S(1))/(F_**v_*WC('b', S(1)) + F_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('a', S(1))), x_), cons1099, cons2, cons3, cons7, cons27, cons1145, cons31, cons168)
rule1987 = ReplacementRule(pattern1987, With1987)
pattern1988 = Pattern(Integral(u_/(F_**v_*WC('b', S(1)) + F_**w_*WC('c', S(1)) + a_), x_), cons1099, cons2, cons3, cons7, cons552, cons1146, cons1147, cons1148)
def replacement1988(v, w, u, b, c, a, x, F):
rubi.append(1988)
return Int(F**v*u/(F**(S(2)*v)*b + F**v*a + c), x)
rule1988 = ReplacementRule(pattern1988, replacement1988)
pattern1989 = Pattern(Integral(F_**((x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1))*WC('g', S(1)))/(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons208, cons4, cons1149)
def replacement1989(g, b, d, a, n, c, x, F, e):
rubi.append(1989)
return Int(ExpandIntegrand(F**(g*(d + e*x)**n), S(1)/(a + b*x + c*x**S(2)), x), x)
rule1989 = ReplacementRule(pattern1989, replacement1989)
pattern1990 = Pattern(Integral(F_**((x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1))*WC('g', S(1)))/(a_ + x_**S(2)*WC('c', S(1))), x_), cons1099, cons2, cons7, cons27, cons48, cons208, cons4, cons1150)
def replacement1990(g, d, c, n, a, x, F, e):
rubi.append(1990)
return Int(ExpandIntegrand(F**(g*(d + e*x)**n), S(1)/(a + c*x**S(2)), x), x)
rule1990 = ReplacementRule(pattern1990, replacement1990)
pattern1991 = Pattern(Integral(F_**((x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1))*WC('g', S(1)))*u_**WC('m', S(1))/(c_*x_**S(2) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons208, cons4, cons804, cons17)
def replacement1991(u, m, g, b, d, a, n, c, x, F, e):
rubi.append(1991)
return Int(ExpandIntegrand(F**(g*(d + e*x)**n), u**m/(a + b*x + c*x**S(2)), x), x)
rule1991 = ReplacementRule(pattern1991, replacement1991)
pattern1992 = Pattern(Integral(F_**((x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1))*WC('g', S(1)))*u_**WC('m', S(1))/(a_ + c_*x_**S(2)), x_), cons1099, cons2, cons7, cons27, cons48, cons208, cons4, cons804, cons17)
def replacement1992(u, m, g, d, a, n, c, x, F, e):
rubi.append(1992)
return Int(ExpandIntegrand(F**(g*(d + e*x)**n), u**m/(a + c*x**S(2)), x), x)
rule1992 = ReplacementRule(pattern1992, replacement1992)
pattern1993 = Pattern(Integral(F_**((x_**S(4)*WC('b', S(1)) + WC('a', S(0)))/x_**S(2)), x_), cons1099, cons2, cons3, cons1151)
def replacement1993(x, a, b, F):
rubi.append(1993)
return -Simp(sqrt(Pi)*Erf((-x**S(2)*sqrt(-b*log(F)) + sqrt(-a*log(F)))/x)*exp(-S(2)*sqrt(-a*log(F))*sqrt(-b*log(F)))/(S(4)*sqrt(-b*log(F))), x) + Simp(sqrt(Pi)*Erf((x**S(2)*sqrt(-b*log(F)) + sqrt(-a*log(F)))/x)*exp(S(2)*sqrt(-a*log(F))*sqrt(-b*log(F)))/(S(4)*sqrt(-b*log(F))), x)
rule1993 = ReplacementRule(pattern1993, replacement1993)
pattern1994 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('m', S(1)) + exp(x_))**n_, x_), cons93, cons168, cons463, cons1152)
def replacement1994(x, m, n):
rubi.append(1994)
return Dist(m, Int(x**(m + S(-1))*(x**m + exp(x))**n, x), x) + Int((x**m + exp(x))**(n + S(1)), x) - Simp((x**m + exp(x))**(n + S(1))/(n + S(1)), x)
rule1994 = ReplacementRule(pattern1994, replacement1994)
pattern1995 = Pattern(Integral(log(a_ + (F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1))))**WC('n', S(1))*WC('b', S(1))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons43)
def replacement1995(b, d, c, n, a, x, F, e):
rubi.append(1995)
return Dist(S(1)/(d*e*n*log(F)), Subst(Int(log(a + b*x)/x, x), x, (F**(e*(c + d*x)))**n), x)
rule1995 = ReplacementRule(pattern1995, replacement1995)
pattern1996 = Pattern(Integral(log(a_ + (F_**((x_*WC('d', S(1)) + WC('c', S(0)))*WC('e', S(1))))**WC('n', S(1))*WC('b', S(1))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons448)
def replacement1996(b, d, c, n, a, x, F, e):
rubi.append(1996)
return -Dist(b*d*e*n*log(F), Int(x*(F**(e*(c + d*x)))**n/(a + b*(F**(e*(c + d*x)))**n), x), x) + Simp(x*log(a + b*(F**(e*(c + d*x)))**n), x)
rule1996 = ReplacementRule(pattern1996, replacement1996)
pattern1997 = Pattern(Integral((F_**v_*WC('a', S(1)))**n_*WC('u', S(1)), x_), cons1099, cons2, cons4, cons23)
def replacement1997(v, u, a, n, x, F):
rubi.append(1997)
return Dist(F**(-n*v)*(F**v*a)**n, Int(F**(n*v)*u, x), x)
rule1997 = ReplacementRule(pattern1997, replacement1997)
def With1998(x, u):
v = FunctionOfExponential(u, x)
rubi.append(1998)
return Dist(v/D(v, x), Subst(Int(FunctionOfExponentialFunction(u, x)/x, x), x, v), x)
pattern1998 = Pattern(Integral(u_, x_), cons1153)
rule1998 = ReplacementRule(pattern1998, With1998)
pattern1999 = Pattern(Integral((F_**v_*WC('a', S(1)) + F_**w_*WC('b', S(1)))**n_*WC('u', S(1)), x_), cons1099, cons2, cons3, cons4, cons196, cons1154)
def replacement1999(v, w, u, b, a, n, x, F):
rubi.append(1999)
return Int(F**(n*v)*u*(F**ExpandToSum(-v + w, x)*b + a)**n, x)
rule1999 = ReplacementRule(pattern1999, replacement1999)
pattern2000 = Pattern(Integral((F_**v_*WC('a', S(1)) + G_**w_*WC('b', S(1)))**n_*WC('u', S(1)), x_), cons1099, cons1137, cons2, cons3, cons4, cons196, cons1154)
def replacement2000(v, w, u, b, G, a, n, x, F):
rubi.append(2000)
return Int(F**(n*v)*u*(a + b*exp(ExpandToSum(-v*log(F) + w*log(G), x)))**n, x)
rule2000 = ReplacementRule(pattern2000, replacement2000)
pattern2001 = Pattern(Integral((F_**v_*WC('a', S(1)) + F_**w_*WC('b', S(1)))**n_*WC('u', S(1)), x_), cons1099, cons2, cons3, cons4, cons23, cons1154)
def replacement2001(v, w, u, b, a, n, x, F):
rubi.append(2001)
return Dist(F**(-n*v)*(F**v*a + F**w*b)**n*(F**ExpandToSum(-v + w, x)*b + a)**(-n), Int(F**(n*v)*u*(F**ExpandToSum(-v + w, x)*b + a)**n, x), x)
rule2001 = ReplacementRule(pattern2001, replacement2001)
pattern2002 = Pattern(Integral((F_**v_*WC('a', S(1)) + G_**w_*WC('b', S(1)))**n_*WC('u', S(1)), x_), cons1099, cons1137, cons2, cons3, cons4, cons23, cons1154)
def replacement2002(v, w, u, b, G, a, n, x, F):
rubi.append(2002)
return Dist(F**(-n*v)*(a + b*exp(ExpandToSum(-v*log(F) + w*log(G), x)))**(-n)*(F**v*a + G**w*b)**n, Int(F**(n*v)*u*(a + b*exp(ExpandToSum(-v*log(F) + w*log(G), x)))**n, x), x)
rule2002 = ReplacementRule(pattern2002, replacement2002)
pattern2003 = Pattern(Integral(F_**v_*G_**w_*WC('u', S(1)), x_), cons1099, cons1137, cons1155)
def replacement2003(v, w, u, G, x, F):
rubi.append(2003)
return Int(u*NormalizeIntegrand(exp(v*log(F) + w*log(G)), x), x)
rule2003 = ReplacementRule(pattern2003, replacement2003)
def With2004(v, w, u, y, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
z = v*y/(D(u, x)*log(F))
if ZeroQ(-w*y + D(z, x)):
return True
return False
pattern2004 = Pattern(Integral(F_**u_*(v_ + w_)*WC('y', S(1)), x_), cons1099, cons1099, CustomConstraint(With2004))
def replacement2004(v, w, u, y, x, F):
z = v*y/(D(u, x)*log(F))
rubi.append(2004)
return Simp(F**u*z, x)
rule2004 = ReplacementRule(pattern2004, replacement2004)
def With2005(v, w, u, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
z = v*D(u, x)*log(F) + (n + S(1))*D(v, x)
if And(Equal(Exponent(w, x), Exponent(z, x)), ZeroQ(w*Coefficient(z, x, Exponent(z, x)) - z*Coefficient(w, x, Exponent(w, x)))):
return True
return False
pattern2005 = Pattern(Integral(F_**u_*v_**WC('n', S(1))*w_, x_), cons1099, cons4, cons804, cons1017, cons1109, CustomConstraint(With2005))
def replacement2005(v, w, u, n, x, F):
z = v*D(u, x)*log(F) + (n + S(1))*D(v, x)
rubi.append(2005)
return Simp(F**u*v**(n + S(1))*Coefficient(w, x, Exponent(w, x))/Coefficient(z, x, Exponent(z, x)), x)
rule2005 = ReplacementRule(pattern2005, replacement2005)
return [rule1901, rule1902, rule1903, rule1904, rule1905, rule1906, rule1907, rule1908, rule1909, rule1910, rule1911, rule1912, rule1913, rule1914, rule1915, rule1916, rule1917, rule1918, rule1919, rule1920, rule1921, rule1922, rule1923, rule1924, rule1925, rule1926, rule1927, rule1928, rule1929, rule1930, rule1931, rule1932, rule1933, rule1934, rule1935, rule1936, rule1937, rule1938, rule1939, rule1940, rule1941, rule1942, rule1943, rule1944, rule1945, rule1946, rule1947, rule1948, rule1949, rule1950, rule1951, rule1952, rule1953, rule1954, rule1955, rule1956, rule1957, rule1958, rule1959, rule1960, rule1961, rule1962, rule1963, rule1964, rule1965, rule1966, rule1967, rule1968, rule1969, rule1970, rule1971, rule1972, rule1973, rule1974, rule1975, rule1976, rule1977, rule1978, rule1979, rule1980, rule1981, rule1982, rule1983, rule1984, rule1985, rule1986, rule1987, rule1988, rule1989, rule1990, rule1991, rule1992, rule1993, rule1994, rule1995, rule1996, rule1997, rule1998, rule1999, rule2000, rule2001, rule2002, rule2003, rule2004, rule2005, ]
|
1ff79596c8df2dc186ae2a1b39bebd81906a02ab73819bfaa452f1a7203458af | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def linear_products(rubi):
from sympy.integrals.rubi.constraints import cons66, cons21, cons67, cons2, cons3, cons68, cons69, cons70, cons7, cons27, cons71, cons72, cons4, cons73, cons74, cons75, cons43, cons76, cons77, cons78, cons79, cons80, cons81, cons82, cons62, cons83, cons84, cons85, cons86, cons87, cons88, cons89, cons90, cons91, cons92, cons23, cons93, cons94, cons95, cons96, cons97, cons98, cons99, cons100, cons101, cons102, cons103, cons104, cons105, cons106, cons31, cons107, cons108, cons109, cons110, cons111, cons112, cons113, cons114, cons18, cons115, cons116, cons117, cons118, cons119, cons120, cons121, cons122, cons123, cons124, cons17, cons48, cons125, cons5, cons126, cons127, cons128, cons129, cons130, cons131, cons132, cons133, cons134, cons135, cons54, cons136, cons13, cons137, cons138, cons12, cons139, cons140, cons141, cons142, cons143, cons144, cons38, cons145, cons146, cons147, cons148, cons149, cons150, cons151, cons152, cons153, cons154, cons155, cons156, cons157, cons158, cons159, cons160, cons161, cons162, cons163, cons164, cons165, cons166, cons167, cons168, cons169, cons170, cons171, cons172, cons173, cons174, cons175, cons176, cons177, cons178, cons179, cons180, cons181, cons182, cons183, cons184, cons185, cons186, cons187, cons188, cons189, cons190, cons191, cons192, cons193, cons194, cons195, cons196, cons197, cons198, cons199, cons200, cons201, cons202, cons203, cons204, cons205, cons206, cons207, cons208, cons209, cons210, cons211, cons212, cons213, cons214, cons215, cons216, cons217, cons218, cons219, cons50, cons220, cons221, cons222, cons223, cons224, cons52
pattern37 = Pattern(Integral(S(1)/x_, x_))
def replacement37(x):
rubi.append(37)
return Simp(log(x), x)
rule37 = ReplacementRule(pattern37, replacement37)
pattern38 = Pattern(Integral(x_**WC('m', S(1)), x_), cons21, cons66)
def replacement38(x, m):
rubi.append(38)
return Simp(x**(m + S(1))/(m + S(1)), x)
rule38 = ReplacementRule(pattern38, replacement38)
pattern39 = Pattern(Integral(S(1)/(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons67)
def replacement39(x, a, b):
rubi.append(39)
return Simp(log(RemoveContent(a + b*x, x))/b, x)
rule39 = ReplacementRule(pattern39, replacement39)
pattern40 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_, x_), cons2, cons3, cons21, cons66)
def replacement40(x, m, a, b):
rubi.append(40)
return Simp((a + b*x)**(m + S(1))/(b*(m + S(1))), x)
rule40 = ReplacementRule(pattern40, replacement40)
pattern41 = Pattern(Integral((u_*WC('b', S(1)) + WC('a', S(0)))**m_, x_), cons2, cons3, cons21, cons68, cons69)
def replacement41(u, m, b, a, x):
rubi.append(41)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x)**m, x), x, u), x)
rule41 = ReplacementRule(pattern41, replacement41)
pattern42 = Pattern(Integral(S(1)/((a_ + x_*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons70)
def replacement42(b, d, c, a, x):
rubi.append(42)
return Int(S(1)/(a*c + b*d*x**S(2)), x)
rule42 = ReplacementRule(pattern42, replacement42)
pattern43 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement43(b, d, c, a, x):
rubi.append(43)
return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*x), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*x), x), x)
rule43 = ReplacementRule(pattern43, replacement43)
pattern44 = Pattern(Integral((c_ + x_*WC('d', S(1)))**n_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons71, cons72, cons66)
def replacement44(m, b, d, a, c, n, x):
rubi.append(44)
return Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))/((m + S(1))*(-a*d + b*c)), x)
rule44 = ReplacementRule(pattern44, replacement44)
pattern45 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**m_, x_), cons2, cons3, cons7, cons27, cons70, cons73)
def replacement45(m, b, d, a, c, x):
rubi.append(45)
return Dist(S(2)*a*c*m/(S(2)*m + S(1)), Int((a + b*x)**(m + S(-1))*(c + d*x)**(m + S(-1)), x), x) + Simp(x*(a + b*x)**m*(c + d*x)**m/(S(2)*m + S(1)), x)
rule45 = ReplacementRule(pattern45, replacement45)
pattern46 = Pattern(Integral(S(1)/((a_ + x_*WC('b', S(1)))**(S(3)/2)*(c_ + x_*WC('d', S(1)))**(S(3)/2)), x_), cons2, cons3, cons7, cons27, cons70)
def replacement46(b, d, c, a, x):
rubi.append(46)
return Simp(x/(a*c*sqrt(a + b*x)*sqrt(c + d*x)), x)
rule46 = ReplacementRule(pattern46, replacement46)
pattern47 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**m_, x_), cons2, cons3, cons7, cons27, cons70, cons74)
def replacement47(m, b, d, a, c, x):
rubi.append(47)
return Dist((S(2)*m + S(3))/(S(2)*a*c*(m + S(1))), Int((a + b*x)**(m + S(1))*(c + d*x)**(m + S(1)), x), x) - Simp(x*(a + b*x)**(m + S(1))*(c + d*x)**(m + S(1))/(S(2)*a*c*(m + S(1))), x)
rule47 = ReplacementRule(pattern47, replacement47)
pattern48 = Pattern(Integral((a_ + x_*WC('b', S(1)))**WC('m', S(1))*(c_ + x_*WC('d', S(1)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons70, cons75)
def replacement48(m, b, d, a, c, x):
rubi.append(48)
return Int((a*c + b*d*x**S(2))**m, x)
rule48 = ReplacementRule(pattern48, replacement48)
pattern49 = Pattern(Integral(S(1)/(sqrt(a_ + x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons70, cons43, cons76)
def replacement49(b, d, c, a, x):
rubi.append(49)
return Simp(acosh(b*x/a)/b, x)
rule49 = ReplacementRule(pattern49, replacement49)
pattern50 = Pattern(Integral(S(1)/(sqrt(a_ + x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons70)
def replacement50(b, d, c, a, x):
rubi.append(50)
return Dist(S(2), Subst(Int(S(1)/(b - d*x**S(2)), x), x, sqrt(a + b*x)/sqrt(c + d*x)), x)
rule50 = ReplacementRule(pattern50, replacement50)
pattern51 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**m_, x_), cons2, cons3, cons7, cons27, cons21, cons70, cons77)
def replacement51(m, b, d, a, c, x):
rubi.append(51)
return Dist((a + b*x)**FracPart(m)*(c + d*x)**FracPart(m)*(a*c + b*d*x**S(2))**(-FracPart(m)), Int((a*c + b*d*x**S(2))**m, x), x)
rule51 = ReplacementRule(pattern51, replacement51)
pattern52 = Pattern(Integral(S(1)/((a_ + x_*WC('b', S(1)))**(S(5)/4)*(c_ + x_*WC('d', S(1)))**(S(1)/4)), x_), cons2, cons3, cons7, cons27, cons70, cons78)
def replacement52(b, d, c, a, x):
rubi.append(52)
return Dist((-a*d + b*c)/(S(2)*b), Int(S(1)/((a + b*x)**(S(5)/4)*(c + d*x)**(S(5)/4)), x), x) + Simp(-S(2)/(b*(a + b*x)**(S(1)/4)*(c + d*x)**(S(1)/4)), x)
rule52 = ReplacementRule(pattern52, replacement52)
pattern53 = Pattern(Integral(S(1)/((a_ + x_*WC('b', S(1)))**(S(9)/4)*(c_ + x_*WC('d', S(1)))**(S(1)/4)), x_), cons2, cons3, cons7, cons27, cons70, cons78)
def replacement53(b, d, c, a, x):
rubi.append(53)
return -Dist(d/(S(5)*b), Int(S(1)/((a + b*x)**(S(5)/4)*(c + d*x)**(S(5)/4)), x), x) + Simp(-S(4)/(S(5)*b*(a + b*x)**(S(5)/4)*(c + d*x)**(S(1)/4)), x)
rule53 = ReplacementRule(pattern53, replacement53)
pattern54 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons2, cons3, cons7, cons27, cons70, cons79, cons80, cons81)
def replacement54(m, b, d, a, c, n, x):
rubi.append(54)
return Dist(S(2)*c*n/(m + n + S(1)), Int((a + b*x)**m*(c + d*x)**(n + S(-1)), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**n/(b*(m + n + S(1))), x)
rule54 = ReplacementRule(pattern54, replacement54)
pattern55 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons2, cons3, cons7, cons27, cons70, cons79, cons80, cons82)
def replacement55(m, b, d, a, c, n, x):
rubi.append(55)
return Dist((m + n + S(2))/(S(2)*a*(m + S(1))), Int((a + b*x)**(m + S(1))*(c + d*x)**n, x), x) - Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))/(S(2)*a*d*(m + S(1))), x)
rule55 = ReplacementRule(pattern55, replacement55)
pattern56 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons71, cons62, cons83)
def replacement56(m, b, d, a, c, n, x):
rubi.append(56)
return Int(ExpandIntegrand((a + b*x)**m*(c + d*x)**n, x), x)
rule56 = ReplacementRule(pattern56, replacement56)
pattern57 = Pattern(Integral((a_ + x_*WC('b', S(1)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons71, cons84, cons85, cons86)
def replacement57(m, b, d, c, n, a, x):
rubi.append(57)
return Int(ExpandIntegrand((a + b*x)**m*(c + d*x)**n, x), x)
rule57 = ReplacementRule(pattern57, replacement57)
pattern58 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**n_/(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons71, cons87, cons88)
def replacement58(b, d, c, a, n, x):
rubi.append(58)
return Dist((-a*d + b*c)/b, Int((c + d*x)**(n + S(-1))/(a + b*x), x), x) + Simp((c + d*x)**n/(b*n), x)
rule58 = ReplacementRule(pattern58, replacement58)
pattern59 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**n_/(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons71, cons87, cons89)
def replacement59(b, d, c, a, n, x):
rubi.append(59)
return Dist(b/(-a*d + b*c), Int((c + d*x)**(n + S(1))/(a + b*x), x), x) - Simp((c + d*x)**(n + S(1))/((n + S(1))*(-a*d + b*c)), x)
rule59 = ReplacementRule(pattern59, replacement59)
def With60(b, d, c, a, x):
q = Rt((-a*d + b*c)/b, S(3))
rubi.append(60)
return Dist(S(3)/(S(2)*b), Subst(Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x, (c + d*x)**(S(1)/3)), x) - Dist(S(3)/(S(2)*b*q), Subst(Int(S(1)/(q - x), x), x, (c + d*x)**(S(1)/3)), x) - Simp(log(RemoveContent(a + b*x, x))/(S(2)*b*q), x)
pattern60 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**(S(1)/3)), x_), cons2, cons3, cons7, cons27, cons90)
rule60 = ReplacementRule(pattern60, With60)
def With61(b, d, c, a, x):
q = Rt(-(-a*d + b*c)/b, S(3))
rubi.append(61)
return Dist(S(3)/(S(2)*b), Subst(Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x, (c + d*x)**(S(1)/3)), x) - Dist(S(3)/(S(2)*b*q), Subst(Int(S(1)/(q + x), x), x, (c + d*x)**(S(1)/3)), x) + Simp(log(RemoveContent(a + b*x, x))/(S(2)*b*q), x)
pattern61 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**(S(1)/3)), x_), cons2, cons3, cons7, cons27, cons91)
rule61 = ReplacementRule(pattern61, With61)
def With62(b, d, c, a, x):
q = Rt((-a*d + b*c)/b, S(3))
rubi.append(62)
return -Dist(S(3)/(S(2)*b*q**S(2)), Subst(Int(S(1)/(q - x), x), x, (c + d*x)**(S(1)/3)), x) - Dist(S(3)/(S(2)*b*q), Subst(Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x, (c + d*x)**(S(1)/3)), x) - Simp(log(RemoveContent(a + b*x, x))/(S(2)*b*q**S(2)), x)
pattern62 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**(S(2)/3)), x_), cons2, cons3, cons7, cons27, cons90)
rule62 = ReplacementRule(pattern62, With62)
def With63(b, d, c, a, x):
q = Rt(-(-a*d + b*c)/b, S(3))
rubi.append(63)
return Dist(S(3)/(S(2)*b*q**S(2)), Subst(Int(S(1)/(q + x), x), x, (c + d*x)**(S(1)/3)), x) + Dist(S(3)/(S(2)*b*q), Subst(Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x, (c + d*x)**(S(1)/3)), x) - Simp(log(RemoveContent(a + b*x, x))/(S(2)*b*q**S(2)), x)
pattern63 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**(S(2)/3)), x_), cons2, cons3, cons7, cons27, cons91)
rule63 = ReplacementRule(pattern63, With63)
def With64(b, d, c, a, n, x):
p = Denominator(n)
rubi.append(64)
return Dist(p, Subst(Int(x**(p*(n + S(1)) + S(-1))/(a*d - b*c + b*x**p), x), x, (c + d*x)**(S(1)/p)), x)
pattern64 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**n_/(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons71, cons87, cons92)
rule64 = ReplacementRule(pattern64, With64)
pattern65 = Pattern(Integral((c_ + x_*WC('d', S(1)))**n_/x_, x_), cons7, cons27, cons4, cons23)
def replacement65(d, x, n, c):
rubi.append(65)
return -Simp((c + d*x)**(n + S(1))*Hypergeometric2F1(S(1), n + S(1), n + S(2), S(1) + d*x/c)/(c*(n + S(1))), x)
rule65 = ReplacementRule(pattern65, replacement65)
pattern66 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons71, cons23)
def replacement66(b, d, c, a, n, x):
rubi.append(66)
return -Simp((c + d*x)**(n + S(1))*Hypergeometric2F1(S(1), n + S(1), n + S(2), TogetherSimplify(b*(c + d*x)/(-a*d + b*c)))/((n + S(1))*(-a*d + b*c)), x)
rule66 = ReplacementRule(pattern66, replacement66)
pattern67 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons71, cons93, cons94, cons88, cons95, cons96, cons97)
def replacement67(m, b, d, c, a, n, x):
rubi.append(67)
return -Dist(d*n/(b*(m + S(1))), Int((a + b*x)**(m + S(1))*(c + d*x)**(n + S(-1)), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**n/(b*(m + S(1))), x)
rule67 = ReplacementRule(pattern67, replacement67)
pattern68 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons71, cons93, cons94, cons98, cons97)
def replacement68(m, b, d, c, a, n, x):
rubi.append(68)
return -Dist(d*(m + n + S(2))/((m + S(1))*(-a*d + b*c)), Int((a + b*x)**(m + S(1))*(c + d*x)**n, x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))/((m + S(1))*(-a*d + b*c)), x)
rule68 = ReplacementRule(pattern68, replacement68)
pattern69 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons71, cons93, cons88, cons99, cons100, cons101, cons97)
def replacement69(m, b, d, c, a, n, x):
rubi.append(69)
return Dist(n*(-a*d + b*c)/(b*(m + n + S(1))), Int((a + b*x)**m*(c + d*x)**(n + S(-1)), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**n/(b*(m + n + S(1))), x)
rule69 = ReplacementRule(pattern69, replacement69)
pattern70 = Pattern(Integral(S(1)/(sqrt(a_ + x_*WC('b', S(1)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons102, cons103)
def replacement70(b, d, c, a, x):
rubi.append(70)
return Int(S(1)/sqrt(a*c - b**S(2)*x**S(2) - b*x*(a - c)), x)
rule70 = ReplacementRule(pattern70, replacement70)
pattern71 = Pattern(Integral(S(1)/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons104, cons105)
def replacement71(b, d, c, a, x):
rubi.append(71)
return Dist(S(2)/sqrt(b), Subst(Int(S(1)/sqrt(-a*d + b*c + d*x**S(2)), x), x, sqrt(a + b*x)), x)
rule71 = ReplacementRule(pattern71, replacement71)
pattern72 = Pattern(Integral(S(1)/(sqrt(c_ + x_*WC('d', S(1)))*sqrt(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons71, cons106)
def replacement72(b, d, c, a, x):
rubi.append(72)
return Dist(S(2)/b, Subst(Int(S(1)/sqrt(-a + c + x**S(2)), x), x, sqrt(a + b*x)), x)
rule72 = ReplacementRule(pattern72, replacement72)
pattern73 = Pattern(Integral(S(1)/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons71)
def replacement73(b, d, c, a, x):
rubi.append(73)
return Dist(S(2), Subst(Int(S(1)/(b - d*x**S(2)), x), x, sqrt(a + b*x)/sqrt(c + d*x)), x)
rule73 = ReplacementRule(pattern73, replacement73)
pattern74 = Pattern(Integral((c_ + x_*WC('d', S(1)))**m_*(x_*WC('b', S(1)) + WC('a', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons71, cons31, cons107, cons108)
def replacement74(m, b, d, a, c, x):
rubi.append(74)
return Dist((a + b*x)**m*(c + d*x)**m*(a*c + b*d*x**S(2) + x*(a*d + b*c))**(-m), Int((a*c + b*d*x**S(2) + x*(a*d + b*c))**m, x), x)
rule74 = ReplacementRule(pattern74, replacement74)
def With75(b, d, c, a, x):
q = Rt(d/b, S(3))
rubi.append(75)
return -Simp(q*log(c + d*x)/(S(2)*d), x) - Simp(S(3)*q*log(q*(a + b*x)**(S(1)/3)/(c + d*x)**(S(1)/3) + S(-1))/(S(2)*d), x) - Simp(sqrt(S(3))*q*ArcTan(S(2)*sqrt(S(3))*q*(a + b*x)**(S(1)/3)/(S(3)*(c + d*x)**(S(1)/3)) + sqrt(S(3))/S(3))/d, x)
pattern75 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))**(S(1)/3)*(x_*WC('d', S(1)) + WC('c', S(0)))**(S(2)/3)), x_), cons2, cons3, cons7, cons27, cons71, cons109)
rule75 = ReplacementRule(pattern75, With75)
def With76(b, d, c, a, x):
q = Rt(-d/b, S(3))
rubi.append(76)
return Simp(q*log(c + d*x)/(S(2)*d), x) + Simp(S(3)*q*log(q*(a + b*x)**(S(1)/3)/(c + d*x)**(S(1)/3) + S(1))/(S(2)*d), x) + Simp(sqrt(S(3))*q*ArcTan(-S(2)*sqrt(S(3))*q*(a + b*x)**(S(1)/3)/(S(3)*(c + d*x)**(S(1)/3)) + sqrt(S(3))/S(3))/d, x)
pattern76 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))**(S(1)/3)*(x_*WC('d', S(1)) + WC('c', S(0)))**(S(2)/3)), x_), cons2, cons3, cons7, cons27, cons71, cons110)
rule76 = ReplacementRule(pattern76, With76)
def With77(m, b, d, c, a, n, x):
p = Denominator(m)
rubi.append(77)
return Dist(p, Subst(Int(x**(p*(m + S(1)) + S(-1))/(b - d*x**p), x), x, (a + b*x)**(S(1)/p)*(c + d*x)**(-S(1)/p)), x)
pattern77 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons71, cons93, cons107, cons111)
rule77 = ReplacementRule(pattern77, With77)
def With78(m, b, d, c, a, n, x):
p = Denominator(m)
rubi.append(78)
return Dist(p/b, Subst(Int(x**(p*(m + S(1)) + S(-1))*(-a*d/b + c + d*x**p/b)**n, x), x, (a + b*x)**(S(1)/p)), x)
pattern78 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons71, cons93, cons107, cons92, cons112, cons97)
rule78 = ReplacementRule(pattern78, With78)
pattern79 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons71, cons113, cons66, cons114)
def replacement79(m, b, d, c, a, n, x):
rubi.append(79)
return -Dist(d*(m + n + S(2))/((m + S(1))*(-a*d + b*c)), Int((a + b*x)**(m + S(1))*(c + d*x)**n, x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))/((m + S(1))*(-a*d + b*c)), x)
rule79 = ReplacementRule(pattern79, replacement79)
pattern80 = Pattern(Integral((x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons3, cons7, cons27, cons21, cons4, cons18, cons115)
def replacement80(m, b, d, c, n, x):
rubi.append(80)
return Simp(c**n*(b*x)**(m + S(1))*Hypergeometric2F1(-n, m + S(1), m + S(2), -d*x/c)/(b*(m + S(1))), x)
rule80 = ReplacementRule(pattern80, replacement80)
pattern81 = Pattern(Integral((x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons3, cons7, cons27, cons21, cons4, cons23, cons116)
def replacement81(m, b, d, c, n, x):
rubi.append(81)
return Simp((-d/(b*c))**(-m)*(c + d*x)**(n + S(1))*Hypergeometric2F1(-m, n + S(1), n + S(2), S(1) + d*x/c)/(d*(n + S(1))), x)
rule81 = ReplacementRule(pattern81, replacement81)
pattern82 = Pattern(Integral((x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons3, cons7, cons27, cons21, cons4, cons18, cons23, cons117, cons118, cons119)
def replacement82(m, b, d, c, n, x):
rubi.append(82)
return Dist(c**IntPart(n)*(S(1) + d*x/c)**(-FracPart(n))*(c + d*x)**FracPart(n), Int((b*x)**m*(S(1) + d*x/c)**n, x), x)
rule82 = ReplacementRule(pattern82, replacement82)
pattern83 = Pattern(Integral((x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons3, cons7, cons27, cons21, cons4, cons18, cons23, cons117, cons118)
def replacement83(m, b, d, c, n, x):
rubi.append(83)
return Dist((b*x)**FracPart(m)*(-b*c/d)**IntPart(m)*(-d*x/c)**(-FracPart(m)), Int((-d*x/c)**m*(c + d*x)**n, x), x)
rule83 = ReplacementRule(pattern83, replacement83)
pattern84 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons2, cons3, cons7, cons27, cons21, cons71, cons18, cons85)
def replacement84(m, b, d, a, c, n, x):
rubi.append(84)
return Simp(b**(-n + S(-1))*(a + b*x)**(m + S(1))*(-a*d + b*c)**n*Hypergeometric2F1(-n, m + S(1), m + S(2), -d*(a + b*x)/(-a*d + b*c))/(m + S(1)), x)
rule84 = ReplacementRule(pattern84, replacement84)
pattern85 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons71, cons18, cons23, cons120, cons121)
def replacement85(m, b, d, a, c, n, x):
rubi.append(85)
return Simp((b/(-a*d + b*c))**(-n)*(a + b*x)**(m + S(1))*Hypergeometric2F1(-n, m + S(1), m + S(2), -d*(a + b*x)/(-a*d + b*c))/(b*(m + S(1))), x)
rule85 = ReplacementRule(pattern85, replacement85)
pattern86 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons71, cons18, cons23, cons122)
def replacement86(m, b, d, a, c, n, x):
rubi.append(86)
return Dist((b/(-a*d + b*c))**(-IntPart(n))*(b*(c + d*x)/(-a*d + b*c))**(-FracPart(n))*(c + d*x)**FracPart(n), Int((a + b*x)**m*(b*c/(-a*d + b*c) + b*d*x/(-a*d + b*c))**n, x), x)
rule86 = ReplacementRule(pattern86, replacement86)
pattern87 = Pattern(Integral((u_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(u_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons68, cons123)
def replacement87(u, m, b, d, a, c, n, x):
rubi.append(87)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x)**m*(c + d*x)**n, x), x, u), x)
rule87 = ReplacementRule(pattern87, replacement87)
pattern88 = Pattern(Integral((a_ + x_*WC('b', S(1)))**WC('m', S(1))*(c_ + x_*WC('d', S(1)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons70, cons124, cons17)
def replacement88(p, m, f, b, d, a, n, c, x, e):
rubi.append(88)
return Int((e + f*x)**p*(a*c + b*d*x**S(2))**m, x)
rule88 = ReplacementRule(pattern88, replacement88)
pattern89 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons126, cons127)
def replacement89(p, f, b, d, c, a, n, x, e):
rubi.append(89)
return Simp(b*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/(d*f*(n + p + S(2))), x)
rule89 = ReplacementRule(pattern89, replacement89)
pattern90 = Pattern(Integral((x_*WC('d', S(1)))**WC('n', S(1))*(a_ + x_*WC('b', S(1)))*(e_ + x_*WC('f', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons128, cons129, cons130)
def replacement90(p, f, b, d, a, n, x, e):
rubi.append(90)
return Int(ExpandIntegrand((d*x)**n*(a + b*x)*(e + f*x)**p, x), x)
rule90 = ReplacementRule(pattern90, replacement90)
pattern91 = Pattern(Integral((x_*WC('d', S(1)))**WC('n', S(1))*(a_ + x_*WC('b', S(1)))*(e_ + x_*WC('f', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons128, cons131, cons132, cons133)
def replacement91(p, f, b, d, a, n, x, e):
rubi.append(91)
return Int(ExpandIntegrand((d*x)**n*(a + b*x)*(e + f*x)**p, x), x)
rule91 = ReplacementRule(pattern91, replacement91)
pattern92 = Pattern(Integral((c_ + x_*WC('d', S(1)))**WC('n', S(1))*(x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons134)
def replacement92(p, f, b, d, a, n, c, x, e):
rubi.append(92)
return Int(ExpandIntegrand((a + b*x)*(c + d*x)**n*(e + f*x)**p, x), x)
rule92 = ReplacementRule(pattern92, replacement92)
pattern93 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons135, cons54, cons136)
def replacement93(p, f, b, d, c, a, n, x, e):
rubi.append(93)
return Dist(b/f, Int((c + d*x)**n*(e + f*x)**(p + S(1)), x), x) - Simp((c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))*(-a*f + b*e)/(f*(p + S(1))*(c*f - d*e)), x)
rule93 = ReplacementRule(pattern93, replacement93)
pattern94 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons126, cons13, cons137, cons138)
def replacement94(p, f, b, d, c, a, n, x, e):
rubi.append(94)
return -Dist((a*d*f*(n + p + S(2)) - b*(c*f*(p + S(1)) + d*e*(n + S(1))))/(f*(p + S(1))*(c*f - d*e)), Int((c + d*x)**n*(e + f*x)**(p + S(1)), x), x) - Simp((c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))*(-a*f + b*e)/(f*(p + S(1))*(c*f - d*e)), x)
rule94 = ReplacementRule(pattern94, replacement94)
pattern95 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons126, cons12, cons139)
def replacement95(p, f, b, d, c, a, n, x, e):
rubi.append(95)
return -Dist((a*d*f*(n + p + S(2)) - b*(c*f*(p + S(1)) + d*e*(n + S(1))))/(f*(p + S(1))*(c*f - d*e)), Int((c + d*x)**n*(e + f*x)**(p + S(1)), x), x) - Simp((c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))*(-a*f + b*e)/(f*(p + S(1))*(c*f - d*e)), x)
rule95 = ReplacementRule(pattern95, replacement95)
pattern96 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons126)
def replacement96(p, f, b, d, c, a, n, x, e):
rubi.append(96)
return Dist((a*d*f*(n + p + S(2)) - b*(c*f*(p + S(1)) + d*e*(n + S(1))))/(d*f*(n + p + S(2))), Int((c + d*x)**n*(e + f*x)**p, x), x) + Simp(b*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/(d*f*(n + p + S(2))), x)
rule96 = ReplacementRule(pattern96, replacement96)
pattern97 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**S(2)*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons126, cons140, cons141)
def replacement97(p, f, b, d, c, a, n, x, e):
rubi.append(97)
return Simp(b*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))*(S(2)*a*d*f*(n + p + S(3)) + b*d*f*x*(n + p + S(2)) - b*(c*f*(p + S(2)) + d*e*(n + S(2))))/(d**S(2)*f**S(2)*(n + p + S(2))*(n + p + S(3))), x)
rule97 = ReplacementRule(pattern97, replacement97)
pattern98 = Pattern(Integral((x_*WC('f', S(1)))**WC('p', S(1))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons21, cons4, cons5, cons70, cons142, cons12, cons143, cons144)
def replacement98(p, m, f, b, d, a, c, n, x):
rubi.append(98)
return Dist(a, Int((f*x)**p*(a + b*x)**n*(c + d*x)**n, x), x) + Dist(b/f, Int((f*x)**(p + S(1))*(a + b*x)**n*(c + d*x)**n, x), x)
rule98 = ReplacementRule(pattern98, replacement98)
pattern99 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons38)
def replacement99(p, f, b, d, c, a, x, e):
rubi.append(99)
return Int(ExpandIntegrand((e + f*x)**p/((a + b*x)*(c + d*x)), x), x)
rule99 = ReplacementRule(pattern99, replacement99)
pattern100 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons13, cons145)
def replacement100(p, f, b, d, c, a, x, e):
rubi.append(100)
return Dist((-a*f + b*e)/(-a*d + b*c), Int((e + f*x)**(p + S(-1))/(a + b*x), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int((e + f*x)**(p + S(-1))/(c + d*x), x), x)
rule100 = ReplacementRule(pattern100, replacement100)
pattern101 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons13, cons146)
def replacement101(p, f, b, d, c, a, x, e):
rubi.append(101)
return Dist(S(1)/(b*d), Int((e + f*x)**(p + S(-2))*(-a*c*f**S(2) + b*d*e**S(2) + f*x*(-a*d*f - b*c*f + S(2)*b*d*e))/((a + b*x)*(c + d*x)), x), x) + Simp(f*(e + f*x)**(p + S(-1))/(b*d*(p + S(-1))), x)
rule101 = ReplacementRule(pattern101, replacement101)
pattern102 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons13, cons137)
def replacement102(p, f, b, d, c, a, x, e):
rubi.append(102)
return Dist(S(1)/((-a*f + b*e)*(-c*f + d*e)), Int((e + f*x)**(p + S(1))*(-a*d*f - b*c*f + b*d*e - b*d*f*x)/((a + b*x)*(c + d*x)), x), x) + Simp(f*(e + f*x)**(p + S(1))/((p + S(1))*(-a*f + b*e)*(-c*f + d*e)), x)
rule102 = ReplacementRule(pattern102, replacement102)
pattern103 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons147)
def replacement103(p, f, b, d, c, a, x, e):
rubi.append(103)
return Dist(b/(-a*d + b*c), Int((e + f*x)**p/(a + b*x), x), x) - Dist(d/(-a*d + b*c), Int((e + f*x)**p/(c + d*x), x), x)
rule103 = ReplacementRule(pattern103, replacement103)
pattern104 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**p_/(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons148, cons149, cons137)
def replacement104(p, f, b, d, c, a, n, x, e):
rubi.append(104)
return Int(ExpandIntegrand((e + f*x)**FractionalPart(p), (c + d*x)**n*(e + f*x)**IntegerPart(p)/(a + b*x), x), x)
rule104 = ReplacementRule(pattern104, replacement104)
pattern105 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons150, cons151)
def replacement105(p, m, f, b, d, a, c, n, x, e):
rubi.append(105)
return Int(ExpandIntegrand((a + b*x)**m*(c + d*x)**n*(e + f*x)**p, x), x)
rule105 = ReplacementRule(pattern105, replacement105)
pattern106 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**S(2)*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons152)
def replacement106(p, f, b, d, c, a, n, x, e):
rubi.append(106)
return -Dist(S(1)/(d**S(2)*(n + S(1))*(-c*f + d*e)), Int((c + d*x)**(n + S(1))*(e + f*x)**p*Simp(a**S(2)*d**S(2)*f*(n + p + S(2)) - S(2)*a*b*d*(c*f*(p + S(1)) + d*e*(n + S(1))) + b**S(2)*c*(c*f*(p + S(1)) + d*e*(n + S(1))) - b**S(2)*d*x*(n + S(1))*(-c*f + d*e), x), x), x) + Simp((c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))*(-a*d + b*c)**S(2)/(d**S(2)*(n + S(1))*(-c*f + d*e)), x)
rule106 = ReplacementRule(pattern106, replacement106)
pattern107 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**S(2)*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons140)
def replacement107(p, f, b, d, c, a, n, x, e):
rubi.append(107)
return Dist(S(1)/(d*f*(n + p + S(3))), Int((c + d*x)**n*(e + f*x)**p*Simp(a**S(2)*d*f*(n + p + S(3)) + b*x*(a*d*f*(n + p + S(4)) - b*(c*f*(p + S(2)) + d*e*(n + S(2)))) - b*(a*(c*f*(p + S(1)) + d*e*(n + S(1))) + b*c*e), x), x), x) + Simp(b*(a + b*x)*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/(d*f*(n + p + S(3))), x)
rule107 = ReplacementRule(pattern107, replacement107)
def With108(f, b, d, c, a, x, e):
q = Rt((-c*f + d*e)/(-a*f + b*e), S(3))
rubi.append(108)
return Simp(q*log(e + f*x)/(-S(2)*c*f + S(2)*d*e), x) - Simp(S(3)*q*log(q*(a + b*x)**(S(1)/3) - (c + d*x)**(S(1)/3))/(-S(2)*c*f + S(2)*d*e), x) - Simp(sqrt(S(3))*q*ArcTan(S(2)*sqrt(S(3))*q*(a + b*x)**(S(1)/3)/(S(3)*(c + d*x)**(S(1)/3)) + sqrt(S(3))/S(3))/(-c*f + d*e), x)
pattern108 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))**(S(1)/3)*(x_*WC('d', S(1)) + WC('c', S(0)))**(S(2)/3)*(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
rule108 = ReplacementRule(pattern108, With108)
pattern109 = Pattern(Integral(S(1)/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons154)
def replacement109(f, b, d, c, a, x, e):
rubi.append(109)
return Dist(b*f, Subst(Int(S(1)/(b*f**S(2)*x**S(2) + d*(-a*f + b*e)**S(2)), x), x, sqrt(a + b*x)*sqrt(c + d*x)), x)
rule109 = ReplacementRule(pattern109, replacement109)
def With110(m, f, b, d, c, a, n, x, e):
q = Denominator(m)
rubi.append(110)
return Dist(q, Subst(Int(x**(q*(m + S(1)) + S(-1))/(-a*f + b*e - x**q*(-c*f + d*e)), x), x, (a + b*x)**(S(1)/q)*(c + d*x)**(-S(1)/q)), x)
pattern110 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons155, cons93, cons107, cons156)
rule110 = ReplacementRule(pattern110, With110)
pattern111 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons157, cons87, cons88, cons158)
def replacement111(p, m, f, b, d, c, a, n, x, e):
rubi.append(111)
return -Dist(n*(-c*f + d*e)/((m + S(1))*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**(n + S(-1))*(e + f*x)**p, x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**(p + S(1))/((m + S(1))*(-a*f + b*e)), x)
rule111 = ReplacementRule(pattern111, replacement111)
pattern112 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons159, cons160, cons66)
def replacement112(p, m, f, b, d, c, a, n, x, e):
rubi.append(112)
return Simp(b*(a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule112 = ReplacementRule(pattern112, replacement112)
pattern113 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons159, cons161)
def replacement113(p, m, f, b, d, c, a, n, x, e):
rubi.append(113)
return Dist((a*d*f*(m + S(1)) + b*c*f*(n + S(1)) + b*d*e*(p + S(1)))/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p, x), x) + Simp(b*(a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule113 = ReplacementRule(pattern113, replacement113)
pattern114 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons162, cons94, cons88, cons163, cons164)
def replacement114(p, m, f, b, d, c, a, n, x, e):
rubi.append(114)
return -Dist(S(1)/(b*(m + S(1))), Int((a + b*x)**(m + S(1))*(c + d*x)**(n + S(-1))*(e + f*x)**(p + S(-1))*Simp(c*f*p + d*e*n + d*f*x*(n + p), x), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p/(b*(m + S(1))), x)
rule114 = ReplacementRule(pattern114, replacement114)
pattern115 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons162, cons94, cons165, cons164)
def replacement115(p, m, f, b, d, c, a, n, x, e):
rubi.append(115)
return Dist(S(1)/(b*(m + S(1))*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**(n + S(-2))*(e + f*x)**p*Simp(a*d*(c*f*(p + S(1)) + d*e*(n + S(-1))) + b*c*(-c*f*(m + p + S(2)) + d*e*(m - n + S(2))) + d*x*(a*d*f*(n + p) + b*(-c*f*(m + n + p + S(1)) + d*e*(m + S(1)))), x), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(-1))*(e + f*x)**(p + S(1))*(-a*d + b*c)/(b*(m + S(1))*(-a*f + b*e)), x)
rule115 = ReplacementRule(pattern115, replacement115)
pattern116 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons162, cons94, cons88, cons164)
def replacement116(p, m, f, b, d, c, a, n, x, e):
rubi.append(116)
return -Dist(S(1)/((m + S(1))*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**(n + S(-1))*(e + f*x)**p*Simp(c*f*(m + p + S(2)) + d*e*n + d*f*x*(m + n + p + S(2)), x), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**(p + S(1))/((m + S(1))*(-a*f + b*e)), x)
rule116 = ReplacementRule(pattern116, replacement116)
pattern117 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons31, cons166, cons167, cons17)
def replacement117(p, m, f, b, d, c, a, n, x, e):
rubi.append(117)
return Dist(S(1)/(d*f*(m + n + p + S(1))), Int((a + b*x)**(m + S(-2))*(c + d*x)**n*(e + f*x)**p*Simp(a**S(2)*d*f*(m + n + p + S(1)) + b*x*(a*d*f*(S(2)*m + n + p) - b*(c*f*(m + p) + d*e*(m + n))) - b*(a*(c*f*(p + S(1)) + d*e*(n + S(1))) + b*c*e*(m + S(-1))), x), x), x) + Simp(b*(a + b*x)**(m + S(-1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/(d*f*(m + n + p + S(1))), x)
rule117 = ReplacementRule(pattern117, replacement117)
pattern118 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons162, cons168, cons88, cons167, cons169)
def replacement118(p, m, f, b, d, a, c, n, x, e):
rubi.append(118)
return -Dist(S(1)/(f*(m + n + p + S(1))), Int((a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1))*(e + f*x)**p*Simp(a*n*(-c*f + d*e) + c*m*(-a*f + b*e) + x*(b*n*(-c*f + d*e) + d*m*(-a*f + b*e)), x), x), x) + Simp((a + b*x)**m*(c + d*x)**n*(e + f*x)**(p + S(1))/(f*(m + n + p + S(1))), x)
rule118 = ReplacementRule(pattern118, replacement118)
pattern119 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons31, cons166, cons167, cons170)
def replacement119(p, m, f, b, d, c, a, n, x, e):
rubi.append(119)
return Dist(S(1)/(d*f*(m + n + p + S(1))), Int((a + b*x)**(m + S(-2))*(c + d*x)**n*(e + f*x)**p*Simp(a**S(2)*d*f*(m + n + p + S(1)) + b*x*(a*d*f*(S(2)*m + n + p) - b*(c*f*(m + p) + d*e*(m + n))) - b*(a*(c*f*(p + S(1)) + d*e*(n + S(1))) + b*c*e*(m + S(-1))), x), x), x) + Simp(b*(a + b*x)**(m + S(-1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/(d*f*(m + n + p + S(1))), x)
rule119 = ReplacementRule(pattern119, replacement119)
pattern120 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons31, cons94, cons17, cons171)
def replacement120(p, m, f, b, d, c, a, n, x, e):
rubi.append(120)
return Dist(S(1)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p*Simp(a*d*f*(m + S(1)) - b*d*f*x*(m + n + p + S(3)) - b*(c*f*(m + p + S(2)) + d*e*(m + n + S(2))), x), x), x) + Simp(b*(a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule120 = ReplacementRule(pattern120, replacement120)
pattern121 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons31, cons94, cons170)
def replacement121(p, m, f, b, d, c, a, n, x, e):
rubi.append(121)
return Dist(S(1)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p*Simp(a*d*f*(m + S(1)) - b*d*f*x*(m + n + p + S(3)) - b*(c*f*(m + p + S(2)) + d*e*(m + n + S(2))), x), x), x) + Simp(b*(a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule121 = ReplacementRule(pattern121, replacement121)
pattern122 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons172, cons173)
def replacement122(m, f, b, d, c, a, n, x, e):
rubi.append(122)
return Dist(b/f, Int((a + b*x)**(m + S(-1))*(c + d*x)**n, x), x) - Dist((-a*f + b*e)/f, Int((a + b*x)**(m + S(-1))*(c + d*x)**n/(e + f*x), x), x)
rule122 = ReplacementRule(pattern122, replacement122)
pattern123 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**(S(1)/4)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons174)
def replacement123(f, b, d, c, a, x, e):
rubi.append(123)
return Dist(S(-4), Subst(Int(x**S(2)/(sqrt(c - d*e/f + d*x**S(4)/f)*(-a*f + b*e - b*x**S(4))), x), x, (e + f*x)**(S(1)/4)), x)
rule123 = ReplacementRule(pattern123, replacement123)
pattern124 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**(S(1)/4)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons175)
def replacement124(f, b, d, c, a, x, e):
rubi.append(124)
return Dist(sqrt(-f*(c + d*x)/(-c*f + d*e))/sqrt(c + d*x), Int(S(1)/((a + b*x)*(e + f*x)**(S(1)/4)*sqrt(-c*f/(-c*f + d*e) - d*f*x/(-c*f + d*e))), x), x)
rule124 = ReplacementRule(pattern124, replacement124)
pattern125 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**(S(3)/4)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons174)
def replacement125(f, b, d, c, a, x, e):
rubi.append(125)
return Dist(S(-4), Subst(Int(S(1)/(sqrt(c - d*e/f + d*x**S(4)/f)*(-a*f + b*e - b*x**S(4))), x), x, (e + f*x)**(S(1)/4)), x)
rule125 = ReplacementRule(pattern125, replacement125)
pattern126 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**(S(3)/4)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons175)
def replacement126(f, b, d, c, a, x, e):
rubi.append(126)
return Dist(sqrt(-f*(c + d*x)/(-c*f + d*e))/sqrt(c + d*x), Int(S(1)/((a + b*x)*(e + f*x)**(S(3)/4)*sqrt(-c*f/(-c*f + d*e) - d*f*x/(-c*f + d*e))), x), x)
rule126 = ReplacementRule(pattern126, replacement126)
pattern127 = Pattern(Integral(sqrt(e_ + x_*WC('f', S(1)))/(sqrt(x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))), x_), cons3, cons7, cons27, cons48, cons125, cons176, cons177, cons178, cons179)
def replacement127(f, b, d, c, x, e):
rubi.append(127)
return Simp(S(2)*sqrt(e)*EllipticE(asin(sqrt(b*x)/(sqrt(c)*Rt(-b/d, S(2)))), c*f/(d*e))*Rt(-b/d, S(2))/b, x)
rule127 = ReplacementRule(pattern127, replacement127)
pattern128 = Pattern(Integral(sqrt(e_ + x_*WC('f', S(1)))/(sqrt(x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))), x_), cons3, cons7, cons27, cons48, cons125, cons176, cons177, cons178, cons180)
def replacement128(f, b, d, c, x, e):
rubi.append(128)
return Dist(sqrt(-b*x)/sqrt(b*x), Int(sqrt(e + f*x)/(sqrt(-b*x)*sqrt(c + d*x)), x), x)
rule128 = ReplacementRule(pattern128, replacement128)
pattern129 = Pattern(Integral(sqrt(e_ + x_*WC('f', S(1)))/(sqrt(x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))), x_), cons3, cons7, cons27, cons48, cons125, cons176, cons181)
def replacement129(f, b, d, c, x, e):
rubi.append(129)
return Dist(sqrt(S(1) + d*x/c)*sqrt(e + f*x)/(sqrt(S(1) + f*x/e)*sqrt(c + d*x)), Int(sqrt(S(1) + f*x/e)/(sqrt(b*x)*sqrt(S(1) + d*x/c)), x), x)
rule129 = ReplacementRule(pattern129, replacement129)
pattern130 = Pattern(Integral(sqrt(x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons120, cons182, cons183, cons184)
def replacement130(f, b, d, a, c, x, e):
rubi.append(130)
return Simp(S(2)*EllipticE(asin(sqrt(a + b*x)/Rt(-(-a*d + b*c)/d, S(2))), f*(-a*d + b*c)/(d*(-a*f + b*e)))*Rt(-(-a*f + b*e)/d, S(2))/b, x)
rule130 = ReplacementRule(pattern130, replacement130)
pattern131 = Pattern(Integral(sqrt(x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons185, cons183)
def replacement131(f, b, d, a, c, x, e):
rubi.append(131)
return Dist(sqrt(b*(c + d*x)/(-a*d + b*c))*sqrt(e + f*x)/(sqrt(b*(e + f*x)/(-a*f + b*e))*sqrt(c + d*x)), Int(sqrt(b*e/(-a*f + b*e) + b*f*x/(-a*f + b*e))/(sqrt(a + b*x)*sqrt(b*c/(-a*d + b*c) + b*d*x/(-a*d + b*c))), x), x)
rule131 = ReplacementRule(pattern131, replacement131)
pattern132 = Pattern(Integral(S(1)/(sqrt(x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))*sqrt(e_ + x_*WC('f', S(1)))), x_), cons3, cons7, cons27, cons48, cons125, cons177, cons178, cons186)
def replacement132(f, b, d, c, x, e):
rubi.append(132)
return Simp(S(2)*EllipticF(asin(sqrt(b*x)/(sqrt(c)*Rt(-b/d, S(2)))), c*f/(d*e))*Rt(-b/d, S(2))/(b*sqrt(e)), x)
rule132 = ReplacementRule(pattern132, replacement132)
pattern133 = Pattern(Integral(S(1)/(sqrt(x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))*sqrt(e_ + x_*WC('f', S(1)))), x_), cons3, cons7, cons27, cons48, cons125, cons177, cons178, cons187)
def replacement133(f, b, d, c, x, e):
rubi.append(133)
return Simp(S(2)*EllipticF(asin(sqrt(b*x)/(sqrt(c)*Rt(-b/d, S(2)))), c*f/(d*e))*Rt(-b/d, S(2))/(b*sqrt(e)), x)
rule133 = ReplacementRule(pattern133, replacement133)
pattern134 = Pattern(Integral(S(1)/(sqrt(x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))*sqrt(e_ + x_*WC('f', S(1)))), x_), cons3, cons7, cons27, cons48, cons125, cons181)
def replacement134(f, b, d, c, x, e):
rubi.append(134)
return Dist(sqrt(S(1) + d*x/c)*sqrt(S(1) + f*x/e)/(sqrt(c + d*x)*sqrt(e + f*x)), Int(S(1)/(sqrt(b*x)*sqrt(S(1) + d*x/c)*sqrt(S(1) + f*x/e)), x), x)
rule134 = ReplacementRule(pattern134, replacement134)
pattern135 = Pattern(Integral(S(1)/(sqrt(a_ + x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))*sqrt(e_ + x_*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons120, cons182, cons156, cons188, cons189)
def replacement135(f, b, d, a, c, x, e):
rubi.append(135)
return Simp(S(2)*sqrt(b**S(2)/((-a*d + b*c)*(-a*f + b*e)))*EllipticF(asin(sqrt(a + b*x)/Rt(-(-a*d + b*c)/d, S(2))), f*(-a*d + b*c)/(d*(-a*f + b*e)))*Rt(-(-a*d + b*c)/d, S(2))/b, x)
rule135 = ReplacementRule(pattern135, replacement135)
pattern136 = Pattern(Integral(S(1)/(sqrt(a_ + x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))*sqrt(e_ + x_*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons120, cons182, cons156, cons188, cons190)
def replacement136(f, b, d, a, c, x, e):
rubi.append(136)
return Simp(S(2)*sqrt(b**S(2)/((-a*d + b*c)*(-a*f + b*e)))*EllipticF(asin(sqrt(a + b*x)/Rt(-(-a*d + b*c)/d, S(2))), f*(-a*d + b*c)/(d*(-a*f + b*e)))*Rt(-(-a*d + b*c)/d, S(2))/b, x)
rule136 = ReplacementRule(pattern136, replacement136)
pattern137 = Pattern(Integral(S(1)/(sqrt(a_ + x_*WC('b', S(1)))*sqrt(c_ + x_*WC('d', S(1)))*sqrt(e_ + x_*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons185, cons156, cons188)
def replacement137(f, b, d, a, c, x, e):
rubi.append(137)
return Dist(sqrt(b*(c + d*x)/(-a*d + b*c))*sqrt(b*(e + f*x)/(-a*f + b*e))/(sqrt(c + d*x)*sqrt(e + f*x)), Int(S(1)/(sqrt(a + b*x)*sqrt(b*c/(-a*d + b*c) + b*d*x/(-a*d + b*c))*sqrt(b*e/(-a*f + b*e) + b*f*x/(-a*f + b*e))), x), x)
rule137 = ReplacementRule(pattern137, replacement137)
def With138(f, b, d, c, a, x, e):
q = Rt(b*(-a*f + b*e)/(-a*d + b*c)**S(2), S(3))
rubi.append(138)
return -Simp(log(a + b*x)/(S(2)*q*(-a*d + b*c)), x) + Simp(S(3)*log(q*(c + d*x)**(S(2)/3) - (e + f*x)**(S(1)/3))/(S(4)*q*(-a*d + b*c)), x) - Simp(sqrt(S(3))*ArcTan(S(2)*sqrt(S(3))*q*(c + d*x)**(S(2)/3)/(S(3)*(e + f*x)**(S(1)/3)) + sqrt(S(3))/S(3))/(S(2)*q*(-a*d + b*c)), x)
pattern138 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))**(S(1)/3)*(x_*WC('f', S(1)) + WC('e', S(0)))**(S(1)/3)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons191)
rule138 = ReplacementRule(pattern138, With138)
pattern139 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_/((x_*WC('d', S(1)) + WC('c', S(0)))**(S(1)/3)*(x_*WC('f', S(1)) + WC('e', S(0)))**(S(1)/3)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons191, cons17, cons94)
def replacement139(m, f, b, d, c, a, x, e):
rubi.append(139)
return Dist(f/(S(6)*(m + S(1))*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(a*d*(S(3)*m + S(1)) - S(3)*b*c*(S(3)*m + S(5)) - S(2)*b*d*x*(S(3)*m + S(7)))/((c + d*x)**(S(1)/3)*(e + f*x)**(S(1)/3)), x), x) + Simp(b*(a + b*x)**(m + S(1))*(c + d*x)**(S(2)/3)*(e + f*x)**(S(2)/3)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule139 = ReplacementRule(pattern139, replacement139)
pattern140 = Pattern(Integral((x_*WC('f', S(1)))**WC('p', S(1))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons21, cons4, cons5, cons70, cons124, cons43, cons177)
def replacement140(p, m, f, b, d, a, c, n, x):
rubi.append(140)
return Int((f*x)**p*(a*c + b*d*x**S(2))**m, x)
rule140 = ReplacementRule(pattern140, replacement140)
pattern141 = Pattern(Integral((x_*WC('f', S(1)))**WC('p', S(1))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons21, cons4, cons5, cons70, cons124)
def replacement141(p, m, f, b, d, a, c, n, x):
rubi.append(141)
return Dist((a + b*x)**FracPart(m)*(c + d*x)**FracPart(m)*(a*c + b*d*x**S(2))**(-FracPart(m)), Int((f*x)**p*(a*c + b*d*x**S(2))**m, x), x)
rule141 = ReplacementRule(pattern141, replacement141)
pattern142 = Pattern(Integral((x_*WC('f', S(1)))**WC('p', S(1))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons21, cons4, cons5, cons70, cons192, cons144)
def replacement142(p, m, f, b, d, a, c, n, x):
rubi.append(142)
return Int(ExpandIntegrand((f*x)**p*(a + b*x)**n*(c + d*x)**n, (a + b*x)**(m - n), x), x)
rule142 = ReplacementRule(pattern142, replacement142)
pattern143 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons193)
def replacement143(p, m, f, b, d, a, c, n, x, e):
rubi.append(143)
return Int(ExpandIntegrand((a + b*x)**m*(c + d*x)**n*(e + f*x)**p, x), x)
rule143 = ReplacementRule(pattern143, replacement143)
pattern144 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons194, cons66, cons195)
def replacement144(p, m, f, b, d, c, a, n, x, e):
rubi.append(144)
return Dist(S(1)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p*Simp(a*d*f*(m + S(1)) - b*d*f*x*(m + n + p + S(3)) - b*(c*f*(m + p + S(2)) + d*e*(m + n + S(2))), x), x), x) + Simp(b*(a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule144 = ReplacementRule(pattern144, replacement144)
pattern145 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons157, cons196)
def replacement145(p, m, f, b, d, c, a, n, x, e):
rubi.append(145)
return Simp((a + b*x)**(m + S(1))*(e + f*x)**(-m + S(-1))*(-a*d + b*c)**n*(-a*f + b*e)**(-n + S(-1))*Hypergeometric2F1(m + S(1), -n, m + S(2), -(a + b*x)*(-c*f + d*e)/((e + f*x)*(-a*d + b*c)))/(m + S(1)), x)
rule145 = ReplacementRule(pattern145, replacement145)
pattern146 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons157, cons23)
def replacement146(p, m, f, b, d, c, a, n, x, e):
rubi.append(146)
return Simp(((c + d*x)*(-a*f + b*e)/((e + f*x)*(-a*d + b*c)))**(-n)*(a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**(p + S(1))*Hypergeometric2F1(m + S(1), -n, m + S(2), -(a + b*x)*(-c*f + d*e)/((e + f*x)*(-a*d + b*c)))/((m + S(1))*(-a*f + b*e)), x)
rule146 = ReplacementRule(pattern146, replacement146)
pattern147 = Pattern(Integral((x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_*(e_ + x_*WC('f', S(1)))**p_, x_), cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons18, cons23, cons177, cons197)
def replacement147(p, m, f, b, d, c, n, x, e):
rubi.append(147)
return Simp(c**n*e**p*(b*x)**(m + S(1))*AppellF1(m + S(1), -n, -p, m + S(2), -d*x/c, -f*x/e)/(b*(m + S(1))), x)
rule147 = ReplacementRule(pattern147, replacement147)
pattern148 = Pattern(Integral((x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_*(e_ + x_*WC('f', S(1)))**p_, x_), cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons18, cons23, cons198, cons199)
def replacement148(p, m, f, b, d, c, n, x, e):
rubi.append(148)
return Simp((d/(-c*f + d*e))**(-p)*(-d/(b*c))**(-m)*(c + d*x)**(n + S(1))*AppellF1(n + S(1), -m, -p, n + S(2), S(1) + d*x/c, -f*(c + d*x)/(-c*f + d*e))/(d*(n + S(1))), x)
rule148 = ReplacementRule(pattern148, replacement148)
pattern149 = Pattern(Integral((x_*WC('b', S(1)))**m_*(c_ + x_*WC('d', S(1)))**n_*(e_ + x_*WC('f', S(1)))**p_, x_), cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons18, cons23, cons117)
def replacement149(p, m, f, b, d, c, n, x, e):
rubi.append(149)
return Dist(c**IntPart(n)*(S(1) + d*x/c)**(-FracPart(n))*(c + d*x)**FracPart(n), Int((b*x)**m*(S(1) + d*x/c)**n*(e + f*x)**p, x), x)
rule149 = ReplacementRule(pattern149, replacement149)
pattern150 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons18, cons23, cons38, cons120, cons200)
def replacement150(p, m, f, b, d, c, a, n, x, e):
rubi.append(150)
return Simp(b**(-p + S(-1))*(b/(-a*d + b*c))**(-n)*(a + b*x)**(m + S(1))*(-a*f + b*e)**p*AppellF1(m + S(1), -n, -p, m + S(2), -d*(a + b*x)/(-a*d + b*c), -f*(a + b*x)/(-a*f + b*e))/(m + S(1)), x)
rule150 = ReplacementRule(pattern150, replacement150)
pattern151 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons18, cons23, cons38, cons201, cons202)
def replacement151(p, m, f, b, d, c, a, n, x, e):
rubi.append(151)
return Dist((b/(-a*d + b*c))**(-IntPart(n))*(b*(c + d*x)/(-a*d + b*c))**(-FracPart(n))*(c + d*x)**FracPart(n), Int((a + b*x)**m*(e + f*x)**p*(b*c/(-a*d + b*c) + b*d*x/(-a*d + b*c))**n, x), x)
rule151 = ReplacementRule(pattern151, replacement151)
pattern152 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons18, cons23, cons147, cons120, cons182, cons203, cons204)
def replacement152(p, m, f, b, d, c, a, n, x, e):
rubi.append(152)
return Simp((b/(-a*d + b*c))**(-n)*(b/(-a*f + b*e))**(-p)*(a + b*x)**(m + S(1))*AppellF1(m + S(1), -n, -p, m + S(2), -d*(a + b*x)/(-a*d + b*c), -f*(a + b*x)/(-a*f + b*e))/(b*(m + S(1))), x)
rule152 = ReplacementRule(pattern152, replacement152)
pattern153 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons18, cons23, cons147, cons120, cons205)
def replacement153(p, m, f, b, d, c, a, n, x, e):
rubi.append(153)
return Dist((b/(-a*f + b*e))**(-IntPart(p))*(b*(e + f*x)/(-a*f + b*e))**(-FracPart(p))*(e + f*x)**FracPart(p), Int((a + b*x)**m*(c + d*x)**n*(b*e/(-a*f + b*e) + b*f*x/(-a*f + b*e))**p, x), x)
rule153 = ReplacementRule(pattern153, replacement153)
pattern154 = Pattern(Integral((a_ + x_*WC('b', S(1)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons18, cons23, cons147, cons201, cons202, cons206)
def replacement154(p, m, f, b, d, c, a, n, x, e):
rubi.append(154)
return Dist((b/(-a*d + b*c))**(-IntPart(n))*(b*(c + d*x)/(-a*d + b*c))**(-FracPart(n))*(c + d*x)**FracPart(n), Int((a + b*x)**m*(e + f*x)**p*(b*c/(-a*d + b*c) + b*d*x/(-a*d + b*c))**n, x), x)
rule154 = ReplacementRule(pattern154, replacement154)
pattern155 = Pattern(Integral((e_ + u_*WC('f', S(1)))**WC('p', S(1))*(u_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(u_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons68, cons69)
def replacement155(p, u, m, f, b, d, a, c, n, x, e):
rubi.append(155)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p, x), x, u), x)
rule155 = ReplacementRule(pattern155, replacement155)
pattern156 = Pattern(Integral((e_ + x_*WC('f', S(1)))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons207)
def replacement156(m, f, b, g, d, a, c, n, x, h, e):
rubi.append(156)
return Int(ExpandIntegrand((a + b*x)**m*(c + d*x)**n*(e + f*x)*(g + h*x), x), x)
rule156 = ReplacementRule(pattern156, replacement156)
pattern157 = Pattern(Integral((e_ + x_*WC('f', S(1)))*(x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons72, cons66, cons210)
def replacement157(m, f, b, g, d, c, a, n, x, h, e):
rubi.append(157)
return Dist((a*d*f*h*m + b*(-c*f*h*(m + S(2)) + d*(e*h + f*g)))/(b**S(2)*d), Int((a + b*x)**(m + S(1))*(c + d*x)**n, x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(-a**S(2)*d*f*h*m - a*b*(-c*f*h*(m + S(1)) + d*(e*h + f*g)) + b**S(2)*d*e*g + b*f*h*x*(m + S(1))*(-a*d + b*c))/(b**S(2)*d*(m + S(1))*(-a*d + b*c)), x)
rule157 = ReplacementRule(pattern157, replacement157)
pattern158 = Pattern(Integral((e_ + x_*WC('f', S(1)))*(x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons93, cons94, cons89)
def replacement158(m, f, b, g, d, c, a, n, x, h, e):
rubi.append(158)
return -Dist((a**S(2)*d**S(2)*f*h*(n**S(2) + S(3)*n + S(2)) + a*b*d*(n + S(1))*(S(2)*c*f*h*(m + S(1)) - d*(e*h + f*g)*(m + n + S(3))) + b**S(2)*(c**S(2)*f*h*(m**S(2) + S(3)*m + S(2)) - c*d*(m + S(1))*(e*h + f*g)*(m + n + S(3)) + d**S(2)*e*g*(m**S(2) + m*(S(2)*n + S(5)) + n**S(2) + S(5)*n + S(6))))/(b*d*(m + S(1))*(n + S(1))*(-a*d + b*c)**S(2)), Int((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1)), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(a**S(2)*c*d*f*h*(n + S(1)) + a*b*(c**S(2)*f*h*(m + S(1)) - c*d*(e*h + f*g)*(m + n + S(2)) + d**S(2)*e*g*(m + S(1))) + b**S(2)*c*d*e*g*(n + S(1)) + x*(a**S(2)*d**S(2)*f*h*(n + S(1)) - a*b*d**S(2)*(n + S(1))*(e*h + f*g) + b**S(2)*(c**S(2)*f*h*(m + S(1)) - c*d*(m + S(1))*(e*h + f*g) + d**S(2)*e*g*(m + n + S(2)))))/(b*d*(m + S(1))*(n + S(1))*(-a*d + b*c)**S(2)), x)
rule158 = ReplacementRule(pattern158, replacement158)
pattern159 = Pattern(Integral((e_ + x_*WC('f', S(1)))*(x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons211)
def replacement159(m, f, b, g, d, c, a, n, x, h, e):
rubi.append(159)
return Dist(-d*(m + n + S(3))*(a**S(2)*d*f*h*(m - n) - a*b*(S(2)*c*f*h*(m + S(1)) - d*(n + S(1))*(e*h + f*g)) + b**S(2)*(c*(m + S(1))*(e*h + f*g) - d*e*g*(m + n + S(2))))/(b**S(2)*(m + S(1))*(m + S(2))*(-a*d + b*c)**S(2)) + f*h/b**S(2), Int((a + b*x)**(m + S(2))*(c + d*x)**n, x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(-a**S(3)*d*f*h*(n + S(2)) - a**S(2)*b*(c*f*h*m - d*(e*h + f*g)*(m + n + S(3))) - a*b**S(2)*(c*(e*h + f*g) + d*e*g*(S(2)*m + n + S(4))) + b**S(3)*c*e*g*(m + S(2)) + b*x*(a**S(2)*d*f*h*(m - n) - a*b*(S(2)*c*f*h*(m + S(1)) - d*(n + S(1))*(e*h + f*g)) + b**S(2)*(c*(m + S(1))*(e*h + f*g) - d*e*g*(m + n + S(2)))))/(b**S(2)*(m + S(1))*(m + S(2))*(-a*d + b*c)**S(2)), x)
rule159 = ReplacementRule(pattern159, replacement159)
pattern160 = Pattern(Integral((e_ + x_*WC('f', S(1)))*(x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons212, cons66, cons213)
def replacement160(m, f, b, g, d, c, a, n, x, h, e):
rubi.append(160)
return -Dist((a**S(2)*d**S(2)*f*h*(n + S(1))*(n + S(2)) + a*b*d*(n + S(1))*(S(2)*c*f*h*(m + S(1)) - d*(e*h + f*g)*(m + n + S(3))) + b**S(2)*(c**S(2)*f*h*(m + S(1))*(m + S(2)) - c*d*(m + S(1))*(e*h + f*g)*(m + n + S(3)) + d**S(2)*e*g*(m + n + S(2))*(m + n + S(3))))/(b**S(2)*d*(m + S(1))*(-a*d + b*c)*(m + n + S(3))), Int((a + b*x)**(m + S(1))*(c + d*x)**n, x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(a**S(2)*d*f*h*(n + S(2)) + a*b*(c*f*h*(m + S(1)) - d*(e*h + f*g)*(m + n + S(3))) + b**S(2)*d*e*g*(m + n + S(3)) + b*f*h*x*(m + S(1))*(-a*d + b*c))/(b**S(2)*d*(m + S(1))*(-a*d + b*c)*(m + n + S(3))), x)
rule160 = ReplacementRule(pattern160, replacement160)
pattern161 = Pattern(Integral((e_ + x_*WC('f', S(1)))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons214, cons213)
def replacement161(m, f, b, g, d, a, c, n, x, h, e):
rubi.append(161)
return Dist((a**S(2)*d**S(2)*f*h*(n + S(1))*(n + S(2)) + a*b*d*(n + S(1))*(S(2)*c*f*h*(m + S(1)) - d*(e*h + f*g)*(m + n + S(3))) + b**S(2)*(c**S(2)*f*h*(m + S(1))*(m + S(2)) - c*d*(m + S(1))*(e*h + f*g)*(m + n + S(3)) + d**S(2)*e*g*(m + n + S(2))*(m + n + S(3))))/(b**S(2)*d**S(2)*(m + n + S(2))*(m + n + S(3))), Int((a + b*x)**m*(c + d*x)**n, x), x) - Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(a*d*f*h*(n + S(2)) + b*c*f*h*(m + S(2)) - b*d*f*h*x*(m + n + S(2)) - b*d*(e*h + f*g)*(m + n + S(3)))/(b**S(2)*d**S(2)*(m + n + S(2))*(m + n + S(3))), x)
rule161 = ReplacementRule(pattern161, replacement161)
pattern162 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons215)
def replacement162(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(162)
return Int(ExpandIntegrand((a + b*x)**m*(c + d*x)**n*(e + f*x)**p*(g + h*x), x), x)
rule162 = ReplacementRule(pattern162, replacement162)
pattern163 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons93, cons94, cons88, cons17)
def replacement163(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(163)
return -Dist(S(1)/(b*(m + S(1))*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**(n + S(-1))*(e + f*x)**p*Simp(b*c*(m + S(1))*(-e*h + f*g) + d*x*(b*(m + S(1))*(-e*h + f*g) + f*(-a*h + b*g)*(n + p + S(1))) + (-a*h + b*g)*(c*f*(p + S(1)) + d*e*n), x), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**(p + S(1))*(-a*h + b*g)/(b*(m + S(1))*(-a*f + b*e)), x)
rule163 = ReplacementRule(pattern163, replacement163)
pattern164 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons93, cons94, cons88, cons170)
def replacement164(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(164)
return -Dist(S(1)/(b*(m + S(1))*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**(n + S(-1))*(e + f*x)**p*Simp(b*c*(m + S(1))*(-e*h + f*g) + d*x*(b*(m + S(1))*(-e*h + f*g) + f*(-a*h + b*g)*(n + p + S(1))) + (-a*h + b*g)*(c*f*(p + S(1)) + d*e*n), x), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**(p + S(1))*(-a*h + b*g)/(b*(m + S(1))*(-a*f + b*e)), x)
rule164 = ReplacementRule(pattern164, replacement164)
pattern165 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons31, cons94, cons17)
def replacement165(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(165)
return Dist(S(1)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p*Simp(-d*f*x*(-a*h + b*g)*(m + n + p + S(3)) + (m + S(1))*(a*d*f*g + b*c*e*h - b*g*(c*f + d*e)) - (-a*h + b*g)*(c*f*(p + S(1)) + d*e*(n + S(1))), x), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))*(-a*h + b*g)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule165 = ReplacementRule(pattern165, replacement165)
pattern166 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons31, cons94, cons170)
def replacement166(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(166)
return Dist(S(1)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p*Simp(-d*f*x*(-a*h + b*g)*(m + n + p + S(3)) + (m + S(1))*(a*d*f*g + b*c*e*h - b*g*(c*f + d*e)) - (-a*h + b*g)*(c*f*(p + S(1)) + d*e*(n + S(1))), x), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))*(-a*h + b*g)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule166 = ReplacementRule(pattern166, replacement166)
pattern167 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons31, cons168, cons144, cons17)
def replacement167(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(167)
return Dist(S(1)/(d*f*(m + n + p + S(2))), Int((a + b*x)**(m + S(-1))*(c + d*x)**n*(e + f*x)**p*Simp(a*d*f*g*(m + n + p + S(2)) - h*(a*(c*f*(p + S(1)) + d*e*(n + S(1))) + b*c*e*m) + x*(b*d*f*g*(m + n + p + S(2)) + h*(a*d*f*m - b*(c*f*(m + p + S(1)) + d*e*(m + n + S(1))))), x), x), x) + Simp(h*(a + b*x)**m*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/(d*f*(m + n + p + S(2))), x)
rule167 = ReplacementRule(pattern167, replacement167)
pattern168 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons31, cons168, cons144, cons170)
def replacement168(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(168)
return Dist(S(1)/(d*f*(m + n + p + S(2))), Int((a + b*x)**(m + S(-1))*(c + d*x)**n*(e + f*x)**p*Simp(a*d*f*g*(m + n + p + S(2)) - h*(a*(c*f*(p + S(1)) + d*e*(n + S(1))) + b*c*e*m) + x*(b*d*f*g*(m + n + p + S(2)) + h*(a*d*f*m - b*(c*f*(m + p + S(1)) + d*e*(m + n + S(1))))), x), x), x) + Simp(h*(a + b*x)**m*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))/(d*f*(m + n + p + S(2))), x)
rule168 = ReplacementRule(pattern168, replacement168)
pattern169 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons194, cons66, cons195)
def replacement169(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(169)
return Dist(S(1)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p*Simp(-d*f*x*(-a*h + b*g)*(m + n + p + S(3)) + (m + S(1))*(a*d*f*g + b*c*e*h - b*g*(c*f + d*e)) - (-a*h + b*g)*(c*f*(p + S(1)) + d*e*(n + S(1))), x), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(n + S(1))*(e + f*x)**(p + S(1))*(-a*h + b*g)/((m + S(1))*(-a*d + b*c)*(-a*f + b*e)), x)
rule169 = ReplacementRule(pattern169, replacement169)
pattern170 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0)))/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement170(p, f, b, g, d, c, a, x, h, e):
rubi.append(170)
return Dist((-a*h + b*g)/(-a*d + b*c), Int((e + f*x)**p/(a + b*x), x), x) - Dist((-c*h + d*g)/(-a*d + b*c), Int((e + f*x)**p/(c + d*x), x), x)
rule170 = ReplacementRule(pattern170, replacement170)
pattern171 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0)))/(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons217)
def replacement171(p, f, b, g, d, c, a, n, x, h, e):
rubi.append(171)
return Dist(h/b, Int((c + d*x)**n*(e + f*x)**p, x), x) + Dist((-a*h + b*g)/b, Int((c + d*x)**n*(e + f*x)**p/(a + b*x), x), x)
rule171 = ReplacementRule(pattern171, replacement171)
pattern172 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))/(sqrt(c_ + x_*WC('d', S(1)))*sqrt(e_ + x_*WC('f', S(1)))*sqrt(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons188, cons218)
def replacement172(f, b, g, d, a, c, x, h, e):
rubi.append(172)
return Dist(h/f, Int(sqrt(e + f*x)/(sqrt(a + b*x)*sqrt(c + d*x)), x), x) + Dist((-e*h + f*g)/f, Int(S(1)/(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(e + f*x)), x), x)
rule172 = ReplacementRule(pattern172, replacement172)
pattern173 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons219)
def replacement173(p, m, f, b, g, d, c, a, n, x, h, e):
rubi.append(173)
return Dist(h/b, Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p, x), x) + Dist((-a*h + b*g)/b, Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p, x), x)
rule173 = ReplacementRule(pattern173, replacement173)
pattern174 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0)))**q_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons50, cons13, cons145)
def replacement174(p, f, b, g, d, c, a, x, h, q, e):
rubi.append(174)
return Dist((-a*f + b*e)/(-a*d + b*c), Int((e + f*x)**(p + S(-1))*(g + h*x)**q/(a + b*x), x), x) - Dist((-c*f + d*e)/(-a*d + b*c), Int((e + f*x)**(p + S(-1))*(g + h*x)**q/(c + d*x), x), x)
rule174 = ReplacementRule(pattern174, replacement174)
pattern175 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement175(f, b, g, d, c, a, x, h, e):
rubi.append(175)
return Simp(-S(2)*sqrt(d*(e + f*x)/(-c*f + d*e))*sqrt(d*(g + h*x)/(-c*h + d*g))*EllipticPi(-b*(-c*f + d*e)/(f*(-a*d + b*c)), asin(sqrt(-f/(-c*f + d*e))*sqrt(c + d*x)), h*(-c*f + d*e)/(f*(-c*h + d*g)))/(sqrt(-f/(-c*f + d*e))*sqrt(e + f*x)*sqrt(g + h*x)*(-a*d + b*c)), x)
rule175 = ReplacementRule(pattern175, replacement175)
pattern176 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**n_/((x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons80)
def replacement176(f, b, g, d, c, a, n, x, h, e):
rubi.append(176)
return Int(ExpandIntegrand(S(1)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), (c + d*x)**(n + S(1)/2)/(a + b*x), x), x)
rule176 = ReplacementRule(pattern176, replacement176)
pattern177 = Pattern(Integral(sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))/((x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement177(f, b, g, d, c, a, x, h, e):
rubi.append(177)
return Dist(b**(S(-2)), Int((-a*f*h + b*e*h + b*f*g + b*f*h*x)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x), x) + Dist((-a*f + b*e)*(-a*h + b*g)/b**S(2), Int(S(1)/((a + b*x)*sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x), x)
rule177 = ReplacementRule(pattern177, replacement177)
pattern178 = Pattern(Integral(S(1)/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement178(f, b, g, d, c, a, x, h, e):
rubi.append(178)
return Dist(-S(2)*sqrt((c + d*x)*(-a*h + b*g)/((a + b*x)*(-c*h + d*g)))*sqrt((e + f*x)*(-a*h + b*g)/((a + b*x)*(-e*h + f*g)))*(a + b*x)/(sqrt(c + d*x)*sqrt(e + f*x)*(-a*h + b*g)), Subst(Int(S(1)/(sqrt(x**S(2)*(-a*d + b*c)/(-c*h + d*g) + S(1))*sqrt(x**S(2)*(-a*f + b*e)/(-e*h + f*g) + S(1))), x), x, sqrt(g + h*x)/sqrt(a + b*x)), x)
rule178 = ReplacementRule(pattern178, replacement178)
pattern179 = Pattern(Integral(sqrt(x_*WC('d', S(1)) + WC('c', S(0)))/((x_*WC('b', S(1)) + WC('a', S(0)))**(S(3)/2)*sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement179(f, b, g, d, c, a, x, h, e):
rubi.append(179)
return Dist(-S(2)*sqrt((c + d*x)*(-a*h + b*g)/((a + b*x)*(-c*h + d*g)))*sqrt((e + f*x)*(-a*h + b*g)/((a + b*x)*(-e*h + f*g)))*(a + b*x)*(-c*h + d*g)/(sqrt(c + d*x)*sqrt(e + f*x)*(-a*h + b*g)**S(2)), Subst(Int(sqrt(x**S(2)*(-a*d + b*c)/(-c*h + d*g) + S(1))/sqrt(x**S(2)*(-a*f + b*e)/(-e*h + f*g) + S(1)), x), x, sqrt(g + h*x)/sqrt(a + b*x)), x)
rule179 = ReplacementRule(pattern179, replacement179)
pattern180 = Pattern(Integral(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))/(sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement180(f, b, g, d, c, a, x, h, e):
rubi.append(180)
return Dist(S(2)*sqrt((c + d*x)*(-a*h + b*g)/((a + b*x)*(-c*h + d*g)))*sqrt((e + f*x)*(-a*h + b*g)/((a + b*x)*(-e*h + f*g)))*(a + b*x)/(sqrt(c + d*x)*sqrt(e + f*x)), Subst(Int(S(1)/((-b*x**S(2) + h)*sqrt(x**S(2)*(-a*d + b*c)/(-c*h + d*g) + S(1))*sqrt(x**S(2)*(-a*f + b*e)/(-e*h + f*g) + S(1))), x), x, sqrt(g + h*x)/sqrt(a + b*x)), x)
rule180 = ReplacementRule(pattern180, replacement180)
pattern181 = Pattern(Integral(S(1)/((x_*WC('b', S(1)) + WC('a', S(0)))**(S(3)/2)*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement181(f, b, g, d, c, a, x, h, e):
rubi.append(181)
return Dist(b/(-a*d + b*c), Int(sqrt(c + d*x)/((a + b*x)**(S(3)/2)*sqrt(e + f*x)*sqrt(g + h*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x), x)
rule181 = ReplacementRule(pattern181, replacement181)
pattern182 = Pattern(Integral(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*sqrt(x_*WC('d', S(1)) + WC('c', S(0)))/(sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement182(f, b, g, d, c, a, x, h, e):
rubi.append(182)
return Dist((a*d*f*h - b*(-c*f*h + d*e*h + d*f*g))/(S(2)*f**S(2)*h), Int(sqrt(e + f*x)/(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(g + h*x)), x), x) + Dist((-c*f + d*e)*(-S(2)*a*f*h + b*e*h + b*f*g)/(S(2)*f**S(2)*h), Int(S(1)/(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x), x) - Dist((-c*f + d*e)*(-e*h + f*g)/(S(2)*f*h), Int(sqrt(a + b*x)/(sqrt(c + d*x)*(e + f*x)**(S(3)/2)*sqrt(g + h*x)), x), x) + Simp(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(g + h*x)/(h*sqrt(e + f*x)), x)
rule182 = ReplacementRule(pattern182, replacement182)
pattern183 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**(S(3)/2)/(sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*sqrt(x_*WC('f', S(1)) + WC('e', S(0)))*sqrt(x_*WC('h', S(1)) + WC('g', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons216)
def replacement183(f, b, g, d, c, a, x, h, e):
rubi.append(183)
return Dist(b/d, Int(sqrt(a + b*x)*sqrt(c + d*x)/(sqrt(e + f*x)*sqrt(g + h*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(a + b*x)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x), x)
rule183 = ReplacementRule(pattern183, replacement183)
pattern184 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons220)
def replacement184(p, m, f, b, g, d, c, a, n, x, h, q, e):
rubi.append(184)
return Int(ExpandIntegrand((a + b*x)**m*(c + d*x)**n*(e + f*x)**p*(g + h*x)**q, x), x)
rule184 = ReplacementRule(pattern184, replacement184)
pattern185 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons221, cons219)
def replacement185(p, m, f, b, g, d, c, a, n, x, h, q, e):
rubi.append(185)
return Dist(h/b, Int((a + b*x)**(m + S(1))*(c + d*x)**n*(e + f*x)**p*(g + h*x)**(q + S(-1)), x), x) + Dist((-a*h + b*g)/b, Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p*(g + h*x)**(q + S(-1)), x), x)
rule185 = ReplacementRule(pattern185, replacement185)
pattern186 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons50, cons222)
def replacement186(p, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(186)
return Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p*(g + h*x)**q, x)
rule186 = ReplacementRule(pattern186, replacement186)
pattern187 = Pattern(Integral((u_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(u_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*(u_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*(u_*WC('h', S(1)) + WC('g', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons50, cons68, cons69)
def replacement187(p, u, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(187)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x)**m*(c + d*x)**n*(e + f*x)**p*(g + h*x)**q, x), x, u), x)
rule187 = ReplacementRule(pattern187, replacement187)
pattern188 = Pattern(Integral(((x_*WC('b', S(1)) + WC('a', S(0)))**m_*(x_*WC('d', S(1)) + WC('c', S(0)))**n_*(x_*WC('f', S(1)) + WC('e', S(0)))**p_*(x_*WC('h', S(1)) + WC('g', S(0)))**q_*WC('i', S(1)))**r_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons21, cons4, cons5, cons50, cons52, cons223)
def replacement188(p, m, f, b, g, r, i, d, c, a, n, x, h, q, e):
rubi.append(188)
return Dist((i*(a + b*x)**m*(c + d*x)**n*(e + f*x)**p*(g + h*x)**q)**r*(a + b*x)**(-m*r)*(c + d*x)**(-n*r)*(e + f*x)**(-p*r)*(g + h*x)**(-q*r), Int((a + b*x)**(m*r)*(c + d*x)**(n*r)*(e + f*x)**(p*r)*(g + h*x)**(q*r), x), x)
rule188 = ReplacementRule(pattern188, replacement188)
return [rule37, rule38, rule39, rule40, rule41, rule42, rule43, rule44, rule45, rule46, rule47, rule48, rule49, rule50, rule51, rule52, rule53, rule54, rule55, rule56, rule57, rule58, rule59, rule60, rule61, rule62, rule63, rule64, rule65, rule66, rule67, rule68, rule69, rule70, rule71, rule72, rule73, rule74, rule75, rule76, rule77, rule78, rule79, rule80, rule81, rule82, rule83, rule84, rule85, rule86, rule87, rule88, rule89, rule90, rule91, rule92, rule93, rule94, rule95, rule96, rule97, rule98, rule99, rule100, rule101, rule102, rule103, rule104, rule105, rule106, rule107, rule108, rule109, rule110, rule111, rule112, rule113, rule114, rule115, rule116, rule117, rule118, rule119, rule120, rule121, rule122, rule123, rule124, rule125, rule126, rule127, rule128, rule129, rule130, rule131, rule132, rule133, rule134, rule135, rule136, rule137, rule138, rule139, rule140, rule141, rule142, rule143, rule144, rule145, rule146, rule147, rule148, rule149, rule150, rule151, rule152, rule153, rule154, rule155, rule156, rule157, rule158, rule159, rule160, rule161, rule162, rule163, rule164, rule165, rule166, rule167, rule168, rule169, rule170, rule171, rule172, rule173, rule174, rule175, rule176, rule177, rule178, rule179, rule180, rule181, rule182, rule183, rule184, rule185, rule186, rule187, rule188, ]
|
b3604f1926512d7617309200212ebf7f6b61da9acbb1f5ead185dbd58c6e300b | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def hyperbolic(rubi):
from sympy.integrals.rubi.constraints import cons31, cons168, cons7, cons27, cons48, cons125, cons94, cons1116, cons176, cons1488, cons21, cons87, cons165, cons3, cons93, cons166, cons85, cons1489, cons245, cons247, cons89, cons1442, cons148, cons1859, cons2, cons1490, cons1439, cons808, cons1491, cons1492, cons1454, cons1440, cons62, cons1267, cons1493, cons810, cons811, cons4, cons1360, cons128, cons38, cons137, cons744, cons63, cons1494, cons196, cons1495, cons5, cons53, cons13, cons596, cons1496, cons17, cons1497, cons376, cons146, cons489, cons1498, cons68, cons69, cons823, cons824, cons1499, cons1501, cons1255, cons1502, cons56, cons150, cons1503, cons1504, cons683, cons367, cons1505, cons356, cons66, cons854, cons23, cons1506, cons54, cons14, cons818, cons1131, cons1132, cons1133, cons1507, cons819, cons528, cons1265, cons1510, cons18, cons1571, cons1572, cons1573, cons1574, cons1575, cons1576, cons1577, cons1578, cons1579, cons1043, cons1580, cons1644, cons1736, cons1645, cons584, cons464, cons1683, cons1408, cons1684, cons1685, cons1860, cons1861, cons1688, cons812, cons813, cons555, cons1862, cons1863, cons25, cons1691, cons1864, cons1099, cons1865, cons1866, cons1395, cons1867, cons1693, cons1868, cons963, cons1869, cons1870, cons208, cons1700, cons1011, cons1551, cons1701, cons1702, cons209, cons224, cons1699, cons1871, cons1703, cons1704, cons1872, cons1706, cons1707, cons1873, cons1874, cons1875, cons1876, cons1877, cons1878, cons1879, cons1880, cons1881, cons1882, cons1883, cons1884, cons1885, cons1886, cons1720, cons1887, cons1888, cons1889, cons1723, cons1890, cons163, cons338, cons162, cons627, cons71, cons1725, cons1726, cons1727, cons88, cons1728, cons1456, cons463, cons1729, cons1478, cons1730, cons1891, cons34, cons35, cons1474, cons1481, cons1733
pattern5643 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons31, cons168)
def replacement5643(m, f, d, c, x, e):
rubi.append(5643)
return -Dist(d*m/f, Int((c + d*x)**(m + S(-1))*cosh(e + f*x), x), x) + Simp((c + d*x)**m*cosh(e + f*x)/f, x)
rule5643 = ReplacementRule(pattern5643, replacement5643)
pattern5644 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons31, cons168)
def replacement5644(m, f, d, c, x, e):
rubi.append(5644)
return -Dist(d*m/f, Int((c + d*x)**(m + S(-1))*sinh(e + f*x), x), x) + Simp((c + d*x)**m*sinh(e + f*x)/f, x)
rule5644 = ReplacementRule(pattern5644, replacement5644)
pattern5645 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sinh(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons31, cons94)
def replacement5645(m, f, d, c, x, e):
rubi.append(5645)
return -Dist(f/(d*(m + S(1))), Int((c + d*x)**(m + S(1))*cosh(e + f*x), x), x) + Simp((c + d*x)**(m + S(1))*sinh(e + f*x)/(d*(m + S(1))), x)
rule5645 = ReplacementRule(pattern5645, replacement5645)
pattern5646 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cosh(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons31, cons94)
def replacement5646(m, f, d, c, x, e):
rubi.append(5646)
return -Dist(f/(d*(m + S(1))), Int((c + d*x)**(m + S(1))*sinh(e + f*x), x), x) + Simp((c + d*x)**(m + S(1))*cosh(e + f*x)/(d*(m + S(1))), x)
rule5646 = ReplacementRule(pattern5646, replacement5646)
pattern5647 = Pattern(Integral(sinh(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons1116)
def replacement5647(f, d, c, x, e):
rubi.append(5647)
return Simp(SinhIntegral(e + f*x)/d, x)
rule5647 = ReplacementRule(pattern5647, replacement5647)
pattern5648 = Pattern(Integral(cosh(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons1116)
def replacement5648(f, d, c, x, e):
rubi.append(5648)
return Simp(CoshIntegral(e + f*x)/d, x)
rule5648 = ReplacementRule(pattern5648, replacement5648)
pattern5649 = Pattern(Integral(sinh(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons176)
def replacement5649(f, d, c, x, e):
rubi.append(5649)
return Dist(sinh((-c*f + d*e)/d), Int(cosh(c*f/d + f*x)/(c + d*x), x), x) + Dist(cosh((-c*f + d*e)/d), Int(sinh(c*f/d + f*x)/(c + d*x), x), x)
rule5649 = ReplacementRule(pattern5649, replacement5649)
pattern5650 = Pattern(Integral(cosh(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons176)
def replacement5650(f, d, c, x, e):
rubi.append(5650)
return Dist(sinh((-c*f + d*e)/d), Int(sinh(c*f/d + f*x)/(c + d*x), x), x) + Dist(cosh((-c*f + d*e)/d), Int(cosh(c*f/d + f*x)/(c + d*x), x), x)
rule5650 = ReplacementRule(pattern5650, replacement5650)
pattern5651 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons21, cons1488)
def replacement5651(m, f, d, c, x, e):
rubi.append(5651)
return -Dist(S(1)/2, Int((c + d*x)**m*exp(-e - f*x), x), x) + Dist(S(1)/2, Int((c + d*x)**m*exp(e + f*x), x), x)
rule5651 = ReplacementRule(pattern5651, replacement5651)
pattern5652 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons21, cons1488)
def replacement5652(m, f, d, c, x, e):
rubi.append(5652)
return Dist(S(1)/2, Int((c + d*x)**m*exp(-e - f*x), x), x) + Dist(S(1)/2, Int((c + d*x)**m*exp(e + f*x), x), x)
rule5652 = ReplacementRule(pattern5652, replacement5652)
pattern5653 = Pattern(Integral((WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons165)
def replacement5653(f, b, d, c, n, x, e):
rubi.append(5653)
return -Dist(b**S(2)*(n + S(-1))/n, Int((b*sinh(e + f*x))**(n + S(-2))*(c + d*x), x), x) - Simp(d*(b*sinh(e + f*x))**n/(f**S(2)*n**S(2)), x) + Simp(b*(b*sinh(e + f*x))**(n + S(-1))*(c + d*x)*cosh(e + f*x)/(f*n), x)
rule5653 = ReplacementRule(pattern5653, replacement5653)
pattern5654 = Pattern(Integral((WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons165)
def replacement5654(f, b, d, c, n, x, e):
rubi.append(5654)
return Dist(b**S(2)*(n + S(-1))/n, Int((b*cosh(e + f*x))**(n + S(-2))*(c + d*x), x), x) - Simp(d*(b*cosh(e + f*x))**n/(f**S(2)*n**S(2)), x) + Simp(b*(b*cosh(e + f*x))**(n + S(-1))*(c + d*x)*sinh(e + f*x)/(f*n), x)
rule5654 = ReplacementRule(pattern5654, replacement5654)
pattern5655 = Pattern(Integral((WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons166)
def replacement5655(m, f, b, d, c, n, x, e):
rubi.append(5655)
return -Dist(b**S(2)*(n + S(-1))/n, Int((b*sinh(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) + Dist(d**S(2)*m*(m + S(-1))/(f**S(2)*n**S(2)), Int((b*sinh(e + f*x))**n*(c + d*x)**(m + S(-2)), x), x) + Simp(b*(b*sinh(e + f*x))**(n + S(-1))*(c + d*x)**m*cosh(e + f*x)/(f*n), x) - Simp(d*m*(b*sinh(e + f*x))**n*(c + d*x)**(m + S(-1))/(f**S(2)*n**S(2)), x)
rule5655 = ReplacementRule(pattern5655, replacement5655)
pattern5656 = Pattern(Integral((WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons166)
def replacement5656(m, f, b, d, c, n, x, e):
rubi.append(5656)
return Dist(b**S(2)*(n + S(-1))/n, Int((b*cosh(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) + Dist(d**S(2)*m*(m + S(-1))/(f**S(2)*n**S(2)), Int((b*cosh(e + f*x))**n*(c + d*x)**(m + S(-2)), x), x) + Simp(b*(b*cosh(e + f*x))**(n + S(-1))*(c + d*x)**m*sinh(e + f*x)/(f*n), x) - Simp(d*m*(b*cosh(e + f*x))**n*(c + d*x)**(m + S(-1))/(f**S(2)*n**S(2)), x)
rule5656 = ReplacementRule(pattern5656, replacement5656)
pattern5657 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sinh(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons7, cons27, cons48, cons125, cons21, cons85, cons165, cons1489)
def replacement5657(m, f, d, c, n, x, e):
rubi.append(5657)
return Int(ExpandTrigReduce((c + d*x)**m, sinh(e + f*x)**n, x), x)
rule5657 = ReplacementRule(pattern5657, replacement5657)
pattern5658 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cosh(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons7, cons27, cons48, cons125, cons21, cons85, cons165, cons1489)
def replacement5658(m, f, d, c, n, x, e):
rubi.append(5658)
return Int(ExpandTrigReduce((c + d*x)**m, cosh(e + f*x)**n, x), x)
rule5658 = ReplacementRule(pattern5658, replacement5658)
pattern5659 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sinh(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons7, cons27, cons48, cons125, cons21, cons85, cons165, cons31, cons245)
def replacement5659(m, f, d, c, n, x, e):
rubi.append(5659)
return -Dist(f*n/(d*(m + S(1))), Int(ExpandTrigReduce((c + d*x)**(m + S(1)), sinh(e + f*x)**(n + S(-1))*cosh(e + f*x), x), x), x) + Simp((c + d*x)**(m + S(1))*sinh(e + f*x)**n/(d*(m + S(1))), x)
rule5659 = ReplacementRule(pattern5659, replacement5659)
pattern5660 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cosh(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons7, cons27, cons48, cons125, cons21, cons85, cons165, cons31, cons245)
def replacement5660(m, f, d, c, n, x, e):
rubi.append(5660)
return -Dist(f*n/(d*(m + S(1))), Int(ExpandTrigReduce((c + d*x)**(m + S(1)), sinh(e + f*x)*cosh(e + f*x)**(n + S(-1)), x), x), x) + Simp((c + d*x)**(m + S(1))*cosh(e + f*x)**n/(d*(m + S(1))), x)
rule5660 = ReplacementRule(pattern5660, replacement5660)
pattern5661 = Pattern(Integral((WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons247)
def replacement5661(m, f, b, d, c, n, x, e):
rubi.append(5661)
return Dist(f**S(2)*n**S(2)/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*sinh(e + f*x))**n*(c + d*x)**(m + S(2)), x), x) + Dist(b**S(2)*f**S(2)*n*(n + S(-1))/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*sinh(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(2)), x), x) + Simp((b*sinh(e + f*x))**n*(c + d*x)**(m + S(1))/(d*(m + S(1))), x) - Simp(b*f*n*(b*sinh(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(2))*cosh(e + f*x)/(d**S(2)*(m + S(1))*(m + S(2))), x)
rule5661 = ReplacementRule(pattern5661, replacement5661)
pattern5662 = Pattern(Integral((WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons247)
def replacement5662(m, f, b, d, c, n, x, e):
rubi.append(5662)
return Dist(f**S(2)*n**S(2)/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*cosh(e + f*x))**n*(c + d*x)**(m + S(2)), x), x) - Dist(b**S(2)*f**S(2)*n*(n + S(-1))/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*cosh(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(2)), x), x) + Simp((b*cosh(e + f*x))**n*(c + d*x)**(m + S(1))/(d*(m + S(1))), x) - Simp(b*f*n*(b*cosh(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(2))*sinh(e + f*x)/(d**S(2)*(m + S(1))*(m + S(2))), x)
rule5662 = ReplacementRule(pattern5662, replacement5662)
pattern5663 = Pattern(Integral((WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons89, cons1442)
def replacement5663(f, b, d, c, n, x, e):
rubi.append(5663)
return -Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*sinh(e + f*x))**(n + S(2))*(c + d*x), x), x) - Simp(d*(b*sinh(e + f*x))**(n + S(2))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x) + Simp((b*sinh(e + f*x))**(n + S(1))*(c + d*x)*cosh(e + f*x)/(b*f*(n + S(1))), x)
rule5663 = ReplacementRule(pattern5663, replacement5663)
pattern5664 = Pattern(Integral((WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons89, cons1442)
def replacement5664(f, b, d, c, n, x, e):
rubi.append(5664)
return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*cosh(e + f*x))**(n + S(2))*(c + d*x), x), x) + Simp(d*(b*cosh(e + f*x))**(n + S(2))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x) - Simp((b*cosh(e + f*x))**(n + S(1))*(c + d*x)*sinh(e + f*x)/(b*f*(n + S(1))), x)
rule5664 = ReplacementRule(pattern5664, replacement5664)
pattern5665 = Pattern(Integral((WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons93, cons89, cons1442, cons166)
def replacement5665(m, f, b, d, c, n, x, e):
rubi.append(5665)
return -Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*sinh(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) + Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), Int((b*sinh(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-2)), x), x) + Simp((b*sinh(e + f*x))**(n + S(1))*(c + d*x)**m*cosh(e + f*x)/(b*f*(n + S(1))), x) - Simp(d*m*(b*sinh(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x)
rule5665 = ReplacementRule(pattern5665, replacement5665)
pattern5666 = Pattern(Integral((WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons93, cons89, cons1442, cons166)
def replacement5666(m, f, b, d, c, n, x, e):
rubi.append(5666)
return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*cosh(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) - Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), Int((b*cosh(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-2)), x), x) - Simp((b*cosh(e + f*x))**(n + S(1))*(c + d*x)**m*sinh(e + f*x)/(b*f*(n + S(1))), x) + Simp(d*m*(b*cosh(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x)
rule5666 = ReplacementRule(pattern5666, replacement5666)
pattern5667 = Pattern(Integral((a_ + WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons148, cons1859)
def replacement5667(m, f, b, d, c, n, a, x, e):
rubi.append(5667)
return Int(ExpandIntegrand((c + d*x)**m, (a + b*sinh(e + f*x))**n, x), x)
rule5667 = ReplacementRule(pattern5667, replacement5667)
pattern5668 = Pattern(Integral((a_ + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons148, cons1490)
def replacement5668(m, f, b, d, c, n, a, x, e):
rubi.append(5668)
return Int(ExpandIntegrand((c + d*x)**m, (a + b*cosh(e + f*x))**n, x), x)
rule5668 = ReplacementRule(pattern5668, replacement5668)
pattern5669 = Pattern(Integral((a_ + WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1439, cons85)
def replacement5669(m, f, b, d, c, n, a, x, e):
rubi.append(5669)
return Dist((S(2)*a)**n, Int((c + d*x)**m*cosh(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule5669 = ReplacementRule(pattern5669, replacement5669)
pattern5670 = Pattern(Integral((a_ + WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1439, cons808, cons1491)
def replacement5670(m, f, b, d, c, a, n, x, e):
rubi.append(5670)
return Dist((S(2)*a)**IntPart(n)*(a + b*sinh(e + f*x))**FracPart(n)*cosh(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*cosh(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule5670 = ReplacementRule(pattern5670, replacement5670)
pattern5671 = Pattern(Integral((a_ + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1492, cons85)
def replacement5671(m, f, b, d, c, n, a, x, e):
rubi.append(5671)
return Dist((S(2)*a)**n, Int((c + d*x)**m*cosh(e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule5671 = ReplacementRule(pattern5671, replacement5671)
pattern5672 = Pattern(Integral((a_ + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1454, cons85)
def replacement5672(m, f, b, d, c, n, a, x, e):
rubi.append(5672)
return Dist((-S(2)*a)**n, Int((c + d*x)**m*sinh(e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule5672 = ReplacementRule(pattern5672, replacement5672)
pattern5673 = Pattern(Integral((a_ + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1492, cons808, cons1491)
def replacement5673(m, f, b, d, c, a, n, x, e):
rubi.append(5673)
return Dist((S(2)*a)**IntPart(n)*(a + b*cosh(e + f*x))**FracPart(n)*cosh(e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*cosh(e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule5673 = ReplacementRule(pattern5673, replacement5673)
pattern5674 = Pattern(Integral((a_ + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1454, cons808, cons1491)
def replacement5674(m, f, b, d, c, a, n, x, e):
rubi.append(5674)
return Dist((-S(2)*a)**IntPart(n)*(a + b*cosh(e + f*x))**FracPart(n)*sinh(e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*sinh(e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule5674 = ReplacementRule(pattern5674, replacement5674)
pattern5675 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440, cons62)
def replacement5675(m, f, b, d, c, a, x, e):
rubi.append(5675)
return Dist(S(-2), Int((c + d*x)**m*exp(e + f*x)/(-S(2)*a*exp(e + f*x) - b*exp(S(2)*e + S(2)*f*x) + b), x), x)
rule5675 = ReplacementRule(pattern5675, replacement5675)
pattern5676 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons62)
def replacement5676(m, f, b, d, c, a, x, e):
rubi.append(5676)
return Dist(S(2), Int((c + d*x)**m*exp(e + f*x)/(S(2)*a*exp(e + f*x) + b*exp(S(2)*e + S(2)*f*x) + b), x), x)
rule5676 = ReplacementRule(pattern5676, replacement5676)
pattern5677 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440, cons62)
def replacement5677(m, f, b, d, c, a, x, e):
rubi.append(5677)
return Dist(a/(a**S(2) + b**S(2)), Int((c + d*x)**m/(a + b*sinh(e + f*x)), x), x) + Dist(b*d*m/(f*(a**S(2) + b**S(2))), Int((c + d*x)**(m + S(-1))*cosh(e + f*x)/(a + b*sinh(e + f*x)), x), x) - Simp(b*(c + d*x)**m*cosh(e + f*x)/(f*(a + b*sinh(e + f*x))*(a**S(2) + b**S(2))), x)
rule5677 = ReplacementRule(pattern5677, replacement5677)
pattern5678 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons62)
def replacement5678(m, f, b, d, c, a, x, e):
rubi.append(5678)
return Dist(a/(a**S(2) - b**S(2)), Int((c + d*x)**m/(a + b*cosh(e + f*x)), x), x) + Dist(b*d*m/(f*(a**S(2) - b**S(2))), Int((c + d*x)**(m + S(-1))*sinh(e + f*x)/(a + b*cosh(e + f*x)), x), x) - Simp(b*(c + d*x)**m*sinh(e + f*x)/(f*(a + b*cosh(e + f*x))*(a**S(2) - b**S(2))), x)
rule5678 = ReplacementRule(pattern5678, replacement5678)
pattern5679 = Pattern(Integral((a_ + WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440, cons1493, cons62)
def replacement5679(m, f, b, d, c, a, n, x, e):
rubi.append(5679)
return Dist(a/(a**S(2) + b**S(2)), Int((a + b*sinh(e + f*x))**(n + S(1))*(c + d*x)**m, x), x) - Dist(b*(n + S(2))/((a**S(2) + b**S(2))*(n + S(1))), Int((a + b*sinh(e + f*x))**(n + S(1))*(c + d*x)**m*sinh(e + f*x), x), x) - Dist(b*d*m/(f*(a**S(2) + b**S(2))*(n + S(1))), Int((a + b*sinh(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1))*cosh(e + f*x), x), x) + Simp(b*(a + b*sinh(e + f*x))**(n + S(1))*(c + d*x)**m*cosh(e + f*x)/(f*(a**S(2) + b**S(2))*(n + S(1))), x)
rule5679 = ReplacementRule(pattern5679, replacement5679)
pattern5680 = Pattern(Integral((a_ + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons1493, cons62)
def replacement5680(m, f, b, d, c, a, n, x, e):
rubi.append(5680)
return Dist(a/(a**S(2) - b**S(2)), Int((a + b*cosh(e + f*x))**(n + S(1))*(c + d*x)**m, x), x) - Dist(b*(n + S(2))/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*cosh(e + f*x))**(n + S(1))*(c + d*x)**m*cosh(e + f*x), x), x) - Dist(b*d*m/(f*(a**S(2) - b**S(2))*(n + S(1))), Int((a + b*cosh(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1))*sinh(e + f*x), x), x) + Simp(b*(a + b*cosh(e + f*x))**(n + S(1))*(c + d*x)**m*sinh(e + f*x)/(f*(a**S(2) - b**S(2))*(n + S(1))), x)
rule5680 = ReplacementRule(pattern5680, replacement5680)
pattern5681 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement5681(v, u, m, b, a, n, x):
rubi.append(5681)
return Int((a + b*sinh(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule5681 = ReplacementRule(pattern5681, replacement5681)
pattern5682 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement5682(v, u, m, b, a, n, x):
rubi.append(5682)
return Int((a + b*cosh(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule5682 = ReplacementRule(pattern5682, replacement5682)
pattern5683 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement5683(m, f, b, d, c, a, n, x, e):
rubi.append(5683)
return Int((a + b*sinh(e + f*x))**n*(c + d*x)**m, x)
rule5683 = ReplacementRule(pattern5683, replacement5683)
pattern5684 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement5684(m, f, b, d, a, n, c, x, e):
rubi.append(5684)
return Int((a + b*cosh(e + f*x))**n*(c + d*x)**m, x)
rule5684 = ReplacementRule(pattern5684, replacement5684)
pattern5685 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons128)
def replacement5685(p, b, d, c, a, n, x):
rubi.append(5685)
return Int(ExpandIntegrand(sinh(c + d*x), (a + b*x**n)**p, x), x)
rule5685 = ReplacementRule(pattern5685, replacement5685)
pattern5686 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons128)
def replacement5686(p, b, d, c, a, n, x):
rubi.append(5686)
return Int(ExpandIntegrand(cosh(c + d*x), (a + b*x**n)**p, x), x)
rule5686 = ReplacementRule(pattern5686, replacement5686)
pattern5687 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons38, cons148, cons137, cons744)
def replacement5687(p, b, d, c, a, n, x):
rubi.append(5687)
return -Dist(d/(b*n*(p + S(1))), Int(x**(-n + S(1))*(a + b*x**n)**(p + S(1))*cosh(c + d*x), x), x) - Dist((-n + S(1))/(b*n*(p + S(1))), Int(x**(-n)*(a + b*x**n)**(p + S(1))*sinh(c + d*x), x), x) + Simp(x**(-n + S(1))*(a + b*x**n)**(p + S(1))*sinh(c + d*x)/(b*n*(p + S(1))), x)
rule5687 = ReplacementRule(pattern5687, replacement5687)
pattern5688 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons38, cons148, cons137, cons744)
def replacement5688(p, b, d, c, a, n, x):
rubi.append(5688)
return -Dist(d/(b*n*(p + S(1))), Int(x**(-n + S(1))*(a + b*x**n)**(p + S(1))*sinh(c + d*x), x), x) - Dist((-n + S(1))/(b*n*(p + S(1))), Int(x**(-n)*(a + b*x**n)**(p + S(1))*cosh(c + d*x), x), x) + Simp(x**(-n + S(1))*(a + b*x**n)**(p + S(1))*cosh(c + d*x)/(b*n*(p + S(1))), x)
rule5688 = ReplacementRule(pattern5688, replacement5688)
pattern5689 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons148, cons1494)
def replacement5689(p, b, d, c, a, n, x):
rubi.append(5689)
return Int(ExpandIntegrand(sinh(c + d*x), (a + b*x**n)**p, x), x)
rule5689 = ReplacementRule(pattern5689, replacement5689)
pattern5690 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons148, cons1494)
def replacement5690(p, b, d, c, a, n, x):
rubi.append(5690)
return Int(ExpandIntegrand(cosh(c + d*x), (a + b*x**n)**p, x), x)
rule5690 = ReplacementRule(pattern5690, replacement5690)
pattern5691 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons196)
def replacement5691(p, b, d, c, a, n, x):
rubi.append(5691)
return Int(x**(n*p)*(a*x**(-n) + b)**p*sinh(c + d*x), x)
rule5691 = ReplacementRule(pattern5691, replacement5691)
pattern5692 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons196)
def replacement5692(p, b, d, c, a, n, x):
rubi.append(5692)
return Int(x**(n*p)*(a*x**(-n) + b)**p*cosh(c + d*x), x)
rule5692 = ReplacementRule(pattern5692, replacement5692)
pattern5693 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement5693(p, b, d, c, a, n, x):
rubi.append(5693)
return Int((a + b*x**n)**p*sinh(c + d*x), x)
rule5693 = ReplacementRule(pattern5693, replacement5693)
pattern5694 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement5694(p, b, d, c, a, n, x):
rubi.append(5694)
return Int((a + b*x**n)**p*cosh(c + d*x), x)
rule5694 = ReplacementRule(pattern5694, replacement5694)
pattern5695 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons128)
def replacement5695(p, m, b, d, c, a, n, x, e):
rubi.append(5695)
return Int(ExpandIntegrand(sinh(c + d*x), (e*x)**m*(a + b*x**n)**p, x), x)
rule5695 = ReplacementRule(pattern5695, replacement5695)
pattern5696 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons128)
def replacement5696(p, m, b, d, c, a, n, x, e):
rubi.append(5696)
return Int(ExpandIntegrand(cosh(c + d*x), (e*x)**m*(a + b*x**n)**p, x), x)
rule5696 = ReplacementRule(pattern5696, replacement5696)
pattern5697 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, cons53, cons13, cons137, cons596)
def replacement5697(p, m, b, d, c, a, n, x, e):
rubi.append(5697)
return -Dist(d*e**m/(b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*cosh(c + d*x), x), x) + Simp(e**m*(a + b*x**n)**(p + S(1))*sinh(c + d*x)/(b*n*(p + S(1))), x)
rule5697 = ReplacementRule(pattern5697, replacement5697)
pattern5698 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, cons53, cons13, cons137, cons596)
def replacement5698(p, m, b, d, c, a, n, x, e):
rubi.append(5698)
return -Dist(d*e**m/(b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*sinh(c + d*x), x), x) + Simp(e**m*(a + b*x**n)**(p + S(1))*cosh(c + d*x)/(b*n*(p + S(1))), x)
rule5698 = ReplacementRule(pattern5698, replacement5698)
pattern5699 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons38, cons148, cons31, cons137, cons1496)
def replacement5699(p, m, b, d, c, a, n, x):
rubi.append(5699)
return -Dist(d/(b*n*(p + S(1))), Int(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*cosh(c + d*x), x), x) - Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*(a + b*x**n)**(p + S(1))*sinh(c + d*x), x), x) + Simp(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*sinh(c + d*x)/(b*n*(p + S(1))), x)
rule5699 = ReplacementRule(pattern5699, replacement5699)
pattern5700 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons38, cons148, cons31, cons137, cons1496)
def replacement5700(p, m, b, d, c, a, n, x):
rubi.append(5700)
return -Dist(d/(b*n*(p + S(1))), Int(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*sinh(c + d*x), x), x) - Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*(a + b*x**n)**(p + S(1))*cosh(c + d*x), x), x) + Simp(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*cosh(c + d*x)/(b*n*(p + S(1))), x)
rule5700 = ReplacementRule(pattern5700, replacement5700)
pattern5701 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons17, cons148, cons1494)
def replacement5701(p, m, b, d, c, a, n, x):
rubi.append(5701)
return Int(ExpandIntegrand(sinh(c + d*x), x**m*(a + b*x**n)**p, x), x)
rule5701 = ReplacementRule(pattern5701, replacement5701)
pattern5702 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons17, cons148, cons1494)
def replacement5702(p, m, b, d, c, a, n, x):
rubi.append(5702)
return Int(ExpandIntegrand(cosh(c + d*x), x**m*(a + b*x**n)**p, x), x)
rule5702 = ReplacementRule(pattern5702, replacement5702)
pattern5703 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons63, cons196)
def replacement5703(p, m, b, d, c, a, n, x):
rubi.append(5703)
return Int(x**(m + n*p)*(a*x**(-n) + b)**p*sinh(c + d*x), x)
rule5703 = ReplacementRule(pattern5703, replacement5703)
pattern5704 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons63, cons196)
def replacement5704(p, m, b, d, c, a, n, x):
rubi.append(5704)
return Int(x**(m + n*p)*(a*x**(-n) + b)**p*cosh(c + d*x), x)
rule5704 = ReplacementRule(pattern5704, replacement5704)
pattern5705 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5705(p, m, b, d, c, a, n, x, e):
rubi.append(5705)
return Int((e*x)**m*(a + b*x**n)**p*sinh(c + d*x), x)
rule5705 = ReplacementRule(pattern5705, replacement5705)
pattern5706 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5706(p, m, b, d, c, a, n, x, e):
rubi.append(5706)
return Int((e*x)**m*(a + b*x**n)**p*cosh(c + d*x), x)
rule5706 = ReplacementRule(pattern5706, replacement5706)
pattern5707 = Pattern(Integral(sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons85, cons165)
def replacement5707(d, c, n, x):
rubi.append(5707)
return -Dist(S(1)/2, Int(exp(-c - d*x**n), x), x) + Dist(S(1)/2, Int(exp(c + d*x**n), x), x)
rule5707 = ReplacementRule(pattern5707, replacement5707)
pattern5708 = Pattern(Integral(cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons85, cons165)
def replacement5708(d, c, n, x):
rubi.append(5708)
return Dist(S(1)/2, Int(exp(-c - d*x**n), x), x) + Dist(S(1)/2, Int(exp(c + d*x**n), x), x)
rule5708 = ReplacementRule(pattern5708, replacement5708)
pattern5709 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons376, cons165, cons146)
def replacement5709(p, b, d, c, a, n, x):
rubi.append(5709)
return Int(ExpandTrigReduce((a + b*sinh(c + d*x**n))**p, x), x)
rule5709 = ReplacementRule(pattern5709, replacement5709)
pattern5710 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons376, cons165, cons146)
def replacement5710(p, b, d, a, c, n, x):
rubi.append(5710)
return Int(ExpandTrigReduce((a + b*cosh(c + d*x**n))**p, x), x)
rule5710 = ReplacementRule(pattern5710, replacement5710)
pattern5711 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons196)
def replacement5711(p, b, d, c, a, n, x):
rubi.append(5711)
return -Subst(Int((a + b*sinh(c + d*x**(-n)))**p/x**S(2), x), x, S(1)/x)
rule5711 = ReplacementRule(pattern5711, replacement5711)
pattern5712 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons196)
def replacement5712(p, b, d, a, c, n, x):
rubi.append(5712)
return -Subst(Int((a + b*cosh(c + d*x**(-n)))**p/x**S(2), x), x, S(1)/x)
rule5712 = ReplacementRule(pattern5712, replacement5712)
def With5713(p, b, d, c, a, n, x):
k = Denominator(n)
rubi.append(5713)
return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*sinh(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x)
pattern5713 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons489)
rule5713 = ReplacementRule(pattern5713, With5713)
def With5714(p, b, d, a, c, n, x):
k = Denominator(n)
rubi.append(5714)
return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*cosh(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x)
pattern5714 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons489)
rule5714 = ReplacementRule(pattern5714, With5714)
pattern5715 = Pattern(Integral(sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons4, cons1498)
def replacement5715(d, c, n, x):
rubi.append(5715)
return -Dist(S(1)/2, Int(exp(-c - d*x**n), x), x) + Dist(S(1)/2, Int(exp(c + d*x**n), x), x)
rule5715 = ReplacementRule(pattern5715, replacement5715)
pattern5716 = Pattern(Integral(cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons4, cons1498)
def replacement5716(d, c, n, x):
rubi.append(5716)
return Dist(S(1)/2, Int(exp(-c - d*x**n), x), x) + Dist(S(1)/2, Int(exp(c + d*x**n), x), x)
rule5716 = ReplacementRule(pattern5716, replacement5716)
pattern5717 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons4, cons128)
def replacement5717(p, b, d, c, a, n, x):
rubi.append(5717)
return Int(ExpandTrigReduce((a + b*sinh(c + d*x**n))**p, x), x)
rule5717 = ReplacementRule(pattern5717, replacement5717)
pattern5718 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons4, cons128)
def replacement5718(p, b, d, a, c, n, x):
rubi.append(5718)
return Int(ExpandTrigReduce((a + b*cosh(c + d*x**n))**p, x), x)
rule5718 = ReplacementRule(pattern5718, replacement5718)
pattern5719 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sinh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons38, cons68, cons69)
def replacement5719(p, u, b, d, c, a, n, x):
rubi.append(5719)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*sinh(c + d*x**n))**p, x), x, u), x)
rule5719 = ReplacementRule(pattern5719, replacement5719)
pattern5720 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cosh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons38, cons68, cons69)
def replacement5720(p, u, b, d, a, c, n, x):
rubi.append(5720)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*cosh(c + d*x**n))**p, x), x, u), x)
rule5720 = ReplacementRule(pattern5720, replacement5720)
pattern5721 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sinh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68)
def replacement5721(p, u, b, d, c, a, n, x):
rubi.append(5721)
return Int((a + b*sinh(c + d*u**n))**p, x)
rule5721 = ReplacementRule(pattern5721, replacement5721)
pattern5722 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cosh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68)
def replacement5722(p, u, b, d, a, c, n, x):
rubi.append(5722)
return Int((a + b*cosh(c + d*u**n))**p, x)
rule5722 = ReplacementRule(pattern5722, replacement5722)
pattern5723 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sinh(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement5723(p, u, b, a, x):
rubi.append(5723)
return Int((a + b*sinh(ExpandToSum(u, x)))**p, x)
rule5723 = ReplacementRule(pattern5723, replacement5723)
pattern5724 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cosh(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement5724(p, u, b, a, x):
rubi.append(5724)
return Int((a + b*cosh(ExpandToSum(u, x)))**p, x)
rule5724 = ReplacementRule(pattern5724, replacement5724)
pattern5725 = Pattern(Integral(sinh(x_**n_*WC('d', S(1)))/x_, x_), cons27, cons4, cons1499)
def replacement5725(d, x, n):
rubi.append(5725)
return Simp(SinhIntegral(d*x**n)/n, x)
rule5725 = ReplacementRule(pattern5725, replacement5725)
pattern5726 = Pattern(Integral(cosh(x_**n_*WC('d', S(1)))/x_, x_), cons27, cons4, cons1499)
def replacement5726(d, x, n):
rubi.append(5726)
return Simp(CoshIntegral(d*x**n)/n, x)
rule5726 = ReplacementRule(pattern5726, replacement5726)
pattern5727 = Pattern(Integral(sinh(c_ + x_**n_*WC('d', S(1)))/x_, x_), cons7, cons27, cons4, cons1498)
def replacement5727(d, x, n, c):
rubi.append(5727)
return Dist(sinh(c), Int(cosh(d*x**n)/x, x), x) + Dist(cosh(c), Int(sinh(d*x**n)/x, x), x)
rule5727 = ReplacementRule(pattern5727, replacement5727)
pattern5728 = Pattern(Integral(cosh(c_ + x_**n_*WC('d', S(1)))/x_, x_), cons7, cons27, cons4, cons1498)
def replacement5728(d, c, n, x):
rubi.append(5728)
return Dist(sinh(c), Int(sinh(d*x**n)/x, x), x) + Dist(cosh(c), Int(cosh(d*x**n)/x, x), x)
rule5728 = ReplacementRule(pattern5728, replacement5728)
def With5729(p, m, b, d, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
mn = (m + S(1))/n
if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))):
return True
return False
pattern5729 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons38, CustomConstraint(With5729))
def replacement5729(p, m, b, d, c, a, n, x):
mn = (m + S(1))/n
rubi.append(5729)
return Dist(S(1)/n, Subst(Int(x**(mn + S(-1))*(a + b*sinh(c + d*x))**p, x), x, x**n), x)
rule5729 = ReplacementRule(pattern5729, replacement5729)
def With5730(p, m, b, d, a, c, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
mn = (m + S(1))/n
if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))):
return True
return False
pattern5730 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons38, CustomConstraint(With5730))
def replacement5730(p, m, b, d, a, c, n, x):
mn = (m + S(1))/n
rubi.append(5730)
return Dist(S(1)/n, Subst(Int(x**(mn + S(-1))*(a + b*cosh(c + d*x))**p, x), x, x**n), x)
rule5730 = ReplacementRule(pattern5730, replacement5730)
def With5731(p, m, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
mn = (m + S(1))/n
if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))):
return True
return False
pattern5731 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, CustomConstraint(With5731))
def replacement5731(p, m, b, d, c, a, n, x, e):
mn = (m + S(1))/n
rubi.append(5731)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sinh(c + d*x**n))**p, x), x)
rule5731 = ReplacementRule(pattern5731, replacement5731)
def With5732(p, m, b, d, a, c, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
mn = (m + S(1))/n
if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))):
return True
return False
pattern5732 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, CustomConstraint(With5732))
def replacement5732(p, m, b, d, a, c, n, x, e):
mn = (m + S(1))/n
rubi.append(5732)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cosh(c + d*x**n))**p, x), x)
rule5732 = ReplacementRule(pattern5732, replacement5732)
pattern5733 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons148, cons31, cons1501)
def replacement5733(m, d, c, n, x, e):
rubi.append(5733)
return -Dist(e**n*(m - n + S(1))/(d*n), Int((e*x)**(m - n)*cosh(c + d*x**n), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*cosh(c + d*x**n)/(d*n), x)
rule5733 = ReplacementRule(pattern5733, replacement5733)
pattern5734 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons148, cons31, cons1501)
def replacement5734(m, d, c, n, x, e):
rubi.append(5734)
return -Dist(e**n*(m - n + S(1))/(d*n), Int((e*x)**(m - n)*sinh(c + d*x**n), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*sinh(c + d*x**n)/(d*n), x)
rule5734 = ReplacementRule(pattern5734, replacement5734)
pattern5735 = Pattern(Integral((x_*WC('e', S(1)))**m_*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons148, cons31, cons94)
def replacement5735(m, d, c, n, x, e):
rubi.append(5735)
return -Dist(d*e**(-n)*n/(m + S(1)), Int((e*x)**(m + n)*cosh(c + d*x**n), x), x) + Simp((e*x)**(m + S(1))*sinh(c + d*x**n)/(e*(m + S(1))), x)
rule5735 = ReplacementRule(pattern5735, replacement5735)
pattern5736 = Pattern(Integral((x_*WC('e', S(1)))**m_*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons148, cons31, cons94)
def replacement5736(m, d, c, n, x, e):
rubi.append(5736)
return -Dist(d*e**(-n)*n/(m + S(1)), Int((e*x)**(m + n)*sinh(c + d*x**n), x), x) + Simp((e*x)**(m + S(1))*cosh(c + d*x**n)/(e*(m + S(1))), x)
rule5736 = ReplacementRule(pattern5736, replacement5736)
pattern5737 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons21, cons148)
def replacement5737(m, d, c, n, x, e):
rubi.append(5737)
return -Dist(S(1)/2, Int((e*x)**m*exp(-c - d*x**n), x), x) + Dist(S(1)/2, Int((e*x)**m*exp(c + d*x**n), x), x)
rule5737 = ReplacementRule(pattern5737, replacement5737)
pattern5738 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons21, cons148)
def replacement5738(m, d, c, n, x, e):
rubi.append(5738)
return Dist(S(1)/2, Int((e*x)**m*exp(-c - d*x**n), x), x) + Dist(S(1)/2, Int((e*x)**m*exp(c + d*x**n), x), x)
rule5738 = ReplacementRule(pattern5738, replacement5738)
pattern5739 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons376, cons1255, cons146, cons1502)
def replacement5739(p, m, b, a, n, x):
rubi.append(5739)
return Dist(b*n*p/(n + S(-1)), Int(sinh(a + b*x**n)**(p + S(-1))*cosh(a + b*x**n), x), x) - Simp(x**(-n + S(1))*sinh(a + b*x**n)**p/(n + S(-1)), x)
rule5739 = ReplacementRule(pattern5739, replacement5739)
pattern5740 = Pattern(Integral(x_**WC('m', S(1))*cosh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons376, cons1255, cons146, cons1502)
def replacement5740(p, m, b, a, n, x):
rubi.append(5740)
return Dist(b*n*p/(n + S(-1)), Int(sinh(a + b*x**n)*cosh(a + b*x**n)**(p + S(-1)), x), x) - Simp(x**(-n + S(1))*cosh(a + b*x**n)**p/(n + S(-1)), x)
rule5740 = ReplacementRule(pattern5740, replacement5740)
pattern5741 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons56, cons13, cons146)
def replacement5741(p, m, b, a, n, x):
rubi.append(5741)
return -Dist((p + S(-1))/p, Int(x**m*sinh(a + b*x**n)**(p + S(-2)), x), x) - Simp(sinh(a + b*x**n)**p/(b**S(2)*n*p**S(2)), x) + Simp(x**n*sinh(a + b*x**n)**(p + S(-1))*cosh(a + b*x**n)/(b*n*p), x)
rule5741 = ReplacementRule(pattern5741, replacement5741)
pattern5742 = Pattern(Integral(x_**WC('m', S(1))*cosh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons56, cons13, cons146)
def replacement5742(p, m, b, a, n, x):
rubi.append(5742)
return Dist((p + S(-1))/p, Int(x**m*cosh(a + b*x**n)**(p + S(-2)), x), x) - Simp(cosh(a + b*x**n)**p/(b**S(2)*n*p**S(2)), x) + Simp(x**n*sinh(a + b*x**n)*cosh(a + b*x**n)**(p + S(-1))/(b*n*p), x)
rule5742 = ReplacementRule(pattern5742, replacement5742)
pattern5743 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons146, cons1503)
def replacement5743(p, m, b, a, n, x):
rubi.append(5743)
return -Dist((p + S(-1))/p, Int(x**m*sinh(a + b*x**n)**(p + S(-2)), x), x) + Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*p**S(2)), Int(x**(m - S(2)*n)*sinh(a + b*x**n)**p, x), x) - Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*sinh(a + b*x**n)**p/(b**S(2)*n**S(2)*p**S(2)), x) + Simp(x**(m - n + S(1))*sinh(a + b*x**n)**(p + S(-1))*cosh(a + b*x**n)/(b*n*p), x)
rule5743 = ReplacementRule(pattern5743, replacement5743)
pattern5744 = Pattern(Integral(x_**WC('m', S(1))*cosh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons146, cons1503)
def replacement5744(p, m, b, a, n, x):
rubi.append(5744)
return Dist((p + S(-1))/p, Int(x**m*cosh(a + b*x**n)**(p + S(-2)), x), x) + Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*p**S(2)), Int(x**(m - S(2)*n)*cosh(a + b*x**n)**p, x), x) - Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*cosh(a + b*x**n)**p/(b**S(2)*n**S(2)*p**S(2)), x) + Simp(x**(m - n + S(1))*sinh(a + b*x**n)*cosh(a + b*x**n)**(p + S(-1))/(b*n*p), x)
rule5744 = ReplacementRule(pattern5744, replacement5744)
pattern5745 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons146, cons1504, cons683)
def replacement5745(p, m, b, a, n, x):
rubi.append(5745)
return Dist(b**S(2)*n**S(2)*p**S(2)/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*sinh(a + b*x**n)**p, x), x) + Dist(b**S(2)*n**S(2)*p*(p + S(-1))/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*sinh(a + b*x**n)**(p + S(-2)), x), x) + Simp(x**(m + S(1))*sinh(a + b*x**n)**p/(m + S(1)), x) - Simp(b*n*p*x**(m + n + S(1))*sinh(a + b*x**n)**(p + S(-1))*cosh(a + b*x**n)/((m + S(1))*(m + n + S(1))), x)
rule5745 = ReplacementRule(pattern5745, replacement5745)
pattern5746 = Pattern(Integral(x_**WC('m', S(1))*cosh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons146, cons1504, cons683)
def replacement5746(p, m, b, a, n, x):
rubi.append(5746)
return Dist(b**S(2)*n**S(2)*p**S(2)/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*cosh(a + b*x**n)**p, x), x) - Dist(b**S(2)*n**S(2)*p*(p + S(-1))/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*cosh(a + b*x**n)**(p + S(-2)), x), x) + Simp(x**(m + S(1))*cosh(a + b*x**n)**p/(m + S(1)), x) - Simp(b*n*p*x**(m + n + S(1))*sinh(a + b*x**n)*cosh(a + b*x**n)**(p + S(-1))/((m + S(1))*(m + n + S(1))), x)
rule5746 = ReplacementRule(pattern5746, replacement5746)
def With5747(p, m, b, d, c, a, n, x, e):
k = Denominator(m)
rubi.append(5747)
return Dist(k/e, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*sinh(c + d*e**(-n)*x**(k*n)))**p, x), x, (e*x)**(S(1)/k)), x)
pattern5747 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons148, cons367)
rule5747 = ReplacementRule(pattern5747, With5747)
def With5748(p, m, b, d, a, c, n, x, e):
k = Denominator(m)
rubi.append(5748)
return Dist(k/e, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*cosh(c + d*e**(-n)*x**(k*n)))**p, x), x, (e*x)**(S(1)/k)), x)
pattern5748 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons148, cons367)
rule5748 = ReplacementRule(pattern5748, With5748)
pattern5749 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons146)
def replacement5749(p, m, b, d, c, a, n, x, e):
rubi.append(5749)
return Int(ExpandTrigReduce((e*x)**m, (a + b*sinh(c + d*x**n))**p, x), x)
rule5749 = ReplacementRule(pattern5749, replacement5749)
pattern5750 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons146)
def replacement5750(p, m, b, d, a, c, n, x, e):
rubi.append(5750)
return Int(ExpandTrigReduce((e*x)**m, (a + b*cosh(c + d*x**n))**p, x), x)
rule5750 = ReplacementRule(pattern5750, replacement5750)
pattern5751 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons56, cons13, cons137, cons1505)
def replacement5751(p, m, b, a, n, x):
rubi.append(5751)
return -Dist((p + S(2))/(p + S(1)), Int(x**m*sinh(a + b*x**n)**(p + S(2)), x), x) - Simp(sinh(a + b*x**n)**(p + S(2))/(b**S(2)*n*(p + S(1))*(p + S(2))), x) + Simp(x**n*sinh(a + b*x**n)**(p + S(1))*cosh(a + b*x**n)/(b*n*(p + S(1))), x)
rule5751 = ReplacementRule(pattern5751, replacement5751)
pattern5752 = Pattern(Integral(x_**WC('m', S(1))*cosh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons56, cons13, cons137, cons1505)
def replacement5752(p, m, b, a, n, x):
rubi.append(5752)
return Dist((p + S(2))/(p + S(1)), Int(x**m*cosh(a + b*x**n)**(p + S(2)), x), x) + Simp(cosh(a + b*x**n)**(p + S(2))/(b**S(2)*n*(p + S(1))*(p + S(2))), x) - Simp(x**n*sinh(a + b*x**n)*cosh(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule5752 = ReplacementRule(pattern5752, replacement5752)
pattern5753 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons137, cons1505, cons1503)
def replacement5753(p, m, b, a, n, x):
rubi.append(5753)
return -Dist((p + S(2))/(p + S(1)), Int(x**m*sinh(a + b*x**n)**(p + S(2)), x), x) + Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**(m - S(2)*n)*sinh(a + b*x**n)**(p + S(2)), x), x) + Simp(x**(m - n + S(1))*sinh(a + b*x**n)**(p + S(1))*cosh(a + b*x**n)/(b*n*(p + S(1))), x) - Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*sinh(a + b*x**n)**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x)
rule5753 = ReplacementRule(pattern5753, replacement5753)
pattern5754 = Pattern(Integral(x_**WC('m', S(1))*cosh(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons137, cons1505, cons1503)
def replacement5754(p, m, b, a, n, x):
rubi.append(5754)
return Dist((p + S(2))/(p + S(1)), Int(x**m*cosh(a + b*x**n)**(p + S(2)), x), x) - Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**(m - S(2)*n)*cosh(a + b*x**n)**(p + S(2)), x), x) - Simp(x**(m - n + S(1))*sinh(a + b*x**n)*cosh(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) + Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*cosh(a + b*x**n)**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x)
rule5754 = ReplacementRule(pattern5754, replacement5754)
pattern5755 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons196, cons17)
def replacement5755(p, m, b, d, c, a, n, x):
rubi.append(5755)
return -Subst(Int(x**(-m + S(-2))*(a + b*sinh(c + d*x**(-n)))**p, x), x, S(1)/x)
rule5755 = ReplacementRule(pattern5755, replacement5755)
pattern5756 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons196, cons17)
def replacement5756(p, m, b, d, a, c, n, x):
rubi.append(5756)
return -Subst(Int(x**(-m + S(-2))*(a + b*cosh(c + d*x**(-n)))**p, x), x, S(1)/x)
rule5756 = ReplacementRule(pattern5756, replacement5756)
def With5757(p, m, b, d, c, a, n, x, e):
k = Denominator(m)
rubi.append(5757)
return -Dist(k/e, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*sinh(c + d*e**(-n)*x**(-k*n)))**p, x), x, (e*x)**(-S(1)/k)), x)
pattern5757 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons196, cons367)
rule5757 = ReplacementRule(pattern5757, With5757)
def With5758(p, m, b, d, a, c, n, x, e):
k = Denominator(m)
rubi.append(5758)
return -Dist(k/e, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*cosh(c + d*e**(-n)*x**(-k*n)))**p, x), x, (e*x)**(-S(1)/k)), x)
pattern5758 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons196, cons367)
rule5758 = ReplacementRule(pattern5758, With5758)
pattern5759 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons196, cons356)
def replacement5759(p, m, b, d, c, a, n, x, e):
rubi.append(5759)
return -Dist((e*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*sinh(c + d*x**(-n)))**p, x), x, S(1)/x), x)
rule5759 = ReplacementRule(pattern5759, replacement5759)
pattern5760 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons196, cons356)
def replacement5760(p, m, b, d, a, c, n, x, e):
rubi.append(5760)
return -Dist((e*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*cosh(c + d*x**(-n)))**p, x), x, S(1)/x), x)
rule5760 = ReplacementRule(pattern5760, replacement5760)
def With5761(p, m, b, d, c, a, n, x):
k = Denominator(n)
rubi.append(5761)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*sinh(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x)
pattern5761 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons38, cons489)
rule5761 = ReplacementRule(pattern5761, With5761)
def With5762(p, m, b, d, a, c, n, x):
k = Denominator(n)
rubi.append(5762)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*cosh(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x)
pattern5762 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons38, cons489)
rule5762 = ReplacementRule(pattern5762, With5762)
pattern5763 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons489)
def replacement5763(p, m, b, d, c, a, n, x, e):
rubi.append(5763)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sinh(c + d*x**n))**p, x), x)
rule5763 = ReplacementRule(pattern5763, replacement5763)
pattern5764 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons489)
def replacement5764(p, m, b, d, a, c, n, x, e):
rubi.append(5764)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cosh(c + d*x**n))**p, x), x)
rule5764 = ReplacementRule(pattern5764, replacement5764)
pattern5765 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons38, cons66, cons854, cons23)
def replacement5765(p, m, b, d, c, a, n, x):
rubi.append(5765)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*sinh(c + d*x**(n/(m + S(1)))))**p, x), x, x**(m + S(1))), x)
rule5765 = ReplacementRule(pattern5765, replacement5765)
pattern5766 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons38, cons66, cons854, cons23)
def replacement5766(p, m, b, d, a, c, n, x):
rubi.append(5766)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*cosh(c + d*x**(n/(m + S(1)))))**p, x), x, x**(m + S(1))), x)
rule5766 = ReplacementRule(pattern5766, replacement5766)
pattern5767 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, cons66, cons854, cons23)
def replacement5767(p, m, b, d, c, a, n, x, e):
rubi.append(5767)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sinh(c + d*x**n))**p, x), x)
rule5767 = ReplacementRule(pattern5767, replacement5767)
pattern5768 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, cons66, cons854, cons23)
def replacement5768(p, m, b, d, a, c, n, x, e):
rubi.append(5768)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cosh(c + d*x**n))**p, x), x)
rule5768 = ReplacementRule(pattern5768, replacement5768)
pattern5769 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons21, cons4, cons1506)
def replacement5769(m, d, c, n, x, e):
rubi.append(5769)
return -Dist(S(1)/2, Int((e*x)**m*exp(-c - d*x**n), x), x) + Dist(S(1)/2, Int((e*x)**m*exp(c + d*x**n), x), x)
rule5769 = ReplacementRule(pattern5769, replacement5769)
pattern5770 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons21, cons4, cons1506)
def replacement5770(m, d, c, n, x, e):
rubi.append(5770)
return Dist(S(1)/2, Int((e*x)**m*exp(-c - d*x**n), x), x) + Dist(S(1)/2, Int((e*x)**m*exp(c + d*x**n), x), x)
rule5770 = ReplacementRule(pattern5770, replacement5770)
pattern5771 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons128)
def replacement5771(p, m, b, d, c, a, n, x, e):
rubi.append(5771)
return Int(ExpandTrigReduce((e*x)**m, (a + b*sinh(c + d*x**n))**p, x), x)
rule5771 = ReplacementRule(pattern5771, replacement5771)
pattern5772 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons128)
def replacement5772(p, m, b, d, a, c, n, x, e):
rubi.append(5772)
return Int(ExpandTrigReduce((e*x)**m, (a + b*cosh(c + d*x**n))**p, x), x)
rule5772 = ReplacementRule(pattern5772, replacement5772)
pattern5773 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69, cons17)
def replacement5773(p, u, m, b, d, c, a, n, x):
rubi.append(5773)
return Dist(Coefficient(u, x, S(1))**(-m + S(-1)), Subst(Int((a + b*sinh(c + d*x**n))**p*(x - Coefficient(u, x, S(0)))**m, x), x, u), x)
rule5773 = ReplacementRule(pattern5773, replacement5773)
pattern5774 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69, cons17)
def replacement5774(p, u, m, b, d, a, c, n, x):
rubi.append(5774)
return Dist(Coefficient(u, x, S(1))**(-m + S(-1)), Subst(Int((a + b*cosh(c + d*x**n))**p*(x - Coefficient(u, x, S(0)))**m, x), x, u), x)
rule5774 = ReplacementRule(pattern5774, replacement5774)
pattern5775 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons68)
def replacement5775(p, u, m, b, d, c, a, n, x, e):
rubi.append(5775)
return Int((e*x)**m*(a + b*sinh(c + d*u**n))**p, x)
rule5775 = ReplacementRule(pattern5775, replacement5775)
pattern5776 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons68)
def replacement5776(p, u, m, b, d, a, c, n, x, e):
rubi.append(5776)
return Int((e*x)**m*(a + b*cosh(c + d*u**n))**p, x)
rule5776 = ReplacementRule(pattern5776, replacement5776)
pattern5777 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sinh(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement5777(p, u, m, b, a, x, e):
rubi.append(5777)
return Int((e*x)**m*(a + b*sinh(ExpandToSum(u, x)))**p, x)
rule5777 = ReplacementRule(pattern5777, replacement5777)
pattern5778 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cosh(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement5778(p, u, m, b, a, x, e):
rubi.append(5778)
return Int((e*x)**m*(a + b*cosh(ExpandToSum(u, x)))**p, x)
rule5778 = ReplacementRule(pattern5778, replacement5778)
pattern5779 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cosh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons4, cons5, cons53, cons54)
def replacement5779(p, m, b, a, n, x):
rubi.append(5779)
return Simp(sinh(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule5779 = ReplacementRule(pattern5779, replacement5779)
pattern5780 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*cosh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons21, cons4, cons5, cons53, cons54)
def replacement5780(p, m, b, a, n, x):
rubi.append(5780)
return Simp(cosh(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule5780 = ReplacementRule(pattern5780, replacement5780)
pattern5781 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cosh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons5, cons93, cons1501, cons54)
def replacement5781(p, m, b, a, n, x):
rubi.append(5781)
return -Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*sinh(a + b*x**n)**(p + S(1)), x), x) + Simp(x**(m - n + S(1))*sinh(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule5781 = ReplacementRule(pattern5781, replacement5781)
pattern5782 = Pattern(Integral(x_**WC('m', S(1))*sinh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*cosh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons5, cons93, cons1501, cons54)
def replacement5782(p, m, b, a, n, x):
rubi.append(5782)
return -Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*cosh(a + b*x**n)**(p + S(1)), x), x) + Simp(x**(m - n + S(1))*cosh(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule5782 = ReplacementRule(pattern5782, replacement5782)
pattern5783 = Pattern(Integral(sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons14)
def replacement5783(x, c, a, b):
rubi.append(5783)
return -Dist(S(1)/2, Int(exp(-a - b*x - c*x**S(2)), x), x) + Dist(S(1)/2, Int(exp(a + b*x + c*x**S(2)), x), x)
rule5783 = ReplacementRule(pattern5783, replacement5783)
pattern5784 = Pattern(Integral(cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons14)
def replacement5784(x, c, a, b):
rubi.append(5784)
return Dist(S(1)/2, Int(exp(-a - b*x - c*x**S(2)), x), x) + Dist(S(1)/2, Int(exp(a + b*x + c*x**S(2)), x), x)
rule5784 = ReplacementRule(pattern5784, replacement5784)
pattern5785 = Pattern(Integral(sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons85, cons165)
def replacement5785(b, c, a, n, x):
rubi.append(5785)
return Int(ExpandTrigReduce(sinh(a + b*x + c*x**S(2))**n, x), x)
rule5785 = ReplacementRule(pattern5785, replacement5785)
pattern5786 = Pattern(Integral(cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons85, cons165)
def replacement5786(b, c, a, n, x):
rubi.append(5786)
return Int(ExpandTrigReduce(cosh(a + b*x + c*x**S(2))**n, x), x)
rule5786 = ReplacementRule(pattern5786, replacement5786)
pattern5787 = Pattern(Integral(sinh(v_)**WC('n', S(1)), x_), cons148, cons818, cons1131)
def replacement5787(v, x, n):
rubi.append(5787)
return Int(sinh(ExpandToSum(v, x))**n, x)
rule5787 = ReplacementRule(pattern5787, replacement5787)
pattern5788 = Pattern(Integral(cosh(v_)**WC('n', S(1)), x_), cons148, cons818, cons1131)
def replacement5788(v, x, n):
rubi.append(5788)
return Int(cosh(ExpandToSum(v, x))**n, x)
rule5788 = ReplacementRule(pattern5788, replacement5788)
pattern5789 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1132)
def replacement5789(b, d, c, a, x, e):
rubi.append(5789)
return Simp(e*cosh(a + b*x + c*x**S(2))/(S(2)*c), x)
rule5789 = ReplacementRule(pattern5789, replacement5789)
pattern5790 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1132)
def replacement5790(b, d, c, a, x, e):
rubi.append(5790)
return Simp(e*sinh(a + b*x + c*x**S(2))/(S(2)*c), x)
rule5790 = ReplacementRule(pattern5790, replacement5790)
pattern5791 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1133)
def replacement5791(b, d, c, a, x, e):
rubi.append(5791)
return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int(sinh(a + b*x + c*x**S(2)), x), x) + Simp(e*cosh(a + b*x + c*x**S(2))/(S(2)*c), x)
rule5791 = ReplacementRule(pattern5791, replacement5791)
pattern5792 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1133)
def replacement5792(b, d, c, a, x, e):
rubi.append(5792)
return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int(cosh(a + b*x + c*x**S(2)), x), x) + Simp(e*sinh(a + b*x + c*x**S(2))/(S(2)*c), x)
rule5792 = ReplacementRule(pattern5792, replacement5792)
pattern5793 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166, cons1132)
def replacement5793(m, b, d, c, a, x, e):
rubi.append(5793)
return -Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*cosh(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*cosh(a + b*x + c*x**S(2))/(S(2)*c), x)
rule5793 = ReplacementRule(pattern5793, replacement5793)
pattern5794 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166, cons1132)
def replacement5794(m, b, d, c, a, x, e):
rubi.append(5794)
return -Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*sinh(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*sinh(a + b*x + c*x**S(2))/(S(2)*c), x)
rule5794 = ReplacementRule(pattern5794, replacement5794)
pattern5795 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166, cons1133)
def replacement5795(m, b, d, c, a, x, e):
rubi.append(5795)
return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int((d + e*x)**(m + S(-1))*sinh(a + b*x + c*x**S(2)), x), x) - Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*cosh(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*cosh(a + b*x + c*x**S(2))/(S(2)*c), x)
rule5795 = ReplacementRule(pattern5795, replacement5795)
pattern5796 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166, cons1133)
def replacement5796(m, b, d, c, a, x, e):
rubi.append(5796)
return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int((d + e*x)**(m + S(-1))*cosh(a + b*x + c*x**S(2)), x), x) - Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*sinh(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*sinh(a + b*x + c*x**S(2))/(S(2)*c), x)
rule5796 = ReplacementRule(pattern5796, replacement5796)
pattern5797 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94, cons1132)
def replacement5797(m, b, d, c, a, x, e):
rubi.append(5797)
return -Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*cosh(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*sinh(a + b*x + c*x**S(2))/(e*(m + S(1))), x)
rule5797 = ReplacementRule(pattern5797, replacement5797)
pattern5798 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94, cons1132)
def replacement5798(m, b, d, c, a, x, e):
rubi.append(5798)
return -Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*sinh(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*cosh(a + b*x + c*x**S(2))/(e*(m + S(1))), x)
rule5798 = ReplacementRule(pattern5798, replacement5798)
pattern5799 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94, cons1133)
def replacement5799(m, b, d, c, a, x, e):
rubi.append(5799)
return -Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*cosh(a + b*x + c*x**S(2)), x), x) - Dist((b*e - S(2)*c*d)/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(1))*cosh(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*sinh(a + b*x + c*x**S(2))/(e*(m + S(1))), x)
rule5799 = ReplacementRule(pattern5799, replacement5799)
pattern5800 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94, cons1133)
def replacement5800(m, b, d, c, a, x, e):
rubi.append(5800)
return -Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*sinh(a + b*x + c*x**S(2)), x), x) - Dist((b*e - S(2)*c*d)/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(1))*sinh(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*cosh(a + b*x + c*x**S(2))/(e*(m + S(1))), x)
rule5800 = ReplacementRule(pattern5800, replacement5800)
pattern5801 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1507)
def replacement5801(m, b, d, a, c, x, e):
rubi.append(5801)
return Int((d + e*x)**m*sinh(a + b*x + c*x**S(2)), x)
rule5801 = ReplacementRule(pattern5801, replacement5801)
pattern5802 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1507)
def replacement5802(m, b, d, c, a, x, e):
rubi.append(5802)
return Int((d + e*x)**m*cosh(a + b*x + c*x**S(2)), x)
rule5802 = ReplacementRule(pattern5802, replacement5802)
pattern5803 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*sinh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons85, cons165)
def replacement5803(m, b, d, a, c, n, x, e):
rubi.append(5803)
return Int(ExpandTrigReduce((d + e*x)**m, sinh(a + b*x + c*x**S(2))**n, x), x)
rule5803 = ReplacementRule(pattern5803, replacement5803)
pattern5804 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cosh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons85, cons165)
def replacement5804(m, b, d, c, a, n, x, e):
rubi.append(5804)
return Int(ExpandTrigReduce((d + e*x)**m, cosh(a + b*x + c*x**S(2))**n, x), x)
rule5804 = ReplacementRule(pattern5804, replacement5804)
pattern5805 = Pattern(Integral(u_**WC('m', S(1))*sinh(v_)**WC('n', S(1)), x_), cons21, cons148, cons68, cons818, cons819)
def replacement5805(v, u, m, n, x):
rubi.append(5805)
return Int(ExpandToSum(u, x)**m*sinh(ExpandToSum(v, x))**n, x)
rule5805 = ReplacementRule(pattern5805, replacement5805)
pattern5806 = Pattern(Integral(u_**WC('m', S(1))*cosh(v_)**WC('n', S(1)), x_), cons21, cons148, cons68, cons818, cons819)
def replacement5806(v, u, m, n, x):
rubi.append(5806)
return Int(ExpandToSum(u, x)**m*cosh(ExpandToSum(v, x))**n, x)
rule5806 = ReplacementRule(pattern5806, replacement5806)
pattern5807 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement5807(m, f, b, a, x, e):
rubi.append(5807)
return Dist(S(2), Int((a + b*x)**m*exp(S(2)*e + S(2)*f*x)/(exp(S(2)*e + S(2)*f*x) + S(1)), x), x) - Simp((a + b*x)**(m + S(1))/(b*(m + S(1))), x)
rule5807 = ReplacementRule(pattern5807, replacement5807)
pattern5808 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons62)
def replacement5808(m, f, b, a, x, e):
rubi.append(5808)
return -Dist(S(2), Int((a + b*x)**m*exp(S(2)*e + S(2)*f*x)/(-exp(S(2)*e + S(2)*f*x) + S(1)), x), x) - Simp((a + b*x)**(m + S(1))/(b*(m + S(1))), x)
rule5808 = ReplacementRule(pattern5808, replacement5808)
pattern5809 = Pattern(Integral((WC('c', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons48, cons125, cons93, cons165, cons168)
def replacement5809(m, f, b, a, c, n, x, e):
rubi.append(5809)
return Dist(c**S(2), Int((c*tanh(e + f*x))**(n + S(-2))*(a + b*x)**m, x), x) + Dist(b*c*m/(f*(n + S(-1))), Int((c*tanh(e + f*x))**(n + S(-1))*(a + b*x)**(m + S(-1)), x), x) - Simp(c*(c*tanh(e + f*x))**(n + S(-1))*(a + b*x)**m/(f*(n + S(-1))), x)
rule5809 = ReplacementRule(pattern5809, replacement5809)
pattern5810 = Pattern(Integral((WC('c', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons48, cons125, cons93, cons165, cons168)
def replacement5810(m, f, b, c, a, n, x, e):
rubi.append(5810)
return Dist(c**S(2), Int((c/tanh(e + f*x))**(n + S(-2))*(a + b*x)**m, x), x) + Dist(b*c*m/(f*(n + S(-1))), Int((c/tanh(e + f*x))**(n + S(-1))*(a + b*x)**(m + S(-1)), x), x) - Simp(c*(c/tanh(e + f*x))**(n + S(-1))*(a + b*x)**m/(f*(n + S(-1))), x)
rule5810 = ReplacementRule(pattern5810, replacement5810)
pattern5811 = Pattern(Integral((WC('c', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons48, cons125, cons93, cons89, cons168)
def replacement5811(m, f, b, a, c, n, x, e):
rubi.append(5811)
return Dist(c**(S(-2)), Int((c*tanh(e + f*x))**(n + S(2))*(a + b*x)**m, x), x) - Dist(b*m/(c*f*(n + S(1))), Int((c*tanh(e + f*x))**(n + S(1))*(a + b*x)**(m + S(-1)), x), x) + Simp((c*tanh(e + f*x))**(n + S(1))*(a + b*x)**m/(c*f*(n + S(1))), x)
rule5811 = ReplacementRule(pattern5811, replacement5811)
pattern5812 = Pattern(Integral((WC('c', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons48, cons125, cons93, cons89, cons168)
def replacement5812(m, f, b, c, a, n, x, e):
rubi.append(5812)
return Dist(c**(S(-2)), Int((c/tanh(e + f*x))**(n + S(2))*(a + b*x)**m, x), x) - Dist(b*m/(c*f*(n + S(1))), Int((c/tanh(e + f*x))**(n + S(1))*(a + b*x)**(m + S(-1)), x), x) + Simp((c/tanh(e + f*x))**(n + S(1))*(a + b*x)**m/(c*f*(n + S(1))), x)
rule5812 = ReplacementRule(pattern5812, replacement5812)
pattern5813 = Pattern(Integral((a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons528)
def replacement5813(m, f, b, d, c, n, a, x, e):
rubi.append(5813)
return Int(ExpandIntegrand((c + d*x)**m, (a + b*tanh(e + f*x))**n, x), x)
rule5813 = ReplacementRule(pattern5813, replacement5813)
pattern5814 = Pattern(Integral((a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons528)
def replacement5814(m, f, b, d, c, n, a, x, e):
rubi.append(5814)
return Int(ExpandIntegrand((c + d*x)**m, (a + b/tanh(e + f*x))**n, x), x)
rule5814 = ReplacementRule(pattern5814, replacement5814)
pattern5815 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons31, cons168)
def replacement5815(m, f, b, d, c, a, x, e):
rubi.append(5815)
return Dist(a*d*m/(S(2)*b*f), Int((c + d*x)**(m + S(-1))/(a + b*tanh(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x) - Simp(a*(c + d*x)**m/(S(2)*b*f*(a + b*tanh(e + f*x))), x)
rule5815 = ReplacementRule(pattern5815, replacement5815)
pattern5816 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons31, cons168)
def replacement5816(m, f, b, d, c, a, x, e):
rubi.append(5816)
return Dist(a*d*m/(S(2)*b*f), Int((c + d*x)**(m + S(-1))/(a + b/tanh(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x) - Simp(a*(c + d*x)**m/(S(2)*b*f*(a + b/tanh(e + f*x))), x)
rule5816 = ReplacementRule(pattern5816, replacement5816)
pattern5817 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265)
def replacement5817(f, b, d, c, a, x, e):
rubi.append(5817)
return Dist(f/(a*d), Int(sinh(S(2)*e + S(2)*f*x)/(c + d*x), x), x) - Dist(f/(b*d), Int(cosh(S(2)*e + S(2)*f*x)/(c + d*x), x), x) - Simp(S(1)/(d*(a + b*tanh(e + f*x))*(c + d*x)), x)
rule5817 = ReplacementRule(pattern5817, replacement5817)
pattern5818 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265)
def replacement5818(f, b, d, c, a, x, e):
rubi.append(5818)
return -Dist(f/(a*d), Int(sinh(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(f/(b*d), Int(cosh(S(2)*e + S(2)*f*x)/(c + d*x), x), x) - Simp(S(1)/(d*(a + b/tanh(e + f*x))*(c + d*x)), x)
rule5818 = ReplacementRule(pattern5818, replacement5818)
pattern5819 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons31, cons94, cons1510)
def replacement5819(m, f, b, d, c, a, x, e):
rubi.append(5819)
return Dist(S(2)*b*f/(a*d*(m + S(1))), Int((c + d*x)**(m + S(1))/(a + b*tanh(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + b*tanh(e + f*x))*(m + S(1))), x) - Simp(f*(c + d*x)**(m + S(2))/(b*d**S(2)*(m + S(1))*(m + S(2))), x)
rule5819 = ReplacementRule(pattern5819, replacement5819)
pattern5820 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons31, cons94, cons1510)
def replacement5820(m, f, b, d, c, a, x, e):
rubi.append(5820)
return Dist(S(2)*b*f/(a*d*(m + S(1))), Int((c + d*x)**(m + S(1))/(a + b/tanh(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + b/tanh(e + f*x))*(m + S(1))), x) - Simp(f*(c + d*x)**(m + S(2))/(b*d**S(2)*(m + S(1))*(m + S(2))), x)
rule5820 = ReplacementRule(pattern5820, replacement5820)
pattern5821 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265)
def replacement5821(f, b, d, c, a, x, e):
rubi.append(5821)
return Dist(S(1)/(S(2)*a), Int(cosh(S(2)*e + S(2)*f*x)/(c + d*x), x), x) - Dist(S(1)/(S(2)*b), Int(sinh(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Simp(log(c + d*x)/(S(2)*a*d), x)
rule5821 = ReplacementRule(pattern5821, replacement5821)
pattern5822 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265)
def replacement5822(f, b, d, c, a, x, e):
rubi.append(5822)
return -Dist(S(1)/(S(2)*a), Int(cosh(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(S(1)/(S(2)*b), Int(sinh(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Simp(log(c + d*x)/(S(2)*a*d), x)
rule5822 = ReplacementRule(pattern5822, replacement5822)
pattern5823 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1265, cons18)
def replacement5823(m, f, b, d, c, a, x, e):
rubi.append(5823)
return Dist(S(1)/(S(2)*a), Int((c + d*x)**m*exp(-S(2)*a*(e + f*x)/b), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x)
rule5823 = ReplacementRule(pattern5823, replacement5823)
pattern5824 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1265, cons18)
def replacement5824(m, f, b, d, c, a, x, e):
rubi.append(5824)
return -Dist(S(1)/(S(2)*a), Int((c + d*x)**m*exp(-S(2)*a*(e + f*x)/b), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x)
rule5824 = ReplacementRule(pattern5824, replacement5824)
pattern5825 = Pattern(Integral((a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons1571)
def replacement5825(m, f, b, d, c, a, n, x, e):
rubi.append(5825)
return Int(ExpandIntegrand((c + d*x)**m, (-sinh(S(2)*e + S(2)*f*x)/(S(2)*b) + cosh(S(2)*e + S(2)*f*x)/(S(2)*a) + S(1)/(S(2)*a))**(-n), x), x)
rule5825 = ReplacementRule(pattern5825, replacement5825)
pattern5826 = Pattern(Integral((a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons1571)
def replacement5826(m, f, b, d, c, a, n, x, e):
rubi.append(5826)
return Int(ExpandIntegrand((c + d*x)**m, (sinh(S(2)*e + S(2)*f*x)/(S(2)*b) - cosh(S(2)*e + S(2)*f*x)/(S(2)*a) + S(1)/(S(2)*a))**(-n), x), x)
rule5826 = ReplacementRule(pattern5826, replacement5826)
pattern5827 = Pattern(Integral((a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1265, cons196)
def replacement5827(m, f, b, d, c, a, n, x, e):
rubi.append(5827)
return Int(ExpandIntegrand((c + d*x)**m, (S(1)/(S(2)*a) + exp(-S(2)*a*(e + f*x)/b)/(S(2)*a))**(-n), x), x)
rule5827 = ReplacementRule(pattern5827, replacement5827)
pattern5828 = Pattern(Integral((a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1265, cons196)
def replacement5828(m, f, b, d, c, a, n, x, e):
rubi.append(5828)
return Int(ExpandIntegrand((c + d*x)**m, (S(1)/(S(2)*a) - exp(-S(2)*a*(e + f*x)/b)/(S(2)*a))**(-n), x), x)
rule5828 = ReplacementRule(pattern5828, replacement5828)
def With5829(m, f, b, d, c, a, n, x, e):
u = IntHide((a + b*tanh(e + f*x))**n, x)
rubi.append(5829)
return -Dist(d*m, Int(Dist((c + d*x)**(m + S(-1)), u, x), x), x) + Dist((c + d*x)**m, u, x)
pattern5829 = Pattern(Integral((a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons1572, cons31, cons168)
rule5829 = ReplacementRule(pattern5829, With5829)
def With5830(m, f, b, d, c, a, n, x, e):
u = IntHide((a + b/tanh(e + f*x))**n, x)
rubi.append(5830)
return -Dist(d*m, Int(Dist((c + d*x)**(m + S(-1)), u, x), x), x) + Dist((c + d*x)**m, u, x)
pattern5830 = Pattern(Integral((a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons1572, cons31, cons168)
rule5830 = ReplacementRule(pattern5830, With5830)
pattern5831 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons62)
def replacement5831(m, f, b, d, c, a, x, e):
rubi.append(5831)
return -Dist(S(2)*b, Int((c + d*x)**m/(a**S(2) - b**S(2) + (a - b)**S(2)*exp(-S(2)*e - S(2)*f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a - b)*(m + S(1))), x)
rule5831 = ReplacementRule(pattern5831, replacement5831)
pattern5832 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons62)
def replacement5832(m, f, b, d, c, a, x, e):
rubi.append(5832)
return Dist(S(2)*b, Int((c + d*x)**m/(a**S(2) - b**S(2) - (a + b)**S(2)*exp(S(2)*e + S(2)*f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + b)*(m + S(1))), x)
rule5832 = ReplacementRule(pattern5832, replacement5832)
pattern5833 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267)
def replacement5833(f, b, d, c, a, x, e):
rubi.append(5833)
return -Dist(S(1)/(f*(a**S(2) - b**S(2))), Int((-S(2)*a*c*f - S(2)*a*d*f*x + b*d)/(a + b*tanh(e + f*x)), x), x) - Simp((c + d*x)**S(2)/(S(2)*d*(a**S(2) - b**S(2))), x) + Simp(b*(c + d*x)/(f*(a + b*tanh(e + f*x))*(a**S(2) - b**S(2))), x)
rule5833 = ReplacementRule(pattern5833, replacement5833)
pattern5834 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267)
def replacement5834(f, b, d, c, a, x, e):
rubi.append(5834)
return -Dist(S(1)/(f*(a**S(2) - b**S(2))), Int((-S(2)*a*c*f - S(2)*a*d*f*x + b*d)/(a + b/tanh(e + f*x)), x), x) - Simp((c + d*x)**S(2)/(S(2)*d*(a**S(2) - b**S(2))), x) + Simp(b*(c + d*x)/(f*(a + b/tanh(e + f*x))*(a**S(2) - b**S(2))), x)
rule5834 = ReplacementRule(pattern5834, replacement5834)
pattern5835 = Pattern(Integral((a_ + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons196, cons62)
def replacement5835(m, f, b, d, c, a, n, x, e):
rubi.append(5835)
return Int(ExpandIntegrand((c + d*x)**m, (-S(2)*b/(a**S(2) - b**S(2) + (a - b)**S(2)*exp(-S(2)*e - S(2)*f*x)) + S(1)/(a - b))**(-n), x), x)
rule5835 = ReplacementRule(pattern5835, replacement5835)
pattern5836 = Pattern(Integral((a_ + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons196, cons62)
def replacement5836(m, f, b, d, c, a, n, x, e):
rubi.append(5836)
return Int(ExpandIntegrand((c + d*x)**m, (S(2)*b/(a**S(2) - b**S(2) - (a + b)**S(2)*exp(S(2)*e + S(2)*f*x)) + S(1)/(a + b))**(-n), x), x)
rule5836 = ReplacementRule(pattern5836, replacement5836)
pattern5837 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tanh(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement5837(v, u, m, b, a, n, x):
rubi.append(5837)
return Int((a + b*tanh(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule5837 = ReplacementRule(pattern5837, replacement5837)
pattern5838 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tanh(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement5838(v, u, m, b, a, n, x):
rubi.append(5838)
return Int((a + b/tanh(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule5838 = ReplacementRule(pattern5838, replacement5838)
pattern5839 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tanh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement5839(m, f, b, d, c, a, n, x, e):
rubi.append(5839)
return Int((a + b*tanh(e + f*x))**n*(c + d*x)**m, x)
rule5839 = ReplacementRule(pattern5839, replacement5839)
pattern5840 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tanh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement5840(m, f, b, d, a, n, c, x, e):
rubi.append(5840)
return Int((a + b/tanh(e + f*x))**n*(c + d*x)**m, x)
rule5840 = ReplacementRule(pattern5840, replacement5840)
pattern5841 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1573, cons38)
def replacement5841(p, b, d, c, a, n, x):
rubi.append(5841)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b*tanh(c + d*x))**p, x), x, x**n), x)
rule5841 = ReplacementRule(pattern5841, replacement5841)
pattern5842 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1573, cons38)
def replacement5842(p, b, d, a, c, n, x):
rubi.append(5842)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b/tanh(c + d*x))**p, x), x, x**n), x)
rule5842 = ReplacementRule(pattern5842, replacement5842)
pattern5843 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement5843(p, b, d, c, a, n, x):
rubi.append(5843)
return Int((a + b*tanh(c + d*x**n))**p, x)
rule5843 = ReplacementRule(pattern5843, replacement5843)
pattern5844 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement5844(p, b, d, a, c, n, x):
rubi.append(5844)
return Int((a + b/tanh(c + d*x**n))**p, x)
rule5844 = ReplacementRule(pattern5844, replacement5844)
pattern5845 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tanh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69)
def replacement5845(p, u, b, d, c, a, n, x):
rubi.append(5845)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*tanh(c + d*x**n))**p, x), x, u), x)
rule5845 = ReplacementRule(pattern5845, replacement5845)
pattern5846 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tanh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69)
def replacement5846(p, u, b, d, a, c, n, x):
rubi.append(5846)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b/tanh(c + d*x**n))**p, x), x, u), x)
rule5846 = ReplacementRule(pattern5846, replacement5846)
pattern5847 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tanh(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement5847(p, u, b, a, x):
rubi.append(5847)
return Int((a + b*tanh(ExpandToSum(u, x)))**p, x)
rule5847 = ReplacementRule(pattern5847, replacement5847)
pattern5848 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tanh(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement5848(p, u, b, a, x):
rubi.append(5848)
return Int((a + b/tanh(ExpandToSum(u, x)))**p, x)
rule5848 = ReplacementRule(pattern5848, replacement5848)
pattern5849 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1574, cons38)
def replacement5849(p, m, b, d, c, a, n, x):
rubi.append(5849)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*tanh(c + d*x))**p, x), x, x**n), x)
rule5849 = ReplacementRule(pattern5849, replacement5849)
pattern5850 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1574, cons38)
def replacement5850(p, m, b, d, a, c, n, x):
rubi.append(5850)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b/tanh(c + d*x))**p, x), x, x**n), x)
rule5850 = ReplacementRule(pattern5850, replacement5850)
pattern5851 = Pattern(Integral(x_**WC('m', S(1))*tanh(x_**n_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons7, cons27, cons21, cons4, cons1575)
def replacement5851(m, d, c, n, x):
rubi.append(5851)
return Dist((m - n + S(1))/(d*n), Int(x**(m - n)*tanh(c + d*x**n), x), x) + Int(x**m, x) - Simp(x**(m - n + S(1))*tanh(c + d*x**n)/(d*n), x)
rule5851 = ReplacementRule(pattern5851, replacement5851)
pattern5852 = Pattern(Integral(x_**WC('m', S(1))/tanh(x_**n_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons7, cons27, cons21, cons4, cons1575)
def replacement5852(m, d, c, n, x):
rubi.append(5852)
return Dist((m - n + S(1))/(d*n), Int(x**(m - n)/tanh(c + d*x**n), x), x) + Int(x**m, x) - Simp(x**(m - n + S(1))/(d*n*tanh(c + d*x**n)), x)
rule5852 = ReplacementRule(pattern5852, replacement5852)
pattern5853 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1576)
def replacement5853(p, m, b, d, c, a, n, x):
rubi.append(5853)
return Int(x**m*(a + b*tanh(c + d*x**n))**p, x)
rule5853 = ReplacementRule(pattern5853, replacement5853)
pattern5854 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1576)
def replacement5854(p, m, b, d, a, c, n, x):
rubi.append(5854)
return Int(x**m*(a + b/tanh(c + d*x**n))**p, x)
rule5854 = ReplacementRule(pattern5854, replacement5854)
pattern5855 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5855(p, m, b, d, c, a, n, x, e):
rubi.append(5855)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*tanh(c + d*x**n))**p, x), x)
rule5855 = ReplacementRule(pattern5855, replacement5855)
pattern5856 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tanh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5856(p, m, b, d, a, c, n, x, e):
rubi.append(5856)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b/tanh(c + d*x**n))**p, x), x)
rule5856 = ReplacementRule(pattern5856, replacement5856)
pattern5857 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tanh(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement5857(p, u, m, b, a, x, e):
rubi.append(5857)
return Int((e*x)**m*(a + b*tanh(ExpandToSum(u, x)))**p, x)
rule5857 = ReplacementRule(pattern5857, replacement5857)
pattern5858 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tanh(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement5858(p, u, m, b, a, x, e):
rubi.append(5858)
return Int((e*x)**m*(a + b/tanh(ExpandToSum(u, x)))**p, x)
rule5858 = ReplacementRule(pattern5858, replacement5858)
pattern5859 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cosh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1))*tanh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons5, cons31, cons85, cons1577, cons1578)
def replacement5859(p, m, b, a, n, x, q):
rubi.append(5859)
return Dist((m - n + S(1))/(b*n*p), Int(x**(m - n)*(S(1)/cosh(a + b*x**n))**p, x), x) - Simp(x**(m - n + S(1))*(S(1)/cosh(a + b*x**n))**p/(b*n*p), x)
rule5859 = ReplacementRule(pattern5859, replacement5859)
pattern5860 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sinh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1))*(S(1)/tanh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('q', S(1)), x_), cons2, cons3, cons5, cons31, cons85, cons1577, cons1578)
def replacement5860(p, m, b, a, n, x, q):
rubi.append(5860)
return Dist((m - n + S(1))/(b*n*p), Int(x**(m - n)*(S(1)/sinh(a + b*x**n))**p, x), x) - Simp(x**(m - n + S(1))*(S(1)/sinh(a + b*x**n))**p/(b*n*p), x)
rule5860 = ReplacementRule(pattern5860, replacement5860)
pattern5861 = Pattern(Integral(tanh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5861(b, c, a, n, x):
rubi.append(5861)
return Int(tanh(a + b*x + c*x**S(2))**n, x)
rule5861 = ReplacementRule(pattern5861, replacement5861)
pattern5862 = Pattern(Integral((S(1)/tanh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement5862(b, c, a, n, x):
rubi.append(5862)
return Int((S(1)/tanh(a + b*x + c*x**S(2)))**n, x)
rule5862 = ReplacementRule(pattern5862, replacement5862)
pattern5863 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*tanh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement5863(b, d, c, a, x, e):
rubi.append(5863)
return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(tanh(a + b*x + c*x**S(2)), x), x) + Simp(e*log(cosh(a + b*x + c*x**S(2)))/(S(2)*c), x)
rule5863 = ReplacementRule(pattern5863, replacement5863)
pattern5864 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))/tanh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement5864(b, d, c, a, x, e):
rubi.append(5864)
return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(S(1)/tanh(a + b*x + c*x**S(2)), x), x) + Simp(e*log(sinh(a + b*x + c*x**S(2)))/(S(2)*c), x)
rule5864 = ReplacementRule(pattern5864, replacement5864)
pattern5865 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*tanh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1580)
def replacement5865(m, b, d, a, c, n, x, e):
rubi.append(5865)
return Int((d + e*x)**m*tanh(a + b*x + c*x**S(2))**n, x)
rule5865 = ReplacementRule(pattern5865, replacement5865)
pattern5866 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(S(1)/tanh(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1580)
def replacement5866(m, b, d, c, a, n, x, e):
rubi.append(5866)
return Int((d + e*x)**m*(S(1)/tanh(a + b*x + c*x**S(2)))**n, x)
rule5866 = ReplacementRule(pattern5866, replacement5866)
pattern5867 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement5867(m, b, d, c, a, x):
rubi.append(5867)
return -Dist(I*d*m/b, Int((c + d*x)**(m + S(-1))*log(-I*exp(a + b*x) + S(1)), x), x) + Dist(I*d*m/b, Int((c + d*x)**(m + S(-1))*log(I*exp(a + b*x) + S(1)), x), x) + Simp(S(2)*(c + d*x)**m*ArcTan(exp(a + b*x))/b, x)
rule5867 = ReplacementRule(pattern5867, replacement5867)
pattern5868 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement5868(m, b, d, c, a, x):
rubi.append(5868)
return -Dist(d*m/b, Int((c + d*x)**(m + S(-1))*log(-exp(a + b*x) + S(1)), x), x) + Dist(d*m/b, Int((c + d*x)**(m + S(-1))*log(exp(a + b*x) + S(1)), x), x) + Simp(-S(2)*(c + d*x)**m*atanh(exp(a + b*x))/b, x)
rule5868 = ReplacementRule(pattern5868, replacement5868)
pattern5869 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/cosh(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons31, cons168)
def replacement5869(m, b, d, c, a, x):
rubi.append(5869)
return -Dist(d*m/b, Int((c + d*x)**(m + S(-1))*tanh(a + b*x), x), x) + Simp((c + d*x)**m*tanh(a + b*x)/b, x)
rule5869 = ReplacementRule(pattern5869, replacement5869)
pattern5870 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/sinh(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons31, cons168)
def replacement5870(m, b, d, c, a, x):
rubi.append(5870)
return Dist(d*m/b, Int((c + d*x)**(m + S(-1))/tanh(a + b*x), x), x) - Simp((c + d*x)**m/(b*tanh(a + b*x)), x)
rule5870 = ReplacementRule(pattern5870, replacement5870)
pattern5871 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons87, cons165, cons1644)
def replacement5871(b, d, c, a, n, x):
rubi.append(5871)
return Dist((n + S(-2))/(n + S(-1)), Int((c + d*x)*(S(1)/cosh(a + b*x))**(n + S(-2)), x), x) + Simp(d*(S(1)/cosh(a + b*x))**(n + S(-2))/(b**S(2)*(n + S(-2))*(n + S(-1))), x) + Simp((c + d*x)*(S(1)/cosh(a + b*x))**(n + S(-2))*tanh(a + b*x)/(b*(n + S(-1))), x)
rule5871 = ReplacementRule(pattern5871, replacement5871)
pattern5872 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons87, cons165, cons1644)
def replacement5872(b, d, c, a, n, x):
rubi.append(5872)
return -Dist((n + S(-2))/(n + S(-1)), Int((c + d*x)*(S(1)/sinh(a + b*x))**(n + S(-2)), x), x) - Simp(d*(S(1)/sinh(a + b*x))**(n + S(-2))/(b**S(2)*(n + S(-2))*(n + S(-1))), x) - Simp((c + d*x)*(S(1)/sinh(a + b*x))**(n + S(-2))/(b*(n + S(-1))*tanh(a + b*x)), x)
rule5872 = ReplacementRule(pattern5872, replacement5872)
pattern5873 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons93, cons165, cons1644, cons166)
def replacement5873(m, b, d, c, a, n, x):
rubi.append(5873)
return Dist((n + S(-2))/(n + S(-1)), Int((c + d*x)**m*(S(1)/cosh(a + b*x))**(n + S(-2)), x), x) - Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*(n + S(-2))*(n + S(-1))), Int((c + d*x)**(m + S(-2))*(S(1)/cosh(a + b*x))**(n + S(-2)), x), x) + Simp((c + d*x)**m*(S(1)/cosh(a + b*x))**(n + S(-2))*tanh(a + b*x)/(b*(n + S(-1))), x) + Simp(d*m*(c + d*x)**(m + S(-1))*(S(1)/cosh(a + b*x))**(n + S(-2))/(b**S(2)*(n + S(-2))*(n + S(-1))), x)
rule5873 = ReplacementRule(pattern5873, replacement5873)
pattern5874 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons93, cons165, cons1644, cons166)
def replacement5874(m, b, d, c, a, n, x):
rubi.append(5874)
return -Dist((n + S(-2))/(n + S(-1)), Int((c + d*x)**m*(S(1)/sinh(a + b*x))**(n + S(-2)), x), x) + Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*(n + S(-2))*(n + S(-1))), Int((c + d*x)**(m + S(-2))*(S(1)/sinh(a + b*x))**(n + S(-2)), x), x) - Simp((c + d*x)**m*(S(1)/sinh(a + b*x))**(n + S(-2))/(b*(n + S(-1))*tanh(a + b*x)), x) - Simp(d*m*(c + d*x)**(m + S(-1))*(S(1)/sinh(a + b*x))**(n + S(-2))/(b**S(2)*(n + S(-2))*(n + S(-1))), x)
rule5874 = ReplacementRule(pattern5874, replacement5874)
pattern5875 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons87, cons89)
def replacement5875(b, d, c, a, n, x):
rubi.append(5875)
return Dist((n + S(1))/n, Int((c + d*x)*(S(1)/cosh(a + b*x))**(n + S(2)), x), x) - Simp(d*(S(1)/cosh(a + b*x))**n/(b**S(2)*n**S(2)), x) - Simp((c + d*x)*(S(1)/cosh(a + b*x))**(n + S(1))*sinh(a + b*x)/(b*n), x)
rule5875 = ReplacementRule(pattern5875, replacement5875)
pattern5876 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons87, cons89)
def replacement5876(b, d, c, a, n, x):
rubi.append(5876)
return -Dist((n + S(1))/n, Int((c + d*x)*(S(1)/sinh(a + b*x))**(n + S(2)), x), x) - Simp(d*(S(1)/sinh(a + b*x))**n/(b**S(2)*n**S(2)), x) - Simp((c + d*x)*(S(1)/sinh(a + b*x))**(n + S(1))*cosh(a + b*x)/(b*n), x)
rule5876 = ReplacementRule(pattern5876, replacement5876)
pattern5877 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons93, cons89, cons166)
def replacement5877(m, b, d, c, a, n, x):
rubi.append(5877)
return Dist((n + S(1))/n, Int((c + d*x)**m*(S(1)/cosh(a + b*x))**(n + S(2)), x), x) + Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*n**S(2)), Int((c + d*x)**(m + S(-2))*(S(1)/cosh(a + b*x))**n, x), x) - Simp((c + d*x)**m*(S(1)/cosh(a + b*x))**(n + S(1))*sinh(a + b*x)/(b*n), x) - Simp(d*m*(c + d*x)**(m + S(-1))*(S(1)/cosh(a + b*x))**n/(b**S(2)*n**S(2)), x)
rule5877 = ReplacementRule(pattern5877, replacement5877)
pattern5878 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons93, cons89, cons166)
def replacement5878(m, b, d, c, a, n, x):
rubi.append(5878)
return -Dist((n + S(1))/n, Int((c + d*x)**m*(S(1)/sinh(a + b*x))**(n + S(2)), x), x) + Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*n**S(2)), Int((c + d*x)**(m + S(-2))*(S(1)/sinh(a + b*x))**n, x), x) - Simp((c + d*x)**m*(S(1)/sinh(a + b*x))**(n + S(1))*cosh(a + b*x)/(b*n), x) - Simp(d*m*(c + d*x)**(m + S(-1))*(S(1)/sinh(a + b*x))**n/(b**S(2)*n**S(2)), x)
rule5878 = ReplacementRule(pattern5878, replacement5878)
pattern5879 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons23)
def replacement5879(m, b, d, c, a, n, x):
rubi.append(5879)
return Dist((S(1)/cosh(a + b*x))**n*cosh(a + b*x)**n, Int((c + d*x)**m*cosh(a + b*x)**(-n), x), x)
rule5879 = ReplacementRule(pattern5879, replacement5879)
pattern5880 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons23)
def replacement5880(m, b, d, c, a, n, x):
rubi.append(5880)
return Dist((S(1)/sinh(a + b*x))**n*sinh(a + b*x)**n, Int((c + d*x)**m*sinh(a + b*x)**(-n), x), x)
rule5880 = ReplacementRule(pattern5880, replacement5880)
pattern5881 = Pattern(Integral((a_ + WC('b', S(1))/cosh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons528)
def replacement5881(m, f, b, d, c, n, a, x, e):
rubi.append(5881)
return Int(ExpandIntegrand((c + d*x)**m, (a + b/cosh(e + f*x))**n, x), x)
rule5881 = ReplacementRule(pattern5881, replacement5881)
pattern5882 = Pattern(Integral((a_ + WC('b', S(1))/sinh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons528)
def replacement5882(m, f, b, d, c, n, a, x, e):
rubi.append(5882)
return Int(ExpandIntegrand((c + d*x)**m, (a + b/sinh(e + f*x))**n, x), x)
rule5882 = ReplacementRule(pattern5882, replacement5882)
pattern5883 = Pattern(Integral((a_ + WC('b', S(1))/cosh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons196, cons62)
def replacement5883(m, f, b, d, c, n, a, x, e):
rubi.append(5883)
return Int(ExpandIntegrand((c + d*x)**m, (a*cosh(e + f*x) + b)**n*cosh(e + f*x)**(-n), x), x)
rule5883 = ReplacementRule(pattern5883, replacement5883)
pattern5884 = Pattern(Integral((a_ + WC('b', S(1))/sinh(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons196, cons62)
def replacement5884(m, f, b, d, c, n, a, x, e):
rubi.append(5884)
return Int(ExpandIntegrand((c + d*x)**m, (a*sinh(e + f*x) + b)**n*sinh(e + f*x)**(-n), x), x)
rule5884 = ReplacementRule(pattern5884, replacement5884)
pattern5885 = Pattern(Integral(u_**WC('m', S(1))*(S(1)/cosh(v_))**WC('n', S(1)), x_), cons21, cons4, cons810, cons811)
def replacement5885(v, u, m, n, x):
rubi.append(5885)
return Int((S(1)/cosh(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule5885 = ReplacementRule(pattern5885, replacement5885)
pattern5886 = Pattern(Integral(u_**WC('m', S(1))*(S(1)/sinh(v_))**WC('n', S(1)), x_), cons21, cons4, cons810, cons811)
def replacement5886(v, u, m, n, x):
rubi.append(5886)
return Int((S(1)/sinh(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule5886 = ReplacementRule(pattern5886, replacement5886)
pattern5887 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1736)
def replacement5887(m, b, d, c, a, n, x):
rubi.append(5887)
return Int((c + d*x)**m*(S(1)/cosh(a + b*x))**n, x)
rule5887 = ReplacementRule(pattern5887, replacement5887)
pattern5888 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1736)
def replacement5888(m, b, d, c, a, n, x):
rubi.append(5888)
return Int((c + d*x)**m*(S(1)/sinh(a + b*x))**n, x)
rule5888 = ReplacementRule(pattern5888, replacement5888)
pattern5889 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1573, cons38)
def replacement5889(p, b, d, c, a, n, x):
rubi.append(5889)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b/cosh(c + d*x))**p, x), x, x**n), x)
rule5889 = ReplacementRule(pattern5889, replacement5889)
pattern5890 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1573, cons38)
def replacement5890(p, b, d, a, c, n, x):
rubi.append(5890)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b/sinh(c + d*x))**p, x), x, x**n), x)
rule5890 = ReplacementRule(pattern5890, replacement5890)
pattern5891 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement5891(p, b, d, c, a, n, x):
rubi.append(5891)
return Int((a + b/cosh(c + d*x**n))**p, x)
rule5891 = ReplacementRule(pattern5891, replacement5891)
pattern5892 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement5892(p, b, d, a, c, n, x):
rubi.append(5892)
return Int((a + b/sinh(c + d*x**n))**p, x)
rule5892 = ReplacementRule(pattern5892, replacement5892)
pattern5893 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cosh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69)
def replacement5893(p, u, b, d, c, a, n, x):
rubi.append(5893)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b/cosh(c + d*x**n))**p, x), x, u), x)
rule5893 = ReplacementRule(pattern5893, replacement5893)
pattern5894 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sinh(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69)
def replacement5894(p, u, b, d, a, c, n, x):
rubi.append(5894)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b/sinh(c + d*x**n))**p, x), x, u), x)
rule5894 = ReplacementRule(pattern5894, replacement5894)
pattern5895 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cosh(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement5895(p, u, b, a, x):
rubi.append(5895)
return Int((a + b/cosh(ExpandToSum(u, x)))**p, x)
rule5895 = ReplacementRule(pattern5895, replacement5895)
pattern5896 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sinh(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement5896(p, u, b, a, x):
rubi.append(5896)
return Int((a + b/sinh(ExpandToSum(u, x)))**p, x)
rule5896 = ReplacementRule(pattern5896, replacement5896)
pattern5897 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1574, cons38)
def replacement5897(p, m, b, d, c, a, n, x):
rubi.append(5897)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b/cosh(c + d*x))**p, x), x, x**n), x)
rule5897 = ReplacementRule(pattern5897, replacement5897)
pattern5898 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1574, cons38)
def replacement5898(p, m, b, d, a, c, n, x):
rubi.append(5898)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b/sinh(c + d*x))**p, x), x, x**n), x)
rule5898 = ReplacementRule(pattern5898, replacement5898)
pattern5899 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1576)
def replacement5899(p, m, b, d, c, a, n, x):
rubi.append(5899)
return Int(x**m*(a + b/cosh(c + d*x**n))**p, x)
rule5899 = ReplacementRule(pattern5899, replacement5899)
pattern5900 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1576)
def replacement5900(p, m, b, d, a, c, n, x):
rubi.append(5900)
return Int(x**m*(a + b/sinh(c + d*x**n))**p, x)
rule5900 = ReplacementRule(pattern5900, replacement5900)
pattern5901 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cosh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5901(p, m, b, d, c, a, n, x, e):
rubi.append(5901)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b/cosh(c + d*x**n))**p, x), x)
rule5901 = ReplacementRule(pattern5901, replacement5901)
pattern5902 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sinh(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement5902(p, m, b, d, a, c, n, x, e):
rubi.append(5902)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b/sinh(c + d*x**n))**p, x), x)
rule5902 = ReplacementRule(pattern5902, replacement5902)
pattern5903 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cosh(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement5903(p, u, m, b, a, x, e):
rubi.append(5903)
return Int((e*x)**m*(a + b/cosh(ExpandToSum(u, x)))**p, x)
rule5903 = ReplacementRule(pattern5903, replacement5903)
pattern5904 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sinh(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement5904(p, u, m, b, a, x, e):
rubi.append(5904)
return Int((e*x)**m*(a + b/sinh(ExpandToSum(u, x)))**p, x)
rule5904 = ReplacementRule(pattern5904, replacement5904)
pattern5905 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cosh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**p_*sinh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons5, cons31, cons85, cons1577, cons1645)
def replacement5905(p, m, b, a, n, x):
rubi.append(5905)
return Dist((m - n + S(1))/(b*n*(p + S(-1))), Int(x**(m - n)*(S(1)/cosh(a + b*x**n))**(p + S(-1)), x), x) - Simp(x**(m - n + S(1))*(S(1)/cosh(a + b*x**n))**(p + S(-1))/(b*n*(p + S(-1))), x)
rule5905 = ReplacementRule(pattern5905, replacement5905)
pattern5906 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sinh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**p_*cosh(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons5, cons31, cons85, cons1577, cons1645)
def replacement5906(p, m, b, a, n, x):
rubi.append(5906)
return Dist((m - n + S(1))/(b*n*(p + S(-1))), Int(x**(m - n)*(S(1)/sinh(a + b*x**n))**(p + S(-1)), x), x) - Simp(x**(m - n + S(1))*(S(1)/sinh(a + b*x**n))**(p + S(-1))/(b*n*(p + S(-1))), x)
rule5906 = ReplacementRule(pattern5906, replacement5906)
pattern5907 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sinh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons62, cons584)
def replacement5907(m, b, d, c, a, n, x):
rubi.append(5907)
return -Dist(d*m/(b*(n + S(1))), Int((c + d*x)**(m + S(-1))*sinh(a + b*x)**(n + S(1)), x), x) + Simp((c + d*x)**m*sinh(a + b*x)**(n + S(1))/(b*(n + S(1))), x)
rule5907 = ReplacementRule(pattern5907, replacement5907)
pattern5908 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sinh(x_*WC('b', S(1)) + WC('a', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons62, cons584)
def replacement5908(m, b, d, c, a, n, x):
rubi.append(5908)
return -Dist(d*m/(b*(n + S(1))), Int((c + d*x)**(m + S(-1))*cosh(a + b*x)**(n + S(1)), x), x) + Simp((c + d*x)**m*cosh(a + b*x)**(n + S(1))/(b*(n + S(1))), x)
rule5908 = ReplacementRule(pattern5908, replacement5908)
pattern5909 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sinh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*cosh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons464)
def replacement5909(p, m, b, d, c, a, n, x):
rubi.append(5909)
return Int(ExpandTrigReduce((c + d*x)**m, sinh(a + b*x)**n*cosh(a + b*x)**p, x), x)
rule5909 = ReplacementRule(pattern5909, replacement5909)
pattern5910 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sinh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons464)
def replacement5910(p, m, b, d, c, a, n, x):
rubi.append(5910)
return Int((c + d*x)**m*sinh(a + b*x)**n*tanh(a + b*x)**(p + S(-2)), x) - Int((c + d*x)**m*sinh(a + b*x)**(n + S(-2))*tanh(a + b*x)**p, x)
rule5910 = ReplacementRule(pattern5910, replacement5910)
pattern5911 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1))*cosh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons464)
def replacement5911(p, m, b, d, c, a, n, x):
rubi.append(5911)
return Int((c + d*x)**m*(S(1)/tanh(a + b*x))**p*cosh(a + b*x)**(n + S(-2)), x) + Int((c + d*x)**m*(S(1)/tanh(a + b*x))**(p + S(-2))*cosh(a + b*x)**n, x)
rule5911 = ReplacementRule(pattern5911, replacement5911)
pattern5912 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1683, cons31, cons168)
def replacement5912(p, m, b, d, c, a, n, x):
rubi.append(5912)
return Dist(d*m/(b*n), Int((c + d*x)**(m + S(-1))*(S(1)/cosh(a + b*x))**n, x), x) - Simp((c + d*x)**m*(S(1)/cosh(a + b*x))**n/(b*n), x)
rule5912 = ReplacementRule(pattern5912, replacement5912)
pattern5913 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1683, cons31, cons168)
def replacement5913(p, m, b, d, c, a, n, x):
rubi.append(5913)
return Dist(d*m/(b*n), Int((c + d*x)**(m + S(-1))*(S(1)/sinh(a + b*x))**n, x), x) - Simp((c + d*x)**m*(S(1)/sinh(a + b*x))**n/(b*n), x)
rule5913 = ReplacementRule(pattern5913, replacement5913)
pattern5914 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/cosh(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons4, cons62, cons584)
def replacement5914(m, b, d, c, a, n, x):
rubi.append(5914)
return -Dist(d*m/(b*(n + S(1))), Int((c + d*x)**(m + S(-1))*tanh(a + b*x)**(n + S(1)), x), x) + Simp((c + d*x)**m*tanh(a + b*x)**(n + S(1))/(b*(n + S(1))), x)
rule5914 = ReplacementRule(pattern5914, replacement5914)
pattern5915 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))/sinh(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons4, cons62, cons584)
def replacement5915(m, b, d, c, a, n, x):
rubi.append(5915)
return Dist(d*m/(b*(n + S(1))), Int((c + d*x)**(m + S(-1))*(S(1)/tanh(a + b*x))**(n + S(1)), x), x) - Simp((c + d*x)**m*(S(1)/tanh(a + b*x))**(n + S(1))/(b*(n + S(1))), x)
rule5915 = ReplacementRule(pattern5915, replacement5915)
pattern5916 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))**p_/cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons1408)
def replacement5916(p, m, b, d, c, a, x):
rubi.append(5916)
return -Int((c + d*x)**m*tanh(a + b*x)**(p + S(-2))/cosh(a + b*x)**S(3), x) + Int((c + d*x)**m*tanh(a + b*x)**(p + S(-2))/cosh(a + b*x), x)
rule5916 = ReplacementRule(pattern5916, replacement5916)
pattern5917 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1408)
def replacement5917(p, m, b, d, c, a, n, x):
rubi.append(5917)
return Int((c + d*x)**m*(S(1)/cosh(a + b*x))**n*tanh(a + b*x)**(p + S(-2)), x) - Int((c + d*x)**m*(S(1)/cosh(a + b*x))**(n + S(2))*tanh(a + b*x)**(p + S(-2)), x)
rule5917 = ReplacementRule(pattern5917, replacement5917)
pattern5918 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0))))**p_/sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons1408)
def replacement5918(p, m, b, d, c, a, x):
rubi.append(5918)
return Int((c + d*x)**m*(S(1)/tanh(a + b*x))**(p + S(-2))/sinh(a + b*x)**S(3), x) + Int((c + d*x)**m*(S(1)/tanh(a + b*x))**(p + S(-2))/sinh(a + b*x), x)
rule5918 = ReplacementRule(pattern5918, replacement5918)
pattern5919 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1408)
def replacement5919(p, m, b, d, c, a, n, x):
rubi.append(5919)
return Int((c + d*x)**m*(S(1)/sinh(a + b*x))**n*(S(1)/tanh(a + b*x))**(p + S(-2)), x) + Int((c + d*x)**m*(S(1)/sinh(a + b*x))**(n + S(2))*(S(1)/tanh(a + b*x))**(p + S(-2)), x)
rule5919 = ReplacementRule(pattern5919, replacement5919)
def With5920(p, m, b, d, c, a, n, x):
u = IntHide((S(1)/cosh(a + b*x))**n*tanh(a + b*x)**p, x)
rubi.append(5920)
return -Dist(d*m, Int(u*(c + d*x)**(m + S(-1)), x), x) + Dist((c + d*x)**m, u, x)
pattern5920 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons62, cons1684)
rule5920 = ReplacementRule(pattern5920, With5920)
def With5921(p, m, b, d, c, a, n, x):
u = IntHide((S(1)/sinh(a + b*x))**n*(S(1)/tanh(a + b*x))**p, x)
rubi.append(5921)
return -Dist(d*m, Int(u*(c + d*x)**(m + S(-1)), x), x) + Dist((c + d*x)**m, u, x)
pattern5921 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/tanh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons62, cons1684)
rule5921 = ReplacementRule(pattern5921, With5921)
pattern5922 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons31, cons85)
def replacement5922(m, b, d, c, a, n, x):
rubi.append(5922)
return Dist(S(2)**n, Int((c + d*x)**m*(S(1)/sinh(S(2)*a + S(2)*b*x))**n, x), x)
rule5922 = ReplacementRule(pattern5922, replacement5922)
def With5923(p, m, b, d, c, a, n, x):
u = IntHide((S(1)/sinh(a + b*x))**n*(S(1)/cosh(a + b*x))**p, x)
rubi.append(5923)
return -Dist(d*m, Int(u*(c + d*x)**(m + S(-1)), x), x) + Dist((c + d*x)**m, u, x)
pattern5923 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sinh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/cosh(x_*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons376, cons31, cons168, cons1685)
rule5923 = ReplacementRule(pattern5923, With5923)
pattern5924 = Pattern(Integral(F_**v_*G_**w_*u_**WC('m', S(1)), x_), cons21, cons4, cons5, cons1860, cons1861, cons1688, cons812, cons813)
def replacement5924(v, w, p, u, m, G, n, x, F):
rubi.append(5924)
return Int(ExpandToSum(u, x)**m*F(ExpandToSum(v, x))**n*G(ExpandToSum(v, x))**p, x)
rule5924 = ReplacementRule(pattern5924, replacement5924)
pattern5925 = Pattern(Integral((a_ + WC('b', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement5925(m, f, b, d, c, n, a, x, e):
rubi.append(5925)
return -Dist(f*m/(b*d*(n + S(1))), Int((a + b*sinh(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) + Simp((a + b*sinh(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule5925 = ReplacementRule(pattern5925, replacement5925)
pattern5926 = Pattern(Integral((a_ + WC('b', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement5926(m, f, b, d, c, n, a, x, e):
rubi.append(5926)
return -Dist(f*m/(b*d*(n + S(1))), Int((a + b*cosh(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) + Simp((a + b*cosh(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule5926 = ReplacementRule(pattern5926, replacement5926)
pattern5927 = Pattern(Integral((a_ + WC('b', S(1))*tanh(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))/cosh(x_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement5927(m, f, b, d, c, n, a, x, e):
rubi.append(5927)
return -Dist(f*m/(b*d*(n + S(1))), Int((a + b*tanh(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) + Simp((a + b*tanh(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule5927 = ReplacementRule(pattern5927, replacement5927)
pattern5928 = Pattern(Integral((a_ + WC('b', S(1))/tanh(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))/sinh(x_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement5928(m, f, b, d, c, n, a, x, e):
rubi.append(5928)
return Dist(f*m/(b*d*(n + S(1))), Int((a + b/tanh(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) - Simp((a + b/tanh(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule5928 = ReplacementRule(pattern5928, replacement5928)
pattern5929 = Pattern(Integral((a_ + WC('b', S(1))/cosh(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*tanh(x_*WC('d', S(1)) + WC('c', S(0)))/cosh(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement5929(m, f, b, d, c, n, a, x, e):
rubi.append(5929)
return Dist(f*m/(b*d*(n + S(1))), Int((a + b/cosh(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) - Simp((a + b/cosh(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule5929 = ReplacementRule(pattern5929, replacement5929)
pattern5930 = Pattern(Integral((a_ + WC('b', S(1))/sinh(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))/(sinh(x_*WC('d', S(1)) + WC('c', S(0)))*tanh(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement5930(m, f, b, d, c, n, a, x, e):
rubi.append(5930)
return Dist(f*m/(b*d*(n + S(1))), Int((a + b/sinh(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) - Simp((a + b/sinh(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule5930 = ReplacementRule(pattern5930, replacement5930)
pattern5931 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sinh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons555, cons17)
def replacement5931(p, m, f, b, d, a, c, x, q, e):
rubi.append(5931)
return Int(ExpandTrigReduce((e + f*x)**m, sinh(a + b*x)**p*sinh(c + d*x)**q, x), x)
rule5931 = ReplacementRule(pattern5931, replacement5931)
pattern5932 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cosh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons555, cons17)
def replacement5932(p, m, f, b, d, c, a, x, q, e):
rubi.append(5932)
return Int(ExpandTrigReduce((e + f*x)**m, cosh(a + b*x)**p*cosh(c + d*x)**q, x), x)
rule5932 = ReplacementRule(pattern5932, replacement5932)
pattern5933 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sinh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons555)
def replacement5933(p, m, f, b, d, c, a, x, q, e):
rubi.append(5933)
return Int(ExpandTrigReduce((e + f*x)**m, sinh(a + b*x)**p*cosh(c + d*x)**q, x), x)
rule5933 = ReplacementRule(pattern5933, replacement5933)
pattern5934 = Pattern(Integral(F_**(x_*WC('b', S(1)) + WC('a', S(0)))*G_**(x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1862, cons1863, cons555, cons25, cons1691)
def replacement5934(p, m, f, b, G, d, a, c, x, q, e, F):
rubi.append(5934)
return Int(ExpandTrigExpand((e + f*x)**m*G(c + d*x)**q, F, c + d*x, p, b/d, x), x)
rule5934 = ReplacementRule(pattern5934, replacement5934)
pattern5935 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sinh(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1864)
def replacement5935(b, d, c, a, x, F, e):
rubi.append(5935)
return Simp(F**(c*(a + b*x))*e*cosh(d + e*x)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*sinh(d + e*x)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)), x)
rule5935 = ReplacementRule(pattern5935, replacement5935)
pattern5936 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cosh(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1864)
def replacement5936(b, d, c, a, x, F, e):
rubi.append(5936)
return Simp(F**(c*(a + b*x))*e*sinh(d + e*x)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*cosh(d + e*x)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)), x)
rule5936 = ReplacementRule(pattern5936, replacement5936)
pattern5937 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1865, cons87, cons165)
def replacement5937(b, d, c, a, n, x, F, e):
rubi.append(5937)
return -Dist(e**S(2)*n*(n + S(-1))/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), Int(F**(c*(a + b*x))*sinh(d + e*x)**(n + S(-2)), x), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*sinh(d + e*x)**n/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x) + Simp(F**(c*(a + b*x))*e*n*sinh(d + e*x)**(n + S(-1))*cosh(d + e*x)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x)
rule5937 = ReplacementRule(pattern5937, replacement5937)
pattern5938 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1865, cons87, cons165)
def replacement5938(b, d, c, a, n, x, F, e):
rubi.append(5938)
return Dist(e**S(2)*n*(n + S(-1))/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), Int(F**(c*(a + b*x))*cosh(d + e*x)**(n + S(-2)), x), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*cosh(d + e*x)**n/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x) + Simp(F**(c*(a + b*x))*e*n*sinh(d + e*x)*cosh(d + e*x)**(n + S(-1))/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x)
rule5938 = ReplacementRule(pattern5938, replacement5938)
pattern5939 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons1866, cons584, cons1395)
def replacement5939(b, d, c, a, n, x, F, e):
rubi.append(5939)
return Simp(F**(c*(a + b*x))*sinh(d + e*x)**(n + S(1))*cosh(d + e*x)/(e*(n + S(1))), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*sinh(d + e*x)**(n + S(2))/(e**S(2)*(n + S(1))*(n + S(2))), x)
rule5939 = ReplacementRule(pattern5939, replacement5939)
pattern5940 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons1866, cons584, cons1395)
def replacement5940(b, d, c, a, n, x, F, e):
rubi.append(5940)
return -Simp(F**(c*(a + b*x))*sinh(d + e*x)*cosh(d + e*x)**(n + S(1))/(e*(n + S(1))), x) + Simp(F**(c*(a + b*x))*b*c*log(F)*cosh(d + e*x)**(n + S(2))/(e**S(2)*(n + S(1))*(n + S(2))), x)
rule5940 = ReplacementRule(pattern5940, replacement5940)
pattern5941 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1867, cons87, cons89, cons1442)
def replacement5941(b, d, c, a, n, x, F, e):
rubi.append(5941)
return -Dist((-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(2))**S(2))/(e**S(2)*(n + S(1))*(n + S(2))), Int(F**(c*(a + b*x))*sinh(d + e*x)**(n + S(2)), x), x) + Simp(F**(c*(a + b*x))*sinh(d + e*x)**(n + S(1))*cosh(d + e*x)/(e*(n + S(1))), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*sinh(d + e*x)**(n + S(2))/(e**S(2)*(n + S(1))*(n + S(2))), x)
rule5941 = ReplacementRule(pattern5941, replacement5941)
pattern5942 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1867, cons87, cons89, cons1442)
def replacement5942(b, d, c, a, n, x, F, e):
rubi.append(5942)
return Dist((-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(2))**S(2))/(e**S(2)*(n + S(1))*(n + S(2))), Int(F**(c*(a + b*x))*cosh(d + e*x)**(n + S(2)), x), x) - Simp(F**(c*(a + b*x))*sinh(d + e*x)*cosh(d + e*x)**(n + S(1))/(e*(n + S(1))), x) + Simp(F**(c*(a + b*x))*b*c*log(F)*cosh(d + e*x)**(n + S(2))/(e**S(2)*(n + S(1))*(n + S(2))), x)
rule5942 = ReplacementRule(pattern5942, replacement5942)
pattern5943 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons23)
def replacement5943(b, d, c, a, n, x, F, e):
rubi.append(5943)
return Dist((exp(S(2)*d + S(2)*e*x) + S(-1))**(-n)*exp(n*(d + e*x))*sinh(d + e*x)**n, Int(F**(c*(a + b*x))*(exp(S(2)*d + S(2)*e*x) + S(-1))**n*exp(-n*(d + e*x)), x), x)
rule5943 = ReplacementRule(pattern5943, replacement5943)
pattern5944 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons23)
def replacement5944(b, d, c, a, n, x, F, e):
rubi.append(5944)
return Dist((exp(S(2)*d + S(2)*e*x) + S(1))**(-n)*exp(n*(d + e*x))*cosh(d + e*x)**n, Int(F**(c*(a + b*x))*(exp(S(2)*d + S(2)*e*x) + S(1))**n*exp(-n*(d + e*x)), x), x)
rule5944 = ReplacementRule(pattern5944, replacement5944)
pattern5945 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*tanh(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons85)
def replacement5945(b, d, c, a, n, x, F, e):
rubi.append(5945)
return Int(ExpandIntegrand(F**(c*(a + b*x))*(exp(S(2)*d + S(2)*e*x) + S(-1))**n*(exp(S(2)*d + S(2)*e*x) + S(1))**(-n), x), x)
rule5945 = ReplacementRule(pattern5945, replacement5945)
pattern5946 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/tanh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons85)
def replacement5946(b, d, c, n, a, x, F, e):
rubi.append(5946)
return Int(ExpandIntegrand(F**(c*(a + b*x))*(exp(S(2)*d + S(2)*e*x) + S(-1))**(-n)*(exp(S(2)*d + S(2)*e*x) + S(1))**n, x), x)
rule5946 = ReplacementRule(pattern5946, replacement5946)
pattern5947 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cosh(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1693, cons87, cons89)
def replacement5947(b, d, c, a, n, x, F, e):
rubi.append(5947)
return Dist(e**S(2)*n*(n + S(1))/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), Int(F**(c*(a + b*x))*(S(1)/cosh(d + e*x))**(n + S(2)), x), x) - Simp(F**(c*(a + b*x))*b*c*(S(1)/cosh(d + e*x))**n*log(F)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x) - Simp(F**(c*(a + b*x))*e*n*(S(1)/cosh(d + e*x))**(n + S(1))*sinh(d + e*x)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x)
rule5947 = ReplacementRule(pattern5947, replacement5947)
pattern5948 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sinh(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1693, cons87, cons89)
def replacement5948(b, d, c, a, n, x, F, e):
rubi.append(5948)
return -Dist(e**S(2)*n*(n + S(1))/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), Int(F**(c*(a + b*x))*(S(1)/sinh(d + e*x))**(n + S(2)), x), x) - Simp(F**(c*(a + b*x))*b*c*(S(1)/sinh(d + e*x))**n*log(F)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x) - Simp(F**(c*(a + b*x))*e*n*(S(1)/sinh(d + e*x))**(n + S(1))*cosh(d + e*x)/(-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x)
rule5948 = ReplacementRule(pattern5948, replacement5948)
pattern5949 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cosh(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons1868, cons1502, cons963)
def replacement5949(b, d, c, a, n, x, F, e):
rubi.append(5949)
return Simp(F**(c*(a + b*x))*(S(1)/cosh(d + e*x))**(n + S(-1))*sinh(d + e*x)/(e*(n + S(-1))), x) + Simp(F**(c*(a + b*x))*b*c*(S(1)/cosh(d + e*x))**(n + S(-2))*log(F)/(e**S(2)*(n + S(-2))*(n + S(-1))), x)
rule5949 = ReplacementRule(pattern5949, replacement5949)
pattern5950 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sinh(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons1868, cons1502, cons963)
def replacement5950(b, d, c, a, n, x, F, e):
rubi.append(5950)
return -Simp(F**(c*(a + b*x))*(S(1)/sinh(d + e*x))**(n + S(-1))*cosh(d + e*x)/(e*(n + S(-1))), x) - Simp(F**(c*(a + b*x))*b*c*(S(1)/sinh(d + e*x))**(n + S(-2))*log(F)/(e**S(2)*(n + S(-2))*(n + S(-1))), x)
rule5950 = ReplacementRule(pattern5950, replacement5950)
pattern5951 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cosh(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1869, cons87, cons165, cons1644)
def replacement5951(b, d, c, a, n, x, F, e):
rubi.append(5951)
return Dist((-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(-2))**S(2))/(e**S(2)*(n + S(-2))*(n + S(-1))), Int(F**(c*(a + b*x))*(S(1)/cosh(d + e*x))**(n + S(-2)), x), x) + Simp(F**(c*(a + b*x))*(S(1)/cosh(d + e*x))**(n + S(-1))*sinh(d + e*x)/(e*(n + S(-1))), x) + Simp(F**(c*(a + b*x))*b*c*(S(1)/cosh(d + e*x))**(n + S(-2))*log(F)/(e**S(2)*(n + S(-2))*(n + S(-1))), x)
rule5951 = ReplacementRule(pattern5951, replacement5951)
pattern5952 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sinh(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1869, cons87, cons165, cons1644)
def replacement5952(b, d, c, a, n, x, F, e):
rubi.append(5952)
return -Dist((-b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(-2))**S(2))/(e**S(2)*(n + S(-2))*(n + S(-1))), Int(F**(c*(a + b*x))*(S(1)/sinh(d + e*x))**(n + S(-2)), x), x) - Simp(F**(c*(a + b*x))*(S(1)/sinh(d + e*x))**(n + S(-1))*cosh(d + e*x)/(e*(n + S(-1))), x) - Simp(F**(c*(a + b*x))*b*c*(S(1)/sinh(d + e*x))**(n + S(-2))*log(F)/(e**S(2)*(n + S(-2))*(n + S(-1))), x)
rule5952 = ReplacementRule(pattern5952, replacement5952)
pattern5953 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cosh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons85)
def replacement5953(b, d, c, a, n, x, F, e):
rubi.append(5953)
return Simp(S(2)**n*F**(c*(a + b*x))*Hypergeometric2F1(n, b*c*log(F)/(S(2)*e) + n/S(2), b*c*log(F)/(S(2)*e) + n/S(2) + S(1), -exp(S(2)*d + S(2)*e*x))*exp(n*(d + e*x))/(b*c*log(F) + e*n), x)
rule5953 = ReplacementRule(pattern5953, replacement5953)
pattern5954 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sinh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons85)
def replacement5954(b, d, c, n, a, x, F, e):
rubi.append(5954)
return Simp((S(-2))**n*F**(c*(a + b*x))*Hypergeometric2F1(n, b*c*log(F)/(S(2)*e) + n/S(2), b*c*log(F)/(S(2)*e) + n/S(2) + S(1), exp(S(2)*d + S(2)*e*x))*exp(n*(d + e*x))/(b*c*log(F) + e*n), x)
rule5954 = ReplacementRule(pattern5954, replacement5954)
pattern5955 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cosh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons23)
def replacement5955(b, d, c, a, n, x, F, e):
rubi.append(5955)
return Dist((exp(S(2)*d + S(2)*e*x) + S(1))**n*(S(1)/cosh(d + e*x))**n*exp(-n*(d + e*x)), Int(SimplifyIntegrand(F**(c*(a + b*x))*(exp(S(2)*d + S(2)*e*x) + S(1))**(-n)*exp(n*(d + e*x)), x), x), x)
rule5955 = ReplacementRule(pattern5955, replacement5955)
pattern5956 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sinh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons23)
def replacement5956(b, d, c, n, a, x, F, e):
rubi.append(5956)
return Dist((-exp(-S(2)*d - S(2)*e*x) + S(1))**n*(S(1)/sinh(d + e*x))**n*exp(n*(d + e*x)), Int(SimplifyIntegrand(F**(c*(a + b*x))*(-exp(-S(2)*d - S(2)*e*x) + S(1))**(-n)*exp(-n*(d + e*x)), x), x), x)
rule5956 = ReplacementRule(pattern5956, replacement5956)
pattern5957 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1870, cons196)
def replacement5957(g, b, f, d, c, a, n, x, F, e):
rubi.append(5957)
return Dist(S(2)**n*f**n, Int(F**(c*(a + b*x))*cosh(-Pi*f/(S(4)*g) + d/S(2) + e*x/S(2))**(S(2)*n), x), x)
rule5957 = ReplacementRule(pattern5957, replacement5957)
pattern5958 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1700, cons196)
def replacement5958(g, b, f, d, c, n, a, x, F, e):
rubi.append(5958)
return Dist(S(2)**n*g**n, Int(F**(c*(a + b*x))*cosh(d/S(2) + e*x/S(2))**(S(2)*n), x), x)
rule5958 = ReplacementRule(pattern5958, replacement5958)
pattern5959 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1011, cons196)
def replacement5959(g, b, f, d, c, n, a, x, F, e):
rubi.append(5959)
return Dist(S(2)**n*g**n, Int(F**(c*(a + b*x))*sinh(d/S(2) + e*x/S(2))**(S(2)*n), x), x)
rule5959 = ReplacementRule(pattern5959, replacement5959)
pattern5960 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1870, cons150, cons1551)
def replacement5960(m, g, b, f, d, c, a, n, x, F, e):
rubi.append(5960)
return Dist(g**n, Int(F**(c*(a + b*x))*tanh(-Pi*f/(S(4)*g) + d/S(2) + e*x/S(2))**m, x), x)
rule5960 = ReplacementRule(pattern5960, replacement5960)
pattern5961 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1700, cons150, cons1551)
def replacement5961(m, g, b, f, d, c, n, a, x, F, e):
rubi.append(5961)
return Dist(g**n, Int(F**(c*(a + b*x))*tanh(d/S(2) + e*x/S(2))**m, x), x)
rule5961 = ReplacementRule(pattern5961, replacement5961)
pattern5962 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1011, cons150, cons1551)
def replacement5962(m, g, b, f, d, c, n, a, x, F, e):
rubi.append(5962)
return Dist(g**n, Int(F**(c*(a + b*x))*(S(1)/tanh(d/S(2) + e*x/S(2)))**m, x), x)
rule5962 = ReplacementRule(pattern5962, replacement5962)
pattern5963 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(h_ + WC('i', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0))))/(f_ + WC('g', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons1870, cons1701, cons1702)
def replacement5963(g, b, f, i, d, c, a, x, h, F, e):
rubi.append(5963)
return Dist(S(2)*i, Int(F**(c*(a + b*x))*cosh(d + e*x)/(f + g*sinh(d + e*x)), x), x) + Int(F**(c*(a + b*x))*(h - i*cosh(d + e*x))/(f + g*sinh(d + e*x)), x)
rule5963 = ReplacementRule(pattern5963, replacement5963)
pattern5964 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(h_ + WC('i', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0))))/(f_ + WC('g', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons1699, cons1871, cons1703)
def replacement5964(g, b, f, i, d, c, a, x, h, F, e):
rubi.append(5964)
return Dist(S(2)*i, Int(F**(c*(a + b*x))*sinh(d + e*x)/(f + g*cosh(d + e*x)), x), x) + Int(F**(c*(a + b*x))*(h - i*sinh(d + e*x))/(f + g*cosh(d + e*x)), x)
rule5964 = ReplacementRule(pattern5964, replacement5964)
pattern5965 = Pattern(Integral(F_**(u_*WC('c', S(1)))*G_**v_, x_), cons1099, cons7, cons4, cons1861, cons810, cons811)
def replacement5965(v, u, G, c, n, x, F):
rubi.append(5965)
return Int(F**(c*ExpandToSum(u, x))*G(ExpandToSum(v, x))**n, x)
rule5965 = ReplacementRule(pattern5965, replacement5965)
def With5966(m, b, d, c, a, n, x, F, e):
u = IntHide(F**(c*(a + b*x))*sinh(d + e*x)**n, x)
rubi.append(5966)
return -Dist(m, Int(u*x**(m + S(-1)), x), x) + Simp(u*x**m, x)
pattern5966 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*x_**WC('m', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons31, cons168, cons148)
rule5966 = ReplacementRule(pattern5966, With5966)
def With5967(m, b, d, c, n, a, x, F, e):
u = IntHide(F**(c*(a + b*x))*cosh(d + e*x)**n, x)
rubi.append(5967)
return -Dist(m, Int(u*x**(m + S(-1)), x), x) + Simp(u*x**m, x)
pattern5967 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*x_**WC('m', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons31, cons168, cons148)
rule5967 = ReplacementRule(pattern5967, With5967)
pattern5968 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cosh(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons528)
def replacement5968(m, f, g, b, d, c, n, a, x, F, e):
rubi.append(5968)
return Int(ExpandTrigReduce(F**(c*(a + b*x)), sinh(d + e*x)**m*cosh(f + g*x)**n, x), x)
rule5968 = ReplacementRule(pattern5968, replacement5968)
pattern5969 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*x_**WC('p', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cosh(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1704)
def replacement5969(p, m, f, g, b, d, c, n, a, x, F, e):
rubi.append(5969)
return Int(ExpandTrigReduce(F**(c*(a + b*x))*x**p, sinh(d + e*x)**m*cosh(f + g*x)**n, x), x)
rule5969 = ReplacementRule(pattern5969, replacement5969)
pattern5970 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*G_**(x_*WC('e', S(1)) + WC('d', S(0)))*H_**(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons528, cons1861, cons1872)
def replacement5970(m, b, G, d, c, a, n, H, x, F, e):
rubi.append(5970)
return Int(ExpandTrigToExp(F**(c*(a + b*x)), G(d + e*x)**m*H(d + e*x)**n, x), x)
rule5970 = ReplacementRule(pattern5970, replacement5970)
pattern5971 = Pattern(Integral(F_**u_*sinh(v_)**WC('n', S(1)), x_), cons1099, cons1706, cons1707, cons148)
def replacement5971(v, u, n, x, F):
rubi.append(5971)
return Int(ExpandTrigToExp(F**u, sinh(v)**n, x), x)
rule5971 = ReplacementRule(pattern5971, replacement5971)
pattern5972 = Pattern(Integral(F_**u_*cosh(v_)**WC('n', S(1)), x_), cons1099, cons1706, cons1707, cons148)
def replacement5972(v, u, n, x, F):
rubi.append(5972)
return Int(ExpandTrigToExp(F**u, cosh(v)**n, x), x)
rule5972 = ReplacementRule(pattern5972, replacement5972)
pattern5973 = Pattern(Integral(F_**u_*sinh(v_)**WC('m', S(1))*cosh(v_)**WC('n', S(1)), x_), cons1099, cons1706, cons1707, cons528)
def replacement5973(v, u, m, n, x, F):
rubi.append(5973)
return Int(ExpandTrigToExp(F**u, sinh(v)**m*cosh(v)**n, x), x)
rule5973 = ReplacementRule(pattern5973, replacement5973)
pattern5974 = Pattern(Integral(sinh(WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons7, cons1873)
def replacement5974(p, b, c, n, x):
rubi.append(5974)
return Int(((c*x**n)**b/S(2) - (c*x**n)**(-b)/S(2))**p, x)
rule5974 = ReplacementRule(pattern5974, replacement5974)
pattern5975 = Pattern(Integral(cosh(WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons7, cons1873)
def replacement5975(p, b, c, n, x):
rubi.append(5975)
return Int(((c*x**n)**b/S(2) + (c*x**n)**(-b)/S(2))**p, x)
rule5975 = ReplacementRule(pattern5975, replacement5975)
pattern5976 = Pattern(Integral(sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1874, cons54)
def replacement5976(p, b, a, n, c, x):
rubi.append(5976)
return -Simp(x*(p + S(2))*sinh(a + b*log(c*x**n))**(p + S(2))/(p + S(1)), x) + Simp(x*sinh(a + b*log(c*x**n))**(p + S(2))/(b*n*(p + S(1))*tanh(a + b*log(c*x**n))), x)
rule5976 = ReplacementRule(pattern5976, replacement5976)
pattern5977 = Pattern(Integral(cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1874, cons54)
def replacement5977(p, b, a, n, c, x):
rubi.append(5977)
return Simp(x*(p + S(2))*cosh(a + b*log(c*x**n))**(p + S(2))/(p + S(1)), x) - Simp(x*cosh(a + b*log(c*x**n))**(p + S(2))*tanh(a + b*log(c*x**n))/(b*n*(p + S(1))), x)
rule5977 = ReplacementRule(pattern5977, replacement5977)
pattern5978 = Pattern(Integral(sqrt(sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))), x_), cons2, cons3, cons7, cons4, cons1875)
def replacement5978(b, a, n, c, x):
rubi.append(5978)
return Dist(x*sqrt(sinh(a + b*log(c*x**n)))/sqrt((c*x**n)**(S(4)/n)*exp(S(2)*a) + S(-1)), Int(sqrt((c*x**n)**(S(4)/n)*exp(S(2)*a) + S(-1))/x, x), x)
rule5978 = ReplacementRule(pattern5978, replacement5978)
pattern5979 = Pattern(Integral(sqrt(cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))), x_), cons2, cons3, cons7, cons4, cons1875)
def replacement5979(b, a, n, c, x):
rubi.append(5979)
return Dist(x*sqrt(cosh(a + b*log(c*x**n)))/sqrt((c*x**n)**(S(4)/n)*exp(S(2)*a) + S(1)), Int(sqrt((c*x**n)**(S(4)/n)*exp(S(2)*a) + S(1))/x, x), x)
rule5979 = ReplacementRule(pattern5979, replacement5979)
pattern5980 = Pattern(Integral(sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons128, cons1876)
def replacement5980(p, b, a, n, c, x):
rubi.append(5980)
return Int(ExpandIntegrand(((c*x**n)**(S(1)/(n*p))*exp(a*b*n*p)/(S(2)*b*n*p) - (c*x**n)**(-S(1)/(n*p))*exp(-a*b*n*p)/(S(2)*b*n*p))**p, x), x)
rule5980 = ReplacementRule(pattern5980, replacement5980)
pattern5981 = Pattern(Integral(cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons128, cons1876)
def replacement5981(p, b, a, n, c, x):
rubi.append(5981)
return Int(ExpandIntegrand(((c*x**n)**(S(1)/(n*p))*exp(a*b*n*p)/S(2) + (c*x**n)**(-S(1)/(n*p))*exp(-a*b*n*p)/S(2))**p, x), x)
rule5981 = ReplacementRule(pattern5981, replacement5981)
pattern5982 = Pattern(Integral(sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1877)
def replacement5982(b, a, n, c, x):
rubi.append(5982)
return -Simp(x*sinh(a + b*log(c*x**n))/(b**S(2)*n**S(2) + S(-1)), x) + Simp(b*n*x*cosh(a + b*log(c*x**n))/(b**S(2)*n**S(2) + S(-1)), x)
rule5982 = ReplacementRule(pattern5982, replacement5982)
pattern5983 = Pattern(Integral(cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1877)
def replacement5983(b, a, n, c, x):
rubi.append(5983)
return -Simp(x*cosh(a + b*log(c*x**n))/(b**S(2)*n**S(2) + S(-1)), x) + Simp(b*n*x*sinh(a + b*log(c*x**n))/(b**S(2)*n**S(2) + S(-1)), x)
rule5983 = ReplacementRule(pattern5983, replacement5983)
pattern5984 = Pattern(Integral(sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons146, cons1878)
def replacement5984(p, b, a, n, c, x):
rubi.append(5984)
return -Dist(b**S(2)*n**S(2)*p*(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + S(-1)), Int(sinh(a + b*log(c*x**n))**(p + S(-2)), x), x) - Simp(x*sinh(a + b*log(c*x**n))**p/(b**S(2)*n**S(2)*p**S(2) + S(-1)), x) + Simp(b*n*p*x*sinh(a + b*log(c*x**n))**(p + S(-1))*cosh(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + S(-1)), x)
rule5984 = ReplacementRule(pattern5984, replacement5984)
pattern5985 = Pattern(Integral(cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons146, cons1878)
def replacement5985(p, b, a, n, c, x):
rubi.append(5985)
return Dist(b**S(2)*n**S(2)*p*(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + S(-1)), Int(cosh(a + b*log(c*x**n))**(p + S(-2)), x), x) - Simp(x*cosh(a + b*log(c*x**n))**p/(b**S(2)*n**S(2)*p**S(2) + S(-1)), x) + Simp(b*n*p*x*sinh(a + b*log(c*x**n))*cosh(a + b*log(c*x**n))**(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + S(-1)), x)
rule5985 = ReplacementRule(pattern5985, replacement5985)
pattern5986 = Pattern(Integral(sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons137, cons1505, cons1879)
def replacement5986(p, b, a, n, c, x):
rubi.append(5986)
return -Dist((b**S(2)*n**S(2)*(p + S(2))**S(2) + S(-1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(sinh(a + b*log(c*x**n))**(p + S(2)), x), x) - Simp(x*sinh(a + b*log(c*x**n))**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x) + Simp(x*sinh(a + b*log(c*x**n))**(p + S(2))/(b*n*(p + S(1))*tanh(a + b*log(c*x**n))), x)
rule5986 = ReplacementRule(pattern5986, replacement5986)
pattern5987 = Pattern(Integral(cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons137, cons1505, cons1879)
def replacement5987(p, b, a, n, c, x):
rubi.append(5987)
return Dist((b**S(2)*n**S(2)*(p + S(2))**S(2) + S(-1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(cosh(a + b*log(c*x**n))**(p + S(2)), x), x) + Simp(x*cosh(a + b*log(c*x**n))**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x) - Simp(x*cosh(a + b*log(c*x**n))**(p + S(2))*tanh(a + b*log(c*x**n))/(b*n*(p + S(1))), x)
rule5987 = ReplacementRule(pattern5987, replacement5987)
pattern5988 = Pattern(Integral(sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1878)
def replacement5988(p, b, a, n, c, x):
rubi.append(5988)
return Simp(x*(S(2) - S(2)*(c*x**n)**(-S(2)*b)*exp(-S(2)*a))**(-p)*((c*x**n)**b*exp(a) - (c*x**n)**(-b)*exp(-a))**p*Hypergeometric2F1(-p, -(b*n*p + S(1))/(S(2)*b*n), S(1) - (b*n*p + S(1))/(S(2)*b*n), (c*x**n)**(-S(2)*b)*exp(-S(2)*a))/(b*n*p + S(1)), x)
rule5988 = ReplacementRule(pattern5988, replacement5988)
pattern5989 = Pattern(Integral(cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1878)
def replacement5989(p, b, a, n, c, x):
rubi.append(5989)
return Simp(x*(S(2) + S(2)*(c*x**n)**(-S(2)*b)*exp(-S(2)*a))**(-p)*((c*x**n)**b*exp(a) + (c*x**n)**(-b)*exp(-a))**p*Hypergeometric2F1(-p, -(b*n*p + S(1))/(S(2)*b*n), S(1) - (b*n*p + S(1))/(S(2)*b*n), -(c*x**n)**(-S(2)*b)*exp(-S(2)*a))/(b*n*p + S(1)), x)
rule5989 = ReplacementRule(pattern5989, replacement5989)
pattern5990 = Pattern(Integral(x_**WC('m', S(1))*sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1880, cons54, cons66)
def replacement5990(p, m, b, c, n, a, x):
rubi.append(5990)
return -Simp(x**(m + S(1))*(p + S(2))*sinh(a + b*log(c*x**n))**(p + S(2))/((m + S(1))*(p + S(1))), x) + Simp(x**(m + S(1))*sinh(a + b*log(c*x**n))**(p + S(2))/(b*n*(p + S(1))*tanh(a + b*log(c*x**n))), x)
rule5990 = ReplacementRule(pattern5990, replacement5990)
pattern5991 = Pattern(Integral(x_**WC('m', S(1))*cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1880, cons54, cons66)
def replacement5991(p, m, b, a, n, c, x):
rubi.append(5991)
return Simp(x**(m + S(1))*(p + S(2))*cosh(a + b*log(c*x**n))**(p + S(2))/((m + S(1))*(p + S(1))), x) - Simp(x**(m + S(1))*cosh(a + b*log(c*x**n))**(p + S(2))*tanh(a + b*log(c*x**n))/(b*n*(p + S(1))), x)
rule5991 = ReplacementRule(pattern5991, replacement5991)
pattern5992 = Pattern(Integral(x_**WC('m', S(1))*sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons128, cons1881)
def replacement5992(p, m, b, c, n, a, x):
rubi.append(5992)
return Dist(S(2)**(-p), Int(ExpandIntegrand(x**m*((c*x**n)**((m + S(1))/(n*p))*(m + S(1))*exp(a*b*n*p/(m + S(1)))/(b*n*p) - (c*x**n)**(-(m + S(1))/(n*p))*(m + S(1))*exp(-a*b*n*p/(m + S(1)))/(b*n*p))**p, x), x), x)
rule5992 = ReplacementRule(pattern5992, replacement5992)
pattern5993 = Pattern(Integral(x_**WC('m', S(1))*cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons128, cons1881)
def replacement5993(p, m, b, a, n, c, x):
rubi.append(5993)
return Dist(S(2)**(-p), Int(ExpandIntegrand(x**m*((c*x**n)**((m + S(1))/(n*p))*exp(a*b*n*p/(m + S(1))) + (c*x**n)**(-(m + S(1))/(n*p))*exp(-a*b*n*p/(m + S(1))))**p, x), x), x)
rule5993 = ReplacementRule(pattern5993, replacement5993)
pattern5994 = Pattern(Integral(x_**WC('m', S(1))*sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons4, cons1882, cons66)
def replacement5994(m, b, c, n, a, x):
rubi.append(5994)
return -Simp(x**(m + S(1))*(m + S(1))*sinh(a + b*log(c*x**n))/(b**S(2)*n**S(2) - (m + S(1))**S(2)), x) + Simp(b*n*x**(m + S(1))*cosh(a + b*log(c*x**n))/(b**S(2)*n**S(2) - (m + S(1))**S(2)), x)
rule5994 = ReplacementRule(pattern5994, replacement5994)
pattern5995 = Pattern(Integral(x_**WC('m', S(1))*cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons4, cons1882, cons66)
def replacement5995(m, b, a, n, c, x):
rubi.append(5995)
return -Simp(x**(m + S(1))*(m + S(1))*cosh(a + b*log(c*x**n))/(b**S(2)*n**S(2) - (m + S(1))**S(2)), x) + Simp(b*n*x**(m + S(1))*sinh(a + b*log(c*x**n))/(b**S(2)*n**S(2) - (m + S(1))**S(2)), x)
rule5995 = ReplacementRule(pattern5995, replacement5995)
pattern5996 = Pattern(Integral(x_**WC('m', S(1))*sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons1883, cons13, cons146, cons66)
def replacement5996(p, m, b, c, n, a, x):
rubi.append(5996)
return -Dist(b**S(2)*n**S(2)*p*(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), Int(x**m*sinh(a + b*log(c*x**n))**(p + S(-2)), x), x) - Simp(x**(m + S(1))*(m + S(1))*sinh(a + b*log(c*x**n))**p/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), x) + Simp(b*n*p*x**(m + S(1))*sinh(a + b*log(c*x**n))**(p + S(-1))*cosh(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), x)
rule5996 = ReplacementRule(pattern5996, replacement5996)
pattern5997 = Pattern(Integral(x_**WC('m', S(1))*cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons1883, cons13, cons146, cons66)
def replacement5997(p, m, b, a, n, c, x):
rubi.append(5997)
return Dist(b**S(2)*n**S(2)*p*(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), Int(x**m*cosh(a + b*log(c*x**n))**(p + S(-2)), x), x) - Simp(x**(m + S(1))*(m + S(1))*cosh(a + b*log(c*x**n))**p/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), x) + Simp(b*n*p*x**(m + S(1))*sinh(a + b*log(c*x**n))*cosh(a + b*log(c*x**n))**(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), x)
rule5997 = ReplacementRule(pattern5997, replacement5997)
pattern5998 = Pattern(Integral(x_**WC('m', S(1))*sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons1884, cons13, cons137, cons1505, cons66)
def replacement5998(p, m, b, c, n, a, x):
rubi.append(5998)
return -Dist((b**S(2)*n**S(2)*(p + S(2))**S(2) - (m + S(1))**S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**m*sinh(a + b*log(c*x**n))**(p + S(2)), x), x) + Simp(x**(m + S(1))*sinh(a + b*log(c*x**n))**(p + S(2))/(b*n*(p + S(1))*tanh(a + b*log(c*x**n))), x) - Simp(x**(m + S(1))*(m + S(1))*sinh(a + b*log(c*x**n))**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x)
rule5998 = ReplacementRule(pattern5998, replacement5998)
pattern5999 = Pattern(Integral(x_**WC('m', S(1))*cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons1884, cons13, cons137, cons1505, cons66)
def replacement5999(p, m, b, a, n, c, x):
rubi.append(5999)
return Dist((b**S(2)*n**S(2)*(p + S(2))**S(2) - (m + S(1))**S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**m*cosh(a + b*log(c*x**n))**(p + S(2)), x), x) - Simp(x**(m + S(1))*cosh(a + b*log(c*x**n))**(p + S(2))*tanh(a + b*log(c*x**n))/(b*n*(p + S(1))), x) + Simp(x**(m + S(1))*(m + S(1))*cosh(a + b*log(c*x**n))**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x)
rule5999 = ReplacementRule(pattern5999, replacement5999)
pattern6000 = Pattern(Integral(x_**WC('m', S(1))*sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1883)
def replacement6000(p, m, b, c, n, a, x):
rubi.append(6000)
return Simp(x**(m + S(1))*(S(2) - S(2)*(c*x**n)**(-S(2)*b)*exp(-S(2)*a))**(-p)*((c*x**n)**b*exp(a) - (c*x**n)**(-b)*exp(-a))**p*Hypergeometric2F1(-p, -(b*n*p + m + S(1))/(S(2)*b*n), S(1) - (b*n*p + m + S(1))/(S(2)*b*n), (c*x**n)**(-S(2)*b)*exp(-S(2)*a))/(b*n*p + m + S(1)), x)
rule6000 = ReplacementRule(pattern6000, replacement6000)
pattern6001 = Pattern(Integral(x_**WC('m', S(1))*cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1883)
def replacement6001(p, m, b, a, n, c, x):
rubi.append(6001)
return Simp(x**(m + S(1))*(S(2) + S(2)*(c*x**n)**(-S(2)*b)*exp(-S(2)*a))**(-p)*((c*x**n)**b*exp(a) + (c*x**n)**(-b)*exp(-a))**p*Hypergeometric2F1(-p, -(b*n*p + m + S(1))/(S(2)*b*n), S(1) - (b*n*p + m + S(1))/(S(2)*b*n), -(c*x**n)**(-S(2)*b)*exp(-S(2)*a))/(b*n*p + m + S(1)), x)
rule6001 = ReplacementRule(pattern6001, replacement6001)
pattern6002 = Pattern(Integral((S(1)/cosh(WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons7, cons1873)
def replacement6002(p, b, c, n, x):
rubi.append(6002)
return Dist(S(2)**p, Int(((c*x**n)**b/((c*x**n)**(S(2)*b) + S(1)))**p, x), x)
rule6002 = ReplacementRule(pattern6002, replacement6002)
pattern6003 = Pattern(Integral((S(1)/sinh(WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons7, cons1873)
def replacement6003(p, b, c, n, x):
rubi.append(6003)
return Dist(S(2)**p, Int(((c*x**n)**b/((c*x**n)**(S(2)*b) + S(-1)))**p, x), x)
rule6003 = ReplacementRule(pattern6003, replacement6003)
pattern6004 = Pattern(Integral(S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1885)
def replacement6004(b, a, n, c, x):
rubi.append(6004)
return Dist(S(2)*exp(-a*b*n), Int((c*x**n)**(S(1)/n)/((c*x**n)**(S(2)/n) + exp(-S(2)*a*b*n)), x), x)
rule6004 = ReplacementRule(pattern6004, replacement6004)
pattern6005 = Pattern(Integral(S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1885)
def replacement6005(b, a, n, c, x):
rubi.append(6005)
return Dist(-S(2)*b*n*exp(-a*b*n), Int((c*x**n)**(S(1)/n)/(-(c*x**n)**(S(2)/n) + exp(-S(2)*a*b*n)), x), x)
rule6005 = ReplacementRule(pattern6005, replacement6005)
pattern6006 = Pattern(Integral((S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1886, cons1645)
def replacement6006(p, b, a, n, c, x):
rubi.append(6006)
return Simp(x*(p + S(-2))*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2))/(p + S(-1)), x) + Simp(x*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2))*tanh(a + b*log(c*x**n))/(b*n*(p + S(-1))), x)
rule6006 = ReplacementRule(pattern6006, replacement6006)
pattern6007 = Pattern(Integral((S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1886, cons1645)
def replacement6007(p, b, a, n, c, x):
rubi.append(6007)
return -Simp(x*(p + S(-2))*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2))/(p + S(-1)), x) - Simp(x*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2))/(b*n*(p + S(-1))*tanh(a + b*log(c*x**n))), x)
rule6007 = ReplacementRule(pattern6007, replacement6007)
pattern6008 = Pattern(Integral((S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons146, cons1720, cons1887)
def replacement6008(p, b, a, n, c, x):
rubi.append(6008)
return Dist((b**S(2)*n**S(2)*(p + S(-2))**S(2) + S(-1))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), Int((S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2)), x), x) + Simp(x*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), x) + Simp(x*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2))*tanh(a + b*log(c*x**n))/(b*n*(p + S(-1))), x)
rule6008 = ReplacementRule(pattern6008, replacement6008)
pattern6009 = Pattern(Integral((S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons146, cons1720, cons1887)
def replacement6009(p, b, a, n, c, x):
rubi.append(6009)
return -Dist((b**S(2)*n**S(2)*(p + S(-2))**S(2) + S(-1))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), Int((S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2)), x), x) - Simp(x*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), x) - Simp(x*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2))/(b*n*(p + S(-1))*tanh(a + b*log(c*x**n))), x)
rule6009 = ReplacementRule(pattern6009, replacement6009)
pattern6010 = Pattern(Integral((S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons137, cons1878)
def replacement6010(p, b, a, n, c, x):
rubi.append(6010)
return Dist(b**S(2)*n**S(2)*p*(p + S(1))/(b**S(2)*n**S(2)*p**S(2) + S(-1)), Int((S(1)/cosh(a + b*log(c*x**n)))**(p + S(2)), x), x) - Simp(x*(S(1)/cosh(a + b*log(c*x**n)))**p/(b**S(2)*n**S(2)*p**S(2) + S(-1)), x) - Simp(b*n*p*x*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(1))*sinh(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + S(-1)), x)
rule6010 = ReplacementRule(pattern6010, replacement6010)
pattern6011 = Pattern(Integral((S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons137, cons1878)
def replacement6011(p, b, a, n, c, x):
rubi.append(6011)
return -Dist(b**S(2)*n**S(2)*p*(p + S(1))/(b**S(2)*n**S(2)*p**S(2) + S(-1)), Int((S(1)/sinh(a + b*log(c*x**n)))**(p + S(2)), x), x) - Simp(x*(S(1)/sinh(a + b*log(c*x**n)))**p/(b**S(2)*n**S(2)*p**S(2) + S(-1)), x) - Simp(b*n*p*x*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(1))*cosh(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + S(-1)), x)
rule6011 = ReplacementRule(pattern6011, replacement6011)
pattern6012 = Pattern(Integral((S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons1878)
def replacement6012(p, b, a, n, c, x):
rubi.append(6012)
return Simp(S(2)**p*x*((c*x**n)**b*exp(a)/((c*x**n)**(S(2)*b)*exp(S(2)*a) + S(1)))**p*((c*x**n)**(S(2)*b)*exp(S(2)*a) + S(1))**p*Hypergeometric2F1(p, (b*n*p + S(1))/(S(2)*b*n), S(1) + (b*n*p + S(1))/(S(2)*b*n), -(c*x**n)**(S(2)*b)*exp(S(2)*a))/(b*n*p + S(1)), x)
rule6012 = ReplacementRule(pattern6012, replacement6012)
pattern6013 = Pattern(Integral((S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons1878)
def replacement6013(p, b, a, n, c, x):
rubi.append(6013)
return Simp(x*((c*x**n)**b*exp(a)/((c*x**n)**(S(2)*b)*exp(S(2)*a) + S(-1)))**p*(-S(2)*(c*x**n)**(S(2)*b)*exp(S(2)*a) + S(2))**p*Hypergeometric2F1(p, (b*n*p + S(1))/(S(2)*b*n), S(1) + (b*n*p + S(1))/(S(2)*b*n), (c*x**n)**(S(2)*b)*exp(S(2)*a))/(b*n*p + S(1)), x)
rule6013 = ReplacementRule(pattern6013, replacement6013)
pattern6014 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cosh(WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons7, cons1888)
def replacement6014(p, m, b, c, n, x):
rubi.append(6014)
return Dist(S(2)**p, Int(x**m*((c*x**n)**b/((c*x**n)**(S(2)*b) + S(1)))**p, x), x)
rule6014 = ReplacementRule(pattern6014, replacement6014)
pattern6015 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sinh(WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons7, cons1888)
def replacement6015(p, m, b, c, n, x):
rubi.append(6015)
return Dist(S(2)**p, Int(x**m*((c*x**n)**b/((c*x**n)**(S(2)*b) + S(-1)))**p, x), x)
rule6015 = ReplacementRule(pattern6015, replacement6015)
pattern6016 = Pattern(Integral(x_**WC('m', S(1))/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons4, cons1889)
def replacement6016(m, b, c, n, a, x):
rubi.append(6016)
return Dist(S(2)*exp(-a*b*n/(m + S(1))), Int(x**m*(c*x**n)**((m + S(1))/n)/((c*x**n)**(S(2)*(m + S(1))/n) + exp(-S(2)*a*b*n/(m + S(1)))), x), x)
rule6016 = ReplacementRule(pattern6016, replacement6016)
pattern6017 = Pattern(Integral(x_**WC('m', S(1))/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons4, cons1889)
def replacement6017(m, b, a, n, c, x):
rubi.append(6017)
return Dist(-S(2)*b*n*exp(-a*b*n/(m + S(1)))/(m + S(1)), Int(x**m*(c*x**n)**((m + S(1))/n)/(-(c*x**n)**(S(2)*(m + S(1))/n) + exp(-S(2)*a*b*n/(m + S(1)))), x), x)
rule6017 = ReplacementRule(pattern6017, replacement6017)
pattern6018 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1723, cons66, cons1645)
def replacement6018(p, m, b, c, n, a, x):
rubi.append(6018)
return Simp(x**(m + S(1))*(p + S(-2))*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2))/((m + S(1))*(p + S(-1))), x) + Simp(x**(m + S(1))*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2))*tanh(a + b*log(c*x**n))/(b*n*(p + S(-1))), x)
rule6018 = ReplacementRule(pattern6018, replacement6018)
pattern6019 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1723, cons66, cons1645)
def replacement6019(p, m, b, a, n, c, x):
rubi.append(6019)
return -Simp(x**(m + S(1))*(p + S(-2))*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2))/((m + S(1))*(p + S(-1))), x) - Simp(x**(m + S(1))*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2))/(b*n*(p + S(-1))*tanh(a + b*log(c*x**n))), x)
rule6019 = ReplacementRule(pattern6019, replacement6019)
pattern6020 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons146, cons1720, cons1890)
def replacement6020(p, m, b, c, n, a, x):
rubi.append(6020)
return Dist((b**S(2)*n**S(2)*(p + S(-2))**S(2) - (m + S(1))**S(2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), Int(x**m*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2)), x), x) + Simp(x**(m + S(1))*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2))*tanh(a + b*log(c*x**n))/(b*n*(p + S(-1))), x) + Simp(x**(m + S(1))*(m + S(1))*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(-2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), x)
rule6020 = ReplacementRule(pattern6020, replacement6020)
pattern6021 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons146, cons1720, cons1890)
def replacement6021(p, m, b, a, n, c, x):
rubi.append(6021)
return -Dist((b**S(2)*n**S(2)*(p + S(-2))**S(2) - (m + S(1))**S(2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), Int(x**m*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2)), x), x) - Simp(x**(m + S(1))*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2))/(b*n*(p + S(-1))*tanh(a + b*log(c*x**n))), x) - Simp(x**(m + S(1))*(m + S(1))*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(-2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), x)
rule6021 = ReplacementRule(pattern6021, replacement6021)
pattern6022 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons137, cons1883)
def replacement6022(p, m, b, c, n, a, x):
rubi.append(6022)
return Dist(b**S(2)*n**S(2)*p*(p + S(1))/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), Int(x**m*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(2)), x), x) - Simp(x**(m + S(1))*(m + S(1))*(S(1)/cosh(a + b*log(c*x**n)))**p/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), x) - Simp(b*n*p*x**(m + S(1))*(S(1)/cosh(a + b*log(c*x**n)))**(p + S(1))*sinh(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), x)
rule6022 = ReplacementRule(pattern6022, replacement6022)
pattern6023 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons137, cons1883)
def replacement6023(p, m, b, a, n, c, x):
rubi.append(6023)
return -Dist(b**S(2)*n**S(2)*p*(p + S(1))/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), Int(x**m*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(2)), x), x) - Simp(x**(m + S(1))*(m + S(1))*(S(1)/sinh(a + b*log(c*x**n)))**p/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), x) - Simp(b*n*p*x**(m + S(1))*(S(1)/sinh(a + b*log(c*x**n)))**(p + S(1))*cosh(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) - (m + S(1))**S(2)), x)
rule6023 = ReplacementRule(pattern6023, replacement6023)
pattern6024 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cosh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1883)
def replacement6024(p, m, b, c, n, a, x):
rubi.append(6024)
return Simp(S(2)**p*x**(m + S(1))*((c*x**n)**b*exp(a)/((c*x**n)**(S(2)*b)*exp(S(2)*a) + S(1)))**p*((c*x**n)**(S(2)*b)*exp(S(2)*a) + S(1))**p*Hypergeometric2F1(p, (b*n*p + m + S(1))/(S(2)*b*n), S(1) + (b*n*p + m + S(1))/(S(2)*b*n), -(c*x**n)**(S(2)*b)*exp(S(2)*a))/(b*n*p + m + S(1)), x)
rule6024 = ReplacementRule(pattern6024, replacement6024)
pattern6025 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sinh(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1883)
def replacement6025(p, m, b, a, n, c, x):
rubi.append(6025)
return Simp(S(2)**p*x**(m + S(1))*((c*x**n)**b*exp(a)/((c*x**n)**(S(2)*b)*exp(S(2)*a) + S(-1)))**p*(-(c*x**n)**(S(2)*b)*exp(S(2)*a) + S(1))**p*Hypergeometric2F1(p, (b*n*p + m + S(1))/(S(2)*b*n), S(1) + (b*n*p + m + S(1))/(S(2)*b*n), (c*x**n)**(S(2)*b)*exp(S(2)*a))/(b*n*p + m + S(1)), x)
rule6025 = ReplacementRule(pattern6025, replacement6025)
pattern6026 = Pattern(Integral(log(x_*WC('b', S(1)))**WC('p', S(1))*sinh(x_*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons13, cons163)
def replacement6026(x, a, p, b):
rubi.append(6026)
return -Dist(p, Int(log(b*x)**(p + S(-1))*sinh(a*x*log(b*x)**p), x), x) + Simp(cosh(a*x*log(b*x)**p)/a, x)
rule6026 = ReplacementRule(pattern6026, replacement6026)
pattern6027 = Pattern(Integral(log(x_*WC('b', S(1)))**WC('p', S(1))*cosh(x_*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons13, cons163)
def replacement6027(x, a, p, b):
rubi.append(6027)
return -Dist(p, Int(log(b*x)**(p + S(-1))*cosh(a*x*log(b*x)**p), x), x) + Simp(sinh(a*x*log(b*x)**p)/a, x)
rule6027 = ReplacementRule(pattern6027, replacement6027)
pattern6028 = Pattern(Integral(log(x_*WC('b', S(1)))**WC('p', S(1))*sinh(x_**n_*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons338, cons163)
def replacement6028(p, b, a, n, x):
rubi.append(6028)
return -Dist(p/n, Int(log(b*x)**(p + S(-1))*sinh(a*x**n*log(b*x)**p), x), x) + Dist((n + S(-1))/(a*n), Int(x**(-n)*cosh(a*x**n*log(b*x)**p), x), x) + Simp(x**(-n + S(1))*cosh(a*x**n*log(b*x)**p)/(a*n), x)
rule6028 = ReplacementRule(pattern6028, replacement6028)
pattern6029 = Pattern(Integral(log(x_*WC('b', S(1)))**WC('p', S(1))*cosh(x_**n_*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons338, cons163)
def replacement6029(p, b, a, n, x):
rubi.append(6029)
return -Dist(p/n, Int(log(b*x)**(p + S(-1))*cosh(a*x**n*log(b*x)**p), x), x) + Dist((n + S(-1))/(a*n), Int(x**(-n)*sinh(a*x**n*log(b*x)**p), x), x) + Simp(x**(-n + S(1))*sinh(a*x**n*log(b*x)**p)/(a*n), x)
rule6029 = ReplacementRule(pattern6029, replacement6029)
pattern6030 = Pattern(Integral(x_**WC('m', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))*sinh(x_**WC('n', S(1))*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons21, cons4, cons53, cons13, cons163)
def replacement6030(p, m, b, a, n, x):
rubi.append(6030)
return -Dist(p/n, Int(x**(n + S(-1))*log(b*x)**(p + S(-1))*sinh(a*x**n*log(b*x)**p), x), x) - Simp(cosh(a*x**n*log(b*x)**p)/(a*n), x)
rule6030 = ReplacementRule(pattern6030, replacement6030)
pattern6031 = Pattern(Integral(x_**WC('m', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))*cosh(x_**WC('n', S(1))*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons21, cons4, cons53, cons13, cons163)
def replacement6031(p, m, b, a, n, x):
rubi.append(6031)
return -Dist(p/n, Int(x**(n + S(-1))*log(b*x)**(p + S(-1))*cosh(a*x**n*log(b*x)**p), x), x) + Simp(sinh(a*x**n*log(b*x)**p)/(a*n), x)
rule6031 = ReplacementRule(pattern6031, replacement6031)
pattern6032 = Pattern(Integral(x_**m_*log(x_*WC('b', S(1)))**WC('p', S(1))*sinh(x_**WC('n', S(1))*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons162, cons163, cons627)
def replacement6032(p, m, b, a, n, x):
rubi.append(6032)
return -Dist(p/n, Int(x**m*log(b*x)**(p + S(-1))*sinh(a*x**n*log(b*x)**p), x), x) - Dist((m - n + S(1))/(a*n), Int(x**(m - n)*cosh(a*x**n*log(b*x)**p), x), x) + Simp(x**(m - n + S(1))*cosh(a*x**n*log(b*x)**p)/(a*n), x)
rule6032 = ReplacementRule(pattern6032, replacement6032)
pattern6033 = Pattern(Integral(x_**m_*log(x_*WC('b', S(1)))**WC('p', S(1))*cosh(x_**WC('n', S(1))*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons162, cons163, cons627)
def replacement6033(p, m, b, a, n, x):
rubi.append(6033)
return -Dist(p/n, Int(x**m*log(b*x)**(p + S(-1))*cosh(a*x**n*log(b*x)**p), x), x) - Dist((m - n + S(1))/(a*n), Int(x**(m - n)*sinh(a*x**n*log(b*x)**p), x), x) + Simp(x**(m - n + S(1))*sinh(a*x**n*log(b*x)**p)/(a*n), x)
rule6033 = ReplacementRule(pattern6033, replacement6033)
pattern6034 = Pattern(Integral(sinh(WC('a', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons7, cons27, cons148)
def replacement6034(d, a, n, c, x):
rubi.append(6034)
return -Dist(S(1)/d, Subst(Int(sinh(a*x)**n/x**S(2), x), x, S(1)/(c + d*x)), x)
rule6034 = ReplacementRule(pattern6034, replacement6034)
pattern6035 = Pattern(Integral(cosh(WC('a', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons7, cons27, cons148)
def replacement6035(d, a, n, c, x):
rubi.append(6035)
return -Dist(S(1)/d, Subst(Int(cosh(a*x)**n/x**S(2), x), x, S(1)/(c + d*x)), x)
rule6035 = ReplacementRule(pattern6035, replacement6035)
pattern6036 = Pattern(Integral(sinh((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons148, cons71)
def replacement6036(b, d, c, a, n, x, e):
rubi.append(6036)
return -Dist(S(1)/d, Subst(Int(sinh(b*e/d - e*x*(-a*d + b*c)/d)**n/x**S(2), x), x, S(1)/(c + d*x)), x)
rule6036 = ReplacementRule(pattern6036, replacement6036)
pattern6037 = Pattern(Integral(cosh((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons148, cons71)
def replacement6037(b, d, c, a, n, x, e):
rubi.append(6037)
return -Dist(S(1)/d, Subst(Int(cosh(b*e/d - e*x*(-a*d + b*c)/d)**n/x**S(2), x), x, S(1)/(c + d*x)), x)
rule6037 = ReplacementRule(pattern6037, replacement6037)
def With6038(x, n, u):
lst = QuotientOfLinearsParts(u, x)
rubi.append(6038)
return Int(sinh((x*Part(lst, S(2)) + Part(lst, S(1)))/(x*Part(lst, S(4)) + Part(lst, S(3))))**n, x)
pattern6038 = Pattern(Integral(sinh(u_)**WC('n', S(1)), x_), cons148, cons1725)
rule6038 = ReplacementRule(pattern6038, With6038)
def With6039(x, n, u):
lst = QuotientOfLinearsParts(u, x)
rubi.append(6039)
return Int(cosh((x*Part(lst, S(2)) + Part(lst, S(1)))/(x*Part(lst, S(4)) + Part(lst, S(3))))**n, x)
pattern6039 = Pattern(Integral(cosh(u_)**WC('n', S(1)), x_), cons148, cons1725)
rule6039 = ReplacementRule(pattern6039, With6039)
pattern6040 = Pattern(Integral(WC('u', S(1))*sinh(v_)**WC('p', S(1))*sinh(w_)**WC('q', S(1)), x_), cons1688)
def replacement6040(v, w, p, u, x, q):
rubi.append(6040)
return Int(u*sinh(v)**(p + q), x)
rule6040 = ReplacementRule(pattern6040, replacement6040)
pattern6041 = Pattern(Integral(WC('u', S(1))*cosh(v_)**WC('p', S(1))*cosh(w_)**WC('q', S(1)), x_), cons1688)
def replacement6041(v, w, p, u, x, q):
rubi.append(6041)
return Int(u*cosh(v)**(p + q), x)
rule6041 = ReplacementRule(pattern6041, replacement6041)
pattern6042 = Pattern(Integral(sinh(v_)**WC('p', S(1))*sinh(w_)**WC('q', S(1)), x_), cons555, cons1726)
def replacement6042(v, w, p, x, q):
rubi.append(6042)
return Int(ExpandTrigReduce(sinh(v)**p*sinh(w)**q, x), x)
rule6042 = ReplacementRule(pattern6042, replacement6042)
pattern6043 = Pattern(Integral(cosh(v_)**WC('p', S(1))*cosh(w_)**WC('q', S(1)), x_), cons555, cons1726)
def replacement6043(v, w, p, x, q):
rubi.append(6043)
return Int(ExpandTrigReduce(cosh(v)**p*cosh(w)**q, x), x)
rule6043 = ReplacementRule(pattern6043, replacement6043)
pattern6044 = Pattern(Integral(x_**WC('m', S(1))*sinh(v_)**WC('p', S(1))*sinh(w_)**WC('q', S(1)), x_), cons1727, cons1726)
def replacement6044(v, w, p, m, x, q):
rubi.append(6044)
return Int(ExpandTrigReduce(x**m, sinh(v)**p*sinh(w)**q, x), x)
rule6044 = ReplacementRule(pattern6044, replacement6044)
pattern6045 = Pattern(Integral(x_**WC('m', S(1))*cosh(v_)**WC('p', S(1))*cosh(w_)**WC('q', S(1)), x_), cons1727, cons1726)
def replacement6045(v, w, p, m, x, q):
rubi.append(6045)
return Int(ExpandTrigReduce(x**m, cosh(v)**p*cosh(w)**q, x), x)
rule6045 = ReplacementRule(pattern6045, replacement6045)
pattern6046 = Pattern(Integral(WC('u', S(1))*sinh(v_)**WC('p', S(1))*cosh(w_)**WC('p', S(1)), x_), cons1688, cons38)
def replacement6046(v, w, p, u, x):
rubi.append(6046)
return Dist(S(2)**(-p), Int(u*sinh(S(2)*v)**p, x), x)
rule6046 = ReplacementRule(pattern6046, replacement6046)
pattern6047 = Pattern(Integral(sinh(v_)**WC('p', S(1))*cosh(w_)**WC('q', S(1)), x_), cons555, cons1726)
def replacement6047(v, w, p, x, q):
rubi.append(6047)
return Int(ExpandTrigReduce(sinh(v)**p*cosh(w)**q, x), x)
rule6047 = ReplacementRule(pattern6047, replacement6047)
pattern6048 = Pattern(Integral(x_**WC('m', S(1))*sinh(v_)**WC('p', S(1))*cosh(w_)**WC('q', S(1)), x_), cons1727, cons1726)
def replacement6048(v, w, p, m, x, q):
rubi.append(6048)
return Int(ExpandTrigReduce(x**m, sinh(v)**p*cosh(w)**q, x), x)
rule6048 = ReplacementRule(pattern6048, replacement6048)
pattern6049 = Pattern(Integral(sinh(v_)*tanh(w_)**WC('n', S(1)), x_), cons87, cons88, cons1728)
def replacement6049(v, w, n, x):
rubi.append(6049)
return -Dist(cosh(v - w), Int(tanh(w)**(n + S(-1))/cosh(w), x), x) + Int(cosh(v)*tanh(w)**(n + S(-1)), x)
rule6049 = ReplacementRule(pattern6049, replacement6049)
pattern6050 = Pattern(Integral((S(1)/tanh(w_))**WC('n', S(1))*cosh(v_), x_), cons87, cons88, cons1728)
def replacement6050(v, w, n, x):
rubi.append(6050)
return Dist(cosh(v - w), Int((S(1)/tanh(w))**(n + S(-1))/sinh(w), x), x) + Int((S(1)/tanh(w))**(n + S(-1))*sinh(v), x)
rule6050 = ReplacementRule(pattern6050, replacement6050)
pattern6051 = Pattern(Integral((S(1)/tanh(w_))**WC('n', S(1))*sinh(v_), x_), cons87, cons88, cons1728)
def replacement6051(v, w, n, x):
rubi.append(6051)
return Dist(sinh(v - w), Int((S(1)/tanh(w))**(n + S(-1))/sinh(w), x), x) + Int((S(1)/tanh(w))**(n + S(-1))*cosh(v), x)
rule6051 = ReplacementRule(pattern6051, replacement6051)
pattern6052 = Pattern(Integral(cosh(v_)*tanh(w_)**WC('n', S(1)), x_), cons87, cons88, cons1728)
def replacement6052(v, w, n, x):
rubi.append(6052)
return -Dist(sinh(v - w), Int(tanh(w)**(n + S(-1))/cosh(w), x), x) + Int(sinh(v)*tanh(w)**(n + S(-1)), x)
rule6052 = ReplacementRule(pattern6052, replacement6052)
pattern6053 = Pattern(Integral((S(1)/cosh(w_))**WC('n', S(1))*sinh(v_), x_), cons87, cons88, cons1728)
def replacement6053(v, w, n, x):
rubi.append(6053)
return Dist(sinh(v - w), Int((S(1)/cosh(w))**(n + S(-1)), x), x) + Dist(cosh(v - w), Int((S(1)/cosh(w))**(n + S(-1))*tanh(w), x), x)
rule6053 = ReplacementRule(pattern6053, replacement6053)
pattern6054 = Pattern(Integral((S(1)/sinh(w_))**WC('n', S(1))*cosh(v_), x_), cons87, cons88, cons1728)
def replacement6054(v, w, n, x):
rubi.append(6054)
return Dist(sinh(v - w), Int((S(1)/sinh(w))**(n + S(-1)), x), x) + Dist(cosh(v - w), Int((S(1)/sinh(w))**(n + S(-1))/tanh(w), x), x)
rule6054 = ReplacementRule(pattern6054, replacement6054)
pattern6055 = Pattern(Integral((S(1)/sinh(w_))**WC('n', S(1))*sinh(v_), x_), cons87, cons88, cons1728)
def replacement6055(v, w, n, x):
rubi.append(6055)
return Dist(sinh(v - w), Int((S(1)/sinh(w))**(n + S(-1))/tanh(w), x), x) + Dist(cosh(v - w), Int((S(1)/sinh(w))**(n + S(-1)), x), x)
rule6055 = ReplacementRule(pattern6055, replacement6055)
pattern6056 = Pattern(Integral((S(1)/cosh(w_))**WC('n', S(1))*cosh(v_), x_), cons87, cons88, cons1728)
def replacement6056(v, w, n, x):
rubi.append(6056)
return Dist(sinh(v - w), Int((S(1)/cosh(w))**(n + S(-1))*tanh(w), x), x) + Dist(cosh(v - w), Int((S(1)/cosh(w))**(n + S(-1)), x), x)
rule6056 = ReplacementRule(pattern6056, replacement6056)
pattern6057 = Pattern(Integral((a_ + WC('b', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement6057(m, f, b, d, c, n, a, x, e):
rubi.append(6057)
return Int((a + b*sinh(S(2)*c + S(2)*d*x)/S(2))**n*(e + f*x)**m, x)
rule6057 = ReplacementRule(pattern6057, replacement6057)
pattern6058 = Pattern(Integral(x_**WC('m', S(1))*(a_ + WC('b', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons1456, cons150, cons168, cons463, cons1729)
def replacement6058(m, b, d, c, a, n, x):
rubi.append(6058)
return Dist(S(2)**(-n), Int(x**m*(S(2)*a + b*cosh(S(2)*c + S(2)*d*x) - b)**n, x), x)
rule6058 = ReplacementRule(pattern6058, replacement6058)
pattern6059 = Pattern(Integral(x_**WC('m', S(1))*(a_ + WC('b', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons1478, cons150, cons168, cons463, cons1729)
def replacement6059(m, b, d, c, a, n, x):
rubi.append(6059)
return Dist(S(2)**(-n), Int(x**m*(S(2)*a + b*cosh(S(2)*c + S(2)*d*x) + b)**n, x), x)
rule6059 = ReplacementRule(pattern6059, replacement6059)
pattern6060 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sinh((c_ + x_*WC('d', S(1)))**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons13)
def replacement6060(p, m, f, b, d, a, c, n, x, e):
rubi.append(6060)
return Dist(d**(-m + S(-1)), Subst(Int((-c*f + d*e + f*x)**m*sinh(a + b*x**n)**p, x), x, c + d*x), x)
rule6060 = ReplacementRule(pattern6060, replacement6060)
pattern6061 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cosh((c_ + x_*WC('d', S(1)))**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons13)
def replacement6061(p, m, f, b, d, a, c, n, x, e):
rubi.append(6061)
return Dist(d**(-m + S(-1)), Subst(Int((-c*f + d*e + f*x)**m*cosh(a + b*x**n)**p, x), x, c + d*x), x)
rule6061 = ReplacementRule(pattern6061, replacement6061)
pattern6062 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/(WC('a', S(0)) + WC('b', S(1))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('c', S(1))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons62, cons1478, cons1730)
def replacement6062(m, f, g, b, d, a, c, x, e):
rubi.append(6062)
return Dist(S(2), Int((f + g*x)**m/(S(2)*a + b - c + (b + c)*cosh(S(2)*d + S(2)*e*x)), x), x)
rule6062 = ReplacementRule(pattern6062, replacement6062)
pattern6063 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/((b_ + WC('c', S(1))*tanh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons3, cons7, cons27, cons48, cons125, cons208, cons62)
def replacement6063(m, f, g, b, d, c, x, e):
rubi.append(6063)
return Dist(S(2), Int((f + g*x)**m/(b - c + (b + c)*cosh(S(2)*d + S(2)*e*x)), x), x)
rule6063 = ReplacementRule(pattern6063, replacement6063)
pattern6064 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/((WC('a', S(1))/cosh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('b', S(0)) + WC('c', S(1))*tanh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))*cosh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons62, cons1478, cons1730)
def replacement6064(m, f, g, b, d, a, c, x, e):
rubi.append(6064)
return Dist(S(2), Int((f + g*x)**m/(S(2)*a + b - c + (b + c)*cosh(S(2)*d + S(2)*e*x)), x), x)
rule6064 = ReplacementRule(pattern6064, replacement6064)
pattern6065 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/((c_ + WC('b', S(1))/tanh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons3, cons7, cons27, cons48, cons125, cons208, cons62)
def replacement6065(m, f, b, g, d, c, x, e):
rubi.append(6065)
return Dist(S(2), Int((f + g*x)**m/(b - c + (b + c)*cosh(S(2)*d + S(2)*e*x)), x), x)
rule6065 = ReplacementRule(pattern6065, replacement6065)
pattern6066 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/((WC('a', S(1))/sinh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('b', S(1))/tanh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('c', S(0)))*sinh(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons62, cons1478, cons1730)
def replacement6066(m, f, b, g, d, c, a, x, e):
rubi.append(6066)
return Dist(S(2), Int((f + g*x)**m/(S(2)*a + b - c + (b + c)*cosh(S(2)*d + S(2)*e*x)), x), x)
rule6066 = ReplacementRule(pattern6066, replacement6066)
pattern6067 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62)
def replacement6067(m, f, b, d, c, a, x, e):
rubi.append(6067)
return Int((e + f*x)**m*exp(c + d*x)/(a + b*exp(c + d*x) - Rt(a**S(2) + b**S(2), S(2))), x) + Int((e + f*x)**m*exp(c + d*x)/(a + b*exp(c + d*x) + Rt(a**S(2) + b**S(2), S(2))), x) - Simp((e + f*x)**(m + S(1))/(b*f*(m + S(1))), x)
rule6067 = ReplacementRule(pattern6067, replacement6067)
pattern6068 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62)
def replacement6068(m, f, b, d, c, a, x, e):
rubi.append(6068)
return Int((e + f*x)**m*exp(c + d*x)/(a + b*exp(c + d*x) - Rt(a**S(2) - b**S(2), S(2))), x) + Int((e + f*x)**m*exp(c + d*x)/(a + b*exp(c + d*x) + Rt(a**S(2) - b**S(2), S(2))), x) - Simp((e + f*x)**(m + S(1))/(b*f*(m + S(1))), x)
rule6068 = ReplacementRule(pattern6068, replacement6068)
pattern6069 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + WC('b', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons85, cons165, cons1439)
def replacement6069(m, f, b, d, c, n, a, x, e):
rubi.append(6069)
return Dist(S(1)/a, Int((e + f*x)**m*cosh(c + d*x)**(n + S(-2)), x), x) + Dist(S(1)/b, Int((e + f*x)**m*sinh(c + d*x)*cosh(c + d*x)**(n + S(-2)), x), x)
rule6069 = ReplacementRule(pattern6069, replacement6069)
pattern6070 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + WC('b', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons85, cons165, cons1265)
def replacement6070(m, f, b, d, c, a, n, x, e):
rubi.append(6070)
return Dist(S(1)/a, Int((e + f*x)**m*sinh(c + d*x)**(n + S(-2)), x), x) + Dist(S(1)/b, Int((e + f*x)**m*sinh(c + d*x)**(n + S(-2))*cosh(c + d*x), x), x)
rule6070 = ReplacementRule(pattern6070, replacement6070)
pattern6071 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + WC('b', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons85, cons165, cons1440)
def replacement6071(m, f, b, d, c, n, a, x, e):
rubi.append(6071)
return Dist(S(1)/b, Int((e + f*x)**m*sinh(c + d*x)*cosh(c + d*x)**(n + S(-2)), x), x) - Dist(a/b**S(2), Int((e + f*x)**m*cosh(c + d*x)**(n + S(-2)), x), x) + Dist((a**S(2) + b**S(2))/b**S(2), Int((e + f*x)**m*cosh(c + d*x)**(n + S(-2))/(a + b*sinh(c + d*x)), x), x)
rule6071 = ReplacementRule(pattern6071, replacement6071)
pattern6072 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + WC('b', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons85, cons165, cons1267)
def replacement6072(m, f, b, d, c, a, n, x, e):
rubi.append(6072)
return Dist(S(1)/b, Int((e + f*x)**m*sinh(c + d*x)**(n + S(-2))*cosh(c + d*x), x), x) - Dist(a/b**S(2), Int((e + f*x)**m*sinh(c + d*x)**(n + S(-2)), x), x) + Dist((a**S(2) - b**S(2))/b**S(2), Int((e + f*x)**m*sinh(c + d*x)**(n + S(-2))/(a + b*cosh(c + d*x)), x), x)
rule6072 = ReplacementRule(pattern6072, replacement6072)
pattern6073 = Pattern(Integral((A_ + WC('B', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0))))*(x_*WC('f', S(1)) + WC('e', S(0)))/(a_ + WC('b', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1891)
def replacement6073(B, f, b, d, c, a, A, x, e):
rubi.append(6073)
return -Dist(B*f/(a*d), Int(cosh(c + d*x)/(a + b*sinh(c + d*x)), x), x) + Simp(B*(e + f*x)*cosh(c + d*x)/(a*d*(a + b*sinh(c + d*x))), x)
rule6073 = ReplacementRule(pattern6073, replacement6073)
pattern6074 = Pattern(Integral((A_ + WC('B', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0))))*(x_*WC('f', S(1)) + WC('e', S(0)))/(a_ + WC('b', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1474)
def replacement6074(B, f, b, d, c, a, A, x, e):
rubi.append(6074)
return -Dist(B*f/(a*d), Int(sinh(c + d*x)/(a + b*cosh(c + d*x)), x), x) + Simp(B*(e + f*x)*sinh(c + d*x)/(a*d*(a + b*cosh(c + d*x))), x)
rule6074 = ReplacementRule(pattern6074, replacement6074)
pattern6075 = Pattern(Integral((a_ + WC('b', S(1))*tanh(v_))**WC('n', S(1))*(S(1)/cosh(v_))**WC('m', S(1)), x_), cons2, cons3, cons150, cons1551, cons1481)
def replacement6075(v, m, b, a, n, x):
rubi.append(6075)
return Int((a*cosh(v) + b*sinh(v))**n, x)
rule6075 = ReplacementRule(pattern6075, replacement6075)
pattern6076 = Pattern(Integral((a_ + WC('b', S(1))/tanh(v_))**WC('n', S(1))*(S(1)/sinh(v_))**WC('m', S(1)), x_), cons2, cons3, cons150, cons1551, cons1481)
def replacement6076(v, m, b, a, n, x):
rubi.append(6076)
return Int((a*sinh(v) + b*cosh(v))**n, x)
rule6076 = ReplacementRule(pattern6076, replacement6076)
pattern6077 = Pattern(Integral(WC('u', S(1))*sinh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*sinh(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons528)
def replacement6077(u, m, b, d, a, c, n, x):
rubi.append(6077)
return Int(ExpandTrigReduce(u, sinh(a + b*x)**m*sinh(c + d*x)**n, x), x)
rule6077 = ReplacementRule(pattern6077, replacement6077)
pattern6078 = Pattern(Integral(WC('u', S(1))*cosh(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*cosh(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons528)
def replacement6078(u, m, b, d, a, c, n, x):
rubi.append(6078)
return Int(ExpandTrigReduce(u, cosh(a + b*x)**m*cosh(c + d*x)**n, x), x)
rule6078 = ReplacementRule(pattern6078, replacement6078)
pattern6079 = Pattern(Integral(S(1)/(cosh(c_ + x_*WC('d', S(1)))*cosh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1733, cons71)
def replacement6079(b, d, c, a, x):
rubi.append(6079)
return Dist(S(1)/sinh((-a*d + b*c)/b), Int(tanh(c + d*x), x), x) - Dist(S(1)/sinh((-a*d + b*c)/d), Int(tanh(a + b*x), x), x)
rule6079 = ReplacementRule(pattern6079, replacement6079)
pattern6080 = Pattern(Integral(S(1)/(sinh(c_ + x_*WC('d', S(1)))*sinh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1733, cons71)
def replacement6080(b, d, c, a, x):
rubi.append(6080)
return Dist(S(1)/sinh((-a*d + b*c)/b), Int(S(1)/tanh(a + b*x), x), x) - Dist(S(1)/sinh((-a*d + b*c)/d), Int(S(1)/tanh(c + d*x), x), x)
rule6080 = ReplacementRule(pattern6080, replacement6080)
pattern6081 = Pattern(Integral(tanh(c_ + x_*WC('d', S(1)))*tanh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1733, cons71)
def replacement6081(b, d, c, a, x):
rubi.append(6081)
return -Dist(b*cosh((-a*d + b*c)/d)/d, Int(S(1)/(cosh(a + b*x)*cosh(c + d*x)), x), x) + Simp(b*x/d, x)
rule6081 = ReplacementRule(pattern6081, replacement6081)
pattern6082 = Pattern(Integral(S(1)/(tanh(c_ + x_*WC('d', S(1)))*tanh(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1733, cons71)
def replacement6082(b, d, c, a, x):
rubi.append(6082)
return Dist(cosh((-a*d + b*c)/d), Int(S(1)/(sinh(a + b*x)*sinh(c + d*x)), x), x) + Simp(b*x/d, x)
rule6082 = ReplacementRule(pattern6082, replacement6082)
pattern6083 = Pattern(Integral((WC('a', S(1))*cosh(v_) + WC('b', S(1))*sinh(v_))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons4, cons1265)
def replacement6083(v, u, b, a, n, x):
rubi.append(6083)
return Int(u*(a*exp(a*v/b))**n, x)
rule6083 = ReplacementRule(pattern6083, replacement6083)
return [rule5643, rule5644, rule5645, rule5646, rule5647, rule5648, rule5649, rule5650, rule5651, rule5652, rule5653, rule5654, rule5655, rule5656, rule5657, rule5658, rule5659, rule5660, rule5661, rule5662, rule5663, rule5664, rule5665, rule5666, rule5667, rule5668, rule5669, rule5670, rule5671, rule5672, rule5673, rule5674, rule5675, rule5676, rule5677, rule5678, rule5679, rule5680, rule5681, rule5682, rule5683, rule5684, rule5685, rule5686, rule5687, rule5688, rule5689, rule5690, rule5691, rule5692, rule5693, rule5694, rule5695, rule5696, rule5697, rule5698, rule5699, rule5700, rule5701, rule5702, rule5703, rule5704, rule5705, rule5706, rule5707, rule5708, rule5709, rule5710, rule5711, rule5712, rule5713, rule5714, rule5715, rule5716, rule5717, rule5718, rule5719, rule5720, rule5721, rule5722, rule5723, rule5724, rule5725, rule5726, rule5727, rule5728, rule5729, rule5730, rule5731, rule5732, rule5733, rule5734, rule5735, rule5736, rule5737, rule5738, rule5739, rule5740, rule5741, rule5742, rule5743, rule5744, rule5745, rule5746, rule5747, rule5748, rule5749, rule5750, rule5751, rule5752, rule5753, rule5754, rule5755, rule5756, rule5757, rule5758, rule5759, rule5760, rule5761, rule5762, rule5763, rule5764, rule5765, rule5766, rule5767, rule5768, rule5769, rule5770, rule5771, rule5772, rule5773, rule5774, rule5775, rule5776, rule5777, rule5778, rule5779, rule5780, rule5781, rule5782, rule5783, rule5784, rule5785, rule5786, rule5787, rule5788, rule5789, rule5790, rule5791, rule5792, rule5793, rule5794, rule5795, rule5796, rule5797, rule5798, rule5799, rule5800, rule5801, rule5802, rule5803, rule5804, rule5805, rule5806, rule5807, rule5808, rule5809, rule5810, rule5811, rule5812, rule5813, rule5814, rule5815, rule5816, rule5817, rule5818, rule5819, rule5820, rule5821, rule5822, rule5823, rule5824, rule5825, rule5826, rule5827, rule5828, rule5829, rule5830, rule5831, rule5832, rule5833, rule5834, rule5835, rule5836, rule5837, rule5838, rule5839, rule5840, rule5841, rule5842, rule5843, rule5844, rule5845, rule5846, rule5847, rule5848, rule5849, rule5850, rule5851, rule5852, rule5853, rule5854, rule5855, rule5856, rule5857, rule5858, rule5859, rule5860, rule5861, rule5862, rule5863, rule5864, rule5865, rule5866, rule5867, rule5868, rule5869, rule5870, rule5871, rule5872, rule5873, rule5874, rule5875, rule5876, rule5877, rule5878, rule5879, rule5880, rule5881, rule5882, rule5883, rule5884, rule5885, rule5886, rule5887, rule5888, rule5889, rule5890, rule5891, rule5892, rule5893, rule5894, rule5895, rule5896, rule5897, rule5898, rule5899, rule5900, rule5901, rule5902, rule5903, rule5904, rule5905, rule5906, rule5907, rule5908, rule5909, rule5910, rule5911, rule5912, rule5913, rule5914, rule5915, rule5916, rule5917, rule5918, rule5919, rule5920, rule5921, rule5922, rule5923, rule5924, rule5925, rule5926, rule5927, rule5928, rule5929, rule5930, rule5931, rule5932, rule5933, rule5934, rule5935, rule5936, rule5937, rule5938, rule5939, rule5940, rule5941, rule5942, rule5943, rule5944, rule5945, rule5946, rule5947, rule5948, rule5949, rule5950, rule5951, rule5952, rule5953, rule5954, rule5955, rule5956, rule5957, rule5958, rule5959, rule5960, rule5961, rule5962, rule5963, rule5964, rule5965, rule5966, rule5967, rule5968, rule5969, rule5970, rule5971, rule5972, rule5973, rule5974, rule5975, rule5976, rule5977, rule5978, rule5979, rule5980, rule5981, rule5982, rule5983, rule5984, rule5985, rule5986, rule5987, rule5988, rule5989, rule5990, rule5991, rule5992, rule5993, rule5994, rule5995, rule5996, rule5997, rule5998, rule5999, rule6000, rule6001, rule6002, rule6003, rule6004, rule6005, rule6006, rule6007, rule6008, rule6009, rule6010, rule6011, rule6012, rule6013, rule6014, rule6015, rule6016, rule6017, rule6018, rule6019, rule6020, rule6021, rule6022, rule6023, rule6024, rule6025, rule6026, rule6027, rule6028, rule6029, rule6030, rule6031, rule6032, rule6033, rule6034, rule6035, rule6036, rule6037, rule6038, rule6039, rule6040, rule6041, rule6042, rule6043, rule6044, rule6045, rule6046, rule6047, rule6048, rule6049, rule6050, rule6051, rule6052, rule6053, rule6054, rule6055, rule6056, rule6057, rule6058, rule6059, rule6060, rule6061, rule6062, rule6063, rule6064, rule6065, rule6066, rule6067, rule6068, rule6069, rule6070, rule6071, rule6072, rule6073, rule6074, rule6075, rule6076, rule6077, rule6078, rule6079, rule6080, rule6081, rule6082, rule6083, ]
|
0d0723e108f47f46ff378158ae5f15e41e43a6d1d6d21edfcf0720ff9fccd604 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def tangent(rubi):
from sympy.integrals.rubi.constraints import cons1508, cons2, cons3, cons48, cons125, cons21, cons4, cons1509, cons17, cons23, cons93, cons165, cons94, cons1510, cons1170, cons87, cons1511, cons89, cons166, cons683, cons31, cons266, cons32, cons1512, cons1513, cons18, cons155, cons1250, cons1514, cons1515, cons1516, cons1517, cons1518, cons1519, cons1520, cons1521, cons1522, cons80, cons79, cons1523, cons1524, cons1525, cons1526, cons27, cons5, cons1527, cons7, cons1261, cons1264, cons1439, cons463, cons1440, cons1528, cons1255, cons1529, cons88, cons1530, cons1531, cons1532, cons515, cons1533, cons1534, cons1535, cons1536, cons1537, cons1538, cons66, cons1539, cons84, cons1228, cons148, cons196, cons1540, cons85, cons70, cons1541, cons71, cons1412, cons267, cons1303, cons168, cons1542, cons1543, cons1322, cons1544, cons1323, cons1545, cons1546, cons1547, cons1548, cons77, cons1549, cons1550, cons1551, cons1423, cons1385, cons111, cons272, cons1325, cons1327, cons1334, cons1552, cons1553, cons1333, cons1554, cons1555, cons1336, cons1556, cons1557, cons808, cons380, cons208, cons147, cons1417, cons50, cons38, cons34, cons35, cons346, cons1418, cons1426, cons1558, cons1559, cons1560, cons1256, cons1561, cons36, cons33, cons1433, cons1562, cons1563, cons1564, cons1565, cons1566, cons1567, cons1568, cons1569, cons1492, cons1456, cons1570, cons1481, cons1479, cons743, cons1497, cons46, cons45, cons226, cons1480, cons62, cons528, cons1571, cons1572, cons810, cons811, cons1360, cons1573, cons1495, cons68, cons69, cons823, cons824, cons1574, cons1575, cons1576, cons1577, cons1578, cons1579, cons47, cons239, cons1580
pattern3398 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1508)
def replacement3398(m, f, b, a, n, x, e):
rubi.append(3398)
return -Simp(b*(a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3398 = ReplacementRule(pattern3398, replacement3398)
pattern3399 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1508)
def replacement3399(m, f, b, a, n, x, e):
rubi.append(3399)
return Simp(b*(a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3399 = ReplacementRule(pattern3399, replacement3399)
pattern3400 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons48, cons125, cons1509)
def replacement3400(m, f, n, x, e):
rubi.append(3400)
return -Dist(S(1)/f, Subst(Int(x**(-n)*(-x**S(2) + S(1))**(m/S(2) + n/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule3400 = ReplacementRule(pattern3400, replacement3400)
pattern3401 = Pattern(Integral((S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons48, cons125, cons1509)
def replacement3401(m, f, n, x, e):
rubi.append(3401)
return Dist(S(1)/f, Subst(Int(x**(-n)*(-x**S(2) + S(1))**(m/S(2) + n/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule3401 = ReplacementRule(pattern3401, replacement3401)
pattern3402 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons3, cons48, cons125, cons4, cons17, cons23)
def replacement3402(m, f, b, n, x, e):
rubi.append(3402)
return Dist(b**(-m), Int((b*tan(e + f*x))**(m + n)*(S(1)/cos(e + f*x))**(-m), x), x)
rule3402 = ReplacementRule(pattern3402, replacement3402)
pattern3403 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons3, cons48, cons125, cons4, cons17, cons23)
def replacement3403(m, f, b, n, x, e):
rubi.append(3403)
return Dist(b**(-m), Int((b/tan(e + f*x))**(m + n)*(S(1)/sin(e + f*x))**(-m), x), x)
rule3403 = ReplacementRule(pattern3403, replacement3403)
pattern3404 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons165, cons94, cons1510, cons1170)
def replacement3404(m, f, b, a, n, x, e):
rubi.append(3404)
return -Dist(b**S(2)*(m + S(2))/(a**S(2)*(n + S(-1))), Int((a*sin(e + f*x))**(m + S(2))*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a*sin(e + f*x))**(m + S(2))*(b*tan(e + f*x))**(n + S(-1))/(a**S(2)*f*(n + S(-1))), x)
rule3404 = ReplacementRule(pattern3404, replacement3404)
pattern3405 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons165, cons94, cons1510, cons1170)
def replacement3405(m, f, b, a, n, x, e):
rubi.append(3405)
return -Dist(b**S(2)*(m + S(2))/(a**S(2)*(n + S(-1))), Int((a*cos(e + f*x))**(m + S(2))*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a*cos(e + f*x))**(m + S(2))*(b/tan(e + f*x))**(n + S(-1))/(a**S(2)*f*(n + S(-1))), x)
rule3405 = ReplacementRule(pattern3405, replacement3405)
pattern3406 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons165, cons1170, cons1511)
def replacement3406(m, f, b, a, n, x, e):
rubi.append(3406)
return -Dist(b**S(2)*(m + n + S(-1))/(n + S(-1)), Int((a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x)
rule3406 = ReplacementRule(pattern3406, replacement3406)
pattern3407 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons165, cons1170, cons1511)
def replacement3407(m, f, b, a, n, x, e):
rubi.append(3407)
return -Dist(b**S(2)*(m + n + S(-1))/(n + S(-1)), Int((a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x)
rule3407 = ReplacementRule(pattern3407, replacement3407)
pattern3408 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons89, cons166, cons1170)
def replacement3408(m, f, b, a, n, x, e):
rubi.append(3408)
return -Dist(a**S(2)*(n + S(1))/(b**S(2)*m), Int((a*sin(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp((a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*m), x)
rule3408 = ReplacementRule(pattern3408, replacement3408)
pattern3409 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons89, cons166, cons1170)
def replacement3409(m, f, b, a, n, x, e):
rubi.append(3409)
return -Dist(a**S(2)*(n + S(1))/(b**S(2)*m), Int((a*cos(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**(n + S(2)), x), x) - Simp((a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*m), x)
rule3409 = ReplacementRule(pattern3409, replacement3409)
pattern3410 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons89, cons683, cons1170)
def replacement3410(m, f, b, a, n, x, e):
rubi.append(3410)
return -Dist((n + S(1))/(b**S(2)*(m + n + S(1))), Int((a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp((a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(1))), x)
rule3410 = ReplacementRule(pattern3410, replacement3410)
pattern3411 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons89, cons683, cons1170)
def replacement3411(m, f, b, a, n, x, e):
rubi.append(3411)
return -Dist((n + S(1))/(b**S(2)*(m + n + S(1))), Int((a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(2)), x), x) - Simp((a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(1))), x)
rule3411 = ReplacementRule(pattern3411, replacement3411)
pattern3412 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons4, cons31, cons266, cons1170)
def replacement3412(m, f, b, a, n, x, e):
rubi.append(3412)
return Dist(a**S(2)*(m + n + S(-1))/m, Int((a*sin(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**n, x), x) - Simp(b*(a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3412 = ReplacementRule(pattern3412, replacement3412)
pattern3413 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons4, cons31, cons266, cons1170)
def replacement3413(m, f, b, a, n, x, e):
rubi.append(3413)
return Dist(a**S(2)*(m + n + S(-1))/m, Int((a*cos(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**n, x), x) + Simp(b*(a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3413 = ReplacementRule(pattern3413, replacement3413)
pattern3414 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons4, cons31, cons32, cons683, cons1170)
def replacement3414(m, f, b, a, n, x, e):
rubi.append(3414)
return Dist((m + S(2))/(a**S(2)*(m + n + S(1))), Int((a*sin(e + f*x))**(m + S(2))*(b*tan(e + f*x))**n, x), x) + Simp(b*(a*sin(e + f*x))**(m + S(2))*(b*tan(e + f*x))**(n + S(-1))/(a**S(2)*f*(m + n + S(1))), x)
rule3414 = ReplacementRule(pattern3414, replacement3414)
pattern3415 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons4, cons31, cons32, cons683, cons1170)
def replacement3415(m, f, b, a, n, x, e):
rubi.append(3415)
return Dist((m + S(2))/(a**S(2)*(m + n + S(1))), Int((a*cos(e + f*x))**(m + S(2))*(b/tan(e + f*x))**n, x), x) - Simp(b*(a*cos(e + f*x))**(m + S(2))*(b/tan(e + f*x))**(n + S(-1))/(a**S(2)*f*(m + n + S(1))), x)
rule3415 = ReplacementRule(pattern3415, replacement3415)
pattern3416 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons48, cons125, cons21, cons1512)
def replacement3416(m, f, a, n, x, e):
rubi.append(3416)
return Dist(S(1)/f, Subst(Int(x**(m + n)*(a**S(2) - x**S(2))**(-n/S(2) + S(-1)/2), x), x, a*sin(e + f*x)), x)
rule3416 = ReplacementRule(pattern3416, replacement3416)
pattern3417 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons48, cons125, cons21, cons1512)
def replacement3417(m, f, a, n, x, e):
rubi.append(3417)
return -Dist(S(1)/f, Subst(Int(x**(m + n)*(a**S(2) - x**S(2))**(-n/S(2) + S(-1)/2), x), x, a*cos(e + f*x)), x)
rule3417 = ReplacementRule(pattern3417, replacement3417)
pattern3418 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1513)
def replacement3418(m, f, b, a, n, x, e):
rubi.append(3418)
return Dist(a**(-S(2)*IntPart(n/S(2) + S(1)/2) + S(1))*b**(S(2)*IntPart(n/S(2) + S(1)/2) + S(-1))*(a*sin(e + f*x))**(-S(2)*FracPart(n/S(2) + S(1)/2))*(b*tan(e + f*x))**(S(2)*FracPart(n/S(2) + S(1)/2))*(cos(e + f*x)**S(2))**FracPart(n/S(2) + S(1)/2)/f, Subst(Int((a*x)**(m + n)*(-x**S(2) + S(1))**(-n/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule3418 = ReplacementRule(pattern3418, replacement3418)
pattern3419 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1513)
def replacement3419(m, f, b, a, n, x, e):
rubi.append(3419)
return -Dist(a**(-S(2)*IntPart(n/S(2) + S(1)/2) + S(1))*b**(S(2)*IntPart(n/S(2) + S(1)/2) + S(-1))*(a*cos(e + f*x))**(-S(2)*FracPart(n/S(2) + S(1)/2))*(b/tan(e + f*x))**(S(2)*FracPart(n/S(2) + S(1)/2))*(sin(e + f*x)**S(2))**FracPart(n/S(2) + S(1)/2)/f, Subst(Int((a*x)**(m + n)*(-x**S(2) + S(1))**(-n/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule3419 = ReplacementRule(pattern3419, replacement3419)
pattern3420 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement3420(m, f, b, a, n, x, e):
rubi.append(3420)
return Dist((S(1)/(a*cos(e + f*x)))**FracPart(m)*(a*cos(e + f*x))**FracPart(m), Int((S(1)/(a*cos(e + f*x)))**(-m)*(b*tan(e + f*x))**n, x), x)
rule3420 = ReplacementRule(pattern3420, replacement3420)
pattern3421 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement3421(m, f, b, a, n, x, e):
rubi.append(3421)
return Dist((S(1)/(a*sin(e + f*x)))**FracPart(m)*(a*sin(e + f*x))**FracPart(m), Int((S(1)/(a*sin(e + f*x)))**(-m)*(b/tan(e + f*x))**n, x), x)
rule3421 = ReplacementRule(pattern3421, replacement3421)
pattern3422 = Pattern(Integral((WC('a', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement3422(m, f, b, a, n, x, e):
rubi.append(3422)
return Dist((a/tan(e + f*x))**m*(b*tan(e + f*x))**m, Int((b*tan(e + f*x))**(-m + n), x), x)
rule3422 = ReplacementRule(pattern3422, replacement3422)
pattern3423 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons4, cons155)
def replacement3423(m, f, b, a, n, x, e):
rubi.append(3423)
return -Simp((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*m), x)
rule3423 = ReplacementRule(pattern3423, replacement3423)
pattern3424 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons4, cons155)
def replacement3424(m, f, b, a, n, x, e):
rubi.append(3424)
return Simp((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*m), x)
rule3424 = ReplacementRule(pattern3424, replacement3424)
pattern3425 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons48, cons125, cons21, cons1250, cons1514)
def replacement3425(m, f, b, a, n, x, e):
rubi.append(3425)
return Dist(a/f, Subst(Int((a*x)**(m + S(-1))*(x**S(2) + S(-1))**(n/S(2) + S(-1)/2), x), x, S(1)/cos(e + f*x)), x)
rule3425 = ReplacementRule(pattern3425, replacement3425)
pattern3426 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons48, cons125, cons21, cons1250, cons1514)
def replacement3426(m, f, b, a, n, x, e):
rubi.append(3426)
return -Dist(a/f, Subst(Int((a*x)**(m + S(-1))*(x**S(2) + S(-1))**(n/S(2) + S(-1)/2), x), x, S(1)/sin(e + f*x)), x)
rule3426 = ReplacementRule(pattern3426, replacement3426)
pattern3427 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons3, cons48, cons125, cons4, cons1515, cons1516)
def replacement3427(m, f, b, n, x, e):
rubi.append(3427)
return Dist(S(1)/f, Subst(Int((b*x)**n*(x**S(2) + S(1))**(m/S(2) + S(-1)), x), x, tan(e + f*x)), x)
rule3427 = ReplacementRule(pattern3427, replacement3427)
pattern3428 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons3, cons48, cons125, cons4, cons1515, cons1516)
def replacement3428(m, f, b, n, x, e):
rubi.append(3428)
return -Dist(S(1)/f, Subst(Int((b*x)**n*(x**S(2) + S(1))**(m/S(2) + S(-1)), x), x, S(1)/tan(e + f*x)), x)
rule3428 = ReplacementRule(pattern3428, replacement3428)
pattern3429 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons89, cons1517, cons1170)
def replacement3429(m, f, b, a, n, x, e):
rubi.append(3429)
return -Dist(a**S(2)*(m + S(-2))/(b**S(2)*(n + S(1))), Int((a/cos(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp(a**S(2)*(a/cos(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x)
rule3429 = ReplacementRule(pattern3429, replacement3429)
pattern3430 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons89, cons1517, cons1170)
def replacement3430(m, f, b, a, n, x, e):
rubi.append(3430)
return -Dist(a**S(2)*(m + S(-2))/(b**S(2)*(n + S(1))), Int((a/sin(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**(n + S(2)), x), x) - Simp(a**S(2)*(a/sin(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x)
rule3430 = ReplacementRule(pattern3430, replacement3430)
pattern3431 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons89, cons1170)
def replacement3431(m, f, b, a, n, x, e):
rubi.append(3431)
return -Dist((m + n + S(1))/(b**S(2)*(n + S(1))), Int((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x)
rule3431 = ReplacementRule(pattern3431, replacement3431)
pattern3432 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons89, cons1170)
def replacement3432(m, f, b, a, n, x, e):
rubi.append(3432)
return -Dist((m + n + S(1))/(b**S(2)*(n + S(1))), Int((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(2)), x), x) - Simp((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x)
rule3432 = ReplacementRule(pattern3432, replacement3432)
pattern3433 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons165, cons1518, cons1170)
def replacement3433(m, f, b, a, n, x, e):
rubi.append(3433)
return -Dist(b**S(2)*(n + S(-1))/(a**S(2)*m), Int((a/cos(e + f*x))**(m + S(2))*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3433 = ReplacementRule(pattern3433, replacement3433)
pattern3434 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons165, cons1518, cons1170)
def replacement3434(m, f, b, a, n, x, e):
rubi.append(3434)
return -Dist(b**S(2)*(n + S(-1))/(a**S(2)*m), Int((a/sin(e + f*x))**(m + S(2))*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3434 = ReplacementRule(pattern3434, replacement3434)
pattern3435 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons165, cons1519, cons1170)
def replacement3435(m, f, b, a, n, x, e):
rubi.append(3435)
return -Dist(b**S(2)*(n + S(-1))/(m + n + S(-1)), Int((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3435 = ReplacementRule(pattern3435, replacement3435)
pattern3436 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons165, cons1519, cons1170)
def replacement3436(m, f, b, a, n, x, e):
rubi.append(3436)
return -Dist(b**S(2)*(n + S(-1))/(m + n + S(-1)), Int((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3436 = ReplacementRule(pattern3436, replacement3436)
pattern3437 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons1520, cons1170)
def replacement3437(m, f, b, a, n, x, e):
rubi.append(3437)
return Dist((m + n + S(1))/(a**S(2)*m), Int((a/cos(e + f*x))**(m + S(2))*(b*tan(e + f*x))**n, x), x) - Simp((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*m), x)
rule3437 = ReplacementRule(pattern3437, replacement3437)
pattern3438 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons1520, cons1170)
def replacement3438(m, f, b, a, n, x, e):
rubi.append(3438)
return Dist((m + n + S(1))/(a**S(2)*m), Int((a/sin(e + f*x))**(m + S(2))*(b/tan(e + f*x))**n, x), x) + Simp((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*m), x)
rule3438 = ReplacementRule(pattern3438, replacement3438)
pattern3439 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons1521, cons1519, cons1170)
def replacement3439(m, f, b, a, n, x, e):
rubi.append(3439)
return Dist(a**S(2)*(m + S(-2))/(m + n + S(-1)), Int((a/cos(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**n, x), x) + Simp(a**S(2)*(a/cos(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(-1))), x)
rule3439 = ReplacementRule(pattern3439, replacement3439)
pattern3440 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons1521, cons1519, cons1170)
def replacement3440(m, f, b, a, n, x, e):
rubi.append(3440)
return Dist(a**S(2)*(m + S(-2))/(m + n + S(-1)), Int((a/sin(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**n, x), x) - Simp(a**S(2)*(a/sin(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(-1))), x)
rule3440 = ReplacementRule(pattern3440, replacement3440)
pattern3441 = Pattern(Integral(S(1)/(sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons48, cons125, cons1522)
def replacement3441(x, f, b, e):
rubi.append(3441)
return Dist(sqrt(sin(e + f*x))/(sqrt(b*tan(e + f*x))*sqrt(cos(e + f*x))), Int(S(1)/(sqrt(sin(e + f*x))*sqrt(cos(e + f*x))), x), x)
rule3441 = ReplacementRule(pattern3441, replacement3441)
pattern3442 = Pattern(Integral(S(1)/(sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons48, cons125, cons1522)
def replacement3442(x, f, b, e):
rubi.append(3442)
return Dist(sqrt(cos(e + f*x))/(sqrt(b/tan(e + f*x))*sqrt(sin(e + f*x))), Int(S(1)/(sqrt(sin(e + f*x))*sqrt(cos(e + f*x))), x), x)
rule3442 = ReplacementRule(pattern3442, replacement3442)
pattern3443 = Pattern(Integral(sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons3, cons48, cons125, cons1522)
def replacement3443(x, f, b, e):
rubi.append(3443)
return Dist(sqrt(b*tan(e + f*x))*sqrt(cos(e + f*x))/sqrt(sin(e + f*x)), Int(sqrt(sin(e + f*x))*sqrt(cos(e + f*x)), x), x)
rule3443 = ReplacementRule(pattern3443, replacement3443)
pattern3444 = Pattern(Integral(sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons3, cons48, cons125, cons1522)
def replacement3444(x, f, b, e):
rubi.append(3444)
return Dist(sqrt(b/tan(e + f*x))*sqrt(sin(e + f*x))/sqrt(cos(e + f*x)), Int(sqrt(sin(e + f*x))*sqrt(cos(e + f*x)), x), x)
rule3444 = ReplacementRule(pattern3444, replacement3444)
pattern3445 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons80, cons79)
def replacement3445(m, f, b, a, n, x, e):
rubi.append(3445)
return Dist(a**(m + n)*(a/cos(e + f*x))**(-n)*(b*sin(e + f*x))**(-n)*(b*tan(e + f*x))**n, Int((b*sin(e + f*x))**n*cos(e + f*x)**(-m - n), x), x)
rule3445 = ReplacementRule(pattern3445, replacement3445)
pattern3446 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons80, cons79)
def replacement3446(m, f, b, a, n, x, e):
rubi.append(3446)
return Dist(a**(m + n)*(a/sin(e + f*x))**(-n)*(b*cos(e + f*x))**(-n)*(b/tan(e + f*x))**n, Int((b*cos(e + f*x))**n*sin(e + f*x)**(-m - n), x), x)
rule3446 = ReplacementRule(pattern3446, replacement3446)
pattern3447 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1523, cons1524)
def replacement3447(m, f, b, a, n, x, e):
rubi.append(3447)
return Simp((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))*(cos(e + f*x)**S(2))**(m/S(2) + n/S(2) + S(1)/2)*Hypergeometric2F1(n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(e + f*x)**S(2))/(b*f*(n + S(1))), x)
rule3447 = ReplacementRule(pattern3447, replacement3447)
pattern3448 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1523, cons1524)
def replacement3448(m, f, b, a, n, x, e):
rubi.append(3448)
return -Simp((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))*(sin(e + f*x)**S(2))**(m/S(2) + n/S(2) + S(1)/2)*Hypergeometric2F1(n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(e + f*x)**S(2))/(b*f*(n + S(1))), x)
rule3448 = ReplacementRule(pattern3448, replacement3448)
pattern3449 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement3449(m, f, b, a, n, x, e):
rubi.append(3449)
return Dist((sin(e + f*x)/a)**FracPart(m)*(a/sin(e + f*x))**FracPart(m), Int((sin(e + f*x)/a)**(-m)*(b*tan(e + f*x))**n, x), x)
rule3449 = ReplacementRule(pattern3449, replacement3449)
pattern3450 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement3450(m, f, b, a, n, x, e):
rubi.append(3450)
return Dist((cos(e + f*x)/a)**FracPart(m)*(a/cos(e + f*x))**FracPart(m), Int((cos(e + f*x)/a)**(-m)*(b/tan(e + f*x))**n, x), x)
rule3450 = ReplacementRule(pattern3450, replacement3450)
pattern3451 = Pattern(Integral(((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons5, cons87, cons165, cons1525, cons1526)
def replacement3451(p, m, f, b, d, a, n, x, e):
rubi.append(3451)
return -Dist(b**S(2)*(n + S(-1))/(m*p + n + S(-1)), Int((a*(d/cos(e + f*x))**p)**m*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a*(d/cos(e + f*x))**p)**m*(b*tan(e + f*x))**(n + S(-1))/(f*(m*p + n + S(-1))), x)
rule3451 = ReplacementRule(pattern3451, replacement3451)
pattern3452 = Pattern(Integral(((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons5, cons87, cons165, cons1525, cons1526)
def replacement3452(p, m, f, b, d, a, n, x, e):
rubi.append(3452)
return -Dist(b**S(2)*(n + S(-1))/(m*p + n + S(-1)), Int((a*(d/sin(e + f*x))**p)**m*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a*(d/sin(e + f*x))**p)**m*(b/tan(e + f*x))**(n + S(-1))/(f*(m*p + n + S(-1))), x)
rule3452 = ReplacementRule(pattern3452, replacement3452)
pattern3453 = Pattern(Integral(((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons5, cons87, cons89, cons1527, cons1526)
def replacement3453(p, m, f, b, d, a, n, x, e):
rubi.append(3453)
return -Dist((m*p + n + S(1))/(b**S(2)*(n + S(1))), Int((a*(d/cos(e + f*x))**p)**m*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp((a*(d/cos(e + f*x))**p)**m*(b*tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x)
rule3453 = ReplacementRule(pattern3453, replacement3453)
pattern3454 = Pattern(Integral(((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons5, cons87, cons89, cons1527, cons1526)
def replacement3454(p, m, f, b, d, a, n, x, e):
rubi.append(3454)
return -Dist((m*p + n + S(1))/(b**S(2)*(n + S(1))), Int((a*(d/sin(e + f*x))**p)**m*(b/tan(e + f*x))**(n + S(2)), x), x) + Simp((a*(d/sin(e + f*x))**p)**m*(b/tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x)
rule3454 = ReplacementRule(pattern3454, replacement3454)
pattern3455 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons87, cons165)
def replacement3455(b, d, c, n, x):
rubi.append(3455)
return -Dist(b**S(2), Int((b*tan(c + d*x))**(n + S(-2)), x), x) + Simp(b*(b*tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x)
rule3455 = ReplacementRule(pattern3455, replacement3455)
pattern3456 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons87, cons165)
def replacement3456(b, d, c, n, x):
rubi.append(3456)
return -Dist(b**S(2), Int((b/tan(c + d*x))**(n + S(-2)), x), x) - Simp(b*(b/tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x)
rule3456 = ReplacementRule(pattern3456, replacement3456)
pattern3457 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons87, cons89)
def replacement3457(b, d, c, n, x):
rubi.append(3457)
return -Dist(b**(S(-2)), Int((b*tan(c + d*x))**(n + S(2)), x), x) + Simp((b*tan(c + d*x))**(n + S(1))/(b*d*(n + S(1))), x)
rule3457 = ReplacementRule(pattern3457, replacement3457)
pattern3458 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons87, cons89)
def replacement3458(b, d, c, n, x):
rubi.append(3458)
return -Dist(b**(S(-2)), Int((b/tan(c + d*x))**(n + S(2)), x), x) - Simp((b/tan(c + d*x))**(n + S(1))/(b*d*(n + S(1))), x)
rule3458 = ReplacementRule(pattern3458, replacement3458)
pattern3459 = Pattern(Integral(tan(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons1261)
def replacement3459(d, c, x):
rubi.append(3459)
return -Simp(log(RemoveContent(cos(c + d*x), x))/d, x)
rule3459 = ReplacementRule(pattern3459, replacement3459)
pattern3460 = Pattern(Integral(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons1261)
def replacement3460(d, c, x):
rubi.append(3460)
return Simp(log(RemoveContent(sin(c + d*x), x))/d, x)
rule3460 = ReplacementRule(pattern3460, replacement3460)
pattern3461 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons4, cons23)
def replacement3461(b, d, c, n, x):
rubi.append(3461)
return Dist(b/d, Subst(Int(x**n/(b**S(2) + x**S(2)), x), x, b*tan(c + d*x)), x)
rule3461 = ReplacementRule(pattern3461, replacement3461)
pattern3462 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons4, cons23)
def replacement3462(b, d, c, n, x):
rubi.append(3462)
return -Dist(b/d, Subst(Int(x**n/(b**S(2) + x**S(2)), x), x, b/tan(c + d*x)), x)
rule3462 = ReplacementRule(pattern3462, replacement3462)
pattern3463 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement3463(b, d, c, a, x):
rubi.append(3463)
return Dist(S(2)*a*b, Int(tan(c + d*x), x), x) + Simp(x*(a**S(2) - b**S(2)), x) + Simp(b**S(2)*tan(c + d*x)/d, x)
rule3463 = ReplacementRule(pattern3463, replacement3463)
pattern3464 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement3464(b, d, c, a, x):
rubi.append(3464)
return Dist(S(2)*a*b, Int(S(1)/tan(c + d*x), x), x) + Simp(x*(a**S(2) - b**S(2)), x) - Simp(b**S(2)/(d*tan(c + d*x)), x)
rule3464 = ReplacementRule(pattern3464, replacement3464)
pattern3465 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1439, cons87, cons165)
def replacement3465(b, d, c, a, n, x):
rubi.append(3465)
return Dist(S(2)*a, Int((a + b*tan(c + d*x))**(n + S(-1)), x), x) + Simp(b*(a + b*tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x)
rule3465 = ReplacementRule(pattern3465, replacement3465)
pattern3466 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1439, cons87, cons165)
def replacement3466(b, d, c, a, n, x):
rubi.append(3466)
return Dist(S(2)*a, Int((a + b/tan(c + d*x))**(n + S(-1)), x), x) - Simp(b*(a + b/tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x)
rule3466 = ReplacementRule(pattern3466, replacement3466)
pattern3467 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1439, cons87, cons463)
def replacement3467(b, d, c, a, n, x):
rubi.append(3467)
return Dist(S(1)/(S(2)*a), Int((a + b*tan(c + d*x))**(n + S(1)), x), x) + Simp(a*(a + b*tan(c + d*x))**n/(S(2)*b*d*n), x)
rule3467 = ReplacementRule(pattern3467, replacement3467)
pattern3468 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1439, cons87, cons463)
def replacement3468(b, d, c, a, n, x):
rubi.append(3468)
return Dist(S(1)/(S(2)*a), Int((a + b/tan(c + d*x))**(n + S(1)), x), x) - Simp(a*(a + b/tan(c + d*x))**n/(S(2)*b*d*n), x)
rule3468 = ReplacementRule(pattern3468, replacement3468)
pattern3469 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1439)
def replacement3469(b, d, c, a, x):
rubi.append(3469)
return Dist(-S(2)*b/d, Subst(Int(S(1)/(S(2)*a - x**S(2)), x), x, sqrt(a + b*tan(c + d*x))), x)
rule3469 = ReplacementRule(pattern3469, replacement3469)
pattern3470 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1439)
def replacement3470(b, d, c, a, x):
rubi.append(3470)
return Dist(S(2)*b/d, Subst(Int(S(1)/(S(2)*a - x**S(2)), x), x, sqrt(a + b/tan(c + d*x))), x)
rule3470 = ReplacementRule(pattern3470, replacement3470)
pattern3471 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1439)
def replacement3471(b, d, c, a, n, x):
rubi.append(3471)
return -Dist(b/d, Subst(Int((a + x)**(n + S(-1))/(a - x), x), x, b*tan(c + d*x)), x)
rule3471 = ReplacementRule(pattern3471, replacement3471)
pattern3472 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1439)
def replacement3472(b, d, c, a, n, x):
rubi.append(3472)
return Dist(b/d, Subst(Int((a + x)**(n + S(-1))/(a - x), x), x, b/tan(c + d*x)), x)
rule3472 = ReplacementRule(pattern3472, replacement3472)
pattern3473 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1440, cons87, cons165)
def replacement3473(b, d, c, a, n, x):
rubi.append(3473)
return Int((a + b*tan(c + d*x))**(n + S(-2))*(a**S(2) + S(2)*a*b*tan(c + d*x) - b**S(2)), x) + Simp(b*(a + b*tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x)
rule3473 = ReplacementRule(pattern3473, replacement3473)
pattern3474 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1440, cons87, cons165)
def replacement3474(b, d, c, a, n, x):
rubi.append(3474)
return Int((a + b/tan(c + d*x))**(n + S(-2))*(a**S(2) + S(2)*a*b/tan(c + d*x) - b**S(2)), x) - Simp(b*(a + b/tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x)
rule3474 = ReplacementRule(pattern3474, replacement3474)
pattern3475 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1440, cons87, cons89)
def replacement3475(b, d, c, a, n, x):
rubi.append(3475)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a - b*tan(c + d*x))*(a + b*tan(c + d*x))**(n + S(1)), x), x) + Simp(b*(a + b*tan(c + d*x))**(n + S(1))/(d*(a**S(2) + b**S(2))*(n + S(1))), x)
rule3475 = ReplacementRule(pattern3475, replacement3475)
pattern3476 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1440, cons87, cons89)
def replacement3476(b, d, c, a, n, x):
rubi.append(3476)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a - b/tan(c + d*x))*(a + b/tan(c + d*x))**(n + S(1)), x), x) - Simp(b*(a + b/tan(c + d*x))**(n + S(1))/(d*(a**S(2) + b**S(2))*(n + S(1))), x)
rule3476 = ReplacementRule(pattern3476, replacement3476)
pattern3477 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440)
def replacement3477(b, d, c, a, x):
rubi.append(3477)
return Dist(b/(a**S(2) + b**S(2)), Int((-a*tan(c + d*x) + b)/(a + b*tan(c + d*x)), x), x) + Simp(a*x/(a**S(2) + b**S(2)), x)
rule3477 = ReplacementRule(pattern3477, replacement3477)
pattern3478 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440)
def replacement3478(b, d, c, a, x):
rubi.append(3478)
return Dist(b/(a**S(2) + b**S(2)), Int((-a/tan(c + d*x) + b)/(a + b/tan(c + d*x)), x), x) + Simp(a*x/(a**S(2) + b**S(2)), x)
rule3478 = ReplacementRule(pattern3478, replacement3478)
pattern3479 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1440)
def replacement3479(b, d, c, a, n, x):
rubi.append(3479)
return Dist(b/d, Subst(Int((a + x)**n/(b**S(2) + x**S(2)), x), x, b*tan(c + d*x)), x)
rule3479 = ReplacementRule(pattern3479, replacement3479)
pattern3480 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1440)
def replacement3480(b, d, c, a, n, x):
rubi.append(3480)
return -Dist(b/d, Subst(Int((a + x)**n/(b**S(2) + x**S(2)), x), x, b/tan(c + d*x)), x)
rule3480 = ReplacementRule(pattern3480, replacement3480)
pattern3481 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1528)
def replacement3481(m, f, b, d, a, x, e):
rubi.append(3481)
return Dist(a, Int((d/cos(e + f*x))**m, x), x) + Simp(b*(d/cos(e + f*x))**m/(f*m), x)
rule3481 = ReplacementRule(pattern3481, replacement3481)
pattern3482 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1528)
def replacement3482(m, f, b, d, a, x, e):
rubi.append(3482)
return Dist(a, Int((d/sin(e + f*x))**m, x), x) - Simp(b*(d/sin(e + f*x))**m/(f*m), x)
rule3482 = ReplacementRule(pattern3482, replacement3482)
pattern3483 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons4, cons1439, cons1515)
def replacement3483(m, f, b, a, n, x, e):
rubi.append(3483)
return Dist(a**(-m + S(2))/(b*f), Subst(Int((a - x)**(m/S(2) + S(-1))*(a + x)**(m/S(2) + n + S(-1)), x), x, b*tan(e + f*x)), x)
rule3483 = ReplacementRule(pattern3483, replacement3483)
pattern3484 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons4, cons1439, cons1515)
def replacement3484(m, f, b, a, n, x, e):
rubi.append(3484)
return -Dist(a**(-m + S(2))/(b*f), Subst(Int((a - x)**(m/S(2) + S(-1))*(a + x)**(m/S(2) + n + S(-1)), x), x, b/tan(e + f*x)), x)
rule3484 = ReplacementRule(pattern3484, replacement3484)
pattern3485 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1255)
def replacement3485(m, f, b, d, a, n, x, e):
rubi.append(3485)
return Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(a*f*m), x)
rule3485 = ReplacementRule(pattern3485, replacement3485)
pattern3486 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1255)
def replacement3486(m, f, b, d, a, n, x, e):
rubi.append(3486)
return -Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(a*f*m), x)
rule3486 = ReplacementRule(pattern3486, replacement3486)
pattern3487 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1439)
def replacement3487(f, b, a, x, e):
rubi.append(3487)
return Dist(-S(2)*a/(b*f), Subst(Int(S(1)/(-a*x**S(2) + S(2)), x), x, S(1)/(sqrt(a + b*tan(e + f*x))*cos(e + f*x))), x)
rule3487 = ReplacementRule(pattern3487, replacement3487)
pattern3488 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1439)
def replacement3488(f, b, a, x, e):
rubi.append(3488)
return Dist(S(2)*a/(b*f), Subst(Int(S(1)/(-a*x**S(2) + S(2)), x), x, S(1)/(sqrt(a + b/tan(e + f*x))*sin(e + f*x))), x)
rule3488 = ReplacementRule(pattern3488, replacement3488)
pattern3489 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons1529, cons93, cons88)
def replacement3489(m, f, b, d, a, n, x, e):
rubi.append(3489)
return Dist(a/(S(2)*d**S(2)), Int((d/cos(e + f*x))**(m + S(2))*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(a*f*m), x)
rule3489 = ReplacementRule(pattern3489, replacement3489)
pattern3490 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons1529, cons93, cons88)
def replacement3490(m, f, b, d, a, n, x, e):
rubi.append(3490)
return Dist(a/(S(2)*d**S(2)), Int((d/sin(e + f*x))**(m + S(2))*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(a*f*m), x)
rule3490 = ReplacementRule(pattern3490, replacement3490)
pattern3491 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons1529, cons93, cons89)
def replacement3491(m, f, b, d, a, n, x, e):
rubi.append(3491)
return Dist(S(2)*d**S(2)/a, Int((d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*d**S(2)*(d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1))/(b*f*(m + S(-2))), x)
rule3491 = ReplacementRule(pattern3491, replacement3491)
pattern3492 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons1529, cons93, cons89)
def replacement3492(m, f, b, d, a, n, x, e):
rubi.append(3492)
return Dist(S(2)*d**S(2)/a, Int((d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*d**S(2)*(d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1))/(b*f*(m + S(-2))), x)
rule3492 = ReplacementRule(pattern3492, replacement3492)
pattern3493 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1529)
def replacement3493(m, f, b, d, a, n, x, e):
rubi.append(3493)
return Dist((a/d)**(S(2)*IntPart(n))*(d/cos(e + f*x))**(-S(2)*FracPart(n))*(a - b*tan(e + f*x))**FracPart(n)*(a + b*tan(e + f*x))**FracPart(n), Int((a - b*tan(e + f*x))**(-n), x), x)
rule3493 = ReplacementRule(pattern3493, replacement3493)
pattern3494 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1529)
def replacement3494(m, f, b, d, a, n, x, e):
rubi.append(3494)
return Dist((a/d)**(S(2)*IntPart(n))*(d/sin(e + f*x))**(-S(2)*FracPart(n))*(a - b/tan(e + f*x))**FracPart(n)*(a + b/tan(e + f*x))**FracPart(n), Int((a - b/tan(e + f*x))**(-n), x), x)
rule3494 = ReplacementRule(pattern3494, replacement3494)
pattern3495 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1530)
def replacement3495(m, f, b, d, a, n, x, e):
rubi.append(3495)
return Simp(S(2)*b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3495 = ReplacementRule(pattern3495, replacement3495)
pattern3496 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1530)
def replacement3496(m, f, b, d, a, n, x, e):
rubi.append(3496)
return Simp(-S(2)*b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3496 = ReplacementRule(pattern3496, replacement3496)
pattern3497 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1531, cons23)
def replacement3497(m, f, b, d, a, n, x, e):
rubi.append(3497)
return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3497 = ReplacementRule(pattern3497, replacement3497)
pattern3498 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1531, cons23)
def replacement3498(m, f, b, d, a, n, x, e):
rubi.append(3498)
return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3498 = ReplacementRule(pattern3498, replacement3498)
pattern3499 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1439)
def replacement3499(f, b, d, a, x, e):
rubi.append(3499)
return Dist(-S(4)*b*d**S(2)/f, Subst(Int(x**S(2)/(a**S(2) + d**S(2)*x**S(4)), x), x, sqrt(a + b*tan(e + f*x))/sqrt(d/cos(e + f*x))), x)
rule3499 = ReplacementRule(pattern3499, replacement3499)
pattern3500 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1439)
def replacement3500(f, b, d, a, x, e):
rubi.append(3500)
return Dist(S(4)*b*d**S(2)/f, Subst(Int(x**S(2)/(a**S(2) + d**S(2)*x**S(4)), x), x, sqrt(a + b/tan(e + f*x))/sqrt(d/sin(e + f*x))), x)
rule3500 = ReplacementRule(pattern3500, replacement3500)
pattern3501 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons93, cons165, cons1532, cons515)
def replacement3501(m, f, b, d, a, n, x, e):
rubi.append(3501)
return -Dist(b**S(2)*(m + S(2)*n + S(-2))/(d**S(2)*m), Int((d/cos(e + f*x))**(m + S(2))*(a + b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(S(2)*b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3501 = ReplacementRule(pattern3501, replacement3501)
pattern3502 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons93, cons165, cons1533, cons515)
def replacement3502(m, f, b, d, a, n, x, e):
rubi.append(3502)
return -Dist(b**S(2)*(m + S(2)*n + S(-2))/(d**S(2)*m), Int((d/sin(e + f*x))**(m + S(2))*(a + b/tan(e + f*x))**(n + S(-2)), x), x) + Simp(-S(2)*b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*m), x)
rule3502 = ReplacementRule(pattern3502, replacement3502)
pattern3503 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons93, cons88, cons94, cons1170)
def replacement3503(m, f, b, d, a, n, x, e):
rubi.append(3503)
return Dist(a*(m + n)/(d**S(2)*m), Int((d/cos(e + f*x))**(m + S(2))*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(a*f*m), x)
rule3503 = ReplacementRule(pattern3503, replacement3503)
pattern3504 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons93, cons88, cons94, cons1170)
def replacement3504(m, f, b, d, a, n, x, e):
rubi.append(3504)
return Dist(a*(m + n)/(d**S(2)*m), Int((d/sin(e + f*x))**(m + S(2))*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(a*f*m), x)
rule3504 = ReplacementRule(pattern3504, replacement3504)
pattern3505 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1439, cons87, cons88, cons1519, cons1170)
def replacement3505(m, f, b, d, a, n, x, e):
rubi.append(3505)
return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3505 = ReplacementRule(pattern3505, replacement3505)
pattern3506 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1439, cons87, cons88, cons1519, cons1170)
def replacement3506(m, f, b, d, a, n, x, e):
rubi.append(3506)
return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3506 = ReplacementRule(pattern3506, replacement3506)
pattern3507 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1439)
def replacement3507(f, b, d, a, x, e):
rubi.append(3507)
return Dist(d/(sqrt(a - b*tan(e + f*x))*sqrt(a + b*tan(e + f*x))*cos(e + f*x)), Int(sqrt(d/cos(e + f*x))*sqrt(a - b*tan(e + f*x)), x), x)
rule3507 = ReplacementRule(pattern3507, replacement3507)
pattern3508 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1439)
def replacement3508(f, b, d, a, x, e):
rubi.append(3508)
return Dist(d/(sqrt(a - b/tan(e + f*x))*sqrt(a + b/tan(e + f*x))*sin(e + f*x)), Int(sqrt(d/sin(e + f*x))*sqrt(a - b/tan(e + f*x)), x), x)
rule3508 = ReplacementRule(pattern3508, replacement3508)
pattern3509 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1439, cons87, cons89, cons1534, cons515)
def replacement3509(m, f, b, d, a, n, x, e):
rubi.append(3509)
return -Dist(d**S(2)*(m + S(-2))/(b**S(2)*(m + S(2)*n)), Int((d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(2)), x), x) + Simp(S(2)*d**S(2)*(d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1))/(b*f*(m + S(2)*n)), x)
rule3509 = ReplacementRule(pattern3509, replacement3509)
pattern3510 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1439, cons87, cons89, cons1534, cons515)
def replacement3510(m, f, b, d, a, n, x, e):
rubi.append(3510)
return -Dist(d**S(2)*(m + S(-2))/(b**S(2)*(m + S(2)*n)), Int((d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(2)), x), x) + Simp(-S(2)*d**S(2)*(d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1))/(b*f*(m + S(2)*n)), x)
rule3510 = ReplacementRule(pattern3510, replacement3510)
pattern3511 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons93, cons463, cons166, cons1535, cons1519, cons1170)
def replacement3511(m, f, b, d, a, n, x, e):
rubi.append(3511)
return Dist(d**S(2)*(m + S(-2))/(a*(m + n + S(-1))), Int((d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1)), x), x) + Simp(d**S(2)*(d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(-1))), x)
rule3511 = ReplacementRule(pattern3511, replacement3511)
pattern3512 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons1439, cons93, cons463, cons166, cons1535, cons1519, cons1170)
def replacement3512(m, f, b, d, a, n, x, e):
rubi.append(3512)
return Dist(d**S(2)*(m + S(-2))/(a*(m + n + S(-1))), Int((d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1)), x), x) - Simp(d**S(2)*(d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(-1))), x)
rule3512 = ReplacementRule(pattern3512, replacement3512)
pattern3513 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1439, cons87, cons463, cons1536, cons1170)
def replacement3513(m, f, b, d, a, n, x, e):
rubi.append(3513)
return Dist((m + n)/(a*(m + S(2)*n)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(1)), x), x) + Simp(a*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(b*f*(m + S(2)*n)), x)
rule3513 = ReplacementRule(pattern3513, replacement3513)
pattern3514 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1439, cons87, cons463, cons1536, cons1170)
def replacement3514(m, f, b, d, a, n, x, e):
rubi.append(3514)
return Dist((m + n)/(a*(m + S(2)*n)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(1)), x), x) - Simp(a*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(b*f*(m + S(2)*n)), x)
rule3514 = ReplacementRule(pattern3514, replacement3514)
pattern3515 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1537, cons87)
def replacement3515(m, f, b, d, a, n, x, e):
rubi.append(3515)
return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3515 = ReplacementRule(pattern3515, replacement3515)
pattern3516 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1537, cons87)
def replacement3516(m, f, b, d, a, n, x, e):
rubi.append(3516)
return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3516 = ReplacementRule(pattern3516, replacement3516)
pattern3517 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1538, cons1536)
def replacement3517(m, f, b, d, a, n, x, e):
rubi.append(3517)
return Dist((m + n)/(a*(m + S(2)*n)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(1)), x), x) + Simp(a*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(b*f*(m + S(2)*n)), x)
rule3517 = ReplacementRule(pattern3517, replacement3517)
pattern3518 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439, cons1538, cons1536)
def replacement3518(m, f, b, d, a, n, x, e):
rubi.append(3518)
return Dist((m + n)/(a*(m + S(2)*n)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(1)), x), x) - Simp(a*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(b*f*(m + S(2)*n)), x)
rule3518 = ReplacementRule(pattern3518, replacement3518)
pattern3519 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439)
def replacement3519(m, f, b, d, a, n, x, e):
rubi.append(3519)
return Dist((d/cos(e + f*x))**m*(a - b*tan(e + f*x))**(-m/S(2))*(a + b*tan(e + f*x))**(-m/S(2)), Int((a - b*tan(e + f*x))**(m/S(2))*(a + b*tan(e + f*x))**(m/S(2) + n), x), x)
rule3519 = ReplacementRule(pattern3519, replacement3519)
pattern3520 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1439)
def replacement3520(m, f, b, d, a, n, x, e):
rubi.append(3520)
return Dist((d/sin(e + f*x))**m*(a - b/tan(e + f*x))**(-m/S(2))*(a + b/tan(e + f*x))**(-m/S(2)), Int((a - b/tan(e + f*x))**(m/S(2))*(a + b/tan(e + f*x))**(m/S(2) + n), x), x)
rule3520 = ReplacementRule(pattern3520, replacement3520)
pattern3521 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons4, cons1440, cons1515)
def replacement3521(m, f, b, a, n, x, e):
rubi.append(3521)
return Dist(S(1)/(b*f), Subst(Int((S(1) + x**S(2)/b**S(2))**(m/S(2) + S(-1))*(a + x)**n, x), x, b*tan(e + f*x)), x)
rule3521 = ReplacementRule(pattern3521, replacement3521)
pattern3522 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons4, cons1440, cons1515)
def replacement3522(m, f, b, a, n, x, e):
rubi.append(3522)
return -Dist(S(1)/(b*f), Subst(Int((S(1) + x**S(2)/b**S(2))**(m/S(2) + S(-1))*(a + x)**n, x), x, b/tan(e + f*x)), x)
rule3522 = ReplacementRule(pattern3522, replacement3522)
pattern3523 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1440)
def replacement3523(f, b, a, x, e):
rubi.append(3523)
return Simp(b**S(2)*atanh(sin(e + f*x))/f, x) + Simp((a**S(2) - b**S(2))*sin(e + f*x)/f, x) - Simp(S(2)*a*b*cos(e + f*x)/f, x)
rule3523 = ReplacementRule(pattern3523, replacement3523)
pattern3524 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)*tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1440)
def replacement3524(f, b, a, x, e):
rubi.append(3524)
return -Simp(b**S(2)*atanh(cos(e + f*x))/f, x) - Simp((a**S(2) - b**S(2))*cos(e + f*x)/f, x) + Simp(S(2)*a*b*sin(e + f*x)/f, x)
rule3524 = ReplacementRule(pattern3524, replacement3524)
pattern3525 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1440, cons66)
def replacement3525(m, f, b, d, a, x, e):
rubi.append(3525)
return Dist(S(1)/(m + S(1)), Int((d/cos(e + f*x))**m*(a**S(2)*(m + S(1)) + a*b*(m + S(2))*tan(e + f*x) - b**S(2)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))/(f*(m + S(1))), x)
rule3525 = ReplacementRule(pattern3525, replacement3525)
pattern3526 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1440, cons66)
def replacement3526(m, f, b, d, a, x, e):
rubi.append(3526)
return Dist(S(1)/(m + S(1)), Int((d/sin(e + f*x))**m*(a**S(2)*(m + S(1)) + a*b*(m + S(2))/tan(e + f*x) - b**S(2)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))/(f*(m + S(1))), x)
rule3526 = ReplacementRule(pattern3526, replacement3526)
pattern3527 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1440)
def replacement3527(f, b, a, x, e):
rubi.append(3527)
return -Dist(S(1)/f, Subst(Int(S(1)/(a**S(2) + b**S(2) - x**S(2)), x), x, (-a*tan(e + f*x) + b)*cos(e + f*x)), x)
rule3527 = ReplacementRule(pattern3527, replacement3527)
pattern3528 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1440)
def replacement3528(f, b, a, x, e):
rubi.append(3528)
return Dist(S(1)/f, Subst(Int(S(1)/(a**S(2) + b**S(2) - x**S(2)), x), x, (-a/tan(e + f*x) + b)*sin(e + f*x)), x)
rule3528 = ReplacementRule(pattern3528, replacement3528)
pattern3529 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1440, cons1539)
def replacement3529(m, f, b, d, a, x, e):
rubi.append(3529)
return -Dist(d**S(2)/b**S(2), Int((d/cos(e + f*x))**(m + S(-2))*(a - b*tan(e + f*x)), x), x) + Dist(d**S(2)*(a**S(2) + b**S(2))/b**S(2), Int((d/cos(e + f*x))**(m + S(-2))/(a + b*tan(e + f*x)), x), x)
rule3529 = ReplacementRule(pattern3529, replacement3529)
pattern3530 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1440, cons1539)
def replacement3530(m, f, b, d, a, x, e):
rubi.append(3530)
return -Dist(d**S(2)/b**S(2), Int((d/sin(e + f*x))**(m + S(-2))*(a - b/tan(e + f*x)), x), x) + Dist(d**S(2)*(a**S(2) + b**S(2))/b**S(2), Int((d/sin(e + f*x))**(m + S(-2))/(a + b/tan(e + f*x)), x), x)
rule3530 = ReplacementRule(pattern3530, replacement3530)
pattern3531 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1440, cons84)
def replacement3531(m, f, b, d, a, x, e):
rubi.append(3531)
return Dist(b**S(2)/(d**S(2)*(a**S(2) + b**S(2))), Int((d/cos(e + f*x))**(m + S(2))/(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((d/cos(e + f*x))**m*(a - b*tan(e + f*x)), x), x)
rule3531 = ReplacementRule(pattern3531, replacement3531)
pattern3532 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1440, cons84)
def replacement3532(m, f, b, d, a, x, e):
rubi.append(3532)
return Dist(b**S(2)/(d**S(2)*(a**S(2) + b**S(2))), Int((d/sin(e + f*x))**(m + S(2))/(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((d/sin(e + f*x))**m*(a - b/tan(e + f*x)), x), x)
rule3532 = ReplacementRule(pattern3532, replacement3532)
pattern3533 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1440, cons1524)
def replacement3533(m, f, b, d, a, n, x, e):
rubi.append(3533)
return Dist(d**(S(2)*IntPart(m/S(2)))*(d/cos(e + f*x))**(S(2)*FracPart(m/S(2)))*(cos(e + f*x)**(S(-2)))**(-FracPart(m/S(2)))/(b*f), Subst(Int((S(1) + x**S(2)/b**S(2))**(m/S(2) + S(-1))*(a + x)**n, x), x, b*tan(e + f*x)), x)
rule3533 = ReplacementRule(pattern3533, replacement3533)
pattern3534 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1440, cons1524)
def replacement3534(m, f, b, d, a, n, x, e):
rubi.append(3534)
return -Dist(d**(S(2)*IntPart(m/S(2)))*(d/sin(e + f*x))**(S(2)*FracPart(m/S(2)))*(sin(e + f*x)**(S(-2)))**(-FracPart(m/S(2)))/(b*f), Subst(Int((S(1) + x**S(2)/b**S(2))**(m/S(2) + S(-1))*(a + x)**n, x), x, b/tan(e + f*x)), x)
rule3534 = ReplacementRule(pattern3534, replacement3534)
pattern3535 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1439)
def replacement3535(f, b, d, a, x, e):
rubi.append(3535)
return Dist(-S(4)*b/f, Subst(Int(x**S(2)/(a**S(2)*d**S(2) + x**S(4)), x), x, sqrt(d*cos(e + f*x))*sqrt(a + b*tan(e + f*x))), x)
rule3535 = ReplacementRule(pattern3535, replacement3535)
pattern3536 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1439)
def replacement3536(f, b, d, a, x, e):
rubi.append(3536)
return Dist(S(4)*b/f, Subst(Int(x**S(2)/(a**S(2)*d**S(2) + x**S(4)), x), x, sqrt(d*sin(e + f*x))*sqrt(a + b/tan(e + f*x))), x)
rule3536 = ReplacementRule(pattern3536, replacement3536)
pattern3537 = Pattern(Integral(S(1)/((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1439)
def replacement3537(f, b, d, a, x, e):
rubi.append(3537)
return Dist(S(1)/(d*sqrt(a - b*tan(e + f*x))*sqrt(a + b*tan(e + f*x))*cos(e + f*x)), Int(sqrt(a - b*tan(e + f*x))/sqrt(d*cos(e + f*x)), x), x)
rule3537 = ReplacementRule(pattern3537, replacement3537)
pattern3538 = Pattern(Integral(S(1)/((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1439)
def replacement3538(f, b, d, a, x, e):
rubi.append(3538)
return Dist(S(1)/(d*sqrt(a - b/tan(e + f*x))*sqrt(a + b/tan(e + f*x))*sin(e + f*x)), Int(sqrt(a - b/tan(e + f*x))/sqrt(d*sin(e + f*x)), x), x)
rule3538 = ReplacementRule(pattern3538, replacement3538)
pattern3539 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons18)
def replacement3539(m, f, b, d, a, n, x, e):
rubi.append(3539)
return Dist((d/cos(e + f*x))**m*(d*cos(e + f*x))**m, Int((d/cos(e + f*x))**(-m)*(a + b*tan(e + f*x))**n, x), x)
rule3539 = ReplacementRule(pattern3539, replacement3539)
pattern3540 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons18)
def replacement3540(m, f, b, d, a, n, x, e):
rubi.append(3540)
return Dist((d/sin(e + f*x))**m*(d*sin(e + f*x))**m, Int((d/sin(e + f*x))**(-m)*(a + b/tan(e + f*x))**n, x), x)
rule3540 = ReplacementRule(pattern3540, replacement3540)
pattern3541 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**m_, x_), cons2, cons3, cons48, cons125, cons4, cons1515)
def replacement3541(m, f, b, a, n, x, e):
rubi.append(3541)
return Dist(b/f, Subst(Int(x**m*(a + x)**n*(b**S(2) + x**S(2))**(-m/S(2) + S(-1)), x), x, b*tan(e + f*x)), x)
rule3541 = ReplacementRule(pattern3541, replacement3541)
pattern3542 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**m_, x_), cons2, cons3, cons48, cons125, cons4, cons1515)
def replacement3542(m, f, b, a, n, x, e):
rubi.append(3542)
return -Dist(b/f, Subst(Int(x**m*(a + x)**n*(b**S(2) + x**S(2))**(-m/S(2) + S(-1)), x), x, b/tan(e + f*x)), x)
rule3542 = ReplacementRule(pattern3542, replacement3542)
pattern3543 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons1228, cons148)
def replacement3543(m, f, b, a, n, x, e):
rubi.append(3543)
return Int((a + b*tan(e + f*x))**n*sin(e + f*x)**m, x)
rule3543 = ReplacementRule(pattern3543, replacement3543)
pattern3544 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons1228, cons148)
def replacement3544(m, f, b, a, n, x, e):
rubi.append(3544)
return Int((a + b/tan(e + f*x))**n*cos(e + f*x)**m, x)
rule3544 = ReplacementRule(pattern3544, replacement3544)
pattern3545 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons1228, cons196, cons1540)
def replacement3545(m, f, b, a, n, x, e):
rubi.append(3545)
return Int((a*cos(e + f*x) + b*sin(e + f*x))**n*sin(e + f*x)**m*cos(e + f*x)**(-n), x)
rule3545 = ReplacementRule(pattern3545, replacement3545)
pattern3546 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons1228, cons196, cons1540)
def replacement3546(m, f, b, a, n, x, e):
rubi.append(3546)
return Int((a*sin(e + f*x) + b*cos(e + f*x))**n*sin(e + f*x)**(-n)*cos(e + f*x)**m, x)
rule3546 = ReplacementRule(pattern3546, replacement3546)
pattern3547 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons18)
def replacement3547(m, f, b, d, a, n, x, e):
rubi.append(3547)
return Dist((sin(e + f*x)/d)**FracPart(m)*(d/sin(e + f*x))**FracPart(m), Int((sin(e + f*x)/d)**(-m)*(a + b*tan(e + f*x))**n, x), x)
rule3547 = ReplacementRule(pattern3547, replacement3547)
pattern3548 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons18)
def replacement3548(m, f, b, d, a, n, x, e):
rubi.append(3548)
return Dist((cos(e + f*x)/d)**FracPart(m)*(d/cos(e + f*x))**FracPart(m), Int((cos(e + f*x)/d)**(-m)*(a + b/tan(e + f*x))**n, x), x)
rule3548 = ReplacementRule(pattern3548, replacement3548)
pattern3549 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons5, cons85)
def replacement3549(p, m, f, b, a, n, x, e):
rubi.append(3549)
return Int((a*cos(e + f*x) + b*sin(e + f*x))**n*sin(e + f*x)**p*cos(e + f*x)**(m - n), x)
rule3549 = ReplacementRule(pattern3549, replacement3549)
pattern3550 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons5, cons85)
def replacement3550(p, m, f, b, a, n, x, e):
rubi.append(3550)
return Int((a*sin(e + f*x) + b*cos(e + f*x))**n*sin(e + f*x)**(m - n)*cos(e + f*x)**p, x)
rule3550 = ReplacementRule(pattern3550, replacement3550)
pattern3551 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1439, cons17, cons1541)
def replacement3551(m, f, b, d, a, n, c, x, e):
rubi.append(3551)
return Dist(a**m*c**m, Int((c + d*tan(e + f*x))**(-m + n)*(S(1)/cos(e + f*x))**(S(2)*m), x), x)
rule3551 = ReplacementRule(pattern3551, replacement3551)
pattern3552 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1439, cons17, cons1541)
def replacement3552(m, f, b, d, a, n, c, x, e):
rubi.append(3552)
return Dist(a**m*c**m, Int((c + d/tan(e + f*x))**(-m + n)*(S(1)/sin(e + f*x))**(S(2)*m), x), x)
rule3552 = ReplacementRule(pattern3552, replacement3552)
pattern3553 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1439)
def replacement3553(m, f, b, d, a, c, n, x, e):
rubi.append(3553)
return Dist(a*c/f, Subst(Int((a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1)), x), x, tan(e + f*x)), x)
rule3553 = ReplacementRule(pattern3553, replacement3553)
pattern3554 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1439)
def replacement3554(m, f, b, d, a, c, n, x, e):
rubi.append(3554)
return -Dist(a*c/f, Subst(Int((a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1)), x), x, S(1)/tan(e + f*x)), x)
rule3554 = ReplacementRule(pattern3554, replacement3554)
pattern3555 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons70)
def replacement3555(f, b, d, a, c, x, e):
rubi.append(3555)
return Simp(x*(a*c - b*d), x) + Simp(b*d*tan(e + f*x)/f, x)
rule3555 = ReplacementRule(pattern3555, replacement3555)
pattern3556 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons70)
def replacement3556(f, b, d, a, c, x, e):
rubi.append(3556)
return Simp(x*(a*c - b*d), x) - Simp(b*d/(f*tan(e + f*x)), x)
rule3556 = ReplacementRule(pattern3556, replacement3556)
pattern3557 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1412)
def replacement3557(f, b, d, c, a, x, e):
rubi.append(3557)
return Dist(a*d + b*c, Int(tan(e + f*x), x), x) + Simp(x*(a*c - b*d), x) + Simp(b*d*tan(e + f*x)/f, x)
rule3557 = ReplacementRule(pattern3557, replacement3557)
pattern3558 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1412)
def replacement3558(f, b, d, c, a, x, e):
rubi.append(3558)
return Dist(a*d + b*c, Int(S(1)/tan(e + f*x), x), x) + Simp(x*(a*c - b*d), x) - Simp(b*d/(f*tan(e + f*x)), x)
rule3558 = ReplacementRule(pattern3558, replacement3558)
pattern3559 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons31, cons267)
def replacement3559(m, f, b, d, c, a, x, e):
rubi.append(3559)
return Dist((a*d + b*c)/(S(2)*a*b), Int((a + b*tan(e + f*x))**(m + S(1)), x), x) - Simp((a + b*tan(e + f*x))**m*(-a*d + b*c)/(S(2)*a*f*m), x)
rule3559 = ReplacementRule(pattern3559, replacement3559)
pattern3560 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons31, cons267)
def replacement3560(m, f, b, d, c, a, x, e):
rubi.append(3560)
return Dist((a*d + b*c)/(S(2)*a*b), Int((a + b/tan(e + f*x))**(m + S(1)), x), x) + Simp((a + b/tan(e + f*x))**m*(-a*d + b*c)/(S(2)*a*f*m), x)
rule3560 = ReplacementRule(pattern3560, replacement3560)
pattern3561 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1439, cons1303)
def replacement3561(m, f, b, d, c, a, x, e):
rubi.append(3561)
return Dist((a*d + b*c)/b, Int((a + b*tan(e + f*x))**m, x), x) + Simp(d*(a + b*tan(e + f*x))**m/(f*m), x)
rule3561 = ReplacementRule(pattern3561, replacement3561)
pattern3562 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1439, cons1303)
def replacement3562(m, f, b, d, c, a, x, e):
rubi.append(3562)
return Dist((a*d + b*c)/b, Int((a + b/tan(e + f*x))**m, x), x) - Simp(d*(a + b/tan(e + f*x))**m/(f*m), x)
rule3562 = ReplacementRule(pattern3562, replacement3562)
pattern3563 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons31, cons168)
def replacement3563(m, f, b, d, c, a, x, e):
rubi.append(3563)
return Int((a + b*tan(e + f*x))**(m + S(-1))*Simp(a*c - b*d + (a*d + b*c)*tan(e + f*x), x), x) + Simp(d*(a + b*tan(e + f*x))**m/(f*m), x)
rule3563 = ReplacementRule(pattern3563, replacement3563)
pattern3564 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons31, cons168)
def replacement3564(m, f, b, d, c, a, x, e):
rubi.append(3564)
return Int((a + b/tan(e + f*x))**(m + S(-1))*Simp(a*c - b*d + (a*d + b*c)/tan(e + f*x), x), x) - Simp(d*(a + b/tan(e + f*x))**m/(f*m), x)
rule3564 = ReplacementRule(pattern3564, replacement3564)
pattern3565 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons31, cons94)
def replacement3565(m, f, b, d, c, a, x, e):
rubi.append(3565)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(a*c + b*d - (-a*d + b*c)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(-a*d + b*c)/(f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3565 = ReplacementRule(pattern3565, replacement3565)
pattern3566 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons31, cons94)
def replacement3566(m, f, b, d, c, a, x, e):
rubi.append(3566)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(a*c + b*d - (-a*d + b*c)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(-a*d + b*c)/(f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3566 = ReplacementRule(pattern3566, replacement3566)
pattern3567 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1542)
def replacement3567(f, b, d, a, c, x, e):
rubi.append(3567)
return Simp(c*log(RemoveContent(a*cos(e + f*x) + b*sin(e + f*x), x))/(b*f), x)
rule3567 = ReplacementRule(pattern3567, replacement3567)
pattern3568 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1542)
def replacement3568(f, b, d, a, c, x, e):
rubi.append(3568)
return -Simp(c*log(RemoveContent(a*sin(e + f*x) + b*cos(e + f*x), x))/(b*f), x)
rule3568 = ReplacementRule(pattern3568, replacement3568)
pattern3569 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1543)
def replacement3569(f, b, d, c, a, x, e):
rubi.append(3569)
return Dist((-a*d + b*c)/(a**S(2) + b**S(2)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) + Simp(x*(a*c + b*d)/(a**S(2) + b**S(2)), x)
rule3569 = ReplacementRule(pattern3569, replacement3569)
pattern3570 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1543)
def replacement3570(f, b, d, c, a, x, e):
rubi.append(3570)
return Dist((-a*d + b*c)/(a**S(2) + b**S(2)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) + Simp(x*(a*c + b*d)/(a**S(2) + b**S(2)), x)
rule3570 = ReplacementRule(pattern3570, replacement3570)
pattern3571 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1322)
def replacement3571(f, b, d, c, x, e):
rubi.append(3571)
return Dist(-S(2)*d**S(2)/f, Subst(Int(S(1)/(b*x**S(2) + S(2)*c*d), x), x, (c - d*tan(e + f*x))/sqrt(b*tan(e + f*x))), x)
rule3571 = ReplacementRule(pattern3571, replacement3571)
pattern3572 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1322)
def replacement3572(f, b, d, c, x, e):
rubi.append(3572)
return Dist(S(2)*d**S(2)/f, Subst(Int(S(1)/(b*x**S(2) + S(2)*c*d), x), x, (c - d/tan(e + f*x))/sqrt(b/tan(e + f*x))), x)
rule3572 = ReplacementRule(pattern3572, replacement3572)
pattern3573 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1544)
def replacement3573(f, b, d, c, x, e):
rubi.append(3573)
return Dist(S(2)*c**S(2)/f, Subst(Int(S(1)/(b*c - d*x**S(2)), x), x, sqrt(b*tan(e + f*x))), x)
rule3573 = ReplacementRule(pattern3573, replacement3573)
pattern3574 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1544)
def replacement3574(f, b, d, c, x, e):
rubi.append(3574)
return Dist(-S(2)*c**S(2)/f, Subst(Int(S(1)/(b*c - d*x**S(2)), x), x, sqrt(b/tan(e + f*x))), x)
rule3574 = ReplacementRule(pattern3574, replacement3574)
pattern3575 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1323, cons1545)
def replacement3575(f, b, d, c, x, e):
rubi.append(3575)
return Dist(S(2)/f, Subst(Int((b*c + d*x**S(2))/(b**S(2) + x**S(4)), x), x, sqrt(b*tan(e + f*x))), x)
rule3575 = ReplacementRule(pattern3575, replacement3575)
pattern3576 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1323, cons1545)
def replacement3576(f, b, d, c, x, e):
rubi.append(3576)
return Dist(-S(2)/f, Subst(Int((b*c + d*x**S(2))/(b**S(2) + x**S(4)), x), x, sqrt(b/tan(e + f*x))), x)
rule3576 = ReplacementRule(pattern3576, replacement3576)
pattern3577 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons1546)
def replacement3577(f, b, d, c, a, x, e):
rubi.append(3577)
return Dist(-S(2)*d**S(2)/f, Subst(Int(S(1)/(-S(4)*a*d**S(2) + S(2)*b*c*d + x**S(2)), x), x, (-S(2)*a*d + b*c - b*d*tan(e + f*x))/sqrt(a + b*tan(e + f*x))), x)
rule3577 = ReplacementRule(pattern3577, replacement3577)
pattern3578 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons1546)
def replacement3578(f, b, d, c, a, x, e):
rubi.append(3578)
return Dist(S(2)*d**S(2)/f, Subst(Int(S(1)/(-S(4)*a*d**S(2) + S(2)*b*c*d + x**S(2)), x), x, (-S(2)*a*d + b*c - b*d/tan(e + f*x))/sqrt(a + b/tan(e + f*x))), x)
rule3578 = ReplacementRule(pattern3578, replacement3578)
def With3579(f, b, d, c, a, x, e):
q = Rt(a**S(2) + b**S(2), S(2))
rubi.append(3579)
return -Dist(S(1)/(S(2)*q), Int((a*c + b*d - c*q + (-a*d + b*c - d*q)*tan(e + f*x))/sqrt(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(S(2)*q), Int((a*c + b*d + c*q + (-a*d + b*c + d*q)*tan(e + f*x))/sqrt(a + b*tan(e + f*x)), x), x)
pattern3579 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons1547, cons1548)
rule3579 = ReplacementRule(pattern3579, With3579)
def With3580(f, b, d, c, a, x, e):
q = Rt(a**S(2) + b**S(2), S(2))
rubi.append(3580)
return -Dist(S(1)/(S(2)*q), Int((a*c + b*d - c*q + (-a*d + b*c - d*q)/tan(e + f*x))/sqrt(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(S(2)*q), Int((a*c + b*d + c*q + (-a*d + b*c + d*q)/tan(e + f*x))/sqrt(a + b/tan(e + f*x)), x), x)
pattern3580 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons1547, cons1548)
rule3580 = ReplacementRule(pattern3580, With3580)
pattern3581 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1440, cons1544)
def replacement3581(m, f, b, d, a, c, x, e):
rubi.append(3581)
return Dist(c*d/f, Subst(Int((a + b*x/d)**m/(c*x + d**S(2)), x), x, d*tan(e + f*x)), x)
rule3581 = ReplacementRule(pattern3581, replacement3581)
pattern3582 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1440, cons1544)
def replacement3582(m, f, b, d, a, c, x, e):
rubi.append(3582)
return -Dist(c*d/f, Subst(Int((a + b*x/d)**m/(c*x + d**S(2)), x), x, d/tan(e + f*x)), x)
rule3582 = ReplacementRule(pattern3582, replacement3582)
pattern3583 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons21, cons1545, cons77)
def replacement3583(m, f, b, d, c, x, e):
rubi.append(3583)
return Dist(c, Int((b*tan(e + f*x))**m, x), x) + Dist(d/b, Int((b*tan(e + f*x))**(m + S(1)), x), x)
rule3583 = ReplacementRule(pattern3583, replacement3583)
pattern3584 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons21, cons1545, cons77)
def replacement3584(m, f, b, d, c, x, e):
rubi.append(3584)
return Dist(c, Int((b/tan(e + f*x))**m, x), x) + Dist(d/b, Int((b/tan(e + f*x))**(m + S(1)), x), x)
rule3584 = ReplacementRule(pattern3584, replacement3584)
pattern3585 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1440, cons1545, cons18)
def replacement3585(m, f, b, d, c, a, x, e):
rubi.append(3585)
return Dist(c/S(2) - I*d/S(2), Int((a + b*tan(e + f*x))**m*(I*tan(e + f*x) + S(1)), x), x) + Dist(c/S(2) + I*d/S(2), Int((a + b*tan(e + f*x))**m*(-I*tan(e + f*x) + S(1)), x), x)
rule3585 = ReplacementRule(pattern3585, replacement3585)
pattern3586 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1440, cons1545, cons18)
def replacement3586(m, f, b, d, c, a, x, e):
rubi.append(3586)
return Dist(c/S(2) - I*d/S(2), Int((S(1) + I/tan(e + f*x))*(a + b/tan(e + f*x))**m, x), x) + Dist(c/S(2) + I*d/S(2), Int((S(1) - I/tan(e + f*x))*(a + b/tan(e + f*x))**m, x), x)
rule3586 = ReplacementRule(pattern3586, replacement3586)
pattern3587 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons32, cons1439)
def replacement3587(m, f, b, d, c, a, x, e):
rubi.append(3587)
return Dist(S(1)/(S(2)*a**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(a*c**S(2) + a*d**S(2) - S(2)*b*c*d - S(2)*b*d**S(2)*tan(e + f*x), x), x), x) - Simp(b*(a + b*tan(e + f*x))**m*(a*c + b*d)**S(2)/(S(2)*a**S(3)*f*m), x)
rule3587 = ReplacementRule(pattern3587, replacement3587)
pattern3588 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons32, cons1439)
def replacement3588(m, f, b, d, c, a, x, e):
rubi.append(3588)
return Dist(S(1)/(S(2)*a**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(a*c**S(2) + a*d**S(2) - S(2)*b*c*d - S(2)*b*d**S(2)/tan(e + f*x), x), x), x) + Simp(b*(a + b/tan(e + f*x))**m*(a*c + b*d)**S(2)/(S(2)*a**S(3)*f*m), x)
rule3588 = ReplacementRule(pattern3588, replacement3588)
pattern3589 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440)
def replacement3589(f, b, d, c, a, x, e):
rubi.append(3589)
return Dist((-a*d + b*c)**S(2)/b**S(2), Int(S(1)/(a + b*tan(e + f*x)), x), x) + Dist(d**S(2)/b, Int(tan(e + f*x), x), x) + Simp(d*x*(-a*d + S(2)*b*c)/b**S(2), x)
rule3589 = ReplacementRule(pattern3589, replacement3589)
pattern3590 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440)
def replacement3590(f, b, d, c, a, x, e):
rubi.append(3590)
return Dist((-a*d + b*c)**S(2)/b**S(2), Int(S(1)/(a + b/tan(e + f*x)), x), x) + Dist(d**S(2)/b, Int(S(1)/tan(e + f*x), x), x) + Simp(d*x*(-a*d + S(2)*b*c)/b**S(2), x)
rule3590 = ReplacementRule(pattern3590, replacement3590)
pattern3591 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons94, cons1440)
def replacement3591(m, f, b, d, c, a, x, e):
rubi.append(3591)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(a*c**S(2) - a*d**S(2) + S(2)*b*c*d - (-S(2)*a*c*d + b*c**S(2) - b*d**S(2))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(-a*d + b*c)**S(2)/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3591 = ReplacementRule(pattern3591, replacement3591)
pattern3592 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons94, cons1440)
def replacement3592(m, f, b, d, c, a, x, e):
rubi.append(3592)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(a*c**S(2) - a*d**S(2) + S(2)*b*c*d - (-S(2)*a*c*d + b*c**S(2) - b*d**S(2))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(-a*d + b*c)**S(2)/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3592 = ReplacementRule(pattern3592, replacement3592)
pattern3593 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1549, cons1550)
def replacement3593(m, f, b, d, c, a, x, e):
rubi.append(3593)
return Int((a + b*tan(e + f*x))**m*Simp(c**S(2) + S(2)*c*d*tan(e + f*x) - d**S(2), x), x) + Simp(d**S(2)*(a + b*tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule3593 = ReplacementRule(pattern3593, replacement3593)
pattern3594 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1549, cons1550)
def replacement3594(m, f, b, d, c, a, x, e):
rubi.append(3594)
return Int((a + b/tan(e + f*x))**m*Simp(c**S(2) + S(2)*c*d/tan(e + f*x) - d**S(2), x), x) - Simp(d**S(2)*(a + b/tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule3594 = ReplacementRule(pattern3594, replacement3594)
pattern3595 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3595(f, b, d, c, a, x, e):
rubi.append(3595)
return Dist(-S(2)*a*b/f, Subst(Int(S(1)/(-S(2)*a**S(2)*x**S(2) + a*c - b*d), x), x, sqrt(c + d*tan(e + f*x))/sqrt(a + b*tan(e + f*x))), x)
rule3595 = ReplacementRule(pattern3595, replacement3595)
pattern3596 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3596(f, b, d, c, a, x, e):
rubi.append(3596)
return Dist(S(2)*a*b/f, Subst(Int(S(1)/(-S(2)*a**S(2)*x**S(2) + a*c - b*d), x), x, sqrt(c + d/tan(e + f*x))/sqrt(a + b/tan(e + f*x))), x)
rule3596 = ReplacementRule(pattern3596, replacement3596)
pattern3597 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons1551, cons1423)
def replacement3597(m, f, b, d, c, a, n, x, e):
rubi.append(3597)
return Dist(S(2)*a**S(2)/(a*c - b*d), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1)), x), x) + Simp(a*b*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))/(f*(m + S(-1))*(a*c - b*d)), x)
rule3597 = ReplacementRule(pattern3597, replacement3597)
pattern3598 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons1551, cons1423)
def replacement3598(m, f, b, d, c, n, a, x, e):
rubi.append(3598)
return Dist(S(2)*a**S(2)/(a*c - b*d), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1)), x), x) - Simp(a*b*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))/(f*(m + S(-1))*(a*c - b*d)), x)
rule3598 = ReplacementRule(pattern3598, replacement3598)
pattern3599 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons1551, cons1385)
def replacement3599(m, f, b, d, c, a, n, x, e):
rubi.append(3599)
return -Dist((a*c - b*d)/(S(2)*b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1)), x), x) + Simp(a*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n/(S(2)*b*f*m), x)
rule3599 = ReplacementRule(pattern3599, replacement3599)
pattern3600 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons1551, cons1385)
def replacement3600(m, f, b, d, c, n, a, x, e):
rubi.append(3600)
return -Dist((a*c - b*d)/(S(2)*b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1)), x), x) - Simp(a*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n/(S(2)*b*f*m), x)
rule3600 = ReplacementRule(pattern3600, replacement3600)
pattern3601 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons111, cons94)
def replacement3601(m, f, b, d, c, a, n, x, e):
rubi.append(3601)
return Dist(S(1)/(S(2)*a), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n, x), x) + Simp(a*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x)
rule3601 = ReplacementRule(pattern3601, replacement3601)
pattern3602 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons111, cons94)
def replacement3602(m, f, b, d, c, n, a, x, e):
rubi.append(3602)
return Dist(S(1)/(S(2)*a), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n, x), x) - Simp(a*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x)
rule3602 = ReplacementRule(pattern3602, replacement3602)
pattern3603 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1439, cons1545, cons155, cons272)
def replacement3603(m, f, b, d, c, a, n, x, e):
rubi.append(3603)
return Dist(a/(a*c - b*d), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1)), x), x) - Simp(d*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(f*m*(c**S(2) + d**S(2))), x)
rule3603 = ReplacementRule(pattern3603, replacement3603)
pattern3604 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1439, cons1545, cons155, cons272)
def replacement3604(m, f, b, d, c, n, a, x, e):
rubi.append(3604)
return Dist(a/(a*c - b*d), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1)), x), x) + Simp(d*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(f*m*(c**S(2) + d**S(2))), x)
rule3604 = ReplacementRule(pattern3604, replacement3604)
pattern3605 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons87, cons1325)
def replacement3605(f, b, d, c, a, n, x, e):
rubi.append(3605)
return Dist(S(1)/(S(2)*a*(-a*d + b*c)), Int((c + d*tan(e + f*x))**(n + S(-1))*Simp(a*c*d*(n + S(-1)) + b*c**S(2) + b*d**S(2)*n - d*(n + S(-1))*(-a*d + b*c)*tan(e + f*x), x), x), x) - Simp((c + d*tan(e + f*x))**n*(a*c + b*d)/(S(2)*f*(a + b*tan(e + f*x))*(-a*d + b*c)), x)
rule3605 = ReplacementRule(pattern3605, replacement3605)
pattern3606 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons87, cons1325)
def replacement3606(f, b, d, c, n, a, x, e):
rubi.append(3606)
return Dist(S(1)/(S(2)*a*(-a*d + b*c)), Int((c + d/tan(e + f*x))**(n + S(-1))*Simp(a*c*d*(n + S(-1)) + b*c**S(2) + b*d**S(2)*n - d*(n + S(-1))*(-a*d + b*c)/tan(e + f*x), x), x), x) + Simp((c + d/tan(e + f*x))**n*(a*c + b*d)/(S(2)*f*(a + b/tan(e + f*x))*(-a*d + b*c)), x)
rule3606 = ReplacementRule(pattern3606, replacement3606)
pattern3607 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons87, cons165)
def replacement3607(f, b, d, c, a, n, x, e):
rubi.append(3607)
return Dist(S(1)/(S(2)*a**S(2)), Int((c + d*tan(e + f*x))**(n + S(-2))*Simp(a*c**S(2) + a*d**S(2)*(n + S(-1)) - b*c*d*n - d*(a*c*(n + S(-2)) + b*d*n)*tan(e + f*x), x), x), x) + Simp((c + d*tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(S(2)*a*f*(a + b*tan(e + f*x))), x)
rule3607 = ReplacementRule(pattern3607, replacement3607)
pattern3608 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons87, cons165)
def replacement3608(f, b, d, c, n, a, x, e):
rubi.append(3608)
return Dist(S(1)/(S(2)*a**S(2)), Int((c + d/tan(e + f*x))**(n + S(-2))*Simp(a*c**S(2) + a*d**S(2)*(n + S(-1)) - b*c*d*n - d*(a*c*(n + S(-2)) + b*d*n)/tan(e + f*x), x), x), x) - Simp((c + d/tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(S(2)*a*f*(a + b/tan(e + f*x))), x)
rule3608 = ReplacementRule(pattern3608, replacement3608)
pattern3609 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3609(f, b, d, c, a, x, e):
rubi.append(3609)
return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*tan(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*tan(e + f*x)), x), x)
rule3609 = ReplacementRule(pattern3609, replacement3609)
pattern3610 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3610(f, b, d, c, a, x, e):
rubi.append(3610)
return Dist(b/(-a*d + b*c), Int(S(1)/(a + b/tan(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d/tan(e + f*x)), x), x)
rule3610 = ReplacementRule(pattern3610, replacement3610)
pattern3611 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1439, cons1545, cons1327)
def replacement3611(f, b, d, c, a, n, x, e):
rubi.append(3611)
return Dist(S(1)/(S(2)*a*(-a*d + b*c)), Int((c + d*tan(e + f*x))**n*Simp(a*d*(n + S(-1)) + b*c - b*d*n*tan(e + f*x), x), x), x) - Simp(a*(c + d*tan(e + f*x))**(n + S(1))/(S(2)*f*(a + b*tan(e + f*x))*(-a*d + b*c)), x)
rule3611 = ReplacementRule(pattern3611, replacement3611)
pattern3612 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1439, cons1545, cons1327)
def replacement3612(f, b, d, c, n, a, x, e):
rubi.append(3612)
return Dist(S(1)/(S(2)*a*(-a*d + b*c)), Int((c + d/tan(e + f*x))**n*Simp(a*d*(n + S(-1)) + b*c - b*d*n/tan(e + f*x), x), x), x) + Simp(a*(c + d/tan(e + f*x))**(n + S(1))/(S(2)*f*(a + b/tan(e + f*x))*(-a*d + b*c)), x)
rule3612 = ReplacementRule(pattern3612, replacement3612)
pattern3613 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons166, cons89, cons1334)
def replacement3613(m, f, b, d, c, a, n, x, e):
rubi.append(3613)
return Dist(a/(d*(n + S(1))*(a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))*Simp(b*(-a*d*(m - S(2)*n + S(-4)) + b*c*(m + S(-2))) + (-a**S(2)*d*(m + n + S(-1)) + a*b*c*(m + S(-2)) + b**S(2)*d*(n + S(1)))*tan(e + f*x), x), x), x) - Simp(a**S(2)*(a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))*(-a*d + b*c)/(d*f*(n + S(1))*(a*d + b*c)), x)
rule3613 = ReplacementRule(pattern3613, replacement3613)
pattern3614 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons166, cons89, cons1334)
def replacement3614(m, f, b, d, c, n, a, x, e):
rubi.append(3614)
return Dist(a/(d*(n + S(1))*(a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))*Simp(b*(-a*d*(m - S(2)*n + S(-4)) + b*c*(m + S(-2))) + (-a**S(2)*d*(m + n + S(-1)) + a*b*c*(m + S(-2)) + b**S(2)*d*(n + S(1)))/tan(e + f*x), x), x), x) + Simp(a**S(2)*(a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))*(-a*d + b*c)/(d*f*(n + S(1))*(a*d + b*c)), x)
rule3614 = ReplacementRule(pattern3614, replacement3614)
pattern3615 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3615(f, b, d, c, a, x, e):
rubi.append(3615)
return Dist(S(2)*a**S(2)/(a*c - b*d), Int(sqrt(a + b*tan(e + f*x)), x), x) - Dist((a*(c**S(2) - d**S(2)) + S(2)*b*c*d)/(a*(c**S(2) + d**S(2))), Int((a - b*tan(e + f*x))*sqrt(a + b*tan(e + f*x))/(c + d*tan(e + f*x)), x), x)
rule3615 = ReplacementRule(pattern3615, replacement3615)
pattern3616 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3616(f, b, d, c, a, x, e):
rubi.append(3616)
return Dist(S(2)*a**S(2)/(a*c - b*d), Int(sqrt(a + b/tan(e + f*x)), x), x) - Dist((a*(c**S(2) - d**S(2)) + S(2)*b*c*d)/(a*(c**S(2) + d**S(2))), Int((a - b/tan(e + f*x))*sqrt(a + b/tan(e + f*x))/(c + d/tan(e + f*x)), x), x)
rule3616 = ReplacementRule(pattern3616, replacement3616)
pattern3617 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3617(f, b, d, c, a, x, e):
rubi.append(3617)
return Dist(S(2)*a, Int(sqrt(a + b*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x), x) + Dist(b/a, Int(sqrt(a + b*tan(e + f*x))*(a*tan(e + f*x) + b)/sqrt(c + d*tan(e + f*x)), x), x)
rule3617 = ReplacementRule(pattern3617, replacement3617)
pattern3618 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3618(f, b, d, c, a, x, e):
rubi.append(3618)
return Dist(S(2)*a, Int(sqrt(a + b/tan(e + f*x))/sqrt(c + d/tan(e + f*x)), x), x) + Dist(b/a, Int(sqrt(a + b/tan(e + f*x))*(a/tan(e + f*x) + b)/sqrt(c + d/tan(e + f*x)), x), x)
rule3618 = ReplacementRule(pattern3618, replacement3618)
pattern3619 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1439, cons1545, cons515, cons166, cons1519, cons1334)
def replacement3619(m, f, b, d, c, a, n, x, e):
rubi.append(3619)
return Dist(a/(d*(m + n + S(-1))), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**n*Simp(a*d*(m + S(2)*n) + b*c*(m + S(-2)) + (a*c*(m + S(-2)) + b*d*(S(3)*m + S(2)*n + S(-4)))*tan(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(-1))), x)
rule3619 = ReplacementRule(pattern3619, replacement3619)
pattern3620 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1439, cons1545, cons515, cons166, cons1519, cons1334)
def replacement3620(m, f, b, d, c, n, a, x, e):
rubi.append(3620)
return Dist(a/(d*(m + n + S(-1))), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**n*Simp(a*d*(m + S(2)*n) + b*c*(m + S(-2)) + (a*c*(m + S(-2)) + b*d*(S(3)*m + S(2)*n + S(-4)))/tan(e + f*x), x), x), x) - Simp(b**S(2)*(a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(-1))), x)
rule3620 = ReplacementRule(pattern3620, replacement3620)
pattern3621 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons31, cons267, cons1552)
def replacement3621(m, f, b, d, c, a, x, e):
rubi.append(3621)
return Dist(S(1)/(S(4)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(S(2)*a*c*m + a*d*(S(2)*m + S(1))*tan(e + f*x) + b*d, x)/sqrt(c + d*tan(e + f*x)), x), x) - Simp(b*(a + b*tan(e + f*x))**m*sqrt(c + d*tan(e + f*x))/(S(2)*a*f*m), x)
rule3621 = ReplacementRule(pattern3621, replacement3621)
pattern3622 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons31, cons267, cons1552)
def replacement3622(m, f, b, d, c, a, x, e):
rubi.append(3622)
return Dist(S(1)/(S(4)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(S(2)*a*c*m + a*d*(S(2)*m + S(1))/tan(e + f*x) + b*d, x)/sqrt(c + d/tan(e + f*x)), x), x) + Simp(b*(a + b/tan(e + f*x))**m*sqrt(c + d/tan(e + f*x))/(S(2)*a*f*m), x)
rule3622 = ReplacementRule(pattern3622, replacement3622)
pattern3623 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons267, cons165, cons1334)
def replacement3623(m, f, b, d, c, a, n, x, e):
rubi.append(3623)
return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-2))*Simp(c*(a*c*m + b*d*(n + S(-1))) - d*(-a*c*(m + n + S(-1)) + b*d*(m - n + S(1)))*tan(e + f*x) - d*(a*d*(n + S(-1)) + b*c*m), x), x), x) - Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(S(2)*a*f*m), x)
rule3623 = ReplacementRule(pattern3623, replacement3623)
pattern3624 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545, cons93, cons267, cons165, cons1334)
def replacement3624(m, f, b, d, c, n, a, x, e):
rubi.append(3624)
return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-2))*Simp(c*(a*c*m + b*d*(n + S(-1))) - d*(-a*c*(m + n + S(-1)) + b*d*(m - n + S(1)))/tan(e + f*x) - d*(a*d*(n + S(-1)) + b*c*m), x), x), x) + Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(S(2)*a*f*m), x)
rule3624 = ReplacementRule(pattern3624, replacement3624)
pattern3625 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1439, cons1545, cons31, cons267, cons1334)
def replacement3625(m, f, b, d, c, a, n, x, e):
rubi.append(3625)
return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(-a*d*(S(2)*m + n + S(1)) + b*c*m + b*d*(m + n + S(1))*tan(e + f*x), x), x), x) + Simp(a*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x)
rule3625 = ReplacementRule(pattern3625, replacement3625)
pattern3626 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1439, cons1545, cons31, cons267, cons1334)
def replacement3626(m, f, b, d, c, n, a, x, e):
rubi.append(3626)
return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(-a*d*(S(2)*m + n + S(1)) + b*c*m + b*d*(m + n + S(1))/tan(e + f*x), x), x), x) - Simp(a*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x)
rule3626 = ReplacementRule(pattern3626, replacement3626)
pattern3627 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1439, cons1545, cons87, cons165, cons1519, cons1553)
def replacement3627(m, f, b, d, c, a, n, x, e):
rubi.append(3627)
return -Dist(S(1)/(a*(m + n + S(-1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(-2))*Simp(-a*c**S(2)*(m + n + S(-1)) + d*(-a*c*(m + S(2)*n + S(-2)) + b*d*m)*tan(e + f*x) + d*(a*d*(n + S(-1)) + b*c*m), x), x), x) + Simp(d*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3627 = ReplacementRule(pattern3627, replacement3627)
pattern3628 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1439, cons1545, cons87, cons165, cons1519, cons1553)
def replacement3628(m, f, b, d, c, a, n, x, e):
rubi.append(3628)
return -Dist(S(1)/(a*(m + n + S(-1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(-2))*Simp(-a*c**S(2)*(m + n + S(-1)) + d*(-a*c*(m + S(2)*n + S(-2)) + b*d*m)/tan(e + f*x) + d*(a*d*(n + S(-1)) + b*c*m), x), x), x) - Simp(d*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x)
rule3628 = ReplacementRule(pattern3628, replacement3628)
pattern3629 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1439, cons1545, cons87, cons89, cons1553)
def replacement3629(m, f, b, d, c, a, n, x, e):
rubi.append(3629)
return -Dist(S(1)/(a*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*Simp(-a*c*(n + S(1)) + a*d*(m + n + S(1))*tan(e + f*x) + b*d*m, x), x), x) + Simp(d*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3629 = ReplacementRule(pattern3629, replacement3629)
pattern3630 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1439, cons1545, cons87, cons89, cons1553)
def replacement3630(m, f, b, d, c, n, a, x, e):
rubi.append(3630)
return -Dist(S(1)/(a*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*Simp(-a*c*(n + S(1)) + a*d*(m + n + S(1))/tan(e + f*x) + b*d*m, x), x), x) - Simp(d*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3630 = ReplacementRule(pattern3630, replacement3630)
pattern3631 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1439, cons1545)
def replacement3631(m, f, b, d, c, a, x, e):
rubi.append(3631)
return Dist(a/(a*c - b*d), Int((a + b*tan(e + f*x))**m, x), x) - Dist(d/(a*c - b*d), Int((a + b*tan(e + f*x))**m*(a*tan(e + f*x) + b)/(c + d*tan(e + f*x)), x), x)
rule3631 = ReplacementRule(pattern3631, replacement3631)
pattern3632 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1439, cons1545)
def replacement3632(m, f, b, d, c, a, x, e):
rubi.append(3632)
return Dist(a/(a*c - b*d), Int((a + b/tan(e + f*x))**m, x), x) - Dist(d/(a*c - b*d), Int((a + b/tan(e + f*x))**m*(a/tan(e + f*x) + b)/(c + d/tan(e + f*x)), x), x)
rule3632 = ReplacementRule(pattern3632, replacement3632)
pattern3633 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3633(f, b, d, c, a, x, e):
rubi.append(3633)
return Dist(d/a, Int(sqrt(a + b*tan(e + f*x))*(a*tan(e + f*x) + b)/sqrt(c + d*tan(e + f*x)), x), x) + Dist((a*c - b*d)/a, Int(sqrt(a + b*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x), x)
rule3633 = ReplacementRule(pattern3633, replacement3633)
pattern3634 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1439, cons1545)
def replacement3634(f, b, d, c, a, x, e):
rubi.append(3634)
return Dist(d/a, Int(sqrt(a + b/tan(e + f*x))*(a/tan(e + f*x) + b)/sqrt(c + d/tan(e + f*x)), x), x) + Dist((a*c - b*d)/a, Int(sqrt(a + b/tan(e + f*x))/sqrt(c + d/tan(e + f*x)), x), x)
rule3634 = ReplacementRule(pattern3634, replacement3634)
pattern3635 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1439, cons1545)
def replacement3635(m, f, b, d, c, a, n, x, e):
rubi.append(3635)
return Dist(a*b/f, Subst(Int((a + x)**(m + S(-1))*(c + d*x/b)**n/(a*x + b**S(2)), x), x, b*tan(e + f*x)), x)
rule3635 = ReplacementRule(pattern3635, replacement3635)
pattern3636 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1439, cons1545)
def replacement3636(m, f, b, d, c, a, n, x, e):
rubi.append(3636)
return -Dist(a*b/f, Subst(Int((a + x)**(m + S(-1))*(c + d*x/b)**n/(a*x + b**S(2)), x), x, b/tan(e + f*x)), x)
rule3636 = ReplacementRule(pattern3636, replacement3636)
pattern3637 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons93, cons1333, cons89, cons515)
def replacement3637(m, f, b, d, c, a, n, x, e):
rubi.append(3637)
return -Dist(S(1)/(d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**(m + S(-3))*(c + d*tan(e + f*x))**(n + S(1))*Simp(a**S(2)*d*(-a*c*(n + S(1)) + b*d*(m + S(-2))) + b*(-S(2)*a*d + b*c)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b*(a*d*(-a*d + S(2)*b*c)*(m + n + S(-1)) - b**S(2)*(c**S(2)*(m + S(-2)) - d**S(2)*(n + S(1))))*tan(e + f*x)**S(2) - d*(n + S(1))*(-a**S(3)*d + S(3)*a**S(2)*b*c + S(3)*a*b**S(2)*d - b**S(3)*c)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))*(-a*d + b*c)**S(2)/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3637 = ReplacementRule(pattern3637, replacement3637)
pattern3638 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons93, cons1333, cons89, cons515)
def replacement3638(m, f, b, d, c, a, n, x, e):
rubi.append(3638)
return -Dist(S(1)/(d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**(m + S(-3))*(c + d/tan(e + f*x))**(n + S(1))*Simp(a**S(2)*d*(-a*c*(n + S(1)) + b*d*(m + S(-2))) + b*(-S(2)*a*d + b*c)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b*(a*d*(-a*d + S(2)*b*c)*(m + n + S(-1)) - b**S(2)*(c**S(2)*(m + S(-2)) - d**S(2)*(n + S(1))))/tan(e + f*x)**S(2) - d*(n + S(1))*(-a**S(3)*d + S(3)*a**S(2)*b*c + S(3)*a*b**S(2)*d - b**S(3)*c)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))*(-a*d + b*c)**S(2)/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3638 = ReplacementRule(pattern3638, replacement3638)
pattern3639 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1440, cons1545, cons515, cons1333, cons1554, cons1555)
def replacement3639(m, f, b, d, c, a, n, x, e):
rubi.append(3639)
return Dist(S(1)/(d*(m + n + S(-1))), Int((a + b*tan(e + f*x))**(m + S(-3))*(c + d*tan(e + f*x))**n*Simp(a**S(3)*d*(m + n + S(-1)) - b**S(2)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b**S(2)*(-a*d*(S(3)*m + S(2)*n + S(-4)) + b*c*(m + S(-2)))*tan(e + f*x)**S(2) + b*d*(S(3)*a**S(2) - b**S(2))*(m + n + S(-1))*tan(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(-1))), x)
rule3639 = ReplacementRule(pattern3639, replacement3639)
pattern3640 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1440, cons1545, cons515, cons1333, cons1554, cons1555)
def replacement3640(m, f, b, d, c, a, n, x, e):
rubi.append(3640)
return Dist(S(1)/(d*(m + n + S(-1))), Int((a + b/tan(e + f*x))**(m + S(-3))*(c + d/tan(e + f*x))**n*Simp(a**S(3)*d*(m + n + S(-1)) - b**S(2)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b**S(2)*(-a*d*(S(3)*m + S(2)*n + S(-4)) + b*c*(m + S(-2)))/tan(e + f*x)**S(2) + b*d*(S(3)*a**S(2) - b**S(2))*(m + n + S(-1))/tan(e + f*x), x), x), x) - Simp(b**S(2)*(a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(-1))), x)
rule3640 = ReplacementRule(pattern3640, replacement3640)
pattern3641 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons93, cons94, cons1336, cons515)
def replacement3641(m, f, b, d, c, a, n, x, e):
rubi.append(3641)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-2))*Simp(a*c**S(2)*(m + S(1)) + a*d**S(2)*(n + S(-1)) + b*c*d*(m - n + S(2)) - d*(m + n)*(-a*d + b*c)*tan(e + f*x)**S(2) - (m + S(1))*(-S(2)*a*c*d + b*c**S(2) - b*d**S(2))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3641 = ReplacementRule(pattern3641, replacement3641)
pattern3642 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons93, cons94, cons1336, cons515)
def replacement3642(m, f, b, d, c, a, n, x, e):
rubi.append(3642)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-2))*Simp(a*c**S(2)*(m + S(1)) + a*d**S(2)*(n + S(-1)) + b*c*d*(m - n + S(2)) - d*(m + n)*(-a*d + b*c)/tan(e + f*x)**S(2) - (m + S(1))*(-S(2)*a*c*d + b*c**S(2) - b*d**S(2))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3642 = ReplacementRule(pattern3642, replacement3642)
pattern3643 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons93, cons94, cons88, cons515)
def replacement3643(m, f, b, d, c, a, n, x, e):
rubi.append(3643)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(a*c*(m + S(1)) - b*d*n - b*d*(m + n + S(1))*tan(e + f*x)**S(2) - (m + S(1))*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp(b*(a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n/(f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3643 = ReplacementRule(pattern3643, replacement3643)
pattern3644 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons93, cons94, cons88, cons515)
def replacement3644(m, f, b, d, c, a, n, x, e):
rubi.append(3644)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(a*c*(m + S(1)) - b*d*n - b*d*(m + n + S(1))/tan(e + f*x)**S(2) - (m + S(1))*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp(b*(a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n/(f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3644 = ReplacementRule(pattern3644, replacement3644)
pattern3645 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1440, cons1545, cons515, cons94, cons1556, cons1557)
def replacement3645(m, f, b, d, c, a, n, x, e):
rubi.append(3645)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))*tan(e + f*x)**S(2) - b**S(2)*d*(m + n + S(2)) - b*(m + S(1))*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(1))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3645 = ReplacementRule(pattern3645, replacement3645)
pattern3646 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1440, cons1545, cons515, cons94, cons1556, cons1557)
def replacement3646(m, f, b, d, c, a, n, x, e):
rubi.append(3646)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2)) - b**S(2)*d*(m + n + S(2))/tan(e + f*x)**S(2) - b*(m + S(1))*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp(b**S(2)*(a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(1))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3646 = ReplacementRule(pattern3646, replacement3646)
pattern3647 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons93, cons166, cons88, cons808)
def replacement3647(m, f, b, d, c, a, n, x, e):
rubi.append(3647)
return Dist(S(1)/(m + n + S(-1)), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(a**S(2)*c*(m + n + S(-1)) - b*(a*d*n + b*c*(m + S(-1))) + b*(a*d*(S(2)*m + n + S(-2)) + b*c*n)*tan(e + f*x)**S(2) + (m + n + S(-1))*(a**S(2)*d + S(2)*a*b*c - b**S(2)*d)*tan(e + f*x), x), x), x) + Simp(b*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**n/(f*(m + n + S(-1))), x)
rule3647 = ReplacementRule(pattern3647, replacement3647)
pattern3648 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545, cons93, cons166, cons88, cons808)
def replacement3648(m, f, b, d, c, a, n, x, e):
rubi.append(3648)
return Dist(S(1)/(m + n + S(-1)), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(a**S(2)*c*(m + n + S(-1)) - b*(a*d*n + b*c*(m + S(-1))) + b*(a*d*(S(2)*m + n + S(-2)) + b*c*n)/tan(e + f*x)**S(2) + (m + n + S(-1))*(a**S(2)*d + S(2)*a*b*c - b**S(2)*d)/tan(e + f*x), x), x), x) - Simp(b*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**n/(f*(m + n + S(-1))), x)
rule3648 = ReplacementRule(pattern3648, replacement3648)
pattern3649 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545)
def replacement3649(f, b, d, c, a, x, e):
rubi.append(3649)
return Dist(b**S(2)/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) - Dist(d**S(2)/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c*tan(e + f*x) + d)/(c + d*tan(e + f*x)), x), x) + Simp(x*(a*c - b*d)/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x)
rule3649 = ReplacementRule(pattern3649, replacement3649)
pattern3650 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545)
def replacement3650(f, b, d, c, a, x, e):
rubi.append(3650)
return Dist(b**S(2)/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) - Dist(d**S(2)/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c/tan(e + f*x) + d)/(c + d/tan(e + f*x)), x), x) + Simp(x*(a*c - b*d)/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x)
rule3650 = ReplacementRule(pattern3650, replacement3650)
pattern3651 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons71, cons1440, cons1545)
def replacement3651(f, b, d, c, a, x, e):
rubi.append(3651)
return -Dist(d*(-a*d + b*c)/(c**S(2) + d**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int(Simp(a*c + b*d + (-a*d + b*c)*tan(e + f*x), x)/sqrt(a + b*tan(e + f*x)), x), x)
rule3651 = ReplacementRule(pattern3651, replacement3651)
pattern3652 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons71, cons1440, cons1545)
def replacement3652(f, b, d, c, a, x, e):
rubi.append(3652)
return -Dist(d*(-a*d + b*c)/(c**S(2) + d**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(sqrt(a + b/tan(e + f*x))*(c + d/tan(e + f*x))), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int(Simp(a*c + b*d + (-a*d + b*c)/tan(e + f*x), x)/sqrt(a + b/tan(e + f*x)), x), x)
rule3652 = ReplacementRule(pattern3652, replacement3652)
pattern3653 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545)
def replacement3653(f, b, d, c, a, x, e):
rubi.append(3653)
return Dist((-a*d + b*c)**S(2)/(c**S(2) + d**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int(Simp(a**S(2)*c + S(2)*a*b*d - b**S(2)*c + (-a**S(2)*d + S(2)*a*b*c + b**S(2)*d)*tan(e + f*x), x)/sqrt(a + b*tan(e + f*x)), x), x)
rule3653 = ReplacementRule(pattern3653, replacement3653)
pattern3654 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1440, cons1545)
def replacement3654(f, b, d, c, a, x, e):
rubi.append(3654)
return Dist((-a*d + b*c)**S(2)/(c**S(2) + d**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(sqrt(a + b/tan(e + f*x))*(c + d/tan(e + f*x))), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int(Simp(a**S(2)*c + S(2)*a*b*d - b**S(2)*c + (-a**S(2)*d + S(2)*a*b*c + b**S(2)*d)/tan(e + f*x), x)/sqrt(a + b/tan(e + f*x)), x), x)
rule3654 = ReplacementRule(pattern3654, replacement3654)
pattern3655 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1440, cons1545, cons18)
def replacement3655(m, f, b, d, c, a, x, e):
rubi.append(3655)
return Dist(d**S(2)/(c**S(2) + d**S(2)), Int((a + b*tan(e + f*x))**m*(tan(e + f*x)**S(2) + S(1))/(c + d*tan(e + f*x)), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int((a + b*tan(e + f*x))**m*(c - d*tan(e + f*x)), x), x)
rule3655 = ReplacementRule(pattern3655, replacement3655)
pattern3656 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1440, cons1545, cons18)
def replacement3656(m, f, b, d, c, a, x, e):
rubi.append(3656)
return Dist(d**S(2)/(c**S(2) + d**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))*(a + b/tan(e + f*x))**m/(c + d/tan(e + f*x)), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int((a + b/tan(e + f*x))**m*(c - d/tan(e + f*x)), x), x)
rule3656 = ReplacementRule(pattern3656, replacement3656)
pattern3657 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1440, cons1545)
def replacement3657(m, f, b, d, a, c, n, x, e):
rubi.append(3657)
return Dist(b/f, Subst(Int((a + x)**m*(c + d*x/b)**n/(b**S(2) + x**S(2)), x), x, b*tan(e + f*x)), x)
rule3657 = ReplacementRule(pattern3657, replacement3657)
pattern3658 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1440, cons1545)
def replacement3658(m, f, b, d, a, c, n, x, e):
rubi.append(3658)
return -Dist(b/f, Subst(Int((a + x)**m*(c + d*x/b)**n/(b**S(2) + x**S(2)), x), x, b/tan(e + f*x)), x)
rule3658 = ReplacementRule(pattern3658, replacement3658)
pattern3659 = Pattern(Integral((WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons23, cons17)
def replacement3659(m, f, b, d, a, n, x, e):
rubi.append(3659)
return Dist(d**m, Int((d/tan(e + f*x))**(-m + n)*(a/tan(e + f*x) + b)**m, x), x)
rule3659 = ReplacementRule(pattern3659, replacement3659)
pattern3660 = Pattern(Integral((WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons23, cons17)
def replacement3660(m, f, b, d, a, n, x, e):
rubi.append(3660)
return Dist(d**m, Int((d*tan(e + f*x))**(-m + n)*(a*tan(e + f*x) + b)**m, x), x)
rule3660 = ReplacementRule(pattern3660, replacement3660)
pattern3661 = Pattern(Integral(((WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons23, cons18)
def replacement3661(p, m, f, b, d, c, a, n, x, e):
rubi.append(3661)
return Dist(c**IntPart(n)*(c*(d*tan(e + f*x))**p)**FracPart(n)*(d*tan(e + f*x))**(-p*FracPart(n)), Int((d*tan(e + f*x))**(n*p)*(a + b*tan(e + f*x))**m, x), x)
rule3661 = ReplacementRule(pattern3661, replacement3661)
pattern3662 = Pattern(Integral(((WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons23, cons18)
def replacement3662(p, m, f, b, d, a, c, n, x, e):
rubi.append(3662)
return Dist(c**IntPart(n)*(c*(d/tan(e + f*x))**p)**FracPart(n)*(d/tan(e + f*x))**(-p*FracPart(n)), Int((d/tan(e + f*x))**(n*p)*(a + b/tan(e + f*x))**m, x), x)
rule3662 = ReplacementRule(pattern3662, replacement3662)
pattern3663 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons380)
def replacement3663(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(3663)
return Int((g*tan(e + f*x))**p*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n, x)
rule3663 = ReplacementRule(pattern3663, replacement3663)
pattern3664 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons380)
def replacement3664(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(3664)
return Int((g/tan(e + f*x))**p*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x)
rule3664 = ReplacementRule(pattern3664, replacement3664)
pattern3665 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons147, cons17, cons85)
def replacement3665(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(3665)
return Dist(g**(m + n), Int((g/tan(e + f*x))**(-m - n + p)*(a/tan(e + f*x) + b)**m*(c/tan(e + f*x) + d)**n, x), x)
rule3665 = ReplacementRule(pattern3665, replacement3665)
pattern3666 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons147, cons17, cons85)
def replacement3666(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(3666)
return Dist(g**(m + n), Int((g*tan(e + f*x))**(-m - n + p)*(a*tan(e + f*x) + b)**m*(c*tan(e + f*x) + d)**n, x), x)
rule3666 = ReplacementRule(pattern3666, replacement3666)
pattern3667 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**q_)**p_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons147, cons1417)
def replacement3667(p, m, f, g, b, d, a, n, c, x, q, e):
rubi.append(3667)
return Dist((g*tan(e + f*x))**(-p*q)*(g*tan(e + f*x)**q)**p, Int((g*tan(e + f*x))**(p*q)*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n, x), x)
rule3667 = ReplacementRule(pattern3667, replacement3667)
pattern3668 = Pattern(Integral(((S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**q_*WC('g', S(1)))**p_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons50, cons147, cons1417)
def replacement3668(p, m, f, b, g, d, a, n, c, x, q, e):
rubi.append(3668)
return Dist((g*(S(1)/tan(e + f*x))**q)**p*(g/tan(e + f*x))**(-p*q), Int((g/tan(e + f*x))**(p*q)*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x)
rule3668 = ReplacementRule(pattern3668, replacement3668)
pattern3669 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons85)
def replacement3669(p, m, g, f, b, d, c, n, a, x, e):
rubi.append(3669)
return Dist(g**n, Int((g*tan(e + f*x))**(-n + p)*(a + b*tan(e + f*x))**m*(c*tan(e + f*x) + d)**n, x), x)
rule3669 = ReplacementRule(pattern3669, replacement3669)
pattern3670 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons85)
def replacement3670(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(3670)
return Dist(g**n, Int((g/tan(e + f*x))**(-n + p)*(a + b/tan(e + f*x))**m*(c/tan(e + f*x) + d)**n, x), x)
rule3670 = ReplacementRule(pattern3670, replacement3670)
pattern3671 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons23, cons17, cons38)
def replacement3671(p, m, f, b, d, c, n, a, x, e):
rubi.append(3671)
return Int((c + d/tan(e + f*x))**n*(a/tan(e + f*x) + b)**m*(S(1)/tan(e + f*x))**(-m - p), x)
rule3671 = ReplacementRule(pattern3671, replacement3671)
pattern3672 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons23, cons17, cons38)
def replacement3672(p, m, f, b, d, a, c, n, x, e):
rubi.append(3672)
return Int((c + d*tan(e + f*x))**n*(a*tan(e + f*x) + b)**m*tan(e + f*x)**(-m - p), x)
rule3672 = ReplacementRule(pattern3672, replacement3672)
pattern3673 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons23, cons17, cons147)
def replacement3673(p, m, f, b, g, d, c, n, a, x, e):
rubi.append(3673)
return Dist((g*tan(e + f*x))**p*(S(1)/tan(e + f*x))**p, Int((c + d/tan(e + f*x))**n*(a/tan(e + f*x) + b)**m*(S(1)/tan(e + f*x))**(-m - p), x), x)
rule3673 = ReplacementRule(pattern3673, replacement3673)
pattern3674 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons23, cons17, cons147)
def replacement3674(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(3674)
return Dist((g/tan(e + f*x))**p*tan(e + f*x)**p, Int((c + d*tan(e + f*x))**n*(a*tan(e + f*x) + b)**m*tan(e + f*x)**(-m - p), x), x)
rule3674 = ReplacementRule(pattern3674, replacement3674)
pattern3675 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons23, cons18)
def replacement3675(p, m, f, g, b, d, c, n, a, x, e):
rubi.append(3675)
return Dist((g*tan(e + f*x))**n*(c + d/tan(e + f*x))**n*(c*tan(e + f*x) + d)**(-n), Int((g*tan(e + f*x))**(-n + p)*(a + b*tan(e + f*x))**m*(c*tan(e + f*x) + d)**n, x), x)
rule3675 = ReplacementRule(pattern3675, replacement3675)
pattern3676 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons23, cons18)
def replacement3676(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(3676)
return Dist((g/tan(e + f*x))**n*(c + d*tan(e + f*x))**n*(c/tan(e + f*x) + d)**(-n), Int((g/tan(e + f*x))**(-n + p)*(a + b/tan(e + f*x))**m*(c/tan(e + f*x) + d)**n, x), x)
rule3676 = ReplacementRule(pattern3676, replacement3676)
pattern3677 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons70, cons1439)
def replacement3677(B, e, m, f, b, d, a, n, c, x, A):
rubi.append(3677)
return Dist(a*c/f, Subst(Int((A + B*x)*(a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1)), x), x, tan(e + f*x)), x)
rule3677 = ReplacementRule(pattern3677, replacement3677)
pattern3678 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons70, cons1439)
def replacement3678(B, m, f, b, d, a, n, c, A, x, e):
rubi.append(3678)
return -Dist(a*c/f, Subst(Int((A + B*x)*(a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1)), x), x, S(1)/tan(e + f*x)), x)
rule3678 = ReplacementRule(pattern3678, replacement3678)
pattern3679 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71)
def replacement3679(B, e, f, b, d, a, c, x, A):
rubi.append(3679)
return Dist(S(1)/b, Int(Simp(A*b*c + (A*b*d + B*(-a*d + b*c))*tan(e + f*x), x)/(a + b*tan(e + f*x)), x), x) + Dist(B*d/b, Int(tan(e + f*x), x), x)
rule3679 = ReplacementRule(pattern3679, replacement3679)
pattern3680 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71)
def replacement3680(B, e, f, b, d, a, c, x, A):
rubi.append(3680)
return Dist(S(1)/b, Int(Simp(A*b*c + (A*b*d + B*(-a*d + b*c))/tan(e + f*x), x)/(a + b/tan(e + f*x)), x), x) + Dist(B*d/b, Int(S(1)/tan(e + f*x), x), x)
rule3680 = ReplacementRule(pattern3680, replacement3680)
pattern3681 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons31, cons94, cons1439)
def replacement3681(B, e, m, f, b, d, c, a, x, A):
rubi.append(3681)
return Dist(S(1)/(S(2)*a*b), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(A*a*d + A*b*c + B*a*c + S(2)*B*a*d*tan(e + f*x) + B*b*d, x), x), x) - Simp((a + b*tan(e + f*x))**m*(A*b - B*a)*(a*c + b*d)/(S(2)*a**S(2)*f*m), x)
rule3681 = ReplacementRule(pattern3681, replacement3681)
pattern3682 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons31, cons94, cons1439)
def replacement3682(B, m, f, b, d, c, a, A, x, e):
rubi.append(3682)
return Dist(S(1)/(S(2)*a*b), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(A*a*d + A*b*c + B*a*c + S(2)*B*a*d/tan(e + f*x) + B*b*d, x), x), x) + Simp((a + b/tan(e + f*x))**m*(A*b - B*a)*(a*c + b*d)/(S(2)*a**S(2)*f*m), x)
rule3682 = ReplacementRule(pattern3682, replacement3682)
pattern3683 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons31, cons94, cons1440)
def replacement3683(B, e, m, f, b, d, a, c, x, A):
rubi.append(3683)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(A*a*c + A*b*d - B*a*d + B*b*c - (-A*a*d + A*b*c - B*a*c - B*b*d)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(A*b - B*a)*(-a*d + b*c)/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3683 = ReplacementRule(pattern3683, replacement3683)
pattern3684 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons31, cons94, cons1440)
def replacement3684(B, e, m, f, b, d, a, c, x, A):
rubi.append(3684)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(A*a*c + A*b*d - B*a*d + B*b*c - (-A*a*d + A*b*c - B*a*c - B*b*d)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(A*b - B*a)*(-a*d + b*c)/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3684 = ReplacementRule(pattern3684, replacement3684)
pattern3685 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1549)
def replacement3685(B, m, f, b, d, a, c, A, x, e):
rubi.append(3685)
return Int((a + b*tan(e + f*x))**m*Simp(A*c - B*d + (A*d + B*c)*tan(e + f*x), x), x) + Simp(B*d*(a + b*tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule3685 = ReplacementRule(pattern3685, replacement3685)
pattern3686 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1549)
def replacement3686(B, m, f, b, d, a, c, A, x, e):
rubi.append(3686)
return Int((a + b/tan(e + f*x))**m*Simp(A*c - B*d + (A*d + B*c)/tan(e + f*x), x), x) - Simp(B*d*(a + b/tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule3686 = ReplacementRule(pattern3686, replacement3686)
pattern3687 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1439, cons93, cons166, cons89)
def replacement3687(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(3687)
return -Dist(a/(d*(n + S(1))*(a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*b*d*(m - n + S(-2)) - B*(a*d*(n + S(1)) + b*c*(m + S(-1))) + (A*a*d*(m + n) - B*(a*c*(m + S(-1)) + b*d*(n + S(1))))*tan(e + f*x), x), x), x) - Simp(a**S(2)*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*(-A*d + B*c)/(d*f*(n + S(1))*(a*d + b*c)), x)
rule3687 = ReplacementRule(pattern3687, replacement3687)
pattern3688 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1439, cons93, cons166, cons89)
def replacement3688(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(3688)
return -Dist(a/(d*(n + S(1))*(a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*b*d*(m - n + S(-2)) - B*(a*d*(n + S(1)) + b*c*(m + S(-1))) + (A*a*d*(m + n) - B*(a*c*(m + S(-1)) + b*d*(n + S(1))))/tan(e + f*x), x), x), x) + Simp(a**S(2)*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*(-A*d + B*c)/(d*f*(n + S(1))*(a*d + b*c)), x)
rule3688 = ReplacementRule(pattern3688, replacement3688)
pattern3689 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1439, cons31, cons166, cons346)
def replacement3689(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(3689)
return Dist(S(1)/(d*(m + n)), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**n*Simp(A*a*d*(m + n) + B*(a*c*(m + S(-1)) - b*d*(n + S(1))) - (B*(m + S(-1))*(-a*d + b*c) - d*(m + n)*(A*b + B*a))*tan(e + f*x), x), x), x) + Simp(B*b*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n)), x)
rule3689 = ReplacementRule(pattern3689, replacement3689)
pattern3690 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1439, cons31, cons166, cons346)
def replacement3690(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(3690)
return Dist(S(1)/(d*(m + n)), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**n*Simp(A*a*d*(m + n) + B*(a*c*(m + S(-1)) - b*d*(n + S(1))) - (B*(m + S(-1))*(-a*d + b*c) - d*(m + n)*(A*b + B*a))/tan(e + f*x), x), x), x) - Simp(B*b*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n)), x)
rule3690 = ReplacementRule(pattern3690, replacement3690)
pattern3691 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1439, cons93, cons267, cons88)
def replacement3691(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(3691)
return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(A*(a*c*m + b*d*n) - B*(a*d*n + b*c*m) - d*(-A*a*(m + n) + B*b*(m - n))*tan(e + f*x), x), x), x) - Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(A*b - B*a)/(S(2)*a*f*m), x)
rule3691 = ReplacementRule(pattern3691, replacement3691)
pattern3692 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1439, cons93, cons267, cons88)
def replacement3692(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(3692)
return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(A*(a*c*m + b*d*n) - B*(a*d*n + b*c*m) - d*(-A*a*(m + n) + B*b*(m - n))/tan(e + f*x), x), x), x) + Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n*(A*b - B*a)/(S(2)*a*f*m), x)
rule3692 = ReplacementRule(pattern3692, replacement3692)
pattern3693 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1439, cons31, cons267, cons1327)
def replacement3693(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(3693)
return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(A*(-a*d*(S(2)*m + n + S(1)) + b*c*m) + B*(a*c*m - b*d*(n + S(1))) + d*(A*b - B*a)*(m + n + S(1))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*a + B*b)/(S(2)*f*m*(-a*d + b*c)), x)
rule3693 = ReplacementRule(pattern3693, replacement3693)
pattern3694 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1439, cons31, cons267, cons1327)
def replacement3694(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(3694)
return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(A*(-a*d*(S(2)*m + n + S(1)) + b*c*m) + B*(a*c*m - b*d*(n + S(1))) + d*(A*b - B*a)*(m + n + S(1))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*a + B*b)/(S(2)*f*m*(-a*d + b*c)), x)
rule3694 = ReplacementRule(pattern3694, replacement3694)
pattern3695 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1439, cons87, cons88)
def replacement3695(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(3695)
return Dist(S(1)/(a*(m + n)), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(-1))*Simp(A*a*c*(m + n) - B*(a*d*n + b*c*m) + (A*a*d*(m + n) - B*(-a*c*n + b*d*m))*tan(e + f*x), x), x), x) + Simp(B*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n/(f*(m + n)), x)
rule3695 = ReplacementRule(pattern3695, replacement3695)
pattern3696 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1439, cons87, cons88)
def replacement3696(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(3696)
return Dist(S(1)/(a*(m + n)), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(-1))*Simp(A*a*c*(m + n) - B*(a*d*n + b*c*m) + (A*a*d*(m + n) - B*(-a*c*n + b*d*m))/tan(e + f*x), x), x), x) - Simp(B*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n/(f*(m + n)), x)
rule3696 = ReplacementRule(pattern3696, replacement3696)
pattern3697 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1439, cons87, cons89)
def replacement3697(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(3697)
return -Dist(S(1)/(a*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*(-a*c*(n + S(1)) + b*d*m) - B*(a*d*(n + S(1)) + b*c*m) - a*(-A*d + B*c)*(m + n + S(1))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d - B*c)/(f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3697 = ReplacementRule(pattern3697, replacement3697)
pattern3698 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1439, cons87, cons89)
def replacement3698(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(3698)
return -Dist(S(1)/(a*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*(-a*c*(n + S(1)) + b*d*m) - B*(a*d*(n + S(1)) + b*c*m) - a*(-A*d + B*c)*(m + n + S(1))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d - B*c)/(f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3698 = ReplacementRule(pattern3698, replacement3698)
pattern3699 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1439, cons1418)
def replacement3699(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(3699)
return Dist(B*b/f, Subst(Int((a + b*x)**(m + S(-1))*(c + d*x)**n, x), x, tan(e + f*x)), x)
rule3699 = ReplacementRule(pattern3699, replacement3699)
pattern3700 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1439, cons1418)
def replacement3700(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(3700)
return -Dist(B*b/f, Subst(Int((a + b*x)**(m + S(-1))*(c + d*x)**n, x), x, S(1)/tan(e + f*x)), x)
rule3700 = ReplacementRule(pattern3700, replacement3700)
pattern3701 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1439, cons1426)
def replacement3701(B, e, m, f, b, d, c, a, x, A):
rubi.append(3701)
return Dist((A*b + B*a)/(a*d + b*c), Int((a + b*tan(e + f*x))**m, x), x) - Dist((-A*d + B*c)/(a*d + b*c), Int((a - b*tan(e + f*x))*(a + b*tan(e + f*x))**m/(c + d*tan(e + f*x)), x), x)
rule3701 = ReplacementRule(pattern3701, replacement3701)
pattern3702 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1439, cons1426)
def replacement3702(B, m, f, b, d, c, a, A, x, e):
rubi.append(3702)
return Dist((A*b + B*a)/(a*d + b*c), Int((a + b/tan(e + f*x))**m, x), x) - Dist((-A*d + B*c)/(a*d + b*c), Int((a - b/tan(e + f*x))*(a + b/tan(e + f*x))**m/(c + d/tan(e + f*x)), x), x)
rule3702 = ReplacementRule(pattern3702, replacement3702)
pattern3703 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1439, cons1426)
def replacement3703(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(3703)
return -Dist(B/b, Int((a - b*tan(e + f*x))*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n, x), x) + Dist((A*b + B*a)/b, Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n, x), x)
rule3703 = ReplacementRule(pattern3703, replacement3703)
pattern3704 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1439, cons1426)
def replacement3704(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(3704)
return -Dist(B/b, Int((a - b/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) + Dist((A*b + B*a)/b, Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x)
rule3704 = ReplacementRule(pattern3704, replacement3704)
pattern3705 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1440, cons18, cons23, cons1513, cons1558)
def replacement3705(B, m, f, b, d, a, c, n, A, x, e):
rubi.append(3705)
return Dist(A**S(2)/f, Subst(Int((a + b*x)**m*(c + d*x)**n/(A - B*x), x), x, tan(e + f*x)), x)
rule3705 = ReplacementRule(pattern3705, replacement3705)
pattern3706 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1440, cons18, cons23, cons1513, cons1558)
def replacement3706(B, m, f, b, d, c, a, n, A, x, e):
rubi.append(3706)
return -Dist(A**S(2)/f, Subst(Int((a + b*x)**m*(c + d*x)**n/(A - B*x), x), x, S(1)/tan(e + f*x)), x)
rule3706 = ReplacementRule(pattern3706, replacement3706)
pattern3707 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1440, cons18, cons23, cons1513, cons1559)
def replacement3707(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3707)
return Dist(A/S(2) - I*B/S(2), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(I*tan(e + f*x) + S(1)), x), x) + Dist(A/S(2) + I*B/S(2), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(-I*tan(e + f*x) + S(1)), x), x)
rule3707 = ReplacementRule(pattern3707, replacement3707)
pattern3708 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1440, cons18, cons23, cons1513, cons1559)
def replacement3708(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3708)
return Dist(A/S(2) - I*B/S(2), Int((S(1) + I/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) + Dist(A/S(2) + I*B/S(2), Int((S(1) - I/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x)
rule3708 = ReplacementRule(pattern3708, replacement3708)
pattern3709 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545, cons93, cons166, cons89, cons1334)
def replacement3709(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3709)
return -Dist(S(1)/(d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*a*d*(-a*c*(n + S(1)) + b*d*(m + S(-1))) - b*(-B*b*(c**S(2)*(m + S(-1)) - d**S(2)*(n + S(1))) + d*(m + n)*(-A*a*d + A*b*c + B*a*c))*tan(e + f*x)**S(2) - d*(n + S(1))*((A*a - B*b)*(-a*d + b*c) + (A*b + B*a)*(a*c + b*d))*tan(e + f*x) + (B*b*c - d*(A*b + B*a))*(a*d*(n + S(1)) + b*c*(m + S(-1))), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*(-A*d + B*c)*(-a*d + b*c)/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3709 = ReplacementRule(pattern3709, replacement3709)
pattern3710 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545, cons93, cons166, cons89, cons1334)
def replacement3710(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3710)
return -Dist(S(1)/(d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*a*d*(-a*c*(n + S(1)) + b*d*(m + S(-1))) - b*(-B*b*(c**S(2)*(m + S(-1)) - d**S(2)*(n + S(1))) + d*(m + n)*(-A*a*d + A*b*c + B*a*c))/tan(e + f*x)**S(2) - d*(n + S(1))*((A*a - B*b)*(-a*d + b*c) + (A*b + B*a)*(a*c + b*d))/tan(e + f*x) + (B*b*c - d*(A*b + B*a))*(a*d*(n + S(1)) + b*c*(m + S(-1))), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*(-A*d + B*c)*(-a*d + b*c)/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3710 = ReplacementRule(pattern3710, replacement3710)
pattern3711 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1440, cons1545, cons31, cons166, cons1334, cons1560)
def replacement3711(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3711)
return Dist(S(1)/(d*(m + n)), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**n*Simp(A*a**S(2)*d*(m + n) - B*b*(a*d*(n + S(1)) + b*c*(m + S(-1))) + d*(m + n)*(S(2)*A*a*b + B*(a**S(2) - b**S(2)))*tan(e + f*x) - (B*b*(m + S(-1))*(-a*d + b*c) - b*d*(m + n)*(A*b + B*a))*tan(e + f*x)**S(2), x), x), x) + Simp(B*b*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n)), x)
rule3711 = ReplacementRule(pattern3711, replacement3711)
pattern3712 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1440, cons1545, cons31, cons166, cons1334, cons1560)
def replacement3712(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3712)
return Dist(S(1)/(d*(m + n)), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**n*Simp(A*a**S(2)*d*(m + n) - B*b*(a*d*(n + S(1)) + b*c*(m + S(-1))) + d*(m + n)*(S(2)*A*a*b + B*(a**S(2) - b**S(2)))/tan(e + f*x) - (B*b*(m + S(-1))*(-a*d + b*c) - b*d*(m + n)*(A*b + B*a))/tan(e + f*x)**S(2), x), x), x) - Simp(B*b*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n)), x)
rule3712 = ReplacementRule(pattern3712, replacement3712)
pattern3713 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545, cons93, cons94, cons1325, cons1334)
def replacement3713(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3713)
return Dist(S(1)/(b*(a**S(2) + b**S(2))*(m + S(1))), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(A*b*(a*c*(m + S(1)) - b*d*n) + B*b*(a*d*n + b*c*(m + S(1))) - b*d*(A*b - B*a)*(m + n + S(1))*tan(e + f*x)**S(2) - b*(m + S(1))*(A*(-a*d + b*c) - B*(a*c + b*d))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*(A*b - B*a)/(f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3713 = ReplacementRule(pattern3713, replacement3713)
pattern3714 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545, cons93, cons94, cons1325, cons1334)
def replacement3714(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3714)
return Dist(S(1)/(b*(a**S(2) + b**S(2))*(m + S(1))), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(A*b*(a*c*(m + S(1)) - b*d*n) + B*b*(a*d*n + b*c*(m + S(1))) - b*d*(A*b - B*a)*(m + n + S(1))/tan(e + f*x)**S(2) - b*(m + S(1))*(A*(-a*d + b*c) - B*(a*c + b*d))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*(A*b - B*a)/(f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3714 = ReplacementRule(pattern3714, replacement3714)
pattern3715 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1440, cons1545, cons31, cons94, cons1334, cons1557)
def replacement3715(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3715)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) + B*b*(a*d*(n + S(1)) + b*c*(m + S(1))) - b*d*(A*b - B*a)*(m + n + S(2))*tan(e + f*x)**S(2) - (m + S(1))*(A*b - B*a)*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp(b*(a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(1))*(A*b - B*a)/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3715 = ReplacementRule(pattern3715, replacement3715)
pattern3716 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1440, cons1545, cons31, cons94, cons1334, cons1557)
def replacement3716(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3716)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) + B*b*(a*d*(n + S(1)) + b*c*(m + S(1))) - b*d*(A*b - B*a)*(m + n + S(2))/tan(e + f*x)**S(2) - (m + S(1))*(A*b - B*a)*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp(b*(a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(1))*(A*b - B*a)/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3716 = ReplacementRule(pattern3716, replacement3716)
pattern3717 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545, cons93, cons1256, cons1325)
def replacement3717(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3717)
return Dist(S(1)/(m + n), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(A*a*c*(m + n) - B*(a*d*n + b*c*m) + (m + n)*(A*a*d + A*b*c + B*a*c - B*b*d)*tan(e + f*x) + (A*b*d*(m + n) + B*(a*d*m + b*c*n))*tan(e + f*x)**S(2), x), x), x) + Simp(B*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n/(f*(m + n)), x)
rule3717 = ReplacementRule(pattern3717, replacement3717)
pattern3718 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545, cons93, cons1256, cons1325)
def replacement3718(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3718)
return Dist(S(1)/(m + n), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(A*a*c*(m + n) - B*(a*d*n + b*c*m) + (m + n)*(A*a*d + A*b*c + B*a*c - B*b*d)/tan(e + f*x) + (A*b*d*(m + n) + B*(a*d*m + b*c*n))/tan(e + f*x)**S(2), x), x), x) - Simp(B*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n/(f*(m + n)), x)
rule3718 = ReplacementRule(pattern3718, replacement3718)
pattern3719 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545)
def replacement3719(B, e, f, b, d, c, a, x, A):
rubi.append(3719)
return Dist(b*(A*b - B*a)/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) + Dist(d*(-A*d + B*c)/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c*tan(e + f*x) + d)/(c + d*tan(e + f*x)), x), x) + Simp(x*(A*(a*c - b*d) + B*(a*d + b*c))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x)
rule3719 = ReplacementRule(pattern3719, replacement3719)
pattern3720 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545)
def replacement3720(B, f, b, d, c, a, A, x, e):
rubi.append(3720)
return Dist(b*(A*b - B*a)/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) + Dist(d*(-A*d + B*c)/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c/tan(e + f*x) + d)/(c + d/tan(e + f*x)), x), x) + Simp(x*(A*(a*c - b*d) + B*(a*d + b*c))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x)
rule3720 = ReplacementRule(pattern3720, replacement3720)
pattern3721 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545)
def replacement3721(B, e, f, b, d, a, c, x, A):
rubi.append(3721)
return -Dist((-A*b + B*a)*(-a*d + b*c)/(a**S(2) + b**S(2)), Int((tan(e + f*x)**S(2) + S(1))/((a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int(Simp(A*(a*c + b*d) + B*(-a*d + b*c) - (A*(-a*d + b*c) - B*(a*c + b*d))*tan(e + f*x), x)/sqrt(c + d*tan(e + f*x)), x), x)
rule3721 = ReplacementRule(pattern3721, replacement3721)
pattern3722 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545)
def replacement3722(B, e, f, b, d, a, c, x, A):
rubi.append(3722)
return -Dist((-A*b + B*a)*(-a*d + b*c)/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/((a + b/tan(e + f*x))*sqrt(c + d/tan(e + f*x))), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int(Simp(A*(a*c + b*d) + B*(-a*d + b*c) - (A*(-a*d + b*c) - B*(a*c + b*d))/tan(e + f*x), x)/sqrt(c + d/tan(e + f*x)), x), x)
rule3722 = ReplacementRule(pattern3722, replacement3722)
pattern3723 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1440, cons1545)
def replacement3723(B, e, f, b, d, a, c, n, x, A):
rubi.append(3723)
return Dist(b*(A*b - B*a)/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*(tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*Simp(A*a + B*b - (A*b - B*a)*tan(e + f*x), x), x), x)
rule3723 = ReplacementRule(pattern3723, replacement3723)
pattern3724 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1440, cons1545)
def replacement3724(B, e, f, b, d, a, c, n, x, A):
rubi.append(3724)
return Dist(b*(A*b - B*a)/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))*(c + d/tan(e + f*x))**n/(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d/tan(e + f*x))**n*Simp(A*a + B*b - (A*b - B*a)/tan(e + f*x), x), x), x)
rule3724 = ReplacementRule(pattern3724, replacement3724)
pattern3725 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545)
def replacement3725(B, e, f, b, d, a, c, x, A):
rubi.append(3725)
return Dist(B*b, Int((tan(e + f*x)**S(2) + S(1))/(sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x), x) + Int(Simp(A*a - B*b + (A*b + B*a)*tan(e + f*x), x)/(sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x)
rule3725 = ReplacementRule(pattern3725, replacement3725)
pattern3726 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1440, cons1545)
def replacement3726(B, e, f, b, d, a, c, x, A):
rubi.append(3726)
return Dist(B*b, Int((S(1) + tan(e + f*x)**(S(-2)))/(sqrt(a + b/tan(e + f*x))*sqrt(c + d/tan(e + f*x))), x), x) + Int(Simp(A*a - B*b + (A*b + B*a)/tan(e + f*x), x)/(sqrt(a + b/tan(e + f*x))*sqrt(c + d/tan(e + f*x))), x)
rule3726 = ReplacementRule(pattern3726, replacement3726)
pattern3727 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1440, cons1558)
def replacement3727(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3727)
return Dist(A**S(2)/f, Subst(Int((a + b*x)**m*(c + d*x)**n/(A - B*x), x), x, tan(e + f*x)), x)
rule3727 = ReplacementRule(pattern3727, replacement3727)
pattern3728 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1440, cons1558)
def replacement3728(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3728)
return -Dist(A**S(2)/f, Subst(Int((a + b*x)**m*(c + d*x)**n/(A - B*x), x), x, S(1)/tan(e + f*x)), x)
rule3728 = ReplacementRule(pattern3728, replacement3728)
pattern3729 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1440, cons1559)
def replacement3729(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3729)
return Dist(A/S(2) - I*B/S(2), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(I*tan(e + f*x) + S(1)), x), x) + Dist(A/S(2) + I*B/S(2), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(-I*tan(e + f*x) + S(1)), x), x)
rule3729 = ReplacementRule(pattern3729, replacement3729)
pattern3730 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1440, cons1559)
def replacement3730(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(3730)
return Dist(A/S(2) - I*B/S(2), Int((S(1) + I/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) + Dist(A/S(2) + I*B/S(2), Int((S(1) - I/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x)
rule3730 = ReplacementRule(pattern3730, replacement3730)
pattern3731 = Pattern(Integral((A_ + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1561)
def replacement3731(C, m, f, b, a, A, x, e):
rubi.append(3731)
return Dist(A/(b*f), Subst(Int((a + x)**m, x), x, b*tan(e + f*x)), x)
rule3731 = ReplacementRule(pattern3731, replacement3731)
pattern3732 = Pattern(Integral((A_ + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1561)
def replacement3732(C, m, f, b, a, A, x, e):
rubi.append(3732)
return -Dist(A/(b*f), Subst(Int((a + x)**m, x), x, b/tan(e + f*x)), x)
rule3732 = ReplacementRule(pattern3732, replacement3732)
pattern3733 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons33)
def replacement3733(B, C, m, f, b, a, A, x, e):
rubi.append(3733)
return Dist(b**(S(-2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b*tan(e + f*x), x), x), x)
rule3733 = ReplacementRule(pattern3733, replacement3733)
pattern3734 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons33)
def replacement3734(B, C, m, f, b, a, A, x, e):
rubi.append(3734)
return Dist(b**(S(-2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b/tan(e + f*x), x), x), x)
rule3734 = ReplacementRule(pattern3734, replacement3734)
pattern3735 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1433)
def replacement3735(C, m, f, b, a, A, x, e):
rubi.append(3735)
return -Dist(C/b**S(2), Int((a - b*tan(e + f*x))*(a + b*tan(e + f*x))**(m + S(1)), x), x)
rule3735 = ReplacementRule(pattern3735, replacement3735)
pattern3736 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1433)
def replacement3736(C, m, f, b, a, A, x, e):
rubi.append(3736)
return -Dist(C/b**S(2), Int((a - b/tan(e + f*x))*(a + b/tan(e + f*x))**(m + S(1)), x), x)
rule3736 = ReplacementRule(pattern3736, replacement3736)
pattern3737 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1562, cons31, cons32, cons1439)
def replacement3737(B, C, m, f, b, a, A, x, e):
rubi.append(3737)
return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(A*a*(S(2)*m + S(1)) + B*b - C*a - (C*b*(m + S(-1)) + (m + S(1))*(A*b - B*a))*tan(e + f*x), x), x), x) - Simp((a + b*tan(e + f*x))**m*(A*a + B*b - C*a)*tan(e + f*x)/(S(2)*a*f*m), x)
rule3737 = ReplacementRule(pattern3737, replacement3737)
pattern3738 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1562, cons31, cons32, cons1439)
def replacement3738(B, C, m, f, b, a, A, x, e):
rubi.append(3738)
return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(A*a*(S(2)*m + S(1)) + B*b - C*a - (C*b*(m + S(-1)) + (m + S(1))*(A*b - B*a))/tan(e + f*x), x), x), x) + Simp((a + b/tan(e + f*x))**m*(A*a + B*b - C*a)/(S(2)*a*f*m*tan(e + f*x)), x)
rule3738 = ReplacementRule(pattern3738, replacement3738)
pattern3739 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1563, cons31, cons32, cons1439)
def replacement3739(C, m, f, b, a, A, x, e):
rubi.append(3739)
return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(A*a*(S(2)*m + S(1)) - C*a - (A*b*(m + S(1)) + C*b*(m + S(-1)))*tan(e + f*x), x), x), x) - Simp((a + b*tan(e + f*x))**m*(A*a - C*a)*tan(e + f*x)/(S(2)*a*f*m), x)
rule3739 = ReplacementRule(pattern3739, replacement3739)
pattern3740 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1563, cons31, cons32, cons1439)
def replacement3740(C, m, f, b, a, A, x, e):
rubi.append(3740)
return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(A*a*(S(2)*m + S(1)) - C*a - (A*b*(m + S(1)) + C*b*(m + S(-1)))/tan(e + f*x), x), x), x) + Simp((a + b/tan(e + f*x))**m*(A*a - C*a)/(S(2)*a*f*m*tan(e + f*x)), x)
rule3740 = ReplacementRule(pattern3740, replacement3740)
pattern3741 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1440, cons1564)
def replacement3741(B, C, f, b, a, A, x, e):
rubi.append(3741)
return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Simp(x*(A*a + B*b - C*a)/(a**S(2) + b**S(2)), x)
rule3741 = ReplacementRule(pattern3741, replacement3741)
pattern3742 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1440, cons1564)
def replacement3742(B, C, f, b, a, A, x, e):
rubi.append(3742)
return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(a + b/tan(e + f*x)), x), x) + Simp(x*(A*a + B*b - C*a)/(a**S(2) + b**S(2)), x)
rule3742 = ReplacementRule(pattern3742, replacement3742)
pattern3743 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons48, cons125, cons34, cons35, cons36, cons1565)
def replacement3743(B, C, f, A, x, e):
rubi.append(3743)
return Dist(A, Int(S(1)/tan(e + f*x), x), x) + Dist(C, Int(tan(e + f*x), x), x) + Simp(B*x, x)
rule3743 = ReplacementRule(pattern3743, replacement3743)
pattern3744 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons48, cons125, cons34, cons35, cons36, cons1565)
def replacement3744(B, C, f, A, x, e):
rubi.append(3744)
return Dist(A, Int(tan(e + f*x), x), x) + Dist(C, Int(S(1)/tan(e + f*x), x), x) + Simp(B*x, x)
rule3744 = ReplacementRule(pattern3744, replacement3744)
pattern3745 = Pattern(Integral((A_ + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons48, cons125, cons34, cons36, cons1565)
def replacement3745(C, e, f, x, A):
rubi.append(3745)
return Dist(A, Int(S(1)/tan(e + f*x), x), x) + Dist(C, Int(tan(e + f*x), x), x)
rule3745 = ReplacementRule(pattern3745, replacement3745)
pattern3746 = Pattern(Integral((A_ + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons48, cons125, cons34, cons36, cons1565)
def replacement3746(C, e, f, x, A):
rubi.append(3746)
return Dist(A, Int(tan(e + f*x), x), x) + Dist(C, Int(S(1)/tan(e + f*x), x), x)
rule3746 = ReplacementRule(pattern3746, replacement3746)
pattern3747 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1562, cons1440, cons1566)
def replacement3747(B, C, f, b, a, A, x, e):
rubi.append(3747)
return -Dist((A*b - B*a - C*b)/(a**S(2) + b**S(2)), Int(tan(e + f*x), x), x) + Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Simp(x*(A*a + B*b - C*a)/(a**S(2) + b**S(2)), x)
rule3747 = ReplacementRule(pattern3747, replacement3747)
pattern3748 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1562, cons1440, cons1566)
def replacement3748(B, C, f, b, a, A, x, e):
rubi.append(3748)
return -Dist((A*b - B*a - C*b)/(a**S(2) + b**S(2)), Int(S(1)/tan(e + f*x), x), x) + Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(a + b/tan(e + f*x)), x), x) + Simp(x*(A*a + B*b - C*a)/(a**S(2) + b**S(2)), x)
rule3748 = ReplacementRule(pattern3748, replacement3748)
pattern3749 = Pattern(Integral((A_ + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1563, cons1440, cons1565)
def replacement3749(C, f, b, a, A, x, e):
rubi.append(3749)
return Dist((A*b**S(2) + C*a**S(2))/(a**S(2) + b**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) - Dist(b*(A - C)/(a**S(2) + b**S(2)), Int(tan(e + f*x), x), x) + Simp(a*x*(A - C)/(a**S(2) + b**S(2)), x)
rule3749 = ReplacementRule(pattern3749, replacement3749)
pattern3750 = Pattern(Integral((A_ + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1563, cons1440, cons1565)
def replacement3750(C, f, b, a, A, x, e):
rubi.append(3750)
return Dist((A*b**S(2) + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(a + b/tan(e + f*x)), x), x) - Dist(b*(A - C)/(a**S(2) + b**S(2)), Int(S(1)/tan(e + f*x), x), x) + Simp(a*x*(A - C)/(a**S(2) + b**S(2)), x)
rule3750 = ReplacementRule(pattern3750, replacement3750)
pattern3751 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1562, cons31, cons94, cons1440)
def replacement3751(B, C, e, m, f, b, a, x, A):
rubi.append(3751)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(B*b + a*(A - C) - (A*b - B*a - C*b)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3751 = ReplacementRule(pattern3751, replacement3751)
pattern3752 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1562, cons31, cons94, cons1440)
def replacement3752(B, C, e, m, f, b, a, x, A):
rubi.append(3752)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(B*b + a*(A - C) - (A*b - B*a - C*b)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3752 = ReplacementRule(pattern3752, replacement3752)
pattern3753 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1563, cons31, cons94, cons1440)
def replacement3753(C, e, m, f, b, a, x, A):
rubi.append(3753)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(a*(A - C) - (A*b - C*b)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3753 = ReplacementRule(pattern3753, replacement3753)
pattern3754 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1563, cons31, cons94, cons1440)
def replacement3754(C, e, m, f, b, a, x, A):
rubi.append(3754)
return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(a*(A - C) - (A*b - C*b)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x)
rule3754 = ReplacementRule(pattern3754, replacement3754)
pattern3755 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1562, cons1549)
def replacement3755(B, C, m, f, b, a, A, x, e):
rubi.append(3755)
return Int((a + b*tan(e + f*x))**m*Simp(A + B*tan(e + f*x) - C, x), x) + Simp(C*(a + b*tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule3755 = ReplacementRule(pattern3755, replacement3755)
pattern3756 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1562, cons1549)
def replacement3756(B, C, m, f, b, a, A, x, e):
rubi.append(3756)
return Int((a + b/tan(e + f*x))**m*Simp(A + B/tan(e + f*x) - C, x), x) - Simp(C*(a + b/tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule3756 = ReplacementRule(pattern3756, replacement3756)
pattern3757 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1563, cons1549)
def replacement3757(C, m, f, b, a, A, x, e):
rubi.append(3757)
return Dist(A - C, Int((a + b*tan(e + f*x))**m, x), x) + Simp(C*(a + b*tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule3757 = ReplacementRule(pattern3757, replacement3757)
pattern3758 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1563, cons1549)
def replacement3758(C, m, f, b, a, A, x, e):
rubi.append(3758)
return Dist(A - C, Int((a + b/tan(e + f*x))**m, x), x) - Simp(C*(a + b/tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule3758 = ReplacementRule(pattern3758, replacement3758)
pattern3759 = Pattern(Integral((A_ + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons1561)
def replacement3759(C, m, f, b, d, a, c, n, A, x, e):
rubi.append(3759)
return Dist(A/f, Subst(Int((a + b*x)**m*(c + d*x)**n, x), x, tan(e + f*x)), x)
rule3759 = ReplacementRule(pattern3759, replacement3759)
pattern3760 = Pattern(Integral((A_ + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons1561)
def replacement3760(C, m, f, b, d, a, c, n, A, x, e):
rubi.append(3760)
return -Dist(A/f, Subst(Int((a + b*x)**m*(c + d*x)**n, x), x, S(1)/tan(e + f*x)), x)
rule3760 = ReplacementRule(pattern3760, replacement3760)
pattern3761 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1545, cons87, cons89)
def replacement3761(B, C, f, b, d, c, a, n, A, x, e):
rubi.append(3761)
return Dist(S(1)/(d*(c**S(2) + d**S(2))), Int((c + d*tan(e + f*x))**(n + S(1))*Simp(C*b*(c**S(2) + d**S(2))*tan(e + f*x)**S(2) + a*d*(A*c + B*d - C*c) + b*(A*d**S(2) - B*c*d + C*c**S(2)) + d*(-A*a*d + A*b*c + B*a*c + B*b*d + C*a*d - C*b*c)*tan(e + f*x), x), x), x) - Simp((c + d*tan(e + f*x))**(n + S(1))*(-a*d + b*c)*(A*d**S(2) - B*c*d + C*c**S(2))/(d**S(2)*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3761 = ReplacementRule(pattern3761, replacement3761)
pattern3762 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1545, cons87, cons89)
def replacement3762(B, C, e, f, b, d, a, c, n, x, A):
rubi.append(3762)
return Dist(S(1)/(d*(c**S(2) + d**S(2))), Int((c + d/tan(e + f*x))**(n + S(1))*Simp(C*b*(c**S(2) + d**S(2))/tan(e + f*x)**S(2) + a*d*(A*c + B*d - C*c) + b*(A*d**S(2) - B*c*d + C*c**S(2)) + d*(-A*a*d + A*b*c + B*a*c + B*b*d + C*a*d - C*b*c)/tan(e + f*x), x), x), x) + Simp((c + d/tan(e + f*x))**(n + S(1))*(-a*d + b*c)*(A*d**S(2) - B*c*d + C*c**S(2))/(d**S(2)*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3762 = ReplacementRule(pattern3762, replacement3762)
pattern3763 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1545, cons87, cons89)
def replacement3763(C, f, b, d, c, a, n, A, x, e):
rubi.append(3763)
return Dist(S(1)/(d*(c**S(2) + d**S(2))), Int((c + d*tan(e + f*x))**(n + S(1))*Simp(C*b*(c**S(2) + d**S(2))*tan(e + f*x)**S(2) + a*d*(A*c - C*c) + b*(A*d**S(2) + C*c**S(2)) + d*(-A*a*d + A*b*c + C*a*d - C*b*c)*tan(e + f*x), x), x), x) - Simp((c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*(-a*d + b*c)/(d**S(2)*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3763 = ReplacementRule(pattern3763, replacement3763)
pattern3764 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1545, cons87, cons89)
def replacement3764(C, e, f, b, d, a, c, n, x, A):
rubi.append(3764)
return Dist(S(1)/(d*(c**S(2) + d**S(2))), Int((c + d/tan(e + f*x))**(n + S(1))*Simp(C*b*(c**S(2) + d**S(2))/tan(e + f*x)**S(2) + a*d*(A*c - C*c) + b*(A*d**S(2) + C*c**S(2)) + d*(-A*a*d + A*b*c + C*a*d - C*b*c)/tan(e + f*x), x), x), x) + Simp((c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*(-a*d + b*c)/(d**S(2)*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3764 = ReplacementRule(pattern3764, replacement3764)
pattern3765 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1545, cons346)
def replacement3765(B, C, f, b, d, c, n, a, A, x, e):
rubi.append(3765)
return -Dist(S(1)/(d*(n + S(2))), Int((c + d*tan(e + f*x))**n*Simp(-A*a*d*(n + S(2)) + C*b*c - d*(n + S(2))*(A*b + B*a - C*b)*tan(e + f*x) - (C*a*d*(n + S(2)) - b*(-B*d*(n + S(2)) + C*c))*tan(e + f*x)**S(2), x), x), x) + Simp(C*b*(c + d*tan(e + f*x))**(n + S(1))*tan(e + f*x)/(d*f*(n + S(2))), x)
rule3765 = ReplacementRule(pattern3765, replacement3765)
pattern3766 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1545, cons346)
def replacement3766(B, C, f, b, d, c, n, a, A, x, e):
rubi.append(3766)
return -Dist(S(1)/(d*(n + S(2))), Int((c + d/tan(e + f*x))**n*Simp(-A*a*d*(n + S(2)) + C*b*c - d*(n + S(2))*(A*b + B*a - C*b)/tan(e + f*x) - (C*a*d*(n + S(2)) - b*(-B*d*(n + S(2)) + C*c))/tan(e + f*x)**S(2), x), x), x) - Simp(C*b*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(n + S(2))*tan(e + f*x)), x)
rule3766 = ReplacementRule(pattern3766, replacement3766)
pattern3767 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1545, cons346)
def replacement3767(C, f, b, d, c, n, a, A, x, e):
rubi.append(3767)
return -Dist(S(1)/(d*(n + S(2))), Int((c + d*tan(e + f*x))**n*Simp(-A*a*d*(n + S(2)) + C*b*c - d*(n + S(2))*(A*b - C*b)*tan(e + f*x) - (C*a*d*(n + S(2)) - C*b*c)*tan(e + f*x)**S(2), x), x), x) + Simp(C*b*(c + d*tan(e + f*x))**(n + S(1))*tan(e + f*x)/(d*f*(n + S(2))), x)
rule3767 = ReplacementRule(pattern3767, replacement3767)
pattern3768 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1545, cons346)
def replacement3768(C, f, b, d, c, n, a, A, x, e):
rubi.append(3768)
return -Dist(S(1)/(d*(n + S(2))), Int((c + d/tan(e + f*x))**n*Simp(-A*a*d*(n + S(2)) + C*b*c - d*(n + S(2))*(A*b - C*b)/tan(e + f*x) - (C*a*d*(n + S(2)) - C*b*c)/tan(e + f*x)**S(2), x), x), x) - Simp(C*b*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(n + S(2))*tan(e + f*x)), x)
rule3768 = ReplacementRule(pattern3768, replacement3768)
pattern3769 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1439, cons1567)
def replacement3769(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3769)
return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(a*(-A*d*(S(2)*m + n + S(1)) + B*c*m + C*d*(n + S(1))) + b*(-B*d*(n + S(1)) + c*m*(A + C)) + (A*b*d*(m + n + S(1)) + C*b*d*(m - n + S(-1)) + a*(-B*d*(m + n + S(1)) + S(2)*C*c*m))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*a + B*b - C*a)/(S(2)*f*m*(-a*d + b*c)), x)
rule3769 = ReplacementRule(pattern3769, replacement3769)
pattern3770 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1439, cons1567)
def replacement3770(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3770)
return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(a*(-A*d*(S(2)*m + n + S(1)) + B*c*m + C*d*(n + S(1))) + b*(-B*d*(n + S(1)) + c*m*(A + C)) + (A*b*d*(m + n + S(1)) + C*b*d*(m - n + S(-1)) + a*(-B*d*(m + n + S(1)) + S(2)*C*c*m))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*a + B*b - C*a)/(S(2)*f*m*(-a*d + b*c)), x)
rule3770 = ReplacementRule(pattern3770, replacement3770)
pattern3771 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1439, cons1567)
def replacement3771(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3771)
return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(a*(-A*d*(S(2)*m + n + S(1)) + C*d*(n + S(1))) + b*c*m*(A + C) + (A*b*d*(m + n + S(1)) + S(2)*C*a*c*m + C*b*d*(m - n + S(-1)))*tan(e + f*x), x), x), x) + Simp(a*(A - C)*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x)
rule3771 = ReplacementRule(pattern3771, replacement3771)
pattern3772 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1439, cons1567)
def replacement3772(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3772)
return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(a*(-A*d*(S(2)*m + n + S(1)) + C*d*(n + S(1))) + b*c*m*(A + C) + (A*b*d*(m + n + S(1)) + S(2)*C*a*c*m + C*b*d*(m - n + S(-1)))/tan(e + f*x), x), x), x) - Simp(a*(A - C)*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x)
rule3772 = ReplacementRule(pattern3772, replacement3772)
pattern3773 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons71, cons1439, cons1303, cons87, cons89, cons1545)
def replacement3773(B, C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3773)
return -Dist(S(1)/(a*d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*Simp(-a*d*(n + S(1))*(A*c + B*d - C*c) - a*(-C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(1)))*tan(e + f*x) + b*m*(A*d**S(2) - B*c*d + C*c**S(2)), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3773 = ReplacementRule(pattern3773, replacement3773)
pattern3774 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons71, cons1439, cons1303, cons87, cons89, cons1545)
def replacement3774(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3774)
return -Dist(S(1)/(a*d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*Simp(-a*d*(n + S(1))*(A*c + B*d - C*c) - a*(-C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(1)))/tan(e + f*x) + b*m*(A*d**S(2) - B*c*d + C*c**S(2)), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3774 = ReplacementRule(pattern3774, replacement3774)
pattern3775 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons71, cons1439, cons1303, cons87, cons89, cons1545)
def replacement3775(C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3775)
return -Dist(S(1)/(a*d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*Simp(-a*d*(n + S(1))*(A*c - C*c) - a*(-A*d**S(2)*(m + n + S(1)) - C*(c**S(2)*m - d**S(2)*(n + S(1))))*tan(e + f*x) + b*m*(A*d**S(2) + C*c**S(2)), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3775 = ReplacementRule(pattern3775, replacement3775)
pattern3776 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons71, cons1439, cons1303, cons87, cons89, cons1545)
def replacement3776(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3776)
return -Dist(S(1)/(a*d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*Simp(-a*d*(n + S(1))*(A*c - C*c) - a*(-A*d**S(2)*(m + n + S(1)) - C*(c**S(2)*m - d**S(2)*(n + S(1))))/tan(e + f*x) + b*m*(A*d**S(2) + C*c**S(2)), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3776 = ReplacementRule(pattern3776, replacement3776)
pattern3777 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons71, cons1439, cons1303, cons683)
def replacement3777(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3777)
return Dist(S(1)/(b*d*(m + n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*Simp(A*b*d*(m + n + S(1)) + C*(a*c*m - b*d*(n + S(1))) - (-B*b*d*(m + n + S(1)) + C*m*(-a*d + b*c))*tan(e + f*x), x), x), x) + Simp(C*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x)
rule3777 = ReplacementRule(pattern3777, replacement3777)
pattern3778 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons71, cons1439, cons1303, cons683)
def replacement3778(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3778)
return Dist(S(1)/(b*d*(m + n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n*Simp(A*b*d*(m + n + S(1)) + C*(a*c*m - b*d*(n + S(1))) - (-B*b*d*(m + n + S(1)) + C*m*(-a*d + b*c))/tan(e + f*x), x), x), x) - Simp(C*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x)
rule3778 = ReplacementRule(pattern3778, replacement3778)
pattern3779 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons71, cons1439, cons1303, cons683)
def replacement3779(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3779)
return Dist(S(1)/(b*d*(m + n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*Simp(A*b*d*(m + n + S(1)) - C*m*(-a*d + b*c)*tan(e + f*x) + C*(a*c*m - b*d*(n + S(1))), x), x), x) + Simp(C*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x)
rule3779 = ReplacementRule(pattern3779, replacement3779)
pattern3780 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons71, cons1439, cons1303, cons683)
def replacement3780(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(3780)
return Dist(S(1)/(b*d*(m + n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n*Simp(A*b*d*(m + n + S(1)) - C*m*(-a*d + b*c)/tan(e + f*x) + C*(a*c*m - b*d*(n + S(1))), x), x), x) - Simp(C*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x)
rule3780 = ReplacementRule(pattern3780, replacement3780)
pattern3781 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1440, cons1545, cons93, cons168, cons89)
def replacement3781(B, C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3781)
return -Dist(S(1)/(d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*d*(-a*c*(n + S(1)) + b*d*m) - b*(-C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(1)))*tan(e + f*x)**S(2) - d*(n + S(1))*(B*(a*c + b*d) + (A - C)*(-a*d + b*c))*tan(e + f*x) + (-B*d + C*c)*(a*d*(n + S(1)) + b*c*m), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) + c*(-B*d + C*c))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3781 = ReplacementRule(pattern3781, replacement3781)
pattern3782 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1440, cons1545, cons93, cons168, cons89)
def replacement3782(B, C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3782)
return -Dist(S(1)/(d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*d*(-a*c*(n + S(1)) + b*d*m) - b*(-C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(1)))/tan(e + f*x)**S(2) - d*(n + S(1))*(B*(a*c + b*d) + (A - C)*(-a*d + b*c))/tan(e + f*x) + (-B*d + C*c)*(a*d*(n + S(1)) + b*c*m), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) + c*(-B*d + C*c))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3782 = ReplacementRule(pattern3782, replacement3782)
pattern3783 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1440, cons1545, cons93, cons168, cons89)
def replacement3783(C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3783)
return -Dist(S(1)/(d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*d*(-a*c*(n + S(1)) + b*d*m) + C*c*(a*d*(n + S(1)) + b*c*m) + b*(A*d**S(2)*(m + n + S(1)) + C*(c**S(2)*m - d**S(2)*(n + S(1))))*tan(e + f*x)**S(2) - d*(A - C)*(n + S(1))*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3783 = ReplacementRule(pattern3783, replacement3783)
pattern3784 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1440, cons1545, cons93, cons168, cons89)
def replacement3784(C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3784)
return -Dist(S(1)/(d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*d*(-a*c*(n + S(1)) + b*d*m) + C*c*(a*d*(n + S(1)) + b*c*m) + b*(A*d**S(2)*(m + n + S(1)) + C*(c**S(2)*m - d**S(2)*(n + S(1))))/tan(e + f*x)**S(2) - d*(A - C)*(n + S(1))*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x)
rule3784 = ReplacementRule(pattern3784, replacement3784)
pattern3785 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1440, cons1545, cons31, cons168, cons1568)
def replacement3785(B, C, m, f, b, d, a, c, n, A, x, e):
rubi.append(3785)
return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) - C*(a*d*(n + S(1)) + b*c*m) + d*(m + n + S(1))*(A*b + B*a - C*b)*tan(e + f*x) - (-B*b*d*(m + n + S(1)) + C*m*(-a*d + b*c))*tan(e + f*x)**S(2), x), x), x) + Simp(C*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x)
rule3785 = ReplacementRule(pattern3785, replacement3785)
pattern3786 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1440, cons1545, cons31, cons168, cons1568)
def replacement3786(B, C, m, f, b, d, a, c, n, A, x, e):
rubi.append(3786)
return Dist(S(1)/(d*(m + n + S(1))), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) - C*(a*d*(n + S(1)) + b*c*m) + d*(m + n + S(1))*(A*b + B*a - C*b)/tan(e + f*x) - (-B*b*d*(m + n + S(1)) + C*m*(-a*d + b*c))/tan(e + f*x)**S(2), x), x), x) - Simp(C*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x)
rule3786 = ReplacementRule(pattern3786, replacement3786)
pattern3787 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1440, cons1545, cons31, cons168, cons1568)
def replacement3787(C, m, f, b, d, a, c, n, A, x, e):
rubi.append(3787)
return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) - C*m*(-a*d + b*c)*tan(e + f*x)**S(2) - C*(a*d*(n + S(1)) + b*c*m) + d*(A*b - C*b)*(m + n + S(1))*tan(e + f*x), x), x), x) + Simp(C*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x)
rule3787 = ReplacementRule(pattern3787, replacement3787)
pattern3788 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1440, cons1545, cons31, cons168, cons1568)
def replacement3788(C, m, f, b, d, a, c, n, A, x, e):
rubi.append(3788)
return Dist(S(1)/(d*(m + n + S(1))), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) - C*m*(-a*d + b*c)/tan(e + f*x)**S(2) - C*(a*d*(n + S(1)) + b*c*m) + d*(A*b - C*b)*(m + n + S(1))/tan(e + f*x), x), x), x) - Simp(C*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x)
rule3788 = ReplacementRule(pattern3788, replacement3788)
pattern3789 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1440, cons1545, cons31, cons94, cons1557)
def replacement3789(B, C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3789)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) - d*(A*b**S(2) - a*(B*b - C*a))*(m + n + S(2))*tan(e + f*x)**S(2) - (m + S(1))*(-a*d + b*c)*(A*b - B*a - C*b)*tan(e + f*x) + (B*b - C*a)*(a*d*(n + S(1)) + b*c*(m + S(1))), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(1))*(A*b**S(2) - a*(B*b - C*a))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3789 = ReplacementRule(pattern3789, replacement3789)
pattern3790 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1440, cons1545, cons31, cons94, cons1557)
def replacement3790(B, C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3790)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) - d*(A*b**S(2) - a*(B*b - C*a))*(m + n + S(2))/tan(e + f*x)**S(2) - (m + S(1))*(-a*d + b*c)*(A*b - B*a - C*b)/tan(e + f*x) + (B*b - C*a)*(a*d*(n + S(1)) + b*c*(m + S(1))), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(1))*(A*b**S(2) - a*(B*b - C*a))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3790 = ReplacementRule(pattern3790, replacement3790)
pattern3791 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1440, cons1545, cons31, cons94, cons1557)
def replacement3791(C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3791)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) - C*a*(a*d*(n + S(1)) + b*c*(m + S(1))) - d*(A*b**S(2) + C*a**S(2))*(m + n + S(2))*tan(e + f*x)**S(2) - (m + S(1))*(A*b - C*b)*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(1))*(A*b**S(2) + C*a**S(2))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3791 = ReplacementRule(pattern3791, replacement3791)
pattern3792 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1440, cons1545, cons31, cons94, cons1557)
def replacement3792(C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3792)
return Dist(S(1)/((a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) - C*a*(a*d*(n + S(1)) + b*c*(m + S(1))) - d*(A*b**S(2) + C*a**S(2))*(m + n + S(2))/tan(e + f*x)**S(2) - (m + S(1))*(A*b - C*b)*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(1))*(A*b**S(2) + C*a**S(2))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3792 = ReplacementRule(pattern3792, replacement3792)
pattern3793 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1440, cons1545)
def replacement3793(B, C, f, b, d, c, a, A, x, e):
rubi.append(3793)
return Dist((A*b**S(2) - B*a*b + C*a**S(2))/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) - Dist((A*d**S(2) - B*c*d + C*c**S(2))/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c*tan(e + f*x) + d)/(c + d*tan(e + f*x)), x), x) + Simp(x*(a*(A*c + B*d - C*c) + b*(-A*d + B*c + C*d))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x)
rule3793 = ReplacementRule(pattern3793, replacement3793)
pattern3794 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1440, cons1545)
def replacement3794(B, C, f, b, d, c, a, A, x, e):
rubi.append(3794)
return Dist((A*b**S(2) - B*a*b + C*a**S(2))/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) - Dist((A*d**S(2) - B*c*d + C*c**S(2))/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c/tan(e + f*x) + d)/(c + d/tan(e + f*x)), x), x) + Simp(x*(a*(A*c + B*d - C*c) + b*(-A*d + B*c + C*d))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x)
rule3794 = ReplacementRule(pattern3794, replacement3794)
pattern3795 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1440, cons1545)
def replacement3795(C, f, b, d, c, a, A, x, e):
rubi.append(3795)
return Dist((A*b**S(2) + C*a**S(2))/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) - Dist((A*d**S(2) + C*c**S(2))/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c*tan(e + f*x) + d)/(c + d*tan(e + f*x)), x), x) + Simp(x*(a*(A*c - C*c) - b*(A*d - C*d))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x)
rule3795 = ReplacementRule(pattern3795, replacement3795)
pattern3796 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1440, cons1545)
def replacement3796(C, f, b, d, c, a, A, x, e):
rubi.append(3796)
return Dist((A*b**S(2) + C*a**S(2))/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) - Dist((A*d**S(2) + C*c**S(2))/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c/tan(e + f*x) + d)/(c + d/tan(e + f*x)), x), x) + Simp(x*(a*(A*c - C*c) - b*(A*d - C*d))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x)
rule3796 = ReplacementRule(pattern3796, replacement3796)
pattern3797 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1440, cons1545, cons1327, cons1569)
def replacement3797(B, C, f, b, d, c, a, n, A, x, e):
rubi.append(3797)
return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*(tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*Simp(B*b + a*(A - C) + (B*a - b*(A - C))*tan(e + f*x), x), x), x)
rule3797 = ReplacementRule(pattern3797, replacement3797)
pattern3798 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1440, cons1545, cons1327, cons1569)
def replacement3798(B, C, e, f, b, d, a, c, n, x, A):
rubi.append(3798)
return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))*(c + d/tan(e + f*x))**n/(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d/tan(e + f*x))**n*Simp(B*b + a*(A - C) + (B*a - b*(A - C))/tan(e + f*x), x), x), x)
rule3798 = ReplacementRule(pattern3798, replacement3798)
pattern3799 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1440, cons1545, cons1327, cons1569)
def replacement3799(C, f, b, d, c, a, n, A, x, e):
rubi.append(3799)
return Dist((A*b**S(2) + C*a**S(2))/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*(tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*Simp(a*(A - C) - (A*b - C*b)*tan(e + f*x), x), x), x)
rule3799 = ReplacementRule(pattern3799, replacement3799)
pattern3800 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1440, cons1545, cons1327, cons1569)
def replacement3800(C, e, f, b, d, a, c, n, x, A):
rubi.append(3800)
return Dist((A*b**S(2) + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))*(c + d/tan(e + f*x))**n/(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d/tan(e + f*x))**n*Simp(a*(A - C) - (A*b - C*b)/tan(e + f*x), x), x), x)
rule3800 = ReplacementRule(pattern3800, replacement3800)
pattern3801 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons71, cons1440, cons1545)
def replacement3801(B, C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3801)
return Dist(S(1)/(b*f), Subst(Int((a + x)**m*(c + d*x/b)**n*(A*b**S(2) + B*b*x + C*x**S(2))/(b**S(2) + x**S(2)), x), x, b*tan(e + f*x)), x)
rule3801 = ReplacementRule(pattern3801, replacement3801)
pattern3802 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons71, cons1440, cons1545)
def replacement3802(B, C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3802)
return -Dist(S(1)/(b*f), Subst(Int((a + x)**m*(c + d*x/b)**n*(A*b**S(2) + B*b*x + C*x**S(2))/(b**S(2) + x**S(2)), x), x, b/tan(e + f*x)), x)
rule3802 = ReplacementRule(pattern3802, replacement3802)
pattern3803 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons71, cons1440, cons1545)
def replacement3803(C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3803)
return Dist(S(1)/(b*f), Subst(Int((a + x)**m*(c + d*x/b)**n*(A*b**S(2) + C*x**S(2))/(b**S(2) + x**S(2)), x), x, b*tan(e + f*x)), x)
rule3803 = ReplacementRule(pattern3803, replacement3803)
pattern3804 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons71, cons1440, cons1545)
def replacement3804(C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3804)
return -Dist(S(1)/(b*f), Subst(Int((a + x)**m*(c + d*x/b)**n*(A*b**S(2) + C*x**S(2))/(b**S(2) + x**S(2)), x), x, b/tan(e + f*x)), x)
rule3804 = ReplacementRule(pattern3804, replacement3804)
pattern3805 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons1492)
def replacement3805(b, d, c, a, x):
rubi.append(3805)
return Dist(S(1)/a, Int(cos(c + d*x)**S(2), x), x)
rule3805 = ReplacementRule(pattern3805, replacement3805)
pattern3806 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons1492)
def replacement3806(b, d, c, a, x):
rubi.append(3806)
return Dist(S(1)/a, Int(sin(c + d*x)**S(2), x), x)
rule3806 = ReplacementRule(pattern3806, replacement3806)
pattern3807 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons1456)
def replacement3807(b, d, c, a, x):
rubi.append(3807)
return -Dist(b/(a - b), Int(S(1)/((a + b*tan(c + d*x)**S(2))*cos(c + d*x)**S(2)), x), x) + Simp(x/(a - b), x)
rule3807 = ReplacementRule(pattern3807, replacement3807)
pattern3808 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons1456)
def replacement3808(b, d, c, a, x):
rubi.append(3808)
return -Dist(b/(a - b), Int(S(1)/((a + b/tan(c + d*x)**S(2))*sin(c + d*x)**S(2)), x), x) + Simp(x/(a - b), x)
rule3808 = ReplacementRule(pattern3808, replacement3808)
pattern3809 = Pattern(Integral((a_ + (WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement3809(p, b, d, c, a, n, x, e):
rubi.append(3809)
return Dist(e/d, Subst(Int((a + b*x**n)**p/(e**S(2) + x**S(2)), x), x, e*tan(c + d*x)), x)
rule3809 = ReplacementRule(pattern3809, replacement3809)
pattern3810 = Pattern(Integral((a_ + (WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1570)
def replacement3810(p, b, d, c, n, a, x, e):
rubi.append(3810)
return -Dist(e/d, Subst(Int((a + b*x**n)**p/(e**S(2) + x**S(2)), x), x, e/tan(c + d*x)), x)
rule3810 = ReplacementRule(pattern3810, replacement3810)
def With3811(p, m, b, d, c, a, n, x, e):
f = FreeFactors(tan(c + d*x), x)
rubi.append(3811)
return Dist(f**(m + S(1))/d, Subst(Int(x**m*(a + b*(e*f*x)**n)**p*(f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)), x), x, tan(c + d*x)/f), x)
pattern3811 = Pattern(Integral((a_ + (WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1515)
rule3811 = ReplacementRule(pattern3811, With3811)
def With3812(p, m, b, d, c, n, a, x, e):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(3812)
return -Dist(f**(m + S(1))/d, Subst(Int(x**m*(a + b*(e*f*x)**n)**p*(f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)), x), x, S(1)/(f*tan(c + d*x))), x)
pattern3812 = Pattern(Integral((a_ + (WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1515)
rule3812 = ReplacementRule(pattern3812, With3812)
def With3813(p, m, b, d, c, a, n, x):
f = FreeFactors(cos(c + d*x), x)
rubi.append(3813)
return -Dist(f/d, Subst(Int((f*x)**(-n*p)*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a*(f*x)**n + b*(-f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, cos(c + d*x)/f), x)
pattern3813 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1481, cons1479, cons38)
rule3813 = ReplacementRule(pattern3813, With3813)
def With3814(p, m, b, d, c, n, a, x):
f = FreeFactors(sin(c + d*x), x)
rubi.append(3814)
return Dist(f/d, Subst(Int((f*x)**(-n*p)*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a*(f*x)**n + b*(-f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, sin(c + d*x)/f), x)
pattern3814 = Pattern(Integral((a_ + (S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1481, cons1479, cons38)
rule3814 = ReplacementRule(pattern3814, With3814)
def With3815(p, m, b, d, c, a, n, x, e):
f = FreeFactors(tan(c + d*x), x)
rubi.append(3815)
return Dist(f/d, Subst(Int((a + b*(e*f*x)**n)**p*(f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)), x), x, tan(c + d*x)/f), x)
pattern3815 = Pattern(Integral((a_ + (WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1515)
rule3815 = ReplacementRule(pattern3815, With3815)
def With3816(p, m, b, d, c, n, a, x, e):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(3816)
return -Dist(f/d, Subst(Int((a + b*(e*f*x)**n)**p*(f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)), x), x, S(1)/(f*tan(c + d*x))), x)
pattern3816 = Pattern(Integral((a_ + (WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1515)
rule3816 = ReplacementRule(pattern3816, With3816)
def With3817(p, m, b, d, c, a, n, x):
f = FreeFactors(sin(c + d*x), x)
rubi.append(3817)
return Dist(f/d, Subst(Int((-f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1)/2)*ExpandToSum(a*(-f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*(f*x)**n, x)**p, x), x, sin(c + d*x)/f), x)
pattern3817 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*(S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1228, cons743, cons38)
rule3817 = ReplacementRule(pattern3817, With3817)
def With3818(p, m, b, d, c, n, a, x):
f = FreeFactors(cos(c + d*x), x)
rubi.append(3818)
return -Dist(f/d, Subst(Int((-f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1)/2)*ExpandToSum(a*(-f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*(f*x)**n, x)**p, x), x, cos(c + d*x)/f), x)
pattern3818 = Pattern(Integral((a_ + (S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1228, cons743, cons38)
rule3818 = ReplacementRule(pattern3818, With3818)
pattern3819 = Pattern(Integral((a_ + (WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement3819(p, m, b, d, c, a, n, x, e):
rubi.append(3819)
return Dist(e/d, Subst(Int((x/e)**m*(a + b*x**n)**p/(e**S(2) + x**S(2)), x), x, e*tan(c + d*x)), x)
rule3819 = ReplacementRule(pattern3819, replacement3819)
pattern3820 = Pattern(Integral((a_ + (WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement3820(p, m, b, d, c, n, a, x, e):
rubi.append(3820)
return -Dist(e/d, Subst(Int((x/e)**m*(a + b*x**n)**p/(e**S(2) + x**S(2)), x), x, e/tan(c + d*x)), x)
rule3820 = ReplacementRule(pattern3820, replacement3820)
pattern3821 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons45, cons38)
def replacement3821(p, b, n2, d, c, n, a, x, e):
rubi.append(3821)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p), x), x)
rule3821 = ReplacementRule(pattern3821, replacement3821)
pattern3822 = Pattern(Integral((a_ + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons45, cons38)
def replacement3822(p, b, n2, d, c, n, a, x, e):
rubi.append(3822)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p), x), x)
rule3822 = ReplacementRule(pattern3822, replacement3822)
pattern3823 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons45, cons147)
def replacement3823(p, b, n2, d, c, n, a, x, e):
rubi.append(3823)
return Dist((b + S(2)*c*tan(d + e*x)**n)**(-S(2)*p)*(a + b*tan(d + e*x)**n + c*tan(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p), x), x)
rule3823 = ReplacementRule(pattern3823, replacement3823)
pattern3824 = Pattern(Integral((a_ + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons45, cons147)
def replacement3824(p, b, n2, d, c, n, a, x, e):
rubi.append(3824)
return Dist((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/tan(d + e*x))**n + c*(S(1)/tan(d + e*x))**(S(2)*n))**p, Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p), x), x)
rule3824 = ReplacementRule(pattern3824, replacement3824)
def With3825(b, n2, d, a, n, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(3825)
return Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*tan(d + e*x)**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*tan(d + e*x)**n + q), x), x)
pattern3825 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226)
rule3825 = ReplacementRule(pattern3825, With3825)
def With3826(b, n2, d, a, n, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(3826)
return Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*(S(1)/tan(d + e*x))**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*(S(1)/tan(d + e*x))**n + q), x), x)
pattern3826 = Pattern(Integral(S(1)/((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226)
rule3826 = ReplacementRule(pattern3826, With3826)
pattern3827 = Pattern(Integral(((WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons46, cons1515)
def replacement3827(p, m, f, b, n2, d, a, n, c, x, e):
rubi.append(3827)
return Dist(f/e, Subst(Int(x**m*(f**S(2) + x**S(2))**(-m/S(2) + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, f*tan(d + e*x)), x)
rule3827 = ReplacementRule(pattern3827, replacement3827)
pattern3828 = Pattern(Integral(((WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons46, cons1515)
def replacement3828(p, m, f, b, n2, d, a, n, c, x, e):
rubi.append(3828)
return -Dist(f/e, Subst(Int(x**m*(f**S(2) + x**S(2))**(-m/S(2) + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, f/tan(d + e*x)), x)
rule3828 = ReplacementRule(pattern3828, replacement3828)
def With3829(p, m, b, n2, d, a, n, c, x, e):
g = FreeFactors(cos(d + e*x), x)
rubi.append(3829)
return -Dist(g/e, Subst(Int((g*x)**(-S(2)*n*p)*(-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a*(g*x)**(S(2)*n) + b*(g*x)**n*(-g**S(2)*x**S(2) + S(1))**(n/S(2)) + c*(-g**S(2)*x**S(2) + S(1))**n, x)**p, x), x, cos(d + e*x)/g), x)
pattern3829 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1481, cons1479, cons38)
rule3829 = ReplacementRule(pattern3829, With3829)
def With3830(p, m, b, n2, d, a, n, c, x, e):
g = FreeFactors(sin(d + e*x), x)
rubi.append(3830)
return Dist(g/e, Subst(Int((g*x)**(-S(2)*n*p)*(-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a*(g*x)**(S(2)*n) + b*(g*x)**n*(-g**S(2)*x**S(2) + S(1))**(n/S(2)) + c*(-g**S(2)*x**S(2) + S(1))**n, x)**p, x), x, sin(d + e*x)/g), x)
pattern3830 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1481, cons1479, cons38)
rule3830 = ReplacementRule(pattern3830, With3830)
pattern3831 = Pattern(Integral(((WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons46, cons1480)
def replacement3831(p, m, f, b, n2, d, a, n, c, x, e):
rubi.append(3831)
return Dist(f**(m + S(1))/e, Subst(Int((f**S(2) + x**S(2))**(-m/S(2) + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, f*tan(d + e*x)), x)
rule3831 = ReplacementRule(pattern3831, replacement3831)
pattern3832 = Pattern(Integral(((WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons46, cons1480)
def replacement3832(p, m, f, b, n2, d, a, n, c, x, e):
rubi.append(3832)
return -Dist(f**(m + S(1))/e, Subst(Int((f**S(2) + x**S(2))**(-m/S(2) + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, f/tan(d + e*x)), x)
rule3832 = ReplacementRule(pattern3832, replacement3832)
def With3833(p, m, b, n2, d, a, n, c, x, e):
g = FreeFactors(sin(d + e*x), x)
rubi.append(3833)
return Dist(g/e, Subst(Int((-g**S(2)*x**S(2) + S(1))**(m/S(2) - n*p + S(-1)/2)*ExpandToSum(a*(-x**S(2) + S(1))**n + b*x**n*(-x**S(2) + S(1))**(n/S(2)) + c*x**(S(2)*n), x)**p, x), x, sin(d + e*x)/g), x)
pattern3833 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1481, cons1479, cons38)
rule3833 = ReplacementRule(pattern3833, With3833)
def With3834(p, m, b, n2, d, a, n, c, x, e):
g = FreeFactors(cos(d + e*x), x)
rubi.append(3834)
return -Dist(g/e, Subst(Int((-g**S(2)*x**S(2) + S(1))**(m/S(2) - n*p + S(-1)/2)*ExpandToSum(a*(-x**S(2) + S(1))**n + b*x**n*(-x**S(2) + S(1))**(n/S(2)) + c*x**(S(2)*n), x)**p, x), x, cos(d + e*x)/g), x)
pattern3834 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1481, cons1479, cons38)
rule3834 = ReplacementRule(pattern3834, With3834)
pattern3835 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons45, cons38)
def replacement3835(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3835)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x)
rule3835 = ReplacementRule(pattern3835, replacement3835)
pattern3836 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons45, cons38)
def replacement3836(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3836)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x)
rule3836 = ReplacementRule(pattern3836, replacement3836)
pattern3837 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons45, cons147)
def replacement3837(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3837)
return Dist((b + S(2)*c*tan(d + e*x)**n)**(-S(2)*p)*(a + b*tan(d + e*x)**n + c*tan(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x)
rule3837 = ReplacementRule(pattern3837, replacement3837)
pattern3838 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons45, cons147)
def replacement3838(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3838)
return Dist((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/tan(d + e*x))**n + c*(S(1)/tan(d + e*x))**(S(2)*n))**p, Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x)
rule3838 = ReplacementRule(pattern3838, replacement3838)
pattern3839 = Pattern(Integral(((WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons46, cons226)
def replacement3839(p, m, f, b, n2, d, a, n, c, x, e):
rubi.append(3839)
return Dist(f/e, Subst(Int((x/f)**m*(a + b*x**n + c*x**(S(2)*n))**p/(f**S(2) + x**S(2)), x), x, f*tan(d + e*x)), x)
rule3839 = ReplacementRule(pattern3839, replacement3839)
pattern3840 = Pattern(Integral(((WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons46, cons226)
def replacement3840(p, m, f, b, n2, d, a, n, c, x, e):
rubi.append(3840)
return -Dist(f/e, Subst(Int((x/f)**m*(a + b*x**n + c*x**(S(2)*n))**p/(f**S(2) + x**S(2)), x), x, f/tan(d + e*x)), x)
rule3840 = ReplacementRule(pattern3840, replacement3840)
pattern3841 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons45, cons38)
def replacement3841(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3841)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x)
rule3841 = ReplacementRule(pattern3841, replacement3841)
pattern3842 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons45, cons38)
def replacement3842(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3842)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p)*tan(d + e*x)**m, x), x)
rule3842 = ReplacementRule(pattern3842, replacement3842)
pattern3843 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons45, cons147)
def replacement3843(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3843)
return Dist((b + S(2)*c*tan(d + e*x)**n)**(-S(2)*p)*(a + b*tan(d + e*x)**n + c*tan(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x)
rule3843 = ReplacementRule(pattern3843, replacement3843)
pattern3844 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons45, cons147)
def replacement3844(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3844)
return Dist((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/tan(d + e*x))**n + c*(S(1)/tan(d + e*x))**(S(2)*n))**p, Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p)*tan(d + e*x)**m, x), x)
rule3844 = ReplacementRule(pattern3844, replacement3844)
def With3845(p, m, b, n2, d, a, c, n, x, e):
g = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(3845)
return Dist(g/e, Subst(Int((g*x)**(m - S(2)*n*p)*(a*(g*x)**(S(2)*n) + b*(g*x)**n + c)**p/(g**S(2)*x**S(2) + S(1)), x), x, S(1)/(g*tan(d + e*x))), x)
pattern3845 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons46, cons226, cons1479)
rule3845 = ReplacementRule(pattern3845, With3845)
def With3846(p, m, b, n2, d, c, a, n, x, e):
g = FreeFactors(tan(d + e*x), x)
rubi.append(3846)
return -Dist(g/e, Subst(Int((g*x)**(m - S(2)*n*p)*(a*(g*x)**(S(2)*n) + b*(g*x)**n + c)**p/(g**S(2)*x**S(2) + S(1)), x), x, tan(d + e*x)/g), x)
pattern3846 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**n2_*WC('c', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons46, cons226, cons1479)
rule3846 = ReplacementRule(pattern3846, With3846)
pattern3847 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons85)
def replacement3847(B, b, d, c, a, n, A, x, e):
rubi.append(3847)
return Dist(S(4)**(-n)*c**(-n), Int((A + B*tan(d + e*x))*(b + S(2)*c*tan(d + e*x))**(S(2)*n), x), x)
rule3847 = ReplacementRule(pattern3847, replacement3847)
pattern3848 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons85)
def replacement3848(B, b, d, c, a, n, A, x, e):
rubi.append(3848)
return Dist(S(4)**(-n)*c**(-n), Int((A + B/tan(d + e*x))*(b + S(2)*c/tan(d + e*x))**(S(2)*n), x), x)
rule3848 = ReplacementRule(pattern3848, replacement3848)
pattern3849 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons23)
def replacement3849(B, b, d, c, a, n, A, x, e):
rubi.append(3849)
return Dist((b + S(2)*c*tan(d + e*x))**(-S(2)*n)*(a + b*tan(d + e*x) + c*tan(d + e*x)**S(2))**n, Int((A + B*tan(d + e*x))*(b + S(2)*c*tan(d + e*x))**(S(2)*n), x), x)
rule3849 = ReplacementRule(pattern3849, replacement3849)
pattern3850 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons23)
def replacement3850(B, b, d, c, a, n, A, x, e):
rubi.append(3850)
return Dist((b + S(2)*c/tan(d + e*x))**(-S(2)*n)*(a + b/tan(d + e*x) + c/tan(d + e*x)**S(2))**n, Int((A + B/tan(d + e*x))*(b + S(2)*c/tan(d + e*x))**(S(2)*n), x), x)
rule3850 = ReplacementRule(pattern3850, replacement3850)
def With3851(B, b, d, a, c, A, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(3851)
return Dist(B - (-S(2)*A*c + B*b)/q, Int(S(1)/Simp(b + S(2)*c*tan(d + e*x) - q, x), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/Simp(b + S(2)*c*tan(d + e*x) + q, x), x), x)
pattern3851 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226)
rule3851 = ReplacementRule(pattern3851, With3851)
def With3852(B, b, d, c, a, A, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(3852)
return Dist(B - (-S(2)*A*c + B*b)/q, Int(S(1)/Simp(b + S(2)*c/tan(d + e*x) - q, x), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/Simp(b + S(2)*c/tan(d + e*x) + q, x), x), x)
pattern3852 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226)
rule3852 = ReplacementRule(pattern3852, With3852)
pattern3853 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226, cons85)
def replacement3853(B, b, d, a, c, n, A, x, e):
rubi.append(3853)
return Int(ExpandTrig((A + B*tan(d + e*x))*(a + b*tan(d + e*x) + c*tan(d + e*x)**S(2))**n, x), x)
rule3853 = ReplacementRule(pattern3853, replacement3853)
pattern3854 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226, cons85)
def replacement3854(B, b, d, c, a, n, A, x, e):
rubi.append(3854)
return Int(ExpandTrig((A + B/tan(d + e*x))*(a + b/tan(d + e*x) + c/tan(d + e*x)**S(2))**n, x), x)
rule3854 = ReplacementRule(pattern3854, replacement3854)
pattern3855 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons62)
def replacement3855(m, f, d, c, x, e):
rubi.append(3855)
return -Dist(S(2)*I, Int((c + d*x)**m*exp(S(2)*I*(e + f*x))/(exp(S(2)*I*(e + f*x)) + S(1)), x), x) + Simp(I*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule3855 = ReplacementRule(pattern3855, replacement3855)
pattern3856 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons62)
def replacement3856(m, f, d, c, x, e):
rubi.append(3856)
return -Dist(S(2)*I, Int((c + d*x)**m*exp(S(2)*I*(e + f*x))/(-exp(S(2)*I*(e + f*x)) + S(1)), x), x) - Simp(I*(c + d*x)**(m + S(1))/(d*(m + S(1))), x)
rule3856 = ReplacementRule(pattern3856, replacement3856)
pattern3857 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons168)
def replacement3857(m, f, b, d, c, n, x, e):
rubi.append(3857)
return -Dist(b**S(2), Int((b*tan(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) - Dist(b*d*m/(f*(n + S(-1))), Int((b*tan(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(-1)), x), x) + Simp(b*(b*tan(e + f*x))**(n + S(-1))*(c + d*x)**m/(f*(n + S(-1))), x)
rule3857 = ReplacementRule(pattern3857, replacement3857)
pattern3858 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons168)
def replacement3858(m, f, b, d, c, n, x, e):
rubi.append(3858)
return -Dist(b**S(2), Int((b/tan(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) + Dist(b*d*m/(f*(n + S(-1))), Int((b/tan(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(-1)), x), x) - Simp(b*(b/tan(e + f*x))**(n + S(-1))*(c + d*x)**m/(f*(n + S(-1))), x)
rule3858 = ReplacementRule(pattern3858, replacement3858)
pattern3859 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons93, cons89, cons168)
def replacement3859(m, f, b, d, c, n, x, e):
rubi.append(3859)
return -Dist(b**(S(-2)), Int((b*tan(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) - Dist(d*m/(b*f*(n + S(1))), Int((b*tan(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1)), x), x) + Simp((b*tan(e + f*x))**(n + S(1))*(c + d*x)**m/(b*f*(n + S(1))), x)
rule3859 = ReplacementRule(pattern3859, replacement3859)
pattern3860 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons93, cons89, cons168)
def replacement3860(m, f, b, d, c, n, x, e):
rubi.append(3860)
return -Dist(b**(S(-2)), Int((b/tan(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) + Dist(d*m/(b*f*(n + S(1))), Int((b/tan(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1)), x), x) - Simp((b/tan(e + f*x))**(n + S(1))*(c + d*x)**m/(b*f*(n + S(1))), x)
rule3860 = ReplacementRule(pattern3860, replacement3860)
pattern3861 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons528)
def replacement3861(m, f, b, d, c, n, a, x, e):
rubi.append(3861)
return Int(ExpandIntegrand((c + d*x)**m, (a + b*tan(e + f*x))**n, x), x)
rule3861 = ReplacementRule(pattern3861, replacement3861)
pattern3862 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons528)
def replacement3862(m, f, b, d, c, n, a, x, e):
rubi.append(3862)
return Int(ExpandIntegrand((c + d*x)**m, (a + b/tan(e + f*x))**n, x), x)
rule3862 = ReplacementRule(pattern3862, replacement3862)
pattern3863 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439, cons31, cons168)
def replacement3863(m, f, b, d, c, a, x, e):
rubi.append(3863)
return Dist(a*d*m/(S(2)*b*f), Int((c + d*x)**(m + S(-1))/(a + b*tan(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x) - Simp(a*(c + d*x)**m/(S(2)*b*f*(a + b*tan(e + f*x))), x)
rule3863 = ReplacementRule(pattern3863, replacement3863)
pattern3864 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439, cons31, cons168)
def replacement3864(m, f, b, d, c, a, x, e):
rubi.append(3864)
return -Dist(a*d*m/(S(2)*b*f), Int((c + d*x)**(m + S(-1))/(a + b/tan(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x) + Simp(a*(c + d*x)**m/(S(2)*b*f*(a + b/tan(e + f*x))), x)
rule3864 = ReplacementRule(pattern3864, replacement3864)
pattern3865 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439)
def replacement3865(f, b, d, c, a, x, e):
rubi.append(3865)
return -Dist(f/(a*d), Int(sin(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(f/(b*d), Int(cos(S(2)*e + S(2)*f*x)/(c + d*x), x), x) - Simp(S(1)/(d*(a + b*tan(e + f*x))*(c + d*x)), x)
rule3865 = ReplacementRule(pattern3865, replacement3865)
pattern3866 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439)
def replacement3866(f, b, d, c, a, x, e):
rubi.append(3866)
return Dist(f/(a*d), Int(sin(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(f/(b*d), Int(cos(S(2)*e + S(2)*f*x)/(c + d*x), x), x) - Simp(S(1)/(d*(a + b/tan(e + f*x))*(c + d*x)), x)
rule3866 = ReplacementRule(pattern3866, replacement3866)
pattern3867 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439, cons31, cons94, cons1510)
def replacement3867(m, f, b, d, c, a, x, e):
rubi.append(3867)
return Dist(S(2)*b*f/(a*d*(m + S(1))), Int((c + d*x)**(m + S(1))/(a + b*tan(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + b*tan(e + f*x))*(m + S(1))), x) + Simp(f*(c + d*x)**(m + S(2))/(b*d**S(2)*(m + S(1))*(m + S(2))), x)
rule3867 = ReplacementRule(pattern3867, replacement3867)
pattern3868 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439, cons31, cons94, cons1510)
def replacement3868(m, f, b, d, c, a, x, e):
rubi.append(3868)
return -Dist(S(2)*b*f/(a*d*(m + S(1))), Int((c + d*x)**(m + S(1))/(a + b/tan(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + b/tan(e + f*x))*(m + S(1))), x) - Simp(f*(c + d*x)**(m + S(2))/(b*d**S(2)*(m + S(1))*(m + S(2))), x)
rule3868 = ReplacementRule(pattern3868, replacement3868)
pattern3869 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439)
def replacement3869(f, b, d, c, a, x, e):
rubi.append(3869)
return Dist(S(1)/(S(2)*a), Int(cos(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(S(1)/(S(2)*b), Int(sin(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Simp(log(c + d*x)/(S(2)*a*d), x)
rule3869 = ReplacementRule(pattern3869, replacement3869)
pattern3870 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439)
def replacement3870(f, b, d, c, a, x, e):
rubi.append(3870)
return -Dist(S(1)/(S(2)*a), Int(cos(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(S(1)/(S(2)*b), Int(sin(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Simp(log(c + d*x)/(S(2)*a*d), x)
rule3870 = ReplacementRule(pattern3870, replacement3870)
pattern3871 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1439, cons18)
def replacement3871(m, f, b, d, c, a, x, e):
rubi.append(3871)
return Dist(S(1)/(S(2)*a), Int((c + d*x)**m*exp(S(2)*a*(e + f*x)/b), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x)
rule3871 = ReplacementRule(pattern3871, replacement3871)
pattern3872 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1439, cons18)
def replacement3872(m, f, b, d, c, a, x, e):
rubi.append(3872)
return -Dist(S(1)/(S(2)*a), Int((c + d*x)**m*exp(-S(2)*a*(e + f*x)/b), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x)
rule3872 = ReplacementRule(pattern3872, replacement3872)
pattern3873 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439, cons1571)
def replacement3873(m, f, b, d, c, a, n, x, e):
rubi.append(3873)
return Int(ExpandIntegrand((c + d*x)**m, (sin(S(2)*e + S(2)*f*x)/(S(2)*b) + cos(S(2)*e + S(2)*f*x)/(S(2)*a) + S(1)/(S(2)*a))**(-n), x), x)
rule3873 = ReplacementRule(pattern3873, replacement3873)
pattern3874 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439, cons1571)
def replacement3874(m, f, b, d, c, a, n, x, e):
rubi.append(3874)
return Int(ExpandIntegrand((c + d*x)**m, (sin(S(2)*e + S(2)*f*x)/(S(2)*b) - cos(S(2)*e + S(2)*f*x)/(S(2)*a) + S(1)/(S(2)*a))**(-n), x), x)
rule3874 = ReplacementRule(pattern3874, replacement3874)
pattern3875 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1439, cons196)
def replacement3875(m, f, b, d, c, a, n, x, e):
rubi.append(3875)
return Int(ExpandIntegrand((c + d*x)**m, (exp(S(2)*a*(e + f*x)/b)/(S(2)*a) + S(1)/(S(2)*a))**(-n), x), x)
rule3875 = ReplacementRule(pattern3875, replacement3875)
pattern3876 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1439, cons196)
def replacement3876(m, f, b, d, c, a, n, x, e):
rubi.append(3876)
return Int(ExpandIntegrand((c + d*x)**m, (S(1)/(S(2)*a) - exp(-S(2)*a*(e + f*x)/b)/(S(2)*a))**(-n), x), x)
rule3876 = ReplacementRule(pattern3876, replacement3876)
def With3877(m, f, b, d, c, a, n, x, e):
u = IntHide((a + b*tan(e + f*x))**n, x)
rubi.append(3877)
return -Dist(d*m, Int(Dist((c + d*x)**(m + S(-1)), u, x), x), x) + Dist((c + d*x)**m, u, x)
pattern3877 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439, cons1572, cons31, cons168)
rule3877 = ReplacementRule(pattern3877, With3877)
def With3878(m, f, b, d, c, a, n, x, e):
u = IntHide((a + b/tan(e + f*x))**n, x)
rubi.append(3878)
return -Dist(d*m, Int(Dist((c + d*x)**(m + S(-1)), u, x), x), x) + Dist((c + d*x)**m, u, x)
pattern3878 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1439, cons1572, cons31, cons168)
rule3878 = ReplacementRule(pattern3878, With3878)
pattern3879 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440, cons62)
def replacement3879(m, f, b, d, c, a, x, e):
rubi.append(3879)
return -Dist(S(2)*I*b, Int((c + d*x)**m/(a**S(2) + b**S(2) + (a - I*b)**S(2)*exp(S(2)*I*(e + f*x))), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a - I*b)*(m + S(1))), x)
rule3879 = ReplacementRule(pattern3879, replacement3879)
pattern3880 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440, cons62)
def replacement3880(m, f, b, d, c, a, x, e):
rubi.append(3880)
return Dist(S(2)*I*b, Int((c + d*x)**m/(a**S(2) + b**S(2) - (a + I*b)**S(2)*exp(S(2)*I*(e + f*x))), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + I*b)*(m + S(1))), x)
rule3880 = ReplacementRule(pattern3880, replacement3880)
pattern3881 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440)
def replacement3881(f, b, d, c, a, x, e):
rubi.append(3881)
return Dist(S(1)/(f*(a**S(2) + b**S(2))), Int((S(2)*a*c*f + S(2)*a*d*f*x + b*d)/(a + b*tan(e + f*x)), x), x) - Simp((c + d*x)**S(2)/(S(2)*d*(a**S(2) + b**S(2))), x) - Simp(b*(c + d*x)/(f*(a + b*tan(e + f*x))*(a**S(2) + b**S(2))), x)
rule3881 = ReplacementRule(pattern3881, replacement3881)
pattern3882 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440)
def replacement3882(f, b, d, c, a, x, e):
rubi.append(3882)
return -Dist(S(1)/(f*(a**S(2) + b**S(2))), Int((-S(2)*a*c*f - S(2)*a*d*f*x + b*d)/(a + b/tan(e + f*x)), x), x) - Simp((c + d*x)**S(2)/(S(2)*d*(a**S(2) + b**S(2))), x) + Simp(b*(c + d*x)/(f*(a + b/tan(e + f*x))*(a**S(2) + b**S(2))), x)
rule3882 = ReplacementRule(pattern3882, replacement3882)
pattern3883 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440, cons196, cons62)
def replacement3883(m, f, b, d, c, a, n, x, e):
rubi.append(3883)
return Int(ExpandIntegrand((c + d*x)**m, (-S(2)*I*b/(a**S(2) + b**S(2) + (a - I*b)**S(2)*exp(S(2)*I*(e + f*x))) + S(1)/(a - I*b))**(-n), x), x)
rule3883 = ReplacementRule(pattern3883, replacement3883)
pattern3884 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1440, cons196, cons62)
def replacement3884(m, f, b, d, c, a, n, x, e):
rubi.append(3884)
return Int(ExpandIntegrand((c + d*x)**m, (S(2)*I*b/(a**S(2) + b**S(2) - (a + I*b)**S(2)*exp(S(2)*I*(e + f*x))) + S(1)/(a + I*b))**(-n), x), x)
rule3884 = ReplacementRule(pattern3884, replacement3884)
pattern3885 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement3885(v, u, m, b, a, n, x):
rubi.append(3885)
return Int((a + b*tan(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule3885 = ReplacementRule(pattern3885, replacement3885)
pattern3886 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement3886(v, u, m, b, a, n, x):
rubi.append(3886)
return Int((a + b/tan(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule3886 = ReplacementRule(pattern3886, replacement3886)
pattern3887 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement3887(m, f, b, d, c, a, n, x, e):
rubi.append(3887)
return Int((a + b*tan(e + f*x))**n*(c + d*x)**m, x)
rule3887 = ReplacementRule(pattern3887, replacement3887)
pattern3888 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement3888(m, f, b, d, a, n, c, x, e):
rubi.append(3888)
return Int((a + b/tan(e + f*x))**n*(c + d*x)**m, x)
rule3888 = ReplacementRule(pattern3888, replacement3888)
pattern3889 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1573, cons38)
def replacement3889(p, b, d, c, a, n, x):
rubi.append(3889)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b*tan(c + d*x))**p, x), x, x**n), x)
rule3889 = ReplacementRule(pattern3889, replacement3889)
pattern3890 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1573, cons38)
def replacement3890(p, b, d, a, c, n, x):
rubi.append(3890)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b/tan(c + d*x))**p, x), x, x**n), x)
rule3890 = ReplacementRule(pattern3890, replacement3890)
pattern3891 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement3891(p, b, d, c, a, n, x):
rubi.append(3891)
return Int((a + b*tan(c + d*x**n))**p, x)
rule3891 = ReplacementRule(pattern3891, replacement3891)
pattern3892 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement3892(p, b, d, a, c, n, x):
rubi.append(3892)
return Int((a + b/tan(c + d*x**n))**p, x)
rule3892 = ReplacementRule(pattern3892, replacement3892)
pattern3893 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69)
def replacement3893(p, u, b, d, c, a, n, x):
rubi.append(3893)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*tan(c + d*x**n))**p, x), x, u), x)
rule3893 = ReplacementRule(pattern3893, replacement3893)
pattern3894 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69)
def replacement3894(p, u, b, d, a, c, n, x):
rubi.append(3894)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b/tan(c + d*x**n))**p, x), x, u), x)
rule3894 = ReplacementRule(pattern3894, replacement3894)
pattern3895 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement3895(p, u, b, a, x):
rubi.append(3895)
return Int((a + b*tan(ExpandToSum(u, x)))**p, x)
rule3895 = ReplacementRule(pattern3895, replacement3895)
pattern3896 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement3896(p, u, b, a, x):
rubi.append(3896)
return Int((a + b/tan(ExpandToSum(u, x)))**p, x)
rule3896 = ReplacementRule(pattern3896, replacement3896)
pattern3897 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1574, cons38)
def replacement3897(p, m, b, d, c, a, n, x):
rubi.append(3897)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*tan(c + d*x))**p, x), x, x**n), x)
rule3897 = ReplacementRule(pattern3897, replacement3897)
pattern3898 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1574, cons38)
def replacement3898(p, m, b, d, a, c, n, x):
rubi.append(3898)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b/tan(c + d*x))**p, x), x, x**n), x)
rule3898 = ReplacementRule(pattern3898, replacement3898)
pattern3899 = Pattern(Integral(x_**WC('m', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons7, cons27, cons21, cons4, cons1575)
def replacement3899(m, d, c, n, x):
rubi.append(3899)
return -Dist((m - n + S(1))/(d*n), Int(x**(m - n)*tan(c + d*x**n), x), x) - Int(x**m, x) + Simp(x**(m - n + S(1))*tan(c + d*x**n)/(d*n), x)
rule3899 = ReplacementRule(pattern3899, replacement3899)
pattern3900 = Pattern(Integral(x_**WC('m', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons7, cons27, cons21, cons4, cons1575)
def replacement3900(m, d, c, n, x):
rubi.append(3900)
return Dist((m - n + S(1))/(d*n), Int(x**(m - n)/tan(c + d*x**n), x), x) - Int(x**m, x) - Simp(x**(m - n + S(1))/(d*n*tan(c + d*x**n)), x)
rule3900 = ReplacementRule(pattern3900, replacement3900)
pattern3901 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1576)
def replacement3901(p, m, b, d, c, a, n, x):
rubi.append(3901)
return Int(x**m*(a + b*tan(c + d*x**n))**p, x)
rule3901 = ReplacementRule(pattern3901, replacement3901)
pattern3902 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1576)
def replacement3902(p, m, b, d, a, c, n, x):
rubi.append(3902)
return Int(x**m*(a + b/tan(c + d*x**n))**p, x)
rule3902 = ReplacementRule(pattern3902, replacement3902)
pattern3903 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement3903(p, m, b, d, c, a, n, x, e):
rubi.append(3903)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*tan(c + d*x**n))**p, x), x)
rule3903 = ReplacementRule(pattern3903, replacement3903)
pattern3904 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement3904(p, m, b, d, a, c, n, x, e):
rubi.append(3904)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b/tan(c + d*x**n))**p, x), x)
rule3904 = ReplacementRule(pattern3904, replacement3904)
pattern3905 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement3905(p, u, m, b, a, x, e):
rubi.append(3905)
return Int((e*x)**m*(a + b*tan(ExpandToSum(u, x)))**p, x)
rule3905 = ReplacementRule(pattern3905, replacement3905)
pattern3906 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement3906(p, u, m, b, a, x, e):
rubi.append(3906)
return Int((e*x)**m*(a + b/tan(ExpandToSum(u, x)))**p, x)
rule3906 = ReplacementRule(pattern3906, replacement3906)
pattern3907 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1))*tan(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons5, cons31, cons85, cons1577, cons1578)
def replacement3907(p, m, b, a, n, x, q):
rubi.append(3907)
return -Dist((m - n + S(1))/(b*n*p), Int(x**(m - n)*(S(1)/cos(a + b*x**n))**p, x), x) + Simp(x**(m - n + S(1))*(S(1)/cos(a + b*x**n))**p/(b*n*p), x)
rule3907 = ReplacementRule(pattern3907, replacement3907)
pattern3908 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1))*(S(1)/tan(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('q', S(1)), x_), cons2, cons3, cons5, cons31, cons85, cons1577, cons1578)
def replacement3908(p, m, b, a, n, x, q):
rubi.append(3908)
return Dist((m - n + S(1))/(b*n*p), Int(x**(m - n)*(S(1)/sin(a + b*x**n))**p, x), x) - Simp(x**(m - n + S(1))*(S(1)/sin(a + b*x**n))**p/(b*n*p), x)
rule3908 = ReplacementRule(pattern3908, replacement3908)
pattern3909 = Pattern(Integral(tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement3909(b, c, a, n, x):
rubi.append(3909)
return Int(tan(a + b*x + c*x**S(2))**n, x)
rule3909 = ReplacementRule(pattern3909, replacement3909)
pattern3910 = Pattern(Integral((S(1)/tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons4, cons1579)
def replacement3910(b, c, a, n, x):
rubi.append(3910)
return Int((S(1)/tan(a + b*x + c*x**S(2)))**n, x)
rule3910 = ReplacementRule(pattern3910, replacement3910)
pattern3911 = Pattern(Integral((d_ + x_*WC('e', S(1)))*tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons47)
def replacement3911(b, d, c, a, x, e):
rubi.append(3911)
return -Simp(e*log(cos(a + b*x + c*x**S(2)))/(S(2)*c), x)
rule3911 = ReplacementRule(pattern3911, replacement3911)
pattern3912 = Pattern(Integral((d_ + x_*WC('e', S(1)))/tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons47)
def replacement3912(b, d, c, a, x, e):
rubi.append(3912)
return Simp(e*log(sin(a + b*x + c*x**S(2)))/(S(2)*c), x)
rule3912 = ReplacementRule(pattern3912, replacement3912)
pattern3913 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons239)
def replacement3913(b, d, c, a, x, e):
rubi.append(3913)
return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(tan(a + b*x + c*x**S(2)), x), x) - Simp(e*log(cos(a + b*x + c*x**S(2)))/(S(2)*c), x)
rule3913 = ReplacementRule(pattern3913, replacement3913)
pattern3914 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))/tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons239)
def replacement3914(b, d, c, a, x, e):
rubi.append(3914)
return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(S(1)/tan(a + b*x + c*x**S(2)), x), x) + Simp(e*log(sin(a + b*x + c*x**S(2)))/(S(2)*c), x)
rule3914 = ReplacementRule(pattern3914, replacement3914)
pattern3915 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1580)
def replacement3915(m, b, d, a, c, n, x, e):
rubi.append(3915)
return Int((d + e*x)**m*tan(a + b*x + c*x**S(2))**n, x)
rule3915 = ReplacementRule(pattern3915, replacement3915)
pattern3916 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(S(1)/tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1580)
def replacement3916(m, b, d, c, a, n, x, e):
rubi.append(3916)
return Int((d + e*x)**m*(S(1)/tan(a + b*x + c*x**S(2)))**n, x)
rule3916 = ReplacementRule(pattern3916, replacement3916)
return [rule3398, rule3399, rule3400, rule3401, rule3402, rule3403, rule3404, rule3405, rule3406, rule3407, rule3408, rule3409, rule3410, rule3411, rule3412, rule3413, rule3414, rule3415, rule3416, rule3417, rule3418, rule3419, rule3420, rule3421, rule3422, rule3423, rule3424, rule3425, rule3426, rule3427, rule3428, rule3429, rule3430, rule3431, rule3432, rule3433, rule3434, rule3435, rule3436, rule3437, rule3438, rule3439, rule3440, rule3441, rule3442, rule3443, rule3444, rule3445, rule3446, rule3447, rule3448, rule3449, rule3450, rule3451, rule3452, rule3453, rule3454, rule3455, rule3456, rule3457, rule3458, rule3459, rule3460, rule3461, rule3462, rule3463, rule3464, rule3465, rule3466, rule3467, rule3468, rule3469, rule3470, rule3471, rule3472, rule3473, rule3474, rule3475, rule3476, rule3477, rule3478, rule3479, rule3480, rule3481, rule3482, rule3483, rule3484, rule3485, rule3486, rule3487, rule3488, rule3489, rule3490, rule3491, rule3492, rule3493, rule3494, rule3495, rule3496, rule3497, rule3498, rule3499, rule3500, rule3501, rule3502, rule3503, rule3504, rule3505, rule3506, rule3507, rule3508, rule3509, rule3510, rule3511, rule3512, rule3513, rule3514, rule3515, rule3516, rule3517, rule3518, rule3519, rule3520, rule3521, rule3522, rule3523, rule3524, rule3525, rule3526, rule3527, rule3528, rule3529, rule3530, rule3531, rule3532, rule3533, rule3534, rule3535, rule3536, rule3537, rule3538, rule3539, rule3540, rule3541, rule3542, rule3543, rule3544, rule3545, rule3546, rule3547, rule3548, rule3549, rule3550, rule3551, rule3552, rule3553, rule3554, rule3555, rule3556, rule3557, rule3558, rule3559, rule3560, rule3561, rule3562, rule3563, rule3564, rule3565, rule3566, rule3567, rule3568, rule3569, rule3570, rule3571, rule3572, rule3573, rule3574, rule3575, rule3576, rule3577, rule3578, rule3579, rule3580, rule3581, rule3582, rule3583, rule3584, rule3585, rule3586, rule3587, rule3588, rule3589, rule3590, rule3591, rule3592, rule3593, rule3594, rule3595, rule3596, rule3597, rule3598, rule3599, rule3600, rule3601, rule3602, rule3603, rule3604, rule3605, rule3606, rule3607, rule3608, rule3609, rule3610, rule3611, rule3612, rule3613, rule3614, rule3615, rule3616, rule3617, rule3618, rule3619, rule3620, rule3621, rule3622, rule3623, rule3624, rule3625, rule3626, rule3627, rule3628, rule3629, rule3630, rule3631, rule3632, rule3633, rule3634, rule3635, rule3636, rule3637, rule3638, rule3639, rule3640, rule3641, rule3642, rule3643, rule3644, rule3645, rule3646, rule3647, rule3648, rule3649, rule3650, rule3651, rule3652, rule3653, rule3654, rule3655, rule3656, rule3657, rule3658, rule3659, rule3660, rule3661, rule3662, rule3663, rule3664, rule3665, rule3666, rule3667, rule3668, rule3669, rule3670, rule3671, rule3672, rule3673, rule3674, rule3675, rule3676, rule3677, rule3678, rule3679, rule3680, rule3681, rule3682, rule3683, rule3684, rule3685, rule3686, rule3687, rule3688, rule3689, rule3690, rule3691, rule3692, rule3693, rule3694, rule3695, rule3696, rule3697, rule3698, rule3699, rule3700, rule3701, rule3702, rule3703, rule3704, rule3705, rule3706, rule3707, rule3708, rule3709, rule3710, rule3711, rule3712, rule3713, rule3714, rule3715, rule3716, rule3717, rule3718, rule3719, rule3720, rule3721, rule3722, rule3723, rule3724, rule3725, rule3726, rule3727, rule3728, rule3729, rule3730, rule3731, rule3732, rule3733, rule3734, rule3735, rule3736, rule3737, rule3738, rule3739, rule3740, rule3741, rule3742, rule3743, rule3744, rule3745, rule3746, rule3747, rule3748, rule3749, rule3750, rule3751, rule3752, rule3753, rule3754, rule3755, rule3756, rule3757, rule3758, rule3759, rule3760, rule3761, rule3762, rule3763, rule3764, rule3765, rule3766, rule3767, rule3768, rule3769, rule3770, rule3771, rule3772, rule3773, rule3774, rule3775, rule3776, rule3777, rule3778, rule3779, rule3780, rule3781, rule3782, rule3783, rule3784, rule3785, rule3786, rule3787, rule3788, rule3789, rule3790, rule3791, rule3792, rule3793, rule3794, rule3795, rule3796, rule3797, rule3798, rule3799, rule3800, rule3801, rule3802, rule3803, rule3804, rule3805, rule3806, rule3807, rule3808, rule3809, rule3810, rule3811, rule3812, rule3813, rule3814, rule3815, rule3816, rule3817, rule3818, rule3819, rule3820, rule3821, rule3822, rule3823, rule3824, rule3825, rule3826, rule3827, rule3828, rule3829, rule3830, rule3831, rule3832, rule3833, rule3834, rule3835, rule3836, rule3837, rule3838, rule3839, rule3840, rule3841, rule3842, rule3843, rule3844, rule3845, rule3846, rule3847, rule3848, rule3849, rule3850, rule3851, rule3852, rule3853, rule3854, rule3855, rule3856, rule3857, rule3858, rule3859, rule3860, rule3861, rule3862, rule3863, rule3864, rule3865, rule3866, rule3867, rule3868, rule3869, rule3870, rule3871, rule3872, rule3873, rule3874, rule3875, rule3876, rule3877, rule3878, rule3879, rule3880, rule3881, rule3882, rule3883, rule3884, rule3885, rule3886, rule3887, rule3888, rule3889, rule3890, rule3891, rule3892, rule3893, rule3894, rule3895, rule3896, rule3897, rule3898, rule3899, rule3900, rule3901, rule3902, rule3903, rule3904, rule3905, rule3906, rule3907, rule3908, rule3909, rule3910, rule3911, rule3912, rule3913, rule3914, rule3915, rule3916, ]
|
d3cb717b958e9214b11a522a8b3a8de00a35b8613fcc1d3e104e858e31d4816f | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def sine(rubi):
from sympy.integrals.rubi.constraints import cons1249, cons72, cons66, cons2, cons3, cons48, cons125, cons21, cons4, cons1250, cons1251, cons1252, cons93, cons166, cons89, cons1253, cons31, cons1170, cons94, cons1254, cons1255, cons1256, cons1257, cons1258, cons1259, cons165, cons1260, cons18, cons23, cons521, cons7, cons27, cons808, cons1261, cons1262, cons1263, cons543, cons1264, cons1265, cons148, cons1266, cons87, cons43, cons448, cons1267, cons1268, cons1269, cons1270, cons1271, cons481, cons482, cons1272, cons1273, cons1274, cons1275, cons1276, cons208, cons5, cons17, cons13, cons137, cons1277, cons1278, cons1279, cons1280, cons1281, cons143, cons1282, cons1283, cons1284, cons1285, cons244, cons168, cons1286, cons1287, cons1288, cons146, cons1289, cons1290, cons1291, cons246, cons1292, cons1293, cons1294, cons1295, cons1296, cons84, cons1297, cons147, cons54, cons1298, cons1299, cons1300, cons1301, cons1302, cons62, cons267, cons1303, cons1304, cons1305, cons1306, cons515, cons272, cons1307, cons1308, cons71, cons70, cons1309, cons1310, cons1311, cons1312, cons346, cons1313, cons155, cons1314, cons1315, cons114, cons1316, cons1317, cons1318, cons1319, cons1320, cons1321, cons77, cons1322, cons1323, cons1324, cons1325, cons1326, cons1327, cons1328, cons463, cons88, cons1329, cons1330, cons85, cons1331, cons1332, cons1333, cons1334, cons1335, cons1336, cons1337, cons1338, cons1339, cons1340, cons1341, cons1342, cons1343, cons1344, cons1345, cons1346, cons1347, cons1348, cons1349, cons1350, cons1351, cons1352, cons1353, cons1354, cons1355, cons1356, cons1357, cons1358, cons1359, cons1360, cons1361, cons1362, cons1363, cons1364, cons142, cons335, cons1365, cons1366, cons1367, cons1368, cons1369, cons1370, cons170, cons1371, cons1372, cons1373, cons1374, cons1375, cons253, cons1376, cons1377, cons1378, cons1379, cons1380, cons358, cons1381, cons1382, cons1383, cons1384, cons1385, cons1386, cons1387, cons1388, cons1389, cons1390, cons1391, cons1392, cons1393, cons1394, cons213, cons584, cons1395, cons1396, cons1397, cons1398, cons1399, cons1400, cons1401, cons1402, cons1403, cons1404, cons1405, cons1406, cons1407, cons1408, cons1409, cons1410, cons1411, cons105, cons1412, cons1413, cons1414, cons1415, cons1416, cons1417, cons38, cons1418, cons34, cons35, cons1419, cons1420, cons1421, cons683, cons1422, cons1423, cons1424, cons1425, cons1426, cons1427, cons1428, cons1429, cons1430, cons1431, cons36, cons1228, cons1432, cons33, cons1433, cons1434, cons1435, cons1436, cons214, cons1437, cons1438, cons1439, cons1440, cons1441, cons1442, cons1443, cons1444, cons1445, cons1446, cons1152, cons1447, cons196, cons128, cons63, cons150, cons375, cons322, cons1448, cons1449, cons1450, cons1451, cons1452, cons1453, cons1454, cons76, cons1455, cons1456, cons1457, cons1458, cons1459, cons1460, cons1461, cons1462, cons1463, cons1464, cons1465, cons1466, cons1467, cons1468, cons1469, cons1470, cons1471, cons1245, cons1472, cons1473, cons1474, cons1475, cons1476, cons1477, cons1043, cons1478, cons1479, cons1480, cons1481, cons1482, cons1483, cons1484, cons1485, cons1486, cons1487, cons46, cons45, cons226, cons376, cons1116, cons176, cons1488, cons1489, cons245, cons247, cons1490, cons1491, cons1492, cons1493, cons810, cons811, cons744, cons1494, cons1495, cons53, cons596, cons1496, cons1497, cons489, cons1498, cons68, cons69, cons823, cons824, cons1499, cons1500, cons1501, cons1502, cons56, cons1503, cons1504, cons367, cons1505, cons356, cons854, cons1506, cons818, cons1131, cons47, cons239, cons1132, cons1133, cons1507, cons819
pattern2164 = Pattern(Integral(u_, x_), cons1249)
def replacement2164(x, u):
rubi.append(2164)
return Int(DeactivateTrig(u, x), x)
rule2164 = ReplacementRule(pattern2164, replacement2164)
pattern2165 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons4, cons72, cons66)
def replacement2165(m, f, b, a, n, x, e):
rubi.append(2165)
return Simp((a*sin(e + f*x))**(m + S(1))*(b*cos(e + f*x))**(n + S(1))/(a*b*f*(m + S(1))), x)
rule2165 = ReplacementRule(pattern2165, replacement2165)
pattern2166 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons48, cons125, cons21, cons1250, cons1251)
def replacement2166(m, f, a, n, x, e):
rubi.append(2166)
return Dist(S(1)/(a*f), Subst(Int(x**m*(S(1) - x**S(2)/a**S(2))**(n/S(2) + S(-1)/2), x), x, a*sin(e + f*x)), x)
rule2166 = ReplacementRule(pattern2166, replacement2166)
pattern2167 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons48, cons125, cons21, cons1250, cons1252)
def replacement2167(m, f, a, n, x, e):
rubi.append(2167)
return -Dist(S(1)/(a*f), Subst(Int(x**m*(S(1) - x**S(2)/a**S(2))**(n/S(2) + S(-1)/2), x), x, a*cos(e + f*x)), x)
rule2167 = ReplacementRule(pattern2167, replacement2167)
pattern2168 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons166, cons89, cons1253)
def replacement2168(m, f, b, a, n, x, e):
rubi.append(2168)
return Dist(a**S(2)*(m + S(-1))/(b**S(2)*(n + S(1))), Int((a*sin(e + f*x))**(m + S(-2))*(b*cos(e + f*x))**(n + S(2)), x), x) - Simp(a*(a*sin(e + f*x))**(m + S(-1))*(b*cos(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x)
rule2168 = ReplacementRule(pattern2168, replacement2168)
pattern2169 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons166, cons89, cons1253)
def replacement2169(m, f, b, a, n, x, e):
rubi.append(2169)
return Dist(a**S(2)*(m + S(-1))/(b**S(2)*(n + S(1))), Int((a*cos(e + f*x))**(m + S(-2))*(b*sin(e + f*x))**(n + S(2)), x), x) + Simp(a*(a*cos(e + f*x))**(m + S(-1))*(b*sin(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x)
rule2169 = ReplacementRule(pattern2169, replacement2169)
pattern2170 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons166, cons1170)
def replacement2170(m, f, b, a, n, x, e):
rubi.append(2170)
return Dist(a**S(2)*(m + S(-1))/(m + n), Int((a*sin(e + f*x))**(m + S(-2))*(b*cos(e + f*x))**n, x), x) - Simp(a*(a*sin(e + f*x))**(m + S(-1))*(b*cos(e + f*x))**(n + S(1))/(b*f*(m + n)), x)
rule2170 = ReplacementRule(pattern2170, replacement2170)
pattern2171 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons166, cons1170)
def replacement2171(m, f, b, a, n, x, e):
rubi.append(2171)
return Dist(a**S(2)*(m + S(-1))/(m + n), Int((a*cos(e + f*x))**(m + S(-2))*(b*sin(e + f*x))**n, x), x) + Simp(a*(a*cos(e + f*x))**(m + S(-1))*(b*sin(e + f*x))**(n + S(1))/(b*f*(m + n)), x)
rule2171 = ReplacementRule(pattern2171, replacement2171)
pattern2172 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons94, cons1170)
def replacement2172(m, f, b, a, n, x, e):
rubi.append(2172)
return Dist((m + n + S(2))/(a**S(2)*(m + S(1))), Int((a*sin(e + f*x))**(m + S(2))*(b*cos(e + f*x))**n, x), x) + Simp((a*sin(e + f*x))**(m + S(1))*(b*cos(e + f*x))**(n + S(1))/(a*b*f*(m + S(1))), x)
rule2172 = ReplacementRule(pattern2172, replacement2172)
pattern2173 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons94, cons1170)
def replacement2173(m, f, b, a, n, x, e):
rubi.append(2173)
return Dist((m + n + S(2))/(a**S(2)*(m + S(1))), Int((a*cos(e + f*x))**(m + S(2))*(b*sin(e + f*x))**n, x), x) - Simp((a*cos(e + f*x))**(m + S(1))*(b*sin(e + f*x))**(n + S(1))/(a*b*f*(m + S(1))), x)
rule2173 = ReplacementRule(pattern2173, replacement2173)
pattern2174 = Pattern(Integral(sqrt(WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1254)
def replacement2174(f, b, a, x, e):
rubi.append(2174)
return Dist(sqrt(a*sin(e + f*x))*sqrt(b*cos(e + f*x))/sqrt(sin(S(2)*e + S(2)*f*x)), Int(sqrt(sin(S(2)*e + S(2)*f*x)), x), x)
rule2174 = ReplacementRule(pattern2174, replacement2174)
pattern2175 = Pattern(Integral(S(1)/(sqrt(WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons1254)
def replacement2175(f, b, a, x, e):
rubi.append(2175)
return Dist(sqrt(sin(S(2)*e + S(2)*f*x))/(sqrt(a*sin(e + f*x))*sqrt(b*cos(e + f*x))), Int(S(1)/sqrt(sin(S(2)*e + S(2)*f*x)), x), x)
rule2175 = ReplacementRule(pattern2175, replacement2175)
def With2176(m, f, b, a, n, x, e):
k = Denominator(m)
rubi.append(2176)
return Dist(a*b*k/f, Subst(Int(x**(k*(m + S(1)) + S(-1))/(a**S(2) + b**S(2)*x**(S(2)*k)), x), x, (a*sin(e + f*x))**(S(1)/k)*(b*cos(e + f*x))**(-S(1)/k)), x)
pattern2176 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons1255, cons31, cons1256)
rule2176 = ReplacementRule(pattern2176, With2176)
def With2177(m, f, b, a, n, x, e):
k = Denominator(m)
rubi.append(2177)
return -Dist(a*b*k/f, Subst(Int(x**(k*(m + S(1)) + S(-1))/(a**S(2) + b**S(2)*x**(S(2)*k)), x), x, (a*cos(e + f*x))**(S(1)/k)*(b*sin(e + f*x))**(-S(1)/k)), x)
pattern2177 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons1255, cons31, cons1256)
rule2177 = ReplacementRule(pattern2177, With2177)
pattern2178 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1257)
def replacement2178(m, f, b, a, n, x, e):
rubi.append(2178)
return Simp(b**(S(2)*IntPart(n/S(2) + S(-1)/2) + S(1))*(a*sin(e + f*x))**(m + S(1))*(b*cos(e + f*x))**(S(2)*FracPart(n/S(2) + S(-1)/2))*(cos(e + f*x)**S(2))**(-FracPart(n/S(2) + S(-1)/2))*Hypergeometric2F1(m/S(2) + S(1)/2, -n/S(2) + S(1)/2, m/S(2) + S(3)/2, sin(e + f*x)**S(2))/(a*f*(m + S(1))), x)
rule2178 = ReplacementRule(pattern2178, replacement2178)
pattern2179 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1258)
def replacement2179(m, f, b, a, n, x, e):
rubi.append(2179)
return -Simp(b**(S(2)*IntPart(n/S(2) + S(-1)/2) + S(1))*(a*cos(e + f*x))**(m + S(1))*(b*sin(e + f*x))**(S(2)*FracPart(n/S(2) + S(-1)/2))*(sin(e + f*x)**S(2))**(-FracPart(n/S(2) + S(-1)/2))*Hypergeometric2F1(m/S(2) + S(1)/2, -n/S(2) + S(1)/2, m/S(2) + S(3)/2, cos(e + f*x)**S(2))/(a*f*(m + S(1))), x)
rule2179 = ReplacementRule(pattern2179, replacement2179)
pattern2180 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1259, cons66)
def replacement2180(m, f, b, a, n, x, e):
rubi.append(2180)
return Simp(b*(a*sin(e + f*x))**(m + S(1))*(b/cos(e + f*x))**(n + S(-1))/(a*f*(m + S(1))), x)
rule2180 = ReplacementRule(pattern2180, replacement2180)
pattern2181 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1259, cons66)
def replacement2181(m, f, b, a, n, x, e):
rubi.append(2181)
return -Simp(b*(a*cos(e + f*x))**(m + S(1))*(b/sin(e + f*x))**(n + S(-1))/(a*f*(m + S(1))), x)
rule2181 = ReplacementRule(pattern2181, replacement2181)
pattern2182 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons166, cons165, cons1170)
def replacement2182(m, f, b, a, n, x, e):
rubi.append(2182)
return -Dist(a**S(2)*b**S(2)*(m + S(-1))/(n + S(-1)), Int((a*sin(e + f*x))**(m + S(-2))*(b/cos(e + f*x))**(n + S(-2)), x), x) + Simp(a*b*(a*sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x)
rule2182 = ReplacementRule(pattern2182, replacement2182)
pattern2183 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons166, cons165, cons1170)
def replacement2183(m, f, b, a, n, x, e):
rubi.append(2183)
return -Dist(a**S(2)*b**S(2)*(m + S(-1))/(n + S(-1)), Int((a*cos(e + f*x))**(m + S(-2))*(b/sin(e + f*x))**(n + S(-2)), x), x) - Simp(a*b*(a*cos(e + f*x))**(m + S(-1))*(b/sin(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x)
rule2183 = ReplacementRule(pattern2183, replacement2183)
pattern2184 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons166, cons1260, cons1170)
def replacement2184(m, f, b, a, n, x, e):
rubi.append(2184)
return Dist(a**S(2)*(m + S(-1))/(m - n), Int((a*sin(e + f*x))**(m + S(-2))*(b/cos(e + f*x))**n, x), x) - Simp(a*b*(a*sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(-1))/(f*(m - n)), x)
rule2184 = ReplacementRule(pattern2184, replacement2184)
pattern2185 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons166, cons1260, cons1170)
def replacement2185(m, f, b, a, n, x, e):
rubi.append(2185)
return Dist(a**S(2)*(m + S(-1))/(m - n), Int((a*cos(e + f*x))**(m + S(-2))*(b/sin(e + f*x))**n, x), x) + Simp(a*b*(a*cos(e + f*x))**(m + S(-1))*(b/sin(e + f*x))**(n + S(-1))/(f*(m - n)), x)
rule2185 = ReplacementRule(pattern2185, replacement2185)
pattern2186 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons94, cons1170)
def replacement2186(m, f, b, a, n, x, e):
rubi.append(2186)
return Dist((m - n + S(2))/(a**S(2)*(m + S(1))), Int((a*sin(e + f*x))**(m + S(2))*(b/cos(e + f*x))**n, x), x) + Simp(b*(a*sin(e + f*x))**(m + S(1))*(b/cos(e + f*x))**(n + S(-1))/(a*f*(m + S(1))), x)
rule2186 = ReplacementRule(pattern2186, replacement2186)
pattern2187 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons4, cons31, cons94, cons1170)
def replacement2187(m, f, b, a, n, x, e):
rubi.append(2187)
return Dist((m - n + S(2))/(a**S(2)*(m + S(1))), Int((a*cos(e + f*x))**(m + S(2))*(b/sin(e + f*x))**n, x), x) - Simp(b*(a*cos(e + f*x))**(m + S(1))*(b/sin(e + f*x))**(n + S(-1))/(a*f*(m + S(1))), x)
rule2187 = ReplacementRule(pattern2187, replacement2187)
pattern2188 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement2188(m, f, b, a, n, x, e):
rubi.append(2188)
return Dist((cos(e + f*x)/b)**(FracPart(n) + S(1))*(b/cos(e + f*x))**(FracPart(n) + S(1)), Int((a*sin(e + f*x))**m*(cos(e + f*x)/b)**(-n), x), x)
rule2188 = ReplacementRule(pattern2188, replacement2188)
pattern2189 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement2189(m, f, b, a, n, x, e):
rubi.append(2189)
return Dist((sin(e + f*x)/b)**(FracPart(n) + S(1))*(b/sin(e + f*x))**(FracPart(n) + S(1)), Int((a*cos(e + f*x))**m*(sin(e + f*x)/b)**(-n), x), x)
rule2189 = ReplacementRule(pattern2189, replacement2189)
pattern2190 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement2190(m, f, b, a, n, x, e):
rubi.append(2190)
return Dist((a*b)**IntPart(n)*(a*sin(e + f*x))**FracPart(n)*(b/sin(e + f*x))**FracPart(n), Int((a*sin(e + f*x))**(m - n), x), x)
rule2190 = ReplacementRule(pattern2190, replacement2190)
pattern2191 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons18, cons23)
def replacement2191(m, f, b, a, n, x, e):
rubi.append(2191)
return Dist((a*b)**IntPart(n)*(a*cos(e + f*x))**FracPart(n)*(b/cos(e + f*x))**FracPart(n), Int((a*cos(e + f*x))**(m - n), x), x)
rule2191 = ReplacementRule(pattern2191, replacement2191)
pattern2192 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons7, cons27, cons521)
def replacement2192(d, c, n, x):
rubi.append(2192)
return -Dist(S(1)/d, Subst(Int((-x**S(2) + S(1))**(n/S(2))/sqrt(-x**S(2) + S(1)), x), x, cos(c + d*x)), x)
rule2192 = ReplacementRule(pattern2192, replacement2192)
pattern2193 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons7, cons27, cons521)
def replacement2193(d, c, n, x):
rubi.append(2193)
return Dist(S(1)/d, Subst(Int((-x**S(2) + S(1))**(n/S(2))/sqrt(-x**S(2) + S(1)), x), x, sin(c + d*x)), x)
rule2193 = ReplacementRule(pattern2193, replacement2193)
pattern2194 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons808, cons165)
def replacement2194(b, d, c, n, x):
rubi.append(2194)
return Dist(b**S(2)*(n + S(-1))/n, Int((b*sin(c + d*x))**(n + S(-2)), x), x) - Simp(b*(b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)/(d*n), x)
rule2194 = ReplacementRule(pattern2194, replacement2194)
pattern2195 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons808, cons165)
def replacement2195(b, d, c, n, x):
rubi.append(2195)
return Dist(b**S(2)*(n + S(-1))/n, Int((b*cos(c + d*x))**(n + S(-2)), x), x) + Simp(b*(b*cos(c + d*x))**(n + S(-1))*sin(c + d*x)/(d*n), x)
rule2195 = ReplacementRule(pattern2195, replacement2195)
pattern2196 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons808, cons89)
def replacement2196(b, d, c, n, x):
rubi.append(2196)
return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*sin(c + d*x))**(n + S(2)), x), x) + Simp((b*sin(c + d*x))**(n + S(1))*cos(c + d*x)/(b*d*(n + S(1))), x)
rule2196 = ReplacementRule(pattern2196, replacement2196)
pattern2197 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons808, cons89)
def replacement2197(b, d, c, n, x):
rubi.append(2197)
return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*cos(c + d*x))**(n + S(2)), x), x) - Simp((b*cos(c + d*x))**(n + S(1))*sin(c + d*x)/(b*d*(n + S(1))), x)
rule2197 = ReplacementRule(pattern2197, replacement2197)
pattern2198 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons1261)
def replacement2198(d, c, x):
rubi.append(2198)
return -Simp(cos(c + d*x)/d, x)
rule2198 = ReplacementRule(pattern2198, replacement2198)
pattern2199 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons7, cons1262)
def replacement2199(d, c, x):
rubi.append(2199)
return Simp(sin(c + d*x)/d, x)
rule2199 = ReplacementRule(pattern2199, replacement2199)
pattern2200 = Pattern(Integral(sqrt(sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons7, cons27, cons1261)
def replacement2200(d, c, x):
rubi.append(2200)
return Simp(S(2)*EllipticE(-Pi/S(4) + c/S(2) + d*x/S(2), S(2))/d, x)
rule2200 = ReplacementRule(pattern2200, replacement2200)
pattern2201 = Pattern(Integral(sqrt(cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons7, cons27, cons1261)
def replacement2201(d, c, x):
rubi.append(2201)
return Simp(S(2)*EllipticE(c/S(2) + d*x/S(2), S(2))/d, x)
rule2201 = ReplacementRule(pattern2201, replacement2201)
pattern2202 = Pattern(Integral(sqrt(b_*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons3, cons7, cons27, cons1263)
def replacement2202(d, c, b, x):
rubi.append(2202)
return Dist(sqrt(b*sin(c + d*x))/sqrt(sin(c + d*x)), Int(sqrt(sin(c + d*x)), x), x)
rule2202 = ReplacementRule(pattern2202, replacement2202)
pattern2203 = Pattern(Integral(sqrt(b_*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons3, cons7, cons27, cons1263)
def replacement2203(d, c, b, x):
rubi.append(2203)
return Dist(sqrt(b*cos(c + d*x))/sqrt(cos(c + d*x)), Int(sqrt(cos(c + d*x)), x), x)
rule2203 = ReplacementRule(pattern2203, replacement2203)
pattern2204 = Pattern(Integral(S(1)/sqrt(sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons7, cons27, cons1261)
def replacement2204(d, c, x):
rubi.append(2204)
return Simp(S(2)*EllipticF(-Pi/S(4) + c/S(2) + d*x/S(2), S(2))/d, x)
rule2204 = ReplacementRule(pattern2204, replacement2204)
pattern2205 = Pattern(Integral(S(1)/sqrt(cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons7, cons27, cons1261)
def replacement2205(d, c, x):
rubi.append(2205)
return Simp(S(2)*EllipticF(c/S(2) + d*x/S(2), S(2))/d, x)
rule2205 = ReplacementRule(pattern2205, replacement2205)
pattern2206 = Pattern(Integral(S(1)/sqrt(b_*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons3, cons7, cons27, cons1263)
def replacement2206(d, c, b, x):
rubi.append(2206)
return Dist(sqrt(sin(c + d*x))/sqrt(b*sin(c + d*x)), Int(S(1)/sqrt(sin(c + d*x)), x), x)
rule2206 = ReplacementRule(pattern2206, replacement2206)
pattern2207 = Pattern(Integral(S(1)/sqrt(b_*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons3, cons7, cons27, cons1263)
def replacement2207(d, c, b, x):
rubi.append(2207)
return Dist(sqrt(cos(c + d*x))/sqrt(b*cos(c + d*x)), Int(S(1)/sqrt(cos(c + d*x)), x), x)
rule2207 = ReplacementRule(pattern2207, replacement2207)
pattern2208 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons4, cons543)
def replacement2208(b, d, c, n, x):
rubi.append(2208)
return Simp((b*sin(c + d*x))**(n + S(1))*Hypergeometric2F1(S(1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(c + d*x)**S(2))*cos(c + d*x)/(b*d*(n + S(1))*sqrt(cos(c + d*x)**S(2))), x)
rule2208 = ReplacementRule(pattern2208, replacement2208)
pattern2209 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons4, cons543)
def replacement2209(b, d, c, n, x):
rubi.append(2209)
return -Simp((b*cos(c + d*x))**(n + S(1))*Hypergeometric2F1(S(1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(c + d*x)**S(2))*sin(c + d*x)/(b*d*(n + S(1))*sqrt(sin(c + d*x)**S(2))), x)
rule2209 = ReplacementRule(pattern2209, replacement2209)
pattern2210 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement2210(b, d, c, a, x):
rubi.append(2210)
return Dist(S(2)*a*b, Int(sin(c + d*x), x), x) + Simp(x*(S(2)*a**S(2) + b**S(2))/S(2), x) - Simp(b**S(2)*sin(c + d*x)*cos(c + d*x)/(S(2)*d), x)
rule2210 = ReplacementRule(pattern2210, replacement2210)
pattern2211 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement2211(b, d, c, a, x):
rubi.append(2211)
return Dist(S(2)*a*b, Int(cos(c + d*x), x), x) + Simp(x*(S(2)*a**S(2) + b**S(2))/S(2), x) + Simp(b**S(2)*sin(c + d*x)*cos(c + d*x)/(S(2)*d), x)
rule2211 = ReplacementRule(pattern2211, replacement2211)
pattern2212 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons148)
def replacement2212(b, d, c, a, n, x):
rubi.append(2212)
return Int(ExpandTrig((a + b*sin(c + d*x))**n, x), x)
rule2212 = ReplacementRule(pattern2212, replacement2212)
pattern2213 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons148)
def replacement2213(b, d, c, a, n, x):
rubi.append(2213)
return Int(ExpandTrig((a + b*cos(c + d*x))**n, x), x)
rule2213 = ReplacementRule(pattern2213, replacement2213)
pattern2214 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement2214(b, d, c, a, x):
rubi.append(2214)
return Simp(-S(2)*b*cos(c + d*x)/(d*sqrt(a + b*sin(c + d*x))), x)
rule2214 = ReplacementRule(pattern2214, replacement2214)
pattern2215 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement2215(b, d, c, a, x):
rubi.append(2215)
return Simp(S(2)*b*sin(c + d*x)/(d*sqrt(a + b*cos(c + d*x))), x)
rule2215 = ReplacementRule(pattern2215, replacement2215)
pattern2216 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1265, cons1266)
def replacement2216(b, d, c, a, n, x):
rubi.append(2216)
return Dist(a*(S(2)*n + S(-1))/n, Int((a + b*sin(c + d*x))**(n + S(-1)), x), x) - Simp(b*(a + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)/(d*n), x)
rule2216 = ReplacementRule(pattern2216, replacement2216)
pattern2217 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1265, cons1266)
def replacement2217(b, d, c, a, n, x):
rubi.append(2217)
return Dist(a*(S(2)*n + S(-1))/n, Int((a + b*cos(c + d*x))**(n + S(-1)), x), x) + Simp(b*(a + b*cos(c + d*x))**(n + S(-1))*sin(c + d*x)/(d*n), x)
rule2217 = ReplacementRule(pattern2217, replacement2217)
pattern2218 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement2218(b, d, c, a, x):
rubi.append(2218)
return -Simp(cos(c + d*x)/(d*(a*sin(c + d*x) + b)), x)
rule2218 = ReplacementRule(pattern2218, replacement2218)
pattern2219 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement2219(b, d, c, a, x):
rubi.append(2219)
return Simp(sin(c + d*x)/(d*(a*cos(c + d*x) + b)), x)
rule2219 = ReplacementRule(pattern2219, replacement2219)
pattern2220 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement2220(b, d, c, a, x):
rubi.append(2220)
return Dist(-S(2)/d, Subst(Int(S(1)/(S(2)*a - x**S(2)), x), x, b*cos(c + d*x)/sqrt(a + b*sin(c + d*x))), x)
rule2220 = ReplacementRule(pattern2220, replacement2220)
pattern2221 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement2221(b, d, c, a, x):
rubi.append(2221)
return Dist(S(2)/d, Subst(Int(S(1)/(S(2)*a - x**S(2)), x), x, b*sin(c + d*x)/sqrt(a + b*cos(c + d*x))), x)
rule2221 = ReplacementRule(pattern2221, replacement2221)
pattern2222 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1265, cons87, cons89, cons808)
def replacement2222(b, d, c, a, n, x):
rubi.append(2222)
return Dist((n + S(1))/(a*(S(2)*n + S(1))), Int((a + b*sin(c + d*x))**(n + S(1)), x), x) + Simp(b*(a + b*sin(c + d*x))**n*cos(c + d*x)/(a*d*(S(2)*n + S(1))), x)
rule2222 = ReplacementRule(pattern2222, replacement2222)
pattern2223 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1265, cons87, cons89, cons808)
def replacement2223(b, d, c, a, n, x):
rubi.append(2223)
return Dist((n + S(1))/(a*(S(2)*n + S(1))), Int((a + b*cos(c + d*x))**(n + S(1)), x), x) - Simp(b*(a + b*cos(c + d*x))**n*sin(c + d*x)/(a*d*(S(2)*n + S(1))), x)
rule2223 = ReplacementRule(pattern2223, replacement2223)
pattern2224 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons543, cons43)
def replacement2224(b, d, c, a, n, x):
rubi.append(2224)
return -Simp(S(2)**(n + S(1)/2)*a**(n + S(-1)/2)*b*Hypergeometric2F1(S(1)/2, -n + S(1)/2, S(3)/2, S(1)/2 - b*sin(c + d*x)/(S(2)*a))*cos(c + d*x)/(d*sqrt(a + b*sin(c + d*x))), x)
rule2224 = ReplacementRule(pattern2224, replacement2224)
pattern2225 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons543, cons43)
def replacement2225(b, d, c, a, n, x):
rubi.append(2225)
return Simp(S(2)**(n + S(1)/2)*a**(n + S(-1)/2)*b*Hypergeometric2F1(S(1)/2, -n + S(1)/2, S(3)/2, S(1)/2 - b*cos(c + d*x)/(S(2)*a))*sin(c + d*x)/(d*sqrt(a + b*cos(c + d*x))), x)
rule2225 = ReplacementRule(pattern2225, replacement2225)
pattern2226 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons543, cons448)
def replacement2226(b, d, c, a, n, x):
rubi.append(2226)
return Dist(a**IntPart(n)*(S(1) + b*sin(c + d*x)/a)**(-FracPart(n))*(a + b*sin(c + d*x))**FracPart(n), Int((S(1) + b*sin(c + d*x)/a)**n, x), x)
rule2226 = ReplacementRule(pattern2226, replacement2226)
pattern2227 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons543, cons448)
def replacement2227(b, d, c, a, n, x):
rubi.append(2227)
return Dist(a**IntPart(n)*(S(1) + b*cos(c + d*x)/a)**(-FracPart(n))*(a + b*cos(c + d*x))**FracPart(n), Int((S(1) + b*cos(c + d*x)/a)**n, x), x)
rule2227 = ReplacementRule(pattern2227, replacement2227)
pattern2228 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1268)
def replacement2228(b, d, c, a, x):
rubi.append(2228)
return Simp(S(2)*sqrt(a + b)*EllipticE(-Pi/S(4) + c/S(2) + d*x/S(2), S(2)*b/(a + b))/d, x)
rule2228 = ReplacementRule(pattern2228, replacement2228)
pattern2229 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1268)
def replacement2229(b, d, c, a, x):
rubi.append(2229)
return Simp(S(2)*sqrt(a + b)*EllipticE(c/S(2) + d*x/S(2), S(2)*b/(a + b))/d, x)
rule2229 = ReplacementRule(pattern2229, replacement2229)
pattern2230 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1269)
def replacement2230(b, d, c, a, x):
rubi.append(2230)
return Simp(S(2)*sqrt(a - b)*EllipticE(Pi/S(4) + c/S(2) + d*x/S(2), -S(2)*b/(a - b))/d, x)
rule2230 = ReplacementRule(pattern2230, replacement2230)
pattern2231 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1269)
def replacement2231(b, d, c, a, x):
rubi.append(2231)
return Simp(S(2)*sqrt(a - b)*EllipticE(Pi/S(2) + c/S(2) + d*x/S(2), -S(2)*b/(a - b))/d, x)
rule2231 = ReplacementRule(pattern2231, replacement2231)
pattern2232 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1270)
def replacement2232(b, d, c, a, x):
rubi.append(2232)
return Dist(sqrt(a + b*sin(c + d*x))/sqrt((a + b*sin(c + d*x))/(a + b)), Int(sqrt(a/(a + b) + b*sin(c + d*x)/(a + b)), x), x)
rule2232 = ReplacementRule(pattern2232, replacement2232)
pattern2233 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1270)
def replacement2233(b, d, c, a, x):
rubi.append(2233)
return Dist(sqrt(a + b*cos(c + d*x))/sqrt((a + b*cos(c + d*x))/(a + b)), Int(sqrt(a/(a + b) + b*cos(c + d*x)/(a + b)), x), x)
rule2233 = ReplacementRule(pattern2233, replacement2233)
pattern2234 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1267, cons87, cons165, cons808)
def replacement2234(b, d, c, a, n, x):
rubi.append(2234)
return Dist(S(1)/n, Int((a + b*sin(c + d*x))**(n + S(-2))*Simp(a**S(2)*n + a*b*(S(2)*n + S(-1))*sin(c + d*x) + b**S(2)*(n + S(-1)), x), x), x) - Simp(b*(a + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)/(d*n), x)
rule2234 = ReplacementRule(pattern2234, replacement2234)
pattern2235 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1267, cons87, cons165, cons808)
def replacement2235(b, d, c, a, n, x):
rubi.append(2235)
return Dist(S(1)/n, Int((a + b*cos(c + d*x))**(n + S(-2))*Simp(a**S(2)*n + a*b*(S(2)*n + S(-1))*cos(c + d*x) + b**S(2)*(n + S(-1)), x), x), x) + Simp(b*(a + b*cos(c + d*x))**(n + S(-1))*sin(c + d*x)/(d*n), x)
rule2235 = ReplacementRule(pattern2235, replacement2235)
def With2236(b, d, c, a, x):
q = Rt(a**S(2) - b**S(2), S(2))
rubi.append(2236)
return Simp(x/q, x) + Simp(S(2)*ArcTan(b*cos(c + d*x)/(a + b*sin(c + d*x) + q))/(d*q), x)
pattern2236 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1271, cons481)
rule2236 = ReplacementRule(pattern2236, With2236)
def With2237(b, d, c, a, x):
q = Rt(a**S(2) - b**S(2), S(2))
rubi.append(2237)
return Simp(x/q, x) - Simp(S(2)*ArcTan(b*sin(c + d*x)/(a + b*cos(c + d*x) + q))/(d*q), x)
pattern2237 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1271, cons481)
rule2237 = ReplacementRule(pattern2237, With2237)
def With2238(b, d, c, a, x):
q = Rt(a**S(2) - b**S(2), S(2))
rubi.append(2238)
return -Simp(x/q, x) - Simp(S(2)*ArcTan(b*cos(c + d*x)/(a + b*sin(c + d*x) - q))/(d*q), x)
pattern2238 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1271, cons482)
rule2238 = ReplacementRule(pattern2238, With2238)
def With2239(b, d, c, a, x):
q = Rt(a**S(2) - b**S(2), S(2))
rubi.append(2239)
return -Simp(x/q, x) + Simp(S(2)*ArcTan(b*sin(c + d*x)/(a + b*cos(c + d*x) - q))/(d*q), x)
pattern2239 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1271, cons482)
rule2239 = ReplacementRule(pattern2239, With2239)
def With2240(b, d, c, a, x):
e = FreeFactors(tan(-Pi/S(4) + c/S(2) + d*x/S(2)), x)
rubi.append(2240)
return Dist(S(2)*e/d, Subst(Int(S(1)/(a + b + e**S(2)*x**S(2)*(a - b)), x), x, tan(-Pi/S(4) + c/S(2) + d*x/S(2))/e), x)
pattern2240 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1272)
rule2240 = ReplacementRule(pattern2240, With2240)
def With2241(b, d, c, a, x):
e = FreeFactors(tan(c/S(2) + d*x/S(2)), x)
rubi.append(2241)
return Dist(S(2)*e/d, Subst(Int(S(1)/(a*e**S(2)*x**S(2) + a + S(2)*b*e*x), x), x, tan(c/S(2) + d*x/S(2))/e), x)
pattern2241 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267)
rule2241 = ReplacementRule(pattern2241, With2241)
def With2242(b, d, c, a, x):
e = FreeFactors(tan(c/S(2) + d*x/S(2)), x)
rubi.append(2242)
return Dist(S(2)*e/d, Subst(Int(S(1)/(a + b + e**S(2)*x**S(2)*(a - b)), x), x, tan(c/S(2) + d*x/S(2))/e), x)
pattern2242 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267)
rule2242 = ReplacementRule(pattern2242, With2242)
pattern2243 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1268)
def replacement2243(b, d, c, a, x):
rubi.append(2243)
return Simp(S(2)*EllipticF(-Pi/S(4) + c/S(2) + d*x/S(2), S(2)*b/(a + b))/(d*sqrt(a + b)), x)
rule2243 = ReplacementRule(pattern2243, replacement2243)
pattern2244 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1268)
def replacement2244(b, d, c, a, x):
rubi.append(2244)
return Simp(S(2)*EllipticF(c/S(2) + d*x/S(2), S(2)*b/(a + b))/(d*sqrt(a + b)), x)
rule2244 = ReplacementRule(pattern2244, replacement2244)
pattern2245 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1269)
def replacement2245(b, d, c, a, x):
rubi.append(2245)
return Simp(S(2)*EllipticF(Pi/S(4) + c/S(2) + d*x/S(2), -S(2)*b/(a - b))/(d*sqrt(a - b)), x)
rule2245 = ReplacementRule(pattern2245, replacement2245)
pattern2246 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1269)
def replacement2246(b, d, c, a, x):
rubi.append(2246)
return Simp(S(2)*EllipticF(Pi/S(2) + c/S(2) + d*x/S(2), -S(2)*b/(a - b))/(d*sqrt(a - b)), x)
rule2246 = ReplacementRule(pattern2246, replacement2246)
pattern2247 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1270)
def replacement2247(b, d, c, a, x):
rubi.append(2247)
return Dist(sqrt((a + b*sin(c + d*x))/(a + b))/sqrt(a + b*sin(c + d*x)), Int(S(1)/sqrt(a/(a + b) + b*sin(c + d*x)/(a + b)), x), x)
rule2247 = ReplacementRule(pattern2247, replacement2247)
pattern2248 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267, cons1270)
def replacement2248(b, d, c, a, x):
rubi.append(2248)
return Dist(sqrt((a + b*cos(c + d*x))/(a + b))/sqrt(a + b*cos(c + d*x)), Int(S(1)/sqrt(a/(a + b) + b*cos(c + d*x)/(a + b)), x), x)
rule2248 = ReplacementRule(pattern2248, replacement2248)
pattern2249 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1267, cons87, cons89, cons808)
def replacement2249(b, d, c, a, n, x):
rubi.append(2249)
return Dist(S(1)/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*sin(c + d*x))**(n + S(1))*Simp(a*(n + S(1)) - b*(n + S(2))*sin(c + d*x), x), x), x) - Simp(b*(a + b*sin(c + d*x))**(n + S(1))*cos(c + d*x)/(d*(a**S(2) - b**S(2))*(n + S(1))), x)
rule2249 = ReplacementRule(pattern2249, replacement2249)
pattern2250 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1267, cons87, cons89, cons808)
def replacement2250(b, d, c, a, n, x):
rubi.append(2250)
return Dist(S(1)/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*cos(c + d*x))**(n + S(1))*Simp(a*(n + S(1)) - b*(n + S(2))*cos(c + d*x), x), x), x) + Simp(b*(a + b*cos(c + d*x))**(n + S(1))*sin(c + d*x)/(d*(a**S(2) - b**S(2))*(n + S(1))), x)
rule2250 = ReplacementRule(pattern2250, replacement2250)
pattern2251 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1267, cons543)
def replacement2251(b, d, c, a, n, x):
rubi.append(2251)
return Dist(cos(c + d*x)/(d*sqrt(-sin(c + d*x) + S(1))*sqrt(sin(c + d*x) + S(1))), Subst(Int((a + b*x)**n/(sqrt(-x + S(1))*sqrt(x + S(1))), x), x, sin(c + d*x)), x)
rule2251 = ReplacementRule(pattern2251, replacement2251)
pattern2252 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1267, cons543)
def replacement2252(b, d, c, a, n, x):
rubi.append(2252)
return -Dist(sin(c + d*x)/(d*sqrt(-cos(c + d*x) + S(1))*sqrt(cos(c + d*x) + S(1))), Subst(Int((a + b*x)**n/(sqrt(-x + S(1))*sqrt(x + S(1))), x), x, cos(c + d*x)), x)
rule2252 = ReplacementRule(pattern2252, replacement2252)
pattern2253 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1273)
def replacement2253(b, d, c, a, n, x):
rubi.append(2253)
return Int((a + b*sin(S(2)*c + S(2)*d*x)/S(2))**n, x)
rule2253 = ReplacementRule(pattern2253, replacement2253)
pattern2254 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1274, cons1265, cons1275)
def replacement2254(p, m, f, b, a, x, e):
rubi.append(2254)
return Dist(b**(-p)/f, Subst(Int((a - x)**(p/S(2) + S(-1)/2)*(a + x)**(m + p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x)
rule2254 = ReplacementRule(pattern2254, replacement2254)
pattern2255 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1274, cons1265, cons1275)
def replacement2255(p, m, f, b, a, x, e):
rubi.append(2255)
return -Dist(b**(-p)/f, Subst(Int((a - x)**(p/S(2) + S(-1)/2)*(a + x)**(m + p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x)
rule2255 = ReplacementRule(pattern2255, replacement2255)
pattern2256 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1274, cons1267)
def replacement2256(p, m, f, b, a, x, e):
rubi.append(2256)
return Dist(b**(-p)/f, Subst(Int((a + x)**m*(b**S(2) - x**S(2))**(p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x)
rule2256 = ReplacementRule(pattern2256, replacement2256)
pattern2257 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1274, cons1267)
def replacement2257(p, m, f, b, a, x, e):
rubi.append(2257)
return -Dist(b**(-p)/f, Subst(Int((a + x)**m*(b**S(2) - x**S(2))**(p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x)
rule2257 = ReplacementRule(pattern2257, replacement2257)
pattern2258 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1276)
def replacement2258(p, f, b, g, a, x, e):
rubi.append(2258)
return Dist(a, Int((g*cos(e + f*x))**p, x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))/(f*g*(p + S(1))), x)
rule2258 = ReplacementRule(pattern2258, replacement2258)
pattern2259 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1276)
def replacement2259(p, f, b, g, a, x, e):
rubi.append(2259)
return Dist(a, Int((g*sin(e + f*x))**p, x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))/(f*g*(p + S(1))), x)
rule2259 = ReplacementRule(pattern2259, replacement2259)
pattern2260 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons17, cons13, cons137, cons1277)
def replacement2260(p, m, f, b, g, a, x, e):
rubi.append(2260)
return Dist((a/g)**(S(2)*m), Int((g*cos(e + f*x))**(S(2)*m + p)*(a - b*sin(e + f*x))**(-m), x), x)
rule2260 = ReplacementRule(pattern2260, replacement2260)
pattern2261 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons17, cons13, cons137, cons1277)
def replacement2261(p, m, f, b, g, a, x, e):
rubi.append(2261)
return Dist((a/g)**(S(2)*m), Int((g*sin(e + f*x))**(S(2)*m + p)*(a - b*cos(e + f*x))**(-m), x), x)
rule2261 = ReplacementRule(pattern2261, replacement2261)
pattern2262 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1278, cons1279)
def replacement2262(p, m, f, b, g, a, x, e):
rubi.append(2262)
return Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*m), x)
rule2262 = ReplacementRule(pattern2262, replacement2262)
pattern2263 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1278, cons1279)
def replacement2263(p, m, f, b, g, a, x, e):
rubi.append(2263)
return -Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*m), x)
rule2263 = ReplacementRule(pattern2263, replacement2263)
pattern2264 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1280, cons1281, cons143)
def replacement2264(p, m, f, b, g, a, x, e):
rubi.append(2264)
return Dist((m + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x)
rule2264 = ReplacementRule(pattern2264, replacement2264)
pattern2265 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1280, cons1281, cons143)
def replacement2265(p, m, f, b, g, a, x, e):
rubi.append(2265)
return Dist((m + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x)
rule2265 = ReplacementRule(pattern2265, replacement2265)
pattern2266 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1282, cons1283)
def replacement2266(p, m, f, b, g, a, x, e):
rubi.append(2266)
return Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(m + S(-1))), x)
rule2266 = ReplacementRule(pattern2266, replacement2266)
pattern2267 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1282, cons1283)
def replacement2267(p, m, f, b, g, a, x, e):
rubi.append(2267)
return -Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(m + S(-1))), x)
rule2267 = ReplacementRule(pattern2267, replacement2267)
pattern2268 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1284, cons1285)
def replacement2268(p, m, f, b, g, a, x, e):
rubi.append(2268)
return Dist(a*(S(2)*m + p + S(-1))/(m + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(m + p)), x)
rule2268 = ReplacementRule(pattern2268, replacement2268)
pattern2269 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1284, cons1285)
def replacement2269(p, m, f, b, g, a, x, e):
rubi.append(2269)
return Dist(a*(S(2)*m + p + S(-1))/(m + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(m + p)), x)
rule2269 = ReplacementRule(pattern2269, replacement2269)
pattern2270 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons244, cons168, cons1286, cons1287)
def replacement2270(p, m, f, b, g, a, x, e):
rubi.append(2270)
return Dist(a*(m + p + S(1))/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*(p + S(1))), x)
rule2270 = ReplacementRule(pattern2270, replacement2270)
pattern2271 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons244, cons168, cons1286, cons1287)
def replacement2271(p, m, f, b, g, a, x, e):
rubi.append(2271)
return Dist(a*(m + p + S(1))/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*(p + S(1))), x)
rule2271 = ReplacementRule(pattern2271, replacement2271)
pattern2272 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons244, cons166, cons137, cons1288)
def replacement2272(p, m, f, b, g, a, x, e):
rubi.append(2272)
return Dist(b**S(2)*(S(2)*m + p + S(-1))/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-2)), x), x) + Simp(-S(2)*b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(p + S(1))), x)
rule2272 = ReplacementRule(pattern2272, replacement2272)
pattern2273 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons244, cons166, cons137, cons1288)
def replacement2273(p, m, f, b, g, a, x, e):
rubi.append(2273)
return Dist(b**S(2)*(S(2)*m + p + S(-1))/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-2)), x), x) + Simp(S(2)*b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(p + S(1))), x)
rule2273 = ReplacementRule(pattern2273, replacement2273)
pattern2274 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265)
def replacement2274(f, b, g, a, x, e):
rubi.append(2274)
return Dist(a*sqrt(a + b*sin(e + f*x))*sqrt(cos(e + f*x) + S(1))/(a*cos(e + f*x) + a + b*sin(e + f*x)), Int(sqrt(cos(e + f*x) + S(1))/sqrt(g*cos(e + f*x)), x), x) + Dist(b*sqrt(a + b*sin(e + f*x))*sqrt(cos(e + f*x) + S(1))/(a*cos(e + f*x) + a + b*sin(e + f*x)), Int(sin(e + f*x)/(sqrt(g*cos(e + f*x))*sqrt(cos(e + f*x) + S(1))), x), x)
rule2274 = ReplacementRule(pattern2274, replacement2274)
pattern2275 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265)
def replacement2275(f, b, g, a, x, e):
rubi.append(2275)
return Dist(a*sqrt(a + b*cos(e + f*x))*sqrt(sin(e + f*x) + S(1))/(a*sin(e + f*x) + a + b*cos(e + f*x)), Int(sqrt(sin(e + f*x) + S(1))/sqrt(g*sin(e + f*x)), x), x) + Dist(b*sqrt(a + b*cos(e + f*x))*sqrt(sin(e + f*x) + S(1))/(a*sin(e + f*x) + a + b*cos(e + f*x)), Int(cos(e + f*x)/(sqrt(g*sin(e + f*x))*sqrt(sin(e + f*x) + S(1))), x), x)
rule2275 = ReplacementRule(pattern2275, replacement2275)
pattern2276 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons31, cons168, cons1285, cons1288)
def replacement2276(p, m, f, b, g, a, x, e):
rubi.append(2276)
return Dist(a*(S(2)*m + p + S(-1))/(m + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(m + p)), x)
rule2276 = ReplacementRule(pattern2276, replacement2276)
pattern2277 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons31, cons168, cons1285, cons1288)
def replacement2277(p, m, f, b, g, a, x, e):
rubi.append(2277)
return Dist(a*(S(2)*m + p + S(-1))/(m + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(m + p)), x)
rule2277 = ReplacementRule(pattern2277, replacement2277)
pattern2278 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons244, cons94, cons146, cons1289, cons1285, cons1288)
def replacement2278(p, m, f, b, g, a, x, e):
rubi.append(2278)
return Dist(g**S(2)*(p + S(-1))/(a*(m + p)), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*(m + p)), x)
rule2278 = ReplacementRule(pattern2278, replacement2278)
pattern2279 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons244, cons94, cons146, cons1289, cons1285, cons1288)
def replacement2279(p, m, f, b, g, a, x, e):
rubi.append(2279)
return Dist(g**S(2)*(p + S(-1))/(a*(m + p)), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*(m + p)), x)
rule2279 = ReplacementRule(pattern2279, replacement2279)
pattern2280 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons244, cons1290, cons146, cons1281, cons1291, cons1288)
def replacement2280(p, m, f, b, g, a, x, e):
rubi.append(2280)
return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(2)), x), x) + Simp(S(2)*g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*(S(2)*m + p + S(1))), x)
rule2280 = ReplacementRule(pattern2280, replacement2280)
pattern2281 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons244, cons1290, cons146, cons1281, cons1291, cons1288)
def replacement2281(p, m, f, b, g, a, x, e):
rubi.append(2281)
return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(2)), x), x) + Simp(-S(2)*g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*(S(2)*m + p + S(1))), x)
rule2281 = ReplacementRule(pattern2281, replacement2281)
pattern2282 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons31, cons94, cons1281, cons1288)
def replacement2282(p, m, f, b, g, a, x, e):
rubi.append(2282)
return Dist((m + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x)
rule2282 = ReplacementRule(pattern2282, replacement2282)
pattern2283 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons31, cons94, cons1281, cons1288)
def replacement2283(p, m, f, b, g, a, x, e):
rubi.append(2283)
return Dist((m + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x)
rule2283 = ReplacementRule(pattern2283, replacement2283)
pattern2284 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons13, cons146, cons246)
def replacement2284(p, f, b, g, a, x, e):
rubi.append(2284)
return Dist(g**S(2)/a, Int((g*cos(e + f*x))**(p + S(-2)), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))/(b*f*(p + S(-1))), x)
rule2284 = ReplacementRule(pattern2284, replacement2284)
pattern2285 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons13, cons146, cons246)
def replacement2285(p, f, b, g, a, x, e):
rubi.append(2285)
return Dist(g**S(2)/a, Int((g*sin(e + f*x))**(p + S(-2)), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))/(b*f*(p + S(-1))), x)
rule2285 = ReplacementRule(pattern2285, replacement2285)
pattern2286 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons1292, cons246)
def replacement2286(p, f, b, g, a, x, e):
rubi.append(2286)
return Dist(p/(a*(p + S(-1))), Int((g*cos(e + f*x))**p, x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))/(a*f*g*(a + b*sin(e + f*x))*(p + S(-1))), x)
rule2286 = ReplacementRule(pattern2286, replacement2286)
pattern2287 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons1292, cons246)
def replacement2287(p, f, b, g, a, x, e):
rubi.append(2287)
return Dist(p/(a*(p + S(-1))), Int((g*sin(e + f*x))**p, x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))/(a*f*g*(a + b*cos(e + f*x))*(p + S(-1))), x)
rule2287 = ReplacementRule(pattern2287, replacement2287)
pattern2288 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265)
def replacement2288(f, b, g, a, x, e):
rubi.append(2288)
return -Dist(g*sqrt(a + b*sin(e + f*x))*sqrt(cos(e + f*x) + S(1))/(a*sin(e + f*x) + b*cos(e + f*x) + b), Int(sin(e + f*x)/(sqrt(g*cos(e + f*x))*sqrt(cos(e + f*x) + S(1))), x), x) + Dist(g*sqrt(a + b*sin(e + f*x))*sqrt(cos(e + f*x) + S(1))/(a*cos(e + f*x) + a + b*sin(e + f*x)), Int(sqrt(cos(e + f*x) + S(1))/sqrt(g*cos(e + f*x)), x), x)
rule2288 = ReplacementRule(pattern2288, replacement2288)
pattern2289 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265)
def replacement2289(f, b, g, a, x, e):
rubi.append(2289)
return Dist(g*sqrt(a + b*cos(e + f*x))*sqrt(sin(e + f*x) + S(1))/(a*sin(e + f*x) + a + b*cos(e + f*x)), Int(sqrt(sin(e + f*x) + S(1))/sqrt(g*sin(e + f*x)), x), x) - Dist(g*sqrt(a + b*cos(e + f*x))*sqrt(sin(e + f*x) + S(1))/(a*cos(e + f*x) + b*sin(e + f*x) + b), Int(cos(e + f*x)/(sqrt(g*sin(e + f*x))*sqrt(sin(e + f*x) + S(1))), x), x)
rule2289 = ReplacementRule(pattern2289, replacement2289)
pattern2290 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265)
def replacement2290(f, b, g, a, x, e):
rubi.append(2290)
return Dist(g**S(2)/(S(2)*a), Int(sqrt(a + b*sin(e + f*x))/sqrt(g*cos(e + f*x)), x), x) + Simp(g*sqrt(g*cos(e + f*x))*sqrt(a + b*sin(e + f*x))/(b*f), x)
rule2290 = ReplacementRule(pattern2290, replacement2290)
pattern2291 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265)
def replacement2291(f, b, g, a, x, e):
rubi.append(2291)
return Dist(g**S(2)/(S(2)*a), Int(sqrt(a + b*cos(e + f*x))/sqrt(g*sin(e + f*x)), x), x) - Simp(g*sqrt(g*sin(e + f*x))*sqrt(a + b*cos(e + f*x))/(b*f), x)
rule2291 = ReplacementRule(pattern2291, replacement2291)
pattern2292 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons13, cons1293, cons246)
def replacement2292(p, f, b, g, a, x, e):
rubi.append(2292)
return Dist(S(2)*a*(p + S(-2))/(S(2)*p + S(-1)), Int((g*cos(e + f*x))**p/(a + b*sin(e + f*x))**(S(3)/2), x), x) + Simp(-S(2)*b*(g*cos(e + f*x))**(p + S(1))/(f*g*(a + b*sin(e + f*x))**(S(3)/2)*(S(2)*p + S(-1))), x)
rule2292 = ReplacementRule(pattern2292, replacement2292)
pattern2293 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons13, cons1293, cons246)
def replacement2293(p, f, b, g, a, x, e):
rubi.append(2293)
return Dist(S(2)*a*(p + S(-2))/(S(2)*p + S(-1)), Int((g*sin(e + f*x))**p/(a + b*cos(e + f*x))**(S(3)/2), x), x) + Simp(S(2)*b*(g*sin(e + f*x))**(p + S(1))/(f*g*(a + b*cos(e + f*x))**(S(3)/2)*(S(2)*p + S(-1))), x)
rule2293 = ReplacementRule(pattern2293, replacement2293)
pattern2294 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons13, cons137, cons246)
def replacement2294(p, f, b, g, a, x, e):
rubi.append(2294)
return Dist(a*(S(2)*p + S(1))/(S(2)*g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))/(a + b*sin(e + f*x))**(S(3)/2), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))/(a*f*g*sqrt(a + b*sin(e + f*x))*(p + S(1))), x)
rule2294 = ReplacementRule(pattern2294, replacement2294)
pattern2295 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1265, cons13, cons137, cons246)
def replacement2295(p, f, b, g, a, x, e):
rubi.append(2295)
return Dist(a*(S(2)*p + S(1))/(S(2)*g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))/(a + b*cos(e + f*x))**(S(3)/2), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))/(a*f*g*sqrt(a + b*cos(e + f*x))*(p + S(1))), x)
rule2295 = ReplacementRule(pattern2295, replacement2295)
pattern2296 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons17)
def replacement2296(p, m, f, b, g, a, x, e):
rubi.append(2296)
return Dist(a**m*(g*cos(e + f*x))**(p + S(1))*(-sin(e + f*x) + S(1))**(-p/S(2) + S(-1)/2)*(sin(e + f*x) + S(1))**(-p/S(2) + S(-1)/2)/(f*g), Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule2296 = ReplacementRule(pattern2296, replacement2296)
pattern2297 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons17)
def replacement2297(p, m, f, b, g, a, x, e):
rubi.append(2297)
return -Dist(a**m*(g*sin(e + f*x))**(p + S(1))*(-cos(e + f*x) + S(1))**(-p/S(2) + S(-1)/2)*(cos(e + f*x) + S(1))**(-p/S(2) + S(-1)/2)/(f*g), Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule2297 = ReplacementRule(pattern2297, replacement2297)
pattern2298 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons18)
def replacement2298(p, m, f, b, g, a, x, e):
rubi.append(2298)
return Dist(a**S(2)*(g*cos(e + f*x))**(p + S(1))*(a - b*sin(e + f*x))**(-p/S(2) + S(-1)/2)*(a + b*sin(e + f*x))**(-p/S(2) + S(-1)/2)/(f*g), Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule2298 = ReplacementRule(pattern2298, replacement2298)
pattern2299 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons18)
def replacement2299(p, m, f, b, g, a, x, e):
rubi.append(2299)
return -Dist(a**S(2)*(g*sin(e + f*x))**(p + S(1))*(a - b*cos(e + f*x))**(-p/S(2) + S(-1)/2)*(a + b*cos(e + f*x))**(-p/S(2) + S(-1)/2)/(f*g), Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule2299 = ReplacementRule(pattern2299, replacement2299)
pattern2300 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons244, cons1256, cons137, cons1294)
def replacement2300(p, m, f, b, g, a, x, e):
rubi.append(2300)
return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1))*(a*(p + S(2)) + b*(m + p + S(2))*sin(e + f*x)), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*sin(e + f*x)/(f*g*(p + S(1))), x)
rule2300 = ReplacementRule(pattern2300, replacement2300)
pattern2301 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons244, cons1256, cons137, cons1294)
def replacement2301(p, m, f, b, g, a, x, e):
rubi.append(2301)
return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1))*(a*(p + S(2)) + b*(m + p + S(2))*cos(e + f*x)), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*cos(e + f*x)/(f*g*(p + S(1))), x)
rule2301 = ReplacementRule(pattern2301, replacement2301)
pattern2302 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons244, cons166, cons137, cons1294)
def replacement2302(p, m, f, b, g, a, x, e):
rubi.append(2302)
return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-2))*(a**S(2)*(p + S(2)) + a*b*(m + p + S(1))*sin(e + f*x) + b**S(2)*(m + S(-1))), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(a*sin(e + f*x) + b)/(f*g*(p + S(1))), x)
rule2302 = ReplacementRule(pattern2302, replacement2302)
pattern2303 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons244, cons166, cons137, cons1294)
def replacement2303(p, m, f, b, g, a, x, e):
rubi.append(2303)
return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-2))*(a**S(2)*(p + S(2)) + a*b*(m + p + S(1))*cos(e + f*x) + b**S(2)*(m + S(-1))), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(a*cos(e + f*x) + b)/(f*g*(p + S(1))), x)
rule2303 = ReplacementRule(pattern2303, replacement2303)
pattern2304 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons31, cons166, cons1285, cons1294)
def replacement2304(p, m, f, b, g, a, x, e):
rubi.append(2304)
return Dist(S(1)/(m + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-2))*(a**S(2)*(m + p) + a*b*(S(2)*m + p + S(-1))*sin(e + f*x) + b**S(2)*(m + S(-1))), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(m + p)), x)
rule2304 = ReplacementRule(pattern2304, replacement2304)
pattern2305 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons31, cons166, cons1285, cons1294)
def replacement2305(p, m, f, b, g, a, x, e):
rubi.append(2305)
return Dist(S(1)/(m + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-2))*(a**S(2)*(m + p) + a*b*(S(2)*m + p + S(-1))*cos(e + f*x) + b**S(2)*(m + S(-1))), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(m + p)), x)
rule2305 = ReplacementRule(pattern2305, replacement2305)
pattern2306 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons244, cons94, cons146, cons1288)
def replacement2306(p, m, f, b, g, a, x, e):
rubi.append(2306)
return Dist(g**S(2)*(p + S(-1))/(b*(m + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1))*sin(e + f*x), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule2306 = ReplacementRule(pattern2306, replacement2306)
pattern2307 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons244, cons94, cons146, cons1288)
def replacement2307(p, m, f, b, g, a, x, e):
rubi.append(2307)
return Dist(g**S(2)*(p + S(-1))/(b*(m + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1))*cos(e + f*x), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x)
rule2307 = ReplacementRule(pattern2307, replacement2307)
pattern2308 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons31, cons94, cons1288)
def replacement2308(p, m, f, b, g, a, x, e):
rubi.append(2308)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*(a*(m + S(1)) - b*(m + p + S(2))*sin(e + f*x)), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(f*g*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2308 = ReplacementRule(pattern2308, replacement2308)
pattern2309 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons31, cons94, cons1288)
def replacement2309(p, m, f, b, g, a, x, e):
rubi.append(2309)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*(a*(m + S(1)) - b*(m + p + S(2))*cos(e + f*x)), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(f*g*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2309 = ReplacementRule(pattern2309, replacement2309)
pattern2310 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons1267, cons13, cons146, cons1285, cons1288)
def replacement2310(p, m, f, b, g, a, x, e):
rubi.append(2310)
return Dist(g**S(2)*(p + S(-1))/(b*(m + p)), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**m*(a*sin(e + f*x) + b), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*(m + p)), x)
rule2310 = ReplacementRule(pattern2310, replacement2310)
pattern2311 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons1267, cons13, cons146, cons1285, cons1288)
def replacement2311(p, m, f, b, g, a, x, e):
rubi.append(2311)
return Dist(g**S(2)*(p + S(-1))/(b*(m + p)), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**m*(a*cos(e + f*x) + b), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*(m + p)), x)
rule2311 = ReplacementRule(pattern2311, replacement2311)
pattern2312 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons1267, cons13, cons137, cons1288)
def replacement2312(p, m, f, b, g, a, x, e):
rubi.append(2312)
return Dist(S(1)/(g**S(2)*(a**S(2) - b**S(2))*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m*(a**S(2)*(p + S(2)) + a*b*(m + p + S(3))*sin(e + f*x) - b**S(2)*(m + p + S(2))), x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*sin(e + f*x) + b)/(f*g*(a**S(2) - b**S(2))*(p + S(1))), x)
rule2312 = ReplacementRule(pattern2312, replacement2312)
pattern2313 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons1267, cons13, cons137, cons1288)
def replacement2313(p, m, f, b, g, a, x, e):
rubi.append(2313)
return Dist(S(1)/(g**S(2)*(a**S(2) - b**S(2))*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m*(a**S(2)*(p + S(2)) + a*b*(m + p + S(3))*cos(e + f*x) - b**S(2)*(m + p + S(2))), x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*cos(e + f*x) + b)/(f*g*(a**S(2) - b**S(2))*(p + S(1))), x)
rule2313 = ReplacementRule(pattern2313, replacement2313)
pattern2314 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
def replacement2314(f, b, g, a, x, e):
rubi.append(2314)
return Dist(S(2)*sqrt(S(2))*sqrt(g*cos(e + f*x))*sqrt((a + b*sin(e + f*x))/((a - b)*(-sin(e + f*x) + S(1))))/(f*g*sqrt((sin(e + f*x) + cos(e + f*x) + S(1))/(-sin(e + f*x) + cos(e + f*x) + S(1)))*sqrt(a + b*sin(e + f*x))), Subst(Int(S(1)/sqrt(x**S(4)*(a + b)/(a - b) + S(1)), x), x, sqrt((sin(e + f*x) + cos(e + f*x) + S(1))/(-sin(e + f*x) + cos(e + f*x) + S(1)))), x)
rule2314 = ReplacementRule(pattern2314, replacement2314)
pattern2315 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
def replacement2315(f, b, g, a, x, e):
rubi.append(2315)
return Dist(-S(2)*sqrt(S(2))*sqrt(g*sin(e + f*x))*sqrt((a + b*cos(e + f*x))/((a - b)*(-cos(e + f*x) + S(1))))/(f*g*sqrt((sin(e + f*x) + cos(e + f*x) + S(1))/(sin(e + f*x) - cos(e + f*x) + S(1)))*sqrt(a + b*cos(e + f*x))), Subst(Int(S(1)/sqrt(x**S(4)*(a + b)/(a - b) + S(1)), x), x, sqrt((sin(e + f*x) + cos(e + f*x) + S(1))/(sin(e + f*x) - cos(e + f*x) + S(1)))), x)
rule2315 = ReplacementRule(pattern2315, replacement2315)
pattern2316 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1267, cons1278)
def replacement2316(p, m, f, b, g, a, x, e):
rubi.append(2316)
return Simp(g*(g*cos(e + f*x))**(p + S(-1))*(-(a - b)*(-sin(e + f*x) + S(1))/((a + b)*(sin(e + f*x) + S(1))))**(m/S(2))*(a + b*sin(e + f*x))**(m + S(1))*(-sin(e + f*x) + S(1))*Hypergeometric2F1(m + S(1), m/S(2) + S(1), m + S(2), S(2)*(a + b*sin(e + f*x))/((a + b)*(sin(e + f*x) + S(1))))/(f*(a + b)*(m + S(1))), x)
rule2316 = ReplacementRule(pattern2316, replacement2316)
pattern2317 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1267, cons1278)
def replacement2317(p, m, f, b, g, a, x, e):
rubi.append(2317)
return -Simp(g*(g*sin(e + f*x))**(p + S(-1))*(-(a - b)*(-cos(e + f*x) + S(1))/((a + b)*(cos(e + f*x) + S(1))))**(m/S(2))*(a + b*cos(e + f*x))**(m + S(1))*(-cos(e + f*x) + S(1))*Hypergeometric2F1(m + S(1), m/S(2) + S(1), m + S(2), S(2)*(a + b*cos(e + f*x))/((a + b)*(cos(e + f*x) + S(1))))/(f*(a + b)*(m + S(1))), x)
rule2317 = ReplacementRule(pattern2317, replacement2317)
pattern2318 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1267, cons1295)
def replacement2318(p, m, f, b, g, a, x, e):
rubi.append(2318)
return Dist(a/(g**S(2)*(a - b)), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m/(-sin(e + f*x) + S(1)), x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(f*g*(a - b)*(p + S(1))), x)
rule2318 = ReplacementRule(pattern2318, replacement2318)
pattern2319 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1267, cons1295)
def replacement2319(p, m, f, b, g, a, x, e):
rubi.append(2319)
return Dist(a/(g**S(2)*(a - b)), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m/(-cos(e + f*x) + S(1)), x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(f*g*(a - b)*(p + S(1))), x)
rule2319 = ReplacementRule(pattern2319, replacement2319)
pattern2320 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1267, cons1296)
def replacement2320(p, m, f, b, g, a, x, e):
rubi.append(2320)
return Dist(a/(g**S(2)*(a - b)), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m/(-sin(e + f*x) + S(1)), x), x) - Dist(b*(m + p + S(2))/(g**S(2)*(a - b)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m, x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(f*g*(a - b)*(p + S(1))), x)
rule2320 = ReplacementRule(pattern2320, replacement2320)
pattern2321 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1267, cons1296)
def replacement2321(p, m, f, b, g, a, x, e):
rubi.append(2321)
return Dist(a/(g**S(2)*(a - b)), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m/(-cos(e + f*x) + S(1)), x), x) - Dist(b*(m + p + S(2))/(g**S(2)*(a - b)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m, x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(f*g*(a - b)*(p + S(1))), x)
rule2321 = ReplacementRule(pattern2321, replacement2321)
def With2322(f, b, g, a, x, e):
q = Rt(-a**S(2) + b**S(2), S(2))
rubi.append(2322)
return -Dist(a*g/(S(2)*b), Int(S(1)/(sqrt(g*cos(e + f*x))*(-b*cos(e + f*x) + q)), x), x) + Dist(a*g/(S(2)*b), Int(S(1)/(sqrt(g*cos(e + f*x))*(b*cos(e + f*x) + q)), x), x) + Dist(b*g/f, Subst(Int(sqrt(x)/(b**S(2)*x**S(2) + g**S(2)*(a**S(2) - b**S(2))), x), x, g*cos(e + f*x)), x)
pattern2322 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
rule2322 = ReplacementRule(pattern2322, With2322)
def With2323(f, b, g, a, x, e):
q = Rt(-a**S(2) + b**S(2), S(2))
rubi.append(2323)
return -Dist(a*g/(S(2)*b), Int(S(1)/(sqrt(g*sin(e + f*x))*(-b*sin(e + f*x) + q)), x), x) + Dist(a*g/(S(2)*b), Int(S(1)/(sqrt(g*sin(e + f*x))*(b*sin(e + f*x) + q)), x), x) - Dist(b*g/f, Subst(Int(sqrt(x)/(b**S(2)*x**S(2) + g**S(2)*(a**S(2) - b**S(2))), x), x, g*sin(e + f*x)), x)
pattern2323 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
rule2323 = ReplacementRule(pattern2323, With2323)
def With2324(f, b, g, a, x, e):
q = Rt(-a**S(2) + b**S(2), S(2))
rubi.append(2324)
return -Dist(a/(S(2)*q), Int(S(1)/(sqrt(g*cos(e + f*x))*(-b*cos(e + f*x) + q)), x), x) - Dist(a/(S(2)*q), Int(S(1)/(sqrt(g*cos(e + f*x))*(b*cos(e + f*x) + q)), x), x) + Dist(b*g/f, Subst(Int(S(1)/(sqrt(x)*(b**S(2)*x**S(2) + g**S(2)*(a**S(2) - b**S(2)))), x), x, g*cos(e + f*x)), x)
pattern2324 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
rule2324 = ReplacementRule(pattern2324, With2324)
def With2325(f, b, g, a, x, e):
q = Rt(-a**S(2) + b**S(2), S(2))
rubi.append(2325)
return -Dist(a/(S(2)*q), Int(S(1)/(sqrt(g*sin(e + f*x))*(-b*sin(e + f*x) + q)), x), x) - Dist(a/(S(2)*q), Int(S(1)/(sqrt(g*sin(e + f*x))*(b*sin(e + f*x) + q)), x), x) - Dist(b*g/f, Subst(Int(S(1)/(sqrt(x)*(b**S(2)*x**S(2) + g**S(2)*(a**S(2) - b**S(2)))), x), x, g*sin(e + f*x)), x)
pattern2325 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
rule2325 = ReplacementRule(pattern2325, With2325)
pattern2326 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons84, cons1297)
def replacement2326(p, m, f, b, g, a, x, e):
rubi.append(2326)
return Simp(g*(g*cos(e + f*x))**(p + S(-1))*(b*(sin(e + f*x) + S(1))/(a + b*sin(e + f*x)))**(-p/S(2) + S(1)/2)*(-b*(-sin(e + f*x) + S(1))/(a + b*sin(e + f*x)))**(-p/S(2) + S(1)/2)*(a + b*sin(e + f*x))**(m + S(1))*AppellF1(-m - p, -p/S(2) + S(1)/2, -p/S(2) + S(1)/2, -m - p + S(1), (a + b)/(a + b*sin(e + f*x)), (a - b)/(a + b*sin(e + f*x)))/(b*f*(m + p)), x)
rule2326 = ReplacementRule(pattern2326, replacement2326)
pattern2327 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons84, cons1297)
def replacement2327(p, m, f, b, g, a, x, e):
rubi.append(2327)
return -Simp(g*(g*sin(e + f*x))**(p + S(-1))*(b*(cos(e + f*x) + S(1))/(a + b*cos(e + f*x)))**(-p/S(2) + S(1)/2)*(-b*(-cos(e + f*x) + S(1))/(a + b*cos(e + f*x)))**(-p/S(2) + S(1)/2)*(a + b*cos(e + f*x))**(m + S(1))*AppellF1(-m - p, -p/S(2) + S(1)/2, -p/S(2) + S(1)/2, -m - p + S(1), (a + b)/(a + b*cos(e + f*x)), (a - b)/(a + b*cos(e + f*x)))/(b*f*(m + p)), x)
rule2327 = ReplacementRule(pattern2327, replacement2327)
pattern2328 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1267, cons143)
def replacement2328(p, m, f, b, g, a, x, e):
rubi.append(2328)
return Dist(g*(g*cos(e + f*x))**(p + S(-1))*(S(1) - (a + b*sin(e + f*x))/(a - b))**(-p/S(2) + S(1)/2)*(S(1) - (a + b*sin(e + f*x))/(a + b))**(-p/S(2) + S(1)/2)/f, Subst(Int((a + b*x)**m*(-b*x/(a - b) - b/(a - b))**(p/S(2) + S(-1)/2)*(-b*x/(a + b) + b/(a + b))**(p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule2328 = ReplacementRule(pattern2328, replacement2328)
pattern2329 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1267, cons143)
def replacement2329(p, m, f, b, g, a, x, e):
rubi.append(2329)
return -Dist(g*(g*sin(e + f*x))**(p + S(-1))*(S(1) - (a + b*cos(e + f*x))/(a - b))**(-p/S(2) + S(1)/2)*(S(1) - (a + b*cos(e + f*x))/(a + b))**(-p/S(2) + S(1)/2)/f, Subst(Int((a + b*x)**m*(-b*x/(a - b) - b/(a - b))**(p/S(2) + S(-1)/2)*(-b*x/(a + b) + b/(a + b))**(p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule2329 = ReplacementRule(pattern2329, replacement2329)
pattern2330 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons147)
def replacement2330(p, m, f, g, b, a, x, e):
rubi.append(2330)
return Dist(g**(S(2)*IntPart(p))*(g/cos(e + f*x))**FracPart(p)*(g*cos(e + f*x))**FracPart(p), Int((g*cos(e + f*x))**(-p)*(a + b*sin(e + f*x))**m, x), x)
rule2330 = ReplacementRule(pattern2330, replacement2330)
pattern2331 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons147)
def replacement2331(p, m, f, b, g, a, x, e):
rubi.append(2331)
return Dist(g**(S(2)*IntPart(p))*(g/sin(e + f*x))**FracPart(p)*(g*sin(e + f*x))**FracPart(p), Int((g*sin(e + f*x))**(-p)*(a + b*cos(e + f*x))**m, x), x)
rule2331 = ReplacementRule(pattern2331, replacement2331)
pattern2332 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons54)
def replacement2332(p, f, b, g, a, x, e):
rubi.append(2332)
return Dist(S(1)/a, Int((g*tan(e + f*x))**p/cos(e + f*x)**S(2), x), x) - Dist(S(1)/(b*g), Int((g*tan(e + f*x))**(p + S(1))/cos(e + f*x), x), x)
rule2332 = ReplacementRule(pattern2332, replacement2332)
pattern2333 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons54)
def replacement2333(p, f, b, g, a, x, e):
rubi.append(2333)
return Dist(S(1)/a, Int((g/tan(e + f*x))**p/sin(e + f*x)**S(2), x), x) - Dist(S(1)/(b*g), Int((g/tan(e + f*x))**(p + S(1))/sin(e + f*x), x), x)
rule2333 = ReplacementRule(pattern2333, replacement2333)
pattern2334 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1298)
def replacement2334(p, m, f, b, a, x, e):
rubi.append(2334)
return Dist(S(1)/f, Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x)
rule2334 = ReplacementRule(pattern2334, replacement2334)
pattern2335 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1298)
def replacement2335(p, m, f, b, a, x, e):
rubi.append(2335)
return -Dist(S(1)/f, Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x)
rule2335 = ReplacementRule(pattern2335, replacement2335)
pattern2336 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons48, cons125, cons1265, cons1299, cons1300)
def replacement2336(p, m, f, b, a, x, e):
rubi.append(2336)
return Dist(a**p, Int((a - b*sin(e + f*x))**(-m)*sin(e + f*x)**p, x), x)
rule2336 = ReplacementRule(pattern2336, replacement2336)
pattern2337 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons48, cons125, cons1265, cons1299, cons1300)
def replacement2337(p, m, f, b, a, x, e):
rubi.append(2337)
return Dist(a**p, Int((a - b*cos(e + f*x))**(-m)*cos(e + f*x)**p, x), x)
rule2337 = ReplacementRule(pattern2337, replacement2337)
pattern2338 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons48, cons125, cons1265, cons1301, cons1302)
def replacement2338(p, m, f, b, a, x, e):
rubi.append(2338)
return Dist(a**p, Int(ExpandIntegrand((a - b*sin(e + f*x))**(-p/S(2))*(a + b*sin(e + f*x))**(m - p/S(2))*sin(e + f*x)**p, x), x), x)
rule2338 = ReplacementRule(pattern2338, replacement2338)
pattern2339 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons48, cons125, cons1265, cons1301, cons1302)
def replacement2339(p, m, f, b, a, x, e):
rubi.append(2339)
return Dist(a**p, Int(ExpandIntegrand((a - b*cos(e + f*x))**(-p/S(2))*(a + b*cos(e + f*x))**(m - p/S(2))*cos(e + f*x)**p, x), x), x)
rule2339 = ReplacementRule(pattern2339, replacement2339)
pattern2340 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons62)
def replacement2340(p, m, f, b, g, a, x, e):
rubi.append(2340)
return Int(ExpandIntegrand((g*tan(e + f*x))**p, (a + b*sin(e + f*x))**m, x), x)
rule2340 = ReplacementRule(pattern2340, replacement2340)
pattern2341 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons62)
def replacement2341(p, m, f, b, g, a, x, e):
rubi.append(2341)
return Int(ExpandIntegrand((g/tan(e + f*x))**p, (a + b*cos(e + f*x))**m, x), x)
rule2341 = ReplacementRule(pattern2341, replacement2341)
pattern2342 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons84)
def replacement2342(p, m, f, b, g, a, x, e):
rubi.append(2342)
return Dist(a**(S(2)*m), Int(ExpandIntegrand((g*tan(e + f*x))**p*(S(1)/cos(e + f*x))**(-m), (a/cos(e + f*x) - b*tan(e + f*x))**(-m), x), x), x)
rule2342 = ReplacementRule(pattern2342, replacement2342)
pattern2343 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons84)
def replacement2343(p, m, f, b, g, a, x, e):
rubi.append(2343)
return Dist(a**(S(2)*m), Int(ExpandIntegrand((g/tan(e + f*x))**p*(S(1)/sin(e + f*x))**(-m), (a/sin(e + f*x) - b/tan(e + f*x))**(-m), x), x), x)
rule2343 = ReplacementRule(pattern2343, replacement2343)
pattern2344 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1265, cons18, cons31, cons267)
def replacement2344(m, f, b, a, x, e):
rubi.append(2344)
return -Dist(S(1)/(a**S(2)*(S(2)*m + S(-1))), Int((a + b*sin(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(-1))*sin(e + f*x))/cos(e + f*x)**S(2), x), x) + Simp(b*(a + b*sin(e + f*x))**m/(a*f*(S(2)*m + S(-1))*cos(e + f*x)), x)
rule2344 = ReplacementRule(pattern2344, replacement2344)
pattern2345 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1265, cons18, cons31, cons267)
def replacement2345(m, f, b, a, x, e):
rubi.append(2345)
return -Dist(S(1)/(a**S(2)*(S(2)*m + S(-1))), Int((a + b*cos(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(-1))*cos(e + f*x))/sin(e + f*x)**S(2), x), x) - Simp(b*(a + b*cos(e + f*x))**m/(a*f*(S(2)*m + S(-1))*sin(e + f*x)), x)
rule2345 = ReplacementRule(pattern2345, replacement2345)
pattern2346 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons18, cons1303)
def replacement2346(m, f, b, a, x, e):
rubi.append(2346)
return Dist(S(1)/(b*m), Int((a + b*sin(e + f*x))**m*(a*sin(e + f*x) + b*(m + S(1)))/cos(e + f*x)**S(2), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))/(b*f*m*cos(e + f*x)), x)
rule2346 = ReplacementRule(pattern2346, replacement2346)
pattern2347 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons18, cons1303)
def replacement2347(m, f, b, a, x, e):
rubi.append(2347)
return Dist(S(1)/(b*m), Int((a + b*cos(e + f*x))**m*(a*cos(e + f*x) + b*(m + S(1)))/sin(e + f*x)**S(2), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))/(b*f*m*sin(e + f*x)), x)
rule2347 = ReplacementRule(pattern2347, replacement2347)
pattern2348 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1304)
def replacement2348(m, f, b, a, x, e):
rubi.append(2348)
return -Int((a + b*sin(e + f*x))**m*(-S(2)*sin(e + f*x)**S(2) + S(1))/cos(e + f*x)**S(4), x) + Int((a + b*sin(e + f*x))**m, x)
rule2348 = ReplacementRule(pattern2348, replacement2348)
pattern2349 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1304)
def replacement2349(m, f, b, a, x, e):
rubi.append(2349)
return -Int((a + b*cos(e + f*x))**m*(-S(2)*cos(e + f*x)**S(2) + S(1))/sin(e + f*x)**S(4), x) + Int((a + b*cos(e + f*x))**m, x)
rule2349 = ReplacementRule(pattern2349, replacement2349)
pattern2350 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1265, cons1304, cons94)
def replacement2350(m, f, b, a, x, e):
rubi.append(2350)
return Dist(b**(S(-2)), Int((a + b*sin(e + f*x))**(m + S(1))*(-a*(m + S(1))*sin(e + f*x) + b*m)/sin(e + f*x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))/(a*f*tan(e + f*x)), x)
rule2350 = ReplacementRule(pattern2350, replacement2350)
pattern2351 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1265, cons1304, cons94)
def replacement2351(m, f, b, a, x, e):
rubi.append(2351)
return Dist(b**(S(-2)), Int((a + b*cos(e + f*x))**(m + S(1))*(-a*(m + S(1))*cos(e + f*x) + b*m)/cos(e + f*x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*tan(e + f*x)/(a*f), x)
rule2351 = ReplacementRule(pattern2351, replacement2351)
pattern2352 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1304, cons1305)
def replacement2352(m, f, b, a, x, e):
rubi.append(2352)
return Dist(S(1)/a, Int((a + b*sin(e + f*x))**m*(-a*(m + S(1))*sin(e + f*x) + b*m)/sin(e + f*x), x), x) - Simp((a + b*sin(e + f*x))**m/(f*tan(e + f*x)), x)
rule2352 = ReplacementRule(pattern2352, replacement2352)
pattern2353 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1304, cons1305)
def replacement2353(m, f, b, a, x, e):
rubi.append(2353)
return Dist(S(1)/a, Int((a + b*cos(e + f*x))**m*(-a*(m + S(1))*cos(e + f*x) + b*m)/cos(e + f*x), x), x) + Simp((a + b*cos(e + f*x))**m*tan(e + f*x)/f, x)
rule2353 = ReplacementRule(pattern2353, replacement2353)
pattern2354 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons1265, cons1304, cons94)
def replacement2354(m, f, b, a, x, e):
rubi.append(2354)
return Dist(a**(S(-2)), Int((a + b*sin(e + f*x))**(m + S(2))*(sin(e + f*x)**S(2) + S(1))/sin(e + f*x)**S(4), x), x) + Dist(-S(2)/(a*b), Int((a + b*sin(e + f*x))**(m + S(2))/sin(e + f*x)**S(3), x), x)
rule2354 = ReplacementRule(pattern2354, replacement2354)
pattern2355 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons1265, cons1304, cons94)
def replacement2355(m, f, b, a, x, e):
rubi.append(2355)
return Dist(a**(S(-2)), Int((a + b*cos(e + f*x))**(m + S(2))*(cos(e + f*x)**S(2) + S(1))/cos(e + f*x)**S(4), x), x) + Dist(-S(2)/(a*b), Int((a + b*cos(e + f*x))**(m + S(2))/cos(e + f*x)**S(3), x), x)
rule2355 = ReplacementRule(pattern2355, replacement2355)
pattern2356 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1304, cons1305)
def replacement2356(m, f, b, a, x, e):
rubi.append(2356)
return Int((a + b*sin(e + f*x))**m*(-S(2)*sin(e + f*x)**S(2) + S(1))/sin(e + f*x)**S(4), x) + Int((a + b*sin(e + f*x))**m, x)
rule2356 = ReplacementRule(pattern2356, replacement2356)
pattern2357 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1304, cons1305)
def replacement2357(m, f, b, a, x, e):
rubi.append(2357)
return Int((a + b*cos(e + f*x))**m*(-S(2)*cos(e + f*x)**S(2) + S(1))/cos(e + f*x)**S(4), x) + Int((a + b*cos(e + f*x))**m, x)
rule2357 = ReplacementRule(pattern2357, replacement2357)
pattern2358 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons18, cons1306)
def replacement2358(p, m, f, b, a, x, e):
rubi.append(2358)
return Dist(sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))/(b*f*cos(e + f*x)), Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x)
rule2358 = ReplacementRule(pattern2358, replacement2358)
pattern2359 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons18, cons1306)
def replacement2359(p, m, f, b, a, x, e):
rubi.append(2359)
return -Dist(sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))/(b*f*sin(e + f*x)), Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x)
rule2359 = ReplacementRule(pattern2359, replacement2359)
pattern2360 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons18, cons147)
def replacement2360(p, m, f, b, g, a, x, e):
rubi.append(2360)
return Dist((b*sin(e + f*x))**(-p + S(-1))*(g*tan(e + f*x))**(p + S(1))*(a - b*sin(e + f*x))**(p/S(2) + S(1)/2)*(a + b*sin(e + f*x))**(p/S(2) + S(1)/2)/(f*g), Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x)
rule2360 = ReplacementRule(pattern2360, replacement2360)
pattern2361 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons18, cons147)
def replacement2361(p, m, f, b, g, a, x, e):
rubi.append(2361)
return -Dist((b*cos(e + f*x))**(-p + S(-1))*(g/tan(e + f*x))**(p + S(1))*(a - b*cos(e + f*x))**(p/S(2) + S(1)/2)*(a + b*cos(e + f*x))**(p/S(2) + S(1)/2)/(f*g), Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x)
rule2361 = ReplacementRule(pattern2361, replacement2361)
pattern2362 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons1298)
def replacement2362(p, m, f, b, a, x, e):
rubi.append(2362)
return Dist(S(1)/f, Subst(Int(x**p*(a + x)**m*(b**S(2) - x**S(2))**(-p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x)
rule2362 = ReplacementRule(pattern2362, replacement2362)
pattern2363 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons1298)
def replacement2363(p, m, f, b, a, x, e):
rubi.append(2363)
return -Dist(S(1)/f, Subst(Int(x**p*(a + x)**m*(b**S(2) - x**S(2))**(-p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x)
rule2363 = ReplacementRule(pattern2363, replacement2363)
pattern2364 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons62)
def replacement2364(p, m, f, b, g, a, x, e):
rubi.append(2364)
return Int(ExpandIntegrand((g*tan(e + f*x))**p, (a + b*sin(e + f*x))**m, x), x)
rule2364 = ReplacementRule(pattern2364, replacement2364)
pattern2365 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons62)
def replacement2365(p, m, f, b, g, a, x, e):
rubi.append(2365)
return Int(ExpandIntegrand((g/tan(e + f*x))**p, (a + b*cos(e + f*x))**m, x), x)
rule2365 = ReplacementRule(pattern2365, replacement2365)
pattern2366 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1267)
def replacement2366(m, f, b, a, x, e):
rubi.append(2366)
return Int((a + b*sin(e + f*x))**m*(-sin(e + f*x)**S(2) + S(1))/sin(e + f*x)**S(2), x)
rule2366 = ReplacementRule(pattern2366, replacement2366)
pattern2367 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1267)
def replacement2367(m, f, b, a, x, e):
rubi.append(2367)
return Int((a + b*cos(e + f*x))**m*(-cos(e + f*x)**S(2) + S(1))/cos(e + f*x)**S(2), x)
rule2367 = ReplacementRule(pattern2367, replacement2367)
pattern2368 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons94, cons515)
def replacement2368(m, f, b, a, x, e):
rubi.append(2368)
return -Dist(S(1)/(S(3)*a**S(2)*b*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(S(6)*a**S(2) + a*b*(m + S(1))*sin(e + f*x) - b**S(2)*(m + S(-2))*(m + S(-1)) - (S(3)*a**S(2) - b**S(2)*m*(m + S(-2)))*sin(e + f*x)**S(2), x)/sin(e + f*x)**S(3), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(S(3)*a*f*sin(e + f*x)**S(3)), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(S(3)*a**S(2) + b**S(2)*(m + S(-2)))*cos(e + f*x)/(S(3)*a**S(2)*b*f*(m + S(1))*sin(e + f*x)**S(2)), x)
rule2368 = ReplacementRule(pattern2368, replacement2368)
pattern2369 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons94, cons515)
def replacement2369(m, f, b, a, x, e):
rubi.append(2369)
return -Dist(S(1)/(S(3)*a**S(2)*b*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(S(6)*a**S(2) + a*b*(m + S(1))*cos(e + f*x) - b**S(2)*(m + S(-2))*(m + S(-1)) - (S(3)*a**S(2) - b**S(2)*m*(m + S(-2)))*cos(e + f*x)**S(2), x)/cos(e + f*x)**S(3), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(S(3)*a*f*cos(e + f*x)**S(3)), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(S(3)*a**S(2) + b**S(2)*(m + S(-2)))*sin(e + f*x)/(S(3)*a**S(2)*b*f*(m + S(1))*cos(e + f*x)**S(2)), x)
rule2369 = ReplacementRule(pattern2369, replacement2369)
pattern2370 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons272, cons515)
def replacement2370(m, f, b, a, x, e):
rubi.append(2370)
return -Dist(S(1)/(S(6)*a**S(2)), Int((a + b*sin(e + f*x))**m*Simp(S(8)*a**S(2) + a*b*m*sin(e + f*x) - b**S(2)*(m + S(-2))*(m + S(-1)) - (S(6)*a**S(2) - b**S(2)*m*(m + S(-2)))*sin(e + f*x)**S(2), x)/sin(e + f*x)**S(2), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(S(3)*a*f*sin(e + f*x)**S(3)), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(1))*(m + S(-2))*cos(e + f*x)/(S(6)*a**S(2)*f*sin(e + f*x)**S(2)), x)
rule2370 = ReplacementRule(pattern2370, replacement2370)
pattern2371 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons272, cons515)
def replacement2371(m, f, b, a, x, e):
rubi.append(2371)
return -Dist(S(1)/(S(6)*a**S(2)), Int((a + b*cos(e + f*x))**m*Simp(S(8)*a**S(2) + a*b*m*cos(e + f*x) - b**S(2)*(m + S(-2))*(m + S(-1)) - (S(6)*a**S(2) - b**S(2)*m*(m + S(-2)))*cos(e + f*x)**S(2), x)/cos(e + f*x)**S(2), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(S(3)*a*f*cos(e + f*x)**S(3)), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(1))*(m + S(-2))*sin(e + f*x)/(S(6)*a**S(2)*f*cos(e + f*x)**S(2)), x)
rule2371 = ReplacementRule(pattern2371, replacement2371)
pattern2372 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(6), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons1283, cons515)
def replacement2372(m, f, b, a, x, e):
rubi.append(2372)
return Dist(S(1)/(S(20)*a**S(2)*b**S(2)*m*(m + S(-1))), Int((a + b*sin(e + f*x))**m*Simp(S(60)*a**S(4) - S(44)*a**S(2)*b**S(2)*m*(m + S(-1)) + a*b*m*(S(20)*a**S(2) - b**S(2)*m*(m + S(-1)))*sin(e + f*x) + b**S(4)*m*(m + S(-4))*(m + S(-3))*(m + S(-1)) - (S(40)*a**S(4) - S(20)*a**S(2)*b**S(2)*(m + S(-1))*(S(2)*m + S(1)) + b**S(4)*m*(m + S(-4))*(m + S(-2))*(m + S(-1)))*sin(e + f*x)**S(2), x)/sin(e + f*x)**S(4), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(S(5)*a*f*sin(e + f*x)**S(5)), x) + Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*m*sin(e + f*x)**S(2)), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(1))*(m + S(-4))*cos(e + f*x)/(S(20)*a**S(2)*f*sin(e + f*x)**S(4)), x) + Simp(a*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b**S(2)*f*m*(m + S(-1))*sin(e + f*x)**S(3)), x)
rule2372 = ReplacementRule(pattern2372, replacement2372)
pattern2373 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(6), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons1283, cons515)
def replacement2373(m, f, b, a, x, e):
rubi.append(2373)
return Dist(S(1)/(S(20)*a**S(2)*b**S(2)*m*(m + S(-1))), Int((a + b*cos(e + f*x))**m*Simp(S(60)*a**S(4) - S(44)*a**S(2)*b**S(2)*m*(m + S(-1)) + a*b*m*(S(20)*a**S(2) - b**S(2)*m*(m + S(-1)))*cos(e + f*x) + b**S(4)*m*(m + S(-4))*(m + S(-3))*(m + S(-1)) - (S(40)*a**S(4) - S(20)*a**S(2)*b**S(2)*(m + S(-1))*(S(2)*m + S(1)) + b**S(4)*m*(m + S(-4))*(m + S(-2))*(m + S(-1)))*cos(e + f*x)**S(2), x)/cos(e + f*x)**S(4), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(S(5)*a*f*cos(e + f*x)**S(5)), x) - Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*m*cos(e + f*x)**S(2)), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(1))*(m + S(-4))*sin(e + f*x)/(S(20)*a**S(2)*f*cos(e + f*x)**S(4)), x) - Simp(a*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b**S(2)*f*m*(m + S(-1))*cos(e + f*x)**S(3)), x)
rule2373 = ReplacementRule(pattern2373, replacement2373)
pattern2374 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons1307, cons146)
def replacement2374(p, f, b, g, a, x, e):
rubi.append(2374)
return Dist(a/(a**S(2) - b**S(2)), Int((g*tan(e + f*x))**p/sin(e + f*x)**S(2), x), x) - Dist(a**S(2)*g**S(2)/(a**S(2) - b**S(2)), Int((g*tan(e + f*x))**(p + S(-2))/(a + b*sin(e + f*x)), x), x) - Dist(b*g/(a**S(2) - b**S(2)), Int((g*tan(e + f*x))**(p + S(-1))/cos(e + f*x), x), x)
rule2374 = ReplacementRule(pattern2374, replacement2374)
pattern2375 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons1307, cons146)
def replacement2375(p, f, b, g, a, x, e):
rubi.append(2375)
return Dist(a/(a**S(2) - b**S(2)), Int((g/tan(e + f*x))**p/cos(e + f*x)**S(2), x), x) - Dist(a**S(2)*g**S(2)/(a**S(2) - b**S(2)), Int((g/tan(e + f*x))**(p + S(-2))/(a + b*cos(e + f*x)), x), x) - Dist(b*g/(a**S(2) - b**S(2)), Int((g/tan(e + f*x))**(p + S(-1))/sin(e + f*x), x), x)
rule2375 = ReplacementRule(pattern2375, replacement2375)
pattern2376 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons1307, cons137)
def replacement2376(p, f, b, g, a, x, e):
rubi.append(2376)
return Dist(S(1)/a, Int((g*tan(e + f*x))**p/cos(e + f*x)**S(2), x), x) - Dist(b/(a**S(2)*g), Int((g*tan(e + f*x))**(p + S(1))/cos(e + f*x), x), x) - Dist((a**S(2) - b**S(2))/(a**S(2)*g**S(2)), Int((g*tan(e + f*x))**(p + S(2))/(a + b*sin(e + f*x)), x), x)
rule2376 = ReplacementRule(pattern2376, replacement2376)
pattern2377 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons1307, cons137)
def replacement2377(p, f, b, g, a, x, e):
rubi.append(2377)
return Dist(S(1)/a, Int((g/tan(e + f*x))**p/sin(e + f*x)**S(2), x), x) - Dist(b/(a**S(2)*g), Int((g/tan(e + f*x))**(p + S(1))/sin(e + f*x), x), x) - Dist((a**S(2) - b**S(2))/(a**S(2)*g**S(2)), Int((g/tan(e + f*x))**(p + S(2))/(a + b*cos(e + f*x)), x), x)
rule2377 = ReplacementRule(pattern2377, replacement2377)
pattern2378 = Pattern(Integral(sqrt(WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
def replacement2378(f, b, g, a, x, e):
rubi.append(2378)
return Dist(sqrt(g*tan(e + f*x))*sqrt(cos(e + f*x))/sqrt(sin(e + f*x)), Int(sqrt(sin(e + f*x))/((a + b*sin(e + f*x))*sqrt(cos(e + f*x))), x), x)
rule2378 = ReplacementRule(pattern2378, replacement2378)
pattern2379 = Pattern(Integral(sqrt(WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
def replacement2379(f, b, g, a, x, e):
rubi.append(2379)
return Dist(sqrt(g/tan(e + f*x))*sqrt(sin(e + f*x))/sqrt(cos(e + f*x)), Int(sqrt(cos(e + f*x))/((a + b*cos(e + f*x))*sqrt(sin(e + f*x))), x), x)
rule2379 = ReplacementRule(pattern2379, replacement2379)
pattern2380 = Pattern(Integral(S(1)/(sqrt(g_*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
def replacement2380(f, b, g, a, x, e):
rubi.append(2380)
return Dist(sqrt(sin(e + f*x))/(sqrt(g*tan(e + f*x))*sqrt(cos(e + f*x))), Int(sqrt(cos(e + f*x))/((a + b*sin(e + f*x))*sqrt(sin(e + f*x))), x), x)
rule2380 = ReplacementRule(pattern2380, replacement2380)
pattern2381 = Pattern(Integral(S(1)/(sqrt(g_/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
def replacement2381(f, b, g, a, x, e):
rubi.append(2381)
return Dist(sqrt(cos(e + f*x))/(sqrt(g/tan(e + f*x))*sqrt(sin(e + f*x))), Int(sqrt(sin(e + f*x))/((a + b*cos(e + f*x))*sqrt(cos(e + f*x))), x), x)
rule2381 = ReplacementRule(pattern2381, replacement2381)
pattern2382 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons48, cons125, cons1267, cons1301)
def replacement2382(p, m, f, b, a, x, e):
rubi.append(2382)
return Int(ExpandIntegrand((a + b*sin(e + f*x))**m*(-sin(e + f*x)**S(2) + S(1))**(-p/S(2))*sin(e + f*x)**p, x), x)
rule2382 = ReplacementRule(pattern2382, replacement2382)
pattern2383 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons48, cons125, cons1267, cons1301)
def replacement2383(p, m, f, b, a, x, e):
rubi.append(2383)
return Int(ExpandIntegrand((a + b*cos(e + f*x))**m*(-cos(e + f*x)**S(2) + S(1))**(-p/S(2))*cos(e + f*x)**p, x), x)
rule2383 = ReplacementRule(pattern2383, replacement2383)
pattern2384 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1308)
def replacement2384(p, m, f, b, g, a, x, e):
rubi.append(2384)
return Int((g*tan(e + f*x))**p*(a + b*sin(e + f*x))**m, x)
rule2384 = ReplacementRule(pattern2384, replacement2384)
pattern2385 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1308)
def replacement2385(p, m, f, b, g, a, x, e):
rubi.append(2385)
return Int((g/tan(e + f*x))**p*(a + b*cos(e + f*x))**m, x)
rule2385 = ReplacementRule(pattern2385, replacement2385)
pattern2386 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons147)
def replacement2386(p, m, f, b, g, a, x, e):
rubi.append(2386)
return Dist(g**(S(2)*IntPart(p))*(g/tan(e + f*x))**FracPart(p)*(g*tan(e + f*x))**FracPart(p), Int((g*tan(e + f*x))**(-p)*(a + b*sin(e + f*x))**m, x), x)
rule2386 = ReplacementRule(pattern2386, replacement2386)
pattern2387 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons147)
def replacement2387(p, m, f, b, g, a, x, e):
rubi.append(2387)
return Dist(g**(S(2)*IntPart(p))*(g/tan(e + f*x))**FracPart(p)*(g*tan(e + f*x))**FracPart(p), Int((g/tan(e + f*x))**(-p)*(a + b*cos(e + f*x))**m, x), x)
rule2387 = ReplacementRule(pattern2387, replacement2387)
pattern2388 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement2388(f, b, d, c, a, x, e):
rubi.append(2388)
return Simp(x*(S(2)*a*c + b*d)/S(2), x) - Simp((a*d + b*c)*cos(e + f*x)/f, x) - Simp(b*d*sin(e + f*x)*cos(e + f*x)/(S(2)*f), x)
rule2388 = ReplacementRule(pattern2388, replacement2388)
pattern2389 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement2389(f, b, d, c, a, x, e):
rubi.append(2389)
return Simp(x*(S(2)*a*c + b*d)/S(2), x) + Simp((a*d + b*c)*sin(e + f*x)/f, x) + Simp(b*d*sin(e + f*x)*cos(e + f*x)/(S(2)*f), x)
rule2389 = ReplacementRule(pattern2389, replacement2389)
pattern2390 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement2390(f, b, d, c, a, x, e):
rubi.append(2390)
return -Dist((-a*d + b*c)/d, Int(S(1)/(c + d*sin(e + f*x)), x), x) + Simp(b*x/d, x)
rule2390 = ReplacementRule(pattern2390, replacement2390)
pattern2391 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement2391(f, b, d, c, a, x, e):
rubi.append(2391)
return -Dist((-a*d + b*c)/d, Int(S(1)/(c + d*cos(e + f*x)), x), x) + Simp(b*x/d, x)
rule2391 = ReplacementRule(pattern2391, replacement2391)
pattern2392 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons17, cons1309)
def replacement2392(m, f, b, d, a, n, c, x, e):
rubi.append(2392)
return Dist(a**m*c**m, Int((c + d*sin(e + f*x))**(-m + n)*cos(e + f*x)**(S(2)*m), x), x)
rule2392 = ReplacementRule(pattern2392, replacement2392)
pattern2393 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons17, cons1309)
def replacement2393(m, f, b, d, a, n, c, x, e):
rubi.append(2393)
return Dist(a**m*c**m, Int((c + d*cos(e + f*x))**(-m + n)*sin(e + f*x)**(S(2)*m), x), x)
rule2393 = ReplacementRule(pattern2393, replacement2393)
pattern2394 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265)
def replacement2394(f, b, d, a, c, x, e):
rubi.append(2394)
return Dist(a*c*cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), Int(cos(e + f*x)/(c + d*sin(e + f*x)), x), x)
rule2394 = ReplacementRule(pattern2394, replacement2394)
pattern2395 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265)
def replacement2395(f, b, d, a, c, x, e):
rubi.append(2395)
return Dist(a*c*sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), Int(sin(e + f*x)/(c + d*cos(e + f*x)), x), x)
rule2395 = ReplacementRule(pattern2395, replacement2395)
pattern2396 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons1310)
def replacement2396(f, b, d, a, c, n, x, e):
rubi.append(2396)
return Simp(-S(2)*b*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(1))), x)
rule2396 = ReplacementRule(pattern2396, replacement2396)
pattern2397 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons1310)
def replacement2397(f, b, d, a, c, n, x, e):
rubi.append(2397)
return Simp(S(2)*b*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(1))), x)
rule2397 = ReplacementRule(pattern2397, replacement2397)
pattern2398 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons1311, cons87, cons89, cons1312)
def replacement2398(m, f, b, d, a, c, n, x, e):
rubi.append(2398)
return -Dist(b*(S(2)*m + S(-1))/(d*(S(2)*n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(S(2)*n + S(1))), x)
rule2398 = ReplacementRule(pattern2398, replacement2398)
pattern2399 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons1311, cons87, cons89, cons1312)
def replacement2399(m, f, b, d, a, c, n, x, e):
rubi.append(2399)
return -Dist(b*(S(2)*m + S(-1))/(d*(S(2)*n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(S(2)*n + S(1))), x)
rule2399 = ReplacementRule(pattern2399, replacement2399)
pattern2400 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons1311, cons346, cons1313, cons1312)
def replacement2400(m, f, b, d, a, c, n, x, e):
rubi.append(2400)
return Dist(a*(S(2)*m + S(-1))/(m + n), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n, x), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n)), x)
rule2400 = ReplacementRule(pattern2400, replacement2400)
pattern2401 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons1311, cons346, cons1313, cons1312)
def replacement2401(m, f, b, d, a, c, n, x, e):
rubi.append(2401)
return Dist(a*(S(2)*m + S(-1))/(m + n), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n, x), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n)), x)
rule2401 = ReplacementRule(pattern2401, replacement2401)
pattern2402 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265)
def replacement2402(f, b, d, a, c, x, e):
rubi.append(2402)
return Dist(cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), Int(S(1)/cos(e + f*x), x), x)
rule2402 = ReplacementRule(pattern2402, replacement2402)
pattern2403 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265)
def replacement2403(f, b, d, a, c, x, e):
rubi.append(2403)
return Dist(sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), Int(S(1)/sin(e + f*x), x), x)
rule2403 = ReplacementRule(pattern2403, replacement2403)
pattern2404 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons155, cons1314)
def replacement2404(m, f, b, d, a, c, n, x, e):
rubi.append(2404)
return Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2404 = ReplacementRule(pattern2404, replacement2404)
pattern2405 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons155, cons1314)
def replacement2405(m, f, b, d, a, c, n, x, e):
rubi.append(2405)
return -Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2405 = ReplacementRule(pattern2405, replacement2405)
pattern2406 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons1315, cons1314, cons114)
def replacement2406(m, f, b, d, a, c, n, x, e):
rubi.append(2406)
return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2406 = ReplacementRule(pattern2406, replacement2406)
pattern2407 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons1315, cons1314, cons114)
def replacement2407(m, f, b, d, a, c, n, x, e):
rubi.append(2407)
return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2407 = ReplacementRule(pattern2407, replacement2407)
pattern2408 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons31, cons94, cons1316, cons1170)
def replacement2408(m, f, b, d, a, c, n, x, e):
rubi.append(2408)
return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2408 = ReplacementRule(pattern2408, replacement2408)
pattern2409 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons31, cons94, cons1316, cons1170)
def replacement2409(m, f, b, d, a, c, n, x, e):
rubi.append(2409)
return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2409 = ReplacementRule(pattern2409, replacement2409)
pattern2410 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons1317)
def replacement2410(m, f, b, d, a, c, n, x, e):
rubi.append(2410)
return Dist(a**IntPart(m)*c**IntPart(m)*(a + b*sin(e + f*x))**FracPart(m)*(c + d*sin(e + f*x))**FracPart(m)*cos(e + f*x)**(-S(2)*FracPart(m)), Int((c + d*sin(e + f*x))**(-m + n)*cos(e + f*x)**(S(2)*m), x), x)
rule2410 = ReplacementRule(pattern2410, replacement2410)
pattern2411 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons1317)
def replacement2411(m, f, b, d, a, c, n, x, e):
rubi.append(2411)
return Dist(a**IntPart(m)*c**IntPart(m)*(a + b*cos(e + f*x))**FracPart(m)*(c + d*cos(e + f*x))**FracPart(m)*sin(e + f*x)**(-S(2)*FracPart(m)), Int((c + d*cos(e + f*x))**(-m + n)*sin(e + f*x)**(S(2)*m), x), x)
rule2411 = ReplacementRule(pattern2411, replacement2411)
pattern2412 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement2412(f, b, d, c, a, x, e):
rubi.append(2412)
return Dist(S(1)/d, Int(Simp(a**S(2)*d - b*(-S(2)*a*d + b*c)*sin(e + f*x), x)/(c + d*sin(e + f*x)), x), x) - Simp(b**S(2)*cos(e + f*x)/(d*f), x)
rule2412 = ReplacementRule(pattern2412, replacement2412)
pattern2413 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement2413(f, b, d, c, a, x, e):
rubi.append(2413)
return Dist(S(1)/d, Int(Simp(a**S(2)*d - b*(-S(2)*a*d + b*c)*cos(e + f*x), x)/(c + d*cos(e + f*x)), x), x) + Simp(b**S(2)*sin(e + f*x)/(d*f), x)
rule2413 = ReplacementRule(pattern2413, replacement2413)
pattern2414 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement2414(f, b, d, c, a, x, e):
rubi.append(2414)
return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*sin(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*sin(e + f*x)), x), x)
rule2414 = ReplacementRule(pattern2414, replacement2414)
pattern2415 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement2415(f, b, d, c, a, x, e):
rubi.append(2415)
return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*cos(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*cos(e + f*x)), x), x)
rule2415 = ReplacementRule(pattern2415, replacement2415)
pattern2416 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons21, cons1318)
def replacement2416(m, f, b, d, c, x, e):
rubi.append(2416)
return Dist(c, Int((b*sin(e + f*x))**m, x), x) + Dist(d/b, Int((b*sin(e + f*x))**(m + S(1)), x), x)
rule2416 = ReplacementRule(pattern2416, replacement2416)
pattern2417 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons21, cons1318)
def replacement2417(m, f, b, d, c, x, e):
rubi.append(2417)
return Dist(c, Int((b*cos(e + f*x))**m, x), x) + Dist(d/b, Int((b*cos(e + f*x))**(m + S(1)), x), x)
rule2417 = ReplacementRule(pattern2417, replacement2417)
pattern2418 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1265, cons1319)
def replacement2418(m, f, b, d, a, c, x, e):
rubi.append(2418)
return -Simp(d*(a + b*sin(e + f*x))**m*cos(e + f*x)/(f*(m + S(1))), x)
rule2418 = ReplacementRule(pattern2418, replacement2418)
pattern2419 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1265, cons1319)
def replacement2419(m, f, b, d, a, c, x, e):
rubi.append(2419)
return Simp(d*(a + b*cos(e + f*x))**m*sin(e + f*x)/(f*(m + S(1))), x)
rule2419 = ReplacementRule(pattern2419, replacement2419)
pattern2420 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons31, cons1320)
def replacement2420(m, f, b, d, c, a, x, e):
rubi.append(2420)
return Dist((a*d*m + b*c*(m + S(1)))/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp((a + b*sin(e + f*x))**m*(-a*d + b*c)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2420 = ReplacementRule(pattern2420, replacement2420)
pattern2421 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons31, cons1320)
def replacement2421(m, f, b, d, c, a, x, e):
rubi.append(2421)
return Dist((a*d*m + b*c*(m + S(1)))/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp((a + b*cos(e + f*x))**m*(-a*d + b*c)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2421 = ReplacementRule(pattern2421, replacement2421)
pattern2422 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1265, cons1321)
def replacement2422(m, f, b, d, c, a, x, e):
rubi.append(2422)
return Dist((a*d*m + b*c*(m + S(1)))/(b*(m + S(1))), Int((a + b*sin(e + f*x))**m, x), x) - Simp(d*(a + b*sin(e + f*x))**m*cos(e + f*x)/(f*(m + S(1))), x)
rule2422 = ReplacementRule(pattern2422, replacement2422)
pattern2423 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1265, cons1321)
def replacement2423(m, f, b, d, c, a, x, e):
rubi.append(2423)
return Dist((a*d*m + b*c*(m + S(1)))/(b*(m + S(1))), Int((a + b*cos(e + f*x))**m, x), x) + Simp(d*(a + b*cos(e + f*x))**m*sin(e + f*x)/(f*(m + S(1))), x)
rule2423 = ReplacementRule(pattern2423, replacement2423)
pattern2424 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement2424(f, b, d, c, a, x, e):
rubi.append(2424)
return Dist(d/b, Int(sqrt(a + b*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/sqrt(a + b*sin(e + f*x)), x), x)
rule2424 = ReplacementRule(pattern2424, replacement2424)
pattern2425 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement2425(f, b, d, c, a, x, e):
rubi.append(2425)
return Dist(d/b, Int(sqrt(a + b*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/sqrt(a + b*cos(e + f*x)), x), x)
rule2425 = ReplacementRule(pattern2425, replacement2425)
pattern2426 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons31, cons168, cons515)
def replacement2426(m, f, b, d, c, a, x, e):
rubi.append(2426)
return Dist(S(1)/(m + S(1)), Int((a + b*sin(e + f*x))**(m + S(-1))*Simp(a*c*(m + S(1)) + b*d*m + (a*d*m + b*c*(m + S(1)))*sin(e + f*x), x), x), x) - Simp(d*(a + b*sin(e + f*x))**m*cos(e + f*x)/(f*(m + S(1))), x)
rule2426 = ReplacementRule(pattern2426, replacement2426)
pattern2427 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons31, cons168, cons515)
def replacement2427(m, f, b, d, c, a, x, e):
rubi.append(2427)
return Dist(S(1)/(m + S(1)), Int((a + b*cos(e + f*x))**(m + S(-1))*Simp(a*c*(m + S(1)) + b*d*m + (a*d*m + b*c*(m + S(1)))*cos(e + f*x), x), x), x) + Simp(d*(a + b*cos(e + f*x))**m*sin(e + f*x)/(f*(m + S(1))), x)
rule2427 = ReplacementRule(pattern2427, replacement2427)
pattern2428 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons31, cons94, cons515)
def replacement2428(m, f, b, d, c, a, x, e):
rubi.append(2428)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp((m + S(1))*(a*c - b*d) - (m + S(2))*(-a*d + b*c)*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c)*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2428 = ReplacementRule(pattern2428, replacement2428)
pattern2429 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons31, cons94, cons515)
def replacement2429(m, f, b, d, c, a, x, e):
rubi.append(2429)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp((m + S(1))*(a*c - b*d) - (m + S(2))*(-a*d + b*c)*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c)*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2429 = ReplacementRule(pattern2429, replacement2429)
pattern2430 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1267, cons77, cons1322)
def replacement2430(m, f, b, d, a, c, x, e):
rubi.append(2430)
return Dist(c*cos(e + f*x)/(f*sqrt(-sin(e + f*x) + S(1))*sqrt(sin(e + f*x) + S(1))), Subst(Int(sqrt(S(1) + d*x/c)*(a + b*x)**m/sqrt(S(1) - d*x/c), x), x, sin(e + f*x)), x)
rule2430 = ReplacementRule(pattern2430, replacement2430)
pattern2431 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1267, cons77, cons1322)
def replacement2431(m, f, b, d, a, c, x, e):
rubi.append(2431)
return -Dist(c*sin(e + f*x)/(f*sqrt(-cos(e + f*x) + S(1))*sqrt(cos(e + f*x) + S(1))), Subst(Int(sqrt(S(1) + d*x/c)*(a + b*x)**m/sqrt(S(1) - d*x/c), x), x, cos(e + f*x)), x)
rule2431 = ReplacementRule(pattern2431, replacement2431)
pattern2432 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1267)
def replacement2432(m, f, b, d, c, a, x, e):
rubi.append(2432)
return Dist(d/b, Int((a + b*sin(e + f*x))**(m + S(1)), x), x) + Dist((-a*d + b*c)/b, Int((a + b*sin(e + f*x))**m, x), x)
rule2432 = ReplacementRule(pattern2432, replacement2432)
pattern2433 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1267)
def replacement2433(m, f, b, d, c, a, x, e):
rubi.append(2433)
return Dist(d/b, Int((a + b*cos(e + f*x))**(m + S(1)), x), x) + Dist((-a*d + b*c)/b, Int((a + b*cos(e + f*x))**m, x), x)
rule2433 = ReplacementRule(pattern2433, replacement2433)
pattern2434 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons62, cons87)
def replacement2434(m, f, b, d, a, n, x, e):
rubi.append(2434)
return Int(ExpandTrig((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x)
rule2434 = ReplacementRule(pattern2434, replacement2434)
pattern2435 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons62, cons87)
def replacement2435(m, f, b, d, a, n, x, e):
rubi.append(2435)
return Int(ExpandTrig((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x)
rule2435 = ReplacementRule(pattern2435, replacement2435)
pattern2436 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1320)
def replacement2436(m, f, b, a, x, e):
rubi.append(2436)
return -Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(1))*sin(e + f*x)), x), x) + Simp(b*(a + b*sin(e + f*x))**m*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2436 = ReplacementRule(pattern2436, replacement2436)
pattern2437 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1320)
def replacement2437(m, f, b, a, x, e):
rubi.append(2437)
return -Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(1))*cos(e + f*x)), x), x) - Simp(b*(a + b*cos(e + f*x))**m*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2437 = ReplacementRule(pattern2437, replacement2437)
pattern2438 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1321)
def replacement2438(m, f, b, a, x, e):
rubi.append(2438)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*(-a*sin(e + f*x) + b*(m + S(1))), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x)
rule2438 = ReplacementRule(pattern2438, replacement2438)
pattern2439 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1321)
def replacement2439(m, f, b, a, x, e):
rubi.append(2439)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*(-a*cos(e + f*x) + b*(m + S(1))), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x)
rule2439 = ReplacementRule(pattern2439, replacement2439)
pattern2440 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons31, cons94)
def replacement2440(m, f, b, d, a, c, x, e):
rubi.append(2440)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(a*c*d*(m + S(-1)) + b*(c**S(2)*(m + S(1)) + d**S(2)) + d*(a*d*(m + S(-1)) + b*c*(m + S(2)))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))*(-a*d + b*c)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2440 = ReplacementRule(pattern2440, replacement2440)
pattern2441 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons31, cons94)
def replacement2441(m, f, b, d, a, c, x, e):
rubi.append(2441)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(a*c*d*(m + S(-1)) + b*(c**S(2)*(m + S(1)) + d**S(2)) + d*(a*d*(m + S(-1)) + b*c*(m + S(2)))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))*(-a*d + b*c)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2441 = ReplacementRule(pattern2441, replacement2441)
pattern2442 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1265, cons272)
def replacement2442(m, f, b, d, a, c, x, e):
rubi.append(2442)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*Simp(b*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))) - d*(a*d - S(2)*b*c*(m + S(2)))*sin(e + f*x), x), x), x) - Simp(d**S(2)*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x)
rule2442 = ReplacementRule(pattern2442, replacement2442)
pattern2443 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1265, cons272)
def replacement2443(m, f, b, d, a, c, x, e):
rubi.append(2443)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*Simp(b*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))) - d*(a*d - S(2)*b*c*(m + S(2)))*cos(e + f*x), x), x), x) + Simp(d**S(2)*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x)
rule2443 = ReplacementRule(pattern2443, replacement2443)
pattern2444 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons93, cons166, cons89, cons1324)
def replacement2444(m, f, b, d, c, a, n, x, e):
rubi.append(2444)
return Dist(b**S(2)/(d*(n + S(1))*(a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*Simp(a*c*(m + S(-2)) - b*d*(m - S(2)*n + S(-4)) - (-a*d*(m + S(2)*n + S(1)) + b*c*(m + S(-1)))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*(-a*d + b*c)*cos(e + f*x)/(d*f*(n + S(1))*(a*d + b*c)), x)
rule2444 = ReplacementRule(pattern2444, replacement2444)
pattern2445 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons93, cons166, cons89, cons1324)
def replacement2445(m, f, b, d, c, n, a, x, e):
rubi.append(2445)
return Dist(b**S(2)/(d*(n + S(1))*(a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*Simp(a*c*(m + S(-2)) - b*d*(m - S(2)*n + S(-4)) - (-a*d*(m + S(2)*n + S(1)) + b*c*(m + S(-1)))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*(-a*d + b*c)*sin(e + f*x)/(d*f*(n + S(1))*(a*d + b*c)), x)
rule2445 = ReplacementRule(pattern2445, replacement2445)
pattern2446 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons31, cons166, cons346, cons1324)
def replacement2446(m, f, b, d, c, a, n, x, e):
rubi.append(2446)
return Dist(S(1)/(d*(m + n)), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**n*Simp(a**S(2)*d*(m + n) + a*b*c*(m + S(-2)) + b**S(2)*d*(n + S(1)) - b*(-a*d*(S(3)*m + S(2)*n + S(-2)) + b*c*(m + S(-1)))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n)), x)
rule2446 = ReplacementRule(pattern2446, replacement2446)
pattern2447 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons31, cons166, cons346, cons1324)
def replacement2447(m, f, b, d, c, n, a, x, e):
rubi.append(2447)
return Dist(S(1)/(d*(m + n)), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**n*Simp(a**S(2)*d*(m + n) + a*b*c*(m + S(-2)) + b**S(2)*d*(n + S(1)) - b*(-a*d*(S(3)*m + S(2)*n + S(-2)) + b*c*(m + S(-1)))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n)), x)
rule2447 = ReplacementRule(pattern2447, replacement2447)
pattern2448 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons93, cons94, cons1325, cons1326)
def replacement2448(m, f, b, d, c, a, n, x, e):
rubi.append(2448)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(a*d*n - b*c*(m + S(1)) - b*d*(m + n + S(1))*sin(e + f*x), x), x), x) + Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2448 = ReplacementRule(pattern2448, replacement2448)
pattern2449 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons93, cons94, cons1325, cons1326)
def replacement2449(m, f, b, d, c, n, a, x, e):
rubi.append(2449)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(a*d*n - b*c*(m + S(1)) - b*d*(m + n + S(1))*cos(e + f*x), x), x), x) - Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2449 = ReplacementRule(pattern2449, replacement2449)
pattern2450 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons93, cons94, cons165, cons1326)
def replacement2450(m, f, b, d, c, a, n, x, e):
rubi.append(2450)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-2))*Simp(a*c*d*(m - n + S(1)) + b*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(-1))) + d*(a*d*(m - n + S(1)) + b*c*(m + n))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(-1))*(-a*d + b*c)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2450 = ReplacementRule(pattern2450, replacement2450)
pattern2451 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons93, cons94, cons165, cons1326)
def replacement2451(m, f, b, d, c, n, a, x, e):
rubi.append(2451)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-2))*Simp(a*c*d*(m - n + S(1)) + b*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(-1))) + d*(a*d*(m - n + S(1)) + b*c*(m + n))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(-1))*(-a*d + b*c)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2451 = ReplacementRule(pattern2451, replacement2451)
pattern2452 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons31, cons94, cons1327, cons1326)
def replacement2452(m, f, b, d, c, a, n, x, e):
rubi.append(2452)
return Dist(S(1)/(a*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(-a*d*(S(2)*m + n + S(2)) + b*c*(m + S(1)) + b*d*(m + n + S(2))*sin(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(a*f*(S(2)*m + S(1))*(-a*d + b*c)), x)
rule2452 = ReplacementRule(pattern2452, replacement2452)
pattern2453 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons31, cons94, cons1327, cons1326)
def replacement2453(m, f, b, d, c, n, a, x, e):
rubi.append(2453)
return Dist(S(1)/(a*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(-a*d*(S(2)*m + n + S(2)) + b*c*(m + S(1)) + b*d*(m + n + S(2))*cos(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(a*f*(S(2)*m + S(1))*(-a*d + b*c)), x)
rule2453 = ReplacementRule(pattern2453, replacement2453)
pattern2454 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons165, cons1328)
def replacement2454(f, b, d, c, a, n, x, e):
rubi.append(2454)
return -Dist(d/(a*b), Int((c + d*sin(e + f*x))**(n + S(-2))*Simp(-a*c*n + b*d*(n + S(-1)) + (-a*d*n + b*c*(n + S(-1)))*sin(e + f*x), x), x), x) - Simp((c + d*sin(e + f*x))**(n + S(-1))*(-a*d + b*c)*cos(e + f*x)/(a*f*(a + b*sin(e + f*x))), x)
rule2454 = ReplacementRule(pattern2454, replacement2454)
pattern2455 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons165, cons1328)
def replacement2455(f, b, d, c, n, a, x, e):
rubi.append(2455)
return -Dist(d/(a*b), Int((c + d*cos(e + f*x))**(n + S(-2))*Simp(-a*c*n + b*d*(n + S(-1)) + (-a*d*n + b*c*(n + S(-1)))*cos(e + f*x), x), x), x) + Simp((c + d*cos(e + f*x))**(n + S(-1))*(-a*d + b*c)*sin(e + f*x)/(a*f*(a + b*cos(e + f*x))), x)
rule2455 = ReplacementRule(pattern2455, replacement2455)
pattern2456 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons463, cons1328)
def replacement2456(f, b, d, c, a, n, x, e):
rubi.append(2456)
return Dist(d/(a*(-a*d + b*c)), Int((c + d*sin(e + f*x))**n*(a*n - b*(n + S(1))*sin(e + f*x)), x), x) - Simp(b**S(2)*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(a*f*(a + b*sin(e + f*x))*(-a*d + b*c)), x)
rule2456 = ReplacementRule(pattern2456, replacement2456)
pattern2457 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons463, cons1328)
def replacement2457(f, b, d, c, n, a, x, e):
rubi.append(2457)
return Dist(d/(a*(-a*d + b*c)), Int((c + d*cos(e + f*x))**n*(a*n - b*(n + S(1))*cos(e + f*x)), x), x) + Simp(b**S(2)*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(a*f*(a + b*cos(e + f*x))*(-a*d + b*c)), x)
rule2457 = ReplacementRule(pattern2457, replacement2457)
pattern2458 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons1328)
def replacement2458(f, b, d, c, a, n, x, e):
rubi.append(2458)
return Dist(d*n/(a*b), Int((a - b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(-1)), x), x) - Simp(b*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(a + b*sin(e + f*x))), x)
rule2458 = ReplacementRule(pattern2458, replacement2458)
pattern2459 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons1328)
def replacement2459(f, b, d, c, n, a, x, e):
rubi.append(2459)
return Dist(d*n/(a*b), Int((a - b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(-1)), x), x) + Simp(b*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(a + b*cos(e + f*x))), x)
rule2459 = ReplacementRule(pattern2459, replacement2459)
pattern2460 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons88, cons808)
def replacement2460(f, b, d, c, a, n, x, e):
rubi.append(2460)
return Dist(S(2)*n*(a*d + b*c)/(b*(S(2)*n + S(1))), Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(-1)), x), x) + Simp(-S(2)*b*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(1))), x)
rule2460 = ReplacementRule(pattern2460, replacement2460)
pattern2461 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons88, cons808)
def replacement2461(f, b, d, c, n, a, x, e):
rubi.append(2461)
return Dist(S(2)*n*(a*d + b*c)/(b*(S(2)*n + S(1))), Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(-1)), x), x) + Simp(S(2)*b*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(1))), x)
rule2461 = ReplacementRule(pattern2461, replacement2461)
pattern2462 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2462(f, b, d, c, a, x, e):
rubi.append(2462)
return Simp(-S(2)*b**S(2)*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))*(a*d + b*c)), x)
rule2462 = ReplacementRule(pattern2462, replacement2462)
pattern2463 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2463(f, b, d, c, a, x, e):
rubi.append(2463)
return Simp(S(2)*b**S(2)*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))*(a*d + b*c)), x)
rule2463 = ReplacementRule(pattern2463, replacement2463)
pattern2464 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons89, cons1329, cons808)
def replacement2464(f, b, d, c, a, n, x, e):
rubi.append(2464)
return Dist((S(2)*n + S(3))*(-a*d + b*c)/(S(2)*b*(c**S(2) - d**S(2))*(n + S(1))), Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(1)), x), x) + Simp((c + d*sin(e + f*x))**(n + S(1))*(-a*d + b*c)*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2464 = ReplacementRule(pattern2464, replacement2464)
pattern2465 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons89, cons808)
def replacement2465(f, b, d, c, n, a, x, e):
rubi.append(2465)
return Dist((S(2)*n + S(3))*(-a*d + b*c)/(S(2)*b*(c**S(2) - d**S(2))*(n + S(1))), Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(1)), x), x) - Simp((c + d*cos(e + f*x))**(n + S(1))*(-a*d + b*c)*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2465 = ReplacementRule(pattern2465, replacement2465)
pattern2466 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2466(f, b, d, c, a, x, e):
rubi.append(2466)
return Dist(-S(2)*b/f, Subst(Int(S(1)/(a*d + b*c - d*x**S(2)), x), x, b*cos(e + f*x)/sqrt(a + b*sin(e + f*x))), x)
rule2466 = ReplacementRule(pattern2466, replacement2466)
pattern2467 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2467(f, b, d, c, a, x, e):
rubi.append(2467)
return Dist(S(2)*b/f, Subst(Int(S(1)/(a*d + b*c - d*x**S(2)), x), x, b*sin(e + f*x)/sqrt(a + b*cos(e + f*x))), x)
rule2467 = ReplacementRule(pattern2467, replacement2467)
pattern2468 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1330)
def replacement2468(f, b, d, a, x, e):
rubi.append(2468)
return Dist(-S(2)/f, Subst(Int(S(1)/sqrt(S(1) - x**S(2)/a), x), x, b*cos(e + f*x)/sqrt(a + b*sin(e + f*x))), x)
rule2468 = ReplacementRule(pattern2468, replacement2468)
pattern2469 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1330)
def replacement2469(f, b, d, a, x, e):
rubi.append(2469)
return Dist(S(2)/f, Subst(Int(S(1)/sqrt(S(1) - x**S(2)/a), x), x, b*sin(e + f*x)/sqrt(a + b*cos(e + f*x))), x)
rule2469 = ReplacementRule(pattern2469, replacement2469)
pattern2470 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2470(f, b, d, c, a, x, e):
rubi.append(2470)
return Dist(-S(2)*b/f, Subst(Int(S(1)/(b + d*x**S(2)), x), x, b*cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))), x)
rule2470 = ReplacementRule(pattern2470, replacement2470)
pattern2471 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2471(f, b, d, c, a, x, e):
rubi.append(2471)
return Dist(S(2)*b/f, Subst(Int(S(1)/(b + d*x**S(2)), x), x, b*sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x)))), x)
rule2471 = ReplacementRule(pattern2471, replacement2471)
pattern2472 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons543)
def replacement2472(f, b, d, c, a, n, x, e):
rubi.append(2472)
return Dist(a**S(2)*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((c + d*x)**n/sqrt(a - b*x), x), x, sin(e + f*x)), x)
rule2472 = ReplacementRule(pattern2472, replacement2472)
pattern2473 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons543)
def replacement2473(f, b, d, c, n, a, x, e):
rubi.append(2473)
return -Dist(a**S(2)*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((c + d*x)**n/sqrt(a - b*x), x), x, cos(e + f*x)), x)
rule2473 = ReplacementRule(pattern2473, replacement2473)
pattern2474 = Pattern(Integral(sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2474(f, b, d, c, a, x, e):
rubi.append(2474)
return Dist(d/b, Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x)
rule2474 = ReplacementRule(pattern2474, replacement2474)
pattern2475 = Pattern(Integral(sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2475(f, b, d, c, a, x, e):
rubi.append(2475)
return Dist(d/b, Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x)
rule2475 = ReplacementRule(pattern2475, replacement2475)
pattern2476 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons165, cons808)
def replacement2476(f, b, d, c, a, n, x, e):
rubi.append(2476)
return -Dist(S(1)/(b*(S(2)*n + S(-1))), Int((c + d*sin(e + f*x))**(n + S(-2))*Simp(a*c*d - b*(c**S(2)*(S(2)*n + S(-1)) + S(2)*d**S(2)*(n + S(-1))) + d*(a*d - b*c*(S(4)*n + S(-3)))*sin(e + f*x), x)/sqrt(a + b*sin(e + f*x)), x), x) + Simp(-S(2)*d*(c + d*sin(e + f*x))**(n + S(-1))*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(-1))), x)
rule2476 = ReplacementRule(pattern2476, replacement2476)
pattern2477 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons165, cons808)
def replacement2477(f, b, d, c, n, a, x, e):
rubi.append(2477)
return -Dist(S(1)/(b*(S(2)*n + S(-1))), Int((c + d*cos(e + f*x))**(n + S(-2))*Simp(a*c*d - b*(c**S(2)*(S(2)*n + S(-1)) + S(2)*d**S(2)*(n + S(-1))) + d*(a*d - b*c*(S(4)*n + S(-3)))*cos(e + f*x), x)/sqrt(a + b*cos(e + f*x)), x), x) + Simp(S(2)*d*(c + d*cos(e + f*x))**(n + S(-1))*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(-1))), x)
rule2477 = ReplacementRule(pattern2477, replacement2477)
pattern2478 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons89, cons808)
def replacement2478(f, b, d, c, a, n, x, e):
rubi.append(2478)
return -Dist(S(1)/(S(2)*b*(c**S(2) - d**S(2))*(n + S(1))), Int((c + d*sin(e + f*x))**(n + S(1))*Simp(a*d - S(2)*b*c*(n + S(1)) + b*d*(S(2)*n + S(3))*sin(e + f*x), x)/sqrt(a + b*sin(e + f*x)), x), x) - Simp(d*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2478 = ReplacementRule(pattern2478, replacement2478)
pattern2479 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323, cons87, cons89, cons808)
def replacement2479(f, b, d, c, n, a, x, e):
rubi.append(2479)
return -Dist(S(1)/(S(2)*b*(c**S(2) - d**S(2))*(n + S(1))), Int((c + d*cos(e + f*x))**(n + S(1))*Simp(a*d - S(2)*b*c*(n + S(1)) + b*d*(S(2)*n + S(3))*cos(e + f*x), x)/sqrt(a + b*cos(e + f*x)), x), x) + Simp(d*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2479 = ReplacementRule(pattern2479, replacement2479)
pattern2480 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2480(f, b, d, c, a, x, e):
rubi.append(2480)
return Dist(b/(-a*d + b*c), Int(S(1)/sqrt(a + b*sin(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x)
rule2480 = ReplacementRule(pattern2480, replacement2480)
pattern2481 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2481(f, b, d, c, a, x, e):
rubi.append(2481)
return Dist(b/(-a*d + b*c), Int(S(1)/sqrt(a + b*cos(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x)
rule2481 = ReplacementRule(pattern2481, replacement2481)
pattern2482 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1330, cons43)
def replacement2482(f, b, d, a, x, e):
rubi.append(2482)
return -Dist(sqrt(S(2))/(sqrt(a)*f), Subst(Int(S(1)/sqrt(-x**S(2) + S(1)), x), x, b*cos(e + f*x)/(a + b*sin(e + f*x))), x)
rule2482 = ReplacementRule(pattern2482, replacement2482)
pattern2483 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1330, cons43)
def replacement2483(f, b, d, a, x, e):
rubi.append(2483)
return Dist(sqrt(S(2))/(sqrt(a)*f), Subst(Int(S(1)/sqrt(-x**S(2) + S(1)), x), x, b*sin(e + f*x)/(a + b*cos(e + f*x))), x)
rule2483 = ReplacementRule(pattern2483, replacement2483)
pattern2484 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2484(f, b, d, c, a, x, e):
rubi.append(2484)
return Dist(-S(2)*a/f, Subst(Int(S(1)/(S(2)*b**S(2) - x**S(2)*(a*c - b*d)), x), x, b*cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))), x)
rule2484 = ReplacementRule(pattern2484, replacement2484)
pattern2485 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement2485(f, b, d, c, a, x, e):
rubi.append(2485)
return Dist(S(2)*a/f, Subst(Int(S(1)/(S(2)*b**S(2) - x**S(2)*(a*c - b*d)), x), x, b*sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x)))), x)
rule2485 = ReplacementRule(pattern2485, replacement2485)
pattern2486 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1265, cons1323, cons87, cons165, cons85)
def replacement2486(m, f, b, d, c, a, n, x, e):
rubi.append(2486)
return Dist(S(1)/(b*(m + n)), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(-2))*Simp(b*c**S(2)*(m + n) + d*(a*c*m + b*d*(n + S(-1))) + d*(a*d*m + b*c*(m + S(2)*n + S(-1)))*sin(e + f*x), x), x), x) - Simp(d*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(-1))*cos(e + f*x)/(f*(m + n)), x)
rule2486 = ReplacementRule(pattern2486, replacement2486)
pattern2487 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1265, cons1323, cons87, cons165, cons85)
def replacement2487(m, f, b, d, c, n, a, x, e):
rubi.append(2487)
return Dist(S(1)/(b*(m + n)), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(-2))*Simp(b*c**S(2)*(m + n) + d*(a*c*m + b*d*(n + S(-1))) + d*(a*d*m + b*c*(m + S(2)*n + S(-1)))*cos(e + f*x), x), x), x) + Simp(d*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(-1))*sin(e + f*x)/(f*(m + n)), x)
rule2487 = ReplacementRule(pattern2487, replacement2487)
pattern2488 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons17)
def replacement2488(m, f, b, d, c, n, a, x, e):
rubi.append(2488)
return Dist(a**m*cos(e + f*x)/(f*sqrt(-sin(e + f*x) + S(1))*sqrt(sin(e + f*x) + S(1))), Subst(Int((S(1) + b*x/a)**(m + S(-1)/2)*(c + d*x)**n/sqrt(S(1) - b*x/a), x), x, sin(e + f*x)), x)
rule2488 = ReplacementRule(pattern2488, replacement2488)
pattern2489 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1265, cons1323, cons17)
def replacement2489(m, f, b, d, c, n, a, x, e):
rubi.append(2489)
return -Dist(a**m*sin(e + f*x)/(f*sqrt(-cos(e + f*x) + S(1))*sqrt(cos(e + f*x) + S(1))), Subst(Int((S(1) + b*x/a)**(m + S(-1)/2)*(c + d*x)**n/sqrt(S(1) - b*x/a), x), x, cos(e + f*x)), x)
rule2489 = ReplacementRule(pattern2489, replacement2489)
pattern2490 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43, cons1331)
def replacement2490(m, f, b, d, a, n, x, e):
rubi.append(2490)
return -Dist(b*(d/b)**n*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((a - x)**n*(S(2)*a - x)**(m + S(-1)/2)/sqrt(x), x), x, a - b*sin(e + f*x)), x)
rule2490 = ReplacementRule(pattern2490, replacement2490)
pattern2491 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43, cons1331)
def replacement2491(m, f, b, d, a, n, x, e):
rubi.append(2491)
return Dist(b*(d/b)**n*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((a - x)**n*(S(2)*a - x)**(m + S(-1)/2)/sqrt(x), x), x, a - b*cos(e + f*x)), x)
rule2491 = ReplacementRule(pattern2491, replacement2491)
pattern2492 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43, cons1332)
def replacement2492(m, f, b, d, a, n, x, e):
rubi.append(2492)
return Dist((d/b)**IntPart(n)*(b*sin(e + f*x))**(-FracPart(n))*(d*sin(e + f*x))**FracPart(n), Int((b*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x)
rule2492 = ReplacementRule(pattern2492, replacement2492)
pattern2493 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43, cons1332)
def replacement2493(m, f, b, d, a, n, x, e):
rubi.append(2493)
return Dist((d/b)**IntPart(n)*(b*cos(e + f*x))**(-FracPart(n))*(d*cos(e + f*x))**FracPart(n), Int((b*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x)
rule2493 = ReplacementRule(pattern2493, replacement2493)
pattern2494 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons448)
def replacement2494(m, f, b, d, a, n, x, e):
rubi.append(2494)
return Dist(a**IntPart(m)*(S(1) + b*sin(e + f*x)/a)**(-FracPart(m))*(a + b*sin(e + f*x))**FracPart(m), Int((d*sin(e + f*x))**n*(S(1) + b*sin(e + f*x)/a)**m, x), x)
rule2494 = ReplacementRule(pattern2494, replacement2494)
pattern2495 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons448)
def replacement2495(m, f, b, d, a, n, x, e):
rubi.append(2495)
return Dist(a**IntPart(m)*(S(1) + b*cos(e + f*x)/a)**(-FracPart(m))*(a + b*cos(e + f*x))**FracPart(m), Int((d*cos(e + f*x))**n*(S(1) + b*cos(e + f*x)/a)**m, x), x)
rule2495 = ReplacementRule(pattern2495, replacement2495)
pattern2496 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1265, cons1323, cons18)
def replacement2496(m, f, b, d, a, n, c, x, e):
rubi.append(2496)
return Dist(a**S(2)*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, sin(e + f*x)), x)
rule2496 = ReplacementRule(pattern2496, replacement2496)
pattern2497 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1265, cons1323, cons18)
def replacement2497(m, f, b, d, a, n, c, x, e):
rubi.append(2497)
return -Dist(a**S(2)*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, cos(e + f*x)), x)
rule2497 = ReplacementRule(pattern2497, replacement2497)
pattern2498 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons3, cons7, cons27, cons48, cons125, cons21, cons1318)
def replacement2498(m, f, b, d, c, x, e):
rubi.append(2498)
return Dist(S(2)*c*d/b, Int((b*sin(e + f*x))**(m + S(1)), x), x) + Int((b*sin(e + f*x))**m*(c**S(2) + d**S(2)*sin(e + f*x)**S(2)), x)
rule2498 = ReplacementRule(pattern2498, replacement2498)
pattern2499 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons3, cons7, cons27, cons48, cons125, cons21, cons1318)
def replacement2499(m, f, b, d, c, x, e):
rubi.append(2499)
return Dist(S(2)*c*d/b, Int((b*cos(e + f*x))**(m + S(1)), x), x) + Int((b*cos(e + f*x))**m*(c**S(2) + d**S(2)*cos(e + f*x)**S(2)), x)
rule2499 = ReplacementRule(pattern2499, replacement2499)
pattern2500 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons31, cons94)
def replacement2500(m, f, b, d, c, a, x, e):
rubi.append(2500)
return -Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(-a*(c**S(2) + d**S(2)) + S(2)*b*c*d) + (a**S(2)*d**S(2) - S(2)*a*b*c*d*(m + S(2)) + b**S(2)*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(a**S(2)*d**S(2) - S(2)*a*b*c*d + b**S(2)*c**S(2))*cos(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2500 = ReplacementRule(pattern2500, replacement2500)
pattern2501 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons31, cons94)
def replacement2501(m, f, b, d, c, a, x, e):
rubi.append(2501)
return -Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(-a*(c**S(2) + d**S(2)) + S(2)*b*c*d) + (a**S(2)*d**S(2) - S(2)*a*b*c*d*(m + S(2)) + b**S(2)*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(a**S(2)*d**S(2) - S(2)*a*b*c*d + b**S(2)*c**S(2))*sin(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2501 = ReplacementRule(pattern2501, replacement2501)
pattern2502 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1267, cons272)
def replacement2502(m, f, b, d, c, a, x, e):
rubi.append(2502)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*Simp(b*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))) - d*(a*d - S(2)*b*c*(m + S(2)))*sin(e + f*x), x), x), x) - Simp(d**S(2)*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x)
rule2502 = ReplacementRule(pattern2502, replacement2502)
pattern2503 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons1267, cons272)
def replacement2503(m, f, b, d, c, a, x, e):
rubi.append(2503)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*Simp(b*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))) - d*(a*d - S(2)*b*c*(m + S(2)))*cos(e + f*x), x), x), x) + Simp(d**S(2)*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x)
rule2503 = ReplacementRule(pattern2503, replacement2503)
pattern2504 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons93, cons1333, cons89, cons1334)
def replacement2504(m, f, b, d, c, a, n, x, e):
rubi.append(2504)
return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-3))*(c + d*sin(e + f*x))**(n + S(1))*Simp(a*d*(n + S(1))*(-S(2)*a*b*d + c*(a**S(2) + b**S(2))) + b*(m + S(-2))*(-a*d + b*c)**S(2) + b*(b**S(2)*(c**S(2) - d**S(2)) + d*n*(S(2)*a*b*c - d*(a**S(2) + b**S(2))) - m*(-a*d + b*c)**S(2))*sin(e + f*x)**S(2) + (-a*(n + S(2))*(-a*d + b*c)**S(2) + b*(n + S(1))*(a*b*c**S(2) - S(3)*a*b*d**S(2) + c*d*(a**S(2) + b**S(2))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*(a**S(2)*d**S(2) - S(2)*a*b*c*d + b**S(2)*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2504 = ReplacementRule(pattern2504, replacement2504)
pattern2505 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons93, cons1333, cons89, cons1334)
def replacement2505(m, f, b, d, c, a, n, x, e):
rubi.append(2505)
return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-3))*(c + d*cos(e + f*x))**(n + S(1))*Simp(a*d*(n + S(1))*(-S(2)*a*b*d + c*(a**S(2) + b**S(2))) + b*(m + S(-2))*(-a*d + b*c)**S(2) + b*(b**S(2)*(c**S(2) - d**S(2)) + d*n*(S(2)*a*b*c - d*(a**S(2) + b**S(2))) - m*(-a*d + b*c)**S(2))*cos(e + f*x)**S(2) + (-a*(n + S(2))*(-a*d + b*c)**S(2) + b*(n + S(1))*(a*b*c**S(2) - S(3)*a*b*d**S(2) + c*d*(a**S(2) + b**S(2))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*(a**S(2)*d**S(2) - S(2)*a*b*c*d + b**S(2)*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2505 = ReplacementRule(pattern2505, replacement2505)
pattern2506 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1267, cons1323, cons31, cons1333, cons1334, cons1335)
def replacement2506(m, f, b, d, c, a, n, x, e):
rubi.append(2506)
return Dist(S(1)/(d*(m + n)), Int((a + b*sin(e + f*x))**(m + S(-3))*(c + d*sin(e + f*x))**n*Simp(a**S(3)*d*(m + n) + b**S(2)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b**S(2)*(-a*d*(S(3)*m + S(2)*n + S(-2)) + b*c*(m + S(-1)))*sin(e + f*x)**S(2) - b*(-S(3)*a**S(2)*d*(m + n) + a*b*c - b**S(2)*d*(m + n + S(-1)))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n)), x)
rule2506 = ReplacementRule(pattern2506, replacement2506)
pattern2507 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1267, cons1323, cons31, cons1333, cons1334, cons1335)
def replacement2507(m, f, b, d, c, a, n, x, e):
rubi.append(2507)
return Dist(S(1)/(d*(m + n)), Int((a + b*cos(e + f*x))**(m + S(-3))*(c + d*cos(e + f*x))**n*Simp(a**S(3)*d*(m + n) + b**S(2)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b**S(2)*(-a*d*(S(3)*m + S(2)*n + S(-2)) + b*c*(m + S(-1)))*cos(e + f*x)**S(2) - b*(-S(3)*a**S(2)*d*(m + n) + a*b*c - b**S(2)*d*(m + n + S(-1)))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n)), x)
rule2507 = ReplacementRule(pattern2507, replacement2507)
pattern2508 = Pattern(Integral(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement2508(f, b, d, a, x, e):
rubi.append(2508)
return -Dist(d**S(2)/(a**S(2) - b**S(2)), Int(sqrt(a + b*sin(e + f*x))/(d*sin(e + f*x))**(S(3)/2), x), x) + Simp(-S(2)*a*d*cos(e + f*x)/(f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*(a**S(2) - b**S(2))), x)
rule2508 = ReplacementRule(pattern2508, replacement2508)
pattern2509 = Pattern(Integral(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement2509(f, b, d, a, x, e):
rubi.append(2509)
return -Dist(d**S(2)/(a**S(2) - b**S(2)), Int(sqrt(a + b*cos(e + f*x))/(d*cos(e + f*x))**(S(3)/2), x), x) + Simp(S(2)*a*d*sin(e + f*x)/(f*sqrt(d*cos(e + f*x))*sqrt(a + b*cos(e + f*x))*(a**S(2) - b**S(2))), x)
rule2509 = ReplacementRule(pattern2509, replacement2509)
pattern2510 = Pattern(Integral(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2510(f, b, d, a, c, x, e):
rubi.append(2510)
return Dist((c - d)/(a - b), Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) - Dist((-a*d + b*c)/(a - b), Int((sin(e + f*x) + S(1))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x)
rule2510 = ReplacementRule(pattern2510, replacement2510)
pattern2511 = Pattern(Integral(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2511(f, b, d, a, c, x, e):
rubi.append(2511)
return Dist((c - d)/(a - b), Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) - Dist((-a*d + b*c)/(a - b), Int((cos(e + f*x) + S(1))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x)
rule2511 = ReplacementRule(pattern2511, replacement2511)
pattern2512 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons93, cons94, cons1325, cons1170)
def replacement2512(m, f, b, d, c, a, n, x, e):
rubi.append(2512)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(a*c*(m + S(1)) + b*d*n - b*d*(m + n + S(2))*sin(e + f*x)**S(2) + (a*d*(m + S(1)) - b*c*(m + S(2)))*sin(e + f*x), x), x), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2512 = ReplacementRule(pattern2512, replacement2512)
pattern2513 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons93, cons94, cons1325, cons1170)
def replacement2513(m, f, b, d, c, a, n, x, e):
rubi.append(2513)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(a*c*(m + S(1)) + b*d*n - b*d*(m + n + S(2))*cos(e + f*x)**S(2) + (a*d*(m + S(1)) - b*c*(m + S(2)))*cos(e + f*x), x), x), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2513 = ReplacementRule(pattern2513, replacement2513)
pattern2514 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement2514(f, b, d, a, x, e):
rubi.append(2514)
return Dist(d/b, Int(sqrt(d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) - Dist(a*d/b, Int(sqrt(d*sin(e + f*x))/(a + b*sin(e + f*x))**(S(3)/2), x), x)
rule2514 = ReplacementRule(pattern2514, replacement2514)
pattern2515 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement2515(f, b, d, a, x, e):
rubi.append(2515)
return Dist(d/b, Int(sqrt(d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) - Dist(a*d/b, Int(sqrt(d*cos(e + f*x))/(a + b*cos(e + f*x))**(S(3)/2), x), x)
rule2515 = ReplacementRule(pattern2515, replacement2515)
pattern2516 = Pattern(Integral((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2516(f, b, d, a, c, x, e):
rubi.append(2516)
return Dist(d**S(2)/b**S(2), Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/b**S(2), Int(Simp(a*d + b*c + S(2)*b*d*sin(e + f*x), x)/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x)
rule2516 = ReplacementRule(pattern2516, replacement2516)
pattern2517 = Pattern(Integral((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2517(f, b, d, a, c, x, e):
rubi.append(2517)
return Dist(d**S(2)/b**S(2), Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/b**S(2), Int(Simp(a*d + b*c + S(2)*b*d*cos(e + f*x), x)/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x)
rule2517 = ReplacementRule(pattern2517, replacement2517)
pattern2518 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons93, cons94, cons1336, cons1170)
def replacement2518(m, f, b, d, c, a, n, x, e):
rubi.append(2518)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-2))*Simp(c*(m + S(1))*(a*c - b*d) + d*(n + S(-1))*(-a*d + b*c) - d*(-a*d + b*c)*(m + n + S(1))*sin(e + f*x)**S(2) + (-c*(m + S(2))*(-a*d + b*c) + d*(m + S(1))*(a*c - b*d))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*(-a*d + b*c)*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2518 = ReplacementRule(pattern2518, replacement2518)
pattern2519 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons93, cons94, cons1336, cons1170)
def replacement2519(m, f, b, d, c, a, n, x, e):
rubi.append(2519)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-2))*Simp(c*(m + S(1))*(a*c - b*d) + d*(n + S(-1))*(-a*d + b*c) - d*(-a*d + b*c)*(m + n + S(1))*cos(e + f*x)**S(2) + (-c*(m + S(2))*(-a*d + b*c) + d*(m + S(1))*(a*c - b*d))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*(-a*d + b*c)*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2519 = ReplacementRule(pattern2519, replacement2519)
pattern2520 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement2520(f, b, d, a, x, e):
rubi.append(2520)
return Dist(d/(a**S(2) - b**S(2)), Int((a*sin(e + f*x) + b)/((d*sin(e + f*x))**(S(3)/2)*sqrt(a + b*sin(e + f*x))), x), x) + Simp(S(2)*b*cos(e + f*x)/(f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*(a**S(2) - b**S(2))), x)
rule2520 = ReplacementRule(pattern2520, replacement2520)
pattern2521 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement2521(f, b, d, a, x, e):
rubi.append(2521)
return Dist(d/(a**S(2) - b**S(2)), Int((a*cos(e + f*x) + b)/((d*cos(e + f*x))**(S(3)/2)*sqrt(a + b*cos(e + f*x))), x), x) + Simp(-S(2)*b*sin(e + f*x)/(f*sqrt(d*cos(e + f*x))*sqrt(a + b*cos(e + f*x))*(a**S(2) - b**S(2))), x)
rule2521 = ReplacementRule(pattern2521, replacement2521)
pattern2522 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2522(f, b, d, c, a, x, e):
rubi.append(2522)
return -Dist(b/(a - b), Int((sin(e + f*x) + S(1))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) + Dist(S(1)/(a - b), Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x)
rule2522 = ReplacementRule(pattern2522, replacement2522)
pattern2523 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2523(f, b, d, c, a, x, e):
rubi.append(2523)
return -Dist(b/(a - b), Int((cos(e + f*x) + S(1))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Dist(S(1)/(a - b), Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x)
rule2523 = ReplacementRule(pattern2523, replacement2523)
pattern2524 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1267, cons1323, cons31, cons94, cons1170, cons1337)
def replacement2524(m, f, b, d, c, a, n, x, e):
rubi.append(2524)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(a*(m + S(1))*(-a*d + b*c) + b**S(2)*d*(m + n + S(2)) - b**S(2)*d*(m + n + S(3))*sin(e + f*x)**S(2) - (b**S(2)*c + b*(m + S(1))*(-a*d + b*c))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule2524 = ReplacementRule(pattern2524, replacement2524)
pattern2525 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons1267, cons1323, cons31, cons94, cons1170, cons1337)
def replacement2525(m, f, b, d, c, a, n, x, e):
rubi.append(2525)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(a*(m + S(1))*(-a*d + b*c) + b**S(2)*d*(m + n + S(2)) - b**S(2)*d*(m + n + S(3))*cos(e + f*x)**S(2) - (b**S(2)*c + b*(m + S(1))*(-a*d + b*c))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule2525 = ReplacementRule(pattern2525, replacement2525)
pattern2526 = Pattern(Integral(sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2526(f, b, d, c, a, x, e):
rubi.append(2526)
return Dist(d/b, Int(S(1)/sqrt(c + d*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/((a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x)
rule2526 = ReplacementRule(pattern2526, replacement2526)
pattern2527 = Pattern(Integral(sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2527(f, b, d, c, a, x, e):
rubi.append(2527)
return Dist(d/b, Int(S(1)/sqrt(c + d*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/((a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x)
rule2527 = ReplacementRule(pattern2527, replacement2527)
pattern2528 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2528(f, b, d, c, a, x, e):
rubi.append(2528)
return Dist(b/d, Int(sqrt(a + b*sin(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x)
rule2528 = ReplacementRule(pattern2528, replacement2528)
pattern2529 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2529(f, b, d, c, a, x, e):
rubi.append(2529)
return Dist(b/d, Int(sqrt(a + b*cos(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x)
rule2529 = ReplacementRule(pattern2529, replacement2529)
pattern2530 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1338)
def replacement2530(f, b, d, c, a, x, e):
rubi.append(2530)
return Simp(S(2)*EllipticPi(S(2)*b/(a + b), -Pi/S(4) + e/S(2) + f*x/S(2), S(2)*d/(c + d))/(f*(a + b)*sqrt(c + d)), x)
rule2530 = ReplacementRule(pattern2530, replacement2530)
pattern2531 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1338)
def replacement2531(f, b, d, c, a, x, e):
rubi.append(2531)
return Simp(S(2)*EllipticPi(S(2)*b/(a + b), e/S(2) + f*x/S(2), S(2)*d/(c + d))/(f*(a + b)*sqrt(c + d)), x)
rule2531 = ReplacementRule(pattern2531, replacement2531)
pattern2532 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1339)
def replacement2532(f, b, d, c, a, x, e):
rubi.append(2532)
return Simp(S(2)*EllipticPi(-S(2)*b/(a - b), Pi/S(4) + e/S(2) + f*x/S(2), -S(2)*d/(c - d))/(f*(a - b)*sqrt(c - d)), x)
rule2532 = ReplacementRule(pattern2532, replacement2532)
pattern2533 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1339)
def replacement2533(f, b, d, c, a, x, e):
rubi.append(2533)
return Simp(S(2)*EllipticPi(-S(2)*b/(a - b), Pi/S(2) + e/S(2) + f*x/S(2), -S(2)*d/(c - d))/(f*(a - b)*sqrt(c - d)), x)
rule2533 = ReplacementRule(pattern2533, replacement2533)
pattern2534 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1340)
def replacement2534(f, b, d, c, a, x, e):
rubi.append(2534)
return Dist(sqrt((c + d*sin(e + f*x))/(c + d))/sqrt(c + d*sin(e + f*x)), Int(S(1)/((a + b*sin(e + f*x))*sqrt(c/(c + d) + d*sin(e + f*x)/(c + d))), x), x)
rule2534 = ReplacementRule(pattern2534, replacement2534)
pattern2535 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1340)
def replacement2535(f, b, d, c, a, x, e):
rubi.append(2535)
return Dist(sqrt((c + d*cos(e + f*x))/(c + d))/sqrt(c + d*cos(e + f*x)), Int(S(1)/((a + b*cos(e + f*x))*sqrt(c/(c + d) + d*cos(e + f*x)/(c + d))), x), x)
rule2535 = ReplacementRule(pattern2535, replacement2535)
pattern2536 = Pattern(Integral(sqrt(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1341, cons1342, cons1343)
def replacement2536(f, b, d, c, x, e):
rubi.append(2536)
return Simp(S(2)*c*sqrt(S(1) - S(1)/sin(e + f*x))*sqrt(S(1) + S(1)/sin(e + f*x))*EllipticPi((c + d)/d, asin(sqrt(c + d*sin(e + f*x))/(sqrt(b*sin(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt(b*(c + d), S(2))*tan(e + f*x)/(d*f*sqrt(c**S(2) - d**S(2))), x)
rule2536 = ReplacementRule(pattern2536, replacement2536)
pattern2537 = Pattern(Integral(sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1341, cons1342, cons1343)
def replacement2537(f, b, d, c, x, e):
rubi.append(2537)
return Simp(-S(2)*c*sqrt(S(1) - S(1)/cos(e + f*x))*sqrt(S(1) + S(1)/cos(e + f*x))*EllipticPi((c + d)/d, asin(sqrt(c + d*cos(e + f*x))/(sqrt(b*cos(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt(b*(c + d), S(2))/(d*f*sqrt(c**S(2) - d**S(2))*tan(e + f*x)), x)
rule2537 = ReplacementRule(pattern2537, replacement2537)
pattern2538 = Pattern(Integral(sqrt(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1323, cons1342)
def replacement2538(f, b, d, c, x, e):
rubi.append(2538)
return Simp(S(2)*b*sqrt(c*(S(1) - S(1)/sin(e + f*x))/(c + d))*sqrt(c*(S(1) + S(1)/sin(e + f*x))/(c - d))*EllipticPi((c + d)/d, asin(sqrt(c + d*sin(e + f*x))/(sqrt(b*sin(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt((c + d)/b, S(2))*tan(e + f*x)/(d*f), x)
rule2538 = ReplacementRule(pattern2538, replacement2538)
pattern2539 = Pattern(Integral(sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1323, cons1342)
def replacement2539(f, b, d, c, x, e):
rubi.append(2539)
return Simp(-S(2)*b*sqrt(c*(S(1) - S(1)/cos(e + f*x))/(c + d))*sqrt(c*(S(1) + S(1)/cos(e + f*x))/(c - d))*EllipticPi((c + d)/d, asin(sqrt(c + d*cos(e + f*x))/(sqrt(b*cos(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt((c + d)/b, S(2))/(d*f*tan(e + f*x)), x)
rule2539 = ReplacementRule(pattern2539, replacement2539)
pattern2540 = Pattern(Integral(sqrt(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1323, cons1344)
def replacement2540(f, b, d, c, x, e):
rubi.append(2540)
return Dist(sqrt(b*sin(e + f*x))/sqrt(-b*sin(e + f*x)), Int(sqrt(-b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x)
rule2540 = ReplacementRule(pattern2540, replacement2540)
pattern2541 = Pattern(Integral(sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons7, cons27, cons48, cons125, cons1323, cons1344)
def replacement2541(f, b, d, c, x, e):
rubi.append(2541)
return Dist(sqrt(b*cos(e + f*x))/sqrt(-b*cos(e + f*x)), Int(sqrt(-b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x)
rule2541 = ReplacementRule(pattern2541, replacement2541)
pattern2542 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1345)
def replacement2542(f, b, d, c, a, x, e):
rubi.append(2542)
return Simp(S(2)*sqrt((-a*d + b*c)*(sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c - d)))*sqrt(-(-a*d + b*c)*(-sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c + d)))*(a + b*sin(e + f*x))*EllipticPi(b*(c + d)/(d*(a + b)), asin(sqrt(c + d*sin(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*sin(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(d*f*Rt((a + b)/(c + d), S(2))*cos(e + f*x)), x)
rule2542 = ReplacementRule(pattern2542, replacement2542)
pattern2543 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1345)
def replacement2543(f, b, d, c, a, x, e):
rubi.append(2543)
return Simp(-S(2)*sqrt((-a*d + b*c)*(cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c - d)))*sqrt(-(-a*d + b*c)*(-cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c + d)))*(a + b*cos(e + f*x))*EllipticPi(b*(c + d)/(d*(a + b)), asin(sqrt(c + d*cos(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*cos(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(d*f*Rt((a + b)/(c + d), S(2))*sin(e + f*x)), x)
rule2543 = ReplacementRule(pattern2543, replacement2543)
pattern2544 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1346)
def replacement2544(f, b, d, c, a, x, e):
rubi.append(2544)
return Dist(sqrt(-c - d*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), Int(sqrt(a + b*sin(e + f*x))/sqrt(-c - d*sin(e + f*x)), x), x)
rule2544 = ReplacementRule(pattern2544, replacement2544)
pattern2545 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1346)
def replacement2545(f, b, d, c, a, x, e):
rubi.append(2545)
return Dist(sqrt(-c - d*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), Int(sqrt(a + b*cos(e + f*x))/sqrt(-c - d*cos(e + f*x)), x), x)
rule2545 = ReplacementRule(pattern2545, replacement2545)
pattern2546 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1347, cons1348, cons1349)
def replacement2546(f, b, d, a, x, e):
rubi.append(2546)
return Simp(-S(2)*d*EllipticF(asin(cos(e + f*x)/(d*sin(e + f*x) + S(1))), -(a - b*d)/(a + b*d))/(f*sqrt(a + b*d)), x)
rule2546 = ReplacementRule(pattern2546, replacement2546)
pattern2547 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1347, cons1348, cons1349)
def replacement2547(f, b, d, a, x, e):
rubi.append(2547)
return Simp(S(2)*d*EllipticF(asin(sin(e + f*x)/(d*cos(e + f*x) + S(1))), -(a - b*d)/(a + b*d))/(f*sqrt(a + b*d)), x)
rule2547 = ReplacementRule(pattern2547, replacement2547)
pattern2548 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1347, cons1350, cons1351)
def replacement2548(f, b, d, a, x, e):
rubi.append(2548)
return Dist(sqrt(sin(e + f*x)*sign(b))/sqrt(d*sin(e + f*x)), Int(S(1)/(sqrt(sin(e + f*x)*sign(b))*sqrt(a + b*sin(e + f*x))), x), x)
rule2548 = ReplacementRule(pattern2548, replacement2548)
pattern2549 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1347, cons1350, cons1351)
def replacement2549(f, b, d, a, x, e):
rubi.append(2549)
return Dist(sqrt(cos(e + f*x)*sign(b))/sqrt(d*cos(e + f*x)), Int(S(1)/(sqrt(cos(e + f*x)*sign(b))*sqrt(a + b*cos(e + f*x))), x), x)
rule2549 = ReplacementRule(pattern2549, replacement2549)
pattern2550 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1271, cons1352, cons1353)
def replacement2550(f, b, d, a, x, e):
rubi.append(2550)
return Simp(-S(2)*sqrt(-S(1)/tan(e + f*x)**S(2))*sqrt(a**S(2))*EllipticF(asin(sqrt(a + b*sin(e + f*x))/(sqrt(d*sin(e + f*x))*Rt((a + b)/d, S(2)))), -(a + b)/(a - b))*Rt((a + b)/d, S(2))*tan(e + f*x)/(a*f*sqrt(a**S(2) - b**S(2))), x)
rule2550 = ReplacementRule(pattern2550, replacement2550)
pattern2551 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1271, cons1352, cons1353)
def replacement2551(f, b, d, a, x, e):
rubi.append(2551)
return Simp(S(2)*sqrt(-tan(e + f*x)**S(2))*sqrt(a**S(2))*EllipticF(asin(sqrt(a + b*cos(e + f*x))/(sqrt(d*cos(e + f*x))*Rt((a + b)/d, S(2)))), -(a + b)/(a - b))*Rt((a + b)/d, S(2))/(a*f*sqrt(a**S(2) - b**S(2))*tan(e + f*x)), x)
rule2551 = ReplacementRule(pattern2551, replacement2551)
pattern2552 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1352)
def replacement2552(f, b, d, a, x, e):
rubi.append(2552)
return Simp(-S(2)*sqrt(a*(S(1) - S(1)/sin(e + f*x))/(a + b))*sqrt(a*(S(1) + S(1)/sin(e + f*x))/(a - b))*EllipticF(asin(sqrt(a + b*sin(e + f*x))/(sqrt(d*sin(e + f*x))*Rt((a + b)/d, S(2)))), -(a + b)/(a - b))*Rt((a + b)/d, S(2))*tan(e + f*x)/(a*f), x)
rule2552 = ReplacementRule(pattern2552, replacement2552)
pattern2553 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1352)
def replacement2553(f, b, d, a, x, e):
rubi.append(2553)
return Simp(S(2)*sqrt(a*(S(1) - S(1)/cos(e + f*x))/(a + b))*sqrt(a*(S(1) + S(1)/cos(e + f*x))/(a - b))*EllipticF(asin(sqrt(a + b*cos(e + f*x))/(sqrt(d*cos(e + f*x))*Rt((a + b)/d, S(2)))), -(a + b)/(a - b))*Rt((a + b)/d, S(2))/(a*f*tan(e + f*x)), x)
rule2553 = ReplacementRule(pattern2553, replacement2553)
pattern2554 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1354)
def replacement2554(f, b, d, a, x, e):
rubi.append(2554)
return Dist(sqrt(-d*sin(e + f*x))/sqrt(d*sin(e + f*x)), Int(S(1)/(sqrt(-d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x)
rule2554 = ReplacementRule(pattern2554, replacement2554)
pattern2555 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1354)
def replacement2555(f, b, d, a, x, e):
rubi.append(2555)
return Dist(sqrt(-d*cos(e + f*x))/sqrt(d*cos(e + f*x)), Int(S(1)/(sqrt(-d*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x)
rule2555 = ReplacementRule(pattern2555, replacement2555)
pattern2556 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1355)
def replacement2556(f, b, d, a, c, x, e):
rubi.append(2556)
return Simp(S(2)*sqrt((-a*d + b*c)*(-sin(e + f*x) + S(1))/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(-a*d + b*c)*(sin(e + f*x) + S(1))/((a - b)*(c + d*sin(e + f*x))))*(c + d*sin(e + f*x))*EllipticF(asin(sqrt(a + b*sin(e + f*x))*Rt((c + d)/(a + b), S(2))/sqrt(c + d*sin(e + f*x))), (a + b)*(c - d)/((a - b)*(c + d)))/(f*(-a*d + b*c)*Rt((c + d)/(a + b), S(2))*cos(e + f*x)), x)
rule2556 = ReplacementRule(pattern2556, replacement2556)
pattern2557 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1355)
def replacement2557(f, b, d, a, c, x, e):
rubi.append(2557)
return Simp(-S(2)*sqrt((-a*d + b*c)*(-cos(e + f*x) + S(1))/((a + b)*(c + d*cos(e + f*x))))*sqrt(-(-a*d + b*c)*(cos(e + f*x) + S(1))/((a - b)*(c + d*cos(e + f*x))))*(c + d*cos(e + f*x))*EllipticF(asin(sqrt(a + b*cos(e + f*x))*Rt((c + d)/(a + b), S(2))/sqrt(c + d*cos(e + f*x))), (a + b)*(c - d)/((a - b)*(c + d)))/(f*(-a*d + b*c)*Rt((c + d)/(a + b), S(2))*sin(e + f*x)), x)
rule2557 = ReplacementRule(pattern2557, replacement2557)
pattern2558 = Pattern(Integral(S(1)/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1356)
def replacement2558(f, b, d, a, c, x, e):
rubi.append(2558)
return Dist(sqrt(-a - b*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), Int(S(1)/(sqrt(-a - b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x)
rule2558 = ReplacementRule(pattern2558, replacement2558)
pattern2559 = Pattern(Integral(S(1)/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons1356)
def replacement2559(f, b, d, a, c, x, e):
rubi.append(2559)
return Dist(sqrt(-a - b*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), Int(S(1)/(sqrt(-a - b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x)
rule2559 = ReplacementRule(pattern2559, replacement2559)
pattern2560 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement2560(f, b, d, a, x, e):
rubi.append(2560)
return Dist(d/(S(2)*b), Int(sqrt(d*sin(e + f*x))*(a + S(2)*b*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) - Dist(a*d/(S(2)*b), Int(sqrt(d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x)
rule2560 = ReplacementRule(pattern2560, replacement2560)
pattern2561 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement2561(f, b, d, a, x, e):
rubi.append(2561)
return Dist(d/(S(2)*b), Int(sqrt(d*cos(e + f*x))*(a + S(2)*b*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) - Dist(a*d/(S(2)*b), Int(sqrt(d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x)
rule2561 = ReplacementRule(pattern2561, replacement2561)
pattern2562 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons93, cons1357, cons1358, cons1359, cons1334)
def replacement2562(m, f, b, d, c, a, n, x, e):
rubi.append(2562)
return Dist(S(1)/(d*(m + n)), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(a**S(2)*c*d*(m + n) + b*d*(a*d*n + b*c*(m + S(-1))) + b*d*(a*d*(S(2)*m + n + S(-1)) + b*c*n)*sin(e + f*x)**S(2) + (a*d*(m + n)*(a*d + S(2)*b*c) - b*d*(a*c - b*d*(m + n + S(-1))))*sin(e + f*x), x), x), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n)), x)
rule2562 = ReplacementRule(pattern2562, replacement2562)
pattern2563 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323, cons93, cons1357, cons1358, cons1359, cons1334)
def replacement2563(m, f, b, d, c, a, n, x, e):
rubi.append(2563)
return Dist(S(1)/(d*(m + n)), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(a**S(2)*c*d*(m + n) + b*d*(a*d*n + b*c*(m + S(-1))) + b*d*(a*d*(S(2)*m + n + S(-1)) + b*c*n)*cos(e + f*x)**S(2) + (a*d*(m + n)*(a*d + S(2)*b*c) - b*d*(a*c - b*d*(m + n + S(-1))))*cos(e + f*x), x), x), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n)), x)
rule2563 = ReplacementRule(pattern2563, replacement2563)
pattern2564 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons62)
def replacement2564(m, f, b, d, c, a, n, x, e):
rubi.append(2564)
return Dist(b/d, Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1)), x), x) - Dist((-a*d + b*c)/d, Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n, x), x)
rule2564 = ReplacementRule(pattern2564, replacement2564)
pattern2565 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons71, cons62)
def replacement2565(m, f, b, d, c, a, n, x, e):
rubi.append(2565)
return Dist(b/d, Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1)), x), x) - Dist((-a*d + b*c)/d, Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n, x), x)
rule2565 = ReplacementRule(pattern2565, replacement2565)
pattern2566 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1267, cons1323)
def replacement2566(m, f, b, d, c, a, n, x, e):
rubi.append(2566)
return Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x)
rule2566 = ReplacementRule(pattern2566, replacement2566)
pattern2567 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1267, cons1323)
def replacement2567(m, f, b, d, c, a, n, x, e):
rubi.append(2567)
return Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x)
rule2567 = ReplacementRule(pattern2567, replacement2567)
pattern2568 = Pattern(Integral(((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons23)
def replacement2568(p, m, f, b, d, c, a, n, x, e):
rubi.append(2568)
return Dist(c**IntPart(n)*(c*(d*sin(e + f*x))**p)**FracPart(n)*(d*sin(e + f*x))**(-p*FracPart(n)), Int((d*sin(e + f*x))**(n*p)*(a + b*sin(e + f*x))**m, x), x)
rule2568 = ReplacementRule(pattern2568, replacement2568)
pattern2569 = Pattern(Integral(((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons23)
def replacement2569(p, m, f, b, d, a, c, n, x, e):
rubi.append(2569)
return Dist(c**IntPart(n)*(c*(d*cos(e + f*x))**p)**FracPart(n)*(d*cos(e + f*x))**(-p*FracPart(n)), Int((d*cos(e + f*x))**(n*p)*(a + b*cos(e + f*x))**m, x), x)
rule2569 = ReplacementRule(pattern2569, replacement2569)
pattern2570 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons85)
def replacement2570(m, f, b, d, c, n, a, x, e):
rubi.append(2570)
return Int((a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-n), x)
rule2570 = ReplacementRule(pattern2570, replacement2570)
pattern2571 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons85)
def replacement2571(m, f, b, d, a, n, c, x, e):
rubi.append(2571)
return Int((a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-n), x)
rule2571 = ReplacementRule(pattern2571, replacement2571)
pattern2572 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons23, cons17)
def replacement2572(m, f, b, d, c, n, a, x, e):
rubi.append(2572)
return Int((c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m*(S(1)/sin(e + f*x))**(-m), x)
rule2572 = ReplacementRule(pattern2572, replacement2572)
pattern2573 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons23, cons17)
def replacement2573(m, f, b, d, a, c, n, x, e):
rubi.append(2573)
return Int((c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m*(S(1)/cos(e + f*x))**(-m), x)
rule2573 = ReplacementRule(pattern2573, replacement2573)
pattern2574 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons23, cons18)
def replacement2574(m, f, b, d, c, n, a, x, e):
rubi.append(2574)
return Dist((c + d/sin(e + f*x))**n*(c*sin(e + f*x) + d)**(-n)*sin(e + f*x)**n, Int((a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-n), x), x)
rule2574 = ReplacementRule(pattern2574, replacement2574)
pattern2575 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons23, cons18)
def replacement2575(m, f, b, d, a, c, n, x, e):
rubi.append(2575)
return Dist((c + d/cos(e + f*x))**n*(c*cos(e + f*x) + d)**(-n)*cos(e + f*x)**n, Int((a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-n), x), x)
rule2575 = ReplacementRule(pattern2575, replacement2575)
pattern2576 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement2576(m, f, b, d, c, n, a, x, e):
rubi.append(2576)
return Dist(S(1)/(b*f), Subst(Int((a + x)**m*(c + d*x/b)**n, x), x, b*sin(e + f*x)), x)
rule2576 = ReplacementRule(pattern2576, replacement2576)
pattern2577 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement2577(m, f, b, d, c, n, a, x, e):
rubi.append(2577)
return -Dist(S(1)/(b*f), Subst(Int((a + x)**m*(c + d*x/b)**n, x), x, b*cos(e + f*x)), x)
rule2577 = ReplacementRule(pattern2577, replacement2577)
pattern2578 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons5, cons1274, cons85, cons1361)
def replacement2578(p, f, b, d, a, n, x, e):
rubi.append(2578)
return Dist(a, Int((d*sin(e + f*x))**n*cos(e + f*x)**p, x), x) + Dist(b/d, Int((d*sin(e + f*x))**(n + S(1))*cos(e + f*x)**p, x), x)
rule2578 = ReplacementRule(pattern2578, replacement2578)
pattern2579 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons5, cons1274, cons85, cons1361)
def replacement2579(p, f, b, d, a, n, x, e):
rubi.append(2579)
return Dist(a, Int((d*cos(e + f*x))**n*sin(e + f*x)**p, x), x) + Dist(b/d, Int((d*cos(e + f*x))**(n + S(1))*sin(e + f*x)**p, x), x)
rule2579 = ReplacementRule(pattern2579, replacement2579)
pattern2580 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons5, cons1274, cons1265, cons85, cons1362)
def replacement2580(p, f, b, d, a, n, x, e):
rubi.append(2580)
return Dist(S(1)/a, Int((d*sin(e + f*x))**n*cos(e + f*x)**(p + S(-2)), x), x) - Dist(S(1)/(b*d), Int((d*sin(e + f*x))**(n + S(1))*cos(e + f*x)**(p + S(-2)), x), x)
rule2580 = ReplacementRule(pattern2580, replacement2580)
pattern2581 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons5, cons1274, cons1265, cons85, cons1362)
def replacement2581(p, f, b, d, a, n, x, e):
rubi.append(2581)
return Dist(S(1)/a, Int((d*cos(e + f*x))**n*sin(e + f*x)**(p + S(-2)), x), x) - Dist(S(1)/(b*d), Int((d*cos(e + f*x))**(n + S(1))*sin(e + f*x)**(p + S(-2)), x), x)
rule2581 = ReplacementRule(pattern2581, replacement2581)
pattern2582 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons48, cons125, cons7, cons27, cons21, cons4, cons1274, cons1265)
def replacement2582(p, m, f, b, d, c, n, a, x, e):
rubi.append(2582)
return Dist(b**(-p)/f, Subst(Int((a - x)**(p/S(2) + S(-1)/2)*(a + x)**(m + p/S(2) + S(-1)/2)*(c + d*x/b)**n, x), x, b*sin(e + f*x)), x)
rule2582 = ReplacementRule(pattern2582, replacement2582)
pattern2583 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons48, cons125, cons7, cons27, cons21, cons4, cons1274, cons1265)
def replacement2583(p, m, f, b, d, c, n, a, x, e):
rubi.append(2583)
return -Dist(b**(-p)/f, Subst(Int((a - x)**(p/S(2) + S(-1)/2)*(a + x)**(m + p/S(2) + S(-1)/2)*(c + d*x/b)**n, x), x, b*cos(e + f*x)), x)
rule2583 = ReplacementRule(pattern2583, replacement2583)
pattern2584 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1274, cons1267)
def replacement2584(p, m, f, b, d, c, n, a, x, e):
rubi.append(2584)
return Dist(b**(-p)/f, Subst(Int((a + x)**m*(b**S(2) - x**S(2))**(p/S(2) + S(-1)/2)*(c + d*x/b)**n, x), x, b*sin(e + f*x)), x)
rule2584 = ReplacementRule(pattern2584, replacement2584)
pattern2585 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1274, cons1267)
def replacement2585(p, m, f, b, d, c, n, a, x, e):
rubi.append(2585)
return -Dist(b**(-p)/f, Subst(Int((a + x)**m*(b**S(2) - x**S(2))**(p/S(2) + S(-1)/2)*(c + d*x/b)**n, x), x, b*cos(e + f*x)), x)
rule2585 = ReplacementRule(pattern2585, replacement2585)
pattern2586 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1363)
def replacement2586(p, f, g, b, d, a, n, x, e):
rubi.append(2586)
return Dist(a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**p, x), x) + Dist(b/d, Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**p, x), x)
rule2586 = ReplacementRule(pattern2586, replacement2586)
pattern2587 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1363)
def replacement2587(p, f, b, g, d, a, n, x, e):
rubi.append(2587)
return Dist(a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**p, x), x) + Dist(b/d, Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**p, x), x)
rule2587 = ReplacementRule(pattern2587, replacement2587)
pattern2588 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1265)
def replacement2588(p, f, g, b, d, a, n, x, e):
rubi.append(2588)
return Dist(g**S(2)/a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2)), x), x) - Dist(g**S(2)/(b*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**(p + S(-2)), x), x)
rule2588 = ReplacementRule(pattern2588, replacement2588)
pattern2589 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1265)
def replacement2589(p, f, b, g, d, a, n, x, e):
rubi.append(2589)
return Dist(g**S(2)/a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2)), x), x) - Dist(g**S(2)/(b*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**(p + S(-2)), x), x)
rule2589 = ReplacementRule(pattern2589, replacement2589)
pattern2590 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons17, cons1364)
def replacement2590(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2590)
return Dist(a**m*c**m*g**(-S(2)*m), Int((g*cos(e + f*x))**(S(2)*m + p)*(c + d*sin(e + f*x))**(-m + n), x), x)
rule2590 = ReplacementRule(pattern2590, replacement2590)
pattern2591 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons17, cons1364)
def replacement2591(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2591)
return Dist(a**m*c**m*g**(-S(2)*m), Int((g*sin(e + f*x))**(S(2)*m + p)*(c + d*cos(e + f*x))**(-m + n), x), x)
rule2591 = ReplacementRule(pattern2591, replacement2591)
pattern2592 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons70, cons1265, cons1306)
def replacement2592(p, m, f, b, d, a, n, c, x, e):
rubi.append(2592)
return Dist(a**(-p/S(2))*c**(-p/S(2)), Int((a + b*sin(e + f*x))**(m + p/S(2))*(c + d*sin(e + f*x))**(n + p/S(2)), x), x)
rule2592 = ReplacementRule(pattern2592, replacement2592)
pattern2593 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons70, cons1265, cons1306)
def replacement2593(p, m, f, b, d, a, n, c, x, e):
rubi.append(2593)
return Dist(a**(-p/S(2))*c**(-p/S(2)), Int((a + b*cos(e + f*x))**(m + p/S(2))*(c + d*cos(e + f*x))**(n + p/S(2)), x), x)
rule2593 = ReplacementRule(pattern2593, replacement2593)
pattern2594 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons70, cons1265)
def replacement2594(p, f, b, g, d, a, c, x, e):
rubi.append(2594)
return Dist(g*cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), Int((g*cos(e + f*x))**(p + S(-1)), x), x)
rule2594 = ReplacementRule(pattern2594, replacement2594)
pattern2595 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons70, cons1265)
def replacement2595(p, f, b, g, d, a, c, x, e):
rubi.append(2595)
return Dist(g*sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), Int((g*sin(e + f*x))**(p + S(-1)), x), x)
rule2595 = ReplacementRule(pattern2595, replacement2595)
pattern2596 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1282, cons142)
def replacement2596(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2596)
return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*cos(e + f*x))**(-S(2)*FracPart(m))*(a + b*sin(e + f*x))**FracPart(m)*(c + d*sin(e + f*x))**FracPart(m), Int((g*cos(e + f*x))**(S(2)*m + p)/(c + d*sin(e + f*x)), x), x)
rule2596 = ReplacementRule(pattern2596, replacement2596)
pattern2597 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1282, cons142)
def replacement2597(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2597)
return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*sin(e + f*x))**(-S(2)*FracPart(m))*(a + b*cos(e + f*x))**FracPart(m)*(c + d*cos(e + f*x))**FracPart(m), Int((g*sin(e + f*x))**(S(2)*m + p)/(c + d*cos(e + f*x)), x), x)
rule2597 = ReplacementRule(pattern2597, replacement2597)
pattern2598 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1282, cons335)
def replacement2598(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2598)
return Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(m - n + S(-1))), x)
rule2598 = ReplacementRule(pattern2598, replacement2598)
pattern2599 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1282, cons335)
def replacement2599(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2599)
return -Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(m - n + S(-1))), x)
rule2599 = ReplacementRule(pattern2599, replacement2599)
pattern2600 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons70, cons1265, cons1284, cons87, cons89, cons1365, cons1366)
def replacement2600(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2600)
return -Dist(b*(S(2)*m + p + S(-1))/(d*(S(2)*n + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(S(2)*n + p + S(1))), x)
rule2600 = ReplacementRule(pattern2600, replacement2600)
pattern2601 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons70, cons1265, cons1284, cons87, cons89, cons1365, cons1366)
def replacement2601(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2601)
return -Dist(b*(S(2)*m + p + S(-1))/(d*(S(2)*n + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(S(2)*n + p + S(1))), x)
rule2601 = ReplacementRule(pattern2601, replacement2601)
pattern2602 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons1284, cons346, cons1367, cons1366)
def replacement2602(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2602)
return Dist(a*(S(2)*m + p + S(-1))/(m + n + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n, x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(m + n + p)), x)
rule2602 = ReplacementRule(pattern2602, replacement2602)
pattern2603 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons1284, cons346, cons1367, cons1366)
def replacement2603(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2603)
return Dist(a*(S(2)*m + p + S(-1))/(m + n + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n, x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(m + n + p)), x)
rule2603 = ReplacementRule(pattern2603, replacement2603)
pattern2604 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons70, cons1265, cons1368)
def replacement2604(p, m, f, b, g, d, a, c, x, e):
rubi.append(2604)
return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*cos(e + f*x))**(-S(2)*FracPart(m))*(a + b*sin(e + f*x))**FracPart(m)*(c + d*sin(e + f*x))**FracPart(m), Int((g*cos(e + f*x))**(S(2)*m + p), x), x)
rule2604 = ReplacementRule(pattern2604, replacement2604)
pattern2605 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons70, cons1265, cons1368)
def replacement2605(p, m, f, b, g, d, a, c, x, e):
rubi.append(2605)
return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*sin(e + f*x))**(-S(2)*FracPart(m))*(a + b*cos(e + f*x))**FracPart(m)*(c + d*cos(e + f*x))**FracPart(m), Int((g*sin(e + f*x))**(S(2)*m + p), x), x)
rule2605 = ReplacementRule(pattern2605, replacement2605)
pattern2606 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1369, cons1260)
def replacement2606(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2606)
return Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n/(a*f*g*(m - n)), x)
rule2606 = ReplacementRule(pattern2606, replacement2606)
pattern2607 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1369, cons1260)
def replacement2607(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2607)
return -Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n/(a*f*g*(m - n)), x)
rule2607 = ReplacementRule(pattern2607, replacement2607)
pattern2608 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1370, cons1281, cons114)
def replacement2608(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2608)
return Dist((m + n + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n/(a*f*g*(S(2)*m + p + S(1))), x)
rule2608 = ReplacementRule(pattern2608, replacement2608)
pattern2609 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1370, cons1281, cons114)
def replacement2609(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2609)
return Dist((m + n + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n/(a*f*g*(S(2)*m + p + S(1))), x)
rule2609 = ReplacementRule(pattern2609, replacement2609)
pattern2610 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons70, cons1265, cons93, cons168, cons89, cons1365, cons170)
def replacement2610(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2610)
return -Dist(b*(S(2)*m + p + S(-1))/(d*(S(2)*n + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(S(2)*n + p + S(1))), x)
rule2610 = ReplacementRule(pattern2610, replacement2610)
pattern2611 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons70, cons1265, cons93, cons168, cons89, cons1365, cons170)
def replacement2611(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2611)
return -Dist(b*(S(2)*m + p + S(-1))/(d*(S(2)*n + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(S(2)*n + p + S(1))), x)
rule2611 = ReplacementRule(pattern2611, replacement2611)
pattern2612 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons31, cons168, cons1371, cons1372, cons170)
def replacement2612(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2612)
return Dist(a*(S(2)*m + p + S(-1))/(m + n + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n, x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(m + n + p)), x)
rule2612 = ReplacementRule(pattern2612, replacement2612)
pattern2613 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons31, cons168, cons1371, cons1372, cons170)
def replacement2613(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2613)
return Dist(a*(S(2)*m + p + S(-1))/(m + n + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n, x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(m + n + p)), x)
rule2613 = ReplacementRule(pattern2613, replacement2613)
pattern2614 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons31, cons94, cons1281, cons1316, cons170)
def replacement2614(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2614)
return Dist((m + n + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n/(a*f*g*(S(2)*m + p + S(1))), x)
rule2614 = ReplacementRule(pattern2614, replacement2614)
pattern2615 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons31, cons94, cons1281, cons1316, cons170)
def replacement2615(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2615)
return Dist((m + n + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n/(a*f*g*(S(2)*m + p + S(1))), x)
rule2615 = ReplacementRule(pattern2615, replacement2615)
pattern2616 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1317)
def replacement2616(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2616)
return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*cos(e + f*x))**(-S(2)*FracPart(m))*(a + b*sin(e + f*x))**FracPart(m)*(c + d*sin(e + f*x))**FracPart(m), Int((g*cos(e + f*x))**(S(2)*m + p)*(c + d*sin(e + f*x))**(-m + n), x), x)
rule2616 = ReplacementRule(pattern2616, replacement2616)
pattern2617 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265, cons1317)
def replacement2617(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2617)
return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*sin(e + f*x))**(-S(2)*FracPart(m))*(a + b*cos(e + f*x))**FracPart(m)*(c + d*cos(e + f*x))**FracPart(m), Int((g*sin(e + f*x))**(S(2)*m + p)*(c + d*cos(e + f*x))**(-m + n), x), x)
rule2617 = ReplacementRule(pattern2617, replacement2617)
pattern2618 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons1265, cons1373)
def replacement2618(p, m, f, b, g, d, c, a, x, e):
rubi.append(2618)
return -Simp(d*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(f*g*(m + p + S(1))), x)
rule2618 = ReplacementRule(pattern2618, replacement2618)
pattern2619 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons1265, cons1373)
def replacement2619(p, m, f, b, g, d, c, a, x, e):
rubi.append(2619)
return Simp(d*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(f*g*(m + p + S(1))), x)
rule2619 = ReplacementRule(pattern2619, replacement2619)
pattern2620 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1265, cons244, cons1374, cons137)
def replacement2620(p, m, f, b, g, d, c, a, x, e):
rubi.append(2620)
return Dist(b*(a*d*m + b*c*(m + p + S(1)))/(a*g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(a*d + b*c)/(a*f*g*(p + S(1))), x)
rule2620 = ReplacementRule(pattern2620, replacement2620)
pattern2621 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1265, cons244, cons1374, cons137)
def replacement2621(p, m, f, b, g, d, c, a, x, e):
rubi.append(2621)
return Dist(b*(a*d*m + b*c*(m + p + S(1)))/(a*g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(a*d + b*c)/(a*f*g*(p + S(1))), x)
rule2621 = ReplacementRule(pattern2621, replacement2621)
pattern2622 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons1265, cons1375, cons253)
def replacement2622(p, m, f, b, g, d, c, a, x, e):
rubi.append(2622)
return Dist((a*d*m + b*c*(m + p + S(1)))/(b*(m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m, x), x) - Simp(d*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(f*g*(m + p + S(1))), x)
rule2622 = ReplacementRule(pattern2622, replacement2622)
pattern2623 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons1265, cons1375, cons253)
def replacement2623(p, m, f, b, g, d, c, a, x, e):
rubi.append(2623)
return Dist((a*d*m + b*c*(m + p + S(1)))/(b*(m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m, x), x) + Simp(d*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(f*g*(m + p + S(1))), x)
rule2623 = ReplacementRule(pattern2623, replacement2623)
pattern2624 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons31, cons1376)
def replacement2624(m, f, b, d, c, a, x, e):
rubi.append(2624)
return Dist(S(1)/(b**S(3)*(S(2)*m + S(3))), Int((a + b*sin(e + f*x))**(m + S(2))*(S(2)*a*d*(m + S(1)) + b*c - b*d*(S(2)*m + S(3))*sin(e + f*x)), x), x) + Simp(S(2)*(a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c)*cos(e + f*x)/(b**S(2)*f*(S(2)*m + S(3))), x)
rule2624 = ReplacementRule(pattern2624, replacement2624)
pattern2625 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons31, cons1376)
def replacement2625(m, f, b, d, c, a, x, e):
rubi.append(2625)
return Dist(S(1)/(b**S(3)*(S(2)*m + S(3))), Int((a + b*cos(e + f*x))**(m + S(2))*(S(2)*a*d*(m + S(1)) + b*c - b*d*(S(2)*m + S(3))*cos(e + f*x)), x), x) + Simp(-S(2)*(a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c)*sin(e + f*x)/(b**S(2)*f*(S(2)*m + S(3))), x)
rule2625 = ReplacementRule(pattern2625, replacement2625)
pattern2626 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons31, cons1377)
def replacement2626(m, f, b, d, c, a, x, e):
rubi.append(2626)
return -Dist(S(1)/(b**S(2)*(m + S(3))), Int((a + b*sin(e + f*x))**(m + S(1))*(-a*c*(m + S(3)) + b*d*(m + S(2)) + (-a*d*(m + S(4)) + b*c*(m + S(3)))*sin(e + f*x)), x), x) + Simp(d*(a + b*sin(e + f*x))**(m + S(2))*cos(e + f*x)/(b**S(2)*f*(m + S(3))), x)
rule2626 = ReplacementRule(pattern2626, replacement2626)
pattern2627 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1265, cons31, cons1377)
def replacement2627(m, f, b, d, c, a, x, e):
rubi.append(2627)
return -Dist(S(1)/(b**S(2)*(m + S(3))), Int((a + b*cos(e + f*x))**(m + S(1))*(-a*c*(m + S(3)) + b*d*(m + S(2)) + (-a*d*(m + S(4)) + b*c*(m + S(3)))*cos(e + f*x)), x), x) - Simp(d*(a + b*cos(e + f*x))**(m + S(2))*sin(e + f*x)/(b**S(2)*f*(m + S(3))), x)
rule2627 = ReplacementRule(pattern2627, replacement2627)
pattern2628 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons1265, cons1378, cons1281)
def replacement2628(p, m, f, b, g, d, c, a, x, e):
rubi.append(2628)
return Dist((a*d*m + b*c*(m + p + S(1)))/(a*b*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(-a*d + b*c)/(a*f*g*(S(2)*m + p + S(1))), x)
rule2628 = ReplacementRule(pattern2628, replacement2628)
pattern2629 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons1265, cons1378, cons1281)
def replacement2629(p, m, f, b, g, d, c, a, x, e):
rubi.append(2629)
return Dist((a*d*m + b*c*(m + p + S(1)))/(a*b*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(-a*d + b*c)/(a*f*g*(S(2)*m + p + S(1))), x)
rule2629 = ReplacementRule(pattern2629, replacement2629)
pattern2630 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons1265, cons253)
def replacement2630(p, m, f, b, g, d, c, a, x, e):
rubi.append(2630)
return Dist((a*d*m + b*c*(m + p + S(1)))/(b*(m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m, x), x) - Simp(d*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(f*g*(m + p + S(1))), x)
rule2630 = ReplacementRule(pattern2630, replacement2630)
pattern2631 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons1265, cons253)
def replacement2631(p, m, f, b, g, d, c, a, x, e):
rubi.append(2631)
return Dist((a*d*m + b*c*(m + p + S(1)))/(b*(m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m, x), x) + Simp(d*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(f*g*(m + p + S(1))), x)
rule2631 = ReplacementRule(pattern2631, replacement2631)
pattern2632 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1267, cons244, cons168, cons137, cons515)
def replacement2632(p, m, f, b, g, d, c, a, x, e):
rubi.append(2632)
return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1))*Simp(a*c*(p + S(2)) + b*c*(m + p + S(2))*sin(e + f*x) + b*d*m, x), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)/(f*g*(p + S(1))), x)
rule2632 = ReplacementRule(pattern2632, replacement2632)
pattern2633 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1267, cons244, cons168, cons137, cons515)
def replacement2633(p, m, f, b, g, d, c, a, x, e):
rubi.append(2633)
return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1))*Simp(a*c*(p + S(2)) + b*c*(m + p + S(2))*cos(e + f*x) + b*d*m, x), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)/(f*g*(p + S(1))), x)
rule2633 = ReplacementRule(pattern2633, replacement2633)
pattern2634 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons1267, cons31, cons168, cons1379, cons515)
def replacement2634(p, m, f, b, g, d, c, a, x, e):
rubi.append(2634)
return Dist(S(1)/(m + p + S(1)), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*Simp(a*c*(m + p + S(1)) + b*d*m + (a*d*m + b*c*(m + p + S(1)))*sin(e + f*x), x), x), x) - Simp(d*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(f*g*(m + p + S(1))), x)
rule2634 = ReplacementRule(pattern2634, replacement2634)
pattern2635 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons1267, cons31, cons168, cons1379, cons515)
def replacement2635(p, m, f, b, g, d, c, a, x, e):
rubi.append(2635)
return Dist(S(1)/(m + p + S(1)), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*Simp(a*c*(m + p + S(1)) + b*d*m + (a*d*m + b*c*(m + p + S(1)))*cos(e + f*x), x), x), x) + Simp(d*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(f*g*(m + p + S(1))), x)
rule2635 = ReplacementRule(pattern2635, replacement2635)
pattern2636 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1267, cons244, cons94, cons146, cons253, cons515)
def replacement2636(p, m, f, b, g, d, c, a, x, e):
rubi.append(2636)
return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(m + S(1))*(m + p + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1))*Simp(b*d*(m + S(1)) + (-a*d*p + b*c*(m + p + S(1)))*sin(e + f*x), x), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*d*p + b*c*(m + p + S(1)) + b*d*(m + S(1))*sin(e + f*x))/(b**S(2)*f*(m + S(1))*(m + p + S(1))), x)
rule2636 = ReplacementRule(pattern2636, replacement2636)
pattern2637 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1267, cons244, cons94, cons146, cons253, cons515)
def replacement2637(p, m, f, b, g, d, c, a, x, e):
rubi.append(2637)
return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(m + S(1))*(m + p + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1))*Simp(b*d*(m + S(1)) + (-a*d*p + b*c*(m + p + S(1)))*cos(e + f*x), x), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*d*p + b*c*(m + p + S(1)) + b*d*(m + S(1))*cos(e + f*x))/(b**S(2)*f*(m + S(1))*(m + p + S(1))), x)
rule2637 = ReplacementRule(pattern2637, replacement2637)
pattern2638 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons1267, cons31, cons94, cons515)
def replacement2638(p, m, f, b, g, d, c, a, x, e):
rubi.append(2638)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*Simp((m + S(1))*(a*c - b*d) - (-a*d + b*c)*(m + p + S(2))*sin(e + f*x), x), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c)/(f*g*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2638 = ReplacementRule(pattern2638, replacement2638)
pattern2639 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons1267, cons31, cons94, cons515)
def replacement2639(p, m, f, b, g, d, c, a, x, e):
rubi.append(2639)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*Simp((m + S(1))*(a*c - b*d) - (-a*d + b*c)*(m + p + S(2))*cos(e + f*x), x), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c)/(f*g*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2639 = ReplacementRule(pattern2639, replacement2639)
pattern2640 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons1267, cons13, cons146, cons1285, cons253, cons515)
def replacement2640(p, m, f, b, g, d, c, a, x, e):
rubi.append(2640)
return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(m + p)*(m + p + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**m*Simp(b*(a*d*m + b*c*(m + p + S(1))) + (a*b*c*(m + p + S(1)) - d*(a**S(2)*p - b**S(2)*(m + p)))*sin(e + f*x), x), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*d*p + b*c*(m + p + S(1)) + b*d*(m + p)*sin(e + f*x))/(b**S(2)*f*(m + p)*(m + p + S(1))), x)
rule2640 = ReplacementRule(pattern2640, replacement2640)
pattern2641 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons1267, cons13, cons146, cons1285, cons253, cons515)
def replacement2641(p, m, f, b, g, d, c, a, x, e):
rubi.append(2641)
return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(m + p)*(m + p + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**m*Simp(b*(a*d*m + b*c*(m + p + S(1))) + (a*b*c*(m + p + S(1)) - d*(a**S(2)*p - b**S(2)*(m + p)))*cos(e + f*x), x), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*d*p + b*c*(m + p + S(1)) + b*d*(m + p)*cos(e + f*x))/(b**S(2)*f*(m + p)*(m + p + S(1))), x)
rule2641 = ReplacementRule(pattern2641, replacement2641)
pattern2642 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons1267, cons13, cons137, cons515)
def replacement2642(p, m, f, b, g, d, c, a, x, e):
rubi.append(2642)
return Dist(S(1)/(g**S(2)*(a**S(2) - b**S(2))*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m*Simp(a*b*d*m + b*(a*c - b*d)*(m + p + S(3))*sin(e + f*x) + c*(a**S(2)*(p + S(2)) - b**S(2)*(m + p + S(2))), x), x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c - (a*c - b*d)*sin(e + f*x))/(f*g*(a**S(2) - b**S(2))*(p + S(1))), x)
rule2642 = ReplacementRule(pattern2642, replacement2642)
pattern2643 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons1267, cons13, cons137, cons515)
def replacement2643(p, m, f, b, g, d, c, a, x, e):
rubi.append(2643)
return Dist(S(1)/(g**S(2)*(a**S(2) - b**S(2))*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m*Simp(a*b*d*m + b*(a*c - b*d)*(m + p + S(3))*cos(e + f*x) + c*(a**S(2)*(p + S(2)) - b**S(2)*(m + p + S(2))), x), x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c - (a*c - b*d)*cos(e + f*x))/(f*g*(a**S(2) - b**S(2))*(p + S(1))), x)
rule2643 = ReplacementRule(pattern2643, replacement2643)
pattern2644 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1267)
def replacement2644(p, f, b, g, d, c, a, x, e):
rubi.append(2644)
return Dist(d/b, Int((g*cos(e + f*x))**p, x), x) + Dist((-a*d + b*c)/b, Int((g*cos(e + f*x))**p/(a + b*sin(e + f*x)), x), x)
rule2644 = ReplacementRule(pattern2644, replacement2644)
pattern2645 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1267)
def replacement2645(p, f, b, g, d, c, a, x, e):
rubi.append(2645)
return Dist(d/b, Int((g*sin(e + f*x))**p, x), x) + Dist((-a*d + b*c)/b, Int((g*sin(e + f*x))**p/(a + b*cos(e + f*x)), x), x)
rule2645 = ReplacementRule(pattern2645, replacement2645)
pattern2646 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1267, cons1322)
def replacement2646(p, m, f, b, g, d, a, c, x, e):
rubi.append(2646)
return Dist(c*g*(g*cos(e + f*x))**(p + S(-1))*(-sin(e + f*x) + S(1))**(-p/S(2) + S(1)/2)*(sin(e + f*x) + S(1))**(-p/S(2) + S(1)/2)/f, Subst(Int((S(1) - d*x/c)**(p/S(2) + S(-1)/2)*(S(1) + d*x/c)**(p/S(2) + S(1)/2)*(a + b*x)**m, x), x, sin(e + f*x)), x)
rule2646 = ReplacementRule(pattern2646, replacement2646)
pattern2647 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons1267, cons1322)
def replacement2647(p, m, f, b, g, d, a, c, x, e):
rubi.append(2647)
return -Dist(c*g*(g*sin(e + f*x))**(p + S(-1))*(-cos(e + f*x) + S(1))**(-p/S(2) + S(1)/2)*(cos(e + f*x) + S(1))**(-p/S(2) + S(1)/2)/f, Subst(Int((S(1) - d*x/c)**(p/S(2) + S(-1)/2)*(S(1) + d*x/c)**(p/S(2) + S(1)/2)*(a + b*x)**m, x), x, cos(e + f*x)), x)
rule2647 = ReplacementRule(pattern2647, replacement2647)
pattern2648 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons1299, cons1380)
def replacement2648(p, m, f, b, d, a, n, x, e):
rubi.append(2648)
return Dist(a**(S(2)*m), Int((d*sin(e + f*x))**n*(a - b*sin(e + f*x))**(-m), x), x)
rule2648 = ReplacementRule(pattern2648, replacement2648)
pattern2649 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons1299, cons1380)
def replacement2649(p, m, f, b, d, a, n, x, e):
rubi.append(2649)
return Dist(a**(S(2)*m), Int((d*cos(e + f*x))**n*(a - b*cos(e + f*x))**(-m), x), x)
rule2649 = ReplacementRule(pattern2649, replacement2649)
pattern2650 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons358)
def replacement2650(p, m, f, b, g, a, x, e):
rubi.append(2650)
return Dist(a/(S(2)*g**S(2)), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(S(2)*b*f*g*(m + S(1))), x)
rule2650 = ReplacementRule(pattern2650, replacement2650)
pattern2651 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons358)
def replacement2651(p, m, f, b, g, a, x, e):
rubi.append(2651)
return Dist(a/(S(2)*g**S(2)), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(S(2)*b*f*g*(m + S(1))), x)
rule2651 = ReplacementRule(pattern2651, replacement2651)
pattern2652 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1278)
def replacement2652(p, m, f, b, g, a, x, e):
rubi.append(2652)
return -Dist(g**(S(-2)), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m, x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*m), x)
rule2652 = ReplacementRule(pattern2652, replacement2652)
pattern2653 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1278)
def replacement2653(p, m, f, b, g, a, x, e):
rubi.append(2653)
return -Dist(g**(S(-2)), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m, x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*m), x)
rule2653 = ReplacementRule(pattern2653, replacement2653)
pattern2654 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1381, cons1382)
def replacement2654(p, m, f, b, d, a, n, x, e):
rubi.append(2654)
return Dist(a**(-p), Int(ExpandTrig((d*sin(e + f*x))**n*(a - b*sin(e + f*x))**(p/S(2))*(a + b*sin(e + f*x))**(m + p/S(2)), x), x), x)
rule2654 = ReplacementRule(pattern2654, replacement2654)
pattern2655 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1381, cons1382)
def replacement2655(p, m, f, b, d, a, n, x, e):
rubi.append(2655)
return Dist(a**(-p), Int(ExpandTrig((d*cos(e + f*x))**n*(a - b*cos(e + f*x))**(p/S(2))*(a + b*cos(e + f*x))**(m + p/S(2)), x), x), x)
rule2655 = ReplacementRule(pattern2655, replacement2655)
pattern2656 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1265, cons62)
def replacement2656(p, m, f, b, g, d, a, n, x, e):
rubi.append(2656)
return Int(ExpandTrig((g*cos(e + f*x))**p, (d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x)
rule2656 = ReplacementRule(pattern2656, replacement2656)
pattern2657 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1265, cons62)
def replacement2657(p, m, f, b, g, d, a, n, x, e):
rubi.append(2657)
return Int(ExpandTrig((g*sin(e + f*x))**p, (d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x)
rule2657 = ReplacementRule(pattern2657, replacement2657)
pattern2658 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1383)
def replacement2658(m, f, b, d, a, n, x, e):
rubi.append(2658)
return Dist(b**(S(-2)), Int((d*sin(e + f*x))**n*(a - b*sin(e + f*x))*(a + b*sin(e + f*x))**(m + S(1)), x), x)
rule2658 = ReplacementRule(pattern2658, replacement2658)
pattern2659 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1383)
def replacement2659(m, f, b, d, a, n, x, e):
rubi.append(2659)
return Dist(b**(S(-2)), Int((d*cos(e + f*x))**n*(a - b*cos(e + f*x))*(a + b*cos(e + f*x))**(m + S(1)), x), x)
rule2659 = ReplacementRule(pattern2659, replacement2659)
pattern2660 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1265, cons84)
def replacement2660(p, m, f, b, g, d, a, n, x, e):
rubi.append(2660)
return Dist((a/g)**(S(2)*m), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(S(2)*m + p)*(a - b*sin(e + f*x))**(-m), x), x)
rule2660 = ReplacementRule(pattern2660, replacement2660)
pattern2661 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1265, cons84)
def replacement2661(p, m, f, b, g, d, a, n, x, e):
rubi.append(2661)
return Dist((a/g)**(S(2)*m), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(S(2)*m + p)*(a - b*cos(e + f*x))**(-m), x), x)
rule2661 = ReplacementRule(pattern2661, replacement2661)
pattern2662 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons1265, cons17, cons13, cons1384)
def replacement2662(p, m, f, b, g, d, a, n, x, e):
rubi.append(2662)
return Dist((a/g)**(S(2)*m), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(S(2)*m + p)*(a - b*sin(e + f*x))**(-m), x), x)
rule2662 = ReplacementRule(pattern2662, replacement2662)
pattern2663 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons1265, cons17, cons13, cons1384)
def replacement2663(p, m, f, b, g, d, a, n, x, e):
rubi.append(2663)
return Dist((a/g)**(S(2)*m), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(S(2)*m + p)*(a - b*cos(e + f*x))**(-m), x), x)
rule2663 = ReplacementRule(pattern2663, replacement2663)
pattern2664 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons31, cons1385, cons1281)
def replacement2664(p, m, f, b, g, a, x, e):
rubi.append(2664)
return -Dist(S(1)/(a**S(2)*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + p + S(1))*sin(e + f*x)), x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x)
rule2664 = ReplacementRule(pattern2664, replacement2664)
pattern2665 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1265, cons31, cons1385, cons1281)
def replacement2665(p, m, f, b, g, a, x, e):
rubi.append(2665)
return -Dist(S(1)/(a**S(2)*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + p + S(1))*cos(e + f*x)), x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x)
rule2665 = ReplacementRule(pattern2665, replacement2665)
pattern2666 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1386)
def replacement2666(p, m, f, b, g, a, x, e):
rubi.append(2666)
return Dist(S(1)/(b*(m + p + S(2))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m*(-a*(p + S(1))*sin(e + f*x) + b*(m + S(1))), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*g*(m + p + S(2))), x)
rule2666 = ReplacementRule(pattern2666, replacement2666)
pattern2667 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1265, cons1386)
def replacement2667(p, m, f, b, g, a, x, e):
rubi.append(2667)
return Dist(S(1)/(b*(m + p + S(2))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m*(-a*(p + S(1))*cos(e + f*x) + b*(m + S(1))), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*g*(m + p + S(2))), x)
rule2667 = ReplacementRule(pattern2667, replacement2667)
pattern2668 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1170)
def replacement2668(m, f, b, d, a, n, x, e):
rubi.append(2668)
return Dist(b**(S(-2)), Int((d*sin(e + f*x))**n*(a - b*sin(e + f*x))*(a + b*sin(e + f*x))**(m + S(1)), x), x)
rule2668 = ReplacementRule(pattern2668, replacement2668)
pattern2669 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1170)
def replacement2669(m, f, b, d, a, n, x, e):
rubi.append(2669)
return Dist(b**(S(-2)), Int((d*cos(e + f*x))**n*(a - b*cos(e + f*x))*(a + b*cos(e + f*x))**(m + S(1)), x), x)
rule2669 = ReplacementRule(pattern2669, replacement2669)
pattern2670 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons31, cons94)
def replacement2670(m, f, b, d, a, n, x, e):
rubi.append(2670)
return Dist(a**(S(-2)), Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**(m + S(2))*(sin(e + f*x)**S(2) + S(1)), x), x) + Dist(-S(2)/(a*b*d), Int((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(2)), x), x)
rule2670 = ReplacementRule(pattern2670, replacement2670)
pattern2671 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons31, cons94)
def replacement2671(m, f, b, d, a, n, x, e):
rubi.append(2671)
return Dist(a**(S(-2)), Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**(m + S(2))*(cos(e + f*x)**S(2) + S(1)), x), x) + Dist(-S(2)/(a*b*d), Int((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(2)), x), x)
rule2671 = ReplacementRule(pattern2671, replacement2671)
pattern2672 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons143)
def replacement2672(m, f, b, d, a, n, x, e):
rubi.append(2672)
return Dist(d**(S(-4)), Int((d*sin(e + f*x))**(n + S(4))*(a + b*sin(e + f*x))**m, x), x) + Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m*(-S(2)*sin(e + f*x)**S(2) + S(1)), x)
rule2672 = ReplacementRule(pattern2672, replacement2672)
pattern2673 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons143)
def replacement2673(m, f, b, d, a, n, x, e):
rubi.append(2673)
return Dist(d**(S(-4)), Int((d*cos(e + f*x))**(n + S(4))*(a + b*cos(e + f*x))**m, x), x) + Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m*(-S(2)*cos(e + f*x)**S(2) + S(1)), x)
rule2673 = ReplacementRule(pattern2673, replacement2673)
pattern2674 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons1306, cons17)
def replacement2674(p, m, f, b, d, a, n, x, e):
rubi.append(2674)
return Dist(a**m*cos(e + f*x)/(f*sqrt(-sin(e + f*x) + S(1))*sqrt(sin(e + f*x) + S(1))), Subst(Int((d*x)**n*(S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule2674 = ReplacementRule(pattern2674, replacement2674)
pattern2675 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons1306, cons17)
def replacement2675(p, m, f, b, d, a, n, x, e):
rubi.append(2675)
return -Dist(a**m*sin(e + f*x)/(f*sqrt(-cos(e + f*x) + S(1))*sqrt(cos(e + f*x) + S(1))), Subst(Int((d*x)**n*(S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule2675 = ReplacementRule(pattern2675, replacement2675)
pattern2676 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1306, cons18)
def replacement2676(p, m, f, b, d, a, n, x, e):
rubi.append(2676)
return Dist(a**(-p + S(2))*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((d*x)**n*(a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule2676 = ReplacementRule(pattern2676, replacement2676)
pattern2677 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1306, cons18)
def replacement2677(p, m, f, b, d, a, n, x, e):
rubi.append(2677)
return -Dist(a**(-p + S(2))*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((d*x)**n*(a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule2677 = ReplacementRule(pattern2677, replacement2677)
pattern2678 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1265, cons62, cons1387)
def replacement2678(p, m, f, b, g, d, a, n, x, e):
rubi.append(2678)
return Int(ExpandTrig((g*cos(e + f*x))**p, (d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x)
rule2678 = ReplacementRule(pattern2678, replacement2678)
pattern2679 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1265, cons62, cons1387)
def replacement2679(p, m, f, b, g, d, a, n, x, e):
rubi.append(2679)
return Int(ExpandTrig((g*sin(e + f*x))**p, (d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x)
rule2679 = ReplacementRule(pattern2679, replacement2679)
pattern2680 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons5, cons1265, cons17)
def replacement2680(p, m, f, b, g, d, a, n, x, e):
rubi.append(2680)
return Dist(a**m*g*(g*cos(e + f*x))**(p + S(-1))*(-sin(e + f*x) + S(1))**(-p/S(2) + S(1)/2)*(sin(e + f*x) + S(1))**(-p/S(2) + S(1)/2)/f, Subst(Int((d*x)**n*(S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule2680 = ReplacementRule(pattern2680, replacement2680)
pattern2681 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons5, cons1265, cons17)
def replacement2681(p, m, f, b, g, d, a, n, x, e):
rubi.append(2681)
return -Dist(a**m*g*(g*sin(e + f*x))**(p + S(-1))*(-cos(e + f*x) + S(1))**(-p/S(2) + S(1)/2)*(cos(e + f*x) + S(1))**(-p/S(2) + S(1)/2)/f, Subst(Int((d*x)**n*(S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule2681 = ReplacementRule(pattern2681, replacement2681)
pattern2682 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons5, cons1265, cons18)
def replacement2682(p, m, f, b, g, d, a, n, x, e):
rubi.append(2682)
return Dist(g*(g*cos(e + f*x))**(p + S(-1))*(a - b*sin(e + f*x))**(-p/S(2) + S(1)/2)*(a + b*sin(e + f*x))**(-p/S(2) + S(1)/2)/f, Subst(Int((d*x)**n*(a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x)
rule2682 = ReplacementRule(pattern2682, replacement2682)
pattern2683 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons5, cons1265, cons18)
def replacement2683(p, m, f, b, g, d, a, n, x, e):
rubi.append(2683)
return -Dist(g*(g*sin(e + f*x))**(p + S(-1))*(a - b*cos(e + f*x))**(-p/S(2) + S(1)/2)*(a + b*cos(e + f*x))**(-p/S(2) + S(1)/2)/f, Subst(Int((d*x)**n*(a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x)
rule2683 = ReplacementRule(pattern2683, replacement2683)
pattern2684 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons31, cons94, cons1388)
def replacement2684(p, m, f, b, g, d, a, x, e):
rubi.append(2684)
return Dist(g**S(2)*(S(2)*m + S(3))/(S(2)*a*(m + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1))/sqrt(d*sin(e + f*x)), x), x) - Simp(g*sqrt(d*sin(e + f*x))*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(a*d*f*(m + S(1))), x)
rule2684 = ReplacementRule(pattern2684, replacement2684)
pattern2685 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons31, cons94, cons1388)
def replacement2685(p, m, f, b, g, d, a, x, e):
rubi.append(2685)
return Dist(g**S(2)*(S(2)*m + S(3))/(S(2)*a*(m + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1))/sqrt(d*cos(e + f*x)), x), x) + Simp(g*sqrt(d*cos(e + f*x))*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(a*d*f*(m + S(1))), x)
rule2685 = ReplacementRule(pattern2685, replacement2685)
pattern2686 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons31, cons168, cons1389)
def replacement2686(p, m, f, b, g, d, a, x, e):
rubi.append(2686)
return Dist(S(2)*a*m/(g**S(2)*(S(2)*m + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1))/sqrt(d*sin(e + f*x)), x), x) + Simp(S(2)*sqrt(d*sin(e + f*x))*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(d*f*g*(S(2)*m + S(1))), x)
rule2686 = ReplacementRule(pattern2686, replacement2686)
pattern2687 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons1267, cons31, cons168, cons1389)
def replacement2687(p, m, f, b, g, d, a, x, e):
rubi.append(2687)
return Dist(S(2)*a*m/(g**S(2)*(S(2)*m + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1))/sqrt(d*cos(e + f*x)), x), x) + Simp(-S(2)*sqrt(d*cos(e + f*x))*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(d*f*g*(S(2)*m + S(1))), x)
rule2687 = ReplacementRule(pattern2687, replacement2687)
pattern2688 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1267, cons1390)
def replacement2688(m, f, b, d, a, n, x, e):
rubi.append(2688)
return Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m*(-sin(e + f*x)**S(2) + S(1)), x)
rule2688 = ReplacementRule(pattern2688, replacement2688)
pattern2689 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1267, cons1390)
def replacement2689(m, f, b, d, a, n, x, e):
rubi.append(2689)
return Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m*(-cos(e + f*x)**S(2) + S(1)), x)
rule2689 = ReplacementRule(pattern2689, replacement2689)
pattern2690 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1170, cons94, cons89)
def replacement2690(m, f, b, d, a, n, x, e):
rubi.append(2690)
return Dist(S(1)/(a**S(2)*b*d*(m + S(1))*(n + S(1))), Int((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(1))*(n + S(2)) + a*b*(m + S(1))*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(1))*(n + S(3)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(a*d*f*(n + S(1))), x) - Simp((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*(a**S(2)*(n + S(1)) - b**S(2)*(m + n + S(2)))*cos(e + f*x)/(a**S(2)*b*d**S(2)*f*(m + S(1))*(n + S(1))), x)
rule2690 = ReplacementRule(pattern2690, replacement2690)
pattern2691 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1170, cons94, cons89)
def replacement2691(m, f, b, d, a, n, x, e):
rubi.append(2691)
return Dist(S(1)/(a**S(2)*b*d*(m + S(1))*(n + S(1))), Int((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(1))*(n + S(2)) + a*b*(m + S(1))*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(1))*(n + S(3)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(a*d*f*(n + S(1))), x) + Simp((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*(a**S(2)*(n + S(1)) - b**S(2)*(m + n + S(2)))*sin(e + f*x)/(a**S(2)*b*d**S(2)*f*(m + S(1))*(n + S(1))), x)
rule2691 = ReplacementRule(pattern2691, replacement2691)
pattern2692 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1267, cons1170, cons94, cons1391, cons1392)
def replacement2692(m, f, b, d, a, n, x, e):
rubi.append(2692)
return -Dist(S(1)/(a**S(2)*b**S(2)*(m + S(1))*(m + S(2))), Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**(m + S(2))*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*(m + S(2))*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(a**S(2) - b**S(2))*cos(e + f*x)/(a*b**S(2)*d*f*(m + S(1))), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(2))*(a**S(2)*(-m + n + S(1)) - b**S(2)*(m + n + S(2)))*cos(e + f*x)/(a**S(2)*b**S(2)*d*f*(m + S(1))*(m + S(2))), x)
rule2692 = ReplacementRule(pattern2692, replacement2692)
pattern2693 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1267, cons1170, cons94, cons1391, cons1392)
def replacement2693(m, f, b, d, a, n, x, e):
rubi.append(2693)
return -Dist(S(1)/(a**S(2)*b**S(2)*(m + S(1))*(m + S(2))), Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**(m + S(2))*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*(m + S(2))*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(a**S(2) - b**S(2))*sin(e + f*x)/(a*b**S(2)*d*f*(m + S(1))), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(2))*(a**S(2)*(-m + n + S(1)) - b**S(2)*(m + n + S(2)))*sin(e + f*x)/(a**S(2)*b**S(2)*d*f*(m + S(1))*(m + S(2))), x)
rule2693 = ReplacementRule(pattern2693, replacement2693)
pattern2694 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1267, cons1170, cons94, cons1391, cons1393)
def replacement2694(m, f, b, d, a, n, x, e):
rubi.append(2694)
return -Dist(S(1)/(a*b**S(2)*(m + S(1))*(m + n + S(4))), Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*(m + S(1))*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(4)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) - Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(2))*cos(e + f*x)/(b**S(2)*d*f*(m + n + S(4))), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(a**S(2) - b**S(2))*cos(e + f*x)/(a*b**S(2)*d*f*(m + S(1))), x)
rule2694 = ReplacementRule(pattern2694, replacement2694)
pattern2695 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1267, cons1170, cons94, cons1391, cons1393)
def replacement2695(m, f, b, d, a, n, x, e):
rubi.append(2695)
return -Dist(S(1)/(a*b**S(2)*(m + S(1))*(m + n + S(4))), Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*(m + S(1))*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(4)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) + Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(2))*sin(e + f*x)/(b**S(2)*d*f*(m + n + S(4))), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(a**S(2) - b**S(2))*sin(e + f*x)/(a*b**S(2)*d*f*(m + S(1))), x)
rule2695 = ReplacementRule(pattern2695, replacement2695)
pattern2696 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1267, cons1390, cons1305, cons87, cons89, cons1394)
def replacement2696(m, f, b, d, a, n, x, e):
rubi.append(2696)
return -Dist(S(1)/(a**S(2)*d**S(2)*(n + S(1))*(n + S(2))), Int((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**m*Simp(a**S(2)*n*(n + S(2)) + a*b*m*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(1))*(n + S(2)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(a*d*f*(n + S(1))), x) - Simp(b*(d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*(m + n + S(2))*cos(e + f*x)/(a**S(2)*d**S(2)*f*(n + S(1))*(n + S(2))), x)
rule2696 = ReplacementRule(pattern2696, replacement2696)
pattern2697 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1267, cons1390, cons1305, cons87, cons89, cons1394)
def replacement2697(m, f, b, d, a, n, x, e):
rubi.append(2697)
return -Dist(S(1)/(a**S(2)*d**S(2)*(n + S(1))*(n + S(2))), Int((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**m*Simp(a**S(2)*n*(n + S(2)) + a*b*m*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(1))*(n + S(2)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(a*d*f*(n + S(1))), x) + Simp(b*(d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*(m + n + S(2))*sin(e + f*x)/(a**S(2)*d**S(2)*f*(n + S(1))*(n + S(2))), x)
rule2697 = ReplacementRule(pattern2697, replacement2697)
pattern2698 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1267, cons1390, cons1305, cons87, cons89, cons1393)
def replacement2698(m, f, b, d, a, n, x, e):
rubi.append(2698)
return Dist(S(1)/(a*b*d*(n + S(1))*(m + n + S(4))), Int((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**m*Simp(a**S(2)*(n + S(1))*(n + S(2)) + a*b*(m + S(3))*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(4)) - (a**S(2)*(n + S(1))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(a*d*f*(n + S(1))), x) - Simp((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*d**S(2)*f*(m + n + S(4))), x)
rule2698 = ReplacementRule(pattern2698, replacement2698)
pattern2699 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1267, cons1390, cons1305, cons87, cons89, cons1393)
def replacement2699(m, f, b, d, a, n, x, e):
rubi.append(2699)
return Dist(S(1)/(a*b*d*(n + S(1))*(m + n + S(4))), Int((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**m*Simp(a**S(2)*(n + S(1))*(n + S(2)) + a*b*(m + S(3))*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(4)) - (a**S(2)*(n + S(1))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(a*d*f*(n + S(1))), x) + Simp((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*d**S(2)*f*(m + n + S(4))), x)
rule2699 = ReplacementRule(pattern2699, replacement2699)
pattern2700 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1267, cons1390, cons1305, cons346, cons213, cons1393)
def replacement2700(m, f, b, d, a, n, x, e):
rubi.append(2700)
return -Dist(S(1)/(b**S(2)*(m + n + S(3))*(m + n + S(4))), Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*m*sin(e + f*x) - b**S(2)*(m + n + S(3))*(m + n + S(4)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(5)))*sin(e + f*x)**S(2), x), x), x) - Simp((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*d**S(2)*f*(m + n + S(4))), x) + Simp(a*(d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(n + S(3))*cos(e + f*x)/(b**S(2)*d*f*(m + n + S(3))*(m + n + S(4))), x)
rule2700 = ReplacementRule(pattern2700, replacement2700)
pattern2701 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1267, cons1390, cons1305, cons346, cons213, cons1393)
def replacement2701(m, f, b, d, a, n, x, e):
rubi.append(2701)
return -Dist(S(1)/(b**S(2)*(m + n + S(3))*(m + n + S(4))), Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*m*cos(e + f*x) - b**S(2)*(m + n + S(3))*(m + n + S(4)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(5)))*cos(e + f*x)**S(2), x), x), x) + Simp((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*d**S(2)*f*(m + n + S(4))), x) - Simp(a*(d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(n + S(3))*sin(e + f*x)/(b**S(2)*d*f*(m + n + S(3))*(m + n + S(4))), x)
rule2701 = ReplacementRule(pattern2701, replacement2701)
pattern2702 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(6), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1267, cons1170, cons584, cons1395, cons1396, cons1397, cons143)
def replacement2702(m, f, b, d, a, n, x, e):
rubi.append(2702)
return Dist(S(1)/(a**S(2)*b**S(2)*d**S(2)*(n + S(1))*(n + S(2))*(m + n + S(5))*(m + n + S(6))), Int((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**m*Simp(a**S(4)*(n + S(1))*(n + S(2))*(n + S(3))*(n + S(5)) - a**S(2)*b**S(2)*(n + S(2))*(S(2)*n + S(1))*(m + n + S(5))*(m + n + S(6)) + a*b*m*(a**S(2)*(n + S(1))*(n + S(2)) - b**S(2)*(m + n + S(5))*(m + n + S(6)))*sin(e + f*x) + b**S(4)*(m + n + S(2))*(m + n + S(3))*(m + n + S(5))*(m + n + S(6)) - (a**S(4)*(n + S(1))*(n + S(2))*(n + S(4))*(n + S(5)) - a**S(2)*b**S(2)*(n + S(1))*(n + S(2))*(m + n + S(5))*(S(2)*m + S(2)*n + S(13)) + b**S(4)*(m + n + S(2))*(m + n + S(4))*(m + n + S(5))*(m + n + S(6)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(a*d*f*(n + S(1))), x) + Simp((d*sin(e + f*x))**(n + S(4))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*d**S(4)*f*(m + n + S(6))), x) - Simp(b*(d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*(m + n + S(2))*cos(e + f*x)/(a**S(2)*d**S(2)*f*(n + S(1))*(n + S(2))), x) - Simp(a*(d*sin(e + f*x))**(n + S(3))*(a + b*sin(e + f*x))**(m + S(1))*(n + S(5))*cos(e + f*x)/(b**S(2)*d**S(3)*f*(m + n + S(5))*(m + n + S(6))), x)
rule2702 = ReplacementRule(pattern2702, replacement2702)
pattern2703 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(6), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1267, cons1170, cons584, cons1395, cons1396, cons1397, cons143)
def replacement2703(m, f, b, d, a, n, x, e):
rubi.append(2703)
return Dist(S(1)/(a**S(2)*b**S(2)*d**S(2)*(n + S(1))*(n + S(2))*(m + n + S(5))*(m + n + S(6))), Int((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**m*Simp(a**S(4)*(n + S(1))*(n + S(2))*(n + S(3))*(n + S(5)) - a**S(2)*b**S(2)*(n + S(2))*(S(2)*n + S(1))*(m + n + S(5))*(m + n + S(6)) + a*b*m*(a**S(2)*(n + S(1))*(n + S(2)) - b**S(2)*(m + n + S(5))*(m + n + S(6)))*cos(e + f*x) + b**S(4)*(m + n + S(2))*(m + n + S(3))*(m + n + S(5))*(m + n + S(6)) - (a**S(4)*(n + S(1))*(n + S(2))*(n + S(4))*(n + S(5)) - a**S(2)*b**S(2)*(n + S(1))*(n + S(2))*(m + n + S(5))*(S(2)*m + S(2)*n + S(13)) + b**S(4)*(m + n + S(2))*(m + n + S(4))*(m + n + S(5))*(m + n + S(6)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(a*d*f*(n + S(1))), x) - Simp((d*cos(e + f*x))**(n + S(4))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*d**S(4)*f*(m + n + S(6))), x) + Simp(b*(d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*(m + n + S(2))*sin(e + f*x)/(a**S(2)*d**S(2)*f*(n + S(1))*(n + S(2))), x) + Simp(a*(d*cos(e + f*x))**(n + S(3))*(a + b*cos(e + f*x))**(m + S(1))*(n + S(5))*sin(e + f*x)/(b**S(2)*d**S(3)*f*(m + n + S(5))*(m + n + S(6))), x)
rule2703 = ReplacementRule(pattern2703, replacement2703)
pattern2704 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1398, cons1399)
def replacement2704(p, m, f, b, d, a, n, x, e):
rubi.append(2704)
return Int(ExpandTrig((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m*(-sin(e + f*x)**S(2) + S(1))**(p/S(2)), x), x)
rule2704 = ReplacementRule(pattern2704, replacement2704)
pattern2705 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1398, cons1399)
def replacement2705(p, m, f, b, d, a, n, x, e):
rubi.append(2705)
return Int(ExpandTrig((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m*(-cos(e + f*x)**S(2) + S(1))**(p/S(2)), x), x)
rule2705 = ReplacementRule(pattern2705, replacement2705)
pattern2706 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**n_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons85, cons1400)
def replacement2706(p, f, b, g, a, n, x, e):
rubi.append(2706)
return Int(ExpandTrig((g*cos(e + f*x))**p, sin(e + f*x)**n/(a + b*sin(e + f*x)), x), x)
rule2706 = ReplacementRule(pattern2706, replacement2706)
pattern2707 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**n_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons1267, cons85, cons1400)
def replacement2707(p, f, b, g, a, n, x, e):
rubi.append(2707)
return Int(ExpandTrig((g*sin(e + f*x))**p, cos(e + f*x)**n/(a + b*cos(e + f*x)), x), x)
rule2707 = ReplacementRule(pattern2707, replacement2707)
pattern2708 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons146, cons1402)
def replacement2708(p, f, b, g, d, a, n, x, e):
rubi.append(2708)
return Dist(g**S(2)/a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2)), x), x) - Dist(b*g**S(2)/(a**S(2)*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**(p + S(-2)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/(a**S(2)*d**S(2)), Int((d*sin(e + f*x))**(n + S(2))*(g*cos(e + f*x))**(p + S(-2))/(a + b*sin(e + f*x)), x), x)
rule2708 = ReplacementRule(pattern2708, replacement2708)
pattern2709 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons146, cons1402)
def replacement2709(p, f, b, g, d, a, n, x, e):
rubi.append(2709)
return Dist(g**S(2)/a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2)), x), x) - Dist(b*g**S(2)/(a**S(2)*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**(p + S(-2)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/(a**S(2)*d**S(2)), Int((d*cos(e + f*x))**(n + S(2))*(g*sin(e + f*x))**(p + S(-2))/(a + b*cos(e + f*x)), x), x)
rule2709 = ReplacementRule(pattern2709, replacement2709)
pattern2710 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons146, cons1403)
def replacement2710(p, f, b, g, d, a, n, x, e):
rubi.append(2710)
return Dist(g**S(2)/(a*b), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2))*(-a*sin(e + f*x) + b), x), x) + Dist(g**S(2)*(a**S(2) - b**S(2))/(a*b*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**(p + S(-2))/(a + b*sin(e + f*x)), x), x)
rule2710 = ReplacementRule(pattern2710, replacement2710)
pattern2711 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons146, cons1403)
def replacement2711(p, f, b, g, d, a, n, x, e):
rubi.append(2711)
return Dist(g**S(2)/(a*b), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2))*(-a*cos(e + f*x) + b), x), x) + Dist(g**S(2)*(a**S(2) - b**S(2))/(a*b*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**(p + S(-2))/(a + b*cos(e + f*x)), x), x)
rule2711 = ReplacementRule(pattern2711, replacement2711)
pattern2712 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons146)
def replacement2712(p, f, b, g, d, a, n, x, e):
rubi.append(2712)
return Dist(g**S(2)/b**S(2), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2))*(a - b*sin(e + f*x)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/b**S(2), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2))/(a + b*sin(e + f*x)), x), x)
rule2712 = ReplacementRule(pattern2712, replacement2712)
pattern2713 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons146)
def replacement2713(p, f, b, g, d, a, n, x, e):
rubi.append(2713)
return Dist(g**S(2)/b**S(2), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2))*(a - b*cos(e + f*x)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/b**S(2), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2))/(a + b*cos(e + f*x)), x), x)
rule2713 = ReplacementRule(pattern2713, replacement2713)
pattern2714 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons137, cons165)
def replacement2714(p, f, b, g, d, a, n, x, e):
rubi.append(2714)
return Dist(a*d**S(2)/(a**S(2) - b**S(2)), Int((d*sin(e + f*x))**(n + S(-2))*(g*cos(e + f*x))**p, x), x) - Dist(b*d/(a**S(2) - b**S(2)), Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**p, x), x) - Dist(a**S(2)*d**S(2)/(g**S(2)*(a**S(2) - b**S(2))), Int((d*sin(e + f*x))**(n + S(-2))*(g*cos(e + f*x))**(p + S(2))/(a + b*sin(e + f*x)), x), x)
rule2714 = ReplacementRule(pattern2714, replacement2714)
pattern2715 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons137, cons165)
def replacement2715(p, f, b, g, d, a, n, x, e):
rubi.append(2715)
return Dist(a*d**S(2)/(a**S(2) - b**S(2)), Int((d*cos(e + f*x))**(n + S(-2))*(g*sin(e + f*x))**p, x), x) - Dist(b*d/(a**S(2) - b**S(2)), Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**p, x), x) - Dist(a**S(2)*d**S(2)/(g**S(2)*(a**S(2) - b**S(2))), Int((d*cos(e + f*x))**(n + S(-2))*(g*sin(e + f*x))**(p + S(2))/(a + b*cos(e + f*x)), x), x)
rule2715 = ReplacementRule(pattern2715, replacement2715)
pattern2716 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons137, cons88)
def replacement2716(p, f, b, g, d, a, n, x, e):
rubi.append(2716)
return -Dist(d/(a**S(2) - b**S(2)), Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**p*(-a*sin(e + f*x) + b), x), x) + Dist(a*b*d/(g**S(2)*(a**S(2) - b**S(2))), Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**(p + S(2))/(a + b*sin(e + f*x)), x), x)
rule2716 = ReplacementRule(pattern2716, replacement2716)
pattern2717 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons137, cons88)
def replacement2717(p, f, b, g, d, a, n, x, e):
rubi.append(2717)
return -Dist(d/(a**S(2) - b**S(2)), Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**p*(-a*cos(e + f*x) + b), x), x) + Dist(a*b*d/(g**S(2)*(a**S(2) - b**S(2))), Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**(p + S(2))/(a + b*cos(e + f*x)), x), x)
rule2717 = ReplacementRule(pattern2717, replacement2717)
pattern2718 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons137)
def replacement2718(p, f, b, g, d, a, n, x, e):
rubi.append(2718)
return -Dist(b**S(2)/(g**S(2)*(a**S(2) - b**S(2))), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(2))/(a + b*sin(e + f*x)), x), x) + Dist(S(1)/(a**S(2) - b**S(2)), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**p*(a - b*sin(e + f*x)), x), x)
rule2718 = ReplacementRule(pattern2718, replacement2718)
pattern2719 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons137)
def replacement2719(p, f, b, g, d, a, n, x, e):
rubi.append(2719)
return -Dist(b**S(2)/(g**S(2)*(a**S(2) - b**S(2))), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(2))/(a + b*cos(e + f*x)), x), x) + Dist(S(1)/(a**S(2) - b**S(2)), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**p*(a - b*cos(e + f*x)), x), x)
rule2719 = ReplacementRule(pattern2719, replacement2719)
pattern2720 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
def replacement2720(f, b, g, a, x, e):
rubi.append(2720)
return Dist(-S(4)*sqrt(S(2))*g/f, Subst(Int(x**S(2)/(sqrt(S(1) - x**S(4)/g**S(2))*(g**S(2)*(a + b) + x**S(4)*(a - b))), x), x, sqrt(g*cos(e + f*x))/sqrt(sin(e + f*x) + S(1))), x)
rule2720 = ReplacementRule(pattern2720, replacement2720)
pattern2721 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons208, cons1267)
def replacement2721(f, b, g, a, x, e):
rubi.append(2721)
return Dist(S(4)*sqrt(S(2))*g/f, Subst(Int(x**S(2)/(sqrt(S(1) - x**S(4)/g**S(2))*(g**S(2)*(a + b) + x**S(4)*(a - b))), x), x, sqrt(g*sin(e + f*x))/sqrt(cos(e + f*x) + S(1))), x)
rule2721 = ReplacementRule(pattern2721, replacement2721)
pattern2722 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(d_*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267)
def replacement2722(f, b, g, d, a, x, e):
rubi.append(2722)
return Dist(sqrt(sin(e + f*x))/sqrt(d*sin(e + f*x)), Int(sqrt(g*cos(e + f*x))/((a + b*sin(e + f*x))*sqrt(sin(e + f*x))), x), x)
rule2722 = ReplacementRule(pattern2722, replacement2722)
pattern2723 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(d_*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267)
def replacement2723(f, b, g, d, a, x, e):
rubi.append(2723)
return Dist(sqrt(cos(e + f*x))/sqrt(d*cos(e + f*x)), Int(sqrt(g*sin(e + f*x))/((a + b*cos(e + f*x))*sqrt(cos(e + f*x))), x), x)
rule2723 = ReplacementRule(pattern2723, replacement2723)
def With2724(f, b, d, a, x, e):
q = Rt(-a**S(2) + b**S(2), S(2))
rubi.append(2724)
return -Dist(S(2)*sqrt(S(2))*d*(b - q)/(f*q), Subst(Int(S(1)/(sqrt(S(1) - x**S(4)/d**S(2))*(a*x**S(2) + d*(b - q))), x), x, sqrt(d*sin(e + f*x))/sqrt(cos(e + f*x) + S(1))), x) + Dist(S(2)*sqrt(S(2))*d*(b + q)/(f*q), Subst(Int(S(1)/(sqrt(S(1) - x**S(4)/d**S(2))*(a*x**S(2) + d*(b + q))), x), x, sqrt(d*sin(e + f*x))/sqrt(cos(e + f*x) + S(1))), x)
pattern2724 = Pattern(Integral(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
rule2724 = ReplacementRule(pattern2724, With2724)
def With2725(f, b, d, a, x, e):
q = Rt(-a**S(2) + b**S(2), S(2))
rubi.append(2725)
return Dist(S(2)*sqrt(S(2))*d*(b - q)/(f*q), Subst(Int(S(1)/(sqrt(S(1) - x**S(4)/d**S(2))*(a*x**S(2) + d*(b - q))), x), x, sqrt(d*cos(e + f*x))/sqrt(sin(e + f*x) + S(1))), x) + Dist(-S(2)*sqrt(S(2))*d*(b + q)/(f*q), Subst(Int(S(1)/(sqrt(S(1) - x**S(4)/d**S(2))*(a*x**S(2) + d*(b + q))), x), x, sqrt(d*cos(e + f*x))/sqrt(sin(e + f*x) + S(1))), x)
pattern2725 = Pattern(Integral(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
rule2725 = ReplacementRule(pattern2725, With2725)
pattern2726 = Pattern(Integral(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267)
def replacement2726(f, b, g, d, a, x, e):
rubi.append(2726)
return Dist(sqrt(cos(e + f*x))/sqrt(g*cos(e + f*x)), Int(sqrt(d*sin(e + f*x))/((a + b*sin(e + f*x))*sqrt(cos(e + f*x))), x), x)
rule2726 = ReplacementRule(pattern2726, replacement2726)
pattern2727 = Pattern(Integral(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267)
def replacement2727(f, b, g, d, a, x, e):
rubi.append(2727)
return Dist(sqrt(sin(e + f*x))/sqrt(g*sin(e + f*x)), Int(sqrt(d*cos(e + f*x))/((a + b*cos(e + f*x))*sqrt(sin(e + f*x))), x), x)
rule2727 = ReplacementRule(pattern2727, replacement2727)
pattern2728 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons1404, cons88)
def replacement2728(p, f, b, g, d, a, n, x, e):
rubi.append(2728)
return Dist(d/b, Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**p, x), x) - Dist(a*d/b, Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**p/(a + b*sin(e + f*x)), x), x)
rule2728 = ReplacementRule(pattern2728, replacement2728)
pattern2729 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons1404, cons88)
def replacement2729(p, f, b, g, d, a, n, x, e):
rubi.append(2729)
return Dist(d/b, Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**p, x), x) - Dist(a*d/b, Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**p/(a + b*cos(e + f*x)), x), x)
rule2729 = ReplacementRule(pattern2729, replacement2729)
pattern2730 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons1404, cons463)
def replacement2730(p, f, b, g, d, a, n, x, e):
rubi.append(2730)
return Dist(S(1)/a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**p, x), x) - Dist(b/(a*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**p/(a + b*sin(e + f*x)), x), x)
rule2730 = ReplacementRule(pattern2730, replacement2730)
pattern2731 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1401, cons1404, cons463)
def replacement2731(p, f, b, g, d, a, n, x, e):
rubi.append(2731)
return Dist(S(1)/a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**p, x), x) - Dist(b/(a*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**p/(a + b*cos(e + f*x)), x), x)
rule2731 = ReplacementRule(pattern2731, replacement2731)
pattern2732 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1267, cons17, cons1405)
def replacement2732(p, m, f, b, g, d, a, n, x, e):
rubi.append(2732)
return Int(ExpandTrig((g*cos(e + f*x))**p, (d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x)
rule2732 = ReplacementRule(pattern2732, replacement2732)
pattern2733 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons4, cons5, cons1267, cons17, cons1405)
def replacement2733(p, m, f, b, g, d, a, n, x, e):
rubi.append(2733)
return Int(ExpandTrig((g*sin(e + f*x))**p, (d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x)
rule2733 = ReplacementRule(pattern2733, replacement2733)
pattern2734 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1406, cons267, cons146, cons1407)
def replacement2734(p, m, f, b, g, d, a, n, x, e):
rubi.append(2734)
return Dist(g**S(2)/a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1)), x), x) - Dist(b*g**S(2)/(a**S(2)*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/(a**S(2)*d**S(2)), Int((d*sin(e + f*x))**(n + S(2))*(g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**m, x), x)
rule2734 = ReplacementRule(pattern2734, replacement2734)
pattern2735 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons208, cons1267, cons1406, cons267, cons146, cons1407)
def replacement2735(p, m, f, b, g, d, a, n, x, e):
rubi.append(2735)
return Dist(g**S(2)/a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Dist(b*g**S(2)/(a**S(2)*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/(a**S(2)*d**S(2)), Int((d*cos(e + f*x))**(n + S(2))*(g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**m, x), x)
rule2735 = ReplacementRule(pattern2735, replacement2735)
pattern2736 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1265, cons1299, cons1380)
def replacement2736(p, m, f, b, d, a, c, n, x, e):
rubi.append(2736)
return Dist(a**(S(2)*m), Int((a - b*sin(e + f*x))**(-m)*(c + d*sin(e + f*x))**n, x), x)
rule2736 = ReplacementRule(pattern2736, replacement2736)
pattern2737 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1265, cons1299, cons1380)
def replacement2737(p, m, f, b, d, a, c, n, x, e):
rubi.append(2737)
return Dist(a**(S(2)*m), Int((a - b*cos(e + f*x))**(-m)*(c + d*cos(e + f*x))**n, x), x)
rule2737 = ReplacementRule(pattern2737, replacement2737)
pattern2738 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1265, cons17, cons13, cons1384)
def replacement2738(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2738)
return Dist((a/g)**(S(2)*m), Int((g*cos(e + f*x))**(S(2)*m + p)*(a - b*sin(e + f*x))**(-m)*(c + d*sin(e + f*x))**n, x), x)
rule2738 = ReplacementRule(pattern2738, replacement2738)
pattern2739 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons1265, cons17, cons13, cons1384)
def replacement2739(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2739)
return Dist((a/g)**(S(2)*m), Int((g*sin(e + f*x))**(S(2)*m + p)*(a - b*cos(e + f*x))**(-m)*(c + d*cos(e + f*x))**n, x), x)
rule2739 = ReplacementRule(pattern2739, replacement2739)
pattern2740 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1265, cons1170)
def replacement2740(m, f, b, d, a, c, n, x, e):
rubi.append(2740)
return Dist(b**(S(-2)), Int((a - b*sin(e + f*x))*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x)
rule2740 = ReplacementRule(pattern2740, replacement2740)
pattern2741 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1265, cons1170)
def replacement2741(m, f, b, d, a, c, n, x, e):
rubi.append(2741)
return Dist(b**(S(-2)), Int((a - b*cos(e + f*x))*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x)
rule2741 = ReplacementRule(pattern2741, replacement2741)
pattern2742 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1265, cons1306, cons17)
def replacement2742(p, m, f, b, d, a, c, n, x, e):
rubi.append(2742)
return Dist(a**m*cos(e + f*x)/(f*sqrt(-sin(e + f*x) + S(1))*sqrt(sin(e + f*x) + S(1))), Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, sin(e + f*x)), x)
rule2742 = ReplacementRule(pattern2742, replacement2742)
pattern2743 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1265, cons1306, cons17)
def replacement2743(p, m, f, b, d, a, c, n, x, e):
rubi.append(2743)
return -Dist(a**m*sin(e + f*x)/(f*sqrt(-cos(e + f*x) + S(1))*sqrt(cos(e + f*x) + S(1))), Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, cos(e + f*x)), x)
rule2743 = ReplacementRule(pattern2743, replacement2743)
pattern2744 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1265, cons1306, cons18)
def replacement2744(p, m, f, b, d, a, c, n, x, e):
rubi.append(2744)
return Dist(a**(-p + S(2))*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, sin(e + f*x)), x)
rule2744 = ReplacementRule(pattern2744, replacement2744)
pattern2745 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1265, cons1306, cons18)
def replacement2745(p, m, f, b, d, a, c, n, x, e):
rubi.append(2745)
return -Dist(a**(-p + S(2))*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, cos(e + f*x)), x)
rule2745 = ReplacementRule(pattern2745, replacement2745)
pattern2746 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons1265, cons62, cons1387)
def replacement2746(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2746)
return Int(ExpandTrig((g*cos(e + f*x))**p, (a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x)
rule2746 = ReplacementRule(pattern2746, replacement2746)
pattern2747 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons1265, cons62, cons1387)
def replacement2747(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2747)
return Int(ExpandTrig((g*sin(e + f*x))**p, (a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x)
rule2747 = ReplacementRule(pattern2747, replacement2747)
pattern2748 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons1265, cons17)
def replacement2748(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2748)
return Dist(a**m*g*(g*cos(e + f*x))**(p + S(-1))*(-sin(e + f*x) + S(1))**(-p/S(2) + S(1)/2)*(sin(e + f*x) + S(1))**(-p/S(2) + S(1)/2)/f, Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, sin(e + f*x)), x)
rule2748 = ReplacementRule(pattern2748, replacement2748)
pattern2749 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons1265, cons17)
def replacement2749(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2749)
return -Dist(a**m*g*(g*sin(e + f*x))**(p + S(-1))*(-cos(e + f*x) + S(1))**(-p/S(2) + S(1)/2)*(cos(e + f*x) + S(1))**(-p/S(2) + S(1)/2)/f, Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, cos(e + f*x)), x)
rule2749 = ReplacementRule(pattern2749, replacement2749)
pattern2750 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons1265, cons18)
def replacement2750(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2750)
return Dist(g*(g*cos(e + f*x))**(p + S(-1))*(a - b*sin(e + f*x))**(-p/S(2) + S(1)/2)*(a + b*sin(e + f*x))**(-p/S(2) + S(1)/2)/f, Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, sin(e + f*x)), x)
rule2750 = ReplacementRule(pattern2750, replacement2750)
pattern2751 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons1265, cons18)
def replacement2751(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2751)
return -Dist(g*(g*sin(e + f*x))**(p + S(-1))*(a - b*cos(e + f*x))**(-p/S(2) + S(1)/2)*(a + b*cos(e + f*x))**(-p/S(2) + S(1)/2)/f, Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, cos(e + f*x)), x)
rule2751 = ReplacementRule(pattern2751, replacement2751)
pattern2752 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1267, cons1390)
def replacement2752(m, f, b, d, a, c, n, x, e):
rubi.append(2752)
return Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*(-sin(e + f*x)**S(2) + S(1)), x)
rule2752 = ReplacementRule(pattern2752, replacement2752)
pattern2753 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1267, cons1390)
def replacement2753(m, f, b, d, a, c, n, x, e):
rubi.append(2753)
return Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*(-cos(e + f*x)**S(2) + S(1)), x)
rule2753 = ReplacementRule(pattern2753, replacement2753)
pattern2754 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1267, cons1408, cons1390)
def replacement2754(p, m, f, b, d, a, c, n, x, e):
rubi.append(2754)
return Int(ExpandTrig((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*(-sin(e + f*x)**S(2) + S(1))**(p/S(2)), x), x)
rule2754 = ReplacementRule(pattern2754, replacement2754)
pattern2755 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1267, cons1408, cons1390)
def replacement2755(p, m, f, b, d, a, c, n, x, e):
rubi.append(2755)
return Int(ExpandTrig((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*(-cos(e + f*x)**S(2) + S(1))**(p/S(2)), x), x)
rule2755 = ReplacementRule(pattern2755, replacement2755)
pattern2756 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons1267, cons1170)
def replacement2756(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2756)
return Int(ExpandTrig((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x)
rule2756 = ReplacementRule(pattern2756, replacement2756)
pattern2757 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons1267, cons1170)
def replacement2757(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2757)
return Int(ExpandTrig((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x)
rule2757 = ReplacementRule(pattern2757, replacement2757)
pattern2758 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons1267)
def replacement2758(p, m, f, b, g, d, c, n, a, x, e):
rubi.append(2758)
return Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x)
rule2758 = ReplacementRule(pattern2758, replacement2758)
pattern2759 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons1267)
def replacement2759(p, m, f, b, g, d, c, n, a, x, e):
rubi.append(2759)
return Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x)
rule2759 = ReplacementRule(pattern2759, replacement2759)
pattern2760 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons147)
def replacement2760(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(2760)
return Dist(g**(S(2)*IntPart(p))*(g/cos(e + f*x))**FracPart(p)*(g*cos(e + f*x))**FracPart(p), Int((g*cos(e + f*x))**(-p)*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x)
rule2760 = ReplacementRule(pattern2760, replacement2760)
pattern2761 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons147)
def replacement2761(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2761)
return Dist(g**(S(2)*IntPart(p))*(g/sin(e + f*x))**FracPart(p)*(g*sin(e + f*x))**FracPart(p), Int((g*sin(e + f*x))**(-p)*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x)
rule2761 = ReplacementRule(pattern2761, replacement2761)
pattern2762 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1409)
def replacement2762(f, g, b, d, a, c, x, e):
rubi.append(2762)
return Dist(g/d, Int(sqrt(a + b*sin(e + f*x))/sqrt(g*sin(e + f*x)), x), x) - Dist(c*g/d, Int(sqrt(a + b*sin(e + f*x))/(sqrt(g*sin(e + f*x))*(c + d*sin(e + f*x))), x), x)
rule2762 = ReplacementRule(pattern2762, replacement2762)
pattern2763 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1409)
def replacement2763(f, b, g, d, a, c, x, e):
rubi.append(2763)
return Dist(g/d, Int(sqrt(a + b*cos(e + f*x))/sqrt(g*cos(e + f*x)), x), x) - Dist(c*g/d, Int(sqrt(a + b*cos(e + f*x))/(sqrt(g*cos(e + f*x))*(c + d*cos(e + f*x))), x), x)
rule2763 = ReplacementRule(pattern2763, replacement2763)
pattern2764 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1323)
def replacement2764(f, g, b, d, a, c, x, e):
rubi.append(2764)
return Dist(b/d, Int(sqrt(g*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x)
rule2764 = ReplacementRule(pattern2764, replacement2764)
pattern2765 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1323)
def replacement2765(f, b, g, d, a, c, x, e):
rubi.append(2765)
return Dist(b/d, Int(sqrt(g*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(g*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x)
rule2765 = ReplacementRule(pattern2765, replacement2765)
pattern2766 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement2766(f, g, b, d, a, c, x, e):
rubi.append(2766)
return Dist(-S(2)*b/f, Subst(Int(S(1)/(a*d + b*c + c*g*x**S(2)), x), x, b*cos(e + f*x)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x)))), x)
rule2766 = ReplacementRule(pattern2766, replacement2766)
pattern2767 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement2767(f, b, g, d, a, c, x, e):
rubi.append(2767)
return Dist(S(2)*b/f, Subst(Int(S(1)/(a*d + b*c + c*g*x**S(2)), x), x, b*sin(e + f*x)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x)))), x)
rule2767 = ReplacementRule(pattern2767, replacement2767)
pattern2768 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1410, cons1411, cons105)
def replacement2768(f, b, d, a, c, x, e):
rubi.append(2768)
return -Simp(sqrt(a + b)*EllipticE(asin(cos(e + f*x)/(sin(e + f*x) + S(1))), -(a - b)/(a + b))/(c*f), x)
rule2768 = ReplacementRule(pattern2768, replacement2768)
pattern2769 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1410, cons1411, cons105)
def replacement2769(f, b, d, a, c, x, e):
rubi.append(2769)
return Simp(sqrt(a + b)*EllipticE(asin(sin(e + f*x)/(cos(e + f*x) + S(1))), -(a - b)/(a + b))/(c*f), x)
rule2769 = ReplacementRule(pattern2769, replacement2769)
pattern2770 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1322)
def replacement2770(f, g, b, d, a, c, x, e):
rubi.append(2770)
return -Simp(sqrt(d*sin(e + f*x)/(c + d*sin(e + f*x)))*sqrt(a + b*sin(e + f*x))*EllipticE(asin(c*cos(e + f*x)/(c + d*sin(e + f*x))), (-a*d + b*c)/(a*d + b*c))/(d*f*sqrt(g*sin(e + f*x))*sqrt(c**S(2)*(a + b*sin(e + f*x))/((c + d*sin(e + f*x))*(a*c + b*d)))), x)
rule2770 = ReplacementRule(pattern2770, replacement2770)
pattern2771 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1322)
def replacement2771(f, b, g, d, a, c, x, e):
rubi.append(2771)
return Simp(sqrt(d*cos(e + f*x)/(c + d*cos(e + f*x)))*sqrt(a + b*cos(e + f*x))*EllipticE(asin(c*sin(e + f*x)/(c + d*cos(e + f*x))), (-a*d + b*c)/(a*d + b*c))/(d*f*sqrt(g*cos(e + f*x))*sqrt(c**S(2)*(a + b*cos(e + f*x))/((c + d*cos(e + f*x))*(a*c + b*d)))), x)
rule2771 = ReplacementRule(pattern2771, replacement2771)
pattern2772 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1323)
def replacement2772(f, g, b, d, a, c, x, e):
rubi.append(2772)
return Dist(a/c, Int(S(1)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) + Dist((-a*d + b*c)/(c*g), Int(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x)
rule2772 = ReplacementRule(pattern2772, replacement2772)
pattern2773 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1323)
def replacement2773(f, b, g, d, a, c, x, e):
rubi.append(2773)
return Dist(a/c, Int(S(1)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) + Dist((-a*d + b*c)/(c*g), Int(sqrt(g*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x)
rule2773 = ReplacementRule(pattern2773, replacement2773)
pattern2774 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement2774(f, b, d, a, c, x, e):
rubi.append(2774)
return Dist(S(1)/c, Int(sqrt(a + b*sin(e + f*x))/sin(e + f*x), x), x) - Dist(d/c, Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x)
rule2774 = ReplacementRule(pattern2774, replacement2774)
pattern2775 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement2775(f, b, d, a, c, x, e):
rubi.append(2775)
return Dist(S(1)/c, Int(sqrt(a + b*cos(e + f*x))/cos(e + f*x), x), x) - Dist(d/c, Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x)
rule2775 = ReplacementRule(pattern2775, replacement2775)
pattern2776 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement2776(f, b, d, a, c, x, e):
rubi.append(2776)
return Dist(a/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x)
rule2776 = ReplacementRule(pattern2776, replacement2776)
pattern2777 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement2777(f, b, d, a, c, x, e):
rubi.append(2777)
return Dist(a/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x)
rule2777 = ReplacementRule(pattern2777, replacement2777)
pattern2778 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1409)
def replacement2778(f, g, b, d, a, c, x, e):
rubi.append(2778)
return -Dist(a*g/(-a*d + b*c), Int(S(1)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) + Dist(c*g/(-a*d + b*c), Int(sqrt(a + b*sin(e + f*x))/(sqrt(g*sin(e + f*x))*(c + d*sin(e + f*x))), x), x)
rule2778 = ReplacementRule(pattern2778, replacement2778)
pattern2779 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1409)
def replacement2779(f, b, g, d, a, c, x, e):
rubi.append(2779)
return -Dist(a*g/(-a*d + b*c), Int(S(1)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) + Dist(c*g/(-a*d + b*c), Int(sqrt(a + b*cos(e + f*x))/(sqrt(g*cos(e + f*x))*(c + d*cos(e + f*x))), x), x)
rule2779 = ReplacementRule(pattern2779, replacement2779)
pattern2780 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1323)
def replacement2780(f, g, b, d, a, c, x, e):
rubi.append(2780)
return Simp(S(2)*sqrt(-S(1)/tan(e + f*x)**S(2))*sqrt(g*sin(e + f*x))*sqrt((a/sin(e + f*x) + b)/(a + b))*EllipticPi(S(2)*c/(c + d), asin(sqrt(S(2))*sqrt(S(1) - S(1)/sin(e + f*x))/S(2)), S(2)*a/(a + b))*tan(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(c + d)), x)
rule2780 = ReplacementRule(pattern2780, replacement2780)
pattern2781 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1323)
def replacement2781(f, b, g, d, a, c, x, e):
rubi.append(2781)
return Simp(-S(2)*sqrt(-tan(e + f*x)**S(2))*sqrt(g*cos(e + f*x))*sqrt((a/cos(e + f*x) + b)/(a + b))*EllipticPi(S(2)*c/(c + d), asin(sqrt(S(2))*sqrt(S(1) - S(1)/cos(e + f*x))/S(2)), S(2)*a/(a + b))/(f*sqrt(a + b*cos(e + f*x))*(c + d)*tan(e + f*x)), x)
rule2781 = ReplacementRule(pattern2781, replacement2781)
pattern2782 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1409)
def replacement2782(f, g, b, d, a, c, x, e):
rubi.append(2782)
return Dist(b/(-a*d + b*c), Int(S(1)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b*sin(e + f*x))/(sqrt(g*sin(e + f*x))*(c + d*sin(e + f*x))), x), x)
rule2782 = ReplacementRule(pattern2782, replacement2782)
pattern2783 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1409)
def replacement2783(f, b, g, d, a, c, x, e):
rubi.append(2783)
return Dist(b/(-a*d + b*c), Int(S(1)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b*cos(e + f*x))/(sqrt(g*cos(e + f*x))*(c + d*cos(e + f*x))), x), x)
rule2783 = ReplacementRule(pattern2783, replacement2783)
pattern2784 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1323)
def replacement2784(f, g, b, d, a, c, x, e):
rubi.append(2784)
return Dist(S(1)/c, Int(S(1)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) - Dist(d/(c*g), Int(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x)
rule2784 = ReplacementRule(pattern2784, replacement2784)
pattern2785 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267, cons1323)
def replacement2785(f, b, g, d, a, c, x, e):
rubi.append(2785)
return Dist(S(1)/c, Int(S(1)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) - Dist(d/(c*g), Int(sqrt(g*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x)
rule2785 = ReplacementRule(pattern2785, replacement2785)
pattern2786 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement2786(f, b, d, a, c, x, e):
rubi.append(2786)
return Dist(S(1)/(c*(-a*d + b*c)), Int((-a*d + b*c - b*d*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x), x) + Dist(d**S(2)/(c*(-a*d + b*c)), Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x)
rule2786 = ReplacementRule(pattern2786, replacement2786)
pattern2787 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement2787(f, b, d, a, c, x, e):
rubi.append(2787)
return Dist(S(1)/(c*(-a*d + b*c)), Int((-a*d + b*c - b*d*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*cos(e + f*x)), x), x) + Dist(d**S(2)/(c*(-a*d + b*c)), Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x)
rule2787 = ReplacementRule(pattern2787, replacement2787)
pattern2788 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement2788(f, b, d, a, c, x, e):
rubi.append(2788)
return Dist(S(1)/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x), x) - Dist(d/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x)
rule2788 = ReplacementRule(pattern2788, replacement2788)
pattern2789 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement2789(f, b, d, a, c, x, e):
rubi.append(2789)
return Dist(S(1)/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*cos(e + f*x)), x), x) - Dist(d/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x)
rule2789 = ReplacementRule(pattern2789, replacement2789)
pattern2790 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons70)
def replacement2790(f, b, d, a, c, x, e):
rubi.append(2790)
return Dist(S(1)/c, Int(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))/sin(e + f*x), x), x) - Dist(d/c, Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x)
rule2790 = ReplacementRule(pattern2790, replacement2790)
pattern2791 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons70)
def replacement2791(f, b, d, a, c, x, e):
rubi.append(2791)
return Dist(S(1)/c, Int(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))/cos(e + f*x), x), x) - Dist(d/c, Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x)
rule2791 = ReplacementRule(pattern2791, replacement2791)
pattern2792 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1412)
def replacement2792(f, b, d, a, c, x, e):
rubi.append(2792)
return Dist(-S(2)*a/f, Subst(Int(S(1)/(-a*c*x**S(2) + S(1)), x), x, cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))), x)
rule2792 = ReplacementRule(pattern2792, replacement2792)
pattern2793 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1412)
def replacement2793(f, b, d, a, c, x, e):
rubi.append(2793)
return Dist(S(2)*a/f, Subst(Int(S(1)/(-a*c*x**S(2) + S(1)), x), x, sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x)))), x)
rule2793 = ReplacementRule(pattern2793, replacement2793)
pattern2794 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1322)
def replacement2794(f, b, d, a, c, x, e):
rubi.append(2794)
return Dist(a/c, Int(sqrt(c + d*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x)
rule2794 = ReplacementRule(pattern2794, replacement2794)
pattern2795 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1322)
def replacement2795(f, b, d, a, c, x, e):
rubi.append(2795)
return Dist(a/c, Int(sqrt(c + d*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x)
rule2795 = ReplacementRule(pattern2795, replacement2795)
pattern2796 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2796(f, b, d, a, c, x, e):
rubi.append(2796)
return Simp(-S(2)*sqrt((-a*d + b*c)*(sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c - d)))*sqrt(-(-a*d + b*c)*(-sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c + d)))*(a + b*sin(e + f*x))*EllipticPi(a*(c + d)/(c*(a + b)), asin(sqrt(c + d*sin(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*sin(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(c*f*Rt((a + b)/(c + d), S(2))*cos(e + f*x)), x)
rule2796 = ReplacementRule(pattern2796, replacement2796)
pattern2797 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement2797(f, b, d, a, c, x, e):
rubi.append(2797)
return Simp(S(2)*sqrt((-a*d + b*c)*(cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c - d)))*sqrt(-(-a*d + b*c)*(-cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c + d)))*(a + b*cos(e + f*x))*EllipticPi(a*(c + d)/(c*(a + b)), asin(sqrt(c + d*cos(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*cos(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(c*f*Rt((a + b)/(c + d), S(2))*sin(e + f*x)), x)
rule2797 = ReplacementRule(pattern2797, replacement2797)
pattern2798 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement2798(f, b, d, a, c, x, e):
rubi.append(2798)
return Dist(cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), Int(S(1)/(sin(e + f*x)*cos(e + f*x)), x), x)
rule2798 = ReplacementRule(pattern2798, replacement2798)
pattern2799 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement2799(f, b, d, a, c, x, e):
rubi.append(2799)
return Dist(sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), Int(S(1)/(sin(e + f*x)*cos(e + f*x)), x), x)
rule2799 = ReplacementRule(pattern2799, replacement2799)
pattern2800 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1413)
def replacement2800(f, b, d, a, c, x, e):
rubi.append(2800)
return Dist(S(1)/a, Int(sqrt(a + b*sin(e + f*x))/(sqrt(c + d*sin(e + f*x))*sin(e + f*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x)
rule2800 = ReplacementRule(pattern2800, replacement2800)
pattern2801 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1413)
def replacement2801(f, b, d, a, c, x, e):
rubi.append(2801)
return Dist(S(1)/a, Int(sqrt(a + b*cos(e + f*x))/(sqrt(c + d*cos(e + f*x))*cos(e + f*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x)
rule2801 = ReplacementRule(pattern2801, replacement2801)
pattern2802 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement2802(f, b, d, a, c, x, e):
rubi.append(2802)
return Dist(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))/cos(e + f*x), Int(S(1)/tan(e + f*x), x), x)
rule2802 = ReplacementRule(pattern2802, replacement2802)
pattern2803 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement2803(f, b, d, a, c, x, e):
rubi.append(2803)
return Dist(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))/sin(e + f*x), Int(tan(e + f*x), x), x)
rule2803 = ReplacementRule(pattern2803, replacement2803)
pattern2804 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1413)
def replacement2804(f, b, d, a, c, x, e):
rubi.append(2804)
return Dist(c, Int(sqrt(a + b*sin(e + f*x))/(sqrt(c + d*sin(e + f*x))*sin(e + f*x)), x), x) + Dist(d, Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x)
rule2804 = ReplacementRule(pattern2804, replacement2804)
pattern2805 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1413)
def replacement2805(f, b, d, a, c, x, e):
rubi.append(2805)
return Dist(c, Int(sqrt(a + b*cos(e + f*x))/(sqrt(c + d*cos(e + f*x))*cos(e + f*x)), x), x) + Dist(d, Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x)
rule2805 = ReplacementRule(pattern2805, replacement2805)
pattern2806 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons70, cons1265, cons1414, cons85)
def replacement2806(p, m, f, b, d, a, n, c, x, e):
rubi.append(2806)
return Dist(a**n*c**n, Int((a + b*sin(e + f*x))**(m - n)*tan(e + f*x)**p, x), x)
rule2806 = ReplacementRule(pattern2806, replacement2806)
pattern2807 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons70, cons1265, cons1414, cons85)
def replacement2807(p, m, f, b, d, a, n, c, x, e):
rubi.append(2807)
return Dist(a**n*c**n, Int((a + b*cos(e + f*x))**(m - n)*(S(1)/tan(e + f*x))**p, x), x)
rule2807 = ReplacementRule(pattern2807, replacement2807)
pattern2808 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons1265, cons1323, cons1304)
def replacement2808(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(2808)
return Dist(sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))/(f*cos(e + f*x)), Subst(Int((g*x)**p*(a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, sin(e + f*x)), x)
rule2808 = ReplacementRule(pattern2808, replacement2808)
pattern2809 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons1265, cons1323, cons1304)
def replacement2809(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2809)
return -Dist(sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))/(f*sin(e + f*x)), Subst(Int((g*x)**p*(a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, cos(e + f*x)), x)
rule2809 = ReplacementRule(pattern2809, replacement2809)
pattern2810 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons71, cons1415, cons1416)
def replacement2810(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(2810)
return Int(ExpandTrig((g*sin(e + f*x))**p*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x)
rule2810 = ReplacementRule(pattern2810, replacement2810)
pattern2811 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons71, cons1415, cons1416)
def replacement2811(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2811)
return Int(ExpandTrig((g*cos(e + f*x))**p*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x)
rule2811 = ReplacementRule(pattern2811, replacement2811)
pattern2812 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons1416)
def replacement2812(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(2812)
return Int((g*sin(e + f*x))**p*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x)
rule2812 = ReplacementRule(pattern2812, replacement2812)
pattern2813 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons1416)
def replacement2813(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2813)
return Int((g*cos(e + f*x))**p*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x)
rule2813 = ReplacementRule(pattern2813, replacement2813)
pattern2814 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons71, cons147, cons17, cons85)
def replacement2814(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2814)
return Dist(g**(m + n), Int((g*sin(e + f*x))**(-m - n + p)*(a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n, x), x)
rule2814 = ReplacementRule(pattern2814, replacement2814)
pattern2815 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons71, cons147, cons17, cons85)
def replacement2815(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(2815)
return Dist(g**(m + n), Int((g*cos(e + f*x))**(-m - n + p)*(a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n, x), x)
rule2815 = ReplacementRule(pattern2815, replacement2815)
pattern2816 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons147, cons1417)
def replacement2816(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2816)
return Dist((g/sin(e + f*x))**p*(g*sin(e + f*x))**p, Int((g/sin(e + f*x))**(-p)*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n, x), x)
rule2816 = ReplacementRule(pattern2816, replacement2816)
pattern2817 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons147, cons1417)
def replacement2817(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(2817)
return Dist((g/cos(e + f*x))**p*(g*cos(e + f*x))**p, Int((g/cos(e + f*x))**(-p)*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n, x), x)
rule2817 = ReplacementRule(pattern2817, replacement2817)
pattern2818 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons85)
def replacement2818(p, m, g, f, b, d, c, n, a, x, e):
rubi.append(2818)
return Dist(g**n, Int((g*sin(e + f*x))**(-n + p)*(a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n, x), x)
rule2818 = ReplacementRule(pattern2818, replacement2818)
pattern2819 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons85)
def replacement2819(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2819)
return Dist(g**n, Int((g*cos(e + f*x))**(-n + p)*(a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n, x), x)
rule2819 = ReplacementRule(pattern2819, replacement2819)
pattern2820 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons23, cons17, cons38)
def replacement2820(p, m, f, b, d, c, n, a, x, e):
rubi.append(2820)
return Int((c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m*(S(1)/sin(e + f*x))**(-m - p), x)
rule2820 = ReplacementRule(pattern2820, replacement2820)
pattern2821 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons23, cons17, cons38)
def replacement2821(p, m, f, b, d, a, c, n, x, e):
rubi.append(2821)
return Int((c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m*(S(1)/cos(e + f*x))**(-m - p), x)
rule2821 = ReplacementRule(pattern2821, replacement2821)
pattern2822 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons23, cons17, cons147)
def replacement2822(p, m, f, b, g, d, c, n, a, x, e):
rubi.append(2822)
return Dist((g*sin(e + f*x))**p*(S(1)/sin(e + f*x))**p, Int((c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m*(S(1)/sin(e + f*x))**(-m - p), x), x)
rule2822 = ReplacementRule(pattern2822, replacement2822)
pattern2823 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons23, cons17, cons147)
def replacement2823(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2823)
return Dist((g*cos(e + f*x))**p*(S(1)/cos(e + f*x))**p, Int((c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m*(S(1)/cos(e + f*x))**(-m - p), x), x)
rule2823 = ReplacementRule(pattern2823, replacement2823)
pattern2824 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons23, cons18)
def replacement2824(p, m, f, g, b, d, c, n, a, x, e):
rubi.append(2824)
return Dist((g*sin(e + f*x))**n*(c + d/sin(e + f*x))**n*(c*sin(e + f*x) + d)**(-n), Int((g*sin(e + f*x))**(-n + p)*(a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n, x), x)
rule2824 = ReplacementRule(pattern2824, replacement2824)
pattern2825 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons23, cons18)
def replacement2825(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2825)
return Dist((g*cos(e + f*x))**n*(c + d/cos(e + f*x))**n*(c*cos(e + f*x) + d)**(-n), Int((g*cos(e + f*x))**(-n + p)*(a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n, x), x)
rule2825 = ReplacementRule(pattern2825, replacement2825)
pattern2826 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons71, cons147, cons17, cons85)
def replacement2826(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(2826)
return Dist(g**(m + n), Int((g/sin(e + f*x))**(-m - n + p)*(a/sin(e + f*x) + b)**m*(c/sin(e + f*x) + d)**n, x), x)
rule2826 = ReplacementRule(pattern2826, replacement2826)
pattern2827 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons71, cons147, cons17, cons85)
def replacement2827(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2827)
return Dist(g**(m + n), Int((g/cos(e + f*x))**(-m - n + p)*(a/cos(e + f*x) + b)**m*(c/cos(e + f*x) + d)**n, x), x)
rule2827 = ReplacementRule(pattern2827, replacement2827)
pattern2828 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons147, cons1417)
def replacement2828(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(2828)
return Dist((g/sin(e + f*x))**p*(g*sin(e + f*x))**p, Int((g*sin(e + f*x))**(-p)*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x)
rule2828 = ReplacementRule(pattern2828, replacement2828)
pattern2829 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons147, cons1417)
def replacement2829(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2829)
return Dist((g/cos(e + f*x))**p*(g*cos(e + f*x))**p, Int((g*cos(e + f*x))**(-p)*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x)
rule2829 = ReplacementRule(pattern2829, replacement2829)
pattern2830 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons17)
def replacement2830(p, m, f, g, b, d, c, n, a, x, e):
rubi.append(2830)
return Dist(g**m, Int((g/sin(e + f*x))**(-m + p)*(c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m, x), x)
rule2830 = ReplacementRule(pattern2830, replacement2830)
pattern2831 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons17)
def replacement2831(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2831)
return Dist(g**m, Int((g/cos(e + f*x))**(-m + p)*(c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m, x), x)
rule2831 = ReplacementRule(pattern2831, replacement2831)
pattern2832 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons18, cons85, cons38)
def replacement2832(p, m, f, b, d, c, n, a, x, e):
rubi.append(2832)
return Int((a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-n - p), x)
rule2832 = ReplacementRule(pattern2832, replacement2832)
pattern2833 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons18, cons85, cons38)
def replacement2833(p, m, f, b, d, a, n, c, x, e):
rubi.append(2833)
return Int((a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-n - p), x)
rule2833 = ReplacementRule(pattern2833, replacement2833)
pattern2834 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons18, cons85, cons147)
def replacement2834(p, m, f, g, b, d, c, n, a, x, e):
rubi.append(2834)
return Dist((g/sin(e + f*x))**p*sin(e + f*x)**p, Int((a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-n - p), x), x)
rule2834 = ReplacementRule(pattern2834, replacement2834)
pattern2835 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons5, cons18, cons85, cons147)
def replacement2835(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(2835)
return Dist((g/cos(e + f*x))**p*cos(e + f*x)**p, Int((a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-n - p), x), x)
rule2835 = ReplacementRule(pattern2835, replacement2835)
pattern2836 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons18, cons23)
def replacement2836(p, m, f, g, b, d, c, n, a, x, e):
rubi.append(2836)
return Dist((g/sin(e + f*x))**m*(a + b*sin(e + f*x))**m*(a/sin(e + f*x) + b)**(-m), Int((g/sin(e + f*x))**(-m + p)*(c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m, x), x)
rule2836 = ReplacementRule(pattern2836, replacement2836)
pattern2837 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons18, cons23)
def replacement2837(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2837)
return Dist((g/cos(e + f*x))**m*(a + b*cos(e + f*x))**m*(a/cos(e + f*x) + b)**(-m), Int((g/cos(e + f*x))**(-m + p)*(c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m, x), x)
rule2837 = ReplacementRule(pattern2837, replacement2837)
pattern2838 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1418, cons1265, cons17, cons85)
def replacement2838(B, m, f, b, a, n, A, x, e):
rubi.append(2838)
return Int(ExpandTrig((A + B*sin(e + f*x))*(a + b*sin(e + f*x))**m*sin(e + f*x)**n, x), x)
rule2838 = ReplacementRule(pattern2838, replacement2838)
pattern2839 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1418, cons1265, cons17, cons85)
def replacement2839(B, e, m, f, b, a, n, x, A):
rubi.append(2839)
return Int(ExpandTrig((A + B*cos(e + f*x))*(a + b*cos(e + f*x))**m*cos(e + f*x)**n, x), x)
rule2839 = ReplacementRule(pattern2839, replacement2839)
pattern2840 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons70, cons1265, cons17, cons1309)
def replacement2840(B, e, m, f, b, d, a, n, c, x, A):
rubi.append(2840)
return Dist(a**m*c**m, Int((A + B*sin(e + f*x))*(c + d*sin(e + f*x))**(-m + n)*cos(e + f*x)**(S(2)*m), x), x)
rule2840 = ReplacementRule(pattern2840, replacement2840)
pattern2841 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons70, cons1265, cons17, cons1309)
def replacement2841(B, m, f, b, d, a, n, c, A, x, e):
rubi.append(2841)
return Dist(a**m*c**m, Int((A + B*cos(e + f*x))*(c + d*cos(e + f*x))**(-m + n)*sin(e + f*x)**(S(2)*m), x), x)
rule2841 = ReplacementRule(pattern2841, replacement2841)
pattern2842 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71)
def replacement2842(B, m, f, b, d, a, c, A, x, e):
rubi.append(2842)
return Int((a + b*sin(e + f*x))**m*(A*c + B*d*sin(e + f*x)**S(2) + (A*d + B*c)*sin(e + f*x)), x)
rule2842 = ReplacementRule(pattern2842, replacement2842)
pattern2843 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71)
def replacement2843(B, m, f, b, d, a, c, A, x, e):
rubi.append(2843)
return Int((a + b*cos(e + f*x))**m*(A*c + B*d*cos(e + f*x)**S(2) + (A*d + B*c)*cos(e + f*x)), x)
rule2843 = ReplacementRule(pattern2843, replacement2843)
pattern2844 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons70, cons1265)
def replacement2844(B, e, f, b, d, a, c, x, A):
rubi.append(2844)
return Dist((A*b + B*a)/(S(2)*a*b), Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Dist((A*d + B*c)/(S(2)*c*d), Int(sqrt(c + d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x)
rule2844 = ReplacementRule(pattern2844, replacement2844)
pattern2845 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons70, cons1265)
def replacement2845(B, f, b, d, a, c, A, x, e):
rubi.append(2845)
return Dist((A*b + B*a)/(S(2)*a*b), Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Dist((A*d + B*c)/(S(2)*c*d), Int(sqrt(c + d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x)
rule2845 = ReplacementRule(pattern2845, replacement2845)
pattern2846 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons70, cons1265, cons1419, cons1314)
def replacement2846(B, e, m, f, b, d, a, n, c, x, A):
rubi.append(2846)
return -Simp(B*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n + S(1))), x)
rule2846 = ReplacementRule(pattern2846, replacement2846)
pattern2847 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons70, cons1265, cons1419, cons1314)
def replacement2847(B, m, f, b, d, a, n, c, A, x, e):
rubi.append(2847)
return Simp(B*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n + S(1))), x)
rule2847 = ReplacementRule(pattern2847, replacement2847)
pattern2848 = Pattern(Integral((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons70, cons1265)
def replacement2848(B, e, f, b, d, a, c, n, x, A):
rubi.append(2848)
return Dist(B/d, Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(1)), x), x) - Dist((-A*d + B*c)/d, Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**n, x), x)
rule2848 = ReplacementRule(pattern2848, replacement2848)
pattern2849 = Pattern(Integral((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons70, cons1265)
def replacement2849(B, e, f, b, d, a, c, n, x, A):
rubi.append(2849)
return Dist(B/d, Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(1)), x), x) - Dist((-A*d + B*c)/d, Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**n, x), x)
rule2849 = ReplacementRule(pattern2849, replacement2849)
pattern2850 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons70, cons1265, cons1420, cons1421)
def replacement2850(B, e, m, f, b, d, a, n, c, x, A):
rubi.append(2850)
return Dist((A*b*(m + n + S(1)) + B*a*(m - n))/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*(A*b - B*a)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2850 = ReplacementRule(pattern2850, replacement2850)
pattern2851 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons70, cons1265, cons1420, cons1421)
def replacement2851(B, m, f, b, d, a, n, c, A, x, e):
rubi.append(2851)
return Dist((A*b*(m + n + S(1)) + B*a*(m - n))/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*(A*b - B*a)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2851 = ReplacementRule(pattern2851, replacement2851)
pattern2852 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons70, cons1265, cons1321, cons683)
def replacement2852(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2852)
return -Dist((-A*d*(m + n + S(1)) + B*c*(m - n))/(d*(m + n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) - Simp(B*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n + S(1))), x)
rule2852 = ReplacementRule(pattern2852, replacement2852)
pattern2853 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons70, cons1265, cons1321, cons683)
def replacement2853(B, m, f, b, d, a, c, n, A, x, e):
rubi.append(2853)
return -Dist((-A*d*(m + n + S(1)) + B*c*(m - n))/(d*(m + n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) + Simp(B*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n + S(1))), x)
rule2853 = ReplacementRule(pattern2853, replacement2853)
pattern2854 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1265, cons1323, cons72, cons1422)
def replacement2854(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(2854)
return Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*cos(e + f*x)/(f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2854 = ReplacementRule(pattern2854, replacement2854)
pattern2855 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1265, cons1323, cons72, cons1422)
def replacement2855(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(2855)
return -Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*sin(e + f*x)/(f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2855 = ReplacementRule(pattern2855, replacement2855)
pattern2856 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323, cons93, cons1423, cons89, cons515, cons1328)
def replacement2856(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(2856)
return -Dist(b/(d*(n + S(1))*(a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*a*d*(m - n + S(-2)) - B*(a*c*(m + S(-1)) + b*d*(n + S(1))) - (A*b*d*(m + n + S(1)) - B*(-a*d*(n + S(1)) + b*c*m))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*cos(e + f*x)/(d*f*(n + S(1))*(a*d + b*c)), x)
rule2856 = ReplacementRule(pattern2856, replacement2856)
pattern2857 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323, cons93, cons1423, cons89, cons515, cons1328)
def replacement2857(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(2857)
return -Dist(b/(d*(n + S(1))*(a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*a*d*(m - n + S(-2)) - B*(a*c*(m + S(-1)) + b*d*(n + S(1))) - (A*b*d*(m + n + S(1)) - B*(-a*d*(n + S(1)) + b*c*m))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*sin(e + f*x)/(d*f*(n + S(1))*(a*d + b*c)), x)
rule2857 = ReplacementRule(pattern2857, replacement2857)
pattern2858 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1265, cons1323, cons31, cons1423, cons346, cons515, cons1328)
def replacement2858(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(2858)
return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) + B*(a*c*(m + S(-1)) + b*d*(n + S(1))) + (A*b*d*(m + n + S(1)) - B*(-a*d*(S(2)*m + n) + b*c*m))*sin(e + f*x), x), x), x) - Simp(B*b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(1))), x)
rule2858 = ReplacementRule(pattern2858, replacement2858)
pattern2859 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1265, cons1323, cons31, cons1423, cons346, cons515, cons1328)
def replacement2859(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(2859)
return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) + B*(a*c*(m + S(-1)) + b*d*(n + S(1))) + (A*b*d*(m + n + S(1)) - B*(-a*d*(S(2)*m + n) + b*c*m))*cos(e + f*x), x), x), x) + Simp(B*b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(1))), x)
rule2859 = ReplacementRule(pattern2859, replacement2859)
pattern2860 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323, cons93, cons1320, cons88, cons515, cons1328)
def replacement2860(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(2860)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(A*(a*d*n - b*c*(m + S(1))) - B*(a*c*m + b*d*n) - d*(A*b*(m + n + S(1)) + B*a*(m - n))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*(A*b - B*a)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2860 = ReplacementRule(pattern2860, replacement2860)
pattern2861 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323, cons93, cons1320, cons88, cons515, cons1328)
def replacement2861(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(2861)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(A*(a*d*n - b*c*(m + S(1))) - B*(a*c*m + b*d*n) - d*(A*b*(m + n + S(1)) + B*a*(m - n))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*(A*b - B*a)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2861 = ReplacementRule(pattern2861, replacement2861)
pattern2862 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1265, cons1323, cons31, cons1320, cons1327, cons515, cons1328)
def replacement2862(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(2862)
return Dist(S(1)/(a*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(-a*d*(S(2)*m + n + S(2)) + b*c*(m + S(1))) + B*(a*c*m + b*d*(n + S(1))) + d*(A*b - B*a)*(m + n + S(2))*sin(e + f*x), x), x), x) + Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*b - B*a)*cos(e + f*x)/(a*f*(S(2)*m + S(1))*(-a*d + b*c)), x)
rule2862 = ReplacementRule(pattern2862, replacement2862)
pattern2863 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1265, cons1323, cons31, cons1320, cons1327, cons515, cons1328)
def replacement2863(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(2863)
return Dist(S(1)/(a*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(-a*d*(S(2)*m + n + S(2)) + b*c*(m + S(1))) + B*(a*c*m + b*d*(n + S(1))) + d*(A*b - B*a)*(m + n + S(2))*cos(e + f*x), x), x), x) - Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*b - B*a)*sin(e + f*x)/(a*f*(S(2)*m + S(1))*(-a*d + b*c)), x)
rule2863 = ReplacementRule(pattern2863, replacement2863)
pattern2864 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1265, cons1323, cons1424)
def replacement2864(B, e, f, b, d, c, a, n, x, A):
rubi.append(2864)
return Simp(-S(2)*B*b*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(3))), x)
rule2864 = ReplacementRule(pattern2864, replacement2864)
pattern2865 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1265, cons1323, cons1424)
def replacement2865(B, f, b, d, c, n, a, A, x, e):
rubi.append(2865)
return Simp(S(2)*B*b*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(3))), x)
rule2865 = ReplacementRule(pattern2865, replacement2865)
pattern2866 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323, cons87, cons89)
def replacement2866(B, e, f, b, d, c, a, n, x, A):
rubi.append(2866)
return Dist((A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))/(S(2)*d*(n + S(1))*(a*d + b*c)), Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(1)), x), x) - Simp(b**S(2)*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))*(n + S(1))*(a*d + b*c)), x)
rule2866 = ReplacementRule(pattern2866, replacement2866)
pattern2867 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323, cons87, cons89)
def replacement2867(B, f, b, d, c, n, a, A, x, e):
rubi.append(2867)
return Dist((A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))/(S(2)*d*(n + S(1))*(a*d + b*c)), Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(1)), x), x) + Simp(b**S(2)*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))*(n + S(1))*(a*d + b*c)), x)
rule2867 = ReplacementRule(pattern2867, replacement2867)
pattern2868 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1265, cons1323, cons346)
def replacement2868(B, e, f, b, d, c, a, n, x, A):
rubi.append(2868)
return Dist((A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))/(b*d*(S(2)*n + S(3))), Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**n, x), x) + Simp(-S(2)*B*b*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(3))), x)
rule2868 = ReplacementRule(pattern2868, replacement2868)
pattern2869 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1265, cons1323, cons346)
def replacement2869(B, f, b, d, c, n, a, A, x, e):
rubi.append(2869)
return Dist((A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))/(b*d*(S(2)*n + S(3))), Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**n, x), x) + Simp(S(2)*B*b*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(3))), x)
rule2869 = ReplacementRule(pattern2869, replacement2869)
pattern2870 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323)
def replacement2870(B, e, f, b, d, c, a, x, A):
rubi.append(2870)
return Dist(B/b, Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Dist((A*b - B*a)/b, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x)
rule2870 = ReplacementRule(pattern2870, replacement2870)
pattern2871 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323)
def replacement2871(B, f, b, d, c, a, A, x, e):
rubi.append(2871)
return Dist(B/b, Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Dist((A*b - B*a)/b, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x)
rule2871 = ReplacementRule(pattern2871, replacement2871)
pattern2872 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1265, cons1323, cons87, cons88, cons1425)
def replacement2872(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(2872)
return Dist(S(1)/(b*(m + n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(-1))*Simp(A*b*c*(m + n + S(1)) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + S(1)) + B*(a*d*m + b*c*n))*sin(e + f*x), x), x), x) - Simp(B*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n + S(1))), x)
rule2872 = ReplacementRule(pattern2872, replacement2872)
pattern2873 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1265, cons1323, cons87, cons88, cons1425)
def replacement2873(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(2873)
return Dist(S(1)/(b*(m + n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(-1))*Simp(A*b*c*(m + n + S(1)) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + S(1)) + B*(a*d*m + b*c*n))*cos(e + f*x), x), x), x) + Simp(B*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n + S(1))), x)
rule2873 = ReplacementRule(pattern2873, replacement2873)
pattern2874 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1265, cons1323, cons87, cons89, cons1425)
def replacement2874(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(2874)
return Dist(S(1)/(b*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*(a*d*m + b*c*(n + S(1))) - B*(a*c*m + b*d*(n + S(1))) + b*(-A*d + B*c)*(m + n + S(2))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*cos(e + f*x)/(f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2874 = ReplacementRule(pattern2874, replacement2874)
pattern2875 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1265, cons1323, cons87, cons89, cons1425)
def replacement2875(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(2875)
return Dist(S(1)/(b*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*(a*d*m + b*c*(n + S(1))) - B*(a*c*m + b*d*(n + S(1))) + b*(-A*d + B*c)*(m + n + S(2))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*sin(e + f*x)/(f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2875 = ReplacementRule(pattern2875, replacement2875)
pattern2876 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323)
def replacement2876(B, e, f, b, d, c, a, x, A):
rubi.append(2876)
return Dist((A*b - B*a)/(-a*d + b*c), Int(S(1)/sqrt(a + b*sin(e + f*x)), x), x) + Dist((-A*d + B*c)/(-a*d + b*c), Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x)
rule2876 = ReplacementRule(pattern2876, replacement2876)
pattern2877 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1265, cons1323)
def replacement2877(B, f, b, d, c, a, A, x, e):
rubi.append(2877)
return Dist((A*b - B*a)/(-a*d + b*c), Int(S(1)/sqrt(a + b*cos(e + f*x)), x), x) + Dist((-A*d + B*c)/(-a*d + b*c), Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x)
rule2877 = ReplacementRule(pattern2877, replacement2877)
pattern2878 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1265, cons1323, cons1314)
def replacement2878(B, e, m, f, b, d, c, a, x, A):
rubi.append(2878)
return Dist(B/d, Int((a + b*sin(e + f*x))**m, x), x) - Dist((-A*d + B*c)/d, Int((a + b*sin(e + f*x))**m/(c + d*sin(e + f*x)), x), x)
rule2878 = ReplacementRule(pattern2878, replacement2878)
pattern2879 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1265, cons1323, cons1314)
def replacement2879(B, m, f, b, d, c, a, A, x, e):
rubi.append(2879)
return Dist(B/d, Int((a + b*cos(e + f*x))**m, x), x) - Dist((-A*d + B*c)/d, Int((a + b*cos(e + f*x))**m/(c + d*cos(e + f*x)), x), x)
rule2879 = ReplacementRule(pattern2879, replacement2879)
pattern2880 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1265, cons1323, cons1426)
def replacement2880(B, e, m, f, b, d, c, a, n, x, A):
rubi.append(2880)
return Dist(B/b, Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Dist((A*b - B*a)/b, Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x)
rule2880 = ReplacementRule(pattern2880, replacement2880)
pattern2881 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1265, cons1323, cons1426)
def replacement2881(B, m, f, b, d, c, n, a, A, x, e):
rubi.append(2881)
return Dist(B/b, Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) + Dist((A*b - B*a)/b, Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x)
rule2881 = ReplacementRule(pattern2881, replacement2881)
pattern2882 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons93, cons166, cons89)
def replacement2882(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2882)
return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*Simp(a*d*(n + S(1))*(A*a*c + B*b*c - d*(A*b + B*a)) + b*(m + S(-1))*(-A*d + B*c)*(-a*d + b*c) + b*(-B*b*(c**S(2)*m + d**S(2)*(n + S(1))) + d*(m + n + S(1))*(-A*a*d + A*b*c + B*a*c))*sin(e + f*x)**S(2) + (-a*(n + S(2))*(-A*d + B*c)*(-a*d + b*c) + b*(n + S(1))*(a*(A*c*d + B*(c**S(2) - S(2)*d**S(2))) + b*d*(-A*d + B*c)))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*(-a*d + b*c)*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2882 = ReplacementRule(pattern2882, replacement2882)
pattern2883 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons93, cons166, cons89)
def replacement2883(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2883)
return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*Simp(a*d*(n + S(1))*(A*a*c + B*b*c - d*(A*b + B*a)) + b*(m + S(-1))*(-A*d + B*c)*(-a*d + b*c) + b*(-B*b*(c**S(2)*m + d**S(2)*(n + S(1))) + d*(m + n + S(1))*(-A*a*d + A*b*c + B*a*c))*cos(e + f*x)**S(2) + (-a*(n + S(2))*(-A*d + B*c)*(-a*d + b*c) + b*(n + S(1))*(a*(A*c*d + B*(c**S(2) - S(2)*d**S(2))) + b*d*(-A*d + B*c)))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*(-a*d + b*c)*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2883 = ReplacementRule(pattern2883, replacement2883)
pattern2884 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1267, cons1323, cons31, cons166, cons1427)
def replacement2884(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2884)
return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**n*Simp(A*a**S(2)*d*(m + n + S(1)) + B*b*(a*d*(n + S(1)) + b*c*(m + S(-1))) + b*(A*b*d*(m + n + S(1)) - B*(-a*d*(S(2)*m + n) + b*c*m))*sin(e + f*x)**S(2) + (-B*b*(a*c - b*d*(m + n)) + a*d*(S(2)*A*b + B*a)*(m + n + S(1)))*sin(e + f*x), x), x), x) - Simp(B*b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(1))), x)
rule2884 = ReplacementRule(pattern2884, replacement2884)
pattern2885 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1267, cons1323, cons31, cons166, cons1427)
def replacement2885(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2885)
return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**n*Simp(A*a**S(2)*d*(m + n + S(1)) + B*b*(a*d*(n + S(1)) + b*c*(m + S(-1))) + b*(A*b*d*(m + n + S(1)) - B*(-a*d*(S(2)*m + n) + b*c*m))*cos(e + f*x)**S(2) + (-B*b*(a*c - b*d*(m + n)) + a*d*(S(2)*A*b + B*a)*(m + n + S(1)))*cos(e + f*x), x), x), x) + Simp(B*b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(1))), x)
rule2885 = ReplacementRule(pattern2885, replacement2885)
pattern2886 = Pattern(Integral(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1323)
def replacement2886(B, e, f, b, d, c, x, A):
rubi.append(2886)
return Dist(B*d/b**S(2), Int(sqrt(b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Int((A*c + (A*d + B*c)*sin(e + f*x))/((b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x)
rule2886 = ReplacementRule(pattern2886, replacement2886)
pattern2887 = Pattern(Integral(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1323)
def replacement2887(B, e, f, b, d, c, x, A):
rubi.append(2887)
return Dist(B*d/b**S(2), Int(sqrt(b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Int((A*c + (A*d + B*c)*cos(e + f*x))/((b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x)
rule2887 = ReplacementRule(pattern2887, replacement2887)
pattern2888 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323)
def replacement2888(B, e, f, b, d, c, a, x, A):
rubi.append(2888)
return Dist(B/b, Int(sqrt(c + d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) + Dist((A*b - B*a)/b, Int(sqrt(c + d*sin(e + f*x))/(a + b*sin(e + f*x))**(S(3)/2), x), x)
rule2888 = ReplacementRule(pattern2888, replacement2888)
pattern2889 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323)
def replacement2889(B, f, b, d, c, a, A, x, e):
rubi.append(2889)
return Dist(B/b, Int(sqrt(c + d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) + Dist((A*b - B*a)/b, Int(sqrt(c + d*cos(e + f*x))/(a + b*cos(e + f*x))**(S(3)/2), x), x)
rule2889 = ReplacementRule(pattern2889, replacement2889)
pattern2890 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1267)
def replacement2890(B, e, f, b, d, a, x, A):
rubi.append(2890)
return Dist(d/(a**S(2) - b**S(2)), Int((A*b - B*a + (A*a - B*b)*sin(e + f*x))/((d*sin(e + f*x))**(S(3)/2)*sqrt(a + b*sin(e + f*x))), x), x) + Simp(S(2)*(A*b - B*a)*cos(e + f*x)/(f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*(a**S(2) - b**S(2))), x)
rule2890 = ReplacementRule(pattern2890, replacement2890)
pattern2891 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1267)
def replacement2891(B, f, b, d, a, A, x, e):
rubi.append(2891)
return Dist(d/(a**S(2) - b**S(2)), Int((A*b - B*a + (A*a - B*b)*cos(e + f*x))/((d*cos(e + f*x))**(S(3)/2)*sqrt(a + b*cos(e + f*x))), x), x) + Simp(-S(2)*(A*b - B*a)*sin(e + f*x)/(f*sqrt(d*cos(e + f*x))*sqrt(a + b*cos(e + f*x))*(a**S(2) - b**S(2))), x)
rule2891 = ReplacementRule(pattern2891, replacement2891)
pattern2892 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1323, cons1428, cons1342)
def replacement2892(B, f, b, d, c, A, x, e):
rubi.append(2892)
return Simp(-S(2)*A*sqrt(c*(S(1) - S(1)/sin(e + f*x))/(c + d))*sqrt(c*(S(1) + S(1)/sin(e + f*x))/(c - d))*(c - d)*EllipticE(asin(sqrt(c + d*sin(e + f*x))/(sqrt(b*sin(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt((c + d)/b, S(2))*tan(e + f*x)/(b*c**S(2)*f), x)
rule2892 = ReplacementRule(pattern2892, replacement2892)
pattern2893 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1323, cons1428, cons1342)
def replacement2893(B, f, b, d, c, A, x, e):
rubi.append(2893)
return Simp(S(2)*A*sqrt(c*(S(1) - S(1)/cos(e + f*x))/(c + d))*sqrt(c*(S(1) + S(1)/cos(e + f*x))/(c - d))*(c - d)*EllipticE(asin(sqrt(c + d*cos(e + f*x))/(sqrt(b*cos(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt((c + d)/b, S(2))/(b*c**S(2)*f*tan(e + f*x)), x)
rule2893 = ReplacementRule(pattern2893, replacement2893)
pattern2894 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1323, cons1428, cons1344)
def replacement2894(B, f, b, d, c, A, x, e):
rubi.append(2894)
return -Dist(sqrt(-b*sin(e + f*x))/sqrt(b*sin(e + f*x)), Int((A + B*sin(e + f*x))/((-b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x)
rule2894 = ReplacementRule(pattern2894, replacement2894)
pattern2895 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1323, cons1428, cons1344)
def replacement2895(B, f, b, d, c, A, x, e):
rubi.append(2895)
return -Dist(sqrt(-b*cos(e + f*x))/sqrt(b*cos(e + f*x)), Int((A + B*cos(e + f*x))/((-b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x)
rule2895 = ReplacementRule(pattern2895, replacement2895)
pattern2896 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons1428, cons1345)
def replacement2896(B, f, b, d, a, c, A, x, e):
rubi.append(2896)
return Simp(-S(2)*A*sqrt((-a*d + b*c)*(sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c - d)))*sqrt(-(-a*d + b*c)*(-sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c + d)))*(a + b*sin(e + f*x))*(c - d)*EllipticE(asin(sqrt(c + d*sin(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*sin(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(f*(-a*d + b*c)**S(2)*Rt((a + b)/(c + d), S(2))*cos(e + f*x)), x)
rule2896 = ReplacementRule(pattern2896, replacement2896)
pattern2897 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons1428, cons1345)
def replacement2897(B, f, b, d, a, c, A, x, e):
rubi.append(2897)
return Simp(S(2)*A*sqrt((-a*d + b*c)*(cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c - d)))*sqrt(-(-a*d + b*c)*(-cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c + d)))*(a + b*cos(e + f*x))*(c - d)*EllipticE(asin(sqrt(c + d*cos(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*cos(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(f*(-a*d + b*c)**S(2)*Rt((a + b)/(c + d), S(2))*sin(e + f*x)), x)
rule2897 = ReplacementRule(pattern2897, replacement2897)
pattern2898 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons1428, cons1346)
def replacement2898(B, f, b, d, a, c, A, x, e):
rubi.append(2898)
return Dist(sqrt(-c - d*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), Int((A + B*sin(e + f*x))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(-c - d*sin(e + f*x))), x), x)
rule2898 = ReplacementRule(pattern2898, replacement2898)
pattern2899 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons1428, cons1346)
def replacement2899(B, f, b, d, a, c, A, x, e):
rubi.append(2899)
return Dist(sqrt(-c - d*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), Int((A + B*cos(e + f*x))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(-c - d*cos(e + f*x))), x), x)
rule2899 = ReplacementRule(pattern2899, replacement2899)
pattern2900 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons1429)
def replacement2900(B, e, f, b, d, a, c, x, A):
rubi.append(2900)
return Dist((A - B)/(a - b), Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) - Dist((A*b - B*a)/(a - b), Int((sin(e + f*x) + S(1))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x)
rule2900 = ReplacementRule(pattern2900, replacement2900)
pattern2901 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons1429)
def replacement2901(B, e, f, b, d, a, c, x, A):
rubi.append(2901)
return Dist((A - B)/(a - b), Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) - Dist((A*b - B*a)/(a - b), Int((cos(e + f*x) + S(1))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x)
rule2901 = ReplacementRule(pattern2901, replacement2901)
pattern2902 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons93, cons94, cons88)
def replacement2902(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2902)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(c*(m + S(1))*(A*a - B*b) + d*n*(A*b - B*a) - d*(A*b - B*a)*(m + n + S(2))*sin(e + f*x)**S(2) + (-c*(m + S(2))*(A*b - B*a) + d*(m + S(1))*(A*a - B*b))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*(-A*b + B*a)*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2902 = ReplacementRule(pattern2902, replacement2902)
pattern2903 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons93, cons94, cons88)
def replacement2903(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2903)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(c*(m + S(1))*(A*a - B*b) + d*n*(A*b - B*a) - d*(A*b - B*a)*(m + n + S(2))*cos(e + f*x)**S(2) + (-c*(m + S(2))*(A*b - B*a) + d*(m + S(1))*(A*a - B*b))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*(-A*b + B*a)*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2903 = ReplacementRule(pattern2903, replacement2903)
pattern2904 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1267, cons1323, cons31, cons94, cons1337)
def replacement2904(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2904)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(b*d*(A*b - B*a)*(m + n + S(2)) - b*d*(A*b - B*a)*(m + n + S(3))*sin(e + f*x)**S(2) + (m + S(1))*(A*a - B*b)*(-a*d + b*c) + (A*b - B*a)*(a*d*(m + S(1)) - b*c*(m + S(2)))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(1))*(A*b**S(2) - B*a*b)*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule2904 = ReplacementRule(pattern2904, replacement2904)
pattern2905 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons71, cons1267, cons1323, cons31, cons94, cons1337)
def replacement2905(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2905)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(b*d*(A*b - B*a)*(m + n + S(2)) - b*d*(A*b - B*a)*(m + n + S(3))*cos(e + f*x)**S(2) + (m + S(1))*(A*a - B*b)*(-a*d + b*c) + (A*b - B*a)*(a*d*(m + S(1)) - b*c*(m + S(2)))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(1))*(A*b**S(2) - B*a*b)*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule2905 = ReplacementRule(pattern2905, replacement2905)
pattern2906 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323)
def replacement2906(B, e, f, b, d, a, c, x, A):
rubi.append(2906)
return Dist((A*b - B*a)/(-a*d + b*c), Int(S(1)/(a + b*sin(e + f*x)), x), x) + Dist((-A*d + B*c)/(-a*d + b*c), Int(S(1)/(c + d*sin(e + f*x)), x), x)
rule2906 = ReplacementRule(pattern2906, replacement2906)
pattern2907 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323)
def replacement2907(B, e, f, b, d, a, c, x, A):
rubi.append(2907)
return Dist((A*b - B*a)/(-a*d + b*c), Int(S(1)/(a + b*cos(e + f*x)), x), x) + Dist((-A*d + B*c)/(-a*d + b*c), Int(S(1)/(c + d*cos(e + f*x)), x), x)
rule2907 = ReplacementRule(pattern2907, replacement2907)
pattern2908 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1267, cons1323)
def replacement2908(B, e, m, f, b, d, a, c, x, A):
rubi.append(2908)
return Dist(B/d, Int((a + b*sin(e + f*x))**m, x), x) - Dist((-A*d + B*c)/d, Int((a + b*sin(e + f*x))**m/(c + d*sin(e + f*x)), x), x)
rule2908 = ReplacementRule(pattern2908, replacement2908)
pattern2909 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons71, cons1267, cons1323)
def replacement2909(B, e, m, f, b, d, a, c, x, A):
rubi.append(2909)
return Dist(B/d, Int((a + b*cos(e + f*x))**m, x), x) - Dist((-A*d + B*c)/d, Int((a + b*cos(e + f*x))**m/(c + d*cos(e + f*x)), x), x)
rule2909 = ReplacementRule(pattern2909, replacement2909)
pattern2910 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons87, cons1430)
def replacement2910(B, e, f, b, d, a, c, n, x, A):
rubi.append(2910)
return Dist(S(1)/(S(2)*n + S(3)), Int((c + d*sin(e + f*x))**(n + S(-1))*Simp(A*a*c*(S(2)*n + S(3)) + B*(S(2)*a*d*n + b*c) + (A*(S(2)*n + S(3))*(a*d + b*c) + B*(S(2)*n + S(1))*(a*c + b*d))*sin(e + f*x) + (A*b*d*(S(2)*n + S(3)) + B*(a*d + S(2)*b*c*n))*sin(e + f*x)**S(2), x)/sqrt(a + b*sin(e + f*x)), x), x) + Simp(-S(2)*B*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(S(2)*n + S(3))), x)
rule2910 = ReplacementRule(pattern2910, replacement2910)
pattern2911 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons87, cons1430)
def replacement2911(B, e, f, b, d, a, c, n, x, A):
rubi.append(2911)
return Dist(S(1)/(S(2)*n + S(3)), Int((c + d*cos(e + f*x))**(n + S(-1))*Simp(A*a*c*(S(2)*n + S(3)) + B*(S(2)*a*d*n + b*c) + (A*(S(2)*n + S(3))*(a*d + b*c) + B*(S(2)*n + S(1))*(a*c + b*d))*cos(e + f*x) + (A*b*d*(S(2)*n + S(3)) + B*(a*d + S(2)*b*c*n))*cos(e + f*x)**S(2), x)/sqrt(a + b*cos(e + f*x)), x), x) + Simp(S(2)*B*sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(S(2)*n + S(3))), x)
rule2911 = ReplacementRule(pattern2911, replacement2911)
pattern2912 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons105, cons1411, cons1428)
def replacement2912(B, f, b, a, A, x, e):
rubi.append(2912)
return Simp(S(4)*A*EllipticPi(S(-1), -asin(cos(e + f*x)/(sin(e + f*x) + S(1))), -(a - b)/(a + b))/(f*sqrt(a + b)), x)
rule2912 = ReplacementRule(pattern2912, replacement2912)
pattern2913 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons105, cons1411, cons1428)
def replacement2913(B, f, b, a, A, x, e):
rubi.append(2913)
return Simp(S(4)*A*EllipticPi(S(-1), asin(sin(e + f*x)/(cos(e + f*x) + S(1))), -(a - b)/(a + b))/(f*sqrt(a + b)), x)
rule2913 = ReplacementRule(pattern2913, replacement2913)
pattern2914 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(d_*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons27, cons34, cons35, cons105, cons1411, cons1428)
def replacement2914(B, f, b, d, a, A, x, e):
rubi.append(2914)
return Dist(sqrt(sin(e + f*x))/sqrt(d*sin(e + f*x)), Int((A + B*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*sqrt(sin(e + f*x))), x), x)
rule2914 = ReplacementRule(pattern2914, replacement2914)
pattern2915 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(d_*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons48, cons125, cons27, cons34, cons35, cons105, cons1411, cons1428)
def replacement2915(B, f, b, d, a, A, x, e):
rubi.append(2915)
return Dist(sqrt(cos(e + f*x))/sqrt(d*cos(e + f*x)), Int((A + B*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*sqrt(cos(e + f*x))), x), x)
rule2915 = ReplacementRule(pattern2915, replacement2915)
pattern2916 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323)
def replacement2916(B, e, f, b, d, c, a, x, A):
rubi.append(2916)
return Dist(B/d, Int(sqrt(c + d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) - Dist((-A*d + B*c)/d, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x)
rule2916 = ReplacementRule(pattern2916, replacement2916)
pattern2917 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323)
def replacement2917(B, f, b, d, c, a, A, x, e):
rubi.append(2917)
return Dist(B/d, Int(sqrt(c + d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) - Dist((-A*d + B*c)/d, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x)
rule2917 = ReplacementRule(pattern2917, replacement2917)
pattern2918 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1267, cons1323)
def replacement2918(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2918)
return Int((A + B*sin(e + f*x))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x)
rule2918 = ReplacementRule(pattern2918, replacement2918)
pattern2919 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons71, cons1267, cons1323)
def replacement2919(B, e, m, f, b, d, a, c, n, x, A):
rubi.append(2919)
return Int((A + B*cos(e + f*x))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x)
rule2919 = ReplacementRule(pattern2919, replacement2919)
pattern2920 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons5, cons70, cons1265)
def replacement2920(B, p, e, m, f, b, d, a, n, c, x, A):
rubi.append(2920)
return Dist(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))/(f*cos(e + f*x)), Subst(Int((A + B*x)**p*(a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2), x), x, sin(e + f*x)), x)
rule2920 = ReplacementRule(pattern2920, replacement2920)
pattern2921 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons5, cons70, cons1265)
def replacement2921(B, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(2921)
return -Dist(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))/(f*sin(e + f*x)), Subst(Int((A + B*x)**p*(a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2), x), x, cos(e + f*x)), x)
rule2921 = ReplacementRule(pattern2921, replacement2921)
pattern2922 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons35, cons36, cons21, cons1431)
def replacement2922(B, C, m, f, b, x, e):
rubi.append(2922)
return Dist(S(1)/b, Int((b*sin(e + f*x))**(m + S(1))*(B + C*sin(e + f*x)), x), x)
rule2922 = ReplacementRule(pattern2922, replacement2922)
pattern2923 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons35, cons36, cons21, cons1431)
def replacement2923(B, C, m, f, b, x, e):
rubi.append(2923)
return Dist(S(1)/b, Int((b*cos(e + f*x))**(m + S(1))*(B + C*cos(e + f*x)), x), x)
rule2923 = ReplacementRule(pattern2923, replacement2923)
pattern2924 = Pattern(Integral((A_ + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons48, cons125, cons34, cons36, cons1228)
def replacement2924(C, m, f, A, x, e):
rubi.append(2924)
return -Dist(S(1)/f, Subst(Int((-x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(A - C*x**S(2) + C), x), x, cos(e + f*x)), x)
rule2924 = ReplacementRule(pattern2924, replacement2924)
pattern2925 = Pattern(Integral((A_ + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons48, cons125, cons34, cons36, cons1228)
def replacement2925(C, m, f, A, x, e):
rubi.append(2925)
return Dist(S(1)/f, Subst(Int((-x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(A - C*x**S(2) + C), x), x, sin(e + f*x)), x)
rule2925 = ReplacementRule(pattern2925, replacement2925)
pattern2926 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(A_ + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons1432)
def replacement2926(C, m, f, b, A, x, e):
rubi.append(2926)
return Simp(A*(b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(1))), x)
rule2926 = ReplacementRule(pattern2926, replacement2926)
pattern2927 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(A_ + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons1432)
def replacement2927(C, m, f, b, A, x, e):
rubi.append(2927)
return -Simp(A*(b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(1))), x)
rule2927 = ReplacementRule(pattern2927, replacement2927)
pattern2928 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(A_ + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons31, cons94)
def replacement2928(C, m, f, b, A, x, e):
rubi.append(2928)
return Dist((A*(m + S(2)) + C*(m + S(1)))/(b**S(2)*(m + S(1))), Int((b*sin(e + f*x))**(m + S(2)), x), x) + Simp(A*(b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(1))), x)
rule2928 = ReplacementRule(pattern2928, replacement2928)
pattern2929 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(A_ + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons31, cons94)
def replacement2929(C, m, f, b, A, x, e):
rubi.append(2929)
return Dist((A*(m + S(2)) + C*(m + S(1)))/(b**S(2)*(m + S(1))), Int((b*cos(e + f*x))**(m + S(2)), x), x) - Simp(A*(b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(1))), x)
rule2929 = ReplacementRule(pattern2929, replacement2929)
pattern2930 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons272)
def replacement2930(C, m, f, b, A, x, e):
rubi.append(2930)
return Dist((A*(m + S(2)) + C*(m + S(1)))/(m + S(2)), Int((b*sin(e + f*x))**m, x), x) - Simp(C*(b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x)
rule2930 = ReplacementRule(pattern2930, replacement2930)
pattern2931 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons272)
def replacement2931(C, m, f, b, A, x, e):
rubi.append(2931)
return Dist((A*(m + S(2)) + C*(m + S(1)))/(m + S(2)), Int((b*cos(e + f*x))**m, x), x) + Simp(C*(b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x)
rule2931 = ReplacementRule(pattern2931, replacement2931)
pattern2932 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons33)
def replacement2932(B, C, e, m, f, b, a, x, A):
rubi.append(2932)
return Dist(b**(S(-2)), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b*sin(e + f*x), x), x), x)
rule2932 = ReplacementRule(pattern2932, replacement2932)
pattern2933 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons33)
def replacement2933(B, C, m, f, b, a, A, x, e):
rubi.append(2933)
return Dist(b**(S(-2)), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b*cos(e + f*x), x), x), x)
rule2933 = ReplacementRule(pattern2933, replacement2933)
pattern2934 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1433)
def replacement2934(C, e, m, f, b, a, x, A):
rubi.append(2934)
return Dist(C/b**S(2), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(-a + b*sin(e + f*x), x), x), x)
rule2934 = ReplacementRule(pattern2934, replacement2934)
pattern2935 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1433)
def replacement2935(C, m, f, b, a, A, x, e):
rubi.append(2935)
return Dist(C/b**S(2), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(-a + b*cos(e + f*x), x), x), x)
rule2935 = ReplacementRule(pattern2935, replacement2935)
pattern2936 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1434, cons77)
def replacement2936(B, C, e, m, f, b, a, x, A):
rubi.append(2936)
return Dist(C, Int((a + b*sin(e + f*x))**m*(sin(e + f*x) + S(1))**S(2), x), x) + Dist(A - C, Int((a + b*sin(e + f*x))**m*(sin(e + f*x) + S(1)), x), x)
rule2936 = ReplacementRule(pattern2936, replacement2936)
pattern2937 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1434, cons77)
def replacement2937(B, C, m, f, b, a, A, x, e):
rubi.append(2937)
return Dist(C, Int((a + b*cos(e + f*x))**m*(cos(e + f*x) + S(1))**S(2), x), x) + Dist(A - C, Int((a + b*cos(e + f*x))**m*(cos(e + f*x) + S(1)), x), x)
rule2937 = ReplacementRule(pattern2937, replacement2937)
pattern2938 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1435, cons77)
def replacement2938(C, e, m, f, b, a, x, A):
rubi.append(2938)
return Dist(C, Int((a + b*sin(e + f*x))**m*(sin(e + f*x) + S(1))**S(2), x), x) + Dist(A - C, Int((a + b*sin(e + f*x))**m*(sin(e + f*x) + S(1)), x), x)
rule2938 = ReplacementRule(pattern2938, replacement2938)
pattern2939 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1435, cons77)
def replacement2939(C, m, f, b, a, A, x, e):
rubi.append(2939)
return Dist(C, Int((a + b*cos(e + f*x))**m*(cos(e + f*x) + S(1))**S(2), x), x) + Dist(A - C, Int((a + b*cos(e + f*x))**m*(cos(e + f*x) + S(1)), x), x)
rule2939 = ReplacementRule(pattern2939, replacement2939)
pattern2940 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons31, cons94, cons1265)
def replacement2940(B, C, e, m, f, b, a, x, A):
rubi.append(2940)
return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(A*a*(m + S(1)) + C*b*(S(2)*m + S(1))*sin(e + f*x) + m*(B*b - C*a), x), x), x) + Simp((a + b*sin(e + f*x))**m*(A*b - B*a + C*b)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2940 = ReplacementRule(pattern2940, replacement2940)
pattern2941 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons31, cons94, cons1265)
def replacement2941(B, C, m, f, b, a, A, x, e):
rubi.append(2941)
return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(A*a*(m + S(1)) + C*b*(S(2)*m + S(1))*cos(e + f*x) + m*(B*b - C*a), x), x), x) - Simp((a + b*cos(e + f*x))**m*(A*b - B*a + C*b)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2941 = ReplacementRule(pattern2941, replacement2941)
pattern2942 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons31, cons94, cons1265)
def replacement2942(C, e, m, f, b, a, x, A):
rubi.append(2942)
return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(A*a*(m + S(1)) - C*a*m + C*b*(S(2)*m + S(1))*sin(e + f*x), x), x), x) + Simp(b*(A + C)*(a + b*sin(e + f*x))**m*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2942 = ReplacementRule(pattern2942, replacement2942)
pattern2943 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons31, cons94, cons1265)
def replacement2943(C, m, f, b, a, A, x, e):
rubi.append(2943)
return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(A*a*(m + S(1)) - C*a*m + C*b*(S(2)*m + S(1))*cos(e + f*x), x), x), x) - Simp(b*(A + C)*(a + b*cos(e + f*x))**m*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule2943 = ReplacementRule(pattern2943, replacement2943)
pattern2944 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons31, cons94, cons1267)
def replacement2944(B, C, e, m, f, b, a, x, A):
rubi.append(2944)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(A*a - B*b + C*a) - (A*b**S(2) - B*a*b + C*a**S(2) + b*(m + S(1))*(A*b - B*a + C*b))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*cos(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2944 = ReplacementRule(pattern2944, replacement2944)
pattern2945 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons31, cons94, cons1267)
def replacement2945(B, C, e, m, f, b, a, x, A):
rubi.append(2945)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(A*a - B*b + C*a) - (A*b**S(2) - B*a*b + C*a**S(2) + b*(m + S(1))*(A*b - B*a + C*b))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*sin(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2945 = ReplacementRule(pattern2945, replacement2945)
pattern2946 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons31, cons94, cons1267)
def replacement2946(C, e, m, f, b, a, x, A):
rubi.append(2946)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(a*b*(A + C)*(m + S(1)) - (A*b**S(2) + C*a**S(2) + b**S(2)*(A + C)*(m + S(1)))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*cos(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2946 = ReplacementRule(pattern2946, replacement2946)
pattern2947 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons31, cons94, cons1267)
def replacement2947(C, m, f, b, a, A, x, e):
rubi.append(2947)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(a*b*(A + C)*(m + S(1)) - (A*b**S(2) + C*a**S(2) + b**S(2)*(A + C)*(m + S(1)))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*sin(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2947 = ReplacementRule(pattern2947, replacement2947)
pattern2948 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons272)
def replacement2948(B, C, m, f, b, a, A, x, e):
rubi.append(2948)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*Simp(A*b*(m + S(2)) + C*b*(m + S(1)) + (B*b*(m + S(2)) - C*a)*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x)
rule2948 = ReplacementRule(pattern2948, replacement2948)
pattern2949 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons272)
def replacement2949(B, C, m, f, b, a, A, x, e):
rubi.append(2949)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*Simp(A*b*(m + S(2)) + C*b*(m + S(1)) + (B*b*(m + S(2)) - C*a)*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x)
rule2949 = ReplacementRule(pattern2949, replacement2949)
pattern2950 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons272)
def replacement2950(C, e, m, f, b, a, x, A):
rubi.append(2950)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*Simp(A*b*(m + S(2)) - C*a*sin(e + f*x) + C*b*(m + S(1)), x), x), x) - Simp(C*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x)
rule2950 = ReplacementRule(pattern2950, replacement2950)
pattern2951 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons272)
def replacement2951(C, m, f, b, a, A, x, e):
rubi.append(2951)
return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*Simp(A*b*(m + S(2)) - C*a*cos(e + f*x) + C*b*(m + S(1)), x), x), x) + Simp(C*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x)
rule2951 = ReplacementRule(pattern2951, replacement2951)
pattern2952 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons5, cons18)
def replacement2952(B, C, e, p, m, f, b, x, A):
rubi.append(2952)
return Dist((b*sin(e + f*x))**(-m*p)*(b*sin(e + f*x)**p)**m, Int((b*sin(e + f*x))**(m*p)*(A + B*sin(e + f*x) + C*sin(e + f*x)**S(2)), x), x)
rule2952 = ReplacementRule(pattern2952, replacement2952)
pattern2953 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons5, cons18)
def replacement2953(B, C, e, p, m, f, b, x, A):
rubi.append(2953)
return Dist((b*cos(e + f*x))**(-m*p)*(b*cos(e + f*x)**p)**m, Int((b*cos(e + f*x))**(m*p)*(A + B*cos(e + f*x) + C*cos(e + f*x)**S(2)), x), x)
rule2953 = ReplacementRule(pattern2953, replacement2953)
pattern2954 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons5, cons18)
def replacement2954(C, e, p, m, f, b, x, A):
rubi.append(2954)
return Dist((b*sin(e + f*x))**(-m*p)*(b*sin(e + f*x)**p)**m, Int((b*sin(e + f*x))**(m*p)*(A + C*sin(e + f*x)**S(2)), x), x)
rule2954 = ReplacementRule(pattern2954, replacement2954)
pattern2955 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons5, cons18)
def replacement2955(C, e, p, m, f, b, x, A):
rubi.append(2955)
return Dist((b*cos(e + f*x))**(-m*p)*(b*cos(e + f*x)**p)**m, Int((b*cos(e + f*x))**(m*p)*(A + C*cos(e + f*x)**S(2)), x), x)
rule2955 = ReplacementRule(pattern2955, replacement2955)
pattern2956 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons31, cons94)
def replacement2956(B, C, m, f, b, d, c, a, A, x, e):
rubi.append(2956)
return -Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(-C*b*d*(a**S(2) - b**S(2))*(m + S(1))*sin(e + f*x)**S(2) + b*(m + S(1))*(-A*b*(a*c - b*d) + (B*b - C*a)*(-a*d + b*c)) + (B*b*(a**S(2)*d - a*b*c*(m + S(2)) + b**S(2)*d*(m + S(1))) + (-a*d + b*c)*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1)))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c)*(A*b**S(2) - B*a*b + C*a**S(2))*cos(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2956 = ReplacementRule(pattern2956, replacement2956)
pattern2957 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons31, cons94)
def replacement2957(B, C, e, m, f, b, d, a, c, x, A):
rubi.append(2957)
return -Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(-C*b*d*(a**S(2) - b**S(2))*(m + S(1))*cos(e + f*x)**S(2) + b*(m + S(1))*(-A*b*(a*c - b*d) + (B*b - C*a)*(-a*d + b*c)) + (B*b*(a**S(2)*d - a*b*c*(m + S(2)) + b**S(2)*d*(m + S(1))) + (-a*d + b*c)*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1)))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c)*(A*b**S(2) - B*a*b + C*a**S(2))*sin(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2957 = ReplacementRule(pattern2957, replacement2957)
pattern2958 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons31, cons94)
def replacement2958(C, m, f, b, d, c, a, A, x, e):
rubi.append(2958)
return Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(C*b*d*(a**S(2) - b**S(2))*(m + S(1))*sin(e + f*x)**S(2) + b*(m + S(1))*(A*b*(a*c - b*d) + C*a*(-a*d + b*c)) - (-a*d + b*c)*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*(-a*d + b*c)*cos(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2958 = ReplacementRule(pattern2958, replacement2958)
pattern2959 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons31, cons94)
def replacement2959(C, e, m, f, b, d, a, c, x, A):
rubi.append(2959)
return Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(C*b*d*(a**S(2) - b**S(2))*(m + S(1))*cos(e + f*x)**S(2) + b*(m + S(1))*(A*b*(a*c - b*d) + C*a*(-a*d + b*c)) - (-a*d + b*c)*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*(-a*d + b*c)*sin(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule2959 = ReplacementRule(pattern2959, replacement2959)
pattern2960 = Pattern(Integral((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons71, cons1267, cons272)
def replacement2960(B, C, m, f, b, d, a, c, A, x, e):
rubi.append(2960)
return Dist(S(1)/(b*(m + S(3))), Int((a + b*sin(e + f*x))**m*Simp(A*b*c*(m + S(3)) + C*a*d + b*(B*c*(m + S(3)) + d*(A*(m + S(3)) + C*(m + S(2))))*sin(e + f*x) - (S(2)*C*a*d - b*(m + S(3))*(B*d + C*c))*sin(e + f*x)**S(2), x), x), x) - Simp(C*d*(a + b*sin(e + f*x))**(m + S(1))*sin(e + f*x)*cos(e + f*x)/(b*f*(m + S(3))), x)
rule2960 = ReplacementRule(pattern2960, replacement2960)
pattern2961 = Pattern(Integral((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons71, cons1267, cons272)
def replacement2961(B, C, m, f, b, d, a, c, A, x, e):
rubi.append(2961)
return Dist(S(1)/(b*(m + S(3))), Int((a + b*cos(e + f*x))**m*Simp(A*b*c*(m + S(3)) + C*a*d + b*(B*c*(m + S(3)) + d*(A*(m + S(3)) + C*(m + S(2))))*cos(e + f*x) - (S(2)*C*a*d - b*(m + S(3))*(B*d + C*c))*cos(e + f*x)**S(2), x), x), x) + Simp(C*d*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)*cos(e + f*x)/(b*f*(m + S(3))), x)
rule2961 = ReplacementRule(pattern2961, replacement2961)
pattern2962 = Pattern(Integral((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons71, cons1267, cons272)
def replacement2962(C, m, f, b, d, a, c, A, x, e):
rubi.append(2962)
return Dist(S(1)/(b*(m + S(3))), Int((a + b*sin(e + f*x))**m*Simp(A*b*c*(m + S(3)) + C*a*d + b*d*(A*(m + S(3)) + C*(m + S(2)))*sin(e + f*x) - (S(2)*C*a*d - C*b*c*(m + S(3)))*sin(e + f*x)**S(2), x), x), x) - Simp(C*d*(a + b*sin(e + f*x))**(m + S(1))*sin(e + f*x)*cos(e + f*x)/(b*f*(m + S(3))), x)
rule2962 = ReplacementRule(pattern2962, replacement2962)
pattern2963 = Pattern(Integral((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons71, cons1267, cons272)
def replacement2963(C, m, f, b, d, a, c, A, x, e):
rubi.append(2963)
return Dist(S(1)/(b*(m + S(3))), Int((a + b*cos(e + f*x))**m*Simp(A*b*c*(m + S(3)) + C*a*d + b*d*(A*(m + S(3)) + C*(m + S(2)))*cos(e + f*x) - (S(2)*C*a*d - C*b*c*(m + S(3)))*cos(e + f*x)**S(2), x), x), x) + Simp(C*d*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)*cos(e + f*x)/(b*f*(m + S(3))), x)
rule2963 = ReplacementRule(pattern2963, replacement2963)
pattern2964 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons70, cons1265, cons1436)
def replacement2964(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2964)
return -Dist(S(1)/(S(2)*b*c*d*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(c**S(2)*(m + S(1)) + d**S(2)*(S(2)*m + n + S(2))) - B*c*d*(m - n + S(-1)) - C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-C*c*(S(3)*m - n) + (A*c + B*d)*(m + n + S(2)))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*a - B*b + C*a)*cos(e + f*x)/(S(2)*b*c*f*(S(2)*m + S(1))), x)
rule2964 = ReplacementRule(pattern2964, replacement2964)
pattern2965 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons70, cons1265, cons1436)
def replacement2965(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2965)
return -Dist(S(1)/(S(2)*b*c*d*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(c**S(2)*(m + S(1)) + d**S(2)*(S(2)*m + n + S(2))) - B*c*d*(m - n + S(-1)) - C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-C*c*(S(3)*m - n) + (A*c + B*d)*(m + n + S(2)))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*a - B*b + C*a)*sin(e + f*x)/(S(2)*b*c*f*(S(2)*m + S(1))), x)
rule2965 = ReplacementRule(pattern2965, replacement2965)
pattern2966 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons70, cons1265, cons1436)
def replacement2966(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2966)
return -Dist(S(1)/(S(2)*b*c*d*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(c**S(2)*(m + S(1)) + d**S(2)*(S(2)*m + n + S(2))) - C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(A*c*(m + n + S(2)) - C*c*(S(3)*m - n))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*a + C*a)*cos(e + f*x)/(S(2)*b*c*f*(S(2)*m + S(1))), x)
rule2966 = ReplacementRule(pattern2966, replacement2966)
pattern2967 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons70, cons1265, cons1436)
def replacement2967(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2967)
return -Dist(S(1)/(S(2)*b*c*d*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(c**S(2)*(m + S(1)) + d**S(2)*(S(2)*m + n + S(2))) - C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(A*c*(m + n + S(2)) - C*c*(S(3)*m - n))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*a + C*a)*sin(e + f*x)/(S(2)*b*c*f*(S(2)*m + S(1))), x)
rule2967 = ReplacementRule(pattern2967, replacement2967)
pattern2968 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons70, cons1265, cons1321)
def replacement2968(B, C, m, f, b, d, a, c, A, x, e):
rubi.append(2968)
return Int((a + b*sin(e + f*x))**m*Simp(A + B*sin(e + f*x) + C, x)/sqrt(c + d*sin(e + f*x)), x) + Simp(-S(2)*C*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*sqrt(c + d*sin(e + f*x))*(S(2)*m + S(3))), x)
rule2968 = ReplacementRule(pattern2968, replacement2968)
pattern2969 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons70, cons1265, cons1321)
def replacement2969(B, C, m, f, b, d, a, c, A, x, e):
rubi.append(2969)
return Int((a + b*cos(e + f*x))**m*Simp(A + B*cos(e + f*x) + C, x)/sqrt(c + d*cos(e + f*x)), x) + Simp(S(2)*C*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*sqrt(c + d*cos(e + f*x))*(S(2)*m + S(3))), x)
rule2969 = ReplacementRule(pattern2969, replacement2969)
pattern2970 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons70, cons1265, cons1321)
def replacement2970(C, m, f, b, d, a, c, A, x, e):
rubi.append(2970)
return Dist(A + C, Int((a + b*sin(e + f*x))**m/sqrt(c + d*sin(e + f*x)), x), x) + Simp(-S(2)*C*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*sqrt(c + d*sin(e + f*x))*(S(2)*m + S(3))), x)
rule2970 = ReplacementRule(pattern2970, replacement2970)
pattern2971 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons70, cons1265, cons1321)
def replacement2971(C, m, f, b, d, a, c, A, x, e):
rubi.append(2971)
return Dist(A + C, Int((a + b*cos(e + f*x))**m/sqrt(c + d*cos(e + f*x)), x), x) + Simp(S(2)*C*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*sqrt(c + d*cos(e + f*x))*(S(2)*m + S(3))), x)
rule2971 = ReplacementRule(pattern2971, replacement2971)
pattern2972 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons70, cons1265, cons1321, cons214)
def replacement2972(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2972)
return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + (B*b*d*(m + n + S(2)) - C*b*c*(S(2)*m + S(1)))*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x)
rule2972 = ReplacementRule(pattern2972, replacement2972)
pattern2973 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons70, cons1265, cons1321, cons214)
def replacement2973(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2973)
return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + (B*b*d*(m + n + S(2)) - C*b*c*(S(2)*m + S(1)))*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x)
rule2973 = ReplacementRule(pattern2973, replacement2973)
pattern2974 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons70, cons1265, cons1321, cons214)
def replacement2974(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2974)
return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) - C*b*c*(S(2)*m + S(1))*sin(e + f*x) + C*(a*c*m + b*d*(n + S(1))), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x)
rule2974 = ReplacementRule(pattern2974, replacement2974)
pattern2975 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons70, cons1265, cons1321, cons214)
def replacement2975(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2975)
return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) - C*b*c*(S(2)*m + S(1))*cos(e + f*x) + C*(a*c*m + b*d*(n + S(1))), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x)
rule2975 = ReplacementRule(pattern2975, replacement2975)
pattern2976 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1265, cons1323, cons31, cons1320)
def replacement2976(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2976)
return Dist(S(1)/(b*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(a*c*(m + S(1)) - b*d*(S(2)*m + n + S(2))) + B*(a*d*(n + S(1)) + b*c*m) - C*(a*c*m + b*d*(n + S(1))) + (C*(-a*d*(m - n + S(-1)) + b*c*(S(2)*m + S(1))) + d*(A*a - B*b)*(m + n + S(2)))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*a - B*b + C*a)*cos(e + f*x)/(f*(S(2)*m + S(1))*(-a*d + b*c)), x)
rule2976 = ReplacementRule(pattern2976, replacement2976)
pattern2977 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1265, cons1323, cons31, cons1320)
def replacement2977(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2977)
return Dist(S(1)/(b*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(a*c*(m + S(1)) - b*d*(S(2)*m + n + S(2))) + B*(a*d*(n + S(1)) + b*c*m) - C*(a*c*m + b*d*(n + S(1))) + (C*(-a*d*(m - n + S(-1)) + b*c*(S(2)*m + S(1))) + d*(A*a - B*b)*(m + n + S(2)))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*a - B*b + C*a)*sin(e + f*x)/(f*(S(2)*m + S(1))*(-a*d + b*c)), x)
rule2977 = ReplacementRule(pattern2977, replacement2977)
pattern2978 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1265, cons1323, cons31, cons1320)
def replacement2978(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2978)
return Dist(S(1)/(b*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(a*c*(m + S(1)) - b*d*(S(2)*m + n + S(2))) - C*(a*c*m + b*d*(n + S(1))) + (A*a*d*(m + n + S(2)) + C*(-a*d*(m - n + S(-1)) + b*c*(S(2)*m + S(1))))*sin(e + f*x), x), x), x) + Simp(a*(A + C)*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(f*(S(2)*m + S(1))*(-a*d + b*c)), x)
rule2978 = ReplacementRule(pattern2978, replacement2978)
pattern2979 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1265, cons1323, cons31, cons1320)
def replacement2979(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2979)
return Dist(S(1)/(b*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(a*c*(m + S(1)) - b*d*(S(2)*m + n + S(2))) - C*(a*c*m + b*d*(n + S(1))) + (A*a*d*(m + n + S(2)) + C*(-a*d*(m - n + S(-1)) + b*c*(S(2)*m + S(1))))*cos(e + f*x), x), x), x) - Simp(a*(A + C)*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(f*(S(2)*m + S(1))*(-a*d + b*c)), x)
rule2979 = ReplacementRule(pattern2979, replacement2979)
pattern2980 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons71, cons1265, cons1323, cons1321, cons1437)
def replacement2980(B, C, m, f, b, d, c, a, n, A, x, e):
rubi.append(2980)
return Dist(S(1)/(b*d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*d*(a*d*m + b*c*(n + S(1))) + b*(-C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(2)))*sin(e + f*x) + (-B*d + C*c)*(a*c*m + b*d*(n + S(1))), x), x), x) - Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2980 = ReplacementRule(pattern2980, replacement2980)
pattern2981 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons71, cons1265, cons1323, cons1321, cons1437)
def replacement2981(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2981)
return Dist(S(1)/(b*d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*d*(a*d*m + b*c*(n + S(1))) + b*(-C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(2)))*cos(e + f*x) + (-B*d + C*c)*(a*c*m + b*d*(n + S(1))), x), x), x) + Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2981 = ReplacementRule(pattern2981, replacement2981)
pattern2982 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons71, cons1265, cons1323, cons1321, cons1437)
def replacement2982(C, m, f, b, d, c, a, n, A, x, e):
rubi.append(2982)
return Dist(S(1)/(b*d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*d*(a*d*m + b*c*(n + S(1))) + C*c*(a*c*m + b*d*(n + S(1))) - b*(A*d**S(2)*(m + n + S(2)) + C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2982 = ReplacementRule(pattern2982, replacement2982)
pattern2983 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons71, cons1265, cons1323, cons1321, cons1437)
def replacement2983(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2983)
return Dist(S(1)/(b*d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*d*(a*d*m + b*c*(n + S(1))) + C*c*(a*c*m + b*d*(n + S(1))) - b*(A*d**S(2)*(m + n + S(2)) + C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2983 = ReplacementRule(pattern2983, replacement2983)
pattern2984 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons71, cons1265, cons1323, cons1321, cons214)
def replacement2984(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2984)
return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + (B*b*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1))))*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x)
rule2984 = ReplacementRule(pattern2984, replacement2984)
pattern2985 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons71, cons1265, cons1323, cons1321, cons214)
def replacement2985(B, C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2985)
return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + (B*b*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1))))*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x)
rule2985 = ReplacementRule(pattern2985, replacement2985)
pattern2986 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons71, cons1265, cons1323, cons1321, cons214)
def replacement2986(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2986)
return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + C*(a*d*m - b*c*(m + S(1)))*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x)
rule2986 = ReplacementRule(pattern2986, replacement2986)
pattern2987 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons71, cons1265, cons1323, cons1321, cons214)
def replacement2987(C, m, f, b, d, c, n, a, A, x, e):
rubi.append(2987)
return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + C*(a*d*m - b*c*(m + S(1)))*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x)
rule2987 = ReplacementRule(pattern2987, replacement2987)
pattern2988 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323, cons93, cons168, cons89)
def replacement2988(B, C, m, f, b, d, c, a, n, A, x, e):
rubi.append(2988)
return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*d*(a*c*(n + S(1)) + b*d*m) + b*(-C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(2)))*sin(e + f*x)**S(2) + (-B*d + C*c)*(a*d*(n + S(1)) + b*c*m) - (-C*(-a*(c**S(2) + d**S(2)*(n + S(1))) + b*c*d*(n + S(1))) + d*(A*(a*d*(n + S(2)) - b*c*(n + S(1))) + B*(-a*c*(n + S(2)) + b*d*(n + S(1)))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2988 = ReplacementRule(pattern2988, replacement2988)
pattern2989 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323, cons93, cons168, cons89)
def replacement2989(B, C, e, m, f, b, d, a, c, n, x, A):
rubi.append(2989)
return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*d*(a*c*(n + S(1)) + b*d*m) + b*(-C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(2)))*cos(e + f*x)**S(2) + (-B*d + C*c)*(a*d*(n + S(1)) + b*c*m) - (-C*(-a*(c**S(2) + d**S(2)*(n + S(1))) + b*c*d*(n + S(1))) + d*(A*(a*d*(n + S(2)) - b*c*(n + S(1))) + B*(-a*c*(n + S(2)) + b*d*(n + S(1)))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2989 = ReplacementRule(pattern2989, replacement2989)
pattern2990 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323, cons93, cons168, cons89)
def replacement2990(C, m, f, b, d, c, a, n, A, x, e):
rubi.append(2990)
return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*d*(a*c*(n + S(1)) + b*d*m) + C*c*(a*d*(n + S(1)) + b*c*m) - b*(A*d**S(2)*(m + n + S(2)) + C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))))*sin(e + f*x)**S(2) - (A*d*(a*d*(n + S(2)) - b*c*(n + S(1))) - C*(-a*(c**S(2) + d**S(2)*(n + S(1))) + b*c*d*(n + S(1))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2990 = ReplacementRule(pattern2990, replacement2990)
pattern2991 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323, cons93, cons168, cons89)
def replacement2991(C, e, m, f, b, d, a, c, n, x, A):
rubi.append(2991)
return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*d*(a*c*(n + S(1)) + b*d*m) + C*c*(a*d*(n + S(1)) + b*c*m) - b*(A*d**S(2)*(m + n + S(2)) + C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))))*cos(e + f*x)**S(2) - (A*d*(a*d*(n + S(2)) - b*c*(n + S(1))) - C*(-a*(c**S(2) + d**S(2)*(n + S(1))) + b*c*d*(n + S(1))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x)
rule2991 = ReplacementRule(pattern2991, replacement2991)
pattern2992 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1267, cons1323, cons31, cons168, cons1438)
def replacement2992(B, C, m, f, b, d, a, c, n, A, x, e):
rubi.append(2992)
return Dist(S(1)/(d*(m + n + S(2))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*Simp(A*a*d*(m + n + S(2)) + C*(a*d*(n + S(1)) + b*c*m) + (-C*(a*c - b*d*(m + n + S(1))) + d*(A*b + B*a)*(m + n + S(2)))*sin(e + f*x) + (B*b*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1))))*sin(e + f*x)**S(2), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x)
rule2992 = ReplacementRule(pattern2992, replacement2992)
pattern2993 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1267, cons1323, cons31, cons168, cons1438)
def replacement2993(B, C, m, f, b, d, a, c, n, A, x, e):
rubi.append(2993)
return Dist(S(1)/(d*(m + n + S(2))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*Simp(A*a*d*(m + n + S(2)) + C*(a*d*(n + S(1)) + b*c*m) + (-C*(a*c - b*d*(m + n + S(1))) + d*(A*b + B*a)*(m + n + S(2)))*cos(e + f*x) + (B*b*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1))))*cos(e + f*x)**S(2), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x)
rule2993 = ReplacementRule(pattern2993, replacement2993)
pattern2994 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1267, cons1323, cons31, cons168, cons1438)
def replacement2994(C, m, f, b, d, a, c, n, A, x, e):
rubi.append(2994)
return Dist(S(1)/(d*(m + n + S(2))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*Simp(A*a*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1)))*sin(e + f*x)**S(2) + C*(a*d*(n + S(1)) + b*c*m) + (A*b*d*(m + n + S(2)) - C*(a*c - b*d*(m + n + S(1))))*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x)
rule2994 = ReplacementRule(pattern2994, replacement2994)
pattern2995 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1267, cons1323, cons31, cons168, cons1438)
def replacement2995(C, m, f, b, d, a, c, n, A, x, e):
rubi.append(2995)
return Dist(S(1)/(d*(m + n + S(2))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*Simp(A*a*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1)))*cos(e + f*x)**S(2) + C*(a*d*(n + S(1)) + b*c*m) + (A*b*d*(m + n + S(2)) - C*(a*c - b*d*(m + n + S(1))))*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x)
rule2995 = ReplacementRule(pattern2995, replacement2995)
pattern2996 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267)
def replacement2996(B, C, e, f, b, d, a, x, A):
rubi.append(2996)
return Dist(S(1)/b, Int((A*b + (B*b - C*a)*sin(e + f*x))/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))**(S(3)/2)), x), x) + Dist(C/(b*d), Int(sqrt(d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x)
rule2996 = ReplacementRule(pattern2996, replacement2996)
pattern2997 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267)
def replacement2997(B, C, f, b, d, a, A, x, e):
rubi.append(2997)
return Dist(S(1)/b, Int((A*b + (B*b - C*a)*cos(e + f*x))/(sqrt(d*cos(e + f*x))*(a + b*cos(e + f*x))**(S(3)/2)), x), x) + Dist(C/(b*d), Int(sqrt(d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x)
rule2997 = ReplacementRule(pattern2997, replacement2997)
pattern2998 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267)
def replacement2998(C, e, f, b, d, a, x, A):
rubi.append(2998)
return Dist(S(1)/b, Int((A*b - C*a*sin(e + f*x))/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))**(S(3)/2)), x), x) + Dist(C/(b*d), Int(sqrt(d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x)
rule2998 = ReplacementRule(pattern2998, replacement2998)
pattern2999 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267)
def replacement2999(C, f, b, d, a, A, x, e):
rubi.append(2999)
return Dist(S(1)/b, Int((A*b - C*a*cos(e + f*x))/(sqrt(d*cos(e + f*x))*(a + b*cos(e + f*x))**(S(3)/2)), x), x) + Dist(C/(b*d), Int(sqrt(d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x)
rule2999 = ReplacementRule(pattern2999, replacement2999)
pattern3000 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323)
def replacement3000(B, C, f, b, d, c, a, A, x, e):
rubi.append(3000)
return Dist(b**(S(-2)), Int((A*b**S(2) - C*a**S(2) + b*(B*b - S(2)*C*a)*sin(e + f*x))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) + Dist(C/b**S(2), Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x)
rule3000 = ReplacementRule(pattern3000, replacement3000)
pattern3001 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323)
def replacement3001(B, C, e, f, b, d, a, c, x, A):
rubi.append(3001)
return Dist(b**(S(-2)), Int((A*b**S(2) - C*a**S(2) + b*(B*b - S(2)*C*a)*cos(e + f*x))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Dist(C/b**S(2), Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x)
rule3001 = ReplacementRule(pattern3001, replacement3001)
pattern3002 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323)
def replacement3002(C, f, b, d, c, a, A, x, e):
rubi.append(3002)
return Dist(b**(S(-2)), Int((A*b**S(2) - C*a**S(2) - S(2)*C*a*b*sin(e + f*x))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) + Dist(C/b**S(2), Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x)
rule3002 = ReplacementRule(pattern3002, replacement3002)
pattern3003 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323)
def replacement3003(C, e, f, b, d, a, c, x, A):
rubi.append(3003)
return Dist(b**(S(-2)), Int((A*b**S(2) - C*a**S(2) - S(2)*C*a*b*cos(e + f*x))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Dist(C/b**S(2), Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x)
rule3003 = ReplacementRule(pattern3003, replacement3003)
pattern3004 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1267, cons1323, cons31, cons94, cons1337)
def replacement3004(B, C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3004)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(d*(m + n + S(2))*(A*b**S(2) - B*a*b + C*a**S(2)) - d*(m + n + S(3))*(A*b**S(2) - B*a*b + C*a**S(2))*sin(e + f*x)**S(2) + (m + S(1))*(-a*d + b*c)*(A*a - B*b + C*a) - (c*(A*b**S(2) - B*a*b + C*a**S(2)) + (m + S(1))*(-a*d + b*c)*(A*b - B*a + C*b))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3004 = ReplacementRule(pattern3004, replacement3004)
pattern3005 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons71, cons1267, cons1323, cons31, cons94, cons1337)
def replacement3005(B, C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3005)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(d*(m + n + S(2))*(A*b**S(2) - B*a*b + C*a**S(2)) - d*(m + n + S(3))*(A*b**S(2) - B*a*b + C*a**S(2))*cos(e + f*x)**S(2) + (m + S(1))*(-a*d + b*c)*(A*a - B*b + C*a) - (c*(A*b**S(2) - B*a*b + C*a**S(2)) + (m + S(1))*(-a*d + b*c)*(A*b - B*a + C*b))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3005 = ReplacementRule(pattern3005, replacement3005)
pattern3006 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1267, cons1323, cons31, cons94, cons1337)
def replacement3006(C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3006)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(a*(A + C)*(m + S(1))*(-a*d + b*c) + d*(A*b**S(2) + C*a**S(2))*(m + n + S(2)) - d*(A*b**S(2) + C*a**S(2))*(m + n + S(3))*sin(e + f*x)**S(2) - (b*(A + C)*(m + S(1))*(-a*d + b*c) + c*(A*b**S(2) + C*a**S(2)))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(1))*(A*b**S(2) + C*a**S(2))*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3006 = ReplacementRule(pattern3006, replacement3006)
pattern3007 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons4, cons71, cons1267, cons1323, cons31, cons94, cons1337)
def replacement3007(C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3007)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(a*(A + C)*(m + S(1))*(-a*d + b*c) + d*(A*b**S(2) + C*a**S(2))*(m + n + S(2)) - d*(A*b**S(2) + C*a**S(2))*(m + n + S(3))*cos(e + f*x)**S(2) - (b*(A + C)*(m + S(1))*(-a*d + b*c) + c*(A*b**S(2) + C*a**S(2)))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(1))*(A*b**S(2) + C*a**S(2))*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x)
rule3007 = ReplacementRule(pattern3007, replacement3007)
pattern3008 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323)
def replacement3008(B, C, f, b, d, c, a, A, x, e):
rubi.append(3008)
return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(b*(-a*d + b*c)), Int(S(1)/(a + b*sin(e + f*x)), x), x) - Dist((A*d**S(2) - B*c*d + C*c**S(2))/(d*(-a*d + b*c)), Int(S(1)/(c + d*sin(e + f*x)), x), x) + Simp(C*x/(b*d), x)
rule3008 = ReplacementRule(pattern3008, replacement3008)
pattern3009 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323)
def replacement3009(B, C, f, b, d, c, a, A, x, e):
rubi.append(3009)
return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(b*(-a*d + b*c)), Int(S(1)/(a + b*cos(e + f*x)), x), x) - Dist((A*d**S(2) - B*c*d + C*c**S(2))/(d*(-a*d + b*c)), Int(S(1)/(c + d*cos(e + f*x)), x), x) + Simp(C*x/(b*d), x)
rule3009 = ReplacementRule(pattern3009, replacement3009)
pattern3010 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323)
def replacement3010(C, f, b, d, c, a, A, x, e):
rubi.append(3010)
return Dist((A*b**S(2) + C*a**S(2))/(b*(-a*d + b*c)), Int(S(1)/(a + b*sin(e + f*x)), x), x) - Dist((A*d**S(2) + C*c**S(2))/(d*(-a*d + b*c)), Int(S(1)/(c + d*sin(e + f*x)), x), x) + Simp(C*x/(b*d), x)
rule3010 = ReplacementRule(pattern3010, replacement3010)
pattern3011 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323)
def replacement3011(C, f, b, d, c, a, A, x, e):
rubi.append(3011)
return Dist((A*b**S(2) + C*a**S(2))/(b*(-a*d + b*c)), Int(S(1)/(a + b*cos(e + f*x)), x), x) - Dist((A*d**S(2) + C*c**S(2))/(d*(-a*d + b*c)), Int(S(1)/(c + d*cos(e + f*x)), x), x) + Simp(C*x/(b*d), x)
rule3011 = ReplacementRule(pattern3011, replacement3011)
pattern3012 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323)
def replacement3012(B, C, f, b, d, c, a, A, x, e):
rubi.append(3012)
return -Dist(S(1)/(b*d), Int(Simp(-A*b*d + C*a*c + (-B*b*d + C*a*d + C*b*c)*sin(e + f*x), x)/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) + Dist(C/(b*d), Int(sqrt(a + b*sin(e + f*x)), x), x)
rule3012 = ReplacementRule(pattern3012, replacement3012)
pattern3013 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323)
def replacement3013(B, C, e, f, b, d, a, c, x, A):
rubi.append(3013)
return -Dist(S(1)/(b*d), Int(Simp(-A*b*d + C*a*c + (-B*b*d + C*a*d + C*b*c)*cos(e + f*x), x)/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) + Dist(C/(b*d), Int(sqrt(a + b*cos(e + f*x)), x), x)
rule3013 = ReplacementRule(pattern3013, replacement3013)
pattern3014 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323)
def replacement3014(C, f, b, d, c, a, A, x, e):
rubi.append(3014)
return -Dist(S(1)/(b*d), Int(Simp(-A*b*d + C*a*c + (C*a*d + C*b*c)*sin(e + f*x), x)/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) + Dist(C/(b*d), Int(sqrt(a + b*sin(e + f*x)), x), x)
rule3014 = ReplacementRule(pattern3014, replacement3014)
pattern3015 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323)
def replacement3015(C, e, f, b, d, a, c, x, A):
rubi.append(3015)
return -Dist(S(1)/(b*d), Int(Simp(-A*b*d + C*a*c + (C*a*d + C*b*c)*cos(e + f*x), x)/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) + Dist(C/(b*d), Int(sqrt(a + b*cos(e + f*x)), x), x)
rule3015 = ReplacementRule(pattern3015, replacement3015)
pattern3016 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323)
def replacement3016(B, C, f, b, d, a, c, A, x, e):
rubi.append(3016)
return Dist(S(1)/(S(2)*d), Int(Simp(S(2)*A*a*d - C*(-a*d + b*c) + (S(2)*B*b*d - C*(a*d + b*c))*sin(e + f*x)**S(2) - S(2)*(C*a*c - d*(A*b + B*a))*sin(e + f*x), x)/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) - Simp(C*sqrt(c + d*sin(e + f*x))*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))), x)
rule3016 = ReplacementRule(pattern3016, replacement3016)
pattern3017 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons71, cons1267, cons1323)
def replacement3017(B, C, e, f, b, d, a, c, x, A):
rubi.append(3017)
return Dist(S(1)/(S(2)*d), Int(Simp(S(2)*A*a*d - C*(-a*d + b*c) + (S(2)*B*b*d - C*(a*d + b*c))*cos(e + f*x)**S(2) - S(2)*(C*a*c - d*(A*b + B*a))*cos(e + f*x), x)/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Simp(C*sqrt(c + d*cos(e + f*x))*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))), x)
rule3017 = ReplacementRule(pattern3017, replacement3017)
pattern3018 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323)
def replacement3018(C, f, b, d, a, c, A, x, e):
rubi.append(3018)
return Dist(S(1)/(S(2)*d), Int(Simp(S(2)*A*a*d - C*(-a*d + b*c) - C*(a*d + b*c)*sin(e + f*x)**S(2) - S(2)*(-A*b*d + C*a*c)*sin(e + f*x), x)/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) - Simp(C*sqrt(c + d*sin(e + f*x))*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))), x)
rule3018 = ReplacementRule(pattern3018, replacement3018)
pattern3019 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons71, cons1267, cons1323)
def replacement3019(C, e, f, b, d, a, c, x, A):
rubi.append(3019)
return Dist(S(1)/(S(2)*d), Int(Simp(S(2)*A*a*d - C*(-a*d + b*c) - C*(a*d + b*c)*cos(e + f*x)**S(2) - S(2)*(-A*b*d + C*a*c)*cos(e + f*x), x)/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Simp(C*sqrt(c + d*cos(e + f*x))*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))), x)
rule3019 = ReplacementRule(pattern3019, replacement3019)
pattern3020 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons71, cons1267, cons1323)
def replacement3020(B, C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3020)
return Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*(A + B*sin(e + f*x) + C*sin(e + f*x)**S(2)), x)
rule3020 = ReplacementRule(pattern3020, replacement3020)
pattern3021 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons71, cons1267, cons1323)
def replacement3021(B, C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3021)
return Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*(A + B*cos(e + f*x) + C*cos(e + f*x)**S(2)), x)
rule3021 = ReplacementRule(pattern3021, replacement3021)
pattern3022 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons71, cons1267, cons1323)
def replacement3022(C, m, f, b, d, c, a, n, A, x, e):
rubi.append(3022)
return Int((A + C*sin(e + f*x)**S(2))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x)
rule3022 = ReplacementRule(pattern3022, replacement3022)
pattern3023 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons71, cons1267, cons1323)
def replacement3023(C, e, m, f, b, d, a, c, n, x, A):
rubi.append(3023)
return Int((A + C*cos(e + f*x)**S(2))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x)
rule3023 = ReplacementRule(pattern3023, replacement3023)
pattern3024 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons18)
def replacement3024(B, C, p, m, f, b, d, c, n, A, x, e):
rubi.append(3024)
return Dist((b*sin(e + f*x))**(-m*p)*(b*sin(e + f*x)**p)**m, Int((b*sin(e + f*x))**(m*p)*(c + d*sin(e + f*x))**n*(A + B*sin(e + f*x) + C*sin(e + f*x)**S(2)), x), x)
rule3024 = ReplacementRule(pattern3024, replacement3024)
pattern3025 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons18)
def replacement3025(B, C, e, p, m, f, b, d, c, n, x, A):
rubi.append(3025)
return Dist((b*cos(e + f*x))**(-m*p)*(b*cos(e + f*x)**p)**m, Int((b*cos(e + f*x))**(m*p)*(c + d*cos(e + f*x))**n*(A + B*cos(e + f*x) + C*cos(e + f*x)**S(2)), x), x)
rule3025 = ReplacementRule(pattern3025, replacement3025)
pattern3026 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons5, cons18)
def replacement3026(C, p, m, f, b, d, c, n, A, x, e):
rubi.append(3026)
return Dist((b*sin(e + f*x))**(-m*p)*(b*sin(e + f*x)**p)**m, Int((b*sin(e + f*x))**(m*p)*(A + C*sin(e + f*x)**S(2))*(c + d*sin(e + f*x))**n, x), x)
rule3026 = ReplacementRule(pattern3026, replacement3026)
pattern3027 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons5, cons18)
def replacement3027(C, e, p, m, f, b, d, c, n, x, A):
rubi.append(3027)
return Dist((b*cos(e + f*x))**(-m*p)*(b*cos(e + f*x)**p)**m, Int((b*cos(e + f*x))**(m*p)*(A + C*cos(e + f*x)**S(2))*(c + d*cos(e + f*x))**n, x), x)
rule3027 = ReplacementRule(pattern3027, replacement3027)
pattern3028 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1439)
def replacement3028(b, d, c, a, n, x):
rubi.append(3028)
return Simp(a*(a*cos(c + d*x) + b*sin(c + d*x))**n/(b*d*n), x)
rule3028 = ReplacementRule(pattern3028, replacement3028)
pattern3029 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1440, cons521)
def replacement3029(b, d, c, a, n, x):
rubi.append(3029)
return -Dist(S(1)/d, Subst(Int((a**S(2) + b**S(2) - x**S(2))**(n/S(2) + S(-1)/2), x), x, -a*sin(c + d*x) + b*cos(c + d*x)), x)
rule3029 = ReplacementRule(pattern3029, replacement3029)
pattern3030 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1440, cons1441, cons87, cons165)
def replacement3030(b, d, c, a, n, x):
rubi.append(3030)
return Dist((a**S(2) + b**S(2))*(n + S(-1))/n, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2)), x), x) - Simp((-a*sin(c + d*x) + b*cos(c + d*x))*(a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))/(d*n), x)
rule3030 = ReplacementRule(pattern3030, replacement3030)
pattern3031 = Pattern(Integral(S(1)/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440)
def replacement3031(b, d, c, a, x):
rubi.append(3031)
return -Dist(S(1)/d, Subst(Int(S(1)/(a**S(2) + b**S(2) - x**S(2)), x), x, -a*sin(c + d*x) + b*cos(c + d*x)), x)
rule3031 = ReplacementRule(pattern3031, replacement3031)
pattern3032 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**(S(-2)), x_), cons2, cons3, cons7, cons27, cons1440)
def replacement3032(b, d, c, a, x):
rubi.append(3032)
return Simp(sin(c + d*x)/(a*d*(a*cos(c + d*x) + b*sin(c + d*x))), x)
rule3032 = ReplacementRule(pattern3032, replacement3032)
pattern3033 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1440, cons87, cons89, cons1442)
def replacement3033(b, d, c, a, n, x):
rubi.append(3033)
return Dist((n + S(2))/((a**S(2) + b**S(2))*(n + S(1))), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2)), x), x) + Simp((-a*sin(c + d*x) + b*cos(c + d*x))*(a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))/(d*(a**S(2) + b**S(2))*(n + S(1))), x)
rule3033 = ReplacementRule(pattern3033, replacement3033)
pattern3034 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1443, cons1444)
def replacement3034(b, d, c, a, n, x):
rubi.append(3034)
return Dist((a**S(2) + b**S(2))**(n/S(2)), Int(cos(c + d*x - ArcTan(a, b))**n, x), x)
rule3034 = ReplacementRule(pattern3034, replacement3034)
pattern3035 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1443, cons1445)
def replacement3035(b, d, c, a, n, x):
rubi.append(3035)
return Dist(((a*cos(c + d*x) + b*sin(c + d*x))/sqrt(a**S(2) + b**S(2)))**(-n)*(a*cos(c + d*x) + b*sin(c + d*x))**n, Int(cos(c + d*x - ArcTan(a, b))**n, x), x)
rule3035 = ReplacementRule(pattern3035, replacement3035)
pattern3036 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1255, cons1439, cons87, cons165)
def replacement3036(m, b, d, c, a, n, x):
rubi.append(3036)
return Dist(S(2)*b, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*sin(c + d*x)**(-n + S(1)), x), x) - Simp(a*(a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*sin(c + d*x)**(-n + S(1))/(d*(n + S(-1))), x)
rule3036 = ReplacementRule(pattern3036, replacement3036)
pattern3037 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1255, cons1439, cons87, cons165)
def replacement3037(m, b, d, c, a, n, x):
rubi.append(3037)
return Dist(S(2)*a, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)**(-n + S(1)), x), x) + Simp(b*(a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)**(-n + S(1))/(d*(n + S(-1))), x)
rule3037 = ReplacementRule(pattern3037, replacement3037)
pattern3038 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1255, cons1439, cons87, cons463)
def replacement3038(m, b, d, c, a, n, x):
rubi.append(3038)
return Dist(S(1)/(S(2)*b), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))*sin(c + d*x)**(-n + S(-1)), x), x) + Simp(a*(a*cos(c + d*x) + b*sin(c + d*x))**n*sin(c + d*x)**(-n)/(S(2)*b*d*n), x)
rule3038 = ReplacementRule(pattern3038, replacement3038)
pattern3039 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1255, cons1439, cons87, cons463)
def replacement3039(m, b, d, c, a, n, x):
rubi.append(3039)
return Dist(S(1)/(S(2)*a), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))*cos(c + d*x)**(-n + S(-1)), x), x) - Simp(b*(a*cos(c + d*x) + b*sin(c + d*x))**n*cos(c + d*x)**(-n)/(S(2)*a*d*n), x)
rule3039 = ReplacementRule(pattern3039, replacement3039)
pattern3040 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1255, cons1439, cons23)
def replacement3040(m, b, d, c, a, n, x):
rubi.append(3040)
return Simp(a*(a*cos(c + d*x) + b*sin(c + d*x))**n*Hypergeometric2F1(S(1), n, n + S(1), (a/tan(c + d*x) + b)/(S(2)*b))*sin(c + d*x)**(-n)/(S(2)*b*d*n), x)
rule3040 = ReplacementRule(pattern3040, replacement3040)
pattern3041 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1255, cons1439, cons23)
def replacement3041(m, b, d, c, a, n, x):
rubi.append(3041)
return -Simp(b*(a*cos(c + d*x) + b*sin(c + d*x))**n*Hypergeometric2F1(S(1), n, n + S(1), (a + b*tan(c + d*x))/(S(2)*a))*cos(c + d*x)**(-n)/(S(2)*a*d*n), x)
rule3041 = ReplacementRule(pattern3041, replacement3041)
pattern3042 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1255, cons85, cons1440)
def replacement3042(m, b, d, c, n, a, x):
rubi.append(3042)
return Int((a/tan(c + d*x) + b)**n, x)
rule3042 = ReplacementRule(pattern3042, replacement3042)
pattern3043 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1255, cons85, cons1440)
def replacement3043(m, b, d, c, n, a, x):
rubi.append(3043)
return Int((a + b*tan(c + d*x))**n, x)
rule3043 = ReplacementRule(pattern3043, replacement3043)
pattern3044 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons85, cons1446, cons1152, cons1447)
def replacement3044(m, b, d, c, a, n, x):
rubi.append(3044)
return Dist(S(1)/d, Subst(Int(x**m*(a + b*x)**n*(x**S(2) + S(1))**(-m/S(2) - n/S(2) + S(-1)), x), x, tan(c + d*x)), x)
rule3044 = ReplacementRule(pattern3044, replacement3044)
pattern3045 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons85, cons1446, cons1152, cons1447)
def replacement3045(m, b, d, c, a, n, x):
rubi.append(3045)
return -Dist(S(1)/d, Subst(Int(x**m*(x**S(2) + S(1))**(-m/S(2) - n/S(2) + S(-1))*(a*x + b)**n, x), x, S(1)/tan(c + d*x)), x)
rule3045 = ReplacementRule(pattern3045, replacement3045)
pattern3046 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons17, cons148)
def replacement3046(m, b, d, c, a, n, x):
rubi.append(3046)
return Int(ExpandTrig((a*cos(c + d*x) + b*sin(c + d*x))**n*sin(c + d*x)**m, x), x)
rule3046 = ReplacementRule(pattern3046, replacement3046)
pattern3047 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons17, cons148)
def replacement3047(m, b, d, c, a, n, x):
rubi.append(3047)
return Int(ExpandTrig((a*cos(c + d*x) + b*sin(c + d*x))**n*cos(c + d*x)**m, x), x)
rule3047 = ReplacementRule(pattern3047, replacement3047)
pattern3048 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons1439, cons196)
def replacement3048(m, b, d, c, a, n, x):
rubi.append(3048)
return Dist(a**n*b**n, Int((a*sin(c + d*x) + b*cos(c + d*x))**(-n)*sin(c + d*x)**m, x), x)
rule3048 = ReplacementRule(pattern3048, replacement3048)
pattern3049 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons1439, cons196)
def replacement3049(m, b, d, c, a, n, x):
rubi.append(3049)
return Dist(a**n*b**n, Int((a*sin(c + d*x) + b*cos(c + d*x))**(-n)*cos(c + d*x)**m, x), x)
rule3049 = ReplacementRule(pattern3049, replacement3049)
pattern3050 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_/sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1440, cons87, cons89)
def replacement3050(b, d, c, a, n, x):
rubi.append(3050)
return Dist(a**(S(-2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2))/sin(c + d*x), x), x) - Dist(b/a**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1)), x), x) - Simp((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))/(a*d*(n + S(1))), x)
rule3050 = ReplacementRule(pattern3050, replacement3050)
pattern3051 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_/cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1440, cons87, cons89)
def replacement3051(b, d, c, a, n, x):
rubi.append(3051)
return Dist(b**(S(-2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2))/cos(c + d*x), x), x) - Dist(a/b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1)), x), x) + Simp((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))/(b*d*(n + S(1))), x)
rule3051 = ReplacementRule(pattern3051, replacement3051)
pattern3052 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1440, cons93, cons165, cons94)
def replacement3052(m, b, d, c, a, n, x):
rubi.append(3052)
return Dist(a**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2))*sin(c + d*x)**m, x), x) + Dist(S(2)*b, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*sin(c + d*x)**(m + S(1)), x), x) - Dist(a**S(2) + b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2))*sin(c + d*x)**(m + S(2)), x), x)
rule3052 = ReplacementRule(pattern3052, replacement3052)
pattern3053 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1440, cons93, cons165, cons94)
def replacement3053(m, b, d, c, a, n, x):
rubi.append(3053)
return Dist(S(2)*a, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)**(m + S(1)), x), x) + Dist(b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2))*cos(c + d*x)**m, x), x) - Dist(a**S(2) + b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2))*cos(c + d*x)**(m + S(2)), x), x)
rule3053 = ReplacementRule(pattern3053, replacement3053)
pattern3054 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440)
def replacement3054(b, d, c, a, x):
rubi.append(3054)
return -Dist(a/(a**S(2) + b**S(2)), Int((-a*sin(c + d*x) + b*cos(c + d*x))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Simp(b*x/(a**S(2) + b**S(2)), x)
rule3054 = ReplacementRule(pattern3054, replacement3054)
pattern3055 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0)))/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440)
def replacement3055(b, d, c, a, x):
rubi.append(3055)
return Dist(b/(a**S(2) + b**S(2)), Int((-a*sin(c + d*x) + b*cos(c + d*x))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Simp(a*x/(a**S(2) + b**S(2)), x)
rule3055 = ReplacementRule(pattern3055, replacement3055)
pattern3056 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440, cons31, cons166)
def replacement3056(m, b, d, c, a, x):
rubi.append(3056)
return Dist(a**S(2)/(a**S(2) + b**S(2)), Int(sin(c + d*x)**(m + S(-2))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Dist(b/(a**S(2) + b**S(2)), Int(sin(c + d*x)**(m + S(-1)), x), x) - Simp(a*sin(c + d*x)**(m + S(-1))/(d*(a**S(2) + b**S(2))*(m + S(-1))), x)
rule3056 = ReplacementRule(pattern3056, replacement3056)
pattern3057 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440, cons31, cons166)
def replacement3057(m, b, d, c, a, x):
rubi.append(3057)
return Dist(a/(a**S(2) + b**S(2)), Int(cos(c + d*x)**(m + S(-1)), x), x) + Dist(b**S(2)/(a**S(2) + b**S(2)), Int(cos(c + d*x)**(m + S(-2))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Simp(b*cos(c + d*x)**(m + S(-1))/(d*(a**S(2) + b**S(2))*(m + S(-1))), x)
rule3057 = ReplacementRule(pattern3057, replacement3057)
pattern3058 = Pattern(Integral(S(1)/((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440)
def replacement3058(b, d, c, a, x):
rubi.append(3058)
return -Dist(S(1)/a, Int((-a*sin(c + d*x) + b*cos(c + d*x))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Dist(S(1)/a, Int(S(1)/tan(c + d*x), x), x)
rule3058 = ReplacementRule(pattern3058, replacement3058)
pattern3059 = Pattern(Integral(S(1)/((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440)
def replacement3059(b, d, c, a, x):
rubi.append(3059)
return Dist(S(1)/b, Int((-a*sin(c + d*x) + b*cos(c + d*x))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Dist(S(1)/b, Int(tan(c + d*x), x), x)
rule3059 = ReplacementRule(pattern3059, replacement3059)
pattern3060 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440, cons31, cons94)
def replacement3060(m, b, d, c, a, x):
rubi.append(3060)
return -Dist(b/a**S(2), Int(sin(c + d*x)**(m + S(1)), x), x) + Dist((a**S(2) + b**S(2))/a**S(2), Int(sin(c + d*x)**(m + S(2))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Simp(sin(c + d*x)**(m + S(1))/(a*d*(m + S(1))), x)
rule3060 = ReplacementRule(pattern3060, replacement3060)
pattern3061 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440, cons31, cons94)
def replacement3061(m, b, d, c, a, x):
rubi.append(3061)
return -Dist(a/b**S(2), Int(cos(c + d*x)**(m + S(1)), x), x) + Dist((a**S(2) + b**S(2))/b**S(2), Int(cos(c + d*x)**(m + S(2))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) - Simp(cos(c + d*x)**(m + S(1))/(b*d*(m + S(1))), x)
rule3061 = ReplacementRule(pattern3061, replacement3061)
pattern3062 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1440, cons93, cons89, cons94)
def replacement3062(m, b, d, c, a, n, x):
rubi.append(3062)
return Dist(a**(S(-2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2))*sin(c + d*x)**m, x), x) - Dist(S(2)*b/a**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))*sin(c + d*x)**(m + S(1)), x), x) + Dist((a**S(2) + b**S(2))/a**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**n*sin(c + d*x)**(m + S(2)), x), x)
rule3062 = ReplacementRule(pattern3062, replacement3062)
pattern3063 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1440, cons93, cons89, cons94)
def replacement3063(m, b, d, c, a, n, x):
rubi.append(3063)
return Dist(b**(S(-2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2))*cos(c + d*x)**m, x), x) - Dist(S(2)*a/b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))*cos(c + d*x)**(m + S(1)), x), x) + Dist((a**S(2) + b**S(2))/b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**n*cos(c + d*x)**(m + S(2)), x), x)
rule3063 = ReplacementRule(pattern3063, replacement3063)
pattern3064 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons128)
def replacement3064(p, m, b, d, c, n, a, x):
rubi.append(3064)
return Int(ExpandTrig((a*cos(c + d*x) + b*sin(c + d*x))**p*sin(c + d*x)**n*cos(c + d*x)**m, x), x)
rule3064 = ReplacementRule(pattern3064, replacement3064)
pattern3065 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1439, cons63)
def replacement3065(p, m, b, d, c, n, a, x):
rubi.append(3065)
return Dist(a**p*b**p, Int((a*sin(c + d*x) + b*cos(c + d*x))**(-p)*sin(c + d*x)**n*cos(c + d*x)**m, x), x)
rule3065 = ReplacementRule(pattern3065, replacement3065)
pattern3066 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1440, cons150, cons168, cons88)
def replacement3066(m, b, d, c, n, a, x):
rubi.append(3066)
return Dist(a/(a**S(2) + b**S(2)), Int(sin(c + d*x)**n*cos(c + d*x)**(m + S(-1)), x), x) + Dist(b/(a**S(2) + b**S(2)), Int(sin(c + d*x)**(n + S(-1))*cos(c + d*x)**m, x), x) - Dist(a*b/(a**S(2) + b**S(2)), Int(sin(c + d*x)**(n + S(-1))*cos(c + d*x)**(m + S(-1))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x)
rule3066 = ReplacementRule(pattern3066, replacement3066)
pattern3067 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons150)
def replacement3067(m, b, d, c, n, a, x):
rubi.append(3067)
return Int(ExpandTrig(sin(c + d*x)**n*cos(c + d*x)**m/(a*cos(c + d*x) + b*sin(c + d*x)), x), x)
rule3067 = ReplacementRule(pattern3067, replacement3067)
pattern3068 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1440, cons375, cons168, cons88, cons322)
def replacement3068(p, m, b, d, c, n, a, x):
rubi.append(3068)
return Dist(a/(a**S(2) + b**S(2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(p + S(1))*sin(c + d*x)**n*cos(c + d*x)**(m + S(-1)), x), x) + Dist(b/(a**S(2) + b**S(2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(p + S(1))*sin(c + d*x)**(n + S(-1))*cos(c + d*x)**m, x), x) - Dist(a*b/(a**S(2) + b**S(2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**p*sin(c + d*x)**(n + S(-1))*cos(c + d*x)**(m + S(-1)), x), x)
rule3068 = ReplacementRule(pattern3068, replacement3068)
pattern3069 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1448)
def replacement3069(b, d, c, a, x, e):
rubi.append(3069)
return Simp(-S(2)*(-b*sin(d + e*x) + c*cos(d + e*x))/(e*sqrt(a + b*cos(d + e*x) + c*sin(d + e*x))), x)
rule3069 = ReplacementRule(pattern3069, replacement3069)
pattern3070 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1448, cons87, cons88)
def replacement3070(b, d, c, a, n, x, e):
rubi.append(3070)
return Dist(a*(S(2)*n + S(-1))/n, Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1)), x), x) - Simp((-b*sin(d + e*x) + c*cos(d + e*x))*(a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))/(e*n), x)
rule3070 = ReplacementRule(pattern3070, replacement3070)
pattern3071 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1448)
def replacement3071(b, d, c, a, x, e):
rubi.append(3071)
return -Simp((-a*sin(d + e*x) + c)/(c*e*(-b*sin(d + e*x) + c*cos(d + e*x))), x)
rule3071 = ReplacementRule(pattern3071, replacement3071)
pattern3072 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1448)
def replacement3072(b, d, c, a, x, e):
rubi.append(3072)
return Int(S(1)/sqrt(a + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))), x)
rule3072 = ReplacementRule(pattern3072, replacement3072)
pattern3073 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1448, cons87, cons89)
def replacement3073(b, d, c, a, n, x, e):
rubi.append(3073)
return Dist((n + S(1))/(a*(S(2)*n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1)), x), x) + Simp((-b*sin(d + e*x) + c*cos(d + e*x))*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(S(2)*n + S(1))), x)
rule3073 = ReplacementRule(pattern3073, replacement3073)
pattern3074 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1449)
def replacement3074(b, d, c, a, x, e):
rubi.append(3074)
return Dist(b/(c*e), Subst(Int(sqrt(a + x)/x, x), x, b*cos(d + e*x) + c*sin(d + e*x)), x)
rule3074 = ReplacementRule(pattern3074, replacement3074)
pattern3075 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1450, cons1451)
def replacement3075(b, d, c, a, x, e):
rubi.append(3075)
return Int(sqrt(a + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))), x)
rule3075 = ReplacementRule(pattern3075, replacement3075)
pattern3076 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1452, cons1450, cons1453)
def replacement3076(b, d, c, a, x, e):
rubi.append(3076)
return Dist(sqrt(a + b*cos(d + e*x) + c*sin(d + e*x))/sqrt((a + b*cos(d + e*x) + c*sin(d + e*x))/(a + sqrt(b**S(2) + c**S(2)))), Int(sqrt(a/(a + sqrt(b**S(2) + c**S(2))) + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))/(a + sqrt(b**S(2) + c**S(2)))), x), x)
rule3076 = ReplacementRule(pattern3076, replacement3076)
pattern3077 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1452, cons87, cons165)
def replacement3077(b, d, c, a, n, x, e):
rubi.append(3077)
return Dist(S(1)/n, Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-2))*Simp(a**S(2)*n + a*b*(S(2)*n + S(-1))*cos(d + e*x) + a*c*(S(2)*n + S(-1))*sin(d + e*x) + (b**S(2) + c**S(2))*(n + S(-1)), x), x), x) - Simp((-b*sin(d + e*x) + c*cos(d + e*x))*(a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))/(e*n), x)
rule3077 = ReplacementRule(pattern3077, replacement3077)
def With3078(b, d, c, a, x, e):
f = FreeFactors(S(1)/tan(d/S(2) + e*x/S(2)), x)
rubi.append(3078)
return -Dist(f/e, Subst(Int(S(1)/(a + c*f*x), x), x, S(1)/(f*tan(d/S(2) + e*x/S(2)))), x)
pattern3078 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1454)
rule3078 = ReplacementRule(pattern3078, With3078)
def With3079(b, d, c, a, x, e):
f = FreeFactors(tan(Pi/S(4) + d/S(2) + e*x/S(2)), x)
rubi.append(3079)
return Dist(f/e, Subst(Int(S(1)/(a + b*f*x), x), x, tan(Pi/S(4) + d/S(2) + e*x/S(2))/f), x)
pattern3079 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons76)
rule3079 = ReplacementRule(pattern3079, With3079)
def With3080(b, d, c, a, x, e):
f = FreeFactors(S(1)/tan(Pi/S(4) + d/S(2) + e*x/S(2)), x)
rubi.append(3080)
return -Dist(f/e, Subst(Int(S(1)/(a + b*f*x), x), x, S(1)/(f*tan(Pi/S(4) + d/S(2) + e*x/S(2)))), x)
pattern3080 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1455, cons1456)
rule3080 = ReplacementRule(pattern3080, With3080)
def With3081(b, d, c, a, x, e):
f = FreeFactors(tan(d/S(2) + e*x/S(2)), x)
rubi.append(3081)
return Dist(S(2)*f/e, Subst(Int(S(1)/(a + b + S(2)*c*f*x + f**S(2)*x**S(2)*(a - b)), x), x, tan(d/S(2) + e*x/S(2))/f), x)
pattern3081 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1452)
rule3081 = ReplacementRule(pattern3081, With3081)
pattern3082 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1449)
def replacement3082(b, d, c, a, x, e):
rubi.append(3082)
return Dist(b/(c*e), Subst(Int(S(1)/(x*sqrt(a + x)), x), x, b*cos(d + e*x) + c*sin(d + e*x)), x)
rule3082 = ReplacementRule(pattern3082, replacement3082)
pattern3083 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1450, cons1451)
def replacement3083(b, d, c, a, x, e):
rubi.append(3083)
return Int(S(1)/sqrt(a + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))), x)
rule3083 = ReplacementRule(pattern3083, replacement3083)
pattern3084 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1452, cons1450, cons1453)
def replacement3084(b, d, c, a, x, e):
rubi.append(3084)
return Dist(sqrt((a + b*cos(d + e*x) + c*sin(d + e*x))/(a + sqrt(b**S(2) + c**S(2))))/sqrt(a + b*cos(d + e*x) + c*sin(d + e*x)), Int(S(1)/sqrt(a/(a + sqrt(b**S(2) + c**S(2))) + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))/(a + sqrt(b**S(2) + c**S(2)))), x), x)
rule3084 = ReplacementRule(pattern3084, replacement3084)
pattern3085 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**(S(-3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons1452)
def replacement3085(b, d, c, a, x, e):
rubi.append(3085)
return Dist(S(1)/(a**S(2) - b**S(2) - c**S(2)), Int(sqrt(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp(S(2)*(-b*sin(d + e*x) + c*cos(d + e*x))/(e*sqrt(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3085 = ReplacementRule(pattern3085, replacement3085)
pattern3086 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons1452, cons87, cons89, cons1457)
def replacement3086(b, d, c, a, n, x, e):
rubi.append(3086)
return Dist(S(1)/((n + S(1))*(a**S(2) - b**S(2) - c**S(2))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*(a*(n + S(1)) - b*(n + S(2))*cos(d + e*x) - c*(n + S(2))*sin(d + e*x)), x), x) + Simp((b*sin(d + e*x) - c*cos(d + e*x))*(a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))/(e*(n + S(1))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3086 = ReplacementRule(pattern3086, replacement3086)
pattern3087 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1449)
def replacement3087(B, C, e, b, d, c, a, x, A):
rubi.append(3087)
return Simp(x*(S(2)*A*a - B*b - C*c)/(S(2)*a**S(2)), x) + Simp((-S(2)*A*a*b**S(2) + a**S(2)*(B*b - C*c) + b**S(2)*(B*b + C*c))*log(RemoveContent(a + b*cos(d + e*x) + c*sin(d + e*x), x))/(S(2)*a**S(2)*b*c*e), x) - Simp((B*b + C*c)*(b*cos(d + e*x) - c*sin(d + e*x))/(S(2)*a*b*c*e), x)
rule3087 = ReplacementRule(pattern3087, replacement3087)
pattern3088 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons1449)
def replacement3088(C, e, b, d, c, a, x, A):
rubi.append(3088)
return Simp(x*(S(2)*A*a - C*c)/(S(2)*a**S(2)), x) - Simp(C*cos(d + e*x)/(S(2)*a*e), x) + Simp((S(2)*A*a*c - C*a**S(2) + C*b**S(2))*log(RemoveContent(a + b*cos(d + e*x) + c*sin(d + e*x), x))/(S(2)*a**S(2)*b*e), x) + Simp(C*c*sin(d + e*x)/(S(2)*a*b*e), x)
rule3088 = ReplacementRule(pattern3088, replacement3088)
pattern3089 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons1449)
def replacement3089(B, b, d, c, a, A, x, e):
rubi.append(3089)
return Simp(x*(S(2)*A*a - B*b)/(S(2)*a**S(2)), x) + Simp(B*sin(d + e*x)/(S(2)*a*e), x) + Simp((-S(2)*A*a*b + B*a**S(2) + B*b**S(2))*log(RemoveContent(a + b*cos(d + e*x) + c*sin(d + e*x), x))/(S(2)*a**S(2)*c*e), x) - Simp(B*b*cos(d + e*x)/(S(2)*a*c*e), x)
rule3089 = ReplacementRule(pattern3089, replacement3089)
pattern3090 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1450, cons1458)
def replacement3090(B, C, b, d, c, a, A, x, e):
rubi.append(3090)
return Simp(x*(B*b + C*c)/(b**S(2) + c**S(2)), x) + Simp((B*c - C*b)*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x)
rule3090 = ReplacementRule(pattern3090, replacement3090)
pattern3091 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons1450, cons1459)
def replacement3091(C, b, d, c, a, A, x, e):
rubi.append(3091)
return Simp(C*c*x/(b**S(2) + c**S(2)), x) - Simp(C*b*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x)
rule3091 = ReplacementRule(pattern3091, replacement3091)
pattern3092 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons1450, cons1460)
def replacement3092(B, b, d, a, c, A, x, e):
rubi.append(3092)
return Simp(B*b*x/(b**S(2) + c**S(2)), x) + Simp(B*c*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x)
rule3092 = ReplacementRule(pattern3092, replacement3092)
pattern3093 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1450, cons1461)
def replacement3093(B, C, b, d, c, a, A, x, e):
rubi.append(3093)
return Dist((A*(b**S(2) + c**S(2)) - a*(B*b + C*c))/(b**S(2) + c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp(x*(B*b + C*c)/(b**S(2) + c**S(2)), x) + Simp((B*c - C*b)*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x)
rule3093 = ReplacementRule(pattern3093, replacement3093)
pattern3094 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons1450, cons1462)
def replacement3094(C, b, d, c, a, A, x, e):
rubi.append(3094)
return Dist((A*(b**S(2) + c**S(2)) - C*a*c)/(b**S(2) + c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) - Simp(C*b*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x) + Simp(C*c*(d + e*x)/(e*(b**S(2) + c**S(2))), x)
rule3094 = ReplacementRule(pattern3094, replacement3094)
pattern3095 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons1450, cons1463)
def replacement3095(B, b, d, a, c, A, x, e):
rubi.append(3095)
return Dist((A*(b**S(2) + c**S(2)) - B*a*b)/(b**S(2) + c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp(B*b*(d + e*x)/(e*(b**S(2) + c**S(2))), x) + Simp(B*c*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x)
rule3095 = ReplacementRule(pattern3095, replacement3095)
pattern3096 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons4, cons584, cons1448, cons1464)
def replacement3096(B, C, b, d, c, n, a, A, x, e):
rubi.append(3096)
return Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**n*(B*a*sin(d + e*x) + B*c - C*a*cos(d + e*x) - C*b)/(a*e*(n + S(1))), x)
rule3096 = ReplacementRule(pattern3096, replacement3096)
pattern3097 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons4, cons584, cons1448, cons1465)
def replacement3097(C, b, d, c, n, a, A, x, e):
rubi.append(3097)
return -Simp((C*a*cos(d + e*x) + C*b)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x)
rule3097 = ReplacementRule(pattern3097, replacement3097)
pattern3098 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons4, cons584, cons1448, cons1466)
def replacement3098(B, b, d, c, n, a, A, x, e):
rubi.append(3098)
return Simp((B*a*sin(d + e*x) + B*c)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x)
rule3098 = ReplacementRule(pattern3098, replacement3098)
pattern3099 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons4, cons584, cons1448, cons1467)
def replacement3099(B, C, b, d, c, n, a, A, x, e):
rubi.append(3099)
return Dist((A*a*(n + S(1)) + n*(B*b + C*c))/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**n, x), x) + Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**n*(B*a*sin(d + e*x) + B*c - C*a*cos(d + e*x) - C*b)/(a*e*(n + S(1))), x)
rule3099 = ReplacementRule(pattern3099, replacement3099)
pattern3100 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons4, cons584, cons1448, cons1468)
def replacement3100(C, b, d, c, n, a, A, x, e):
rubi.append(3100)
return Dist((A*a*(n + S(1)) + C*c*n)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**n, x), x) - Simp((C*a*cos(d + e*x) + C*b)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x)
rule3100 = ReplacementRule(pattern3100, replacement3100)
pattern3101 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons4, cons584, cons1448, cons1469)
def replacement3101(B, b, d, c, n, a, A, x, e):
rubi.append(3101)
return Dist((A*a*(n + S(1)) + B*b*n)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**n, x), x) + Simp((B*a*sin(d + e*x) + B*c)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x)
rule3101 = ReplacementRule(pattern3101, replacement3101)
pattern3102 = Pattern(Integral((WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons3, cons7, cons27, cons48, cons35, cons36, cons584, cons1450, cons1470)
def replacement3102(B, C, b, d, c, n, x, e):
rubi.append(3102)
return Simp((B*c - C*b)*(b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))/(e*(b**S(2) + c**S(2))*(n + S(1))), x)
rule3102 = ReplacementRule(pattern3102, replacement3102)
pattern3103 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons87, cons88, cons1452)
def replacement3103(B, C, b, d, c, n, a, A, x, e):
rubi.append(3103)
return Dist(S(1)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))*Simp(A*a**S(2)*(n + S(1)) + a*n*(B*b + C*c) + (A*a*b*(n + S(1)) + n*(B*a**S(2) - B*c**S(2) + C*b*c))*cos(d + e*x) + (A*a*c*(n + S(1)) + n*(B*b*c + C*a**S(2) - C*b**S(2)))*sin(d + e*x), x), x), x) + Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**n*(B*a*sin(d + e*x) + B*c - C*a*cos(d + e*x) - C*b)/(a*e*(n + S(1))), x)
rule3103 = ReplacementRule(pattern3103, replacement3103)
pattern3104 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons87, cons88, cons1452)
def replacement3104(C, b, d, c, n, a, A, x, e):
rubi.append(3104)
return Dist(S(1)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))*Simp(A*a**S(2)*(n + S(1)) + C*a*c*n + (A*a*b*(n + S(1)) + C*b*c*n)*cos(d + e*x) + (A*a*c*(n + S(1)) + C*a**S(2)*n - C*b**S(2)*n)*sin(d + e*x), x), x), x) - Simp((C*a*cos(d + e*x) + C*b)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x)
rule3104 = ReplacementRule(pattern3104, replacement3104)
pattern3105 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons87, cons88, cons1452)
def replacement3105(B, b, d, c, n, a, A, x, e):
rubi.append(3105)
return Dist(S(1)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))*Simp(A*a**S(2)*(n + S(1)) + B*a*b*n + (A*a*c*(n + S(1)) + B*b*c*n)*sin(d + e*x) + (A*a*b*(n + S(1)) + B*a**S(2)*n - B*c**S(2)*n)*cos(d + e*x), x), x), x) + Simp((B*a*sin(d + e*x) + B*c)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x)
rule3105 = ReplacementRule(pattern3105, replacement3105)
pattern3106 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1471, cons1245)
def replacement3106(B, C, e, b, d, c, a, x, A):
rubi.append(3106)
return Dist(B/b, Int(sqrt(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Dist((A*b - B*a)/b, Int(S(1)/sqrt(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x)
rule3106 = ReplacementRule(pattern3106, replacement3106)
pattern3107 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1452, cons1472)
def replacement3107(B, C, b, d, c, a, A, x, e):
rubi.append(3107)
return Simp((B*c - C*b + (-A*b + B*a)*sin(d + e*x) - (-A*c + C*a)*cos(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3107 = ReplacementRule(pattern3107, replacement3107)
pattern3108 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons1452, cons1473)
def replacement3108(C, b, d, c, a, A, x, e):
rubi.append(3108)
return -Simp((A*b*sin(d + e*x) + C*b + (-A*c + C*a)*cos(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3108 = ReplacementRule(pattern3108, replacement3108)
pattern3109 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons1452, cons1474)
def replacement3109(B, b, d, a, c, A, x, e):
rubi.append(3109)
return Simp((A*c*cos(d + e*x) + B*c + (-A*b + B*a)*sin(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3109 = ReplacementRule(pattern3109, replacement3109)
pattern3110 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1452, cons1475)
def replacement3110(B, C, b, d, c, a, A, x, e):
rubi.append(3110)
return Dist((A*a - B*b - C*c)/(a**S(2) - b**S(2) - c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp((B*c - C*b + (-A*b + B*a)*sin(d + e*x) - (-A*c + C*a)*cos(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3110 = ReplacementRule(pattern3110, replacement3110)
pattern3111 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons1452, cons1476)
def replacement3111(C, b, d, c, a, A, x, e):
rubi.append(3111)
return Dist((A*a - C*c)/(a**S(2) - b**S(2) - c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) - Simp((A*b*sin(d + e*x) + C*b + (-A*c + C*a)*cos(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3111 = ReplacementRule(pattern3111, replacement3111)
pattern3112 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons1452, cons1477)
def replacement3112(B, b, d, a, c, A, x, e):
rubi.append(3112)
return Dist((A*a - B*b)/(a**S(2) - b**S(2) - c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp((A*c*cos(d + e*x) + B*c + (-A*b + B*a)*sin(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3112 = ReplacementRule(pattern3112, replacement3112)
pattern3113 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons87, cons89, cons1452, cons1442)
def replacement3113(B, C, b, d, c, a, n, A, x, e):
rubi.append(3113)
return Dist(S(1)/((n + S(1))*(a**S(2) - b**S(2) - c**S(2))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*Simp((n + S(1))*(A*a - B*b - C*c) + (n + S(2))*(-A*b + B*a)*cos(d + e*x) + (n + S(2))*(-A*c + C*a)*sin(d + e*x), x), x), x) - Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*(B*c - C*b + (-A*b + B*a)*sin(d + e*x) - (-A*c + C*a)*cos(d + e*x))/(e*(n + S(1))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3113 = ReplacementRule(pattern3113, replacement3113)
pattern3114 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons87, cons89, cons1452, cons1442)
def replacement3114(C, b, d, c, a, n, A, x, e):
rubi.append(3114)
return Dist(S(1)/((n + S(1))*(a**S(2) - b**S(2) - c**S(2))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*Simp(-A*b*(n + S(2))*cos(d + e*x) + (n + S(1))*(A*a - C*c) + (n + S(2))*(-A*c + C*a)*sin(d + e*x), x), x), x) + Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*(A*b*sin(d + e*x) + C*b + (-A*c + C*a)*cos(d + e*x))/(e*(n + S(1))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3114 = ReplacementRule(pattern3114, replacement3114)
pattern3115 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons87, cons89, cons1452, cons1442)
def replacement3115(B, b, d, a, c, n, A, x, e):
rubi.append(3115)
return Dist(S(1)/((n + S(1))*(a**S(2) - b**S(2) - c**S(2))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*Simp(-A*c*(n + S(2))*sin(d + e*x) + (n + S(1))*(A*a - B*b) + (n + S(2))*(-A*b + B*a)*cos(d + e*x), x), x), x) - Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*(A*c*cos(d + e*x) + B*c + (-A*b + B*a)*sin(d + e*x))/(e*(n + S(1))*(a**S(2) - b**S(2) - c**S(2))), x)
rule3115 = ReplacementRule(pattern3115, replacement3115)
pattern3116 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement3116(b, d, c, a, x, e):
rubi.append(3116)
return Int(cos(d + e*x)/(a*cos(d + e*x) + b + c*sin(d + e*x)), x)
rule3116 = ReplacementRule(pattern3116, replacement3116)
pattern3117 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement3117(b, d, c, a, x, e):
rubi.append(3117)
return Int(sin(d + e*x)/(a*sin(d + e*x) + b + c*cos(d + e*x)), x)
rule3117 = ReplacementRule(pattern3117, replacement3117)
pattern3118 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons85)
def replacement3118(b, d, a, n, c, x, e):
rubi.append(3118)
return Int((a*cos(d + e*x) + b + c*sin(d + e*x))**n, x)
rule3118 = ReplacementRule(pattern3118, replacement3118)
pattern3119 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons85)
def replacement3119(b, d, c, a, n, x, e):
rubi.append(3119)
return Int((a*sin(d + e*x) + b + c*cos(d + e*x))**n, x)
rule3119 = ReplacementRule(pattern3119, replacement3119)
pattern3120 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**n_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons23)
def replacement3120(b, d, c, a, n, x, e):
rubi.append(3120)
return Dist((a + b/cos(d + e*x) + c*tan(d + e*x))**n*(a*cos(d + e*x) + b + c*sin(d + e*x))**(-n)*cos(d + e*x)**n, Int((a*cos(d + e*x) + b + c*sin(d + e*x))**n, x), x)
rule3120 = ReplacementRule(pattern3120, replacement3120)
pattern3121 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**n_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons23)
def replacement3121(b, d, c, a, n, x, e):
rubi.append(3121)
return Dist((a + b/sin(d + e*x) + c/tan(d + e*x))**n*(a*sin(d + e*x) + b + c*cos(d + e*x))**(-n)*sin(d + e*x)**n, Int((a*sin(d + e*x) + b + c*cos(d + e*x))**n, x), x)
rule3121 = ReplacementRule(pattern3121, replacement3121)
pattern3122 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1255, cons85)
def replacement3122(m, b, d, a, n, c, x, e):
rubi.append(3122)
return Int((a*cos(d + e*x) + b + c*sin(d + e*x))**(-n), x)
rule3122 = ReplacementRule(pattern3122, replacement3122)
pattern3123 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1255, cons85)
def replacement3123(m, b, d, a, n, c, x, e):
rubi.append(3123)
return Int((a*sin(d + e*x) + b + c*cos(d + e*x))**(-n), x)
rule3123 = ReplacementRule(pattern3123, replacement3123)
pattern3124 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1255, cons23)
def replacement3124(m, b, d, a, n, c, x, e):
rubi.append(3124)
return Dist((a + b/cos(d + e*x) + c*tan(d + e*x))**(-n)*(a*cos(d + e*x) + b + c*sin(d + e*x))**n*(S(1)/cos(d + e*x))**n, Int((a*cos(d + e*x) + b + c*sin(d + e*x))**(-n), x), x)
rule3124 = ReplacementRule(pattern3124, replacement3124)
pattern3125 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons1255, cons23)
def replacement3125(m, b, d, a, n, c, x, e):
rubi.append(3125)
return Dist((a + b/sin(d + e*x) + c/tan(d + e*x))**(-n)*(a*sin(d + e*x) + b + c*cos(d + e*x))**n*(S(1)/sin(d + e*x))**n, Int((a*sin(d + e*x) + b + c*cos(d + e*x))**(-n), x), x)
rule3125 = ReplacementRule(pattern3125, replacement3125)
pattern3126 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons3, cons7, cons27, cons13, cons146)
def replacement3126(p, b, d, c, x):
rubi.append(3126)
return Dist(b*(S(2)*p + S(-1))/(S(2)*p), Int((b*sin(c + d*x)**S(2))**(p + S(-1)), x), x) - Simp((b*sin(c + d*x)**S(2))**p/(S(2)*d*p*tan(c + d*x)), x)
rule3126 = ReplacementRule(pattern3126, replacement3126)
pattern3127 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons3, cons7, cons27, cons13, cons146)
def replacement3127(p, b, d, c, x):
rubi.append(3127)
return Dist(b*(S(2)*p + S(-1))/(S(2)*p), Int((b*cos(c + d*x)**S(2))**(p + S(-1)), x), x) + Simp((b*cos(c + d*x)**S(2))**p*tan(c + d*x)/(S(2)*d*p), x)
rule3127 = ReplacementRule(pattern3127, replacement3127)
pattern3128 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons3, cons7, cons27, cons13, cons137)
def replacement3128(p, b, d, c, x):
rubi.append(3128)
return Dist(S(2)*(p + S(1))/(b*(S(2)*p + S(1))), Int((b*sin(c + d*x)**S(2))**(p + S(1)), x), x) + Simp((b*sin(c + d*x)**S(2))**(p + S(1))/(b*d*(S(2)*p + S(1))*tan(c + d*x)), x)
rule3128 = ReplacementRule(pattern3128, replacement3128)
pattern3129 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons3, cons7, cons27, cons13, cons137)
def replacement3129(p, b, d, c, x):
rubi.append(3129)
return Dist(S(2)*(p + S(1))/(b*(S(2)*p + S(1))), Int((b*cos(c + d*x)**S(2))**(p + S(1)), x), x) - Simp((b*cos(c + d*x)**S(2))**(p + S(1))*tan(c + d*x)/(b*d*(S(2)*p + S(1))), x)
rule3129 = ReplacementRule(pattern3129, replacement3129)
pattern3130 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons1454, cons38)
def replacement3130(p, b, d, c, a, x):
rubi.append(3130)
return Dist(a**p, Int(cos(c + d*x)**(S(2)*p), x), x)
rule3130 = ReplacementRule(pattern3130, replacement3130)
pattern3131 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons1454, cons38)
def replacement3131(p, b, d, c, a, x):
rubi.append(3131)
return Dist(a**p, Int(sin(c + d*x)**(S(2)*p), x), x)
rule3131 = ReplacementRule(pattern3131, replacement3131)
pattern3132 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1454, cons147)
def replacement3132(p, b, d, c, a, x):
rubi.append(3132)
return Int((a*cos(c + d*x)**S(2))**p, x)
rule3132 = ReplacementRule(pattern3132, replacement3132)
pattern3133 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1454, cons147)
def replacement3133(p, b, d, c, a, x):
rubi.append(3133)
return Int((a*sin(c + d*x)**S(2))**p, x)
rule3133 = ReplacementRule(pattern3133, replacement3133)
def With3134(p, b, d, c, a, x):
e = FreeFactors(tan(c + d*x), x)
rubi.append(3134)
return Dist(e/d, Subst(Int((a + e**S(2)*x**S(2)*(a + b))**p*(e**S(2)*x**S(2) + S(1))**(-p + S(-1)), x), x, tan(c + d*x)/e), x)
pattern3134 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons38)
rule3134 = ReplacementRule(pattern3134, With3134)
def With3135(p, b, d, c, a, x):
e = FreeFactors(tan(c + d*x), x)
rubi.append(3135)
return Dist(e/d, Subst(Int((e**S(2)*x**S(2) + S(1))**(-p + S(-1))*(a*e**S(2)*x**S(2) + a + b)**p, x), x, tan(c + d*x)/e), x)
pattern3135 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons38)
rule3135 = ReplacementRule(pattern3135, With3135)
pattern3136 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1478, cons147)
def replacement3136(p, b, d, c, a, x):
rubi.append(3136)
return Dist(S(2)**(-p), Int((S(2)*a - b*cos(S(2)*c + S(2)*d*x) + b)**p, x), x)
rule3136 = ReplacementRule(pattern3136, replacement3136)
pattern3137 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1478, cons147)
def replacement3137(p, b, d, c, a, x):
rubi.append(3137)
return Dist(S(2)**(-p), Int((S(2)*a + b*cos(S(2)*c + S(2)*d*x) + b)**p, x), x)
rule3137 = ReplacementRule(pattern3137, replacement3137)
pattern3138 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_, x_), cons2, cons3, cons7, cons27, cons85, cons128)
def replacement3138(p, b, d, c, a, n, x):
rubi.append(3138)
return Int((a + b*sin(c + d*x)**n)**p, x)
rule3138 = ReplacementRule(pattern3138, replacement3138)
pattern3139 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_, x_), cons2, cons3, cons7, cons27, cons85, cons128)
def replacement3139(p, b, d, c, n, a, x):
rubi.append(3139)
return Int((a + b*cos(c + d*x)**n)**p, x)
rule3139 = ReplacementRule(pattern3139, replacement3139)
def With3140(p, b, d, c, a, n, x):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(3140)
return -Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, S(1)/(f*tan(c + d*x))), x)
pattern3140 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_, x_), cons2, cons3, cons7, cons27, cons1479, cons38, cons137)
rule3140 = ReplacementRule(pattern3140, With3140)
def With3141(p, b, d, c, n, a, x):
f = FreeFactors(tan(c + d*x), x)
rubi.append(3141)
return Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, tan(c + d*x)/f), x)
pattern3141 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_, x_), cons2, cons3, cons7, cons27, cons1479, cons38, cons137)
rule3141 = ReplacementRule(pattern3141, With3141)
pattern3142 = Pattern(Integral(u_*(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons1454, cons38)
def replacement3142(p, u, b, d, c, a, x):
rubi.append(3142)
return Dist(a**p, Int(ActivateTrig(u)*cos(c + d*x)**(S(2)*p), x), x)
rule3142 = ReplacementRule(pattern3142, replacement3142)
pattern3143 = Pattern(Integral(u_*(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons1454, cons38)
def replacement3143(p, u, b, d, c, a, x):
rubi.append(3143)
return Dist(a**p, Int(ActivateTrig(u)*sin(c + d*x)**(S(2)*p), x), x)
rule3143 = ReplacementRule(pattern3143, replacement3143)
pattern3144 = Pattern(Integral(u_*(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1454, cons147)
def replacement3144(p, u, b, d, c, a, x):
rubi.append(3144)
return Dist((a*cos(c + d*x)**S(2))**p*cos(c + d*x)**(-S(2)*p), Int(ActivateTrig(u)*cos(c + d*x)**(S(2)*p), x), x)
rule3144 = ReplacementRule(pattern3144, replacement3144)
pattern3145 = Pattern(Integral(u_*(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1454, cons147)
def replacement3145(p, u, b, d, c, a, x):
rubi.append(3145)
return Dist((a*sin(c + d*x)**S(2))**p*sin(c + d*x)**(-S(2)*p), Int(ActivateTrig(u)*sin(c + d*x)**(S(2)*p), x), x)
rule3145 = ReplacementRule(pattern3145, replacement3145)
def With3146(p, m, b, d, c, a, n, x):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(3146)
return -Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, S(1)/(f*tan(c + d*x))), x)
pattern3146 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1479, cons1480, cons38)
rule3146 = ReplacementRule(pattern3146, With3146)
def With3147(p, m, b, d, c, n, a, x):
f = FreeFactors(tan(c + d*x), x)
rubi.append(3147)
return Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, tan(c + d*x)/f), x)
pattern3147 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1479, cons1480, cons38)
rule3147 = ReplacementRule(pattern3147, With3147)
def With3148(p, m, b, d, c, a, n, x):
f = FreeFactors(cos(c + d*x), x)
rubi.append(3148)
return -Dist(f/d, Subst(Int((-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a + b*(-f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, cos(c + d*x)/f), x)
pattern3148 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1479, cons1481)
rule3148 = ReplacementRule(pattern3148, With3148)
def With3149(p, m, b, d, c, n, a, x):
f = FreeFactors(sin(c + d*x), x)
rubi.append(3149)
return Dist(f/d, Subst(Int((-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a + b*(-f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, sin(c + d*x)/f), x)
pattern3149 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1479, cons1481)
rule3149 = ReplacementRule(pattern3149, With3149)
pattern3150 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons375)
def replacement3150(p, m, b, d, c, a, n, x):
rubi.append(3150)
return Int(ExpandTrig((a + b*sin(c + d*x)**n)**p*sin(c + d*x)**m, x), x)
rule3150 = ReplacementRule(pattern3150, replacement3150)
pattern3151 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons375)
def replacement3151(p, m, b, d, c, n, a, x):
rubi.append(3151)
return Int(ExpandTrig((a + b*cos(c + d*x)**n)**p*cos(c + d*x)**m, x), x)
rule3151 = ReplacementRule(pattern3151, replacement3151)
def With3152(p, m, b, d, c, a, n, x):
f = FreeFactors(tan(c + d*x), x)
rubi.append(3152)
return Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x)**p, x), x, tan(c + d*x)/f), x)
pattern3152 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1480, cons1479, cons38)
rule3152 = ReplacementRule(pattern3152, With3152)
def With3153(p, m, b, d, c, n, a, x):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(3153)
return -Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x)**p, x), x, S(1)/(f*tan(c + d*x))), x)
pattern3153 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1480, cons1479, cons38)
rule3153 = ReplacementRule(pattern3153, With3153)
pattern3154 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1480, cons1482, cons38, cons168)
def replacement3154(p, m, b, d, c, a, n, x):
rubi.append(3154)
return Int((a + b*sin(c + d*x)**n)**p*(-sin(c + d*x)**S(2) + S(1))**(m/S(2)), x)
rule3154 = ReplacementRule(pattern3154, replacement3154)
pattern3155 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1480, cons1482, cons38, cons168)
def replacement3155(p, m, b, d, c, n, a, x):
rubi.append(3155)
return Int((a + b*cos(c + d*x)**n)**p*(-cos(c + d*x)**S(2) + S(1))**(m/S(2)), x)
rule3155 = ReplacementRule(pattern3155, replacement3155)
pattern3156 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1480, cons1482, cons38, cons267, cons137)
def replacement3156(p, m, b, d, c, a, n, x):
rubi.append(3156)
return Int(ExpandTrig((a + b*sin(c + d*x)**n)**p*(-sin(c + d*x)**S(2) + S(1))**(m/S(2)), x), x)
rule3156 = ReplacementRule(pattern3156, replacement3156)
pattern3157 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons1480, cons1482, cons38, cons267, cons137)
def replacement3157(p, m, b, d, c, n, a, x):
rubi.append(3157)
return Int(ExpandTrig((a + b*cos(c + d*x)**n)**p*(-cos(c + d*x)**S(2) + S(1))**(m/S(2)), x), x)
rule3157 = ReplacementRule(pattern3157, replacement3157)
def With3158(p, m, b, d, c, a, n, x, e):
f = FreeFactors(sin(c + d*x), x)
rubi.append(3158)
return Dist(f/d, Subst(Int((a + b*(e*f*x)**n)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, sin(c + d*x)/f), x)
pattern3158 = Pattern(Integral((a_ + (WC('e', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1481)
rule3158 = ReplacementRule(pattern3158, With3158)
def With3159(p, m, b, d, c, n, a, x, e):
f = FreeFactors(cos(c + d*x), x)
rubi.append(3159)
return -Dist(f/d, Subst(Int((a + b*(e*f*x)**n)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, cos(c + d*x)/f), x)
pattern3159 = Pattern(Integral((a_ + (WC('e', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1481)
rule3159 = ReplacementRule(pattern3159, With3159)
def With3160(p, m, b, d, c, a, n, x, e):
f = FreeFactors(sin(c + d*x), x)
rubi.append(3160)
return Dist(f**(m + S(1))/d, Subst(Int(x**m*(a + b*(e*f*x)**n)**p*(-f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)/2), x), x, sin(c + d*x)/f), x)
pattern3160 = Pattern(Integral((a_ + (WC('e', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1481, cons246)
rule3160 = ReplacementRule(pattern3160, With3160)
def With3161(p, m, b, d, c, n, a, x, e):
f = FreeFactors(cos(c + d*x), x)
rubi.append(3161)
return -Dist(f**(m + S(1))/d, Subst(Int(x**m*(a + b*(e*f*x)**n)**p*(-f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)/2), x), x, cos(c + d*x)/f), x)
pattern3161 = Pattern(Integral((a_ + (WC('e', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1481, cons246)
rule3161 = ReplacementRule(pattern3161, With3161)
def With3162(p, m, b, d, c, a, n, x):
f = FreeFactors(tan(c + d*x), x)
rubi.append(3162)
return Dist(f**(m + S(1))/d, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x)**p, x), x, tan(c + d*x)/f), x)
pattern3162 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons21, cons1483, cons1479, cons38)
rule3162 = ReplacementRule(pattern3162, With3162)
def With3163(p, m, b, d, c, n, a, x):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(3163)
return -Dist(f**(m + S(1))/d, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x)**p, x), x, S(1)/(f*tan(c + d*x))), x)
pattern3163 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons21, cons1483, cons1479, cons38)
rule3163 = ReplacementRule(pattern3163, With3163)
def With3164(p, m, b, d, c, a, n, x):
f = FreeFactors(tan(c + d*x), x)
rubi.append(3164)
return Dist(f**(m + S(1))/d, Subst(Int(x**m*((f**S(2)*x**S(2) + S(1))**(-n/S(2))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x))**p/(f**S(2)*x**S(2) + S(1)), x), x, tan(c + d*x)/f), x)
pattern3164 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons1483, cons1479, cons147)
rule3164 = ReplacementRule(pattern3164, With3164)
def With3165(p, m, b, d, c, n, a, x):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(3165)
return -Dist(f**(m + S(1))/d, Subst(Int(x**m*((f**S(2)*x**S(2) + S(1))**(-n/S(2))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x))**p/(f**S(2)*x**S(2) + S(1)), x), x, S(1)/(f*tan(c + d*x))), x)
pattern3165 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons1483, cons1479, cons147)
rule3165 = ReplacementRule(pattern3165, With3165)
def With3166(p, m, b, d, c, a, n, x, q, e):
f = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(3166)
return -Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*q/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(q/S(2)) + b*(f**S(2)*x**S(2) + S(1))**(-p/S(2) + q/S(2)) + c, x)**n, x), x, S(1)/(f*tan(d + e*x))), x)
pattern3166 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**p_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**q_)**n_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons1480, cons1484, cons1485, cons85, cons1486)
rule3166 = ReplacementRule(pattern3166, With3166)
def With3167(p, m, b, d, c, a, n, x, q, e):
f = FreeFactors(tan(d + e*x), x)
rubi.append(3167)
return Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*q/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(q/S(2)) + b*(f**S(2)*x**S(2) + S(1))**(-p/S(2) + q/S(2)) + c, x)**n, x), x, tan(d + e*x)/f), x)
pattern3167 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**p_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**q_)**n_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons1480, cons1484, cons1485, cons85, cons1486)
rule3167 = ReplacementRule(pattern3167, With3167)
def With3168(p, m, b, d, c, a, n, x, q, e):
f = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(3168)
return -Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(p/S(2)) + b*f**p*x**p + c*(f**S(2)*x**S(2) + S(1))**(p/S(2) - q/S(2)), x)**n, x), x, S(1)/(f*tan(d + e*x))), x)
pattern3168 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**p_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**q_)**n_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons1480, cons1484, cons1485, cons85, cons1487)
rule3168 = ReplacementRule(pattern3168, With3168)
def With3169(p, m, b, d, c, a, n, x, q, e):
f = FreeFactors(tan(d + e*x), x)
rubi.append(3169)
return Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(p/S(2)) + b*f**p*x**p + c*(f**S(2)*x**S(2) + S(1))**(p/S(2) - q/S(2)), x)**n, x), x, tan(d + e*x)/f), x)
pattern3169 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**p_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**q_)**n_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons1480, cons1484, cons1485, cons85, cons1487)
rule3169 = ReplacementRule(pattern3169, With3169)
pattern3170 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons45, cons38)
def replacement3170(p, b, n2, d, a, n, c, x, e):
rubi.append(3170)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p), x), x)
rule3170 = ReplacementRule(pattern3170, replacement3170)
pattern3171 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons45, cons38)
def replacement3171(p, b, n2, d, a, n, c, x, e):
rubi.append(3171)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p), x), x)
rule3171 = ReplacementRule(pattern3171, replacement3171)
pattern3172 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons46, cons45, cons147)
def replacement3172(p, b, n2, d, a, n, c, x, e):
rubi.append(3172)
return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int(u*(b + S(2)*c*sin(d + e*x)**n)**(S(2)*p), x), x)
rule3172 = ReplacementRule(pattern3172, replacement3172)
pattern3173 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons46, cons45, cons147)
def replacement3173(p, b, n2, d, a, n, c, x, e):
rubi.append(3173)
return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int(u*(b + S(2)*c*cos(d + e*x)**n)**(S(2)*p), x), x)
rule3173 = ReplacementRule(pattern3173, replacement3173)
def With3174(b, n2, d, a, n, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(3174)
return Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*sin(d + e*x)**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*sin(d + e*x)**n + q), x), x)
pattern3174 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226)
rule3174 = ReplacementRule(pattern3174, With3174)
def With3175(b, n2, d, a, n, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(3175)
return Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*cos(d + e*x)**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*cos(d + e*x)**n + q), x), x)
pattern3175 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226)
rule3175 = ReplacementRule(pattern3175, With3175)
pattern3176 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons45, cons38)
def replacement3176(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3176)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*sin(d + e*x)**m, x), x)
rule3176 = ReplacementRule(pattern3176, replacement3176)
pattern3177 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons45, cons38)
def replacement3177(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3177)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*cos(d + e*x)**m, x), x)
rule3177 = ReplacementRule(pattern3177, replacement3177)
pattern3178 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons45, cons147)
def replacement3178(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3178)
return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*sin(d + e*x)**m, x), x)
rule3178 = ReplacementRule(pattern3178, replacement3178)
pattern3179 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons45, cons147)
def replacement3179(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3179)
return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*cos(d + e*x)**m, x), x)
rule3179 = ReplacementRule(pattern3179, replacement3179)
def With3180(p, m, b, n2, d, c, a, n, x, e):
f = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(3180)
return -Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*(x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, S(1)/(f*tan(d + e*x))), x)
pattern3180 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons1479, cons38)
rule3180 = ReplacementRule(pattern3180, With3180)
def With3181(p, m, b, n2, d, a, c, n, x, e):
f = FreeFactors(tan(d + e*x), x)
rubi.append(3181)
return Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*(x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, tan(d + e*x)/f), x)
pattern3181 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons1479, cons38)
rule3181 = ReplacementRule(pattern3181, With3181)
pattern3182 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons226, cons375)
def replacement3182(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3182)
return Int(ExpandTrig((a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p*sin(d + e*x)**m, x), x)
rule3182 = ReplacementRule(pattern3182, replacement3182)
pattern3183 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons226, cons375)
def replacement3183(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3183)
return Int(ExpandTrig((a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p*cos(d + e*x)**m, x), x)
rule3183 = ReplacementRule(pattern3183, replacement3183)
def With3184(p, m, f, b, n2, d, a, n, c, x, e):
g = FreeFactors(sin(d + e*x), x)
rubi.append(3184)
return Dist(g/e, Subst(Int((-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, sin(d + e*x)/g), x)
pattern3184 = Pattern(Integral(((WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons46, cons1481)
rule3184 = ReplacementRule(pattern3184, With3184)
def With3185(p, m, f, b, n2, d, a, n, c, x, e):
g = FreeFactors(cos(d + e*x), x)
rubi.append(3185)
return -Dist(g/e, Subst(Int((-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, cos(d + e*x)/g), x)
pattern3185 = Pattern(Integral(((WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons46, cons1481)
rule3185 = ReplacementRule(pattern3185, With3185)
pattern3186 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons1483, cons45, cons38)
def replacement3186(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3186)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*cos(d + e*x)**m, x), x)
rule3186 = ReplacementRule(pattern3186, replacement3186)
pattern3187 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons1483, cons45, cons38)
def replacement3187(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3187)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*sin(d + e*x)**m, x), x)
rule3187 = ReplacementRule(pattern3187, replacement3187)
pattern3188 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons1483, cons45, cons147)
def replacement3188(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3188)
return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*cos(d + e*x)**m, x), x)
rule3188 = ReplacementRule(pattern3188, replacement3188)
pattern3189 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons1483, cons45, cons147)
def replacement3189(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3189)
return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*sin(d + e*x)**m, x), x)
rule3189 = ReplacementRule(pattern3189, replacement3189)
def With3190(p, m, b, n2, d, c, a, n, x, e):
f = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(3190)
return -Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*(x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, S(1)/(f*tan(d + e*x))), x)
pattern3190 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons1479, cons38)
rule3190 = ReplacementRule(pattern3190, With3190)
def With3191(p, m, b, n2, d, c, a, n, x, e):
f = FreeFactors(tan(d + e*x), x)
rubi.append(3191)
return Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*(x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, tan(d + e*x)/f), x)
pattern3191 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons1479, cons38)
rule3191 = ReplacementRule(pattern3191, With3191)
pattern3192 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons376)
def replacement3192(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3192)
return Int(ExpandTrig((-sin(d + e*x)**S(2) + S(1))**(m/S(2))*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, x), x)
rule3192 = ReplacementRule(pattern3192, replacement3192)
pattern3193 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons376)
def replacement3193(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3193)
return Int(ExpandTrig((-cos(d + e*x)**S(2) + S(1))**(m/S(2))*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, x), x)
rule3193 = ReplacementRule(pattern3193, replacement3193)
def With3194(p, m, f, b, n2, d, c, a, n, x, e):
g = FreeFactors(sin(d + e*x), x)
rubi.append(3194)
return Dist(g**(m + S(1))/e, Subst(Int(x**m*(-g**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, sin(d + e*x)/g), x)
pattern3194 = Pattern(Integral((a_ + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1481, cons246)
rule3194 = ReplacementRule(pattern3194, With3194)
def With3195(p, m, f, b, n2, d, c, n, a, x, e):
g = FreeFactors(cos(d + e*x), x)
rubi.append(3195)
return -Dist(g**(m + S(1))/e, Subst(Int(x**m*(-g**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, cos(d + e*x)/g), x)
pattern3195 = Pattern(Integral((a_ + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1481, cons246)
rule3195 = ReplacementRule(pattern3195, With3195)
pattern3196 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons1483, cons45, cons38)
def replacement3196(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3196)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x)
rule3196 = ReplacementRule(pattern3196, replacement3196)
pattern3197 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons1483, cons45, cons38)
def replacement3197(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3197)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x)
rule3197 = ReplacementRule(pattern3197, replacement3197)
pattern3198 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons1483, cons45, cons147)
def replacement3198(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3198)
return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x)
rule3198 = ReplacementRule(pattern3198, replacement3198)
pattern3199 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons1483, cons45, cons147)
def replacement3199(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3199)
return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x)
rule3199 = ReplacementRule(pattern3199, replacement3199)
def With3200(p, m, b, n2, d, c, a, n, x, e):
f = FreeFactors(tan(d + e*x), x)
rubi.append(3200)
return Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*x**n*(x**S(2) + S(1))**(n/S(2)) + c*x**(S(2)*n), x)**p, x), x, tan(d + e*x)/f), x)
pattern3200 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons46, cons1483, cons226, cons1479, cons38)
rule3200 = ReplacementRule(pattern3200, With3200)
def With3201(p, m, b, n2, d, a, c, n, x, e):
f = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(3201)
return -Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*x**n*(x**S(2) + S(1))**(n/S(2)) + c*x**(S(2)*n), x)**p, x), x, S(1)/(f*tan(d + e*x))), x)
pattern3201 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons46, cons1483, cons226, cons1479, cons38)
rule3201 = ReplacementRule(pattern3201, With3201)
pattern3202 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons376)
def replacement3202(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3202)
return Int(ExpandTrig((-sin(d + e*x)**S(2) + S(1))**(-m/S(2))*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p*sin(d + e*x)**m, x), x)
rule3202 = ReplacementRule(pattern3202, replacement3202)
pattern3203 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons376)
def replacement3203(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3203)
return Int(ExpandTrig((-cos(d + e*x)**S(2) + S(1))**(-m/S(2))*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p*cos(d + e*x)**m, x), x)
rule3203 = ReplacementRule(pattern3203, replacement3203)
def With3204(p, m, f, b, n2, d, c, a, n, x, e):
g = FreeFactors(sin(d + e*x), x)
rubi.append(3204)
return Dist(g**(m + S(1))/e, Subst(Int(x**(-m)*(-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, sin(d + e*x)/g), x)
pattern3204 = Pattern(Integral((a_ + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1481, cons246)
rule3204 = ReplacementRule(pattern3204, With3204)
def With3205(p, m, f, b, n2, d, c, n, a, x, e):
g = FreeFactors(cos(d + e*x), x)
rubi.append(3205)
return -Dist(g**(m + S(1))/e, Subst(Int(x**(-m)*(-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, cos(d + e*x)/g), x)
pattern3205 = Pattern(Integral((a_ + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons1481, cons246)
rule3205 = ReplacementRule(pattern3205, With3205)
pattern3206 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons1483, cons45, cons38)
def replacement3206(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3206)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x)
rule3206 = ReplacementRule(pattern3206, replacement3206)
pattern3207 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons1483, cons45, cons38)
def replacement3207(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3207)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x)
rule3207 = ReplacementRule(pattern3207, replacement3207)
pattern3208 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons1483, cons45, cons147)
def replacement3208(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3208)
return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x)
rule3208 = ReplacementRule(pattern3208, replacement3208)
pattern3209 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons1483, cons45, cons147)
def replacement3209(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3209)
return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x)
rule3209 = ReplacementRule(pattern3209, replacement3209)
def With3210(p, m, b, n2, d, c, a, n, x, e):
f = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(3210)
return -Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**n + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, S(1)/(f*tan(d + e*x))), x)
pattern3210 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons46, cons1479, cons38)
rule3210 = ReplacementRule(pattern3210, With3210)
def With3211(p, m, b, n2, d, c, n, a, x, e):
f = FreeFactors(tan(d + e*x), x)
rubi.append(3211)
return Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**n + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, tan(d + e*x)/f), x)
pattern3211 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons46, cons1479, cons38)
rule3211 = ReplacementRule(pattern3211, With3211)
pattern3212 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons376)
def replacement3212(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3212)
return Int(ExpandTrig((-sin(d + e*x)**S(2) + S(1))**(m/S(2))*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p*sin(d + e*x)**(-m), x), x)
rule3212 = ReplacementRule(pattern3212, replacement3212)
pattern3213 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons226, cons376)
def replacement3213(p, m, b, n2, d, a, n, c, x, e):
rubi.append(3213)
return Int(ExpandTrig((-cos(d + e*x)**S(2) + S(1))**(m/S(2))*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p*cos(d + e*x)**(-m), x), x)
rule3213 = ReplacementRule(pattern3213, replacement3213)
pattern3214 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons85)
def replacement3214(B, b, d, c, a, n, A, x, e):
rubi.append(3214)
return Dist(S(4)**(-n)*c**(-n), Int((A + B*sin(d + e*x))*(b + S(2)*c*sin(d + e*x))**(S(2)*n), x), x)
rule3214 = ReplacementRule(pattern3214, replacement3214)
pattern3215 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons85)
def replacement3215(B, b, d, c, a, n, A, x, e):
rubi.append(3215)
return Dist(S(4)**(-n)*c**(-n), Int((A + B*cos(d + e*x))*(b + S(2)*c*cos(d + e*x))**(S(2)*n), x), x)
rule3215 = ReplacementRule(pattern3215, replacement3215)
pattern3216 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons23)
def replacement3216(B, b, d, c, a, n, A, x, e):
rubi.append(3216)
return Dist((b + S(2)*c*sin(d + e*x))**(-S(2)*n)*(a + b*sin(d + e*x) + c*sin(d + e*x)**S(2))**n, Int((A + B*sin(d + e*x))*(b + S(2)*c*sin(d + e*x))**(S(2)*n), x), x)
rule3216 = ReplacementRule(pattern3216, replacement3216)
pattern3217 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons23)
def replacement3217(B, b, d, c, a, n, A, x, e):
rubi.append(3217)
return Dist((b + S(2)*c*cos(d + e*x))**(-S(2)*n)*(a + b*cos(d + e*x) + c*cos(d + e*x)**S(2))**n, Int((A + B*cos(d + e*x))*(b + S(2)*c*cos(d + e*x))**(S(2)*n), x), x)
rule3217 = ReplacementRule(pattern3217, replacement3217)
def With3218(B, b, d, a, c, A, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(3218)
return Dist(B - (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c*sin(d + e*x) - q), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c*sin(d + e*x) + q), x), x)
pattern3218 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226)
rule3218 = ReplacementRule(pattern3218, With3218)
def With3219(B, b, d, c, a, A, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(3219)
return Dist(B - (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c*cos(d + e*x) - q), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c*cos(d + e*x) + q), x), x)
pattern3219 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226)
rule3219 = ReplacementRule(pattern3219, With3219)
pattern3220 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226, cons85)
def replacement3220(B, b, d, a, c, n, A, x, e):
rubi.append(3220)
return Int(ExpandTrig((A + B*sin(d + e*x))*(a + b*sin(d + e*x) + c*sin(d + e*x)**S(2))**n, x), x)
rule3220 = ReplacementRule(pattern3220, replacement3220)
pattern3221 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226, cons85)
def replacement3221(B, b, d, c, a, n, A, x, e):
rubi.append(3221)
return Int(ExpandTrig((A + B*cos(d + e*x))*(a + b*cos(d + e*x) + c*cos(d + e*x)**S(2))**n, x), x)
rule3221 = ReplacementRule(pattern3221, replacement3221)
pattern3222 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons31, cons168)
def replacement3222(m, f, d, c, x, e):
rubi.append(3222)
return Dist(d*m/f, Int((c + d*x)**(m + S(-1))*cos(e + f*x), x), x) - Simp((c + d*x)**m*cos(e + f*x)/f, x)
rule3222 = ReplacementRule(pattern3222, replacement3222)
pattern3223 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons31, cons168)
def replacement3223(m, f, d, c, x, e):
rubi.append(3223)
return -Dist(d*m/f, Int((c + d*x)**(m + S(-1))*sin(e + f*x), x), x) + Simp((c + d*x)**m*sin(e + f*x)/f, x)
rule3223 = ReplacementRule(pattern3223, replacement3223)
pattern3224 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons31, cons94)
def replacement3224(m, f, d, c, x, e):
rubi.append(3224)
return -Dist(f/(d*(m + S(1))), Int((c + d*x)**(m + S(1))*cos(e + f*x), x), x) + Simp((c + d*x)**(m + S(1))*sin(e + f*x)/(d*(m + S(1))), x)
rule3224 = ReplacementRule(pattern3224, replacement3224)
pattern3225 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons31, cons94)
def replacement3225(m, f, d, c, x, e):
rubi.append(3225)
return Dist(f/(d*(m + S(1))), Int((c + d*x)**(m + S(1))*sin(e + f*x), x), x) + Simp((c + d*x)**(m + S(1))*cos(e + f*x)/(d*(m + S(1))), x)
rule3225 = ReplacementRule(pattern3225, replacement3225)
pattern3226 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons1116)
def replacement3226(f, d, c, x, e):
rubi.append(3226)
return Simp(SinIntegral(e + f*x)/d, x)
rule3226 = ReplacementRule(pattern3226, replacement3226)
pattern3227 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons1116)
def replacement3227(f, d, c, x, e):
rubi.append(3227)
return Simp(CosIntegral(e + f*x)/d, x)
rule3227 = ReplacementRule(pattern3227, replacement3227)
pattern3228 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons176)
def replacement3228(f, d, c, x, e):
rubi.append(3228)
return Dist(sin((-c*f + d*e)/d), Int(cos(c*f/d + f*x)/(c + d*x), x), x) + Dist(cos((-c*f + d*e)/d), Int(sin(c*f/d + f*x)/(c + d*x), x), x)
rule3228 = ReplacementRule(pattern3228, replacement3228)
pattern3229 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons176)
def replacement3229(f, d, c, x, e):
rubi.append(3229)
return -Dist(sin((-c*f + d*e)/d), Int(sin(c*f/d + f*x)/(c + d*x), x), x) + Dist(cos((-c*f + d*e)/d), Int(cos(c*f/d + f*x)/(c + d*x), x), x)
rule3229 = ReplacementRule(pattern3229, replacement3229)
pattern3230 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons1116)
def replacement3230(f, d, c, x, e):
rubi.append(3230)
return Dist(S(2)/d, Subst(Int(sin(f*x**S(2)/d), x), x, sqrt(c + d*x)), x)
rule3230 = ReplacementRule(pattern3230, replacement3230)
pattern3231 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons1116)
def replacement3231(f, d, c, x, e):
rubi.append(3231)
return Dist(S(2)/d, Subst(Int(cos(f*x**S(2)/d), x), x, sqrt(c + d*x)), x)
rule3231 = ReplacementRule(pattern3231, replacement3231)
pattern3232 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons176)
def replacement3232(f, d, c, x, e):
rubi.append(3232)
return Dist(sin((-c*f + d*e)/d), Int(cos(c*f/d + f*x)/sqrt(c + d*x), x), x) + Dist(cos((-c*f + d*e)/d), Int(sin(c*f/d + f*x)/sqrt(c + d*x), x), x)
rule3232 = ReplacementRule(pattern3232, replacement3232)
pattern3233 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons125, cons176)
def replacement3233(f, d, c, x, e):
rubi.append(3233)
return -Dist(sin((-c*f + d*e)/d), Int(sin(c*f/d + f*x)/sqrt(c + d*x), x), x) + Dist(cos((-c*f + d*e)/d), Int(cos(c*f/d + f*x)/sqrt(c + d*x), x), x)
rule3233 = ReplacementRule(pattern3233, replacement3233)
pattern3234 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons21, cons1488)
def replacement3234(m, f, d, c, x, e):
rubi.append(3234)
return Dist(I/S(2), Int((c + d*x)**m*exp(-I*(e + f*x)), x), x) - Dist(I/S(2), Int((c + d*x)**m*exp(I*(e + f*x)), x), x)
rule3234 = ReplacementRule(pattern3234, replacement3234)
pattern3235 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons21, cons1488)
def replacement3235(m, f, d, c, x, e):
rubi.append(3235)
return Dist(S(1)/2, Int((c + d*x)**m*exp(-I*(e + f*x)), x), x) + Dist(S(1)/2, Int((c + d*x)**m*exp(I*(e + f*x)), x), x)
rule3235 = ReplacementRule(pattern3235, replacement3235)
pattern3236 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons165)
def replacement3236(f, b, d, c, n, x, e):
rubi.append(3236)
return Dist(b**S(2)*(n + S(-1))/n, Int((b*sin(e + f*x))**(n + S(-2))*(c + d*x), x), x) + Simp(d*(b*sin(e + f*x))**n/(f**S(2)*n**S(2)), x) - Simp(b*(b*sin(e + f*x))**(n + S(-1))*(c + d*x)*cos(e + f*x)/(f*n), x)
rule3236 = ReplacementRule(pattern3236, replacement3236)
pattern3237 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons165)
def replacement3237(f, b, d, c, n, x, e):
rubi.append(3237)
return Dist(b**S(2)*(n + S(-1))/n, Int((b*cos(e + f*x))**(n + S(-2))*(c + d*x), x), x) + Simp(d*(b*cos(e + f*x))**n/(f**S(2)*n**S(2)), x) + Simp(b*(b*cos(e + f*x))**(n + S(-1))*(c + d*x)*sin(e + f*x)/(f*n), x)
rule3237 = ReplacementRule(pattern3237, replacement3237)
pattern3238 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons166)
def replacement3238(m, f, b, d, c, n, x, e):
rubi.append(3238)
return Dist(b**S(2)*(n + S(-1))/n, Int((b*sin(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) - Dist(d**S(2)*m*(m + S(-1))/(f**S(2)*n**S(2)), Int((b*sin(e + f*x))**n*(c + d*x)**(m + S(-2)), x), x) - Simp(b*(b*sin(e + f*x))**(n + S(-1))*(c + d*x)**m*cos(e + f*x)/(f*n), x) + Simp(d*m*(b*sin(e + f*x))**n*(c + d*x)**(m + S(-1))/(f**S(2)*n**S(2)), x)
rule3238 = ReplacementRule(pattern3238, replacement3238)
pattern3239 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons166)
def replacement3239(m, f, b, d, c, n, x, e):
rubi.append(3239)
return Dist(b**S(2)*(n + S(-1))/n, Int((b*cos(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) - Dist(d**S(2)*m*(m + S(-1))/(f**S(2)*n**S(2)), Int((b*cos(e + f*x))**n*(c + d*x)**(m + S(-2)), x), x) + Simp(b*(b*cos(e + f*x))**(n + S(-1))*(c + d*x)**m*sin(e + f*x)/(f*n), x) + Simp(d*m*(b*cos(e + f*x))**n*(c + d*x)**(m + S(-1))/(f**S(2)*n**S(2)), x)
rule3239 = ReplacementRule(pattern3239, replacement3239)
pattern3240 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons7, cons27, cons48, cons125, cons21, cons85, cons165, cons1489)
def replacement3240(m, f, d, c, n, x, e):
rubi.append(3240)
return Int(ExpandTrigReduce((c + d*x)**m, sin(e + f*x)**n, x), x)
rule3240 = ReplacementRule(pattern3240, replacement3240)
pattern3241 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons7, cons27, cons48, cons125, cons21, cons85, cons165, cons1489)
def replacement3241(m, f, d, c, n, x, e):
rubi.append(3241)
return Int(ExpandTrigReduce((c + d*x)**m, cos(e + f*x)**n, x), x)
rule3241 = ReplacementRule(pattern3241, replacement3241)
pattern3242 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons7, cons27, cons48, cons125, cons21, cons85, cons165, cons31, cons245)
def replacement3242(m, f, d, c, n, x, e):
rubi.append(3242)
return -Dist(f*n/(d*(m + S(1))), Int(ExpandTrigReduce((c + d*x)**(m + S(1)), sin(e + f*x)**(n + S(-1))*cos(e + f*x), x), x), x) + Simp((c + d*x)**(m + S(1))*sin(e + f*x)**n/(d*(m + S(1))), x)
rule3242 = ReplacementRule(pattern3242, replacement3242)
pattern3243 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons7, cons27, cons48, cons125, cons21, cons85, cons165, cons31, cons245)
def replacement3243(m, f, d, c, n, x, e):
rubi.append(3243)
return Dist(f*n/(d*(m + S(1))), Int(ExpandTrigReduce((c + d*x)**(m + S(1)), sin(e + f*x)*cos(e + f*x)**(n + S(-1)), x), x), x) + Simp((c + d*x)**(m + S(1))*cos(e + f*x)**n/(d*(m + S(1))), x)
rule3243 = ReplacementRule(pattern3243, replacement3243)
pattern3244 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons247)
def replacement3244(m, f, b, d, c, n, x, e):
rubi.append(3244)
return -Dist(f**S(2)*n**S(2)/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*sin(e + f*x))**n*(c + d*x)**(m + S(2)), x), x) + Dist(b**S(2)*f**S(2)*n*(n + S(-1))/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*sin(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(2)), x), x) + Simp((b*sin(e + f*x))**n*(c + d*x)**(m + S(1))/(d*(m + S(1))), x) - Simp(b*f*n*(b*sin(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(2))*cos(e + f*x)/(d**S(2)*(m + S(1))*(m + S(2))), x)
rule3244 = ReplacementRule(pattern3244, replacement3244)
pattern3245 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons247)
def replacement3245(m, f, b, d, c, n, x, e):
rubi.append(3245)
return -Dist(f**S(2)*n**S(2)/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*cos(e + f*x))**n*(c + d*x)**(m + S(2)), x), x) + Dist(b**S(2)*f**S(2)*n*(n + S(-1))/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*cos(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(2)), x), x) + Simp((b*cos(e + f*x))**n*(c + d*x)**(m + S(1))/(d*(m + S(1))), x) + Simp(b*f*n*(b*cos(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(2))*sin(e + f*x)/(d**S(2)*(m + S(1))*(m + S(2))), x)
rule3245 = ReplacementRule(pattern3245, replacement3245)
pattern3246 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons89, cons1442)
def replacement3246(f, b, d, c, n, x, e):
rubi.append(3246)
return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*sin(e + f*x))**(n + S(2))*(c + d*x), x), x) - Simp(d*(b*sin(e + f*x))**(n + S(2))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x) + Simp((b*sin(e + f*x))**(n + S(1))*(c + d*x)*cos(e + f*x)/(b*f*(n + S(1))), x)
rule3246 = ReplacementRule(pattern3246, replacement3246)
pattern3247 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons89, cons1442)
def replacement3247(f, b, d, c, n, x, e):
rubi.append(3247)
return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*cos(e + f*x))**(n + S(2))*(c + d*x), x), x) - Simp(d*(b*cos(e + f*x))**(n + S(2))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x) - Simp((b*cos(e + f*x))**(n + S(1))*(c + d*x)*sin(e + f*x)/(b*f*(n + S(1))), x)
rule3247 = ReplacementRule(pattern3247, replacement3247)
pattern3248 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons93, cons89, cons1442, cons166)
def replacement3248(m, f, b, d, c, n, x, e):
rubi.append(3248)
return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*sin(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) + Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), Int((b*sin(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-2)), x), x) + Simp((b*sin(e + f*x))**(n + S(1))*(c + d*x)**m*cos(e + f*x)/(b*f*(n + S(1))), x) - Simp(d*m*(b*sin(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x)
rule3248 = ReplacementRule(pattern3248, replacement3248)
pattern3249 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons93, cons89, cons1442, cons166)
def replacement3249(m, f, b, d, c, n, x, e):
rubi.append(3249)
return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*cos(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) + Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), Int((b*cos(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-2)), x), x) - Simp((b*cos(e + f*x))**(n + S(1))*(c + d*x)**m*sin(e + f*x)/(b*f*(n + S(1))), x) - Simp(d*m*(b*cos(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x)
rule3249 = ReplacementRule(pattern3249, replacement3249)
pattern3250 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons148, cons1490)
def replacement3250(m, f, b, d, c, n, a, x, e):
rubi.append(3250)
return Int(ExpandIntegrand((c + d*x)**m, (a + b*sin(e + f*x))**n, x), x)
rule3250 = ReplacementRule(pattern3250, replacement3250)
pattern3251 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons148, cons1490)
def replacement3251(m, f, b, d, c, n, a, x, e):
rubi.append(3251)
return Int(ExpandIntegrand((c + d*x)**m, (a + b*cos(e + f*x))**n, x), x)
rule3251 = ReplacementRule(pattern3251, replacement3251)
pattern3252 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1265, cons85)
def replacement3252(m, f, b, d, c, n, a, x, e):
rubi.append(3252)
return Dist((S(2)*a)**n, Int((c + d*x)**m*cos(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule3252 = ReplacementRule(pattern3252, replacement3252)
pattern3253 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1265, cons808, cons1491)
def replacement3253(m, f, b, d, c, a, n, x, e):
rubi.append(3253)
return Dist((S(2)*a)**IntPart(n)*(a + b*sin(e + f*x))**FracPart(n)*cos(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*cos(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule3253 = ReplacementRule(pattern3253, replacement3253)
pattern3254 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1492, cons85)
def replacement3254(m, f, b, d, c, n, a, x, e):
rubi.append(3254)
return Dist((S(2)*a)**n, Int((c + d*x)**m*cos(e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule3254 = ReplacementRule(pattern3254, replacement3254)
pattern3255 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1454, cons85)
def replacement3255(m, f, b, d, c, n, a, x, e):
rubi.append(3255)
return Dist((S(2)*a)**n, Int((c + d*x)**m*sin(e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule3255 = ReplacementRule(pattern3255, replacement3255)
pattern3256 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1492, cons808, cons1491)
def replacement3256(m, f, b, d, c, a, n, x, e):
rubi.append(3256)
return Dist((S(2)*a)**IntPart(n)*(a + b*cos(e + f*x))**FracPart(n)*cos(e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*cos(e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule3256 = ReplacementRule(pattern3256, replacement3256)
pattern3257 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1454, cons808, cons1491)
def replacement3257(m, f, b, d, c, a, n, x, e):
rubi.append(3257)
return Dist((S(2)*a)**IntPart(n)*(a + b*cos(e + f*x))**FracPart(n)*sin(e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*sin(e/S(2) + f*x/S(2))**(S(2)*n), x), x)
rule3257 = ReplacementRule(pattern3257, replacement3257)
pattern3258 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons62)
def replacement3258(m, f, b, d, c, a, x, e):
rubi.append(3258)
return Dist(S(2), Int((c + d*x)**m*exp(I*(e + f*x))/(S(2)*a*exp(I*(e + f*x)) - I*b*exp(S(2)*I*(e + f*x)) + I*b), x), x)
rule3258 = ReplacementRule(pattern3258, replacement3258)
pattern3259 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons62)
def replacement3259(m, f, b, d, c, a, x, e):
rubi.append(3259)
return Dist(S(2), Int((c + d*x)**m*exp(I*(e + f*x))/(S(2)*a*exp(I*(e + f*x)) + b*exp(S(2)*I*(e + f*x)) + b), x), x)
rule3259 = ReplacementRule(pattern3259, replacement3259)
pattern3260 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons62)
def replacement3260(m, f, b, d, c, a, x, e):
rubi.append(3260)
return Dist(a/(a**S(2) - b**S(2)), Int((c + d*x)**m/(a + b*sin(e + f*x)), x), x) - Dist(b*d*m/(f*(a**S(2) - b**S(2))), Int((c + d*x)**(m + S(-1))*cos(e + f*x)/(a + b*sin(e + f*x)), x), x) + Simp(b*(c + d*x)**m*cos(e + f*x)/(f*(a + b*sin(e + f*x))*(a**S(2) - b**S(2))), x)
rule3260 = ReplacementRule(pattern3260, replacement3260)
pattern3261 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons62)
def replacement3261(m, f, b, d, c, a, x, e):
rubi.append(3261)
return Dist(a/(a**S(2) - b**S(2)), Int((c + d*x)**m/(a + b*cos(e + f*x)), x), x) + Dist(b*d*m/(f*(a**S(2) - b**S(2))), Int((c + d*x)**(m + S(-1))*sin(e + f*x)/(a + b*cos(e + f*x)), x), x) - Simp(b*(c + d*x)**m*sin(e + f*x)/(f*(a + b*cos(e + f*x))*(a**S(2) - b**S(2))), x)
rule3261 = ReplacementRule(pattern3261, replacement3261)
pattern3262 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons1493, cons62)
def replacement3262(m, f, b, d, c, a, n, x, e):
rubi.append(3262)
return Dist(a/(a**S(2) - b**S(2)), Int((a + b*sin(e + f*x))**(n + S(1))*(c + d*x)**m, x), x) - Dist(b*(n + S(2))/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(n + S(1))*(c + d*x)**m*sin(e + f*x), x), x) + Dist(b*d*m/(f*(a**S(2) - b**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1))*cos(e + f*x), x), x) - Simp(b*(a + b*sin(e + f*x))**(n + S(1))*(c + d*x)**m*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(n + S(1))), x)
rule3262 = ReplacementRule(pattern3262, replacement3262)
pattern3263 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1267, cons1493, cons62)
def replacement3263(m, f, b, d, c, a, n, x, e):
rubi.append(3263)
return Dist(a/(a**S(2) - b**S(2)), Int((a + b*cos(e + f*x))**(n + S(1))*(c + d*x)**m, x), x) - Dist(b*(n + S(2))/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(n + S(1))*(c + d*x)**m*cos(e + f*x), x), x) - Dist(b*d*m/(f*(a**S(2) - b**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1))*sin(e + f*x), x), x) + Simp(b*(a + b*cos(e + f*x))**(n + S(1))*(c + d*x)**m*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(n + S(1))), x)
rule3263 = ReplacementRule(pattern3263, replacement3263)
pattern3264 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement3264(v, u, m, b, a, n, x):
rubi.append(3264)
return Int((a + b*sin(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule3264 = ReplacementRule(pattern3264, replacement3264)
pattern3265 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement3265(v, u, m, b, a, n, x):
rubi.append(3265)
return Int((a + b*cos(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule3265 = ReplacementRule(pattern3265, replacement3265)
pattern3266 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement3266(m, f, b, d, c, a, n, x, e):
rubi.append(3266)
return Int((a + b*sin(e + f*x))**n*(c + d*x)**m, x)
rule3266 = ReplacementRule(pattern3266, replacement3266)
pattern3267 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement3267(m, f, b, d, a, n, c, x, e):
rubi.append(3267)
return Int((a + b*cos(e + f*x))**n*(c + d*x)**m, x)
rule3267 = ReplacementRule(pattern3267, replacement3267)
pattern3268 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons128)
def replacement3268(p, b, d, c, a, n, x):
rubi.append(3268)
return Int(ExpandIntegrand(sin(c + d*x), (a + b*x**n)**p, x), x)
rule3268 = ReplacementRule(pattern3268, replacement3268)
pattern3269 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons128)
def replacement3269(p, b, d, c, a, n, x):
rubi.append(3269)
return Int(ExpandIntegrand(cos(c + d*x), (a + b*x**n)**p, x), x)
rule3269 = ReplacementRule(pattern3269, replacement3269)
pattern3270 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons38, cons148, cons137, cons744)
def replacement3270(p, b, d, c, a, n, x):
rubi.append(3270)
return -Dist(d/(b*n*(p + S(1))), Int(x**(-n + S(1))*(a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) - Dist((-n + S(1))/(b*n*(p + S(1))), Int(x**(-n)*(a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) + Simp(x**(-n + S(1))*(a + b*x**n)**(p + S(1))*sin(c + d*x)/(b*n*(p + S(1))), x)
rule3270 = ReplacementRule(pattern3270, replacement3270)
pattern3271 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons38, cons148, cons137, cons744)
def replacement3271(p, b, d, c, a, n, x):
rubi.append(3271)
return Dist(d/(b*n*(p + S(1))), Int(x**(-n + S(1))*(a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) - Dist((-n + S(1))/(b*n*(p + S(1))), Int(x**(-n)*(a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) + Simp(x**(-n + S(1))*(a + b*x**n)**(p + S(1))*cos(c + d*x)/(b*n*(p + S(1))), x)
rule3271 = ReplacementRule(pattern3271, replacement3271)
pattern3272 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons148, cons1494)
def replacement3272(p, b, d, c, a, n, x):
rubi.append(3272)
return Int(ExpandIntegrand(sin(c + d*x), (a + b*x**n)**p, x), x)
rule3272 = ReplacementRule(pattern3272, replacement3272)
pattern3273 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons148, cons1494)
def replacement3273(p, b, d, c, a, n, x):
rubi.append(3273)
return Int(ExpandIntegrand(cos(c + d*x), (a + b*x**n)**p, x), x)
rule3273 = ReplacementRule(pattern3273, replacement3273)
pattern3274 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons196)
def replacement3274(p, b, d, c, a, n, x):
rubi.append(3274)
return Int(x**(n*p)*(a*x**(-n) + b)**p*sin(c + d*x), x)
rule3274 = ReplacementRule(pattern3274, replacement3274)
pattern3275 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons196)
def replacement3275(p, b, d, c, a, n, x):
rubi.append(3275)
return Int(x**(n*p)*(a*x**(-n) + b)**p*cos(c + d*x), x)
rule3275 = ReplacementRule(pattern3275, replacement3275)
pattern3276 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement3276(p, b, d, c, a, n, x):
rubi.append(3276)
return Int((a + b*x**n)**p*sin(c + d*x), x)
rule3276 = ReplacementRule(pattern3276, replacement3276)
pattern3277 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement3277(p, b, d, c, a, n, x):
rubi.append(3277)
return Int((a + b*x**n)**p*cos(c + d*x), x)
rule3277 = ReplacementRule(pattern3277, replacement3277)
pattern3278 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons128)
def replacement3278(p, m, b, d, c, a, n, x, e):
rubi.append(3278)
return Int(ExpandIntegrand(sin(c + d*x), (e*x)**m*(a + b*x**n)**p, x), x)
rule3278 = ReplacementRule(pattern3278, replacement3278)
pattern3279 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons128)
def replacement3279(p, m, b, d, c, a, n, x, e):
rubi.append(3279)
return Int(ExpandIntegrand(cos(c + d*x), (e*x)**m*(a + b*x**n)**p, x), x)
rule3279 = ReplacementRule(pattern3279, replacement3279)
pattern3280 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, cons53, cons13, cons137, cons596)
def replacement3280(p, m, b, d, c, a, n, x, e):
rubi.append(3280)
return -Dist(d*e**m/(b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) + Simp(e**m*(a + b*x**n)**(p + S(1))*sin(c + d*x)/(b*n*(p + S(1))), x)
rule3280 = ReplacementRule(pattern3280, replacement3280)
pattern3281 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, cons53, cons13, cons137, cons596)
def replacement3281(p, m, b, d, c, a, n, x, e):
rubi.append(3281)
return Dist(d*e**m/(b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) + Simp(e**m*(a + b*x**n)**(p + S(1))*cos(c + d*x)/(b*n*(p + S(1))), x)
rule3281 = ReplacementRule(pattern3281, replacement3281)
pattern3282 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons38, cons148, cons31, cons137, cons1496)
def replacement3282(p, m, b, d, c, a, n, x):
rubi.append(3282)
return -Dist(d/(b*n*(p + S(1))), Int(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) - Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*(a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) + Simp(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*sin(c + d*x)/(b*n*(p + S(1))), x)
rule3282 = ReplacementRule(pattern3282, replacement3282)
pattern3283 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons38, cons148, cons31, cons137, cons1496)
def replacement3283(p, m, b, d, c, a, n, x):
rubi.append(3283)
return Dist(d/(b*n*(p + S(1))), Int(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) - Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*(a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) + Simp(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*cos(c + d*x)/(b*n*(p + S(1))), x)
rule3283 = ReplacementRule(pattern3283, replacement3283)
pattern3284 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons17, cons148, cons1494)
def replacement3284(p, m, b, d, c, a, n, x):
rubi.append(3284)
return Int(ExpandIntegrand(sin(c + d*x), x**m*(a + b*x**n)**p, x), x)
rule3284 = ReplacementRule(pattern3284, replacement3284)
pattern3285 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons63, cons17, cons148, cons1494)
def replacement3285(p, m, b, d, c, a, n, x):
rubi.append(3285)
return Int(ExpandIntegrand(cos(c + d*x), x**m*(a + b*x**n)**p, x), x)
rule3285 = ReplacementRule(pattern3285, replacement3285)
pattern3286 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons63, cons196)
def replacement3286(p, m, b, d, c, a, n, x):
rubi.append(3286)
return Int(x**(m + n*p)*(a*x**(-n) + b)**p*sin(c + d*x), x)
rule3286 = ReplacementRule(pattern3286, replacement3286)
pattern3287 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons63, cons196)
def replacement3287(p, m, b, d, c, a, n, x):
rubi.append(3287)
return Int(x**(m + n*p)*(a*x**(-n) + b)**p*cos(c + d*x), x)
rule3287 = ReplacementRule(pattern3287, replacement3287)
pattern3288 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement3288(p, m, b, d, c, a, n, x, e):
rubi.append(3288)
return Int((e*x)**m*(a + b*x**n)**p*sin(c + d*x), x)
rule3288 = ReplacementRule(pattern3288, replacement3288)
pattern3289 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement3289(p, m, b, d, c, a, n, x, e):
rubi.append(3289)
return Int((e*x)**m*(a + b*x**n)**p*cos(c + d*x), x)
rule3289 = ReplacementRule(pattern3289, replacement3289)
pattern3290 = Pattern(Integral(sin(x_**S(2)*WC('d', S(1))), x_), cons27, cons27)
def replacement3290(d, x):
rubi.append(3290)
return Simp(sqrt(S(2))*sqrt(Pi)*FresnelS(sqrt(S(2))*x*sqrt(S(1)/Pi)*Rt(d, S(2)))/(S(2)*Rt(d, S(2))), x)
rule3290 = ReplacementRule(pattern3290, replacement3290)
pattern3291 = Pattern(Integral(cos(x_**S(2)*WC('d', S(1))), x_), cons27, cons27)
def replacement3291(d, x):
rubi.append(3291)
return Simp(sqrt(S(2))*sqrt(Pi)*FresnelC(sqrt(S(2))*x*sqrt(S(1)/Pi)*Rt(d, S(2)))/(S(2)*Rt(d, S(2))), x)
rule3291 = ReplacementRule(pattern3291, replacement3291)
pattern3292 = Pattern(Integral(sin(c_ + x_**S(2)*WC('d', S(1))), x_), cons7, cons27, cons1261)
def replacement3292(d, c, x):
rubi.append(3292)
return Dist(sin(c), Int(cos(d*x**S(2)), x), x) + Dist(cos(c), Int(sin(d*x**S(2)), x), x)
rule3292 = ReplacementRule(pattern3292, replacement3292)
pattern3293 = Pattern(Integral(cos(c_ + x_**S(2)*WC('d', S(1))), x_), cons7, cons27, cons1261)
def replacement3293(d, c, x):
rubi.append(3293)
return -Dist(sin(c), Int(sin(d*x**S(2)), x), x) + Dist(cos(c), Int(cos(d*x**S(2)), x), x)
rule3293 = ReplacementRule(pattern3293, replacement3293)
pattern3294 = Pattern(Integral(sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons85, cons744)
def replacement3294(d, c, n, x):
rubi.append(3294)
return Dist(I/S(2), Int(exp(-I*c - I*d*x**n), x), x) - Dist(I/S(2), Int(exp(I*c + I*d*x**n), x), x)
rule3294 = ReplacementRule(pattern3294, replacement3294)
pattern3295 = Pattern(Integral(cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons85, cons744)
def replacement3295(d, c, n, x):
rubi.append(3295)
return Dist(S(1)/2, Int(exp(-I*c - I*d*x**n), x), x) + Dist(S(1)/2, Int(exp(I*c + I*d*x**n), x), x)
rule3295 = ReplacementRule(pattern3295, replacement3295)
pattern3296 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons376, cons165, cons146)
def replacement3296(p, b, d, c, a, n, x):
rubi.append(3296)
return Int(ExpandTrigReduce((a + b*sin(c + d*x**n))**p, x), x)
rule3296 = ReplacementRule(pattern3296, replacement3296)
pattern3297 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons376, cons165, cons146)
def replacement3297(p, b, d, a, c, n, x):
rubi.append(3297)
return Int(ExpandTrigReduce((a + b*cos(c + d*x**n))**p, x), x)
rule3297 = ReplacementRule(pattern3297, replacement3297)
pattern3298 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons196)
def replacement3298(p, b, d, c, a, n, x):
rubi.append(3298)
return -Subst(Int((a + b*sin(c + d*x**(-n)))**p/x**S(2), x), x, S(1)/x)
rule3298 = ReplacementRule(pattern3298, replacement3298)
pattern3299 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons196)
def replacement3299(p, b, d, a, c, n, x):
rubi.append(3299)
return -Subst(Int((a + b*cos(c + d*x**(-n)))**p/x**S(2), x), x, S(1)/x)
rule3299 = ReplacementRule(pattern3299, replacement3299)
def With3300(p, b, d, c, a, n, x):
k = Denominator(n)
rubi.append(3300)
return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*sin(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x)
pattern3300 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons489)
rule3300 = ReplacementRule(pattern3300, With3300)
def With3301(p, b, d, a, c, n, x):
k = Denominator(n)
rubi.append(3301)
return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*cos(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x)
pattern3301 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons489)
rule3301 = ReplacementRule(pattern3301, With3301)
pattern3302 = Pattern(Integral(sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons4, cons1498)
def replacement3302(d, c, n, x):
rubi.append(3302)
return Dist(I/S(2), Int(exp(-I*c - I*d*x**n), x), x) - Dist(I/S(2), Int(exp(I*c + I*d*x**n), x), x)
rule3302 = ReplacementRule(pattern3302, replacement3302)
pattern3303 = Pattern(Integral(cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons4, cons1498)
def replacement3303(d, c, n, x):
rubi.append(3303)
return Dist(S(1)/2, Int(exp(-I*c - I*d*x**n), x), x) + Dist(S(1)/2, Int(exp(I*c + I*d*x**n), x), x)
rule3303 = ReplacementRule(pattern3303, replacement3303)
pattern3304 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons4, cons128)
def replacement3304(p, b, d, c, a, n, x):
rubi.append(3304)
return Int(ExpandTrigReduce((a + b*sin(c + d*x**n))**p, x), x)
rule3304 = ReplacementRule(pattern3304, replacement3304)
pattern3305 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons4, cons128)
def replacement3305(p, b, d, a, c, n, x):
rubi.append(3305)
return Int(ExpandTrigReduce((a + b*cos(c + d*x**n))**p, x), x)
rule3305 = ReplacementRule(pattern3305, replacement3305)
pattern3306 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons38, cons68, cons69)
def replacement3306(p, u, b, d, c, a, n, x):
rubi.append(3306)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*sin(c + d*x**n))**p, x), x, u), x)
rule3306 = ReplacementRule(pattern3306, replacement3306)
pattern3307 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons38, cons68, cons69)
def replacement3307(p, u, b, d, a, c, n, x):
rubi.append(3307)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*cos(c + d*x**n))**p, x), x, u), x)
rule3307 = ReplacementRule(pattern3307, replacement3307)
pattern3308 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68)
def replacement3308(p, u, b, d, c, a, n, x):
rubi.append(3308)
return Int((a + b*sin(c + d*u**n))**p, x)
rule3308 = ReplacementRule(pattern3308, replacement3308)
pattern3309 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68)
def replacement3309(p, u, b, d, a, c, n, x):
rubi.append(3309)
return Int((a + b*cos(c + d*u**n))**p, x)
rule3309 = ReplacementRule(pattern3309, replacement3309)
pattern3310 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement3310(p, u, b, a, x):
rubi.append(3310)
return Int((a + b*sin(ExpandToSum(u, x)))**p, x)
rule3310 = ReplacementRule(pattern3310, replacement3310)
pattern3311 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement3311(p, u, b, a, x):
rubi.append(3311)
return Int((a + b*cos(ExpandToSum(u, x)))**p, x)
rule3311 = ReplacementRule(pattern3311, replacement3311)
pattern3312 = Pattern(Integral(sin(x_**n_*WC('d', S(1)))/x_, x_), cons27, cons4, cons1499)
def replacement3312(d, x, n):
rubi.append(3312)
return Simp(SinIntegral(d*x**n)/n, x)
rule3312 = ReplacementRule(pattern3312, replacement3312)
pattern3313 = Pattern(Integral(cos(x_**n_*WC('d', S(1)))/x_, x_), cons27, cons4, cons1499)
def replacement3313(d, x, n):
rubi.append(3313)
return Simp(CosIntegral(d*x**n)/n, x)
rule3313 = ReplacementRule(pattern3313, replacement3313)
pattern3314 = Pattern(Integral(sin(c_ + x_**n_*WC('d', S(1)))/x_, x_), cons7, cons27, cons4, cons1498)
def replacement3314(d, x, n, c):
rubi.append(3314)
return Dist(sin(c), Int(cos(d*x**n)/x, x), x) + Dist(cos(c), Int(sin(d*x**n)/x, x), x)
rule3314 = ReplacementRule(pattern3314, replacement3314)
pattern3315 = Pattern(Integral(cos(c_ + x_**n_*WC('d', S(1)))/x_, x_), cons7, cons27, cons4, cons1498)
def replacement3315(d, c, n, x):
rubi.append(3315)
return -Dist(sin(c), Int(sin(d*x**n)/x, x), x) + Dist(cos(c), Int(cos(d*x**n)/x, x), x)
rule3315 = ReplacementRule(pattern3315, replacement3315)
def With3316(p, m, b, d, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
mn = (m + S(1))/n
if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))):
return True
return False
pattern3316 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons38, CustomConstraint(With3316))
def replacement3316(p, m, b, d, c, a, n, x):
mn = (m + S(1))/n
rubi.append(3316)
return Dist(S(1)/n, Subst(Int(x**(mn + S(-1))*(a + b*sin(c + d*x))**p, x), x, x**n), x)
rule3316 = ReplacementRule(pattern3316, replacement3316)
def With3317(p, m, b, d, a, c, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
mn = (m + S(1))/n
if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))):
return True
return False
pattern3317 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons38, CustomConstraint(With3317))
def replacement3317(p, m, b, d, a, c, n, x):
mn = (m + S(1))/n
rubi.append(3317)
return Dist(S(1)/n, Subst(Int(x**(mn + S(-1))*(a + b*cos(c + d*x))**p, x), x, x**n), x)
rule3317 = ReplacementRule(pattern3317, replacement3317)
def With3318(p, m, b, d, c, a, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
mn = (m + S(1))/n
if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))):
return True
return False
pattern3318 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, CustomConstraint(With3318))
def replacement3318(p, m, b, d, c, a, n, x, e):
mn = (m + S(1))/n
rubi.append(3318)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sin(c + d*x**n))**p, x), x)
rule3318 = ReplacementRule(pattern3318, replacement3318)
def With3319(p, m, b, d, a, c, n, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
mn = (m + S(1))/n
if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))):
return True
return False
pattern3319 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, CustomConstraint(With3319))
def replacement3319(p, m, b, d, a, c, n, x, e):
mn = (m + S(1))/n
rubi.append(3319)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cos(c + d*x**n))**p, x), x)
rule3319 = ReplacementRule(pattern3319, replacement3319)
pattern3320 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons4, cons1500)
def replacement3320(m, b, a, n, x):
rubi.append(3320)
return Dist(S(2)/n, Subst(Int(sin(a + b*x**S(2)), x), x, x**(n/S(2))), x)
rule3320 = ReplacementRule(pattern3320, replacement3320)
pattern3321 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons4, cons1500)
def replacement3321(m, b, a, n, x):
rubi.append(3321)
return Dist(S(2)/n, Subst(Int(cos(a + b*x**S(2)), x), x, x**(n/S(2))), x)
rule3321 = ReplacementRule(pattern3321, replacement3321)
pattern3322 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons148, cons31, cons1501)
def replacement3322(m, d, c, n, x, e):
rubi.append(3322)
return Dist(e**n*(m - n + S(1))/(d*n), Int((e*x)**(m - n)*cos(c + d*x**n), x), x) - Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*cos(c + d*x**n)/(d*n), x)
rule3322 = ReplacementRule(pattern3322, replacement3322)
pattern3323 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons148, cons31, cons1501)
def replacement3323(m, d, c, n, x, e):
rubi.append(3323)
return -Dist(e**n*(m - n + S(1))/(d*n), Int((e*x)**(m - n)*sin(c + d*x**n), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*sin(c + d*x**n)/(d*n), x)
rule3323 = ReplacementRule(pattern3323, replacement3323)
pattern3324 = Pattern(Integral((x_*WC('e', S(1)))**m_*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons148, cons31, cons94)
def replacement3324(m, d, c, n, x, e):
rubi.append(3324)
return -Dist(d*e**(-n)*n/(m + S(1)), Int((e*x)**(m + n)*cos(c + d*x**n), x), x) + Simp((e*x)**(m + S(1))*sin(c + d*x**n)/(e*(m + S(1))), x)
rule3324 = ReplacementRule(pattern3324, replacement3324)
pattern3325 = Pattern(Integral((x_*WC('e', S(1)))**m_*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons148, cons31, cons94)
def replacement3325(m, d, c, n, x, e):
rubi.append(3325)
return Dist(d*e**(-n)*n/(m + S(1)), Int((e*x)**(m + n)*sin(c + d*x**n), x), x) + Simp((e*x)**(m + S(1))*cos(c + d*x**n)/(e*(m + S(1))), x)
rule3325 = ReplacementRule(pattern3325, replacement3325)
pattern3326 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons21, cons148)
def replacement3326(m, d, c, n, x, e):
rubi.append(3326)
return Dist(I/S(2), Int((e*x)**m*exp(-I*c - I*d*x**n), x), x) - Dist(I/S(2), Int((e*x)**m*exp(I*c + I*d*x**n), x), x)
rule3326 = ReplacementRule(pattern3326, replacement3326)
pattern3327 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons21, cons148)
def replacement3327(m, d, c, n, x, e):
rubi.append(3327)
return Dist(S(1)/2, Int((e*x)**m*exp(-I*c - I*d*x**n), x), x) + Dist(S(1)/2, Int((e*x)**m*exp(I*c + I*d*x**n), x), x)
rule3327 = ReplacementRule(pattern3327, replacement3327)
pattern3328 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons376, cons1255, cons146, cons1502)
def replacement3328(p, m, b, a, n, x):
rubi.append(3328)
return Dist(b*n*p/(n + S(-1)), Int(sin(a + b*x**n)**(p + S(-1))*cos(a + b*x**n), x), x) - Simp(x**(-n + S(1))*sin(a + b*x**n)**p/(n + S(-1)), x)
rule3328 = ReplacementRule(pattern3328, replacement3328)
pattern3329 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons376, cons1255, cons146, cons1502)
def replacement3329(p, m, b, a, n, x):
rubi.append(3329)
return -Dist(b*n*p/(n + S(-1)), Int(sin(a + b*x**n)*cos(a + b*x**n)**(p + S(-1)), x), x) - Simp(x**(-n + S(1))*cos(a + b*x**n)**p/(n + S(-1)), x)
rule3329 = ReplacementRule(pattern3329, replacement3329)
pattern3330 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons56, cons13, cons146)
def replacement3330(p, m, b, a, n, x):
rubi.append(3330)
return Dist((p + S(-1))/p, Int(x**m*sin(a + b*x**n)**(p + S(-2)), x), x) + Simp(sin(a + b*x**n)**p/(b**S(2)*n*p**S(2)), x) - Simp(x**n*sin(a + b*x**n)**(p + S(-1))*cos(a + b*x**n)/(b*n*p), x)
rule3330 = ReplacementRule(pattern3330, replacement3330)
pattern3331 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons56, cons13, cons146)
def replacement3331(p, m, b, a, n, x):
rubi.append(3331)
return Dist((p + S(-1))/p, Int(x**m*cos(a + b*x**n)**(p + S(-2)), x), x) + Simp(cos(a + b*x**n)**p/(b**S(2)*n*p**S(2)), x) + Simp(x**n*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(-1))/(b*n*p), x)
rule3331 = ReplacementRule(pattern3331, replacement3331)
pattern3332 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons146, cons1503)
def replacement3332(p, m, b, a, n, x):
rubi.append(3332)
return Dist((p + S(-1))/p, Int(x**m*sin(a + b*x**n)**(p + S(-2)), x), x) - Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*p**S(2)), Int(x**(m - S(2)*n)*sin(a + b*x**n)**p, x), x) + Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*sin(a + b*x**n)**p/(b**S(2)*n**S(2)*p**S(2)), x) - Simp(x**(m - n + S(1))*sin(a + b*x**n)**(p + S(-1))*cos(a + b*x**n)/(b*n*p), x)
rule3332 = ReplacementRule(pattern3332, replacement3332)
pattern3333 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons146, cons1503)
def replacement3333(p, m, b, a, n, x):
rubi.append(3333)
return Dist((p + S(-1))/p, Int(x**m*cos(a + b*x**n)**(p + S(-2)), x), x) - Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*p**S(2)), Int(x**(m - S(2)*n)*cos(a + b*x**n)**p, x), x) + Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*cos(a + b*x**n)**p/(b**S(2)*n**S(2)*p**S(2)), x) + Simp(x**(m - n + S(1))*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(-1))/(b*n*p), x)
rule3333 = ReplacementRule(pattern3333, replacement3333)
pattern3334 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons146, cons1504, cons683)
def replacement3334(p, m, b, a, n, x):
rubi.append(3334)
return -Dist(b**S(2)*n**S(2)*p**S(2)/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*sin(a + b*x**n)**p, x), x) + Dist(b**S(2)*n**S(2)*p*(p + S(-1))/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*sin(a + b*x**n)**(p + S(-2)), x), x) + Simp(x**(m + S(1))*sin(a + b*x**n)**p/(m + S(1)), x) - Simp(b*n*p*x**(m + n + S(1))*sin(a + b*x**n)**(p + S(-1))*cos(a + b*x**n)/((m + S(1))*(m + n + S(1))), x)
rule3334 = ReplacementRule(pattern3334, replacement3334)
pattern3335 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons146, cons1504, cons683)
def replacement3335(p, m, b, a, n, x):
rubi.append(3335)
return -Dist(b**S(2)*n**S(2)*p**S(2)/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*cos(a + b*x**n)**p, x), x) + Dist(b**S(2)*n**S(2)*p*(p + S(-1))/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*cos(a + b*x**n)**(p + S(-2)), x), x) + Simp(x**(m + S(1))*cos(a + b*x**n)**p/(m + S(1)), x) + Simp(b*n*p*x**(m + n + S(1))*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(-1))/((m + S(1))*(m + n + S(1))), x)
rule3335 = ReplacementRule(pattern3335, replacement3335)
def With3336(p, m, b, d, c, a, n, x, e):
k = Denominator(m)
rubi.append(3336)
return Dist(k/e, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*sin(c + d*e**(-n)*x**(k*n)))**p, x), x, (e*x)**(S(1)/k)), x)
pattern3336 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons148, cons367)
rule3336 = ReplacementRule(pattern3336, With3336)
def With3337(p, m, b, d, a, c, n, x, e):
k = Denominator(m)
rubi.append(3337)
return Dist(k/e, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*cos(c + d*e**(-n)*x**(k*n)))**p, x), x, (e*x)**(S(1)/k)), x)
pattern3337 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons148, cons367)
rule3337 = ReplacementRule(pattern3337, With3337)
pattern3338 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons146)
def replacement3338(p, m, b, d, c, a, n, x, e):
rubi.append(3338)
return Int(ExpandTrigReduce((e*x)**m, (a + b*sin(c + d*x**n))**p, x), x)
rule3338 = ReplacementRule(pattern3338, replacement3338)
pattern3339 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons148, cons146)
def replacement3339(p, m, b, d, a, c, n, x, e):
rubi.append(3339)
return Int(ExpandTrigReduce((e*x)**m, (a + b*cos(c + d*x**n))**p, x), x)
rule3339 = ReplacementRule(pattern3339, replacement3339)
pattern3340 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons56, cons13, cons137, cons1505)
def replacement3340(p, m, b, a, n, x):
rubi.append(3340)
return Dist((p + S(2))/(p + S(1)), Int(x**m*sin(a + b*x**n)**(p + S(2)), x), x) - Simp(sin(a + b*x**n)**(p + S(2))/(b**S(2)*n*(p + S(1))*(p + S(2))), x) + Simp(x**n*sin(a + b*x**n)**(p + S(1))*cos(a + b*x**n)/(b*n*(p + S(1))), x)
rule3340 = ReplacementRule(pattern3340, replacement3340)
pattern3341 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons21, cons4, cons56, cons13, cons137, cons1505)
def replacement3341(p, m, b, a, n, x):
rubi.append(3341)
return Dist((p + S(2))/(p + S(1)), Int(x**m*cos(a + b*x**n)**(p + S(2)), x), x) - Simp(cos(a + b*x**n)**(p + S(2))/(b**S(2)*n*(p + S(1))*(p + S(2))), x) - Simp(x**n*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule3341 = ReplacementRule(pattern3341, replacement3341)
pattern3342 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons137, cons1505, cons1503)
def replacement3342(p, m, b, a, n, x):
rubi.append(3342)
return Dist((p + S(2))/(p + S(1)), Int(x**m*sin(a + b*x**n)**(p + S(2)), x), x) + Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**(m - S(2)*n)*sin(a + b*x**n)**(p + S(2)), x), x) + Simp(x**(m - n + S(1))*sin(a + b*x**n)**(p + S(1))*cos(a + b*x**n)/(b*n*(p + S(1))), x) - Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*sin(a + b*x**n)**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x)
rule3342 = ReplacementRule(pattern3342, replacement3342)
pattern3343 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons150, cons13, cons137, cons1505, cons1503)
def replacement3343(p, m, b, a, n, x):
rubi.append(3343)
return Dist((p + S(2))/(p + S(1)), Int(x**m*cos(a + b*x**n)**(p + S(2)), x), x) + Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**(m - S(2)*n)*cos(a + b*x**n)**(p + S(2)), x), x) - Simp(x**(m - n + S(1))*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) - Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*cos(a + b*x**n)**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x)
rule3343 = ReplacementRule(pattern3343, replacement3343)
pattern3344 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons196, cons17)
def replacement3344(p, m, b, d, c, a, n, x):
rubi.append(3344)
return -Subst(Int(x**(-m + S(-2))*(a + b*sin(c + d*x**(-n)))**p, x), x, S(1)/x)
rule3344 = ReplacementRule(pattern3344, replacement3344)
pattern3345 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons38, cons196, cons17)
def replacement3345(p, m, b, d, a, c, n, x):
rubi.append(3345)
return -Subst(Int(x**(-m + S(-2))*(a + b*cos(c + d*x**(-n)))**p, x), x, S(1)/x)
rule3345 = ReplacementRule(pattern3345, replacement3345)
def With3346(p, m, b, d, c, a, n, x, e):
k = Denominator(m)
rubi.append(3346)
return -Dist(k/e, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*sin(c + d*e**(-n)*x**(-k*n)))**p, x), x, (e*x)**(-S(1)/k)), x)
pattern3346 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons196, cons367)
rule3346 = ReplacementRule(pattern3346, With3346)
def With3347(p, m, b, d, a, c, n, x, e):
k = Denominator(m)
rubi.append(3347)
return -Dist(k/e, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*cos(c + d*e**(-n)*x**(-k*n)))**p, x), x, (e*x)**(-S(1)/k)), x)
pattern3347 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons38, cons196, cons367)
rule3347 = ReplacementRule(pattern3347, With3347)
pattern3348 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons196, cons356)
def replacement3348(p, m, b, d, c, a, n, x, e):
rubi.append(3348)
return -Dist((e*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*sin(c + d*x**(-n)))**p, x), x, S(1)/x), x)
rule3348 = ReplacementRule(pattern3348, replacement3348)
pattern3349 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons196, cons356)
def replacement3349(p, m, b, d, a, c, n, x, e):
rubi.append(3349)
return -Dist((e*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*cos(c + d*x**(-n)))**p, x), x, S(1)/x), x)
rule3349 = ReplacementRule(pattern3349, replacement3349)
def With3350(p, m, b, d, c, a, n, x):
k = Denominator(n)
rubi.append(3350)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*sin(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x)
pattern3350 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons38, cons489)
rule3350 = ReplacementRule(pattern3350, With3350)
def With3351(p, m, b, d, a, c, n, x):
k = Denominator(n)
rubi.append(3351)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*cos(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x)
pattern3351 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons38, cons489)
rule3351 = ReplacementRule(pattern3351, With3351)
pattern3352 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons489)
def replacement3352(p, m, b, d, c, a, n, x, e):
rubi.append(3352)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sin(c + d*x**n))**p, x), x)
rule3352 = ReplacementRule(pattern3352, replacement3352)
pattern3353 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons38, cons489)
def replacement3353(p, m, b, d, a, c, n, x, e):
rubi.append(3353)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cos(c + d*x**n))**p, x), x)
rule3353 = ReplacementRule(pattern3353, replacement3353)
pattern3354 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons38, cons66, cons854, cons23)
def replacement3354(p, m, b, d, c, a, n, x):
rubi.append(3354)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*sin(c + d*x**(n/(m + S(1)))))**p, x), x, x**(m + S(1))), x)
rule3354 = ReplacementRule(pattern3354, replacement3354)
pattern3355 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons38, cons66, cons854, cons23)
def replacement3355(p, m, b, d, a, c, n, x):
rubi.append(3355)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*cos(c + d*x**(n/(m + S(1)))))**p, x), x, x**(m + S(1))), x)
rule3355 = ReplacementRule(pattern3355, replacement3355)
pattern3356 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, cons66, cons854, cons23)
def replacement3356(p, m, b, d, c, a, n, x, e):
rubi.append(3356)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sin(c + d*x**n))**p, x), x)
rule3356 = ReplacementRule(pattern3356, replacement3356)
pattern3357 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons38, cons66, cons854, cons23)
def replacement3357(p, m, b, d, a, c, n, x, e):
rubi.append(3357)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cos(c + d*x**n))**p, x), x)
rule3357 = ReplacementRule(pattern3357, replacement3357)
pattern3358 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons21, cons4, cons1506)
def replacement3358(m, d, c, n, x, e):
rubi.append(3358)
return Dist(I/S(2), Int((e*x)**m*exp(-I*c - I*d*x**n), x), x) - Dist(I/S(2), Int((e*x)**m*exp(I*c + I*d*x**n), x), x)
rule3358 = ReplacementRule(pattern3358, replacement3358)
pattern3359 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons48, cons21, cons4, cons1506)
def replacement3359(m, d, c, n, x, e):
rubi.append(3359)
return Dist(S(1)/2, Int((e*x)**m*exp(-I*c - I*d*x**n), x), x) + Dist(S(1)/2, Int((e*x)**m*exp(I*c + I*d*x**n), x), x)
rule3359 = ReplacementRule(pattern3359, replacement3359)
pattern3360 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons128)
def replacement3360(p, m, b, d, c, a, n, x, e):
rubi.append(3360)
return Int(ExpandTrigReduce((e*x)**m, (a + b*sin(c + d*x**n))**p, x), x)
rule3360 = ReplacementRule(pattern3360, replacement3360)
pattern3361 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons128)
def replacement3361(p, m, b, d, a, c, n, x, e):
rubi.append(3361)
return Int(ExpandTrigReduce((e*x)**m, (a + b*cos(c + d*x**n))**p, x), x)
rule3361 = ReplacementRule(pattern3361, replacement3361)
pattern3362 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69, cons17)
def replacement3362(p, u, m, b, d, c, a, n, x):
rubi.append(3362)
return Dist(Coefficient(u, x, S(1))**(-m + S(-1)), Subst(Int((a + b*sin(c + d*x**n))**p*(x - Coefficient(u, x, S(0)))**m, x), x, u), x)
rule3362 = ReplacementRule(pattern3362, replacement3362)
pattern3363 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69, cons17)
def replacement3363(p, u, m, b, d, a, c, n, x):
rubi.append(3363)
return Dist(Coefficient(u, x, S(1))**(-m + S(-1)), Subst(Int((a + b*cos(c + d*x**n))**p*(x - Coefficient(u, x, S(0)))**m, x), x, u), x)
rule3363 = ReplacementRule(pattern3363, replacement3363)
pattern3364 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons68)
def replacement3364(p, u, m, b, d, c, a, n, x, e):
rubi.append(3364)
return Int((e*x)**m*(a + b*sin(c + d*u**n))**p, x)
rule3364 = ReplacementRule(pattern3364, replacement3364)
pattern3365 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons68)
def replacement3365(p, u, m, b, d, a, c, n, x, e):
rubi.append(3365)
return Int((e*x)**m*(a + b*cos(c + d*u**n))**p, x)
rule3365 = ReplacementRule(pattern3365, replacement3365)
pattern3366 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement3366(p, u, m, b, a, x, e):
rubi.append(3366)
return Int((e*x)**m*(a + b*sin(ExpandToSum(u, x)))**p, x)
rule3366 = ReplacementRule(pattern3366, replacement3366)
pattern3367 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement3367(p, u, m, b, a, x, e):
rubi.append(3367)
return Int((e*x)**m*(a + b*cos(ExpandToSum(u, x)))**p, x)
rule3367 = ReplacementRule(pattern3367, replacement3367)
pattern3368 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons4, cons5, cons53, cons54)
def replacement3368(p, m, b, a, n, x):
rubi.append(3368)
return Simp(sin(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule3368 = ReplacementRule(pattern3368, replacement3368)
pattern3369 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons21, cons4, cons5, cons53, cons54)
def replacement3369(p, m, b, a, n, x):
rubi.append(3369)
return -Simp(cos(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule3369 = ReplacementRule(pattern3369, replacement3369)
pattern3370 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons5, cons93, cons1501, cons54)
def replacement3370(p, m, b, a, n, x):
rubi.append(3370)
return -Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*sin(a + b*x**n)**(p + S(1)), x), x) + Simp(x**(m - n + S(1))*sin(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule3370 = ReplacementRule(pattern3370, replacement3370)
pattern3371 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons5, cons93, cons1501, cons54)
def replacement3371(p, m, b, a, n, x):
rubi.append(3371)
return Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*cos(a + b*x**n)**(p + S(1)), x), x) - Simp(x**(m - n + S(1))*cos(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule3371 = ReplacementRule(pattern3371, replacement3371)
pattern3372 = Pattern(Integral(sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons45)
def replacement3372(x, c, a, b):
rubi.append(3372)
return Int(sin((b + S(2)*c*x)**S(2)/(S(4)*c)), x)
rule3372 = ReplacementRule(pattern3372, replacement3372)
pattern3373 = Pattern(Integral(cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons45)
def replacement3373(x, c, a, b):
rubi.append(3373)
return Int(cos((b + S(2)*c*x)**S(2)/(S(4)*c)), x)
rule3373 = ReplacementRule(pattern3373, replacement3373)
pattern3374 = Pattern(Integral(sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons226)
def replacement3374(x, c, a, b):
rubi.append(3374)
return -Dist(sin((-S(4)*a*c + b**S(2))/(S(4)*c)), Int(cos((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x) + Dist(cos((-S(4)*a*c + b**S(2))/(S(4)*c)), Int(sin((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x)
rule3374 = ReplacementRule(pattern3374, replacement3374)
pattern3375 = Pattern(Integral(cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons226)
def replacement3375(x, c, a, b):
rubi.append(3375)
return Dist(sin((-S(4)*a*c + b**S(2))/(S(4)*c)), Int(sin((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x) + Dist(cos((-S(4)*a*c + b**S(2))/(S(4)*c)), Int(cos((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x)
rule3375 = ReplacementRule(pattern3375, replacement3375)
pattern3376 = Pattern(Integral(sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons85, cons165)
def replacement3376(b, c, a, n, x):
rubi.append(3376)
return Int(ExpandTrigReduce(sin(a + b*x + c*x**S(2))**n, x), x)
rule3376 = ReplacementRule(pattern3376, replacement3376)
pattern3377 = Pattern(Integral(cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons85, cons165)
def replacement3377(b, c, a, n, x):
rubi.append(3377)
return Int(ExpandTrigReduce(cos(a + b*x + c*x**S(2))**n, x), x)
rule3377 = ReplacementRule(pattern3377, replacement3377)
pattern3378 = Pattern(Integral(sin(v_)**WC('n', S(1)), x_), cons148, cons818, cons1131)
def replacement3378(v, x, n):
rubi.append(3378)
return Int(sin(ExpandToSum(v, x))**n, x)
rule3378 = ReplacementRule(pattern3378, replacement3378)
pattern3379 = Pattern(Integral(cos(v_)**WC('n', S(1)), x_), cons148, cons818, cons1131)
def replacement3379(v, x, n):
rubi.append(3379)
return Int(cos(ExpandToSum(v, x))**n, x)
rule3379 = ReplacementRule(pattern3379, replacement3379)
pattern3380 = Pattern(Integral((d_ + x_*WC('e', S(1)))*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons47)
def replacement3380(b, d, c, a, x, e):
rubi.append(3380)
return -Simp(e*cos(a + b*x + c*x**S(2))/(S(2)*c), x)
rule3380 = ReplacementRule(pattern3380, replacement3380)
pattern3381 = Pattern(Integral((d_ + x_*WC('e', S(1)))*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons47)
def replacement3381(b, d, c, a, x, e):
rubi.append(3381)
return Simp(e*sin(a + b*x + c*x**S(2))/(S(2)*c), x)
rule3381 = ReplacementRule(pattern3381, replacement3381)
pattern3382 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons239)
def replacement3382(b, d, c, a, x, e):
rubi.append(3382)
return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(sin(a + b*x + c*x**S(2)), x), x) - Simp(e*cos(a + b*x + c*x**S(2))/(S(2)*c), x)
rule3382 = ReplacementRule(pattern3382, replacement3382)
pattern3383 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons239)
def replacement3383(b, d, c, a, x, e):
rubi.append(3383)
return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(cos(a + b*x + c*x**S(2)), x), x) + Simp(e*sin(a + b*x + c*x**S(2))/(S(2)*c), x)
rule3383 = ReplacementRule(pattern3383, replacement3383)
pattern3384 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166, cons1132)
def replacement3384(m, b, d, c, a, x, e):
rubi.append(3384)
return Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*cos(a + b*x + c*x**S(2)), x), x) - Simp(e*(d + e*x)**(m + S(-1))*cos(a + b*x + c*x**S(2))/(S(2)*c), x)
rule3384 = ReplacementRule(pattern3384, replacement3384)
pattern3385 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166, cons1132)
def replacement3385(m, b, d, c, a, x, e):
rubi.append(3385)
return -Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*sin(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*sin(a + b*x + c*x**S(2))/(S(2)*c), x)
rule3385 = ReplacementRule(pattern3385, replacement3385)
pattern3386 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166, cons1133)
def replacement3386(m, b, d, c, a, x, e):
rubi.append(3386)
return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int((d + e*x)**(m + S(-1))*sin(a + b*x + c*x**S(2)), x), x) + Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*cos(a + b*x + c*x**S(2)), x), x) - Simp(e*(d + e*x)**(m + S(-1))*cos(a + b*x + c*x**S(2))/(S(2)*c), x)
rule3386 = ReplacementRule(pattern3386, replacement3386)
pattern3387 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166, cons1133)
def replacement3387(m, b, d, c, a, x, e):
rubi.append(3387)
return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int((d + e*x)**(m + S(-1))*cos(a + b*x + c*x**S(2)), x), x) - Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*sin(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*sin(a + b*x + c*x**S(2))/(S(2)*c), x)
rule3387 = ReplacementRule(pattern3387, replacement3387)
pattern3388 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94, cons1132)
def replacement3388(m, b, d, c, a, x, e):
rubi.append(3388)
return -Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*cos(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*sin(a + b*x + c*x**S(2))/(e*(m + S(1))), x)
rule3388 = ReplacementRule(pattern3388, replacement3388)
pattern3389 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94, cons1132)
def replacement3389(m, b, d, c, a, x, e):
rubi.append(3389)
return Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*sin(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*cos(a + b*x + c*x**S(2))/(e*(m + S(1))), x)
rule3389 = ReplacementRule(pattern3389, replacement3389)
pattern3390 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94, cons1133)
def replacement3390(m, b, d, c, a, x, e):
rubi.append(3390)
return -Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*cos(a + b*x + c*x**S(2)), x), x) - Dist((b*e - S(2)*c*d)/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(1))*cos(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*sin(a + b*x + c*x**S(2))/(e*(m + S(1))), x)
rule3390 = ReplacementRule(pattern3390, replacement3390)
pattern3391 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94, cons1133)
def replacement3391(m, b, d, c, a, x, e):
rubi.append(3391)
return Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*sin(a + b*x + c*x**S(2)), x), x) + Dist((b*e - S(2)*c*d)/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(1))*sin(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*cos(a + b*x + c*x**S(2))/(e*(m + S(1))), x)
rule3391 = ReplacementRule(pattern3391, replacement3391)
pattern3392 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1507)
def replacement3392(m, b, d, a, c, x, e):
rubi.append(3392)
return Int((d + e*x)**m*sin(a + b*x + c*x**S(2)), x)
rule3392 = ReplacementRule(pattern3392, replacement3392)
pattern3393 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1507)
def replacement3393(m, b, d, c, a, x, e):
rubi.append(3393)
return Int((d + e*x)**m*cos(a + b*x + c*x**S(2)), x)
rule3393 = ReplacementRule(pattern3393, replacement3393)
pattern3394 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons85, cons165)
def replacement3394(m, b, d, a, c, n, x, e):
rubi.append(3394)
return Int(ExpandTrigReduce((d + e*x)**m, sin(a + b*x + c*x**S(2))**n, x), x)
rule3394 = ReplacementRule(pattern3394, replacement3394)
pattern3395 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons85, cons165)
def replacement3395(m, b, d, c, a, n, x, e):
rubi.append(3395)
return Int(ExpandTrigReduce((d + e*x)**m, cos(a + b*x + c*x**S(2))**n, x), x)
rule3395 = ReplacementRule(pattern3395, replacement3395)
pattern3396 = Pattern(Integral(u_**WC('m', S(1))*sin(v_)**WC('n', S(1)), x_), cons21, cons148, cons68, cons818, cons819)
def replacement3396(v, u, m, n, x):
rubi.append(3396)
return Int(ExpandToSum(u, x)**m*sin(ExpandToSum(v, x))**n, x)
rule3396 = ReplacementRule(pattern3396, replacement3396)
pattern3397 = Pattern(Integral(u_**WC('m', S(1))*cos(v_)**WC('n', S(1)), x_), cons21, cons148, cons68, cons818, cons819)
def replacement3397(v, u, m, n, x):
rubi.append(3397)
return Int(ExpandToSum(u, x)**m*cos(ExpandToSum(v, x))**n, x)
rule3397 = ReplacementRule(pattern3397, replacement3397)
return [rule2164, rule2165, rule2166, rule2167, rule2168, rule2169, rule2170, rule2171, rule2172, rule2173, rule2174, rule2175, rule2176, rule2177, rule2178, rule2179, rule2180, rule2181, rule2182, rule2183, rule2184, rule2185, rule2186, rule2187, rule2188, rule2189, rule2190, rule2191, rule2192, rule2193, rule2194, rule2195, rule2196, rule2197, rule2198, rule2199, rule2200, rule2201, rule2202, rule2203, rule2204, rule2205, rule2206, rule2207, rule2208, rule2209, rule2210, rule2211, rule2212, rule2213, rule2214, rule2215, rule2216, rule2217, rule2218, rule2219, rule2220, rule2221, rule2222, rule2223, rule2224, rule2225, rule2226, rule2227, rule2228, rule2229, rule2230, rule2231, rule2232, rule2233, rule2234, rule2235, rule2236, rule2237, rule2238, rule2239, rule2240, rule2241, rule2242, rule2243, rule2244, rule2245, rule2246, rule2247, rule2248, rule2249, rule2250, rule2251, rule2252, rule2253, rule2254, rule2255, rule2256, rule2257, rule2258, rule2259, rule2260, rule2261, rule2262, rule2263, rule2264, rule2265, rule2266, rule2267, rule2268, rule2269, rule2270, rule2271, rule2272, rule2273, rule2274, rule2275, rule2276, rule2277, rule2278, rule2279, rule2280, rule2281, rule2282, rule2283, rule2284, rule2285, rule2286, rule2287, rule2288, rule2289, rule2290, rule2291, rule2292, rule2293, rule2294, rule2295, rule2296, rule2297, rule2298, rule2299, rule2300, rule2301, rule2302, rule2303, rule2304, rule2305, rule2306, rule2307, rule2308, rule2309, rule2310, rule2311, rule2312, rule2313, rule2314, rule2315, rule2316, rule2317, rule2318, rule2319, rule2320, rule2321, rule2322, rule2323, rule2324, rule2325, rule2326, rule2327, rule2328, rule2329, rule2330, rule2331, rule2332, rule2333, rule2334, rule2335, rule2336, rule2337, rule2338, rule2339, rule2340, rule2341, rule2342, rule2343, rule2344, rule2345, rule2346, rule2347, rule2348, rule2349, rule2350, rule2351, rule2352, rule2353, rule2354, rule2355, rule2356, rule2357, rule2358, rule2359, rule2360, rule2361, rule2362, rule2363, rule2364, rule2365, rule2366, rule2367, rule2368, rule2369, rule2370, rule2371, rule2372, rule2373, rule2374, rule2375, rule2376, rule2377, rule2378, rule2379, rule2380, rule2381, rule2382, rule2383, rule2384, rule2385, rule2386, rule2387, rule2388, rule2389, rule2390, rule2391, rule2392, rule2393, rule2394, rule2395, rule2396, rule2397, rule2398, rule2399, rule2400, rule2401, rule2402, rule2403, rule2404, rule2405, rule2406, rule2407, rule2408, rule2409, rule2410, rule2411, rule2412, rule2413, rule2414, rule2415, rule2416, rule2417, rule2418, rule2419, rule2420, rule2421, rule2422, rule2423, rule2424, rule2425, rule2426, rule2427, rule2428, rule2429, rule2430, rule2431, rule2432, rule2433, rule2434, rule2435, rule2436, rule2437, rule2438, rule2439, rule2440, rule2441, rule2442, rule2443, rule2444, rule2445, rule2446, rule2447, rule2448, rule2449, rule2450, rule2451, rule2452, rule2453, rule2454, rule2455, rule2456, rule2457, rule2458, rule2459, rule2460, rule2461, rule2462, rule2463, rule2464, rule2465, rule2466, rule2467, rule2468, rule2469, rule2470, rule2471, rule2472, rule2473, rule2474, rule2475, rule2476, rule2477, rule2478, rule2479, rule2480, rule2481, rule2482, rule2483, rule2484, rule2485, rule2486, rule2487, rule2488, rule2489, rule2490, rule2491, rule2492, rule2493, rule2494, rule2495, rule2496, rule2497, rule2498, rule2499, rule2500, rule2501, rule2502, rule2503, rule2504, rule2505, rule2506, rule2507, rule2508, rule2509, rule2510, rule2511, rule2512, rule2513, rule2514, rule2515, rule2516, rule2517, rule2518, rule2519, rule2520, rule2521, rule2522, rule2523, rule2524, rule2525, rule2526, rule2527, rule2528, rule2529, rule2530, rule2531, rule2532, rule2533, rule2534, rule2535, rule2536, rule2537, rule2538, rule2539, rule2540, rule2541, rule2542, rule2543, rule2544, rule2545, rule2546, rule2547, rule2548, rule2549, rule2550, rule2551, rule2552, rule2553, rule2554, rule2555, rule2556, rule2557, rule2558, rule2559, rule2560, rule2561, rule2562, rule2563, rule2564, rule2565, rule2566, rule2567, rule2568, rule2569, rule2570, rule2571, rule2572, rule2573, rule2574, rule2575, rule2576, rule2577, rule2578, rule2579, rule2580, rule2581, rule2582, rule2583, rule2584, rule2585, rule2586, rule2587, rule2588, rule2589, rule2590, rule2591, rule2592, rule2593, rule2594, rule2595, rule2596, rule2597, rule2598, rule2599, rule2600, rule2601, rule2602, rule2603, rule2604, rule2605, rule2606, rule2607, rule2608, rule2609, rule2610, rule2611, rule2612, rule2613, rule2614, rule2615, rule2616, rule2617, rule2618, rule2619, rule2620, rule2621, rule2622, rule2623, rule2624, rule2625, rule2626, rule2627, rule2628, rule2629, rule2630, rule2631, rule2632, rule2633, rule2634, rule2635, rule2636, rule2637, rule2638, rule2639, rule2640, rule2641, rule2642, rule2643, rule2644, rule2645, rule2646, rule2647, rule2648, rule2649, rule2650, rule2651, rule2652, rule2653, rule2654, rule2655, rule2656, rule2657, rule2658, rule2659, rule2660, rule2661, rule2662, rule2663, rule2664, rule2665, rule2666, rule2667, rule2668, rule2669, rule2670, rule2671, rule2672, rule2673, rule2674, rule2675, rule2676, rule2677, rule2678, rule2679, rule2680, rule2681, rule2682, rule2683, rule2684, rule2685, rule2686, rule2687, rule2688, rule2689, rule2690, rule2691, rule2692, rule2693, rule2694, rule2695, rule2696, rule2697, rule2698, rule2699, rule2700, rule2701, rule2702, rule2703, rule2704, rule2705, rule2706, rule2707, rule2708, rule2709, rule2710, rule2711, rule2712, rule2713, rule2714, rule2715, rule2716, rule2717, rule2718, rule2719, rule2720, rule2721, rule2722, rule2723, rule2724, rule2725, rule2726, rule2727, rule2728, rule2729, rule2730, rule2731, rule2732, rule2733, rule2734, rule2735, rule2736, rule2737, rule2738, rule2739, rule2740, rule2741, rule2742, rule2743, rule2744, rule2745, rule2746, rule2747, rule2748, rule2749, rule2750, rule2751, rule2752, rule2753, rule2754, rule2755, rule2756, rule2757, rule2758, rule2759, rule2760, rule2761, rule2762, rule2763, rule2764, rule2765, rule2766, rule2767, rule2768, rule2769, rule2770, rule2771, rule2772, rule2773, rule2774, rule2775, rule2776, rule2777, rule2778, rule2779, rule2780, rule2781, rule2782, rule2783, rule2784, rule2785, rule2786, rule2787, rule2788, rule2789, rule2790, rule2791, rule2792, rule2793, rule2794, rule2795, rule2796, rule2797, rule2798, rule2799, rule2800, rule2801, rule2802, rule2803, rule2804, rule2805, rule2806, rule2807, rule2808, rule2809, rule2810, rule2811, rule2812, rule2813, rule2814, rule2815, rule2816, rule2817, rule2818, rule2819, rule2820, rule2821, rule2822, rule2823, rule2824, rule2825, rule2826, rule2827, rule2828, rule2829, rule2830, rule2831, rule2832, rule2833, rule2834, rule2835, rule2836, rule2837, rule2838, rule2839, rule2840, rule2841, rule2842, rule2843, rule2844, rule2845, rule2846, rule2847, rule2848, rule2849, rule2850, rule2851, rule2852, rule2853, rule2854, rule2855, rule2856, rule2857, rule2858, rule2859, rule2860, rule2861, rule2862, rule2863, rule2864, rule2865, rule2866, rule2867, rule2868, rule2869, rule2870, rule2871, rule2872, rule2873, rule2874, rule2875, rule2876, rule2877, rule2878, rule2879, rule2880, rule2881, rule2882, rule2883, rule2884, rule2885, rule2886, rule2887, rule2888, rule2889, rule2890, rule2891, rule2892, rule2893, rule2894, rule2895, rule2896, rule2897, rule2898, rule2899, rule2900, rule2901, rule2902, rule2903, rule2904, rule2905, rule2906, rule2907, rule2908, rule2909, rule2910, rule2911, rule2912, rule2913, rule2914, rule2915, rule2916, rule2917, rule2918, rule2919, rule2920, rule2921, rule2922, rule2923, rule2924, rule2925, rule2926, rule2927, rule2928, rule2929, rule2930, rule2931, rule2932, rule2933, rule2934, rule2935, rule2936, rule2937, rule2938, rule2939, rule2940, rule2941, rule2942, rule2943, rule2944, rule2945, rule2946, rule2947, rule2948, rule2949, rule2950, rule2951, rule2952, rule2953, rule2954, rule2955, rule2956, rule2957, rule2958, rule2959, rule2960, rule2961, rule2962, rule2963, rule2964, rule2965, rule2966, rule2967, rule2968, rule2969, rule2970, rule2971, rule2972, rule2973, rule2974, rule2975, rule2976, rule2977, rule2978, rule2979, rule2980, rule2981, rule2982, rule2983, rule2984, rule2985, rule2986, rule2987, rule2988, rule2989, rule2990, rule2991, rule2992, rule2993, rule2994, rule2995, rule2996, rule2997, rule2998, rule2999, rule3000, rule3001, rule3002, rule3003, rule3004, rule3005, rule3006, rule3007, rule3008, rule3009, rule3010, rule3011, rule3012, rule3013, rule3014, rule3015, rule3016, rule3017, rule3018, rule3019, rule3020, rule3021, rule3022, rule3023, rule3024, rule3025, rule3026, rule3027, rule3028, rule3029, rule3030, rule3031, rule3032, rule3033, rule3034, rule3035, rule3036, rule3037, rule3038, rule3039, rule3040, rule3041, rule3042, rule3043, rule3044, rule3045, rule3046, rule3047, rule3048, rule3049, rule3050, rule3051, rule3052, rule3053, rule3054, rule3055, rule3056, rule3057, rule3058, rule3059, rule3060, rule3061, rule3062, rule3063, rule3064, rule3065, rule3066, rule3067, rule3068, rule3069, rule3070, rule3071, rule3072, rule3073, rule3074, rule3075, rule3076, rule3077, rule3078, rule3079, rule3080, rule3081, rule3082, rule3083, rule3084, rule3085, rule3086, rule3087, rule3088, rule3089, rule3090, rule3091, rule3092, rule3093, rule3094, rule3095, rule3096, rule3097, rule3098, rule3099, rule3100, rule3101, rule3102, rule3103, rule3104, rule3105, rule3106, rule3107, rule3108, rule3109, rule3110, rule3111, rule3112, rule3113, rule3114, rule3115, rule3116, rule3117, rule3118, rule3119, rule3120, rule3121, rule3122, rule3123, rule3124, rule3125, rule3126, rule3127, rule3128, rule3129, rule3130, rule3131, rule3132, rule3133, rule3134, rule3135, rule3136, rule3137, rule3138, rule3139, rule3140, rule3141, rule3142, rule3143, rule3144, rule3145, rule3146, rule3147, rule3148, rule3149, rule3150, rule3151, rule3152, rule3153, rule3154, rule3155, rule3156, rule3157, rule3158, rule3159, rule3160, rule3161, rule3162, rule3163, rule3164, rule3165, rule3166, rule3167, rule3168, rule3169, rule3170, rule3171, rule3172, rule3173, rule3174, rule3175, rule3176, rule3177, rule3178, rule3179, rule3180, rule3181, rule3182, rule3183, rule3184, rule3185, rule3186, rule3187, rule3188, rule3189, rule3190, rule3191, rule3192, rule3193, rule3194, rule3195, rule3196, rule3197, rule3198, rule3199, rule3200, rule3201, rule3202, rule3203, rule3204, rule3205, rule3206, rule3207, rule3208, rule3209, rule3210, rule3211, rule3212, rule3213, rule3214, rule3215, rule3216, rule3217, rule3218, rule3219, rule3220, rule3221, rule3222, rule3223, rule3224, rule3225, rule3226, rule3227, rule3228, rule3229, rule3230, rule3231, rule3232, rule3233, rule3234, rule3235, rule3236, rule3237, rule3238, rule3239, rule3240, rule3241, rule3242, rule3243, rule3244, rule3245, rule3246, rule3247, rule3248, rule3249, rule3250, rule3251, rule3252, rule3253, rule3254, rule3255, rule3256, rule3257, rule3258, rule3259, rule3260, rule3261, rule3262, rule3263, rule3264, rule3265, rule3266, rule3267, rule3268, rule3269, rule3270, rule3271, rule3272, rule3273, rule3274, rule3275, rule3276, rule3277, rule3278, rule3279, rule3280, rule3281, rule3282, rule3283, rule3284, rule3285, rule3286, rule3287, rule3288, rule3289, rule3290, rule3291, rule3292, rule3293, rule3294, rule3295, rule3296, rule3297, rule3298, rule3299, rule3300, rule3301, rule3302, rule3303, rule3304, rule3305, rule3306, rule3307, rule3308, rule3309, rule3310, rule3311, rule3312, rule3313, rule3314, rule3315, rule3316, rule3317, rule3318, rule3319, rule3320, rule3321, rule3322, rule3323, rule3324, rule3325, rule3326, rule3327, rule3328, rule3329, rule3330, rule3331, rule3332, rule3333, rule3334, rule3335, rule3336, rule3337, rule3338, rule3339, rule3340, rule3341, rule3342, rule3343, rule3344, rule3345, rule3346, rule3347, rule3348, rule3349, rule3350, rule3351, rule3352, rule3353, rule3354, rule3355, rule3356, rule3357, rule3358, rule3359, rule3360, rule3361, rule3362, rule3363, rule3364, rule3365, rule3366, rule3367, rule3368, rule3369, rule3370, rule3371, rule3372, rule3373, rule3374, rule3375, rule3376, rule3377, rule3378, rule3379, rule3380, rule3381, rule3382, rule3383, rule3384, rule3385, rule3386, rule3387, rule3388, rule3389, rule3390, rule3391, rule3392, rule3393, rule3394, rule3395, rule3396, rule3397, ]
|
614ae478a3f078e7f8701a6a682f48dbb7c64bf5d441839652a73f8eaa4c1226 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def logarithms(rubi):
from sympy.integrals.rubi.constraints import cons1156, cons7, cons27, cons48, cons125, cons5, cons50, cons87, cons88, cons2, cons3, cons1157, cons415, cons1158, cons1159, cons89, cons543, cons1160, cons208, cons209, cons584, cons4, cons66, cons21, cons1161, cons1162, cons1163, cons1164, cons1165, cons1166, cons1167, cons1168, cons1169, cons148, cons1170, cons1171, cons62, cons93, cons168, cons810, cons811, cons222, cons1172, cons224, cons796, cons79, cons1173, cons17, cons1174, cons1175, cons1176, cons1177, cons1178, cons1179, cons1180, cons1181, cons1182, cons1183, cons1184, cons1185, cons1186, cons1187, cons1188, cons1189, cons1190, cons797, cons1191, cons52, cons925, cons1192, cons1193, cons1194, cons1195, cons1196, cons1197, cons1198, cons1199, cons38, cons552, cons1200, cons1201, cons1202, cons25, cons652, cons1203, cons71, cons128, cons1204, cons1205, cons1206, cons1207, cons1208, cons146, cons1209, cons1210, cons13, cons163, cons1211, cons137, cons1212, cons1213, cons1214, cons1215, cons1216, cons1217, cons1218, cons1219, cons1220, cons1221, cons1222, cons70, cons1223, cons1224, cons806, cons840, cons1225, cons1226, cons68, cons1125, cons1227, cons1228, cons1229, cons1230, cons463, cons1231, cons1232, cons1233, cons1234, cons1235, cons1236, cons31, cons1099, cons1237, cons1055, cons515, cons816, cons817, cons1238, cons1239, cons1240, cons1241, cons1242, cons1243, cons1244, cons1245, cons34, cons35, cons1246, cons1247, cons1248
pattern2006 = Pattern(Integral(log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))), x_), cons7, cons27, cons48, cons125, cons5, cons50, cons1156)
def replacement2006(p, f, d, c, x, q, e):
rubi.append(2006)
return Simp((e + f*x)*log(c*(d*(e + f*x)**p)**q)/f, x) - Simp(p*q*x, x)
rule2006 = ReplacementRule(pattern2006, replacement2006)
pattern2007 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons87, cons88)
def replacement2007(p, f, b, d, a, c, n, x, q, e):
rubi.append(2007)
return -Dist(b*n*p*q, Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(-1)), x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)/f, x)
rule2007 = ReplacementRule(pattern2007, replacement2007)
pattern2008 = Pattern(Integral(S(1)/log((x_*WC('f', S(1)) + WC('e', S(0)))*WC('d', S(1))), x_), cons27, cons48, cons125, cons1157)
def replacement2008(d, x, f, e):
rubi.append(2008)
return Simp(LogIntegral(d*(e + f*x))/(d*f), x)
rule2008 = ReplacementRule(pattern2008, replacement2008)
pattern2009 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons415)
def replacement2009(p, f, b, d, a, c, x, q, e):
rubi.append(2009)
return Simp((c*(d*(e + f*x)**p)**q)**(-S(1)/(p*q))*(e + f*x)*ExpIntegralEi((a + b*log(c*(d*(e + f*x)**p)**q))/(b*p*q))*exp(-a/(b*p*q))/(b*f*p*q), x)
rule2009 = ReplacementRule(pattern2009, replacement2009)
pattern2010 = Pattern(Integral(S(1)/sqrt(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons1158)
def replacement2010(p, f, b, d, a, c, x, q, e):
rubi.append(2010)
return Simp(sqrt(Pi)*(c*(d*(e + f*x)**p)**q)**(-S(1)/(p*q))*(e + f*x)*Erfi(sqrt(a + b*log(c*(d*(e + f*x)**p)**q))/Rt(b*p*q, S(2)))*Rt(b*p*q, S(2))*exp(-a/(b*p*q))/(b*f*p*q), x)
rule2010 = ReplacementRule(pattern2010, replacement2010)
pattern2011 = Pattern(Integral(S(1)/sqrt(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons1159)
def replacement2011(p, f, b, d, a, c, x, q, e):
rubi.append(2011)
return Simp(sqrt(Pi)*(c*(d*(e + f*x)**p)**q)**(-S(1)/(p*q))*(e + f*x)*Erf(sqrt(a + b*log(c*(d*(e + f*x)**p)**q))/Rt(-b*p*q, S(2)))*Rt(-b*p*q, S(2))*exp(-a/(b*p*q))/(b*f*p*q), x)
rule2011 = ReplacementRule(pattern2011, replacement2011)
pattern2012 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons87, cons89)
def replacement2012(p, f, b, d, a, c, n, x, q, e):
rubi.append(2012)
return -Dist(S(1)/(b*p*q*(n + S(1))), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(1)), x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(1))*(e + f*x)/(b*f*p*q*(n + S(1))), x)
rule2012 = ReplacementRule(pattern2012, replacement2012)
pattern2013 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons543)
def replacement2013(p, f, b, d, a, c, n, x, q, e):
rubi.append(2013)
return Simp((c*(d*(e + f*x)**p)**q)**(-S(1)/(p*q))*(-(a + b*log(c*(d*(e + f*x)**p)**q))/(b*p*q))**(-n)*(a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)*Gamma(n + S(1), -(a + b*log(c*(d*(e + f*x)**p)**q))/(b*p*q))*exp(-a/(b*p*q))/f, x)
rule2013 = ReplacementRule(pattern2013, replacement2013)
pattern2014 = Pattern(Integral(S(1)/((x_*WC('h', S(1)) + WC('g', S(0)))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons1160)
def replacement2014(p, f, b, g, d, a, c, x, h, q, e):
rubi.append(2014)
return Simp(log(RemoveContent(a + b*log(c*(d*(e + f*x)**p)**q), x))/(b*h*p*q), x)
rule2014 = ReplacementRule(pattern2014, replacement2014)
pattern2015 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons50, cons1160, cons584)
def replacement2015(p, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2015)
return Simp((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(1))/(b*h*p*q*(n + S(1))), x)
rule2015 = ReplacementRule(pattern2015, replacement2015)
pattern2016 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons1160, cons66, cons87, cons88)
def replacement2016(p, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2016)
return -Dist(b*n*p*q/(m + S(1)), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(-1))*(g + h*x)**m, x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**n*(g + h*x)**(m + S(1))/(h*(m + S(1))), x)
rule2016 = ReplacementRule(pattern2016, replacement2016)
pattern2017 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))/log((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1))), x_), cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons1161, cons1160, cons1162)
def replacement2017(p, m, f, g, d, x, h, e):
rubi.append(2017)
return Simp((h/f)**(p + S(-1))*LogIntegral(d*(e + f*x)**p)/(d*f*p), x)
rule2017 = ReplacementRule(pattern2017, replacement2017)
pattern2018 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**m_/log((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1))), x_), cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons1161, cons1160, cons1163)
def replacement2018(p, m, f, g, d, x, h, e):
rubi.append(2018)
return Dist((e + f*x)**(-p + S(1))*(g + h*x)**(p + S(-1)), Int((e + f*x)**(p + S(-1))/log(d*(e + f*x)**p), x), x)
rule2018 = ReplacementRule(pattern2018, replacement2018)
pattern2019 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))/(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons1160, cons66)
def replacement2019(p, m, f, b, g, d, a, c, x, h, q, e):
rubi.append(2019)
return Simp((c*(d*(e + f*x)**p)**q)**(-(m + S(1))/(p*q))*(g + h*x)**(m + S(1))*ExpIntegralEi((a + b*log(c*(d*(e + f*x)**p)**q))*(m + S(1))/(b*p*q))*exp(-a*(m + S(1))/(b*p*q))/(b*h*p*q), x)
rule2019 = ReplacementRule(pattern2019, replacement2019)
pattern2020 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))/sqrt(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons1160, cons66, cons1164)
def replacement2020(p, m, f, b, g, d, a, c, x, h, q, e):
rubi.append(2020)
return Simp(sqrt(Pi)*(c*(d*(e + f*x)**p)**q)**(-(m + S(1))/(p*q))*(g + h*x)**(m + S(1))*Erfi(sqrt(a + b*log(c*(d*(e + f*x)**p)**q))*Rt((m + S(1))/(b*p*q), S(2)))*exp(-a*(m + S(1))/(b*p*q))/(b*h*p*q*Rt((m + S(1))/(b*p*q), S(2))), x)
rule2020 = ReplacementRule(pattern2020, replacement2020)
pattern2021 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))/sqrt(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons1160, cons66, cons1165)
def replacement2021(p, m, f, b, g, d, a, c, x, h, q, e):
rubi.append(2021)
return Simp(sqrt(Pi)*(c*(d*(e + f*x)**p)**q)**(-(m + S(1))/(p*q))*(g + h*x)**(m + S(1))*Erf(sqrt(a + b*log(c*(d*(e + f*x)**p)**q))*Rt(-(m + S(1))/(b*p*q), S(2)))*exp(-a*(m + S(1))/(b*p*q))/(b*h*p*q*Rt(-(m + S(1))/(b*p*q), S(2))), x)
rule2021 = ReplacementRule(pattern2021, replacement2021)
pattern2022 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons1160, cons66, cons87, cons89)
def replacement2022(p, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2022)
return -Dist((m + S(1))/(b*p*q*(n + S(1))), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(1))*(g + h*x)**m, x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(1))*(g + h*x)**(m + S(1))/(b*h*p*q*(n + S(1))), x)
rule2022 = ReplacementRule(pattern2022, replacement2022)
pattern2023 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons1160, cons66)
def replacement2023(p, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2023)
return Simp((c*(d*(e + f*x)**p)**q)**(-(m + S(1))/(p*q))*(-(a + b*log(c*(d*(e + f*x)**p)**q))*(m + S(1))/(b*p*q))**(-n)*(a + b*log(c*(d*(e + f*x)**p)**q))**n*(g + h*x)**(m + S(1))*Gamma(n + S(1), -(a + b*log(c*(d*(e + f*x)**p)**q))*(m + S(1))/(b*p*q))*exp(-a*(m + S(1))/(b*p*q))/(h*(m + S(1))), x)
rule2023 = ReplacementRule(pattern2023, replacement2023)
pattern2024 = Pattern(Integral(log((x_*WC('f', S(1)) + WC('e', S(0)))*WC('c', S(1)))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons7, cons48, cons125, cons208, cons209, cons1166)
def replacement2024(f, g, c, x, h, e):
rubi.append(2024)
return -Simp(PolyLog(S(2), -(g + h*x)*Together(c*f/h))/h, x)
rule2024 = ReplacementRule(pattern2024, replacement2024)
pattern2025 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log((x_*WC('f', S(1)) + WC('e', S(0)))*WC('c', S(1))))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons48, cons125, cons208, cons209, cons1167, cons1168)
def replacement2025(f, b, g, a, c, x, h, e):
rubi.append(2025)
return Dist(b, Int(log(-h*(e + f*x)/(-e*h + f*g))/(g + h*x), x), x) + Simp((a + b*log(c*(e - f*g/h)))*log(g + h*x)/h, x)
rule2025 = ReplacementRule(pattern2025, replacement2025)
pattern2026 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons1169, cons148)
def replacement2026(p, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2026)
return -Dist(b*f*n*p*q/h, Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(-1))*log(f*(g + h*x)/(-e*h + f*g))/(e + f*x), x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**n*log(f*(g + h*x)/(-e*h + f*g))/h, x)
rule2026 = ReplacementRule(pattern2026, replacement2026)
pattern2027 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons1169, cons66)
def replacement2027(p, m, f, b, g, d, a, c, x, h, q, e):
rubi.append(2027)
return -Dist(b*f*p*q/(h*(m + S(1))), Int((g + h*x)**(m + S(1))/(e + f*x), x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))*(g + h*x)**(m + S(1))/(h*(m + S(1))), x)
rule2027 = ReplacementRule(pattern2027, replacement2027)
pattern2028 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_/(x_*WC('h', S(1)) + WC('g', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons1169, cons87, cons88)
def replacement2028(p, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2028)
return -Dist(b*f*n*p*q/(-e*h + f*g), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(-1))/(g + h*x), x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)/((g + h*x)*(-e*h + f*g)), x)
rule2028 = ReplacementRule(pattern2028, replacement2028)
pattern2029 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons5, cons50, cons1169, cons87, cons88, cons66, cons1170, cons1171)
def replacement2029(p, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2029)
return -Dist(b*f*n*p*q/(h*(m + S(1))), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(-1))*(g + h*x)**(m + S(1))/(e + f*x), x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**n*(g + h*x)**(m + S(1))/(h*(m + S(1))), x)
rule2029 = ReplacementRule(pattern2029, replacement2029)
pattern2030 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))/(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons1169, cons62)
def replacement2030(p, m, f, b, g, d, a, c, x, h, q, e):
rubi.append(2030)
return Int(ExpandIntegrand((g + h*x)**m/(a + b*log(c*(d*(e + f*x)**p)**q)), x), x)
rule2030 = ReplacementRule(pattern2030, replacement2030)
pattern2031 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons1169, cons93, cons89, cons168)
def replacement2031(p, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2031)
return -Dist((m + S(1))/(b*p*q*(n + S(1))), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(1))*(g + h*x)**m, x), x) + Dist(m*(-e*h + f*g)/(b*f*p*q*(n + S(1))), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(1))*(g + h*x)**(m + S(-1)), x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(1))*(e + f*x)*(g + h*x)**m/(b*f*p*q*(n + S(1))), x)
rule2031 = ReplacementRule(pattern2031, replacement2031)
pattern2032 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons50, cons1169, cons62)
def replacement2032(p, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2032)
return Int(ExpandIntegrand((a + b*log(c*(d*(e + f*x)**p)**q))**n*(g + h*x)**m, x), x)
rule2032 = ReplacementRule(pattern2032, replacement2032)
pattern2033 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log((v_**p_*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons50, cons810, cons811)
def replacement2033(v, p, u, m, b, d, a, c, n, x, q):
rubi.append(2033)
return Int((a + b*log(c*(d*ExpandToSum(v, x)**p)**q))**n*ExpandToSum(u, x)**m, x)
rule2033 = ReplacementRule(pattern2033, replacement2033)
pattern2034 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons50, cons222)
def replacement2034(p, m, f, b, g, d, a, c, n, x, h, q, e):
rubi.append(2034)
return Int((a + b*log(c*(d*(e + f*x)**p)**q))**n*(g + h*x)**m, x)
rule2034 = ReplacementRule(pattern2034, replacement2034)
pattern2035 = Pattern(Integral(log(WC('c', S(1))/(x_*WC('f', S(1)) + WC('e', S(0))))/((x_*WC('h', S(1)) + WC('g', S(0)))*(x_*WC('j', S(1)) + WC('i', S(0)))), x_), cons7, cons48, cons125, cons208, cons209, cons224, cons796, cons1160, cons1172)
def replacement2035(j, f, g, i, c, x, h, e):
rubi.append(2035)
return Simp(f*PolyLog(S(2), f*(i + j*x)/(j*(e + f*x)))/(h*(-e*j + f*i)), x)
rule2035 = ReplacementRule(pattern2035, replacement2035)
pattern2036 = Pattern(Integral((a_ + WC('b', S(1))*log(WC('c', S(1))/(x_*WC('f', S(1)) + WC('e', S(0)))))/((x_*WC('h', S(1)) + WC('g', S(0)))*(x_*WC('j', S(1)) + WC('i', S(0)))), x_), cons2, cons3, cons7, cons48, cons125, cons208, cons209, cons224, cons796, cons1160, cons1172)
def replacement2036(j, f, b, g, i, c, a, x, h, e):
rubi.append(2036)
return Dist(a, Int(S(1)/((g + h*x)*(i + j*x)), x), x) + Dist(b, Int(log(c/(e + f*x))/((g + h*x)*(i + j*x)), x), x)
rule2036 = ReplacementRule(pattern2036, replacement2036)
def With2037(p, j, m, f, b, g, i, d, a, c, x, h, q, e):
u = IntHide((i + j*x)**m/(g + h*x), x)
rubi.append(2037)
return -Dist(b*h*p*q, Int(SimplifyIntegrand(u/(g + h*x), x), x), x) + Dist(a + b*log(c*(d*(e + f*x)**p)**q), u, x)
pattern2037 = Pattern(Integral((x_*WC('j', S(1)) + WC('i', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons796, cons5, cons50, cons1160, cons79)
rule2037 = ReplacementRule(pattern2037, With2037)
pattern2038 = Pattern(Integral((x_*WC('j', S(1)) + WC('i', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log((x_*WC('f', S(1)) + WC('e', S(0)))*WC('c', S(1))))**WC('n', S(1))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons48, cons125, cons208, cons209, cons224, cons796, cons4, cons1160, cons62, cons1173)
def replacement2038(j, m, f, b, g, i, a, c, n, x, h, e):
rubi.append(2038)
return Dist(c**(-m)*f**(-m)/h, Subst(Int((a + b*x)**n*(-c*e*j + c*f*i + j*exp(x))**m, x), x, log(c*(e + f*x))), x)
rule2038 = ReplacementRule(pattern2038, replacement2038)
def With2039(p, j, m, f, b, g, i, d, a, c, n, x, h, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((a + b*log(c*(d*(e + f*x)**p)**q))**n, (i + j*x)**m/(g + h*x), x)
if SumQ(u):
return True
return False
pattern2039 = Pattern(Integral((x_*WC('j', S(1)) + WC('i', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons796, cons5, cons50, cons17, cons148, CustomConstraint(With2039))
def replacement2039(p, j, m, f, b, g, i, d, a, c, n, x, h, q, e):
u = ExpandIntegrand((a + b*log(c*(d*(e + f*x)**p)**q))**n, (i + j*x)**m/(g + h*x), x)
rubi.append(2039)
return Int(u, x)
rule2039 = ReplacementRule(pattern2039, replacement2039)
pattern2040 = Pattern(Integral((x_*WC('j', S(1)) + WC('i', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons796, cons21, cons4, cons5, cons50, cons1174)
def replacement2040(p, j, m, f, b, g, i, d, a, c, n, x, h, q, e):
rubi.append(2040)
return Int((a + b*log(c*(d*(e + f*x)**p)**q))**n*(i + j*x)**m/(g + h*x), x)
rule2040 = ReplacementRule(pattern2040, replacement2040)
pattern2041 = Pattern(Integral(log(WC('c', S(1))/(x_*WC('f', S(1)) + WC('e', S(0))))/(g_ + x_**S(2)*WC('h', S(1))), x_), cons7, cons48, cons125, cons208, cons209, cons1175, cons1176)
def replacement2041(f, g, c, x, h, e):
rubi.append(2041)
return -Simp(f*PolyLog(S(2), (-e + f*x)/(e + f*x))/(S(2)*e*h), x)
rule2041 = ReplacementRule(pattern2041, replacement2041)
pattern2042 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(WC('c', S(1))/(x_*WC('f', S(1)) + WC('e', S(0)))))/(g_ + x_**S(2)*WC('h', S(1))), x_), cons7, cons48, cons125, cons208, cons209, cons1175, cons1177, cons1178)
def replacement2042(f, b, g, a, c, x, h, e):
rubi.append(2042)
return Dist(b, Int(log(S(2)*e/(e + f*x))/(g + h*x**S(2)), x), x) + Dist(a + b*log(c/(S(2)*e)), Int(S(1)/(g + h*x**S(2)), x), x)
rule2042 = ReplacementRule(pattern2042, replacement2042)
pattern2043 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))/(x_**S(2)*WC('i', S(1)) + x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons5, cons50, cons1179)
def replacement2043(p, f, b, g, i, d, a, c, x, h, q, e):
rubi.append(2043)
return Dist(e*f, Int((a + b*log(c*(d*(e + f*x)**p)**q))/((e + f*x)*(e*i*x + f*g)), x), x)
rule2043 = ReplacementRule(pattern2043, replacement2043)
pattern2044 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((e_ + x_*WC('f', S(1)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))/(g_ + x_**S(2)*WC('i', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons224, cons5, cons50, cons1180)
def replacement2044(p, f, b, g, i, d, a, c, x, q, e):
rubi.append(2044)
return Dist(e*f, Int((a + b*log(c*(d*(e + f*x)**p)**q))/((e + f*x)*(e*i*x + f*g)), x), x)
rule2044 = ReplacementRule(pattern2044, replacement2044)
def With2045(p, f, b, g, d, a, c, x, h, q, e):
u = IntHide(S(1)/sqrt(g + h*x**S(2)), x)
rubi.append(2045)
return -Dist(b*f*p*q, Int(SimplifyIntegrand(u/(e + f*x), x), x), x) + Simp(u*(a + b*log(c*(d*(e + f*x)**p)**q)), x)
pattern2045 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))/sqrt(g_ + x_**S(2)*WC('h', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons1181)
rule2045 = ReplacementRule(pattern2045, With2045)
def With2046(p, f, b, d, h1, a, c, h2, g2, x, g1, q, e):
u = IntHide(S(1)/sqrt(g1*g2 + h1*h2*x**S(2)), x)
rubi.append(2046)
return -Dist(b*f*p*q, Int(SimplifyIntegrand(u/(e + f*x), x), x), x) + Simp(u*(a + b*log(c*(d*(e + f*x)**p)**q)), x)
pattern2046 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))/(sqrt(g1_ + x_*WC('h1', S(1)))*sqrt(g2_ + x_*WC('h2', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1185, cons1186, cons1187, cons1188, cons5, cons50, cons1182, cons1183, cons1184)
rule2046 = ReplacementRule(pattern2046, With2046)
pattern2047 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))/sqrt(g_ + x_**S(2)*WC('h', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons1189)
def replacement2047(p, f, b, g, d, a, c, x, h, q, e):
rubi.append(2047)
return Dist(sqrt(S(1) + h*x**S(2)/g)/sqrt(g + h*x**S(2)), Int((a + b*log(c*(d*(e + f*x)**p)**q))/sqrt(S(1) + h*x**S(2)/g), x), x)
rule2047 = ReplacementRule(pattern2047, replacement2047)
pattern2048 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))/(sqrt(g1_ + x_*WC('h1', S(1)))*sqrt(g2_ + x_*WC('h2', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1185, cons1186, cons1187, cons1188, cons5, cons50, cons1182)
def replacement2048(p, f, b, d, h1, a, c, h2, g2, x, g1, q, e):
rubi.append(2048)
return Dist(sqrt(S(1) + h1*h2*x**S(2)/(g1*g2))/(sqrt(g1 + h1*x)*sqrt(g2 + h2*x)), Int((a + b*log(c*(d*(e + f*x)**p)**q))/sqrt(S(1) + h1*h2*x**S(2)/(g1*g2)), x), x)
rule2048 = ReplacementRule(pattern2048, replacement2048)
pattern2049 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('k', S(1)) + WC('j', S(0)))*WC('i', S(1)))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons796, cons797, cons5, cons50, cons87, cons88, cons1190)
def replacement2049(p, j, k, f, b, g, i, d, a, c, n, x, h, q, e):
rubi.append(2049)
return Dist(b*f*n*p*q/h, Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(-1))*PolyLog(S(2), Together(-i*(j + k*x) + S(1)))/(e + f*x), x), x) - Simp((a + b*log(c*(d*(e + f*x)**p)**q))**n*PolyLog(S(2), Together(-i*(j + k*x) + S(1)))/h, x)
rule2049 = ReplacementRule(pattern2049, replacement2049)
pattern2050 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))*log((x_*WC('k', S(1)) + WC('j', S(0)))**WC('m', S(1))*WC('i', S(1)) + S(1))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons796, cons797, cons21, cons5, cons50, cons87, cons88, cons1191)
def replacement2050(p, j, k, m, f, b, g, i, d, a, c, n, x, h, q, e):
rubi.append(2050)
return Dist(b*f*n*p*q/(h*m), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(-1))*PolyLog(S(2), -i*(j + k*x)**m)/(e + f*x), x), x) - Simp((a + b*log(c*(d*(e + f*x)**p)**q))**n*PolyLog(S(2), -i*(j + k*x)**m)/(h*m), x)
rule2050 = ReplacementRule(pattern2050, replacement2050)
pattern2051 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))*PolyLog(r_, (x_*WC('k', S(1)) + WC('j', S(0)))**WC('m', S(1))*WC('i', S(1)))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons796, cons797, cons21, cons5, cons50, cons52, cons87, cons88, cons1191)
def replacement2051(p, j, k, m, f, b, g, r, i, d, a, c, n, x, h, q, e):
rubi.append(2051)
return -Dist(b*f*n*p*q/(h*m), Int((a + b*log(c*(d*(e + f*x)**p)**q))**(n + S(-1))*PolyLog(r + S(1), i*(j + k*x)**m)/(e + f*x), x), x) + Simp((a + b*log(c*(d*(e + f*x)**p)**q))**n*PolyLog(r + S(1), i*(j + k*x)**m)/(h*m), x)
rule2051 = ReplacementRule(pattern2051, replacement2051)
def With2052(p, m, f, b, g, Px, d, a, c, x, h, q, e, F):
u = IntHide(Px*F(g + h*x)**m, x)
rubi.append(2052)
return -Dist(b*f*p*q, Int(SimplifyIntegrand(u/(e + f*x), x), x), x) + Dist(a + b*log(c*(d*(e + f*x)**p)**q), u, x)
pattern2052 = Pattern(Integral(F_**(x_*WC('h', S(1)) + WC('g', S(0)))*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))*WC('Px', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons5, cons50, cons925, cons62, cons1192)
rule2052 = ReplacementRule(pattern2052, With2052)
pattern2053 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((e_ + x_**m_*WC('f', S(1)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))/x_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons148)
def replacement2053(p, m, f, b, d, a, c, n, x, q, e):
rubi.append(2053)
return Dist(S(1)/m, Subst(Int((a + b*log(c*(d*(e + f*x)**p)**q))**n/x, x), x, x**m), x)
rule2053 = ReplacementRule(pattern2053, replacement2053)
pattern2054 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_**m_*(f_ + x_**WC('r', S(1))*WC('e', S(1))))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))/x_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons1193, cons148)
def replacement2054(p, m, f, b, r, d, a, c, n, x, q, e):
rubi.append(2054)
return Dist(S(1)/m, Subst(Int((a + b*log(c*(d*(e + f*x)**p)**q))**n/x, x), x, x**m), x)
rule2054 = ReplacementRule(pattern2054, replacement2054)
pattern2055 = Pattern(Integral(x_**WC('r1', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_**r_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons50, cons52, cons1194)
def replacement2055(p, r1, f, b, r, d, a, c, n, x, q, e):
rubi.append(2055)
return Dist(S(1)/r, Subst(Int((a + b*log(c*(d*(e + f*x)**p)**q))**n, x), x, x**r), x)
rule2055 = ReplacementRule(pattern2055, replacement2055)
pattern2056 = Pattern(Integral(x_**WC('r1', S(1))*(x_**r_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(((x_**r_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons50, cons52, cons1194)
def replacement2056(p, r1, m, f, b, g, r, d, a, c, n, x, h, q, e):
rubi.append(2056)
return Dist(S(1)/r, Subst(Int((a + b*log(c*(d*(e + f*x)**p)**q))**n*(g + h*x)**m, x), x, x**r), x)
rule2056 = ReplacementRule(pattern2056, replacement2056)
def With2057(b, d, a, n, c, x, e):
u = IntHide(S(1)/(d + e*x**S(2)), x)
rubi.append(2057)
return -Dist(b*n, Int(u/x, x), x) + Dist(a + b*log(c*x**n), u, x)
pattern2057 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1195)
rule2057 = ReplacementRule(pattern2057, With2057)
pattern2058 = Pattern(Integral(log((x_**mn_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))/(x_*(d_ + x_**WC('n', S(1))*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1196, cons1197)
def replacement2058(mn, b, d, c, n, a, x, e):
rubi.append(2058)
return Simp(PolyLog(S(2), -Together(b*c*x**(-n)*(d + e*x**n)/d))/(d*n), x)
rule2058 = ReplacementRule(pattern2058, replacement2058)
pattern2059 = Pattern(Integral(log(x_**mn_*(x_**WC('n', S(1))*WC('a', S(1)) + WC('b', S(0)))*WC('c', S(1)))/(x_*(d_ + x_**WC('n', S(1))*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1196, cons1197)
def replacement2059(mn, b, d, c, n, a, x, e):
rubi.append(2059)
return Simp(PolyLog(S(2), -Together(b*c*x**(-n)*(d + e*x**n)/d))/(d*n), x)
rule2059 = ReplacementRule(pattern2059, replacement2059)
pattern2060 = Pattern(Integral(Px_*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons50, cons925)
def replacement2060(p, f, b, Px, d, a, c, n, x, q, e):
rubi.append(2060)
return Int(ExpandIntegrand(Px*(a + b*log(c*(d*(e + f*x)**p)**q))**n, x), x)
rule2060 = ReplacementRule(pattern2060, replacement2060)
def With2061(p, RFx, f, b, d, a, c, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((a + b*log(c*(d*(e + f*x)**p)**q))**n, RFx, x)
if SumQ(u):
return True
return False
pattern2061 = Pattern(Integral(RFx_*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons1198, cons148, CustomConstraint(With2061))
def replacement2061(p, RFx, f, b, d, a, c, n, x, q, e):
u = ExpandIntegrand((a + b*log(c*(d*(e + f*x)**p)**q))**n, RFx, x)
rubi.append(2061)
return Int(u, x)
rule2061 = ReplacementRule(pattern2061, replacement2061)
def With2062(p, RFx, f, b, d, a, c, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(RFx*(a + b*log(c*(d*(e + f*x)**p)**q))**n, x)
if SumQ(u):
return True
return False
pattern2062 = Pattern(Integral(RFx_*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons1198, cons148, CustomConstraint(With2062))
def replacement2062(p, RFx, f, b, d, a, c, n, x, q, e):
u = ExpandIntegrand(RFx*(a + b*log(c*(d*(e + f*x)**p)**q))**n, x)
rubi.append(2062)
return Int(u, x)
rule2062 = ReplacementRule(pattern2062, replacement2062)
pattern2063 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_**S(2)*WC('g', S(1)) + x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons50, cons4, cons1199, cons38)
def replacement2063(p, u, f, g, b, d, a, c, n, x, q, e):
rubi.append(2063)
return Int(u*(a + b*log(c*(S(4)**(-p)*d*g**(-p)*(f + S(2)*g*x)**(S(2)*p))**q))**n, x)
rule2063 = ReplacementRule(pattern2063, replacement2063)
pattern2064 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log((v_**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons50, cons552, cons1200)
def replacement2064(v, p, u, b, d, a, c, n, x, q):
rubi.append(2064)
return Int(u*(a + b*log(c*(d*ExpandToSum(v, x)**p)**q))**n, x)
rule2064 = ReplacementRule(pattern2064, replacement2064)
pattern2065 = Pattern(Integral(log(((x_**WC('n', S(1))*WC('c', S(1)))**p_*WC('b', S(1)))**q_*WC('a', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons50, cons52, cons1201)
def replacement2065(p, b, r, c, a, n, x, q):
rubi.append(2065)
return Subst(Int(log(x**(n*p*q))**r, x), x**(n*p*q), a*(b*(c*x**n)**p)**q)
rule2065 = ReplacementRule(pattern2065, replacement2065)
pattern2066 = Pattern(Integral(x_**WC('m', S(1))*log(((x_**WC('n', S(1))*WC('c', S(1)))**p_*WC('b', S(1)))**q_*WC('a', S(1)))**WC('r', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons50, cons52, cons66, cons1202)
def replacement2066(p, m, b, r, c, a, n, x, q):
rubi.append(2066)
return Subst(Int(x**m*log(x**(n*p*q))**r, x), x**(n*p*q), a*(b*(c*x**n)**p)**q)
rule2066 = ReplacementRule(pattern2066, replacement2066)
pattern2067 = Pattern(Integral(WC('u', S(1))*log(((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e1', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons652, cons25)
def replacement2067(p, u, b, d, c, a, n, e1, x, e):
rubi.append(2067)
return Dist(log(e*(b*e1/d)**n)**p, Int(u, x), x)
rule2067 = ReplacementRule(pattern2067, replacement2067)
pattern2068 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons652, cons1204, cons1203, cons71, cons128)
def replacement2068(p, n1, b, n2, d, a, c, n, e1, x, e):
rubi.append(2068)
return -Dist(n*n1*p*(-a*d + b*c)/b, Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**(p + S(-1))/(c + d*x), x), x) + Simp((a + b*x)*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/b, x)
rule2068 = ReplacementRule(pattern2068, replacement2068)
pattern2069 = Pattern(Integral(log((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))/(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1205, cons1206)
def replacement2069(f, b, g, d, c, a, x, e):
rubi.append(2069)
return Simp(PolyLog(S(2), Together(-a*e + c)/(c + d*x))/g, x)
rule2069 = ReplacementRule(pattern2069, replacement2069)
pattern2070 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1205, cons128)
def replacement2070(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2070)
return Dist(n*n1*p*(-a*d + b*c)/g, Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**(p + S(-1))*log((-a*d + b*c)/(b*(c + d*x)))/((a + b*x)*(c + d*x)), x), x) - Simp(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p*log((-a*d + b*c)/(b*(c + d*x)))/g, x)
rule2070 = ReplacementRule(pattern2070, replacement2070)
pattern2071 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1207, cons128)
def replacement2071(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2071)
return Dist(n*n1*p*(-a*d + b*c)/g, Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**(p + S(-1))*log(-(-a*d + b*c)/(d*(a + b*x)))/((a + b*x)*(c + d*x)), x), x) - Simp(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p*log(-(-a*d + b*c)/(d*(a + b*x)))/g, x)
rule2071 = ReplacementRule(pattern2071, replacement2071)
pattern2072 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))/(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1208)
def replacement2072(n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2072)
return -Dist(n*n1*(-a*d + b*c)/g, Int(log(f + g*x)/((a + b*x)*(c + d*x)), x), x) + Simp(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)*log(f + g*x)/g, x)
rule2072 = ReplacementRule(pattern2072, replacement2072)
pattern2073 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**p_/(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1208, cons38, cons146)
def replacement2073(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2073)
return Dist(d/g, Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/(c + d*x), x), x) - Dist((-c*g + d*f)/g, Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/((c + d*x)*(f + g*x)), x), x)
rule2073 = ReplacementRule(pattern2073, replacement2073)
pattern2074 = Pattern(Integral(S(1)/((x_*WC('g', S(1)) + WC('f', S(0)))**S(2)*log((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1205)
def replacement2074(f, b, g, d, c, a, x, e):
rubi.append(2074)
return Simp(d**S(2)*LogIntegral(e*(a + b*x)/(c + d*x))/(e*g**S(2)*(-a*d + b*c)), x)
rule2074 = ReplacementRule(pattern2074, replacement2074)
pattern2075 = Pattern(Integral(S(1)/((x_*WC('g', S(1)) + WC('f', S(0)))**S(2)*log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1205)
def replacement2075(n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2075)
return Simp(d**S(2)*(e*(e1*(a + b*x)**n1*(c + d*x)**n2)**n)**(-S(1)/(n*n1))*(a + b*x)*ExpIntegralEi(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)/(n*n1))/(g**S(2)*n*n1*(c + d*x)*(-a*d + b*c)), x)
rule2075 = ReplacementRule(pattern2075, replacement2075)
pattern2076 = Pattern(Integral(S(1)/((x_*WC('g', S(1)) + WC('f', S(0)))**S(2)*log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1207)
def replacement2076(n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2076)
return Simp(b**S(2)*(e*(e1*(a + b*x)**n1*(c + d*x)**n2)**n)**(S(1)/(n*n1))*(c + d*x)*ExpIntegralEi(-log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)/(n*n1))/(g**S(2)*n*n1*(a + b*x)*(-a*d + b*c)), x)
rule2076 = ReplacementRule(pattern2076, replacement2076)
pattern2077 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/(x_*WC('g', S(1)) + WC('f', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1209, cons128)
def replacement2077(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2077)
return -Dist(n*n1*p*(-a*d + b*c)/(-a*g + b*f), Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**(p + S(-1))/((c + d*x)*(f + g*x)), x), x) + Simp((a + b*x)*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/((f + g*x)*(-a*g + b*f)), x)
rule2077 = ReplacementRule(pattern2077, replacement2077)
pattern2078 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/(x_*WC('g', S(1)) + WC('f', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1208, cons128)
def replacement2078(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2078)
return -Dist(n*n1*p*(-a*d + b*c)/(-c*g + d*f), Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**(p + S(-1))/((a + b*x)*(f + g*x)), x), x) + Simp((c + d*x)*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/((f + g*x)*(-c*g + d*f)), x)
rule2078 = ReplacementRule(pattern2078, replacement2078)
pattern2079 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**p_/(x_*WC('g', S(1)) + WC('f', S(0)))**S(3), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons652, cons1204, cons1203, cons71, cons1209, cons1205)
def replacement2079(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2079)
return Dist(b/(-a*g + b*f), Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/(f + g*x)**S(2), x), x) - Dist(g/(-a*g + b*f), Int((a + b*x)*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/(f + g*x)**S(3), x), x)
rule2079 = ReplacementRule(pattern2079, replacement2079)
pattern2080 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**p_/(x_*WC('g', S(1)) + WC('f', S(0)))**S(3), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons652, cons1204, cons1203, cons71, cons1208, cons1207)
def replacement2080(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2080)
return Dist(d/(-c*g + d*f), Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/(f + g*x)**S(2), x), x) - Dist(g/(-c*g + d*f), Int((c + d*x)*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/(f + g*x)**S(3), x), x)
rule2080 = ReplacementRule(pattern2080, replacement2080)
pattern2081 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons128, cons17, cons66)
def replacement2081(p, m, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2081)
return -Dist(n*n1*p*(-a*d + b*c)/(g*(m + S(1))), Int((f + g*x)**(m + S(1))*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) + Simp((f + g*x)**(m + S(1))*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/(g*(m + S(1))), x)
rule2081 = ReplacementRule(pattern2081, replacement2081)
pattern2082 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m2', S(1))*log(u_**n_*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1211, cons1210, cons71, cons66, cons13, cons163)
def replacement2082(p, u, m2, m, b, d, a, c, n, x, e):
rubi.append(2082)
return -Dist(n*p/(m + S(1)), Int((a + b*x)**m*(c + d*x)**(-m + S(-2))*log(e*u**n)**(p + S(-1)), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(-m + S(-1))*log(e*u**n)**p/((m + S(1))*(-a*d + b*c)), x)
rule2082 = ReplacementRule(pattern2082, replacement2082)
pattern2083 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m2', S(1))*log(u_)**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons1211, cons1210, cons71, cons66, cons13, cons163)
def replacement2083(p, u, m2, m, b, d, a, c, x):
rubi.append(2083)
return -Dist(p/(m + S(1)), Int((a + b*x)**m*(c + d*x)**(-m + S(-2))*log(u)**(p + S(-1)), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(-m + S(-1))*log(u)**p/((m + S(1))*(-a*d + b*c)), x)
rule2083 = ReplacementRule(pattern2083, replacement2083)
pattern2084 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m2', S(1))/log(u_**n_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1211, cons1210, cons71, cons66)
def replacement2084(u, m2, m, b, d, a, c, n, x, e):
rubi.append(2084)
return Simp((e*u**n)**(-(m + S(1))/n)*(a + b*x)**(m + S(1))*(c + d*x)**(-m + S(-1))*ExpIntegralEi((m + S(1))*log(e*u**n)/n)/(n*(-a*d + b*c)), x)
rule2084 = ReplacementRule(pattern2084, replacement2084)
pattern2085 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m2', S(1))/log(u_), x_), cons2, cons3, cons7, cons27, cons1211, cons1210, cons71, cons66)
def replacement2085(u, m2, m, b, d, a, c, x):
rubi.append(2085)
return Simp(u**(-m + S(-1))*(a + b*x)**(m + S(1))*(c + d*x)**(-m + S(-1))*ExpIntegralEi((m + S(1))*log(u))/(-a*d + b*c), x)
rule2085 = ReplacementRule(pattern2085, replacement2085)
pattern2086 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m2', S(1))*log(u_**n_*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1211, cons1210, cons71, cons66, cons13, cons137)
def replacement2086(p, u, m2, m, b, d, a, c, n, x, e):
rubi.append(2086)
return -Dist((m + S(1))/(n*(p + S(1))), Int((a + b*x)**m*(c + d*x)**(-m + S(-2))*log(e*u**n)**(p + S(1)), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(-m + S(-1))*log(e*u**n)**(p + S(1))/(n*(p + S(1))*(-a*d + b*c)), x)
rule2086 = ReplacementRule(pattern2086, replacement2086)
pattern2087 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m2', S(1))*log(u_)**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons1211, cons1210, cons71, cons66, cons13, cons137)
def replacement2087(p, u, m2, m, b, d, a, c, x):
rubi.append(2087)
return -Dist((m + S(1))/(p + S(1)), Int((a + b*x)**m*(c + d*x)**(-m + S(-2))*log(u)**(p + S(1)), x), x) + Simp((a + b*x)**(m + S(1))*(c + d*x)**(-m + S(-1))*log(u)**(p + S(1))/((p + S(1))*(-a*d + b*c)), x)
rule2087 = ReplacementRule(pattern2087, replacement2087)
pattern2088 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/((x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1209, cons1205)
def replacement2088(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2088)
return Dist(d/g, Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/(c + d*x)**S(2), x), x)
rule2088 = ReplacementRule(pattern2088, replacement2088)
pattern2089 = Pattern(Integral(log((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))/((x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1209, cons1208, cons1212)
def replacement2089(f, b, g, d, c, a, x, e):
rubi.append(2089)
return Simp(PolyLog(S(2), -(f + g*x)*(a*e - c)/(f*(c + d*x)))/(-c*g + d*f), x)
rule2089 = ReplacementRule(pattern2089, replacement2089)
pattern2090 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/((x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons652, cons1204, cons1203, cons71, cons1209, cons1208, cons128)
def replacement2090(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2090)
return Dist(n*n1*p*(-a*d + b*c)/(-c*g + d*f), Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**(p + S(-1))*log((f + g*x)*(-a*d + b*c)/((c + d*x)*(-a*g + b*f)))/((a + b*x)*(c + d*x)), x), x) - Simp(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p*log((f + g*x)*(-a*d + b*c)/((c + d*x)*(-a*g + b*f)))/(-c*g + d*f), x)
rule2090 = ReplacementRule(pattern2090, replacement2090)
pattern2091 = Pattern(Integral(log((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))/(f_ + x_**S(2)*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1213, cons1214)
def replacement2091(g, b, f, d, c, a, x, e):
rubi.append(2091)
return Simp(c*PolyLog(S(2), -(c - d*x)*(a*e - c)/(c*(c + d*x)))/(S(2)*d*f), x)
rule2091 = ReplacementRule(pattern2091, replacement2091)
pattern2092 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/(x_**S(2)*WC('h', S(1)) + x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons652, cons1204, cons1203, cons71, cons1215)
def replacement2092(p, n1, b, f, g, n2, d, a, c, n, e1, x, h, e):
rubi.append(2092)
return Dist(d**S(2), Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/((c + d*x)*(-c*h + d*g + d*h*x)), x), x)
rule2092 = ReplacementRule(pattern2092, replacement2092)
pattern2093 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/(x_**S(2)*WC('h', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons209, cons4, cons5, cons652, cons1204, cons1203, cons71, cons1216)
def replacement2093(p, n1, b, f, n2, d, a, c, n, e1, x, h, e):
rubi.append(2093)
return -Dist(d**S(2)/h, Int(log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/((c - d*x)*(c + d*x)), x), x)
rule2093 = ReplacementRule(pattern2093, replacement2093)
pattern2094 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/((x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('g', S(1)) + WC('f', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons652, cons1204, cons1203, cons71, cons1208, cons1207)
def replacement2094(p, n1, b, f, g, n2, d, a, c, n, e1, x, e):
rubi.append(2094)
return Dist(b/(g*n*n1*(-a*d + b*c)), Subst(Int(x**p, x), x, log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)), x)
rule2094 = ReplacementRule(pattern2094, replacement2094)
pattern2095 = Pattern(Integral(log(v_)*log(u_**n_*WC('e', S(1)))**WC('p', S(1))/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1217, cons1211, cons71, cons13, cons163)
def replacement2095(v, p, u, b, d, c, a, n, x, e):
rubi.append(2095)
return Dist(n*p, Int(PolyLog(S(2), Together(-v + S(1)))*log(e*u**n)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) - Simp(PolyLog(S(2), Together(-v + S(1)))*log(e*u**n)**p/(-a*d + b*c), x)
rule2095 = ReplacementRule(pattern2095, replacement2095)
pattern2096 = Pattern(Integral(log(u_)**WC('p', S(1))*log(v_)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1217, cons1211, cons71, cons13, cons163)
def replacement2096(v, p, u, b, d, c, a, x):
rubi.append(2096)
return Dist(p, Int(PolyLog(S(2), Together(-v + S(1)))*log(u)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) - Simp(PolyLog(S(2), Together(-v + S(1)))*log(u)**p/(-a*d + b*c), x)
rule2096 = ReplacementRule(pattern2096, replacement2096)
def With2097(v, p, u, b, d, c, a, n, x, e):
f = (-v + S(1))/u
rubi.append(2097)
return Dist(f/(n*(p + S(1))), Int(log(e*u**n)**(p + S(1))/((c + d*x)*(-a*f - b*f + c + d)), x), x) + Simp(log(v)*log(e*u**n)**(p + S(1))/(n*(p + S(1))*(-a*d + b*c)), x)
pattern2097 = Pattern(Integral(log(v_)*log(u_**n_*WC('e', S(1)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1217, cons1211, cons71, cons13, cons137)
rule2097 = ReplacementRule(pattern2097, With2097)
def With2098(v, p, u, b, d, c, a, x):
f = (-v + S(1))/u
rubi.append(2098)
return Dist(f/(p + S(1)), Int(log(u)**(p + S(1))/((c + d*x)*(-a*f - b*f + c + d)), x), x) + Simp(log(u)**(p + S(1))*log(v)/((p + S(1))*(-a*d + b*c)), x)
pattern2098 = Pattern(Integral(log(u_)**p_*log(v_)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1217, cons1211, cons71, cons13, cons137)
rule2098 = ReplacementRule(pattern2098, With2098)
pattern2099 = Pattern(Integral(log(v_)*log(u_**n_*WC('e', S(1)))**WC('p', S(1))/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1218, cons1211, cons71, cons13, cons163)
def replacement2099(v, p, u, b, d, c, a, n, x, e):
rubi.append(2099)
return -Dist(n*p, Int(PolyLog(S(2), Together(-v + S(1)))*log(e*u**n)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) + Simp(PolyLog(S(2), Together(-v + S(1)))*log(e*u**n)**p/(-a*d + b*c), x)
rule2099 = ReplacementRule(pattern2099, replacement2099)
pattern2100 = Pattern(Integral(log(u_)**WC('p', S(1))*log(v_)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1218, cons1211, cons71, cons13, cons163)
def replacement2100(v, p, u, b, d, c, a, x):
rubi.append(2100)
return -Dist(p, Int(PolyLog(S(2), Together(-v + S(1)))*log(u)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) + Simp(PolyLog(S(2), Together(-v + S(1)))*log(u)**p/(-a*d + b*c), x)
rule2100 = ReplacementRule(pattern2100, replacement2100)
def With2101(v, p, u, b, d, c, a, n, x, e):
f = u*(-v + S(1))
rubi.append(2101)
return -Dist(f/(n*(p + S(1))), Int(log(e*u**n)**(p + S(1))/((a + b*x)*(a + b - c*f - d*f)), x), x) + Simp(log(v)*log(e*u**n)**(p + S(1))/(n*(p + S(1))*(-a*d + b*c)), x)
pattern2101 = Pattern(Integral(log(v_)*log(u_**n_*WC('e', S(1)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1218, cons1211, cons71, cons13, cons137)
rule2101 = ReplacementRule(pattern2101, With2101)
def With2102(v, p, u, b, d, c, a, x):
f = u*(-v + S(1))
rubi.append(2102)
return -Dist(f/(p + S(1)), Int(log(u)**(p + S(1))/((a + b*x)*(a + b - c*f - d*f)), x), x) + Simp(log(u)**(p + S(1))*log(v)/((p + S(1))*(-a*d + b*c)), x)
pattern2102 = Pattern(Integral(log(u_)**p_*log(v_)/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1218, cons1211, cons71, cons13, cons137)
rule2102 = ReplacementRule(pattern2102, With2102)
pattern2103 = Pattern(Integral(PolyLog(q_, v_)*log(u_**n_*WC('e', S(1)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons1219, cons1211, cons71, cons13, cons146)
def replacement2103(v, p, u, b, d, c, a, n, x, q, e):
rubi.append(2103)
return -Dist(n*p, Int(PolyLog(q + S(1), v)*log(e*u**n)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) + Simp(PolyLog(q + S(1), v)*log(e*u**n)**p/(-a*d + b*c), x)
rule2103 = ReplacementRule(pattern2103, replacement2103)
pattern2104 = Pattern(Integral(PolyLog(q_, v_)*log(u_)**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons50, cons1219, cons1211, cons71, cons13, cons146)
def replacement2104(v, p, u, b, d, c, a, x, q):
rubi.append(2104)
return -Dist(p, Int(PolyLog(q + S(1), v)*log(u)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) + Simp(PolyLog(q + S(1), v)*log(u)**p/(-a*d + b*c), x)
rule2104 = ReplacementRule(pattern2104, replacement2104)
pattern2105 = Pattern(Integral(PolyLog(q_, v_)*log(u_**n_*WC('e', S(1)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons1219, cons1211, cons71, cons13, cons137)
def replacement2105(v, p, u, b, d, c, a, n, x, q, e):
rubi.append(2105)
return -Dist(S(1)/(n*(p + S(1))), Int(PolyLog(q + S(-1), v)*log(e*u**n)**(p + S(1))/((a + b*x)*(c + d*x)), x), x) + Simp(PolyLog(q, v)*log(e*u**n)**(p + S(1))/(n*(p + S(1))*(-a*d + b*c)), x)
rule2105 = ReplacementRule(pattern2105, replacement2105)
pattern2106 = Pattern(Integral(PolyLog(q_, v_)*log(u_)**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons50, cons1219, cons1211, cons71, cons13, cons137)
def replacement2106(v, p, u, b, d, c, a, x, q):
rubi.append(2106)
return -Dist(S(1)/(p + S(1)), Int(PolyLog(q + S(-1), v)*log(u)**(p + S(1))/((a + b*x)*(c + d*x)), x), x) + Simp(PolyLog(q, v)*log(u)**(p + S(1))/((p + S(1))*(-a*d + b*c)), x)
rule2106 = ReplacementRule(pattern2106, replacement2106)
pattern2107 = Pattern(Integral(PolyLog(q_, v_)*log(u_**n_*WC('e', S(1)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons1220, cons1211, cons71, cons13, cons146)
def replacement2107(v, p, u, b, d, c, a, n, x, q, e):
rubi.append(2107)
return Dist(n*p, Int(PolyLog(q + S(1), v)*log(e*u**n)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) - Simp(PolyLog(q + S(1), v)*log(e*u**n)**p/(-a*d + b*c), x)
rule2107 = ReplacementRule(pattern2107, replacement2107)
pattern2108 = Pattern(Integral(PolyLog(q_, v_)*log(u_)**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons50, cons1220, cons1211, cons71, cons13, cons146)
def replacement2108(v, p, u, b, d, c, a, x, q):
rubi.append(2108)
return Dist(p, Int(PolyLog(q + S(1), v)*log(u)**(p + S(-1))/((a + b*x)*(c + d*x)), x), x) - Simp(PolyLog(q + S(1), v)*log(u)**p/(-a*d + b*c), x)
rule2108 = ReplacementRule(pattern2108, replacement2108)
pattern2109 = Pattern(Integral(PolyLog(q_, v_)*log(u_**n_*WC('e', S(1)))**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons1220, cons1211, cons71, cons13, cons137)
def replacement2109(v, p, u, b, d, c, a, n, x, q, e):
rubi.append(2109)
return Dist(S(1)/(n*(p + S(1))), Int(PolyLog(q + S(-1), v)*log(e*u**n)**(p + S(1))/((a + b*x)*(c + d*x)), x), x) + Simp(PolyLog(q, v)*log(e*u**n)**(p + S(1))/(n*(p + S(1))*(-a*d + b*c)), x)
rule2109 = ReplacementRule(pattern2109, replacement2109)
pattern2110 = Pattern(Integral(PolyLog(q_, v_)*log(u_)**p_/((x_*WC('b', S(1)) + WC('a', S(0)))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons50, cons1220, cons1211, cons71, cons13, cons137)
def replacement2110(v, p, u, b, d, c, a, x, q):
rubi.append(2110)
return Dist(S(1)/(p + S(1)), Int(PolyLog(q + S(-1), v)*log(u)**(p + S(1))/((a + b*x)*(c + d*x)), x), x) + Simp(PolyLog(q, v)*log(u)**(p + S(1))/((p + S(1))*(-a*d + b*c)), x)
rule2110 = ReplacementRule(pattern2110, replacement2110)
pattern2111 = Pattern(Integral(WC('u', S(1))*log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/(x_**S(2)*WC('h', S(1)) + x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons5, cons652, cons1204, cons1203, cons71, cons1221, cons1222)
def replacement2111(p, u, n1, b, f, g, n2, d, a, c, n, e1, x, h, e):
rubi.append(2111)
return Dist(b*d/h, Int(u*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/((a + b*x)*(c + d*x)), x), x)
rule2111 = ReplacementRule(pattern2111, replacement2111)
pattern2112 = Pattern(Integral(WC('u', S(1))*log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1))/(x_**S(2)*WC('h', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons209, cons4, cons5, cons652, cons1204, cons1203, cons71, cons1221, cons70)
def replacement2112(p, u, n1, b, f, n2, d, a, c, n, e1, x, h, e):
rubi.append(2112)
return Dist(b*d/h, Int(u*log(e*(e1*(a + b*x)**n1*(c + d*x)**(-n1))**n)**p/((a + b*x)*(c + d*x)), x), x)
rule2112 = ReplacementRule(pattern2112, replacement2112)
def With2113(n1, b, g, n2, f, d, a, c, n, e1, x, h, e):
u = IntHide(S(1)/(f + g*x + h*x**S(2)), x)
rubi.append(2113)
return -Dist(n*(-a*d + b*c), Int(u/((a + b*x)*(c + d*x)), x), x) + Simp(u*log(e*(e1*(a + b*x)**n1*(c + d*x)**n2)**n), x)
pattern2113 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))/(f_ + x_**S(2)*WC('h', S(1)) + x_*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons652, cons125, cons208, cons209, cons4, cons1204, cons1203)
rule2113 = ReplacementRule(pattern2113, With2113)
def With2114(n1, b, n2, f, d, a, c, n, e1, x, h, e):
u = IntHide(S(1)/(f + h*x**S(2)), x)
rubi.append(2114)
return -Dist(n*(-a*d + b*c), Int(u/((a + b*x)*(c + d*x)), x), x) + Simp(u*log(e*(e1*(a + b*x)**n1*(c + d*x)**n2)**n), x)
pattern2114 = Pattern(Integral(log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))/(f_ + x_**S(2)*WC('h', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons652, cons125, cons209, cons4, cons1204, cons1203)
rule2114 = ReplacementRule(pattern2114, With2114)
def With2115(p, RFx, n1, b, n2, d, a, c, n, e1, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(log(e*(e1*(a + b*x)**n1*(c + d*x)**n2)**n)**p, RFx, x)
if SumQ(u):
return True
return False
pattern2115 = Pattern(Integral(RFx_*log(((x_*WC('b', S(1)) + WC('a', S(0)))**WC('n1', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**n2_*WC('e1', S(1)))**WC('n', S(1))*WC('e', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons652, cons1204, cons1203, cons1198, cons128, CustomConstraint(With2115))
def replacement2115(p, RFx, n1, b, n2, d, a, c, n, e1, x, e):
u = ExpandIntegrand(log(e*(e1*(a + b*x)**n1*(c + d*x)**n2)**n)**p, RFx, x)
rubi.append(2115)
return Int(u, x)
rule2115 = ReplacementRule(pattern2115, replacement2115)
def With2116(v, x, p, u):
if isinstance(x, (int, Integer, float, Float)):
return False
lst = QuotientOfLinearsParts(v, x)
if Not(And(OneQ(p), ZeroQ(Part(lst, S(3))))):
return True
return False
pattern2116 = Pattern(Integral(WC('u', S(1))*log(v_)**WC('p', S(1)), x_), cons5, cons1223, cons1224, CustomConstraint(With2116))
def replacement2116(v, x, p, u):
lst = QuotientOfLinearsParts(v, x)
rubi.append(2116)
return Int(u*log((x*Part(lst, S(2)) + Part(lst, S(1)))/(x*Part(lst, S(4)) + Part(lst, S(3))))**p, x)
rule2116 = ReplacementRule(pattern2116, replacement2116)
pattern2117 = Pattern(Integral(log((x_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons4, cons5, cons806)
def replacement2117(p, b, c, a, n, x):
rubi.append(2117)
return -Dist(b*n*p, Int(x**n/(a + b*x**n), x), x) + Simp(x*log(c*(a + b*x**n)**p), x)
rule2117 = ReplacementRule(pattern2117, replacement2117)
pattern2118 = Pattern(Integral(log(v_**WC('p', S(1))*WC('c', S(1))), x_), cons7, cons5, cons840, cons1225)
def replacement2118(v, c, p, x):
rubi.append(2118)
return Int(log(c*ExpandToSum(v, x)**p), x)
rule2118 = ReplacementRule(pattern2118, replacement2118)
pattern2119 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log((x_**n_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*WC('c', S(1))))/(x_*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons1226)
def replacement2119(p, f, b, g, d, a, c, n, x, e):
rubi.append(2119)
return -Dist(b*e*n*p/g, Int(x**(n + S(-1))*log(f + g*x)/(d + e*x**n), x), x) + Simp((a + b*log(c*(d + e*x**n)**p))*log(f + g*x)/g, x)
rule2119 = ReplacementRule(pattern2119, replacement2119)
pattern2120 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log((x_**n_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons66)
def replacement2120(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(2120)
return -Dist(b*e*n*p/(g*(m + S(1))), Int(x**(n + S(-1))*(f + g*x)**(m + S(1))/(d + e*x**n), x), x) + Simp((a + b*log(c*(d + e*x**n)**p))*(f + g*x)**(m + S(1))/(g*(m + S(1))), x)
rule2120 = ReplacementRule(pattern2120, replacement2120)
pattern2121 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(v_**WC('p', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons5, cons68, cons840, cons1125)
def replacement2121(v, p, u, m, b, a, c, x):
rubi.append(2121)
return Int((a + b*log(c*ExpandToSum(v, x)**p))*ExpandToSum(u, x)**m, x)
rule2121 = ReplacementRule(pattern2121, replacement2121)
def With2122(p, m, f, g, b, d, c, a, n, x, e):
w = IntHide(asin(f + g*x)**m, x)
rubi.append(2122)
return -Dist(b*e*n*p, Int(SimplifyIntegrand(w*x**(n + S(-1))/(d + e*x**n), x), x), x) + Dist(a + b*log(c*(d + e*x**n)**p), w, x)
pattern2122 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log((x_**n_*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*WC('c', S(1))))*asin(x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons62)
rule2122 = ReplacementRule(pattern2122, With2122)
def With2123(p, f, b, g, d, a, c, x, e):
u = IntHide(S(1)/(f + g*x**S(2)), x)
rubi.append(2123)
return -Dist(S(2)*b*e*p, Int(u*x/(d + e*x**S(2)), x), x) + Simp(u*(a + b*log(c*(d + e*x**S(2))**p)), x)
pattern2123 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log((x_**S(2)*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*WC('c', S(1))))/(x_**S(2)*WC('g', S(1)) + WC('f', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons1227)
rule2123 = ReplacementRule(pattern2123, With2123)
pattern2124 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons148)
def replacement2124(p, b, d, a, c, n, x, e):
rubi.append(2124)
return -Dist(S(2)*b*e*n*p, Int(x**S(2)*(a + b*log(c*(d + e*x**S(2))**p))**(n + S(-1))/(d + e*x**S(2)), x), x) + Simp(x*(a + b*log(c*(d + e*x**S(2))**p))**n, x)
rule2124 = ReplacementRule(pattern2124, replacement2124)
pattern2125 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons148, cons1228)
def replacement2125(p, m, b, d, a, c, n, x, e):
rubi.append(2125)
return Dist(S(1)/2, Subst(Int(x**(m/S(2) + S(-1)/2)*(a + b*log(c*(d + e*x)**p))**n, x), x, x**S(2)), x)
rule2125 = ReplacementRule(pattern2125, replacement2125)
pattern2126 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log((d_ + x_**S(2)*WC('e', S(1)))**WC('p', S(1))*WC('c', S(1))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons148, cons1229)
def replacement2126(p, m, b, d, a, c, n, x, e):
rubi.append(2126)
return -Dist(S(2)*b*e*n*p/(m + S(1)), Int(x**(m + S(2))*(a + b*log(c*(d + e*x**S(2))**p))**(n + S(-1))/(d + e*x**S(2)), x), x) + Simp(x**(m + S(1))*(a + b*log(c*(d + e*x**S(2))**p))**n/(m + S(1)), x)
rule2126 = ReplacementRule(pattern2126, replacement2126)
def With2127(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
w = DerivativeDivides(v, u*(-v + S(1)), x)
res = Not(FalseQ(w))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern2127 = Pattern(Integral(u_*log(v_), x_), CustomConstraint(With2127))
def replacement2127(v, x, u):
w = DerivativeDivides(v, u*(-v + S(1)), x)
rubi.append(2127)
return Simp(w*PolyLog(S(2), Together(-v + S(1))), x)
rule2127 = ReplacementRule(pattern2127, replacement2127)
def With2128(v, w, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
z = DerivativeDivides(v, w*(-v + S(1)), x)
res = Not(FalseQ(z))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern2128 = Pattern(Integral(w_*(WC('a', S(0)) + WC('b', S(1))*log(u_))*log(v_), x_), cons2, cons3, cons1230, CustomConstraint(With2128))
def replacement2128(v, w, u, b, a, x):
z = DerivativeDivides(v, w*(-v + S(1)), x)
rubi.append(2128)
return -Dist(b, Int(SimplifyIntegrand(z*D(u, x)*PolyLog(S(2), Together(-v + S(1)))/u, x), x), x) + Simp(z*(a + b*log(u))*PolyLog(S(2), Together(-v + S(1))), x)
rule2128 = ReplacementRule(pattern2128, replacement2128)
pattern2129 = Pattern(Integral(log((a_ + (x_*WC('e', S(1)) + WC('d', S(0)))**n_*WC('b', S(1)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons87, cons463)
def replacement2129(p, b, d, c, a, n, x, e):
rubi.append(2129)
return -Dist(b*n*p, Int(S(1)/(a*(d + e*x)**(-n) + b), x), x) + Simp((d + e*x)*log(c*(a + b*(d + e*x)**n)**p)/e, x)
rule2129 = ReplacementRule(pattern2129, replacement2129)
pattern2130 = Pattern(Integral(log((a_ + (x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons1231)
def replacement2130(p, b, d, c, n, a, x, e):
rubi.append(2130)
return Dist(a*n*p, Int(S(1)/(a + b*(d + e*x)**n), x), x) + Simp((d + e*x)*log(c*(a + b*(d + e*x)**n)**p)/e, x) - Simp(n*p*x, x)
rule2130 = ReplacementRule(pattern2130, replacement2130)
pattern2131 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log((d_ + WC('e', S(1))/(x_*WC('g', S(1)) + WC('f', S(0))))**WC('p', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons148)
def replacement2131(p, f, g, b, d, a, c, n, x, e):
rubi.append(2131)
return -Dist(b*e*n*p/(d*g), Subst(Int((a + b*log(c*(d + e*x)**p))**(n + S(-1))/x, x), x, S(1)/(f + g*x)), x) + Simp((a + b*log(c*(d + e/(f + g*x))**p))**n*(d*(f + g*x) + e)/(d*g), x)
rule2131 = ReplacementRule(pattern2131, replacement2131)
pattern2132 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(RFx_**WC('p', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons5, cons1198, cons148)
def replacement2132(p, RFx, b, a, c, n, x):
rubi.append(2132)
return -Dist(b*n*p, Int(SimplifyIntegrand(x*(a + b*log(RFx**p*c))**(n + S(-1))*D(RFx, x)/RFx, x), x), x) + Simp(x*(a + b*log(RFx**p*c))**n, x)
rule2132 = ReplacementRule(pattern2132, replacement2132)
pattern2133 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(RFx_**WC('p', S(1))*WC('c', S(1))))**WC('n', S(1))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1198, cons148)
def replacement2133(p, RFx, b, d, a, c, n, x, e):
rubi.append(2133)
return -Dist(b*n*p/e, Int((a + b*log(RFx**p*c))**(n + S(-1))*D(RFx, x)*log(d + e*x)/RFx, x), x) + Simp((a + b*log(RFx**p*c))**n*log(d + e*x)/e, x)
rule2133 = ReplacementRule(pattern2133, replacement2133)
pattern2134 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*log(RFx_**WC('p', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons1198, cons148, cons1232, cons66)
def replacement2134(p, RFx, m, b, d, a, c, n, x, e):
rubi.append(2134)
return -Dist(b*n*p/(e*(m + S(1))), Int(SimplifyIntegrand((a + b*log(RFx**p*c))**(n + S(-1))*(d + e*x)**(m + S(1))*D(RFx, x)/RFx, x), x), x) + Simp((a + b*log(RFx**p*c))**n*(d + e*x)**(m + S(1))/(e*(m + S(1))), x)
rule2134 = ReplacementRule(pattern2134, replacement2134)
def With2135(RFx, d, c, n, x, e):
u = IntHide(S(1)/(d + e*x**S(2)), x)
rubi.append(2135)
return -Dist(n, Int(SimplifyIntegrand(u*D(RFx, x)/RFx, x), x), x) + Simp(u*log(RFx**n*c), x)
pattern2135 = Pattern(Integral(log(RFx_**WC('n', S(1))*WC('c', S(1)))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons7, cons27, cons48, cons4, cons1198, cons1233)
rule2135 = ReplacementRule(pattern2135, With2135)
def With2136(Px, c, n, Qx, x):
u = IntHide(S(1)/Qx, x)
rubi.append(2136)
return -Dist(n, Int(SimplifyIntegrand(u*D(Px, x)/Px, x), x), x) + Simp(u*log(Px**n*c), x)
pattern2136 = Pattern(Integral(log(Px_**WC('n', S(1))*WC('c', S(1)))/Qx_, x_), cons7, cons4, cons1234, cons1235)
rule2136 = ReplacementRule(pattern2136, With2136)
def With2137(p, RFx, b, a, c, n, RGx, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand((a + b*log(RFx**p*c))**n, RGx, x)
if SumQ(u):
return True
return False
pattern2137 = Pattern(Integral(RGx_*(WC('a', S(0)) + WC('b', S(1))*log(RFx_**WC('p', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons5, cons1198, cons1236, cons148, CustomConstraint(With2137))
def replacement2137(p, RFx, b, a, c, n, RGx, x):
u = ExpandIntegrand((a + b*log(RFx**p*c))**n, RGx, x)
rubi.append(2137)
return Int(u, x)
rule2137 = ReplacementRule(pattern2137, replacement2137)
def With2138(p, RFx, b, a, c, n, RGx, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = ExpandIntegrand(RGx*(a + b*log(RFx**p*c))**n, x)
if SumQ(u):
return True
return False
pattern2138 = Pattern(Integral(RGx_*(WC('a', S(0)) + WC('b', S(1))*log(RFx_**WC('p', S(1))*WC('c', S(1))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons5, cons1198, cons1236, cons148, CustomConstraint(With2138))
def replacement2138(p, RFx, b, a, c, n, RGx, x):
u = ExpandIntegrand(RGx*(a + b*log(RFx**p*c))**n, x)
rubi.append(2138)
return Int(u, x)
rule2138 = ReplacementRule(pattern2138, replacement2138)
def With2139(RFx, u, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = SubstForFractionalPowerOfLinear(RFx*(a + b*log(u)), x)
res = Not(FalseQ(lst))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern2139 = Pattern(Integral(RFx_*(WC('a', S(0)) + WC('b', S(1))*log(u_)), x_), cons2, cons3, cons1198, CustomConstraint(With2139))
def replacement2139(RFx, u, b, a, x):
lst = SubstForFractionalPowerOfLinear(RFx*(a + b*log(u)), x)
rubi.append(2139)
return Dist(Part(lst, S(2))*Part(lst, S(4)), Subst(Int(Part(lst, S(1)), x), x, Part(lst, S(3))**(S(1)/Part(lst, S(2)))), x)
rule2139 = ReplacementRule(pattern2139, replacement2139)
pattern2140 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*log((F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('n', S(1))*WC('e', S(1)) + S(1)), x_), cons1099, cons2, cons3, cons7, cons48, cons125, cons208, cons4, cons31, cons168)
def replacement2140(m, f, b, g, c, n, a, x, F, e):
rubi.append(2140)
return Dist(g*m/(b*c*n*log(F)), Int((f + g*x)**(m + S(-1))*PolyLog(S(2), -e*(F**(c*(a + b*x)))**n), x), x) - Simp((f + g*x)**m*PolyLog(S(2), -e*(F**(c*(a + b*x)))**n)/(b*c*n*log(F)), x)
rule2140 = ReplacementRule(pattern2140, replacement2140)
pattern2141 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))*log(d_ + (F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('n', S(1))*WC('e', S(1))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons31, cons168, cons1237)
def replacement2141(m, f, b, g, d, c, n, a, x, F, e):
rubi.append(2141)
return Int((f + g*x)**m*log(S(1) + e*(F**(c*(a + b*x)))**n/d), x) - Simp((f + g*x)**(m + S(1))*log(S(1) + e*(F**(c*(a + b*x)))**n/d)/(g*(m + S(1))), x) + Simp((f + g*x)**(m + S(1))*log(d + e*(F**(c*(a + b*x)))**n)/(g*(m + S(1))), x)
rule2141 = ReplacementRule(pattern2141, replacement2141)
pattern2142 = Pattern(Integral(log(x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1055)
def replacement2142(f, b, d, a, c, x, e):
rubi.append(2142)
return Dist(f**S(2)*(-S(4)*a*c + b**S(2))/S(2), Int(x/(-f*sqrt(a + b*x + c*x**S(2))*(-S(2)*a*e + b*d + x*(-b*e + S(2)*c*d)) + (-b*f**S(2) + S(2)*d*e)*(a + b*x + c*x**S(2))), x), x) + Simp(x*log(d + e*x + f*sqrt(a + b*x + c*x**S(2))), x)
rule2142 = ReplacementRule(pattern2142, replacement2142)
pattern2143 = Pattern(Integral(log(x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons1055)
def replacement2143(f, d, a, c, x, e):
rubi.append(2143)
return -Dist(a*c*f**S(2), Int(x/(d*e*(a + c*x**S(2)) + f*sqrt(a + c*x**S(2))*(a*e - c*d*x)), x), x) + Simp(x*log(d + e*x + f*sqrt(a + c*x**S(2))), x)
rule2143 = ReplacementRule(pattern2143, replacement2143)
pattern2144 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*log(x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons1055, cons66, cons515)
def replacement2144(m, f, b, g, d, a, c, x, e):
rubi.append(2144)
return Dist(f**S(2)*(-S(4)*a*c + b**S(2))/(S(2)*g*(m + S(1))), Int((g*x)**(m + S(1))/(-f*sqrt(a + b*x + c*x**S(2))*(-S(2)*a*e + b*d + x*(-b*e + S(2)*c*d)) + (-b*f**S(2) + S(2)*d*e)*(a + b*x + c*x**S(2))), x), x) + Simp((g*x)**(m + S(1))*log(d + e*x + f*sqrt(a + b*x + c*x**S(2)))/(g*(m + S(1))), x)
rule2144 = ReplacementRule(pattern2144, replacement2144)
pattern2145 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*log(x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons21, cons1055, cons66, cons515)
def replacement2145(m, f, g, d, a, c, x, e):
rubi.append(2145)
return -Dist(a*c*f**S(2)/(g*(m + S(1))), Int((g*x)**(m + S(1))/(d*e*(a + c*x**S(2)) + f*sqrt(a + c*x**S(2))*(a*e - c*d*x)), x), x) + Simp((g*x)**(m + S(1))*log(d + e*x + f*sqrt(a + c*x**S(2)))/(g*(m + S(1))), x)
rule2145 = ReplacementRule(pattern2145, replacement2145)
pattern2146 = Pattern(Integral(WC('v', S(1))*log(sqrt(u_)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0))), x_), cons27, cons48, cons125, cons816, cons817, cons1238)
def replacement2146(v, u, f, d, x, e):
rubi.append(2146)
return Int(v*log(d + e*x + f*sqrt(ExpandToSum(u, x))), x)
rule2146 = ReplacementRule(pattern2146, replacement2146)
pattern2147 = Pattern(Integral(log(u_), x_), cons1230)
def replacement2147(x, u):
rubi.append(2147)
return -Int(SimplifyIntegrand(x*D(u, x)/u, x), x) + Simp(x*log(u), x)
rule2147 = ReplacementRule(pattern2147, replacement2147)
pattern2148 = Pattern(Integral(log(u_)/(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons1239, cons1240)
def replacement2148(x, a, b, u):
rubi.append(2148)
return -Dist(S(1)/b, Int(SimplifyIntegrand(D(u, x)*log(a + b*x)/u, x), x), x) + Simp(log(u)*log(a + b*x)/b, x)
rule2148 = ReplacementRule(pattern2148, replacement2148)
pattern2149 = Pattern(Integral((x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*log(u_), x_), cons2, cons3, cons21, cons1230, cons66)
def replacement2149(u, m, b, a, x):
rubi.append(2149)
return -Dist(S(1)/(b*(m + S(1))), Int(SimplifyIntegrand((a + b*x)**(m + S(1))*D(u, x)/u, x), x), x) + Simp((a + b*x)**(m + S(1))*log(u)/(b*(m + S(1))), x)
rule2149 = ReplacementRule(pattern2149, replacement2149)
def With2150(x, Qx, u):
v = IntHide(S(1)/Qx, x)
rubi.append(2150)
return -Int(SimplifyIntegrand(v*D(u, x)/u, x), x) + Simp(v*log(u), x)
pattern2150 = Pattern(Integral(log(u_)/Qx_, x_), cons1241, cons1230)
rule2150 = ReplacementRule(pattern2150, With2150)
pattern2151 = Pattern(Integral(u_**(x_*WC('a', S(1)))*log(u_), x_), cons2, cons1230)
def replacement2151(x, a, u):
rubi.append(2151)
return -Int(SimplifyIntegrand(u**(a*x + S(-1))*x*D(u, x), x), x) + Simp(u**(a*x)/a, x)
rule2151 = ReplacementRule(pattern2151, replacement2151)
def With2152(v, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
w = IntHide(v, x)
if InverseFunctionFreeQ(w, x):
return True
return False
pattern2152 = Pattern(Integral(v_*log(u_), x_), cons1230, CustomConstraint(With2152))
def replacement2152(v, x, u):
w = IntHide(v, x)
rubi.append(2152)
return Dist(log(u), w, x) - Int(SimplifyIntegrand(w*D(u, x)/u, x), x)
rule2152 = ReplacementRule(pattern2152, replacement2152)
pattern2153 = Pattern(Integral(log(v_)*log(w_), x_), cons1242, cons1243)
def replacement2153(v, w, x):
rubi.append(2153)
return -Int(SimplifyIntegrand(x*D(v, x)*log(w)/v, x), x) - Int(SimplifyIntegrand(x*D(w, x)*log(v)/w, x), x) + Simp(x*log(v)*log(w), x)
rule2153 = ReplacementRule(pattern2153, replacement2153)
def With2154(v, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
z = IntHide(u, x)
if InverseFunctionFreeQ(z, x):
return True
return False
pattern2154 = Pattern(Integral(u_*log(v_)*log(w_), x_), cons1242, cons1243, CustomConstraint(With2154))
def replacement2154(v, w, u, x):
z = IntHide(u, x)
rubi.append(2154)
return Dist(log(v)*log(w), z, x) - Int(SimplifyIntegrand(z*D(v, x)*log(w)/v, x), x) - Int(SimplifyIntegrand(z*D(w, x)*log(v)/w, x), x)
rule2154 = ReplacementRule(pattern2154, replacement2154)
pattern2155 = Pattern(Integral(log(WC('a', S(1))*log(x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons4, cons5, cons1244)
def replacement2155(p, b, a, n, x):
rubi.append(2155)
return -Dist(n*p, Int(S(1)/log(b*x**n), x), x) + Simp(x*log(a*log(b*x**n)**p), x)
rule2155 = ReplacementRule(pattern2155, replacement2155)
pattern2156 = Pattern(Integral(log(WC('a', S(1))*log(x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1)))/x_, x_), cons2, cons3, cons4, cons5, cons1244)
def replacement2156(p, b, a, n, x):
rubi.append(2156)
return Simp((-p + log(a*log(b*x**n)**p))*log(b*x**n)/n, x)
rule2156 = ReplacementRule(pattern2156, replacement2156)
pattern2157 = Pattern(Integral(x_**WC('m', S(1))*log(WC('a', S(1))*log(x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons21, cons4, cons5, cons66)
def replacement2157(p, m, b, a, n, x):
rubi.append(2157)
return -Dist(n*p/(m + S(1)), Int(x**m/log(b*x**n), x), x) + Simp(x**(m + S(1))*log(a*log(b*x**n)**p)/(m + S(1)), x)
rule2157 = ReplacementRule(pattern2157, replacement2157)
pattern2158 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*log(x_*WC('d', S(1)) + WC('c', S(0))))/sqrt(a_ + WC('b', S(1))*log(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons1245)
def replacement2158(B, b, d, c, a, x, A):
rubi.append(2158)
return Dist(B/b, Int(sqrt(a + b*log(c + d*x)), x), x) + Dist((A*b - B*a)/b, Int(S(1)/sqrt(a + b*log(c + d*x)), x), x)
rule2158 = ReplacementRule(pattern2158, replacement2158)
pattern2159 = Pattern(Integral(f_**(WC('a', S(1))*log(u_)), x_), cons2, cons125, cons1246)
def replacement2159(x, a, f, u):
rubi.append(2159)
return Int(u**(a*log(f)), x)
rule2159 = ReplacementRule(pattern2159, replacement2159)
def With2160(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
lst = FunctionOfLog(u*x, x)
res = Not(FalseQ(lst))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern2160 = Pattern(Integral(u_, x_), cons1247, CustomConstraint(With2160))
def replacement2160(x, u):
lst = FunctionOfLog(u*x, x)
rubi.append(2160)
return Dist(S(1)/Part(lst, S(3)), Subst(Int(Part(lst, S(1)), x), x, log(Part(lst, S(2)))), x)
rule2160 = ReplacementRule(pattern2160, replacement2160)
pattern2161 = Pattern(Integral(WC('u', S(1))*log(Gamma(v_)), x_))
def replacement2161(v, x, u):
rubi.append(2161)
return Dist(-LogGamma(v) + log(Gamma(v)), Int(u, x), x) + Int(u*LogGamma(v), x)
rule2161 = ReplacementRule(pattern2161, replacement2161)
pattern2162 = Pattern(Integral((w_*WC('a', S(1)) + w_*WC('b', S(1))*log(v_)**WC('n', S(1)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons4, cons38)
def replacement2162(v, w, p, u, b, a, n, x):
rubi.append(2162)
return Int(u*w**p*(a + b*log(v)**n)**p, x)
rule2162 = ReplacementRule(pattern2162, replacement2162)
pattern2163 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons50, cons1248)
def replacement2163(p, u, f, b, d, a, c, n, x, q, e):
rubi.append(2163)
return Int(u*(a + b*log(c*(d*(e + f*x)**p)**q))**n, x)
rule2163 = ReplacementRule(pattern2163, replacement2163)
return [rule2006, rule2007, rule2008, rule2009, rule2010, rule2011, rule2012, rule2013, rule2014, rule2015, rule2016, rule2017, rule2018, rule2019, rule2020, rule2021, rule2022, rule2023, rule2024, rule2025, rule2026, rule2027, rule2028, rule2029, rule2030, rule2031, rule2032, rule2033, rule2034, rule2035, rule2036, rule2037, rule2038, rule2039, rule2040, rule2041, rule2042, rule2043, rule2044, rule2045, rule2046, rule2047, rule2048, rule2049, rule2050, rule2051, rule2052, rule2053, rule2054, rule2055, rule2056, rule2057, rule2058, rule2059, rule2060, rule2061, rule2062, rule2063, rule2064, rule2065, rule2066, rule2067, rule2068, rule2069, rule2070, rule2071, rule2072, rule2073, rule2074, rule2075, rule2076, rule2077, rule2078, rule2079, rule2080, rule2081, rule2082, rule2083, rule2084, rule2085, rule2086, rule2087, rule2088, rule2089, rule2090, rule2091, rule2092, rule2093, rule2094, rule2095, rule2096, rule2097, rule2098, rule2099, rule2100, rule2101, rule2102, rule2103, rule2104, rule2105, rule2106, rule2107, rule2108, rule2109, rule2110, rule2111, rule2112, rule2113, rule2114, rule2115, rule2116, rule2117, rule2118, rule2119, rule2120, rule2121, rule2122, rule2123, rule2124, rule2125, rule2126, rule2127, rule2128, rule2129, rule2130, rule2131, rule2132, rule2133, rule2134, rule2135, rule2136, rule2137, rule2138, rule2139, rule2140, rule2141, rule2142, rule2143, rule2144, rule2145, rule2146, rule2147, rule2148, rule2149, rule2150, rule2151, rule2152, rule2153, rule2154, rule2155, rule2156, rule2157, rule2158, rule2159, rule2160, rule2161, rule2162, rule2163, ]
|
b8abda9dc2cbb0c543ffc3df7c4ac241bd7960ff079a67688b17942eb39da8d9 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def trinomial_products(rubi):
from sympy.integrals.rubi.constraints import cons46, cons87, cons463, cons38, cons2, cons3, cons7, cons489, cons5, cons45, cons147, cons664, cons4, cons665, cons584, cons666, cons13, cons163, cons667, cons314, cons668, cons462, cons196, cons669, cons670, cons146, cons671, cons672, cons338, cons137, cons226, cons128, cons246, cons673, cons674, cons413, cons675, cons293, cons676, cons677, cons484, cons177, cons678, cons679, cons680, cons585, cons681, cons68, cons69, cons53, cons21, cons501, cons27, cons63, cons502, cons682, cons155, cons683, cons684, cons225, cons56, cons243, cons148, cons244, cons685, cons17, cons686, cons687, cons510, cons688, cons689, cons690, cons529, cons31, cons530, cons691, cons94, cons367, cons356, cons500, cons692, cons693, cons694, cons695, cons696, cons697, cons698, cons699, cons700, cons701, cons541, cons23, cons702, cons552, cons553, cons554, cons220, cons48, cons50, cons703, cons704, cons256, cons257, cons279, cons221, cons280, cons395, cons396, cons705, cons706, cons707, cons708, cons709, cons85, cons710, cons711, cons712, cons713, cons586, cons386, cons149, cons714, cons43, cons715, cons448, cons716, cons400, cons717, cons718, cons719, cons347, cons564, cons720, cons268, cons721, cons722, cons723, cons724, cons725, cons726, cons727, cons728, cons125, cons208, cons52, cons593, cons729, cons730, cons731, cons652, cons732, cons654, cons34, cons35, cons733, cons18, cons734, cons464, cons735, cons168, cons267, cons736, cons737, cons738, cons739, cons740, cons81, cons434, cons741, cons742, cons743, cons744, cons611, cons403, cons745, cons746, cons747, cons748, cons749, cons750, cons751, cons752, cons753, cons754, cons755, cons756, cons757, cons758, cons759, cons760, cons606, cons761, cons762, cons763, cons764, cons765, cons766, cons767, cons768, cons769, cons770, cons771, cons772, cons773, cons774, cons775, cons776, cons777, cons778, cons779, cons780, cons781, cons782, cons783, cons784, cons785, cons786, cons787, cons788, cons789, cons790, cons791, cons792, cons793, cons794, cons795, cons796, cons797
pattern1076 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons46, cons87, cons463, cons38)
def replacement1076(p, b, n2, c, a, n, x):
rubi.append(1076)
return Int(x**(S(2)*n*p)*(a*x**(-S(2)*n) + b*x**(-n) + c)**p, x)
rule1076 = ReplacementRule(pattern1076, replacement1076)
def With1077(p, b, n2, c, a, n, x):
k = Denominator(n)
rubi.append(1077)
return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*x**(k*n) + c*x**(S(2)*k*n))**p, x), x, x**(S(1)/k)), x)
pattern1077 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons46, cons489)
rule1077 = ReplacementRule(pattern1077, With1077)
pattern1078 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons46, cons45, cons147, cons664)
def replacement1078(p, b, n2, c, a, n, x):
rubi.append(1078)
return Simp(x*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*a), x)
rule1078 = ReplacementRule(pattern1078, replacement1078)
pattern1079 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons46, cons45, cons147, cons665, cons584)
def replacement1079(p, b, n2, c, a, n, x):
rubi.append(1079)
return -Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(a*(S(2)*p + S(1))), x) + Simp(x*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*a*(n + S(1))), x)
rule1079 = ReplacementRule(pattern1079, replacement1079)
pattern1080 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons46, cons45, cons666, cons13, cons163, cons667)
def replacement1080(p, b, n2, c, a, n, x):
rubi.append(1080)
return Dist(sqrt(a + b*x**n + c*x**(S(2)*n))/(b + S(2)*c*x**n), Int((b + S(2)*c*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)/2), x), x)
rule1080 = ReplacementRule(pattern1080, replacement1080)
pattern1081 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons46, cons45, cons666, cons13, cons163, cons314)
def replacement1081(p, b, n2, c, a, n, x):
rubi.append(1081)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((b + S(2)*c*x**n)**(S(2)*p), x), x)
rule1081 = ReplacementRule(pattern1081, replacement1081)
pattern1082 = Pattern(Integral(sqrt(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons4, cons46, cons45, cons584, cons668, cons462)
def replacement1082(b, n2, c, n, a, x):
rubi.append(1082)
return Simp(x*sqrt(a + b*x**n + c*x**(S(2)*n))/(n + S(1)), x) + Simp(b*n*x*sqrt(a + b*x**n + c*x**(S(2)*n))/((b + S(2)*c*x**n)*(n + S(1))), x)
rule1082 = ReplacementRule(pattern1082, replacement1082)
pattern1083 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons46, cons45, cons147, cons196)
def replacement1083(p, b, n2, c, a, n, x):
rubi.append(1083)
return -Subst(Int((a + b*x**(-n) + c*x**(-S(2)*n))**p/x**S(2), x), x, S(1)/x)
rule1083 = ReplacementRule(pattern1083, replacement1083)
pattern1084 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons46, cons45, cons147, cons669, cons670, cons13, cons146)
def replacement1084(p, b, n2, c, a, n, x):
rubi.append(1084)
return Dist(S(2)*a*n**S(2)*p*(S(2)*p + S(-1))/((S(2)*n*p + S(1))*(n*(S(2)*p + S(-1)) + S(1))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) + Simp(x*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*n*p + S(1)), x) + Simp(n*p*x*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))/((S(2)*n*p + S(1))*(n*(S(2)*p + S(-1)) + S(1))), x)
rule1084 = ReplacementRule(pattern1084, replacement1084)
pattern1085 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons46, cons45, cons147, cons671, cons672, cons338, cons137)
def replacement1085(p, b, n2, c, a, n, x):
rubi.append(1085)
return Dist((S(2)*n*(p + S(1)) + S(1))*(n*(S(2)*p + S(1)) + S(1))/(S(2)*a*n**S(2)*(p + S(1))*(S(2)*p + S(1))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1)), x), x) - Simp(x*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*a*n*(S(2)*p + S(1))), x) - Simp(x*(n*(S(2)*p + S(1)) + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(S(2)*a*n**S(2)*(p + S(1))*(S(2)*p + S(1))), x)
rule1085 = ReplacementRule(pattern1085, replacement1085)
pattern1086 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons46, cons45, cons147)
def replacement1086(p, b, n2, c, a, n, x):
rubi.append(1086)
return Dist(c**(-IntPart(p))*(b/S(2) + c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((b/S(2) + c*x**n)**(S(2)*p), x), x)
rule1086 = ReplacementRule(pattern1086, replacement1086)
pattern1087 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons46, cons196)
def replacement1087(p, b, n2, c, a, n, x):
rubi.append(1087)
return -Subst(Int((a + b*x**(-n) + c*x**(-S(2)*n))**p/x**S(2), x), x, S(1)/x)
rule1087 = ReplacementRule(pattern1087, replacement1087)
pattern1088 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons46, cons226, cons128)
def replacement1088(p, b, n2, c, a, n, x):
rubi.append(1088)
return Int(ExpandIntegrand((a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1088 = ReplacementRule(pattern1088, replacement1088)
pattern1089 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons46, cons226, cons13, cons163, cons669, cons246, cons673)
def replacement1089(p, b, n2, c, a, n, x):
rubi.append(1089)
return Dist(n*p/(S(2)*n*p + S(1)), Int((S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) + Simp(x*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*n*p + S(1)), x)
rule1089 = ReplacementRule(pattern1089, replacement1089)
pattern1090 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons46, cons226, cons13, cons137, cons246, cons673)
def replacement1090(p, b, n2, c, a, n, x):
rubi.append(1090)
return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c + b**S(2) + b*c*x**n*(n*(S(2)*p + S(3)) + S(1)) + n*(p + S(1))*(-S(4)*a*c + b**S(2))), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c + b**S(2) + b*c*x**n)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1090 = ReplacementRule(pattern1090, replacement1090)
def With1091(b, n2, c, n, a, x):
q = Rt(a/c, S(2))
r = Rt(-b/c + S(2)*q, S(2))
rubi.append(1091)
return Dist(1/(2*c*q*r), Int((r - x**(n/2))/(q - r*x**(n/2) + x**n), x), x) + Dist(1/(2*c*q*r), Int((r + x**(n/2))/(q + r*x**(n/2) + x**n), x), x)
pattern1091 = Pattern(Integral(S(1)/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons46, cons226, cons674, cons413)
rule1091 = ReplacementRule(pattern1091, With1091)
def With1092(b, n2, c, n, a, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1092)
return Dist(c/q, Int(S(1)/(b/S(2) + c*x**n - q/S(2)), x), x) - Dist(c/q, Int(S(1)/(b/S(2) + c*x**n + q/S(2)), x), x)
pattern1092 = Pattern(Integral(S(1)/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons46, cons226)
rule1092 = ReplacementRule(pattern1092, With1092)
def With1093(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1093)
return Dist(S(2)*sqrt(-c), Int(S(1)/(sqrt(-b - S(2)*c*x**S(2) + q)*sqrt(b + S(2)*c*x**S(2) + q)), x), x)
pattern1093 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, cons293)
rule1093 = ReplacementRule(pattern1093, With1093)
def With1094(x, c, b, a):
q = Rt(c/a, S(4))
rubi.append(1094)
return Simp(sqrt((a + b*x**S(2) + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(q**S(2)*x**S(2) + S(1))*EllipticF(S(2)*ArcTan(q*x), -b*q**S(2)/(S(4)*c) + S(1)/2)/(S(2)*q*sqrt(a + b*x**S(2) + c*x**S(4))), x)
pattern1094 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, cons676, cons677)
rule1094 = ReplacementRule(pattern1094, With1094)
def With1095(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if IntegerQ(q):
return True
return False
pattern1095 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, cons484, cons177, CustomConstraint(With1095))
def replacement1095(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1095)
return Simp(sqrt((S(2)*a + x**S(2)*(b + q))/q)*sqrt(-S(2)*a - x**S(2)*(b - q))*EllipticF(asin(sqrt(S(2))*x/sqrt((S(2)*a + x**S(2)*(b + q))/q)), (b + q)/(S(2)*q))/(S(2)*sqrt(-a)*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1095 = ReplacementRule(pattern1095, replacement1095)
def With1096(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1096)
return Simp(sqrt((S(2)*a + x**S(2)*(b + q))/q)*sqrt((S(2)*a + x**S(2)*(b - q))/(S(2)*a + x**S(2)*(b + q)))*EllipticF(asin(sqrt(S(2))*x/sqrt((S(2)*a + x**S(2)*(b + q))/q)), (b + q)/(S(2)*q))/(S(2)*sqrt(a/(S(2)*a + x**S(2)*(b + q)))*sqrt(a + b*x**S(2) + c*x**S(4))), x)
pattern1096 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, cons484, cons177)
rule1096 = ReplacementRule(pattern1096, With1096)
def With1097(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if And(PosQ((b + q)/a), Not(And(PosQ((b - q)/a), SimplerSqrtQ((b - q)/(S(2)*a), (b + q)/(S(2)*a))))):
return True
return False
pattern1097 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, CustomConstraint(With1097))
def replacement1097(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1097)
return Simp(sqrt((S(2)*a + x**S(2)*(b - q))/(S(2)*a + x**S(2)*(b + q)))*(S(2)*a + x**S(2)*(b + q))*EllipticF(ArcTan(x*Rt((b + q)/(S(2)*a), S(2))), S(2)*q/(b + q))/(S(2)*a*sqrt(a + b*x**S(2) + c*x**S(4))*Rt((b + q)/(S(2)*a), S(2))), x)
rule1097 = ReplacementRule(pattern1097, replacement1097)
def With1098(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if PosQ((b - q)/a):
return True
return False
pattern1098 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, CustomConstraint(With1098))
def replacement1098(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1098)
return Simp(sqrt((S(2)*a + x**S(2)*(b + q))/(S(2)*a + x**S(2)*(b - q)))*(S(2)*a + x**S(2)*(b - q))*EllipticF(ArcTan(x*Rt((b - q)/(S(2)*a), S(2))), -S(2)*q/(b - q))/(S(2)*a*sqrt(a + b*x**S(2) + c*x**S(4))*Rt((b - q)/(S(2)*a), S(2))), x)
rule1098 = ReplacementRule(pattern1098, replacement1098)
def With1099(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if And(NegQ((b + q)/a), Not(And(NegQ((b - q)/a), SimplerSqrtQ(-(b - q)/(S(2)*a), -(b + q)/(S(2)*a))))):
return True
return False
pattern1099 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, CustomConstraint(With1099))
def replacement1099(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1099)
return Simp(sqrt(S(1) + x**S(2)*(b - q)/(S(2)*a))*sqrt(S(1) + x**S(2)*(b + q)/(S(2)*a))*EllipticF(asin(x*Rt(-(b + q)/(S(2)*a), S(2))), (b - q)/(b + q))/(sqrt(a + b*x**S(2) + c*x**S(4))*Rt(-(b + q)/(S(2)*a), S(2))), x)
rule1099 = ReplacementRule(pattern1099, replacement1099)
def With1100(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if NegQ((b - q)/a):
return True
return False
pattern1100 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, CustomConstraint(With1100))
def replacement1100(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1100)
return Simp(sqrt(S(1) + x**S(2)*(b - q)/(S(2)*a))*sqrt(S(1) + x**S(2)*(b + q)/(S(2)*a))*EllipticF(asin(x*Rt(-(b - q)/(S(2)*a), S(2))), (b + q)/(b - q))/(sqrt(a + b*x**S(2) + c*x**S(4))*Rt(-(b - q)/(S(2)*a), S(2))), x)
rule1100 = ReplacementRule(pattern1100, replacement1100)
def With1101(x, c, b, a):
q = Rt(c/a, S(4))
rubi.append(1101)
return Simp(sqrt((a + b*x**S(2) + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(q**S(2)*x**S(2) + S(1))*EllipticF(S(2)*ArcTan(q*x), -b*q**S(2)/(S(4)*c) + S(1)/2)/(S(2)*q*sqrt(a + b*x**S(2) + c*x**S(4))), x)
pattern1101 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons226, cons678)
rule1101 = ReplacementRule(pattern1101, With1101)
def With1102(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1102)
return Dist(sqrt(S(2)*c*x**S(2)/(b - q) + S(1))*sqrt(S(2)*c*x**S(2)/(b + q) + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), Int(S(1)/(sqrt(S(2)*c*x**S(2)/(b - q) + S(1))*sqrt(S(2)*c*x**S(2)/(b + q) + S(1))), x), x)
pattern1102 = Pattern(Integral(S(1)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons226, cons679)
rule1102 = ReplacementRule(pattern1102, With1102)
pattern1103 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons680)
def replacement1103(p, b, n2, c, a, n, x):
rubi.append(1103)
return Dist(a**IntPart(p)*(S(2)*c*x**n/(b - Rt(-S(4)*a*c + b**S(2), S(2))) + S(1))**(-FracPart(p))*(S(2)*c*x**n/(b + Rt(-S(4)*a*c + b**S(2), S(2))) + S(1))**(-FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((S(2)*c*x**n/(b - sqrt(-S(4)*a*c + b**S(2))) + S(1))**p*(S(2)*c*x**n/(b + sqrt(-S(4)*a*c + b**S(2))) + S(1))**p, x), x)
rule1103 = ReplacementRule(pattern1103, replacement1103)
pattern1104 = Pattern(Integral((a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons585, cons38, cons681)
def replacement1104(mn, p, b, c, n, a, x):
rubi.append(1104)
return Int(x**(-n*p)*(a*x**n + b + c*x**(S(2)*n))**p, x)
rule1104 = ReplacementRule(pattern1104, replacement1104)
pattern1105 = Pattern(Integral((a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons585, cons147, cons681)
def replacement1105(mn, p, b, c, n, a, x):
rubi.append(1105)
return Dist(x**(n*FracPart(p))*(a + b*x**(-n) + c*x**n)**FracPart(p)*(a*x**n + b + c*x**(S(2)*n))**(-FracPart(p)), Int(x**(-n*p)*(a*x**n + b + c*x**(S(2)*n))**p, x), x)
rule1105 = ReplacementRule(pattern1105, replacement1105)
pattern1106 = Pattern(Integral((a_ + u_**n_*WC('b', S(1)) + u_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons46, cons68, cons69)
def replacement1106(p, u, b, n2, c, a, n, x):
rubi.append(1106)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*x**n + c*x**(S(2)*n))**p, x), x, u), x)
rule1106 = ReplacementRule(pattern1106, replacement1106)
pattern1107 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons680, cons53)
def replacement1107(p, m, b, n2, c, a, n, x):
rubi.append(1107)
return Dist(S(1)/n, Subst(Int((a + b*x + c*x**S(2))**p, x), x, x**n), x)
rule1107 = ReplacementRule(pattern1107, replacement1107)
pattern1108 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons680, cons128, cons501)
def replacement1108(p, m, b, n2, d, c, a, n, x):
rubi.append(1108)
return Int(ExpandIntegrand((d*x)**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1108 = ReplacementRule(pattern1108, replacement1108)
pattern1109 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons680, cons63, cons502)
def replacement1109(p, m, b, n2, c, a, n, x):
rubi.append(1109)
return Int(x**(m + S(2)*n*p)*(a*x**(-S(2)*n) + b*x**(-n) + c)**p, x)
rule1109 = ReplacementRule(pattern1109, replacement1109)
pattern1110 = Pattern(Integral(sqrt(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))/x_, x_), cons2, cons3, cons7, cons4, cons680, cons45)
def replacement1110(b, n2, c, a, n, x):
rubi.append(1110)
return Simp(sqrt(a + b*x**n + c*x**(S(2)*n))/n, x) + Simp(b*sqrt(a + b*x**n + c*x**(S(2)*n))*log(x)/(b + S(2)*c*x**n), x)
rule1110 = ReplacementRule(pattern1110, replacement1110)
pattern1111 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_/x_, x_), cons2, cons3, cons7, cons4, cons680, cons45, cons13, cons146)
def replacement1111(p, b, n2, c, a, n, x):
rubi.append(1111)
return Dist(a, Int((a + b*x**n + c*x**(S(2)*n))**(p + S(-1))/x, x), x) + Simp((a + b*x**n + c*x**(S(2)*n))**p/(S(2)*n*p), x) + Simp((S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))/(S(2)*n*(S(2)*p + S(-1))), x)
rule1111 = ReplacementRule(pattern1111, replacement1111)
pattern1112 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_/x_, x_), cons2, cons3, cons7, cons4, cons680, cons45, cons13, cons137)
def replacement1112(p, b, n2, c, a, n, x):
rubi.append(1112)
return Dist(S(1)/a, Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))/x, x), x) - Simp((a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(S(2)*a*n*(p + S(1))), x) - Simp((S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*a*n*(S(2)*p + S(1))), x)
rule1112 = ReplacementRule(pattern1112, replacement1112)
pattern1113 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_/x_, x_), cons2, cons3, cons7, cons4, cons5, cons680, cons45, cons147)
def replacement1113(p, b, n2, c, a, n, x):
rubi.append(1113)
return Dist(c**(-IntPart(p))*(b/S(2) + c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((b/S(2) + c*x**n)**(S(2)*p)/x, x), x)
rule1113 = ReplacementRule(pattern1113, replacement1113)
pattern1114 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons680, cons45, cons682)
def replacement1114(p, m, b, n2, d, c, a, n, x):
rubi.append(1114)
return Simp((d*x)**(m + S(1))*(b + S(2)*c*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(b*d*(m + S(1))), x)
rule1114 = ReplacementRule(pattern1114, replacement1114)
pattern1115 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*sqrt(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons680, cons45, cons155)
def replacement1115(m, b, n2, d, c, a, n, x):
rubi.append(1115)
return Dist(sqrt(a + b*x**n + c*x**(S(2)*n))/(b + S(2)*c*x**n), Int((d*x)**m*(b + S(2)*c*x**n), x), x)
rule1115 = ReplacementRule(pattern1115, replacement1115)
pattern1116 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*sqrt(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons680, cons45, cons683)
def replacement1116(m, b, n2, d, c, a, n, x):
rubi.append(1116)
return Simp((d*x)**(m + S(1))*sqrt(a + b*x**n + c*x**(S(2)*n))/(d*(m + n + S(1))), x) + Simp(b*n*(d*x)**(m + S(1))*sqrt(a + b*x**n + c*x**(S(2)*n))/(d*(b + S(2)*c*x**n)*(m + S(1))*(m + n + S(1))), x)
rule1116 = ReplacementRule(pattern1116, replacement1116)
pattern1117 = Pattern(Integral(x_**WC('m', S(1))/sqrt(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons21, cons4, cons680, cons45, cons155)
def replacement1117(m, b, n2, c, a, n, x):
rubi.append(1117)
return -Dist(b/(S(2)*a), Int(S(1)/(x*sqrt(a + b*x**n + c*x**(S(2)*n))), x), x) - Simp(x**(m + S(1))*sqrt(a + b*x**n + c*x**(S(2)*n))/(a*n), x)
rule1117 = ReplacementRule(pattern1117, replacement1117)
pattern1118 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons680, cons45, cons684, cons225)
def replacement1118(p, m, b, n2, d, c, a, n, x):
rubi.append(1118)
return -Simp((d*x)**(m + S(1))*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*a*d*n*(S(2)*p + S(1))), x) + Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(S(2)*a*d*n*(p + S(1))*(S(2)*p + S(1))), x)
rule1118 = ReplacementRule(pattern1118, replacement1118)
pattern1119 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons680, cons45, cons56, cons243)
def replacement1119(p, m, b, n2, c, a, n, x):
rubi.append(1119)
return -Dist(b/(S(2)*c), Int(x**(n + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x) + Simp((a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(S(2)*c*n*(p + S(1))), x)
rule1119 = ReplacementRule(pattern1119, replacement1119)
pattern1120 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons680, cons45, cons148, cons244, cons146, cons685, cons246, cons17)
def replacement1120(p, m, b, n2, d, c, a, n, x):
rubi.append(1120)
return -Dist(b*d**(-n)*n**S(2)*p*(S(2)*p + S(-1))/((m + S(1))*(m + S(2)*n*p + S(1))), Int((d*x)**(m + n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) + Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**p/(d*(m + S(2)*n*p + S(1))), x) + Simp(n*p*(d*x)**(m + S(1))*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))/(d*(m + S(1))*(m + S(2)*n*p + S(1))), x)
rule1120 = ReplacementRule(pattern1120, replacement1120)
pattern1121 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons680, cons45, cons148, cons244, cons146, cons686, cons687, cons246, cons17)
def replacement1121(p, m, b, n2, d, c, a, n, x):
rubi.append(1121)
return Dist(S(2)*c*d**(-S(2)*n)*n**S(2)*p*(S(2)*p + S(-1))/((m + S(1))*(m + n + S(1))), Int((d*x)**(m + S(2)*n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) + Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**p*(m - n*(S(2)*p + S(-1)) + S(1))/(d*(m + S(1))*(m + n + S(1))), x) + Simp(n*p*(d*x)**(m + S(1))*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))/(d*(m + S(1))*(m + n + S(1))), x)
rule1121 = ReplacementRule(pattern1121, replacement1121)
pattern1122 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons680, cons45, cons148, cons13, cons146, cons510, cons688, cons687, cons689, cons246)
def replacement1122(p, m, b, n2, d, c, a, n, x):
rubi.append(1122)
return Dist(S(2)*a*n**S(2)*p*(S(2)*p + S(-1))/((m + S(2)*n*p + S(1))*(m + n*(S(2)*p + S(-1)) + S(1))), Int((d*x)**m*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) + Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**p/(d*(m + S(2)*n*p + S(1))), x) + Simp(n*p*(d*x)**(m + S(1))*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))/(d*(m + S(2)*n*p + S(1))*(m + n*(S(2)*p + S(-1)) + S(1))), x)
rule1122 = ReplacementRule(pattern1122, replacement1122)
pattern1123 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons680, cons45, cons148, cons244, cons137, cons690, cons246)
def replacement1123(p, m, b, n2, d, c, a, n, x):
rubi.append(1123)
return -Dist(d**n*(m - n + S(1))*(m + n*(S(2)*p + S(1)) + S(1))/(b*n**S(2)*(p + S(1))*(S(2)*p + S(1))), Int((d*x)**(m - n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(1)), x), x) - Simp((d*x)**(m + S(1))*(b + S(2)*c*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(b*d*n*(S(2)*p + S(1))), x) + Simp(d**(n + S(-1))*(d*x)**(m - n + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(m + n*(S(2)*p + S(1)) + S(1))/(b*n**S(2)*(p + S(1))*(S(2)*p + S(1))), x)
rule1123 = ReplacementRule(pattern1123, replacement1123)
pattern1124 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons680, cons45, cons148, cons244, cons137, cons529, cons246)
def replacement1124(p, m, b, n2, d, c, a, n, x):
rubi.append(1124)
return Dist(d**(S(2)*n)*(m - S(2)*n + S(1))*(m - n + S(1))/(S(2)*c*n**S(2)*(p + S(1))*(S(2)*p + S(1))), Int((d*x)**(m - S(2)*n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(1)), x), x) - Simp(d**(S(2)*n + S(-1))*(d*x)**(m - S(2)*n + S(1))*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*c*n*(S(2)*p + S(1))), x) - Simp(d**(S(2)*n + S(-1))*(d*x)**(m - S(2)*n + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(m - S(2)*n*p - S(3)*n + S(1))/(S(2)*c*n**S(2)*(p + S(1))*(S(2)*p + S(1))), x)
rule1124 = ReplacementRule(pattern1124, replacement1124)
pattern1125 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons680, cons45, cons148, cons244, cons137, cons246)
def replacement1125(p, m, b, n2, d, c, a, n, x):
rubi.append(1125)
return Dist((m + S(2)*n*(p + S(1)) + S(1))*(m + n*(S(2)*p + S(1)) + S(1))/(S(2)*a*n**S(2)*(p + S(1))*(S(2)*p + S(1))), Int((d*x)**m*(a + b*x**n + c*x**(S(2)*n))**(p + S(1)), x), x) - Simp((d*x)**(m + S(1))*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*a*d*n*(S(2)*p + S(1))), x) - Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(m + n*(S(2)*p + S(1)) + S(1))/(S(2)*a*d*n**S(2)*(p + S(1))*(S(2)*p + S(1))), x)
rule1125 = ReplacementRule(pattern1125, replacement1125)
pattern1126 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons680, cons45, cons148, cons31, cons530, cons510, cons691)
def replacement1126(p, m, b, n2, d, c, a, n, x):
rubi.append(1126)
return -Dist(b*d**n*(m - n + S(1))/(S(2)*c*(m + S(2)*n*p + S(1))), Int((d*x)**(m - n)*(a + b*x**n + c*x**(S(2)*n))**p, x), x) + Simp(d**(n + S(-1))*(d*x)**(m - n + S(1))*(b + S(2)*c*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(S(2)*c*(m + S(2)*n*p + S(1))), x)
rule1126 = ReplacementRule(pattern1126, replacement1126)
pattern1127 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons680, cons45, cons148, cons31, cons94, cons691)
def replacement1127(p, m, b, n2, d, c, a, n, x):
rubi.append(1127)
return -Dist(S(2)*c*d**(-n)*(m + n*(S(2)*p + S(1)) + S(1))/(b*(m + S(1))), Int((d*x)**(m + n)*(a + b*x**n + c*x**(S(2)*n))**p, x), x) + Simp((d*x)**(m + S(1))*(b + S(2)*c*x**n)*(a + b*x**n + c*x**(S(2)*n))**p/(b*d*(m + S(1))), x)
rule1127 = ReplacementRule(pattern1127, replacement1127)
pattern1128 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons680, cons45, cons196, cons17)
def replacement1128(p, m, b, n2, c, a, n, x):
rubi.append(1128)
return -Subst(Int(x**(-m + S(-2))*(a + b*x**(-n) + c*x**(-S(2)*n))**p, x), x, S(1)/x)
rule1128 = ReplacementRule(pattern1128, replacement1128)
def With1129(p, m, b, n2, d, c, a, n, x):
k = Denominator(m)
rubi.append(1129)
return -Dist(k/d, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*d**(-n)*x**(-k*n) + c*d**(-S(2)*n)*x**(-S(2)*k*n))**p, x), x, (d*x)**(-S(1)/k)), x)
pattern1129 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons680, cons45, cons196, cons367)
rule1129 = ReplacementRule(pattern1129, With1129)
pattern1130 = Pattern(Integral((x_*WC('d', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons680, cons45, cons196, cons356)
def replacement1130(p, m, b, n2, d, c, a, n, x):
rubi.append(1130)
return -Dist(d**IntPart(m)*(d*x)**FracPart(m)*(S(1)/x)**FracPart(m), Subst(Int(x**(-m + S(-2))*(a + b*x**(-n) + c*x**(-S(2)*n))**p, x), x, S(1)/x), x)
rule1130 = ReplacementRule(pattern1130, replacement1130)
pattern1131 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons680, cons45, cons147)
def replacement1131(p, m, b, n2, d, c, a, n, x):
rubi.append(1131)
return Dist(c**(-IntPart(p))*(b/S(2) + c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((d*x)**m*(b/S(2) + c*x**n)**(S(2)*p), x), x)
rule1131 = ReplacementRule(pattern1131, replacement1131)
pattern1132 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons680, cons226, cons500)
def replacement1132(p, m, b, n2, c, a, n, x):
rubi.append(1132)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x + c*x**S(2))**p, x), x, x**n), x)
rule1132 = ReplacementRule(pattern1132, replacement1132)
pattern1133 = Pattern(Integral((d_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons680, cons226, cons500)
def replacement1133(p, m, b, n2, d, c, a, n, x):
rubi.append(1133)
return Dist(d**IntPart(m)*x**(-FracPart(m))*(d*x)**FracPart(m), Int(x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1133 = ReplacementRule(pattern1133, replacement1133)
def With1134(p, m, b, n2, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
pattern1134 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons680, cons226, cons148, cons17, CustomConstraint(With1134))
def replacement1134(p, m, b, n2, c, a, n, x):
k = GCD(m + S(1), n)
rubi.append(1134)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a + b*x**(n/k) + c*x**(S(2)*n/k))**p, x), x, x**k), x)
rule1134 = ReplacementRule(pattern1134, replacement1134)
def With1135(p, m, b, n2, d, c, a, n, x):
k = Denominator(m)
rubi.append(1135)
return Dist(k/d, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*d**(-n)*x**(k*n) + c*d**(-S(2)*n)*x**(S(2)*k*n))**p, x), x, (d*x)**(S(1)/k)), x)
pattern1135 = Pattern(Integral((x_*WC('d', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons680, cons226, cons148, cons367, cons38)
rule1135 = ReplacementRule(pattern1135, With1135)
pattern1136 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons680, cons226, cons148, cons244, cons163, cons530, cons692, cons693, cons694)
def replacement1136(p, m, b, n2, d, c, a, n, x):
rubi.append(1136)
return -Dist(d**n*n*p/(c*(m + S(2)*n*p + S(1))*(m + n*(S(2)*p + S(-1)) + S(1))), Int((d*x)**(m - n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))*Simp(a*b*(m - n + S(1)) - x**n*(S(2)*a*c*(m + n*(S(2)*p + S(-1)) + S(1)) - b**S(2)*(m + n*(p + S(-1)) + S(1))), x), x), x) + Simp(d**(n + S(-1))*(d*x)**(m - n + S(1))*(b*n*p + c*x**n*(m + n*(S(2)*p + S(-1)) + S(1)))*(a + b*x**n + c*x**(S(2)*n))**p/(c*(m + S(2)*n*p + S(1))*(m + n*(S(2)*p + S(-1)) + S(1))), x)
rule1136 = ReplacementRule(pattern1136, replacement1136)
pattern1137 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons680, cons226, cons148, cons244, cons163, cons94, cons694)
def replacement1137(p, m, b, n2, d, c, a, n, x):
rubi.append(1137)
return -Dist(d**(-n)*n*p/(m + S(1)), Int((d*x)**(m + n)*(b + S(2)*c*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) + Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**p/(d*(m + S(1))), x)
rule1137 = ReplacementRule(pattern1137, replacement1137)
pattern1138 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons680, cons226, cons148, cons13, cons163, cons510, cons694)
def replacement1138(p, m, b, n2, d, c, a, n, x):
rubi.append(1138)
return Dist(n*p/(m + S(2)*n*p + S(1)), Int((d*x)**m*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) + Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**p/(d*(m + S(2)*n*p + S(1))), x)
rule1138 = ReplacementRule(pattern1138, replacement1138)
pattern1139 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons680, cons226, cons148, cons244, cons137, cons690, cons694)
def replacement1139(p, m, b, n2, d, c, a, n, x):
rubi.append(1139)
return -Dist(d**n/(n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((d*x)**(m - n)*(b*(m - n + S(1)) + S(2)*c*x**n*(m + S(2)*n*(p + S(1)) + S(1)))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1)), x), x) + Simp(d**(n + S(-1))*(d*x)**(m - n + S(1))*(b + S(2)*c*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1139 = ReplacementRule(pattern1139, replacement1139)
pattern1140 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons680, cons226, cons148, cons244, cons137, cons529, cons694)
def replacement1140(p, m, b, n2, d, c, a, n, x):
rubi.append(1140)
return Dist(d**(S(2)*n)/(n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((d*x)**(m - S(2)*n)*(S(2)*a*(m - S(2)*n + S(1)) + b*x**n*(m + n*(S(2)*p + S(1)) + S(1)))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1)), x), x) - Simp(d**(S(2)*n + S(-1))*(d*x)**(m - S(2)*n + S(1))*(S(2)*a + b*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1140 = ReplacementRule(pattern1140, replacement1140)
pattern1141 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons680, cons226, cons148, cons13, cons137, cons694)
def replacement1141(p, m, b, n2, d, c, a, n, x):
rubi.append(1141)
return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((d*x)**m*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(-S(2)*a*c*(m + S(2)*n*(p + S(1)) + S(1)) + b**S(2)*(m + n*(p + S(1)) + S(1)) + b*c*x**n*(m + S(2)*n*p + S(3)*n + S(1)), x), x), x) - Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c + b**S(2) + b*c*x**n)/(a*d*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1141 = ReplacementRule(pattern1141, replacement1141)
pattern1142 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons680, cons226, cons148, cons31, cons529, cons510, cons694)
def replacement1142(p, m, b, n2, d, c, a, n, x):
rubi.append(1142)
return -Dist(d**(S(2)*n)/(c*(m + S(2)*n*p + S(1))), Int((d*x)**(m - S(2)*n)*(a + b*x**n + c*x**(S(2)*n))**p*Simp(a*(m - S(2)*n + S(1)) + b*x**n*(m + n*(p + S(-1)) + S(1)), x), x), x) + Simp(d**(S(2)*n + S(-1))*(d*x)**(m - S(2)*n + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(c*(m + S(2)*n*p + S(1))), x)
rule1142 = ReplacementRule(pattern1142, replacement1142)
pattern1143 = Pattern(Integral((x_*WC('d', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons680, cons226, cons148, cons31, cons94, cons694)
def replacement1143(p, m, b, n2, d, c, a, n, x):
rubi.append(1143)
return -Dist(d**(-n)/(a*(m + S(1))), Int((d*x)**(m + n)*(b*(m + n*(p + S(1)) + S(1)) + c*x**n*(m + S(2)*n*(p + S(1)) + S(1)))*(a + b*x**n + c*x**(S(2)*n))**p, x), x) + Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(a*d*(m + S(1))), x)
rule1143 = ReplacementRule(pattern1143, replacement1143)
pattern1144 = Pattern(Integral((x_*WC('d', S(1)))**m_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons680, cons226, cons148, cons31, cons94)
def replacement1144(m, b, n2, d, c, a, n, x):
rubi.append(1144)
return -Dist(d**(-n)/a, Int((d*x)**(m + n)*(b + c*x**n)/(a + b*x**n + c*x**(S(2)*n)), x), x) + Simp((d*x)**(m + S(1))/(a*d*(m + S(1))), x)
rule1144 = ReplacementRule(pattern1144, replacement1144)
pattern1145 = Pattern(Integral(x_**m_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons680, cons226, cons148, cons17, cons695)
def replacement1145(m, b, n2, c, a, n, x):
rubi.append(1145)
return Int(PolynomialDivide(x**m, a + b*x**n + c*x**(S(2)*n), x), x)
rule1145 = ReplacementRule(pattern1145, replacement1145)
pattern1146 = Pattern(Integral((x_*WC('d', S(1)))**m_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons680, cons226, cons148, cons31, cons529)
def replacement1146(m, b, n2, d, c, a, n, x):
rubi.append(1146)
return -Dist(d**(S(2)*n)/c, Int((d*x)**(m - S(2)*n)*(a + b*x**n)/(a + b*x**n + c*x**(S(2)*n)), x), x) + Simp(d**(S(2)*n + S(-1))*(d*x)**(m - S(2)*n + S(1))/(c*(m - S(2)*n + S(1))), x)
rule1146 = ReplacementRule(pattern1146, replacement1146)
def With1147(x, c, b, a):
q = Rt(a/c, S(2))
rubi.append(1147)
return -Dist(S(1)/2, Int((q - x**S(2))/(a + b*x**S(2) + c*x**S(4)), x), x) + Dist(S(1)/2, Int((q + x**S(2))/(a + b*x**S(2) + c*x**S(4)), x), x)
pattern1147 = Pattern(Integral(x_**S(2)/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons696, cons697)
rule1147 = ReplacementRule(pattern1147, With1147)
def With1148(m, b, n2, c, a, n, x):
q = Rt(a/c, S(2))
r = Rt(-b/c + S(2)*q, S(2))
rubi.append(1148)
return -Dist(1/(2*c*r), Int(x**(m - 3*n/2)*(q - r*x**(n/2))/(q - r*x**(n/2) + x**n), x), x) + Dist(1/(2*c*r), Int(x**(m - 3*n/2)*(q + r*x**(n/2))/(q + r*x**(n/2) + x**n), x), x)
pattern1148 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons46, cons226, cons698, cons699, cons413)
rule1148 = ReplacementRule(pattern1148, With1148)
def With1149(m, b, n2, c, a, n, x):
q = Rt(a/c, S(2))
r = Rt(-b/c + S(2)*q, S(2))
rubi.append(1149)
return Dist(1/(2*c*r), Int(x**(m - n/2)/(q - r*x**(n/2) + x**n), x), x) - Dist(1/(2*c*r), Int(x**(m - n/2)/(q + r*x**(n/2) + x**n), x), x)
pattern1149 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons46, cons226, cons698, cons700, cons413)
rule1149 = ReplacementRule(pattern1149, With1149)
def With1150(m, b, n2, d, c, a, n, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1150)
return -Dist(d**n*(b/q + S(-1))/S(2), Int((d*x)**(m - n)/(b/S(2) + c*x**n - q/S(2)), x), x) + Dist(d**n*(b/q + S(1))/S(2), Int((d*x)**(m - n)/(b/S(2) + c*x**n + q/S(2)), x), x)
pattern1150 = Pattern(Integral((x_*WC('d', S(1)))**m_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons680, cons226, cons148, cons31, cons701)
rule1150 = ReplacementRule(pattern1150, With1150)
def With1151(m, b, n2, d, c, a, n, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1151)
return Dist(c/q, Int((d*x)**m/(b/S(2) + c*x**n - q/S(2)), x), x) - Dist(c/q, Int((d*x)**m/(b/S(2) + c*x**n + q/S(2)), x), x)
pattern1151 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons680, cons226, cons148)
rule1151 = ReplacementRule(pattern1151, With1151)
def With1152(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1152)
return Dist(S(2)*sqrt(-c), Int(x**S(2)/(sqrt(-b - S(2)*c*x**S(2) + q)*sqrt(b + S(2)*c*x**S(2) + q)), x), x)
pattern1152 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, cons293)
rule1152 = ReplacementRule(pattern1152, With1152)
def With1153(x, c, b, a):
q = Rt(c/a, S(2))
rubi.append(1153)
return -Dist(S(1)/q, Int((-q*x**S(2) + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist(S(1)/q, Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
pattern1153 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, cons676, cons677)
rule1153 = ReplacementRule(pattern1153, With1153)
def With1154(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1154)
return Dist(S(1)/(S(2)*c), Int((b + S(2)*c*x**S(2) - q)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) - Dist((b - q)/(S(2)*c), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
pattern1154 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, cons484, cons177)
rule1154 = ReplacementRule(pattern1154, With1154)
def With1155(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if And(PosQ((b + q)/a), Not(And(PosQ((b - q)/a), SimplerSqrtQ((b - q)/(S(2)*a), (b + q)/(S(2)*a))))):
return True
return False
pattern1155 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, CustomConstraint(With1155))
def replacement1155(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1155)
return Simp(x*(b + S(2)*c*x**S(2) + q)/(S(2)*c*sqrt(a + b*x**S(2) + c*x**S(4))), x) - Simp(sqrt((S(2)*a + x**S(2)*(b - q))/(S(2)*a + x**S(2)*(b + q)))*(S(2)*a + x**S(2)*(b + q))*EllipticE(ArcTan(x*Rt((b + q)/(S(2)*a), S(2))), S(2)*q/(b + q))*Rt((b + q)/(S(2)*a), S(2))/(S(2)*c*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1155 = ReplacementRule(pattern1155, replacement1155)
def With1156(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if PosQ((b - q)/a):
return True
return False
pattern1156 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, CustomConstraint(With1156))
def replacement1156(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1156)
return Simp(x*(b + S(2)*c*x**S(2) - q)/(S(2)*c*sqrt(a + b*x**S(2) + c*x**S(4))), x) - Simp(sqrt((S(2)*a + x**S(2)*(b + q))/(S(2)*a + x**S(2)*(b - q)))*(S(2)*a + x**S(2)*(b - q))*EllipticE(ArcTan(x*Rt((b - q)/(S(2)*a), S(2))), -S(2)*q/(b - q))*Rt((b - q)/(S(2)*a), S(2))/(S(2)*c*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1156 = ReplacementRule(pattern1156, replacement1156)
def With1157(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if And(NegQ((b + q)/a), Not(And(NegQ((b - q)/a), SimplerSqrtQ(-(b - q)/(S(2)*a), -(b + q)/(S(2)*a))))):
return True
return False
pattern1157 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, CustomConstraint(With1157))
def replacement1157(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1157)
return Dist(S(1)/(S(2)*c), Int((b + S(2)*c*x**S(2) + q)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) - Dist((b + q)/(S(2)*c), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
rule1157 = ReplacementRule(pattern1157, replacement1157)
def With1158(x, c, b, a):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if NegQ((b - q)/a):
return True
return False
pattern1158 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons675, CustomConstraint(With1158))
def replacement1158(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1158)
return Dist(S(1)/(S(2)*c), Int((b + S(2)*c*x**S(2) - q)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) - Dist((b - q)/(S(2)*c), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
rule1158 = ReplacementRule(pattern1158, replacement1158)
def With1159(x, c, b, a):
q = Rt(c/a, S(2))
rubi.append(1159)
return -Dist(S(1)/q, Int((-q*x**S(2) + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist(S(1)/q, Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
pattern1159 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons226, cons678)
rule1159 = ReplacementRule(pattern1159, With1159)
def With1160(x, c, b, a):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1160)
return Dist(sqrt(S(2)*c*x**S(2)/(b - q) + S(1))*sqrt(S(2)*c*x**S(2)/(b + q) + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), Int(x**S(2)/(sqrt(S(2)*c*x**S(2)/(b - q) + S(1))*sqrt(S(2)*c*x**S(2)/(b + q) + S(1))), x), x)
pattern1160 = Pattern(Integral(x_**S(2)/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons226, cons679)
rule1160 = ReplacementRule(pattern1160, With1160)
pattern1161 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons680, cons226, cons196, cons17)
def replacement1161(p, m, b, n2, c, a, n, x):
rubi.append(1161)
return -Subst(Int(x**(-m + S(-2))*(a + b*x**(-n) + c*x**(-S(2)*n))**p, x), x, S(1)/x)
rule1161 = ReplacementRule(pattern1161, replacement1161)
def With1162(p, m, b, n2, d, c, a, n, x):
k = Denominator(m)
rubi.append(1162)
return -Dist(k/d, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*d**(-n)*x**(-k*n) + c*d**(-S(2)*n)*x**(-S(2)*k*n))**p, x), x, (d*x)**(-S(1)/k)), x)
pattern1162 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons680, cons226, cons196, cons367)
rule1162 = ReplacementRule(pattern1162, With1162)
pattern1163 = Pattern(Integral((x_*WC('d', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons680, cons226, cons196, cons356)
def replacement1163(p, m, b, n2, d, c, a, n, x):
rubi.append(1163)
return -Dist(d**IntPart(m)*(d*x)**FracPart(m)*(S(1)/x)**FracPart(m), Subst(Int(x**(-m + S(-2))*(a + b*x**(-n) + c*x**(-S(2)*n))**p, x), x, S(1)/x), x)
rule1163 = ReplacementRule(pattern1163, replacement1163)
def With1164(p, m, b, n2, c, a, n, x):
k = Denominator(n)
rubi.append(1164)
return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*x**(k*n) + c*x**(S(2)*k*n))**p, x), x, x**(S(1)/k)), x)
pattern1164 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons680, cons226, cons489)
rule1164 = ReplacementRule(pattern1164, With1164)
pattern1165 = Pattern(Integral((d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons680, cons226, cons489)
def replacement1165(p, m, b, n2, d, c, a, n, x):
rubi.append(1165)
return Dist(d**IntPart(m)*x**(-FracPart(m))*(d*x)**FracPart(m), Int(x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1165 = ReplacementRule(pattern1165, replacement1165)
pattern1166 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons680, cons226, cons541, cons23)
def replacement1166(p, m, b, n2, c, a, n, x):
rubi.append(1166)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))) + c*x**(S(2)*n/(m + S(1))))**p, x), x, x**(m + S(1))), x)
rule1166 = ReplacementRule(pattern1166, replacement1166)
pattern1167 = Pattern(Integral((d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons680, cons226, cons541, cons23)
def replacement1167(p, m, b, n2, d, c, a, n, x):
rubi.append(1167)
return Dist(d**IntPart(m)*x**(-FracPart(m))*(d*x)**FracPart(m), Int(x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1167 = ReplacementRule(pattern1167, replacement1167)
def With1168(m, b, n2, d, c, a, n, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1168)
return Dist(S(2)*c/q, Int((d*x)**m/(b + S(2)*c*x**n - q), x), x) - Dist(S(2)*c/q, Int((d*x)**m/(b + S(2)*c*x**n + q), x), x)
pattern1168 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons680, cons226)
rule1168 = ReplacementRule(pattern1168, With1168)
pattern1169 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons680, cons226, cons702)
def replacement1169(p, m, b, n2, d, c, a, n, x):
rubi.append(1169)
return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((d*x)**m*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(-S(2)*a*c*(m + S(2)*n*(p + S(1)) + S(1)) + b**S(2)*(m + n*(p + S(1)) + S(1)) + b*c*x**n*(m + S(2)*n*p + S(3)*n + S(1)), x), x), x) - Simp((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c + b**S(2) + b*c*x**n)/(a*d*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1169 = ReplacementRule(pattern1169, replacement1169)
pattern1170 = Pattern(Integral((x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons680)
def replacement1170(p, m, b, n2, d, c, a, n, x):
rubi.append(1170)
return Dist(a**IntPart(p)*(S(2)*c*x**n/(b - Rt(-S(4)*a*c + b**S(2), S(2))) + S(1))**(-FracPart(p))*(S(2)*c*x**n/(b + Rt(-S(4)*a*c + b**S(2), S(2))) + S(1))**(-FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((d*x)**m*(S(2)*c*x**n/(b - sqrt(-S(4)*a*c + b**S(2))) + S(1))**p*(S(2)*c*x**n/(b + sqrt(-S(4)*a*c + b**S(2))) + S(1))**p, x), x)
rule1170 = ReplacementRule(pattern1170, replacement1170)
pattern1171 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons585, cons38, cons681)
def replacement1171(mn, p, m, b, c, n, a, x):
rubi.append(1171)
return Int(x**(m - n*p)*(a*x**n + b + c*x**(S(2)*n))**p, x)
rule1171 = ReplacementRule(pattern1171, replacement1171)
pattern1172 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons585, cons147, cons681)
def replacement1172(mn, p, m, b, c, n, a, x):
rubi.append(1172)
return Dist(x**(n*FracPart(p))*(a + b*x**(-n) + c*x**n)**FracPart(p)*(a*x**n + b + c*x**(S(2)*n))**(-FracPart(p)), Int(x**(m - n*p)*(a*x**n + b + c*x**(S(2)*n))**p, x), x)
rule1172 = ReplacementRule(pattern1172, replacement1172)
pattern1173 = Pattern(Integral((d_*x_)**WC('m', S(1))*(a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons585)
def replacement1173(mn, p, m, b, d, c, n, a, x):
rubi.append(1173)
return Dist(d**IntPart(m)*x**(-FracPart(m))*(d*x)**FracPart(m), Int(x**m*(a + b*x**(-n) + c*x**n)**p, x), x)
rule1173 = ReplacementRule(pattern1173, replacement1173)
pattern1174 = Pattern(Integral(x_**WC('m', S(1))*(v_**n_*WC('b', S(1)) + v_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons680, cons552, cons17, cons553)
def replacement1174(v, p, m, b, n2, a, c, n, x):
rubi.append(1174)
return Dist(Coefficient(v, x, S(1))**(-m + S(-1)), Subst(Int(SimplifyIntegrand((x - Coefficient(v, x, S(0)))**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x), x, v), x)
rule1174 = ReplacementRule(pattern1174, replacement1174)
pattern1175 = Pattern(Integral(u_**WC('m', S(1))*(v_**n_*WC('b', S(1)) + v_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons680, cons554)
def replacement1175(v, p, u, m, b, n2, c, a, n, x):
rubi.append(1175)
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x, v), x)
rule1175 = ReplacementRule(pattern1175, replacement1175)
pattern1176 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons220, cons502)
def replacement1176(p, b, n2, d, c, a, n, x, q, e):
rubi.append(1176)
return Int(x**(n*(S(2)*p + q))*(d*x**(-n) + e)**q*(a*x**(-S(2)*n) + b*x**(-n) + c)**p, x)
rule1176 = ReplacementRule(pattern1176, replacement1176)
pattern1177 = Pattern(Integral((a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons4, cons46, cons220, cons502)
def replacement1177(p, n2, d, c, a, n, x, q, e):
rubi.append(1177)
return Int(x**(n*(S(2)*p + q))*(a*x**(-S(2)*n) + c)**p*(d*x**(-n) + e)**q, x)
rule1177 = ReplacementRule(pattern1177, replacement1177)
pattern1178 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons46, cons196)
def replacement1178(p, b, n2, d, c, a, n, x, q, e):
rubi.append(1178)
return -Subst(Int((d + e*x**(-n))**q*(a + b*x**(-n) + c*x**(-S(2)*n))**p/x**S(2), x), x, S(1)/x)
rule1178 = ReplacementRule(pattern1178, replacement1178)
pattern1179 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons5, cons50, cons46, cons196)
def replacement1179(p, n2, d, c, n, a, x, q, e):
rubi.append(1179)
return -Subst(Int((a + c*x**(-S(2)*n))**p*(d + e*x**(-n))**q/x**S(2), x), x, S(1)/x)
rule1179 = ReplacementRule(pattern1179, replacement1179)
def With1180(p, b, n2, d, a, c, n, x, q, e):
g = Denominator(n)
rubi.append(1180)
return Dist(g, Subst(Int(x**(g + S(-1))*(d + e*x**(g*n))**q*(a + b*x**(g*n) + c*x**(S(2)*g*n))**p, x), x, x**(S(1)/g)), x)
pattern1180 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons46, cons489)
rule1180 = ReplacementRule(pattern1180, With1180)
def With1181(p, n2, d, c, a, n, x, q, e):
g = Denominator(n)
rubi.append(1181)
return Dist(g, Subst(Int(x**(g + S(-1))*(a + c*x**(S(2)*g*n))**p*(d + e*x**(g*n))**q, x), x, x**(S(1)/g)), x)
pattern1181 = Pattern(Integral((a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons5, cons50, cons46, cons489)
rule1181 = ReplacementRule(pattern1181, With1181)
pattern1182 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons3, cons7, cons27, cons48, cons4, cons5, cons46, cons147, cons664)
def replacement1182(p, b, n2, d, c, n, x, e):
rubi.append(1182)
return Dist(e/c, Int(x**(-n)*(b*x**n + c*x**(S(2)*n))**(p + S(1)), x), x) + Simp(x**(-S(2)*n*(p + S(1)))*(b*e - c*d)*(b*x**n + c*x**(S(2)*n))**(p + S(1))/(b*c*n*(p + S(1))), x)
rule1182 = ReplacementRule(pattern1182, replacement1182)
pattern1183 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons3, cons7, cons27, cons48, cons4, cons5, cons46, cons147, cons671, cons703)
def replacement1183(p, b, n2, d, c, n, x, e):
rubi.append(1183)
return Simp(e*x**(-n + S(1))*(b*x**n + c*x**(S(2)*n))**(p + S(1))/(c*(n*(S(2)*p + S(1)) + S(1))), x)
rule1183 = ReplacementRule(pattern1183, replacement1183)
pattern1184 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons3, cons7, cons27, cons48, cons4, cons5, cons46, cons147, cons671, cons704)
def replacement1184(p, b, n2, d, c, n, x, e):
rubi.append(1184)
return -Dist((b*e*(n*p + S(1)) - c*d*(n*(S(2)*p + S(1)) + S(1)))/(c*(n*(S(2)*p + S(1)) + S(1))), Int((b*x**n + c*x**(S(2)*n))**p, x), x) + Simp(e*x**(-n + S(1))*(b*x**n + c*x**(S(2)*n))**(p + S(1))/(c*(n*(S(2)*p + S(1)) + S(1))), x)
rule1184 = ReplacementRule(pattern1184, replacement1184)
pattern1185 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons3, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons147)
def replacement1185(p, b, n2, d, c, n, x, q, e):
rubi.append(1185)
return Dist(x**(-n*FracPart(p))*(b + c*x**n)**(-FracPart(p))*(b*x**n + c*x**(S(2)*n))**FracPart(p), Int(x**(n*p)*(b + c*x**n)**p*(d + e*x**n)**q, x), x)
rule1185 = ReplacementRule(pattern1185, replacement1185)
pattern1186 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons45, cons147)
def replacement1186(p, b, n2, d, c, n, a, x, q, e):
rubi.append(1186)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((b + S(2)*c*x**n)**(S(2)*p)*(d + e*x**n)**q, x), x)
rule1186 = ReplacementRule(pattern1186, replacement1186)
pattern1187 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons46, cons226, cons256, cons38)
def replacement1187(p, b, n2, d, c, n, a, x, q, e):
rubi.append(1187)
return Int((d + e*x**n)**(p + q)*(a/d + c*x**n/e)**p, x)
rule1187 = ReplacementRule(pattern1187, replacement1187)
pattern1188 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons4, cons50, cons46, cons257, cons38)
def replacement1188(p, n2, d, c, n, a, x, q, e):
rubi.append(1188)
return Int((d + e*x**n)**(p + q)*(a/d + c*x**n/e)**p, x)
rule1188 = ReplacementRule(pattern1188, replacement1188)
pattern1189 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons226, cons256, cons147)
def replacement1189(p, b, n2, d, c, n, a, x, q, e):
rubi.append(1189)
return Dist((d + e*x**n)**(-FracPart(p))*(a/d + c*x**n/e)**(-FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((d + e*x**n)**(p + q)*(a/d + c*x**n/e)**p, x), x)
rule1189 = ReplacementRule(pattern1189, replacement1189)
pattern1190 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons257, cons147)
def replacement1190(p, n2, d, c, n, a, x, q, e):
rubi.append(1190)
return Dist((a + c*x**(S(2)*n))**FracPart(p)*(d + e*x**n)**(-FracPart(p))*(a/d + c*x**n/e)**(-FracPart(p)), Int((d + e*x**n)**(p + q)*(a/d + c*x**n/e)**p, x), x)
rule1190 = ReplacementRule(pattern1190, replacement1190)
pattern1191 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226, cons279, cons221)
def replacement1191(b, n2, d, c, n, a, x, q, e):
rubi.append(1191)
return Int(ExpandIntegrand((d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1191 = ReplacementRule(pattern1191, replacement1191)
pattern1192 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons4, cons46, cons280, cons221)
def replacement1192(n2, d, c, n, a, x, q, e):
rubi.append(1192)
return Int(ExpandIntegrand((a + c*x**(S(2)*n))*(d + e*x**n)**q, x), x)
rule1192 = ReplacementRule(pattern1192, replacement1192)
pattern1193 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226, cons279, cons395, cons396)
def replacement1193(b, n2, d, c, n, a, x, q, e):
rubi.append(1193)
return Dist(S(1)/(d*e**S(2)*n*(q + S(1))), Int((d + e*x**n)**(q + S(1))*Simp(a*e**S(2)*(n*(q + S(1)) + S(1)) - b*d*e + c*d**S(2) + c*d*e*n*x**n*(q + S(1)), x), x), x) - Simp(x*(d + e*x**n)**(q + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))/(d*e**S(2)*n*(q + S(1))), x)
rule1193 = ReplacementRule(pattern1193, replacement1193)
pattern1194 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons4, cons46, cons280, cons395, cons396)
def replacement1194(n2, d, c, n, a, x, q, e):
rubi.append(1194)
return Dist(S(1)/(d*e**S(2)*n*(q + S(1))), Int((d + e*x**n)**(q + S(1))*Simp(a*e**S(2)*(n*(q + S(1)) + S(1)) + c*d**S(2) + c*d*e*n*x**n*(q + S(1)), x), x), x) - Simp(x*(d + e*x**n)**(q + S(1))*(a*e**S(2) + c*d**S(2))/(d*e**S(2)*n*(q + S(1))), x)
rule1194 = ReplacementRule(pattern1194, replacement1194)
pattern1195 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons46, cons226, cons279)
def replacement1195(b, n2, d, c, n, a, x, q, e):
rubi.append(1195)
return Dist(S(1)/(e*(n*(q + S(2)) + S(1))), Int((d + e*x**n)**q*(a*e*(n*(q + S(2)) + S(1)) - x**n*(-b*e*(n*(q + S(2)) + S(1)) + c*d*(n + S(1)))), x), x) + Simp(c*x**(n + S(1))*(d + e*x**n)**(q + S(1))/(e*(n*(q + S(2)) + S(1))), x)
rule1195 = ReplacementRule(pattern1195, replacement1195)
pattern1196 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons4, cons50, cons46, cons280)
def replacement1196(n2, d, c, n, a, x, q, e):
rubi.append(1196)
return Dist(S(1)/(e*(n*(q + S(2)) + S(1))), Int((d + e*x**n)**q*(a*e*(n*(q + S(2)) + S(1)) - c*d*x**n*(n + S(1))), x), x) + Simp(c*x**(n + S(1))*(d + e*x**n)**(q + S(1))/(e*(n*(q + S(2)) + S(1))), x)
rule1196 = ReplacementRule(pattern1196, replacement1196)
def With1197(n2, d, c, n, a, x, e):
q = Rt(S(2)*d*e, S(2))
rubi.append(1197)
return Dist(e**S(2)/(S(2)*c), Int(S(1)/(d + e*x**n - q*x**(n/S(2))), x), x) + Dist(e**S(2)/(S(2)*c), Int(S(1)/(d + e*x**n + q*x**(n/S(2))), x), x)
pattern1197 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons46, cons705, cons674, cons706)
rule1197 = ReplacementRule(pattern1197, With1197)
def With1198(n2, d, c, n, a, x, e):
q = Rt(-S(2)*d*e, S(2))
rubi.append(1198)
return Dist(d/(S(2)*a), Int((d - q*x**(n/S(2)))/(d - e*x**n - q*x**(n/S(2))), x), x) + Dist(d/(S(2)*a), Int((d + q*x**(n/S(2)))/(d - e*x**n + q*x**(n/S(2))), x), x)
pattern1198 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons46, cons705, cons674, cons707)
rule1198 = ReplacementRule(pattern1198, With1198)
def With1199(d, c, a, x, e):
q = Rt(a*c, S(2))
rubi.append(1199)
return Dist((-a*e + d*q)/(S(2)*a*c), Int((-c*x**S(2) + q)/(a + c*x**S(4)), x), x) + Dist((a*e + d*q)/(S(2)*a*c), Int((c*x**S(2) + q)/(a + c*x**S(4)), x), x)
pattern1199 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons708, cons697)
rule1199 = ReplacementRule(pattern1199, With1199)
def With1200(n2, d, c, n, a, x, e):
q = Rt(a/c, S(4))
rubi.append(1200)
return Dist(sqrt(S(2))/(S(4)*c*q**S(3)), Int((sqrt(S(2))*d*q - x**(n/S(2))*(d - e*q**S(2)))/(q**S(2) - sqrt(S(2))*q*x**(n/S(2)) + x**n), x), x) + Dist(sqrt(S(2))/(S(4)*c*q**S(3)), Int((sqrt(S(2))*d*q + x**(n/S(2))*(d - e*q**S(2)))/(q**S(2) + sqrt(S(2))*q*x**(n/S(2)) + x**n), x), x)
pattern1200 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons46, cons280, cons708, cons674, cons697)
rule1200 = ReplacementRule(pattern1200, With1200)
def With1201(d, c, a, x, e):
q = Rt(c/a, S(6))
rubi.append(1201)
return Dist(S(1)/(S(6)*a*q**S(2)), Int((S(2)*d*q**S(2) - x*(sqrt(S(3))*d*q**S(3) - e))/(q**S(2)*x**S(2) - sqrt(S(3))*q*x + S(1)), x), x) + Dist(S(1)/(S(6)*a*q**S(2)), Int((S(2)*d*q**S(2) + x*(sqrt(S(3))*d*q**S(3) + e))/(q**S(2)*x**S(2) + sqrt(S(3))*q*x + S(1)), x), x) + Dist(S(1)/(S(3)*a*q**S(2)), Int((d*q**S(2) - e*x)/(q**S(2)*x**S(2) + S(1)), x), x)
pattern1201 = Pattern(Integral((d_ + x_**S(3)*WC('e', S(1)))/(a_ + x_**S(6)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons678)
rule1201 = ReplacementRule(pattern1201, With1201)
def With1202(n2, d, c, n, a, x, e):
q = Rt(-a/c, S(2))
rubi.append(1202)
return Dist(d/S(2) - e*q/S(2), Int(S(1)/(a - c*q*x**n), x), x) + Dist(d/S(2) + e*q/S(2), Int(S(1)/(a + c*q*x**n), x), x)
pattern1202 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons4, cons46, cons280, cons709, cons85)
rule1202 = ReplacementRule(pattern1202, With1202)
pattern1203 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons4, cons46, cons280, cons710)
def replacement1203(n2, d, c, n, a, x, e):
rubi.append(1203)
return Dist(d, Int(S(1)/(a + c*x**(S(2)*n)), x), x) + Dist(e, Int(x**n/(a + c*x**(S(2)*n)), x), x)
rule1203 = ReplacementRule(pattern1203, replacement1203)
def With1204(b, n2, d, c, n, a, x, e):
q = Rt(-b/c + S(2)*d/e, S(2))
rubi.append(1204)
return Dist(e**S(2)/(S(2)*c), Int(S(1)/(d - e*q*x**(n/S(2)) + e*x**n), x), x) + Dist(e**S(2)/(S(2)*c), Int(S(1)/(d + e*q*x**(n/S(2)) + e*x**n), x), x)
pattern1204 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons226, cons705, cons674, cons711)
rule1204 = ReplacementRule(pattern1204, With1204)
def With1205(b, n2, d, c, n, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1205)
return Dist(e/S(2) - (-b*e + S(2)*c*d)/(S(2)*q), Int(S(1)/(b/S(2) + c*x**n + q/S(2)), x), x) + Dist(e/S(2) + (-b*e + S(2)*c*d)/(S(2)*q), Int(S(1)/(b/S(2) + c*x**n - q/S(2)), x), x)
pattern1205 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226, cons705, cons674, cons675)
rule1205 = ReplacementRule(pattern1205, With1205)
def With1206(b, n2, d, c, n, a, x, e):
q = Rt(a/c, S(2))
r = Rt(-b/c + S(2)*q, S(2))
rubi.append(1206)
return Dist(1/(2*c*q*r), Int((d*r - x**(n/2)*(d - e*q))/(q - r*x**(n/2) + x**n), x), x) + Dist(1/(2*c*q*r), Int((d*r + x**(n/2)*(d - e*q))/(q + r*x**(n/2) + x**n), x), x)
pattern1206 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons226, cons705, cons674, cons712)
rule1206 = ReplacementRule(pattern1206, With1206)
def With1207(b, n2, d, c, n, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1207)
return Dist(e/S(2) - (-b*e + S(2)*c*d)/(S(2)*q), Int(S(1)/(b/S(2) + c*x**n + q/S(2)), x), x) + Dist(e/S(2) + (-b*e + S(2)*c*d)/(S(2)*q), Int(S(1)/(b/S(2) + c*x**n - q/S(2)), x), x)
pattern1207 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226, cons279, cons713)
rule1207 = ReplacementRule(pattern1207, With1207)
def With1208(b, n2, d, c, n, a, x, e):
q = Rt(a/c, S(2))
r = Rt(-b/c + S(2)*q, S(2))
rubi.append(1208)
return Dist(1/(2*c*q*r), Int((d*r - x**(n/2)*(d - e*q))/(q - r*x**(n/2) + x**n), x), x) + Dist(1/(2*c*q*r), Int((d*r + x**(n/2)*(d - e*q))/(q + r*x**(n/2) + x**n), x), x)
pattern1208 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons226, cons279, cons674, cons413)
rule1208 = ReplacementRule(pattern1208, With1208)
pattern1209 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226, cons279, cons586)
def replacement1209(b, n2, d, c, n, a, x, q, e):
rubi.append(1209)
return Int(ExpandIntegrand((d + e*x**n)**q/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1209 = ReplacementRule(pattern1209, replacement1209)
pattern1210 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons4, cons46, cons280, cons586)
def replacement1210(n2, d, c, n, a, x, q, e):
rubi.append(1210)
return Int(ExpandIntegrand((d + e*x**n)**q/(a + c*x**(S(2)*n)), x), x)
rule1210 = ReplacementRule(pattern1210, replacement1210)
pattern1211 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226, cons279, cons386, cons395, cons396)
def replacement1211(b, n2, d, c, n, a, x, q, e):
rubi.append(1211)
return Dist(e**S(2)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((d + e*x**n)**q, x), x) + Dist(S(1)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((d + e*x**n)**(q + S(1))*(-b*e + c*d - c*e*x**n)/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1211 = ReplacementRule(pattern1211, replacement1211)
pattern1212 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons4, cons46, cons280, cons386, cons395, cons396)
def replacement1212(n2, d, c, n, a, x, q, e):
rubi.append(1212)
return Dist(c/(a*e**S(2) + c*d**S(2)), Int((d - e*x**n)*(d + e*x**n)**(q + S(1))/(a + c*x**(S(2)*n)), x), x) + Dist(e**S(2)/(a*e**S(2) + c*d**S(2)), Int((d + e*x**n)**q, x), x)
rule1212 = ReplacementRule(pattern1212, replacement1212)
def With1213(b, n2, d, c, n, a, x, q, e):
r = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1213)
return Dist(S(2)*c/r, Int((d + e*x**n)**q/(b + S(2)*c*x**n - r), x), x) - Dist(S(2)*c/r, Int((d + e*x**n)**q/(b + S(2)*c*x**n + r), x), x)
pattern1213 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons46, cons226, cons279, cons386)
rule1213 = ReplacementRule(pattern1213, With1213)
def With1214(n2, d, c, n, a, x, q, e):
r = Rt(-a*c, S(2))
rubi.append(1214)
return -Dist(c/(S(2)*r), Int((d + e*x**n)**q/(-c*x**n + r), x), x) - Dist(c/(S(2)*r), Int((d + e*x**n)**q/(c*x**n + r), x), x)
pattern1214 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons4, cons50, cons46, cons280, cons386)
rule1214 = ReplacementRule(pattern1214, With1214)
pattern1215 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226, cons149, cons163, cons669, cons714, cons246, cons673)
def replacement1215(p, b, n2, d, c, n, a, x, e):
rubi.append(1215)
return Dist(n*p/(c*(S(2)*n*p + S(1))*(S(2)*n*p + n + S(1))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(-1))*Simp(-a*b*e + S(2)*a*c*d*(S(2)*n*p + n + S(1)) + x**n*(S(2)*a*c*e*(S(2)*n*p + S(1)) - b**S(2)*e*(n*p + S(1)) + b*c*d*(S(2)*n*p + n + S(1))), x), x), x) + Simp(x*(a + b*x**n + c*x**(S(2)*n))**p*(b*e*n*p + c*d*(S(2)*n*p + n + S(1)) + c*e*x**n*(S(2)*n*p + S(1)))/(c*(S(2)*n*p + S(1))*(S(2)*n*p + n + S(1))), x)
rule1215 = ReplacementRule(pattern1215, replacement1215)
pattern1216 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons4, cons46, cons149, cons163, cons669, cons714, cons246, cons673)
def replacement1216(p, n2, d, c, n, a, x, e):
rubi.append(1216)
return Dist(S(2)*a*n*p/((S(2)*n*p + S(1))*(S(2)*n*p + n + S(1))), Int((a + c*x**(S(2)*n))**(p + S(-1))*(d*(S(2)*n*p + n + S(1)) + e*x**n*(S(2)*n*p + S(1))), x), x) + Simp(x*(a + c*x**(S(2)*n))**p*(d*(S(2)*n*p + n + S(1)) + e*x**n*(S(2)*n*p + S(1)))/((S(2)*n*p + S(1))*(S(2)*n*p + n + S(1))), x)
rule1216 = ReplacementRule(pattern1216, replacement1216)
pattern1217 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226, cons13, cons137, cons246, cons673)
def replacement1217(p, b, n2, d, c, n, a, x, e):
rubi.append(1217)
return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(-a*b*e - S(2)*a*c*d*(S(2)*n*p + S(2)*n + S(1)) + b**S(2)*d*(n*p + n + S(1)) + c*x**n*(-S(2)*a*e + b*d)*(S(2)*n*p + S(3)*n + S(1)), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*e - S(2)*a*c*d + b**S(2)*d + c*x**n*(-S(2)*a*e + b*d))/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1217 = ReplacementRule(pattern1217, replacement1217)
pattern1218 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons4, cons46, cons13, cons137, cons246, cons673)
def replacement1218(p, n2, d, c, n, a, x, e):
rubi.append(1218)
return Dist(S(1)/(S(2)*a*n*(p + S(1))), Int((a + c*x**(S(2)*n))**(p + S(1))*(d*(S(2)*n*p + S(2)*n + S(1)) + e*x**n*(S(2)*n*p + S(3)*n + S(1))), x), x) - Simp(x*(a + c*x**(S(2)*n))**(p + S(1))*(d + e*x**n)/(S(2)*a*n*(p + S(1))), x)
rule1218 = ReplacementRule(pattern1218, replacement1218)
def With1219(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1219)
return Dist(S(2)*sqrt(-c), Int((d + e*x**S(2))/(sqrt(-b - S(2)*c*x**S(2) + q)*sqrt(b + S(2)*c*x**S(2) + q)), x), x)
pattern1219 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, cons293)
rule1219 = ReplacementRule(pattern1219, With1219)
def With1220(d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1220)
return Dist(sqrt(-c), Int((d + e*x**S(2))/(sqrt(-c*x**S(2) + q)*sqrt(c*x**S(2) + q)), x), x)
pattern1220 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons43, cons293)
rule1220 = ReplacementRule(pattern1220, With1220)
def With1221(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(c/a, S(4))
if ZeroQ(d*q**S(2) + e):
return True
return False
pattern1221 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, cons676, cons677, CustomConstraint(With1221))
def replacement1221(b, d, c, a, x, e):
q = Rt(c/a, S(4))
rubi.append(1221)
return -Simp(d*x*sqrt(a + b*x**S(2) + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))), x) + Simp(d*sqrt((a + b*x**S(2) + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(q**S(2)*x**S(2) + S(1))*EllipticE(S(2)*ArcTan(q*x), -b*q**S(2)/(S(4)*c) + S(1)/2)/(q*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1221 = ReplacementRule(pattern1221, replacement1221)
def With1222(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(c/a, S(2))
if NonzeroQ(d*q + e):
return True
return False
pattern1222 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, cons676, cons677, CustomConstraint(With1222))
def replacement1222(b, d, c, a, x, e):
q = Rt(c/a, S(2))
rubi.append(1222)
return -Dist(e/q, Int((-q*x**S(2) + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((d*q + e)/q, Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
rule1222 = ReplacementRule(pattern1222, replacement1222)
def With1223(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if ZeroQ(S(2)*c*d - e*(b - q)):
return True
return False
pattern1223 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, cons484, cons177, CustomConstraint(With1223))
def replacement1223(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1223)
return Simp(e*x*(b + S(2)*c*x**S(2) + q)/(S(2)*c*sqrt(a + b*x**S(2) + c*x**S(4))), x) - Simp(e*q*sqrt((S(2)*a + x**S(2)*(b + q))/q)*sqrt((S(2)*a + x**S(2)*(b - q))/(S(2)*a + x**S(2)*(b + q)))*EllipticE(asin(sqrt(S(2))*x/sqrt((S(2)*a + x**S(2)*(b + q))/q)), (b + q)/(S(2)*q))/(S(2)*c*sqrt(a/(S(2)*a + x**S(2)*(b + q)))*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1223 = ReplacementRule(pattern1223, replacement1223)
def With1224(d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-a*c, S(2))
if And(ZeroQ(c*d + e*q), IntegerQ(q)):
return True
return False
pattern1224 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons484, cons177, CustomConstraint(With1224))
def replacement1224(d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1224)
return Simp(e*x*(c*x**S(2) + q)/(c*sqrt(a + c*x**S(4))), x) - Simp(sqrt(S(2))*e*q*sqrt((a + q*x**S(2))/q)*sqrt(-a + q*x**S(2))*EllipticE(asin(sqrt(S(2))*x/sqrt((a + q*x**S(2))/q)), S(1)/2)/(c*sqrt(-a)*sqrt(a + c*x**S(4))), x)
rule1224 = ReplacementRule(pattern1224, replacement1224)
def With1225(d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-a*c, S(2))
if ZeroQ(c*d + e*q):
return True
return False
pattern1225 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons484, cons177, CustomConstraint(With1225))
def replacement1225(d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1225)
return Simp(e*x*(c*x**S(2) + q)/(c*sqrt(a + c*x**S(4))), x) - Simp(sqrt(S(2))*e*q*sqrt((a + q*x**S(2))/q)*sqrt((a - q*x**S(2))/(a + q*x**S(2)))*EllipticE(asin(sqrt(S(2))*x/sqrt((a + q*x**S(2))/q)), S(1)/2)/(c*sqrt(a/(a + q*x**S(2)))*sqrt(a + c*x**S(4))), x)
rule1225 = ReplacementRule(pattern1225, replacement1225)
def With1226(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if NonzeroQ(S(2)*c*d - e*(b - q)):
return True
return False
pattern1226 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, cons484, cons177, CustomConstraint(With1226))
def replacement1226(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1226)
return Dist(e/(S(2)*c), Int((b + S(2)*c*x**S(2) - q)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((S(2)*c*d - e*(b - q))/(S(2)*c), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
rule1226 = ReplacementRule(pattern1226, replacement1226)
def With1227(d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-a*c, S(2))
if NonzeroQ(c*d + e*q):
return True
return False
pattern1227 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons484, cons177, CustomConstraint(With1227))
def replacement1227(d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1227)
return -Dist(e/c, Int((-c*x**S(2) + q)/sqrt(a + c*x**S(4)), x), x) + Dist((c*d + e*q)/c, Int(S(1)/sqrt(a + c*x**S(4)), x), x)
rule1227 = ReplacementRule(pattern1227, replacement1227)
def With1228(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if Or(PosQ((b + q)/a), PosQ((b - q)/a)):
return True
return False
pattern1228 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, CustomConstraint(With1228))
def replacement1228(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1228)
return Dist(d, Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist(e, Int(x**S(2)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
rule1228 = ReplacementRule(pattern1228, replacement1228)
pattern1229 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons715)
def replacement1229(d, c, a, x, e):
rubi.append(1229)
return Dist(d, Int(S(1)/sqrt(a + c*x**S(4)), x), x) + Dist(e, Int(x**S(2)/sqrt(a + c*x**S(4)), x), x)
rule1229 = ReplacementRule(pattern1229, replacement1229)
def With1230(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if And(NegQ((b + q)/a), ZeroQ(S(2)*c*d - e*(b + q)), Not(SimplerSqrtQ(-(b - q)/(S(2)*a), -(b + q)/(S(2)*a)))):
return True
return False
pattern1230 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, CustomConstraint(With1230))
def replacement1230(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1230)
return -Simp(a*e*sqrt(S(1) + x**S(2)*(b - q)/(S(2)*a))*sqrt(S(1) + x**S(2)*(b + q)/(S(2)*a))*EllipticE(asin(x*Rt(-(b + q)/(S(2)*a), S(2))), (b - q)/(b + q))*Rt(-(b + q)/(S(2)*a), S(2))/(c*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1230 = ReplacementRule(pattern1230, replacement1230)
def With1231(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if And(NegQ((b + q)/a), NonzeroQ(S(2)*c*d - e*(b + q)), Not(SimplerSqrtQ(-(b - q)/(S(2)*a), -(b + q)/(S(2)*a)))):
return True
return False
pattern1231 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, CustomConstraint(With1231))
def replacement1231(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1231)
return Dist(e/(S(2)*c), Int((b + S(2)*c*x**S(2) + q)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((S(2)*c*d - e*(b + q))/(S(2)*c), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
rule1231 = ReplacementRule(pattern1231, replacement1231)
def With1232(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if And(NegQ((b - q)/a), ZeroQ(S(2)*c*d - e*(b - q))):
return True
return False
pattern1232 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, CustomConstraint(With1232))
def replacement1232(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1232)
return -Simp(a*e*sqrt(S(1) + x**S(2)*(b - q)/(S(2)*a))*sqrt(S(1) + x**S(2)*(b + q)/(S(2)*a))*EllipticE(asin(x*Rt(-(b - q)/(S(2)*a), S(2))), (b + q)/(b - q))*Rt(-(b - q)/(S(2)*a), S(2))/(c*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1232 = ReplacementRule(pattern1232, replacement1232)
def With1233(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if And(NegQ((b - q)/a), NonzeroQ(S(2)*c*d - e*(b - q))):
return True
return False
pattern1233 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons675, CustomConstraint(With1233))
def replacement1233(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1233)
return Dist(e/(S(2)*c), Int((b + S(2)*c*x**S(2) - q)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((S(2)*c*d - e*(b - q))/(S(2)*c), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
rule1233 = ReplacementRule(pattern1233, replacement1233)
def With1234(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(c/a, S(4))
if ZeroQ(d*q**S(2) + e):
return True
return False
pattern1234 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons678, CustomConstraint(With1234))
def replacement1234(b, d, c, a, x, e):
q = Rt(c/a, S(4))
rubi.append(1234)
return -Simp(d*x*sqrt(a + b*x**S(2) + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))), x) + Simp(d*sqrt((a + b*x**S(2) + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(q**S(2)*x**S(2) + S(1))*EllipticE(S(2)*ArcTan(q*x), -b*q**S(2)/(S(4)*c) + S(1)/2)/(q*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1234 = ReplacementRule(pattern1234, replacement1234)
def With1235(d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(c/a, S(4))
if ZeroQ(d*q**S(2) + e):
return True
return False
pattern1235 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons678, CustomConstraint(With1235))
def replacement1235(d, c, a, x, e):
q = Rt(c/a, S(4))
rubi.append(1235)
return -Simp(d*x*sqrt(a + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))), x) + Simp(d*sqrt((a + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(q**S(2)*x**S(2) + S(1))*EllipticE(S(2)*ArcTan(q*x), S(1)/2)/(q*sqrt(a + c*x**S(4))), x)
rule1235 = ReplacementRule(pattern1235, replacement1235)
def With1236(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(c/a, S(2))
if NonzeroQ(d*q + e):
return True
return False
pattern1236 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons678, CustomConstraint(With1236))
def replacement1236(b, d, c, a, x, e):
q = Rt(c/a, S(2))
rubi.append(1236)
return -Dist(e/q, Int((-q*x**S(2) + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((d*q + e)/q, Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x)
rule1236 = ReplacementRule(pattern1236, replacement1236)
def With1237(d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(c/a, S(2))
if NonzeroQ(d*q + e):
return True
return False
pattern1237 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons678, CustomConstraint(With1237))
def replacement1237(d, c, a, x, e):
q = Rt(c/a, S(2))
rubi.append(1237)
return -Dist(e/q, Int((-q*x**S(2) + S(1))/sqrt(a + c*x**S(4)), x), x) + Dist((d*q + e)/q, Int(S(1)/sqrt(a + c*x**S(4)), x), x)
rule1237 = ReplacementRule(pattern1237, replacement1237)
pattern1238 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons679, cons257, cons43)
def replacement1238(d, c, a, x, e):
rubi.append(1238)
return Dist(d/sqrt(a), Int(sqrt(S(1) + e*x**S(2)/d)/sqrt(S(1) - e*x**S(2)/d), x), x)
rule1238 = ReplacementRule(pattern1238, replacement1238)
pattern1239 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons679, cons257, cons448)
def replacement1239(d, c, a, x, e):
rubi.append(1239)
return Dist(sqrt(S(1) + c*x**S(4)/a)/sqrt(a + c*x**S(4)), Int((d + e*x**S(2))/sqrt(S(1) + c*x**S(4)/a), x), x)
rule1239 = ReplacementRule(pattern1239, replacement1239)
def With1240(d, c, a, x, e):
q = Rt(-c/a, S(2))
rubi.append(1240)
return Dist(e/q, Int((q*x**S(2) + S(1))/sqrt(a + c*x**S(4)), x), x) + Dist((d*q - e)/q, Int(S(1)/sqrt(a + c*x**S(4)), x), x)
pattern1240 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons679, cons280)
rule1240 = ReplacementRule(pattern1240, With1240)
def With1241(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1241)
return Dist(sqrt(S(2)*c*x**S(2)/(b - q) + S(1))*sqrt(S(2)*c*x**S(2)/(b + q) + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), Int((d + e*x**S(2))/(sqrt(S(2)*c*x**S(2)/(b - q) + S(1))*sqrt(S(2)*c*x**S(2)/(b + q) + S(1))), x), x)
pattern1241 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))/sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons679)
rule1241 = ReplacementRule(pattern1241, With1241)
pattern1242 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226)
def replacement1242(p, b, n2, d, c, n, a, x, e):
rubi.append(1242)
return Int(ExpandIntegrand((d + e*x**n)*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1242 = ReplacementRule(pattern1242, replacement1242)
pattern1243 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons4, cons46)
def replacement1243(p, n2, d, c, n, a, x, e):
rubi.append(1243)
return Int(ExpandIntegrand((a + c*x**(S(2)*n))**p*(d + e*x**n), x), x)
rule1243 = ReplacementRule(pattern1243, replacement1243)
pattern1244 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons46, cons226, cons128, cons716, cons148, cons400)
def replacement1244(p, b, n2, d, c, n, a, x, q, e):
rubi.append(1244)
return Int((d + e*x**n)**q*ExpandToSum(-c**p*d*x**(S(2)*n*p - n)*(S(2)*n*p - n + S(1))/(e*(S(2)*n*p + n*q + S(1))) - c**p*x**(S(2)*n*p) + (a + b*x**n + c*x**(S(2)*n))**p, x), x) + Simp(c**p*x**(S(2)*n*p - n + S(1))*(d + e*x**n)**(q + S(1))/(e*(S(2)*n*p + n*q + S(1))), x)
rule1244 = ReplacementRule(pattern1244, replacement1244)
pattern1245 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons4, cons50, cons46, cons128, cons716, cons148, cons400)
def replacement1245(p, n2, d, c, n, a, x, q, e):
rubi.append(1245)
return Int((d + e*x**n)**q*ExpandToSum(-c**p*d*x**(S(2)*n*p - n)*(S(2)*n*p - n + S(1))/(e*(S(2)*n*p + n*q + S(1))) - c**p*x**(S(2)*n*p) + (a + c*x**(S(2)*n))**p, x), x) + Simp(c**p*x**(S(2)*n*p - n + S(1))*(d + e*x**n)**(q + S(1))/(e*(S(2)*n*p + n*q + S(1))), x)
rule1245 = ReplacementRule(pattern1245, replacement1245)
pattern1246 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279)
def replacement1246(b, d, c, a, x, e):
rubi.append(1246)
return -Dist(e**(S(-2)), Int((-b*e + c*d - c*e*x**S(2))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((a*e**S(2) - b*d*e + c*d**S(2))/e**S(2), Int(S(1)/((d + e*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x)
rule1246 = ReplacementRule(pattern1246, replacement1246)
pattern1247 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('c', S(1)))/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons280)
def replacement1247(d, c, a, x, e):
rubi.append(1247)
return -Dist(c/e**S(2), Int((d - e*x**S(2))/sqrt(a + c*x**S(4)), x), x) + Dist((a*e**S(2) + c*d**S(2))/e**S(2), Int(S(1)/(sqrt(a + c*x**S(4))*(d + e*x**S(2))), x), x)
rule1247 = ReplacementRule(pattern1247, replacement1247)
def With1248(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1248)
return -Dist(e**(S(-4)), Int(Simp(-c**S(2)*e**S(3)*x**S(6) + c*e**S(2)*x**S(4)*(-S(2)*b*e + c*d) - S(2)*c*(a*e**S(2) - b*d*e + c*d**S(2))**S(2)/(S(2)*c*d - e*(b + q)) - e*x**S(2)*(b**S(2)*e**S(2) + c**S(2)*d**S(2) - S(2)*c*e*(-a*e + b*d)) + (-b*e + c*d)*(S(2)*a*e**S(2) - b*d*e + c*d**S(2)), x)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) - Dist((a*e**S(2) - b*d*e + c*d**S(2))**S(2)/(e**S(3)*(S(2)*c*d - e*(b + q))), Int((b + S(2)*c*x**S(2) + q)/((d + e*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x)
pattern1248 = Pattern(Integral((a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons675)
rule1248 = ReplacementRule(pattern1248, With1248)
def With1249(d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1249)
return -Dist(c/e**S(4), Int(Simp(c*d*e**S(2)*x**S(4) - c*e**S(3)*x**S(6) + d*(S(2)*a*e**S(2) + c*d**S(2)) - e*x**S(2)*(S(2)*a*e**S(2) + c*d**S(2)) - (a*e**S(2) + c*d**S(2))**S(2)/(c*d - e*q), x)/sqrt(a + c*x**S(4)), x), x) - Dist((a*e**S(2) + c*d**S(2))**S(2)/(e**S(3)*(c*d - e*q)), Int((c*x**S(2) + q)/(sqrt(a + c*x**S(4))*(d + e*x**S(2))), x), x)
pattern1249 = Pattern(Integral((a_ + x_**S(4)*WC('c', S(1)))**(S(3)/2)/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons715)
rule1249 = ReplacementRule(pattern1249, With1249)
pattern1250 = Pattern(Integral((a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**p_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons717)
def replacement1250(p, b, d, c, a, x, e):
rubi.append(1250)
return Dist(a, Int((a + b*x**S(2) + c*x**S(4))**(p + S(-1))/(d + e*x**S(2)), x), x) + Dist(b, Int(x**S(2)*(a + b*x**S(2) + c*x**S(4))**(p + S(-1))/(d + e*x**S(2)), x), x) + Dist(c, Int(x**S(4)*(a + b*x**S(2) + c*x**S(4))**(p + S(-1))/(d + e*x**S(2)), x), x)
rule1250 = ReplacementRule(pattern1250, replacement1250)
pattern1251 = Pattern(Integral((a_ + x_**S(4)*WC('c', S(1)))**p_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons717)
def replacement1251(p, d, c, a, x, e):
rubi.append(1251)
return Dist(a, Int((a + c*x**S(4))**(p + S(-1))/(d + e*x**S(2)), x), x) + Dist(c, Int(x**S(4)*(a + c*x**S(4))**(p + S(-1))/(d + e*x**S(2)), x), x)
rule1251 = ReplacementRule(pattern1251, replacement1251)
def With1252(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1252)
return Dist(S(2)*sqrt(-c), Int(S(1)/((d + e*x**S(2))*sqrt(-b - S(2)*c*x**S(2) + q)*sqrt(b + S(2)*c*x**S(2) + q)), x), x)
pattern1252 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons675, cons293)
rule1252 = ReplacementRule(pattern1252, With1252)
def With1253(d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1253)
return Dist(sqrt(-c), Int(S(1)/((d + e*x**S(2))*sqrt(-c*x**S(2) + q)*sqrt(c*x**S(2) + q)), x), x)
pattern1253 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons43, cons293)
rule1253 = ReplacementRule(pattern1253, With1253)
def With1254(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1254)
return Dist(S(2)*c/(S(2)*c*d - e*(b - q)), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) - Dist(e/(S(2)*c*d - e*(b - q)), Int((b + S(2)*c*x**S(2) - q)/((d + e*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x)
pattern1254 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons675, cons718)
rule1254 = ReplacementRule(pattern1254, With1254)
def With1255(d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1255)
return Dist(c/(c*d + e*q), Int(S(1)/sqrt(a + c*x**S(4)), x), x) + Dist(e/(c*d + e*q), Int((-c*x**S(2) + q)/(sqrt(a + c*x**S(4))*(d + e*x**S(2))), x), x)
pattern1255 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons715, cons718)
rule1255 = ReplacementRule(pattern1255, With1255)
def With1256(b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(c/a, S(4))
if NonzeroQ(-d*q**S(2) + e):
return True
return False
pattern1256 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons678, CustomConstraint(With1256))
def replacement1256(b, d, c, a, x, e):
q = Rt(c/a, S(4))
rubi.append(1256)
return -Dist(q**S(2)/(-d*q**S(2) + e), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Simp(ArcTan(x*sqrt((a*e**S(2) - b*d*e + c*d**S(2))/(d*e))/sqrt(a + b*x**S(2) + c*x**S(4)))/(S(2)*d*sqrt((a*e**S(2) - b*d*e + c*d**S(2))/(d*e))), x) + Simp(sqrt((a + b*x**S(2) + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(d*q**S(2) + e)*(q**S(2)*x**S(2) + S(1))*EllipticPi(-(-d*q**S(2) + e)**S(2)/(S(4)*d*e*q**S(2)), S(2)*ArcTan(q*x), -b*q**S(2)/(S(4)*c) + S(1)/2)/(S(4)*d*q*(-d*q**S(2) + e)*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1256 = ReplacementRule(pattern1256, replacement1256)
def With1257(d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(c/a, S(4))
if NonzeroQ(-d*q**S(2) + e):
return True
return False
pattern1257 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons280, cons678, CustomConstraint(With1257))
def replacement1257(d, c, a, x, e):
q = Rt(c/a, S(4))
rubi.append(1257)
return -Dist(q**S(2)/(-d*q**S(2) + e), Int(S(1)/sqrt(a + c*x**S(4)), x), x) + Simp(ArcTan(x*sqrt((a*e**S(2) + c*d**S(2))/(d*e))/sqrt(a + c*x**S(4)))/(S(2)*d*sqrt((a*e**S(2) + c*d**S(2))/(d*e))), x) + Simp(sqrt((a + c*x**S(4))/(a*(q**S(2)*x**S(2) + S(1))**S(2)))*(d*q**S(2) + e)*(q**S(2)*x**S(2) + S(1))*EllipticPi(-(-d*q**S(2) + e)**S(2)/(S(4)*d*e*q**S(2)), S(2)*ArcTan(q*x), S(1)/2)/(S(4)*d*q*sqrt(a + c*x**S(4))*(-d*q**S(2) + e)), x)
rule1257 = ReplacementRule(pattern1257, replacement1257)
def With1258(d, c, a, x, e):
q = Rt(-c/a, S(4))
rubi.append(1258)
return Simp(EllipticPi(-e/(d*q**S(2)), asin(q*x), S(-1))/(sqrt(a)*d*q), x)
pattern1258 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons679, cons43)
rule1258 = ReplacementRule(pattern1258, With1258)
pattern1259 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons679, cons448)
def replacement1259(d, c, a, x, e):
rubi.append(1259)
return Dist(sqrt(S(1) + c*x**S(4)/a)/sqrt(a + c*x**S(4)), Int(S(1)/(sqrt(S(1) + c*x**S(4)/a)*(d + e*x**S(2))), x), x)
rule1259 = ReplacementRule(pattern1259, replacement1259)
def With1260(b, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1260)
return Dist(sqrt(S(2)*c*x**S(2)/(b - q) + S(1))*sqrt(S(2)*c*x**S(2)/(b + q) + S(1))/sqrt(a + b*x**S(2) + c*x**S(4)), Int(S(1)/((d + e*x**S(2))*sqrt(S(2)*c*x**S(2)/(b - q) + S(1))*sqrt(S(2)*c*x**S(2)/(b + q) + S(1))), x), x)
pattern1260 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons679)
rule1260 = ReplacementRule(pattern1260, With1260)
pattern1261 = Pattern(Integral((a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**p_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279, cons719)
def replacement1261(p, b, d, c, a, x, e):
rubi.append(1261)
return -Dist(S(1)/(S(2)*a*(p + S(1))*(-S(4)*a*c + b**S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), Int((a + b*x**S(2) + c*x**S(4))**(p + S(1))*Simp(-a*b*c*d*e*(S(8)*p + S(11)) + S(2)*a*c*(S(4)*a*e**S(2)*(p + S(1)) + c*d**S(2)*(S(4)*p + S(5))) + b**S(3)*d*e*(S(2)*p + S(3)) - b**S(2)*(S(2)*a*e**S(2)*(p + S(1)) + c*d**S(2)*(S(2)*p + S(3))) - c*e*x**S(4)*(S(4)*p + S(7))*(S(2)*a*c*e - b**S(2)*e + b*c*d) - x**S(2)*(S(4)*a*c**S(2)*d*e - b**S(3)*e**S(2)*(S(2)*p + S(3)) - S(2)*b**S(2)*c*d*e*(p + S(2)) + b*c*(a*e**S(2)*(S(8)*p + S(11)) + c*d**S(2)*(S(4)*p + S(7)))), x)/(d + e*x**S(2)), x), x) - Simp(x*(a + b*x**S(2) + c*x**S(4))**(p + S(1))*(S(3)*a*b*c*e - S(2)*a*c**S(2)*d - b**S(3)*e + b**S(2)*c*d + c*x**S(2)*(S(2)*a*c*e - b**S(2)*e + b*c*d))/(S(2)*a*(p + S(1))*(-S(4)*a*c + b**S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule1261 = ReplacementRule(pattern1261, replacement1261)
pattern1262 = Pattern(Integral((a_ + x_**S(4)*WC('c', S(1)))**p_/(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons280, cons719)
def replacement1262(p, d, c, a, x, e):
rubi.append(1262)
return -Dist(-S(1)/(S(8)*a**S(2)*c*(p + S(1))*(a*e**S(2) + c*d**S(2))), Int((a + c*x**S(4))**(p + S(1))*Simp(-S(4)*a*c**S(2)*d*e*x**S(2) - S(2)*a*c**S(2)*e**S(2)*x**S(4)*(S(4)*p + S(7)) + S(2)*a*c*(S(4)*a*e**S(2)*(p + S(1)) + c*d**S(2)*(S(4)*p + S(5))), x)/(d + e*x**S(2)), x), x) - Simp(-x*(a + c*x**S(4))**(p + S(1))*(-S(2)*a*c**S(2)*d + S(2)*a*c**S(2)*e*x**S(2))/(S(8)*a**S(2)*c*(p + S(1))*(a*e**S(2) + c*d**S(2))), x)
rule1262 = ReplacementRule(pattern1262, replacement1262)
pattern1263 = Pattern(Integral(S(1)/((d_ + x_**S(2)*WC('e', S(1)))**S(2)*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons226, cons279)
def replacement1263(b, d, c, a, x, e):
rubi.append(1263)
return -Dist(c/(S(2)*d*(a*e**S(2) - b*d*e + c*d**S(2))), Int((d + e*x**S(2))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((a*e**S(2) - S(2)*b*d*e + S(3)*c*d**S(2))/(S(2)*d*(a*e**S(2) - b*d*e + c*d**S(2))), Int(S(1)/((d + e*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x) + Simp(e**S(2)*x*sqrt(a + b*x**S(2) + c*x**S(4))/(S(2)*d*(d + e*x**S(2))*(a*e**S(2) - b*d*e + c*d**S(2))), x)
rule1263 = ReplacementRule(pattern1263, replacement1263)
pattern1264 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))**S(2)), x_), cons2, cons7, cons27, cons48, cons280)
def replacement1264(d, c, a, x, e):
rubi.append(1264)
return -Dist(c/(S(2)*d*(a*e**S(2) + c*d**S(2))), Int((d + e*x**S(2))/sqrt(a + c*x**S(4)), x), x) + Dist((a*e**S(2) + S(3)*c*d**S(2))/(S(2)*d*(a*e**S(2) + c*d**S(2))), Int(S(1)/(sqrt(a + c*x**S(4))*(d + e*x**S(2))), x), x) + Simp(e**S(2)*x*sqrt(a + c*x**S(4))/(S(2)*d*(d + e*x**S(2))*(a*e**S(2) + c*d**S(2))), x)
rule1264 = ReplacementRule(pattern1264, replacement1264)
def With1265(p, b, d, c, a, x, q, e):
r = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1265)
return Dist(a**IntPart(p)*(S(2)*c*x**S(2)/(b - r) + S(1))**(-FracPart(p))*(S(2)*c*x**S(2)/(b + r) + S(1))**(-FracPart(p))*(a + b*x**S(2) + c*x**S(4))**FracPart(p), Int((d + e*x**S(2))**q*(S(2)*c*x**S(2)/(b - r) + S(1))**p*(S(2)*c*x**S(2)/(b + r) + S(1))**p, x), x)
pattern1265 = Pattern(Integral((d_ + x_**S(2)*WC('e', S(1)))**q_*(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons50, cons226, cons279, cons347, cons564)
rule1265 = ReplacementRule(pattern1265, With1265)
def With1266(p, d, c, a, x, q, e):
r = Rt(-a*c, S(2))
rubi.append(1266)
return Dist(a**IntPart(p)*(a + c*x**S(4))**FracPart(p)*(-c*x**S(2)/r + S(1))**(-FracPart(p))*(c*x**S(2)/r + S(1))**(-FracPart(p)), Int((d + e*x**S(2))**q*(-c*x**S(2)/r + S(1))**p*(c*x**S(2)/r + S(1))**p, x), x)
pattern1266 = Pattern(Integral((a_ + x_**S(4)*WC('c', S(1)))**p_*(d_ + x_**S(2)*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons50, cons280, cons347, cons564)
rule1266 = ReplacementRule(pattern1266, With1266)
pattern1267 = Pattern(Integral(S(1)/(sqrt(d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons720, cons43, cons268)
def replacement1267(b, d, c, a, x, e):
rubi.append(1267)
return Simp(EllipticF(S(2)*asin(x*Rt(-e/d, S(2))), b*d/(S(4)*a*e))/(S(2)*sqrt(a)*sqrt(d)*Rt(-e/d, S(2))), x)
rule1267 = ReplacementRule(pattern1267, replacement1267)
pattern1268 = Pattern(Integral(S(1)/(sqrt(d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons720, cons721)
def replacement1268(b, d, c, a, x, e):
rubi.append(1268)
return Dist(sqrt((a + b*x**S(2) + c*x**S(4))/a)*sqrt((d + e*x**S(2))/d)/(sqrt(d + e*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), Int(S(1)/(sqrt(S(1) + e*x**S(2)/d)*sqrt(S(1) + b*x**S(2)/a + c*x**S(4)/a)), x), x)
rule1268 = ReplacementRule(pattern1268, replacement1268)
pattern1269 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons720, cons43, cons268)
def replacement1269(b, d, c, a, x, e):
rubi.append(1269)
return Simp(sqrt(a)*EllipticE(S(2)*asin(x*Rt(-e/d, S(2))), b*d/(S(4)*a*e))/(S(2)*sqrt(d)*Rt(-e/d, S(2))), x)
rule1269 = ReplacementRule(pattern1269, replacement1269)
pattern1270 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))/sqrt(d_ + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons720, cons721)
def replacement1270(b, d, c, a, x, e):
rubi.append(1270)
return Dist(sqrt((d + e*x**S(2))/d)*sqrt(a + b*x**S(2) + c*x**S(4))/(sqrt((a + b*x**S(2) + c*x**S(4))/a)*sqrt(d + e*x**S(2))), Int(sqrt(S(1) + b*x**S(2)/a + c*x**S(4)/a)/sqrt(S(1) + e*x**S(2)/d), x), x)
rule1270 = ReplacementRule(pattern1270, replacement1270)
pattern1271 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons226, cons279, cons722)
def replacement1271(p, b, n2, d, c, n, a, x, q, e):
rubi.append(1271)
return Int(ExpandIntegrand((d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1271 = ReplacementRule(pattern1271, replacement1271)
pattern1272 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons280, cons722)
def replacement1272(p, n2, d, c, n, a, x, q, e):
rubi.append(1272)
return Int(ExpandIntegrand((a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x)
rule1272 = ReplacementRule(pattern1272, replacement1272)
pattern1273 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons4, cons5, cons46, cons280, cons564, cons723)
def replacement1273(p, n2, d, c, n, a, x, q, e):
rubi.append(1273)
return Int(ExpandIntegrand((a + c*x**(S(2)*n))**p, (d/(d**S(2) - e**S(2)*x**(S(2)*n)) - e*x**n/(d**S(2) - e**S(2)*x**(S(2)*n)))**(-q), x), x)
rule1273 = ReplacementRule(pattern1273, replacement1273)
pattern1274 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons724)
def replacement1274(p, b, n2, d, c, n, a, x, q, e):
rubi.append(1274)
return Int((d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x)
rule1274 = ReplacementRule(pattern1274, replacement1274)
pattern1275 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons724)
def replacement1275(p, n2, d, c, n, a, x, q, e):
rubi.append(1275)
return Int((a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x)
rule1275 = ReplacementRule(pattern1275, replacement1275)
pattern1276 = Pattern(Integral((d_ + u_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + u_**n2_*WC('c', S(1)) + u_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons68, cons69)
def replacement1276(p, u, b, n2, d, c, n, a, x, q, e):
rubi.append(1276)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x, u), x)
rule1276 = ReplacementRule(pattern1276, replacement1276)
pattern1277 = Pattern(Integral((a_ + u_**n2_*WC('c', S(1)))**WC('p', S(1))*(d_ + u_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons4, cons5, cons50, cons46, cons68, cons69)
def replacement1277(p, u, n2, d, c, n, a, x, q, e):
rubi.append(1277)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x, u), x)
rule1277 = ReplacementRule(pattern1277, replacement1277)
pattern1278 = Pattern(Integral((d_ + x_**WC('mn', S(1))*WC('e', S(1)))**WC('q', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons680, cons585, cons586)
def replacement1278(mn, p, b, n2, d, a, n, c, x, q, e):
rubi.append(1278)
return Int(x**(-n*q)*(d*x**n + e)**q*(a + b*x**n + c*x**(S(2)*n))**p, x)
rule1278 = ReplacementRule(pattern1278, replacement1278)
pattern1279 = Pattern(Integral((a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons726, cons5, cons725, cons586)
def replacement1279(mn, p, n2, d, c, a, x, q, e):
rubi.append(1279)
return Int(x**(mn*q)*(a + c*x**n2)**p*(d*x**(-mn) + e)**q, x)
rule1279 = ReplacementRule(pattern1279, replacement1279)
pattern1280 = Pattern(Integral((d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons680, cons585, cons386, cons38)
def replacement1280(mn, p, b, n2, d, a, n, c, x, q, e):
rubi.append(1280)
return Int(x**(S(2)*n*p)*(d + e*x**(-n))**q*(a*x**(-S(2)*n) + b*x**(-n) + c)**p, x)
rule1280 = ReplacementRule(pattern1280, replacement1280)
pattern1281 = Pattern(Integral((a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons726, cons50, cons725, cons386, cons38)
def replacement1281(mn, p, n2, d, c, a, x, q, e):
rubi.append(1281)
return Int(x**(-S(2)*mn*p)*(d + e*x**mn)**q*(a*x**(S(2)*mn) + c)**p, x)
rule1281 = ReplacementRule(pattern1281, replacement1281)
pattern1282 = Pattern(Integral((d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons50, cons680, cons585, cons386, cons147, cons681)
def replacement1282(mn, p, b, n2, d, a, n, c, x, q, e):
rubi.append(1282)
return Dist(x**(n*FracPart(q))*(d + e*x**(-n))**FracPart(q)*(d*x**n + e)**(-FracPart(q)), Int(x**(-n*q)*(d*x**n + e)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1282 = ReplacementRule(pattern1282, replacement1282)
pattern1283 = Pattern(Integral((a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons726, cons5, cons50, cons725, cons386, cons147, cons727)
def replacement1283(mn, p, n2, d, c, a, x, q, e):
rubi.append(1283)
return Dist(x**(-mn*FracPart(q))*(d + e*x**mn)**FracPart(q)*(d*x**(-mn) + e)**(-FracPart(q)), Int(x**(mn*q)*(a + c*x**n2)**p*(d*x**(-mn) + e)**q, x), x)
rule1283 = ReplacementRule(pattern1283, replacement1283)
pattern1284 = Pattern(Integral((d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons680, cons585, cons386, cons147, cons502)
def replacement1284(mn, p, b, n2, d, a, n, c, x, q, e):
rubi.append(1284)
return Dist(x**(-S(2)*n*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p)*(a*x**(-S(2)*n) + b*x**(-n) + c)**(-FracPart(p)), Int(x**(S(2)*n*p)*(d + e*x**(-n))**q*(a*x**(-S(2)*n) + b*x**(-n) + c)**p, x), x)
rule1284 = ReplacementRule(pattern1284, replacement1284)
pattern1285 = Pattern(Integral((a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons726, cons50, cons725, cons386, cons147, cons728)
def replacement1285(mn, p, n2, d, c, a, x, q, e):
rubi.append(1285)
return Dist(x**(-n2*FracPart(p))*(a + c*x**n2)**FracPart(p)*(a*x**(S(2)*mn) + c)**(-FracPart(p)), Int(x**(n2*p)*(d + e*x**mn)**q*(a*x**(S(2)*mn) + c)**p, x), x)
rule1285 = ReplacementRule(pattern1285, replacement1285)
pattern1286 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons50, cons585, cons38)
def replacement1286(mn, p, b, d, c, n, a, x, q, e):
rubi.append(1286)
return Int(x**(-n*p)*(d + e*x**n)**q*(a*x**n + b + c*x**(S(2)*n))**p, x)
rule1286 = ReplacementRule(pattern1286, replacement1286)
pattern1287 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons50, cons585, cons147)
def replacement1287(mn, p, b, d, c, n, a, x, q, e):
rubi.append(1287)
return Dist(x**(n*FracPart(p))*(a + b*x**(-n) + c*x**n)**FracPart(p)*(a*x**n + b + c*x**(S(2)*n))**(-FracPart(p)), Int(x**(-n*p)*(d + e*x**n)**q*(a*x**n + b + c*x**(S(2)*n))**p, x), x)
rule1287 = ReplacementRule(pattern1287, replacement1287)
pattern1288 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(f_ + x_**n_*WC('g', S(1)))**WC('r', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons50, cons52, cons46, cons45, cons147)
def replacement1288(p, g, b, r, f, n2, d, c, n, a, x, q, e):
rubi.append(1288)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((b + S(2)*c*x**n)**(S(2)*p)*(d + e*x**n)**q*(f + g*x**n)**r, x), x)
rule1288 = ReplacementRule(pattern1288, replacement1288)
pattern1289 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(f_ + x_**n_*WC('g', S(1)))**WC('r', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons50, cons52, cons46, cons226, cons256, cons38)
def replacement1289(p, g, b, r, f, n2, d, c, n, a, x, q, e):
rubi.append(1289)
return Int((d + e*x**n)**(p + q)*(f + g*x**n)**r*(a/d + c*x**n/e)**p, x)
rule1289 = ReplacementRule(pattern1289, replacement1289)
pattern1290 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(f_ + x_**n_*WC('g', S(1)))**WC('r', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons4, cons50, cons52, cons46, cons257, cons38)
def replacement1290(p, g, f, r, n2, d, c, n, a, x, q, e):
rubi.append(1290)
return Int((d + e*x**n)**(p + q)*(f + g*x**n)**r*(a/d + c*x**n/e)**p, x)
rule1290 = ReplacementRule(pattern1290, replacement1290)
pattern1291 = Pattern(Integral((d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(f_ + x_**n_*WC('g', S(1)))**WC('r', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons50, cons52, cons46, cons226, cons256, cons147)
def replacement1291(p, g, b, r, f, n2, d, c, n, a, x, q, e):
rubi.append(1291)
return Dist((d + e*x**n)**(-FracPart(p))*(a/d + c*x**n/e)**(-FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((d + e*x**n)**(p + q)*(f + g*x**n)**r*(a/d + c*x**n/e)**p, x), x)
rule1291 = ReplacementRule(pattern1291, replacement1291)
pattern1292 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(f_ + x_**n_*WC('g', S(1)))**WC('r', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons50, cons52, cons46, cons257, cons147)
def replacement1292(p, g, f, r, n2, d, c, n, a, x, q, e):
rubi.append(1292)
return Dist((a + c*x**(S(2)*n))**FracPart(p)*(d + e*x**n)**(-FracPart(p))*(a/d + c*x**n/e)**(-FracPart(p)), Int((d + e*x**n)**(p + q)*(f + g*x**n)**r*(a/d + c*x**n/e)**p, x), x)
rule1292 = ReplacementRule(pattern1292, replacement1292)
def With1293(f, b, g, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-S(4)*a*c + b**S(2), S(2))
if NonzeroQ(S(2)*c*f - g*(b - q)):
return True
return False
pattern1293 = Pattern(Integral((x_**S(2)*WC('g', S(1)) + WC('f', S(0)))/((d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons675, cons279, cons718, CustomConstraint(With1293))
def replacement1293(f, b, g, d, c, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1293)
return Dist((S(2)*c*f - g*(b - q))/(S(2)*c*d - e*(b - q)), Int(S(1)/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) - Dist((-d*g + e*f)/(S(2)*c*d - e*(b - q)), Int((b + S(2)*c*x**S(2) - q)/((d + e*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x)
rule1293 = ReplacementRule(pattern1293, replacement1293)
def With1294(g, f, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-a*c, S(2))
if NonzeroQ(c*f + g*q):
return True
return False
pattern1294 = Pattern(Integral((f_ + x_**S(2)*WC('g', S(1)))/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons715, cons280, cons718, CustomConstraint(With1294))
def replacement1294(g, f, d, c, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1294)
return Dist((c*f + g*q)/(c*d + e*q), Int(S(1)/sqrt(a + c*x**S(4)), x), x) + Dist((-d*g + e*f)/(c*d + e*q), Int((-c*x**S(2) + q)/(sqrt(a + c*x**S(4))*(d + e*x**S(2))), x), x)
rule1294 = ReplacementRule(pattern1294, replacement1294)
pattern1295 = Pattern(Integral((d1_ + x_**WC('non2', S(1))*WC('e1', S(1)))**WC('q', S(1))*(d2_ + x_**WC('non2', S(1))*WC('e2', S(1)))**WC('q', S(1))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons5, cons50, cons46, cons593, cons729, cons730)
def replacement1295(p, d2, b, n2, e2, a, c, non2, d1, e1, n, x, q):
rubi.append(1295)
return Int((d1*d2 + e1*e2*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x)
rule1295 = ReplacementRule(pattern1295, replacement1295)
pattern1296 = Pattern(Integral((d1_ + x_**WC('non2', S(1))*WC('e1', S(1)))**WC('q', S(1))*(d2_ + x_**WC('non2', S(1))*WC('e2', S(1)))**WC('q', S(1))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons4, cons5, cons50, cons46, cons593, cons729)
def replacement1296(p, d2, b, n2, e2, a, c, non2, d1, e1, n, x, q):
rubi.append(1296)
return Dist((d1 + e1*x**(n/S(2)))**FracPart(q)*(d2 + e2*x**(n/S(2)))**FracPart(q)*(d1*d2 + e1*e2*x**n)**(-FracPart(q)), Int((d1*d2 + e1*e2*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1296 = ReplacementRule(pattern1296, replacement1296)
pattern1297 = Pattern(Integral((A_ + x_**WC('m', S(1))*WC('B', S(1)))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons21, cons4, cons5, cons50, cons46, cons53)
def replacement1297(B, p, m, b, n2, d, c, n, a, A, x, q, e):
rubi.append(1297)
return Dist(A, Int((d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x) + Dist(B, Int(x**m*(d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1297 = ReplacementRule(pattern1297, replacement1297)
pattern1298 = Pattern(Integral((A_ + x_**WC('m', S(1))*WC('B', S(1)))*(a_ + x_**n2_*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons34, cons35, cons21, cons4, cons5, cons50, cons46, cons53)
def replacement1298(B, p, m, n2, d, c, n, a, A, x, q, e):
rubi.append(1298)
return Dist(A, Int((a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x) + Dist(B, Int(x**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x)
rule1298 = ReplacementRule(pattern1298, replacement1298)
pattern1299 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)))**q_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons733, cons500)
def replacement1299(p, m, f, b, n2, c, n, a, x, q, e):
rubi.append(1299)
return Dist(e**(S(1) - (m + S(1))/n)*f**m/n, Subst(Int((e*x)**(q + S(-1) + (m + S(1))/n)*(a + b*x + c*x**S(2))**p, x), x, x**n), x)
rule1299 = ReplacementRule(pattern1299, replacement1299)
pattern1300 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)))**q_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons733, cons500)
def replacement1300(p, m, f, n2, c, n, a, x, q, e):
rubi.append(1300)
return Dist(e**(S(1) - (m + S(1))/n)*f**m/n, Subst(Int((e*x)**(q + S(-1) + (m + S(1))/n)*(a + c*x**S(2))**p, x), x, x**n), x)
rule1300 = ReplacementRule(pattern1300, replacement1300)
pattern1301 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)))**q_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons733, cons501)
def replacement1301(p, m, f, b, n2, c, n, a, x, q, e):
rubi.append(1301)
return Dist(e**IntPart(q)*f**m*x**(-n*FracPart(q))*(e*x**n)**FracPart(q), Int(x**(m + n*q)*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1301 = ReplacementRule(pattern1301, replacement1301)
pattern1302 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)))**q_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons733, cons501)
def replacement1302(p, m, f, n2, c, n, a, x, q, e):
rubi.append(1302)
return Dist(e**IntPart(q)*f**m*x**(-n*FracPart(q))*(e*x**n)**FracPart(q), Int(x**(m + n*q)*(a + c*x**(S(2)*n))**p, x), x)
rule1302 = ReplacementRule(pattern1302, replacement1302)
pattern1303 = Pattern(Integral((f_*x_)**WC('m', S(1))*(x_**n_*WC('e', S(1)))**q_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons18)
def replacement1303(p, m, f, b, n2, c, n, a, x, q, e):
rubi.append(1303)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1303 = ReplacementRule(pattern1303, replacement1303)
pattern1304 = Pattern(Integral((f_*x_)**WC('m', S(1))*(x_**n_*WC('e', S(1)))**q_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons18)
def replacement1304(p, m, f, n2, c, n, a, x, q, e):
rubi.append(1304)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(e*x**n)**q*(a + c*x**(S(2)*n))**p, x), x)
rule1304 = ReplacementRule(pattern1304, replacement1304)
pattern1305 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons46, cons53)
def replacement1305(p, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1305)
return Dist(S(1)/n, Subst(Int((d + e*x)**q*(a + b*x + c*x**S(2))**p, x), x, x**n), x)
rule1305 = ReplacementRule(pattern1305, replacement1305)
pattern1306 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons46, cons53)
def replacement1306(p, m, n2, d, c, a, n, x, q, e):
rubi.append(1306)
return Dist(S(1)/n, Subst(Int((a + c*x**S(2))**p*(d + e*x)**q, x), x, x**n), x)
rule1306 = ReplacementRule(pattern1306, replacement1306)
pattern1307 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons220, cons502)
def replacement1307(p, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1307)
return Int(x**(m + n*(S(2)*p + q))*(d*x**(-n) + e)**q*(a*x**(-S(2)*n) + b*x**(-n) + c)**p, x)
rule1307 = ReplacementRule(pattern1307, replacement1307)
pattern1308 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons4, cons46, cons220, cons502)
def replacement1308(p, m, n2, d, c, a, n, x, q, e):
rubi.append(1308)
return Int(x**(m + n*(S(2)*p + q))*(a*x**(-S(2)*n) + c)**p*(d*x**(-n) + e)**q, x)
rule1308 = ReplacementRule(pattern1308, replacement1308)
pattern1309 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons46, cons45, cons147, cons734)
def replacement1309(p, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1309)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(d + e*x)**q*(a + b*x + c*x**S(2))**p, x), x, x**n), x)
rule1309 = ReplacementRule(pattern1309, replacement1309)
pattern1310 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons45, cons147)
def replacement1310(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1310)
return Dist(c**(-IntPart(p))*(b/S(2) + c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((f*x)**m*(b/S(2) + c*x**n)**(S(2)*p)*(d + e*x**n)**q, x), x)
rule1310 = ReplacementRule(pattern1310, replacement1310)
pattern1311 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons46, cons500)
def replacement1311(p, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1311)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(d + e*x)**q*(a + b*x + c*x**S(2))**p, x), x, x**n), x)
rule1311 = ReplacementRule(pattern1311, replacement1311)
pattern1312 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons46, cons500)
def replacement1312(p, m, n2, d, c, a, n, x, q, e):
rubi.append(1312)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + c*x**S(2))**p*(d + e*x)**q, x), x, x**n), x)
rule1312 = ReplacementRule(pattern1312, replacement1312)
pattern1313 = Pattern(Integral((f_*x_)**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons500)
def replacement1313(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1313)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1313 = ReplacementRule(pattern1313, replacement1313)
pattern1314 = Pattern(Integral((f_*x_)**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons500)
def replacement1314(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1314)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x)
rule1314 = ReplacementRule(pattern1314, replacement1314)
pattern1315 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons50, cons46, cons226, cons256, cons38)
def replacement1315(p, m, f, b, n2, d, c, n, a, x, q, e):
rubi.append(1315)
return Int((f*x)**m*(d + e*x**n)**(p + q)*(a/d + c*x**n/e)**p, x)
rule1315 = ReplacementRule(pattern1315, replacement1315)
pattern1316 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons50, cons21, cons4, cons50, cons46, cons257, cons38)
def replacement1316(p, m, f, n2, d, c, n, a, x, q, e):
rubi.append(1316)
return Int((f*x)**m*(d + e*x**n)**(p + q)*(a/d + c*x**n/e)**p, x)
rule1316 = ReplacementRule(pattern1316, replacement1316)
pattern1317 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons226, cons256, cons147)
def replacement1317(p, m, f, b, n2, d, c, n, a, x, q, e):
rubi.append(1317)
return Dist((d + e*x**n)**(-FracPart(p))*(a/d + c*x**n/e)**(-FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int((f*x)**m*(d + e*x**n)**(p + q)*(a/d + c*x**n/e)**p, x), x)
rule1317 = ReplacementRule(pattern1317, replacement1317)
pattern1318 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons257, cons147)
def replacement1318(p, m, f, n2, d, c, n, a, x, q, e):
rubi.append(1318)
return Dist((a + c*x**(S(2)*n))**FracPart(p)*(d + e*x**n)**(-FracPart(p))*(a/d + c*x**n/e)**(-FracPart(p)), Int((f*x)**m*(d + e*x**n)**(p + q)*(a/d + c*x**n/e)**p, x), x)
rule1318 = ReplacementRule(pattern1318, replacement1318)
pattern1319 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons226, cons464, cons735, cons396, cons168)
def replacement1319(p, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1319)
return Dist(e**(-S(2)*p - (m - Mod(m, n))/n)/(n*(q + S(1))), Int(x**Mod(m, n)*(d + e*x**n)**(q + S(1))*ExpandToSum(Together((e**(S(2)*p + (m - Mod(m, n))/n)*n*x**(m - Mod(m, n))*(q + S(1))*(a + b*x**n + c*x**(S(2)*n))**p - (-d)**(S(-1) + (m - Mod(m, n))/n)*(d*(Mod(m, n) + S(1)) + e*x**n*(n*(q + S(1)) + Mod(m, n) + S(1)))*(a*e**S(2) - b*d*e + c*d**S(2))**p)/(d + e*x**n)), x), x), x) + Simp(e**(-S(2)*p - (m - Mod(m, n))/n)*x**(Mod(m, n) + S(1))*(-d)**(S(-1) + (m - Mod(m, n))/n)*(d + e*x**n)**(q + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))**p/(n*(q + S(1))), x)
rule1319 = ReplacementRule(pattern1319, replacement1319)
pattern1320 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons46, cons464, cons735, cons396, cons168)
def replacement1320(p, m, n2, d, c, a, n, x, q, e):
rubi.append(1320)
return Dist(e**(-S(2)*p - (m - Mod(m, n))/n)/(n*(q + S(1))), Int(x**Mod(m, n)*(d + e*x**n)**(q + S(1))*ExpandToSum(Together((e**(S(2)*p + (m - Mod(m, n))/n)*n*x**(m - Mod(m, n))*(a + c*x**(S(2)*n))**p*(q + S(1)) - (-d)**(S(-1) + (m - Mod(m, n))/n)*(a*e**S(2) + c*d**S(2))**p*(d*(Mod(m, n) + S(1)) + e*x**n*(n*(q + S(1)) + Mod(m, n) + S(1))))/(d + e*x**n)), x), x), x) + Simp(e**(-S(2)*p - (m - Mod(m, n))/n)*x**(Mod(m, n) + S(1))*(-d)**(S(-1) + (m - Mod(m, n))/n)*(d + e*x**n)**(q + S(1))*(a*e**S(2) + c*d**S(2))**p/(n*(q + S(1))), x)
rule1320 = ReplacementRule(pattern1320, replacement1320)
pattern1321 = Pattern(Integral(x_**m_*(d_ + x_**n_*WC('e', S(1)))**q_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons226, cons464, cons735, cons396, cons267)
def replacement1321(p, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1321)
return Dist(e**(-S(2)*p)*(-d)**(S(-1) + (m - Mod(m, n))/n)/(n*(q + S(1))), Int(x**m*(d + e*x**n)**(q + S(1))*ExpandToSum(Together((e**(S(2)*p)*n*(-d)**(S(1) - (m - Mod(m, n))/n)*(q + S(1))*(a + b*x**n + c*x**(S(2)*n))**p - e**(-(m - Mod(m, n))/n)*x**(-m + Mod(m, n))*(d*(Mod(m, n) + S(1)) + e*x**n*(n*(q + S(1)) + Mod(m, n) + S(1)))*(a*e**S(2) - b*d*e + c*d**S(2))**p)/(d + e*x**n)), x), x), x) + Simp(e**(-S(2)*p - (m - Mod(m, n))/n)*x**(Mod(m, n) + S(1))*(-d)**(S(-1) + (m - Mod(m, n))/n)*(d + e*x**n)**(q + S(1))*(a*e**S(2) - b*d*e + c*d**S(2))**p/(n*(q + S(1))), x)
rule1321 = ReplacementRule(pattern1321, replacement1321)
pattern1322 = Pattern(Integral(x_**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons46, cons464, cons735, cons396, cons267)
def replacement1322(p, m, n2, d, c, a, n, x, q, e):
rubi.append(1322)
return Dist(e**(-S(2)*p)*(-d)**(S(-1) + (m - Mod(m, n))/n)/(n*(q + S(1))), Int(x**m*(d + e*x**n)**(q + S(1))*ExpandToSum(Together((e**(S(2)*p)*n*(-d)**(S(1) - (m - Mod(m, n))/n)*(a + c*x**(S(2)*n))**p*(q + S(1)) - e**(-(m - Mod(m, n))/n)*x**(-m + Mod(m, n))*(a*e**S(2) + c*d**S(2))**p*(d*(Mod(m, n) + S(1)) + e*x**n*(n*(q + S(1)) + Mod(m, n) + S(1))))/(d + e*x**n)), x), x), x) + Simp(e**(-S(2)*p - (m - Mod(m, n))/n)*x**(Mod(m, n) + S(1))*(-d)**(S(-1) + (m - Mod(m, n))/n)*(d + e*x**n)**(q + S(1))*(a*e**S(2) + c*d**S(2))**p/(n*(q + S(1))), x)
rule1322 = ReplacementRule(pattern1322, replacement1322)
pattern1323 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons50, cons46, cons226, cons464, cons736, cons386, cons737)
def replacement1323(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1323)
return Dist(S(1)/(e*(m + S(2)*n*p + n*q + S(1))), Int((f*x)**m*(d + e*x**n)**q*ExpandToSum(-c**p*d*x**(S(2)*n*p - n)*(m + S(2)*n*p - n + S(1)) + e*(-c**p*x**(S(2)*n*p) + (a + b*x**n + c*x**(S(2)*n))**p)*(m + S(2)*n*p + n*q + S(1)), x), x), x) + Simp(c**p*f**(-S(2)*n*p + n + S(-1))*(f*x)**(m + S(2)*n*p - n + S(1))*(d + e*x**n)**(q + S(1))/(e*(m + S(2)*n*p + n*q + S(1))), x)
rule1323 = ReplacementRule(pattern1323, replacement1323)
pattern1324 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons50, cons46, cons464, cons736, cons386, cons737)
def replacement1324(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1324)
return Dist(S(1)/(e*(m + S(2)*n*p + n*q + S(1))), Int((f*x)**m*(d + e*x**n)**q*ExpandToSum(-c**p*d*x**(S(2)*n*p - n)*(m + S(2)*n*p - n + S(1)) + e*(-c**p*x**(S(2)*n*p) + (a + c*x**(S(2)*n))**p)*(m + S(2)*n*p + n*q + S(1)), x), x), x) + Simp(c**p*f**(-S(2)*n*p + n + S(-1))*(f*x)**(m + S(2)*n*p - n + S(1))*(d + e*x**n)**(q + S(1))/(e*(m + S(2)*n*p + n*q + S(1))), x)
rule1324 = ReplacementRule(pattern1324, replacement1324)
pattern1325 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons50, cons46, cons464)
def replacement1325(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1325)
return Int(ExpandIntegrand((f*x)**m*(d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1325 = ReplacementRule(pattern1325, replacement1325)
pattern1326 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons50, cons46, cons46, cons464)
def replacement1326(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1326)
return Int(ExpandIntegrand((f*x)**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x)
rule1326 = ReplacementRule(pattern1326, replacement1326)
def With1327(p, m, b, n2, d, c, a, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
pattern1327 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons46, cons226, cons148, cons17, CustomConstraint(With1327))
def replacement1327(p, m, b, n2, d, c, a, n, x, q, e):
k = GCD(m + S(1), n)
rubi.append(1327)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(d + e*x**(n/k))**q*(a + b*x**(n/k) + c*x**(S(2)*n/k))**p, x), x, x**k), x)
rule1327 = ReplacementRule(pattern1327, replacement1327)
def With1328(p, m, n2, d, c, a, n, x, q, e):
if isinstance(x, (int, Integer, float, Float)):
return False
k = GCD(m + S(1), n)
if Unequal(k, S(1)):
return True
return False
pattern1328 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons5, cons50, cons46, cons148, cons17, CustomConstraint(With1328))
def replacement1328(p, m, n2, d, c, a, n, x, q, e):
k = GCD(m + S(1), n)
rubi.append(1328)
return Dist(S(1)/k, Subst(Int(x**(S(-1) + (m + S(1))/k)*(a + c*x**(S(2)*n/k))**p*(d + e*x**(n/k))**q, x), x, x**k), x)
rule1328 = ReplacementRule(pattern1328, replacement1328)
def With1329(p, m, f, b, n2, d, c, a, n, x, q, e):
k = Denominator(m)
rubi.append(1329)
return Dist(k/f, Subst(Int(x**(k*(m + S(1)) + S(-1))*(d + e*f**(-n)*x**(k*n))**q*(a + b*f**(-n)*x**(k*n) + c*f**(-S(2)*n)*x**(S(2)*k*n))**p, x), x, (f*x)**(S(1)/k)), x)
pattern1329 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons46, cons226, cons148, cons367, cons38)
rule1329 = ReplacementRule(pattern1329, With1329)
def With1330(p, m, f, n2, d, c, a, n, x, q, e):
k = Denominator(m)
rubi.append(1330)
return Dist(k/f, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + c*x**(S(2)*k*n)/f)**p*(d + e*x**(k*n)/f)**q, x), x, (f*x)**(S(1)/k)), x)
pattern1330 = Pattern(Integral((x_*WC('f', S(1)))**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons5, cons50, cons46, cons148, cons367, cons38)
rule1330 = ReplacementRule(pattern1330, With1330)
pattern1331 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons244, cons163, cons94, cons738, cons694)
def replacement1331(p, m, f, b, n2, d, c, n, a, x, e):
rubi.append(1331)
return Dist(f**(-n)*n*p/((m + S(1))*(m + n*(S(2)*p + S(1)) + S(1))), Int((f*x)**(m + n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))*Simp(S(2)*a*e*(m + S(1)) - b*d*(m + n*(S(2)*p + S(1)) + S(1)) + x**n*(b*e*(m + S(1)) - S(2)*c*d*(m + n*(S(2)*p + S(1)) + S(1))), x), x), x) + Simp((f*x)**(m + S(1))*(d*(m + n*(S(2)*p + S(1)) + S(1)) + e*x**n*(m + S(1)))*(a + b*x**n + c*x**(S(2)*n))**p/(f*(m + S(1))*(m + n*(S(2)*p + S(1)) + S(1))), x)
rule1331 = ReplacementRule(pattern1331, replacement1331)
pattern1332 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons244, cons163, cons94, cons738, cons694)
def replacement1332(p, m, f, n2, d, c, n, a, x, e):
rubi.append(1332)
return Dist(S(2)*f**(-n)*n*p/((m + S(1))*(m + n*(S(2)*p + S(1)) + S(1))), Int((f*x)**(m + n)*(a + c*x**(S(2)*n))**(p + S(-1))*(a*e*(m + S(1)) - c*d*x**n*(m + n*(S(2)*p + S(1)) + S(1))), x), x) + Simp((f*x)**(m + S(1))*(a + c*x**(S(2)*n))**p*(d*(m + n*(S(2)*p + S(1)) + S(1)) + e*x**n*(m + S(1)))/(f*(m + S(1))*(m + n*(S(2)*p + S(1)) + S(1))), x)
rule1332 = ReplacementRule(pattern1332, replacement1332)
pattern1333 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons46, cons226, cons148, cons13, cons163, cons510, cons739, cons694)
def replacement1333(p, m, f, b, n2, d, c, n, a, x, e):
rubi.append(1333)
return Dist(n*p/(c*(m + S(2)*n*p + S(1))*(m + n*(S(2)*p + S(1)) + S(1))), Int((f*x)**m*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))*Simp(-a*b*e*(m + S(1)) + S(2)*a*c*d*(m + n*(S(2)*p + S(1)) + S(1)) + x**n*(S(2)*a*c*e*(m + S(2)*n*p + S(1)) - b**S(2)*e*(m + n*p + S(1)) + b*c*d*(m + n*(S(2)*p + S(1)) + S(1))), x), x), x) + Simp((f*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**p*(b*e*n*p + c*d*(m + n*(S(2)*p + S(1)) + S(1)) + c*e*x**n*(m + S(2)*n*p + S(1)))/(c*f*(m + S(2)*n*p + S(1))*(m + n*(S(2)*p + S(1)) + S(1))), x)
rule1333 = ReplacementRule(pattern1333, replacement1333)
pattern1334 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons46, cons148, cons13, cons163, cons510, cons739, cons694)
def replacement1334(p, m, f, n2, d, c, n, a, x, e):
rubi.append(1334)
return Dist(S(2)*a*n*p/((m + S(2)*n*p + S(1))*(m + n*(S(2)*p + S(1)) + S(1))), Int((f*x)**m*(a + c*x**(S(2)*n))**(p + S(-1))*Simp(d*(m + n*(S(2)*p + S(1)) + S(1)) + e*x**n*(m + S(2)*n*p + S(1)), x), x), x) + Simp((f*x)**(m + S(1))*(a + c*x**(S(2)*n))**p*(c*d*(m + n*(S(2)*p + S(1)) + S(1)) + c*e*x**n*(m + S(2)*n*p + S(1)))/(c*f*(m + S(2)*n*p + S(1))*(m + n*(S(2)*p + S(1)) + S(1))), x)
rule1334 = ReplacementRule(pattern1334, replacement1334)
pattern1335 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons244, cons137, cons530, cons694)
def replacement1335(p, m, f, b, n2, d, c, n, a, x, e):
rubi.append(1335)
return Dist(f**n/(n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((f*x)**(m - n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(x**n*(b*e - S(2)*c*d)*(m + S(2)*n*p + S(2)*n + S(1)) + (-S(2)*a*e + b*d)*(-m + n + S(-1)), x), x), x) + Simp(f**(n + S(-1))*(f*x)**(m - n + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*e + b*d - x**n*(b*e - S(2)*c*d))/(n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1335 = ReplacementRule(pattern1335, replacement1335)
pattern1336 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons244, cons137, cons530, cons694)
def replacement1336(p, m, f, n2, d, c, n, a, x, e):
rubi.append(1336)
return Dist(f**n/(S(2)*a*c*n*(p + S(1))), Int((f*x)**(m - n)*(a + c*x**(S(2)*n))**(p + S(1))*(a*e*(-m + n + S(-1)) + c*d*x**n*(m + S(2)*n*p + S(2)*n + S(1))), x), x) + Simp(f**(n + S(-1))*(f*x)**(m - n + S(1))*(a + c*x**(S(2)*n))**(p + S(1))*(a*e - c*d*x**n)/(S(2)*a*c*n*(p + S(1))), x)
rule1336 = ReplacementRule(pattern1336, replacement1336)
pattern1337 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons46, cons226, cons148, cons13, cons137, cons694)
def replacement1337(p, m, f, b, n2, d, c, n, a, x, e):
rubi.append(1337)
return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((f*x)**m*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(-a*b*e*(m + S(1)) + c*x**n*(-S(2)*a*e + b*d)*(m + n*(S(2)*p + S(3)) + S(1)) + d*(-S(2)*a*c*(m + S(2)*n*(p + S(1)) + S(1)) + b**S(2)*(m + n*(p + S(1)) + S(1))), x), x), x) - Simp((f*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*e + c*x**n*(-S(2)*a*e + b*d) + d*(-S(2)*a*c + b**S(2)))/(a*f*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1337 = ReplacementRule(pattern1337, replacement1337)
pattern1338 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons46, cons148, cons13, cons137, cons694)
def replacement1338(p, m, f, n2, d, c, n, a, x, e):
rubi.append(1338)
return Dist(S(1)/(S(2)*a*n*(p + S(1))), Int((f*x)**m*(a + c*x**(S(2)*n))**(p + S(1))*Simp(d*(m + S(2)*n*(p + S(1)) + S(1)) + e*x**n*(m + n*(S(2)*p + S(3)) + S(1)), x), x), x) - Simp((f*x)**(m + S(1))*(a + c*x**(S(2)*n))**(p + S(1))*(d + e*x**n)/(S(2)*a*f*n*(p + S(1))), x)
rule1338 = ReplacementRule(pattern1338, replacement1338)
pattern1339 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons46, cons226, cons148, cons31, cons530, cons739, cons694)
def replacement1339(p, m, f, b, n2, d, c, n, a, x, e):
rubi.append(1339)
return -Dist(f**n/(c*(m + n*(S(2)*p + S(1)) + S(1))), Int((f*x)**(m - n)*(a + b*x**n + c*x**(S(2)*n))**p*Simp(a*e*(m - n + S(1)) + x**n*(b*e*(m + n*p + S(1)) - c*d*(m + n*(S(2)*p + S(1)) + S(1))), x), x), x) + Simp(e*f**(n + S(-1))*(f*x)**(m - n + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(c*(m + n*(S(2)*p + S(1)) + S(1))), x)
rule1339 = ReplacementRule(pattern1339, replacement1339)
pattern1340 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons5, cons46, cons148, cons31, cons530, cons739, cons694)
def replacement1340(p, m, f, n2, d, c, n, a, x, e):
rubi.append(1340)
return -Dist(f**n/(c*(m + n*(S(2)*p + S(1)) + S(1))), Int((f*x)**(m - n)*(a + c*x**(S(2)*n))**p*(a*e*(m - n + S(1)) - c*d*x**n*(m + n*(S(2)*p + S(1)) + S(1))), x), x) + Simp(e*f**(n + S(-1))*(f*x)**(m - n + S(1))*(a + c*x**(S(2)*n))**(p + S(1))/(c*(m + n*(S(2)*p + S(1)) + S(1))), x)
rule1340 = ReplacementRule(pattern1340, replacement1340)
pattern1341 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons46, cons226, cons148, cons31, cons94, cons694)
def replacement1341(p, m, f, b, n2, d, c, n, a, x, e):
rubi.append(1341)
return Dist(f**(-n)/(a*(m + S(1))), Int((f*x)**(m + n)*(a + b*x**n + c*x**(S(2)*n))**p*Simp(a*e*(m + S(1)) - b*d*(m + n*(p + S(1)) + S(1)) - c*d*x**n*(m + S(2)*n*(p + S(1)) + S(1)), x), x), x) + Simp(d*(f*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(a*f*(m + S(1))), x)
rule1341 = ReplacementRule(pattern1341, replacement1341)
pattern1342 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons5, cons46, cons148, cons31, cons94, cons694)
def replacement1342(p, m, f, n2, d, c, n, a, x, e):
rubi.append(1342)
return Dist(f**(-n)/(a*(m + S(1))), Int((f*x)**(m + n)*(a + c*x**(S(2)*n))**p*(a*e*(m + S(1)) - c*d*x**n*(m + S(2)*n*(p + S(1)) + S(1))), x), x) + Simp(d*(f*x)**(m + S(1))*(a + c*x**(S(2)*n))**(p + S(1))/(a*f*(m + S(1))), x)
rule1342 = ReplacementRule(pattern1342, replacement1342)
def With1343(m, f, b, n2, d, c, n, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(a*c, S(2))
r = Rt(-b*c + S(2)*c*q, S(2))
if Not(NegativeQ(-b*c + S(2)*c*q)):
return True
return False
pattern1343 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons696, cons740, cons81, cons697, CustomConstraint(With1343))
def replacement1343(m, f, b, n2, d, c, n, a, x, e):
q = Rt(a*c, S(2))
r = Rt(-b*c + S(2)*c*q, S(2))
rubi.append(1343)
return Dist(c/(2*q*r), Int((f*x)**m*Simp(d*r - x**(n/2)*(c*d - e*q), x)/(c*x**n + q - r*x**(n/2)), x), x) + Dist(c/(2*q*r), Int((f*x)**m*Simp(d*r + x**(n/2)*(c*d - e*q), x)/(c*x**n + q + r*x**(n/2)), x), x)
rule1343 = ReplacementRule(pattern1343, replacement1343)
def With1344(m, f, n2, d, c, n, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(a*c, S(2))
r = Rt(S(2)*c*q, S(2))
if Not(NegativeQ(S(2)*c*q)):
return True
return False
pattern1344 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons434, cons740, cons81, CustomConstraint(With1344))
def replacement1344(m, f, n2, d, c, n, a, x, e):
q = Rt(a*c, S(2))
r = Rt(S(2)*c*q, S(2))
rubi.append(1344)
return Dist(c/(2*q*r), Int((f*x)**m*Simp(d*r - x**(n/2)*(c*d - e*q), x)/(c*x**n + q - r*x**(n/2)), x), x) + Dist(c/(2*q*r), Int((f*x)**m*Simp(d*r + x**(n/2)*(c*d - e*q), x)/(c*x**n + q + r*x**(n/2)), x), x)
rule1344 = ReplacementRule(pattern1344, replacement1344)
def With1345(m, f, b, d, c, a, x, e):
r = Rt(c*(-b*e + S(2)*c*d)/e, S(2))
rubi.append(1345)
return Dist(e/S(2), Int((f*x)**m/(c*d/e + c*x**S(2) - r*x), x), x) + Dist(e/S(2), Int((f*x)**m/(c*d/e + c*x**S(2) + r*x), x), x)
pattern1345 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons226, cons705, cons741, cons742)
rule1345 = ReplacementRule(pattern1345, With1345)
def With1346(m, f, d, c, a, x, e):
r = Rt(S(2)*c**S(2)*d/e, S(2))
rubi.append(1346)
return Dist(e/S(2), Int((f*x)**m/(c*d/e + c*x**S(2) - r*x), x), x) + Dist(e/S(2), Int((f*x)**m/(c*d/e + c*x**S(2) + r*x), x), x)
pattern1346 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons705, cons741)
rule1346 = ReplacementRule(pattern1346, With1346)
def With1347(m, f, b, n2, d, c, n, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(a*c, S(2))
r = Rt(-b*c + S(2)*c*q, S(2))
if Not(NegativeQ(-b*c + S(2)*c*q)):
return True
return False
pattern1347 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons46, cons696, cons743, cons744, cons697, CustomConstraint(With1347))
def replacement1347(m, f, b, n2, d, c, n, a, x, e):
q = Rt(a*c, S(2))
r = Rt(-b*c + S(2)*c*q, S(2))
rubi.append(1347)
return Dist(c/(2*q*r), Int((f*x)**m*(d*r - x**(n/2)*(c*d - e*q))/(c*x**n + q - r*x**(n/2)), x), x) + Dist(c/(2*q*r), Int((f*x)**m*(d*r + x**(n/2)*(c*d - e*q))/(c*x**n + q + r*x**(n/2)), x), x)
rule1347 = ReplacementRule(pattern1347, replacement1347)
def With1348(m, f, n2, d, c, n, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(a*c, S(2))
r = Rt(S(2)*c*q, S(2))
if Not(NegativeQ(S(2)*c*q)):
return True
return False
pattern1348 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons46, cons743, cons744, cons434, CustomConstraint(With1348))
def replacement1348(m, f, n2, d, c, n, a, x, e):
q = Rt(a*c, S(2))
r = Rt(S(2)*c*q, S(2))
rubi.append(1348)
return Dist(c/(2*q*r), Int((f*x)**m*(d*r - x**(n/2)*(c*d - e*q))/(c*x**n + q - r*x**(n/2)), x), x) + Dist(c/(2*q*r), Int((f*x)**m*(d*r + x**(n/2)*(c*d - e*q))/(c*x**n + q + r*x**(n/2)), x), x)
rule1348 = ReplacementRule(pattern1348, replacement1348)
def With1349(m, f, b, n2, d, c, n, a, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1349)
return Dist(e/S(2) - (-b*e + S(2)*c*d)/(S(2)*q), Int((f*x)**m/(b/S(2) + c*x**n + q/S(2)), x), x) + Dist(e/S(2) + (-b*e + S(2)*c*d)/(S(2)*q), Int((f*x)**m/(b/S(2) + c*x**n - q/S(2)), x), x)
pattern1349 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons46, cons226, cons148)
rule1349 = ReplacementRule(pattern1349, With1349)
def With1350(m, f, n2, d, c, n, a, x, e):
q = Rt(-a*c, S(2))
rubi.append(1350)
return Dist(-c*d/(S(2)*q) + e/S(2), Int((f*x)**m/(c*x**n + q), x), x) - Dist(c*d/(S(2)*q) + e/S(2), Int((f*x)**m/(-c*x**n + q), x), x)
pattern1350 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons46, cons148)
rule1350 = ReplacementRule(pattern1350, With1350)
pattern1351 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons46, cons226, cons148, cons586, cons17)
def replacement1351(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1351)
return Int(ExpandIntegrand((f*x)**m*(d + e*x**n)**q/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1351 = ReplacementRule(pattern1351, replacement1351)
pattern1352 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons46, cons148, cons586, cons17)
def replacement1352(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1352)
return Int(ExpandIntegrand((f*x)**m*(d + e*x**n)**q/(a + c*x**(S(2)*n)), x), x)
rule1352 = ReplacementRule(pattern1352, replacement1352)
pattern1353 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons46, cons226, cons148, cons586, cons18)
def replacement1353(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1353)
return Int(ExpandIntegrand((f*x)**m, (d + e*x**n)**q/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1353 = ReplacementRule(pattern1353, replacement1353)
pattern1354 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons46, cons148, cons586, cons18)
def replacement1354(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1354)
return Int(ExpandIntegrand((f*x)**m, (d + e*x**n)**q/(a + c*x**(S(2)*n)), x), x)
rule1354 = ReplacementRule(pattern1354, replacement1354)
pattern1355 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons386, cons611, cons403, cons529)
def replacement1355(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1355)
return Dist(f**(S(2)*n)/c**S(2), Int((f*x)**(m - S(2)*n)*(d + e*x**n)**(q + S(-1))*(-b*e + c*d + c*e*x**n), x), x) - Dist(f**(S(2)*n)/c**S(2), Int((f*x)**(m - S(2)*n)*(d + e*x**n)**(q + S(-1))*Simp(a*(-b*e + c*d) + x**n*(a*c*e - b**S(2)*e + b*c*d), x)/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1355 = ReplacementRule(pattern1355, replacement1355)
pattern1356 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons50, cons46, cons148, cons386, cons31, cons529)
def replacement1356(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1356)
return Dist(f**(S(2)*n)/c, Int((f*x)**(m - S(2)*n)*(d + e*x**n)**q, x), x) - Dist(a*f**(S(2)*n)/c, Int((f*x)**(m - S(2)*n)*(d + e*x**n)**q/(a + c*x**(S(2)*n)), x), x)
rule1356 = ReplacementRule(pattern1356, replacement1356)
pattern1357 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons386, cons611, cons403, cons690)
def replacement1357(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1357)
return -Dist(f**n/c, Int((f*x)**(m - n)*(d + e*x**n)**(q + S(-1))*Simp(a*e - x**n*(-b*e + c*d), x)/(a + b*x**n + c*x**(S(2)*n)), x), x) + Dist(e*f**n/c, Int((f*x)**(m - n)*(d + e*x**n)**(q + S(-1)), x), x)
rule1357 = ReplacementRule(pattern1357, replacement1357)
pattern1358 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons386, cons611, cons403, cons690)
def replacement1358(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1358)
return -Dist(f**n/c, Int((f*x)**(m - n)*(d + e*x**n)**(q + S(-1))*Simp(a*e - c*d*x**n, x)/(a + c*x**(S(2)*n)), x), x) + Dist(e*f**n/c, Int((f*x)**(m - n)*(d + e*x**n)**(q + S(-1)), x), x)
rule1358 = ReplacementRule(pattern1358, replacement1358)
pattern1359 = Pattern(Integral((x_*WC('f', S(1)))**m_*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons386, cons611, cons403, cons267)
def replacement1359(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1359)
return Dist(d/a, Int((f*x)**m*(d + e*x**n)**(q + S(-1)), x), x) - Dist(f**(-n)/a, Int((f*x)**(m + n)*(d + e*x**n)**(q + S(-1))*Simp(-a*e + b*d + c*d*x**n, x)/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1359 = ReplacementRule(pattern1359, replacement1359)
pattern1360 = Pattern(Integral((x_*WC('f', S(1)))**m_*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons386, cons611, cons403, cons267)
def replacement1360(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1360)
return Dist(d/a, Int((f*x)**m*(d + e*x**n)**(q + S(-1)), x), x) + Dist(f**(-n)/a, Int((f*x)**(m + n)*(d + e*x**n)**(q + S(-1))*Simp(a*e - c*d*x**n, x)/(a + c*x**(S(2)*n)), x), x)
rule1360 = ReplacementRule(pattern1360, replacement1360)
pattern1361 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons386, cons611, cons396, cons529)
def replacement1361(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1361)
return -Dist(f**(S(2)*n)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**(m - S(2)*n)*(d + e*x**n)**(q + S(1))*Simp(a*d + x**n*(-a*e + b*d), x)/(a + b*x**n + c*x**(S(2)*n)), x), x) + Dist(d**S(2)*f**(S(2)*n)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**(m - S(2)*n)*(d + e*x**n)**q, x), x)
rule1361 = ReplacementRule(pattern1361, replacement1361)
pattern1362 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons386, cons611, cons396, cons529)
def replacement1362(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1362)
return -Dist(a*f**(S(2)*n)/(a*e**S(2) + c*d**S(2)), Int((f*x)**(m - S(2)*n)*(d - e*x**n)*(d + e*x**n)**(q + S(1))/(a + c*x**(S(2)*n)), x), x) + Dist(d**S(2)*f**(S(2)*n)/(a*e**S(2) + c*d**S(2)), Int((f*x)**(m - S(2)*n)*(d + e*x**n)**q, x), x)
rule1362 = ReplacementRule(pattern1362, replacement1362)
pattern1363 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons386, cons611, cons396, cons690)
def replacement1363(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1363)
return Dist(f**n/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**(m - n)*(d + e*x**n)**(q + S(1))*Simp(a*e + c*d*x**n, x)/(a + b*x**n + c*x**(S(2)*n)), x), x) - Dist(d*e*f**n/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**(m - n)*(d + e*x**n)**q, x), x)
rule1363 = ReplacementRule(pattern1363, replacement1363)
pattern1364 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('e', S(1)) + WC('d', S(0)))**q_/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons386, cons611, cons396, cons690)
def replacement1364(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1364)
return Dist(f**n/(a*e**S(2) + c*d**S(2)), Int((f*x)**(m - n)*(d + e*x**n)**(q + S(1))*Simp(a*e + c*d*x**n, x)/(a + c*x**(S(2)*n)), x), x) - Dist(d*e*f**n/(a*e**S(2) + c*d**S(2)), Int((f*x)**(m - n)*(d + e*x**n)**q, x), x)
rule1364 = ReplacementRule(pattern1364, replacement1364)
pattern1365 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons46, cons226, cons148, cons386, cons395, cons396)
def replacement1365(m, f, b, n2, d, c, n, a, x, q, e):
rubi.append(1365)
return Dist(e**S(2)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**m*(d + e*x**n)**q, x), x) + Dist(S(1)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**m*(d + e*x**n)**(q + S(1))*Simp(-b*e + c*d - c*e*x**n, x)/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1365 = ReplacementRule(pattern1365, replacement1365)
pattern1366 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n2_*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons46, cons148, cons386, cons395, cons396)
def replacement1366(m, f, n2, d, c, n, a, x, q, e):
rubi.append(1366)
return Dist(c/(a*e**S(2) + c*d**S(2)), Int((f*x)**m*(d - e*x**n)*(d + e*x**n)**(q + S(1))/(a + c*x**(S(2)*n)), x), x) + Dist(e**S(2)/(a*e**S(2) + c*d**S(2)), Int((f*x)**m*(d + e*x**n)**q, x), x)
rule1366 = ReplacementRule(pattern1366, replacement1366)
pattern1367 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons50, cons4, cons46, cons226, cons148, cons386, cons17)
def replacement1367(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1367)
return Int(ExpandIntegrand((d + e*x**n)**q, (f*x)**m/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1367 = ReplacementRule(pattern1367, replacement1367)
pattern1368 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons50, cons4, cons46, cons148, cons386, cons17)
def replacement1368(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1368)
return Int(ExpandIntegrand((d + e*x**n)**q, (f*x)**m/(a + c*x**(S(2)*n)), x), x)
rule1368 = ReplacementRule(pattern1368, replacement1368)
pattern1369 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons50, cons4, cons46, cons226, cons148, cons386, cons18)
def replacement1369(m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1369)
return Int(ExpandIntegrand((f*x)**m*(d + e*x**n)**q, S(1)/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1369 = ReplacementRule(pattern1369, replacement1369)
pattern1370 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons50, cons4, cons46, cons148, cons386, cons18)
def replacement1370(m, f, n2, d, c, a, n, x, q, e):
rubi.append(1370)
return Int(ExpandIntegrand((f*x)**m*(d + e*x**n)**q, S(1)/(a + c*x**(S(2)*n)), x), x)
rule1370 = ReplacementRule(pattern1370, replacement1370)
pattern1371 = Pattern(Integral((x_*WC('f', S(1)))**m_*(x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))/(x_**n_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons244, cons163, cons745)
def replacement1371(p, m, f, b, n2, d, a, c, n, x, e):
rubi.append(1371)
return Dist(d**(S(-2)), Int((f*x)**m*(a*d + x**n*(-a*e + b*d))*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) + Dist(f**(-S(2)*n)*(a*e**S(2) - b*d*e + c*d**S(2))/d**S(2), Int((f*x)**(m + S(2)*n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))/(d + e*x**n), x), x)
rule1371 = ReplacementRule(pattern1371, replacement1371)
pattern1372 = Pattern(Integral((x_*WC('f', S(1)))**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))/(x_**n_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons244, cons163, cons745)
def replacement1372(p, m, f, n2, d, c, a, n, x, e):
rubi.append(1372)
return Dist(a/d**S(2), Int((f*x)**m*(a + c*x**(S(2)*n))**(p + S(-1))*(d - e*x**n), x), x) + Dist(f**(-S(2)*n)*(a*e**S(2) + c*d**S(2))/d**S(2), Int((f*x)**(m + S(2)*n)*(a + c*x**(S(2)*n))**(p + S(-1))/(d + e*x**n), x), x)
rule1372 = ReplacementRule(pattern1372, replacement1372)
pattern1373 = Pattern(Integral((x_*WC('f', S(1)))**m_*(x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))/(x_**n_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons244, cons163, cons267)
def replacement1373(p, m, f, b, n2, d, a, c, n, x, e):
rubi.append(1373)
return Dist(S(1)/(d*e), Int((f*x)**m*(a*e + c*d*x**n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1)), x), x) - Dist(f**(-n)*(a*e**S(2) - b*d*e + c*d**S(2))/(d*e), Int((f*x)**(m + n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(-1))/(d + e*x**n), x), x)
rule1373 = ReplacementRule(pattern1373, replacement1373)
pattern1374 = Pattern(Integral((x_*WC('f', S(1)))**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))/(x_**n_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons244, cons163, cons267)
def replacement1374(p, m, f, n2, d, c, a, n, x, e):
rubi.append(1374)
return Dist(S(1)/(d*e), Int((f*x)**m*(a + c*x**(S(2)*n))**(p + S(-1))*(a*e + c*d*x**n), x), x) - Dist(f**(-n)*(a*e**S(2) + c*d**S(2))/(d*e), Int((f*x)**(m + n)*(a + c*x**(S(2)*n))**(p + S(-1))/(d + e*x**n), x), x)
rule1374 = ReplacementRule(pattern1374, replacement1374)
pattern1375 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_/(x_**n_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons244, cons137, cons746)
def replacement1375(p, m, f, b, n2, d, c, a, n, x, e):
rubi.append(1375)
return -Dist(f**(S(2)*n)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**(m - S(2)*n)*(a*d + x**n*(-a*e + b*d))*(a + b*x**n + c*x**(S(2)*n))**p, x), x) + Dist(d**S(2)*f**(S(2)*n)/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**(m - S(2)*n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(d + e*x**n), x), x)
rule1375 = ReplacementRule(pattern1375, replacement1375)
pattern1376 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_/(x_**n_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons244, cons137, cons746)
def replacement1376(p, m, f, n2, d, c, a, n, x, e):
rubi.append(1376)
return -Dist(a*f**(S(2)*n)/(a*e**S(2) + c*d**S(2)), Int((f*x)**(m - S(2)*n)*(a + c*x**(S(2)*n))**p*(d - e*x**n), x), x) + Dist(d**S(2)*f**(S(2)*n)/(a*e**S(2) + c*d**S(2)), Int((f*x)**(m - S(2)*n)*(a + c*x**(S(2)*n))**(p + S(1))/(d + e*x**n), x), x)
rule1376 = ReplacementRule(pattern1376, replacement1376)
pattern1377 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_/(x_**n_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons46, cons226, cons148, cons244, cons137, cons168)
def replacement1377(p, m, f, b, n2, d, c, a, n, x, e):
rubi.append(1377)
return Dist(f**n/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**(m - n)*(a*e + c*d*x**n)*(a + b*x**n + c*x**(S(2)*n))**p, x), x) - Dist(d*e*f**n/(a*e**S(2) - b*d*e + c*d**S(2)), Int((f*x)**(m - n)*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(d + e*x**n), x), x)
rule1377 = ReplacementRule(pattern1377, replacement1377)
pattern1378 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_/(x_**n_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons7, cons27, cons48, cons125, cons46, cons148, cons244, cons137, cons168)
def replacement1378(p, m, f, n2, d, c, a, n, x, e):
rubi.append(1378)
return Dist(f**n/(a*e**S(2) + c*d**S(2)), Int((f*x)**(m - n)*(a + c*x**(S(2)*n))**p*(a*e + c*d*x**n), x), x) - Dist(d*e*f**n/(a*e**S(2) + c*d**S(2)), Int((f*x)**(m - n)*(a + c*x**(S(2)*n))**(p + S(1))/(d + e*x**n), x), x)
rule1378 = ReplacementRule(pattern1378, replacement1378)
pattern1379 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons50, cons46, cons226, cons148, cons747)
def replacement1379(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1379)
return Int(ExpandIntegrand((a + b*x**n + c*x**(S(2)*n))**p, (f*x)**m*(d + e*x**n)**q, x), x)
rule1379 = ReplacementRule(pattern1379, replacement1379)
pattern1380 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons50, cons46, cons148, cons747)
def replacement1380(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1380)
return Int(ExpandIntegrand((a + c*x**(S(2)*n))**p, (f*x)**m*(d + e*x**n)**q, x), x)
rule1380 = ReplacementRule(pattern1380, replacement1380)
pattern1381 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons46, cons226, cons196, cons17)
def replacement1381(p, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1381)
return -Subst(Int(x**(-m + S(-2))*(d + e*x**(-n))**q*(a + b*x**(-n) + c*x**(-S(2)*n))**p, x), x, S(1)/x)
rule1381 = ReplacementRule(pattern1381, replacement1381)
pattern1382 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons5, cons50, cons46, cons196, cons17)
def replacement1382(p, m, n2, d, c, a, n, x, q, e):
rubi.append(1382)
return -Subst(Int(x**(-m + S(-2))*(a + c*x**(-S(2)*n))**p*(d + e*x**(-n))**q, x), x, S(1)/x)
rule1382 = ReplacementRule(pattern1382, replacement1382)
def With1383(p, m, f, b, n2, d, c, a, n, x, q, e):
g = Denominator(m)
rubi.append(1383)
return -Dist(g/f, Subst(Int(x**(-g*(m + S(1)) + S(-1))*(d + e*f**(-n)*x**(-g*n))**q*(a + b*f**(-n)*x**(-g*n) + c*f**(-S(2)*n)*x**(-S(2)*g*n))**p, x), x, (f*x)**(-S(1)/g)), x)
pattern1383 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons5, cons50, cons46, cons226, cons196, cons367)
rule1383 = ReplacementRule(pattern1383, With1383)
def With1384(p, m, f, n2, d, c, a, n, x, q, e):
g = Denominator(m)
rubi.append(1384)
return -Dist(g/f, Subst(Int(x**(-g*(m + S(1)) + S(-1))*(a + c*f**(-S(2)*n)*x**(-S(2)*g*n))**p*(d + e*f**(-n)*x**(-g*n))**q, x), x, (f*x)**(-S(1)/g)), x)
pattern1384 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons5, cons50, cons46, cons196, cons367)
rule1384 = ReplacementRule(pattern1384, With1384)
pattern1385 = Pattern(Integral((x_*WC('f', S(1)))**m_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons50, cons46, cons226, cons196, cons356)
def replacement1385(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1385)
return -Dist(f**IntPart(m)*(f*x)**FracPart(m)*(S(1)/x)**FracPart(m), Subst(Int(x**(-m + S(-2))*(d + e*x**(-n))**q*(a + b*x**(-n) + c*x**(-S(2)*n))**p, x), x, S(1)/x), x)
rule1385 = ReplacementRule(pattern1385, replacement1385)
pattern1386 = Pattern(Integral((x_*WC('f', S(1)))**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons5, cons50, cons46, cons196, cons356)
def replacement1386(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1386)
return -Dist(f**IntPart(m)*(f*x)**FracPart(m)*(S(1)/x)**FracPart(m), Subst(Int(x**(-m + S(-2))*(a + c*x**(-S(2)*n))**p*(d + e*x**(-n))**q, x), x, S(1)/x), x)
rule1386 = ReplacementRule(pattern1386, replacement1386)
def With1387(p, m, b, n2, d, c, a, n, x, q, e):
g = Denominator(n)
rubi.append(1387)
return Dist(g, Subst(Int(x**(g*(m + S(1)) + S(-1))*(d + e*x**(g*n))**q*(a + b*x**(g*n) + c*x**(S(2)*g*n))**p, x), x, x**(S(1)/g)), x)
pattern1387 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons5, cons50, cons46, cons226, cons489)
rule1387 = ReplacementRule(pattern1387, With1387)
def With1388(p, m, n2, d, c, a, n, x, q, e):
g = Denominator(n)
rubi.append(1388)
return Dist(g, Subst(Int(x**(g*(m + S(1)) + S(-1))*(a + c*x**(S(2)*g*n))**p*(d + e*x**(g*n))**q, x), x, x**(S(1)/g)), x)
pattern1388 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons5, cons50, cons46, cons489)
rule1388 = ReplacementRule(pattern1388, With1388)
pattern1389 = Pattern(Integral((f_*x_)**m_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons50, cons46, cons226, cons489)
def replacement1389(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1389)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1389 = ReplacementRule(pattern1389, replacement1389)
pattern1390 = Pattern(Integral((f_*x_)**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons5, cons50, cons46, cons489)
def replacement1390(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1390)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x)
rule1390 = ReplacementRule(pattern1390, replacement1390)
pattern1391 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons46, cons226, cons541, cons23)
def replacement1391(p, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1391)
return Dist(S(1)/(m + S(1)), Subst(Int((d + e*x**(n/(m + S(1))))**q*(a + b*x**(n/(m + S(1))) + c*x**(S(2)*n/(m + S(1))))**p, x), x, x**(m + S(1))), x)
rule1391 = ReplacementRule(pattern1391, replacement1391)
pattern1392 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons46, cons541, cons23)
def replacement1392(p, m, n2, d, c, a, n, x, q, e):
rubi.append(1392)
return Dist(S(1)/(m + S(1)), Subst(Int((a + c*x**(S(2)*n/(m + S(1))))**p*(d + e*x**(n/(m + S(1))))**q, x), x, x**(m + S(1))), x)
rule1392 = ReplacementRule(pattern1392, replacement1392)
pattern1393 = Pattern(Integral((f_*x_)**m_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons50, cons46, cons226, cons541, cons23)
def replacement1393(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1393)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1393 = ReplacementRule(pattern1393, replacement1393)
pattern1394 = Pattern(Integral((f_*x_)**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons5, cons50, cons46, cons541, cons23)
def replacement1394(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1394)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x)
rule1394 = ReplacementRule(pattern1394, replacement1394)
def With1395(m, f, b, n2, d, c, a, n, x, q, e):
r = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1395)
return Dist(S(2)*c/r, Int((f*x)**m*(d + e*x**n)**q/(b + S(2)*c*x**n - r), x), x) - Dist(S(2)*c/r, Int((f*x)**m*(d + e*x**n)**q/(b + S(2)*c*x**n + r), x), x)
pattern1395 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons50, cons46, cons226)
rule1395 = ReplacementRule(pattern1395, With1395)
def With1396(m, f, n2, d, c, a, n, x, q, e):
r = Rt(-a*c, S(2))
rubi.append(1396)
return -Dist(c/(S(2)*r), Int((f*x)**m*(d + e*x**n)**q/(-c*x**n + r), x), x) - Dist(c/(S(2)*r), Int((f*x)**m*(d + e*x**n)**q/(c*x**n + r), x), x)
pattern1396 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**q_/(a_ + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons4, cons50, cons46)
rule1396 = ReplacementRule(pattern1396, With1396)
pattern1397 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons46, cons226, cons702)
def replacement1397(p, m, f, b, n2, d, c, n, a, x, e):
rubi.append(1397)
return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((f*x)**m*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(-a*b*e*(m + S(1)) + c*x**n*(-S(2)*a*e + b*d)*(m + n*(S(2)*p + S(3)) + S(1)) + d*(-S(2)*a*c*(m + S(2)*n*(p + S(1)) + S(1)) + b**S(2)*(m + n*(p + S(1)) + S(1))), x), x), x) - Simp((f*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*e + c*x**n*(-S(2)*a*e + b*d) + d*(-S(2)*a*c + b**S(2)))/(a*f*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1397 = ReplacementRule(pattern1397, replacement1397)
pattern1398 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons4, cons46, cons702)
def replacement1398(p, m, f, n2, d, c, n, a, x, e):
rubi.append(1398)
return Dist(S(1)/(S(2)*a*n*(p + S(1))), Int((f*x)**m*(a + c*x**(S(2)*n))**(p + S(1))*Simp(d*(m + S(2)*n*(p + S(1)) + S(1)) + e*x**n*(m + n*(S(2)*p + S(3)) + S(1)), x), x), x) - Simp((f*x)**(m + S(1))*(a + c*x**(S(2)*n))**(p + S(1))*(d + e*x**n)/(S(2)*a*f*n*(p + S(1))), x)
rule1398 = ReplacementRule(pattern1398, replacement1398)
pattern1399 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons226, cons748)
def replacement1399(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1399)
return Int(ExpandIntegrand((f*x)**m*(d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1399 = ReplacementRule(pattern1399, replacement1399)
pattern1400 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons748)
def replacement1400(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1400)
return Int(ExpandIntegrand((f*x)**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x)
rule1400 = ReplacementRule(pattern1400, replacement1400)
pattern1401 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons564, cons733)
def replacement1401(p, m, f, n2, d, c, n, a, x, q, e):
rubi.append(1401)
return Dist(f**m, Int(ExpandIntegrand(x**m*(a + c*x**(S(2)*n))**p, (d/(d**S(2) - e**S(2)*x**(S(2)*n)) - e*x**n/(d**S(2) - e**S(2)*x**(S(2)*n)))**(-q), x), x), x)
rule1401 = ReplacementRule(pattern1401, replacement1401)
pattern1402 = Pattern(Integral((x_*WC('f', S(1)))**m_*(a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n_*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46, cons564, cons749)
def replacement1402(p, m, f, n2, d, c, n, a, x, q, e):
rubi.append(1402)
return Dist(x**(-m)*(f*x)**m, Int(x**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x)
rule1402 = ReplacementRule(pattern1402, replacement1402)
pattern1403 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46)
def replacement1403(p, m, f, b, n2, d, c, a, n, x, q, e):
rubi.append(1403)
return Int((f*x)**m*(d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x)
rule1403 = ReplacementRule(pattern1403, replacement1403)
pattern1404 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons46)
def replacement1404(p, m, f, n2, d, c, a, n, x, q, e):
rubi.append(1404)
return Int((f*x)**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x)
rule1404 = ReplacementRule(pattern1404, replacement1404)
pattern1405 = Pattern(Integral(u_**WC('m', S(1))*(d_ + v_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + v_**n_*WC('b', S(1)) + v_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons46, cons554)
def replacement1405(v, p, u, m, b, n2, d, c, a, n, x, q, e):
rubi.append(1405)
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(d + e*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x, v), x)
rule1405 = ReplacementRule(pattern1405, replacement1405)
pattern1406 = Pattern(Integral(u_**WC('m', S(1))*(a_ + v_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + v_**n_*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons554)
def replacement1406(v, p, u, m, n2, d, c, a, n, x, q, e):
rubi.append(1406)
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + c*x**(S(2)*n))**p*(d + e*x**n)**q, x), x, v), x)
rule1406 = ReplacementRule(pattern1406, replacement1406)
pattern1407 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**WC('q', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons680, cons585, cons586)
def replacement1407(mn, p, m, b, n2, d, a, n, c, x, q, e):
rubi.append(1407)
return Int(x**(m - n*q)*(d*x**n + e)**q*(a + b*x**n + c*x**(S(2)*n))**p, x)
rule1407 = ReplacementRule(pattern1407, replacement1407)
pattern1408 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons21, cons726, cons5, cons725, cons586)
def replacement1408(mn, p, m, n2, d, c, a, x, q, e):
rubi.append(1408)
return Int(x**(m + mn*q)*(a + c*x**n2)**p*(d*x**(-mn) + e)**q, x)
rule1408 = ReplacementRule(pattern1408, replacement1408)
pattern1409 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons50, cons680, cons585, cons386, cons38)
def replacement1409(mn, p, m, b, n2, d, a, n, c, x, q, e):
rubi.append(1409)
return Int(x**(m + S(2)*n*p)*(d + e*x**(-n))**q*(a*x**(-S(2)*n) + b*x**(-n) + c)**p, x)
rule1409 = ReplacementRule(pattern1409, replacement1409)
pattern1410 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons21, cons726, cons50, cons725, cons386, cons38)
def replacement1410(mn, p, m, n2, d, c, a, x, q, e):
rubi.append(1410)
return Int(x**(m - S(2)*mn*p)*(d + e*x**mn)**q*(a*x**(S(2)*mn) + c)**p, x)
rule1410 = ReplacementRule(pattern1410, replacement1410)
pattern1411 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons680, cons585, cons386, cons147)
def replacement1411(mn, p, m, b, n2, d, a, n, c, x, q, e):
rubi.append(1411)
return Dist(x**(n*FracPart(q))*(d + e*x**(-n))**FracPart(q)*(d*x**n + e)**(-FracPart(q)), Int(x**(m - n*q)*(d*x**n + e)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1411 = ReplacementRule(pattern1411, replacement1411)
pattern1412 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**q_, x_), cons2, cons7, cons27, cons48, cons21, cons726, cons5, cons50, cons725, cons386, cons147)
def replacement1412(mn, p, m, n2, d, c, a, x, q, e):
rubi.append(1412)
return Dist(x**(-mn*FracPart(q))*(d + e*x**mn)**FracPart(q)*(d*x**(-mn) + e)**(-FracPart(q)), Int(x**(m + mn*q)*(a + c*x**n2)**p*(d*x**(-mn) + e)**q, x), x)
rule1412 = ReplacementRule(pattern1412, replacement1412)
pattern1413 = Pattern(Integral((f_*x_)**m_*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**WC('q', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons680, cons585)
def replacement1413(mn, p, m, f, b, n2, d, a, n, c, x, q, e):
rubi.append(1413)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(d + e*x**mn)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1413 = ReplacementRule(pattern1413, replacement1413)
pattern1414 = Pattern(Integral((f_*x_)**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)))**p_*(d_ + x_**WC('mn', S(1))*WC('e', S(1)))**WC('q', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons21, cons726, cons5, cons50, cons725)
def replacement1414(mn, p, m, f, n2, d, c, a, x, q, e):
rubi.append(1414)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(a + c*x**(S(2)*n))**p*(d + e*x**mn)**q, x), x)
rule1414 = ReplacementRule(pattern1414, replacement1414)
pattern1415 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons50, cons585, cons38)
def replacement1415(mn, p, m, b, d, c, n, a, x, q, e):
rubi.append(1415)
return Int(x**(m - n*p)*(d + e*x**n)**q*(a*x**n + b + c*x**(S(2)*n))**p, x)
rule1415 = ReplacementRule(pattern1415, replacement1415)
pattern1416 = Pattern(Integral(x_**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons50, cons585, cons147)
def replacement1416(mn, p, m, b, d, c, n, a, x, q, e):
rubi.append(1416)
return Dist(x**(n*FracPart(p))*(a + b*x**(-n) + c*x**n)**FracPart(p)*(a*x**n + b + c*x**(S(2)*n))**(-FracPart(p)), Int(x**(m - n*p)*(d + e*x**n)**q*(a*x**n + b + c*x**(S(2)*n))**p, x), x)
rule1416 = ReplacementRule(pattern1416, replacement1416)
pattern1417 = Pattern(Integral((f_*x_)**WC('m', S(1))*(d_ + x_**n_*WC('e', S(1)))**WC('q', S(1))*(a_ + x_**mn_*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons585)
def replacement1417(mn, p, m, f, b, d, c, n, a, x, q, e):
rubi.append(1417)
return Dist(f**IntPart(m)*x**(-FracPart(m))*(f*x)**FracPart(m), Int(x**m*(d + e*x**n)**q*(a + b*x**(-n) + c*x**n)**p, x), x)
rule1417 = ReplacementRule(pattern1417, replacement1417)
pattern1418 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d1_ + x_**WC('non2', S(1))*WC('e1', S(1)))**WC('q', S(1))*(d2_ + x_**WC('non2', S(1))*WC('e2', S(1)))**WC('q', S(1))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons4, cons5, cons50, cons46, cons593, cons729, cons730)
def replacement1418(p, d2, m, f, b, n2, e2, a, c, non2, d1, e1, n, x, q):
rubi.append(1418)
return Int((f*x)**m*(d1*d2 + e1*e2*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x)
rule1418 = ReplacementRule(pattern1418, replacement1418)
pattern1419 = Pattern(Integral((x_*WC('f', S(1)))**WC('m', S(1))*(d1_ + x_**WC('non2', S(1))*WC('e1', S(1)))**WC('q', S(1))*(d2_ + x_**WC('non2', S(1))*WC('e2', S(1)))**WC('q', S(1))*(x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons731, cons652, cons732, cons654, cons125, cons4, cons5, cons50, cons46, cons593, cons729)
def replacement1419(p, d2, m, f, b, n2, e2, a, c, non2, d1, e1, n, x, q):
rubi.append(1419)
return Dist((d1 + e1*x**(n/S(2)))**FracPart(q)*(d2 + e2*x**(n/S(2)))**FracPart(q)*(d1*d2 + e1*e2*x**n)**(-FracPart(q)), Int((f*x)**m*(d1*d2 + e1*e2*x**n)**q*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1419 = ReplacementRule(pattern1419, replacement1419)
pattern1420 = Pattern(Integral((x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons750, cons751)
def replacement1420(p, b, r, c, a, n, x, q):
rubi.append(1420)
return Int((x**n*(a + b + c))**p, x)
rule1420 = ReplacementRule(pattern1420, replacement1420)
pattern1421 = Pattern(Integral((x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons50, cons752, cons753, cons38)
def replacement1421(p, b, r, c, a, n, x, q):
rubi.append(1421)
return Int(x**(p*q)*(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))**p, x)
rule1421 = ReplacementRule(pattern1421, replacement1421)
pattern1422 = Pattern(Integral(sqrt(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons4, cons50, cons752, cons753)
def replacement1422(b, r, c, a, n, x, q):
rubi.append(1422)
return Dist(x**(-q/S(2))*sqrt(a*x**q + b*x**n + c*x**(S(2)*n - q))/sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q)), Int(x**(q/S(2))*sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q)), x), x)
rule1422 = ReplacementRule(pattern1422, replacement1422)
pattern1423 = Pattern(Integral(S(1)/sqrt(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons4, cons50, cons752, cons753)
def replacement1423(b, r, c, a, n, x, q):
rubi.append(1423)
return Dist(x**(q/S(2))*sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))/sqrt(a*x**q + b*x**n + c*x**(S(2)*n - q)), Int(x**(-q/S(2))/sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q)), x), x)
rule1423 = ReplacementRule(pattern1423, replacement1423)
pattern1424 = Pattern(Integral((x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons50, cons752, cons753, cons147, cons226, cons13, cons163, cons754)
def replacement1424(p, b, r, c, a, n, x, q):
rubi.append(1424)
return Dist(p*(n - q)/(p*(S(2)*n - q) + S(1)), Int(x**q*(S(2)*a + b*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(-1)), x), x) + Simp(x*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p/(p*(S(2)*n - q) + S(1)), x)
rule1424 = ReplacementRule(pattern1424, replacement1424)
pattern1425 = Pattern(Integral((x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons50, cons752, cons753, cons147, cons226, cons13, cons137)
def replacement1425(p, b, r, c, a, n, x, q):
rubi.append(1425)
return Dist(S(1)/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), Int(x**(-q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))*(b*c*x**(n - q)*(p*q + (n - q)*(S(2)*p + S(3)) + S(1)) + (n - q)*(p + S(1))*(-S(4)*a*c + b**S(2)) + (-S(2)*a*c + b**S(2))*(p*q + S(1))), x), x) - Simp(x**(-q + S(1))*(-S(2)*a*c + b**S(2) + b*c*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1425 = ReplacementRule(pattern1425, replacement1425)
pattern1426 = Pattern(Integral((x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons50, cons752, cons753, cons147)
def replacement1426(p, b, r, c, a, n, x, q):
rubi.append(1426)
return Dist(x**(-p*q)*(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))**(-p)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, Int(x**(p*q)*(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))**p, x), x)
rule1426 = ReplacementRule(pattern1426, replacement1426)
pattern1427 = Pattern(Integral((x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons50, cons752)
def replacement1427(p, b, r, c, a, n, x, q):
rubi.append(1427)
return Int((a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x)
rule1427 = ReplacementRule(pattern1427, replacement1427)
pattern1428 = Pattern(Integral((u_**WC('n', S(1))*WC('b', S(1)) + u_**WC('q', S(1))*WC('a', S(1)) + u_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons50, cons752, cons68, cons69)
def replacement1428(p, u, b, r, c, a, n, x, q):
rubi.append(1428)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x), x, u), x)
rule1428 = ReplacementRule(pattern1428, replacement1428)
pattern1429 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons755, cons751)
def replacement1429(p, m, b, r, a, n, c, x, q):
rubi.append(1429)
return Int(x**m*(x**n*(a + b + c))**p, x)
rule1429 = ReplacementRule(pattern1429, replacement1429)
pattern1430 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons50, cons752, cons38, cons753)
def replacement1430(p, m, b, r, a, n, c, x, q):
rubi.append(1430)
return Int(x**(m + p*q)*(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))**p, x)
rule1430 = ReplacementRule(pattern1430, replacement1430)
pattern1431 = Pattern(Integral(x_**WC('m', S(1))/sqrt(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons21, cons4, cons50, cons752, cons753, cons756)
def replacement1431(m, b, r, a, n, c, x, q):
rubi.append(1431)
return Dist(x**(q/S(2))*sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))/sqrt(a*x**q + b*x**n + c*x**(S(2)*n - q)), Int(x**(m - q/S(2))/sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q)), x), x)
rule1431 = ReplacementRule(pattern1431, replacement1431)
pattern1432 = Pattern(Integral(x_**WC('m', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons4, cons757, cons758, cons759, cons226)
def replacement1432(m, b, r, a, n, c, x, q):
rubi.append(1432)
return Simp(-S(2)*x**(n/S(2) + S(-1)/2)*(b + S(2)*c*x)/((-S(4)*a*c + b**S(2))*sqrt(a*x**(n + S(-1)) + b*x**n + c*x**(n + S(1)))), x)
rule1432 = ReplacementRule(pattern1432, replacement1432)
pattern1433 = Pattern(Integral(x_**WC('m', S(1))/(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons4, cons760, cons758, cons759, cons226)
def replacement1433(m, b, r, a, n, c, x, q):
rubi.append(1433)
return Simp(x**(n/S(2) + S(-1)/2)*(S(4)*a + S(2)*b*x)/((-S(4)*a*c + b**S(2))*sqrt(a*x**(n + S(-1)) + b*x**n + c*x**(n + S(1)))), x)
rule1433 = ReplacementRule(pattern1433, replacement1433)
pattern1434 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons761)
def replacement1434(p, m, b, r, a, n, c, x, q):
rubi.append(1434)
return -Dist(b/(S(2)*c), Int(x**(m + S(-1))*(a*x**(n + S(-1)) + b*x**n + c*x**(n + S(1)))**p, x), x) + Simp(x**(m - n)*(a*x**(n + S(-1)) + b*x**n + c*x**(n + S(1)))**(p + S(1))/(S(2)*c*(p + S(1))), x)
rule1434 = ReplacementRule(pattern1434, replacement1434)
pattern1435 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons163, cons762)
def replacement1435(p, m, b, r, a, n, c, x, q):
rubi.append(1435)
return -Dist(p*(-S(4)*a*c + b**S(2))/(S(2)*c*(S(2)*p + S(1))), Int(x**(m + q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(-1)), x), x) + Simp(x**(m - n + q + S(1))*(b + S(2)*c*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p/(S(2)*c*(n - q)*(S(2)*p + S(1))), x)
rule1435 = ReplacementRule(pattern1435, replacement1435)
pattern1436 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons163, cons763, cons764, cons765)
def replacement1436(p, m, b, r, a, n, c, x, q):
rubi.append(1436)
return Dist(p*(n - q)/(c*(m + p*(S(2)*n - q) + S(1))*(m + p*q + (n - q)*(S(2)*p + S(-1)) + S(1))), Int(x**(m - n + S(2)*q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(-1))*Simp(-a*b*(m - n + p*q + q + S(1)) + x**(n - q)*(S(2)*a*c*(m + p*q + (n - q)*(S(2)*p + S(-1)) + S(1)) - b**S(2)*(m + p*q + (n - q)*(p + S(-1)) + S(1))), x), x), x) + Simp(x**(m - n + q + S(1))*(b*p*(n - q) + c*x**(n - q)*(m + p*q + (n - q)*(S(2)*p + S(-1)) + S(1)))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p/(c*(m + p*(S(2)*n - q) + S(1))*(m + p*q + (n - q)*(S(2)*p + S(-1)) + S(1))), x)
rule1436 = ReplacementRule(pattern1436, replacement1436)
pattern1437 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons163, cons766, cons767)
def replacement1437(p, m, b, r, a, n, c, x, q):
rubi.append(1437)
return -Dist(p*(n - q)/(m + p*q + S(1)), Int(x**(m + n)*(b + S(2)*c*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(-1)), x), x) + Simp(x**(m + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p/(m + p*q + S(1)), x)
rule1437 = ReplacementRule(pattern1437, replacement1437)
pattern1438 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons163, cons768, cons764)
def replacement1438(p, m, b, r, a, n, c, x, q):
rubi.append(1438)
return Dist(p*(n - q)/(m + p*(S(2)*n - q) + S(1)), Int(x**(m + q)*(S(2)*a + b*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(-1)), x), x) + Simp(x**(m + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p/(m + p*(S(2)*n - q) + S(1)), x)
rule1438 = ReplacementRule(pattern1438, replacement1438)
pattern1439 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons137, cons769)
def replacement1439(p, m, b, r, a, n, c, x, q):
rubi.append(1439)
return Dist((S(2)*a*c - b**S(2)*(p + S(2)))/(a*(p + S(1))*(-S(4)*a*c + b**S(2))), Int(x**(m - q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1)), x), x) - Simp(x**(m - q + S(1))*(-S(2)*a*c + b**S(2) + b*c*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1439 = ReplacementRule(pattern1439, replacement1439)
pattern1440 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons137, cons770)
def replacement1440(p, m, b, r, a, n, c, x, q):
rubi.append(1440)
return Dist(S(1)/((n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), Int(x**(m - S(2)*n + q)*(S(2)*a*(m - S(2)*n + p*q + S(2)*q + S(1)) + b*x**(n - q)*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1)), x), x) - Simp(x**(m - S(2)*n + q + S(1))*(S(2)*a + b*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/((n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1440 = ReplacementRule(pattern1440, replacement1440)
pattern1441 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons137, cons771)
def replacement1441(p, m, b, r, a, n, c, x, q):
rubi.append(1441)
return Dist(S(1)/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), Int(x**(m - q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))*(-S(2)*a*c*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + b**S(2)*(m + p*q + (n - q)*(p + S(1)) + S(1)) + b*c*x**(n - q)*(m + p*q + (n - q)*(S(2)*p + S(3)) + S(1))), x), x) - Simp(x**(m - q + S(1))*(-S(2)*a*c + b**S(2) + b*c*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1441 = ReplacementRule(pattern1441, replacement1441)
pattern1442 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons137, cons772)
def replacement1442(p, m, b, r, a, n, c, x, q):
rubi.append(1442)
return -Dist(S(1)/((n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), Int(x**(m - n)*(b*(m - n + p*q + q + S(1)) + S(2)*c*x**(n - q)*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1)), x), x) + Simp(x**(m - n + S(1))*(b + S(2)*c*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/((n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1442 = ReplacementRule(pattern1442, replacement1442)
pattern1443 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons773, cons774)
def replacement1443(p, m, b, r, a, n, c, x, q):
rubi.append(1443)
return -Dist(b/(S(2)*c), Int(x**(m - n + q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x), x) + Simp(x**(m - S(2)*n + q + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(S(2)*c*(n - q)*(p + S(1))), x)
rule1443 = ReplacementRule(pattern1443, replacement1443)
pattern1444 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons773, cons775)
def replacement1444(p, m, b, r, a, n, c, x, q):
rubi.append(1444)
return -Dist(b/(S(2)*a), Int(x**(m + n - q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x), x) - Simp(x**(m - q + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(S(2)*a*(n - q)*(p + S(1))), x)
rule1444 = ReplacementRule(pattern1444, replacement1444)
pattern1445 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons773, cons770)
def replacement1445(p, m, b, r, a, n, c, x, q):
rubi.append(1445)
return -Dist(S(1)/(c*(m + p*q + S(2)*p*(n - q) + S(1))), Int(x**(m - S(2)*n + S(2)*q)*(a*(m - S(2)*n + p*q + S(2)*q + S(1)) + b*x**(n - q)*(m + p*q + (n - q)*(p + S(-1)) + S(1)))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x), x) + Simp(x**(m - S(2)*n + q + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(c*(m + p*q + S(2)*p*(n - q) + S(1))), x)
rule1445 = ReplacementRule(pattern1445, replacement1445)
pattern1446 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons752, cons753, cons147, cons226, cons148, cons606, cons773, cons776)
def replacement1446(p, m, b, r, a, n, c, x, q):
rubi.append(1446)
return -Dist(S(1)/(a*(m + p*q + S(1))), Int(x**(m + n - q)*(b*(m + p*q + (n - q)*(p + S(1)) + S(1)) + c*x**(n - q)*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x), x) + Simp(x**(m - q + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(a*(m + p*q + S(1))), x)
rule1446 = ReplacementRule(pattern1446, replacement1446)
pattern1447 = Pattern(Integral(x_**WC('m', S(1))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons50, cons752, cons147, cons753)
def replacement1447(p, m, b, r, a, n, c, x, q):
rubi.append(1447)
return Dist(x**(-p*q)*(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))**(-p)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, Int(x**(m + p*q)*(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))**p, x), x)
rule1447 = ReplacementRule(pattern1447, replacement1447)
pattern1448 = Pattern(Integral(u_**WC('m', S(1))*(u_**WC('n', S(1))*WC('b', S(1)) + u_**WC('q', S(1))*WC('a', S(1)) + u_**WC('r', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons50, cons752, cons68, cons69)
def replacement1448(p, u, m, b, r, a, n, c, x, q):
rubi.append(1448)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int(x**m*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x), x, u), x)
rule1448 = ReplacementRule(pattern1448, replacement1448)
pattern1449 = Pattern(Integral((A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons50, cons777, cons778, cons38, cons753)
def replacement1449(B, p, j, b, r, c, n, a, x, q, A):
rubi.append(1449)
return Int(x**(p*q)*(A + B*x**(n - q))*(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))**p, x)
rule1449 = ReplacementRule(pattern1449, replacement1449)
pattern1450 = Pattern(Integral((A_ + x_**WC('j', S(1))*WC('B', S(1)))/sqrt(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons50, cons779, cons752, cons753, cons780, cons781)
def replacement1450(B, j, b, r, a, n, c, x, q, A):
rubi.append(1450)
return Dist(x**(q/S(2))*sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))/sqrt(a*x**q + b*x**n + c*x**(S(2)*n - q)), Int(x**(-q/S(2))*(A + B*x**(n - q))/sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q)), x), x)
rule1450 = ReplacementRule(pattern1450, replacement1450)
pattern1451 = Pattern(Integral((A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons34, cons35, cons4, cons50, cons777, cons778, cons147, cons226, cons13, cons163, cons754, cons782)
def replacement1451(B, p, j, b, r, c, n, a, x, q, A):
rubi.append(1451)
return Dist(p*(n - q)/(c*(p*(S(2)*n - q) + S(1))*(p*q + (n - q)*(S(2)*p + S(1)) + S(1))), Int(x**q*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(-1))*(S(2)*A*a*c*(p*q + (n - q)*(S(2)*p + S(1)) + S(1)) - B*a*b*(p*q + S(1)) + x**(n - q)*(A*b*c*(p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + S(2)*B*a*c*(p*(S(2)*n - q) + S(1)) - B*b**S(2)*(p*q + p*(n - q) + S(1)))), x), x) + Simp(x*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p*(A*c*(p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*b*p*(n - q) + B*c*x**(n - q)*(p*(S(2)*n - q) + S(1)))/(c*(p*(S(2)*n - q) + S(1))*(p*q + (n - q)*(S(2)*p + S(1)) + S(1))), x)
rule1451 = ReplacementRule(pattern1451, replacement1451)
def With1452(B, p, j, r, c, a, x, q, A):
if isinstance(x, (int, Integer, float, Float)):
return False
n = q + r
if And(ZeroQ(j - S(2)*n + q), NonzeroQ(p*(S(2)*n - q) + S(1)), NonzeroQ(p*q + (n - q)*(S(2)*p + S(1)) + S(1))):
return True
return False
pattern1452 = Pattern(Integral((A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**p_, x_), cons2, cons7, cons34, cons35, cons50, cons147, cons13, cons163, CustomConstraint(With1452))
def replacement1452(B, p, j, r, c, a, x, q, A):
n = q + r
rubi.append(1452)
return Dist(p*(n - q)/((p*(S(2)*n - q) + S(1))*(p*q + (n - q)*(S(2)*p + S(1)) + S(1))), Int(x**q*(a*x**q + c*x**(S(2)*n - q))**(p + S(-1))*(S(2)*A*a*(p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + S(2)*B*a*x**(n - q)*(p*(S(2)*n - q) + S(1))), x), x) + Simp(x*(A*(p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*x**(n - q)*(p*(S(2)*n - q) + S(1)))*(a*x**q + c*x**(S(2)*n - q))**p/((p*(S(2)*n - q) + S(1))*(p*q + (n - q)*(S(2)*p + S(1)) + S(1))), x)
rule1452 = ReplacementRule(pattern1452, replacement1452)
pattern1453 = Pattern(Integral((A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons34, cons35, cons4, cons50, cons777, cons778, cons147, cons226, cons13, cons137)
def replacement1453(B, p, j, b, r, c, n, a, x, q, A):
rubi.append(1453)
return Dist(S(1)/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), Int(x**(-q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))*(-S(2)*A*a*c*(p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + A*b**S(2)*(p*q + (n - q)*(p + S(1)) + S(1)) - B*a*b*(p*q + S(1)) + c*x**(n - q)*(A*b - S(2)*B*a)*(p*q + (n - q)*(S(2)*p + S(3)) + S(1))), x), x) - Simp(x**(-q + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))*(-S(2)*A*a*c + A*b**S(2) - B*a*b + c*x**(n - q)*(A*b - S(2)*B*a))/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1453 = ReplacementRule(pattern1453, replacement1453)
def With1454(B, p, j, r, c, a, x, q, A):
if isinstance(x, (int, Integer, float, Float)):
return False
n = q + r
if ZeroQ(j - S(2)*n + q):
return True
return False
pattern1454 = Pattern(Integral((A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**p_, x_), cons2, cons7, cons34, cons35, cons50, cons147, cons13, cons137, CustomConstraint(With1454))
def replacement1454(B, p, j, r, c, a, x, q, A):
n = q + r
rubi.append(1454)
return Dist(S(1)/(S(2)*a**S(2)*c*(n - q)*(p + S(1))), Int(x**(-q)*(a*x**q + c*x**(S(2)*n - q))**(p + S(1))*(A*a*c*(p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + B*a*c*x**(n - q)*(p*q + (n - q)*(S(2)*p + S(3)) + S(1))), x), x) - Simp(x**(-q + S(1))*(a*x**q + c*x**(S(2)*n - q))**(p + S(1))*(A*a*c + B*a*c*x**(n - q))/(S(2)*a**S(2)*c*(n - q)*(p + S(1))), x)
rule1454 = ReplacementRule(pattern1454, replacement1454)
pattern1455 = Pattern(Integral((A_ + x_**WC('j', S(1))*WC('B', S(1)))*(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons5, cons50, cons779, cons752)
def replacement1455(B, p, j, b, r, a, n, c, x, q, A):
rubi.append(1455)
return Int((A + B*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x)
rule1455 = ReplacementRule(pattern1455, replacement1455)
pattern1456 = Pattern(Integral((A_ + u_**WC('j', S(1))*WC('B', S(1)))*(u_**WC('n', S(1))*WC('b', S(1)) + u_**WC('q', S(1))*WC('a', S(1)) + u_**WC('r', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons5, cons50, cons779, cons752, cons68, cons69)
def replacement1456(B, p, u, j, b, r, a, n, c, x, q, A):
rubi.append(1456)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((A + B*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x), x, u), x)
rule1456 = ReplacementRule(pattern1456, replacement1456)
pattern1457 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons21, cons4, cons50, cons777, cons778, cons38, cons753)
def replacement1457(B, p, j, m, b, r, c, n, a, x, q, A):
rubi.append(1457)
return Int(x**(m + p*q)*(A + B*x**(n - q))*(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))**p, x)
rule1457 = ReplacementRule(pattern1457, replacement1457)
pattern1458 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons777, cons778, cons147, cons226, cons148, cons606, cons163, cons783, cons784, cons785)
def replacement1458(B, p, j, m, b, r, c, n, a, x, q, A):
rubi.append(1458)
return Dist(p*(n - q)/((m + p*q + S(1))*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), Int(x**(m + n)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(-1))*Simp(-A*b*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + S(2)*B*a*(m + p*q + S(1)) + x**(n - q)*(-S(2)*A*c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*b*(m + p*q + S(1))), x), x), x) + Simp(x**(m + S(1))*(A*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*x**(n - q)*(m + p*q + S(1)))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p/((m + p*q + S(1))*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), x)
rule1458 = ReplacementRule(pattern1458, replacement1458)
def With1459(B, p, j, m, r, c, a, x, q, A):
if isinstance(x, (int, Integer, float, Float)):
return False
n = q + r
if And(ZeroQ(j - S(2)*n + q), PositiveIntegerQ(n), LessEqual(m + p*q, -n + q), Unequal(m + p*q + S(1), S(0)), Unequal(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1), S(0))):
return True
return False
pattern1459 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons34, cons35, cons147, cons606, cons163, CustomConstraint(With1459))
def replacement1459(B, p, j, m, r, c, a, x, q, A):
n = q + r
rubi.append(1459)
return Dist(S(2)*p*(n - q)/((m + p*q + S(1))*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), Int(x**(m + n)*(a*x**q + c*x**(S(2)*n - q))**(p + S(-1))*Simp(-A*c*x**(n - q)*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*a*(m + p*q + S(1)), x), x), x) + Simp(x**(m + S(1))*(A*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*x**(n - q)*(m + p*q + S(1)))*(a*x**q + c*x**(S(2)*n - q))**p/((m + p*q + S(1))*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), x)
rule1459 = ReplacementRule(pattern1459, replacement1459)
pattern1460 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons777, cons778, cons147, cons226, cons148, cons606, cons137, cons786)
def replacement1460(B, p, j, m, b, r, c, n, a, x, q, A):
rubi.append(1460)
return Dist(S(1)/((n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), Int(x**(m - n)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))*Simp(x**(n - q)*(-S(2)*A*c + B*b)*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + (-A*b + S(2)*B*a)*(m - n + p*q + q + S(1)), x), x), x) + Simp(x**(m - n + S(1))*(A*b - S(2)*B*a - x**(n - q)*(-S(2)*A*c + B*b))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/((n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1460 = ReplacementRule(pattern1460, replacement1460)
def With1461(B, p, j, m, r, c, a, x, q, A):
if isinstance(x, (int, Integer, float, Float)):
return False
n = q + r
if And(ZeroQ(j - S(2)*n + q), PositiveIntegerQ(n), Greater(m + p*q, n - q + S(-1))):
return True
return False
pattern1461 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons34, cons35, cons147, cons606, cons137, CustomConstraint(With1461))
def replacement1461(B, p, j, m, r, c, a, x, q, A):
n = q + r
rubi.append(1461)
return -Dist(S(1)/(S(2)*a*c*(n - q)*(p + S(1))), Int(x**(m - n)*(a*x**q + c*x**(S(2)*n - q))**(p + S(1))*Simp(-A*c*x**(n - q)*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + B*a*(m - n + p*q + q + S(1)), x), x), x) + Simp(x**(m - n + S(1))*(a*x**q + c*x**(S(2)*n - q))**(p + S(1))*(-A*c*x**(n - q) + B*a)/(S(2)*a*c*(n - q)*(p + S(1))), x)
rule1461 = ReplacementRule(pattern1461, replacement1461)
pattern1462 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons777, cons778, cons147, cons226, cons148, cons606, cons163, cons787, cons764, cons785)
def replacement1462(B, p, j, m, b, r, c, n, a, x, q, A):
rubi.append(1462)
return Dist(p*(n - q)/(c*(m + p*(S(2)*n - q) + S(1))*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), Int(x**(m + q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(-1))*Simp(S(2)*A*a*c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) - B*a*b*(m + p*q + S(1)) + x**(n - q)*(A*b*c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + S(2)*B*a*c*(m + p*q + S(2)*p*(n - q) + S(1)) - B*b**S(2)*(m + p*q + p*(n - q) + S(1))), x), x), x) + Simp(x**(m + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p*(A*c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*b*p*(n - q) + B*c*x**(n - q)*(m + p*q + S(2)*p*(n - q) + S(1)))/(c*(m + p*(S(2)*n - q) + S(1))*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), x)
rule1462 = ReplacementRule(pattern1462, replacement1462)
def With1463(B, p, j, m, r, c, a, x, q, A):
if isinstance(x, (int, Integer, float, Float)):
return False
n = q + r
if And(ZeroQ(j - S(2)*n + q), PositiveIntegerQ(n), Greater(m + p*q, -n + q), Unequal(m + p*q + S(2)*p*(n - q) + S(1), S(0)), Unequal(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1), S(0)), Unequal(m + S(1), n)):
return True
return False
pattern1463 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons34, cons35, cons147, cons606, cons163, CustomConstraint(With1463))
def replacement1463(B, p, j, m, r, c, a, x, q, A):
n = q + r
rubi.append(1463)
return Dist(p*(n - q)/((m + p*(S(2)*n - q) + S(1))*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), Int(x**(m + q)*(a*x**q + c*x**(S(2)*n - q))**(p + S(-1))*Simp(S(2)*A*a*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + S(2)*B*a*x**(n - q)*(m + p*q + S(2)*p*(n - q) + S(1)), x), x), x) + Simp(x**(m + S(1))*(A*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*x**(n - q)*(m + p*q + S(2)*p*(n - q) + S(1)))*(a*x**q + c*x**(S(2)*n - q))**p/((m + p*(S(2)*n - q) + S(1))*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), x)
rule1463 = ReplacementRule(pattern1463, replacement1463)
pattern1464 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons777, cons778, cons147, cons226, cons148, cons606, cons137, cons788)
def replacement1464(B, p, j, m, b, r, c, n, a, x, q, A):
rubi.append(1464)
return Dist(S(1)/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), Int(x**(m - q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))*Simp(-S(2)*A*a*c*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + A*b**S(2)*(m + p*q + (n - q)*(p + S(1)) + S(1)) - B*a*b*(m + p*q + S(1)) + c*x**(n - q)*(A*b - S(2)*B*a)*(m + p*q + (n - q)*(S(2)*p + S(3)) + S(1)), x), x), x) - Simp(x**(m - q + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))*(-S(2)*A*a*c + A*b**S(2) - B*a*b + c*x**(n - q)*(A*b - S(2)*B*a))/(a*(n - q)*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1464 = ReplacementRule(pattern1464, replacement1464)
def With1465(B, p, j, m, r, c, a, x, q, A):
if isinstance(x, (int, Integer, float, Float)):
return False
n = q + r
if And(ZeroQ(j - S(2)*n + q), PositiveIntegerQ(n), Less(m + p*q, n - q + S(-1))):
return True
return False
pattern1465 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons34, cons35, cons147, cons606, cons137, CustomConstraint(With1465))
def replacement1465(B, p, j, m, r, c, a, x, q, A):
n = q + r
rubi.append(1465)
return Dist(S(1)/(S(2)*a*c*(n - q)*(p + S(1))), Int(x**(m - q)*(a*x**q + c*x**(S(2)*n - q))**(p + S(1))*Simp(A*c*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + B*c*x**(n - q)*(m + p*q + (n - q)*(S(2)*p + S(3)) + S(1)), x), x), x) - Simp(x**(m - q + S(1))*(A*c + B*c*x**(n - q))*(a*x**q + c*x**(S(2)*n - q))**(p + S(1))/(S(2)*a*c*(n - q)*(p + S(1))), x)
rule1465 = ReplacementRule(pattern1465, replacement1465)
pattern1466 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons777, cons778, cons147, cons226, cons148, cons606, cons773, cons789, cons785)
def replacement1466(B, p, j, m, b, r, c, n, a, x, q, A):
rubi.append(1466)
return -Dist(S(1)/(c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), Int(x**(m - n + q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p*Simp(B*a*(m - n + p*q + q + S(1)) + x**(n - q)*(-A*c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*b*(m + p*q + p*(n - q) + S(1))), x), x), x) + Simp(B*x**(m - n + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), x)
rule1466 = ReplacementRule(pattern1466, replacement1466)
def With1467(B, p, j, m, r, c, a, x, q, A):
if isinstance(x, (int, Integer, float, Float)):
return False
n = q + r
if And(ZeroQ(j - S(2)*n + q), PositiveIntegerQ(n), GreaterEqual(m + p*q, n - q + S(-1)), Unequal(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1), S(0))):
return True
return False
pattern1467 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons34, cons35, cons147, cons606, cons773, CustomConstraint(With1467))
def replacement1467(B, p, j, m, r, c, a, x, q, A):
n = q + r
rubi.append(1467)
return -Dist(S(1)/(c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), Int(x**(m - n + q)*(a*x**q + c*x**(S(2)*n - q))**p*Simp(-A*c*x**(n - q)*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1)) + B*a*(m - n + p*q + q + S(1)), x), x), x) + Simp(B*x**(m - n + S(1))*(a*x**q + c*x**(S(2)*n - q))**(p + S(1))/(c*(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1))), x)
rule1467 = ReplacementRule(pattern1467, replacement1467)
pattern1468 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons777, cons778, cons147, cons226, cons148, cons606, cons790, cons783, cons784)
def replacement1468(B, p, j, m, b, r, c, n, a, x, q, A):
rubi.append(1468)
return Dist(S(1)/(a*(m + p*q + S(1))), Int(x**(m + n - q)*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p*Simp(-A*b*(m + p*q + (n - q)*(p + S(1)) + S(1)) - A*c*x**(n - q)*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + B*a*(m + p*q + S(1)), x), x), x) + Simp(A*x**(m - q + S(1))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**(p + S(1))/(a*(m + p*q + S(1))), x)
rule1468 = ReplacementRule(pattern1468, replacement1468)
def With1469(B, p, j, m, r, c, a, x, q, A):
if isinstance(x, (int, Integer, float, Float)):
return False
n = q + r
if And(ZeroQ(j - S(2)*n + q), PositiveIntegerQ(n), Or(Inequality(S(-1), LessEqual, p, Less, S(0)), Equal(m + p*q + (n - q)*(S(2)*p + S(1)) + S(1), S(0))), LessEqual(m + p*q, -n + q), Unequal(m + p*q + S(1), S(0))):
return True
return False
pattern1469 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('r', S(1))*WC('B', S(1)))*(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('q', S(1))*WC('a', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons34, cons35, cons147, cons606, CustomConstraint(With1469))
def replacement1469(B, p, j, m, r, c, a, x, q, A):
n = q + r
rubi.append(1469)
return Dist(S(1)/(a*(m + p*q + S(1))), Int(x**(m + n - q)*(a*x**q + c*x**(S(2)*n - q))**p*Simp(-A*c*x**(n - q)*(m + p*q + S(2)*(n - q)*(p + S(1)) + S(1)) + B*a*(m + p*q + S(1)), x), x), x) + Simp(A*x**(m - q + S(1))*(a*x**q + c*x**(S(2)*n - q))**(p + S(1))/(a*(m + p*q + S(1))), x)
rule1469 = ReplacementRule(pattern1469, replacement1469)
pattern1470 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('j', S(1))*WC('B', S(1)))/sqrt(x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('q', S(1))*WC('a', S(1)) + x_**WC('r', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons34, cons35, cons21, cons4, cons50, cons779, cons752, cons753, cons791, cons780, cons792)
def replacement1470(B, j, m, b, r, a, n, c, x, q, A):
rubi.append(1470)
return Dist(x**(q/S(2))*sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q))/sqrt(a*x**q + b*x**n + c*x**(S(2)*n - q)), Int(x**(m - q/S(2))*(A + B*x**(n - q))/sqrt(a + b*x**(n - q) + c*x**(S(2)*n - S(2)*q)), x), x)
rule1470 = ReplacementRule(pattern1470, replacement1470)
pattern1471 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**q_*WC('B', S(1)))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons34, cons35, cons796, cons797, cons21, cons5, cons793, cons794, cons147, cons795)
def replacement1471(B, p, j, k, m, b, a, c, n, x, q, A):
rubi.append(1471)
return Dist(x**(-j*p)*(a + b*x**(-j + k) + c*x**(-S(2)*j + S(2)*k))**(-p)*(a*x**j + b*x**k + c*x**n)**p, Int(x**(j*p + m)*(A + B*x**(-j + k))*(a + b*x**(-j + k) + c*x**(-S(2)*j + S(2)*k))**p, x), x)
rule1471 = ReplacementRule(pattern1471, replacement1471)
pattern1472 = Pattern(Integral(u_**WC('m', S(1))*(A_ + u_**WC('j', S(1))*WC('B', S(1)))*(u_**WC('n', S(1))*WC('b', S(1)) + u_**WC('q', S(1))*WC('a', S(1)) + u_**WC('r', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons21, cons4, cons5, cons50, cons779, cons752, cons68, cons69)
def replacement1472(B, p, u, j, m, b, r, a, n, c, x, q, A):
rubi.append(1472)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int(x**m*(A + B*x**(n - q))*(a*x**q + b*x**n + c*x**(S(2)*n - q))**p, x), x, u), x)
rule1472 = ReplacementRule(pattern1472, replacement1472)
return [rule1076, rule1077, rule1078, rule1079, rule1080, rule1081, rule1082, rule1083, rule1084, rule1085, rule1086, rule1087, rule1088, rule1089, rule1090, rule1091, rule1092, rule1093, rule1094, rule1095, rule1096, rule1097, rule1098, rule1099, rule1100, rule1101, rule1102, rule1103, rule1104, rule1105, rule1106, rule1107, rule1108, rule1109, rule1110, rule1111, rule1112, rule1113, rule1114, rule1115, rule1116, rule1117, rule1118, rule1119, rule1120, rule1121, rule1122, rule1123, rule1124, rule1125, rule1126, rule1127, rule1128, rule1129, rule1130, rule1131, rule1132, rule1133, rule1134, rule1135, rule1136, rule1137, rule1138, rule1139, rule1140, rule1141, rule1142, rule1143, rule1144, rule1145, rule1146, rule1147, rule1148, rule1149, rule1150, rule1151, rule1152, rule1153, rule1154, rule1155, rule1156, rule1157, rule1158, rule1159, rule1160, rule1161, rule1162, rule1163, rule1164, rule1165, rule1166, rule1167, rule1168, rule1169, rule1170, rule1171, rule1172, rule1173, rule1174, rule1175, rule1176, rule1177, rule1178, rule1179, rule1180, rule1181, rule1182, rule1183, rule1184, rule1185, rule1186, rule1187, rule1188, rule1189, rule1190, rule1191, rule1192, rule1193, rule1194, rule1195, rule1196, rule1197, rule1198, rule1199, rule1200, rule1201, rule1202, rule1203, rule1204, rule1205, rule1206, rule1207, rule1208, rule1209, rule1210, rule1211, rule1212, rule1213, rule1214, rule1215, rule1216, rule1217, rule1218, rule1219, rule1220, rule1221, rule1222, rule1223, rule1224, rule1225, rule1226, rule1227, rule1228, rule1229, rule1230, rule1231, rule1232, rule1233, rule1234, rule1235, rule1236, rule1237, rule1238, rule1239, rule1240, rule1241, rule1242, rule1243, rule1244, rule1245, rule1246, rule1247, rule1248, rule1249, rule1250, rule1251, rule1252, rule1253, rule1254, rule1255, rule1256, rule1257, rule1258, rule1259, rule1260, rule1261, rule1262, rule1263, rule1264, rule1265, rule1266, rule1267, rule1268, rule1269, rule1270, rule1271, rule1272, rule1273, rule1274, rule1275, rule1276, rule1277, rule1278, rule1279, rule1280, rule1281, rule1282, rule1283, rule1284, rule1285, rule1286, rule1287, rule1288, rule1289, rule1290, rule1291, rule1292, rule1293, rule1294, rule1295, rule1296, rule1297, rule1298, rule1299, rule1300, rule1301, rule1302, rule1303, rule1304, rule1305, rule1306, rule1307, rule1308, rule1309, rule1310, rule1311, rule1312, rule1313, rule1314, rule1315, rule1316, rule1317, rule1318, rule1319, rule1320, rule1321, rule1322, rule1323, rule1324, rule1325, rule1326, rule1327, rule1328, rule1329, rule1330, rule1331, rule1332, rule1333, rule1334, rule1335, rule1336, rule1337, rule1338, rule1339, rule1340, rule1341, rule1342, rule1343, rule1344, rule1345, rule1346, rule1347, rule1348, rule1349, rule1350, rule1351, rule1352, rule1353, rule1354, rule1355, rule1356, rule1357, rule1358, rule1359, rule1360, rule1361, rule1362, rule1363, rule1364, rule1365, rule1366, rule1367, rule1368, rule1369, rule1370, rule1371, rule1372, rule1373, rule1374, rule1375, rule1376, rule1377, rule1378, rule1379, rule1380, rule1381, rule1382, rule1383, rule1384, rule1385, rule1386, rule1387, rule1388, rule1389, rule1390, rule1391, rule1392, rule1393, rule1394, rule1395, rule1396, rule1397, rule1398, rule1399, rule1400, rule1401, rule1402, rule1403, rule1404, rule1405, rule1406, rule1407, rule1408, rule1409, rule1410, rule1411, rule1412, rule1413, rule1414, rule1415, rule1416, rule1417, rule1418, rule1419, rule1420, rule1421, rule1422, rule1423, rule1424, rule1425, rule1426, rule1427, rule1428, rule1429, rule1430, rule1431, rule1432, rule1433, rule1434, rule1435, rule1436, rule1437, rule1438, rule1439, rule1440, rule1441, rule1442, rule1443, rule1444, rule1445, rule1446, rule1447, rule1448, rule1449, rule1450, rule1451, rule1452, rule1453, rule1454, rule1455, rule1456, rule1457, rule1458, rule1459, rule1460, rule1461, rule1462, rule1463, rule1464, rule1465, rule1466, rule1467, rule1468, rule1469, rule1470, rule1471, rule1472, ]
|
3bb45004e105dba442e97b3aefa71dc943b3f3e72398f9f5b90159e7c2e0d997 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def secant(rubi):
from sympy.integrals.rubi.constraints import cons1581, cons1502, cons2, cons3, cons48, cons125, cons21, cons4, cons1582, cons1512, cons1583, cons18, cons93, cons166, cons89, cons1170, cons94, cons165, cons31, cons1584, cons87, cons1359, cons23, cons1255, cons1258, cons674, cons7, cons27, cons808, cons1261, cons1585, cons1264, cons1265, cons1586, cons543, cons43, cons448, cons1267, cons744, cons1587, cons1254, cons62, cons1423, cons515, cons1320, cons1321, cons1588, cons1589, cons1590, cons1330, cons111, cons155, cons1591, cons1519, cons1336, cons1592, cons463, cons85, cons1593, cons77, cons168, cons272, cons1333, cons1594, cons1334, cons1595, cons1325, cons1596, cons1597, cons1598, cons1599, cons1553, cons1600, cons1357, cons1601, cons1602, cons1603, cons1604, cons17, cons208, cons5, cons1274, cons1605, cons1606, cons1308, cons147, cons1228, cons1507, cons148, cons1515, cons1607, cons196, cons1311, cons1608, cons1609, cons1580, cons70, cons1610, cons1611, cons79, cons1612, cons1613, cons1304, cons71, cons1412, cons1409, cons1323, cons1322, cons1614, cons80, cons1360, cons1421, cons1315, cons1231, cons1615, cons1314, cons1266, cons1616, cons150, cons1617, cons1618, cons1619, cons1620, cons1621, cons38, cons1622, cons1415, cons380, cons1428, cons34, cons35, cons1245, cons1569, cons1623, cons1624, cons1625, cons1626, cons1627, cons32, cons1549, cons1628, cons1629, cons346, cons88, cons1327, cons1630, cons1631, cons1256, cons1632, cons375, cons33, cons36, cons1433, cons1633, cons1634, cons1431, cons1635, cons1636, cons1637, cons1638, cons1639, cons1640, cons683, cons1641, cons1642, cons1643, cons1454, cons1478, cons54, cons1480, cons1479, cons1481, cons376, cons46, cons45, cons226, cons1644, cons528, cons810, cons811, cons1573, cons1495, cons68, cons69, cons823, cons824, cons1574, cons1576, cons1497, cons1577, cons1645
pattern3917 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1581, cons1502)
def replacement3917(m, f, b, a, n, x, e):
rubi.append(3917)
return Simp(a*b*(a/sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x)
rule3917 = ReplacementRule(pattern3917, replacement3917)
pattern3918 = Pattern(Integral((S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons48, cons125, cons1582)
def replacement3918(m, f, n, x, e):
rubi.append(3918)
return Dist(S(1)/f, Subst(Int(x**(-m)*(x**S(2) + S(1))**(m/S(2) + n/S(2) + S(-1)), x), x, tan(e + f*x)), x)
rule3918 = ReplacementRule(pattern3918, replacement3918)
pattern3919 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons48, cons125, cons21, cons1512, cons1583, cons18)
def replacement3919(m, f, a, n, x, e):
rubi.append(3919)
return -Dist(a**(-n + S(1))/f, Subst(Int((a*x)**(m + n + S(-1))*(x**S(2) + S(-1))**(-n/S(2) + S(-1)/2), x), x, S(1)/sin(e + f*x)), x)
rule3919 = ReplacementRule(pattern3919, replacement3919)
pattern3920 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons48, cons125, cons21, cons1512, cons1583, cons18)
def replacement3920(m, f, a, n, x, e):
rubi.append(3920)
return Dist(a**(-n + S(1))/f, Subst(Int((a*x)**(m + n + S(-1))*(x**S(2) + S(-1))**(-n/S(2) + S(-1)/2), x), x, S(1)/cos(e + f*x)), x)
rule3920 = ReplacementRule(pattern3920, replacement3920)
pattern3921 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons166, cons89, cons1170)
def replacement3921(m, f, b, a, n, x, e):
rubi.append(3921)
return Dist(a**S(2)*(n + S(1))/(b**S(2)*(m + S(-1))), Int((a/sin(e + f*x))**(m + S(-2))*(b/cos(e + f*x))**(n + S(2)), x), x) - Simp(a*(a/sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(1))/(b*f*(m + S(-1))), x)
rule3921 = ReplacementRule(pattern3921, replacement3921)
pattern3922 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons93, cons94, cons165, cons1170)
def replacement3922(m, f, b, a, n, x, e):
rubi.append(3922)
return Dist(b**S(2)*(m + S(1))/(a**S(2)*(n + S(-1))), Int((a/sin(e + f*x))**(m + S(2))*(b/cos(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a/sin(e + f*x))**(m + S(1))*(b/cos(e + f*x))**(n + S(-1))/(a*f*(n + S(-1))), x)
rule3922 = ReplacementRule(pattern3922, replacement3922)
pattern3923 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons4, cons31, cons166, cons1170, cons1584)
def replacement3923(m, f, b, a, n, x, e):
rubi.append(3923)
return Dist(a**S(2)*(m + n + S(-2))/(m + S(-1)), Int((a/sin(e + f*x))**(m + S(-2))*(b/cos(e + f*x))**n, x), x) - Simp(a*b*(a/sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(-1))/(f*(m + S(-1))), x)
rule3923 = ReplacementRule(pattern3923, replacement3923)
pattern3924 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons165, cons1170)
def replacement3924(m, f, b, a, n, x, e):
rubi.append(3924)
return Dist(b**S(2)*(m + n + S(-2))/(n + S(-1)), Int((a/sin(e + f*x))**m*(b/cos(e + f*x))**(n + S(-2)), x), x) + Simp(a*b*(a/sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x)
rule3924 = ReplacementRule(pattern3924, replacement3924)
pattern3925 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons48, cons125, cons4, cons31, cons94, cons1359, cons1170)
def replacement3925(m, f, b, a, n, x, e):
rubi.append(3925)
return Dist((m + S(1))/(a**S(2)*(m + n)), Int((a/sin(e + f*x))**(m + S(2))*(b/cos(e + f*x))**n, x), x) + Simp(b*(a/sin(e + f*x))**(m + S(1))*(b/cos(e + f*x))**(n + S(-1))/(a*f*(m + n)), x)
rule3925 = ReplacementRule(pattern3925, replacement3925)
pattern3926 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons87, cons89, cons1359, cons1170)
def replacement3926(m, f, b, a, n, x, e):
rubi.append(3926)
return Dist((n + S(1))/(b**S(2)*(m + n)), Int((a/sin(e + f*x))**m*(b/cos(e + f*x))**(n + S(2)), x), x) - Simp(a*(a/sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(1))/(b*f*(m + n)), x)
rule3926 = ReplacementRule(pattern3926, replacement3926)
pattern3927 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons23, cons1255)
def replacement3927(m, f, b, a, n, x, e):
rubi.append(3927)
return Dist((a/sin(e + f*x))**m*(b/cos(e + f*x))**n*tan(e + f*x)**(-n), Int(tan(e + f*x)**n, x), x)
rule3927 = ReplacementRule(pattern3927, replacement3927)
pattern3928 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons48, cons125, cons21, cons4, cons1258)
def replacement3928(m, f, b, a, n, x, e):
rubi.append(3928)
return Dist((a/sin(e + f*x))**m*(a*sin(e + f*x))**m*(b/cos(e + f*x))**n*(b*cos(e + f*x))**n, Int((a*sin(e + f*x))**(-m)*(b*cos(e + f*x))**(-n), x), x)
rule3928 = ReplacementRule(pattern3928, replacement3928)
pattern3929 = Pattern(Integral((S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons7, cons27, cons674)
def replacement3929(d, c, n, x):
rubi.append(3929)
return Dist(S(1)/d, Subst(Int(ExpandIntegrand((x**S(2) + S(1))**(n/S(2) + S(-1)), x), x), x, tan(c + d*x)), x)
rule3929 = ReplacementRule(pattern3929, replacement3929)
pattern3930 = Pattern(Integral((S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons7, cons27, cons674)
def replacement3930(d, c, n, x):
rubi.append(3930)
return -Dist(S(1)/d, Subst(Int(ExpandIntegrand((x**S(2) + S(1))**(n/S(2) + S(-1)), x), x), x, S(1)/tan(c + d*x)), x)
rule3930 = ReplacementRule(pattern3930, replacement3930)
pattern3931 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons165, cons808)
def replacement3931(b, d, c, n, x):
rubi.append(3931)
return Dist(b**S(2)*(n + S(-2))/(n + S(-1)), Int((b/cos(c + d*x))**(n + S(-2)), x), x) + Simp(b*(b/cos(c + d*x))**(n + S(-1))*sin(c + d*x)/(d*(n + S(-1))), x)
rule3931 = ReplacementRule(pattern3931, replacement3931)
pattern3932 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons165, cons808)
def replacement3932(b, d, c, n, x):
rubi.append(3932)
return Dist(b**S(2)*(n + S(-2))/(n + S(-1)), Int((b/sin(c + d*x))**(n + S(-2)), x), x) - Simp(b*(b/sin(c + d*x))**(n + S(-1))*cos(c + d*x)/(d*(n + S(-1))), x)
rule3932 = ReplacementRule(pattern3932, replacement3932)
pattern3933 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons89, cons808)
def replacement3933(b, d, c, n, x):
rubi.append(3933)
return Dist((n + S(1))/(b**S(2)*n), Int((b/cos(c + d*x))**(n + S(2)), x), x) - Simp((b/cos(c + d*x))**(n + S(1))*sin(c + d*x)/(b*d*n), x)
rule3933 = ReplacementRule(pattern3933, replacement3933)
pattern3934 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons89, cons808)
def replacement3934(b, d, c, n, x):
rubi.append(3934)
return Dist((n + S(1))/(b**S(2)*n), Int((b/sin(c + d*x))**(n + S(2)), x), x) + Simp((b/sin(c + d*x))**(n + S(1))*cos(c + d*x)/(b*d*n), x)
rule3934 = ReplacementRule(pattern3934, replacement3934)
pattern3935 = Pattern(Integral(S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons1261)
def replacement3935(d, c, x):
rubi.append(3935)
return Simp(atanh(sin(c + d*x))/d, x)
rule3935 = ReplacementRule(pattern3935, replacement3935)
pattern3936 = Pattern(Integral(S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons7, cons27, cons1261)
def replacement3936(d, c, x):
rubi.append(3936)
return -Simp(atanh(cos(c + d*x))/d, x)
rule3936 = ReplacementRule(pattern3936, replacement3936)
pattern3937 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons1585)
def replacement3937(b, d, c, n, x):
rubi.append(3937)
return Dist((b/cos(c + d*x))**n*cos(c + d*x)**n, Int(cos(c + d*x)**(-n), x), x)
rule3937 = ReplacementRule(pattern3937, replacement3937)
pattern3938 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons1585)
def replacement3938(b, d, c, n, x):
rubi.append(3938)
return Dist((b/sin(c + d*x))**n*sin(c + d*x)**n, Int(sin(c + d*x)**(-n), x), x)
rule3938 = ReplacementRule(pattern3938, replacement3938)
pattern3939 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons4, cons23)
def replacement3939(b, d, c, n, x):
rubi.append(3939)
return Simp((cos(c + d*x)/b)**(n + S(-1))*(b/cos(c + d*x))**(n + S(-1))*Int((cos(c + d*x)/b)**(-n), x), x)
rule3939 = ReplacementRule(pattern3939, replacement3939)
pattern3940 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons7, cons27, cons4, cons23)
def replacement3940(b, d, c, n, x):
rubi.append(3940)
return Simp((sin(c + d*x)/b)**(n + S(-1))*(b/sin(c + d*x))**(n + S(-1))*Int((sin(c + d*x)/b)**(-n), x), x)
rule3940 = ReplacementRule(pattern3940, replacement3940)
pattern3941 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement3941(b, d, c, a, x):
rubi.append(3941)
return Dist(b**S(2), Int(cos(c + d*x)**(S(-2)), x), x) + Dist(S(2)*a*b, Int(S(1)/cos(c + d*x), x), x) + Simp(a**S(2)*x, x)
rule3941 = ReplacementRule(pattern3941, replacement3941)
pattern3942 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement3942(b, d, c, a, x):
rubi.append(3942)
return Dist(b**S(2), Int(sin(c + d*x)**(S(-2)), x), x) + Dist(S(2)*a*b, Int(S(1)/sin(c + d*x), x), x) + Simp(a**S(2)*x, x)
rule3942 = ReplacementRule(pattern3942, replacement3942)
pattern3943 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement3943(b, d, c, a, x):
rubi.append(3943)
return Dist(S(2)*b/d, Subst(Int(S(1)/(a + x**S(2)), x), x, b*tan(c + d*x)/sqrt(a + b/cos(c + d*x))), x)
rule3943 = ReplacementRule(pattern3943, replacement3943)
pattern3944 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement3944(b, d, c, a, x):
rubi.append(3944)
return Dist(-S(2)*b/d, Subst(Int(S(1)/(a + x**S(2)), x), x, b/(sqrt(a + b/sin(c + d*x))*tan(c + d*x))), x)
rule3944 = ReplacementRule(pattern3944, replacement3944)
pattern3945 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1265, cons87, cons165, cons808)
def replacement3945(b, d, c, a, n, x):
rubi.append(3945)
return Dist(a/(n + S(-1)), Int((a + b/cos(c + d*x))**(n + S(-2))*(a*(n + S(-1)) + b*(S(3)*n + S(-4))/cos(c + d*x)), x), x) + Simp(b**S(2)*(a + b/cos(c + d*x))**(n + S(-2))*tan(c + d*x)/(d*(n + S(-1))), x)
rule3945 = ReplacementRule(pattern3945, replacement3945)
pattern3946 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1265, cons87, cons165, cons808)
def replacement3946(b, d, c, a, n, x):
rubi.append(3946)
return Dist(a/(n + S(-1)), Int((a + b/sin(c + d*x))**(n + S(-2))*(a*(n + S(-1)) + b*(S(3)*n + S(-4))/sin(c + d*x)), x), x) - Simp(b**S(2)*(a + b/sin(c + d*x))**(n + S(-2))/(d*(n + S(-1))*tan(c + d*x)), x)
rule3946 = ReplacementRule(pattern3946, replacement3946)
pattern3947 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement3947(b, d, c, a, x):
rubi.append(3947)
return Dist(S(1)/a, Int(sqrt(a + b/cos(c + d*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(a + b/cos(c + d*x))*cos(c + d*x)), x), x)
rule3947 = ReplacementRule(pattern3947, replacement3947)
pattern3948 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1265)
def replacement3948(b, d, c, a, x):
rubi.append(3948)
return Dist(S(1)/a, Int(sqrt(a + b/sin(c + d*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(a + b/sin(c + d*x))*sin(c + d*x)), x), x)
rule3948 = ReplacementRule(pattern3948, replacement3948)
pattern3949 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1265, cons87, cons1586, cons808)
def replacement3949(b, d, c, a, n, x):
rubi.append(3949)
return Dist(S(1)/(a**S(2)*(S(2)*n + S(1))), Int((a + b/cos(c + d*x))**(n + S(1))*(a*(S(2)*n + S(1)) - b*(n + S(1))/cos(c + d*x)), x), x) + Simp((a + b/cos(c + d*x))**n*tan(c + d*x)/(d*(S(2)*n + S(1))), x)
rule3949 = ReplacementRule(pattern3949, replacement3949)
pattern3950 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1265, cons87, cons1586, cons808)
def replacement3950(b, d, c, a, n, x):
rubi.append(3950)
return Dist(S(1)/(a**S(2)*(S(2)*n + S(1))), Int((a + b/sin(c + d*x))**(n + S(1))*(a*(S(2)*n + S(1)) - b*(n + S(1))/sin(c + d*x)), x), x) - Simp((a + b/sin(c + d*x))**n/(d*(S(2)*n + S(1))*tan(c + d*x)), x)
rule3950 = ReplacementRule(pattern3950, replacement3950)
pattern3951 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons543, cons43)
def replacement3951(b, d, c, a, n, x):
rubi.append(3951)
return -Dist(a**n*tan(c + d*x)/(d*sqrt(S(1) - S(1)/cos(c + d*x))*sqrt(S(1) + S(1)/cos(c + d*x))), Subst(Int((S(1) + b*x/a)**(n + S(-1)/2)/(x*sqrt(S(1) - b*x/a)), x), x, S(1)/cos(c + d*x)), x)
rule3951 = ReplacementRule(pattern3951, replacement3951)
pattern3952 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons543, cons43)
def replacement3952(b, d, c, a, n, x):
rubi.append(3952)
return Dist(a**n/(d*sqrt(S(1) - S(1)/sin(c + d*x))*sqrt(S(1) + S(1)/sin(c + d*x))*tan(c + d*x)), Subst(Int((S(1) + b*x/a)**(n + S(-1)/2)/(x*sqrt(S(1) - b*x/a)), x), x, S(1)/sin(c + d*x)), x)
rule3952 = ReplacementRule(pattern3952, replacement3952)
pattern3953 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons543, cons448)
def replacement3953(b, d, c, a, n, x):
rubi.append(3953)
return Dist(a**IntPart(n)*(S(1) + b/(a*cos(c + d*x)))**(-FracPart(n))*(a + b/cos(c + d*x))**FracPart(n), Int((S(1) + b/(a*cos(c + d*x)))**n, x), x)
rule3953 = ReplacementRule(pattern3953, replacement3953)
pattern3954 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1265, cons543, cons448)
def replacement3954(b, d, c, a, n, x):
rubi.append(3954)
return Dist(a**IntPart(n)*(S(1) + b/(a*sin(c + d*x)))**(-FracPart(n))*(a + b/sin(c + d*x))**FracPart(n), Int((S(1) + b/(a*sin(c + d*x)))**n, x), x)
rule3954 = ReplacementRule(pattern3954, replacement3954)
pattern3955 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement3955(b, d, c, a, x):
rubi.append(3955)
return Simp(-S(2)*sqrt(b*(S(1) + S(1)/cos(c + d*x))/(a + b/cos(c + d*x)))*sqrt(-b*(S(1) - S(1)/cos(c + d*x))/(a + b/cos(c + d*x)))*(a + b/cos(c + d*x))*EllipticPi(a/(a + b), asin(Rt(a + b, S(2))/sqrt(a + b/cos(c + d*x))), (a - b)/(a + b))/(d*Rt(a + b, S(2))*tan(c + d*x)), x)
rule3955 = ReplacementRule(pattern3955, replacement3955)
pattern3956 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement3956(b, d, c, a, x):
rubi.append(3956)
return Simp(S(2)*sqrt(b*(S(1) + S(1)/sin(c + d*x))/(a + b/sin(c + d*x)))*sqrt(-b*(S(1) - S(1)/sin(c + d*x))/(a + b/sin(c + d*x)))*(a + b/sin(c + d*x))*EllipticPi(a/(a + b), asin(Rt(a + b, S(2))/sqrt(a + b/sin(c + d*x))), (a - b)/(a + b))*tan(c + d*x)/(d*Rt(a + b, S(2))), x)
rule3956 = ReplacementRule(pattern3956, replacement3956)
pattern3957 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement3957(b, d, c, a, x):
rubi.append(3957)
return Dist(b**S(2), Int((S(1) + S(1)/cos(c + d*x))/(sqrt(a + b/cos(c + d*x))*cos(c + d*x)), x), x) + Int((a**S(2) + b*(S(2)*a - b)/cos(c + d*x))/sqrt(a + b/cos(c + d*x)), x)
rule3957 = ReplacementRule(pattern3957, replacement3957)
pattern3958 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement3958(b, d, c, a, x):
rubi.append(3958)
return Dist(b**S(2), Int((S(1) + S(1)/sin(c + d*x))/(sqrt(a + b/sin(c + d*x))*sin(c + d*x)), x), x) + Int((a**S(2) + b*(S(2)*a - b)/sin(c + d*x))/sqrt(a + b/sin(c + d*x)), x)
rule3958 = ReplacementRule(pattern3958, replacement3958)
pattern3959 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1267, cons87, cons744, cons808)
def replacement3959(b, d, c, a, n, x):
rubi.append(3959)
return Dist(S(1)/(n + S(-1)), Int((a + b/cos(c + d*x))**(n + S(-3))*Simp(a**S(3)*(n + S(-1)) + a*b**S(2)*(S(3)*n + S(-4))/cos(c + d*x)**S(2) + b*(S(3)*a**S(2)*(n + S(-1)) + b**S(2)*(n + S(-2)))/cos(c + d*x), x), x), x) + Simp(b**S(2)*(a + b/cos(c + d*x))**(n + S(-2))*tan(c + d*x)/(d*(n + S(-1))), x)
rule3959 = ReplacementRule(pattern3959, replacement3959)
pattern3960 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1267, cons87, cons744, cons808)
def replacement3960(b, d, c, a, n, x):
rubi.append(3960)
return Dist(S(1)/(n + S(-1)), Int((a + b/sin(c + d*x))**(n + S(-3))*Simp(a**S(3)*(n + S(-1)) + a*b**S(2)*(S(3)*n + S(-4))/sin(c + d*x)**S(2) + b*(S(3)*a**S(2)*(n + S(-1)) + b**S(2)*(n + S(-2)))/sin(c + d*x), x), x), x) - Simp(b**S(2)*(a + b/sin(c + d*x))**(n + S(-2))/(d*(n + S(-1))*tan(c + d*x)), x)
rule3960 = ReplacementRule(pattern3960, replacement3960)
pattern3961 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement3961(b, d, c, a, x):
rubi.append(3961)
return -Dist(S(1)/a, Int(S(1)/(a*cos(c + d*x)/b + S(1)), x), x) + Simp(x/a, x)
rule3961 = ReplacementRule(pattern3961, replacement3961)
pattern3962 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement3962(b, d, c, a, x):
rubi.append(3962)
return -Dist(S(1)/a, Int(S(1)/(a*sin(c + d*x)/b + S(1)), x), x) + Simp(x/a, x)
rule3962 = ReplacementRule(pattern3962, replacement3962)
pattern3963 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement3963(b, d, c, a, x):
rubi.append(3963)
return Simp(-S(2)*sqrt(b*(S(1) - S(1)/cos(c + d*x))/(a + b))*sqrt(-b*(S(1) + S(1)/cos(c + d*x))/(a - b))*EllipticPi((a + b)/a, asin(sqrt(a + b/cos(c + d*x))/Rt(a + b, S(2))), (a + b)/(a - b))*Rt(a + b, S(2))/(a*d*tan(c + d*x)), x)
rule3963 = ReplacementRule(pattern3963, replacement3963)
pattern3964 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement3964(b, d, c, a, x):
rubi.append(3964)
return Simp(S(2)*sqrt(b*(S(1) - S(1)/sin(c + d*x))/(a + b))*sqrt(-b*(S(1) + S(1)/sin(c + d*x))/(a - b))*EllipticPi((a + b)/a, asin(sqrt(a + b/sin(c + d*x))/Rt(a + b, S(2))), (a + b)/(a - b))*Rt(a + b, S(2))*tan(c + d*x)/(a*d), x)
rule3964 = ReplacementRule(pattern3964, replacement3964)
pattern3965 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1267, cons87, cons89, cons808)
def replacement3965(b, d, c, a, n, x):
rubi.append(3965)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(n + S(1))), Int((a + b/cos(c + d*x))**(n + S(1))*Simp(-a*b*(n + S(1))/cos(c + d*x) + b**S(2)*(n + S(2))/cos(c + d*x)**S(2) + (a**S(2) - b**S(2))*(n + S(1)), x), x), x) - Simp(b**S(2)*(a + b/cos(c + d*x))**(n + S(1))*tan(c + d*x)/(a*d*(a**S(2) - b**S(2))*(n + S(1))), x)
rule3965 = ReplacementRule(pattern3965, replacement3965)
pattern3966 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons1267, cons87, cons89, cons808)
def replacement3966(b, d, c, a, n, x):
rubi.append(3966)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(n + S(1))), Int((a + b/sin(c + d*x))**(n + S(1))*Simp(-a*b*(n + S(1))/sin(c + d*x) + b**S(2)*(n + S(2))/sin(c + d*x)**S(2) + (a**S(2) - b**S(2))*(n + S(1)), x), x), x) + Simp(b**S(2)*(a + b/sin(c + d*x))**(n + S(1))/(a*d*(a**S(2) - b**S(2))*(n + S(1))*tan(c + d*x)), x)
rule3966 = ReplacementRule(pattern3966, replacement3966)
pattern3967 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1267, cons543)
def replacement3967(b, d, c, a, n, x):
rubi.append(3967)
return Int((a + b/cos(c + d*x))**n, x)
rule3967 = ReplacementRule(pattern3967, replacement3967)
pattern3968 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons4, cons1267, cons543)
def replacement3968(b, d, c, a, n, x):
rubi.append(3968)
return Int((a + b/sin(c + d*x))**n, x)
rule3968 = ReplacementRule(pattern3968, replacement3968)
pattern3969 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1587)
def replacement3969(f, b, d, a, n, x, e):
rubi.append(3969)
return Dist(a, Int((d/cos(e + f*x))**n, x), x) + Dist(b/d, Int((d/cos(e + f*x))**(n + S(1)), x), x)
rule3969 = ReplacementRule(pattern3969, replacement3969)
pattern3970 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1587)
def replacement3970(f, b, d, a, n, x, e):
rubi.append(3970)
return Dist(a, Int((d/sin(e + f*x))**n, x), x) + Dist(b/d, Int((d/sin(e + f*x))**(n + S(1)), x), x)
rule3970 = ReplacementRule(pattern3970, replacement3970)
pattern3971 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1587)
def replacement3971(f, b, d, a, n, x, e):
rubi.append(3971)
return Dist(S(2)*a*b/d, Int((d/cos(e + f*x))**(n + S(1)), x), x) + Int((d/cos(e + f*x))**n*(a**S(2) + b**S(2)/cos(e + f*x)**S(2)), x)
rule3971 = ReplacementRule(pattern3971, replacement3971)
pattern3972 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1587)
def replacement3972(f, b, d, a, n, x, e):
rubi.append(3972)
return Dist(S(2)*a*b/d, Int((d/sin(e + f*x))**(n + S(1)), x), x) + Int((d/sin(e + f*x))**n*(a**S(2) + b**S(2)/sin(e + f*x)**S(2)), x)
rule3972 = ReplacementRule(pattern3972, replacement3972)
pattern3973 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons1254)
def replacement3973(f, b, a, x, e):
rubi.append(3973)
return Dist(S(1)/b, Int(S(1)/cos(e + f*x), x), x) - Dist(a/b, Int(S(1)/((a + b/cos(e + f*x))*cos(e + f*x)), x), x)
rule3973 = ReplacementRule(pattern3973, replacement3973)
pattern3974 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons1254)
def replacement3974(f, b, a, x, e):
rubi.append(3974)
return Dist(S(1)/b, Int(S(1)/sin(e + f*x), x), x) - Dist(a/b, Int(S(1)/((a + b/sin(e + f*x))*sin(e + f*x)), x), x)
rule3974 = ReplacementRule(pattern3974, replacement3974)
pattern3975 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(3)), x_), cons2, cons3, cons48, cons125, cons1254)
def replacement3975(f, b, a, x, e):
rubi.append(3975)
return -Dist(a/b, Int(S(1)/((a + b/cos(e + f*x))*cos(e + f*x)**S(2)), x), x) + Simp(tan(e + f*x)/(b*f), x)
rule3975 = ReplacementRule(pattern3975, replacement3975)
pattern3976 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(3)), x_), cons2, cons3, cons48, cons125, cons1254)
def replacement3976(f, b, a, x, e):
rubi.append(3976)
return -Dist(a/b, Int(S(1)/((a + b/sin(e + f*x))*sin(e + f*x)**S(2)), x), x) - Simp(S(1)/(b*f*tan(e + f*x)), x)
rule3976 = ReplacementRule(pattern3976, replacement3976)
pattern3977 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons62, cons87)
def replacement3977(m, f, b, d, a, n, x, e):
rubi.append(3977)
return Int(ExpandTrig((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m, x), x)
rule3977 = ReplacementRule(pattern3977, replacement3977)
pattern3978 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons62, cons87)
def replacement3978(m, f, b, d, a, n, x, e):
rubi.append(3978)
return Int(ExpandTrig((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m, x), x)
rule3978 = ReplacementRule(pattern3978, replacement3978)
pattern3979 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1265)
def replacement3979(f, b, a, x, e):
rubi.append(3979)
return Simp(S(2)*b*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))), x)
rule3979 = ReplacementRule(pattern3979, replacement3979)
pattern3980 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1265)
def replacement3980(f, b, a, x, e):
rubi.append(3980)
return Simp(-S(2)*b/(f*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), x)
rule3980 = ReplacementRule(pattern3980, replacement3980)
pattern3981 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1423, cons515)
def replacement3981(m, f, b, a, x, e):
rubi.append(3981)
return Dist(a*(S(2)*m + S(-1))/m, Int((a + b/cos(e + f*x))**(m + S(-1))/cos(e + f*x), x), x) + Simp(b*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*m), x)
rule3981 = ReplacementRule(pattern3981, replacement3981)
pattern3982 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1423, cons515)
def replacement3982(m, f, b, a, x, e):
rubi.append(3982)
return Dist(a*(S(2)*m + S(-1))/m, Int((a + b/sin(e + f*x))**(m + S(-1))/sin(e + f*x), x), x) - Simp(b*(a + b/sin(e + f*x))**(m + S(-1))/(f*m*tan(e + f*x)), x)
rule3982 = ReplacementRule(pattern3982, replacement3982)
pattern3983 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1265)
def replacement3983(f, b, a, x, e):
rubi.append(3983)
return Simp(tan(e + f*x)/(f*(a/cos(e + f*x) + b)), x)
rule3983 = ReplacementRule(pattern3983, replacement3983)
pattern3984 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1265)
def replacement3984(f, b, a, x, e):
rubi.append(3984)
return -Simp(S(1)/(f*(a/sin(e + f*x) + b)*tan(e + f*x)), x)
rule3984 = ReplacementRule(pattern3984, replacement3984)
pattern3985 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1265)
def replacement3985(f, b, a, x, e):
rubi.append(3985)
return Dist(S(2)/f, Subst(Int(S(1)/(S(2)*a + x**S(2)), x), x, b*tan(e + f*x)/sqrt(a + b/cos(e + f*x))), x)
rule3985 = ReplacementRule(pattern3985, replacement3985)
pattern3986 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1265)
def replacement3986(f, b, a, x, e):
rubi.append(3986)
return Dist(-S(2)/f, Subst(Int(S(1)/(S(2)*a + x**S(2)), x), x, b/(sqrt(a + b/sin(e + f*x))*tan(e + f*x))), x)
rule3986 = ReplacementRule(pattern3986, replacement3986)
pattern3987 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1320, cons515)
def replacement3987(m, f, b, a, x, e):
rubi.append(3987)
return Dist((m + S(1))/(a*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))/cos(e + f*x), x), x) - Simp(b*(a + b/cos(e + f*x))**m*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule3987 = ReplacementRule(pattern3987, replacement3987)
pattern3988 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1320, cons515)
def replacement3988(m, f, b, a, x, e):
rubi.append(3988)
return Dist((m + S(1))/(a*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))/sin(e + f*x), x), x) + Simp(b*(a + b/sin(e + f*x))**m/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule3988 = ReplacementRule(pattern3988, replacement3988)
pattern3989 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1320)
def replacement3989(m, f, b, a, x, e):
rubi.append(3989)
return Dist(m/(b*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(S(2)*m + S(1))), x)
rule3989 = ReplacementRule(pattern3989, replacement3989)
pattern3990 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1320)
def replacement3990(m, f, b, a, x, e):
rubi.append(3990)
return Dist(m/(b*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**m/(f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule3990 = ReplacementRule(pattern3990, replacement3990)
pattern3991 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1321)
def replacement3991(m, f, b, a, x, e):
rubi.append(3991)
return Dist(a*m/(b*(m + S(1))), Int((a + b/cos(e + f*x))**m/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule3991 = ReplacementRule(pattern3991, replacement3991)
pattern3992 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1321)
def replacement3992(m, f, b, a, x, e):
rubi.append(3992)
return Dist(a*m/(b*(m + S(1))), Int((a + b/sin(e + f*x))**m/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule3992 = ReplacementRule(pattern3992, replacement3992)
pattern3993 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(3), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1320)
def replacement3993(m, f, b, a, x, e):
rubi.append(3993)
return -Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(1))/cos(e + f*x))/cos(e + f*x), x), x) - Simp(b*(a + b/cos(e + f*x))**m*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule3993 = ReplacementRule(pattern3993, replacement3993)
pattern3994 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(3), x_), cons2, cons3, cons48, cons125, cons1265, cons31, cons1320)
def replacement3994(m, f, b, a, x, e):
rubi.append(3994)
return -Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(1))/sin(e + f*x))/sin(e + f*x), x), x) + Simp(b*(a + b/sin(e + f*x))**m/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule3994 = ReplacementRule(pattern3994, replacement3994)
pattern3995 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(3), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1321)
def replacement3995(m, f, b, a, x, e):
rubi.append(3995)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/cos(e + f*x))**m*(-a/cos(e + f*x) + b*(m + S(1)))/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + S(2))), x)
rule3995 = ReplacementRule(pattern3995, replacement3995)
pattern3996 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(3), x_), cons2, cons3, cons48, cons125, cons21, cons1265, cons1321)
def replacement3996(m, f, b, a, x, e):
rubi.append(3996)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/sin(e + f*x))**m*(-a/sin(e + f*x) + b*(m + S(1)))/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + S(2))*tan(e + f*x)), x)
rule3996 = ReplacementRule(pattern3996, replacement3996)
pattern3997 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1588)
def replacement3997(f, b, d, a, x, e):
rubi.append(3997)
return Dist(S(2)*a*sqrt(a*d/b)/(b*f), Subst(Int(S(1)/sqrt(S(1) + x**S(2)/a), x), x, b*tan(e + f*x)/sqrt(a + b/cos(e + f*x))), x)
rule3997 = ReplacementRule(pattern3997, replacement3997)
pattern3998 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1588)
def replacement3998(f, b, d, a, x, e):
rubi.append(3998)
return Dist(-S(2)*a*sqrt(a*d/b)/(b*f), Subst(Int(S(1)/sqrt(S(1) + x**S(2)/a), x), x, b/(sqrt(a + b/sin(e + f*x))*tan(e + f*x))), x)
rule3998 = ReplacementRule(pattern3998, replacement3998)
pattern3999 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1589)
def replacement3999(f, b, d, a, x, e):
rubi.append(3999)
return Dist(S(2)*b*d/f, Subst(Int(S(1)/(b - d*x**S(2)), x), x, b*tan(e + f*x)/(sqrt(d/cos(e + f*x))*sqrt(a + b/cos(e + f*x)))), x)
rule3999 = ReplacementRule(pattern3999, replacement3999)
pattern4000 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1589)
def replacement4000(f, b, d, a, x, e):
rubi.append(4000)
return Dist(-S(2)*b*d/f, Subst(Int(S(1)/(b - d*x**S(2)), x), x, b/(sqrt(d/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x))), x)
rule4000 = ReplacementRule(pattern4000, replacement4000)
pattern4001 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons165, cons808)
def replacement4001(f, b, d, a, n, x, e):
rubi.append(4001)
return Dist(S(2)*a*d*(n + S(-1))/(b*(S(2)*n + S(-1))), Int((d/cos(e + f*x))**(n + S(-1))*sqrt(a + b/cos(e + f*x)), x), x) + Simp(S(2)*b*d*(d/cos(e + f*x))**(n + S(-1))*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(-1))), x)
rule4001 = ReplacementRule(pattern4001, replacement4001)
pattern4002 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons165, cons808)
def replacement4002(f, b, d, a, n, x, e):
rubi.append(4002)
return Dist(S(2)*a*d*(n + S(-1))/(b*(S(2)*n + S(-1))), Int((d/sin(e + f*x))**(n + S(-1))*sqrt(a + b/sin(e + f*x)), x), x) + Simp(-S(2)*b*d*(d/sin(e + f*x))**(n + S(-1))/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(-1))*tan(e + f*x)), x)
rule4002 = ReplacementRule(pattern4002, replacement4002)
pattern4003 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265)
def replacement4003(f, b, d, a, x, e):
rubi.append(4003)
return Simp(S(2)*a*tan(e + f*x)/(f*sqrt(d/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), x)
rule4003 = ReplacementRule(pattern4003, replacement4003)
pattern4004 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265)
def replacement4004(f, b, d, a, x, e):
rubi.append(4004)
return Simp(-S(2)*a/(f*sqrt(d/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), x)
rule4004 = ReplacementRule(pattern4004, replacement4004)
pattern4005 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons1590, cons808)
def replacement4005(f, b, d, a, n, x, e):
rubi.append(4005)
return Dist(a*(S(2)*n + S(1))/(S(2)*b*d*n), Int((d/cos(e + f*x))**(n + S(1))*sqrt(a + b/cos(e + f*x)), x), x) - Simp(a*(d/cos(e + f*x))**n*tan(e + f*x)/(f*n*sqrt(a + b/cos(e + f*x))), x)
rule4005 = ReplacementRule(pattern4005, replacement4005)
pattern4006 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons1590, cons808)
def replacement4006(f, b, d, a, n, x, e):
rubi.append(4006)
return Dist(a*(S(2)*n + S(1))/(S(2)*b*d*n), Int((d/sin(e + f*x))**(n + S(1))*sqrt(a + b/sin(e + f*x)), x), x) + Simp(a*(d/sin(e + f*x))**n/(f*n*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), x)
rule4006 = ReplacementRule(pattern4006, replacement4006)
pattern4007 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265)
def replacement4007(f, b, d, a, n, x, e):
rubi.append(4007)
return -Dist(a**S(2)*d*tan(e + f*x)/(f*sqrt(a - b/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), Subst(Int((d*x)**(n + S(-1))/sqrt(a - b*x), x), x, S(1)/cos(e + f*x)), x)
rule4007 = ReplacementRule(pattern4007, replacement4007)
pattern4008 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265)
def replacement4008(f, b, d, a, n, x, e):
rubi.append(4008)
return Dist(a**S(2)*d/(f*sqrt(a - b/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), Subst(Int((d*x)**(n + S(-1))/sqrt(a - b*x), x), x, S(1)/sin(e + f*x)), x)
rule4008 = ReplacementRule(pattern4008, replacement4008)
pattern4009 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1330, cons43)
def replacement4009(f, b, d, a, x, e):
rubi.append(4009)
return Dist(sqrt(S(2))*sqrt(a)/(b*f), Subst(Int(S(1)/sqrt(x**S(2) + S(1)), x), x, b*tan(e + f*x)/(a + b/cos(e + f*x))), x)
rule4009 = ReplacementRule(pattern4009, replacement4009)
pattern4010 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons1330, cons43)
def replacement4010(f, b, d, a, x, e):
rubi.append(4010)
return -Dist(sqrt(S(2))*sqrt(a)/(b*f), Subst(Int(S(1)/sqrt(x**S(2) + S(1)), x), x, b/((a + b/sin(e + f*x))*tan(e + f*x))), x)
rule4010 = ReplacementRule(pattern4010, replacement4010)
pattern4011 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265)
def replacement4011(f, b, d, a, x, e):
rubi.append(4011)
return Dist(S(2)*b*d/(a*f), Subst(Int(S(1)/(S(2)*b - d*x**S(2)), x), x, b*tan(e + f*x)/(sqrt(d/cos(e + f*x))*sqrt(a + b/cos(e + f*x)))), x)
rule4011 = ReplacementRule(pattern4011, replacement4011)
pattern4012 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265)
def replacement4012(f, b, d, a, x, e):
rubi.append(4012)
return Dist(-S(2)*b*d/(a*f), Subst(Int(S(1)/(S(2)*b - d*x**S(2)), x), x, b/(sqrt(d/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x))), x)
rule4012 = ReplacementRule(pattern4012, replacement4012)
pattern4013 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1255, cons31, cons1423, cons515)
def replacement4013(m, f, b, d, a, n, x, e):
rubi.append(4013)
return Dist(b*(S(2)*m + S(-1))/(d*m), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-1)), x), x) + Simp(a*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*m), x)
rule4013 = ReplacementRule(pattern4013, replacement4013)
pattern4014 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1255, cons31, cons1423, cons515)
def replacement4014(m, f, b, d, a, n, x, e):
rubi.append(4014)
return Dist(b*(S(2)*m + S(-1))/(d*m), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-1)), x), x) - Simp(a*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1))/(f*m*tan(e + f*x)), x)
rule4014 = ReplacementRule(pattern4014, replacement4014)
pattern4015 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1255, cons31, cons1320, cons515)
def replacement4015(m, f, b, d, a, n, x, e):
rubi.append(4015)
return Dist(d*(m + S(1))/(b*(S(2)*m + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1)), x), x) - Simp(b*d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**m*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4015 = ReplacementRule(pattern4015, replacement4015)
pattern4016 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons1255, cons31, cons1320, cons515)
def replacement4016(m, f, b, d, a, n, x, e):
rubi.append(4016)
return Dist(d*(m + S(1))/(b*(S(2)*m + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1)), x), x) + Simp(b*d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**m/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4016 = ReplacementRule(pattern4016, replacement4016)
pattern4017 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons93, cons111, cons1320)
def replacement4017(m, f, b, d, a, n, x, e):
rubi.append(4017)
return Dist(m/(a*(S(2)*m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1)), x), x) + Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(S(2)*m + S(1))), x)
rule4017 = ReplacementRule(pattern4017, replacement4017)
pattern4018 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons93, cons111, cons1320)
def replacement4018(m, f, b, d, a, n, x, e):
rubi.append(4018)
return Dist(m/(a*(S(2)*m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1)), x), x) - Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4018 = ReplacementRule(pattern4018, replacement4018)
pattern4019 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons155, cons1321)
def replacement4019(m, f, b, d, a, n, x, e):
rubi.append(4019)
return Dist(a*m/(b*d*(m + S(1))), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m, x), x) + Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4019 = ReplacementRule(pattern4019, replacement4019)
pattern4020 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons155, cons1321)
def replacement4020(m, f, b, d, a, n, x, e):
rubi.append(4020)
return Dist(a*m/(b*d*(m + S(1))), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m, x), x) - Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4020 = ReplacementRule(pattern4020, replacement4020)
pattern4021 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons93, cons166, cons1591, cons515)
def replacement4021(m, f, b, d, a, n, x, e):
rubi.append(4021)
return -Dist(a/(d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-2))*(-a*(m + S(2)*n + S(-1))/cos(e + f*x) + b*(m - S(2)*n + S(-2))), x), x) - Simp(b**S(2)*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-2))*tan(e + f*x)/(f*n), x)
rule4021 = ReplacementRule(pattern4021, replacement4021)
pattern4022 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons93, cons166, cons1591, cons515)
def replacement4022(m, f, b, d, a, n, x, e):
rubi.append(4022)
return -Dist(a/(d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-2))*(-a*(m + S(2)*n + S(-1))/sin(e + f*x) + b*(m - S(2)*n + S(-2))), x), x) + Simp(b**S(2)*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-2))/(f*n*tan(e + f*x)), x)
rule4022 = ReplacementRule(pattern4022, replacement4022)
pattern4023 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons31, cons166, cons1519, cons515)
def replacement4023(m, f, b, d, a, n, x, e):
rubi.append(4023)
return Dist(b/(m + n + S(-1)), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-2))*(a*(S(3)*m + S(2)*n + S(-4))/cos(e + f*x) + b*(m + S(2)*n + S(-1))), x), x) + Simp(b**S(2)*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-2))*tan(e + f*x)/(f*(m + n + S(-1))), x)
rule4023 = ReplacementRule(pattern4023, replacement4023)
pattern4024 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons31, cons166, cons1519, cons515)
def replacement4024(m, f, b, d, a, n, x, e):
rubi.append(4024)
return Dist(b/(m + n + S(-1)), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-2))*(a*(S(3)*m + S(2)*n + S(-4))/sin(e + f*x) + b*(m + S(2)*n + S(-1))), x), x) - Simp(b**S(2)*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-2))/(f*(m + n + S(-1))*tan(e + f*x)), x)
rule4024 = ReplacementRule(pattern4024, replacement4024)
pattern4025 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons93, cons94, cons1336, cons1592)
def replacement4025(m, f, b, d, a, n, x, e):
rubi.append(4025)
return -Dist(d/(a*b*(S(2)*m + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*(a*(n + S(-1)) - b*(m + n)/cos(e + f*x)), x), x) - Simp(b*d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**m*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4025 = ReplacementRule(pattern4025, replacement4025)
pattern4026 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons93, cons94, cons1336, cons1592)
def replacement4026(m, f, b, d, a, n, x, e):
rubi.append(4026)
return -Dist(d/(a*b*(S(2)*m + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*(a*(n + S(-1)) - b*(m + n)/sin(e + f*x)), x), x) + Simp(b*d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**m/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4026 = ReplacementRule(pattern4026, replacement4026)
pattern4027 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons93, cons94, cons744, cons1592)
def replacement4027(m, f, b, d, a, n, x, e):
rubi.append(4027)
return Dist(d**S(2)/(a*b*(S(2)*m + S(1))), Int((d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**(m + S(1))*(a*(m - n + S(2))/cos(e + f*x) + b*(n + S(-2))), x), x) + Simp(d**S(2)*(d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(S(2)*m + S(1))), x)
rule4027 = ReplacementRule(pattern4027, replacement4027)
pattern4028 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons93, cons94, cons744, cons1592)
def replacement4028(m, f, b, d, a, n, x, e):
rubi.append(4028)
return Dist(d**S(2)/(a*b*(S(2)*m + S(1))), Int((d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**(m + S(1))*(a*(m - n + S(2))/sin(e + f*x) + b*(n + S(-2))), x), x) - Simp(d**S(2)*(d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**m/(f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4028 = ReplacementRule(pattern4028, replacement4028)
pattern4029 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons31, cons94, cons1592)
def replacement4029(m, f, b, d, a, n, x, e):
rubi.append(4029)
return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*(a*(S(2)*m + n + S(1)) - b*(m + n + S(1))/cos(e + f*x)), x), x) + Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(S(2)*m + S(1))), x)
rule4029 = ReplacementRule(pattern4029, replacement4029)
pattern4030 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265, cons31, cons94, cons1592)
def replacement4030(m, f, b, d, a, n, x, e):
rubi.append(4030)
return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*(a*(S(2)*m + n + S(1)) - b*(m + n + S(1))/sin(e + f*x)), x), x) - Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4030 = ReplacementRule(pattern4030, replacement4030)
pattern4031 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons165)
def replacement4031(f, b, d, a, n, x, e):
rubi.append(4031)
return -Dist(d**S(2)/(a*b), Int((d/cos(e + f*x))**(n + S(-2))*(-a*(n + S(-1))/cos(e + f*x) + b*(n + S(-2))), x), x) - Simp(d**S(2)*(d/cos(e + f*x))**(n + S(-2))*tan(e + f*x)/(f*(a + b/cos(e + f*x))), x)
rule4031 = ReplacementRule(pattern4031, replacement4031)
pattern4032 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons165)
def replacement4032(f, b, d, a, n, x, e):
rubi.append(4032)
return -Dist(d**S(2)/(a*b), Int((d/sin(e + f*x))**(n + S(-2))*(-a*(n + S(-1))/sin(e + f*x) + b*(n + S(-2))), x), x) + Simp(d**S(2)*(d/sin(e + f*x))**(n + S(-2))/(f*(a + b/sin(e + f*x))*tan(e + f*x)), x)
rule4032 = ReplacementRule(pattern4032, replacement4032)
pattern4033 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons463)
def replacement4033(f, b, d, a, n, x, e):
rubi.append(4033)
return -Dist(a**(S(-2)), Int((d/cos(e + f*x))**n*(a*(n + S(-1)) - b*n/cos(e + f*x)), x), x) - Simp((d/cos(e + f*x))**n*tan(e + f*x)/(f*(a + b/cos(e + f*x))), x)
rule4033 = ReplacementRule(pattern4033, replacement4033)
pattern4034 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons463)
def replacement4034(f, b, d, a, n, x, e):
rubi.append(4034)
return -Dist(a**(S(-2)), Int((d/sin(e + f*x))**n*(a*(n + S(-1)) - b*n/sin(e + f*x)), x), x) + Simp((d/sin(e + f*x))**n/(f*(a + b/sin(e + f*x))*tan(e + f*x)), x)
rule4034 = ReplacementRule(pattern4034, replacement4034)
pattern4035 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265)
def replacement4035(f, b, d, a, n, x, e):
rubi.append(4035)
return Dist(d*(n + S(-1))/(a*b), Int((d/cos(e + f*x))**(n + S(-1))*(a - b/cos(e + f*x)), x), x) + Simp(b*d*(d/cos(e + f*x))**(n + S(-1))*tan(e + f*x)/(a*f*(a + b/cos(e + f*x))), x)
rule4035 = ReplacementRule(pattern4035, replacement4035)
pattern4036 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1265)
def replacement4036(f, b, d, a, n, x, e):
rubi.append(4036)
return Dist(d*(n + S(-1))/(a*b), Int((d/sin(e + f*x))**(n + S(-1))*(a - b/sin(e + f*x)), x), x) - Simp(b*d*(d/sin(e + f*x))**(n + S(-1))/(a*f*(a + b/sin(e + f*x))*tan(e + f*x)), x)
rule4036 = ReplacementRule(pattern4036, replacement4036)
pattern4037 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265)
def replacement4037(f, b, d, a, x, e):
rubi.append(4037)
return Dist(d/b, Int(sqrt(d/cos(e + f*x))*sqrt(a + b/cos(e + f*x)), x), x) - Dist(a*d/b, Int(sqrt(d/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x)
rule4037 = ReplacementRule(pattern4037, replacement4037)
pattern4038 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265)
def replacement4038(f, b, d, a, x, e):
rubi.append(4038)
return Dist(d/b, Int(sqrt(d/sin(e + f*x))*sqrt(a + b/sin(e + f*x)), x), x) - Dist(a*d/b, Int(sqrt(d/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x)
rule4038 = ReplacementRule(pattern4038, replacement4038)
pattern4039 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons744, cons808)
def replacement4039(f, b, d, a, n, x, e):
rubi.append(4039)
return Dist(d**S(2)/(b*(S(2)*n + S(-3))), Int((d/cos(e + f*x))**(n + S(-2))*(-a/cos(e + f*x) + S(2)*b*(n + S(-2)))/sqrt(a + b/cos(e + f*x)), x), x) + Simp(S(2)*d**S(2)*(d/cos(e + f*x))**(n + S(-2))*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(-3))), x)
rule4039 = ReplacementRule(pattern4039, replacement4039)
pattern4040 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons744, cons808)
def replacement4040(f, b, d, a, n, x, e):
rubi.append(4040)
return Dist(d**S(2)/(b*(S(2)*n + S(-3))), Int((d/sin(e + f*x))**(n + S(-2))*(-a/sin(e + f*x) + S(2)*b*(n + S(-2)))/sqrt(a + b/sin(e + f*x)), x), x) + Simp(-S(2)*d**S(2)*(d/sin(e + f*x))**(n + S(-2))/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(-3))*tan(e + f*x)), x)
rule4040 = ReplacementRule(pattern4040, replacement4040)
pattern4041 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons463, cons808)
def replacement4041(f, b, d, a, n, x, e):
rubi.append(4041)
return Dist(S(1)/(S(2)*b*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b*(S(2)*n + S(1))/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) - Simp((d/cos(e + f*x))**n*tan(e + f*x)/(f*n*sqrt(a + b/cos(e + f*x))), x)
rule4041 = ReplacementRule(pattern4041, replacement4041)
pattern4042 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1265, cons87, cons463, cons808)
def replacement4042(f, b, d, a, n, x, e):
rubi.append(4042)
return Dist(S(1)/(S(2)*b*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b*(S(2)*n + S(1))/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) + Simp((d/sin(e + f*x))**n/(f*n*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), x)
rule4042 = ReplacementRule(pattern4042, replacement4042)
pattern4043 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1265, cons87, cons744, cons1519, cons85)
def replacement4043(m, f, b, d, a, n, x, e):
rubi.append(4043)
return Dist(d**S(2)/(b*(m + n + S(-1))), Int((d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**m*(a*m/cos(e + f*x) + b*(n + S(-2))), x), x) + Simp(d**S(2)*(d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + n + S(-1))), x)
rule4043 = ReplacementRule(pattern4043, replacement4043)
pattern4044 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1265, cons87, cons744, cons1519, cons85)
def replacement4044(m, f, b, d, a, n, x, e):
rubi.append(4044)
return Dist(d**S(2)/(b*(m + n + S(-1))), Int((d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**m*(a*m/sin(e + f*x) + b*(n + S(-2))), x), x) - Simp(d**S(2)*(d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**m/(f*(m + n + S(-1))*tan(e + f*x)), x)
rule4044 = ReplacementRule(pattern4044, replacement4044)
pattern4045 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43, cons23, cons1588)
def replacement4045(m, f, b, d, a, n, x, e):
rubi.append(4045)
return Dist(a**(-n + S(2))*(a*d/b)**n*tan(e + f*x)/(f*sqrt(a - b/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), Subst(Int((a - x)**(n + S(-1))*(S(2)*a - x)**(m + S(-1)/2)/sqrt(x), x), x, a - b/cos(e + f*x)), x)
rule4045 = ReplacementRule(pattern4045, replacement4045)
pattern4046 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43, cons23, cons1588)
def replacement4046(m, f, b, d, a, n, x, e):
rubi.append(4046)
return -Dist(a**(-n + S(2))*(a*d/b)**n/(f*sqrt(a - b/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), Subst(Int((a - x)**(n + S(-1))*(S(2)*a - x)**(m + S(-1)/2)/sqrt(x), x), x, a - b/sin(e + f*x)), x)
rule4046 = ReplacementRule(pattern4046, replacement4046)
pattern4047 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43, cons23, cons1593)
def replacement4047(m, f, b, d, a, n, x, e):
rubi.append(4047)
return Dist(a**(-n + S(1))*(-a*d/b)**n*tan(e + f*x)/(f*sqrt(a - b/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), Subst(Int(x**(m + S(-1)/2)*(a - x)**(n + S(-1))/sqrt(S(2)*a - x), x), x, a + b/cos(e + f*x)), x)
rule4047 = ReplacementRule(pattern4047, replacement4047)
pattern4048 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43, cons23, cons1593)
def replacement4048(m, f, b, d, a, n, x, e):
rubi.append(4048)
return -Dist(a**(-n + S(1))*(-a*d/b)**n/(f*sqrt(a - b/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), Subst(Int(x**(m + S(-1)/2)*(a - x)**(n + S(-1))/sqrt(S(2)*a - x), x), x, a + b/sin(e + f*x)), x)
rule4048 = ReplacementRule(pattern4048, replacement4048)
pattern4049 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43)
def replacement4049(m, f, b, d, a, n, x, e):
rubi.append(4049)
return -Dist(a**S(2)*d*tan(e + f*x)/(f*sqrt(a - b/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), Subst(Int((d*x)**(n + S(-1))*(a + b*x)**(m + S(-1)/2)/sqrt(a - b*x), x), x, S(1)/cos(e + f*x)), x)
rule4049 = ReplacementRule(pattern4049, replacement4049)
pattern4050 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons43)
def replacement4050(m, f, b, d, a, n, x, e):
rubi.append(4050)
return Dist(a**S(2)*d/(f*sqrt(a - b/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), Subst(Int((d*x)**(n + S(-1))*(a + b*x)**(m + S(-1)/2)/sqrt(a - b*x), x), x, S(1)/sin(e + f*x)), x)
rule4050 = ReplacementRule(pattern4050, replacement4050)
pattern4051 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons448)
def replacement4051(m, f, b, d, a, n, x, e):
rubi.append(4051)
return Dist(a**IntPart(m)*(S(1) + b/(a*cos(e + f*x)))**(-FracPart(m))*(a + b/cos(e + f*x))**FracPart(m), Int((d/cos(e + f*x))**n*(S(1) + b/(a*cos(e + f*x)))**m, x), x)
rule4051 = ReplacementRule(pattern4051, replacement4051)
pattern4052 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1265, cons18, cons448)
def replacement4052(m, f, b, d, a, n, x, e):
rubi.append(4052)
return Dist(a**IntPart(m)*(S(1) + b/(a*sin(e + f*x)))**(-FracPart(m))*(a + b/sin(e + f*x))**FracPart(m), Int((d/sin(e + f*x))**n*(S(1) + b/(a*sin(e + f*x)))**m, x), x)
rule4052 = ReplacementRule(pattern4052, replacement4052)
pattern4053 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4053(f, b, a, x, e):
rubi.append(4053)
return Dist(b, Int((S(1) + S(1)/cos(e + f*x))/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) + Dist(a - b, Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x)
rule4053 = ReplacementRule(pattern4053, replacement4053)
pattern4054 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4054(f, b, a, x, e):
rubi.append(4054)
return Dist(b, Int((S(1) + S(1)/sin(e + f*x))/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) + Dist(a - b, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x)
rule4054 = ReplacementRule(pattern4054, replacement4054)
pattern4055 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons166, cons515)
def replacement4055(m, f, b, a, x, e):
rubi.append(4055)
return Dist(S(1)/m, Int((a + b/cos(e + f*x))**(m + S(-2))*(a**S(2)*m + a*b*(S(2)*m + S(-1))/cos(e + f*x) + b**S(2)*(m + S(-1)))/cos(e + f*x), x), x) + Simp(b*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*m), x)
rule4055 = ReplacementRule(pattern4055, replacement4055)
pattern4056 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons166, cons515)
def replacement4056(m, f, b, a, x, e):
rubi.append(4056)
return Dist(S(1)/m, Int((a + b/sin(e + f*x))**(m + S(-2))*(a**S(2)*m + a*b*(S(2)*m + S(-1))/sin(e + f*x) + b**S(2)*(m + S(-1)))/sin(e + f*x), x), x) - Simp(b*(a + b/sin(e + f*x))**(m + S(-1))/(f*m*tan(e + f*x)), x)
rule4056 = ReplacementRule(pattern4056, replacement4056)
pattern4057 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4057(f, b, a, x, e):
rubi.append(4057)
return Dist(S(1)/b, Int(S(1)/(a*cos(e + f*x)/b + S(1)), x), x)
rule4057 = ReplacementRule(pattern4057, replacement4057)
pattern4058 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4058(f, b, a, x, e):
rubi.append(4058)
return Dist(S(1)/b, Int(S(1)/(a*sin(e + f*x)/b + S(1)), x), x)
rule4058 = ReplacementRule(pattern4058, replacement4058)
pattern4059 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4059(f, b, a, x, e):
rubi.append(4059)
return Simp(S(2)*sqrt(b*(S(1) - S(1)/cos(e + f*x))/(a + b))*sqrt(-b*(S(1) + S(1)/cos(e + f*x))/(a - b))*EllipticF(asin(sqrt(a + b/cos(e + f*x))/Rt(a + b, S(2))), (a + b)/(a - b))*Rt(a + b, S(2))/(b*f*tan(e + f*x)), x)
rule4059 = ReplacementRule(pattern4059, replacement4059)
pattern4060 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4060(f, b, a, x, e):
rubi.append(4060)
return Simp(-S(2)*sqrt(b*(S(1) - S(1)/sin(e + f*x))/(a + b))*sqrt(-b*(S(1) + S(1)/sin(e + f*x))/(a - b))*EllipticF(asin(sqrt(a + b/sin(e + f*x))/Rt(a + b, S(2))), (a + b)/(a - b))*Rt(a + b, S(2))*tan(e + f*x)/(b*f), x)
rule4060 = ReplacementRule(pattern4060, replacement4060)
pattern4061 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons94, cons515)
def replacement4061(m, f, b, a, x, e):
rubi.append(4061)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*(a*(m + S(1)) - b*(m + S(2))/cos(e + f*x))/cos(e + f*x), x), x) + Simp(b*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4061 = ReplacementRule(pattern4061, replacement4061)
pattern4062 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons94, cons515)
def replacement4062(m, f, b, a, x, e):
rubi.append(4062)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*(a*(m + S(1)) - b*(m + S(2))/sin(e + f*x))/sin(e + f*x), x), x) - Simp(b*(a + b/sin(e + f*x))**(m + S(1))/(f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4062 = ReplacementRule(pattern4062, replacement4062)
pattern4063 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons77)
def replacement4063(m, f, b, a, x, e):
rubi.append(4063)
return -Dist(tan(e + f*x)/(f*sqrt(S(1) - S(1)/cos(e + f*x))*sqrt(S(1) + S(1)/cos(e + f*x))), Subst(Int((a + b*x)**m/(sqrt(-x + S(1))*sqrt(x + S(1))), x), x, S(1)/cos(e + f*x)), x)
rule4063 = ReplacementRule(pattern4063, replacement4063)
pattern4064 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons77)
def replacement4064(m, f, b, a, x, e):
rubi.append(4064)
return Dist(S(1)/(f*sqrt(S(1) - S(1)/sin(e + f*x))*sqrt(S(1) + S(1)/sin(e + f*x))*tan(e + f*x)), Subst(Int((a + b*x)**m/(sqrt(-x + S(1))*sqrt(x + S(1))), x), x, S(1)/sin(e + f*x)), x)
rule4064 = ReplacementRule(pattern4064, replacement4064)
pattern4065 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons168)
def replacement4065(m, f, b, a, x, e):
rubi.append(4065)
return Dist(m/(m + S(1)), Int((a + b/cos(e + f*x))**(m + S(-1))*(a/cos(e + f*x) + b)/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4065 = ReplacementRule(pattern4065, replacement4065)
pattern4066 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons168)
def replacement4066(m, f, b, a, x, e):
rubi.append(4066)
return Dist(m/(m + S(1)), Int((a + b/sin(e + f*x))**(m + S(-1))*(a/sin(e + f*x) + b)/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4066 = ReplacementRule(pattern4066, replacement4066)
pattern4067 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons94)
def replacement4067(m, f, b, a, x, e):
rubi.append(4067)
return -Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*(-a*(m + S(2))/cos(e + f*x) + b*(m + S(1)))/cos(e + f*x), x), x) - Simp(a*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4067 = ReplacementRule(pattern4067, replacement4067)
pattern4068 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons94)
def replacement4068(m, f, b, a, x, e):
rubi.append(4068)
return -Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*(-a*(m + S(2))/sin(e + f*x) + b*(m + S(1)))/sin(e + f*x), x), x) + Simp(a*(a + b/sin(e + f*x))**(m + S(1))/(f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4068 = ReplacementRule(pattern4068, replacement4068)
pattern4069 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4069(f, b, a, x, e):
rubi.append(4069)
return -Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x) + Int((S(1) + S(1)/cos(e + f*x))/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x)
rule4069 = ReplacementRule(pattern4069, replacement4069)
pattern4070 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4070(f, b, a, x, e):
rubi.append(4070)
return -Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x) + Int((S(1) + S(1)/sin(e + f*x))/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x)
rule4070 = ReplacementRule(pattern4070, replacement4070)
pattern4071 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1267)
def replacement4071(m, f, b, a, x, e):
rubi.append(4071)
return Dist(S(1)/b, Int((a + b/cos(e + f*x))**(m + S(1))/cos(e + f*x), x), x) - Dist(a/b, Int((a + b/cos(e + f*x))**m/cos(e + f*x), x), x)
rule4071 = ReplacementRule(pattern4071, replacement4071)
pattern4072 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1267)
def replacement4072(m, f, b, a, x, e):
rubi.append(4072)
return Dist(S(1)/b, Int((a + b/sin(e + f*x))**(m + S(1))/sin(e + f*x), x), x) - Dist(a/b, Int((a + b/sin(e + f*x))**m/sin(e + f*x), x), x)
rule4072 = ReplacementRule(pattern4072, replacement4072)
pattern4073 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(3), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons94)
def replacement4073(m, f, b, a, x, e):
rubi.append(4073)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(a*b*(m + S(1)) - (a**S(2) + b**S(2)*(m + S(1)))/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp(a**S(2)*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4073 = ReplacementRule(pattern4073, replacement4073)
pattern4074 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(3), x_), cons2, cons3, cons48, cons125, cons1267, cons31, cons94)
def replacement4074(m, f, b, a, x, e):
rubi.append(4074)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(a*b*(m + S(1)) - (a**S(2) + b**S(2)*(m + S(1)))/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp(a**S(2)*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4074 = ReplacementRule(pattern4074, replacement4074)
pattern4075 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(3), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons272)
def replacement4075(m, f, b, a, x, e):
rubi.append(4075)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/cos(e + f*x))**m*(-a/cos(e + f*x) + b*(m + S(1)))/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + S(2))), x)
rule4075 = ReplacementRule(pattern4075, replacement4075)
pattern4076 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(3), x_), cons2, cons3, cons48, cons125, cons21, cons1267, cons272)
def replacement4076(m, f, b, a, x, e):
rubi.append(4076)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/sin(e + f*x))**m*(-a/sin(e + f*x) + b*(m + S(1)))/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + S(2))*tan(e + f*x)), x)
rule4076 = ReplacementRule(pattern4076, replacement4076)
pattern4077 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons1333, cons1594)
def replacement4077(m, f, b, d, a, n, x, e):
rubi.append(4077)
return -Dist(S(1)/(d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-3))*Simp(a**S(2)*b*(m - S(2)*n + S(-2)) - a*(a**S(2)*(n + S(1)) + S(3)*b**S(2)*n)/cos(e + f*x) - b*(a**S(2)*(m + n + S(-1)) + b**S(2)*n)/cos(e + f*x)**S(2), x), x), x) - Simp(a**S(2)*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-2))*tan(e + f*x)/(f*n), x)
rule4077 = ReplacementRule(pattern4077, replacement4077)
pattern4078 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons1333, cons1594)
def replacement4078(m, f, b, d, a, n, x, e):
rubi.append(4078)
return -Dist(S(1)/(d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-3))*Simp(a**S(2)*b*(m - S(2)*n + S(-2)) - a*(a**S(2)*(n + S(1)) + S(3)*b**S(2)*n)/sin(e + f*x) - b*(a**S(2)*(m + n + S(-1)) + b**S(2)*n)/sin(e + f*x)**S(2), x), x), x) + Simp(a**S(2)*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-2))/(f*n*tan(e + f*x)), x)
rule4078 = ReplacementRule(pattern4078, replacement4078)
pattern4079 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1267, cons31, cons1333, cons1334, cons1595)
def replacement4079(m, f, b, d, a, n, x, e):
rubi.append(4079)
return Dist(S(1)/(d*(m + n + S(-1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-3))*Simp(a**S(3)*d*(m + n + S(-1)) + a*b**S(2)*d*n + a*b**S(2)*d*(S(3)*m + S(2)*n + S(-4))/cos(e + f*x)**S(2) + b*(S(3)*a**S(2)*d*(m + n + S(-1)) + b**S(2)*d*(m + n + S(-2)))/cos(e + f*x), x), x), x) + Simp(b**S(2)*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-2))*tan(e + f*x)/(f*(m + n + S(-1))), x)
rule4079 = ReplacementRule(pattern4079, replacement4079)
pattern4080 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1267, cons31, cons1333, cons1334, cons1595)
def replacement4080(m, f, b, d, a, n, x, e):
rubi.append(4080)
return Dist(S(1)/(d*(m + n + S(-1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-3))*Simp(a**S(3)*d*(m + n + S(-1)) + a*b**S(2)*d*n + a*b**S(2)*d*(S(3)*m + S(2)*n + S(-4))/sin(e + f*x)**S(2) + b*(S(3)*a**S(2)*d*(m + n + S(-1)) + b**S(2)*d*(m + n + S(-2)))/sin(e + f*x), x), x), x) - Simp(b**S(2)*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-2))/(f*(m + n + S(-1))*tan(e + f*x)), x)
rule4080 = ReplacementRule(pattern4080, replacement4080)
pattern4081 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons94, cons1325, cons1170)
def replacement4081(m, f, b, d, a, n, x, e):
rubi.append(4081)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*Simp(a*d*(m + S(1))/cos(e + f*x) + b*d*(n + S(-1)) - b*d*(m + n + S(1))/cos(e + f*x)**S(2), x), x), x) + Simp(b*d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4081 = ReplacementRule(pattern4081, replacement4081)
pattern4082 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons94, cons1325, cons1170)
def replacement4082(m, f, b, d, a, n, x, e):
rubi.append(4082)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*Simp(a*d*(m + S(1))/sin(e + f*x) + b*d*(n + S(-1)) - b*d*(m + n + S(1))/sin(e + f*x)**S(2), x), x), x) - Simp(b*d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))/(f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4082 = ReplacementRule(pattern4082, replacement4082)
pattern4083 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons94, cons1336, cons1170)
def replacement4083(m, f, b, d, a, n, x, e):
rubi.append(4083)
return -Dist(d**S(2)/((a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**(m + S(1))*(-a*(m + n)/cos(e + f*x)**S(2) + a*(n + S(-2)) + b*(m + S(1))/cos(e + f*x)), x), x) - Simp(a*d**S(2)*(d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4083 = ReplacementRule(pattern4083, replacement4083)
pattern4084 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons94, cons1336, cons1170)
def replacement4084(m, f, b, d, a, n, x, e):
rubi.append(4084)
return -Dist(d**S(2)/((a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**(m + S(1))*(-a*(m + n)/sin(e + f*x)**S(2) + a*(n + S(-2)) + b*(m + S(1))/sin(e + f*x)), x), x) + Simp(a*d**S(2)*(d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**(m + S(1))/(f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4084 = ReplacementRule(pattern4084, replacement4084)
pattern4085 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons94, cons1596)
def replacement4085(m, f, b, d, a, n, x, e):
rubi.append(4085)
return Dist(d**S(3)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**(n + S(-3))*(a + b/cos(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(-3)) + a*b*(m + S(1))/cos(e + f*x) - (a**S(2)*(n + S(-2)) + b**S(2)*(m + S(1)))/cos(e + f*x)**S(2), x), x), x) + Simp(a**S(2)*d**S(3)*(d/cos(e + f*x))**(n + S(-3))*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4085 = ReplacementRule(pattern4085, replacement4085)
pattern4086 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons94, cons1596)
def replacement4086(m, f, b, d, a, n, x, e):
rubi.append(4086)
return Dist(d**S(3)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**(n + S(-3))*(a + b/sin(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(-3)) + a*b*(m + S(1))/sin(e + f*x) - (a**S(2)*(n + S(-2)) + b**S(2)*(m + S(1)))/sin(e + f*x)**S(2), x), x), x) - Simp(a**S(2)*d**S(3)*(d/sin(e + f*x))**(n + S(-3))*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4086 = ReplacementRule(pattern4086, replacement4086)
pattern4087 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1597)
def replacement4087(m, f, b, d, a, n, x, e):
rubi.append(4087)
return -Dist(S(1)/(a*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m*Simp(-a*(n + S(1))/cos(e + f*x) + b*(m + n + S(1)) - b*(m + n + S(2))/cos(e + f*x)**S(2), x), x), x) - Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(a*f*n), x)
rule4087 = ReplacementRule(pattern4087, replacement4087)
pattern4088 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons1597)
def replacement4088(m, f, b, d, a, n, x, e):
rubi.append(4088)
return -Dist(S(1)/(a*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m*Simp(-a*(n + S(1))/sin(e + f*x) + b*(m + n + S(1)) - b*(m + n + S(2))/sin(e + f*x)**S(2), x), x), x) + Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))/(a*f*n*tan(e + f*x)), x)
rule4088 = ReplacementRule(pattern4088, replacement4088)
pattern4089 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1267, cons31, cons94, cons1170)
def replacement4089(m, f, b, d, a, n, x, e):
rubi.append(4089)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*(a**S(2)*(m + S(1)) - a*b*(m + S(1))/cos(e + f*x) - b**S(2)*(m + n + S(1)) + b**S(2)*(m + n + S(2))/cos(e + f*x)**S(2)), x), x) - Simp(b**S(2)*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(a*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4089 = ReplacementRule(pattern4089, replacement4089)
pattern4090 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons1267, cons31, cons94, cons1170)
def replacement4090(m, f, b, d, a, n, x, e):
rubi.append(4090)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*(a**S(2)*(m + S(1)) - a*b*(m + S(1))/sin(e + f*x) - b**S(2)*(m + n + S(1)) + b**S(2)*(m + n + S(2))/sin(e + f*x)**S(2)), x), x) + Simp(b**S(2)*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))/(a*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4090 = ReplacementRule(pattern4090, replacement4090)
pattern4091 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4091(f, b, d, a, x, e):
rubi.append(4091)
return Dist(sqrt(d/cos(e + f*x))*sqrt(d*cos(e + f*x))/d, Int(sqrt(d*cos(e + f*x))/(a*cos(e + f*x) + b), x), x)
rule4091 = ReplacementRule(pattern4091, replacement4091)
pattern4092 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4092(f, b, d, a, x, e):
rubi.append(4092)
return Dist(sqrt(d/sin(e + f*x))*sqrt(d*sin(e + f*x))/d, Int(sqrt(d*sin(e + f*x))/(a*sin(e + f*x) + b), x), x)
rule4092 = ReplacementRule(pattern4092, replacement4092)
pattern4093 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4093(f, b, d, a, x, e):
rubi.append(4093)
return Dist(d*sqrt(d/cos(e + f*x))*sqrt(d*cos(e + f*x)), Int(S(1)/(sqrt(d*cos(e + f*x))*(a*cos(e + f*x) + b)), x), x)
rule4093 = ReplacementRule(pattern4093, replacement4093)
pattern4094 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4094(f, b, d, a, x, e):
rubi.append(4094)
return Dist(d*sqrt(d/sin(e + f*x))*sqrt(d*sin(e + f*x)), Int(S(1)/(sqrt(d*sin(e + f*x))*(a*sin(e + f*x) + b)), x), x)
rule4094 = ReplacementRule(pattern4094, replacement4094)
pattern4095 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(5)/2)/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4095(f, b, d, a, x, e):
rubi.append(4095)
return Dist(d/b, Int((d/cos(e + f*x))**(S(3)/2), x), x) - Dist(a*d/b, Int((d/cos(e + f*x))**(S(3)/2)/(a + b/cos(e + f*x)), x), x)
rule4095 = ReplacementRule(pattern4095, replacement4095)
pattern4096 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(5)/2)/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4096(f, b, d, a, x, e):
rubi.append(4096)
return Dist(d/b, Int((d/sin(e + f*x))**(S(3)/2), x), x) - Dist(a*d/b, Int((d/sin(e + f*x))**(S(3)/2)/(a + b/sin(e + f*x)), x), x)
rule4096 = ReplacementRule(pattern4096, replacement4096)
pattern4097 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons1598)
def replacement4097(f, b, d, a, n, x, e):
rubi.append(4097)
return Dist(d**S(3)/(b*(n + S(-2))), Int((d/cos(e + f*x))**(n + S(-3))*Simp(a*(n + S(-3)) - a*(n + S(-2))/cos(e + f*x)**S(2) + b*(n + S(-3))/cos(e + f*x), x)/(a + b/cos(e + f*x)), x), x) + Simp(d**S(3)*(d/cos(e + f*x))**(n + S(-3))*tan(e + f*x)/(b*f*(n + S(-2))), x)
rule4097 = ReplacementRule(pattern4097, replacement4097)
pattern4098 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons1598)
def replacement4098(f, b, d, a, n, x, e):
rubi.append(4098)
return Dist(d**S(3)/(b*(n + S(-2))), Int((d/sin(e + f*x))**(n + S(-3))*Simp(a*(n + S(-3)) - a*(n + S(-2))/sin(e + f*x)**S(2) + b*(n + S(-3))/sin(e + f*x), x)/(a + b/sin(e + f*x)), x), x) - Simp(d**S(3)*(d/sin(e + f*x))**(n + S(-3))/(b*f*(n + S(-2))*tan(e + f*x)), x)
rule4098 = ReplacementRule(pattern4098, replacement4098)
pattern4099 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4099(f, b, d, a, x, e):
rubi.append(4099)
return Dist(a**(S(-2)), Int((a - b/cos(e + f*x))/sqrt(d/cos(e + f*x)), x), x) + Dist(b**S(2)/(a**S(2)*d**S(2)), Int((d/cos(e + f*x))**(S(3)/2)/(a + b/cos(e + f*x)), x), x)
rule4099 = ReplacementRule(pattern4099, replacement4099)
pattern4100 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4100(f, b, d, a, x, e):
rubi.append(4100)
return Dist(a**(S(-2)), Int((a - b/sin(e + f*x))/sqrt(d/sin(e + f*x)), x), x) + Dist(b**S(2)/(a**S(2)*d**S(2)), Int((d/sin(e + f*x))**(S(3)/2)/(a + b/sin(e + f*x)), x), x)
rule4100 = ReplacementRule(pattern4100, replacement4100)
pattern4101 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons1586, cons808)
def replacement4101(f, b, d, a, n, x, e):
rubi.append(4101)
return -Dist(S(1)/(a*d*n), Int((d/cos(e + f*x))**(n + S(1))*Simp(-a*(n + S(1))/cos(e + f*x) + b*n - b*(n + S(1))/cos(e + f*x)**S(2), x)/(a + b/cos(e + f*x)), x), x) - Simp((d/cos(e + f*x))**n*tan(e + f*x)/(a*f*n), x)
rule4101 = ReplacementRule(pattern4101, replacement4101)
pattern4102 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons1586, cons808)
def replacement4102(f, b, d, a, n, x, e):
rubi.append(4102)
return -Dist(S(1)/(a*d*n), Int((d/sin(e + f*x))**(n + S(1))*Simp(-a*(n + S(1))/sin(e + f*x) + b*n - b*(n + S(1))/sin(e + f*x)**S(2), x)/(a + b/sin(e + f*x)), x), x) + Simp((d/sin(e + f*x))**n/(a*f*n*tan(e + f*x)), x)
rule4102 = ReplacementRule(pattern4102, replacement4102)
pattern4103 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4103(f, b, d, a, x, e):
rubi.append(4103)
return Dist(a, Int(sqrt(d/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) + Dist(b/d, Int((d/cos(e + f*x))**(S(3)/2)/sqrt(a + b/cos(e + f*x)), x), x)
rule4103 = ReplacementRule(pattern4103, replacement4103)
pattern4104 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4104(f, b, d, a, x, e):
rubi.append(4104)
return Dist(a, Int(sqrt(d/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) + Dist(b/d, Int((d/sin(e + f*x))**(S(3)/2)/sqrt(a + b/sin(e + f*x)), x), x)
rule4104 = ReplacementRule(pattern4104, replacement4104)
pattern4105 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons165, cons808)
def replacement4105(f, b, d, a, n, x, e):
rubi.append(4105)
return Dist(d**S(2)/(S(2)*n + S(-1)), Int((d/cos(e + f*x))**(n + S(-2))*Simp(S(2)*a*(n + S(-2)) + a/cos(e + f*x)**S(2) + b*(S(2)*n + S(-3))/cos(e + f*x), x)/sqrt(a + b/cos(e + f*x)), x), x) + Simp(S(2)*d*(d/cos(e + f*x))**(n + S(-1))*sqrt(a + b/cos(e + f*x))*sin(e + f*x)/(f*(S(2)*n + S(-1))), x)
rule4105 = ReplacementRule(pattern4105, replacement4105)
pattern4106 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons165, cons808)
def replacement4106(f, b, d, a, n, x, e):
rubi.append(4106)
return Dist(d**S(2)/(S(2)*n + S(-1)), Int((d/sin(e + f*x))**(n + S(-2))*Simp(S(2)*a*(n + S(-2)) + a/sin(e + f*x)**S(2) + b*(S(2)*n + S(-3))/sin(e + f*x), x)/sqrt(a + b/sin(e + f*x)), x), x) + Simp(-S(2)*d*(d/sin(e + f*x))**(n + S(-1))*sqrt(a + b/sin(e + f*x))*cos(e + f*x)/(f*(S(2)*n + S(-1))), x)
rule4106 = ReplacementRule(pattern4106, replacement4106)
pattern4107 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4107(f, b, d, a, x, e):
rubi.append(4107)
return Dist(sqrt(a + b/cos(e + f*x))/(sqrt(d/cos(e + f*x))*sqrt(a*cos(e + f*x) + b)), Int(sqrt(a*cos(e + f*x) + b), x), x)
rule4107 = ReplacementRule(pattern4107, replacement4107)
pattern4108 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4108(f, b, d, a, x, e):
rubi.append(4108)
return Dist(sqrt(a + b/sin(e + f*x))/(sqrt(d/sin(e + f*x))*sqrt(a*sin(e + f*x) + b)), Int(sqrt(a*sin(e + f*x) + b), x), x)
rule4108 = ReplacementRule(pattern4108, replacement4108)
pattern4109 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons1586, cons808)
def replacement4109(f, b, d, a, n, x, e):
rubi.append(4109)
return -Dist(S(1)/(S(2)*d*n), Int((d/cos(e + f*x))**(n + S(1))*Simp(-S(2)*a*(n + S(1))/cos(e + f*x) - b*(S(2)*n + S(3))/cos(e + f*x)**S(2) + b, x)/sqrt(a + b/cos(e + f*x)), x), x) - Simp((d/cos(e + f*x))**n*sqrt(a + b/cos(e + f*x))*tan(e + f*x)/(f*n), x)
rule4109 = ReplacementRule(pattern4109, replacement4109)
pattern4110 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons1586, cons808)
def replacement4110(f, b, d, a, n, x, e):
rubi.append(4110)
return -Dist(S(1)/(S(2)*d*n), Int((d/sin(e + f*x))**(n + S(1))*Simp(-S(2)*a*(n + S(1))/sin(e + f*x) - b*(S(2)*n + S(3))/sin(e + f*x)**S(2) + b, x)/sqrt(a + b/sin(e + f*x)), x), x) + Simp((d/sin(e + f*x))**n*sqrt(a + b/sin(e + f*x))/(f*n*tan(e + f*x)), x)
rule4110 = ReplacementRule(pattern4110, replacement4110)
pattern4111 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4111(f, b, d, a, x, e):
rubi.append(4111)
return Dist(sqrt(d/cos(e + f*x))*sqrt(a*cos(e + f*x) + b)/sqrt(a + b/cos(e + f*x)), Int(S(1)/sqrt(a*cos(e + f*x) + b), x), x)
rule4111 = ReplacementRule(pattern4111, replacement4111)
pattern4112 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4112(f, b, d, a, x, e):
rubi.append(4112)
return Dist(sqrt(d/sin(e + f*x))*sqrt(a*sin(e + f*x) + b)/sqrt(a + b/sin(e + f*x)), Int(S(1)/sqrt(a*sin(e + f*x) + b), x), x)
rule4112 = ReplacementRule(pattern4112, replacement4112)
pattern4113 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4113(f, b, d, a, x, e):
rubi.append(4113)
return Dist(d*sqrt(d/cos(e + f*x))*sqrt(a*cos(e + f*x) + b)/sqrt(a + b/cos(e + f*x)), Int(S(1)/(sqrt(a*cos(e + f*x) + b)*cos(e + f*x)), x), x)
rule4113 = ReplacementRule(pattern4113, replacement4113)
pattern4114 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4114(f, b, d, a, x, e):
rubi.append(4114)
return Dist(d*sqrt(d/sin(e + f*x))*sqrt(a*sin(e + f*x) + b)/sqrt(a + b/sin(e + f*x)), Int(S(1)/(sqrt(a*sin(e + f*x) + b)*sin(e + f*x)), x), x)
rule4114 = ReplacementRule(pattern4114, replacement4114)
pattern4115 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons744, cons808)
def replacement4115(f, b, d, a, n, x, e):
rubi.append(4115)
return Dist(d**S(3)/(b*(S(2)*n + S(-3))), Int((d/cos(e + f*x))**(n + S(-3))*Simp(S(2)*a*(n + S(-3)) - S(2)*a*(n + S(-2))/cos(e + f*x)**S(2) + b*(S(2)*n + S(-5))/cos(e + f*x), x)/sqrt(a + b/cos(e + f*x)), x), x) + Simp(S(2)*d**S(2)*(d/cos(e + f*x))**(n + S(-2))*sqrt(a + b/cos(e + f*x))*sin(e + f*x)/(b*f*(S(2)*n + S(-3))), x)
rule4115 = ReplacementRule(pattern4115, replacement4115)
pattern4116 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons744, cons808)
def replacement4116(f, b, d, a, n, x, e):
rubi.append(4116)
return Dist(d**S(3)/(b*(S(2)*n + S(-3))), Int((d/sin(e + f*x))**(n + S(-3))*Simp(S(2)*a*(n + S(-3)) - S(2)*a*(n + S(-2))/sin(e + f*x)**S(2) + b*(S(2)*n + S(-5))/sin(e + f*x), x)/sqrt(a + b/sin(e + f*x)), x), x) + Simp(-S(2)*d**S(2)*(d/sin(e + f*x))**(n + S(-2))*sqrt(a + b/sin(e + f*x))*cos(e + f*x)/(b*f*(S(2)*n + S(-3))), x)
rule4116 = ReplacementRule(pattern4116, replacement4116)
pattern4117 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4117(f, b, a, x, e):
rubi.append(4117)
return -Dist(b/(S(2)*a), Int((S(1) + cos(e + f*x)**(S(-2)))/sqrt(a + b/cos(e + f*x)), x), x) + Simp(sqrt(a + b/cos(e + f*x))*sin(e + f*x)/(a*f), x)
rule4117 = ReplacementRule(pattern4117, replacement4117)
pattern4118 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons1267)
def replacement4118(f, b, a, x, e):
rubi.append(4118)
return -Dist(b/(S(2)*a), Int((S(1) + sin(e + f*x)**(S(-2)))/sqrt(a + b/sin(e + f*x)), x), x) - Simp(sqrt(a + b/sin(e + f*x))*cos(e + f*x)/(a*f), x)
rule4118 = ReplacementRule(pattern4118, replacement4118)
pattern4119 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4119(f, b, d, a, x, e):
rubi.append(4119)
return Dist(S(1)/a, Int(sqrt(a + b/cos(e + f*x))/sqrt(d/cos(e + f*x)), x), x) - Dist(b/(a*d), Int(sqrt(d/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x)
rule4119 = ReplacementRule(pattern4119, replacement4119)
pattern4120 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4120(f, b, d, a, x, e):
rubi.append(4120)
return Dist(S(1)/a, Int(sqrt(a + b/sin(e + f*x))/sqrt(d/sin(e + f*x)), x), x) - Dist(b/(a*d), Int(sqrt(d/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x)
rule4120 = ReplacementRule(pattern4120, replacement4120)
pattern4121 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons89, cons808)
def replacement4121(f, b, d, a, n, x, e):
rubi.append(4121)
return Dist(S(1)/(S(2)*a*d*n), Int((d/cos(e + f*x))**(n + S(1))*Simp(S(2)*a*(n + S(1))/cos(e + f*x) - b*(S(2)*n + S(1)) + b*(S(2)*n + S(3))/cos(e + f*x)**S(2), x)/sqrt(a + b/cos(e + f*x)), x), x) - Simp((d/cos(e + f*x))**(n + S(1))*sqrt(a + b/cos(e + f*x))*sin(e + f*x)/(a*d*f*n), x)
rule4121 = ReplacementRule(pattern4121, replacement4121)
pattern4122 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons89, cons808)
def replacement4122(f, b, d, a, n, x, e):
rubi.append(4122)
return Dist(S(1)/(S(2)*a*d*n), Int((d/sin(e + f*x))**(n + S(1))*Simp(S(2)*a*(n + S(1))/sin(e + f*x) - b*(S(2)*n + S(1)) + b*(S(2)*n + S(3))/sin(e + f*x)**S(2), x)/sqrt(a + b/sin(e + f*x)), x), x) + Simp((d/sin(e + f*x))**(n + S(1))*sqrt(a + b/sin(e + f*x))*cos(e + f*x)/(a*d*f*n), x)
rule4122 = ReplacementRule(pattern4122, replacement4122)
pattern4123 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons1586, cons1599)
def replacement4123(f, b, d, a, n, x, e):
rubi.append(4123)
return Dist(S(1)/(S(2)*d*n), Int((d/cos(e + f*x))**(n + S(1))*Simp(a*b*(S(2)*n + S(-1)) + a*b*(S(2)*n + S(3))/cos(e + f*x)**S(2) + S(2)*(a**S(2)*(n + S(1)) + b**S(2)*n)/cos(e + f*x), x)/sqrt(a + b/cos(e + f*x)), x), x) - Simp(a*(d/cos(e + f*x))**n*sqrt(a + b/cos(e + f*x))*tan(e + f*x)/(f*n), x)
rule4123 = ReplacementRule(pattern4123, replacement4123)
pattern4124 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons87, cons1586, cons1599)
def replacement4124(f, b, d, a, n, x, e):
rubi.append(4124)
return Dist(S(1)/(S(2)*d*n), Int((d/sin(e + f*x))**(n + S(1))*Simp(a*b*(S(2)*n + S(-1)) + a*b*(S(2)*n + S(3))/sin(e + f*x)**S(2) + S(2)*(a**S(2)*(n + S(1)) + b**S(2)*n)/sin(e + f*x), x)/sqrt(a + b/sin(e + f*x)), x), x) + Simp(a*(d/sin(e + f*x))**n*sqrt(a + b/sin(e + f*x))/(f*n*tan(e + f*x)), x)
rule4124 = ReplacementRule(pattern4124, replacement4124)
pattern4125 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1267, cons87, cons1598, cons1553, cons1600)
def replacement4125(m, f, b, d, a, n, x, e):
rubi.append(4125)
return Dist(d**S(3)/(b*(m + n + S(-1))), Int((d/cos(e + f*x))**(n + S(-3))*(a + b/cos(e + f*x))**m*Simp(a*(n + S(-3)) - a*(n + S(-2))/cos(e + f*x)**S(2) + b*(m + n + S(-2))/cos(e + f*x), x), x), x) + Simp(d**S(3)*(d/cos(e + f*x))**(n + S(-3))*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + n + S(-1))), x)
rule4125 = ReplacementRule(pattern4125, replacement4125)
pattern4126 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons1267, cons87, cons1598, cons1553, cons1600)
def replacement4126(m, f, b, d, a, n, x, e):
rubi.append(4126)
return Dist(d**S(3)/(b*(m + n + S(-1))), Int((d/sin(e + f*x))**(n + S(-3))*(a + b/sin(e + f*x))**m*Simp(a*(n + S(-3)) - a*(n + S(-2))/sin(e + f*x)**S(2) + b*(m + n + S(-2))/sin(e + f*x), x), x), x) - Simp(d**S(3)*(d/sin(e + f*x))**(n + S(-3))*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + n + S(-1))*tan(e + f*x)), x)
rule4126 = ReplacementRule(pattern4126, replacement4126)
pattern4127 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons1357, cons1601, cons1519, cons1334)
def replacement4127(m, f, b, d, a, n, x, e):
rubi.append(4127)
return Dist(d/(m + n + S(-1)), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(-2))*Simp(a*b*(n + S(-1)) + a*b*(S(2)*m + n + S(-2))/cos(e + f*x)**S(2) + (a**S(2)*(m + n + S(-1)) + b**S(2)*(m + n + S(-2)))/cos(e + f*x), x), x), x) + Simp(b*d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*(m + n + S(-1))), x)
rule4127 = ReplacementRule(pattern4127, replacement4127)
pattern4128 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons1357, cons1601, cons1519, cons1334)
def replacement4128(m, f, b, d, a, n, x, e):
rubi.append(4128)
return Dist(d/(m + n + S(-1)), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(-2))*Simp(a*b*(n + S(-1)) + a*b*(S(2)*m + n + S(-2))/sin(e + f*x)**S(2) + (a**S(2)*(m + n + S(-1)) + b**S(2)*(m + n + S(-2)))/sin(e + f*x), x), x), x) - Simp(b*d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(-1))/(f*(m + n + S(-1))*tan(e + f*x)), x)
rule4128 = ReplacementRule(pattern4128, replacement4128)
pattern4129 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons1602, cons1603, cons1519, cons1553)
def replacement4129(m, f, b, d, a, n, x, e):
rubi.append(4129)
return Dist(d**S(2)/(b*(m + n + S(-1))), Int((d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**(m + S(-1))*Simp(a*b*m/cos(e + f*x)**S(2) + a*b*(n + S(-2)) + b**S(2)*(m + n + S(-2))/cos(e + f*x), x), x), x) + Simp(d**S(2)*(d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + n + S(-1))), x)
rule4129 = ReplacementRule(pattern4129, replacement4129)
pattern4130 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons1267, cons93, cons1602, cons1603, cons1519, cons1553)
def replacement4130(m, f, b, d, a, n, x, e):
rubi.append(4130)
return Dist(d**S(2)/(b*(m + n + S(-1))), Int((d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**(m + S(-1))*Simp(a*b*m/sin(e + f*x)**S(2) + a*b*(n + S(-2)) + b**S(2)*(m + n + S(-2))/sin(e + f*x), x), x), x) - Simp(d**S(2)*(d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**m/(f*(m + n + S(-1))*tan(e + f*x)), x)
rule4130 = ReplacementRule(pattern4130, replacement4130)
pattern4131 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4131(f, b, d, a, x, e):
rubi.append(4131)
return Dist(a, Int(sqrt(a + b/cos(e + f*x))/sqrt(d/cos(e + f*x)), x), x) + Dist(b/d, Int(sqrt(d/cos(e + f*x))*sqrt(a + b/cos(e + f*x)), x), x)
rule4131 = ReplacementRule(pattern4131, replacement4131)
pattern4132 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons1267)
def replacement4132(f, b, d, a, x, e):
rubi.append(4132)
return Dist(a, Int(sqrt(a + b/sin(e + f*x))/sqrt(d/sin(e + f*x)), x), x) + Dist(b/d, Int(sqrt(d/sin(e + f*x))*sqrt(a + b/sin(e + f*x)), x), x)
rule4132 = ReplacementRule(pattern4132, replacement4132)
pattern4133 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons62)
def replacement4133(m, f, b, d, a, n, x, e):
rubi.append(4133)
return Dist(a, Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1)), x), x) + Dist(b/d, Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-1)), x), x)
rule4133 = ReplacementRule(pattern4133, replacement4133)
pattern4134 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons4, cons62)
def replacement4134(m, f, b, d, a, n, x, e):
rubi.append(4134)
return Dist(a, Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1)), x), x) + Dist(b/d, Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-1)), x), x)
rule4134 = ReplacementRule(pattern4134, replacement4134)
pattern4135 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1604)
def replacement4135(m, f, b, d, a, n, x, e):
rubi.append(4135)
return Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m, x)
rule4135 = ReplacementRule(pattern4135, replacement4135)
pattern4136 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons21, cons4, cons1604)
def replacement4136(m, f, b, d, a, n, x, e):
rubi.append(4136)
return Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m, x)
rule4136 = ReplacementRule(pattern4136, replacement4136)
pattern4137 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons17)
def replacement4137(p, m, f, b, g, a, x, e):
rubi.append(4137)
return Int((g*sin(e + f*x))**p*(a*cos(e + f*x) + b)**m*cos(e + f*x)**(-m), x)
rule4137 = ReplacementRule(pattern4137, replacement4137)
pattern4138 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons5, cons17)
def replacement4138(p, m, f, b, g, a, x, e):
rubi.append(4138)
return Int((g*cos(e + f*x))**p*(a*sin(e + f*x) + b)**m*sin(e + f*x)**(-m), x)
rule4138 = ReplacementRule(pattern4138, replacement4138)
pattern4139 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1274, cons1265)
def replacement4139(p, m, f, b, a, x, e):
rubi.append(4139)
return Dist(b**(-p + S(1))/f, Subst(Int(x**(-p + S(-1))*(-a + b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, S(1)/cos(e + f*x)), x)
rule4139 = ReplacementRule(pattern4139, replacement4139)
pattern4140 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1274, cons1265)
def replacement4140(p, m, f, b, a, x, e):
rubi.append(4140)
return -Dist(b**(-p + S(1))/f, Subst(Int(x**(-p + S(-1))*(-a + b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, S(1)/sin(e + f*x)), x)
rule4140 = ReplacementRule(pattern4140, replacement4140)
pattern4141 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1274, cons1267)
def replacement4141(p, m, f, b, a, x, e):
rubi.append(4141)
return Dist(S(1)/f, Subst(Int(x**(-p + S(-1))*(a + b*x)**m*(x + S(-1))**(p/S(2) + S(-1)/2)*(x + S(1))**(p/S(2) + S(-1)/2), x), x, S(1)/cos(e + f*x)), x)
rule4141 = ReplacementRule(pattern4141, replacement4141)
pattern4142 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons48, cons125, cons21, cons1274, cons1267)
def replacement4142(p, m, f, b, a, x, e):
rubi.append(4142)
return -Dist(S(1)/f, Subst(Int(x**(-p + S(-1))*(a + b*x)**m*(x + S(-1))**(p/S(2) + S(-1)/2)*(x + S(1))**(p/S(2) + S(-1)/2), x), x, S(1)/sin(e + f*x)), x)
rule4142 = ReplacementRule(pattern4142, replacement4142)
pattern4143 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1605)
def replacement4143(m, f, b, a, x, e):
rubi.append(4143)
return Dist(b*m, Int((a + b/cos(e + f*x))**(m + S(-1))/cos(e + f*x), x), x) - Simp((a + b/cos(e + f*x))**m/(f*tan(e + f*x)), x)
rule4143 = ReplacementRule(pattern4143, replacement4143)
pattern4144 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons21, cons1605)
def replacement4144(m, f, b, a, x, e):
rubi.append(4144)
return Dist(b*m, Int((a + b/sin(e + f*x))**(m + S(-1))/sin(e + f*x), x), x) + Simp((a + b/sin(e + f*x))**m*tan(e + f*x)/f, x)
rule4144 = ReplacementRule(pattern4144, replacement4144)
pattern4145 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1606)
def replacement4145(p, m, f, b, g, a, x, e):
rubi.append(4145)
return Dist((a + b/cos(e + f*x))**FracPart(m)*(a*cos(e + f*x) + b)**(-FracPart(m))*cos(e + f*x)**FracPart(m), Int((g*sin(e + f*x))**p*(a*cos(e + f*x) + b)**m*cos(e + f*x)**(-m), x), x)
rule4145 = ReplacementRule(pattern4145, replacement4145)
pattern4146 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1606)
def replacement4146(p, m, f, b, g, a, x, e):
rubi.append(4146)
return Dist((a + b/sin(e + f*x))**FracPart(m)*(a*sin(e + f*x) + b)**(-FracPart(m))*sin(e + f*x)**FracPart(m), Int((g*cos(e + f*x))**p*(a*sin(e + f*x) + b)**m*sin(e + f*x)**(-m), x), x)
rule4146 = ReplacementRule(pattern4146, replacement4146)
pattern4147 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1308)
def replacement4147(p, m, f, b, g, a, x, e):
rubi.append(4147)
return Int((g*sin(e + f*x))**p*(a + b/cos(e + f*x))**m, x)
rule4147 = ReplacementRule(pattern4147, replacement4147)
pattern4148 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons1308)
def replacement4148(p, m, f, b, g, a, x, e):
rubi.append(4148)
return Int((g*cos(e + f*x))**p*(a + b/sin(e + f*x))**m, x)
rule4148 = ReplacementRule(pattern4148, replacement4148)
pattern4149 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons147)
def replacement4149(p, m, f, b, g, a, x, e):
rubi.append(4149)
return Dist((g/sin(e + f*x))**p*(g*sin(e + f*x))**p, Int((g*sin(e + f*x))**(-p)*(a + b/cos(e + f*x))**m, x), x)
rule4149 = ReplacementRule(pattern4149, replacement4149)
pattern4150 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons48, cons125, cons208, cons21, cons5, cons147)
def replacement4150(p, m, f, b, g, a, x, e):
rubi.append(4150)
return Dist((g/cos(e + f*x))**p*(g*cos(e + f*x))**p, Int((g*cos(e + f*x))**(-p)*(a + b/sin(e + f*x))**m, x), x)
rule4150 = ReplacementRule(pattern4150, replacement4150)
pattern4151 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1228, cons1265, cons85)
def replacement4151(m, b, d, c, n, a, x):
rubi.append(4151)
return -Dist(a**(-m + n + S(1))*b**(-n)/d, Subst(Int(x**(-m - n)*(a - b*x)**(m/S(2) + S(-1)/2)*(a + b*x)**(m/S(2) + n + S(-1)/2), x), x, cos(c + d*x)), x)
rule4151 = ReplacementRule(pattern4151, replacement4151)
pattern4152 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1228, cons1265, cons85)
def replacement4152(m, b, d, c, n, a, x):
rubi.append(4152)
return Dist(a**(-m + n + S(1))*b**(-n)/d, Subst(Int(x**(-m - n)*(a - b*x)**(m/S(2) + S(-1)/2)*(a + b*x)**(m/S(2) + n + S(-1)/2), x), x, sin(c + d*x)), x)
rule4152 = ReplacementRule(pattern4152, replacement4152)
pattern4153 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1228, cons1265, cons23)
def replacement4153(m, b, d, c, a, n, x):
rubi.append(4153)
return Dist(b**(-m + S(1))/d, Subst(Int((-a + b*x)**(m/S(2) + S(-1)/2)*(a + b*x)**(m/S(2) + n + S(-1)/2)/x, x), x, S(1)/cos(c + d*x)), x)
rule4153 = ReplacementRule(pattern4153, replacement4153)
pattern4154 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1228, cons1265, cons23)
def replacement4154(m, b, d, c, a, n, x):
rubi.append(4154)
return -Dist(b**(-m + S(1))/d, Subst(Int((-a + b*x)**(m/S(2) + S(-1)/2)*(a + b*x)**(m/S(2) + n + S(-1)/2)/x, x), x, S(1)/sin(c + d*x)), x)
rule4154 = ReplacementRule(pattern4154, replacement4154)
pattern4155 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166)
def replacement4155(m, b, d, c, a, x, e):
rubi.append(4155)
return -Dist(e**S(2)/m, Int((e*tan(c + d*x))**(m + S(-2))*(a*m + b*(m + S(-1))/cos(c + d*x)), x), x) + Simp(e*(e*tan(c + d*x))**(m + S(-1))*(a*m + b*(m + S(-1))/cos(c + d*x))/(d*m*(m + S(-1))), x)
rule4155 = ReplacementRule(pattern4155, replacement4155)
pattern4156 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons166)
def replacement4156(m, b, d, c, a, x, e):
rubi.append(4156)
return -Dist(e**S(2)/m, Int((e/tan(c + d*x))**(m + S(-2))*(a*m + b*(m + S(-1))/sin(c + d*x)), x), x) - Simp(e*(e/tan(c + d*x))**(m + S(-1))*(a*m + b*(m + S(-1))/sin(c + d*x))/(d*m*(m + S(-1))), x)
rule4156 = ReplacementRule(pattern4156, replacement4156)
pattern4157 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94)
def replacement4157(m, b, d, c, a, x, e):
rubi.append(4157)
return -Dist(S(1)/(e**S(2)*(m + S(1))), Int((e*tan(c + d*x))**(m + S(2))*(a*(m + S(1)) + b*(m + S(2))/cos(c + d*x)), x), x) + Simp((e*tan(c + d*x))**(m + S(1))*(a + b/cos(c + d*x))/(d*e*(m + S(1))), x)
rule4157 = ReplacementRule(pattern4157, replacement4157)
pattern4158 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons31, cons94)
def replacement4158(m, b, d, c, a, x, e):
rubi.append(4158)
return -Dist(S(1)/(e**S(2)*(m + S(1))), Int((e/tan(c + d*x))**(m + S(2))*(a*(m + S(1)) + b*(m + S(2))/sin(c + d*x)), x), x) - Simp((e/tan(c + d*x))**(m + S(1))*(a + b/sin(c + d*x))/(d*e*(m + S(1))), x)
rule4158 = ReplacementRule(pattern4158, replacement4158)
pattern4159 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))/tan(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement4159(b, d, c, a, x):
rubi.append(4159)
return Int((a*cos(c + d*x) + b)/sin(c + d*x), x)
rule4159 = ReplacementRule(pattern4159, replacement4159)
pattern4160 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))*tan(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement4160(b, d, c, a, x):
rubi.append(4160)
return Int((a*sin(c + d*x) + b)/cos(c + d*x), x)
rule4160 = ReplacementRule(pattern4160, replacement4160)
pattern4161 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1507)
def replacement4161(m, b, d, c, a, x, e):
rubi.append(4161)
return Dist(a, Int((e*tan(c + d*x))**m, x), x) + Dist(b, Int((e*tan(c + d*x))**m/cos(c + d*x), x), x)
rule4161 = ReplacementRule(pattern4161, replacement4161)
pattern4162 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1507)
def replacement4162(m, b, d, c, a, x, e):
rubi.append(4162)
return Dist(a, Int((e/tan(c + d*x))**m, x), x) + Dist(b, Int((e/tan(c + d*x))**m/sin(c + d*x), x), x)
rule4162 = ReplacementRule(pattern4162, replacement4162)
pattern4163 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1228, cons1267)
def replacement4163(m, b, d, c, a, n, x):
rubi.append(4163)
return Dist((S(-1))**(m/S(2) + S(-1)/2)*b**(-m + S(1))/d, Subst(Int((a + x)**n*(b**S(2) - x**S(2))**(m/S(2) + S(-1)/2)/x, x), x, b/cos(c + d*x)), x)
rule4163 = ReplacementRule(pattern4163, replacement4163)
pattern4164 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1228, cons1267)
def replacement4164(m, b, d, c, a, n, x):
rubi.append(4164)
return -Dist((S(-1))**(m/S(2) + S(-1)/2)*b**(-m + S(1))/d, Subst(Int((a + x)**n*(b**S(2) - x**S(2))**(m/S(2) + S(-1)/2)/x, x), x, b/sin(c + d*x)), x)
rule4164 = ReplacementRule(pattern4164, replacement4164)
pattern4165 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons148)
def replacement4165(m, b, d, c, a, n, x, e):
rubi.append(4165)
return Int(ExpandIntegrand((e*tan(c + d*x))**m, (a + b/cos(c + d*x))**n, x), x)
rule4165 = ReplacementRule(pattern4165, replacement4165)
pattern4166 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons148)
def replacement4166(m, b, d, c, a, n, x, e):
rubi.append(4166)
return Int(ExpandIntegrand((e/tan(c + d*x))**m, (a + b/sin(c + d*x))**n, x), x)
rule4166 = ReplacementRule(pattern4166, replacement4166)
pattern4167 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1265, cons1515, cons1607)
def replacement4167(m, b, d, c, n, a, x):
rubi.append(4167)
return Dist(S(2)*a**(m/S(2) + n + S(1)/2)/d, Subst(Int(x**m*(a*x**S(2) + S(2))**(m/S(2) + n + S(-1)/2)/(a*x**S(2) + S(1)), x), x, tan(c + d*x)/sqrt(a + b/cos(c + d*x))), x)
rule4167 = ReplacementRule(pattern4167, replacement4167)
pattern4168 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1265, cons1515, cons1607)
def replacement4168(m, b, d, c, n, a, x):
rubi.append(4168)
return Dist(-S(2)*a**(m/S(2) + n + S(1)/2)/d, Subst(Int(x**m*(a*x**S(2) + S(2))**(m/S(2) + n + S(-1)/2)/(a*x**S(2) + S(1)), x), x, S(1)/(sqrt(a + b/sin(c + d*x))*tan(c + d*x))), x)
rule4168 = ReplacementRule(pattern4168, replacement4168)
pattern4169 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1265, cons196)
def replacement4169(m, b, d, c, a, n, x, e):
rubi.append(4169)
return Dist(a**(S(2)*n)*e**(-S(2)*n), Int((e*tan(c + d*x))**(m + S(2)*n)*(-a + b/cos(c + d*x))**(-n), x), x)
rule4169 = ReplacementRule(pattern4169, replacement4169)
pattern4170 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1265, cons196)
def replacement4170(m, b, d, c, a, n, x, e):
rubi.append(4170)
return Dist(a**(S(2)*n)*e**(-S(2)*n), Int((e/tan(c + d*x))**(m + S(2)*n)*(-a + b/sin(c + d*x))**(-n), x), x)
rule4170 = ReplacementRule(pattern4170, replacement4170)
pattern4171 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1265, cons23)
def replacement4171(m, b, d, c, a, n, x, e):
rubi.append(4171)
return Simp(S(2)**(m + n + S(1))*(a/(a + b/cos(c + d*x)))**(m + n + S(1))*(e*tan(c + d*x))**(m + S(1))*(a + b/cos(c + d*x))**n*AppellF1(m/S(2) + S(1)/2, m + n, S(1), m/S(2) + S(3)/2, -(a - b/cos(c + d*x))/(a + b/cos(c + d*x)), (a - b/cos(c + d*x))/(a + b/cos(c + d*x)))/(d*e*(m + S(1))), x)
rule4171 = ReplacementRule(pattern4171, replacement4171)
pattern4172 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1265, cons23)
def replacement4172(m, b, d, c, a, n, x, e):
rubi.append(4172)
return -Simp(S(2)**(m + n + S(1))*(a/(a + b/sin(c + d*x)))**(m + n + S(1))*(e/tan(c + d*x))**(m + S(1))*(a + b/sin(c + d*x))**n*AppellF1(m/S(2) + S(1)/2, m + n, S(1), m/S(2) + S(3)/2, -(a - b/sin(c + d*x))/(a + b/sin(c + d*x)), (a - b/sin(c + d*x))/(a + b/sin(c + d*x)))/(d*e*(m + S(1))), x)
rule4172 = ReplacementRule(pattern4172, replacement4172)
pattern4173 = Pattern(Integral(sqrt(WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))/(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1267)
def replacement4173(b, d, c, a, x, e):
rubi.append(4173)
return Dist(S(1)/a, Int(sqrt(e*tan(c + d*x)), x), x) - Dist(b/a, Int(sqrt(e*tan(c + d*x))/(a*cos(c + d*x) + b), x), x)
rule4173 = ReplacementRule(pattern4173, replacement4173)
pattern4174 = Pattern(Integral(sqrt(WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))/(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1267)
def replacement4174(b, d, c, a, x, e):
rubi.append(4174)
return Dist(S(1)/a, Int(sqrt(e/tan(c + d*x)), x), x) - Dist(b/a, Int(sqrt(e/tan(c + d*x))/(a*sin(c + d*x) + b), x), x)
rule4174 = ReplacementRule(pattern4174, replacement4174)
pattern4175 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_/(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1267, cons1311)
def replacement4175(m, b, d, c, a, x, e):
rubi.append(4175)
return -Dist(e**S(2)/b**S(2), Int((e*tan(c + d*x))**(m + S(-2))*(a - b/cos(c + d*x)), x), x) + Dist(e**S(2)*(a**S(2) - b**S(2))/b**S(2), Int((e*tan(c + d*x))**(m + S(-2))/(a + b/cos(c + d*x)), x), x)
rule4175 = ReplacementRule(pattern4175, replacement4175)
pattern4176 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_/(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1267, cons1311)
def replacement4176(m, b, d, c, a, x, e):
rubi.append(4176)
return -Dist(e**S(2)/b**S(2), Int((e/tan(c + d*x))**(m + S(-2))*(a - b/sin(c + d*x)), x), x) + Dist(e**S(2)*(a**S(2) - b**S(2))/b**S(2), Int((e/tan(c + d*x))**(m + S(-2))/(a + b/sin(c + d*x)), x), x)
rule4176 = ReplacementRule(pattern4176, replacement4176)
pattern4177 = Pattern(Integral(S(1)/(sqrt(WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons1267)
def replacement4177(b, d, c, a, x, e):
rubi.append(4177)
return Dist(S(1)/a, Int(S(1)/sqrt(e*tan(c + d*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(e*tan(c + d*x))*(a*cos(c + d*x) + b)), x), x)
rule4177 = ReplacementRule(pattern4177, replacement4177)
pattern4178 = Pattern(Integral(S(1)/(sqrt(WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons1267)
def replacement4178(b, d, c, a, x, e):
rubi.append(4178)
return Dist(S(1)/a, Int(S(1)/sqrt(e/tan(c + d*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(e/tan(c + d*x))*(a*sin(c + d*x) + b)), x), x)
rule4178 = ReplacementRule(pattern4178, replacement4178)
pattern4179 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_/(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1267, cons1608)
def replacement4179(m, b, d, c, a, x, e):
rubi.append(4179)
return Dist(b**S(2)/(e**S(2)*(a**S(2) - b**S(2))), Int((e*tan(c + d*x))**(m + S(2))/(a + b/cos(c + d*x)), x), x) + Dist(S(1)/(a**S(2) - b**S(2)), Int((e*tan(c + d*x))**m*(a - b/cos(c + d*x)), x), x)
rule4179 = ReplacementRule(pattern4179, replacement4179)
pattern4180 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_/(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons1267, cons1608)
def replacement4180(m, b, d, c, a, x, e):
rubi.append(4180)
return Dist(b**S(2)/(e**S(2)*(a**S(2) - b**S(2))), Int((e/tan(c + d*x))**(m + S(2))/(a + b/sin(c + d*x)), x), x) + Dist(S(1)/(a**S(2) - b**S(2)), Int((e/tan(c + d*x))**m*(a - b/sin(c + d*x)), x), x)
rule4180 = ReplacementRule(pattern4180, replacement4180)
pattern4181 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement4181(b, d, c, a, n, x):
rubi.append(4181)
return Int((S(-1) + cos(c + d*x)**(S(-2)))*(a + b/cos(c + d*x))**n, x)
rule4181 = ReplacementRule(pattern4181, replacement4181)
pattern4182 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_/tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons1267)
def replacement4182(b, d, c, a, n, x):
rubi.append(4182)
return Int((S(-1) + sin(c + d*x)**(S(-2)))*(a + b/sin(c + d*x))**n, x)
rule4182 = ReplacementRule(pattern4182, replacement4182)
pattern4183 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1267, cons148)
def replacement4183(m, b, d, c, a, n, x, e):
rubi.append(4183)
return Int(ExpandIntegrand((e*tan(c + d*x))**m, (a + b/cos(c + d*x))**n, x), x)
rule4183 = ReplacementRule(pattern4183, replacement4183)
pattern4184 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons1267, cons148)
def replacement4184(m, b, d, c, a, n, x, e):
rubi.append(4184)
return Int(ExpandIntegrand((e/tan(c + d*x))**m, (a + b/sin(c + d*x))**n, x), x)
rule4184 = ReplacementRule(pattern4184, replacement4184)
pattern4185 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1267, cons85, cons17, cons1609)
def replacement4185(m, b, d, c, a, n, x):
rubi.append(4185)
return Int((a*cos(c + d*x) + b)**n*sin(c + d*x)**m*cos(c + d*x)**(-m - n), x)
rule4185 = ReplacementRule(pattern4185, replacement4185)
pattern4186 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1267, cons85, cons17, cons1609)
def replacement4186(m, b, d, c, a, n, x):
rubi.append(4186)
return Int((a*sin(c + d*x) + b)**n*sin(c + d*x)**(-m - n)*cos(c + d*x)**m, x)
rule4186 = ReplacementRule(pattern4186, replacement4186)
pattern4187 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1580)
def replacement4187(m, b, d, c, a, n, x, e):
rubi.append(4187)
return Int((e*tan(c + d*x))**m*(a + b/cos(c + d*x))**n, x)
rule4187 = ReplacementRule(pattern4187, replacement4187)
pattern4188 = Pattern(Integral((WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1580)
def replacement4188(m, b, d, a, n, c, x, e):
rubi.append(4188)
return Int((e/tan(c + d*x))**m*(a + b/sin(c + d*x))**n, x)
rule4188 = ReplacementRule(pattern4188, replacement4188)
pattern4189 = Pattern(Integral((WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**p_)**m_*(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons18)
def replacement4189(p, m, b, d, c, n, a, x, e):
rubi.append(4189)
return Dist((e*tan(c + d*x))**(-m*p)*(e*tan(c + d*x)**p)**m, Int((e*tan(c + d*x))**(m*p)*(a + b/cos(c + d*x))**n, x), x)
rule4189 = ReplacementRule(pattern4189, replacement4189)
pattern4190 = Pattern(Integral(((S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**p_*WC('e', S(1)))**m_*(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons18)
def replacement4190(p, m, b, d, c, n, a, x, e):
rubi.append(4190)
return Dist((e*(S(1)/tan(c + d*x))**p)**m*(e/tan(c + d*x))**(-m*p), Int((e/tan(c + d*x))**(m*p)*(a + b/sin(c + d*x))**n, x), x)
rule4190 = ReplacementRule(pattern4190, replacement4190)
pattern4191 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons62, cons196, cons1610)
def replacement4191(m, f, b, d, a, c, n, x, e):
rubi.append(4191)
return Dist(c**n, Int(ExpandTrig((S(1) + d/(c*cos(e + f*x)))**n, (a + b/cos(e + f*x))**m, x), x), x)
rule4191 = ReplacementRule(pattern4191, replacement4191)
pattern4192 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons62, cons196, cons1610)
def replacement4192(m, f, b, d, a, c, n, x, e):
rubi.append(4192)
return Dist(c**n, Int(ExpandTrig((S(1) + d/(c*sin(e + f*x)))**n, (a + b/sin(e + f*x))**m, x), x), x)
rule4192 = ReplacementRule(pattern4192, replacement4192)
pattern4193 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons17, cons87, cons1611)
def replacement4193(m, f, b, d, a, n, c, x, e):
rubi.append(4193)
return Dist((-a*c)**m, Int((c + d/cos(e + f*x))**(-m + n)*tan(e + f*x)**(S(2)*m), x), x)
rule4193 = ReplacementRule(pattern4193, replacement4193)
pattern4194 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons17, cons87, cons1611)
def replacement4194(m, f, b, d, a, n, c, x, e):
rubi.append(4194)
return Dist((-a*c)**m, Int((c + d/sin(e + f*x))**(-m + n)*(S(1)/tan(e + f*x))**(S(2)*m), x), x)
rule4194 = ReplacementRule(pattern4194, replacement4194)
pattern4195 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons70, cons1265, cons79)
def replacement4195(m, f, b, d, a, c, x, e):
rubi.append(4195)
return Dist((-a*c)**(m + S(1)/2)*tan(e + f*x)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), Int(tan(e + f*x)**(S(2)*m), x), x)
rule4195 = ReplacementRule(pattern4195, replacement4195)
pattern4196 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons70, cons1265, cons79)
def replacement4196(m, f, b, d, a, c, x, e):
rubi.append(4196)
return Dist((-a*c)**(m + S(1)/2)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), Int((S(1)/tan(e + f*x))**(S(2)*m), x), x)
rule4196 = ReplacementRule(pattern4196, replacement4196)
pattern4197 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons87, cons1612)
def replacement4197(f, b, d, a, n, c, x, e):
rubi.append(4197)
return Dist(c, Int(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))**(n + S(-1)), x), x) + Simp(-S(2)*a*c*(c + d/cos(e + f*x))**(n + S(-1))*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(-1))), x)
rule4197 = ReplacementRule(pattern4197, replacement4197)
pattern4198 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons87, cons1612)
def replacement4198(f, b, d, a, n, c, x, e):
rubi.append(4198)
return Dist(c, Int(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))**(n + S(-1)), x), x) + Simp(S(2)*a*c*(c + d/sin(e + f*x))**(n + S(-1))/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(-1))*tan(e + f*x)), x)
rule4198 = ReplacementRule(pattern4198, replacement4198)
pattern4199 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons87, cons1590)
def replacement4199(f, b, d, a, c, n, x, e):
rubi.append(4199)
return Dist(S(1)/c, Int(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*a*(c + d/cos(e + f*x))**n*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(1))), x)
rule4199 = ReplacementRule(pattern4199, replacement4199)
pattern4200 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons87, cons1590)
def replacement4200(f, b, d, a, c, n, x, e):
rubi.append(4200)
return Dist(S(1)/c, Int(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*a*(c + d/sin(e + f*x))**n/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(1))*tan(e + f*x)), x)
rule4200 = ReplacementRule(pattern4200, replacement4200)
pattern4201 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons87, cons1590)
def replacement4201(f, b, d, a, n, c, x, e):
rubi.append(4201)
return Dist(a/c, Int(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))**(n + S(1)), x), x) + Simp(S(4)*a**S(2)*(c + d/cos(e + f*x))**n*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(1))), x)
rule4201 = ReplacementRule(pattern4201, replacement4201)
pattern4202 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons87, cons1590)
def replacement4202(f, b, d, a, n, c, x, e):
rubi.append(4202)
return Dist(a/c, Int(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))**(n + S(1)), x), x) + Simp(-S(4)*a**S(2)*(c + d/sin(e + f*x))**n/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(1))*tan(e + f*x)), x)
rule4202 = ReplacementRule(pattern4202, replacement4202)
pattern4203 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons1613)
def replacement4203(f, b, d, a, n, c, x, e):
rubi.append(4203)
return Dist(a, Int(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))**n, x), x) + Simp(S(2)*a**S(2)*(c + d/cos(e + f*x))**n*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(1))), x)
rule4203 = ReplacementRule(pattern4203, replacement4203)
pattern4204 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons1613)
def replacement4204(f, b, d, a, n, c, x, e):
rubi.append(4204)
return Dist(a, Int(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))**n, x), x) + Simp(-S(2)*a**S(2)*(c + d/sin(e + f*x))**n/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(1))*tan(e + f*x)), x)
rule4204 = ReplacementRule(pattern4204, replacement4204)
pattern4205 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(5)/2)*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons87, cons1590)
def replacement4205(f, b, d, a, n, c, x, e):
rubi.append(4205)
return Dist(a**S(2)/c**S(2), Int(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))**(n + S(2)), x), x) + Simp(S(8)*a**S(3)*(c + d/cos(e + f*x))**n*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(1))), x)
rule4205 = ReplacementRule(pattern4205, replacement4205)
pattern4206 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(5)/2)*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons87, cons1590)
def replacement4206(f, b, d, a, n, c, x, e):
rubi.append(4206)
return Dist(a**S(2)/c**S(2), Int(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))**(n + S(2)), x), x) + Simp(-S(8)*a**S(3)*(c + d/sin(e + f*x))**n/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(1))*tan(e + f*x)), x)
rule4206 = ReplacementRule(pattern4206, replacement4206)
pattern4207 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons1304, cons1255)
def replacement4207(m, f, b, d, a, c, n, x, e):
rubi.append(4207)
return Dist(a*c*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), Subst(Int(x**(-m - n)*(a*x + b)**(m + S(-1)/2)*(c*x + d)**(n + S(-1)/2), x), x, cos(e + f*x)), x)
rule4207 = ReplacementRule(pattern4207, replacement4207)
pattern4208 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons1304, cons1255)
def replacement4208(m, f, b, d, a, c, n, x, e):
rubi.append(4208)
return -Dist(a*c/(f*sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), Subst(Int(x**(-m - n)*(a*x + b)**(m + S(-1)/2)*(c*x + d)**(n + S(-1)/2), x), x, sin(e + f*x)), x)
rule4208 = ReplacementRule(pattern4208, replacement4208)
pattern4209 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265)
def replacement4209(m, f, b, d, a, n, c, x, e):
rubi.append(4209)
return -Dist(a*c*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2)/x, x), x, S(1)/cos(e + f*x)), x)
rule4209 = ReplacementRule(pattern4209, replacement4209)
pattern4210 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265)
def replacement4210(m, f, b, d, a, n, c, x, e):
rubi.append(4210)
return Dist(a*c/(f*sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2)/x, x), x, S(1)/sin(e + f*x)), x)
rule4210 = ReplacementRule(pattern4210, replacement4210)
pattern4211 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70)
def replacement4211(f, b, d, a, c, x, e):
rubi.append(4211)
return Dist(b*d, Int(cos(e + f*x)**(S(-2)), x), x) + Simp(a*c*x, x)
rule4211 = ReplacementRule(pattern4211, replacement4211)
pattern4212 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70)
def replacement4212(f, b, d, a, c, x, e):
rubi.append(4212)
return Dist(b*d, Int(sin(e + f*x)**(S(-2)), x), x) + Simp(a*c*x, x)
rule4212 = ReplacementRule(pattern4212, replacement4212)
pattern4213 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1412)
def replacement4213(f, b, d, a, c, x, e):
rubi.append(4213)
return Dist(b*d, Int(cos(e + f*x)**(S(-2)), x), x) + Dist(a*d + b*c, Int(S(1)/cos(e + f*x), x), x) + Simp(a*c*x, x)
rule4213 = ReplacementRule(pattern4213, replacement4213)
pattern4214 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1412)
def replacement4214(f, b, d, a, c, x, e):
rubi.append(4214)
return Dist(b*d, Int(sin(e + f*x)**(S(-2)), x), x) + Dist(a*d + b*c, Int(S(1)/sin(e + f*x), x), x) + Simp(a*c*x, x)
rule4214 = ReplacementRule(pattern4214, replacement4214)
pattern4215 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement4215(f, b, d, a, c, x, e):
rubi.append(4215)
return Dist(c, Int(sqrt(a + b/cos(e + f*x)), x), x) + Dist(d, Int(sqrt(a + b/cos(e + f*x))/cos(e + f*x), x), x)
rule4215 = ReplacementRule(pattern4215, replacement4215)
pattern4216 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement4216(f, b, d, a, c, x, e):
rubi.append(4216)
return Dist(c, Int(sqrt(a + b/sin(e + f*x)), x), x) + Dist(d, Int(sqrt(a + b/sin(e + f*x))/sin(e + f*x), x), x)
rule4216 = ReplacementRule(pattern4216, replacement4216)
pattern4217 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement4217(f, b, d, a, c, x, e):
rubi.append(4217)
return Dist(a*c, Int(S(1)/sqrt(a + b/cos(e + f*x)), x), x) + Int((a*d + b*c + b*d/cos(e + f*x))/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x)
rule4217 = ReplacementRule(pattern4217, replacement4217)
pattern4218 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement4218(f, b, d, a, c, x, e):
rubi.append(4218)
return Dist(a*c, Int(S(1)/sqrt(a + b/sin(e + f*x)), x), x) + Int((a*d + b*c + b*d/sin(e + f*x))/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x)
rule4218 = ReplacementRule(pattern4218, replacement4218)
pattern4219 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons166, cons1265, cons515)
def replacement4219(m, f, b, d, a, c, x, e):
rubi.append(4219)
return Dist(S(1)/m, Int((a + b/cos(e + f*x))**(m + S(-1))*Simp(a*c*m + (a*d*(S(2)*m + S(-1)) + b*c*m)/cos(e + f*x), x), x), x) + Simp(b*d*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*m), x)
rule4219 = ReplacementRule(pattern4219, replacement4219)
pattern4220 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons166, cons1265, cons515)
def replacement4220(m, f, b, d, a, c, x, e):
rubi.append(4220)
return Dist(S(1)/m, Int((a + b/sin(e + f*x))**(m + S(-1))*Simp(a*c*m + (a*d*(S(2)*m + S(-1)) + b*c*m)/sin(e + f*x), x), x), x) - Simp(b*d*(a + b/sin(e + f*x))**(m + S(-1))/(f*m*tan(e + f*x)), x)
rule4220 = ReplacementRule(pattern4220, replacement4220)
pattern4221 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons166, cons1267, cons515)
def replacement4221(m, f, b, d, a, c, x, e):
rubi.append(4221)
return Dist(S(1)/m, Int((a + b/cos(e + f*x))**(m + S(-2))*Simp(a**S(2)*c*m + b*(a*d*(S(2)*m + S(-1)) + b*c*m)/cos(e + f*x)**S(2) + (a**S(2)*d*m + S(2)*a*b*c*m + b**S(2)*d*(m + S(-1)))/cos(e + f*x), x), x), x) + Simp(b*d*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*m), x)
rule4221 = ReplacementRule(pattern4221, replacement4221)
pattern4222 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons166, cons1267, cons515)
def replacement4222(m, f, b, d, a, c, x, e):
rubi.append(4222)
return Dist(S(1)/m, Int((a + b/sin(e + f*x))**(m + S(-2))*Simp(a**S(2)*c*m + b*(a*d*(S(2)*m + S(-1)) + b*c*m)/sin(e + f*x)**S(2) + (a**S(2)*d*m + S(2)*a*b*c*m + b**S(2)*d*(m + S(-1)))/sin(e + f*x), x), x), x) - Simp(b*d*(a + b/sin(e + f*x))**(m + S(-1))/(f*m*tan(e + f*x)), x)
rule4222 = ReplacementRule(pattern4222, replacement4222)
pattern4223 = Pattern(Integral((c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement4223(f, b, d, a, c, x, e):
rubi.append(4223)
return -Dist((-a*d + b*c)/a, Int(S(1)/((a + b/cos(e + f*x))*cos(e + f*x)), x), x) + Simp(c*x/a, x)
rule4223 = ReplacementRule(pattern4223, replacement4223)
pattern4224 = Pattern(Integral((c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement4224(f, b, d, a, c, x, e):
rubi.append(4224)
return -Dist((-a*d + b*c)/a, Int(S(1)/((a + b/sin(e + f*x))*sin(e + f*x)), x), x) + Simp(c*x/a, x)
rule4224 = ReplacementRule(pattern4224, replacement4224)
pattern4225 = Pattern(Integral((c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement4225(f, b, d, a, c, x, e):
rubi.append(4225)
return Dist(c/a, Int(sqrt(a + b/cos(e + f*x)), x), x) - Dist((-a*d + b*c)/a, Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x)
rule4225 = ReplacementRule(pattern4225, replacement4225)
pattern4226 = Pattern(Integral((c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement4226(f, b, d, a, c, x, e):
rubi.append(4226)
return Dist(c/a, Int(sqrt(a + b/sin(e + f*x)), x), x) - Dist((-a*d + b*c)/a, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x)
rule4226 = ReplacementRule(pattern4226, replacement4226)
pattern4227 = Pattern(Integral((c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement4227(f, b, d, a, c, x, e):
rubi.append(4227)
return Dist(c, Int(S(1)/sqrt(a + b/cos(e + f*x)), x), x) + Dist(d, Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x)
rule4227 = ReplacementRule(pattern4227, replacement4227)
pattern4228 = Pattern(Integral((c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement4228(f, b, d, a, c, x, e):
rubi.append(4228)
return Dist(c, Int(S(1)/sqrt(a + b/sin(e + f*x)), x), x) + Dist(d, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x)
rule4228 = ReplacementRule(pattern4228, replacement4228)
pattern4229 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons94, cons1265, cons515)
def replacement4229(m, f, b, d, a, c, x, e):
rubi.append(4229)
return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(a*c*(S(2)*m + S(1)) - (m + S(1))*(-a*d + b*c)/cos(e + f*x), x), x), x) + Simp((a + b/cos(e + f*x))**m*(-a*d + b*c)*tan(e + f*x)/(b*f*(S(2)*m + S(1))), x)
rule4229 = ReplacementRule(pattern4229, replacement4229)
pattern4230 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons94, cons1265, cons515)
def replacement4230(m, f, b, d, a, c, x, e):
rubi.append(4230)
return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(a*c*(S(2)*m + S(1)) - (m + S(1))*(-a*d + b*c)/sin(e + f*x), x), x), x) - Simp((a + b/sin(e + f*x))**m*(-a*d + b*c)/(b*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4230 = ReplacementRule(pattern4230, replacement4230)
pattern4231 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons94, cons1267, cons515)
def replacement4231(m, f, b, d, a, c, x, e):
rubi.append(4231)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(-a*(m + S(1))*(-a*d + b*c)/cos(e + f*x) + b*(m + S(2))*(-a*d + b*c)/cos(e + f*x)**S(2) + c*(a**S(2) - b**S(2))*(m + S(1)), x), x), x) - Simp(b*(a + b/cos(e + f*x))**(m + S(1))*(-a*d + b*c)*tan(e + f*x)/(a*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4231 = ReplacementRule(pattern4231, replacement4231)
pattern4232 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons31, cons94, cons1267, cons515)
def replacement4232(m, f, b, d, a, c, x, e):
rubi.append(4232)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(-a*(m + S(1))*(-a*d + b*c)/sin(e + f*x) + b*(m + S(2))*(-a*d + b*c)/sin(e + f*x)**S(2) + c*(a**S(2) - b**S(2))*(m + S(1)), x), x), x) + Simp(b*(a + b/sin(e + f*x))**(m + S(1))*(-a*d + b*c)/(a*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4232 = ReplacementRule(pattern4232, replacement4232)
pattern4233 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons77)
def replacement4233(m, f, b, d, a, c, x, e):
rubi.append(4233)
return Dist(c, Int((a + b/cos(e + f*x))**m, x), x) + Dist(d, Int((a + b/cos(e + f*x))**m/cos(e + f*x), x), x)
rule4233 = ReplacementRule(pattern4233, replacement4233)
pattern4234 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons71, cons77)
def replacement4234(m, f, b, d, a, c, x, e):
rubi.append(4234)
return Dist(c, Int((a + b/sin(e + f*x))**m, x), x) + Dist(d, Int((a + b/sin(e + f*x))**m/sin(e + f*x), x), x)
rule4234 = ReplacementRule(pattern4234, replacement4234)
pattern4235 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4235(f, b, d, a, c, x, e):
rubi.append(4235)
return Dist(S(1)/c, Int(sqrt(a + b/cos(e + f*x)), x), x) - Dist(d/c, Int(sqrt(a + b/cos(e + f*x))/((c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4235 = ReplacementRule(pattern4235, replacement4235)
pattern4236 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4236(f, b, d, a, c, x, e):
rubi.append(4236)
return Dist(S(1)/c, Int(sqrt(a + b/sin(e + f*x)), x), x) - Dist(d/c, Int(sqrt(a + b/sin(e + f*x))/((c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4236 = ReplacementRule(pattern4236, replacement4236)
pattern4237 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4237(f, b, d, a, c, x, e):
rubi.append(4237)
return Dist(a/c, Int(S(1)/sqrt(a + b/cos(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4237 = ReplacementRule(pattern4237, replacement4237)
pattern4238 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4238(f, b, d, a, c, x, e):
rubi.append(4238)
return Dist(a/c, Int(S(1)/sqrt(a + b/sin(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4238 = ReplacementRule(pattern4238, replacement4238)
pattern4239 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4239(f, b, d, a, c, x, e):
rubi.append(4239)
return Dist(a/c, Int(sqrt(a + b/cos(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(sqrt(a + b/cos(e + f*x))/((c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4239 = ReplacementRule(pattern4239, replacement4239)
pattern4240 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4240(f, b, d, a, c, x, e):
rubi.append(4240)
return Dist(a/c, Int(sqrt(a + b/sin(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(sqrt(a + b/sin(e + f*x))/((c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4240 = ReplacementRule(pattern4240, replacement4240)
pattern4241 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4241(f, b, d, a, c, x, e):
rubi.append(4241)
return Dist(S(1)/(c*d), Int((a**S(2)*d + b**S(2)*c/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) - Dist((-a*d + b*c)**S(2)/(c*d), Int(S(1)/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4241 = ReplacementRule(pattern4241, replacement4241)
pattern4242 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4242(f, b, d, a, c, x, e):
rubi.append(4242)
return Dist(S(1)/(c*d), Int((a**S(2)*d + b**S(2)*c/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) - Dist((-a*d + b*c)**S(2)/(c*d), Int(S(1)/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4242 = ReplacementRule(pattern4242, replacement4242)
pattern4243 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4243(f, b, d, a, c, x, e):
rubi.append(4243)
return Dist(S(1)/(c*(-a*d + b*c)), Int((-a*d + b*c - b*d/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) + Dist(d**S(2)/(c*(-a*d + b*c)), Int(sqrt(a + b/cos(e + f*x))/((c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4243 = ReplacementRule(pattern4243, replacement4243)
pattern4244 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4244(f, b, d, a, c, x, e):
rubi.append(4244)
return Dist(S(1)/(c*(-a*d + b*c)), Int((-a*d + b*c - b*d/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) + Dist(d**S(2)/(c*(-a*d + b*c)), Int(sqrt(a + b/sin(e + f*x))/((c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4244 = ReplacementRule(pattern4244, replacement4244)
pattern4245 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement4245(f, b, d, a, c, x, e):
rubi.append(4245)
return Dist(S(1)/c, Int(S(1)/sqrt(a + b/cos(e + f*x)), x), x) - Dist(d/c, Int(S(1)/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4245 = ReplacementRule(pattern4245, replacement4245)
pattern4246 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267)
def replacement4246(f, b, d, a, c, x, e):
rubi.append(4246)
return Dist(S(1)/c, Int(S(1)/sqrt(a + b/sin(e + f*x)), x), x) - Dist(d/c, Int(S(1)/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4246 = ReplacementRule(pattern4246, replacement4246)
pattern4247 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement4247(f, b, d, a, c, x, e):
rubi.append(4247)
return Dist(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))/tan(e + f*x), Int(tan(e + f*x), x), x)
rule4247 = ReplacementRule(pattern4247, replacement4247)
pattern4248 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement4248(f, b, d, a, c, x, e):
rubi.append(4248)
return Dist(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x), Int(S(1)/tan(e + f*x), x), x)
rule4248 = ReplacementRule(pattern4248, replacement4248)
pattern4249 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement4249(f, b, d, a, c, x, e):
rubi.append(4249)
return Dist(c, Int(sqrt(a + b/cos(e + f*x))/sqrt(c + d/cos(e + f*x)), x), x) + Dist(d, Int(sqrt(a + b/cos(e + f*x))/(sqrt(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4249 = ReplacementRule(pattern4249, replacement4249)
pattern4250 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement4250(f, b, d, a, c, x, e):
rubi.append(4250)
return Dist(c, Int(sqrt(a + b/sin(e + f*x))/sqrt(c + d/sin(e + f*x)), x), x) + Dist(d, Int(sqrt(a + b/sin(e + f*x))/(sqrt(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4250 = ReplacementRule(pattern4250, replacement4250)
pattern4251 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement4251(f, b, d, a, c, x, e):
rubi.append(4251)
return Dist(S(1)/c, Int(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x)), x), x) - Dist(d/c, Int(sqrt(a + b/cos(e + f*x))/(sqrt(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4251 = ReplacementRule(pattern4251, replacement4251)
pattern4252 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement4252(f, b, d, a, c, x, e):
rubi.append(4252)
return Dist(S(1)/c, Int(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x)), x), x) - Dist(d/c, Int(sqrt(a + b/sin(e + f*x))/(sqrt(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4252 = ReplacementRule(pattern4252, replacement4252)
pattern4253 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement4253(f, b, d, a, c, x, e):
rubi.append(4253)
return Dist(S(2)*a/f, Subst(Int(S(1)/(a*c*x**S(2) + S(1)), x), x, tan(e + f*x)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x)))), x)
rule4253 = ReplacementRule(pattern4253, replacement4253)
pattern4254 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement4254(f, b, d, a, c, x, e):
rubi.append(4254)
return Dist(-S(2)*a/f, Subst(Int(S(1)/(a*c*x**S(2) + S(1)), x), x, S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x))), x)
rule4254 = ReplacementRule(pattern4254, replacement4254)
pattern4255 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1322)
def replacement4255(f, b, d, a, c, x, e):
rubi.append(4255)
return Dist(a/c, Int(sqrt(c + d/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4255 = ReplacementRule(pattern4255, replacement4255)
pattern4256 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1322)
def replacement4256(f, b, d, a, c, x, e):
rubi.append(4256)
return Dist(a/c, Int(sqrt(c + d/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4256 = ReplacementRule(pattern4256, replacement4256)
pattern4257 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4257(f, b, d, a, c, x, e):
rubi.append(4257)
return Simp(-S(2)*sqrt((S(1) + S(1)/cos(e + f*x))*(-a*d + b*c)/((a + b/cos(e + f*x))*(c - d)))*sqrt(-(S(1) - S(1)/cos(e + f*x))*(-a*d + b*c)/((a + b/cos(e + f*x))*(c + d)))*(a + b/cos(e + f*x))*EllipticPi(a*(c + d)/(c*(a + b)), asin(sqrt(c + d/cos(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b/cos(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(c*f*Rt((a + b)/(c + d), S(2))*tan(e + f*x)), x)
rule4257 = ReplacementRule(pattern4257, replacement4257)
pattern4258 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4258(f, b, d, a, c, x, e):
rubi.append(4258)
return Simp(S(2)*sqrt((S(1) + S(1)/sin(e + f*x))*(-a*d + b*c)/((a + b/sin(e + f*x))*(c - d)))*sqrt(-(S(1) - S(1)/sin(e + f*x))*(-a*d + b*c)/((a + b/sin(e + f*x))*(c + d)))*(a + b/sin(e + f*x))*EllipticPi(a*(c + d)/(c*(a + b)), asin(sqrt(c + d/sin(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b/sin(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))*tan(e + f*x)/(c*f*Rt((a + b)/(c + d), S(2))), x)
rule4258 = ReplacementRule(pattern4258, replacement4258)
pattern4259 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement4259(f, b, d, a, c, x, e):
rubi.append(4259)
return Dist(tan(e + f*x)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), Int(S(1)/tan(e + f*x), x), x)
rule4259 = ReplacementRule(pattern4259, replacement4259)
pattern4260 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1322)
def replacement4260(f, b, d, a, c, x, e):
rubi.append(4260)
return Dist(S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), Int(tan(e + f*x), x), x)
rule4260 = ReplacementRule(pattern4260, replacement4260)
pattern4261 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement4261(f, b, d, a, c, x, e):
rubi.append(4261)
return Dist(S(1)/a, Int(sqrt(a + b/cos(e + f*x))/sqrt(c + d/cos(e + f*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4261 = ReplacementRule(pattern4261, replacement4261)
pattern4262 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement4262(f, b, d, a, c, x, e):
rubi.append(4262)
return Dist(S(1)/a, Int(sqrt(a + b/sin(e + f*x))/sqrt(c + d/sin(e + f*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4262 = ReplacementRule(pattern4262, replacement4262)
pattern4263 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1323)
def replacement4263(f, b, d, a, c, x, e):
rubi.append(4263)
return Dist(S(1)/c, Int(sqrt(a + b/cos(e + f*x))/sqrt(c + d/cos(e + f*x)), x), x) - Dist(d/c, Int(sqrt(a + b/cos(e + f*x))/((c + d/cos(e + f*x))**(S(3)/2)*cos(e + f*x)), x), x)
rule4263 = ReplacementRule(pattern4263, replacement4263)
pattern4264 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1323)
def replacement4264(f, b, d, a, c, x, e):
rubi.append(4264)
return Dist(S(1)/c, Int(sqrt(a + b/sin(e + f*x))/sqrt(c + d/sin(e + f*x)), x), x) - Dist(d/c, Int(sqrt(a + b/sin(e + f*x))/((c + d/sin(e + f*x))**(S(3)/2)*sin(e + f*x)), x), x)
rule4264 = ReplacementRule(pattern4264, replacement4264)
pattern4265 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1265, cons1323, cons1304)
def replacement4265(m, f, b, d, a, n, c, x, e):
rubi.append(4265)
return -Dist(a**S(2)*tan(e + f*x)/(f*sqrt(a - b/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**n/(x*sqrt(a - b*x)), x), x, S(1)/cos(e + f*x)), x)
rule4265 = ReplacementRule(pattern4265, replacement4265)
pattern4266 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1265, cons1323, cons1304)
def replacement4266(m, f, b, d, a, n, c, x, e):
rubi.append(4266)
return Dist(a**S(2)*cos(e + f*x)/(f*sqrt(a - b/sin(e + f*x))*sqrt(a + b/sin(e + f*x))), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**n/(x*sqrt(a - b*x)), x), x, S(1)/sin(e + f*x)), x)
rule4266 = ReplacementRule(pattern4266, replacement4266)
pattern4267 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons17, cons85, cons1614)
def replacement4267(m, f, b, d, a, c, n, x, e):
rubi.append(4267)
return Int((a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-m - n), x)
rule4267 = ReplacementRule(pattern4267, replacement4267)
pattern4268 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons17, cons85, cons1614)
def replacement4268(m, f, b, d, a, c, n, x, e):
rubi.append(4268)
return Int((a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-m - n), x)
rule4268 = ReplacementRule(pattern4268, replacement4268)
pattern4269 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons79, cons80, cons1614)
def replacement4269(m, f, b, d, a, c, n, x, e):
rubi.append(4269)
return Dist(sqrt(a + b/cos(e + f*x))*sqrt(c*cos(e + f*x) + d)/(sqrt(c + d/cos(e + f*x))*sqrt(a*cos(e + f*x) + b)), Int((a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-m - n), x), x)
rule4269 = ReplacementRule(pattern4269, replacement4269)
pattern4270 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons79, cons80, cons1614)
def replacement4270(m, f, b, d, a, c, n, x, e):
rubi.append(4270)
return Dist(sqrt(a + b/sin(e + f*x))*sqrt(c*sin(e + f*x) + d)/(sqrt(c + d/sin(e + f*x))*sqrt(a*sin(e + f*x) + b)), Int((a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-m - n), x), x)
rule4270 = ReplacementRule(pattern4270, replacement4270)
pattern4271 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1255, cons77)
def replacement4271(m, f, b, d, a, c, n, x, e):
rubi.append(4271)
return Dist((a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n*(a*cos(e + f*x) + b)**(-m)*(c*cos(e + f*x) + d)**(-n)*cos(e + f*x)**(m + n), Int((a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-m - n), x), x)
rule4271 = ReplacementRule(pattern4271, replacement4271)
pattern4272 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1255, cons77)
def replacement4272(m, f, b, d, a, c, n, x, e):
rubi.append(4272)
return Dist((a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n*(a*sin(e + f*x) + b)**(-m)*(c*sin(e + f*x) + d)**(-n)*sin(e + f*x)**(m + n), Int((a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-m - n), x), x)
rule4272 = ReplacementRule(pattern4272, replacement4272)
pattern4273 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons148)
def replacement4273(m, f, b, d, a, c, n, x, e):
rubi.append(4273)
return Int(ExpandTrig((a + b/cos(e + f*x))**m, (c + d/cos(e + f*x))**n, x), x)
rule4273 = ReplacementRule(pattern4273, replacement4273)
pattern4274 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons148)
def replacement4274(m, f, b, d, a, c, n, x, e):
rubi.append(4274)
return Int(ExpandTrig((a + b/sin(e + f*x))**m, (c + d/sin(e + f*x))**n, x), x)
rule4274 = ReplacementRule(pattern4274, replacement4274)
pattern4275 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement4275(m, f, b, d, a, n, c, x, e):
rubi.append(4275)
return Int((a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n, x)
rule4275 = ReplacementRule(pattern4275, replacement4275)
pattern4276 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement4276(m, f, b, d, a, n, c, x, e):
rubi.append(4276)
return Int((a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n, x)
rule4276 = ReplacementRule(pattern4276, replacement4276)
pattern4277 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons23, cons17)
def replacement4277(m, f, b, d, a, n, x, e):
rubi.append(4277)
return Dist(d**m, Int((d*cos(e + f*x))**(-m + n)*(a*cos(e + f*x) + b)**m, x), x)
rule4277 = ReplacementRule(pattern4277, replacement4277)
pattern4278 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons27, cons48, cons125, cons4, cons23, cons17)
def replacement4278(m, f, b, d, a, n, x, e):
rubi.append(4278)
return Dist(d**m, Int((d*sin(e + f*x))**(-m + n)*(a*sin(e + f*x) + b)**m, x), x)
rule4278 = ReplacementRule(pattern4278, replacement4278)
pattern4279 = Pattern(Integral(((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons23)
def replacement4279(p, m, f, b, d, c, a, n, x, e):
rubi.append(4279)
return Dist(c**IntPart(n)*(c*(d/cos(e + f*x))**p)**FracPart(n)*(d/cos(e + f*x))**(-p*FracPart(n)), Int((d/cos(e + f*x))**(n*p)*(a + b/cos(e + f*x))**m, x), x)
rule4279 = ReplacementRule(pattern4279, replacement4279)
pattern4280 = Pattern(Integral(((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons23)
def replacement4280(p, m, f, b, d, a, c, n, x, e):
rubi.append(4280)
return Dist(c**IntPart(n)*(c*(d/sin(e + f*x))**p)**FracPart(n)*(d/sin(e + f*x))**(-p*FracPart(n)), Int((d*cos(e + f*x))**(n*p)*(a + b*cos(e + f*x))**m, x), x)
rule4280 = ReplacementRule(pattern4280, replacement4280)
pattern4281 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons155, cons1421)
def replacement4281(m, f, b, d, a, n, c, x, e):
rubi.append(4281)
return -Simp(b*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4281 = ReplacementRule(pattern4281, replacement4281)
pattern4282 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons155, cons1421)
def replacement4282(m, f, b, d, a, n, c, x, e):
rubi.append(4282)
return Simp(b*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4282 = ReplacementRule(pattern4282, replacement4282)
pattern4283 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons1315, cons1421, cons1231, cons1615)
def replacement4283(m, f, b, d, a, n, c, x, e):
rubi.append(4283)
return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*(c + d/cos(e + f*x))**n/cos(e + f*x), x), x) - Simp(b*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4283 = ReplacementRule(pattern4283, replacement4283)
pattern4284 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265, cons1315, cons1421, cons1231, cons1615)
def replacement4284(m, f, b, d, a, n, c, x, e):
rubi.append(4284)
return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*(c + d/sin(e + f*x))**n/sin(e + f*x), x), x) + Simp(b*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4284 = ReplacementRule(pattern4284, replacement4284)
pattern4285 = Pattern(Integral(sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265)
def replacement4285(f, b, d, a, c, x, e):
rubi.append(4285)
return -Simp(a*c*log(S(1) + b/(a*cos(e + f*x)))*tan(e + f*x)/(b*f*sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), x)
rule4285 = ReplacementRule(pattern4285, replacement4285)
pattern4286 = Pattern(Integral(sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265)
def replacement4286(f, b, d, a, c, x, e):
rubi.append(4286)
return Simp(a*c*log(S(1) + b/(a*sin(e + f*x)))/(b*f*sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), x)
rule4286 = ReplacementRule(pattern4286, replacement4286)
pattern4287 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons70, cons1265, cons1314)
def replacement4287(m, f, b, d, a, c, x, e):
rubi.append(4287)
return Simp(-S(2)*a*c*(a + b/cos(e + f*x))**m*tan(e + f*x)/(b*f*sqrt(c + d/cos(e + f*x))*(S(2)*m + S(1))), x)
rule4287 = ReplacementRule(pattern4287, replacement4287)
pattern4288 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons70, cons1265, cons1314)
def replacement4288(m, f, b, d, a, c, x, e):
rubi.append(4288)
return Simp(S(2)*a*c*(a + b/sin(e + f*x))**m/(b*f*sqrt(c + d/sin(e + f*x))*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4288 = ReplacementRule(pattern4288, replacement4288)
pattern4289 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons1266, cons31, cons1320)
def replacement4289(m, f, b, d, a, n, c, x, e):
rubi.append(4289)
return -Dist(d*(S(2)*n + S(-1))/(b*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*(c + d/cos(e + f*x))**(n + S(-1))/cos(e + f*x), x), x) + Simp(-S(2)*a*c*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**(n + S(-1))*tan(e + f*x)/(b*f*(S(2)*m + S(1))), x)
rule4289 = ReplacementRule(pattern4289, replacement4289)
pattern4290 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons1266, cons31, cons1320)
def replacement4290(m, f, b, d, a, n, c, x, e):
rubi.append(4290)
return -Dist(d*(S(2)*n + S(-1))/(b*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*(c + d/sin(e + f*x))**(n + S(-1))/sin(e + f*x), x), x) + Simp(S(2)*a*c*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**(n + S(-1))/(b*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4290 = ReplacementRule(pattern4290, replacement4290)
pattern4291 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons70, cons1265, cons1266, cons1321, cons1616)
def replacement4291(m, f, b, d, a, c, n, x, e):
rubi.append(4291)
return Dist(c*(S(2)*n + S(-1))/(m + n), Int((a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**(n + S(-1))/cos(e + f*x), x), x) + Simp(d*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**(n + S(-1))*tan(e + f*x)/(f*(m + n)), x)
rule4291 = ReplacementRule(pattern4291, replacement4291)
pattern4292 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons70, cons1265, cons1266, cons1321, cons1616)
def replacement4292(m, f, b, d, a, c, n, x, e):
rubi.append(4292)
return Dist(c*(S(2)*n + S(-1))/(m + n), Int((a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**(n + S(-1))/sin(e + f*x), x), x) - Simp(d*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**(n + S(-1))/(f*(m + n)*tan(e + f*x)), x)
rule4292 = ReplacementRule(pattern4292, replacement4292)
pattern4293 = Pattern(Integral((c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons148)
def replacement4293(f, b, d, a, n, c, x, e):
rubi.append(4293)
return Dist(S(2)*c, Int((c + d/cos(e + f*x))**(n + S(-1))/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) + Simp(S(2)*d*(c + d/cos(e + f*x))**(n + S(-1))*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(-1))), x)
rule4293 = ReplacementRule(pattern4293, replacement4293)
pattern4294 = Pattern(Integral((c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons148)
def replacement4294(f, b, d, a, n, c, x, e):
rubi.append(4294)
return Dist(S(2)*c, Int((c + d/sin(e + f*x))**(n + S(-1))/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) + Simp(-S(2)*d*(c + d/sin(e + f*x))**(n + S(-1))/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(-1))*tan(e + f*x)), x)
rule4294 = ReplacementRule(pattern4294, replacement4294)
pattern4295 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons148, cons1320, cons515)
def replacement4295(m, f, b, d, a, n, c, x, e):
rubi.append(4295)
return -Dist(d*(S(2)*n + S(-1))/(b*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*(c + d/cos(e + f*x))**(n + S(-1))/cos(e + f*x), x), x) + Simp(-S(2)*a*c*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**(n + S(-1))*tan(e + f*x)/(b*f*(S(2)*m + S(1))), x)
rule4295 = ReplacementRule(pattern4295, replacement4295)
pattern4296 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons148, cons1320, cons515)
def replacement4296(m, f, b, d, a, n, c, x, e):
rubi.append(4296)
return -Dist(d*(S(2)*n + S(-1))/(b*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*(c + d/sin(e + f*x))**(n + S(-1))/sin(e + f*x), x), x) + Simp(S(2)*a*c*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**(n + S(-1))/(b*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4296 = ReplacementRule(pattern4296, replacement4296)
pattern4297 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons150, cons1617, cons1618)
def replacement4297(m, f, b, d, a, n, c, x, e):
rubi.append(4297)
return Dist((-a*c)**m, Int(ExpandTrig(tan(e + f*x)**(S(2)*m)/cos(e + f*x), (c + d/cos(e + f*x))**(-m + n), x), x), x)
rule4297 = ReplacementRule(pattern4297, replacement4297)
pattern4298 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons150, cons1617, cons1618)
def replacement4298(m, f, b, d, a, n, c, x, e):
rubi.append(4298)
return Dist((-a*c)**m, Int(ExpandTrig((S(1)/tan(e + f*x))**(S(2)*m)/sin(e + f*x), (c + d/sin(e + f*x))**(-m + n), x), x), x)
rule4298 = ReplacementRule(pattern4298, replacement4298)
pattern4299 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons79)
def replacement4299(m, f, b, d, a, c, x, e):
rubi.append(4299)
return Dist((-a*c)**(m + S(1)/2)*tan(e + f*x)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), Int(tan(e + f*x)**(S(2)*m)/cos(e + f*x), x), x)
rule4299 = ReplacementRule(pattern4299, replacement4299)
pattern4300 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons70, cons1265, cons79)
def replacement4300(m, f, b, d, a, c, x, e):
rubi.append(4300)
return Dist((-a*c)**(m + S(1)/2)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), Int((S(1)/tan(e + f*x))**(S(2)*m)/sin(e + f*x), x), x)
rule4300 = ReplacementRule(pattern4300, replacement4300)
pattern4301 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons1619)
def replacement4301(m, f, b, d, a, c, n, x, e):
rubi.append(4301)
return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*(c + d/cos(e + f*x))**n/cos(e + f*x), x), x) - Simp(b*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4301 = ReplacementRule(pattern4301, replacement4301)
pattern4302 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons70, cons1265, cons1619)
def replacement4302(m, f, b, d, a, c, n, x, e):
rubi.append(4302)
return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*(c + d/sin(e + f*x))**n/sin(e + f*x), x), x) + Simp(b*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4302 = ReplacementRule(pattern4302, replacement4302)
pattern4303 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265)
def replacement4303(m, f, b, d, a, c, n, x, e):
rubi.append(4303)
return -Dist(a*c*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2), x), x, S(1)/cos(e + f*x)), x)
rule4303 = ReplacementRule(pattern4303, replacement4303)
pattern4304 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons70, cons1265)
def replacement4304(m, f, b, d, a, c, n, x, e):
rubi.append(4304)
return Dist(a*c/(f*sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2), x), x, S(1)/sin(e + f*x)), x)
rule4304 = ReplacementRule(pattern4304, replacement4304)
pattern4305 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons150, cons1617, cons1618)
def replacement4305(p, m, f, g, b, d, a, n, c, x, e):
rubi.append(4305)
return Dist((-a*c)**m, Int(ExpandTrig((g/cos(e + f*x))**p*tan(e + f*x)**(S(2)*m), (c + d/cos(e + f*x))**(-m + n), x), x), x)
rule4305 = ReplacementRule(pattern4305, replacement4305)
pattern4306 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons70, cons1265, cons150, cons1617, cons1618)
def replacement4306(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4306)
return Dist((-a*c)**m, Int(ExpandTrig((g/sin(e + f*x))**p*(S(1)/tan(e + f*x))**(S(2)*m), (c + d/sin(e + f*x))**(-m + n), x), x), x)
rule4306 = ReplacementRule(pattern4306, replacement4306)
pattern4307 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons70, cons1265, cons79)
def replacement4307(p, m, f, g, b, d, a, c, x, e):
rubi.append(4307)
return Dist((-a*c)**(m + S(1)/2)*tan(e + f*x)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), Int((g/cos(e + f*x))**p*tan(e + f*x)**(S(2)*m), x), x)
rule4307 = ReplacementRule(pattern4307, replacement4307)
pattern4308 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons70, cons1265, cons79)
def replacement4308(p, m, f, b, g, d, a, c, x, e):
rubi.append(4308)
return Dist((-a*c)**(m + S(1)/2)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), Int((g/sin(e + f*x))**p*(S(1)/tan(e + f*x))**(S(2)*m), x), x)
rule4308 = ReplacementRule(pattern4308, replacement4308)
pattern4309 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265)
def replacement4309(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(4309)
return -Dist(a*c*g*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))), Subst(Int((g*x)**(p + S(-1))*(a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2), x), x, S(1)/cos(e + f*x)), x)
rule4309 = ReplacementRule(pattern4309, replacement4309)
pattern4310 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons70, cons1265)
def replacement4310(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4310)
return Dist(a*c*g/(f*sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x)), Subst(Int((g*x)**(p + S(-1))*(a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2), x), x, S(1)/sin(e + f*x)), x)
rule4310 = ReplacementRule(pattern4310, replacement4310)
pattern4311 = Pattern(Integral(sqrt(WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement4311(f, g, b, d, a, c, x, e):
rubi.append(4311)
return Dist(S(2)*b*g/f, Subst(Int(S(1)/(a*d + b*c - c*g*x**S(2)), x), x, b*tan(e + f*x)/(sqrt(g/cos(e + f*x))*sqrt(a + b/cos(e + f*x)))), x)
rule4311 = ReplacementRule(pattern4311, replacement4311)
pattern4312 = Pattern(Integral(sqrt(WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement4312(f, b, g, d, a, c, x, e):
rubi.append(4312)
return Dist(-S(2)*b*g/f, Subst(Int(S(1)/(a*d + b*c - c*g*x**S(2)), x), x, b/(sqrt(g/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x))), x)
rule4312 = ReplacementRule(pattern4312, replacement4312)
pattern4313 = Pattern(Integral(sqrt(WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267)
def replacement4313(f, g, b, d, a, c, x, e):
rubi.append(4313)
return Dist(a/c, Int(sqrt(g/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) + Dist((-a*d + b*c)/(c*g), Int((g/cos(e + f*x))**(S(3)/2)/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))), x), x)
rule4313 = ReplacementRule(pattern4313, replacement4313)
pattern4314 = Pattern(Integral(sqrt(WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267)
def replacement4314(f, b, g, d, a, c, x, e):
rubi.append(4314)
return Dist(a/c, Int(sqrt(g/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) + Dist((-a*d + b*c)/(c*g), Int((g/sin(e + f*x))**(S(3)/2)/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))), x), x)
rule4314 = ReplacementRule(pattern4314, replacement4314)
pattern4315 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement4315(f, b, d, a, c, x, e):
rubi.append(4315)
return Dist(S(2)*b/f, Subst(Int(S(1)/(a*d + b*c + d*x**S(2)), x), x, b*tan(e + f*x)/sqrt(a + b/cos(e + f*x))), x)
rule4315 = ReplacementRule(pattern4315, replacement4315)
pattern4316 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265)
def replacement4316(f, b, d, a, c, x, e):
rubi.append(4316)
return Dist(-S(2)*b/f, Subst(Int(S(1)/(a*d + b*c + d*x**S(2)), x), x, b/(sqrt(a + b/sin(e + f*x))*tan(e + f*x))), x)
rule4316 = ReplacementRule(pattern4316, replacement4316)
pattern4317 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1322)
def replacement4317(f, b, d, a, c, x, e):
rubi.append(4317)
return Simp(sqrt(c/(c + d/cos(e + f*x)))*sqrt(a + b/cos(e + f*x))*EllipticE(asin(c*tan(e + f*x)/(c + d/cos(e + f*x))), -(-a*d + b*c)/(a*d + b*c))/(d*f*sqrt(c*d*(a + b/cos(e + f*x))/((c + d/cos(e + f*x))*(a*d + b*c)))), x)
rule4317 = ReplacementRule(pattern4317, replacement4317)
pattern4318 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1322)
def replacement4318(f, b, d, a, c, x, e):
rubi.append(4318)
return -Simp(sqrt(c/(c + d/sin(e + f*x)))*sqrt(a + b/sin(e + f*x))*EllipticE(asin(c/((c + d/sin(e + f*x))*tan(e + f*x))), -(-a*d + b*c)/(a*d + b*c))/(d*f*sqrt(c*d*(a + b/sin(e + f*x))/((c + d/sin(e + f*x))*(a*d + b*c)))), x)
rule4318 = ReplacementRule(pattern4318, replacement4318)
pattern4319 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4319(f, b, d, a, c, x, e):
rubi.append(4319)
return Dist(b/d, Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4319 = ReplacementRule(pattern4319, replacement4319)
pattern4320 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4320(f, b, d, a, c, x, e):
rubi.append(4320)
return Dist(b/d, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4320 = ReplacementRule(pattern4320, replacement4320)
pattern4321 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement4321(f, g, b, d, a, c, x, e):
rubi.append(4321)
return Dist(g/d, Int(sqrt(g/cos(e + f*x))*sqrt(a + b/cos(e + f*x)), x), x) - Dist(c*g/d, Int(sqrt(g/cos(e + f*x))*sqrt(a + b/cos(e + f*x))/(c + d/cos(e + f*x)), x), x)
rule4321 = ReplacementRule(pattern4321, replacement4321)
pattern4322 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement4322(f, b, g, d, a, c, x, e):
rubi.append(4322)
return Dist(g/d, Int(sqrt(g/sin(e + f*x))*sqrt(a + b/sin(e + f*x)), x), x) - Dist(c*g/d, Int(sqrt(g/sin(e + f*x))*sqrt(a + b/sin(e + f*x))/(c + d/sin(e + f*x)), x), x)
rule4322 = ReplacementRule(pattern4322, replacement4322)
pattern4323 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267)
def replacement4323(f, g, b, d, a, c, x, e):
rubi.append(4323)
return Dist(b/d, Int((g/cos(e + f*x))**(S(3)/2)/sqrt(a + b/cos(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int((g/cos(e + f*x))**(S(3)/2)/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))), x), x)
rule4323 = ReplacementRule(pattern4323, replacement4323)
pattern4324 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267)
def replacement4324(f, b, g, d, a, c, x, e):
rubi.append(4324)
return Dist(b/d, Int((g/sin(e + f*x))**(S(3)/2)/sqrt(a + b/sin(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int((g/sin(e + f*x))**(S(3)/2)/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))), x), x)
rule4324 = ReplacementRule(pattern4324, replacement4324)
pattern4325 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4325(f, b, d, a, c, x, e):
rubi.append(4325)
return Dist(b/(-a*d + b*c), Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b/cos(e + f*x))/((c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4325 = ReplacementRule(pattern4325, replacement4325)
pattern4326 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4326(f, b, d, a, c, x, e):
rubi.append(4326)
return Dist(b/(-a*d + b*c), Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b/sin(e + f*x))/((c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4326 = ReplacementRule(pattern4326, replacement4326)
pattern4327 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4327(f, b, d, a, c, x, e):
rubi.append(4327)
return Simp(S(2)*sqrt((a + b/cos(e + f*x))/(a + b))*EllipticPi(S(2)*d/(c + d), asin(sqrt(S(2))*sqrt(S(1) - S(1)/cos(e + f*x))/S(2)), S(2)*b/(a + b))*tan(e + f*x)/(f*sqrt(-tan(e + f*x)**S(2))*sqrt(a + b/cos(e + f*x))*(c + d)), x)
rule4327 = ReplacementRule(pattern4327, replacement4327)
pattern4328 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4328(f, b, d, a, c, x, e):
rubi.append(4328)
return Simp(-S(2)*sqrt((a + b/sin(e + f*x))/(a + b))*EllipticPi(S(2)*d/(c + d), asin(sqrt(S(2))*sqrt(S(1) - S(1)/sin(e + f*x))/S(2)), S(2)*b/(a + b))/(f*sqrt(-S(1)/tan(e + f*x)**S(2))*sqrt(a + b/sin(e + f*x))*(c + d)*tan(e + f*x)), x)
rule4328 = ReplacementRule(pattern4328, replacement4328)
pattern4329 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement4329(f, g, b, d, a, c, x, e):
rubi.append(4329)
return -Dist(a*g/(-a*d + b*c), Int(sqrt(g/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) + Dist(c*g/(-a*d + b*c), Int(sqrt(g/cos(e + f*x))*sqrt(a + b/cos(e + f*x))/(c + d/cos(e + f*x)), x), x)
rule4329 = ReplacementRule(pattern4329, replacement4329)
pattern4330 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement4330(f, b, g, d, a, c, x, e):
rubi.append(4330)
return -Dist(a*g/(-a*d + b*c), Int(sqrt(g/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) + Dist(c*g/(-a*d + b*c), Int(sqrt(g/sin(e + f*x))*sqrt(a + b/sin(e + f*x))/(c + d/sin(e + f*x)), x), x)
rule4330 = ReplacementRule(pattern4330, replacement4330)
pattern4331 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267)
def replacement4331(f, g, b, d, a, c, x, e):
rubi.append(4331)
return Dist(g*sqrt(g/cos(e + f*x))*sqrt(a*cos(e + f*x) + b)/sqrt(a + b/cos(e + f*x)), Int(S(1)/(sqrt(a*cos(e + f*x) + b)*(c*cos(e + f*x) + d)), x), x)
rule4331 = ReplacementRule(pattern4331, replacement4331)
pattern4332 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267)
def replacement4332(f, b, g, d, a, c, x, e):
rubi.append(4332)
return Dist(g*sqrt(g/sin(e + f*x))*sqrt(a*sin(e + f*x) + b)/sqrt(a + b/sin(e + f*x)), Int(S(1)/(sqrt(a*sin(e + f*x) + b)*(c*sin(e + f*x) + d)), x), x)
rule4332 = ReplacementRule(pattern4332, replacement4332)
pattern4333 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4333(f, b, d, a, c, x, e):
rubi.append(4333)
return -Dist(a/(-a*d + b*c), Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) + Dist(c/(-a*d + b*c), Int(sqrt(a + b/cos(e + f*x))/((c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4333 = ReplacementRule(pattern4333, replacement4333)
pattern4334 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1409)
def replacement4334(f, b, d, a, c, x, e):
rubi.append(4334)
return -Dist(a/(-a*d + b*c), Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) + Dist(c/(-a*d + b*c), Int(sqrt(a + b/sin(e + f*x))/((c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4334 = ReplacementRule(pattern4334, replacement4334)
pattern4335 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4335(f, b, d, a, c, x, e):
rubi.append(4335)
return Dist(S(1)/d, Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) - Dist(c/d, Int(S(1)/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4335 = ReplacementRule(pattern4335, replacement4335)
pattern4336 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4336(f, b, d, a, c, x, e):
rubi.append(4336)
return Dist(S(1)/d, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) - Dist(c/d, Int(S(1)/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4336 = ReplacementRule(pattern4336, replacement4336)
pattern4337 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(5)/2)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement4337(f, g, b, d, a, c, x, e):
rubi.append(4337)
return Dist(g**S(2)/(d*(-a*d + b*c)), Int(sqrt(g/cos(e + f*x))*(a*c + (-a*d + b*c)/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) - Dist(c**S(2)*g**S(2)/(d*(-a*d + b*c)), Int(sqrt(g/cos(e + f*x))*sqrt(a + b/cos(e + f*x))/(c + d/cos(e + f*x)), x), x)
rule4337 = ReplacementRule(pattern4337, replacement4337)
pattern4338 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(5)/2)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1265)
def replacement4338(f, b, g, d, a, c, x, e):
rubi.append(4338)
return Dist(g**S(2)/(d*(-a*d + b*c)), Int(sqrt(g/sin(e + f*x))*(a*c + (-a*d + b*c)/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) - Dist(c**S(2)*g**S(2)/(d*(-a*d + b*c)), Int(sqrt(g/sin(e + f*x))*sqrt(a + b/sin(e + f*x))/(c + d/sin(e + f*x)), x), x)
rule4338 = ReplacementRule(pattern4338, replacement4338)
pattern4339 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(5)/2)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267)
def replacement4339(f, g, b, d, a, c, x, e):
rubi.append(4339)
return Dist(g/d, Int((g/cos(e + f*x))**(S(3)/2)/sqrt(a + b/cos(e + f*x)), x), x) - Dist(c*g/d, Int((g/cos(e + f*x))**(S(3)/2)/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))), x), x)
rule4339 = ReplacementRule(pattern4339, replacement4339)
pattern4340 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(5)/2)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons71, cons1267)
def replacement4340(f, b, g, d, a, c, x, e):
rubi.append(4340)
return Dist(g/d, Int((g/sin(e + f*x))**(S(3)/2)/sqrt(a + b/sin(e + f*x)), x), x) - Dist(c*g/d, Int((g/sin(e + f*x))**(S(3)/2)/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))), x), x)
rule4340 = ReplacementRule(pattern4340, replacement4340)
pattern4341 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement4341(f, b, d, a, c, x, e):
rubi.append(4341)
return Dist(S(2)*b/f, Subst(Int(S(1)/(-b*d*x**S(2) + S(1)), x), x, tan(e + f*x)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x)))), x)
rule4341 = ReplacementRule(pattern4341, replacement4341)
pattern4342 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement4342(f, b, d, a, c, x, e):
rubi.append(4342)
return Dist(-S(2)*b/f, Subst(Int(S(1)/(-b*d*x**S(2) + S(1)), x), x, S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x))), x)
rule4342 = ReplacementRule(pattern4342, replacement4342)
pattern4343 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1322)
def replacement4343(f, b, d, a, c, x, e):
rubi.append(4343)
return Dist(b/d, Int(sqrt(c + d/cos(e + f*x))/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4343 = ReplacementRule(pattern4343, replacement4343)
pattern4344 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1322)
def replacement4344(f, b, d, a, c, x, e):
rubi.append(4344)
return Dist(b/d, Int(sqrt(c + d/sin(e + f*x))/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4344 = ReplacementRule(pattern4344, replacement4344)
pattern4345 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4345(f, b, d, a, c, x, e):
rubi.append(4345)
return Simp(S(2)*sqrt((S(1) + S(1)/cos(e + f*x))*(-a*d + b*c)/((a + b/cos(e + f*x))*(c - d)))*sqrt(-(S(1) - S(1)/cos(e + f*x))*(-a*d + b*c)/((a + b/cos(e + f*x))*(c + d)))*(a + b/cos(e + f*x))*EllipticPi(b*(c + d)/(d*(a + b)), asin(sqrt((a + b)/(c + d))*sqrt(c + d/cos(e + f*x))/sqrt(a + b/cos(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(d*f*sqrt((a + b)/(c + d))*tan(e + f*x)), x)
rule4345 = ReplacementRule(pattern4345, replacement4345)
pattern4346 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4346(f, b, d, a, c, x, e):
rubi.append(4346)
return Simp(-S(2)*sqrt((S(1) + S(1)/sin(e + f*x))*(-a*d + b*c)/((a + b/sin(e + f*x))*(c - d)))*sqrt(-(S(1) - S(1)/sin(e + f*x))*(-a*d + b*c)/((a + b/sin(e + f*x))*(c + d)))*(a + b/sin(e + f*x))*EllipticPi(b*(c + d)/(d*(a + b)), asin(sqrt((a + b)/(c + d))*sqrt(c + d/sin(e + f*x))/sqrt(a + b/sin(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))*tan(e + f*x)/(d*f*sqrt((a + b)/(c + d))), x)
rule4346 = ReplacementRule(pattern4346, replacement4346)
pattern4347 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement4347(f, b, d, a, c, x, e):
rubi.append(4347)
return Dist(S(2)*a/(b*f), Subst(Int(S(1)/(x**S(2)*(a*c - b*d) + S(2)), x), x, tan(e + f*x)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x)))), x)
rule4347 = ReplacementRule(pattern4347, replacement4347)
pattern4348 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1265, cons1323)
def replacement4348(f, b, d, a, c, x, e):
rubi.append(4348)
return Dist(-S(2)*a/(b*f), Subst(Int(S(1)/(x**S(2)*(a*c - b*d) + S(2)), x), x, S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*tan(e + f*x))), x)
rule4348 = ReplacementRule(pattern4348, replacement4348)
pattern4349 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4349(f, b, d, a, c, x, e):
rubi.append(4349)
return Simp(S(2)*sqrt((S(1) - S(1)/cos(e + f*x))*(-a*d + b*c)/((a + b)*(c + d/cos(e + f*x))))*sqrt(-(S(1) + S(1)/cos(e + f*x))*(-a*d + b*c)/((a - b)*(c + d/cos(e + f*x))))*(c + d/cos(e + f*x))*EllipticF(asin(sqrt(a + b/cos(e + f*x))*Rt((c + d)/(a + b), S(2))/sqrt(c + d/cos(e + f*x))), (a + b)*(c - d)/((a - b)*(c + d)))/(f*(-a*d + b*c)*Rt((c + d)/(a + b), S(2))*tan(e + f*x)), x)
rule4349 = ReplacementRule(pattern4349, replacement4349)
pattern4350 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4350(f, b, d, a, c, x, e):
rubi.append(4350)
return Simp(-S(2)*sqrt((S(1) - S(1)/sin(e + f*x))*(-a*d + b*c)/((a + b)*(c + d/sin(e + f*x))))*sqrt(-(S(1) + S(1)/sin(e + f*x))*(-a*d + b*c)/((a - b)*(c + d/sin(e + f*x))))*(c + d/sin(e + f*x))*EllipticF(asin(sqrt(a + b/sin(e + f*x))*Rt((c + d)/(a + b), S(2))/sqrt(c + d/sin(e + f*x))), (a + b)*(c - d)/((a - b)*(c + d)))*tan(e + f*x)/(f*(-a*d + b*c)*Rt((c + d)/(a + b), S(2))), x)
rule4350 = ReplacementRule(pattern4350, replacement4350)
pattern4351 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement4351(f, b, d, a, c, x, e):
rubi.append(4351)
return Dist(S(1)/b, Int(sqrt(a + b/cos(e + f*x))/(sqrt(c + d/cos(e + f*x))*cos(e + f*x)), x), x) - Dist(a/b, Int(S(1)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))*cos(e + f*x)), x), x)
rule4351 = ReplacementRule(pattern4351, replacement4351)
pattern4352 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71)
def replacement4352(f, b, d, a, c, x, e):
rubi.append(4352)
return Dist(S(1)/b, Int(sqrt(a + b/sin(e + f*x))/(sqrt(c + d/sin(e + f*x))*sin(e + f*x)), x), x) - Dist(a/b, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*sin(e + f*x)), x), x)
rule4352 = ReplacementRule(pattern4352, replacement4352)
pattern4353 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4353(f, b, d, a, c, x, e):
rubi.append(4353)
return Dist((a - b)/(c - d), Int(S(1)/(sqrt(a + b/cos(e + f*x))*sqrt(c + d/cos(e + f*x))*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/(c - d), Int((S(1) + S(1)/cos(e + f*x))/(sqrt(a + b/cos(e + f*x))*(c + d/cos(e + f*x))**(S(3)/2)*cos(e + f*x)), x), x)
rule4353 = ReplacementRule(pattern4353, replacement4353)
pattern4354 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1267, cons1323)
def replacement4354(f, b, d, a, c, x, e):
rubi.append(4354)
return Dist((a - b)/(c - d), Int(S(1)/(sqrt(a + b/sin(e + f*x))*sqrt(c + d/sin(e + f*x))*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/(c - d), Int((S(1) + S(1)/sin(e + f*x))/(sqrt(a + b/sin(e + f*x))*(c + d/sin(e + f*x))**(S(3)/2)*sin(e + f*x)), x), x)
rule4354 = ReplacementRule(pattern4354, replacement4354)
pattern4355 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons1265, cons1323, cons1620)
def replacement4355(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(4355)
return -Dist(a**S(2)*g*tan(e + f*x)/(f*sqrt(a - b/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), Subst(Int((g*x)**(p + S(-1))*(a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, S(1)/cos(e + f*x)), x)
rule4355 = ReplacementRule(pattern4355, replacement4355)
pattern4356 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons1265, cons1323, cons1620)
def replacement4356(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4356)
return Dist(a**S(2)*g/(f*sqrt(a - b/sin(e + f*x))*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), Subst(Int((g*x)**(p + S(-1))*(a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, S(1)/sin(e + f*x)), x)
rule4356 = ReplacementRule(pattern4356, replacement4356)
pattern4357 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons71, cons17, cons85)
def replacement4357(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(4357)
return Dist(g**(-m - n), Int((g/cos(e + f*x))**(m + n + p)*(a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n, x), x)
rule4357 = ReplacementRule(pattern4357, replacement4357)
pattern4358 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons71, cons17, cons85)
def replacement4358(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4358)
return Dist(g**(-m - n), Int((g/sin(e + f*x))**(m + n + p)*(a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n, x), x)
rule4358 = ReplacementRule(pattern4358, replacement4358)
pattern4359 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons71, cons1621, cons17)
def replacement4359(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(4359)
return Dist(g**(-m)*(g/cos(e + f*x))**(m + p)*(c + d/cos(e + f*x))**n*(c*cos(e + f*x) + d)**(-n), Int((a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n, x), x)
rule4359 = ReplacementRule(pattern4359, replacement4359)
pattern4360 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons71, cons1621, cons17)
def replacement4360(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4360)
return Dist(g**(-m)*(g/sin(e + f*x))**(m + p)*(c + d/sin(e + f*x))**n*(c*sin(e + f*x) + d)**(-n), Int((a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n, x), x)
rule4360 = ReplacementRule(pattern4360, replacement4360)
pattern4361 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons1621, cons18)
def replacement4361(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(4361)
return Dist((g/cos(e + f*x))**p*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n*(a*cos(e + f*x) + b)**(-m)*(c*cos(e + f*x) + d)**(-n), Int((a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n, x), x)
rule4361 = ReplacementRule(pattern4361, replacement4361)
pattern4362 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons1621, cons18)
def replacement4362(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4362)
return Dist((g/sin(e + f*x))**p*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n*(a*sin(e + f*x) + b)**(-m)*(c*sin(e + f*x) + d)**(-n), Int((a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n, x), x)
rule4362 = ReplacementRule(pattern4362, replacement4362)
pattern4363 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1304, cons1607, cons38, cons1622)
def replacement4363(p, m, f, b, d, a, c, n, x, e):
rubi.append(4363)
return Dist(sqrt(a + b/cos(e + f*x))*sqrt(c*cos(e + f*x) + d)/(sqrt(c + d/cos(e + f*x))*sqrt(a*cos(e + f*x) + b)), Int((a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-m - n - p), x), x)
rule4363 = ReplacementRule(pattern4363, replacement4363)
pattern4364 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons71, cons1304, cons1607, cons38, cons1622)
def replacement4364(p, m, f, b, d, a, c, n, x, e):
rubi.append(4364)
return Dist(sqrt(a + b/sin(e + f*x))*sqrt(c*sin(e + f*x) + d)/(sqrt(c + d/sin(e + f*x))*sqrt(a*sin(e + f*x) + b)), Int((a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-m - n - p), x), x)
rule4364 = ReplacementRule(pattern4364, replacement4364)
pattern4365 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons1415)
def replacement4365(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(4365)
return Int(ExpandTrig((g/cos(e + f*x))**p*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n, x), x)
rule4365 = ReplacementRule(pattern4365, replacement4365)
pattern4366 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons71, cons1415)
def replacement4366(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4366)
return Int(ExpandTrig((g/sin(e + f*x))**p*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n, x), x)
rule4366 = ReplacementRule(pattern4366, replacement4366)
pattern4367 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons380)
def replacement4367(p, m, f, g, b, d, a, c, n, x, e):
rubi.append(4367)
return Int((g/cos(e + f*x))**p*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n, x)
rule4367 = ReplacementRule(pattern4367, replacement4367)
pattern4368 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons380)
def replacement4368(p, m, f, b, g, d, c, a, n, x, e):
rubi.append(4368)
return Int((g/sin(e + f*x))**p*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n, x)
rule4368 = ReplacementRule(pattern4368, replacement4368)
pattern4369 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons1428)
def replacement4369(B, f, b, d, a, c, A, x, e):
rubi.append(4369)
return Simp(S(2)*A*sqrt((S(1) - S(1)/cos(e + f*x))*(-a*d + b*c)/((a + b)*(c + d/cos(e + f*x))))*(S(1) + S(1)/cos(e + f*x))*EllipticE(asin(sqrt(a + b/cos(e + f*x))*Rt((c + d)/(a + b), S(2))/sqrt(c + d/cos(e + f*x))), (a + b)*(c - d)/((a - b)*(c + d)))/(f*sqrt(-(S(1) + S(1)/cos(e + f*x))*(-a*d + b*c)/((a - b)*(c + d/cos(e + f*x))))*(-a*d + b*c)*Rt((c + d)/(a + b), S(2))*tan(e + f*x)), x)
rule4369 = ReplacementRule(pattern4369, replacement4369)
pattern4370 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons71, cons1267, cons1323, cons1428)
def replacement4370(B, f, b, d, a, c, A, x, e):
rubi.append(4370)
return Simp(-S(2)*A*sqrt((S(1) - S(1)/sin(e + f*x))*(-a*d + b*c)/((a + b)*(c + d/sin(e + f*x))))*(S(1) + S(1)/sin(e + f*x))*EllipticE(asin(sqrt(a + b/sin(e + f*x))*Rt((c + d)/(a + b), S(2))/sqrt(c + d/sin(e + f*x))), (a + b)*(c - d)/((a - b)*(c + d)))*tan(e + f*x)/(f*sqrt(-(S(1) + S(1)/sin(e + f*x))*(-a*d + b*c)/((a - b)*(c + d/sin(e + f*x))))*(-a*d + b*c)*Rt((c + d)/(a + b), S(2))), x)
rule4370 = ReplacementRule(pattern4370, replacement4370)
pattern4371 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons87, cons1586)
def replacement4371(B, f, b, d, a, n, A, x, e):
rubi.append(4371)
return Dist(S(1)/(d*n), Int((d/cos(e + f*x))**(n + S(1))*Simp(n*(A*b + B*a) + (A*a*(n + S(1)) + B*b*n)/cos(e + f*x), x), x), x) - Simp(A*a*(d/cos(e + f*x))**n*tan(e + f*x)/(f*n), x)
rule4371 = ReplacementRule(pattern4371, replacement4371)
pattern4372 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons87, cons1586)
def replacement4372(B, f, b, d, a, n, A, x, e):
rubi.append(4372)
return Dist(S(1)/(d*n), Int((d/sin(e + f*x))**(n + S(1))*Simp(n*(A*b + B*a) + (A*a*(n + S(1)) + B*b*n)/sin(e + f*x), x), x), x) + Simp(A*a*(d/sin(e + f*x))**n/(f*n*tan(e + f*x)), x)
rule4372 = ReplacementRule(pattern4372, replacement4372)
pattern4373 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1569)
def replacement4373(B, f, b, d, a, n, A, x, e):
rubi.append(4373)
return Dist(S(1)/(n + S(1)), Int((d/cos(e + f*x))**n*Simp(A*a*(n + S(1)) + B*b*n + (n + S(1))*(A*b + B*a)/cos(e + f*x), x), x), x) + Simp(B*b*(d/cos(e + f*x))**n*tan(e + f*x)/(f*(n + S(1))), x)
rule4373 = ReplacementRule(pattern4373, replacement4373)
pattern4374 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1569)
def replacement4374(B, f, b, d, a, n, A, x, e):
rubi.append(4374)
return Dist(S(1)/(n + S(1)), Int((d/sin(e + f*x))**n*Simp(A*a*(n + S(1)) + B*b*n + (n + S(1))*(A*b + B*a)/sin(e + f*x), x), x), x) - Simp(B*b*(d/sin(e + f*x))**n/(f*(n + S(1))*tan(e + f*x)), x)
rule4374 = ReplacementRule(pattern4374, replacement4374)
pattern4375 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1245)
def replacement4375(B, f, b, a, A, x, e):
rubi.append(4375)
return Dist(B/b, Int(S(1)/cos(e + f*x), x), x) + Dist((A*b - B*a)/b, Int(S(1)/((a + b/cos(e + f*x))*cos(e + f*x)), x), x)
rule4375 = ReplacementRule(pattern4375, replacement4375)
pattern4376 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1245)
def replacement4376(B, f, b, a, A, x, e):
rubi.append(4376)
return Dist(B/b, Int(S(1)/sin(e + f*x), x), x) + Dist((A*b - B*a)/b, Int(S(1)/((a + b/sin(e + f*x))*sin(e + f*x)), x), x)
rule4376 = ReplacementRule(pattern4376, replacement4376)
pattern4377 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons21, cons1245, cons1265, cons1623)
def replacement4377(B, m, f, b, a, A, x, e):
rubi.append(4377)
return Simp(B*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4377 = ReplacementRule(pattern4377, replacement4377)
pattern4378 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons21, cons1245, cons1265, cons1623)
def replacement4378(B, m, f, b, a, A, x, e):
rubi.append(4378)
return -Simp(B*(a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4378 = ReplacementRule(pattern4378, replacement4378)
pattern4379 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons1245, cons1265, cons1624, cons31, cons1320)
def replacement4379(B, m, f, b, a, A, x, e):
rubi.append(4379)
return Dist((A*b*(m + S(1)) + B*a*m)/(a*b*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))/cos(e + f*x), x), x) - Simp((a + b/cos(e + f*x))**m*(A*b - B*a)*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4379 = ReplacementRule(pattern4379, replacement4379)
pattern4380 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons1245, cons1265, cons1624, cons31, cons1320)
def replacement4380(B, m, f, b, a, A, x, e):
rubi.append(4380)
return Dist((A*b*(m + S(1)) + B*a*m)/(a*b*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))/sin(e + f*x), x), x) + Simp((a + b/sin(e + f*x))**m*(A*b - B*a)/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4380 = ReplacementRule(pattern4380, replacement4380)
pattern4381 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons21, cons1245, cons1265, cons1624, cons1321)
def replacement4381(B, m, f, b, a, A, x, e):
rubi.append(4381)
return Dist((A*b*(m + S(1)) + B*a*m)/(b*(m + S(1))), Int((a + b/cos(e + f*x))**m/cos(e + f*x), x), x) + Simp(B*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4381 = ReplacementRule(pattern4381, replacement4381)
pattern4382 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons21, cons1245, cons1265, cons1624, cons1321)
def replacement4382(B, m, f, b, a, A, x, e):
rubi.append(4382)
return Dist((A*b*(m + S(1)) + B*a*m)/(b*(m + S(1))), Int((a + b/sin(e + f*x))**m/sin(e + f*x), x), x) - Simp(B*(a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4382 = ReplacementRule(pattern4382, replacement4382)
pattern4383 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons1245, cons1267, cons31, cons168)
def replacement4383(B, m, f, b, a, A, x, e):
rubi.append(4383)
return Dist(S(1)/(m + S(1)), Int((a + b/cos(e + f*x))**(m + S(-1))*Simp(A*a*(m + S(1)) + B*b*m + (A*b*(m + S(1)) + B*a*m)/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp(B*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4383 = ReplacementRule(pattern4383, replacement4383)
pattern4384 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons1245, cons1267, cons31, cons168)
def replacement4384(B, m, f, b, a, A, x, e):
rubi.append(4384)
return Dist(S(1)/(m + S(1)), Int((a + b/sin(e + f*x))**(m + S(-1))*Simp(A*a*(m + S(1)) + B*b*m + (A*b*(m + S(1)) + B*a*m)/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp(B*(a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4384 = ReplacementRule(pattern4384, replacement4384)
pattern4385 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons1245, cons1267, cons31, cons94)
def replacement4385(B, m, f, b, a, A, x, e):
rubi.append(4385)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp((m + S(1))*(A*a - B*b) - (m + S(2))*(A*b - B*a)/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**(m + S(1))*(A*b - B*a)*tan(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4385 = ReplacementRule(pattern4385, replacement4385)
pattern4386 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons1245, cons1267, cons31, cons94)
def replacement4386(B, m, f, b, a, A, x, e):
rubi.append(4386)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp((m + S(1))*(A*a - B*b) - (m + S(2))*(A*b - B*a)/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**(m + S(1))*(A*b - B*a)/(f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4386 = ReplacementRule(pattern4386, replacement4386)
pattern4387 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1267, cons1625)
def replacement4387(B, f, b, a, A, x, e):
rubi.append(4387)
return Simp(S(2)*sqrt(b*(S(1) - S(1)/cos(e + f*x))/(a + b))*sqrt(-b*(S(1) + S(1)/cos(e + f*x))/(a - b))*(A*b - B*a)*EllipticE(asin(sqrt(a + b/cos(e + f*x))/Rt(a + B*b/A, S(2))), (A*a + B*b)/(A*a - B*b))*Rt(a + B*b/A, S(2))/(b**S(2)*f*tan(e + f*x)), x)
rule4387 = ReplacementRule(pattern4387, replacement4387)
pattern4388 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1267, cons1625)
def replacement4388(B, f, b, a, A, x, e):
rubi.append(4388)
return Simp(-S(2)*sqrt(b*(S(1) - S(1)/sin(e + f*x))/(a + b))*sqrt(-b*(S(1) + S(1)/sin(e + f*x))/(a - b))*(A*b - B*a)*EllipticE(asin(sqrt(a + b/sin(e + f*x))/Rt(a + B*b/A, S(2))), (A*a + B*b)/(A*a - B*b))*Rt(a + B*b/A, S(2))*tan(e + f*x)/(b**S(2)*f), x)
rule4388 = ReplacementRule(pattern4388, replacement4388)
pattern4389 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1267, cons1626)
def replacement4389(B, f, b, a, A, x, e):
rubi.append(4389)
return Dist(B, Int((S(1) + S(1)/cos(e + f*x))/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) + Dist(A - B, Int(S(1)/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x)
rule4389 = ReplacementRule(pattern4389, replacement4389)
pattern4390 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1267, cons1626)
def replacement4390(B, f, b, a, A, x, e):
rubi.append(4390)
return Dist(B, Int((S(1) + S(1)/sin(e + f*x))/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) + Dist(A - B, Int(S(1)/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x)
rule4390 = ReplacementRule(pattern4390, replacement4390)
pattern4391 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons1245, cons1267, cons1625, cons77)
def replacement4391(B, m, f, b, a, A, x, e):
rubi.append(4391)
return Simp(-S(2)*sqrt(S(2))*A*sqrt((A + B/cos(e + f*x))/A)*(A*(a + b/cos(e + f*x))/(A*a + B*b))**(-m)*(A - B/cos(e + f*x))*(a + b/cos(e + f*x))**m*AppellF1(S(1)/2, S(-1)/2, -m, S(3)/2, (A - B/cos(e + f*x))/(S(2)*A), b*(A - B/cos(e + f*x))/(A*b + B*a))/(B*f*tan(e + f*x)), x)
rule4391 = ReplacementRule(pattern4391, replacement4391)
pattern4392 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons1245, cons1267, cons1625, cons77)
def replacement4392(B, m, f, b, a, A, x, e):
rubi.append(4392)
return Simp(S(2)*sqrt(S(2))*A*sqrt((A + B/sin(e + f*x))/A)*(A*(a + b/sin(e + f*x))/(A*a + B*b))**(-m)*(A - B/sin(e + f*x))*(a + b/sin(e + f*x))**m*AppellF1(S(1)/2, S(-1)/2, -m, S(3)/2, (A - B/sin(e + f*x))/(S(2)*A), b*(A - B/sin(e + f*x))/(A*b + B*a))*tan(e + f*x)/(B*f), x)
rule4392 = ReplacementRule(pattern4392, replacement4392)
pattern4393 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons21, cons1245, cons1267)
def replacement4393(B, m, f, b, a, A, x, e):
rubi.append(4393)
return Dist(B/b, Int((a + b/cos(e + f*x))**(m + S(1))/cos(e + f*x), x), x) + Dist((A*b - B*a)/b, Int((a + b/cos(e + f*x))**m/cos(e + f*x), x), x)
rule4393 = ReplacementRule(pattern4393, replacement4393)
pattern4394 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons34, cons35, cons48, cons125, cons21, cons1245, cons1267)
def replacement4394(B, m, f, b, a, A, x, e):
rubi.append(4394)
return Dist(B/b, Int((a + b/sin(e + f*x))**(m + S(1))/sin(e + f*x), x), x) + Dist((A*b - B*a)/b, Int((a + b/sin(e + f*x))**m/sin(e + f*x), x), x)
rule4394 = ReplacementRule(pattern4394, replacement4394)
pattern4395 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1245, cons1265, cons31, cons1320)
def replacement4395(B, m, f, b, a, A, x, e):
rubi.append(4395)
return Dist(S(1)/(b**S(2)*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(A*b*m - B*a*m + B*b*(S(2)*m + S(1))/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**m*(A*b - B*a)*tan(e + f*x)/(b*f*(S(2)*m + S(1))), x)
rule4395 = ReplacementRule(pattern4395, replacement4395)
pattern4396 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1245, cons1265, cons31, cons1320)
def replacement4396(B, m, f, b, a, A, x, e):
rubi.append(4396)
return Dist(S(1)/(b**S(2)*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(A*b*m - B*a*m + B*b*(S(2)*m + S(1))/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**m*(A*b - B*a)/(b*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4396 = ReplacementRule(pattern4396, replacement4396)
pattern4397 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1245, cons1267, cons31, cons94)
def replacement4397(B, m, f, b, a, A, x, e):
rubi.append(4397)
return -Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(A*b - B*a) - (A*a*b*(m + S(2)) - B*(a**S(2) + b**S(2)*(m + S(1))))/cos(e + f*x), x)/cos(e + f*x), x), x) - Simp(a*(a + b/cos(e + f*x))**(m + S(1))*(A*b - B*a)*tan(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4397 = ReplacementRule(pattern4397, replacement4397)
pattern4398 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons1245, cons1267, cons31, cons94)
def replacement4398(B, m, f, b, a, A, x, e):
rubi.append(4398)
return -Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(A*b - B*a) - (A*a*b*(m + S(2)) - B*(a**S(2) + b**S(2)*(m + S(1))))/sin(e + f*x), x)/sin(e + f*x), x), x) + Simp(a*(a + b/sin(e + f*x))**(m + S(1))*(A*b - B*a)/(b*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4398 = ReplacementRule(pattern4398, replacement4398)
pattern4399 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons21, cons1245, cons272)
def replacement4399(B, m, f, b, a, A, x, e):
rubi.append(4399)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/cos(e + f*x))**m*Simp(B*b*(m + S(1)) + (A*b*(m + S(2)) - B*a)/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp(B*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + S(2))), x)
rule4399 = ReplacementRule(pattern4399, replacement4399)
pattern4400 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons21, cons1245, cons272)
def replacement4400(B, m, f, b, a, A, x, e):
rubi.append(4400)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/sin(e + f*x))**m*Simp(B*b*(m + S(1)) + (A*b*(m + S(2)) - B*a)/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp(B*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + S(2))*tan(e + f*x)), x)
rule4400 = ReplacementRule(pattern4400, replacement4400)
pattern4401 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons1245, cons1265, cons155, cons1627)
def replacement4401(B, m, f, b, d, a, n, A, x, e):
rubi.append(4401)
return -Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*n), x)
rule4401 = ReplacementRule(pattern4401, replacement4401)
pattern4402 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons1245, cons1265, cons155, cons1627)
def replacement4402(B, m, f, b, d, a, n, A, x, e):
rubi.append(4402)
return Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*n*tan(e + f*x)), x)
rule4402 = ReplacementRule(pattern4402, replacement4402)
pattern4403 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons155, cons31, cons32)
def replacement4403(B, m, f, b, d, a, n, A, x, e):
rubi.append(4403)
return Dist((A*a*m + B*b*(m + S(1)))/(a**S(2)*(S(2)*m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1)), x), x) + Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*(A*b - B*a)*tan(e + f*x)/(b*f*(S(2)*m + S(1))), x)
rule4403 = ReplacementRule(pattern4403, replacement4403)
pattern4404 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons155, cons31, cons32)
def replacement4404(B, m, f, b, d, a, n, A, x, e):
rubi.append(4404)
return Dist((A*a*m + B*b*(m + S(1)))/(a**S(2)*(S(2)*m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1)), x), x) - Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m*(A*b - B*a)/(b*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4404 = ReplacementRule(pattern4404, replacement4404)
pattern4405 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons1245, cons1265, cons155, cons1549)
def replacement4405(B, m, f, b, d, a, n, A, x, e):
rubi.append(4405)
return -Dist((A*a*m - B*b*n)/(b*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m, x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*n), x)
rule4405 = ReplacementRule(pattern4405, replacement4405)
pattern4406 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons1245, cons1265, cons155, cons1549)
def replacement4406(B, m, f, b, d, a, n, A, x, e):
rubi.append(4406)
return -Dist((A*a*m - B*b*n)/(b*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m, x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*n*tan(e + f*x)), x)
rule4406 = ReplacementRule(pattern4406, replacement4406)
pattern4407 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons1628)
def replacement4407(B, f, b, d, a, n, A, x, e):
rubi.append(4407)
return Simp(S(2)*B*b*(d/cos(e + f*x))**n*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(1))), x)
rule4407 = ReplacementRule(pattern4407, replacement4407)
pattern4408 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons1628)
def replacement4408(B, f, b, d, a, n, A, x, e):
rubi.append(4408)
return Simp(-S(2)*B*b*(d/sin(e + f*x))**n/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(1))*tan(e + f*x)), x)
rule4408 = ReplacementRule(pattern4408, replacement4408)
pattern4409 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1265, cons1629, cons87, cons463)
def replacement4409(B, f, b, d, a, n, A, x, e):
rubi.append(4409)
return Dist((A*b*(S(2)*n + S(1)) + S(2)*B*a*n)/(S(2)*a*d*n), Int((d/cos(e + f*x))**(n + S(1))*sqrt(a + b/cos(e + f*x)), x), x) - Simp(A*b**S(2)*(d/cos(e + f*x))**n*tan(e + f*x)/(a*f*n*sqrt(a + b/cos(e + f*x))), x)
rule4409 = ReplacementRule(pattern4409, replacement4409)
pattern4410 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1265, cons1629, cons87, cons463)
def replacement4410(B, f, b, d, a, n, A, x, e):
rubi.append(4410)
return Dist((A*b*(S(2)*n + S(1)) + S(2)*B*a*n)/(S(2)*a*d*n), Int((d/sin(e + f*x))**(n + S(1))*sqrt(a + b/sin(e + f*x)), x), x) + Simp(A*b**S(2)*(d/sin(e + f*x))**n/(a*f*n*sqrt(a + b/sin(e + f*x))*tan(e + f*x)), x)
rule4410 = ReplacementRule(pattern4410, replacement4410)
pattern4411 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons1629, cons1231)
def replacement4411(B, f, b, d, a, n, A, x, e):
rubi.append(4411)
return Dist((A*b*(S(2)*n + S(1)) + S(2)*B*a*n)/(b*(S(2)*n + S(1))), Int((d/cos(e + f*x))**n*sqrt(a + b/cos(e + f*x)), x), x) + Simp(S(2)*B*b*(d/cos(e + f*x))**n*tan(e + f*x)/(f*sqrt(a + b/cos(e + f*x))*(S(2)*n + S(1))), x)
rule4411 = ReplacementRule(pattern4411, replacement4411)
pattern4412 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons1629, cons1231)
def replacement4412(B, f, b, d, a, n, A, x, e):
rubi.append(4412)
return Dist((A*b*(S(2)*n + S(1)) + S(2)*B*a*n)/(b*(S(2)*n + S(1))), Int((d/sin(e + f*x))**n*sqrt(a + b/sin(e + f*x)), x), x) + Simp(-S(2)*B*b*(d/sin(e + f*x))**n/(f*sqrt(a + b/sin(e + f*x))*(S(2)*n + S(1))*tan(e + f*x)), x)
rule4412 = ReplacementRule(pattern4412, replacement4412)
pattern4413 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1265, cons93, cons1423, cons89)
def replacement4413(B, m, f, b, d, a, n, A, x, e):
rubi.append(4413)
return -Dist(b/(a*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-1))*Simp(A*a*(m - n + S(-1)) - B*b*n - (A*b*(m + n) + B*a*n)/cos(e + f*x), x), x), x) - Simp(A*a*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*n), x)
rule4413 = ReplacementRule(pattern4413, replacement4413)
pattern4414 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1265, cons93, cons1423, cons89)
def replacement4414(B, m, f, b, d, a, n, A, x, e):
rubi.append(4414)
return -Dist(b/(a*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-1))*Simp(A*a*(m - n + S(-1)) - B*b*n - (A*b*(m + n) + B*a*n)/sin(e + f*x), x), x), x) + Simp(A*a*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1))/(f*n*tan(e + f*x)), x)
rule4414 = ReplacementRule(pattern4414, replacement4414)
pattern4415 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons31, cons1423, cons346)
def replacement4415(B, m, f, b, d, a, n, A, x, e):
rubi.append(4415)
return Dist(S(1)/(d*(m + n)), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1))*Simp(A*a*d*(m + n) + B*b*d*n + (A*b*d*(m + n) + B*a*d*(S(2)*m + n + S(-1)))/cos(e + f*x), x), x), x) + Simp(B*b*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*(m + n)), x)
rule4415 = ReplacementRule(pattern4415, replacement4415)
pattern4416 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons31, cons1423, cons346)
def replacement4416(B, m, f, b, d, a, n, A, x, e):
rubi.append(4416)
return Dist(S(1)/(d*(m + n)), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1))*Simp(A*a*d*(m + n) + B*b*d*n + (A*b*d*(m + n) + B*a*d*(S(2)*m + n + S(-1)))/sin(e + f*x), x), x), x) - Simp(B*b*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1))/(f*(m + n)*tan(e + f*x)), x)
rule4416 = ReplacementRule(pattern4416, replacement4416)
pattern4417 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1265, cons93, cons1320, cons88)
def replacement4417(B, m, f, b, d, a, n, A, x, e):
rubi.append(4417)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*Simp(A*a*d*(n + S(-1)) - B*b*d*(n + S(-1)) - d*(A*b*(m + n) + B*a*(m - n + S(1)))/cos(e + f*x), x), x), x) - Simp(d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**m*(A*b - B*a)*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4417 = ReplacementRule(pattern4417, replacement4417)
pattern4418 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1265, cons93, cons1320, cons88)
def replacement4418(B, m, f, b, d, a, n, A, x, e):
rubi.append(4418)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*Simp(A*a*d*(n + S(-1)) - B*b*d*(n + S(-1)) - d*(A*b*(m + n) + B*a*(m - n + S(1)))/sin(e + f*x), x), x), x) + Simp(d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**m*(A*b - B*a)/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4418 = ReplacementRule(pattern4418, replacement4418)
pattern4419 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons31, cons1320, cons1327)
def replacement4419(B, m, f, b, d, a, n, A, x, e):
rubi.append(4419)
return -Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*Simp(-A*a*(S(2)*m + n + S(1)) + B*b*n + (A*b - B*a)*(m + n + S(1))/cos(e + f*x), x), x), x) + Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*(A*b - B*a)*tan(e + f*x)/(b*f*(S(2)*m + S(1))), x)
rule4419 = ReplacementRule(pattern4419, replacement4419)
pattern4420 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1265, cons31, cons1320, cons1327)
def replacement4420(B, m, f, b, d, a, n, A, x, e):
rubi.append(4420)
return -Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*Simp(-A*a*(S(2)*m + n + S(1)) + B*b*n + (A*b - B*a)*(m + n + S(1))/sin(e + f*x), x), x), x) - Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m*(A*b - B*a)/(b*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4420 = ReplacementRule(pattern4420, replacement4420)
pattern4421 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1265, cons87, cons165)
def replacement4421(B, m, f, b, d, a, n, A, x, e):
rubi.append(4421)
return Dist(d/(b*(m + n)), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**m*Simp(B*b*(n + S(-1)) + (A*b*(m + n) + B*a*m)/cos(e + f*x), x), x), x) + Simp(B*d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + n)), x)
rule4421 = ReplacementRule(pattern4421, replacement4421)
pattern4422 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1265, cons87, cons165)
def replacement4422(B, m, f, b, d, a, n, A, x, e):
rubi.append(4422)
return Dist(d/(b*(m + n)), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**m*Simp(B*b*(n + S(-1)) + (A*b*(m + n) + B*a*m)/sin(e + f*x), x), x), x) - Simp(B*d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**m/(f*(m + n)*tan(e + f*x)), x)
rule4422 = ReplacementRule(pattern4422, replacement4422)
pattern4423 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1265, cons87, cons463)
def replacement4423(B, m, f, b, d, a, n, A, x, e):
rubi.append(4423)
return -Dist(S(1)/(b*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m*Simp(A*a*m - A*b*(m + n + S(1))/cos(e + f*x) - B*b*n, x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*n), x)
rule4423 = ReplacementRule(pattern4423, replacement4423)
pattern4424 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1265, cons87, cons463)
def replacement4424(B, m, f, b, d, a, n, A, x, e):
rubi.append(4424)
return -Dist(S(1)/(b*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m*Simp(A*a*m - A*b*(m + n + S(1))/sin(e + f*x) - B*b*n, x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*n*tan(e + f*x)), x)
rule4424 = ReplacementRule(pattern4424, replacement4424)
pattern4425 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1265)
def replacement4425(B, m, f, b, d, a, n, A, x, e):
rubi.append(4425)
return Dist(B/b, Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1)), x), x) + Dist((A*b - B*a)/b, Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m, x), x)
rule4425 = ReplacementRule(pattern4425, replacement4425)
pattern4426 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1265)
def replacement4426(B, m, f, b, d, a, n, A, x, e):
rubi.append(4426)
return Dist(B/b, Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1)), x), x) + Dist((A*b - B*a)/b, Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m, x), x)
rule4426 = ReplacementRule(pattern4426, replacement4426)
pattern4427 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons166, cons1586)
def replacement4427(B, m, f, b, d, a, n, A, x, e):
rubi.append(4427)
return Dist(S(1)/(d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-2))*Simp(a*(-A*b*(m - n + S(-1)) + B*a*n) + b*(A*a*(m + n) + B*b*n)/cos(e + f*x)**S(2) + (A*(a**S(2)*(n + S(1)) + b**S(2)*n) + S(2)*B*a*b*n)/cos(e + f*x), x), x), x) - Simp(A*a*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*n), x)
rule4427 = ReplacementRule(pattern4427, replacement4427)
pattern4428 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons166, cons1586)
def replacement4428(B, m, f, b, d, a, n, A, x, e):
rubi.append(4428)
return Dist(S(1)/(d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-2))*Simp(a*(-A*b*(m - n + S(-1)) + B*a*n) + b*(A*a*(m + n) + B*b*n)/sin(e + f*x)**S(2) + (A*(a**S(2)*(n + S(1)) + b**S(2)*n) + S(2)*B*a*b*n)/sin(e + f*x), x), x), x) + Simp(A*a*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1))/(f*n*tan(e + f*x)), x)
rule4428 = ReplacementRule(pattern4428, replacement4428)
pattern4429 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1267, cons31, cons166, cons1630)
def replacement4429(B, m, f, b, d, a, n, A, x, e):
rubi.append(4429)
return Dist(S(1)/(m + n), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-2))*Simp(A*a**S(2)*(m + n) + B*a*b*n + b*(A*b*(m + n) + B*a*(S(2)*m + n + S(-1)))/cos(e + f*x)**S(2) + (B*b**S(2)*(m + n + S(-1)) + a*(m + n)*(S(2)*A*b + B*a))/cos(e + f*x), x), x), x) + Simp(B*b*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1))*tan(e + f*x)/(f*(m + n)), x)
rule4429 = ReplacementRule(pattern4429, replacement4429)
pattern4430 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1267, cons31, cons166, cons1630)
def replacement4430(B, m, f, b, d, a, n, A, x, e):
rubi.append(4430)
return Dist(S(1)/(m + n), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-2))*Simp(A*a**S(2)*(m + n) + B*a*b*n + b*(A*b*(m + n) + B*a*(S(2)*m + n + S(-1)))/sin(e + f*x)**S(2) + (B*b**S(2)*(m + n + S(-1)) + a*(m + n)*(S(2)*A*b + B*a))/sin(e + f*x), x), x), x) - Simp(B*b*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1))/(f*(m + n)*tan(e + f*x)), x)
rule4430 = ReplacementRule(pattern4430, replacement4430)
pattern4431 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons94, cons1325)
def replacement4431(B, m, f, b, d, a, n, A, x, e):
rubi.append(4431)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*Simp(d*(m + S(1))*(A*a - B*b)/cos(e + f*x) + d*(n + S(-1))*(A*b - B*a) - d*(A*b - B*a)*(m + n + S(1))/cos(e + f*x)**S(2), x), x), x) + Simp(d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*(A*b - B*a)*tan(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4431 = ReplacementRule(pattern4431, replacement4431)
pattern4432 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons94, cons1325)
def replacement4432(B, m, f, b, d, a, n, A, x, e):
rubi.append(4432)
return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*Simp(d*(m + S(1))*(A*a - B*b)/sin(e + f*x) + d*(n + S(-1))*(A*b - B*a) - d*(A*b - B*a)*(m + n + S(1))/sin(e + f*x)**S(2), x), x), x) - Simp(d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*(A*b - B*a)/(f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4432 = ReplacementRule(pattern4432, replacement4432)
pattern4433 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons94, cons165)
def replacement4433(B, m, f, b, d, a, n, A, x, e):
rubi.append(4433)
return -Dist(d/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**(m + S(1))*Simp(a*d*(n + S(-2))*(A*b - B*a) + b*d*(m + S(1))*(A*b - B*a)/cos(e + f*x) - (A*a*b*d*(m + n) - B*d*(a**S(2)*(n + S(-1)) + b**S(2)*(m + S(1))))/cos(e + f*x)**S(2), x), x), x) - Simp(a*d**S(2)*(d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**(m + S(1))*(A*b - B*a)*tan(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4433 = ReplacementRule(pattern4433, replacement4433)
pattern4434 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons94, cons165)
def replacement4434(B, m, f, b, d, a, n, A, x, e):
rubi.append(4434)
return -Dist(d/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**(m + S(1))*Simp(a*d*(n + S(-2))*(A*b - B*a) + b*d*(m + S(1))*(A*b - B*a)/sin(e + f*x) - (A*a*b*d*(m + n) - B*d*(a**S(2)*(n + S(-1)) + b**S(2)*(m + S(1))))/sin(e + f*x)**S(2), x), x), x) + Simp(a*d**S(2)*(d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**(m + S(1))*(A*b - B*a)/(b*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4434 = ReplacementRule(pattern4434, replacement4434)
pattern4435 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1267, cons31, cons94, cons1631)
def replacement4435(B, m, f, b, d, a, n, A, x, e):
rubi.append(4435)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*Simp(A*(a**S(2)*(m + S(1)) - b**S(2)*(m + n + S(1))) + B*a*b*n - a*(m + S(1))*(A*b - B*a)/cos(e + f*x) + b*(A*b - B*a)*(m + n + S(2))/cos(e + f*x)**S(2), x), x), x) - Simp(b*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*(A*b - B*a)*tan(e + f*x)/(a*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4435 = ReplacementRule(pattern4435, replacement4435)
pattern4436 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1267, cons31, cons94, cons1631)
def replacement4436(B, m, f, b, d, a, n, A, x, e):
rubi.append(4436)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*Simp(A*(a**S(2)*(m + S(1)) - b**S(2)*(m + n + S(1))) + B*a*b*n - a*(m + S(1))*(A*b - B*a)/sin(e + f*x) + b*(A*b - B*a)*(m + n + S(2))/sin(e + f*x)**S(2), x), x), x) + Simp(b*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*(A*b - B*a)/(a*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4436 = ReplacementRule(pattern4436, replacement4436)
pattern4437 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons1256, cons88)
def replacement4437(B, m, f, b, d, a, n, A, x, e):
rubi.append(4437)
return Dist(d/(m + n), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(-1))*Simp(B*a*(n + S(-1)) + (A*a*(m + n) + B*b*(m + n + S(-1)))/cos(e + f*x) + (A*b*(m + n) + B*a*m)/cos(e + f*x)**S(2), x), x), x) + Simp(B*d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + n)), x)
rule4437 = ReplacementRule(pattern4437, replacement4437)
pattern4438 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons1256, cons88)
def replacement4438(B, m, f, b, d, a, n, A, x, e):
rubi.append(4438)
return Dist(d/(m + n), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(-1))*Simp(B*a*(n + S(-1)) + (A*a*(m + n) + B*b*(m + n + S(-1)))/sin(e + f*x) + (A*b*(m + n) + B*a*m)/sin(e + f*x)**S(2), x), x), x) - Simp(B*d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**m/(f*(m + n)*tan(e + f*x)), x)
rule4438 = ReplacementRule(pattern4438, replacement4438)
pattern4439 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons1256, cons1586)
def replacement4439(B, m, f, b, d, a, n, A, x, e):
rubi.append(4439)
return -Dist(S(1)/(d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-1))*Simp(A*b*m - A*b*(m + n + S(1))/cos(e + f*x)**S(2) - B*a*n - (A*a*(n + S(1)) + B*b*n)/cos(e + f*x), x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*n), x)
rule4439 = ReplacementRule(pattern4439, replacement4439)
pattern4440 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267, cons93, cons1256, cons1586)
def replacement4440(B, m, f, b, d, a, n, A, x, e):
rubi.append(4440)
return -Dist(S(1)/(d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-1))*Simp(A*b*m - A*b*(m + n + S(1))/sin(e + f*x)**S(2) - B*a*n - (A*a*(n + S(1)) + B*b*n)/sin(e + f*x), x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*n*tan(e + f*x)), x)
rule4440 = ReplacementRule(pattern4440, replacement4440)
pattern4441 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1267, cons87, cons165, cons1359, cons1632)
def replacement4441(B, m, f, b, d, a, n, A, x, e):
rubi.append(4441)
return Dist(d**S(2)/(b*(m + n)), Int((d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**m*Simp(B*a*(n + S(-2)) + B*b*(m + n + S(-1))/cos(e + f*x) + (A*b*(m + n) - B*a*(n + S(-1)))/cos(e + f*x)**S(2), x), x), x) + Simp(B*d**S(2)*(d/cos(e + f*x))**(n + S(-2))*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + n)), x)
rule4441 = ReplacementRule(pattern4441, replacement4441)
pattern4442 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1267, cons87, cons165, cons1359, cons1632)
def replacement4442(B, m, f, b, d, a, n, A, x, e):
rubi.append(4442)
return Dist(d**S(2)/(b*(m + n)), Int((d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**m*Simp(B*a*(n + S(-2)) + B*b*(m + n + S(-1))/sin(e + f*x) + (A*b*(m + n) - B*a*(n + S(-1)))/sin(e + f*x)**S(2), x), x), x) - Simp(B*d**S(2)*(d/sin(e + f*x))**(n + S(-2))*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + n)*tan(e + f*x)), x)
rule4442 = ReplacementRule(pattern4442, replacement4442)
pattern4443 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1267, cons87, cons1586)
def replacement4443(B, m, f, b, d, a, n, A, x, e):
rubi.append(4443)
return Dist(S(1)/(a*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m*Simp(A*a*(n + S(1))/cos(e + f*x) - A*b*(m + n + S(1)) + A*b*(m + n + S(2))/cos(e + f*x)**S(2) + B*a*n, x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(a*f*n), x)
rule4443 = ReplacementRule(pattern4443, replacement4443)
pattern4444 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons1245, cons1267, cons87, cons1586)
def replacement4444(B, m, f, b, d, a, n, A, x, e):
rubi.append(4444)
return Dist(S(1)/(a*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m*Simp(A*a*(n + S(1))/sin(e + f*x) - A*b*(m + n + S(1)) + A*b*(m + n + S(2))/sin(e + f*x)**S(2) + B*a*n, x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))/(a*f*n*tan(e + f*x)), x)
rule4444 = ReplacementRule(pattern4444, replacement4444)
pattern4445 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267)
def replacement4445(B, f, b, d, a, A, x, e):
rubi.append(4445)
return Dist(A/a, Int(sqrt(a + b/cos(e + f*x))/sqrt(d/cos(e + f*x)), x), x) - Dist((A*b - B*a)/(a*d), Int(sqrt(d/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x)
rule4445 = ReplacementRule(pattern4445, replacement4445)
pattern4446 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267)
def replacement4446(B, f, b, d, a, A, x, e):
rubi.append(4446)
return Dist(A/a, Int(sqrt(a + b/sin(e + f*x))/sqrt(d/sin(e + f*x)), x), x) - Dist((A*b - B*a)/(a*d), Int(sqrt(d/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x)
rule4446 = ReplacementRule(pattern4446, replacement4446)
pattern4447 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267)
def replacement4447(B, f, b, d, a, A, x, e):
rubi.append(4447)
return Dist(A, Int(sqrt(d/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x), x) + Dist(B/d, Int((d/cos(e + f*x))**(S(3)/2)/sqrt(a + b/cos(e + f*x)), x), x)
rule4447 = ReplacementRule(pattern4447, replacement4447)
pattern4448 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267)
def replacement4448(B, f, b, d, a, A, x, e):
rubi.append(4448)
return Dist(A, Int(sqrt(d/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x), x) + Dist(B/d, Int((d/sin(e + f*x))**(S(3)/2)/sqrt(a + b/sin(e + f*x)), x), x)
rule4448 = ReplacementRule(pattern4448, replacement4448)
pattern4449 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267)
def replacement4449(B, f, b, d, a, A, x, e):
rubi.append(4449)
return Dist(A, Int(sqrt(a + b/cos(e + f*x))/sqrt(d/cos(e + f*x)), x), x) + Dist(B/d, Int(sqrt(d/cos(e + f*x))*sqrt(a + b/cos(e + f*x)), x), x)
rule4449 = ReplacementRule(pattern4449, replacement4449)
pattern4450 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons1245, cons1267)
def replacement4450(B, f, b, d, a, A, x, e):
rubi.append(4450)
return Dist(A, Int(sqrt(a + b/sin(e + f*x))/sqrt(d/sin(e + f*x)), x), x) + Dist(B/d, Int(sqrt(d/sin(e + f*x))*sqrt(a + b/sin(e + f*x)), x), x)
rule4450 = ReplacementRule(pattern4450, replacement4450)
pattern4451 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1267)
def replacement4451(B, f, b, d, a, n, A, x, e):
rubi.append(4451)
return Dist(A/a, Int((d/cos(e + f*x))**n, x), x) - Dist((A*b - B*a)/(a*d), Int((d/cos(e + f*x))**(n + S(1))/(a + b/cos(e + f*x)), x), x)
rule4451 = ReplacementRule(pattern4451, replacement4451)
pattern4452 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons4, cons1245, cons1267)
def replacement4452(B, f, b, d, a, n, A, x, e):
rubi.append(4452)
return Dist(A/a, Int((d/sin(e + f*x))**n, x), x) - Dist((A*b - B*a)/(a*d), Int((d/sin(e + f*x))**(n + S(1))/(a + b/sin(e + f*x)), x), x)
rule4452 = ReplacementRule(pattern4452, replacement4452)
pattern4453 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons1245, cons1267)
def replacement4453(B, m, f, b, d, a, n, A, x, e):
rubi.append(4453)
return Int((d/cos(e + f*x))**n*(A + B/cos(e + f*x))*(a + b/cos(e + f*x))**m, x)
rule4453 = ReplacementRule(pattern4453, replacement4453)
pattern4454 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons21, cons4, cons1245, cons1267)
def replacement4454(B, m, f, b, d, a, n, A, x, e):
rubi.append(4454)
return Int((d/sin(e + f*x))**n*(A + B/sin(e + f*x))*(a + b/sin(e + f*x))**m, x)
rule4454 = ReplacementRule(pattern4454, replacement4454)
pattern4455 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons5, cons70, cons1265, cons375)
def replacement4455(B, p, e, m, f, b, d, a, n, c, x, A):
rubi.append(4455)
return Dist((-a*c)**m, Int((A*cos(e + f*x) + B)**p*(c*cos(e + f*x) + d)**(-m + n)*sin(e + f*x)**(S(2)*m)*cos(e + f*x)**(-m - n - p), x), x)
rule4455 = ReplacementRule(pattern4455, replacement4455)
pattern4456 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons4, cons5, cons70, cons1265, cons375)
def replacement4456(B, p, m, f, b, d, a, n, c, A, x, e):
rubi.append(4456)
return Dist((-a*c)**m, Int((A*sin(e + f*x) + B)**p*(c*sin(e + f*x) + d)**(-m + n)*sin(e + f*x)**(-m - n - p)*cos(e + f*x)**(S(2)*m), x), x)
rule4456 = ReplacementRule(pattern4456, replacement4456)
pattern4457 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons33)
def replacement4457(B, C, e, m, f, b, a, x, A):
rubi.append(4457)
return Dist(b**(S(-2)), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b/cos(e + f*x), x), x), x)
rule4457 = ReplacementRule(pattern4457, replacement4457)
pattern4458 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons33)
def replacement4458(B, C, m, f, b, a, A, x, e):
rubi.append(4458)
return Dist(b**(S(-2)), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b/sin(e + f*x), x), x), x)
rule4458 = ReplacementRule(pattern4458, replacement4458)
pattern4459 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1433)
def replacement4459(C, e, m, f, b, a, x, A):
rubi.append(4459)
return Dist(C/b**S(2), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(-a + b/cos(e + f*x), x), x), x)
rule4459 = ReplacementRule(pattern4459, replacement4459)
pattern4460 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1433)
def replacement4460(C, m, f, b, a, A, x, e):
rubi.append(4460)
return Dist(C/b**S(2), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(-a + b/sin(e + f*x), x), x), x)
rule4460 = ReplacementRule(pattern4460, replacement4460)
pattern4461 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons1633)
def replacement4461(C, m, f, b, A, x, e):
rubi.append(4461)
return -Simp(A*(b/cos(e + f*x))**m*tan(e + f*x)/(f*m), x)
rule4461 = ReplacementRule(pattern4461, replacement4461)
pattern4462 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons1633)
def replacement4462(C, m, f, b, A, x, e):
rubi.append(4462)
return Simp(A*(b/sin(e + f*x))**m/(f*m*tan(e + f*x)), x)
rule4462 = ReplacementRule(pattern4462, replacement4462)
pattern4463 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons1634, cons31, cons32)
def replacement4463(C, m, f, b, A, x, e):
rubi.append(4463)
return Dist((A*(m + S(1)) + C*m)/(b**S(2)*m), Int((b/cos(e + f*x))**(m + S(2)), x), x) - Simp(A*(b/cos(e + f*x))**m*tan(e + f*x)/(f*m), x)
rule4463 = ReplacementRule(pattern4463, replacement4463)
pattern4464 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons1634, cons31, cons32)
def replacement4464(C, m, f, b, A, x, e):
rubi.append(4464)
return Dist((A*(m + S(1)) + C*m)/(b**S(2)*m), Int((b/sin(e + f*x))**(m + S(2)), x), x) + Simp(A*(b/sin(e + f*x))**m/(f*m*tan(e + f*x)), x)
rule4464 = ReplacementRule(pattern4464, replacement4464)
pattern4465 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons1634, cons1549)
def replacement4465(C, m, f, b, A, x, e):
rubi.append(4465)
return Dist((A*(m + S(1)) + C*m)/(m + S(1)), Int((b/cos(e + f*x))**m, x), x) + Simp(C*(b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4465 = ReplacementRule(pattern4465, replacement4465)
pattern4466 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons1634, cons1549)
def replacement4466(C, m, f, b, A, x, e):
rubi.append(4466)
return Dist((A*(m + S(1)) + C*m)/(m + S(1)), Int((b/sin(e + f*x))**m, x), x) - Simp(C*(b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4466 = ReplacementRule(pattern4466, replacement4466)
pattern4467 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons35, cons36, cons21, cons1431)
def replacement4467(B, C, m, f, b, x, e):
rubi.append(4467)
return Dist(B/b, Int((b/cos(e + f*x))**(m + S(1)), x), x) + Dist(C/b**S(2), Int((b/cos(e + f*x))**(m + S(2)), x), x)
rule4467 = ReplacementRule(pattern4467, replacement4467)
pattern4468 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons35, cons36, cons21, cons1431)
def replacement4468(B, C, m, f, b, x, e):
rubi.append(4468)
return Dist(B/b, Int((b/sin(e + f*x))**(m + S(1)), x), x) + Dist(C/b**S(2), Int((b/sin(e + f*x))**(m + S(2)), x), x)
rule4468 = ReplacementRule(pattern4468, replacement4468)
pattern4469 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1635)
def replacement4469(B, C, m, f, b, A, x, e):
rubi.append(4469)
return Dist(B/b, Int((b/cos(e + f*x))**(m + S(1)), x), x) + Int((b/cos(e + f*x))**m*(A + C/cos(e + f*x)**S(2)), x)
rule4469 = ReplacementRule(pattern4469, replacement4469)
pattern4470 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1635)
def replacement4470(B, C, m, f, b, A, x, e):
rubi.append(4470)
return Dist(B/b, Int((b/sin(e + f*x))**(m + S(1)), x), x) + Int((b/sin(e + f*x))**m*(A + C/sin(e + f*x)**S(2)), x)
rule4470 = ReplacementRule(pattern4470, replacement4470)
pattern4471 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1636)
def replacement4471(B, C, e, f, b, a, x, A):
rubi.append(4471)
return Dist(S(1)/2, Int(Simp(S(2)*A*a + (S(2)*B*a + b*(S(2)*A + C))/cos(e + f*x) + S(2)*(B*b + C*a)/cos(e + f*x)**S(2), x), x), x) + Simp(C*b*tan(e + f*x)/(S(2)*f*cos(e + f*x)), x)
rule4471 = ReplacementRule(pattern4471, replacement4471)
pattern4472 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1636)
def replacement4472(B, C, f, b, a, A, x, e):
rubi.append(4472)
return Dist(S(1)/2, Int(Simp(S(2)*A*a + (S(2)*B*a + b*(S(2)*A + C))/sin(e + f*x) + S(2)*(B*b + C*a)/sin(e + f*x)**S(2), x), x), x) - Simp(C*b/(S(2)*f*sin(e + f*x)*tan(e + f*x)), x)
rule4472 = ReplacementRule(pattern4472, replacement4472)
pattern4473 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1637)
def replacement4473(C, e, f, b, a, x, A):
rubi.append(4473)
return Dist(S(1)/2, Int(Simp(S(2)*A*a + S(2)*C*a/cos(e + f*x)**S(2) + b*(S(2)*A + C)/cos(e + f*x), x), x), x) + Simp(C*b*tan(e + f*x)/(S(2)*f*cos(e + f*x)), x)
rule4473 = ReplacementRule(pattern4473, replacement4473)
pattern4474 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1637)
def replacement4474(C, f, b, a, A, x, e):
rubi.append(4474)
return Dist(S(1)/2, Int(Simp(S(2)*A*a + S(2)*C*a/sin(e + f*x)**S(2) + b*(S(2)*A + C)/sin(e + f*x), x), x), x) - Simp(C*b/(S(2)*f*sin(e + f*x)*tan(e + f*x)), x)
rule4474 = ReplacementRule(pattern4474, replacement4474)
pattern4475 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1636)
def replacement4475(B, C, e, f, b, a, x, A):
rubi.append(4475)
return Dist(S(1)/b, Int((A*b + (B*b - C*a)/cos(e + f*x))/(a + b/cos(e + f*x)), x), x) + Dist(C/b, Int(S(1)/cos(e + f*x), x), x)
rule4475 = ReplacementRule(pattern4475, replacement4475)
pattern4476 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1636)
def replacement4476(B, C, f, b, a, A, x, e):
rubi.append(4476)
return Dist(S(1)/b, Int((A*b + (B*b - C*a)/sin(e + f*x))/(a + b/sin(e + f*x)), x), x) + Dist(C/b, Int(S(1)/sin(e + f*x), x), x)
rule4476 = ReplacementRule(pattern4476, replacement4476)
pattern4477 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1637)
def replacement4477(C, e, f, b, a, x, A):
rubi.append(4477)
return Dist(S(1)/b, Int((A*b - C*a/cos(e + f*x))/(a + b/cos(e + f*x)), x), x) + Dist(C/b, Int(S(1)/cos(e + f*x), x), x)
rule4477 = ReplacementRule(pattern4477, replacement4477)
pattern4478 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1637)
def replacement4478(C, f, b, a, A, x, e):
rubi.append(4478)
return Dist(S(1)/b, Int((A*b - C*a/sin(e + f*x))/(a + b/sin(e + f*x)), x), x) + Dist(C/b, Int(S(1)/sin(e + f*x), x), x)
rule4478 = ReplacementRule(pattern4478, replacement4478)
pattern4479 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1265, cons31, cons1320)
def replacement4479(B, C, e, m, f, b, a, x, A):
rubi.append(4479)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(A*b*(S(2)*m + S(1)) + (B*b*(m + S(1)) - a*(A*(m + S(1)) - C*m))/cos(e + f*x), x), x), x) + Simp((a + b/cos(e + f*x))**m*(A*a - B*b + C*a)*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4479 = ReplacementRule(pattern4479, replacement4479)
pattern4480 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1265, cons31, cons1320)
def replacement4480(B, C, m, f, b, a, A, x, e):
rubi.append(4480)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(A*b*(S(2)*m + S(1)) + (B*b*(m + S(1)) - a*(A*(m + S(1)) - C*m))/sin(e + f*x), x), x), x) - Simp((a + b/sin(e + f*x))**m*(A*a - B*b + C*a)/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4480 = ReplacementRule(pattern4480, replacement4480)
pattern4481 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1265, cons31, cons1320)
def replacement4481(C, e, m, f, b, a, x, A):
rubi.append(4481)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(A*b*(S(2)*m + S(1)) - a*(A*(m + S(1)) - C*m)/cos(e + f*x), x), x), x) + Simp((A + C)*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(S(2)*m + S(1))), x)
rule4481 = ReplacementRule(pattern4481, replacement4481)
pattern4482 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1265, cons31, cons1320)
def replacement4482(C, m, f, b, a, A, x, e):
rubi.append(4482)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(A*b*(S(2)*m + S(1)) - a*(A*(m + S(1)) - C*m)/sin(e + f*x), x), x), x) - Simp((A + C)*(a + b/sin(e + f*x))**m/(f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4482 = ReplacementRule(pattern4482, replacement4482)
pattern4483 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1265, cons1321)
def replacement4483(B, C, e, m, f, b, a, x, A):
rubi.append(4483)
return Dist(S(1)/(b*(m + S(1))), Int((a + b/cos(e + f*x))**m*Simp(A*b*(m + S(1)) + (B*b*(m + S(1)) + C*a*m)/cos(e + f*x), x), x), x) + Simp(C*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4483 = ReplacementRule(pattern4483, replacement4483)
pattern4484 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1265, cons1321)
def replacement4484(B, C, m, f, b, a, A, x, e):
rubi.append(4484)
return Dist(S(1)/(b*(m + S(1))), Int((a + b/sin(e + f*x))**m*Simp(A*b*(m + S(1)) + (B*b*(m + S(1)) + C*a*m)/sin(e + f*x), x), x), x) - Simp(C*(a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4484 = ReplacementRule(pattern4484, replacement4484)
pattern4485 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1265, cons1321)
def replacement4485(C, e, m, f, b, a, x, A):
rubi.append(4485)
return Dist(S(1)/(b*(m + S(1))), Int((a + b/cos(e + f*x))**m*Simp(A*b*(m + S(1)) + C*a*m/cos(e + f*x), x), x), x) + Simp(C*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4485 = ReplacementRule(pattern4485, replacement4485)
pattern4486 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1265, cons1321)
def replacement4486(C, m, f, b, a, A, x, e):
rubi.append(4486)
return Dist(S(1)/(b*(m + S(1))), Int((a + b/sin(e + f*x))**m*Simp(A*b*(m + S(1)) + C*a*m/sin(e + f*x), x), x), x) - Simp(C*(a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4486 = ReplacementRule(pattern4486, replacement4486)
pattern4487 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1267, cons1638)
def replacement4487(B, C, e, m, f, b, a, x, A):
rubi.append(4487)
return Dist(S(1)/(m + S(1)), Int((a + b/cos(e + f*x))**(m + S(-1))*Simp(A*a*(m + S(1)) + (B*b*(m + S(1)) + C*a*m)/cos(e + f*x)**S(2) + (C*b*m + (m + S(1))*(A*b + B*a))/cos(e + f*x), x), x), x) + Simp(C*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4487 = ReplacementRule(pattern4487, replacement4487)
pattern4488 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1267, cons1638)
def replacement4488(B, C, m, f, b, a, A, x, e):
rubi.append(4488)
return Dist(S(1)/(m + S(1)), Int((a + b/sin(e + f*x))**(m + S(-1))*Simp(A*a*(m + S(1)) + (B*b*(m + S(1)) + C*a*m)/sin(e + f*x)**S(2) + (C*b*m + (m + S(1))*(A*b + B*a))/sin(e + f*x), x), x), x) - Simp(C*(a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4488 = ReplacementRule(pattern4488, replacement4488)
pattern4489 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1267, cons1638)
def replacement4489(C, e, m, f, b, a, x, A):
rubi.append(4489)
return Dist(S(1)/(m + S(1)), Int((a + b/cos(e + f*x))**(m + S(-1))*Simp(A*a*(m + S(1)) + C*a*m/cos(e + f*x)**S(2) + (A*b*(m + S(1)) + C*b*m)/cos(e + f*x), x), x), x) + Simp(C*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + S(1))), x)
rule4489 = ReplacementRule(pattern4489, replacement4489)
pattern4490 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1267, cons1638)
def replacement4490(C, m, f, b, a, A, x, e):
rubi.append(4490)
return Dist(S(1)/(m + S(1)), Int((a + b/sin(e + f*x))**(m + S(-1))*Simp(A*a*(m + S(1)) + C*a*m/sin(e + f*x)**S(2) + (A*b*(m + S(1)) + C*b*m)/sin(e + f*x), x), x), x) - Simp(C*(a + b/sin(e + f*x))**m/(f*(m + S(1))*tan(e + f*x)), x)
rule4490 = ReplacementRule(pattern4490, replacement4490)
pattern4491 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1267)
def replacement4491(B, C, e, f, b, a, x, A):
rubi.append(4491)
return Dist(C, Int((S(1) + S(1)/cos(e + f*x))/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) + Int((A + (B - C)/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x)
rule4491 = ReplacementRule(pattern4491, replacement4491)
pattern4492 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1267)
def replacement4492(B, C, f, b, a, A, x, e):
rubi.append(4492)
return Dist(C, Int((S(1) + S(1)/sin(e + f*x))/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) + Int((A + (B - C)/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x)
rule4492 = ReplacementRule(pattern4492, replacement4492)
pattern4493 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1267)
def replacement4493(C, e, f, b, a, x, A):
rubi.append(4493)
return Dist(C, Int((S(1) + S(1)/cos(e + f*x))/(sqrt(a + b/cos(e + f*x))*cos(e + f*x)), x), x) + Int((A - C/cos(e + f*x))/sqrt(a + b/cos(e + f*x)), x)
rule4493 = ReplacementRule(pattern4493, replacement4493)
pattern4494 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1267)
def replacement4494(C, f, b, a, A, x, e):
rubi.append(4494)
return Dist(C, Int((S(1) + S(1)/sin(e + f*x))/(sqrt(a + b/sin(e + f*x))*sin(e + f*x)), x), x) + Int((A - C/sin(e + f*x))/sqrt(a + b/sin(e + f*x)), x)
rule4494 = ReplacementRule(pattern4494, replacement4494)
pattern4495 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1267, cons515, cons94)
def replacement4495(B, C, e, m, f, b, a, x, A):
rubi.append(4495)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(A*(a**S(2) - b**S(2))*(m + S(1)) - a*(m + S(1))*(A*b - B*a + C*b)/cos(e + f*x) + (m + S(2))*(A*b**S(2) - B*a*b + C*a**S(2))/cos(e + f*x)**S(2), x), x), x) - Simp((a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*tan(e + f*x)/(a*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4495 = ReplacementRule(pattern4495, replacement4495)
pattern4496 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1267, cons31, cons94)
def replacement4496(B, C, m, f, b, a, A, x, e):
rubi.append(4496)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(A*(a**S(2) - b**S(2))*(m + S(1)) - a*(m + S(1))*(A*b - B*a + C*b)/sin(e + f*x) + (m + S(2))*(A*b**S(2) - B*a*b + C*a**S(2))/sin(e + f*x)**S(2), x), x), x) + Simp((a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(a*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4496 = ReplacementRule(pattern4496, replacement4496)
pattern4497 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1267, cons515, cons94)
def replacement4497(C, e, m, f, b, a, x, A):
rubi.append(4497)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(A*(a**S(2) - b**S(2))*(m + S(1)) - a*b*(A + C)*(m + S(1))/cos(e + f*x) + (m + S(2))*(A*b**S(2) + C*a**S(2))/cos(e + f*x)**S(2), x), x), x) - Simp((a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*tan(e + f*x)/(a*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4497 = ReplacementRule(pattern4497, replacement4497)
pattern4498 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1267, cons515, cons94)
def replacement4498(C, m, f, b, a, A, x, e):
rubi.append(4498)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(A*(a**S(2) - b**S(2))*(m + S(1)) - a*b*(A + C)*(m + S(1))/sin(e + f*x) + (m + S(2))*(A*b**S(2) + C*a**S(2))/sin(e + f*x)**S(2), x), x), x) + Simp((a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(a*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4498 = ReplacementRule(pattern4498, replacement4498)
pattern4499 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1267, cons77)
def replacement4499(B, C, e, m, f, b, a, x, A):
rubi.append(4499)
return Dist(S(1)/b, Int((a + b/cos(e + f*x))**m*(A*b + (B*b - C*a)/cos(e + f*x)), x), x) + Dist(C/b, Int((a + b/cos(e + f*x))**(m + S(1))/cos(e + f*x), x), x)
rule4499 = ReplacementRule(pattern4499, replacement4499)
pattern4500 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1267, cons77)
def replacement4500(B, C, m, f, b, a, A, x, e):
rubi.append(4500)
return Dist(S(1)/b, Int((a + b/sin(e + f*x))**m*(A*b + (B*b - C*a)/sin(e + f*x)), x), x) + Dist(C/b, Int((a + b/sin(e + f*x))**(m + S(1))/sin(e + f*x), x), x)
rule4500 = ReplacementRule(pattern4500, replacement4500)
pattern4501 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1267, cons77)
def replacement4501(C, e, m, f, b, a, x, A):
rubi.append(4501)
return Dist(S(1)/b, Int((a + b/cos(e + f*x))**m*(A*b - C*a/cos(e + f*x)), x), x) + Dist(C/b, Int((a + b/cos(e + f*x))**(m + S(1))/cos(e + f*x), x), x)
rule4501 = ReplacementRule(pattern4501, replacement4501)
pattern4502 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1267, cons77)
def replacement4502(C, m, f, b, a, A, x, e):
rubi.append(4502)
return Dist(S(1)/b, Int((a + b/sin(e + f*x))**m*(A*b - C*a/sin(e + f*x)), x), x) + Dist(C/b, Int((a + b/sin(e + f*x))**(m + S(1))/sin(e + f*x), x), x)
rule4502 = ReplacementRule(pattern4502, replacement4502)
pattern4503 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons18)
def replacement4503(B, C, e, m, f, b, x, A):
rubi.append(4503)
return Dist(b**S(2), Int((b*cos(e + f*x))**(m + S(-2))*(A*cos(e + f*x)**S(2) + B*cos(e + f*x) + C), x), x)
rule4503 = ReplacementRule(pattern4503, replacement4503)
pattern4504 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons18)
def replacement4504(B, C, e, m, f, b, x, A):
rubi.append(4504)
return Dist(b**S(2), Int((b*sin(e + f*x))**(m + S(-2))*(A*sin(e + f*x)**S(2) + B*sin(e + f*x) + C), x), x)
rule4504 = ReplacementRule(pattern4504, replacement4504)
pattern4505 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons18)
def replacement4505(C, e, m, f, b, x, A):
rubi.append(4505)
return Dist(b**S(2), Int((b*cos(e + f*x))**(m + S(-2))*(A*cos(e + f*x)**S(2) + C), x), x)
rule4505 = ReplacementRule(pattern4505, replacement4505)
pattern4506 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons48, cons125, cons34, cons36, cons21, cons18)
def replacement4506(C, e, m, f, b, x, A):
rubi.append(4506)
return Dist(b**S(2), Int((b*sin(e + f*x))**(m + S(-2))*(A*sin(e + f*x)**S(2) + C), x), x)
rule4506 = ReplacementRule(pattern4506, replacement4506)
pattern4507 = Pattern(Integral(((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons5, cons18)
def replacement4507(B, C, e, p, m, f, b, a, x, A):
rubi.append(4507)
return Dist(a**IntPart(m)*(a*(b/cos(e + f*x))**p)**FracPart(m)*(b/cos(e + f*x))**(-p*FracPart(m)), Int((b/cos(e + f*x))**(m*p)*(A + B/cos(e + f*x) + C/cos(e + f*x)**S(2)), x), x)
rule4507 = ReplacementRule(pattern4507, replacement4507)
pattern4508 = Pattern(Integral(((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons5, cons18)
def replacement4508(B, C, e, p, m, f, b, a, x, A):
rubi.append(4508)
return Dist(a**IntPart(m)*(a*(b/sin(e + f*x))**p)**FracPart(m)*(b/sin(e + f*x))**(-p*FracPart(m)), Int((b/sin(e + f*x))**(m*p)*(A + B/sin(e + f*x) + C/sin(e + f*x)**S(2)), x), x)
rule4508 = ReplacementRule(pattern4508, replacement4508)
pattern4509 = Pattern(Integral(((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons5, cons18)
def replacement4509(C, e, p, m, f, b, a, x, A):
rubi.append(4509)
return Dist(a**IntPart(m)*(a*(b/cos(e + f*x))**p)**FracPart(m)*(b/cos(e + f*x))**(-p*FracPart(m)), Int((b/cos(e + f*x))**(m*p)*(A + C/cos(e + f*x)**S(2)), x), x)
rule4509 = ReplacementRule(pattern4509, replacement4509)
pattern4510 = Pattern(Integral(((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons5, cons18)
def replacement4510(C, e, p, m, f, b, a, x, A):
rubi.append(4510)
return Dist(a**IntPart(m)*(a*(b/sin(e + f*x))**p)**FracPart(m)*(b/sin(e + f*x))**(-p*FracPart(m)), Int((b/sin(e + f*x))**(m*p)*(A + C/sin(e + f*x)**S(2)), x), x)
rule4510 = ReplacementRule(pattern4510, replacement4510)
pattern4511 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons87, cons89)
def replacement4511(B, C, e, f, b, d, a, n, x, A):
rubi.append(4511)
return Dist(S(1)/(d*n), Int((d/cos(e + f*x))**(n + S(1))*Simp(C*b*n/cos(e + f*x)**S(2) + n*(A*b + B*a) + (A*a*(n + S(1)) + n*(B*b + C*a))/cos(e + f*x), x), x), x) - Simp(A*a*(d/cos(e + f*x))**n*tan(e + f*x)/(f*n), x)
rule4511 = ReplacementRule(pattern4511, replacement4511)
pattern4512 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons87, cons89)
def replacement4512(B, C, f, b, d, a, n, A, x, e):
rubi.append(4512)
return Dist(S(1)/(d*n), Int((d/sin(e + f*x))**(n + S(1))*Simp(C*b*n/sin(e + f*x)**S(2) + n*(A*b + B*a) + (A*a*(n + S(1)) + n*(B*b + C*a))/sin(e + f*x), x), x), x) + Simp(A*a*(d/sin(e + f*x))**n/(f*n*tan(e + f*x)), x)
rule4512 = ReplacementRule(pattern4512, replacement4512)
pattern4513 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons87, cons89)
def replacement4513(C, e, f, b, d, a, n, x, A):
rubi.append(4513)
return Dist(S(1)/(d*n), Int((d/cos(e + f*x))**(n + S(1))*Simp(A*b*n + C*b*n/cos(e + f*x)**S(2) + a*(A*(n + S(1)) + C*n)/cos(e + f*x), x), x), x) - Simp(A*a*(d/cos(e + f*x))**n*tan(e + f*x)/(f*n), x)
rule4513 = ReplacementRule(pattern4513, replacement4513)
pattern4514 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons87, cons89)
def replacement4514(C, f, b, d, a, n, A, x, e):
rubi.append(4514)
return Dist(S(1)/(d*n), Int((d/sin(e + f*x))**(n + S(1))*Simp(A*b*n + C*b*n/sin(e + f*x)**S(2) + a*(A*(n + S(1)) + C*n)/sin(e + f*x), x), x), x) + Simp(A*a*(d/sin(e + f*x))**n/(f*n*tan(e + f*x)), x)
rule4514 = ReplacementRule(pattern4514, replacement4514)
pattern4515 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons346)
def replacement4515(B, C, f, b, d, a, n, A, x, e):
rubi.append(4515)
return Dist(S(1)/(n + S(2)), Int((d/cos(e + f*x))**n*Simp(A*a*(n + S(2)) + (n + S(2))*(B*b + C*a)/cos(e + f*x)**S(2) + (B*a*(n + S(2)) + b*(A*(n + S(2)) + C*(n + S(1))))/cos(e + f*x), x), x), x) + Simp(C*b*(d/cos(e + f*x))**n*tan(e + f*x)/(f*(n + S(2))*cos(e + f*x)), x)
rule4515 = ReplacementRule(pattern4515, replacement4515)
pattern4516 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons346)
def replacement4516(B, C, f, b, d, a, n, A, x, e):
rubi.append(4516)
return Dist(S(1)/(n + S(2)), Int((d/sin(e + f*x))**n*Simp(A*a*(n + S(2)) + (n + S(2))*(B*b + C*a)/sin(e + f*x)**S(2) + (B*a*(n + S(2)) + b*(A*(n + S(2)) + C*(n + S(1))))/sin(e + f*x), x), x), x) - Simp(C*b*(d/sin(e + f*x))**n/(f*(n + S(2))*sin(e + f*x)*tan(e + f*x)), x)
rule4516 = ReplacementRule(pattern4516, replacement4516)
pattern4517 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons346)
def replacement4517(C, f, b, d, a, n, A, x, e):
rubi.append(4517)
return Dist(S(1)/(n + S(2)), Int((d/cos(e + f*x))**n*Simp(A*a*(n + S(2)) + C*a*(n + S(2))/cos(e + f*x)**S(2) + b*(A*(n + S(2)) + C*(n + S(1)))/cos(e + f*x), x), x), x) + Simp(C*b*(d/cos(e + f*x))**n*tan(e + f*x)/(f*(n + S(2))*cos(e + f*x)), x)
rule4517 = ReplacementRule(pattern4517, replacement4517)
pattern4518 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons346)
def replacement4518(C, f, b, d, a, n, A, x, e):
rubi.append(4518)
return Dist(S(1)/(n + S(2)), Int((d/sin(e + f*x))**n*Simp(A*a*(n + S(2)) + C*a*(n + S(2))/sin(e + f*x)**S(2) + b*(A*(n + S(2)) + C*(n + S(1)))/sin(e + f*x), x), x), x) - Simp(C*b*(d/sin(e + f*x))**n/(f*(n + S(2))*sin(e + f*x)*tan(e + f*x)), x)
rule4518 = ReplacementRule(pattern4518, replacement4518)
pattern4519 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons31, cons94, cons1265)
def replacement4519(B, C, e, m, f, b, a, x, A):
rubi.append(4519)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(-S(2)*A*b*(m + S(1)) + B*a - C*b - (B*b*(m + S(2)) - a*(A*(m + S(2)) - C*(m + S(-1))))/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**m*(A*a - B*b + C*a)*tan(e + f*x)/(a*f*(S(2)*m + S(1))*cos(e + f*x)), x)
rule4519 = ReplacementRule(pattern4519, replacement4519)
pattern4520 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons31, cons94, cons1265)
def replacement4520(B, C, e, m, f, b, a, x, A):
rubi.append(4520)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(-S(2)*A*b*(m + S(1)) + B*a - C*b - (B*b*(m + S(2)) - a*(A*(m + S(2)) - C*(m + S(-1))))/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**m*(A*a - B*b + C*a)/(a*f*(S(2)*m + S(1))*sin(e + f*x)*tan(e + f*x)), x)
rule4520 = ReplacementRule(pattern4520, replacement4520)
pattern4521 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons31, cons94, cons1265)
def replacement4521(C, e, m, f, b, a, x, A):
rubi.append(4521)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(-S(2)*A*b*(m + S(1)) - C*b + a*(A*(m + S(2)) - C*(m + S(-1)))/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp((A + C)*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(S(2)*m + S(1))*cos(e + f*x)), x)
rule4521 = ReplacementRule(pattern4521, replacement4521)
pattern4522 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons31, cons94, cons1265)
def replacement4522(C, e, m, f, b, a, x, A):
rubi.append(4522)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(-S(2)*A*b*(m + S(1)) - C*b + a*(A*(m + S(2)) - C*(m + S(-1)))/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp((A + C)*(a + b/sin(e + f*x))**m/(f*(S(2)*m + S(1))*sin(e + f*x)*tan(e + f*x)), x)
rule4522 = ReplacementRule(pattern4522, replacement4522)
pattern4523 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons31, cons94, cons1267)
def replacement4523(B, C, e, m, f, b, a, x, A):
rubi.append(4523)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(A*a - B*b + C*a) - (A*b**S(2) - B*a*b + C*a**S(2) + b*(m + S(1))*(A*b - B*a + C*b))/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*tan(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4523 = ReplacementRule(pattern4523, replacement4523)
pattern4524 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons31, cons94, cons1267)
def replacement4524(B, C, e, m, f, b, a, x, A):
rubi.append(4524)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(A*a - B*b + C*a) - (A*b**S(2) - B*a*b + C*a**S(2) + b*(m + S(1))*(A*b - B*a + C*b))/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(b*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4524 = ReplacementRule(pattern4524, replacement4524)
pattern4525 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons31, cons94, cons1267)
def replacement4525(C, e, m, f, b, a, x, A):
rubi.append(4525)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(a*b*(A + C)*(m + S(1)) - (A*b**S(2) + C*a**S(2) + b*(m + S(1))*(A*b + C*b))/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp((a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*tan(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4525 = ReplacementRule(pattern4525, replacement4525)
pattern4526 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons31, cons94, cons1267)
def replacement4526(C, e, m, f, b, a, x, A):
rubi.append(4526)
return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(a*b*(A + C)*(m + S(1)) - (A*b**S(2) + C*a**S(2) + b*(m + S(1))*(A*b + C*b))/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp((a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(b*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4526 = ReplacementRule(pattern4526, replacement4526)
pattern4527 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons272)
def replacement4527(B, C, e, m, f, b, a, x, A):
rubi.append(4527)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/cos(e + f*x))**m*Simp(A*b*(m + S(2)) + C*b*(m + S(1)) + (B*b*(m + S(2)) - C*a)/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp(C*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + S(2))), x)
rule4527 = ReplacementRule(pattern4527, replacement4527)
pattern4528 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons272)
def replacement4528(B, C, e, m, f, b, a, x, A):
rubi.append(4528)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/sin(e + f*x))**m*Simp(A*b*(m + S(2)) + C*b*(m + S(1)) + (B*b*(m + S(2)) - C*a)/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp(C*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + S(2))*tan(e + f*x)), x)
rule4528 = ReplacementRule(pattern4528, replacement4528)
pattern4529 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons272)
def replacement4529(C, e, m, f, b, a, x, A):
rubi.append(4529)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/cos(e + f*x))**m*Simp(A*b*(m + S(2)) - C*a/cos(e + f*x) + C*b*(m + S(1)), x)/cos(e + f*x), x), x) + Simp(C*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + S(2))), x)
rule4529 = ReplacementRule(pattern4529, replacement4529)
pattern4530 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons272)
def replacement4530(C, e, m, f, b, a, x, A):
rubi.append(4530)
return Dist(S(1)/(b*(m + S(2))), Int((a + b/sin(e + f*x))**m*Simp(A*b*(m + S(2)) - C*a/sin(e + f*x) + C*b*(m + S(1)), x)/sin(e + f*x), x), x) - Simp(C*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + S(2))*tan(e + f*x)), x)
rule4530 = ReplacementRule(pattern4530, replacement4530)
pattern4531 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons1265, cons31, cons1320)
def replacement4531(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4531)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*Simp(-A*b*(S(2)*m + n + S(1)) + B*a*n - C*b*n - (B*b*(m + n + S(1)) - a*(A*(m + n + S(1)) - C*(m - n)))/cos(e + f*x), x), x), x) + Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*(A*a - B*b + C*a)*tan(e + f*x)/(a*f*(S(2)*m + S(1))), x)
rule4531 = ReplacementRule(pattern4531, replacement4531)
pattern4532 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons1265, cons31, cons1320)
def replacement4532(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4532)
return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*Simp(-A*b*(S(2)*m + n + S(1)) + B*a*n - C*b*n - (B*b*(m + n + S(1)) - a*(A*(m + n + S(1)) - C*(m - n)))/sin(e + f*x), x), x), x) - Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m*(A*a - B*b + C*a)/(a*f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4532 = ReplacementRule(pattern4532, replacement4532)
pattern4533 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons1265, cons31, cons1320)
def replacement4533(C, e, m, f, b, d, a, n, x, A):
rubi.append(4533)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*Simp(A*b*(S(2)*m + n + S(1)) + C*b*n - a*(A*(m + n + S(1)) - C*(m - n))/cos(e + f*x), x), x), x) + Simp((d/cos(e + f*x))**n*(A + C)*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(S(2)*m + S(1))), x)
rule4533 = ReplacementRule(pattern4533, replacement4533)
pattern4534 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons1265, cons31, cons1320)
def replacement4534(C, m, f, b, d, a, n, A, x, e):
rubi.append(4534)
return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*Simp(A*b*(S(2)*m + n + S(1)) + C*b*n - a*(A*(m + n + S(1)) - C*(m - n))/sin(e + f*x), x), x), x) - Simp((d/sin(e + f*x))**n*(A + C)*(a + b/sin(e + f*x))**m/(f*(S(2)*m + S(1))*tan(e + f*x)), x)
rule4534 = ReplacementRule(pattern4534, replacement4534)
pattern4535 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons1265, cons1321, cons1639)
def replacement4535(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4535)
return -Dist(S(1)/(b*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m*Simp(A*a*m - B*b*n - b*(A*(m + n + S(1)) + C*n)/cos(e + f*x), x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*n), x)
rule4535 = ReplacementRule(pattern4535, replacement4535)
pattern4536 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons1265, cons1321, cons1639)
def replacement4536(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4536)
return -Dist(S(1)/(b*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m*Simp(A*a*m - B*b*n - b*(A*(m + n + S(1)) + C*n)/sin(e + f*x), x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*n*tan(e + f*x)), x)
rule4536 = ReplacementRule(pattern4536, replacement4536)
pattern4537 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons1265, cons1321, cons1639)
def replacement4537(C, e, m, f, b, d, a, n, x, A):
rubi.append(4537)
return -Dist(S(1)/(b*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m*Simp(A*a*m - b*(A*(m + n + S(1)) + C*n)/cos(e + f*x), x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*n), x)
rule4537 = ReplacementRule(pattern4537, replacement4537)
pattern4538 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons1265, cons1321, cons1639)
def replacement4538(C, m, f, b, d, a, n, A, x, e):
rubi.append(4538)
return -Dist(S(1)/(b*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m*Simp(A*a*m - b*(A*(m + n + S(1)) + C*n)/sin(e + f*x), x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*n*tan(e + f*x)), x)
rule4538 = ReplacementRule(pattern4538, replacement4538)
pattern4539 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons1265, cons1321, cons1640, cons683)
def replacement4539(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4539)
return Dist(S(1)/(b*(m + n + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*Simp(A*b*(m + n + S(1)) + C*b*n + (B*b*(m + n + S(1)) + C*a*m)/cos(e + f*x), x), x), x) + Simp(C*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + n + S(1))), x)
rule4539 = ReplacementRule(pattern4539, replacement4539)
pattern4540 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons1265, cons1321, cons1640, cons683)
def replacement4540(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4540)
return Dist(S(1)/(b*(m + n + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m*Simp(A*b*(m + n + S(1)) + C*b*n + (B*b*(m + n + S(1)) + C*a*m)/sin(e + f*x), x), x), x) - Simp(C*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*(m + n + S(1))*tan(e + f*x)), x)
rule4540 = ReplacementRule(pattern4540, replacement4540)
pattern4541 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons1265, cons1321, cons1640, cons683)
def replacement4541(C, e, m, f, b, d, a, n, x, A):
rubi.append(4541)
return Dist(S(1)/(b*(m + n + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*Simp(A*b*(m + n + S(1)) + C*a*m/cos(e + f*x) + C*b*n, x), x), x) + Simp(C*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + n + S(1))), x)
rule4541 = ReplacementRule(pattern4541, replacement4541)
pattern4542 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons1265, cons1321, cons1640, cons683)
def replacement4542(C, m, f, b, d, a, n, A, x, e):
rubi.append(4542)
return Dist(S(1)/(b*(m + n + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m*Simp(A*b*(m + n + S(1)) + C*a*m/sin(e + f*x) + C*b*n, x), x), x) - Simp(C*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*(m + n + S(1))*tan(e + f*x)), x)
rule4542 = ReplacementRule(pattern4542, replacement4542)
pattern4543 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1267, cons31, cons94)
def replacement4543(B, C, e, m, f, b, a, x, A):
rubi.append(4543)
return -Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(-C*b*(a**S(2) - b**S(2))*(m + S(1))/cos(e + f*x)**S(2) + b*(m + S(1))*(A*b**S(2) - a*(B*b - C*a)) + (B*b*(a**S(2) + b**S(2)*(m + S(1))) - a*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1)))))/cos(e + f*x), x)/cos(e + f*x), x), x) - Simp(a*(a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*tan(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4543 = ReplacementRule(pattern4543, replacement4543)
pattern4544 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons1267, cons31, cons94)
def replacement4544(B, C, e, m, f, b, a, x, A):
rubi.append(4544)
return -Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(-C*b*(a**S(2) - b**S(2))*(m + S(1))/sin(e + f*x)**S(2) + b*(m + S(1))*(A*b**S(2) - a*(B*b - C*a)) + (B*b*(a**S(2) + b**S(2)*(m + S(1))) - a*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1)))))/sin(e + f*x), x)/sin(e + f*x), x), x) + Simp(a*(a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4544 = ReplacementRule(pattern4544, replacement4544)
pattern4545 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1267, cons31, cons94)
def replacement4545(C, e, m, f, b, a, x, A):
rubi.append(4545)
return -Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/cos(e + f*x))**(m + S(1))*Simp(-C*b*(a**S(2) - b**S(2))*(m + S(1))/cos(e + f*x)**S(2) - a*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1))))/cos(e + f*x) + b*(m + S(1))*(A*b**S(2) + C*a**S(2)), x)/cos(e + f*x), x), x) - Simp(a*(a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*tan(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4545 = ReplacementRule(pattern4545, replacement4545)
pattern4546 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons1267, cons31, cons94)
def replacement4546(C, e, m, f, b, a, x, A):
rubi.append(4546)
return -Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b/sin(e + f*x))**(m + S(1))*Simp(-C*b*(a**S(2) - b**S(2))*(m + S(1))/sin(e + f*x)**S(2) - a*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1))))/sin(e + f*x) + b*(m + S(1))*(A*b**S(2) + C*a**S(2)), x)/sin(e + f*x), x), x) + Simp(a*(a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4546 = ReplacementRule(pattern4546, replacement4546)
pattern4547 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1267, cons272)
def replacement4547(B, C, e, m, f, b, a, x, A):
rubi.append(4547)
return Dist(S(1)/(b*(m + S(3))), Int((a + b/cos(e + f*x))**m*Simp(C*a + b*(A*(m + S(3)) + C*(m + S(2)))/cos(e + f*x) - (-B*b*(m + S(3)) + S(2)*C*a)/cos(e + f*x)**S(2), x)/cos(e + f*x), x), x) + Simp(C*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + S(3))*cos(e + f*x)), x)
rule4547 = ReplacementRule(pattern4547, replacement4547)
pattern4548 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons35, cons36, cons21, cons1267, cons272)
def replacement4548(B, C, e, m, f, b, a, x, A):
rubi.append(4548)
return Dist(S(1)/(b*(m + S(3))), Int((a + b/sin(e + f*x))**m*Simp(C*a + b*(A*(m + S(3)) + C*(m + S(2)))/sin(e + f*x) - (-B*b*(m + S(3)) + S(2)*C*a)/sin(e + f*x)**S(2), x)/sin(e + f*x), x), x) - Simp(C*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + S(3))*sin(e + f*x)*tan(e + f*x)), x)
rule4548 = ReplacementRule(pattern4548, replacement4548)
pattern4549 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1267, cons272)
def replacement4549(C, e, m, f, b, a, x, A):
rubi.append(4549)
return Dist(S(1)/(b*(m + S(3))), Int((a + b/cos(e + f*x))**m*Simp(C*a - S(2)*C*a/cos(e + f*x)**S(2) + b*(A*(m + S(3)) + C*(m + S(2)))/cos(e + f*x), x)/cos(e + f*x), x), x) + Simp(C*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + S(3))*cos(e + f*x)), x)
rule4549 = ReplacementRule(pattern4549, replacement4549)
pattern4550 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons48, cons125, cons34, cons36, cons21, cons1267, cons272)
def replacement4550(C, e, m, f, b, a, x, A):
rubi.append(4550)
return Dist(S(1)/(b*(m + S(3))), Int((a + b/sin(e + f*x))**m*Simp(C*a - S(2)*C*a/sin(e + f*x)**S(2) + b*(A*(m + S(3)) + C*(m + S(2)))/sin(e + f*x), x)/sin(e + f*x), x), x) - Simp(C*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + S(3))*sin(e + f*x)*tan(e + f*x)), x)
rule4550 = ReplacementRule(pattern4550, replacement4550)
pattern4551 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267, cons93, cons168, cons1586)
def replacement4551(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4551)
return -Dist(S(1)/(d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-1))*Simp(A*b*m - B*a*n - b*(A*(m + n + S(1)) + C*n)/cos(e + f*x)**S(2) - (B*b*n + a*(A*(n + S(1)) + C*n))/cos(e + f*x), x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*n), x)
rule4551 = ReplacementRule(pattern4551, replacement4551)
pattern4552 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267, cons93, cons168, cons1586)
def replacement4552(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4552)
return -Dist(S(1)/(d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-1))*Simp(A*b*m - B*a*n - b*(A*(m + n + S(1)) + C*n)/sin(e + f*x)**S(2) - (B*b*n + a*(A*(n + S(1)) + C*n))/sin(e + f*x), x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*n*tan(e + f*x)), x)
rule4552 = ReplacementRule(pattern4552, replacement4552)
pattern4553 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267, cons93, cons168, cons1586)
def replacement4553(C, e, m, f, b, d, a, n, x, A):
rubi.append(4553)
return -Dist(S(1)/(d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**(m + S(-1))*Simp(A*b*m - a*(A*(n + S(1)) + C*n)/cos(e + f*x) - b*(A*(m + n + S(1)) + C*n)/cos(e + f*x)**S(2), x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*n), x)
rule4553 = ReplacementRule(pattern4553, replacement4553)
pattern4554 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267, cons93, cons168, cons1586)
def replacement4554(C, m, f, b, d, a, n, A, x, e):
rubi.append(4554)
return -Dist(S(1)/(d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**(m + S(-1))*Simp(A*b*m - a*(A*(n + S(1)) + C*n)/sin(e + f*x) - b*(A*(m + n + S(1)) + C*n)/sin(e + f*x)**S(2), x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*n*tan(e + f*x)), x)
rule4554 = ReplacementRule(pattern4554, replacement4554)
pattern4555 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons1267, cons31, cons168, cons1569)
def replacement4555(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4555)
return Dist(S(1)/(m + n + S(1)), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1))*Simp(A*a*(m + n + S(1)) + C*a*n + (B*b*(m + n + S(1)) + C*a*m)/cos(e + f*x)**S(2) + (C*b*(m + n) + (A*b + B*a)*(m + n + S(1)))/cos(e + f*x), x), x), x) + Simp(C*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + n + S(1))), x)
rule4555 = ReplacementRule(pattern4555, replacement4555)
pattern4556 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons1267, cons31, cons168, cons1569)
def replacement4556(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4556)
return Dist(S(1)/(m + n + S(1)), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1))*Simp(A*a*(m + n + S(1)) + C*a*n + (B*b*(m + n + S(1)) + C*a*m)/sin(e + f*x)**S(2) + (C*b*(m + n) + (A*b + B*a)*(m + n + S(1)))/sin(e + f*x), x), x), x) - Simp(C*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*(m + n + S(1))*tan(e + f*x)), x)
rule4556 = ReplacementRule(pattern4556, replacement4556)
pattern4557 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons1267, cons31, cons168, cons1569)
def replacement4557(C, e, m, f, b, d, a, n, x, A):
rubi.append(4557)
return Dist(S(1)/(m + n + S(1)), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(-1))*Simp(A*a*(m + n + S(1)) + C*a*m/cos(e + f*x)**S(2) + C*a*n + b*(A*(m + n + S(1)) + C*(m + n))/cos(e + f*x), x), x), x) + Simp(C*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*tan(e + f*x)/(f*(m + n + S(1))), x)
rule4557 = ReplacementRule(pattern4557, replacement4557)
pattern4558 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons1267, cons31, cons168, cons1569)
def replacement4558(C, m, f, b, d, a, n, A, x, e):
rubi.append(4558)
return Dist(S(1)/(m + n + S(1)), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(-1))*Simp(A*a*(m + n + S(1)) + C*a*m/sin(e + f*x)**S(2) + C*a*n + b*(A*(m + n + S(1)) + C*(m + n))/sin(e + f*x), x), x), x) - Simp(C*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m/(f*(m + n + S(1))*tan(e + f*x)), x)
rule4558 = ReplacementRule(pattern4558, replacement4558)
pattern4559 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267, cons93, cons94, cons88)
def replacement4559(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4559)
return Dist(d/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*Simp(A*b**S(2)*(n + S(-1)) - a*(n + S(-1))*(B*b - C*a) + b*(m + S(1))*(A*a - B*b + C*a)/cos(e + f*x) - (C*(a**S(2)*n + b**S(2)*(m + S(1))) + b*(A*b - B*a)*(m + n + S(1)))/cos(e + f*x)**S(2), x), x), x) + Simp(d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*tan(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4559 = ReplacementRule(pattern4559, replacement4559)
pattern4560 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267, cons93, cons94, cons88)
def replacement4560(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4560)
return Dist(d/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*Simp(A*b**S(2)*(n + S(-1)) - a*(n + S(-1))*(B*b - C*a) + b*(m + S(1))*(A*a - B*b + C*a)/sin(e + f*x) - (C*(a**S(2)*n + b**S(2)*(m + S(1))) + b*(A*b - B*a)*(m + n + S(1)))/sin(e + f*x)**S(2), x), x), x) - Simp(d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(b*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4560 = ReplacementRule(pattern4560, replacement4560)
pattern4561 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267, cons93, cons94, cons88)
def replacement4561(C, e, m, f, b, d, a, n, x, A):
rubi.append(4561)
return Dist(d/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*Simp(A*b**S(2)*(n + S(-1)) + C*a**S(2)*(n + S(-1)) + a*b*(A + C)*(m + S(1))/cos(e + f*x) - (A*b**S(2)*(m + n + S(1)) + C*(a**S(2)*n + b**S(2)*(m + S(1))))/cos(e + f*x)**S(2), x), x), x) + Simp(d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*tan(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4561 = ReplacementRule(pattern4561, replacement4561)
pattern4562 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267, cons93, cons94, cons88)
def replacement4562(C, m, f, b, d, a, n, A, x, e):
rubi.append(4562)
return Dist(d/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*Simp(A*b**S(2)*(n + S(-1)) + C*a**S(2)*(n + S(-1)) + a*b*(A + C)*(m + S(1))/sin(e + f*x) - (A*b**S(2)*(m + n + S(1)) + C*(a**S(2)*n + b**S(2)*(m + S(1))))/sin(e + f*x)**S(2), x), x), x) - Simp(d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(b*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4562 = ReplacementRule(pattern4562, replacement4562)
pattern4563 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons1267, cons31, cons94, cons1631)
def replacement4563(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4563)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*Simp(a*(m + S(1))*(A*a - B*b + C*a) - a*(m + S(1))*(A*b - B*a + C*b)/cos(e + f*x) - (m + n + S(1))*(A*b**S(2) - B*a*b + C*a**S(2)) + (m + n + S(2))*(A*b**S(2) - B*a*b + C*a**S(2))/cos(e + f*x)**S(2), x), x), x) - Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*tan(e + f*x)/(a*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4563 = ReplacementRule(pattern4563, replacement4563)
pattern4564 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons1267, cons31, cons94, cons1631)
def replacement4564(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4564)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*Simp(a*(m + S(1))*(A*a - B*b + C*a) - a*(m + S(1))*(A*b - B*a + C*b)/sin(e + f*x) - (m + n + S(1))*(A*b**S(2) - B*a*b + C*a**S(2)) + (m + n + S(2))*(A*b**S(2) - B*a*b + C*a**S(2))/sin(e + f*x)**S(2), x), x), x) + Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(a*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4564 = ReplacementRule(pattern4564, replacement4564)
pattern4565 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons1267, cons31, cons94, cons1631)
def replacement4565(C, e, m, f, b, d, a, n, x, A):
rubi.append(4565)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*Simp(a**S(2)*(A + C)*(m + S(1)) - a*b*(A + C)*(m + S(1))/cos(e + f*x) - (A*b**S(2) + C*a**S(2))*(m + n + S(1)) + (A*b**S(2) + C*a**S(2))*(m + n + S(2))/cos(e + f*x)**S(2), x), x), x) - Simp((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*tan(e + f*x)/(a*f*(a**S(2) - b**S(2))*(m + S(1))), x)
rule4565 = ReplacementRule(pattern4565, replacement4565)
pattern4566 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons1267, cons31, cons94, cons1631)
def replacement4566(C, m, f, b, d, a, n, A, x, e):
rubi.append(4566)
return Dist(S(1)/(a*(a**S(2) - b**S(2))*(m + S(1))), Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*Simp(a**S(2)*(A + C)*(m + S(1)) - a*b*(A + C)*(m + S(1))/sin(e + f*x) - (A*b**S(2) + C*a**S(2))*(m + n + S(1)) + (A*b**S(2) + C*a**S(2))*(m + n + S(2))/sin(e + f*x)**S(2), x), x), x) + Simp((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(a*f*(a**S(2) - b**S(2))*(m + S(1))*tan(e + f*x)), x)
rule4566 = ReplacementRule(pattern4566, replacement4566)
pattern4567 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons1267, cons87, cons88)
def replacement4567(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4567)
return Dist(d/(b*(m + n + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**m*Simp(C*a*(n + S(-1)) + (A*b*(m + n + S(1)) + C*b*(m + n))/cos(e + f*x) + (B*b*(m + n + S(1)) - C*a*n)/cos(e + f*x)**S(2), x), x), x) + Simp(C*d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + n + S(1))), x)
rule4567 = ReplacementRule(pattern4567, replacement4567)
pattern4568 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons1267, cons87, cons88)
def replacement4568(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4568)
return Dist(d/(b*(m + n + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**m*Simp(C*a*(n + S(-1)) + (A*b*(m + n + S(1)) + C*b*(m + n))/sin(e + f*x) + (B*b*(m + n + S(1)) - C*a*n)/sin(e + f*x)**S(2), x), x), x) - Simp(C*d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + n + S(1))*tan(e + f*x)), x)
rule4568 = ReplacementRule(pattern4568, replacement4568)
pattern4569 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons1267, cons87, cons88)
def replacement4569(C, e, m, f, b, d, a, n, x, A):
rubi.append(4569)
return Dist(d/(b*(m + n + S(1))), Int((d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**m*Simp(-C*a*n/cos(e + f*x)**S(2) + C*a*(n + S(-1)) + (A*b*(m + n + S(1)) + C*b*(m + n))/cos(e + f*x), x), x), x) + Simp(C*d*(d/cos(e + f*x))**(n + S(-1))*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(b*f*(m + n + S(1))), x)
rule4569 = ReplacementRule(pattern4569, replacement4569)
pattern4570 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons1267, cons87, cons88)
def replacement4570(C, m, f, b, d, a, n, A, x, e):
rubi.append(4570)
return Dist(d/(b*(m + n + S(1))), Int((d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**m*Simp(-C*a*n/sin(e + f*x)**S(2) + C*a*(n + S(-1)) + (A*b*(m + n + S(1)) + C*b*(m + n))/sin(e + f*x), x), x), x) - Simp(C*d*(d/sin(e + f*x))**(n + S(-1))*(a + b/sin(e + f*x))**(m + S(1))/(b*f*(m + n + S(1))*tan(e + f*x)), x)
rule4570 = ReplacementRule(pattern4570, replacement4570)
pattern4571 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons1267, cons87, cons1586)
def replacement4571(B, C, e, m, f, b, d, a, n, x, A):
rubi.append(4571)
return Dist(S(1)/(a*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m*Simp(-A*b*(m + n + S(1)) + A*b*(m + n + S(2))/cos(e + f*x)**S(2) + B*a*n + a*(A*n + A + C*n)/cos(e + f*x), x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(a*f*n), x)
rule4571 = ReplacementRule(pattern4571, replacement4571)
pattern4572 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons1267, cons87, cons1586)
def replacement4572(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4572)
return Dist(S(1)/(a*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m*Simp(-A*b*(m + n + S(1)) + A*b*(m + n + S(2))/sin(e + f*x)**S(2) + B*a*n + a*(A*n + A + C*n)/sin(e + f*x), x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))/(a*f*n*tan(e + f*x)), x)
rule4572 = ReplacementRule(pattern4572, replacement4572)
pattern4573 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons1267, cons87, cons1586)
def replacement4573(C, e, m, f, b, d, a, n, x, A):
rubi.append(4573)
return Dist(S(1)/(a*d*n), Int((d/cos(e + f*x))**(n + S(1))*(a + b/cos(e + f*x))**m*Simp(-A*b*(m + n + S(1)) + A*b*(m + n + S(2))/cos(e + f*x)**S(2) + a*(A*n + A + C*n)/cos(e + f*x), x), x), x) - Simp(A*(d/cos(e + f*x))**n*(a + b/cos(e + f*x))**(m + S(1))*tan(e + f*x)/(a*f*n), x)
rule4573 = ReplacementRule(pattern4573, replacement4573)
pattern4574 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons1267, cons87, cons1586)
def replacement4574(C, m, f, b, d, a, n, A, x, e):
rubi.append(4574)
return Dist(S(1)/(a*d*n), Int((d/sin(e + f*x))**(n + S(1))*(a + b/sin(e + f*x))**m*Simp(-A*b*(m + n + S(1)) + A*b*(m + n + S(2))/sin(e + f*x)**S(2) + a*(A*n + A + C*n)/sin(e + f*x), x), x), x) + Simp(A*(d/sin(e + f*x))**n*(a + b/sin(e + f*x))**(m + S(1))/(a*f*n*tan(e + f*x)), x)
rule4574 = ReplacementRule(pattern4574, replacement4574)
pattern4575 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267)
def replacement4575(B, C, e, f, b, d, a, x, A):
rubi.append(4575)
return Dist(a**(S(-2)), Int((A*a - (A*b - B*a)/cos(e + f*x))/sqrt(d/cos(e + f*x)), x), x) + Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2)*d**S(2)), Int((d/cos(e + f*x))**(S(3)/2)/(a + b/cos(e + f*x)), x), x)
rule4575 = ReplacementRule(pattern4575, replacement4575)
pattern4576 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267)
def replacement4576(B, C, f, b, d, a, A, x, e):
rubi.append(4576)
return Dist(a**(S(-2)), Int((A*a - (A*b - B*a)/sin(e + f*x))/sqrt(d/sin(e + f*x)), x), x) + Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2)*d**S(2)), Int((d/sin(e + f*x))**(S(3)/2)/(a + b/sin(e + f*x)), x), x)
rule4576 = ReplacementRule(pattern4576, replacement4576)
pattern4577 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267)
def replacement4577(C, e, f, b, d, a, x, A):
rubi.append(4577)
return Dist(a**(S(-2)), Int((A*a - A*b/cos(e + f*x))/sqrt(d/cos(e + f*x)), x), x) + Dist((A*b**S(2) + C*a**S(2))/(a**S(2)*d**S(2)), Int((d/cos(e + f*x))**(S(3)/2)/(a + b/cos(e + f*x)), x), x)
rule4577 = ReplacementRule(pattern4577, replacement4577)
pattern4578 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267)
def replacement4578(C, f, b, d, a, A, x, e):
rubi.append(4578)
return Dist(a**(S(-2)), Int((A*a - A*b/sin(e + f*x))/sqrt(d/sin(e + f*x)), x), x) + Dist((A*b**S(2) + C*a**S(2))/(a**S(2)*d**S(2)), Int((d/sin(e + f*x))**(S(3)/2)/(a + b/sin(e + f*x)), x), x)
rule4578 = ReplacementRule(pattern4578, replacement4578)
pattern4579 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267)
def replacement4579(B, C, e, f, b, d, a, x, A):
rubi.append(4579)
return Dist(C/d**S(2), Int((d/cos(e + f*x))**(S(3)/2)/sqrt(a + b/cos(e + f*x)), x), x) + Int((A + B/cos(e + f*x))/(sqrt(d/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), x)
rule4579 = ReplacementRule(pattern4579, replacement4579)
pattern4580 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons1267)
def replacement4580(B, C, f, b, d, a, A, x, e):
rubi.append(4580)
return Dist(C/d**S(2), Int((d/sin(e + f*x))**(S(3)/2)/sqrt(a + b/sin(e + f*x)), x), x) + Int((A + B/sin(e + f*x))/(sqrt(d/sin(e + f*x))*sqrt(a + b/sin(e + f*x))), x)
rule4580 = ReplacementRule(pattern4580, replacement4580)
pattern4581 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267)
def replacement4581(C, e, f, b, d, a, x, A):
rubi.append(4581)
return Dist(A, Int(S(1)/(sqrt(d/cos(e + f*x))*sqrt(a + b/cos(e + f*x))), x), x) + Dist(C/d**S(2), Int((d/cos(e + f*x))**(S(3)/2)/sqrt(a + b/cos(e + f*x)), x), x)
rule4581 = ReplacementRule(pattern4581, replacement4581)
pattern4582 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons1267)
def replacement4582(C, f, b, d, a, A, x, e):
rubi.append(4582)
return Dist(A, Int(S(1)/(sqrt(d/sin(e + f*x))*sqrt(a + b/sin(e + f*x))), x), x) + Dist(C/d**S(2), Int((d/sin(e + f*x))**(S(3)/2)/sqrt(a + b/sin(e + f*x)), x), x)
rule4582 = ReplacementRule(pattern4582, replacement4582)
pattern4583 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons1641)
def replacement4583(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4583)
return Int((d/cos(e + f*x))**n*(a + b/cos(e + f*x))**m*(A + B/cos(e + f*x) + C/cos(e + f*x)**S(2)), x)
rule4583 = ReplacementRule(pattern4583, replacement4583)
pattern4584 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons1641)
def replacement4584(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4584)
return Int((d/sin(e + f*x))**n*(a + b/sin(e + f*x))**m*(A + B/sin(e + f*x) + C/sin(e + f*x)**S(2)), x)
rule4584 = ReplacementRule(pattern4584, replacement4584)
pattern4585 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons1642)
def replacement4585(C, m, f, b, d, a, n, A, x, e):
rubi.append(4585)
return Int((d/cos(e + f*x))**n*(A + C/cos(e + f*x)**S(2))*(a + b/cos(e + f*x))**m, x)
rule4585 = ReplacementRule(pattern4585, replacement4585)
pattern4586 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons1642)
def replacement4586(C, m, f, b, d, a, n, A, x, e):
rubi.append(4586)
return Int((d/sin(e + f*x))**n*(A + C/sin(e + f*x)**S(2))*(a + b/sin(e + f*x))**m, x)
rule4586 = ReplacementRule(pattern4586, replacement4586)
pattern4587 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons23, cons17)
def replacement4587(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4587)
return Dist(d**(m + S(2)), Int((d*cos(e + f*x))**(-m + n + S(-2))*(a*cos(e + f*x) + b)**m*(A*cos(e + f*x)**S(2) + B*cos(e + f*x) + C), x), x)
rule4587 = ReplacementRule(pattern4587, replacement4587)
pattern4588 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons35, cons36, cons4, cons23, cons17)
def replacement4588(B, C, m, f, b, d, a, n, A, x, e):
rubi.append(4588)
return Dist(d**(m + S(2)), Int((d*sin(e + f*x))**(-m + n + S(-2))*(a*sin(e + f*x) + b)**m*(A*sin(e + f*x)**S(2) + B*sin(e + f*x) + C), x), x)
rule4588 = ReplacementRule(pattern4588, replacement4588)
pattern4589 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons23, cons17)
def replacement4589(C, m, f, b, d, a, n, A, x, e):
rubi.append(4589)
return Dist(d**(m + S(2)), Int((d*cos(e + f*x))**(-m + n + S(-2))*(A*cos(e + f*x)**S(2) + C)*(a*cos(e + f*x) + b)**m, x), x)
rule4589 = ReplacementRule(pattern4589, replacement4589)
pattern4590 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons27, cons48, cons125, cons34, cons36, cons4, cons23, cons17)
def replacement4590(C, m, f, b, d, a, n, A, x, e):
rubi.append(4590)
return Dist(d**(m + S(2)), Int((d*sin(e + f*x))**(-m + n + S(-2))*(A*sin(e + f*x)**S(2) + C)*(a*sin(e + f*x) + b)**m, x), x)
rule4590 = ReplacementRule(pattern4590, replacement4590)
pattern4591 = Pattern(Integral(((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons23)
def replacement4591(B, C, p, m, f, b, d, c, n, a, A, x, e):
rubi.append(4591)
return Dist(c**IntPart(n)*(c*(d/cos(e + f*x))**p)**FracPart(n)*(d/cos(e + f*x))**(-p*FracPart(n)), Int((d/cos(e + f*x))**(n*p)*(a + b/cos(e + f*x))**m*(A + B/cos(e + f*x) + C/cos(e + f*x)**S(2)), x), x)
rule4591 = ReplacementRule(pattern4591, replacement4591)
pattern4592 = Pattern(Integral(((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons36, cons21, cons4, cons5, cons23)
def replacement4592(B, C, p, m, f, b, d, c, n, a, A, x, e):
rubi.append(4592)
return Dist(c**IntPart(n)*(c*(d/sin(e + f*x))**p)**FracPart(n)*(d/sin(e + f*x))**(-p*FracPart(n)), Int((d/sin(e + f*x))**(n*p)*(a + b/sin(e + f*x))**m*(A + B/sin(e + f*x) + C/sin(e + f*x)**S(2)), x), x)
rule4592 = ReplacementRule(pattern4592, replacement4592)
pattern4593 = Pattern(Integral(((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons5, cons23)
def replacement4593(C, p, m, f, b, d, c, n, a, A, x, e):
rubi.append(4593)
return Dist(c**IntPart(n)*(c*(d/cos(e + f*x))**p)**FracPart(n)*(d/cos(e + f*x))**(-p*FracPart(n)), Int((d/cos(e + f*x))**(n*p)*(A + C/cos(e + f*x)**S(2))*(a + b/cos(e + f*x))**m, x), x)
rule4593 = ReplacementRule(pattern4593, replacement4593)
pattern4594 = Pattern(Integral(((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons36, cons21, cons4, cons5, cons23)
def replacement4594(C, p, m, f, b, d, c, n, a, A, x, e):
rubi.append(4594)
return Dist(c**IntPart(n)*(c*(d/sin(e + f*x))**p)**FracPart(n)*(d/sin(e + f*x))**(-p*FracPart(n)), Int((d/sin(e + f*x))**(n*p)*(A + C/sin(e + f*x)**S(2))*(a + b/sin(e + f*x))**m, x), x)
rule4594 = ReplacementRule(pattern4594, replacement4594)
pattern4595 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**n_, x_), cons3, cons7, cons27, cons4, cons1643)
def replacement4595(b, d, c, n, x):
rubi.append(4595)
return Dist(b/d, Subst(Int((b*x**S(2) + b)**(n + S(-1)), x), x, tan(c + d*x)), x)
rule4595 = ReplacementRule(pattern4595, replacement4595)
pattern4596 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**n_, x_), cons3, cons7, cons27, cons4, cons1643)
def replacement4596(b, d, c, n, x):
rubi.append(4596)
return -Dist(b/d, Subst(Int((b*x**S(2) + b)**(n + S(-1)), x), x, S(1)/tan(c + d*x)), x)
rule4596 = ReplacementRule(pattern4596, replacement4596)
pattern4597 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1454)
def replacement4597(p, b, d, c, a, x):
rubi.append(4597)
return Int((-a*tan(c + d*x)**S(2))**p, x)
rule4597 = ReplacementRule(pattern4597, replacement4597)
pattern4598 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1454)
def replacement4598(p, b, d, c, a, x):
rubi.append(4598)
return Int((-a/tan(c + d*x)**S(2))**p, x)
rule4598 = ReplacementRule(pattern4598, replacement4598)
pattern4599 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons1478)
def replacement4599(b, d, c, a, x):
rubi.append(4599)
return -Dist(b/a, Int(S(1)/(a*cos(c + d*x)**S(2) + b), x), x) + Simp(x/a, x)
rule4599 = ReplacementRule(pattern4599, replacement4599)
pattern4600 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons1478)
def replacement4600(b, d, c, a, x):
rubi.append(4600)
return -Dist(b/a, Int(S(1)/(a*sin(c + d*x)**S(2) + b), x), x) + Simp(x/a, x)
rule4600 = ReplacementRule(pattern4600, replacement4600)
pattern4601 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1478, cons54)
def replacement4601(p, b, d, c, a, x):
rubi.append(4601)
return Dist(S(1)/d, Subst(Int((a + b*x**S(2) + b)**p/(x**S(2) + S(1)), x), x, tan(c + d*x)), x)
rule4601 = ReplacementRule(pattern4601, replacement4601)
pattern4602 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons1478, cons54)
def replacement4602(p, b, d, c, a, x):
rubi.append(4602)
return -Dist(S(1)/d, Subst(Int((a + b*x**S(2) + b)**p/(x**S(2) + S(1)), x), x, S(1)/tan(c + d*x)), x)
rule4602 = ReplacementRule(pattern4602, replacement4602)
def With4603(p, m, b, d, c, a, n, x):
f = FreeFactors(tan(c + d*x), x)
rubi.append(4603)
return Dist(f**(m + S(1))/d, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1))*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, tan(c + d*x)/f), x)
pattern4603 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons5, cons1480, cons1479)
rule4603 = ReplacementRule(pattern4603, With4603)
def With4604(p, m, b, d, c, n, a, x):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(4604)
return -Dist(f**(m + S(1))/d, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1))*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, S(1)/(f*tan(c + d*x))), x)
pattern4604 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons5, cons1480, cons1479)
rule4604 = ReplacementRule(pattern4604, With4604)
def With4605(p, m, b, d, c, a, n, x):
f = FreeFactors(cos(c + d*x), x)
rubi.append(4605)
return -Dist(f/d, Subst(Int((f*x)**(-n*p)*(a*(f*x)**n + b)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, cos(c + d*x)/f), x)
pattern4605 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1481, cons376)
rule4605 = ReplacementRule(pattern4605, With4605)
def With4606(p, m, b, d, c, n, a, x):
f = FreeFactors(sin(c + d*x), x)
rubi.append(4606)
return Dist(f/d, Subst(Int((f*x)**(-n*p)*(a*(f*x)**n + b)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, sin(c + d*x)/f), x)
pattern4606 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1481, cons376)
rule4606 = ReplacementRule(pattern4606, With4606)
def With4607(p, m, b, d, c, a, n, x):
f = FreeFactors(tan(c + d*x), x)
rubi.append(4607)
return Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1))*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, tan(c + d*x)/f), x)
pattern4607 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*(S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons5, cons1480, cons1479)
rule4607 = ReplacementRule(pattern4607, With4607)
def With4608(p, m, b, d, c, n, a, x):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(4608)
return -Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1))*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, S(1)/(f*tan(c + d*x))), x)
pattern4608 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*(S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons7, cons27, cons5, cons1480, cons1479)
rule4608 = ReplacementRule(pattern4608, With4608)
def With4609(p, m, b, d, c, a, n, x):
f = FreeFactors(sin(c + d*x), x)
rubi.append(4609)
return Dist(f/d, Subst(Int((-f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1)/2)*ExpandToSum(a*(-f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, sin(c + d*x)/f), x)
pattern4609 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*(S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1481, cons1479, cons38)
rule4609 = ReplacementRule(pattern4609, With4609)
def With4610(p, m, b, d, c, n, a, x):
f = FreeFactors(cos(c + d*x), x)
rubi.append(4610)
return -Dist(f/d, Subst(Int((-f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1)/2)*ExpandToSum(a*(-f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, cos(c + d*x)/f), x)
pattern4610 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*(S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1481, cons1479, cons38)
rule4610 = ReplacementRule(pattern4610, With4610)
pattern4611 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*(S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons375)
def replacement4611(p, m, b, d, c, a, n, x):
rubi.append(4611)
return Int(ExpandTrig((a + b*(S(1)/cos(c + d*x))**n)**p*(S(1)/cos(c + d*x))**m, x), x)
rule4611 = ReplacementRule(pattern4611, replacement4611)
pattern4612 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*(S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons375)
def replacement4612(p, m, b, d, c, n, a, x):
rubi.append(4612)
return Int(ExpandTrig((a + b*(S(1)/sin(c + d*x))**n)**p*(S(1)/sin(c + d*x))**m, x), x)
rule4612 = ReplacementRule(pattern4612, replacement4612)
def With4613(p, m, b, d, c, a, n, x):
f = FreeFactors(cos(c + d*x), x)
rubi.append(4613)
return -Dist(f**(-m - n*p + S(1))/d, Subst(Int(x**(-m - n*p)*(a*(f*x)**n + b)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, cos(c + d*x)/f), x)
pattern4613 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1481, cons85, cons38)
rule4613 = ReplacementRule(pattern4613, With4613)
def With4614(p, m, b, d, c, n, a, x):
f = FreeFactors(sin(c + d*x), x)
rubi.append(4614)
return Dist(f**(-m - n*p + S(1))/d, Subst(Int(x**(-m - n*p)*(a*(f*x)**n + b)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, sin(c + d*x)/f), x)
pattern4614 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1481, cons85, cons38)
rule4614 = ReplacementRule(pattern4614, With4614)
def With4615(p, m, b, d, c, a, n, x):
f = FreeFactors(tan(c + d*x), x)
rubi.append(4615)
return Dist(f**(m + S(1))/d, Subst(Int(x**m*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p/(f**S(2)*x**S(2) + S(1)), x), x, tan(c + d*x)/f), x)
pattern4615 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1480, cons1479)
rule4615 = ReplacementRule(pattern4615, With4615)
def With4616(p, m, b, d, c, n, a, x):
f = FreeFactors(S(1)/tan(c + d*x), x)
rubi.append(4616)
return -Dist(f**(m + S(1))/d, Subst(Int(x**m*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p/(f**S(2)*x**S(2) + S(1)), x), x, S(1)/(f*tan(c + d*x))), x)
pattern4616 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons1480, cons1479)
rule4616 = ReplacementRule(pattern4616, With4616)
pattern4617 = Pattern(Integral(((S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons45, cons38)
def replacement4617(p, b, n2, d, a, n, c, x, e):
rubi.append(4617)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/cos(d + e*x))**n)**(S(2)*p), x), x)
rule4617 = ReplacementRule(pattern4617, replacement4617)
pattern4618 = Pattern(Integral(((S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons45, cons38)
def replacement4618(p, b, n2, d, a, n, c, x, e):
rubi.append(4618)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/sin(d + e*x))**n)**(S(2)*p), x), x)
rule4618 = ReplacementRule(pattern4618, replacement4618)
pattern4619 = Pattern(Integral(((S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons46, cons45, cons147)
def replacement4619(p, b, n2, d, a, n, c, x, e):
rubi.append(4619)
return Dist((b + S(2)*c*(S(1)/cos(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/cos(d + e*x))**n + c*(S(1)/cos(d + e*x))**(S(2)*n))**p, Int(u*(b + S(2)*c*(S(1)/cos(d + e*x))**n)**(S(2)*p), x), x)
rule4619 = ReplacementRule(pattern4619, replacement4619)
pattern4620 = Pattern(Integral(((S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons46, cons45, cons147)
def replacement4620(p, b, n2, d, a, n, c, x, e):
rubi.append(4620)
return Dist((b + S(2)*c*(S(1)/sin(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/sin(d + e*x))**n + c*(S(1)/sin(d + e*x))**(S(2)*n))**p, Int(u*(b + S(2)*c*(S(1)/sin(d + e*x))**n)**(S(2)*p), x), x)
rule4620 = ReplacementRule(pattern4620, replacement4620)
def With4621(b, n2, d, a, n, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(4621)
return Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*(S(1)/cos(d + e*x))**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*(S(1)/cos(d + e*x))**n + q), x), x)
pattern4621 = Pattern(Integral(S(1)/((S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226)
rule4621 = ReplacementRule(pattern4621, With4621)
def With4622(b, n2, d, a, n, c, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(4622)
return Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*(S(1)/sin(d + e*x))**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*(S(1)/sin(d + e*x))**n + q), x), x)
pattern4622 = Pattern(Integral(S(1)/((S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons226)
rule4622 = ReplacementRule(pattern4622, With4622)
def With4623(p, m, b, n2, d, a, n, c, x, e):
f = FreeFactors(cos(d + e*x), x)
rubi.append(4623)
return -Dist(f/e, Subst(Int((f*x)**(-n*p)*(a*(f*x)**n + b)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, cos(d + e*x)/f), x)
pattern4623 = Pattern(Integral(((S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**n2_*WC('c', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1481, cons376)
rule4623 = ReplacementRule(pattern4623, With4623)
def With4624(p, m, b, n2, d, a, n, c, x, e):
f = FreeFactors(sin(d + e*x), x)
rubi.append(4624)
return Dist(f/e, Subst(Int((f*x)**(-n*p)*(a*(f*x)**n + b)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, sin(d + e*x)/f), x)
pattern4624 = Pattern(Integral(((S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**n2_*WC('c', S(1)) + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1481, cons376)
rule4624 = ReplacementRule(pattern4624, With4624)
def With4625(p, m, b, n2, d, c, a, n, x, e):
f = FreeFactors(tan(d + e*x), x)
rubi.append(4625)
return Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1))*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + c*(f**S(2)*x**S(2) + S(1))**n, x)**p, x), x, tan(d + e*x)/f), x)
pattern4625 = Pattern(Integral(((S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**n2_*WC('c', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons46, cons1480, cons1479)
rule4625 = ReplacementRule(pattern4625, With4625)
def With4626(p, m, b, n2, d, c, a, n, x, e):
f = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(4626)
return -Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1))*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + c*(f**S(2)*x**S(2) + S(1))**n, x)**p, x), x, S(1)/(f*tan(d + e*x))), x)
pattern4626 = Pattern(Integral(((S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**n2_*WC('c', S(1)) + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons46, cons1480, cons1479)
rule4626 = ReplacementRule(pattern4626, With4626)
pattern4627 = Pattern(Integral(((S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons45, cons38)
def replacement4627(p, m, b, n2, d, a, n, c, x, e):
rubi.append(4627)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/cos(d + e*x))**n)**(S(2)*p)*(S(1)/cos(d + e*x))**m, x), x)
rule4627 = ReplacementRule(pattern4627, replacement4627)
pattern4628 = Pattern(Integral(((S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons46, cons45, cons38)
def replacement4628(p, m, b, n2, d, a, n, c, x, e):
rubi.append(4628)
return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/sin(d + e*x))**n)**(S(2)*p)*(S(1)/sin(d + e*x))**m, x), x)
rule4628 = ReplacementRule(pattern4628, replacement4628)
pattern4629 = Pattern(Integral(((S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons45, cons147)
def replacement4629(p, m, b, n2, d, a, n, c, x, e):
rubi.append(4629)
return Dist((b + S(2)*c*(S(1)/cos(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/cos(d + e*x))**n + c*(S(1)/cos(d + e*x))**(S(2)*n))**p, Int((b + S(2)*c*(S(1)/cos(d + e*x))**n)**(S(2)*p)*(S(1)/cos(d + e*x))**m, x), x)
rule4629 = ReplacementRule(pattern4629, replacement4629)
pattern4630 = Pattern(Integral(((S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons45, cons147)
def replacement4630(p, m, b, n2, d, a, n, c, x, e):
rubi.append(4630)
return Dist((b + S(2)*c*(S(1)/sin(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/sin(d + e*x))**n + c*(S(1)/sin(d + e*x))**(S(2)*n))**p, Int((b + S(2)*c*(S(1)/sin(d + e*x))**n)**(S(2)*p)*(S(1)/sin(d + e*x))**m, x), x)
rule4630 = ReplacementRule(pattern4630, replacement4630)
pattern4631 = Pattern(Integral(((S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons375)
def replacement4631(p, m, b, n2, d, a, n, c, x, e):
rubi.append(4631)
return Int(ExpandTrig((a + b*(S(1)/cos(d + e*x))**n + c*(S(1)/cos(d + e*x))**(S(2)*n))**p*(S(1)/cos(d + e*x))**m, x), x)
rule4631 = ReplacementRule(pattern4631, replacement4631)
pattern4632 = Pattern(Integral(((S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons375)
def replacement4632(p, m, b, n2, d, a, n, c, x, e):
rubi.append(4632)
return Int(ExpandTrig((a + b*(S(1)/sin(d + e*x))**n + c*(S(1)/sin(d + e*x))**(S(2)*n))**p*(S(1)/sin(d + e*x))**m, x), x)
rule4632 = ReplacementRule(pattern4632, replacement4632)
def With4633(p, m, b, n2, d, c, n, a, x, e):
f = FreeFactors(cos(d + e*x), x)
rubi.append(4633)
return -Dist(f**(-m - n*p + S(1))/e, Subst(Int(x**(-m - S(2)*n*p)*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(b*(f*x)**n + c*(f*x)**(S(2)*n) + c)**p, x), x, cos(d + e*x)/f), x)
pattern4633 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons1481, cons85, cons38)
rule4633 = ReplacementRule(pattern4633, With4633)
def With4634(p, m, b, n2, d, c, n, a, x, e):
f = FreeFactors(sin(d + e*x), x)
rubi.append(4634)
return Dist(f**(-m - n*p + S(1))/e, Subst(Int(x**(-m - S(2)*n*p)*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(b*(f*x)**n + c*(f*x)**(S(2)*n) + c)**p, x), x, sin(d + e*x)/f), x)
pattern4634 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons46, cons1481, cons85, cons38)
rule4634 = ReplacementRule(pattern4634, With4634)
def With4635(p, m, b, n2, d, c, a, n, x, e):
f = FreeFactors(tan(d + e*x), x)
rubi.append(4635)
return Dist(f**(m + S(1))/e, Subst(Int(x**m*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + c*(f**S(2)*x**S(2) + S(1))**n, x)**p/(f**S(2)*x**S(2) + S(1)), x), x, tan(d + e*x)/f), x)
pattern4635 = Pattern(Integral((a_ + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons1479)
rule4635 = ReplacementRule(pattern4635, With4635)
def With4636(p, m, b, n2, d, c, n, a, x, e):
f = FreeFactors(S(1)/tan(d + e*x), x)
rubi.append(4636)
return -Dist(f**(m + S(1))/e, Subst(Int(x**m*ExpandToSum(a + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + c*(f**S(2)*x**S(2) + S(1))**n, x)**p/(f**S(2)*x**S(2) + S(1)), x), x, S(1)/(f*tan(d + e*x))), x)
pattern4636 = Pattern(Integral((a_ + (S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons46, cons1480, cons1479)
rule4636 = ReplacementRule(pattern4636, With4636)
pattern4637 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons85)
def replacement4637(B, b, d, c, a, n, A, x, e):
rubi.append(4637)
return Dist(S(4)**(-n)*c**(-n), Int((A + B/cos(d + e*x))*(b + S(2)*c/cos(d + e*x))**(S(2)*n), x), x)
rule4637 = ReplacementRule(pattern4637, replacement4637)
pattern4638 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons85)
def replacement4638(B, b, d, c, a, n, A, x, e):
rubi.append(4638)
return Dist(S(4)**(-n)*c**(-n), Int((A + B/sin(d + e*x))*(b + S(2)*c/sin(d + e*x))**(S(2)*n), x), x)
rule4638 = ReplacementRule(pattern4638, replacement4638)
pattern4639 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons23)
def replacement4639(B, b, d, c, a, n, A, x, e):
rubi.append(4639)
return Dist((b + S(2)*c/cos(d + e*x))**(-S(2)*n)*(a + b/cos(d + e*x) + c/cos(d + e*x)**S(2))**n, Int((A + B/cos(d + e*x))*(b + S(2)*c/cos(d + e*x))**(S(2)*n), x), x)
rule4639 = ReplacementRule(pattern4639, replacement4639)
pattern4640 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons45, cons23)
def replacement4640(B, b, d, c, a, n, A, x, e):
rubi.append(4640)
return Dist((b + S(2)*c/sin(d + e*x))**(-S(2)*n)*(a + b/sin(d + e*x) + c/sin(d + e*x)**S(2))**n, Int((A + B/sin(d + e*x))*(b + S(2)*c/sin(d + e*x))**(S(2)*n), x), x)
rule4640 = ReplacementRule(pattern4640, replacement4640)
def With4641(B, b, d, a, c, A, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(4641)
return Dist(B - (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c/cos(d + e*x) - q), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c/cos(d + e*x) + q), x), x)
pattern4641 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226)
rule4641 = ReplacementRule(pattern4641, With4641)
def With4642(B, b, d, c, a, A, x, e):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(4642)
return Dist(B - (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c/sin(d + e*x) - q), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c/sin(d + e*x) + q), x), x)
pattern4642 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226)
rule4642 = ReplacementRule(pattern4642, With4642)
pattern4643 = Pattern(Integral((A_ + WC('B', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226, cons85)
def replacement4643(B, b, d, a, c, n, A, x, e):
rubi.append(4643)
return Int(ExpandTrig((A + B/cos(d + e*x))*(a + b/cos(d + e*x) + c/cos(d + e*x)**S(2))**n, x), x)
rule4643 = ReplacementRule(pattern4643, replacement4643)
pattern4644 = Pattern(Integral((A_ + WC('B', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons226, cons85)
def replacement4644(B, b, d, c, a, n, A, x, e):
rubi.append(4644)
return Int(ExpandTrig((A + B/sin(d + e*x))*(a + b/sin(d + e*x) + c/sin(d + e*x)**S(2))**n, x), x)
rule4644 = ReplacementRule(pattern4644, replacement4644)
pattern4645 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons62)
def replacement4645(m, f, d, c, x, e):
rubi.append(4645)
return -Dist(d*m/f, Int((c + d*x)**(m + S(-1))*log(-I*exp(I*(e + f*x)) + S(1)), x), x) + Dist(d*m/f, Int((c + d*x)**(m + S(-1))*log(I*exp(I*(e + f*x)) + S(1)), x), x) + Simp(-S(2)*I*(c + d*x)**m*ArcTan(exp(I*e + I*f*x))/f, x)
rule4645 = ReplacementRule(pattern4645, replacement4645)
pattern4646 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons7, cons27, cons48, cons125, cons62)
def replacement4646(m, f, d, c, x, e):
rubi.append(4646)
return -Dist(d*m/f, Int((c + d*x)**(m + S(-1))*log(-exp(I*(e + f*x)) + S(1)), x), x) + Dist(d*m/f, Int((c + d*x)**(m + S(-1))*log(exp(I*(e + f*x)) + S(1)), x), x) + Simp(-S(2)*(c + d*x)**m*atanh(exp(I*e + I*f*x))/f, x)
rule4646 = ReplacementRule(pattern4646, replacement4646)
pattern4647 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons7, cons27, cons48, cons125, cons31, cons168)
def replacement4647(m, f, d, c, x, e):
rubi.append(4647)
return -Dist(d*m/f, Int((c + d*x)**(m + S(-1))*tan(e + f*x), x), x) + Simp((c + d*x)**m*tan(e + f*x)/f, x)
rule4647 = ReplacementRule(pattern4647, replacement4647)
pattern4648 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons7, cons27, cons48, cons125, cons31, cons168)
def replacement4648(m, f, d, c, x, e):
rubi.append(4648)
return Dist(d*m/f, Int((c + d*x)**(m + S(-1))/tan(e + f*x), x), x) - Simp((c + d*x)**m/(f*tan(e + f*x)), x)
rule4648 = ReplacementRule(pattern4648, replacement4648)
pattern4649 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons165, cons1644)
def replacement4649(f, b, d, c, n, x, e):
rubi.append(4649)
return Dist(b**S(2)*(n + S(-2))/(n + S(-1)), Int((b/cos(e + f*x))**(n + S(-2))*(c + d*x), x), x) - Simp(b**S(2)*d*(b/cos(e + f*x))**(n + S(-2))/(f**S(2)*(n + S(-2))*(n + S(-1))), x) + Simp(b**S(2)*(b/cos(e + f*x))**(n + S(-2))*(c + d*x)*tan(e + f*x)/(f*(n + S(-1))), x)
rule4649 = ReplacementRule(pattern4649, replacement4649)
pattern4650 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons165, cons1644)
def replacement4650(f, b, d, c, n, x, e):
rubi.append(4650)
return Dist(b**S(2)*(n + S(-2))/(n + S(-1)), Int((b/sin(e + f*x))**(n + S(-2))*(c + d*x), x), x) - Simp(b**S(2)*d*(b/sin(e + f*x))**(n + S(-2))/(f**S(2)*(n + S(-2))*(n + S(-1))), x) - Simp(b**S(2)*(b/sin(e + f*x))**(n + S(-2))*(c + d*x)/(f*(n + S(-1))*tan(e + f*x)), x)
rule4650 = ReplacementRule(pattern4650, replacement4650)
pattern4651 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons1644, cons166)
def replacement4651(m, f, b, d, c, n, x, e):
rubi.append(4651)
return Dist(b**S(2)*(n + S(-2))/(n + S(-1)), Int((b/cos(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) + Dist(b**S(2)*d**S(2)*m*(m + S(-1))/(f**S(2)*(n + S(-2))*(n + S(-1))), Int((b/cos(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(-2)), x), x) + Simp(b**S(2)*(b/cos(e + f*x))**(n + S(-2))*(c + d*x)**m*tan(e + f*x)/(f*(n + S(-1))), x) - Simp(b**S(2)*d*m*(b/cos(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(-1))/(f**S(2)*(n + S(-2))*(n + S(-1))), x)
rule4651 = ReplacementRule(pattern4651, replacement4651)
pattern4652 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons165, cons1644, cons166)
def replacement4652(m, f, b, d, c, n, x, e):
rubi.append(4652)
return Dist(b**S(2)*(n + S(-2))/(n + S(-1)), Int((b/sin(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) + Dist(b**S(2)*d**S(2)*m*(m + S(-1))/(f**S(2)*(n + S(-2))*(n + S(-1))), Int((b/sin(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(-2)), x), x) - Simp(b**S(2)*(b/sin(e + f*x))**(n + S(-2))*(c + d*x)**m/(f*(n + S(-1))*tan(e + f*x)), x) - Simp(b**S(2)*d*m*(b/sin(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(-1))/(f**S(2)*(n + S(-2))*(n + S(-1))), x)
rule4652 = ReplacementRule(pattern4652, replacement4652)
pattern4653 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons89)
def replacement4653(f, b, d, c, n, x, e):
rubi.append(4653)
return Dist((n + S(1))/(b**S(2)*n), Int((b/cos(e + f*x))**(n + S(2))*(c + d*x), x), x) + Simp(d*(b/cos(e + f*x))**n/(f**S(2)*n**S(2)), x) - Simp((b/cos(e + f*x))**(n + S(1))*(c + d*x)*sin(e + f*x)/(b*f*n), x)
rule4653 = ReplacementRule(pattern4653, replacement4653)
pattern4654 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons7, cons27, cons48, cons125, cons87, cons89)
def replacement4654(f, b, d, c, n, x, e):
rubi.append(4654)
return Dist((n + S(1))/(b**S(2)*n), Int((b/sin(e + f*x))**(n + S(2))*(c + d*x), x), x) + Simp(d*(b/sin(e + f*x))**n/(f**S(2)*n**S(2)), x) + Simp((b/sin(e + f*x))**(n + S(1))*(c + d*x)*cos(e + f*x)/(b*f*n), x)
rule4654 = ReplacementRule(pattern4654, replacement4654)
pattern4655 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons89, cons166)
def replacement4655(m, f, b, d, c, n, x, e):
rubi.append(4655)
return Dist((n + S(1))/(b**S(2)*n), Int((b/cos(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) - Dist(d**S(2)*m*(m + S(-1))/(f**S(2)*n**S(2)), Int((b/cos(e + f*x))**n*(c + d*x)**(m + S(-2)), x), x) - Simp((b/cos(e + f*x))**(n + S(1))*(c + d*x)**m*sin(e + f*x)/(b*f*n), x) + Simp(d*m*(b/cos(e + f*x))**n*(c + d*x)**(m + S(-1))/(f**S(2)*n**S(2)), x)
rule4655 = ReplacementRule(pattern4655, replacement4655)
pattern4656 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons7, cons27, cons48, cons125, cons93, cons89, cons166)
def replacement4656(m, f, b, d, c, n, x, e):
rubi.append(4656)
return Dist((n + S(1))/(b**S(2)*n), Int((b/sin(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) - Dist(d**S(2)*m*(m + S(-1))/(f**S(2)*n**S(2)), Int((b/sin(e + f*x))**n*(c + d*x)**(m + S(-2)), x), x) + Simp((b/sin(e + f*x))**(n + S(1))*(c + d*x)**m*cos(e + f*x)/(b*f*n), x) + Simp(d*m*(b/sin(e + f*x))**n*(c + d*x)**(m + S(-1))/(f**S(2)*n**S(2)), x)
rule4656 = ReplacementRule(pattern4656, replacement4656)
pattern4657 = Pattern(Integral((WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons23)
def replacement4657(m, f, b, d, c, n, x, e):
rubi.append(4657)
return Dist((b/cos(e + f*x))**n*(b*cos(e + f*x))**n, Int((b*cos(e + f*x))**(-n)*(c + d*x)**m, x), x)
rule4657 = ReplacementRule(pattern4657, replacement4657)
pattern4658 = Pattern(Integral((WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons23)
def replacement4658(m, f, b, d, c, n, x, e):
rubi.append(4658)
return Dist((b/sin(e + f*x))**n*(b*sin(e + f*x))**n, Int((b*sin(e + f*x))**(-n)*(c + d*x)**m, x), x)
rule4658 = ReplacementRule(pattern4658, replacement4658)
pattern4659 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons528)
def replacement4659(m, f, b, d, c, n, a, x, e):
rubi.append(4659)
return Int(ExpandIntegrand((c + d*x)**m, (a + b/cos(e + f*x))**n, x), x)
rule4659 = ReplacementRule(pattern4659, replacement4659)
pattern4660 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons528)
def replacement4660(m, f, b, d, c, n, a, x, e):
rubi.append(4660)
return Int(ExpandIntegrand((c + d*x)**m, (a + b/sin(e + f*x))**n, x), x)
rule4660 = ReplacementRule(pattern4660, replacement4660)
pattern4661 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons196, cons62)
def replacement4661(m, f, b, d, c, n, a, x, e):
rubi.append(4661)
return Int(ExpandIntegrand((c + d*x)**m, (a*cos(e + f*x) + b)**n*cos(e + f*x)**(-n), x), x)
rule4661 = ReplacementRule(pattern4661, replacement4661)
pattern4662 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons196, cons62)
def replacement4662(m, f, b, d, c, n, a, x, e):
rubi.append(4662)
return Int(ExpandIntegrand((c + d*x)**m, (a*sin(e + f*x) + b)**n*sin(e + f*x)**(-n), x), x)
rule4662 = ReplacementRule(pattern4662, replacement4662)
pattern4663 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement4663(v, u, m, b, a, n, x):
rubi.append(4663)
return Int((a + b/cos(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule4663 = ReplacementRule(pattern4663, replacement4663)
pattern4664 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(v_))**WC('n', S(1)), x_), cons2, cons3, cons21, cons4, cons810, cons811)
def replacement4664(v, u, m, b, a, n, x):
rubi.append(4664)
return Int((a + b/sin(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x)
rule4664 = ReplacementRule(pattern4664, replacement4664)
pattern4665 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement4665(m, f, b, d, c, a, n, x, e):
rubi.append(4665)
return Int((a + b/cos(e + f*x))**n*(c + d*x)**m, x)
rule4665 = ReplacementRule(pattern4665, replacement4665)
pattern4666 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement4666(m, f, b, d, a, n, c, x, e):
rubi.append(4666)
return Int((a + b/sin(e + f*x))**n*(c + d*x)**m, x)
rule4666 = ReplacementRule(pattern4666, replacement4666)
pattern4667 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1573, cons38)
def replacement4667(p, b, d, c, a, n, x):
rubi.append(4667)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b/cos(c + d*x))**p, x), x, x**n), x)
rule4667 = ReplacementRule(pattern4667, replacement4667)
pattern4668 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons1573, cons38)
def replacement4668(p, b, d, a, c, n, x):
rubi.append(4668)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b/sin(c + d*x))**p, x), x, x**n), x)
rule4668 = ReplacementRule(pattern4668, replacement4668)
pattern4669 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement4669(p, b, d, c, a, n, x):
rubi.append(4669)
return Int((a + b/cos(c + d*x**n))**p, x)
rule4669 = ReplacementRule(pattern4669, replacement4669)
pattern4670 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1495)
def replacement4670(p, b, d, a, c, n, x):
rubi.append(4670)
return Int((a + b/sin(c + d*x**n))**p, x)
rule4670 = ReplacementRule(pattern4670, replacement4670)
pattern4671 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69)
def replacement4671(p, u, b, d, c, a, n, x):
rubi.append(4671)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b/cos(c + d*x**n))**p, x), x, u), x)
rule4671 = ReplacementRule(pattern4671, replacement4671)
pattern4672 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons68, cons69)
def replacement4672(p, u, b, d, a, c, n, x):
rubi.append(4672)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b/sin(c + d*x**n))**p, x), x, u), x)
rule4672 = ReplacementRule(pattern4672, replacement4672)
pattern4673 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement4673(p, u, b, a, x):
rubi.append(4673)
return Int((a + b/cos(ExpandToSum(u, x)))**p, x)
rule4673 = ReplacementRule(pattern4673, replacement4673)
pattern4674 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons823, cons824)
def replacement4674(p, u, b, a, x):
rubi.append(4674)
return Int((a + b/sin(ExpandToSum(u, x)))**p, x)
rule4674 = ReplacementRule(pattern4674, replacement4674)
pattern4675 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1574, cons38)
def replacement4675(p, m, b, d, c, a, n, x):
rubi.append(4675)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b/cos(c + d*x))**p, x), x, x**n), x)
rule4675 = ReplacementRule(pattern4675, replacement4675)
pattern4676 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1574, cons38)
def replacement4676(p, m, b, d, a, c, n, x):
rubi.append(4676)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b/sin(c + d*x))**p, x), x, x**n), x)
rule4676 = ReplacementRule(pattern4676, replacement4676)
pattern4677 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1576)
def replacement4677(p, m, b, d, c, a, n, x):
rubi.append(4677)
return Int(x**m*(a + b/cos(c + d*x**n))**p, x)
rule4677 = ReplacementRule(pattern4677, replacement4677)
pattern4678 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons1576)
def replacement4678(p, m, b, d, a, c, n, x):
rubi.append(4678)
return Int(x**m*(a + b/sin(c + d*x**n))**p, x)
rule4678 = ReplacementRule(pattern4678, replacement4678)
pattern4679 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement4679(p, m, b, d, c, a, n, x, e):
rubi.append(4679)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b/cos(c + d*x**n))**p, x), x)
rule4679 = ReplacementRule(pattern4679, replacement4679)
pattern4680 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons1497)
def replacement4680(p, m, b, d, a, c, n, x, e):
rubi.append(4680)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b/sin(c + d*x**n))**p, x), x)
rule4680 = ReplacementRule(pattern4680, replacement4680)
pattern4681 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement4681(p, u, m, b, a, x, e):
rubi.append(4681)
return Int((e*x)**m*(a + b/cos(ExpandToSum(u, x)))**p, x)
rule4681 = ReplacementRule(pattern4681, replacement4681)
pattern4682 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(u_))**WC('p', S(1)), x_), cons2, cons3, cons48, cons21, cons5, cons823, cons824)
def replacement4682(p, u, m, b, a, x, e):
rubi.append(4682)
return Int((e*x)**m*(a + b/sin(ExpandToSum(u, x)))**p, x)
rule4682 = ReplacementRule(pattern4682, replacement4682)
pattern4683 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**p_*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons5, cons31, cons85, cons1577, cons1645)
def replacement4683(p, m, b, a, n, x):
rubi.append(4683)
return -Dist((m - n + S(1))/(b*n*(p + S(-1))), Int(x**(m - n)*(S(1)/cos(a + b*x**n))**(p + S(-1)), x), x) + Simp(x**(m - n + S(1))*(S(1)/cos(a + b*x**n))**(p + S(-1))/(b*n*(p + S(-1))), x)
rule4683 = ReplacementRule(pattern4683, replacement4683)
pattern4684 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**p_*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons5, cons31, cons85, cons1577, cons1645)
def replacement4684(p, m, b, a, n, x):
rubi.append(4684)
return Dist((m - n + S(1))/(b*n*(p + S(-1))), Int(x**(m - n)*(S(1)/sin(a + b*x**n))**(p + S(-1)), x), x) - Simp(x**(m - n + S(1))*(S(1)/sin(a + b*x**n))**(p + S(-1))/(b*n*(p + S(-1))), x)
rule4684 = ReplacementRule(pattern4684, replacement4684)
return [rule3917, rule3918, rule3919, rule3920, rule3921, rule3922, rule3923, rule3924, rule3925, rule3926, rule3927, rule3928, rule3929, rule3930, rule3931, rule3932, rule3933, rule3934, rule3935, rule3936, rule3937, rule3938, rule3939, rule3940, rule3941, rule3942, rule3943, rule3944, rule3945, rule3946, rule3947, rule3948, rule3949, rule3950, rule3951, rule3952, rule3953, rule3954, rule3955, rule3956, rule3957, rule3958, rule3959, rule3960, rule3961, rule3962, rule3963, rule3964, rule3965, rule3966, rule3967, rule3968, rule3969, rule3970, rule3971, rule3972, rule3973, rule3974, rule3975, rule3976, rule3977, rule3978, rule3979, rule3980, rule3981, rule3982, rule3983, rule3984, rule3985, rule3986, rule3987, rule3988, rule3989, rule3990, rule3991, rule3992, rule3993, rule3994, rule3995, rule3996, rule3997, rule3998, rule3999, rule4000, rule4001, rule4002, rule4003, rule4004, rule4005, rule4006, rule4007, rule4008, rule4009, rule4010, rule4011, rule4012, rule4013, rule4014, rule4015, rule4016, rule4017, rule4018, rule4019, rule4020, rule4021, rule4022, rule4023, rule4024, rule4025, rule4026, rule4027, rule4028, rule4029, rule4030, rule4031, rule4032, rule4033, rule4034, rule4035, rule4036, rule4037, rule4038, rule4039, rule4040, rule4041, rule4042, rule4043, rule4044, rule4045, rule4046, rule4047, rule4048, rule4049, rule4050, rule4051, rule4052, rule4053, rule4054, rule4055, rule4056, rule4057, rule4058, rule4059, rule4060, rule4061, rule4062, rule4063, rule4064, rule4065, rule4066, rule4067, rule4068, rule4069, rule4070, rule4071, rule4072, rule4073, rule4074, rule4075, rule4076, rule4077, rule4078, rule4079, rule4080, rule4081, rule4082, rule4083, rule4084, rule4085, rule4086, rule4087, rule4088, rule4089, rule4090, rule4091, rule4092, rule4093, rule4094, rule4095, rule4096, rule4097, rule4098, rule4099, rule4100, rule4101, rule4102, rule4103, rule4104, rule4105, rule4106, rule4107, rule4108, rule4109, rule4110, rule4111, rule4112, rule4113, rule4114, rule4115, rule4116, rule4117, rule4118, rule4119, rule4120, rule4121, rule4122, rule4123, rule4124, rule4125, rule4126, rule4127, rule4128, rule4129, rule4130, rule4131, rule4132, rule4133, rule4134, rule4135, rule4136, rule4137, rule4138, rule4139, rule4140, rule4141, rule4142, rule4143, rule4144, rule4145, rule4146, rule4147, rule4148, rule4149, rule4150, rule4151, rule4152, rule4153, rule4154, rule4155, rule4156, rule4157, rule4158, rule4159, rule4160, rule4161, rule4162, rule4163, rule4164, rule4165, rule4166, rule4167, rule4168, rule4169, rule4170, rule4171, rule4172, rule4173, rule4174, rule4175, rule4176, rule4177, rule4178, rule4179, rule4180, rule4181, rule4182, rule4183, rule4184, rule4185, rule4186, rule4187, rule4188, rule4189, rule4190, rule4191, rule4192, rule4193, rule4194, rule4195, rule4196, rule4197, rule4198, rule4199, rule4200, rule4201, rule4202, rule4203, rule4204, rule4205, rule4206, rule4207, rule4208, rule4209, rule4210, rule4211, rule4212, rule4213, rule4214, rule4215, rule4216, rule4217, rule4218, rule4219, rule4220, rule4221, rule4222, rule4223, rule4224, rule4225, rule4226, rule4227, rule4228, rule4229, rule4230, rule4231, rule4232, rule4233, rule4234, rule4235, rule4236, rule4237, rule4238, rule4239, rule4240, rule4241, rule4242, rule4243, rule4244, rule4245, rule4246, rule4247, rule4248, rule4249, rule4250, rule4251, rule4252, rule4253, rule4254, rule4255, rule4256, rule4257, rule4258, rule4259, rule4260, rule4261, rule4262, rule4263, rule4264, rule4265, rule4266, rule4267, rule4268, rule4269, rule4270, rule4271, rule4272, rule4273, rule4274, rule4275, rule4276, rule4277, rule4278, rule4279, rule4280, rule4281, rule4282, rule4283, rule4284, rule4285, rule4286, rule4287, rule4288, rule4289, rule4290, rule4291, rule4292, rule4293, rule4294, rule4295, rule4296, rule4297, rule4298, rule4299, rule4300, rule4301, rule4302, rule4303, rule4304, rule4305, rule4306, rule4307, rule4308, rule4309, rule4310, rule4311, rule4312, rule4313, rule4314, rule4315, rule4316, rule4317, rule4318, rule4319, rule4320, rule4321, rule4322, rule4323, rule4324, rule4325, rule4326, rule4327, rule4328, rule4329, rule4330, rule4331, rule4332, rule4333, rule4334, rule4335, rule4336, rule4337, rule4338, rule4339, rule4340, rule4341, rule4342, rule4343, rule4344, rule4345, rule4346, rule4347, rule4348, rule4349, rule4350, rule4351, rule4352, rule4353, rule4354, rule4355, rule4356, rule4357, rule4358, rule4359, rule4360, rule4361, rule4362, rule4363, rule4364, rule4365, rule4366, rule4367, rule4368, rule4369, rule4370, rule4371, rule4372, rule4373, rule4374, rule4375, rule4376, rule4377, rule4378, rule4379, rule4380, rule4381, rule4382, rule4383, rule4384, rule4385, rule4386, rule4387, rule4388, rule4389, rule4390, rule4391, rule4392, rule4393, rule4394, rule4395, rule4396, rule4397, rule4398, rule4399, rule4400, rule4401, rule4402, rule4403, rule4404, rule4405, rule4406, rule4407, rule4408, rule4409, rule4410, rule4411, rule4412, rule4413, rule4414, rule4415, rule4416, rule4417, rule4418, rule4419, rule4420, rule4421, rule4422, rule4423, rule4424, rule4425, rule4426, rule4427, rule4428, rule4429, rule4430, rule4431, rule4432, rule4433, rule4434, rule4435, rule4436, rule4437, rule4438, rule4439, rule4440, rule4441, rule4442, rule4443, rule4444, rule4445, rule4446, rule4447, rule4448, rule4449, rule4450, rule4451, rule4452, rule4453, rule4454, rule4455, rule4456, rule4457, rule4458, rule4459, rule4460, rule4461, rule4462, rule4463, rule4464, rule4465, rule4466, rule4467, rule4468, rule4469, rule4470, rule4471, rule4472, rule4473, rule4474, rule4475, rule4476, rule4477, rule4478, rule4479, rule4480, rule4481, rule4482, rule4483, rule4484, rule4485, rule4486, rule4487, rule4488, rule4489, rule4490, rule4491, rule4492, rule4493, rule4494, rule4495, rule4496, rule4497, rule4498, rule4499, rule4500, rule4501, rule4502, rule4503, rule4504, rule4505, rule4506, rule4507, rule4508, rule4509, rule4510, rule4511, rule4512, rule4513, rule4514, rule4515, rule4516, rule4517, rule4518, rule4519, rule4520, rule4521, rule4522, rule4523, rule4524, rule4525, rule4526, rule4527, rule4528, rule4529, rule4530, rule4531, rule4532, rule4533, rule4534, rule4535, rule4536, rule4537, rule4538, rule4539, rule4540, rule4541, rule4542, rule4543, rule4544, rule4545, rule4546, rule4547, rule4548, rule4549, rule4550, rule4551, rule4552, rule4553, rule4554, rule4555, rule4556, rule4557, rule4558, rule4559, rule4560, rule4561, rule4562, rule4563, rule4564, rule4565, rule4566, rule4567, rule4568, rule4569, rule4570, rule4571, rule4572, rule4573, rule4574, rule4575, rule4576, rule4577, rule4578, rule4579, rule4580, rule4581, rule4582, rule4583, rule4584, rule4585, rule4586, rule4587, rule4588, rule4589, rule4590, rule4591, rule4592, rule4593, rule4594, rule4595, rule4596, rule4597, rule4598, rule4599, rule4600, rule4601, rule4602, rule4603, rule4604, rule4605, rule4606, rule4607, rule4608, rule4609, rule4610, rule4611, rule4612, rule4613, rule4614, rule4615, rule4616, rule4617, rule4618, rule4619, rule4620, rule4621, rule4622, rule4623, rule4624, rule4625, rule4626, rule4627, rule4628, rule4629, rule4630, rule4631, rule4632, rule4633, rule4634, rule4635, rule4636, rule4637, rule4638, rule4639, rule4640, rule4641, rule4642, rule4643, rule4644, rule4645, rule4646, rule4647, rule4648, rule4649, rule4650, rule4651, rule4652, rule4653, rule4654, rule4655, rule4656, rule4657, rule4658, rule4659, rule4660, rule4661, rule4662, rule4663, rule4664, rule4665, rule4666, rule4667, rule4668, rule4669, rule4670, rule4671, rule4672, rule4673, rule4674, rule4675, rule4676, rule4677, rule4678, rule4679, rule4680, rule4681, rule4682, rule4683, rule4684, ]
|
04cbf6a1e56c071357b5d1204da78a9c3352475421727852eba875b3e410294b | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def special_functions(rubi):
from sympy.integrals.rubi.constraints import cons67, cons2, cons3, cons66, cons21, cons1264, cons7, cons27, cons17, cons166, cons1957, cons1958, cons94, cons261, cons1959, cons1832, cons62, cons1960, cons1961, cons1962, cons247, cons1963, cons1964, cons1965, cons1831, cons4, cons1255, cons18, cons1359, cons1966, cons1967, cons168, cons1968, cons1969, cons31, cons1970, cons1971, cons1972, cons800, cons87, cons88, cons5, cons50, cons89, cons383, cons48, cons1973, cons1974, cons1975, cons52, cons1976, cons1099, cons125, cons1243, cons13, cons137, cons1379, cons1977, cons1978, cons196, cons1979, cons1980, cons1981, cons150, cons463, cons1765, cons163, cons948, cons949, cons1982, cons1983, cons803, cons1984, cons1985, cons1986, cons1987, cons338, cons1988, cons1989, cons1990, cons1991, cons1992, cons1993, cons38, cons1994, cons347, cons1995, cons1996, cons1997, cons1998, cons1999, cons2000, cons2001
pattern6739 = Pattern(Integral(Erf(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6739(x, a, b):
rubi.append(6739)
return Simp(exp(-(a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erf(a + b*x)/b, x)
rule6739 = ReplacementRule(pattern6739, replacement6739)
pattern6740 = Pattern(Integral(Erfc(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6740(x, a, b):
rubi.append(6740)
return -Simp(exp(-(a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erfc(a + b*x)/b, x)
rule6740 = ReplacementRule(pattern6740, replacement6740)
pattern6741 = Pattern(Integral(Erfi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6741(x, a, b):
rubi.append(6741)
return -Simp(exp((a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erfi(a + b*x)/b, x)
rule6741 = ReplacementRule(pattern6741, replacement6741)
pattern6742 = Pattern(Integral(Erf(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6742(x, b):
rubi.append(6742)
return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -b**S(2)*x**S(2))/sqrt(Pi), x)
rule6742 = ReplacementRule(pattern6742, replacement6742)
pattern6743 = Pattern(Integral(Erfc(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6743(x, b):
rubi.append(6743)
return -Int(Erf(b*x)/x, x) + Simp(log(x), x)
rule6743 = ReplacementRule(pattern6743, replacement6743)
pattern6744 = Pattern(Integral(Erfi(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6744(x, b):
rubi.append(6744)
return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), b**S(2)*x**S(2))/sqrt(Pi), x)
rule6744 = ReplacementRule(pattern6744, replacement6744)
pattern6745 = Pattern(Integral(x_**WC('m', S(1))*Erf(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6745(x, m, a, b):
rubi.append(6745)
return -Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-(a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erf(a + b*x)/(m + S(1)), x)
rule6745 = ReplacementRule(pattern6745, replacement6745)
pattern6746 = Pattern(Integral(x_**WC('m', S(1))*Erfc(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6746(x, m, a, b):
rubi.append(6746)
return Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-(a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erfc(a + b*x)/(m + S(1)), x)
rule6746 = ReplacementRule(pattern6746, replacement6746)
pattern6747 = Pattern(Integral(x_**WC('m', S(1))*Erfi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6747(x, m, a, b):
rubi.append(6747)
return -Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp((a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erfi(a + b*x)/(m + S(1)), x)
rule6747 = ReplacementRule(pattern6747, replacement6747)
pattern6748 = Pattern(Integral(x_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6748(b, d, c, a, x):
rubi.append(6748)
return -Dist(b/(sqrt(Pi)*d), Int(exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(Erf(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
rule6748 = ReplacementRule(pattern6748, replacement6748)
pattern6749 = Pattern(Integral(x_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6749(b, d, c, a, x):
rubi.append(6749)
return Dist(b/(sqrt(Pi)*d), Int(exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(Erfc(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
rule6749 = ReplacementRule(pattern6749, replacement6749)
pattern6750 = Pattern(Integral(x_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6750(b, d, c, a, x):
rubi.append(6750)
return -Dist(b/(sqrt(Pi)*d), Int(exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(Erfi(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
rule6750 = ReplacementRule(pattern6750, replacement6750)
pattern6751 = Pattern(Integral(x_**m_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons166)
def replacement6751(m, b, d, c, a, x):
rubi.append(6751)
return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erf(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(-1))*Erf(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
rule6751 = ReplacementRule(pattern6751, replacement6751)
pattern6752 = Pattern(Integral(x_**m_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons166)
def replacement6752(m, b, d, c, a, x):
rubi.append(6752)
return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erfc(a + b*x)*exp(c + d*x**S(2)), x), x) + Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(-1))*Erfc(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
rule6752 = ReplacementRule(pattern6752, replacement6752)
pattern6753 = Pattern(Integral(x_**m_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons166)
def replacement6753(m, b, d, c, a, x):
rubi.append(6753)
return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erfi(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(x**(m + S(-1))*Erfi(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x)
rule6753 = ReplacementRule(pattern6753, replacement6753)
pattern6754 = Pattern(Integral(Erf(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1957)
def replacement6754(d, c, b, x):
rubi.append(6754)
return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)), List(S(3)/2, S(3)/2), d*x**S(2))*exp(c)/sqrt(Pi), x)
rule6754 = ReplacementRule(pattern6754, replacement6754)
pattern6755 = Pattern(Integral(Erfc(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1957)
def replacement6755(d, c, b, x):
rubi.append(6755)
return Int(exp(c + d*x**S(2))/x, x) - Int(Erf(b*x)*exp(c + d*x**S(2))/x, x)
rule6755 = ReplacementRule(pattern6755, replacement6755)
pattern6756 = Pattern(Integral(Erfi(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1958)
def replacement6756(d, c, b, x):
rubi.append(6756)
return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)), List(S(3)/2, S(3)/2), d*x**S(2))*exp(c)/sqrt(Pi), x)
rule6756 = ReplacementRule(pattern6756, replacement6756)
pattern6757 = Pattern(Integral(x_**m_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6757(m, b, d, c, a, x):
rubi.append(6757)
return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erf(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(1))*Erf(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x)
rule6757 = ReplacementRule(pattern6757, replacement6757)
pattern6758 = Pattern(Integral(x_**m_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6758(m, b, d, c, a, x):
rubi.append(6758)
return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erfc(a + b*x)*exp(c + d*x**S(2)), x), x) + Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(1))*Erfc(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x)
rule6758 = ReplacementRule(pattern6758, replacement6758)
pattern6759 = Pattern(Integral(x_**m_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6759(m, b, d, c, a, x):
rubi.append(6759)
return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erfi(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(x**(m + S(1))*Erfi(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x)
rule6759 = ReplacementRule(pattern6759, replacement6759)
pattern6760 = Pattern(Integral(Erf(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6760(x, a, b):
rubi.append(6760)
return -Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erf(a + b*x)*exp(-(a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erf(a + b*x)**S(2)/b, x)
rule6760 = ReplacementRule(pattern6760, replacement6760)
pattern6761 = Pattern(Integral(Erfc(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6761(x, a, b):
rubi.append(6761)
return Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erfc(a + b*x)*exp(-(a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erfc(a + b*x)**S(2)/b, x)
rule6761 = ReplacementRule(pattern6761, replacement6761)
pattern6762 = Pattern(Integral(Erfi(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6762(x, a, b):
rubi.append(6762)
return -Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erfi(a + b*x)*exp((a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erfi(a + b*x)**S(2)/b, x)
rule6762 = ReplacementRule(pattern6762, replacement6762)
pattern6763 = Pattern(Integral(x_**WC('m', S(1))*Erf(x_*WC('b', S(1)))**S(2), x_), cons3, cons17, cons261, cons1959)
def replacement6763(x, m, b):
rubi.append(6763)
return -Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erf(b*x)*exp(-b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erf(b*x)**S(2)/(m + S(1)), x)
rule6763 = ReplacementRule(pattern6763, replacement6763)
pattern6764 = Pattern(Integral(x_**WC('m', S(1))*Erfc(x_*WC('b', S(1)))**S(2), x_), cons3, cons17, cons1832, cons1959)
def replacement6764(x, m, b):
rubi.append(6764)
return Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erfc(b*x)*exp(-b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erfc(b*x)**S(2)/(m + S(1)), x)
rule6764 = ReplacementRule(pattern6764, replacement6764)
pattern6765 = Pattern(Integral(x_**WC('m', S(1))*Erfi(x_*WC('b', S(1)))**S(2), x_), cons3, cons17, cons1832, cons1959)
def replacement6765(x, m, b):
rubi.append(6765)
return -Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erfi(b*x)*exp(b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erfi(b*x)**S(2)/(m + S(1)), x)
rule6765 = ReplacementRule(pattern6765, replacement6765)
pattern6766 = Pattern(Integral(x_**WC('m', S(1))*Erf(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons62)
def replacement6766(x, m, b, a):
rubi.append(6766)
return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erf(x)**S(2), x), x, a + b*x), x)
rule6766 = ReplacementRule(pattern6766, replacement6766)
pattern6767 = Pattern(Integral(x_**WC('m', S(1))*Erfc(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons62)
def replacement6767(x, m, b, a):
rubi.append(6767)
return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erfc(x)**S(2), x), x, a + b*x), x)
rule6767 = ReplacementRule(pattern6767, replacement6767)
pattern6768 = Pattern(Integral(x_**WC('m', S(1))*Erfi(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons62)
def replacement6768(x, m, b, a):
rubi.append(6768)
return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erfi(x)**S(2), x), x, a + b*x), x)
rule6768 = ReplacementRule(pattern6768, replacement6768)
pattern6769 = Pattern(Integral(FresnelS(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6769(x, a, b):
rubi.append(6769)
return Simp(cos(Pi*(a + b*x)**S(2)/S(2))/(Pi*b), x) + Simp((a + b*x)*FresnelS(a + b*x)/b, x)
rule6769 = ReplacementRule(pattern6769, replacement6769)
pattern6770 = Pattern(Integral(FresnelC(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6770(x, a, b):
rubi.append(6770)
return -Simp(sin(Pi*(a + b*x)**S(2)/S(2))/(Pi*b), x) + Simp((a + b*x)*FresnelC(a + b*x)/b, x)
rule6770 = ReplacementRule(pattern6770, replacement6770)
pattern6771 = Pattern(Integral(FresnelS(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6771(x, b):
rubi.append(6771)
return Simp(I*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x) - Simp(I*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x)
rule6771 = ReplacementRule(pattern6771, replacement6771)
pattern6772 = Pattern(Integral(FresnelC(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6772(x, b):
rubi.append(6772)
return Simp(b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x)
rule6772 = ReplacementRule(pattern6772, replacement6772)
pattern6773 = Pattern(Integral(x_**WC('m', S(1))*FresnelS(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6773(x, m, a, b):
rubi.append(6773)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sin(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(a + b*x)/(m + S(1)), x)
rule6773 = ReplacementRule(pattern6773, replacement6773)
pattern6774 = Pattern(Integral(x_**WC('m', S(1))*FresnelC(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6774(x, m, a, b):
rubi.append(6774)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cos(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(a + b*x)/(m + S(1)), x)
rule6774 = ReplacementRule(pattern6774, replacement6774)
pattern6775 = Pattern(Integral(FresnelS(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6775(x, a, b):
rubi.append(6775)
return -Dist(S(2), Int((a + b*x)*FresnelS(a + b*x)*sin(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp((a + b*x)*FresnelS(a + b*x)**S(2)/b, x)
rule6775 = ReplacementRule(pattern6775, replacement6775)
pattern6776 = Pattern(Integral(FresnelC(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6776(x, a, b):
rubi.append(6776)
return -Dist(S(2), Int((a + b*x)*FresnelC(a + b*x)*cos(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp((a + b*x)*FresnelC(a + b*x)**S(2)/b, x)
rule6776 = ReplacementRule(pattern6776, replacement6776)
pattern6777 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))**S(2), x_), cons3, cons17, cons1832, cons1960)
def replacement6777(x, m, b):
rubi.append(6777)
return -Dist(S(2)*b/(m + S(1)), Int(x**(m + S(1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(b*x)**S(2)/(m + S(1)), x)
rule6777 = ReplacementRule(pattern6777, replacement6777)
pattern6778 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))**S(2), x_), cons3, cons17, cons1832, cons1960)
def replacement6778(x, m, b):
rubi.append(6778)
return -Dist(S(2)*b/(m + S(1)), Int(x**(m + S(1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(b*x)**S(2)/(m + S(1)), x)
rule6778 = ReplacementRule(pattern6778, replacement6778)
pattern6779 = Pattern(Integral(x_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961)
def replacement6779(x, c, b):
rubi.append(6779)
return Dist(S(1)/(S(2)*Pi*b), Int(sin(Pi*b**S(2)*x**S(2)), x), x) - Simp(FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
rule6779 = ReplacementRule(pattern6779, replacement6779)
pattern6780 = Pattern(Integral(x_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961)
def replacement6780(x, c, b):
rubi.append(6780)
return -Dist(S(1)/(S(2)*Pi*b), Int(sin(Pi*b**S(2)*x**S(2)), x), x) + Simp(FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
rule6780 = ReplacementRule(pattern6780, replacement6780)
pattern6781 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961, cons17, cons166, cons1962)
def replacement6781(x, m, c, b):
rubi.append(6781)
return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*sin(Pi*b**S(2)*x**S(2)), x), x) + Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(x**(m + S(-1))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
rule6781 = ReplacementRule(pattern6781, replacement6781)
pattern6782 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961, cons17, cons166, cons1962)
def replacement6782(x, m, c, b):
rubi.append(6782)
return -Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*sin(Pi*b**S(2)*x**S(2)), x), x) - Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(-1))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
rule6782 = ReplacementRule(pattern6782, replacement6782)
pattern6783 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961, cons17, cons247, cons1963)
def replacement6783(x, m, c, b):
rubi.append(6783)
return Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*cos(Pi*b**S(2)*x**S(2)), x), x) - Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(b*x**(m + S(2))/(S(2)*(m + S(1))*(m + S(2))), x) + Simp(x**(m + S(1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x)
rule6783 = ReplacementRule(pattern6783, replacement6783)
pattern6784 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961, cons17, cons247, cons1963)
def replacement6784(x, m, c, b):
rubi.append(6784)
return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*cos(Pi*b**S(2)*x**S(2)), x), x) + Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(b*x**(m + S(2))/(S(2)*(m + S(1))*(m + S(2))), x) + Simp(x**(m + S(1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x)
rule6784 = ReplacementRule(pattern6784, replacement6784)
pattern6785 = Pattern(Integral(x_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961)
def replacement6785(x, c, b):
rubi.append(6785)
return Dist(S(1)/(S(2)*Pi*b), Int(cos(Pi*b**S(2)*x**S(2)), x), x) - Simp(x/(S(2)*Pi*b), x) + Simp(FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
rule6785 = ReplacementRule(pattern6785, replacement6785)
pattern6786 = Pattern(Integral(x_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961)
def replacement6786(x, c, b):
rubi.append(6786)
return Dist(S(1)/(S(2)*Pi*b), Int(cos(Pi*b**S(2)*x**S(2)), x), x) + Simp(x/(S(2)*Pi*b), x) - Simp(FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
rule6786 = ReplacementRule(pattern6786, replacement6786)
pattern6787 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961, cons17, cons166, cons1964)
def replacement6787(x, m, c, b):
rubi.append(6787)
return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*cos(Pi*b**S(2)*x**S(2)), x), x) - Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(x**m/(S(2)*Pi*b*m), x) + Simp(x**(m + S(-1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
rule6787 = ReplacementRule(pattern6787, replacement6787)
pattern6788 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961, cons17, cons166, cons1964)
def replacement6788(x, m, c, b):
rubi.append(6788)
return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*cos(Pi*b**S(2)*x**S(2)), x), x) + Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**m/(S(2)*Pi*b*m), x) - Simp(x**(m + S(-1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x)
rule6788 = ReplacementRule(pattern6788, replacement6788)
pattern6789 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961, cons17, cons94, cons1965)
def replacement6789(x, m, c, b):
rubi.append(6789)
return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*sin(Pi*b**S(2)*x**S(2)), x), x) + Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x)
rule6789 = ReplacementRule(pattern6789, replacement6789)
pattern6790 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons7, cons1961, cons17, cons94, cons1965)
def replacement6790(x, m, c, b):
rubi.append(6790)
return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*sin(Pi*b**S(2)*x**S(2)), x), x) - Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x)
rule6790 = ReplacementRule(pattern6790, replacement6790)
pattern6791 = Pattern(Integral(ExpIntegralE(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1831)
def replacement6791(x, a, n, b):
rubi.append(6791)
return -Simp(ExpIntegralE(n + S(1), a + b*x)/b, x)
rule6791 = ReplacementRule(pattern6791, replacement6791)
pattern6792 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons1255, cons62)
def replacement6792(x, m, n, b):
rubi.append(6792)
return Dist(m/b, Int(x**(m + S(-1))*ExpIntegralE(n + S(1), b*x), x), x) - Simp(x**m*ExpIntegralE(n + S(1), b*x)/b, x)
rule6792 = ReplacementRule(pattern6792, replacement6792)
pattern6793 = Pattern(Integral(ExpIntegralE(S(1), x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6793(x, b):
rubi.append(6793)
return -Simp(EulerGamma*log(x), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x), x) - Simp(log(b*x)**S(2)/S(2), x)
rule6793 = ReplacementRule(pattern6793, replacement6793)
pattern6794 = Pattern(Integral(x_**m_*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons1255, cons17, cons94)
def replacement6794(x, m, n, b):
rubi.append(6794)
return Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralE(n + S(-1), b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralE(n, b*x)/(m + S(1)), x)
rule6794 = ReplacementRule(pattern6794, replacement6794)
pattern6795 = Pattern(Integral(x_**m_*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons21, cons4, cons1255, cons18)
def replacement6795(x, m, n, b):
rubi.append(6795)
return -Simp(x**(m + S(1))*HypergeometricPFQ(List(m + S(1), m + S(1)), List(m + S(2), m + S(2)), -b*x)/(m + S(1))**S(2), x) + Simp(x**m*(b*x)**(-m)*Gamma(m + S(1))*log(x)/b, x)
rule6795 = ReplacementRule(pattern6795, replacement6795)
pattern6796 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons21, cons4, cons1359)
def replacement6796(x, m, n, b):
rubi.append(6796)
return -Simp(x**(m + S(1))*ExpIntegralE(-m, b*x)/(m + n), x) + Simp(x**(m + S(1))*ExpIntegralE(n, b*x)/(m + n), x)
rule6796 = ReplacementRule(pattern6796, replacement6796)
pattern6797 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons21, cons4, cons1966)
def replacement6797(m, b, a, n, x):
rubi.append(6797)
return Dist(m/b, Int(x**(m + S(-1))*ExpIntegralE(n + S(1), a + b*x), x), x) - Simp(x**m*ExpIntegralE(n + S(1), a + b*x)/b, x)
rule6797 = ReplacementRule(pattern6797, replacement6797)
pattern6798 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons21, cons1967, cons66)
def replacement6798(m, b, a, n, x):
rubi.append(6798)
return Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralE(n + S(-1), a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralE(n, a + b*x)/(m + S(1)), x)
rule6798 = ReplacementRule(pattern6798, replacement6798)
pattern6799 = Pattern(Integral(ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6799(x, a, b):
rubi.append(6799)
return -Simp(exp(a + b*x)/b, x) + Simp((a + b*x)*ExpIntegralEi(a + b*x)/b, x)
rule6799 = ReplacementRule(pattern6799, replacement6799)
pattern6800 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6800(x, m, a, b):
rubi.append(6800)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*exp(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(a + b*x)/(m + S(1)), x)
rule6800 = ReplacementRule(pattern6800, replacement6800)
pattern6801 = Pattern(Integral(ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6801(x, a, b):
rubi.append(6801)
return -Dist(S(2), Int(ExpIntegralEi(a + b*x)*exp(a + b*x), x), x) + Simp((a + b*x)*ExpIntegralEi(a + b*x)**S(2)/b, x)
rule6801 = ReplacementRule(pattern6801, replacement6801)
pattern6802 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('b', S(1)))**S(2), x_), cons3, cons62)
def replacement6802(x, m, b):
rubi.append(6802)
return -Dist(S(2)/(m + S(1)), Int(x**m*ExpIntegralEi(b*x)*exp(b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(b*x)**S(2)/(m + S(1)), x)
rule6802 = ReplacementRule(pattern6802, replacement6802)
pattern6803 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons62)
def replacement6803(x, m, b, a):
rubi.append(6803)
return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*ExpIntegralEi(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*ExpIntegralEi(a + b*x)*exp(a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*ExpIntegralEi(a + b*x)**S(2)/(b*(m + S(1))), x)
rule6803 = ReplacementRule(pattern6803, replacement6803)
pattern6804 = Pattern(Integral(ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6804(b, d, c, a, x):
rubi.append(6804)
return -Dist(d/b, Int(exp(a + c + x*(b + d))/(c + d*x), x), x) + Simp(ExpIntegralEi(c + d*x)*exp(a + b*x)/b, x)
rule6804 = ReplacementRule(pattern6804, replacement6804)
pattern6805 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement6805(m, b, d, c, a, x):
rubi.append(6805)
return -Dist(d/b, Int(x**m*exp(a + c + x*(b + d))/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*ExpIntegralEi(c + d*x)*exp(a + b*x), x), x) + Simp(x**m*ExpIntegralEi(c + d*x)*exp(a + b*x)/b, x)
rule6805 = ReplacementRule(pattern6805, replacement6805)
pattern6806 = Pattern(Integral(x_**m_*ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6806(m, b, d, c, a, x):
rubi.append(6806)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralEi(c + d*x)*exp(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*exp(a + c + x*(b + d))/(c + d*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(c + d*x)*exp(a + b*x)/(m + S(1)), x)
rule6806 = ReplacementRule(pattern6806, replacement6806)
pattern6807 = Pattern(Integral(LogIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6807(x, a, b):
rubi.append(6807)
return -Simp(ExpIntegralEi(S(2)*log(a + b*x))/b, x) + Simp((a + b*x)*LogIntegral(a + b*x)/b, x)
rule6807 = ReplacementRule(pattern6807, replacement6807)
pattern6808 = Pattern(Integral(LogIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6808(x, b):
rubi.append(6808)
return -Simp(b*x, x) + Simp(LogIntegral(b*x)*log(b*x), x)
rule6808 = ReplacementRule(pattern6808, replacement6808)
pattern6809 = Pattern(Integral(x_**WC('m', S(1))*LogIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6809(x, m, a, b):
rubi.append(6809)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))/log(a + b*x), x), x) + Simp(x**(m + S(1))*LogIntegral(a + b*x)/(m + S(1)), x)
rule6809 = ReplacementRule(pattern6809, replacement6809)
pattern6810 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6810(x, a, b):
rubi.append(6810)
return Simp(cos(a + b*x)/b, x) + Simp((a + b*x)*SinIntegral(a + b*x)/b, x)
rule6810 = ReplacementRule(pattern6810, replacement6810)
pattern6811 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6811(x, a, b):
rubi.append(6811)
return -Simp(sin(a + b*x)/b, x) + Simp((a + b*x)*CosIntegral(a + b*x)/b, x)
rule6811 = ReplacementRule(pattern6811, replacement6811)
pattern6812 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6812(x, b):
rubi.append(6812)
return Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -I*b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), I*b*x)/S(2), x)
rule6812 = ReplacementRule(pattern6812, replacement6812)
pattern6813 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6813(x, b):
rubi.append(6813)
return Simp(EulerGamma*log(x), x) - Simp(I*b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -I*b*x)/S(2), x) + Simp(I*b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), I*b*x)/S(2), x) + Simp(log(b*x)**S(2)/S(2), x)
rule6813 = ReplacementRule(pattern6813, replacement6813)
pattern6814 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6814(x, m, b, a):
rubi.append(6814)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(a + b*x)/(m + S(1)), x)
rule6814 = ReplacementRule(pattern6814, replacement6814)
pattern6815 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6815(x, m, a, b):
rubi.append(6815)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cos(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(a + b*x)/(m + S(1)), x)
rule6815 = ReplacementRule(pattern6815, replacement6815)
pattern6816 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6816(x, a, b):
rubi.append(6816)
return -Dist(S(2), Int(SinIntegral(a + b*x)*sin(a + b*x), x), x) + Simp((a + b*x)*SinIntegral(a + b*x)**S(2)/b, x)
rule6816 = ReplacementRule(pattern6816, replacement6816)
pattern6817 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6817(x, a, b):
rubi.append(6817)
return -Dist(S(2), Int(CosIntegral(a + b*x)*cos(a + b*x), x), x) + Simp((a + b*x)*CosIntegral(a + b*x)**S(2)/b, x)
rule6817 = ReplacementRule(pattern6817, replacement6817)
pattern6818 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons62)
def replacement6818(x, m, b):
rubi.append(6818)
return -Dist(S(2)/(m + S(1)), Int(x**m*SinIntegral(b*x)*sin(b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(b*x)**S(2)/(m + S(1)), x)
rule6818 = ReplacementRule(pattern6818, replacement6818)
pattern6819 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons62)
def replacement6819(x, m, b):
rubi.append(6819)
return -Dist(S(2)/(m + S(1)), Int(x**m*CosIntegral(b*x)*cos(b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(b*x)**S(2)/(m + S(1)), x)
rule6819 = ReplacementRule(pattern6819, replacement6819)
pattern6820 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons62)
def replacement6820(x, m, b, a):
rubi.append(6820)
return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*SinIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*SinIntegral(a + b*x)*sin(a + b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*SinIntegral(a + b*x)**S(2)/(b*(m + S(1))), x)
rule6820 = ReplacementRule(pattern6820, replacement6820)
pattern6821 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons62)
def replacement6821(x, m, b, a):
rubi.append(6821)
return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*CosIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*CosIntegral(a + b*x)*cos(a + b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*CosIntegral(a + b*x)**S(2)/(b*(m + S(1))), x)
rule6821 = ReplacementRule(pattern6821, replacement6821)
pattern6822 = Pattern(Integral(SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6822(b, d, c, a, x):
rubi.append(6822)
return Dist(d/b, Int(sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) - Simp(SinIntegral(c + d*x)*cos(a + b*x)/b, x)
rule6822 = ReplacementRule(pattern6822, replacement6822)
pattern6823 = Pattern(Integral(CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6823(b, d, c, a, x):
rubi.append(6823)
return -Dist(d/b, Int(sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(CosIntegral(c + d*x)*sin(a + b*x)/b, x)
rule6823 = ReplacementRule(pattern6823, replacement6823)
pattern6824 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement6824(m, b, d, c, a, x):
rubi.append(6824)
return Dist(d/b, Int(x**m*sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) + Dist(m/b, Int(x**(m + S(-1))*SinIntegral(c + d*x)*cos(a + b*x), x), x) - Simp(x**m*SinIntegral(c + d*x)*cos(a + b*x)/b, x)
rule6824 = ReplacementRule(pattern6824, replacement6824)
pattern6825 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement6825(m, b, d, c, a, x):
rubi.append(6825)
return -Dist(d/b, Int(x**m*sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CosIntegral(c + d*x)*sin(a + b*x), x), x) + Simp(x**m*CosIntegral(c + d*x)*sin(a + b*x)/b, x)
rule6825 = ReplacementRule(pattern6825, replacement6825)
pattern6826 = Pattern(Integral(x_**m_*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6826(m, b, d, c, a, x):
rubi.append(6826)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinIntegral(c + d*x)*cos(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinIntegral(c + d*x)*sin(a + b*x)/(m + S(1)), x)
rule6826 = ReplacementRule(pattern6826, replacement6826)
pattern6827 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6827(m, b, d, c, a, x):
rubi.append(6827)
return Dist(b/(m + S(1)), Int(x**(m + S(1))*CosIntegral(c + d*x)*sin(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CosIntegral(c + d*x)*cos(a + b*x)/(m + S(1)), x)
rule6827 = ReplacementRule(pattern6827, replacement6827)
pattern6828 = Pattern(Integral(SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6828(b, d, c, a, x):
rubi.append(6828)
return -Dist(d/b, Int(sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) + Simp(SinIntegral(c + d*x)*sin(a + b*x)/b, x)
rule6828 = ReplacementRule(pattern6828, replacement6828)
pattern6829 = Pattern(Integral(CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6829(b, d, c, a, x):
rubi.append(6829)
return Dist(d/b, Int(cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) - Simp(CosIntegral(c + d*x)*cos(a + b*x)/b, x)
rule6829 = ReplacementRule(pattern6829, replacement6829)
pattern6830 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement6830(m, b, d, a, c, x):
rubi.append(6830)
return -Dist(d/b, Int(x**m*sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinIntegral(c + d*x)*sin(a + b*x), x), x) + Simp(x**m*SinIntegral(c + d*x)*sin(a + b*x)/b, x)
rule6830 = ReplacementRule(pattern6830, replacement6830)
pattern6831 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement6831(m, b, d, c, a, x):
rubi.append(6831)
return Dist(d/b, Int(x**m*cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Dist(m/b, Int(x**(m + S(-1))*CosIntegral(c + d*x)*cos(a + b*x), x), x) - Simp(x**m*CosIntegral(c + d*x)*cos(a + b*x)/b, x)
rule6831 = ReplacementRule(pattern6831, replacement6831)
pattern6832 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6832(m, b, d, a, c, x):
rubi.append(6832)
return Dist(b/(m + S(1)), Int(x**(m + S(1))*SinIntegral(c + d*x)*sin(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinIntegral(c + d*x)*cos(a + b*x)/(m + S(1)), x)
rule6832 = ReplacementRule(pattern6832, replacement6832)
pattern6833 = Pattern(Integral(x_**m_*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6833(m, b, d, c, a, x):
rubi.append(6833)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CosIntegral(c + d*x)*cos(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CosIntegral(c + d*x)*sin(a + b*x)/(m + S(1)), x)
rule6833 = ReplacementRule(pattern6833, replacement6833)
pattern6834 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6834(x, a, b):
rubi.append(6834)
return -Simp(cosh(a + b*x)/b, x) + Simp((a + b*x)*SinhIntegral(a + b*x)/b, x)
rule6834 = ReplacementRule(pattern6834, replacement6834)
pattern6835 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6835(x, a, b):
rubi.append(6835)
return -Simp(sinh(a + b*x)/b, x) + Simp((a + b*x)*CoshIntegral(a + b*x)/b, x)
rule6835 = ReplacementRule(pattern6835, replacement6835)
pattern6836 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6836(x, b):
rubi.append(6836)
return Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), b*x)/S(2), x)
rule6836 = ReplacementRule(pattern6836, replacement6836)
pattern6837 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3)
def replacement6837(x, b):
rubi.append(6837)
return Simp(EulerGamma*log(x), x) - Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), b*x)/S(2), x) + Simp(log(b*x)**S(2)/S(2), x)
rule6837 = ReplacementRule(pattern6837, replacement6837)
pattern6838 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6838(x, m, b, a):
rubi.append(6838)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(a + b*x)/(m + S(1)), x)
rule6838 = ReplacementRule(pattern6838, replacement6838)
pattern6839 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons21, cons66)
def replacement6839(x, m, a, b):
rubi.append(6839)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cosh(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(a + b*x)/(m + S(1)), x)
rule6839 = ReplacementRule(pattern6839, replacement6839)
pattern6840 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6840(x, a, b):
rubi.append(6840)
return -Dist(S(2), Int(SinhIntegral(a + b*x)*sinh(a + b*x), x), x) + Simp((a + b*x)*SinhIntegral(a + b*x)**S(2)/b, x)
rule6840 = ReplacementRule(pattern6840, replacement6840)
pattern6841 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons67)
def replacement6841(x, a, b):
rubi.append(6841)
return -Dist(S(2), Int(CoshIntegral(a + b*x)*cosh(a + b*x), x), x) + Simp((a + b*x)*CoshIntegral(a + b*x)**S(2)/b, x)
rule6841 = ReplacementRule(pattern6841, replacement6841)
pattern6842 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons62)
def replacement6842(x, m, b):
rubi.append(6842)
return -Dist(S(2)/(m + S(1)), Int(x**m*SinhIntegral(b*x)*sinh(b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(b*x)**S(2)/(m + S(1)), x)
rule6842 = ReplacementRule(pattern6842, replacement6842)
pattern6843 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons62)
def replacement6843(x, m, b):
rubi.append(6843)
return -Dist(S(2)/(m + S(1)), Int(x**m*CoshIntegral(b*x)*cosh(b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(b*x)**S(2)/(m + S(1)), x)
rule6843 = ReplacementRule(pattern6843, replacement6843)
pattern6844 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons62)
def replacement6844(x, m, b, a):
rubi.append(6844)
return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*SinhIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*SinhIntegral(a + b*x)*sinh(a + b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*SinhIntegral(a + b*x)**S(2)/(b*(m + S(1))), x)
rule6844 = ReplacementRule(pattern6844, replacement6844)
pattern6845 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons62)
def replacement6845(x, m, b, a):
rubi.append(6845)
return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*CoshIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*CoshIntegral(a + b*x)*cosh(a + b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*CoshIntegral(a + b*x)**S(2)/(b*(m + S(1))), x)
rule6845 = ReplacementRule(pattern6845, replacement6845)
pattern6846 = Pattern(Integral(SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6846(b, d, c, a, x):
rubi.append(6846)
return -Dist(d/b, Int(sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) + Simp(SinhIntegral(c + d*x)*cosh(a + b*x)/b, x)
rule6846 = ReplacementRule(pattern6846, replacement6846)
pattern6847 = Pattern(Integral(CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6847(b, d, c, a, x):
rubi.append(6847)
return -Dist(d/b, Int(sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(CoshIntegral(c + d*x)*sinh(a + b*x)/b, x)
rule6847 = ReplacementRule(pattern6847, replacement6847)
pattern6848 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons168)
def replacement6848(m, b, d, c, a, x):
rubi.append(6848)
return -Dist(d/b, Int(x**m*sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinhIntegral(c + d*x)*cosh(a + b*x), x), x) + Simp(x**m*SinhIntegral(c + d*x)*cosh(a + b*x)/b, x)
rule6848 = ReplacementRule(pattern6848, replacement6848)
pattern6849 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons168)
def replacement6849(m, b, d, c, a, x):
rubi.append(6849)
return -Dist(d/b, Int(x**m*sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CoshIntegral(c + d*x)*sinh(a + b*x), x), x) + Simp(x**m*CoshIntegral(c + d*x)*sinh(a + b*x)/b, x)
rule6849 = ReplacementRule(pattern6849, replacement6849)
pattern6850 = Pattern(Integral(x_**m_*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6850(m, b, d, c, a, x):
rubi.append(6850)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinhIntegral(c + d*x)*cosh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(c + d*x)*sinh(a + b*x)/(m + S(1)), x)
rule6850 = ReplacementRule(pattern6850, replacement6850)
pattern6851 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6851(m, b, d, c, a, x):
rubi.append(6851)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CoshIntegral(c + d*x)*sinh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(c + d*x)*cosh(a + b*x)/(m + S(1)), x)
rule6851 = ReplacementRule(pattern6851, replacement6851)
pattern6852 = Pattern(Integral(SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6852(b, d, c, a, x):
rubi.append(6852)
return -Dist(d/b, Int(sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) + Simp(SinhIntegral(c + d*x)*sinh(a + b*x)/b, x)
rule6852 = ReplacementRule(pattern6852, replacement6852)
pattern6853 = Pattern(Integral(CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1264)
def replacement6853(b, d, c, a, x):
rubi.append(6853)
return -Dist(d/b, Int(cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(CoshIntegral(c + d*x)*cosh(a + b*x)/b, x)
rule6853 = ReplacementRule(pattern6853, replacement6853)
pattern6854 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement6854(m, b, d, a, c, x):
rubi.append(6854)
return -Dist(d/b, Int(x**m*sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinhIntegral(c + d*x)*sinh(a + b*x), x), x) + Simp(x**m*SinhIntegral(c + d*x)*sinh(a + b*x)/b, x)
rule6854 = ReplacementRule(pattern6854, replacement6854)
pattern6855 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons62)
def replacement6855(m, b, d, c, a, x):
rubi.append(6855)
return -Dist(d/b, Int(x**m*cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CoshIntegral(c + d*x)*cosh(a + b*x), x), x) + Simp(x**m*CoshIntegral(c + d*x)*cosh(a + b*x)/b, x)
rule6855 = ReplacementRule(pattern6855, replacement6855)
pattern6856 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6856(m, b, d, a, c, x):
rubi.append(6856)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinhIntegral(c + d*x)*sinh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(c + d*x)*cosh(a + b*x)/(m + S(1)), x)
rule6856 = ReplacementRule(pattern6856, replacement6856)
pattern6857 = Pattern(Integral(x_**m_*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons17, cons94)
def replacement6857(m, b, d, c, a, x):
rubi.append(6857)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CoshIntegral(c + d*x)*cosh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(c + d*x)*sinh(a + b*x)/(m + S(1)), x)
rule6857 = ReplacementRule(pattern6857, replacement6857)
pattern6858 = Pattern(Integral(Gamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6858(x, a, n, b):
rubi.append(6858)
return -Simp(Gamma(n + S(1), a + b*x)/b, x) + Simp((a + b*x)*Gamma(n, a + b*x)/b, x)
rule6858 = ReplacementRule(pattern6858, replacement6858)
pattern6859 = Pattern(Integral(Gamma(n_, b_*x_)/x_, x_), cons3, cons4, cons1968)
def replacement6859(x, n, b):
rubi.append(6859)
return Simp(Gamma(n)*log(x), x) - Simp((b*x)**n*HypergeometricPFQ(List(n, n), List(n + S(1), n + S(1)), -b*x)/n**S(2), x)
rule6859 = ReplacementRule(pattern6859, replacement6859)
pattern6860 = Pattern(Integral(x_**WC('m', S(1))*Gamma(n_, b_*x_), x_), cons3, cons21, cons4, cons66)
def replacement6860(x, m, n, b):
rubi.append(6860)
return Simp(x**(m + S(1))*Gamma(n, b*x)/(m + S(1)), x) - Simp(x**m*(b*x)**(-m)*Gamma(m + n + S(1), b*x)/(b*(m + S(1))), x)
rule6860 = ReplacementRule(pattern6860, replacement6860)
def With6861(m, b, a, n, x):
_UseGamma = True
rubi.append(6861)
return Dist(b/(m + S(1)), Int(x**(m + S(1))*(a + b*x)**(n + S(-1))*exp(-a - b*x), x), x) + Simp(x**(m + S(1))*Gamma(n, a + b*x)/(m + S(1)), x)
pattern6861 = Pattern(Integral(x_**WC('m', S(1))*Gamma(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons21, cons4, cons1969, cons66)
rule6861 = ReplacementRule(pattern6861, With6861)
pattern6862 = Pattern(Integral(LogGamma(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6862(x, a, b):
rubi.append(6862)
return Simp(PolyGamma(S(-2), a + b*x)/b, x)
rule6862 = ReplacementRule(pattern6862, replacement6862)
pattern6863 = Pattern(Integral(x_**WC('m', S(1))*LogGamma(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons31, cons168)
def replacement6863(x, m, a, b):
rubi.append(6863)
return -Dist(m/b, Int(x**(m + S(-1))*PolyGamma(S(-2), a + b*x), x), x) + Simp(x**m*PolyGamma(S(-2), a + b*x)/b, x)
rule6863 = ReplacementRule(pattern6863, replacement6863)
pattern6864 = Pattern(Integral(PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1831)
def replacement6864(x, a, n, b):
rubi.append(6864)
return Simp(PolyGamma(n + S(-1), a + b*x)/b, x)
rule6864 = ReplacementRule(pattern6864, replacement6864)
pattern6865 = Pattern(Integral(x_**WC('m', S(1))*PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons31, cons168)
def replacement6865(m, b, a, n, x):
rubi.append(6865)
return -Dist(m/b, Int(x**(m + S(-1))*PolyGamma(n + S(-1), a + b*x), x), x) + Simp(x**m*PolyGamma(n + S(-1), a + b*x)/b, x)
rule6865 = ReplacementRule(pattern6865, replacement6865)
pattern6866 = Pattern(Integral(x_**WC('m', S(1))*PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons31, cons94)
def replacement6866(m, b, a, n, x):
rubi.append(6866)
return -Dist(b/(m + S(1)), Int(x**(m + S(1))*PolyGamma(n + S(1), a + b*x), x), x) + Simp(x**(m + S(1))*PolyGamma(n, a + b*x)/(m + S(1)), x)
rule6866 = ReplacementRule(pattern6866, replacement6866)
pattern6867 = Pattern(Integral(Gamma(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*PolyGamma(S(0), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1831)
def replacement6867(x, a, n, b):
rubi.append(6867)
return Simp(Gamma(a + b*x)**n/(b*n), x)
rule6867 = ReplacementRule(pattern6867, replacement6867)
pattern6868 = Pattern(Integral(Factorial(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*PolyGamma(S(0), x_*WC('b', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons4, cons1970)
def replacement6868(b, c, a, n, x):
rubi.append(6868)
return Simp(Factorial(a + b*x)**n/(b*n), x)
rule6868 = ReplacementRule(pattern6868, replacement6868)
pattern6869 = Pattern(Integral(Zeta(S(2), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons67)
def replacement6869(x, a, b):
rubi.append(6869)
return Int(PolyGamma(S(1), a + b*x), x)
rule6869 = ReplacementRule(pattern6869, replacement6869)
pattern6870 = Pattern(Integral(Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons800, cons1971, cons1972)
def replacement6870(x, a, b, s):
rubi.append(6870)
return -Simp(Zeta(s + S(-1), a + b*x)/(b*(s + S(-1))), x)
rule6870 = ReplacementRule(pattern6870, replacement6870)
pattern6871 = Pattern(Integral(x_**WC('m', S(1))*Zeta(S(2), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons31)
def replacement6871(x, m, b, a):
rubi.append(6871)
return Int(x**m*PolyGamma(S(1), a + b*x), x)
rule6871 = ReplacementRule(pattern6871, replacement6871)
pattern6872 = Pattern(Integral(x_**WC('m', S(1))*Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons800, cons1971, cons1972, cons31, cons168)
def replacement6872(m, b, a, x, s):
rubi.append(6872)
return Dist(m/(b*(s + S(-1))), Int(x**(m + S(-1))*Zeta(s + S(-1), a + b*x), x), x) - Simp(x**m*Zeta(s + S(-1), a + b*x)/(b*(s + S(-1))), x)
rule6872 = ReplacementRule(pattern6872, replacement6872)
pattern6873 = Pattern(Integral(x_**WC('m', S(1))*Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons800, cons1971, cons1972, cons31, cons94)
def replacement6873(m, b, a, x, s):
rubi.append(6873)
return Dist(b*s/(m + S(1)), Int(x**(m + S(1))*Zeta(s + S(1), a + b*x), x), x) + Simp(x**(m + S(1))*Zeta(s, a + b*x)/(m + S(1)), x)
rule6873 = ReplacementRule(pattern6873, replacement6873)
pattern6874 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons5, cons50, cons87, cons88)
def replacement6874(p, b, a, n, x, q):
rubi.append(6874)
return -Dist(p*q, Int(PolyLog(n + S(-1), a*(b*x**p)**q), x), x) + Simp(x*PolyLog(n, a*(b*x**p)**q), x)
rule6874 = ReplacementRule(pattern6874, replacement6874)
pattern6875 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons5, cons50, cons87, cons89)
def replacement6875(p, b, a, n, x, q):
rubi.append(6875)
return -Dist(S(1)/(p*q), Int(PolyLog(n + S(1), a*(b*x**p)**q), x), x) + Simp(x*PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x)
rule6875 = ReplacementRule(pattern6875, replacement6875)
pattern6876 = Pattern(Integral(PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons383)
def replacement6876(p, b, d, c, a, n, x, e):
rubi.append(6876)
return Simp(PolyLog(n + S(1), c*(a + b*x)**p)/(e*p), x)
rule6876 = ReplacementRule(pattern6876, replacement6876)
pattern6877 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1)))/x_, x_), cons2, cons3, cons4, cons5, cons50, cons1973)
def replacement6877(p, b, a, n, x, q):
rubi.append(6877)
return Simp(PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x)
rule6877 = ReplacementRule(pattern6877, replacement6877)
pattern6878 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons21, cons5, cons50, cons66, cons87, cons88)
def replacement6878(p, m, b, a, n, x, q):
rubi.append(6878)
return -Dist(p*q/(m + S(1)), Int(x**m*PolyLog(n + S(-1), a*(b*x**p)**q), x), x) + Simp(x**(m + S(1))*PolyLog(n, a*(b*x**p)**q)/(m + S(1)), x)
rule6878 = ReplacementRule(pattern6878, replacement6878)
pattern6879 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons21, cons5, cons50, cons66, cons87, cons89)
def replacement6879(p, m, b, a, n, x, q):
rubi.append(6879)
return -Dist((m + S(1))/(p*q), Int(x**m*PolyLog(n + S(1), a*(b*x**p)**q), x), x) + Simp(x**(m + S(1))*PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x)
rule6879 = ReplacementRule(pattern6879, replacement6879)
pattern6880 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1)))*log(x_**WC('m', S(1))*WC('c', S(1)))**WC('r', S(1))/x_, x_), cons2, cons3, cons7, cons21, cons4, cons50, cons52, cons1974, cons1975)
def replacement6880(p, m, b, r, c, a, n, x, q):
rubi.append(6880)
return -Dist(m*r/(p*q), Int(PolyLog(n + S(1), a*(b*x**p)**q)*log(c*x**m)**(r + S(-1))/x, x), x) + Simp(PolyLog(n + S(1), a*(b*x**p)**q)*log(c*x**m)**r/(p*q), x)
rule6880 = ReplacementRule(pattern6880, replacement6880)
pattern6881 = Pattern(Integral(PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons5, cons87, cons88)
def replacement6881(p, b, c, a, n, x):
rubi.append(6881)
return -Dist(p, Int(PolyLog(n + S(-1), c*(a + b*x)**p), x), x) + Dist(a*p, Int(PolyLog(n + S(-1), c*(a + b*x)**p)/(a + b*x), x), x) + Simp(x*PolyLog(n, c*(a + b*x)**p), x)
rule6881 = ReplacementRule(pattern6881, replacement6881)
pattern6882 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons21, cons5, cons87, cons88, cons62)
def replacement6882(p, m, b, c, a, n, x):
rubi.append(6882)
return -Dist(b*p/(m + S(1)), Int(x**(m + S(1))*PolyLog(n + S(-1), c*(a + b*x)**p)/(a + b*x), x), x) + Simp(x**(m + S(1))*PolyLog(n, c*(a + b*x)**p)/(m + S(1)), x)
rule6882 = ReplacementRule(pattern6882, replacement6882)
pattern6883 = Pattern(Integral(PolyLog(n_, (F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('p', S(1))*WC('d', S(1))), x_), cons1099, cons2, cons3, cons7, cons27, cons4, cons5, cons1976)
def replacement6883(p, b, d, c, a, n, x, F):
rubi.append(6883)
return Simp(PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p)/(b*c*p*log(F)), x)
rule6883 = ReplacementRule(pattern6883, replacement6883)
pattern6884 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*PolyLog(n_, (F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('p', S(1))*WC('d', S(1))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons31, cons168)
def replacement6884(p, m, f, b, d, c, a, n, x, F, e):
rubi.append(6884)
return -Dist(f*m/(b*c*p*log(F)), Int((e + f*x)**(m + S(-1))*PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p), x), x) + Simp((e + f*x)**m*PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p)/(b*c*p*log(F)), x)
rule6884 = ReplacementRule(pattern6884, replacement6884)
def With6885(v, x, n, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
w = DerivativeDivides(v, u*v, x)
res = Not(FalseQ(w))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6885 = Pattern(Integral(u_*PolyLog(n_, v_), x_), cons4, cons4, CustomConstraint(With6885))
def replacement6885(v, x, n, u):
w = DerivativeDivides(v, u*v, x)
rubi.append(6885)
return Simp(w*PolyLog(n + S(1), v), x)
rule6885 = ReplacementRule(pattern6885, replacement6885)
def With6886(v, w, u, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
z = DerivativeDivides(v, u*v, x)
res = Not(FalseQ(z))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern6886 = Pattern(Integral(u_*PolyLog(n_, v_)*log(w_), x_), cons4, cons1243, CustomConstraint(With6886))
def replacement6886(v, w, u, n, x):
z = DerivativeDivides(v, u*v, x)
rubi.append(6886)
return -Int(SimplifyIntegrand(z*D(w, x)*PolyLog(n + S(1), v)/w, x), x) + Simp(z*PolyLog(n + S(1), v)*log(w), x)
rule6886 = ReplacementRule(pattern6886, replacement6886)
pattern6887 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons13, cons137)
def replacement6887(p, b, c, a, x):
rubi.append(6887)
return Dist(p/(c*(p + S(1))), Int((c*ProductLog(a + b*x))**(p + S(1))/(ProductLog(a + b*x) + S(1)), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/(b*(p + S(1))), x)
rule6887 = ReplacementRule(pattern6887, replacement6887)
pattern6888 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons1379)
def replacement6888(p, b, c, a, x):
rubi.append(6888)
return -Dist(p, Int((c*ProductLog(a + b*x))**p/(ProductLog(a + b*x) + S(1)), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/b, x)
rule6888 = ReplacementRule(pattern6888, replacement6888)
pattern6889 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(a_ + x_*WC('b', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons5, cons62)
def replacement6889(p, m, b, c, a, x):
rubi.append(6889)
return Dist(S(1)/b, Subst(Int(ExpandIntegrand((c*ProductLog(x))**p, (-a/b + x/b)**m, x), x), x, a + b*x), x)
rule6889 = ReplacementRule(pattern6889, replacement6889)
pattern6890 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons4, cons5, cons1977)
def replacement6890(p, c, a, n, x):
rubi.append(6890)
return -Dist(n*p, Int((c*ProductLog(a*x**n))**p/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x*(c*ProductLog(a*x**n))**p, x)
rule6890 = ReplacementRule(pattern6890, replacement6890)
pattern6891 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons4, cons1978)
def replacement6891(p, c, a, n, x):
rubi.append(6891)
return Dist(n*p/(c*(n*p + S(1))), Int((c*ProductLog(a*x**n))**(p + S(1))/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x*(c*ProductLog(a*x**n))**p/(n*p + S(1)), x)
rule6891 = ReplacementRule(pattern6891, replacement6891)
pattern6892 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons5, cons196)
def replacement6892(p, c, a, n, x):
rubi.append(6892)
return -Subst(Int((c*ProductLog(a*x**(-n)))**p/x**S(2), x), x, S(1)/x)
rule6892 = ReplacementRule(pattern6892, replacement6892)
pattern6893 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons21, cons4, cons5, cons66, cons1979)
def replacement6893(p, m, c, n, a, x):
rubi.append(6893)
return -Dist(n*p/(m + S(1)), Int(x**m*(c*ProductLog(a*x**n))**p/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(m + S(1)), x)
rule6893 = ReplacementRule(pattern6893, replacement6893)
pattern6894 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons21, cons4, cons5, cons1980)
def replacement6894(p, m, c, n, a, x):
rubi.append(6894)
return Dist(n*p/(c*(m + n*p + S(1))), Int(x**m*(c*ProductLog(a*x**n))**(p + S(1))/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(m + n*p + S(1)), x)
rule6894 = ReplacementRule(pattern6894, replacement6894)
pattern6895 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons21, cons1981)
def replacement6895(p, m, c, a, x):
rubi.append(6895)
return Dist(S(1)/c, Int(x**m*(c*ProductLog(a*x))**(p + S(1))/(ProductLog(a*x) + S(1)), x), x) + Int(x**m*(c*ProductLog(a*x))**p/(ProductLog(a*x) + S(1)), x)
rule6895 = ReplacementRule(pattern6895, replacement6895)
pattern6896 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons7, cons5, cons150, cons463, cons66)
def replacement6896(p, m, c, a, n, x):
rubi.append(6896)
return -Subst(Int(x**(-m + S(-2))*(c*ProductLog(a*x**(-n)))**p, x), x, S(1)/x)
rule6896 = ReplacementRule(pattern6896, replacement6896)
pattern6897 = Pattern(Integral(S(1)/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons27, cons1765)
def replacement6897(d, a, b, x):
rubi.append(6897)
return Simp((a + b*x)/(b*d*ProductLog(a + b*x)), x)
rule6897 = ReplacementRule(pattern6897, replacement6897)
pattern6898 = Pattern(Integral(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons27, cons1765)
def replacement6898(d, a, b, x):
rubi.append(6898)
return -Int(S(1)/(d*ProductLog(a + b*x) + d), x) + Simp(d*x, x)
rule6898 = ReplacementRule(pattern6898, replacement6898)
pattern6899 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons13, cons163)
def replacement6899(p, b, d, c, a, x):
rubi.append(6899)
return -Dist(c*p, Int((c*ProductLog(a + b*x))**(p + S(-1))/(d*ProductLog(a + b*x) + d), x), x) + Simp(c*(c*ProductLog(a + b*x))**(p + S(-1))*(a + b*x)/(b*d), x)
rule6899 = ReplacementRule(pattern6899, replacement6899)
pattern6900 = Pattern(Integral(S(1)/((d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))*ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons27, cons1765)
def replacement6900(d, a, b, x):
rubi.append(6900)
return Simp(ExpIntegralEi(ProductLog(a + b*x))/(b*d), x)
rule6900 = ReplacementRule(pattern6900, replacement6900)
pattern6901 = Pattern(Integral(S(1)/(sqrt(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons948)
def replacement6901(b, d, c, a, x):
rubi.append(6901)
return Simp(Erfi(sqrt(c*ProductLog(a + b*x))/Rt(c, S(2)))*Rt(Pi*c, S(2))/(b*c*d), x)
rule6901 = ReplacementRule(pattern6901, replacement6901)
pattern6902 = Pattern(Integral(S(1)/(sqrt(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons949)
def replacement6902(b, d, c, a, x):
rubi.append(6902)
return Simp(Erf(sqrt(c*ProductLog(a + b*x))/Rt(-c, S(2)))*Rt(-Pi*c, S(2))/(b*c*d), x)
rule6902 = ReplacementRule(pattern6902, replacement6902)
pattern6903 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons13, cons137)
def replacement6903(p, b, d, c, a, x):
rubi.append(6903)
return -Dist(S(1)/(c*(p + S(1))), Int((c*ProductLog(a + b*x))**(p + S(1))/(d*ProductLog(a + b*x) + d), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/(b*d*(p + S(1))), x)
rule6903 = ReplacementRule(pattern6903, replacement6903)
pattern6904 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons5, cons1982)
def replacement6904(p, b, d, c, a, x):
rubi.append(6904)
return Simp((-ProductLog(a + b*x))**(-p)*(c*ProductLog(a + b*x))**p*Gamma(p + S(1), -ProductLog(a + b*x))/(b*d), x)
rule6904 = ReplacementRule(pattern6904, replacement6904)
pattern6905 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(a_ + x_*WC('b', S(1)))*WC('d', S(1))), x_), cons2, cons3, cons27, cons62)
def replacement6905(m, b, d, a, x):
rubi.append(6905)
return Dist(S(1)/b, Subst(Int(ExpandIntegrand(S(1)/(d*ProductLog(x) + d), (-a/b + x/b)**m, x), x), x, a + b*x), x)
rule6905 = ReplacementRule(pattern6905, replacement6905)
pattern6906 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(a_ + x_*WC('b', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(a_ + x_*WC('b', S(1)))*WC('d', S(1))), x_), cons2, cons3, cons7, cons27, cons5, cons62)
def replacement6906(p, m, b, d, c, a, x):
rubi.append(6906)
return Dist(S(1)/b, Subst(Int(ExpandIntegrand((c*ProductLog(x))**p/(d*ProductLog(x) + d), (-a/b + x/b)**m, x), x), x, a + b*x), x)
rule6906 = ReplacementRule(pattern6906, replacement6906)
pattern6907 = Pattern(Integral(S(1)/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons27, cons196)
def replacement6907(d, a, n, x):
rubi.append(6907)
return -Subst(Int(S(1)/(x**S(2)*(d*ProductLog(a*x**(-n)) + d)), x), x, S(1)/x)
rule6907 = ReplacementRule(pattern6907, replacement6907)
pattern6908 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons4, cons5, cons1983)
def replacement6908(p, d, c, n, a, x):
rubi.append(6908)
return Simp(c*x*(c*ProductLog(a*x**n))**(p + S(-1))/d, x)
rule6908 = ReplacementRule(pattern6908, replacement6908)
pattern6909 = Pattern(Integral(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons27, cons803, cons1984)
def replacement6909(p, d, a, n, x):
rubi.append(6909)
return Simp(a**p*ExpIntegralEi(-p*ProductLog(a*x**n))/(d*n), x)
rule6909 = ReplacementRule(pattern6909, replacement6909)
pattern6910 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons803, cons1985, cons1986)
def replacement6910(p, d, c, n, a, x):
rubi.append(6910)
return Simp(a**(-S(1)/n)*c**(-S(1)/n)*Erfi(sqrt(c*ProductLog(a*x**n))/Rt(c*n, S(2)))*Rt(Pi*c*n, S(2))/(d*n), x)
rule6910 = ReplacementRule(pattern6910, replacement6910)
pattern6911 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons803, cons1985, cons1987)
def replacement6911(p, d, c, n, a, x):
rubi.append(6911)
return Simp(a**(-S(1)/n)*c**(-S(1)/n)*Erf(sqrt(c*ProductLog(a*x**n))/Rt(-c*n, S(2)))*Rt(-Pi*c*n, S(2))/(d*n), x)
rule6911 = ReplacementRule(pattern6911, replacement6911)
pattern6912 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons338, cons88, cons1988)
def replacement6912(p, d, c, n, a, x):
rubi.append(6912)
return -Dist(c*(n*(p + S(-1)) + S(1)), Int((c*ProductLog(a*x**n))**(p + S(-1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(c*x*(c*ProductLog(a*x**n))**(p + S(-1))/d, x)
rule6912 = ReplacementRule(pattern6912, replacement6912)
pattern6913 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons338, cons88, cons1989)
def replacement6913(p, d, c, n, a, x):
rubi.append(6913)
return -Dist(S(1)/(c*(n*p + S(1))), Int((c*ProductLog(a*x**n))**(p + S(1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(x*(c*ProductLog(a*x**n))**p/(d*(n*p + S(1))), x)
rule6913 = ReplacementRule(pattern6913, replacement6913)
pattern6914 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons5, cons196)
def replacement6914(p, d, c, a, n, x):
rubi.append(6914)
return -Subst(Int((c*ProductLog(a*x**(-n)))**p/(x**S(2)*(d*ProductLog(a*x**(-n)) + d)), x), x, S(1)/x)
rule6914 = ReplacementRule(pattern6914, replacement6914)
pattern6915 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons27, cons31, cons168)
def replacement6915(d, m, x, a):
rubi.append(6915)
return -Dist(m/(m + S(1)), Int(x**m/((d*ProductLog(a*x) + d)*ProductLog(a*x)), x), x) + Simp(x**(m + S(1))/(d*(m + S(1))*ProductLog(a*x)), x)
rule6915 = ReplacementRule(pattern6915, replacement6915)
pattern6916 = Pattern(Integral(S(1)/(x_*(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons27, cons1990)
def replacement6916(d, a, x):
rubi.append(6916)
return Simp(log(ProductLog(a*x))/d, x)
rule6916 = ReplacementRule(pattern6916, replacement6916)
pattern6917 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons27, cons31, cons94)
def replacement6917(d, m, x, a):
rubi.append(6917)
return -Int(x**m*ProductLog(a*x)/(d*ProductLog(a*x) + d), x) + Simp(x**(m + S(1))/(d*(m + S(1))), x)
rule6917 = ReplacementRule(pattern6917, replacement6917)
pattern6918 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons27, cons21, cons18)
def replacement6918(d, m, x, a):
rubi.append(6918)
return Simp(x**m*(-(m + S(1))*ProductLog(a*x))**(-m)*Gamma(m + S(1), -(m + S(1))*ProductLog(a*x))*exp(-m*ProductLog(a*x))/(a*d*(m + S(1))), x)
rule6918 = ReplacementRule(pattern6918, replacement6918)
pattern6919 = Pattern(Integral(S(1)/(x_*(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons27, cons4, cons1991)
def replacement6919(d, a, n, x):
rubi.append(6919)
return Simp(log(ProductLog(a*x**n))/(d*n), x)
rule6919 = ReplacementRule(pattern6919, replacement6919)
pattern6920 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons27, cons150, cons463, cons66)
def replacement6920(m, d, a, n, x):
rubi.append(6920)
return -Subst(Int(x**(-m + S(-2))/(d*ProductLog(a*x**(-n)) + d), x), x, S(1)/x)
rule6920 = ReplacementRule(pattern6920, replacement6920)
pattern6921 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(x_*(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons7, cons27, cons4, cons5, cons1992)
def replacement6921(p, d, c, n, a, x):
rubi.append(6921)
return Simp((c*ProductLog(a*x**n))**p/(d*n*p), x)
rule6921 = ReplacementRule(pattern6921, replacement6921)
pattern6922 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons66, cons1993)
def replacement6922(p, m, d, c, n, a, x):
rubi.append(6922)
return Simp(c*x**(m + S(1))*(c*ProductLog(a*x**n))**(p + S(-1))/(d*(m + S(1))), x)
rule6922 = ReplacementRule(pattern6922, replacement6922)
pattern6923 = Pattern(Integral(x_**WC('m', S(1))*ProductLog(x_**WC('n', S(1))*WC('a', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons27, cons21, cons4, cons38, cons1994)
def replacement6923(p, m, d, a, n, x):
rubi.append(6923)
return Simp(a**p*ExpIntegralEi(-p*ProductLog(a*x**n))/(d*n), x)
rule6923 = ReplacementRule(pattern6923, replacement6923)
pattern6924 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons21, cons4, cons66, cons347, cons1995, cons1996)
def replacement6924(p, m, d, c, n, a, x):
rubi.append(6924)
return Simp(a**(p + S(-1)/2)*c**(p + S(-1)/2)*Erf(sqrt(c*ProductLog(a*x**n))/Rt(c/(p + S(-1)/2), S(2)))*Rt(Pi*c/(p + S(-1)/2), S(2))/(d*n), x)
rule6924 = ReplacementRule(pattern6924, replacement6924)
pattern6925 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons21, cons4, cons66, cons347, cons1995, cons1997)
def replacement6925(p, m, d, c, n, a, x):
rubi.append(6925)
return Simp(a**(p + S(-1)/2)*c**(p + S(-1)/2)*Erfi(sqrt(c*ProductLog(a*x**n))/Rt(-c/(p + S(-1)/2), S(2)))*Rt(-Pi*c/(p + S(-1)/2), S(2))/(d*n), x)
rule6925 = ReplacementRule(pattern6925, replacement6925)
pattern6926 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons66, cons1998, cons1999)
def replacement6926(p, m, d, c, n, a, x):
rubi.append(6926)
return -Dist(c*(m + n*(p + S(-1)) + S(1))/(m + S(1)), Int(x**m*(c*ProductLog(a*x**n))**(p + S(-1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(c*x**(m + S(1))*(c*ProductLog(a*x**n))**(p + S(-1))/(d*(m + S(1))), x)
rule6926 = ReplacementRule(pattern6926, replacement6926)
pattern6927 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons21, cons4, cons5, cons66, cons1998, cons2000)
def replacement6927(p, m, d, c, n, a, x):
rubi.append(6927)
return -Dist((m + S(1))/(c*(m + n*p + S(1))), Int(x**m*(c*ProductLog(a*x**n))**(p + S(1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(d*(m + n*p + S(1))), x)
rule6927 = ReplacementRule(pattern6927, replacement6927)
pattern6928 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons21, cons5, cons66)
def replacement6928(p, m, d, c, a, x):
rubi.append(6928)
return Simp(x**m*(c*ProductLog(a*x))**p*(-(m + S(1))*ProductLog(a*x))**(-m - p)*Gamma(m + p + S(1), -(m + S(1))*ProductLog(a*x))*exp(-m*ProductLog(a*x))/(a*d*(m + S(1))), x)
rule6928 = ReplacementRule(pattern6928, replacement6928)
pattern6929 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons7, cons27, cons5, cons66, cons150, cons463)
def replacement6929(p, m, d, c, n, a, x):
rubi.append(6929)
return -Subst(Int(x**(-m + S(-2))*(c*ProductLog(a*x**(-n)))**p/(d*ProductLog(a*x**(-n)) + d), x), x, S(1)/x)
rule6929 = ReplacementRule(pattern6929, replacement6929)
pattern6930 = Pattern(Integral(u_, x_), cons2001)
def replacement6930(x, u):
rubi.append(6930)
return Subst(Int(SimplifyIntegrand((x + S(1))*SubstFor(ProductLog(x), u, x)*exp(x), x), x), x, ProductLog(x))
rule6930 = ReplacementRule(pattern6930, replacement6930)
return [rule6739, rule6740, rule6741, rule6742, rule6743, rule6744, rule6745, rule6746, rule6747, rule6748, rule6749, rule6750, rule6751, rule6752, rule6753, rule6754, rule6755, rule6756, rule6757, rule6758, rule6759, rule6760, rule6761, rule6762, rule6763, rule6764, rule6765, rule6766, rule6767, rule6768, rule6769, rule6770, rule6771, rule6772, rule6773, rule6774, rule6775, rule6776, rule6777, rule6778, rule6779, rule6780, rule6781, rule6782, rule6783, rule6784, rule6785, rule6786, rule6787, rule6788, rule6789, rule6790, rule6791, rule6792, rule6793, rule6794, rule6795, rule6796, rule6797, rule6798, rule6799, rule6800, rule6801, rule6802, rule6803, rule6804, rule6805, rule6806, rule6807, rule6808, rule6809, rule6810, rule6811, rule6812, rule6813, rule6814, rule6815, rule6816, rule6817, rule6818, rule6819, rule6820, rule6821, rule6822, rule6823, rule6824, rule6825, rule6826, rule6827, rule6828, rule6829, rule6830, rule6831, rule6832, rule6833, rule6834, rule6835, rule6836, rule6837, rule6838, rule6839, rule6840, rule6841, rule6842, rule6843, rule6844, rule6845, rule6846, rule6847, rule6848, rule6849, rule6850, rule6851, rule6852, rule6853, rule6854, rule6855, rule6856, rule6857, rule6858, rule6859, rule6860, rule6861, rule6862, rule6863, rule6864, rule6865, rule6866, rule6867, rule6868, rule6869, rule6870, rule6871, rule6872, rule6873, rule6874, rule6875, rule6876, rule6877, rule6878, rule6879, rule6880, rule6881, rule6882, rule6883, rule6884, rule6885, rule6886, rule6887, rule6888, rule6889, rule6890, rule6891, rule6892, rule6893, rule6894, rule6895, rule6896, rule6897, rule6898, rule6899, rule6900, rule6901, rule6902, rule6903, rule6904, rule6905, rule6906, rule6907, rule6908, rule6909, rule6910, rule6911, rule6912, rule6913, rule6914, rule6915, rule6916, rule6917, rule6918, rule6919, rule6920, rule6921, rule6922, rule6923, rule6924, rule6925, rule6926, rule6927, rule6928, rule6929, rule6930, ]
|
5e2abd8f8cdab5153884efa0a8afc39deccc7c37e82b88fddaeaa85f5fe23ab6 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def miscellaneous_algebraic(rubi):
from sympy.integrals.rubi.constraints import cons798, cons2, cons3, cons7, cons50, cons4, cons5, cons17, cons21, cons799, cons27, cons48, cons125, cons52, cons800, cons25, cons801, cons802, cons149, cons803, cons500, cons804, cons648, cons805, cons806, cons18, cons46, cons807, cons808, cons68, cons809, cons810, cons811, cons812, cons813, cons814, cons815, cons816, cons817, cons818, cons819, cons820, cons821, cons452, cons822, cons823, cons824, cons825, cons826, cons827, cons828, cons829, cons830, cons831, cons832, cons833, cons834, cons835, cons836, cons837, cons838, cons839, cons840, cons841, cons842, cons843, cons844, cons845, cons846, cons847, cons848, cons849, cons850, cons851, cons852, cons208, cons209, cons64, cons853, cons66, cons854, cons855, cons464, cons856, cons857, cons858, cons53, cons13, cons137, cons859, cons860, cons148, cons244, cons163, cons861, cons521, cons862, cons863, cons864, cons84, cons865, cons34, cons35, cons866, cons468, cons469, cons867, cons868, cons36, cons869, cons870, cons871, cons872, cons873, cons874, cons875, cons876, cons877, cons878, cons879, cons880, cons881, cons882, cons883, cons884, cons885, cons886, cons887, cons888, cons889, cons890, cons891, cons892, cons893, cons894, cons895, cons896, cons897, cons898, cons899, cons900, cons901, cons902, cons903, cons904, cons674, cons905, cons481, cons906, cons907, cons482, cons908, cons909, cons910, cons911, cons912, cons913, cons914, cons915, cons916, cons85, cons31, cons94, cons917, cons196, cons367, cons356, cons489, cons541, cons23, cons918, cons554, cons919, cons552, cons55, cons494, cons57, cons58, cons59, cons60, cons920, cons921, cons922, cons923, cons924, cons595, cons71, cons925, cons586, cons87, cons128, cons926, cons927, cons928, cons929, cons930, cons45, cons314, cons226, cons931, cons932, cons933, cons934, cons935, cons936, cons937, cons938, cons939, cons940, cons941, cons942, cons943, cons944, cons945, cons946, cons282, cons947, cons63, cons719, cons948, cons949, cons950, cons73, cons951, cons702, cons147, cons952, cons953, cons796, cons954, cons955, cons956, cons957, cons958, cons959, cons960, cons961, cons962, cons963, cons964, cons965, cons966, cons69, cons967, cons968, cons969, cons970, cons971, cons972, cons973, cons974, cons975, cons512, cons976, cons977, cons978, cons979, cons980, cons667, cons981, cons982, cons797, cons983, cons984, cons985, cons986, cons987, cons988, cons93, cons88, cons989, cons990, cons991, cons992, cons993, cons994, cons995, cons996, cons997, cons998, cons38, cons999, cons1000, cons1001, cons1002, cons1003, cons1004, cons1005, cons1006, cons1007, cons1008, cons1009, cons1010, cons383, cons1011, cons1012, cons1013, cons1014, cons1015, cons1016, cons1017, cons1018, cons357, cons1019, cons246, cons1020, cons1021, cons1022, cons1023, cons1024, cons1025, cons1026, cons1027, cons1028, cons1029, cons1030, cons1031, cons1032, cons1033, cons1034, cons1035, cons1036, cons1037, cons1038, cons1039, cons1040, cons1041, cons1042, cons1043, cons297, cons1044, cons1045, cons1046, cons1047, cons1048, cons705, cons382, cons1049, cons1050, cons697, cons709, cons153, cons1051, cons1052, cons1053, cons1054, cons1055, cons1056, cons1057, cons1058, cons1059, cons224, cons1060, cons515, cons1061, cons1062, cons1063, cons1064, cons1065, cons1066, cons1067, cons1068, cons1069, cons1070, cons1071, cons43, cons479, cons480, cons1072, cons1073, cons1074, cons1075, cons1076, cons1077, cons1078, cons1079, cons1080, cons1081, cons1082, cons1083, cons1084, cons1085, cons1086, cons1087, cons1088, cons1089
pattern1473 = Pattern(Integral(((x_**n_*WC('c', S(1)))**q_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons50, cons4, cons5, cons798)
def replacement1473(p, b, c, a, n, x, q):
rubi.append(1473)
return Dist(x*(c*x**n)**(-S(1)/n), Subst(Int((a + b*x**(n*q))**p, x), x, (c*x**n)**(S(1)/n)), x)
rule1473 = ReplacementRule(pattern1473, replacement1473)
pattern1474 = Pattern(Integral(x_**WC('m', S(1))*((x_**n_*WC('c', S(1)))**q_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons50, cons798, cons17)
def replacement1474(p, m, b, c, a, n, x, q):
rubi.append(1474)
return Dist(x**(m + S(1))*(c*x**n)**(-(m + S(1))/n), Subst(Int(x**m*(a + b*x**(n*q))**p, x), x, (c*x**n)**(S(1)/n)), x)
rule1474 = ReplacementRule(pattern1474, replacement1474)
pattern1475 = Pattern(Integral(x_**WC('m', S(1))*((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('r', S(1))*WC('e', S(1)))**p_*((c_ + x_**WC('n', S(1))*WC('d', S(1)))**s_*WC('f', S(1)))**q_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons5, cons50, cons52, cons800, cons799)
def replacement1475(p, q, m, f, b, r, d, a, n, c, x, s, e):
rubi.append(1475)
return Dist((e*(a + b*x**n)**r)**p*(f*(c + d*x**n)**s)**q*(a + b*x**n)**(-p*r)*(c + d*x**n)**(-q*s), Int(x**m*(a + b*x**n)**(p*r)*(c + d*x**n)**(q*s), x), x)
rule1475 = ReplacementRule(pattern1475, replacement1475)
pattern1476 = Pattern(Integral(((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons25)
def replacement1476(p, u, b, d, a, n, c, x, e):
rubi.append(1476)
return Dist((b*e/d)**p, Int(u, x), x)
rule1476 = ReplacementRule(pattern1476, replacement1476)
pattern1477 = Pattern(Integral(((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons801, cons802)
def replacement1477(p, u, b, d, a, n, c, x, e):
rubi.append(1477)
return Int(u*(e*(a + b*x**n))**p*(c + d*x**n)**(-p), x)
rule1477 = ReplacementRule(pattern1477, replacement1477)
def With1478(p, b, d, a, n, c, x, e):
q = Denominator(p)
rubi.append(1478)
return Dist(e*q*(-a*d + b*c)/n, Subst(Int(x**(q*(p + S(1)) + S(-1))*(-a*e + c*x**q)**(S(-1) + S(1)/n)*(b*e - d*x**q)**(S(-1) - S(1)/n), x), x, (e*(a + b*x**n)/(c + d*x**n))**(S(1)/q)), x)
pattern1478 = Pattern(Integral(((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons149, cons803)
rule1478 = ReplacementRule(pattern1478, With1478)
def With1479(p, m, b, d, a, n, c, x, e):
q = Denominator(p)
rubi.append(1479)
return Dist(e*q*(-a*d + b*c)/n, Subst(Int(x**(q*(p + S(1)) + S(-1))*(-a*e + c*x**q)**(S(-1) + (m + S(1))/n)*(b*e - d*x**q)**(S(-1) - (m + S(1))/n), x), x, (e*(a + b*x**n)/(c + d*x**n))**(S(1)/q)), x)
pattern1479 = Pattern(Integral(x_**WC('m', S(1))*((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons149, cons500)
rule1479 = ReplacementRule(pattern1479, With1479)
def With1480(p, u, b, r, d, a, n, c, x, e):
q = Denominator(p)
rubi.append(1480)
return Dist(e*q*(-a*d + b*c)/n, Subst(Int(SimplifyIntegrand(x**(q*(p + S(1)) + S(-1))*(-a*e + c*x**q)**(S(-1) + S(1)/n)*(b*e - d*x**q)**(S(-1) - S(1)/n)*ReplaceAll(u, Rule(x, (-a*e + c*x**q)**(S(1)/n)*(b*e - d*x**q)**(-S(1)/n)))**r, x), x), x, (e*(a + b*x**n)/(c + d*x**n))**(S(1)/q)), x)
pattern1480 = Pattern(Integral(u_**WC('r', S(1))*((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons804, cons149, cons803, cons648)
rule1480 = ReplacementRule(pattern1480, With1480)
def With1481(p, u, m, b, r, d, a, n, c, x, e):
q = Denominator(p)
rubi.append(1481)
return Dist(e*q*(-a*d + b*c)/n, Subst(Int(SimplifyIntegrand(x**(q*(p + S(1)) + S(-1))*(-a*e + c*x**q)**(S(-1) + (m + S(1))/n)*(b*e - d*x**q)**(S(-1) - (m + S(1))/n)*ReplaceAll(u, Rule(x, (-a*e + c*x**q)**(S(1)/n)*(b*e - d*x**q)**(-S(1)/n)))**r, x), x), x, (e*(a + b*x**n)/(c + d*x**n))**(S(1)/q)), x)
pattern1481 = Pattern(Integral(u_**WC('r', S(1))*x_**WC('m', S(1))*((x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(c_ + x_**WC('n', S(1))*WC('d', S(1))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons804, cons149, cons803, cons805)
rule1481 = ReplacementRule(pattern1481, With1481)
pattern1482 = Pattern(Integral(((WC('c', S(1))/x_)**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons806)
def replacement1482(p, b, c, a, n, x):
rubi.append(1482)
return -Dist(c, Subst(Int((a + b*x**n)**p/x**S(2), x), x, c/x), x)
rule1482 = ReplacementRule(pattern1482, replacement1482)
pattern1483 = Pattern(Integral(x_**WC('m', S(1))*((WC('c', S(1))/x_)**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons17)
def replacement1483(p, m, b, c, a, n, x):
rubi.append(1483)
return -Dist(c**(m + S(1)), Subst(Int(x**(-m + S(-2))*(a + b*x**n)**p, x), x, c/x), x)
rule1483 = ReplacementRule(pattern1483, replacement1483)
pattern1484 = Pattern(Integral((x_*WC('d', S(1)))**m_*((WC('c', S(1))/x_)**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons18)
def replacement1484(p, m, b, d, c, a, n, x):
rubi.append(1484)
return -Dist(c*(c/x)**m*(d*x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**n)**p, x), x, c/x), x)
rule1484 = ReplacementRule(pattern1484, replacement1484)
pattern1485 = Pattern(Integral(((WC('d', S(1))/x_)**n_*WC('b', S(1)) + (WC('d', S(1))/x_)**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons46)
def replacement1485(p, b, n2, d, a, c, n, x):
rubi.append(1485)
return -Dist(d, Subst(Int((a + b*x**n + c*x**(S(2)*n))**p/x**S(2), x), x, d/x), x)
rule1485 = ReplacementRule(pattern1485, replacement1485)
pattern1486 = Pattern(Integral(x_**WC('m', S(1))*(a_ + (WC('d', S(1))/x_)**n_*WC('b', S(1)) + (WC('d', S(1))/x_)**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons46, cons17)
def replacement1486(p, m, b, n2, d, c, a, n, x):
rubi.append(1486)
return -Dist(d**(m + S(1)), Subst(Int(x**(-m + S(-2))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, d/x), x)
rule1486 = ReplacementRule(pattern1486, replacement1486)
pattern1487 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + (WC('d', S(1))/x_)**n_*WC('b', S(1)) + (WC('d', S(1))/x_)**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons5, cons46, cons18)
def replacement1487(p, m, b, n2, d, c, a, n, x, e):
rubi.append(1487)
return -Dist(d*(d/x)**m*(e*x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, d/x), x)
rule1487 = ReplacementRule(pattern1487, replacement1487)
pattern1488 = Pattern(Integral((x_**WC('n2', S(1))*WC('c', S(1)) + (WC('d', S(1))/x_)**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons807, cons808)
def replacement1488(p, b, n2, d, c, a, n, x):
rubi.append(1488)
return -Dist(d, Subst(Int((a + b*x**n + c*d**(-S(2)*n)*x**(S(2)*n))**p/x**S(2), x), x, d/x), x)
rule1488 = ReplacementRule(pattern1488, replacement1488)
pattern1489 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**WC('n2', S(1))*WC('c', S(1)) + (WC('d', S(1))/x_)**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons807, cons808, cons17)
def replacement1489(p, m, b, n2, d, c, a, n, x):
rubi.append(1489)
return -Dist(d**(m + S(1)), Subst(Int(x**(-m + S(-2))*(a + b*x**n + c*d**(-S(2)*n)*x**(S(2)*n))**p, x), x, d/x), x)
rule1489 = ReplacementRule(pattern1489, replacement1489)
pattern1490 = Pattern(Integral((x_*WC('e', S(1)))**m_*(a_ + x_**WC('n2', S(1))*WC('c', S(1)) + (WC('d', S(1))/x_)**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons5, cons807, cons18, cons808)
def replacement1490(p, m, b, n2, d, c, a, n, x, e):
rubi.append(1490)
return -Dist(d*(d/x)**m*(e*x)**m, Subst(Int(x**(-m + S(-2))*(a + b*x**n + c*d**(-S(2)*n)*x**(S(2)*n))**p, x), x, d/x), x)
rule1490 = ReplacementRule(pattern1490, replacement1490)
pattern1491 = Pattern(Integral(u_**m_, x_), cons21, cons68, cons809)
def replacement1491(x, m, u):
rubi.append(1491)
return Int(ExpandToSum(u, x)**m, x)
rule1491 = ReplacementRule(pattern1491, replacement1491)
pattern1492 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('n', S(1)), x_), cons21, cons4, cons810, cons811)
def replacement1492(v, u, m, n, x):
rubi.append(1492)
return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**n, x)
rule1492 = ReplacementRule(pattern1492, replacement1492)
pattern1493 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('n', S(1))*w_**WC('p', S(1)), x_), cons21, cons4, cons5, cons812, cons813)
def replacement1493(v, w, p, u, m, n, x):
rubi.append(1493)
return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**n*ExpandToSum(w, x)**p, x)
rule1493 = ReplacementRule(pattern1493, replacement1493)
pattern1494 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('n', S(1))*w_**WC('p', S(1))*z_**WC('q', S(1)), x_), cons21, cons4, cons5, cons50, cons814, cons815)
def replacement1494(v, z, w, p, u, m, n, x, q):
rubi.append(1494)
return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**n*ExpandToSum(w, x)**p*ExpandToSum(z, x)**q, x)
rule1494 = ReplacementRule(pattern1494, replacement1494)
pattern1495 = Pattern(Integral(u_**p_, x_), cons5, cons816, cons817)
def replacement1495(x, p, u):
rubi.append(1495)
return Int(ExpandToSum(u, x)**p, x)
rule1495 = ReplacementRule(pattern1495, replacement1495)
pattern1496 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('p', S(1)), x_), cons21, cons5, cons68, cons818, cons819)
def replacement1496(v, p, u, m, x):
rubi.append(1496)
return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**p, x)
rule1496 = ReplacementRule(pattern1496, replacement1496)
pattern1497 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('n', S(1))*w_**WC('p', S(1)), x_), cons21, cons4, cons5, cons810, cons820, cons821)
def replacement1497(v, w, p, u, m, n, x):
rubi.append(1497)
return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**n*ExpandToSum(w, x)**p, x)
rule1497 = ReplacementRule(pattern1497, replacement1497)
pattern1498 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1)), x_), cons5, cons50, cons452, cons822)
def replacement1498(v, p, u, x, q):
rubi.append(1498)
return Int(ExpandToSum(u, x)**p*ExpandToSum(v, x)**q, x)
rule1498 = ReplacementRule(pattern1498, replacement1498)
pattern1499 = Pattern(Integral(u_**p_, x_), cons5, cons823, cons824)
def replacement1499(x, p, u):
rubi.append(1499)
return Int(ExpandToSum(u, x)**p, x)
rule1499 = ReplacementRule(pattern1499, replacement1499)
pattern1500 = Pattern(Integral(u_**WC('p', S(1))*(x_*WC('c', S(1)))**WC('m', S(1)), x_), cons7, cons21, cons5, cons823, cons824)
def replacement1500(p, u, m, c, x):
rubi.append(1500)
return Int((c*x)**m*ExpandToSum(u, x)**p, x)
rule1500 = ReplacementRule(pattern1500, replacement1500)
pattern1501 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1)), x_), cons5, cons50, cons825, cons826, cons827)
def replacement1501(v, p, u, x, q):
rubi.append(1501)
return Int(ExpandToSum(u, x)**p*ExpandToSum(v, x)**q, x)
rule1501 = ReplacementRule(pattern1501, replacement1501)
pattern1502 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1))*x_**WC('m', S(1)), x_), cons21, cons5, cons50, cons825, cons826, cons827)
def replacement1502(v, p, u, m, x, q):
rubi.append(1502)
return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(v, x)**q, x)
rule1502 = ReplacementRule(pattern1502, replacement1502)
pattern1503 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('p', S(1))*w_**WC('q', S(1)), x_), cons21, cons5, cons50, cons828, cons826, cons829, cons830)
def replacement1503(v, w, p, u, m, x, q):
rubi.append(1503)
return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**p*ExpandToSum(w, x)**q, x)
rule1503 = ReplacementRule(pattern1503, replacement1503)
pattern1504 = Pattern(Integral(u_**WC('p', S(1))*v_**WC('q', S(1))*x_**WC('m', S(1))*z_**WC('r', S(1)), x_), cons21, cons5, cons50, cons52, cons831, cons826, cons832, cons833)
def replacement1504(v, z, p, u, m, r, x, q):
rubi.append(1504)
return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(v, x)**q*ExpandToSum(z, x)**r, x)
rule1504 = ReplacementRule(pattern1504, replacement1504)
pattern1505 = Pattern(Integral(u_**p_, x_), cons5, cons834, cons835)
def replacement1505(x, p, u):
rubi.append(1505)
return Int(ExpandToSum(u, x)**p, x)
rule1505 = ReplacementRule(pattern1505, replacement1505)
pattern1506 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1)), x_), cons21, cons5, cons834, cons835)
def replacement1506(x, m, p, u):
rubi.append(1506)
return Int(x**m*ExpandToSum(u, x)**p, x)
rule1506 = ReplacementRule(pattern1506, replacement1506)
pattern1507 = Pattern(Integral(u_**p_, x_), cons5, cons836, cons837)
def replacement1507(x, p, u):
rubi.append(1507)
return Int(ExpandToSum(u, x)**p, x)
rule1507 = ReplacementRule(pattern1507, replacement1507)
pattern1508 = Pattern(Integral(u_**WC('p', S(1))*(x_*WC('d', S(1)))**WC('m', S(1)), x_), cons27, cons21, cons5, cons836, cons837)
def replacement1508(p, u, m, d, x):
rubi.append(1508)
return Int((d*x)**m*ExpandToSum(u, x)**p, x)
rule1508 = ReplacementRule(pattern1508, replacement1508)
pattern1509 = Pattern(Integral(u_**WC('q', S(1))*v_**WC('p', S(1)), x_), cons5, cons50, cons823, cons838, cons839)
def replacement1509(v, p, u, x, q):
rubi.append(1509)
return Int(ExpandToSum(u, x)**q*ExpandToSum(v, x)**p, x)
rule1509 = ReplacementRule(pattern1509, replacement1509)
pattern1510 = Pattern(Integral(u_**WC('q', S(1))*v_**WC('p', S(1)), x_), cons5, cons50, cons823, cons840, cons841)
def replacement1510(v, p, u, x, q):
rubi.append(1510)
return Int(ExpandToSum(u, x)**q*ExpandToSum(v, x)**p, x)
rule1510 = ReplacementRule(pattern1510, replacement1510)
pattern1511 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1))*z_**WC('q', S(1)), x_), cons21, cons5, cons50, cons842, cons836, cons843)
def replacement1511(z, p, u, m, x, q):
rubi.append(1511)
return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(z, x)**q, x)
rule1511 = ReplacementRule(pattern1511, replacement1511)
pattern1512 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1))*z_**WC('q', S(1)), x_), cons21, cons5, cons50, cons842, cons823, cons844)
def replacement1512(z, p, u, m, x, q):
rubi.append(1512)
return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(z, x)**q, x)
rule1512 = ReplacementRule(pattern1512, replacement1512)
pattern1513 = Pattern(Integral(u_**p_, x_), cons5, cons845, cons846)
def replacement1513(x, p, u):
rubi.append(1513)
return Int(ExpandToSum(u, x)**p, x)
rule1513 = ReplacementRule(pattern1513, replacement1513)
pattern1514 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1)), x_), cons21, cons5, cons845, cons846)
def replacement1514(x, m, p, u):
rubi.append(1514)
return Int(x**m*ExpandToSum(u, x)**p, x)
rule1514 = ReplacementRule(pattern1514, replacement1514)
pattern1515 = Pattern(Integral(u_**WC('p', S(1))*z_, x_), cons5, cons842, cons845, cons847, cons848)
def replacement1515(x, z, p, u):
rubi.append(1515)
return Int(ExpandToSum(u, x)**p*ExpandToSum(z, x), x)
rule1515 = ReplacementRule(pattern1515, replacement1515)
pattern1516 = Pattern(Integral(u_**WC('p', S(1))*x_**WC('m', S(1))*z_, x_), cons21, cons5, cons842, cons845, cons847, cons848)
def replacement1516(z, p, u, m, x):
rubi.append(1516)
return Int(x**m*ExpandToSum(u, x)**p*ExpandToSum(z, x), x)
rule1516 = ReplacementRule(pattern1516, replacement1516)
pattern1517 = Pattern(Integral(x_**WC('m', S(1))*(e_ + x_**WC('n', S(1))*WC('h', S(1)) + x_**WC('q', S(1))*WC('f', S(1)) + x_**WC('r', S(1))*WC('g', S(1)))/(a_ + x_**WC('n', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons7, cons48, cons125, cons208, cons209, cons21, cons4, cons849, cons850, cons851, cons852)
def replacement1517(m, f, g, r, c, n, a, x, h, q, e):
rubi.append(1517)
return -Simp((S(2)*a*g + S(4)*a*h*x**(n/S(4)) - S(2)*c*f*x**(n/S(2)))/(a*c*n*sqrt(a + c*x**n)), x)
rule1517 = ReplacementRule(pattern1517, replacement1517)
pattern1518 = Pattern(Integral((d_*x_)**WC('m', S(1))*(e_ + x_**WC('n', S(1))*WC('h', S(1)) + x_**WC('q', S(1))*WC('f', S(1)) + x_**WC('r', S(1))*WC('g', S(1)))/(a_ + x_**WC('n', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons851, cons849, cons850, cons852)
def replacement1518(m, f, g, r, d, c, n, a, x, h, q, e):
rubi.append(1518)
return Dist(x**(-m)*(d*x)**m, Int(x**m*(e + f*x**(n/S(4)) + g*x**(S(3)*n/S(4)) + h*x**n)/(a + c*x**n)**(S(3)/2), x), x)
rule1518 = ReplacementRule(pattern1518, replacement1518)
def With1519(p, m, b, Pq, c, a, x):
n = Denominator(p)
rubi.append(1519)
return Dist(n/b, Subst(Int(x**(n*p + n + S(-1))*(-a*c/b + c*x**n/b)**m*ReplaceAll(Pq, Rule(x, -a/b + x**n/b)), x), x, (a + b*x)**(S(1)/n)), x)
pattern1519 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**m_*(a_ + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons64, cons149, cons853)
rule1519 = ReplacementRule(pattern1519, With1519)
pattern1520 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons21, cons4, cons5, cons66, cons854, cons855)
def replacement1520(p, m, b, Pq, a, n, x):
rubi.append(1520)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p*SubstFor(x**(m + S(1)), Pq, x), x), x, x**(m + S(1))), x)
rule1520 = ReplacementRule(pattern1520, replacement1520)
pattern1521 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons64, cons464, cons856)
def replacement1521(p, b, Pq, a, n, x):
rubi.append(1521)
return Int((a + b*x**n)**p*ExpandToSum(Pq - x**(n + S(-1))*Coeff(Pq, x, n + S(-1)), x), x) + Simp((a + b*x**n)**(p + S(1))*Coeff(Pq, x, n + S(-1))/(b*n*(p + S(1))), x)
rule1521 = ReplacementRule(pattern1521, replacement1521)
pattern1522 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons64, cons857)
def replacement1522(p, m, b, Pq, c, n, a, x):
rubi.append(1522)
return Int(ExpandIntegrand(Pq*(c*x)**m*(a + b*x**n)**p, x), x)
rule1522 = ReplacementRule(pattern1522, replacement1522)
pattern1523 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons4, cons64, cons857)
def replacement1523(p, b, Pq, a, n, x):
rubi.append(1523)
return Int(ExpandIntegrand(Pq*(a + b*x**n)**p, x), x)
rule1523 = ReplacementRule(pattern1523, replacement1523)
pattern1524 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons21, cons4, cons5, cons858, cons500)
def replacement1524(p, m, b, Pq, a, n, x):
rubi.append(1524)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x)**p*SubstFor(x**n, Pq, x), x), x, x**n), x)
rule1524 = ReplacementRule(pattern1524, replacement1524)
pattern1525 = Pattern(Integral(Pq_*(c_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons858, cons500)
def replacement1525(p, m, b, Pq, c, a, n, x):
rubi.append(1525)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(Pq*x**m*(a + b*x**n)**p, x), x)
rule1525 = ReplacementRule(pattern1525, replacement1525)
pattern1526 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons64, cons53, cons13, cons137)
def replacement1526(p, m, b, Pq, a, n, x):
rubi.append(1526)
return -Dist(S(1)/(b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*D(Pq, x), x), x) + Simp(Pq*(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule1526 = ReplacementRule(pattern1526, replacement1526)
pattern1527 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons27, cons21, cons4, cons5, cons64, cons859)
def replacement1527(p, m, b, Pq, d, a, n, x):
rubi.append(1527)
return Dist(S(1)/d, Int((d*x)**(m + S(1))*(a + b*x**n)**p*ExpandToSum(Pq/x, x), x), x)
rule1527 = ReplacementRule(pattern1527, replacement1527)
pattern1528 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons64, cons859, cons860)
def replacement1528(p, b, Pq, a, n, x):
rubi.append(1528)
return Int(x*(a + b*x**n)**p*ExpandToSum(Pq/x, x), x)
rule1528 = ReplacementRule(pattern1528, replacement1528)
def With1529(p, m, b, Pq, a, n, x):
u = IntHide(Pq*x**m, x)
rubi.append(1529)
return -Dist(b*n*p, Int(x**(m + n)*(a + b*x**n)**(p + S(-1))*ExpandToSum(u*x**(-m + S(-1)), x), x), x) + Simp(u*(a + b*x**n)**p, x)
pattern1529 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons64, cons148, cons244, cons163, cons861)
rule1529 = ReplacementRule(pattern1529, With1529)
def With1530(p, m, b, Pq, c, n, a, x):
q = Expon(Pq, x)
i = Symbol('i')
rubi.append(1530)
return Dist(a*n*p, Int((c*x)**m*(a + b*x**n)**(p + S(-1))*Sum_doit(x**i*Coeff(Pq, x, i)/(i + m + n*p + S(1)), List(i, S(0), q)), x), x) + Simp((c*x)**m*(a + b*x**n)**p*Sum_doit(x**(i + S(1))*Coeff(Pq, x, i)/(i + m + n*p + S(1)), List(i, S(0), q)), x)
pattern1530 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons64, cons521, cons13, cons163)
rule1530 = ReplacementRule(pattern1530, With1530)
def With1531(p, b, Pq, a, n, x):
q = Expon(Pq, x)
i = Symbol('i')
rubi.append(1531)
return Dist(a*n*p, Int((a + b*x**n)**(p + S(-1))*Sum_doit(x**i*Coeff(Pq, x, i)/(i + n*p + S(1)), List(i, S(0), q)), x), x) + Simp((a + b*x**n)**p*Sum_doit(x**(i + S(1))*Coeff(Pq, x, i)/(i + n*p + S(1)), List(i, S(0), q)), x)
pattern1531 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons64, cons521, cons13, cons163)
rule1531 = ReplacementRule(pattern1531, With1531)
def With1532(p, b, Pq, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
i = Symbol('i')
if Equal(q, n + S(-1)):
return True
return False
pattern1532 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons64, cons148, cons13, cons137, CustomConstraint(With1532))
def replacement1532(p, b, Pq, a, n, x):
q = Expon(Pq, x)
i = Symbol('i')
rubi.append(1532)
return Dist(S(1)/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*Sum_doit(x**i*(i + n*(p + S(1)) + S(1))*Coeff(Pq, x, i), List(i, S(0), q + S(-1))), x), x) + Simp((a + b*x**n)**(p + S(1))*(a*Coeff(Pq, x, q) - b*x*ExpandToSum(Pq - x**q*Coeff(Pq, x, q), x))/(a*b*n*(p + S(1))), x)
rule1532 = ReplacementRule(pattern1532, replacement1532)
pattern1533 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons64, cons148, cons13, cons137, cons862)
def replacement1533(p, b, Pq, a, n, x):
rubi.append(1533)
return Dist(S(1)/(a*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*ExpandToSum(Pq*n*(p + S(1)) + D(Pq*x, x), x), x), x) - Simp(Pq*x*(a + b*x**n)**(p + S(1))/(a*n*(p + S(1))), x)
rule1533 = ReplacementRule(pattern1533, replacement1533)
pattern1534 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)) + x_*WC('e', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons863)
def replacement1534(f, b, g, d, a, x, e):
rubi.append(1534)
return -Simp((S(2)*a*f + S(4)*a*g*x - S(2)*b*e*x**S(2))/(S(4)*a*b*sqrt(a + b*x**S(4))), x)
rule1534 = ReplacementRule(pattern1534, replacement1534)
pattern1535 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons27, cons125, cons208, cons863)
def replacement1535(f, b, g, d, a, x):
rubi.append(1535)
return -Simp((f + S(2)*g*x)/(S(2)*b*sqrt(a + b*x**S(4))), x)
rule1535 = ReplacementRule(pattern1535, replacement1535)
pattern1536 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_*WC('e', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons208, cons863)
def replacement1536(g, b, d, a, x, e):
rubi.append(1536)
return -Simp(x*(S(2)*a*g - b*e*x)/(S(2)*a*b*sqrt(a + b*x**S(4))), x)
rule1536 = ReplacementRule(pattern1536, replacement1536)
pattern1537 = Pattern(Integral(x_**S(2)*(x_**S(4)*WC('h', S(1)) + x_*WC('f', S(1)) + WC('e', S(0)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons48, cons125, cons209, cons864)
def replacement1537(f, b, a, x, h, e):
rubi.append(1537)
return -Simp((f - S(2)*h*x**S(3))/(S(2)*b*sqrt(a + b*x**S(4))), x)
rule1537 = ReplacementRule(pattern1537, replacement1537)
pattern1538 = Pattern(Integral(x_**S(2)*(x_**S(4)*WC('h', S(1)) + WC('e', S(0)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons48, cons209, cons864)
def replacement1538(b, a, x, h, e):
rubi.append(1538)
return Simp(h*x**S(3)/(b*sqrt(a + b*x**S(4))), x)
rule1538 = ReplacementRule(pattern1538, replacement1538)
pattern1539 = Pattern(Integral((d_ + x_**S(6)*WC('h', S(1)) + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons125, cons208, cons209, cons864, cons863)
def replacement1539(f, b, g, d, a, x, h, e):
rubi.append(1539)
return -Simp((a*f - S(2)*a*h*x**S(3) - S(2)*b*d*x)/(S(2)*a*b*sqrt(a + b*x**S(4))), x)
rule1539 = ReplacementRule(pattern1539, replacement1539)
pattern1540 = Pattern(Integral((d_ + x_**S(6)*WC('h', S(1)) + x_**S(4)*WC('g', S(1)) + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons27, cons48, cons208, cons209, cons864, cons863)
def replacement1540(g, b, d, a, x, h, e):
rubi.append(1540)
return Simp(x*(a*h*x**S(2) + b*d)/(a*b*sqrt(a + b*x**S(4))), x)
rule1540 = ReplacementRule(pattern1540, replacement1540)
def With1541(p, b, Pq, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Q = PolynomialQuotient(Pq*b**(Floor((q + S(-1))/n) + S(1)), a + b*x**n, x)
R = PolynomialRemainder(Pq*b**(Floor((q + S(-1))/n) + S(1)), a + b*x**n, x)
if GreaterEqual(q, n):
return True
return False
pattern1541 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons64, cons148, cons13, cons137, CustomConstraint(With1541))
def replacement1541(p, b, Pq, a, n, x):
q = Expon(Pq, x)
Q = PolynomialQuotient(Pq*b**(Floor((q + S(-1))/n) + S(1)), a + b*x**n, x)
R = PolynomialRemainder(Pq*b**(Floor((q + S(-1))/n) + S(1)), a + b*x**n, x)
rubi.append(1541)
return Dist(b**(-Floor((q - 1)/n) - 1)/(a*n*(p + 1)), Int((a + b*x**n)**(p + 1)*ExpandToSum(Q*a*n*(p + 1) + R*n*(p + 1) + D(R*x, x), x), x), x) - Simp(R*b**(-Floor((q - 1)/n) - 1)*x*(a + b*x**n)**(p + 1)/(a*n*(p + 1)), x)
rule1541 = ReplacementRule(pattern1541, replacement1541)
def With1542(p, m, b, Pq, a, n, x):
q = Expon(Pq, x)
Q = PolynomialQuotient(Pq*a*b**(Floor((q + S(-1))/n) + S(1))*x**m, a + b*x**n, x)
R = PolynomialRemainder(Pq*a*b**(Floor((q + S(-1))/n) + S(1))*x**m, a + b*x**n, x)
rubi.append(1542)
return Dist(b**(-Floor((q - 1)/n) - 1)/(a*n*(p + 1)), Int(x**m*(a + b*x**n)**(p + 1)*ExpandToSum(Q*n*x**(-m)*(p + 1) + Sum_doit(x**(i - m)*(i + n*(p + 1) + 1)*Coeff(R, x, i)/a, List(i, 0, n - 1)), x), x), x) - Simp(R*b**(-Floor((q - 1)/n) - 1)*x*(a + b*x**n)**(p + 1)/(a**2*n*(p + 1)), x)
pattern1542 = Pattern(Integral(Pq_*x_**m_*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons64, cons148, cons13, cons137, cons84)
rule1542 = ReplacementRule(pattern1542, With1542)
def With1543(p, m, b, Pq, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
g = GCD(m + S(1), n)
if Unequal(g, S(1)):
return True
return False
pattern1543 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons858, cons148, cons17, CustomConstraint(With1543))
def replacement1543(p, m, b, Pq, a, n, x):
g = GCD(m + S(1), n)
rubi.append(1543)
return Dist(S(1)/g, Subst(Int(x**(S(-1) + (m + S(1))/g)*(a + b*x**(n/g))**p*ReplaceAll(Pq, Rule(x, x**(S(1)/g))), x), x, x**g), x)
rule1543 = ReplacementRule(pattern1543, replacement1543)
pattern1544 = Pattern(Integral((A_ + x_*WC('B', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons865)
def replacement1544(B, b, a, x, A):
rubi.append(1544)
return Dist(B**S(3)/b, Int(S(1)/(A**S(2) - A*B*x + B**S(2)*x**S(2)), x), x)
rule1544 = ReplacementRule(pattern1544, replacement1544)
def With1545(B, b, a, x, A):
r = Numerator(Rt(a/b, S(3)))
s = Denominator(Rt(a/b, S(3)))
rubi.append(1545)
return Dist(r/(S(3)*a*s), Int((r*(S(2)*A*s + B*r) + s*x*(-A*s + B*r))/(r**S(2) - r*s*x + s**S(2)*x**S(2)), x), x) - Dist(r*(-A*s + B*r)/(S(3)*a*s), Int(S(1)/(r + s*x), x), x)
pattern1545 = Pattern(Integral((A_ + x_*WC('B', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons866, cons468)
rule1545 = ReplacementRule(pattern1545, With1545)
def With1546(B, b, a, x, A):
r = Numerator(Rt(-a/b, S(3)))
s = Denominator(Rt(-a/b, S(3)))
rubi.append(1546)
return -Dist(r/(S(3)*a*s), Int((r*(-S(2)*A*s + B*r) - s*x*(A*s + B*r))/(r**S(2) + r*s*x + s**S(2)*x**S(2)), x), x) + Dist(r*(A*s + B*r)/(S(3)*a*s), Int(S(1)/(r - s*x), x), x)
pattern1546 = Pattern(Integral((A_ + x_*WC('B', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons866, cons469)
rule1546 = ReplacementRule(pattern1546, With1546)
pattern1547 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons867, cons868)
def replacement1547(B, C, b, a, x, A):
rubi.append(1547)
return -Dist(C**S(2)/b, Int(S(1)/(B - C*x), x), x)
rule1547 = ReplacementRule(pattern1547, replacement1547)
def With1548(B, C, b, a, x, A):
q = a**(S(1)/3)/b**(S(1)/3)
rubi.append(1548)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1548 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons869)
rule1548 = ReplacementRule(pattern1548, With1548)
def With1549(B, C, b, a, x):
q = a**(S(1)/3)/b**(S(1)/3)
rubi.append(1549)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1549 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons870)
rule1549 = ReplacementRule(pattern1549, With1549)
def With1550(C, b, a, x, A):
q = a**(S(1)/3)/b**(S(1)/3)
rubi.append(1550)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist(C*q/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1550 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons871)
rule1550 = ReplacementRule(pattern1550, With1550)
def With1551(B, C, b, a, x, A):
q = (-a)**(S(1)/3)/(-b)**(S(1)/3)
rubi.append(1551)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1551 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons872)
rule1551 = ReplacementRule(pattern1551, With1551)
def With1552(B, C, b, a, x):
q = (-a)**(S(1)/3)/(-b)**(S(1)/3)
rubi.append(1552)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1552 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons873)
rule1552 = ReplacementRule(pattern1552, With1552)
def With1553(C, b, a, x, A):
q = (-a)**(S(1)/3)/(-b)**(S(1)/3)
rubi.append(1553)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist(C*q/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1553 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons874)
rule1553 = ReplacementRule(pattern1553, With1553)
def With1554(B, C, b, a, x, A):
q = (-a)**(S(1)/3)/b**(S(1)/3)
rubi.append(1554)
return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1554 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons875)
rule1554 = ReplacementRule(pattern1554, With1554)
def With1555(B, C, b, a, x):
q = (-a)**(S(1)/3)/b**(S(1)/3)
rubi.append(1555)
return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1555 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons876)
rule1555 = ReplacementRule(pattern1555, With1555)
def With1556(C, b, a, x, A):
q = (-a)**(S(1)/3)/b**(S(1)/3)
rubi.append(1556)
return -Dist(C/b, Int(S(1)/(q - x), x), x) - Dist(C*q/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1556 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons877)
rule1556 = ReplacementRule(pattern1556, With1556)
def With1557(B, C, b, a, x, A):
q = a**(S(1)/3)/(-b)**(S(1)/3)
rubi.append(1557)
return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1557 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons878)
rule1557 = ReplacementRule(pattern1557, With1557)
def With1558(B, C, b, a, x):
q = a**(S(1)/3)/(-b)**(S(1)/3)
rubi.append(1558)
return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1558 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons879)
rule1558 = ReplacementRule(pattern1558, With1558)
def With1559(C, b, a, x, A):
q = a**(S(1)/3)/(-b)**(S(1)/3)
rubi.append(1559)
return -Dist(C/b, Int(S(1)/(q - x), x), x) - Dist(C*q/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1559 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons880)
rule1559 = ReplacementRule(pattern1559, With1559)
def With1560(B, C, b, a, x, A):
q = (a/b)**(S(1)/3)
rubi.append(1560)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1560 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons881)
rule1560 = ReplacementRule(pattern1560, With1560)
def With1561(B, C, b, a, x):
q = (a/b)**(S(1)/3)
rubi.append(1561)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1561 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons882)
rule1561 = ReplacementRule(pattern1561, With1561)
def With1562(C, b, a, x, A):
q = (a/b)**(S(1)/3)
rubi.append(1562)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist(C*q/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1562 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons883)
rule1562 = ReplacementRule(pattern1562, With1562)
def With1563(B, C, b, a, x, A):
q = Rt(a/b, S(3))
rubi.append(1563)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1563 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons884)
rule1563 = ReplacementRule(pattern1563, With1563)
def With1564(B, C, b, a, x):
q = Rt(a/b, S(3))
rubi.append(1564)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist((B + C*q)/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1564 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons885)
rule1564 = ReplacementRule(pattern1564, With1564)
def With1565(C, b, a, x, A):
q = Rt(a/b, S(3))
rubi.append(1565)
return Dist(C/b, Int(S(1)/(q + x), x), x) + Dist(C*q/b, Int(S(1)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1565 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons886)
rule1565 = ReplacementRule(pattern1565, With1565)
def With1566(B, C, b, a, x, A):
q = (-a/b)**(S(1)/3)
rubi.append(1566)
return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1566 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons887)
rule1566 = ReplacementRule(pattern1566, With1566)
def With1567(B, C, b, a, x):
q = (-a/b)**(S(1)/3)
rubi.append(1567)
return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1567 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons888)
rule1567 = ReplacementRule(pattern1567, With1567)
def With1568(C, b, a, x, A):
q = (-a/b)**(S(1)/3)
rubi.append(1568)
return -Dist(C/b, Int(S(1)/(q - x), x), x) - Dist(C*q/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1568 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons889)
rule1568 = ReplacementRule(pattern1568, With1568)
def With1569(B, C, b, a, x, A):
q = Rt(-a/b, S(3))
rubi.append(1569)
return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1569 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons890)
rule1569 = ReplacementRule(pattern1569, With1569)
def With1570(B, C, b, a, x):
q = Rt(-a/b, S(3))
rubi.append(1570)
return -Dist(C/b, Int(S(1)/(q - x), x), x) + Dist((B - C*q)/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1570 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons891)
rule1570 = ReplacementRule(pattern1570, With1570)
def With1571(C, b, a, x, A):
q = Rt(-a/b, S(3))
rubi.append(1571)
return -Dist(C/b, Int(S(1)/(q - x), x), x) - Dist(C*q/b, Int(S(1)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1571 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons892)
rule1571 = ReplacementRule(pattern1571, With1571)
pattern1572 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons893)
def replacement1572(B, C, b, a, x, A):
rubi.append(1572)
return Dist(C, Int(x**S(2)/(a + b*x**S(3)), x), x) + Int((A + B*x)/(a + b*x**S(3)), x)
rule1572 = ReplacementRule(pattern1572, replacement1572)
pattern1573 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons894)
def replacement1573(B, C, b, a, x):
rubi.append(1573)
return Dist(B, Int(x/(a + b*x**S(3)), x), x) + Dist(C, Int(x**S(2)/(a + b*x**S(3)), x), x)
rule1573 = ReplacementRule(pattern1573, replacement1573)
pattern1574 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons895)
def replacement1574(C, b, a, x, A):
rubi.append(1574)
return Dist(A, Int(S(1)/(a + b*x**S(3)), x), x) + Dist(C, Int(x**S(2)/(a + b*x**S(3)), x), x)
rule1574 = ReplacementRule(pattern1574, replacement1574)
def With1575(B, C, b, a, x, A):
q = (a/b)**(S(1)/3)
rubi.append(1575)
return Dist(q**S(2)/a, Int((A + C*q*x)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1575 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons896)
rule1575 = ReplacementRule(pattern1575, With1575)
def With1576(B, C, b, a, x):
q = (a/b)**(S(1)/3)
rubi.append(1576)
return Dist(C*q**S(3)/a, Int(x/(q**S(2) - q*x + x**S(2)), x), x)
pattern1576 = Pattern(Integral(x_*(x_*WC('C', S(1)) + WC('B', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons897)
rule1576 = ReplacementRule(pattern1576, With1576)
def With1577(C, b, a, x, A):
q = (a/b)**(S(1)/3)
rubi.append(1577)
return Dist(q**S(2)/a, Int((A + C*q*x)/(q**S(2) - q*x + x**S(2)), x), x)
pattern1577 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons898)
rule1577 = ReplacementRule(pattern1577, With1577)
def With1578(B, C, b, a, x, A):
q = (-a/b)**(S(1)/3)
rubi.append(1578)
return Dist(q/a, Int((A*q + x*(A + B*q))/(q**S(2) + q*x + x**S(2)), x), x)
pattern1578 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons899)
rule1578 = ReplacementRule(pattern1578, With1578)
def With1579(B, C, b, a, x):
q = (-a/b)**(S(1)/3)
rubi.append(1579)
return Dist(B*q**S(2)/a, Int(x/(q**S(2) + q*x + x**S(2)), x), x)
pattern1579 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons900)
rule1579 = ReplacementRule(pattern1579, With1579)
def With1580(C, b, a, x, A):
q = (-a/b)**(S(1)/3)
rubi.append(1580)
return Dist(A*q/a, Int((q + x)/(q**S(2) + q*x + x**S(2)), x), x)
pattern1580 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons901)
rule1580 = ReplacementRule(pattern1580, With1580)
def With1581(B, C, b, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
q = (a/b)**(S(1)/3)
if NonzeroQ(A - B*q + C*q**S(2)):
return True
return False
pattern1581 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons866, cons902, cons903, CustomConstraint(With1581))
def replacement1581(B, C, b, a, x, A):
q = (a/b)**(S(1)/3)
rubi.append(1581)
return Dist(q/(S(3)*a), Int((q*(S(2)*A + B*q - C*q**S(2)) - x*(A - B*q - S(2)*C*q**S(2)))/(q**S(2) - q*x + x**S(2)), x), x) + Dist(q*(A - B*q + C*q**S(2))/(S(3)*a), Int(S(1)/(q + x), x), x)
rule1581 = ReplacementRule(pattern1581, replacement1581)
def With1582(B, C, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = (a/b)**(S(1)/3)
if NonzeroQ(B*q - C*q**S(2)):
return True
return False
pattern1582 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons902, cons903, CustomConstraint(With1582))
def replacement1582(B, C, b, a, x):
q = (a/b)**(S(1)/3)
rubi.append(1582)
return Dist(q/(S(3)*a), Int((q*(B*q - C*q**S(2)) + x*(B*q + S(2)*C*q**S(2)))/(q**S(2) - q*x + x**S(2)), x), x) - Dist(q*(B*q - C*q**S(2))/(S(3)*a), Int(S(1)/(q + x), x), x)
rule1582 = ReplacementRule(pattern1582, replacement1582)
def With1583(C, b, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
q = (a/b)**(S(1)/3)
if NonzeroQ(A + C*q**S(2)):
return True
return False
pattern1583 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons902, cons903, CustomConstraint(With1583))
def replacement1583(C, b, a, x, A):
q = (a/b)**(S(1)/3)
rubi.append(1583)
return Dist(q/(S(3)*a), Int((q*(S(2)*A - C*q**S(2)) - x*(A - S(2)*C*q**S(2)))/(q**S(2) - q*x + x**S(2)), x), x) + Dist(q*(A + C*q**S(2))/(S(3)*a), Int(S(1)/(q + x), x), x)
rule1583 = ReplacementRule(pattern1583, replacement1583)
def With1584(B, C, b, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
q = (-a/b)**(S(1)/3)
if NonzeroQ(A + B*q + C*q**S(2)):
return True
return False
pattern1584 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons866, cons902, cons904, CustomConstraint(With1584))
def replacement1584(B, C, b, a, x, A):
q = (-a/b)**(S(1)/3)
rubi.append(1584)
return Dist(q/(S(3)*a), Int((q*(S(2)*A - B*q - C*q**S(2)) + x*(A + B*q - S(2)*C*q**S(2)))/(q**S(2) + q*x + x**S(2)), x), x) + Dist(q*(A + B*q + C*q**S(2))/(S(3)*a), Int(S(1)/(q - x), x), x)
rule1584 = ReplacementRule(pattern1584, replacement1584)
def With1585(B, C, b, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = (-a/b)**(S(1)/3)
if NonzeroQ(B*q + C*q**S(2)):
return True
return False
pattern1585 = Pattern(Integral(x_*(B_ + x_*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons35, cons36, cons902, cons904, CustomConstraint(With1585))
def replacement1585(B, C, b, a, x):
q = (-a/b)**(S(1)/3)
rubi.append(1585)
return Dist(q/(S(3)*a), Int((-q*(B*q + C*q**S(2)) + x*(B*q - S(2)*C*q**S(2)))/(q**S(2) + q*x + x**S(2)), x), x) + Dist(q*(B*q + C*q**S(2))/(S(3)*a), Int(S(1)/(q - x), x), x)
rule1585 = ReplacementRule(pattern1585, replacement1585)
def With1586(C, b, a, x, A):
if isinstance(x, (int, Integer, float, Float)):
return False
q = (-a/b)**(S(1)/3)
if NonzeroQ(A + C*q**S(2)):
return True
return False
pattern1586 = Pattern(Integral((A_ + x_**S(2)*WC('C', S(1)))/(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons34, cons36, cons902, cons904, CustomConstraint(With1586))
def replacement1586(C, b, a, x, A):
q = (-a/b)**(S(1)/3)
rubi.append(1586)
return Dist(q/(S(3)*a), Int((q*(S(2)*A - C*q**S(2)) + x*(A - S(2)*C*q**S(2)))/(q**S(2) + q*x + x**S(2)), x), x) + Dist(q*(A + C*q**S(2))/(S(3)*a), Int(S(1)/(q - x), x), x)
rule1586 = ReplacementRule(pattern1586, replacement1586)
def With1587(m, b, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
v = Sum_doit(c**(-ii)*(c*x)**(ii + m)*(x**(n/S(2))*Coeff(Pq, x, ii + n/S(2)) + Coeff(Pq, x, ii))/(a + b*x**n), List(ii, S(0), n/S(2) + S(-1)))
if SumQ(v):
return True
return False
pattern1587 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons21, cons64, cons674, cons905, CustomConstraint(With1587))
def replacement1587(m, b, Pq, c, a, n, x):
v = Sum_doit(c**(-ii)*(c*x)**(ii + m)*(x**(n/S(2))*Coeff(Pq, x, ii + n/S(2)) + Coeff(Pq, x, ii))/(a + b*x**n), List(ii, S(0), n/S(2) + S(-1)))
rubi.append(1587)
return Int(v, x)
rule1587 = ReplacementRule(pattern1587, replacement1587)
def With1588(b, Pq, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
v = Sum_doit(x**ii*(x**(n/S(2))*Coeff(Pq, x, ii + n/S(2)) + Coeff(Pq, x, ii))/(a + b*x**n), List(ii, S(0), n/S(2) + S(-1)))
if SumQ(v):
return True
return False
pattern1588 = Pattern(Integral(Pq_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons64, cons674, cons905, CustomConstraint(With1588))
def replacement1588(b, Pq, a, n, x):
v = Sum_doit(x**ii*(x**(n/S(2))*Coeff(Pq, x, ii + n/S(2)) + Coeff(Pq, x, ii))/(a + b*x**n), List(ii, S(0), n/S(2) + S(-1)))
rubi.append(1588)
return Int(v, x)
rule1588 = ReplacementRule(pattern1588, replacement1588)
def With1589(b, d, c, a, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(1589)
return Simp(S(2)*d*s**S(3)*sqrt(a + b*x**S(3))/(a*r**S(2)*(r*x + s*(S(1) + sqrt(S(3))))), x) - Simp(S(3)**(S(1)/4)*d*s*sqrt((r**S(2)*x**S(2) - r*s*x + s**S(2))/(r*x + s*(S(1) + sqrt(S(3))))**S(2))*sqrt(-sqrt(S(3)) + S(2))*(r*x + s)*EllipticE(asin((r*x + s*(-sqrt(S(3)) + S(1)))/(r*x + s*(S(1) + sqrt(S(3))))), S(-7) - S(4)*sqrt(S(3)))/(r**S(2)*sqrt(s*(r*x + s)/(r*x + s*(S(1) + sqrt(S(3))))**S(2))*sqrt(a + b*x**S(3))), x)
pattern1589 = Pattern(Integral((c_ + x_*WC('d', S(1)))/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons481, cons906)
rule1589 = ReplacementRule(pattern1589, With1589)
def With1590(b, d, c, a, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(1590)
return Dist(d/r, Int((r*x + s*(-sqrt(S(3)) + S(1)))/sqrt(a + b*x**S(3)), x), x) + Dist((c*r - d*s*(-sqrt(S(3)) + S(1)))/r, Int(S(1)/sqrt(a + b*x**S(3)), x), x)
pattern1590 = Pattern(Integral((c_ + x_*WC('d', S(1)))/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons481, cons907)
rule1590 = ReplacementRule(pattern1590, With1590)
def With1591(b, d, c, a, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(1591)
return Simp(S(2)*d*s**S(3)*sqrt(a + b*x**S(3))/(a*r**S(2)*(r*x + s*(-sqrt(S(3)) + S(1)))), x) + Simp(S(3)**(S(1)/4)*d*s*sqrt((r**S(2)*x**S(2) - r*s*x + s**S(2))/(r*x + s*(-sqrt(S(3)) + S(1)))**S(2))*sqrt(sqrt(S(3)) + S(2))*(r*x + s)*EllipticE(asin((r*x + s*(S(1) + sqrt(S(3))))/(r*x + s*(-sqrt(S(3)) + S(1)))), S(-7) + S(4)*sqrt(S(3)))/(r**S(2)*sqrt(-s*(r*x + s)/(r*x + s*(-sqrt(S(3)) + S(1)))**S(2))*sqrt(a + b*x**S(3))), x)
pattern1591 = Pattern(Integral((c_ + x_*WC('d', S(1)))/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons482, cons908)
rule1591 = ReplacementRule(pattern1591, With1591)
def With1592(b, d, c, a, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(1592)
return Dist(d/r, Int((r*x + s*(S(1) + sqrt(S(3))))/sqrt(a + b*x**S(3)), x), x) + Dist((c*r - d*s*(S(1) + sqrt(S(3))))/r, Int(S(1)/sqrt(a + b*x**S(3)), x), x)
pattern1592 = Pattern(Integral((c_ + x_*WC('d', S(1)))/sqrt(a_ + x_**S(3)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons482, cons909)
rule1592 = ReplacementRule(pattern1592, With1592)
def With1593(b, d, c, a, x):
r = Numer(Rt(b/a, S(3)))
s = Denom(Rt(b/a, S(3)))
rubi.append(1593)
return Simp(d*s**S(3)*x*(S(1) + sqrt(S(3)))*sqrt(a + b*x**S(6))/(S(2)*a*r**S(2)*(r*x**S(2)*(S(1) + sqrt(S(3))) + s)), x) - Simp(S(3)**(S(1)/4)*d*s*x*sqrt((r**S(2)*x**S(4) - r*s*x**S(2) + s**S(2))/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)**S(2))*(r*x**S(2) + s)*EllipticE(acos((r*x**S(2)*(-sqrt(S(3)) + S(1)) + s)/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)), sqrt(S(3))/S(4) + S(1)/2)/(S(2)*r**S(2)*sqrt(r*x**S(2)*(r*x**S(2) + s)/(r*x**S(2)*(S(1) + sqrt(S(3))) + s)**S(2))*sqrt(a + b*x**S(6))), x)
pattern1593 = Pattern(Integral((c_ + x_**S(4)*WC('d', S(1)))/sqrt(a_ + x_**S(6)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons910)
rule1593 = ReplacementRule(pattern1593, With1593)
def With1594(b, d, c, a, x):
q = Rt(b/a, S(3))
rubi.append(1594)
return Dist(d/(S(2)*q**S(2)), Int((S(2)*q**S(2)*x**S(4) - sqrt(S(3)) + S(1))/sqrt(a + b*x**S(6)), x), x) + Dist((S(2)*c*q**S(2) - d*(-sqrt(S(3)) + S(1)))/(S(2)*q**S(2)), Int(S(1)/sqrt(a + b*x**S(6)), x), x)
pattern1594 = Pattern(Integral((c_ + x_**S(4)*WC('d', S(1)))/sqrt(a_ + x_**S(6)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons911)
rule1594 = ReplacementRule(pattern1594, With1594)
pattern1595 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))/sqrt(a_ + x_**S(8)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons912)
def replacement1595(b, d, c, a, x):
rubi.append(1595)
return -Simp(c*d*x**S(3)*sqrt(-(c - d*x**S(2))**S(2)/(c*d*x**S(2)))*sqrt(-d**S(2)*(a + b*x**S(8))/(b*c**S(2)*x**S(4)))*EllipticF(asin(sqrt((sqrt(S(2))*c**S(2) + S(2)*c*d*x**S(2) + sqrt(S(2))*d**S(2)*x**S(4))/(c*d*x**S(2)))/S(2)), S(-2) + S(2)*sqrt(S(2)))/(sqrt(sqrt(S(2)) + S(2))*sqrt(a + b*x**S(8))*(c - d*x**S(2))), x)
rule1595 = ReplacementRule(pattern1595, replacement1595)
pattern1596 = Pattern(Integral((c_ + x_**S(2)*WC('d', S(1)))/sqrt(a_ + x_**S(8)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons913)
def replacement1596(b, d, c, a, x):
rubi.append(1596)
return -Dist((-c*Rt(b/a, S(4)) + d)/(S(2)*Rt(b/a, S(4))), Int((-x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x) + Dist((c*Rt(b/a, S(4)) + d)/(S(2)*Rt(b/a, S(4))), Int((x**S(2)*Rt(b/a, S(4)) + S(1))/sqrt(a + b*x**S(8)), x), x)
rule1596 = ReplacementRule(pattern1596, replacement1596)
pattern1597 = Pattern(Integral(Pq_/(x_*sqrt(a_ + x_**n_*WC('b', S(1)))), x_), cons2, cons3, cons64, cons148, cons914)
def replacement1597(b, Pq, a, n, x):
rubi.append(1597)
return Dist(Coeff(Pq, x, S(0)), Int(S(1)/(x*sqrt(a + b*x**n)), x), x) + Int(ExpandToSum((Pq - Coeff(Pq, x, S(0)))/x, x)/sqrt(a + b*x**n), x)
rule1597 = ReplacementRule(pattern1597, replacement1597)
def With1598(p, m, b, Pq, c, a, n, x):
q = Expon(Pq, x)
j = Symbol('j')
k = Symbol('k')
rubi.append(1598)
return Int(Sum_doit(c**(-j)*(c*x)**(j + m)*(a + b*x**n)**p*Sum_doit(x**(k*n/S(2))*Coeff(Pq, x, j + k*n/S(2)), List(k, S(0), S(1) + S(2)*(-j + q)/n)), List(j, S(0), n/S(2) + S(-1))), x)
pattern1598 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons64, cons674, cons915)
rule1598 = ReplacementRule(pattern1598, With1598)
def With1599(p, b, Pq, a, n, x):
q = Expon(Pq, x)
j = Symbol('j')
k = Symbol('k')
rubi.append(1599)
return Int(Sum_doit(x**j*(a + b*x**n)**p*Sum_doit(x**(k*n/S(2))*Coeff(Pq, x, j + k*n/S(2)), List(k, S(0), S(1) + S(2)*(-j + q)/n)), List(j, S(0), n/S(2) + S(-1))), x)
pattern1599 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons64, cons674, cons915)
rule1599 = ReplacementRule(pattern1599, With1599)
pattern1600 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons64, cons148, cons916)
def replacement1600(p, b, Pq, a, n, x):
rubi.append(1600)
return Dist(Coeff(Pq, x, n + S(-1)), Int(x**(n + S(-1))*(a + b*x**n)**p, x), x) + Int((a + b*x**n)**p*ExpandToSum(Pq - x**(n + S(-1))*Coeff(Pq, x, n + S(-1)), x), x)
rule1600 = ReplacementRule(pattern1600, replacement1600)
pattern1601 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons7, cons21, cons64, cons85)
def replacement1601(m, b, Pq, c, a, n, x):
rubi.append(1601)
return Int(ExpandIntegrand(Pq*(c*x)**m/(a + b*x**n), x), x)
rule1601 = ReplacementRule(pattern1601, replacement1601)
pattern1602 = Pattern(Integral(Pq_/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons64, cons85)
def replacement1602(b, Pq, a, n, x):
rubi.append(1602)
return Int(ExpandIntegrand(Pq/(a + b*x**n), x), x)
rule1602 = ReplacementRule(pattern1602, replacement1602)
def With1603(p, m, b, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
Pq0 = Coeff(Pq, x, S(0))
if NonzeroQ(Pq0):
return True
return False
pattern1603 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons64, cons148, cons31, cons94, cons917, CustomConstraint(With1603))
def replacement1603(p, m, b, Pq, c, a, n, x):
Pq0 = Coeff(Pq, x, S(0))
rubi.append(1603)
return Dist(S(1)/(S(2)*a*c*(m + S(1))), Int((c*x)**(m + S(1))*(a + b*x**n)**p*ExpandToSum(-S(2)*Pq0*b*x**(n + S(-1))*(m + n*(p + S(1)) + S(1)) + S(2)*a*(Pq - Pq0)*(m + S(1))/x, x), x), x) + Simp(Pq0*(c*x)**(m + S(1))*(a + b*x**n)**(p + S(1))/(a*c*(m + S(1))), x)
rule1603 = ReplacementRule(pattern1603, replacement1603)
def With1604(p, m, b, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
if And(NonzeroQ(m + n*p + q + S(1)), GreaterEqual(-n + q, S(0)), Or(IntegerQ(S(2)*p), IntegerQ(p + (q + S(1))/(S(2)*n)))):
return True
return False
pattern1604 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons64, cons148, CustomConstraint(With1604))
def replacement1604(p, m, b, Pq, c, a, n, x):
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
rubi.append(1604)
return Dist(1/(b*(m + n*p + q + 1)), Int((c*x)**m*(a + b*x**n)**p*ExpandToSum(-Pqq*a*x**(-n + q)*(m - n + q + 1) + b*(Pq - Pqq*x**q)*(m + n*p + q + 1), x), x), x) + Simp(Pqq*c**(n - q - 1)*(c*x)**(m - n + q + 1)*(a + b*x**n)**(p + 1)/(b*(m + n*p + q + 1)), x)
rule1604 = ReplacementRule(pattern1604, replacement1604)
def With1605(p, b, Pq, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
if And(NonzeroQ(n*p + q + S(1)), GreaterEqual(-n + q, S(0)), Or(IntegerQ(S(2)*p), IntegerQ(p + (q + S(1))/(S(2)*n)))):
return True
return False
pattern1605 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons64, cons148, CustomConstraint(With1605))
def replacement1605(p, b, Pq, a, n, x):
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
rubi.append(1605)
return Dist(1/(b*(n*p + q + 1)), Int((a + b*x**n)**p*ExpandToSum(-Pqq*a*x**(-n + q)*(-n + q + 1) + b*(Pq - Pqq*x**q)*(n*p + q + 1), x), x), x) + Simp(Pqq*x**(-n + q + 1)*(a + b*x**n)**(p + 1)/(b*(n*p + q + 1)), x)
rule1605 = ReplacementRule(pattern1605, replacement1605)
def With1606(p, m, b, Pq, a, n, x):
q = Expon(Pq, x)
rubi.append(1606)
return -Subst(Int(x**(-m - q + S(-2))*(a + b*x**(-n))**p*ExpandToSum(x**q*ReplaceAll(Pq, Rule(x, S(1)/x)), x), x), x, S(1)/x)
pattern1606 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons64, cons196, cons17)
rule1606 = ReplacementRule(pattern1606, With1606)
def With1607(p, m, b, Pq, c, a, n, x):
g = Denominator(m)
q = Expon(Pq, x)
rubi.append(1607)
return -Dist(g/c, Subst(Int(x**(-g*(m + q + S(1)) + S(-1))*(a + b*c**(-n)*x**(-g*n))**p*ExpandToSum(x**(g*q)*ReplaceAll(Pq, Rule(x, x**(-g)/c)), x), x), x, (c*x)**(-S(1)/g)), x)
pattern1607 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons64, cons196, cons367)
rule1607 = ReplacementRule(pattern1607, With1607)
def With1608(p, m, b, Pq, c, a, n, x):
q = Expon(Pq, x)
rubi.append(1608)
return -Dist((c*x)**m*(S(1)/x)**m, Subst(Int(x**(-m - q + S(-2))*(a + b*x**(-n))**p*ExpandToSum(x**q*ReplaceAll(Pq, Rule(x, S(1)/x)), x), x), x, S(1)/x), x)
pattern1608 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons64, cons196, cons356)
rule1608 = ReplacementRule(pattern1608, With1608)
def With1609(p, m, b, Pq, a, n, x):
g = Denominator(n)
rubi.append(1609)
return Dist(g, Subst(Int(x**(g*(m + S(1)) + S(-1))*(a + b*x**(g*n))**p*ReplaceAll(Pq, Rule(x, x**g)), x), x, x**(S(1)/g)), x)
pattern1609 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons5, cons64, cons489)
rule1609 = ReplacementRule(pattern1609, With1609)
def With1610(p, b, Pq, a, n, x):
g = Denominator(n)
rubi.append(1610)
return Dist(g, Subst(Int(x**(g + S(-1))*(a + b*x**(g*n))**p*ReplaceAll(Pq, Rule(x, x**g)), x), x, x**(S(1)/g)), x)
pattern1610 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons5, cons64, cons489)
rule1610 = ReplacementRule(pattern1610, With1610)
pattern1611 = Pattern(Integral(Pq_*(c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons64, cons489)
def replacement1611(p, m, b, Pq, c, a, n, x):
rubi.append(1611)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(Pq*x**m*(a + b*x**n)**p, x), x)
rule1611 = ReplacementRule(pattern1611, replacement1611)
pattern1612 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons5, cons858, cons541, cons23)
def replacement1612(p, m, b, Pq, a, n, x):
rubi.append(1612)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))))**p*ReplaceAll(SubstFor(x**n, Pq, x), Rule(x, x**(n/(m + S(1))))), x), x, x**(m + S(1))), x)
rule1612 = ReplacementRule(pattern1612, replacement1612)
pattern1613 = Pattern(Integral(Pq_*(c_*x_)**m_*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons858, cons541, cons23)
def replacement1613(p, m, b, Pq, c, a, n, x):
rubi.append(1613)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(Pq*x**m*(a + b*x**n)**p, x), x)
rule1613 = ReplacementRule(pattern1613, replacement1613)
pattern1614 = Pattern(Integral((A_ + x_**WC('m', S(1))*WC('B', S(1)))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons34, cons35, cons21, cons4, cons5, cons53)
def replacement1614(B, p, m, b, a, n, x, A):
rubi.append(1614)
return Dist(A, Int((a + b*x**n)**p, x), x) + Dist(B, Int(x**m*(a + b*x**n)**p, x), x)
rule1614 = ReplacementRule(pattern1614, replacement1614)
pattern1615 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons918)
def replacement1615(p, m, b, Pq, c, a, n, x):
rubi.append(1615)
return Int(ExpandIntegrand(Pq*(c*x)**m*(a + b*x**n)**p, x), x)
rule1615 = ReplacementRule(pattern1615, replacement1615)
pattern1616 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons4, cons5, cons918)
def replacement1616(p, b, Pq, a, n, x):
rubi.append(1616)
return Int(ExpandIntegrand(Pq*(a + b*x**n)**p, x), x)
rule1616 = ReplacementRule(pattern1616, replacement1616)
pattern1617 = Pattern(Integral(Pq_*u_**WC('m', S(1))*(a_ + v_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons5, cons554, cons919)
def replacement1617(v, p, u, m, b, Pq, a, n, x):
rubi.append(1617)
return Dist(u**m*v**(-m)/Coeff(v, x, S(1)), Subst(Int(x**m*(a + b*x**n)**p*SubstFor(v, Pq, x), x), x, v), x)
rule1617 = ReplacementRule(pattern1617, replacement1617)
pattern1618 = Pattern(Integral(Pq_*(a_ + v_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons4, cons5, cons552, cons919)
def replacement1618(v, p, b, Pq, a, n, x):
rubi.append(1618)
return Dist(S(1)/Coeff(v, x, S(1)), Subst(Int((a + b*x**n)**p*SubstFor(v, Pq, x), x), x, v), x)
rule1618 = ReplacementRule(pattern1618, replacement1618)
pattern1619 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons5, cons64, cons55, cons494)
def replacement1619(p, a2, m, b2, b1, Pq, c, n, x, a1):
rubi.append(1619)
return Int(Pq*(c*x)**m*(a1*a2 + b1*b2*x**(S(2)*n))**p, x)
rule1619 = ReplacementRule(pattern1619, replacement1619)
pattern1620 = Pattern(Integral(Pq_*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons57, cons58, cons59, cons60, cons4, cons5, cons64, cons55, cons494)
def replacement1620(p, a2, b2, Pq, b1, n, x, a1):
rubi.append(1620)
return Int(Pq*(a1*a2 + b1*b2*x**(S(2)*n))**p, x)
rule1620 = ReplacementRule(pattern1620, replacement1620)
pattern1621 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons57, cons58, cons59, cons60, cons7, cons21, cons4, cons5, cons64, cons55)
def replacement1621(p, a2, m, b2, b1, Pq, c, n, x, a1):
rubi.append(1621)
return Dist((a1 + b1*x**n)**FracPart(p)*(a2 + b2*x**n)**FracPart(p)*(a1*a2 + b1*b2*x**(S(2)*n))**(-FracPart(p)), Int(Pq*(c*x)**m*(a1*a2 + b1*b2*x**(S(2)*n))**p, x), x)
rule1621 = ReplacementRule(pattern1621, replacement1621)
pattern1622 = Pattern(Integral(Pq_*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**WC('p', S(1))*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**WC('p', S(1)), x_), cons57, cons58, cons59, cons60, cons4, cons5, cons64, cons55)
def replacement1622(p, a2, b2, Pq, b1, n, x, a1):
rubi.append(1622)
return Dist((a1 + b1*x**n)**FracPart(p)*(a2 + b2*x**n)**FracPart(p)*(a1*a2 + b1*b2*x**(S(2)*n))**(-FracPart(p)), Int(Pq*(a1*a2 + b1*b2*x**(S(2)*n))**p, x), x)
rule1622 = ReplacementRule(pattern1622, replacement1622)
pattern1623 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('p', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)) + x_**WC('n2', S(1))*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons46, cons920, cons921)
def replacement1623(p, f, b, g, n2, d, a, n, c, x, e):
rubi.append(1623)
return Simp(e*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(p + S(1))/(a*c), x)
rule1623 = ReplacementRule(pattern1623, replacement1623)
pattern1624 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('p', S(1))*(e_ + x_**WC('n2', S(1))*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons208, cons4, cons5, cons46, cons922, cons921)
def replacement1624(p, g, b, n2, d, a, n, c, x, e):
rubi.append(1624)
return Simp(e*x*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(p + S(1))/(a*c), x)
rule1624 = ReplacementRule(pattern1624, replacement1624)
pattern1625 = Pattern(Integral((x_*WC('h', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('p', S(1))*(e_ + x_**WC('n', S(1))*WC('f', S(1)) + x_**WC('n2', S(1))*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons5, cons46, cons923, cons924, cons66)
def replacement1625(p, m, f, b, g, n2, d, a, n, c, x, h, e):
rubi.append(1625)
return Simp(e*(h*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(p + S(1))/(a*c*h*(m + S(1))), x)
rule1625 = ReplacementRule(pattern1625, replacement1625)
pattern1626 = Pattern(Integral((x_*WC('h', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('p', S(1))*(e_ + x_**WC('n2', S(1))*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons208, cons209, cons21, cons4, cons5, cons46, cons595, cons924, cons66)
def replacement1626(p, m, g, b, n2, d, a, n, c, x, h, e):
rubi.append(1626)
return Simp(e*(h*x)**(m + S(1))*(a + b*x**n)**(p + S(1))*(c + d*x**n)**(p + S(1))/(a*c*h*(m + S(1))), x)
rule1626 = ReplacementRule(pattern1626, replacement1626)
pattern1627 = Pattern(Integral((A_ + x_**WC('m', S(1))*WC('B', S(1)))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(x_**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons21, cons4, cons5, cons50, cons71, cons53)
def replacement1627(B, p, m, b, d, a, n, c, x, q, A):
rubi.append(1627)
return Dist(A, Int((a + b*x**n)**p*(c + d*x**n)**q, x), x) + Dist(B, Int(x**m*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule1627 = ReplacementRule(pattern1627, replacement1627)
def With1628(p, b, Px, d, a, c, n, x, q):
k = Denominator(n)
rubi.append(1628)
return Dist(k/d, Subst(Int(SimplifyIntegrand(x**(k + S(-1))*(a + b*x**(k*n))**p*ReplaceAll(Px, Rule(x, -c/d + x**k/d))**q, x), x), x, (c + d*x)**(S(1)/k)), x)
pattern1628 = Pattern(Integral(Px_**WC('q', S(1))*((c_ + x_*WC('d', S(1)))**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons925, cons586, cons87)
rule1628 = ReplacementRule(pattern1628, With1628)
pattern1629 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons46, cons858, cons53)
def replacement1629(p, m, b, n2, Pq, c, a, n, x):
rubi.append(1629)
return Dist(S(1)/n, Subst(Int((a + b*x + c*x**S(2))**p*SubstFor(x**n, Pq, x), x), x, x**n), x)
rule1629 = ReplacementRule(pattern1629, replacement1629)
pattern1630 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons46, cons64, cons128)
def replacement1630(p, m, b, n2, Pq, d, c, n, a, x):
rubi.append(1630)
return Int(ExpandIntegrand(Pq*(d*x)**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1630 = ReplacementRule(pattern1630, replacement1630)
pattern1631 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons46, cons64, cons128)
def replacement1631(p, b, n2, Pq, c, n, a, x):
rubi.append(1631)
return Int(ExpandIntegrand(Pq*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1631 = ReplacementRule(pattern1631, replacement1631)
pattern1632 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('n', S(1))*WC('e', S(1)) + x_**WC('n2', S(1))*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons5, cons46, cons926, cons927)
def replacement1632(p, f, b, n2, d, c, n, a, x, e):
rubi.append(1632)
return Simp(d*x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/a, x)
rule1632 = ReplacementRule(pattern1632, replacement1632)
pattern1633 = Pattern(Integral((d_ + x_**WC('n2', S(1))*WC('f', S(1)))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons4, cons5, cons46, cons922, cons928)
def replacement1633(p, f, b, n2, d, c, n, a, x):
rubi.append(1633)
return Simp(d*x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/a, x)
rule1633 = ReplacementRule(pattern1633, replacement1633)
pattern1634 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('n', S(1))*WC('e', S(1)) + x_**WC('n2', S(1))*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons46, cons929, cons930, cons66)
def replacement1634(p, m, g, b, n2, f, d, c, n, a, x, e):
rubi.append(1634)
return Simp(d*(g*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(a*g*(m + S(1))), x)
rule1634 = ReplacementRule(pattern1634, replacement1634)
pattern1635 = Pattern(Integral((x_*WC('g', S(1)))**WC('m', S(1))*(d_ + x_**WC('n2', S(1))*WC('f', S(1)))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons21, cons4, cons5, cons46, cons595, cons928, cons66)
def replacement1635(p, m, g, b, n2, f, d, c, n, a, x):
rubi.append(1635)
return Simp(d*(g*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))/(a*g*(m + S(1))), x)
rule1635 = ReplacementRule(pattern1635, replacement1635)
pattern1636 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons46, cons64, cons45, cons314)
def replacement1636(p, m, b, n2, Pq, d, c, n, a, x):
rubi.append(1636)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int(Pq*(d*x)**m*(b + S(2)*c*x**n)**(S(2)*p), x), x)
rule1636 = ReplacementRule(pattern1636, replacement1636)
pattern1637 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons46, cons64, cons45, cons314)
def replacement1637(p, b, n2, Pq, c, n, a, x):
rubi.append(1637)
return Dist((S(4)*c)**(-IntPart(p))*(b + S(2)*c*x**n)**(-S(2)*FracPart(p))*(a + b*x**n + c*x**(S(2)*n))**FracPart(p), Int(Pq*(b + S(2)*c*x**n)**(S(2)*p), x), x)
rule1637 = ReplacementRule(pattern1637, replacement1637)
pattern1638 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons46, cons858, cons226, cons500)
def replacement1638(p, m, b, n2, Pq, c, a, n, x):
rubi.append(1638)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*x + c*x**S(2))**p*SubstFor(x**n, Pq, x), x), x, x**n), x)
rule1638 = ReplacementRule(pattern1638, replacement1638)
pattern1639 = Pattern(Integral(Pq_*(d_*x_)**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons46, cons858, cons226, cons500)
def replacement1639(p, m, b, n2, Pq, d, c, a, n, x):
rubi.append(1639)
return Dist(x**(-m)*(d*x)**m, Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1639 = ReplacementRule(pattern1639, replacement1639)
pattern1640 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons46, cons64, cons859)
def replacement1640(p, m, b, n2, Pq, d, c, n, a, x):
rubi.append(1640)
return Dist(S(1)/d, Int((d*x)**(m + S(1))*(a + b*x**n + c*x**(S(2)*n))**p*ExpandToSum(Pq/x, x), x), x)
rule1640 = ReplacementRule(pattern1640, replacement1640)
pattern1641 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons46, cons64, cons859, cons860)
def replacement1641(p, b, n2, Pq, c, n, a, x):
rubi.append(1641)
return Int(x*(a + b*x**n + c*x**(S(2)*n))**p*ExpandToSum(Pq/x, x), x)
rule1641 = ReplacementRule(pattern1641, replacement1641)
pattern1642 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)) + x_**WC('n2', S(1))*WC('f', S(1)) + x_**WC('n3', S(1))*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons46, cons931, cons226, cons932, cons933)
def replacement1642(p, f, b, n2, g, d, n3, c, a, n, x, e):
rubi.append(1642)
return Simp(x*(S(3)*a*d - x**S(2)*(-a*e + S(2)*b*d*p + S(3)*b*d))*(a + b*x**S(2) + c*x**S(4))**(p + S(1))/(S(3)*a**S(2)), x)
rule1642 = ReplacementRule(pattern1642, replacement1642)
pattern1643 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**WC('n2', S(1))*WC('f', S(1)) + x_**WC('n3', S(1))*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons4, cons5, cons46, cons931, cons226, cons934, cons935)
def replacement1643(p, f, g, n2, b, d, n3, c, a, n, x):
rubi.append(1643)
return Simp(x*(S(3)*a*d - x**S(2)*(S(2)*b*d*p + S(3)*b*d))*(a + b*x**S(2) + c*x**S(4))**(p + S(1))/(S(3)*a**S(2)), x)
rule1643 = ReplacementRule(pattern1643, replacement1643)
pattern1644 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(d_ + x_**n_*WC('e', S(1)) + x_**WC('n3', S(1))*WC('g', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons208, cons4, cons5, cons46, cons931, cons226, cons932, cons936)
def replacement1644(p, g, b, n2, d, n3, c, a, n, x, e):
rubi.append(1644)
return Simp(x*(S(3)*a*d - x**S(2)*(-a*e + S(2)*b*d*p + S(3)*b*d))*(a + b*x**S(2) + c*x**S(4))**(p + S(1))/(S(3)*a**S(2)), x)
rule1644 = ReplacementRule(pattern1644, replacement1644)
pattern1645 = Pattern(Integral((d_ + x_**WC('n3', S(1))*WC('g', S(1)))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons208, cons4, cons5, cons46, cons931, cons226, cons934, cons937)
def replacement1645(p, g, b, n2, d, n3, c, a, n, x):
rubi.append(1645)
return Simp(x*(S(3)*a*d - x**S(2)*(S(2)*b*d*p + S(3)*b*d))*(a + b*x**S(2) + c*x**S(4))**(p + S(1))/(S(3)*a**S(2)), x)
rule1645 = ReplacementRule(pattern1645, replacement1645)
pattern1646 = Pattern(Integral(x_**WC('m', S(1))*(e_ + x_**WC('q', S(1))*WC('f', S(1)) + x_**WC('r', S(1))*WC('g', S(1)) + x_**WC('s', S(1))*WC('h', S(1)))/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons48, cons125, cons208, cons209, cons21, cons4, cons46, cons938, cons939, cons940, cons226, cons941, cons852)
def replacement1646(s, m, f, b, n2, g, r, c, n, a, x, h, q, e):
rubi.append(1646)
return -Simp((S(2)*c*x**n*(-b*g + S(2)*c*f) + S(2)*c*(-S(2)*a*g + b*f) + S(2)*h*x**(n/S(2))*(-S(4)*a*c + b**S(2)))/(c*n*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**n + c*x**(S(2)*n))), x)
rule1646 = ReplacementRule(pattern1646, replacement1646)
pattern1647 = Pattern(Integral((d_*x_)**WC('m', S(1))*(e_ + x_**WC('q', S(1))*WC('f', S(1)) + x_**WC('r', S(1))*WC('g', S(1)) + x_**WC('s', S(1))*WC('h', S(1)))/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons21, cons4, cons46, cons938, cons939, cons940, cons226, cons941, cons852)
def replacement1647(s, m, f, b, n2, g, r, d, c, n, a, x, h, q, e):
rubi.append(1647)
return Dist(x**(-m)*(d*x)**m, Int(x**m*(e + f*x**(n/S(2)) + g*x**(S(3)*n/S(2)) + h*x**(S(2)*n))/(a + b*x**n + c*x**(S(2)*n))**(S(3)/2), x), x)
rule1647 = ReplacementRule(pattern1647, replacement1647)
def With1648(p, b, n2, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
i = Symbol('i')
if Less(q, S(2)*n):
return True
return False
pattern1648 = Pattern(Integral(Pq_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons46, cons64, cons226, cons148, cons13, cons137, CustomConstraint(With1648))
def replacement1648(p, b, n2, Pq, c, a, n, x):
q = Expon(Pq, x)
i = Symbol('i')
rubi.append(1648)
return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Sum_doit(c*x**(i + n)*(-S(2)*a*Coeff(Pq, x, i + n) + b*Coeff(Pq, x, i))*(i + n*(S(2)*p + S(3)) + S(1)) + x**i*(-a*b*(i + S(1))*Coeff(Pq, x, i + n) + (-S(2)*a*c*(i + S(2)*n*(p + S(1)) + S(1)) + b**S(2)*(i + n*(p + S(1)) + S(1)))*Coeff(Pq, x, i)), List(i, S(0), n + S(-1))), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Sum_doit(c*x**(i + n)*(-S(2)*a*Coeff(Pq, x, i + n) + b*Coeff(Pq, x, i)) + x**i*(-a*b*Coeff(Pq, x, i + n) + (-S(2)*a*c + b**S(2))*Coeff(Pq, x, i)), List(i, S(0), n + S(-1)))/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1648 = ReplacementRule(pattern1648, replacement1648)
pattern1649 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)) + x_*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons226, cons942)
def replacement1649(f, b, g, d, c, a, x, e):
rubi.append(1649)
return -Simp((c*x**S(2)*(-b*f + S(2)*c*e) + c*(-S(2)*a*f + b*e) + g*x*(-S(4)*a*c + b**S(2)))/(c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1649 = ReplacementRule(pattern1649, replacement1649)
pattern1650 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons226, cons942)
def replacement1650(f, b, g, d, c, a, x):
rubi.append(1650)
return Simp((S(2)*a*c*f + b*c*f*x**S(2) - g*x*(-S(4)*a*c + b**S(2)))/(c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1650 = ReplacementRule(pattern1650, replacement1650)
pattern1651 = Pattern(Integral((d_ + x_**S(4)*WC('g', S(1)) + x_*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons208, cons226, cons942)
def replacement1651(g, b, d, c, a, x, e):
rubi.append(1651)
return -Simp((b*c*e + S(2)*c**S(2)*e*x**S(2) + g*x*(-S(4)*a*c + b**S(2)))/(c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1651 = ReplacementRule(pattern1651, replacement1651)
pattern1652 = Pattern(Integral(x_**S(2)*(x_**S(4)*WC('h', S(1)) + x_**S(2)*WC('g', S(1)) + x_*WC('f', S(1)) + WC('e', S(0)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons48, cons125, cons208, cons209, cons226, cons943, cons944)
def replacement1652(f, b, g, c, a, x, h, e):
rubi.append(1652)
return Simp((S(2)*a**S(2)*c*f + a*b*c*f*x**S(2) + a*h*x**S(3)*(-S(4)*a*c + b**S(2)))/(a*c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1652 = ReplacementRule(pattern1652, replacement1652)
pattern1653 = Pattern(Integral(x_**S(2)*(x_**S(4)*WC('h', S(1)) + x_**S(2)*WC('g', S(1)) + WC('e', S(0)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons48, cons208, cons209, cons226, cons943, cons944)
def replacement1653(g, b, c, a, x, h, e):
rubi.append(1653)
return Simp(h*x**S(3)/(c*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1653 = ReplacementRule(pattern1653, replacement1653)
pattern1654 = Pattern(Integral((d_ + x_**S(6)*WC('h', S(1)) + x_**S(4)*WC('g', S(1)) + x_**S(3)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons226, cons943, cons945)
def replacement1654(f, b, g, d, c, a, x, h, e):
rubi.append(1654)
return Simp((S(2)*a**S(2)*c*f + a*b*c*f*x**S(2) + a*h*x**S(3)*(-S(4)*a*c + b**S(2)) + c*d*x*(-S(4)*a*c + b**S(2)))/(a*c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1654 = ReplacementRule(pattern1654, replacement1654)
pattern1655 = Pattern(Integral((d_ + x_**S(6)*WC('h', S(1)) + x_**S(3)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))/(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))**(S(3)/2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons209, cons226, cons943, cons946)
def replacement1655(f, b, d, c, a, x, h, e):
rubi.append(1655)
return Simp((S(2)*a**S(2)*c*f + a*b*c*f*x**S(2) + a*h*x**S(3)*(-S(4)*a*c + b**S(2)) + c*d*x*(-S(4)*a*c + b**S(2)))/(a*c*(-S(4)*a*c + b**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1655 = ReplacementRule(pattern1655, replacement1655)
def With1656(p, b, n2, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Q = PolynomialQuotient(Pq*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x)
R = PolynomialRemainder(Pq*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x)
if GreaterEqual(q, S(2)*n):
return True
return False
pattern1656 = Pattern(Integral(Pq_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons46, cons64, cons226, cons148, cons13, cons137, CustomConstraint(With1656))
def replacement1656(p, b, n2, Pq, c, a, n, x):
q = Expon(Pq, x)
Q = PolynomialQuotient(Pq*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x)
R = PolynomialRemainder(Pq*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x)
rubi.append(1656)
return Dist((b*c)**(-Floor((q - 1)/n) - 1)/(a*n*(p + 1)*(-4*a*c + b**2)), Int((a + b*x**n + c*x**(2*n))**(p + 1)*ExpandToSum(Q*a*n*(p + 1)*(-4*a*c + b**2) + Sum_doit(c*x**(i + n)*(-2*a*Coeff(R, x, i + n) + b*Coeff(R, x, i))*(i + n*(2*p + 3) + 1) + x**i*(-a*b*(i + 1)*Coeff(R, x, i + n) + (-2*a*c*(i + 2*n*(p + 1) + 1) + b**2*(i + n*(p + 1) + 1))*Coeff(R, x, i)), List(i, 0, n - 1)), x), x), x) - Simp(x*(b*c)**(-Floor((q - 1)/n) - 1)*(a + b*x**n + c*x**(2*n))**(p + 1)*Sum_doit(c*x**(i + n)*(-2*a*Coeff(R, x, i + n) + b*Coeff(R, x, i)) + x**i*(-a*b*Coeff(R, x, i + n) + (-2*a*c + b**2)*Coeff(R, x, i)), List(i, 0, n - 1))/(a*n*(p + 1)*(-4*a*c + b**2)), x)
rule1656 = ReplacementRule(pattern1656, replacement1656)
def With1657(p, m, b, n2, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Q = PolynomialQuotient(Pq*a*x**m*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x)
R = PolynomialRemainder(Pq*a*x**m*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x)
if GreaterEqual(q, S(2)*n):
return True
return False
pattern1657 = Pattern(Integral(Pq_*x_**m_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons46, cons64, cons226, cons148, cons13, cons137, cons84, CustomConstraint(With1657))
def replacement1657(p, m, b, n2, Pq, c, a, n, x):
q = Expon(Pq, x)
Q = PolynomialQuotient(Pq*a*x**m*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x)
R = PolynomialRemainder(Pq*a*x**m*(b*c)**(Floor((q + S(-1))/n) + S(1)), a + b*x**n + c*x**(S(2)*n), x)
rubi.append(1657)
return Dist((b*c)**(-Floor((q - 1)/n) - 1)/(a*n*(p + 1)*(-4*a*c + b**2)), Int(x**m*(a + b*x**n + c*x**(2*n))**(p + 1)*ExpandToSum(Q*n*x**(-m)*(p + 1)*(-4*a*c + b**2) + Sum_doit(c*x**(i - m + n)*(-2*Coeff(R, x, i + n) + b*Coeff(R, x, i)/a)*(i + n*(2*p + 3) + 1) + x**(i - m)*(-b*(i + 1)*Coeff(R, x, i + n) + (-2*c*(i + 2*n*(p + 1) + 1) + b**2*(i + n*(p + 1) + 1)/a)*Coeff(R, x, i)), List(i, 0, n - 1)), x), x), x) - Simp(x*(b*c)**(-Floor((q - 1)/n) - 1)*(a + b*x**n + c*x**(2*n))**(p + 1)*Sum_doit(c*x**(i + n)*(-2*a*Coeff(R, x, i + n) + b*Coeff(R, x, i)) + x**i*(-a*b*Coeff(R, x, i + n) + (-2*a*c + b**2)*Coeff(R, x, i)), List(i, 0, n - 1))/(a**2*n*(p + 1)*(-4*a*c + b**2)), x)
rule1657 = ReplacementRule(pattern1657, replacement1657)
def With1658(p, m, b, n2, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
g = GCD(m + S(1), n)
if Unequal(g, S(1)):
return True
return False
pattern1658 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons46, cons858, cons226, cons148, cons17, CustomConstraint(With1658))
def replacement1658(p, m, b, n2, Pq, c, a, n, x):
g = GCD(m + S(1), n)
rubi.append(1658)
return Dist(S(1)/g, Subst(Int(x**(S(-1) + (m + S(1))/g)*(a + b*x**(n/g) + c*x**(S(2)*n/g))**p*ReplaceAll(Pq, Rule(x, x**(S(1)/g))), x), x, x**g), x)
rule1658 = ReplacementRule(pattern1658, replacement1658)
pattern1659 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))/(a_ + x_**n2_*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons46, cons858, cons226, cons148, cons282)
def replacement1659(m, b, n2, Pq, d, c, n, a, x):
rubi.append(1659)
return Int(ExpandIntegrand(Pq*(d*x)**m/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1659 = ReplacementRule(pattern1659, replacement1659)
pattern1660 = Pattern(Integral(Pq_/(a_ + x_**n2_*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons7, cons46, cons858, cons226, cons148, cons947)
def replacement1660(b, n2, Pq, c, n, a, x):
rubi.append(1660)
return Int(ExpandIntegrand(Pq/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1660 = ReplacementRule(pattern1660, replacement1660)
def With1661(p, b, Pq, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
if Equal(S(2)*p + q + S(1), S(0)):
return True
return False
pattern1661 = Pattern(Integral(Pq_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons64, cons226, cons63, CustomConstraint(With1661))
def replacement1661(p, b, Pq, c, a, x):
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
rubi.append(1661)
return Dist(S.Half, Int((a + b*x + c*x**2)**p*ExpandToSum(2*Pq - Pqq*c**p*(b + 2*c*x)*(a + b*x + c*x**2)**(-p - 1), x), x), x) + Simp(Pqq*c**p*log(a + b*x + c*x**2)/2, x)
rule1661 = ReplacementRule(pattern1661, replacement1661)
def With1662(p, b, Pq, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
if Equal(S(2)*p + q + S(1), S(0)):
return True
return False
pattern1662 = Pattern(Integral(Pq_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons64, cons226, cons719, cons948, CustomConstraint(With1662))
def replacement1662(p, b, Pq, c, a, x):
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
rubi.append(1662)
return Int((a + b*x + c*x**2)**p*ExpandToSum(Pq - Pqq*c**(p + S.Half)*(a + b*x + c*x**2)**(-p - S.Half), x), x) + Simp(Pqq*c**p*atanh((b + 2*c*x)/(2*sqrt(a + b*x + c*x**2)*Rt(c, 2))), x)
rule1662 = ReplacementRule(pattern1662, replacement1662)
def With1663(p, b, Pq, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
if Equal(S(2)*p + q + S(1), S(0)):
return True
return False
pattern1663 = Pattern(Integral(Pq_*(a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons64, cons226, cons719, cons949, CustomConstraint(With1663))
def replacement1663(p, b, Pq, c, a, x):
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
rubi.append(1663)
return Int((a + b*x + c*x**2)**p*ExpandToSum(Pq - Pqq*(-c)**(p + S.Half)*(a + b*x + c*x**2)**(-p - S.Half), x), x) - Simp(Pqq*(-c)**p*ArcTan((b + 2*c*x)/(2*sqrt(a + b*x + c*x**2)*Rt(-c, 2))), x)
rule1663 = ReplacementRule(pattern1663, replacement1663)
def With1664(p, m, b, n2, Pq, d, c, n, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
if And(GreaterEqual(q, S(2)*n), Unequal(m + S(2)*n*p + q + S(1), S(0)), Or(IntegerQ(S(2)*p), And(Equal(n, S(1)), IntegerQ(S(4)*p)), IntegerQ(p + (q + S(1))/(S(2)*n)))):
return True
return False
pattern1664 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons46, cons858, cons226, cons148, CustomConstraint(With1664))
def replacement1664(p, m, b, n2, Pq, d, c, n, a, x):
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
rubi.append(1664)
return Int((d*x)**m*(a + b*x**n + c*x**(2*n))**p*ExpandToSum(Pq - Pqq*x**q - Pqq*(a*x**(-2*n + q)*(m - 2*n + q + 1) + b*x**(-n + q)*(m + n*(p - 1) + q + 1))/(c*(m + 2*n*p + q + 1)), x), x) + Simp(Pqq*d**(2*n - q - 1)*(d*x)**(m - 2*n + q + 1)*(a + b*x**n + c*x**(2*n))**(p + 1)/(c*(m + 2*n*p + q + 1)), x)
rule1664 = ReplacementRule(pattern1664, replacement1664)
def With1665(p, b, n2, Pq, c, n, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
if And(GreaterEqual(q, S(2)*n), Unequal(S(2)*n*p + q + S(1), S(0)), Or(IntegerQ(S(2)*p), And(Equal(n, S(1)), IntegerQ(S(4)*p)), IntegerQ(p + (q + S(1))/(S(2)*n)))):
return True
return False
pattern1665 = Pattern(Integral(Pq_*(a_ + x_**n2_*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons46, cons858, cons226, cons148, CustomConstraint(With1665))
def replacement1665(p, b, n2, Pq, c, n, a, x):
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
rubi.append(1665)
return Int((a + b*x**n + c*x**(2*n))**p*ExpandToSum(Pq - Pqq*x**q - Pqq*(a*x**(-2*n + q)*(-2*n + q + 1) + b*x**(-n + q)*(n*(p - 1) + q + 1))/(c*(2*n*p + q + 1)), x), x) + Simp(Pqq*x**(-2*n + q + 1)*(a + b*x**n + c*x**(2*n))**(p + 1)/(c*(2*n*p + q + 1)), x)
rule1665 = ReplacementRule(pattern1665, replacement1665)
def With1666(p, m, b, n2, Pq, d, c, a, n, x):
q = Expon(Pq, x)
j = Symbol('j')
k = Symbol('k')
rubi.append(1666)
return Int(Sum_doit(d**(-j)*(d*x)**(j + m)*(a + b*x**n + c*x**(S(2)*n))**p*Sum_doit(x**(k*n)*Coeff(Pq, x, j + k*n), List(k, S(0), S(1) + (-j + q)/n)), List(j, S(0), n + S(-1))), x)
pattern1666 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons46, cons64, cons226, cons148, cons950)
rule1666 = ReplacementRule(pattern1666, With1666)
def With1667(p, b, n2, Pq, c, a, n, x):
q = Expon(Pq, x)
j = Symbol('j')
k = Symbol('k')
rubi.append(1667)
return Int(Sum_doit(x**j*(a + b*x**n + c*x**(S(2)*n))**p*Sum_doit(x**(k*n)*Coeff(Pq, x, j + k*n), List(k, S(0), S(1) + (-j + q)/n)), List(j, S(0), n + S(-1))), x)
pattern1667 = Pattern(Integral(Pq_*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons46, cons64, cons226, cons148, cons950)
rule1667 = ReplacementRule(pattern1667, With1667)
pattern1668 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons46, cons64, cons226, cons148)
def replacement1668(m, b, n2, Pq, d, c, n, a, x):
rubi.append(1668)
return Int(RationalFunctionExpand(Pq*(d*x)**m/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1668 = ReplacementRule(pattern1668, replacement1668)
pattern1669 = Pattern(Integral(Pq_/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons46, cons64, cons226, cons148)
def replacement1669(b, n2, Pq, c, n, a, x):
rubi.append(1669)
return Int(RationalFunctionExpand(Pq/(a + b*x**n + c*x**(S(2)*n)), x), x)
rule1669 = ReplacementRule(pattern1669, replacement1669)
def With1670(p, m, b, n2, Pq, c, a, n, x):
q = Expon(Pq, x)
rubi.append(1670)
return -Subst(Int(x**(-m - q + S(-2))*(a + b*x**(-n) + c*x**(-S(2)*n))**p*ExpandToSum(x**q*ReplaceAll(Pq, Rule(x, S(1)/x)), x), x), x, S(1)/x)
pattern1670 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons46, cons64, cons226, cons196, cons17)
rule1670 = ReplacementRule(pattern1670, With1670)
def With1671(p, m, b, n2, Pq, d, c, a, n, x):
g = Denominator(m)
q = Expon(Pq, x)
rubi.append(1671)
return -Dist(g/d, Subst(Int(x**(-g*(m + q + S(1)) + S(-1))*(a + b*d**(-n)*x**(-g*n) + c*d**(-S(2)*n)*x**(-S(2)*g*n))**p*ExpandToSum(x**(g*q)*ReplaceAll(Pq, Rule(x, x**(-g)/d)), x), x), x, (d*x)**(-S(1)/g)), x)
pattern1671 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons46, cons64, cons226, cons196, cons367)
rule1671 = ReplacementRule(pattern1671, With1671)
def With1672(p, m, b, n2, Pq, d, c, a, n, x):
q = Expon(Pq, x)
rubi.append(1672)
return -Dist((d*x)**m*(S(1)/x)**m, Subst(Int(x**(-m - q + S(-2))*(a + b*x**(-n) + c*x**(-S(2)*n))**p*ExpandToSum(x**q*ReplaceAll(Pq, Rule(x, S(1)/x)), x), x), x, S(1)/x), x)
pattern1672 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons46, cons64, cons226, cons196, cons356)
rule1672 = ReplacementRule(pattern1672, With1672)
def With1673(p, m, b, n2, Pq, c, a, n, x):
g = Denominator(n)
rubi.append(1673)
return Dist(g, Subst(Int(x**(g*(m + S(1)) + S(-1))*(a + b*x**(g*n) + c*x**(S(2)*g*n))**p*ReplaceAll(Pq, Rule(x, x**g)), x), x, x**(S(1)/g)), x)
pattern1673 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons46, cons64, cons226, cons489)
rule1673 = ReplacementRule(pattern1673, With1673)
def With1674(p, b, n2, Pq, c, a, n, x):
g = Denominator(n)
rubi.append(1674)
return Dist(g, Subst(Int(x**(g + S(-1))*(a + b*x**(g*n) + c*x**(S(2)*g*n))**p*ReplaceAll(Pq, Rule(x, x**g)), x), x, x**(S(1)/g)), x)
pattern1674 = Pattern(Integral(Pq_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons5, cons46, cons64, cons226, cons489)
rule1674 = ReplacementRule(pattern1674, With1674)
pattern1675 = Pattern(Integral(Pq_*(d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons46, cons64, cons226, cons489, cons73)
def replacement1675(p, m, b, n2, Pq, d, c, a, n, x):
rubi.append(1675)
return Dist(d**(m + S(-1)/2)*sqrt(d*x)/sqrt(x), Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1675 = ReplacementRule(pattern1675, replacement1675)
pattern1676 = Pattern(Integral(Pq_*(d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons46, cons64, cons226, cons489, cons951)
def replacement1676(p, m, b, n2, Pq, d, c, a, n, x):
rubi.append(1676)
return Dist(d**(m + S(1)/2)*sqrt(x)/sqrt(d*x), Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1676 = ReplacementRule(pattern1676, replacement1676)
pattern1677 = Pattern(Integral(Pq_*(d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons46, cons64, cons226, cons489)
def replacement1677(p, m, b, n2, Pq, d, c, a, n, x):
rubi.append(1677)
return Dist(x**(-m)*(d*x)**m, Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1677 = ReplacementRule(pattern1677, replacement1677)
pattern1678 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons46, cons858, cons226, cons541, cons23)
def replacement1678(p, m, b, n2, Pq, c, a, n, x):
rubi.append(1678)
return Dist(S(1)/(m + S(1)), Subst(Int((a + b*x**(n/(m + S(1))) + c*x**(S(2)*n/(m + S(1))))**p*ReplaceAll(SubstFor(x**n, Pq, x), Rule(x, x**(n/(m + S(1))))), x), x, x**(m + S(1))), x)
rule1678 = ReplacementRule(pattern1678, replacement1678)
pattern1679 = Pattern(Integral(Pq_*(d_*x_)**m_*(a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons5, cons46, cons858, cons226, cons541, cons23)
def replacement1679(p, m, b, n2, Pq, d, c, a, n, x):
rubi.append(1679)
return Dist(x**(-m)*(d*x)**m, Int(Pq*x**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1679 = ReplacementRule(pattern1679, replacement1679)
def With1680(m, b, n2, Pq, d, c, n, a, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1680)
return Dist(S(2)*c/q, Int(Pq*(d*x)**m/(b + S(2)*c*x**n - q), x), x) - Dist(S(2)*c/q, Int(Pq*(d*x)**m/(b + S(2)*c*x**n + q), x), x)
pattern1680 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons46, cons64, cons226)
rule1680 = ReplacementRule(pattern1680, With1680)
def With1681(b, n2, Pq, c, n, a, x):
q = Rt(-S(4)*a*c + b**S(2), S(2))
rubi.append(1681)
return Dist(S(2)*c/q, Int(Pq/(b + S(2)*c*x**n - q), x), x) - Dist(S(2)*c/q, Int(Pq/(b + S(2)*c*x**n + q), x), x)
pattern1681 = Pattern(Integral(Pq_/(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons4, cons46, cons64, cons226)
rule1681 = ReplacementRule(pattern1681, With1681)
pattern1682 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons46, cons64, cons702)
def replacement1682(p, m, b, n2, Pq, d, c, n, a, x):
rubi.append(1682)
return Int(ExpandIntegrand(Pq*(d*x)**m*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1682 = ReplacementRule(pattern1682, replacement1682)
pattern1683 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons7, cons4, cons46, cons64, cons702)
def replacement1683(p, b, n2, Pq, c, n, a, x):
rubi.append(1683)
return Int(ExpandIntegrand(Pq*(a + b*x**n + c*x**(S(2)*n))**p, x), x)
rule1683 = ReplacementRule(pattern1683, replacement1683)
pattern1684 = Pattern(Integral(Pq_*(x_*WC('d', S(1)))**WC('m', S(1))*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons46, cons918)
def replacement1684(p, m, b, n2, Pq, d, c, n, a, x):
rubi.append(1684)
return Int(Pq*(d*x)**m*(a + b*x**n + c*x**(S(2)*n))**p, x)
rule1684 = ReplacementRule(pattern1684, replacement1684)
pattern1685 = Pattern(Integral(Pq_*(a_ + x_**WC('n', S(1))*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons46, cons918)
def replacement1685(p, b, n2, Pq, c, n, a, x):
rubi.append(1685)
return Int(Pq*(a + b*x**n + c*x**(S(2)*n))**p, x)
rule1685 = ReplacementRule(pattern1685, replacement1685)
pattern1686 = Pattern(Integral(Pq_*u_**WC('m', S(1))*(a_ + v_**n_*WC('b', S(1)) + v_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons46, cons554, cons919)
def replacement1686(v, p, u, m, b, n2, Pq, c, a, n, x):
rubi.append(1686)
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a + b*x**n + c*x**(S(2)*n))**p*SubstFor(v, Pq, x), x), x, v), x)
rule1686 = ReplacementRule(pattern1686, replacement1686)
pattern1687 = Pattern(Integral(Pq_*(a_ + v_**n_*WC('b', S(1)) + v_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons46, cons552, cons919)
def replacement1687(v, p, b, n2, Pq, c, a, n, x):
rubi.append(1687)
return Dist(S(1)/Coefficient(v, x, S(1)), Subst(Int((a + b*x**n + c*x**(S(2)*n))**p*SubstFor(v, Pq, x), x), x, v), x)
rule1687 = ReplacementRule(pattern1687, replacement1687)
pattern1688 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons5, cons147, cons952, cons953)
def replacement1688(p, j, b, a, n, x):
rubi.append(1688)
return Simp(x**(-n + S(1))*(a*x**j + b*x**n)**(p + S(1))/(b*(-j + n)*(p + S(1))), x)
rule1688 = ReplacementRule(pattern1688, replacement1688)
pattern1689 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons147, cons952, cons954, cons13, cons137)
def replacement1689(p, j, b, a, n, x):
rubi.append(1689)
return Dist((-j + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int(x**(-j)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(x**(-j + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x)
rule1689 = ReplacementRule(pattern1689, replacement1689)
pattern1690 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons5, cons147, cons952, cons954, cons955)
def replacement1690(p, j, b, a, n, x):
rubi.append(1690)
return -Dist(b*(-j + n*p + n + S(1))/(a*(j*p + S(1))), Int(x**(-j + n)*(a*x**j + b*x**n)**p, x), x) + Simp(x**(-j + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(j*p + S(1))), x)
rule1690 = ReplacementRule(pattern1690, replacement1690)
pattern1691 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons147, cons956, cons957, cons163, cons958)
def replacement1691(p, j, b, a, n, x):
rubi.append(1691)
return -Dist(b*p*(-j + n)/(j*p + S(1)), Int(x**n*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp(x*(a*x**j + b*x**n)**p/(j*p + S(1)), x)
rule1691 = ReplacementRule(pattern1691, replacement1691)
pattern1692 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons147, cons956, cons957, cons163, cons959)
def replacement1692(p, j, b, a, n, x):
rubi.append(1692)
return Dist(a*p*(-j + n)/(n*p + S(1)), Int(x**j*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp(x*(a*x**j + b*x**n)**p/(n*p + S(1)), x)
rule1692 = ReplacementRule(pattern1692, replacement1692)
pattern1693 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons147, cons956, cons957, cons137, cons960)
def replacement1693(p, j, b, a, n, x):
rubi.append(1693)
return -Dist((j*p + j - n + S(1))/(b*(-j + n)*(p + S(1))), Int(x**(-n)*(a*x**j + b*x**n)**(p + S(1)), x), x) + Simp(x**(-n + S(1))*(a*x**j + b*x**n)**(p + S(1))/(b*(-j + n)*(p + S(1))), x)
rule1693 = ReplacementRule(pattern1693, replacement1693)
pattern1694 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons147, cons956, cons957, cons137)
def replacement1694(p, j, b, a, n, x):
rubi.append(1694)
return Dist((-j + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int(x**(-j)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(x**(-j + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x)
rule1694 = ReplacementRule(pattern1694, replacement1694)
pattern1695 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons961, cons952, cons962)
def replacement1695(p, j, b, a, n, x):
rubi.append(1695)
return Dist(a, Int(x**j*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp(x*(a*x**j + b*x**n)**p/(p*(-j + n)), x)
rule1695 = ReplacementRule(pattern1695, replacement1695)
pattern1696 = Pattern(Integral(S(1)/sqrt(x_**S(2)*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons4, cons963)
def replacement1696(x, a, n, b):
rubi.append(1696)
return Dist(S(2)/(-n + S(2)), Subst(Int(S(1)/(-a*x**S(2) + S(1)), x), x, x/sqrt(a*x**S(2) + b*x**n)), x)
rule1696 = ReplacementRule(pattern1696, replacement1696)
pattern1697 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons719, cons952, cons962)
def replacement1697(p, j, b, a, n, x):
rubi.append(1697)
return Dist((-j + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int(x**(-j)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(x**(-j + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x)
rule1697 = ReplacementRule(pattern1697, replacement1697)
pattern1698 = Pattern(Integral(S(1)/sqrt(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons964, cons965)
def replacement1698(j, b, a, n, x):
rubi.append(1698)
return -Dist(a*(-j + S(2)*n + S(-2))/(b*(n + S(-2))), Int(x**(j - n)/sqrt(a*x**j + b*x**n), x), x) + Simp(-S(2)*x**(-n + S(1))*sqrt(a*x**j + b*x**n)/(b*(n + S(-2))), x)
rule1698 = ReplacementRule(pattern1698, replacement1698)
pattern1699 = Pattern(Integral((x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons5, cons147, cons952, cons966)
def replacement1699(p, j, b, a, n, x):
rubi.append(1699)
return Dist(x**(-j*FracPart(p))*(a + b*x**(-j + n))**(-FracPart(p))*(a*x**j + b*x**n)**FracPart(p), Int(x**(j*p)*(a + b*x**(-j + n))**p, x), x)
rule1699 = ReplacementRule(pattern1699, replacement1699)
pattern1700 = Pattern(Integral((u_**WC('j', S(1))*WC('a', S(1)) + u_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons5, cons68, cons69)
def replacement1700(p, u, j, b, a, n, x):
rubi.append(1700)
return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a*x**j + b*x**n)**p, x), x, u), x)
rule1700 = ReplacementRule(pattern1700, replacement1700)
pattern1701 = Pattern(Integral(x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons796, cons21, cons4, cons5, cons147, cons952, cons967, cons53)
def replacement1701(p, j, m, b, a, n, x):
rubi.append(1701)
return Dist(S(1)/n, Subst(Int((a*x**(j/n) + b*x)**p, x), x, x**n), x)
rule1701 = ReplacementRule(pattern1701, replacement1701)
pattern1702 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons147, cons952, cons968, cons969)
def replacement1702(p, j, m, b, c, a, n, x):
rubi.append(1702)
return -Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x)
rule1702 = ReplacementRule(pattern1702, replacement1702)
pattern1703 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons147, cons952, cons970, cons13, cons137, cons969)
def replacement1703(p, j, m, b, c, a, n, x):
rubi.append(1703)
return Dist(c**j*(-j + m + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int((c*x)**(-j + m)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x)
rule1703 = ReplacementRule(pattern1703, replacement1703)
pattern1704 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons147, cons952, cons970, cons971, cons972)
def replacement1704(p, j, m, b, c, a, n, x):
rubi.append(1704)
return -Dist(b*c**(j - n)*(-j + m + n*p + n + S(1))/(a*(j*p + m + S(1))), Int((c*x)**(-j + m + n)*(a*x**j + b*x**n)**p, x), x) + Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(j*p + m + S(1))), x)
rule1704 = ReplacementRule(pattern1704, replacement1704)
pattern1705 = Pattern(Integral((c_*x_)**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons147, cons952, cons970)
def replacement1705(p, j, m, b, a, n, c, x):
rubi.append(1705)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a*x**j + b*x**n)**p, x), x)
rule1705 = ReplacementRule(pattern1705, replacement1705)
pattern1706 = Pattern(Integral(x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons796, cons21, cons4, cons5, cons147, cons952, cons967, cons500, cons973)
def replacement1706(p, j, m, b, a, n, x):
rubi.append(1706)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a*x**(j/n) + b*x)**p, x), x, x**n), x)
rule1706 = ReplacementRule(pattern1706, replacement1706)
pattern1707 = Pattern(Integral((c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons147, cons952, cons967, cons500, cons973)
def replacement1707(p, j, m, b, c, a, n, x):
rubi.append(1707)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a*x**j + b*x**n)**p, x), x)
rule1707 = ReplacementRule(pattern1707, replacement1707)
pattern1708 = Pattern(Integral((x_*WC('c', S(1)))**m_*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons147, cons974, cons957, cons972, cons163, cons975)
def replacement1708(p, j, m, b, c, n, a, x):
rubi.append(1708)
return -Dist(b*c**(-n)*p*(-j + n)/(j*p + m + S(1)), Int((c*x)**(m + n)*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a*x**j + b*x**n)**p/(c*(j*p + m + S(1))), x)
rule1708 = ReplacementRule(pattern1708, replacement1708)
pattern1709 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons147, cons956, cons957, cons972, cons163, cons512)
def replacement1709(p, j, m, b, c, a, n, x):
rubi.append(1709)
return Dist(a*c**(-j)*p*(-j + n)/(m + n*p + S(1)), Int((c*x)**(j + m)*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a*x**j + b*x**n)**p/(c*(m + n*p + S(1))), x)
rule1709 = ReplacementRule(pattern1709, replacement1709)
pattern1710 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons147, cons974, cons957, cons972, cons137, cons976)
def replacement1710(p, j, m, b, c, a, n, x):
rubi.append(1710)
return -Dist(c**n*(j*p + j + m - n + S(1))/(b*(-j + n)*(p + S(1))), Int((c*x)**(m - n)*(a*x**j + b*x**n)**(p + S(1)), x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a*x**j + b*x**n)**(p + S(1))/(b*(-j + n)*(p + S(1))), x)
rule1710 = ReplacementRule(pattern1710, replacement1710)
pattern1711 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons147, cons956, cons957, cons972, cons137)
def replacement1711(p, j, m, b, c, a, n, x):
rubi.append(1711)
return Dist(c**j*(-j + m + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int((c*x)**(-j + m)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x)
rule1711 = ReplacementRule(pattern1711, replacement1711)
pattern1712 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons147, cons964, cons957, cons972, cons977, cons512)
def replacement1712(p, j, m, b, c, a, n, x):
rubi.append(1712)
return -Dist(a*c**(-j + n)*(j*p + j + m - n + S(1))/(b*(m + n*p + S(1))), Int((c*x)**(j + m - n)*(a*x**j + b*x**n)**p, x), x) + Simp(c**(n + S(-1))*(c*x)**(m - n + S(1))*(a*x**j + b*x**n)**(p + S(1))/(b*(m + n*p + S(1))), x)
rule1712 = ReplacementRule(pattern1712, replacement1712)
pattern1713 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons147, cons964, cons957, cons972, cons978)
def replacement1713(p, j, m, b, c, a, n, x):
rubi.append(1713)
return -Dist(b*c**(j - n)*(-j + m + n*p + n + S(1))/(a*(j*p + m + S(1))), Int((c*x)**(-j + m + n)*(a*x**j + b*x**n)**p, x), x) + Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(j*p + m + S(1))), x)
rule1713 = ReplacementRule(pattern1713, replacement1713)
pattern1714 = Pattern(Integral(x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons796, cons21, cons4, cons5, cons147, cons952, cons967, cons66, cons541, cons23)
def replacement1714(p, j, m, b, a, n, x):
rubi.append(1714)
return Dist(S(1)/(m + S(1)), Subst(Int((a*x**(j/(m + S(1))) + b*x**(n/(m + S(1))))**p, x), x, x**(m + S(1))), x)
rule1714 = ReplacementRule(pattern1714, replacement1714)
pattern1715 = Pattern(Integral((c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons147, cons952, cons967, cons66, cons541, cons23)
def replacement1715(p, j, m, b, c, a, n, x):
rubi.append(1715)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a*x**j + b*x**n)**p, x), x)
rule1715 = ReplacementRule(pattern1715, replacement1715)
pattern1716 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons961, cons952, cons979, cons969)
def replacement1716(p, j, m, b, c, a, n, x):
rubi.append(1716)
return Dist(a*c**(-j), Int((c*x)**(j + m)*(a*x**j + b*x**n)**(p + S(-1)), x), x) + Simp((c*x)**(m + S(1))*(a*x**j + b*x**n)**p/(c*p*(-j + n)), x)
rule1716 = ReplacementRule(pattern1716, replacement1716)
pattern1717 = Pattern(Integral(x_**WC('m', S(1))/sqrt(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1))), x_), cons2, cons3, cons796, cons4, cons980, cons952)
def replacement1717(j, m, b, a, n, x):
rubi.append(1717)
return Dist(-S(2)/(-j + n), Subst(Int(S(1)/(-a*x**S(2) + S(1)), x), x, x**(j/S(2))/sqrt(a*x**j + b*x**n)), x)
rule1717 = ReplacementRule(pattern1717, replacement1717)
pattern1718 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons719, cons952, cons979, cons969)
def replacement1718(p, j, m, b, c, a, n, x):
rubi.append(1718)
return Dist(c**j*(-j + m + n*p + n + S(1))/(a*(-j + n)*(p + S(1))), Int((c*x)**(-j + m)*(a*x**j + b*x**n)**(p + S(1)), x), x) - Simp(c**(j + S(-1))*(c*x)**(-j + m + S(1))*(a*x**j + b*x**n)**(p + S(1))/(a*(-j + n)*(p + S(1))), x)
rule1718 = ReplacementRule(pattern1718, replacement1718)
pattern1719 = Pattern(Integral((c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons667, cons952, cons979)
def replacement1719(p, j, m, b, c, a, n, x):
rubi.append(1719)
return Dist(c**IntPart(m)*x**(-FracPart(m))*(c*x)**FracPart(m), Int(x**m*(a*x**j + b*x**n)**p, x), x)
rule1719 = ReplacementRule(pattern1719, replacement1719)
pattern1720 = Pattern(Integral((x_*WC('c', S(1)))**WC('m', S(1))*(x_**WC('j', S(1))*WC('a', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons147, cons952, cons966)
def replacement1720(p, j, m, b, c, a, n, x):
rubi.append(1720)
return Dist(c**IntPart(m)*x**(-j*FracPart(p) - FracPart(m))*(c*x)**FracPart(m)*(a + b*x**(-j + n))**(-FracPart(p))*(a*x**j + b*x**n)**FracPart(p), Int(x**(j*p + m)*(a + b*x**(-j + n))**p, x), x)
rule1720 = ReplacementRule(pattern1720, replacement1720)
pattern1721 = Pattern(Integral(u_**WC('m', S(1))*(v_**WC('j', S(1))*WC('a', S(1)) + v_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons796, cons21, cons4, cons5, cons554)
def replacement1721(v, p, u, j, m, b, a, n, x):
rubi.append(1721)
return Dist(u**m*v**(-m)/Coefficient(v, x, S(1)), Subst(Int(x**m*(a*x**j + b*x**n)**p, x), x, v), x)
rule1721 = ReplacementRule(pattern1721, replacement1721)
pattern1722 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**n_*WC('d', S(1)))**WC('q', S(1))*(x_**j_*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons796, cons797, cons21, cons4, cons5, cons50, cons147, cons981, cons967, cons982, cons500, cons973)
def replacement1722(p, j, k, m, b, d, a, c, n, x, q):
rubi.append(1722)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(c + d*x)**q*(a*x**(j/n) + b*x**(k/n))**p, x), x, x**n), x)
rule1722 = ReplacementRule(pattern1722, replacement1722)
pattern1723 = Pattern(Integral((e_*x_)**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**j_*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons796, cons797, cons21, cons4, cons5, cons50, cons147, cons981, cons967, cons982, cons500, cons973)
def replacement1723(p, j, k, m, b, d, a, n, c, x, q, e):
rubi.append(1723)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(c + d*x**n)**q*(a*x**j + b*x**k)**p, x), x)
rule1723 = ReplacementRule(pattern1723, replacement1723)
pattern1724 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons796, cons21, cons4, cons5, cons983, cons147, cons71, cons984, cons985, cons971)
def replacement1724(p, j, m, b, d, a, n, jn, c, x, e):
rubi.append(1724)
return Simp(c*e**(j + S(-1))*(e*x)**(-j + m + S(1))*(a*x**j + b*x**(j + n))**(p + S(1))/(a*(j*p + m + S(1))), x)
rule1724 = ReplacementRule(pattern1724, replacement1724)
pattern1725 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons796, cons21, cons4, cons983, cons147, cons71, cons986, cons137, cons987, cons988)
def replacement1725(p, j, m, b, d, a, n, jn, c, x, e):
rubi.append(1725)
return -Dist(e**j*(a*d*(j*p + m + S(1)) - b*c*(m + n + p*(j + n) + S(1)))/(a*b*n*(p + S(1))), Int((e*x)**(-j + m)*(a*x**j + b*x**(j + n))**(p + S(1)), x), x) - Simp(e**(j + S(-1))*(e*x)**(-j + m + S(1))*(-a*d + b*c)*(a*x**j + b*x**(j + n))**(p + S(1))/(a*b*n*(p + S(1))), x)
rule1725 = ReplacementRule(pattern1725, replacement1725)
pattern1726 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons796, cons5, cons983, cons147, cons71, cons93, cons88, cons989, cons990, cons971, cons991)
def replacement1726(p, j, m, b, d, a, n, jn, c, x, e):
rubi.append(1726)
return Dist(e**(-n)*(a*d*(j*p + m + S(1)) - b*c*(m + n + p*(j + n) + S(1)))/(a*(j*p + m + S(1))), Int((e*x)**(m + n)*(a*x**j + b*x**(j + n))**p, x), x) + Simp(c*e**(j + S(-1))*(e*x)**(-j + m + S(1))*(a*x**j + b*x**(j + n))**(p + S(1))/(a*(j*p + m + S(1))), x)
rule1726 = ReplacementRule(pattern1726, replacement1726)
pattern1727 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons796, cons21, cons4, cons5, cons983, cons147, cons71, cons992, cons988)
def replacement1727(p, j, m, b, d, a, n, jn, c, x, e):
rubi.append(1727)
return -Dist((a*d*(j*p + m + S(1)) - b*c*(m + n + p*(j + n) + S(1)))/(b*(m + n + p*(j + n) + S(1))), Int((e*x)**m*(a*x**j + b*x**(j + n))**p, x), x) + Simp(d*e**(j + S(-1))*(e*x)**(-j + m + S(1))*(a*x**j + b*x**(j + n))**(p + S(1))/(b*(m + n + p*(j + n) + S(1))), x)
rule1727 = ReplacementRule(pattern1727, replacement1727)
pattern1728 = Pattern(Integral(x_**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**j_*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons796, cons797, cons21, cons4, cons5, cons50, cons147, cons981, cons967, cons982, cons66, cons541, cons23)
def replacement1728(p, j, k, m, b, d, a, n, c, x, q):
rubi.append(1728)
return Dist(S(1)/(m + S(1)), Subst(Int((c + d*x**(n/(m + S(1))))**q*(a*x**(j/(m + S(1))) + b*x**(k/(m + S(1))))**p, x), x, x**(m + S(1))), x)
rule1728 = ReplacementRule(pattern1728, replacement1728)
pattern1729 = Pattern(Integral((e_*x_)**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**j_*WC('a', S(1)) + x_**WC('k', S(1))*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons796, cons797, cons21, cons4, cons5, cons50, cons147, cons981, cons967, cons982, cons66, cons541, cons23)
def replacement1729(p, j, k, m, b, d, a, n, c, x, q, e):
rubi.append(1729)
return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(c + d*x**n)**q*(a*x**j + b*x**k)**p, x), x)
rule1729 = ReplacementRule(pattern1729, replacement1729)
pattern1730 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(c_ + x_**WC('n', S(1))*WC('d', S(1)))**WC('q', S(1))*(x_**WC('jn', S(1))*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons796, cons21, cons4, cons5, cons50, cons983, cons147, cons71, cons993)
def replacement1730(p, j, m, b, d, a, n, jn, c, x, q, e):
rubi.append(1730)
return Dist(e**IntPart(m)*x**(-j*FracPart(p) - FracPart(m))*(e*x)**FracPart(m)*(a + b*x**n)**(-FracPart(p))*(a*x**j + b*x**(j + n))**FracPart(p), Int(x**(j*p + m)*(a + b*x**n)**p*(c + d*x**n)**q, x), x)
rule1730 = ReplacementRule(pattern1730, replacement1730)
def With1731(p, j, b, Pq, a, n, x):
d = Denominator(n)
rubi.append(1731)
return Dist(d, Subst(Int(x**(d + S(-1))*(a*x**(d*j) + b*x**(d*n))**p*ReplaceAll(SubstFor(x**n, Pq, x), Rule(x, x**(d*n))), x), x, x**(S(1)/d)), x)
pattern1731 = Pattern(Integral(Pq_*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons5, cons858, cons147, cons952, cons964, cons967, cons994)
rule1731 = ReplacementRule(pattern1731, With1731)
pattern1732 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons796, cons21, cons4, cons5, cons858, cons147, cons952, cons967, cons500)
def replacement1732(p, j, m, b, Pq, a, n, x):
rubi.append(1732)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a*x**(j/n) + b*x)**p*SubstFor(x**n, Pq, x), x), x, x**n), x)
rule1732 = ReplacementRule(pattern1732, replacement1732)
pattern1733 = Pattern(Integral(Pq_*(c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons4, cons5, cons858, cons147, cons952, cons967, cons500, cons31, cons995)
def replacement1733(p, j, m, b, Pq, a, c, n, x):
rubi.append(1733)
return Dist(c**(Quotient(m, sign(m))*sign(m))*x**(-Mod(m, sign(m)))*(c*x)**Mod(m, sign(m)), Int(Pq*x**m*(a*x**j + b*x**n)**p, x), x)
rule1733 = ReplacementRule(pattern1733, replacement1733)
pattern1734 = Pattern(Integral(Pq_*(c_*x_)**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons858, cons147, cons952, cons967, cons500)
def replacement1734(p, j, m, b, Pq, a, c, n, x):
rubi.append(1734)
return Dist(x**(-m)*(c*x)**m, Int(Pq*x**m*(a*x**j + b*x**n)**p, x), x)
rule1734 = ReplacementRule(pattern1734, replacement1734)
def With1735(p, j, m, b, Pq, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
g = GCD(m + S(1), n)
if Unequal(g, S(1)):
return True
return False
pattern1735 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons5, cons858, cons147, cons996, cons17, CustomConstraint(With1735))
def replacement1735(p, j, m, b, Pq, a, n, x):
g = GCD(m + S(1), n)
rubi.append(1735)
return Dist(S(1)/g, Subst(Int(x**(S(-1) + (m + S(1))/g)*(a*x**(j/g) + b*x**(n/g))**p*ReplaceAll(Pq, Rule(x, x**(S(1)/g))), x), x, x**g), x)
rule1735 = ReplacementRule(pattern1735, replacement1735)
def With1736(p, j, m, b, Pq, c, a, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
if And(Greater(q, n + S(-1)), Unequal(m + n*p + q + S(1), S(0)), Or(IntegerQ(S(2)*p), IntegerQ(p + (q + S(1))/(S(2)*n)))):
return True
return False
pattern1736 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons21, cons5, cons64, cons147, cons997, cons998, CustomConstraint(With1736))
def replacement1736(p, j, m, b, Pq, c, a, n, x):
q = Expon(Pq, x)
Pqq = Coeff(Pq, x, q)
rubi.append(1736)
return Int((c*x)**m*(a*x**j + b*x**n)**p*ExpandToSum(Pq - Pqq*a*x**(-n + q)*(m - n + q + 1)/(b*(m + n*p + q + 1)) - Pqq*x**q, x), x) + Simp(Pqq*c**(n - q - 1)*(c*x)**(m - n + q + 1)*(a*x**j + b*x**n)**(p + 1)/(b*(m + n*p + q + 1)), x)
rule1736 = ReplacementRule(pattern1736, replacement1736)
pattern1737 = Pattern(Integral(Pq_*x_**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons796, cons21, cons4, cons5, cons858, cons147, cons952, cons967, cons541, cons23)
def replacement1737(p, j, m, b, Pq, a, n, x):
rubi.append(1737)
return Dist(S(1)/(m + S(1)), Subst(Int((a*x**(j/(m + S(1))) + b*x**(n/(m + S(1))))**p*ReplaceAll(SubstFor(x**n, Pq, x), Rule(x, x**(n/(m + S(1))))), x), x, x**(m + S(1))), x)
rule1737 = ReplacementRule(pattern1737, replacement1737)
pattern1738 = Pattern(Integral(Pq_*(c_*x_)**m_*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons4, cons5, cons858, cons147, cons952, cons967, cons541, cons23, cons31, cons995)
def replacement1738(p, j, m, b, Pq, c, a, n, x):
rubi.append(1738)
return Dist(c**(Quotient(m, sign(m))*sign(m))*x**(-Mod(m, sign(m)))*(c*x)**Mod(m, sign(m)), Int(Pq*x**m*(a*x**j + b*x**n)**p, x), x)
rule1738 = ReplacementRule(pattern1738, replacement1738)
pattern1739 = Pattern(Integral(Pq_*(c_*x_)**m_*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons858, cons147, cons952, cons967, cons541, cons23)
def replacement1739(p, j, m, b, Pq, c, a, n, x):
rubi.append(1739)
return Dist(x**(-m)*(c*x)**m, Int(Pq*x**m*(a*x**j + b*x**n)**p, x), x)
rule1739 = ReplacementRule(pattern1739, replacement1739)
pattern1740 = Pattern(Integral(Pq_*(x_*WC('c', S(1)))**WC('m', S(1))*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons7, cons796, cons21, cons4, cons5, cons918, cons147, cons952)
def replacement1740(p, j, m, b, Pq, c, a, n, x):
rubi.append(1740)
return Int(ExpandIntegrand(Pq*(c*x)**m*(a*x**j + b*x**n)**p, x), x)
rule1740 = ReplacementRule(pattern1740, replacement1740)
pattern1741 = Pattern(Integral(Pq_*(x_**n_*WC('b', S(1)) + x_**WC('j', S(1))*WC('a', S(1)))**p_, x_), cons2, cons3, cons796, cons4, cons5, cons918, cons147, cons952)
def replacement1741(p, j, b, Pq, a, n, x):
rubi.append(1741)
return Int(ExpandIntegrand(Pq*(a*x**j + b*x**n)**p, x), x)
rule1741 = ReplacementRule(pattern1741, replacement1741)
pattern1742 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons38, cons999)
def replacement1742(p, b, d, a, x):
rubi.append(1742)
return Dist(S(3)**(-S(3)*p)*a**(-S(2)*p), Int((S(3)*a - b*x)**p*(S(3)*a + S(2)*b*x)**(S(2)*p), x), x)
rule1742 = ReplacementRule(pattern1742, replacement1742)
pattern1743 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons128, cons1000)
def replacement1743(p, b, d, a, x):
rubi.append(1743)
return Int(ExpandToSum((a + b*x + d*x**S(3))**p, x), x)
rule1743 = ReplacementRule(pattern1743, replacement1743)
def With1744(p, b, d, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = Factor(a + b*x + d*x**S(3))
if ProductQ(NonfreeFactors(u, x)):
return True
return False
pattern1744 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons63, cons1000, CustomConstraint(With1744))
def replacement1744(p, b, d, a, x):
u = Factor(a + b*x + d*x**S(3))
rubi.append(1744)
return Dist(FreeFactors(u, x)**p, Int(DistributeDegree(NonfreeFactors(u, x), p), x), x)
rule1744 = ReplacementRule(pattern1744, replacement1744)
def With1745(p, b, d, a, x):
r = Rt(-S(27)*a*d**S(2) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*b**S(3)*d), S(3))
rubi.append(1745)
return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p, x), x)
pattern1745 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons63, cons1000)
rule1745 = ReplacementRule(pattern1745, With1745)
pattern1746 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons5, cons147, cons999)
def replacement1746(p, b, d, a, x):
rubi.append(1746)
return Dist((S(3)*a - b*x)**(-p)*(S(3)*a + S(2)*b*x)**(-S(2)*p)*(a + b*x + d*x**S(3))**p, Int((S(3)*a - b*x)**p*(S(3)*a + S(2)*b*x)**(S(2)*p), x), x)
rule1746 = ReplacementRule(pattern1746, replacement1746)
def With1747(p, b, d, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = NonfreeFactors(Factor(a + b*x + d*x**S(3)), x)
if ProductQ(u):
return True
return False
pattern1747 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons5, cons147, cons1000, CustomConstraint(With1747))
def replacement1747(p, b, d, a, x):
u = NonfreeFactors(Factor(a + b*x + d*x**S(3)), x)
rubi.append(1747)
return Dist((a + b*x + d*x**S(3))**p/DistributeDegree(u, p), Int(DistributeDegree(u, p), x), x)
rule1747 = ReplacementRule(pattern1747, replacement1747)
def With1748(p, b, d, a, x):
r = Rt(-S(27)*a*d**S(2) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*b**S(3)*d), S(3))
rubi.append(1748)
return Dist((-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**(-p)*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**(-p)*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(a + b*x + d*x**S(3))**p, Int((-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p, x), x)
pattern1748 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons5, cons147, cons1000)
rule1748 = ReplacementRule(pattern1748, With1748)
pattern1749 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons38, cons999)
def replacement1749(p, m, f, b, d, a, x, e):
rubi.append(1749)
return Dist(S(3)**(-S(3)*p)*a**(-S(2)*p), Int((S(3)*a - b*x)**p*(S(3)*a + S(2)*b*x)**(S(2)*p)*(e + f*x)**m, x), x)
rule1749 = ReplacementRule(pattern1749, replacement1749)
pattern1750 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons128, cons1000)
def replacement1750(p, m, f, b, d, a, x, e):
rubi.append(1750)
return Int(ExpandIntegrand((e + f*x)**m*(a + b*x + d*x**S(3))**p, x), x)
rule1750 = ReplacementRule(pattern1750, replacement1750)
def With1751(p, m, f, b, d, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = Factor(a + b*x + d*x**S(3))
if ProductQ(NonfreeFactors(u, x)):
return True
return False
pattern1751 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons63, cons1000, CustomConstraint(With1751))
def replacement1751(p, m, f, b, d, a, x, e):
u = Factor(a + b*x + d*x**S(3))
rubi.append(1751)
return Dist(FreeFactors(u, x)**p, Int((e + f*x)**m*DistributeDegree(NonfreeFactors(u, x), p), x), x)
rule1751 = ReplacementRule(pattern1751, replacement1751)
def With1752(p, m, f, b, d, a, x, e):
r = Rt(-S(27)*a*d**S(2) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*b**S(3)*d), S(3))
rubi.append(1752)
return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((e + f*x)**m*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p, x), x)
pattern1752 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons63, cons1000)
rule1752 = ReplacementRule(pattern1752, With1752)
pattern1753 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons5, cons147, cons999)
def replacement1753(p, m, f, b, d, a, x, e):
rubi.append(1753)
return Dist((S(3)*a - b*x)**(-p)*(S(3)*a + S(2)*b*x)**(-S(2)*p)*(a + b*x + d*x**S(3))**p, Int((S(3)*a - b*x)**p*(S(3)*a + S(2)*b*x)**(S(2)*p)*(e + f*x)**m, x), x)
rule1753 = ReplacementRule(pattern1753, replacement1753)
def With1754(p, m, f, b, d, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = NonfreeFactors(Factor(a + b*x + d*x**S(3)), x)
if ProductQ(u):
return True
return False
pattern1754 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons5, cons147, cons1000, CustomConstraint(With1754))
def replacement1754(p, m, f, b, d, a, x, e):
u = NonfreeFactors(Factor(a + b*x + d*x**S(3)), x)
rubi.append(1754)
return Dist((a + b*x + d*x**S(3))**p/DistributeDegree(u, p), Int((e + f*x)**m*DistributeDegree(u, p), x), x)
rule1754 = ReplacementRule(pattern1754, replacement1754)
def With1755(p, m, f, b, d, a, x, e):
r = Rt(-S(27)*a*d**S(2) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*b**S(3)*d), S(3))
rubi.append(1755)
return Dist((-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**(-p)*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**(-p)*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(a + b*x + d*x**S(3))**p, Int((e + f*x)**m*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) - sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(-S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p*(S(3)*d*x + S(2)**(S(1)/3)*(S(6)*b*d - S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p, x), x)
pattern1755 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons27, cons48, cons125, cons21, cons5, cons147, cons1000)
rule1755 = ReplacementRule(pattern1755, With1755)
pattern1756 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons38, cons1001)
def replacement1756(p, d, a, c, x):
rubi.append(1756)
return -Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((c - S(3)*d*x)**p*(S(2)*c + S(3)*d*x)**(S(2)*p), x), x)
rule1756 = ReplacementRule(pattern1756, replacement1756)
pattern1757 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons128, cons1002)
def replacement1757(p, d, a, c, x):
rubi.append(1757)
return Int(ExpandToSum((a + c*x**S(2) + d*x**S(3))**p, x), x)
rule1757 = ReplacementRule(pattern1757, replacement1757)
def With1758(p, d, a, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = Factor(a + c*x**S(2) + d*x**S(3))
if ProductQ(NonfreeFactors(u, x)):
return True
return False
pattern1758 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons63, cons1002, CustomConstraint(With1758))
def replacement1758(p, d, a, c, x):
u = Factor(a + c*x**S(2) + d*x**S(3))
rubi.append(1758)
return Dist(FreeFactors(u, x)**p, Int(DistributeDegree(NonfreeFactors(u, x), p), x), x)
rule1758 = ReplacementRule(pattern1758, replacement1758)
def With1759(p, d, a, c, x):
r = Rt(-S(27)*a*d**S(2) - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*a*c**S(3)), S(3))
rubi.append(1759)
return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p, x), x)
pattern1759 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons63, cons1002)
rule1759 = ReplacementRule(pattern1759, With1759)
pattern1760 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons5, cons147, cons1001)
def replacement1760(p, d, a, c, x):
rubi.append(1760)
return Dist((c - S(3)*d*x)**(-p)*(S(2)*c + S(3)*d*x)**(-S(2)*p)*(a + c*x**S(2) + d*x**S(3))**p, Int((c - S(3)*d*x)**p*(S(2)*c + S(3)*d*x)**(S(2)*p), x), x)
rule1760 = ReplacementRule(pattern1760, replacement1760)
def With1761(p, d, a, c, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = NonfreeFactors(Factor(a + c*x**S(2) + d*x**S(3)), x)
if ProductQ(u):
return True
return False
pattern1761 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons5, cons147, cons1002, CustomConstraint(With1761))
def replacement1761(p, d, a, c, x):
u = NonfreeFactors(Factor(a + c*x**S(2) + d*x**S(3)), x)
rubi.append(1761)
return Dist((a + c*x**S(2) + d*x**S(3))**p/DistributeDegree(u, p), Int(DistributeDegree(u, p), x), x)
rule1761 = ReplacementRule(pattern1761, replacement1761)
def With1762(p, d, a, c, x):
r = Rt(-S(27)*a*d**S(2) - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*a*c**S(3)), S(3))
rubi.append(1762)
return Dist((a + c*x**S(2) + d*x**S(3))**p*(c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**(-p), Int((c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p, x), x)
pattern1762 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons5, cons147, cons1002)
rule1762 = ReplacementRule(pattern1762, With1762)
pattern1763 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons38, cons1001)
def replacement1763(p, m, f, d, c, a, x, e):
rubi.append(1763)
return -Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((c - S(3)*d*x)**p*(S(2)*c + S(3)*d*x)**(S(2)*p)*(e + f*x)**m, x), x)
rule1763 = ReplacementRule(pattern1763, replacement1763)
pattern1764 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons128, cons1002)
def replacement1764(p, m, f, d, c, a, x, e):
rubi.append(1764)
return Int(ExpandIntegrand((e + f*x)**m*(a + c*x**S(2) + d*x**S(3))**p, x), x)
rule1764 = ReplacementRule(pattern1764, replacement1764)
def With1765(p, m, f, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = Factor(a + c*x**S(2) + d*x**S(3))
if ProductQ(NonfreeFactors(u, x)):
return True
return False
pattern1765 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons63, cons1002, CustomConstraint(With1765))
def replacement1765(p, m, f, d, c, a, x, e):
u = Factor(a + c*x**S(2) + d*x**S(3))
rubi.append(1765)
return Dist(FreeFactors(u, x)**p, Int((e + f*x)**m*DistributeDegree(NonfreeFactors(u, x), p), x), x)
rule1765 = ReplacementRule(pattern1765, replacement1765)
def With1766(p, m, f, d, c, a, x, e):
r = Rt(-S(27)*a*d**S(2) - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*a*c**S(3)), S(3))
rubi.append(1766)
return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Int((e + f*x)**m*(c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p, x), x)
pattern1766 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons63, cons1002)
rule1766 = ReplacementRule(pattern1766, With1766)
pattern1767 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons5, cons147, cons1001)
def replacement1767(p, m, f, d, c, a, x, e):
rubi.append(1767)
return Dist((c - S(3)*d*x)**(-p)*(S(2)*c + S(3)*d*x)**(-S(2)*p)*(a + c*x**S(2) + d*x**S(3))**p, Int((c - S(3)*d*x)**p*(S(2)*c + S(3)*d*x)**(S(2)*p)*(e + f*x)**m, x), x)
rule1767 = ReplacementRule(pattern1767, replacement1767)
def With1768(p, m, f, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = NonfreeFactors(Factor(a + c*x**S(2) + d*x**S(3)), x)
if ProductQ(u):
return True
return False
pattern1768 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons5, cons147, cons1002, CustomConstraint(With1768))
def replacement1768(p, m, f, d, c, a, x, e):
u = NonfreeFactors(Factor(a + c*x**S(2) + d*x**S(3)), x)
rubi.append(1768)
return Dist((a + c*x**S(2) + d*x**S(3))**p/DistributeDegree(u, p), Int((e + f*x)**m*DistributeDegree(u, p), x), x)
rule1768 = ReplacementRule(pattern1768, replacement1768)
def With1769(p, m, f, d, c, a, x, e):
r = Rt(-S(27)*a*d**S(2) - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) + S(4)*a*c**S(3)), S(3))
rubi.append(1769)
return Dist((a + c*x**S(2) + d*x**S(3))**p*(c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**(-p), Int((e + f*x)**m*(c + S(3)*d*x - S(2)**(S(1)/3)*(S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) + sqrt(S(3))*I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) + S(2)**(S(1)/3)*r**S(2)*(S(1) - sqrt(S(3))*I))/(S(4)*r))**p, x), x)
pattern1769 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons7, cons27, cons48, cons125, cons21, cons5, cons147, cons1002)
rule1769 = ReplacementRule(pattern1769, With1769)
pattern1770 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons38, cons1003, cons1004)
def replacement1770(p, b, d, c, a, x):
rubi.append(1770)
return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((b + c*x)**(S(3)*p), x), x)
rule1770 = ReplacementRule(pattern1770, replacement1770)
pattern1771 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons38, cons1003, cons1005)
def replacement1771(p, b, d, c, a, x):
rubi.append(1771)
return Dist(S(3)**(-p)*b**(-p)*c**(-p), Subst(Int((S(3)*a*b*c - b**S(3) + c**S(3)*x**S(3))**p, x), x, c/(S(3)*d) + x), x)
rule1771 = ReplacementRule(pattern1771, replacement1771)
def With1772(p, b, d, c, a, x):
r = Rt(-S(3)*b*c*d + c**S(3), S(3))
rubi.append(1772)
return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((b + x*(c - r))**p*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**p*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**p, x), x)
pattern1772 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons38, cons1006, cons1004)
rule1772 = ReplacementRule(pattern1772, With1772)
pattern1773 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons128, cons1006, cons1005)
def replacement1773(p, b, d, c, a, x):
rubi.append(1773)
return Int(ExpandToSum((a + b*x + c*x**S(2) + d*x**S(3))**p, x), x)
rule1773 = ReplacementRule(pattern1773, replacement1773)
def With1774(p, b, d, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = Factor(a + b*x + c*x**S(2) + d*x**S(3))
if ProductQ(NonfreeFactors(u, x)):
return True
return False
pattern1774 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons63, cons1006, cons1005, CustomConstraint(With1774))
def replacement1774(p, b, d, c, a, x):
u = Factor(a + b*x + c*x**S(2) + d*x**S(3))
rubi.append(1774)
return Dist(FreeFactors(u, x)**p, Int(DistributeDegree(NonfreeFactors(u, x), p), x), x)
rule1774 = ReplacementRule(pattern1774, replacement1774)
pattern1775 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons63, cons1006, cons1005)
def replacement1775(p, b, d, c, a, x):
rubi.append(1775)
return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Subst(Int((S(27)*a*d**S(2) - S(9)*b*c*d + S(2)*c**S(3) + S(27)*d**S(3)*x**S(3) - S(9)*d*x*(-S(3)*b*d + c**S(2)))**p, x), x, c/(S(3)*d) + x), x)
rule1775 = ReplacementRule(pattern1775, replacement1775)
pattern1776 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons147, cons1003, cons1004)
def replacement1776(p, b, d, c, a, x):
rubi.append(1776)
return Dist((b + c*x)**(-S(3)*p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + c*x)**(S(3)*p), x), x)
rule1776 = ReplacementRule(pattern1776, replacement1776)
def With1777(p, b, d, c, a, x):
r = Rt(-S(3)*a*b*c + b**S(3), S(3))
rubi.append(1777)
return Dist((b + c*x - r)**(-p)*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**(-p)*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + c*x - r)**p*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**p*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**p, x), x)
pattern1777 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons147, cons1003, cons1005)
rule1777 = ReplacementRule(pattern1777, With1777)
def With1778(p, b, d, c, a, x):
r = Rt(-S(3)*b*c*d + c**S(3), S(3))
rubi.append(1778)
return Dist((b + x*(c - r))**(-p)*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**(-p)*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + x*(c - r))**p*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**p*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**p, x), x)
pattern1778 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons147, cons1006, cons1004)
rule1778 = ReplacementRule(pattern1778, With1778)
def With1779(p, b, d, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
u = NonfreeFactors(Factor(a + b*x + c*x**S(2) + d*x**S(3)), x)
if ProductQ(u):
return True
return False
pattern1779 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons147, cons1006, cons1005, CustomConstraint(With1779))
def replacement1779(p, b, d, c, a, x):
u = NonfreeFactors(Factor(a + b*x + c*x**S(2) + d*x**S(3)), x)
rubi.append(1779)
return Dist((a + b*x + c*x**S(2) + d*x**S(3))**p/DistributeDegree(u, p), Int(DistributeDegree(u, p), x), x)
rule1779 = ReplacementRule(pattern1779, replacement1779)
def With1780(p, b, d, c, a, x):
r = Rt(-S(27)*a*d**S(2) + S(9)*b*c*d - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) - S(18)*a*b*c*d + S(4)*a*c**S(3) + S(4)*b**S(3)*d - b**S(2)*c**S(2)), S(3))
rubi.append(1780)
return Dist((c + S(3)*d*x - S(2)**(S(1)/3)*(-S(6)*b*d + S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) - sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) - I))/(S(4)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) + sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) + I))/(S(4)*r))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((c + S(3)*d*x - S(2)**(S(1)/3)*(-S(6)*b*d + S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) - sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) - I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) + sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) + I))/(S(4)*r))**p, x), x)
pattern1780 = Pattern(Integral((x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons5, cons147, cons1006, cons1005)
rule1780 = ReplacementRule(pattern1780, With1780)
pattern1781 = Pattern(Integral(u_**p_, x_), cons5, cons1007, cons1008)
def replacement1781(x, p, u):
rubi.append(1781)
return Int(ExpandToSum(u, x)**p, x)
rule1781 = ReplacementRule(pattern1781, replacement1781)
pattern1782 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons38, cons1003, cons1004)
def replacement1782(p, m, f, b, d, a, c, x, e):
rubi.append(1782)
return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((b + c*x)**(S(3)*p)*(e + f*x)**m, x), x)
rule1782 = ReplacementRule(pattern1782, replacement1782)
def With1783(p, m, f, b, d, a, c, x, e):
r = Rt(-S(3)*a*b*c + b**S(3), S(3))
rubi.append(1783)
return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((e + f*x)**m*(b + c*x - r)**p*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**p*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**p, x), x)
pattern1783 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons38, cons1003, cons1005)
rule1783 = ReplacementRule(pattern1783, With1783)
def With1784(p, m, f, b, d, a, c, x, e):
r = Rt(-S(3)*b*c*d + c**S(3), S(3))
rubi.append(1784)
return Dist(S(3)**(-p)*b**(-p)*c**(-p), Int((b + x*(c - r))**p*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**p*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**p*(e + f*x)**m, x), x)
pattern1784 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons38, cons1006, cons1004)
rule1784 = ReplacementRule(pattern1784, With1784)
pattern1785 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons128, cons1006, cons1005)
def replacement1785(p, m, f, b, d, a, c, x, e):
rubi.append(1785)
return Int(ExpandIntegrand((e + f*x)**m*(a + b*x + c*x**S(2) + d*x**S(3))**p, x), x)
rule1785 = ReplacementRule(pattern1785, replacement1785)
def With1786(p, m, f, b, d, a, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = Factor(a + b*x + c*x**S(2) + d*x**S(3))
if ProductQ(NonfreeFactors(u, x)):
return True
return False
pattern1786 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons63, cons1006, cons1005, CustomConstraint(With1786))
def replacement1786(p, m, f, b, d, a, c, x, e):
u = Factor(a + b*x + c*x**S(2) + d*x**S(3))
rubi.append(1786)
return Dist(FreeFactors(u, x)**p, Int((e + f*x)**m*DistributeDegree(NonfreeFactors(u, x), p), x), x)
rule1786 = ReplacementRule(pattern1786, replacement1786)
pattern1787 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons63, cons1006, cons1005)
def replacement1787(p, m, f, b, d, a, c, x, e):
rubi.append(1787)
return Dist(S(3)**(-S(3)*p)*d**(-S(2)*p), Subst(Int((S(27)*a*d**S(2) - S(9)*b*c*d + S(2)*c**S(3) + S(27)*d**S(3)*x**S(3) - S(9)*d*x*(-S(3)*b*d + c**S(2)))**p, x), x, c/(S(3)*d) + x), x)
rule1787 = ReplacementRule(pattern1787, replacement1787)
pattern1788 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons147, cons1003, cons1004)
def replacement1788(p, m, f, b, d, a, c, x, e):
rubi.append(1788)
return Dist((b + c*x)**(-S(3)*p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + c*x)**(S(3)*p)*(e + f*x)**m, x), x)
rule1788 = ReplacementRule(pattern1788, replacement1788)
def With1789(p, m, f, b, d, a, c, x, e):
r = Rt(-S(3)*a*b*c + b**S(3), S(3))
rubi.append(1789)
return Dist((b + c*x - r)**(-p)*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**(-p)*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((e + f*x)**m*(b + c*x - r)**p*(b + c*x + r*(S(1) - sqrt(S(3))*I)/S(2))**p*(b + c*x + r*(S(1) + sqrt(S(3))*I)/S(2))**p, x), x)
pattern1789 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons147, cons1003, cons1005)
rule1789 = ReplacementRule(pattern1789, With1789)
def With1790(p, m, f, b, d, a, c, x, e):
r = Rt(-S(3)*b*c*d + c**S(3), S(3))
rubi.append(1790)
return Dist((b + x*(c - r))**(-p)*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**(-p)*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((b + x*(c - r))**p*(b + x*(c + r*(S(1) - sqrt(S(3))*I)/S(2)))**p*(b + x*(c + r*(S(1) + sqrt(S(3))*I)/S(2)))**p*(e + f*x)**m, x), x)
pattern1790 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons147, cons1006, cons1004)
rule1790 = ReplacementRule(pattern1790, With1790)
def With1791(p, m, f, b, d, a, c, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
u = NonfreeFactors(Factor(a + b*x + c*x**S(2) + d*x**S(3)), x)
if ProductQ(u):
return True
return False
pattern1791 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons147, cons1006, cons1005, CustomConstraint(With1791))
def replacement1791(p, m, f, b, d, a, c, x, e):
u = NonfreeFactors(Factor(a + b*x + c*x**S(2) + d*x**S(3)), x)
rubi.append(1791)
return Dist((a + b*x + c*x**S(2) + d*x**S(3))**p/DistributeDegree(u, p), Int((e + f*x)**m*DistributeDegree(u, p), x), x)
rule1791 = ReplacementRule(pattern1791, replacement1791)
def With1792(p, m, f, b, d, a, c, x, e):
r = Rt(-S(27)*a*d**S(2) + S(9)*b*c*d - S(2)*c**S(3) + S(3)*sqrt(S(3))*d*sqrt(S(27)*a**S(2)*d**S(2) - S(18)*a*b*c*d + S(4)*a*c**S(3) + S(4)*b**S(3)*d - b**S(2)*c**S(2)), S(3))
rubi.append(1792)
return Dist((c + S(3)*d*x - S(2)**(S(1)/3)*(-S(6)*b*d + S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) - sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) - I))/(S(4)*r))**(-p)*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) + sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) + I))/(S(4)*r))**(-p)*(a + b*x + c*x**S(2) + d*x**S(3))**p, Int((e + f*x)**m*(c + S(3)*d*x - S(2)**(S(1)/3)*(-S(6)*b*d + S(2)*c**S(2) + S(2)**(S(1)/3)*r**S(2))/(S(2)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) - sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) - sqrt(S(3))*I) + S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) - I))/(S(4)*r))**p*(c + S(3)*d*x + S(2)**(S(1)/3)*(-S(6)*b*d*(S(1) + sqrt(S(3))*I) + S(2)*c**S(2)*(S(1) + sqrt(S(3))*I) - S(2)**(S(1)/3)*I*r**S(2)*(sqrt(S(3)) + I))/(S(4)*r))**p, x), x)
pattern1792 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*(x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons5, cons147, cons1006, cons1005)
rule1792 = ReplacementRule(pattern1792, With1792)
pattern1793 = Pattern(Integral(u_**WC('m', S(1))*v_**WC('p', S(1)), x_), cons21, cons5, cons68, cons1009, cons1010)
def replacement1793(v, p, u, m, x):
rubi.append(1793)
return Int(ExpandToSum(u, x)**m*ExpandToSum(v, x)**p, x)
rule1793 = ReplacementRule(pattern1793, replacement1793)
pattern1794 = Pattern(Integral((f_ + x_**S(2)*WC('g', S(1)))/((d_ + x_**S(2)*WC('d', S(1)) + x_*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('a', S(1)) + x_**S(3)*WC('b', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons383, cons1011, cons1012)
def replacement1794(g, b, f, d, c, a, x, e):
rubi.append(1794)
return Simp(a*f*ArcTan((a*b*x**S(2) + a*b + x*(S(4)*a**S(2) - S(2)*a*c + b**S(2)))/(S(2)*sqrt(a*x**S(4) + a + b*x**S(3) + b*x + c*x**S(2))*Rt(a**S(2)*(S(2)*a - c), S(2))))/(d*Rt(a**S(2)*(S(2)*a - c), S(2))), x)
rule1794 = ReplacementRule(pattern1794, replacement1794)
pattern1795 = Pattern(Integral((f_ + x_**S(2)*WC('g', S(1)))/((d_ + x_**S(2)*WC('d', S(1)) + x_*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('a', S(1)) + x_**S(3)*WC('b', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons383, cons1011, cons1013)
def replacement1795(g, b, f, d, c, a, x, e):
rubi.append(1795)
return -Simp(a*f*atanh((a*b*x**S(2) + a*b + x*(S(4)*a**S(2) - S(2)*a*c + b**S(2)))/(S(2)*sqrt(a*x**S(4) + a + b*x**S(3) + b*x + c*x**S(2))*Rt(-a**S(2)*(S(2)*a - c), S(2))))/(d*Rt(-a**S(2)*(S(2)*a - c), S(2))), x)
rule1795 = ReplacementRule(pattern1795, replacement1795)
pattern1796 = Pattern(Integral((x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1014, cons1015, cons1016)
def replacement1796(p, b, d, c, a, x, e):
rubi.append(1796)
return Subst(Int(SimplifyIntegrand((a - b*d/(S(8)*e) + d**S(4)/(S(256)*e**S(3)) + e*x**S(4) + x**S(2)*(c - S(3)*d**S(2)/(S(8)*e)))**p, x), x), x, d/(S(4)*e) + x)
rule1796 = ReplacementRule(pattern1796, replacement1796)
def With1797(v, x, p):
if isinstance(x, (int, Integer, float, Float)):
return False
a = Coefficient(v, x, S(0))
b = Coefficient(v, x, S(1))
c = Coefficient(v, x, S(2))
d = Coefficient(v, x, S(3))
e = Coefficient(v, x, S(4))
if And(ZeroQ(S(8)*b*e**S(2) - S(4)*c*d*e + d**S(3)), NonzeroQ(d)):
return True
return False
pattern1797 = Pattern(Integral(v_**p_, x_), cons5, cons1017, cons1018, cons1015, cons1016, CustomConstraint(With1797))
def replacement1797(v, x, p):
a = Coefficient(v, x, S(0))
b = Coefficient(v, x, S(1))
c = Coefficient(v, x, S(2))
d = Coefficient(v, x, S(3))
e = Coefficient(v, x, S(4))
rubi.append(1797)
return Subst(Int(SimplifyIntegrand((a - b*d/(S(8)*e) + d**S(4)/(S(256)*e**S(3)) + e*x**S(4) + x**S(2)*(c - S(3)*d**S(2)/(S(8)*e)))**p, x), x), x, d/(S(4)*e) + x)
rule1797 = ReplacementRule(pattern1797, replacement1797)
pattern1798 = Pattern(Integral(u_*(x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons5, cons804, cons1014, cons357)
def replacement1798(p, u, b, d, c, a, x, e):
rubi.append(1798)
return Subst(Int(SimplifyIntegrand((a - b*d/(S(8)*e) + d**S(4)/(S(256)*e**S(3)) + e*x**S(4) + x**S(2)*(c - S(3)*d**S(2)/(S(8)*e)))**p*ReplaceAll(u, Rule(x, -d/(S(4)*e) + x)), x), x), x, d/(S(4)*e) + x)
rule1798 = ReplacementRule(pattern1798, replacement1798)
def With1799(v, x, p, u):
if isinstance(x, (int, Integer, float, Float)):
return False
a = Coefficient(v, x, S(0))
b = Coefficient(v, x, S(1))
c = Coefficient(v, x, S(2))
d = Coefficient(v, x, S(3))
e = Coefficient(v, x, S(4))
if And(ZeroQ(S(8)*b*e**S(2) - S(4)*c*d*e + d**S(3)), NonzeroQ(d)):
return True
return False
pattern1799 = Pattern(Integral(u_*v_**p_, x_), cons5, cons804, cons1017, cons1018, cons357, CustomConstraint(With1799))
def replacement1799(v, x, p, u):
a = Coefficient(v, x, S(0))
b = Coefficient(v, x, S(1))
c = Coefficient(v, x, S(2))
d = Coefficient(v, x, S(3))
e = Coefficient(v, x, S(4))
rubi.append(1799)
return Subst(Int(SimplifyIntegrand((a - b*d/(S(8)*e) + d**S(4)/(S(256)*e**S(3)) + e*x**S(4) + x**S(2)*(c - S(3)*d**S(2)/(S(8)*e)))**p*ReplaceAll(u, Rule(x, -d/(S(4)*e) + x)), x), x), x, d/(S(4)*e) + x)
rule1799 = ReplacementRule(pattern1799, replacement1799)
pattern1800 = Pattern(Integral((a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons1019, cons246)
def replacement1800(p, b, d, c, a, x, e):
rubi.append(1800)
return Dist(-S(16)*a**S(2), Subst(Int((a*(S(256)*a**S(4)*x**S(4) + S(256)*a**S(3)*e - S(64)*a**S(2)*b*d - S(32)*a**S(2)*x**S(2)*(-S(8)*a*c + S(3)*b**S(2)) + S(16)*a*b**S(2)*c - S(3)*b**S(4))/(-S(4)*a*x + b)**S(4))**p/(-S(4)*a*x + b)**S(2), x), x, S(1)/x + b/(S(4)*a)), x)
rule1800 = ReplacementRule(pattern1800, replacement1800)
def With1801(v, x, p):
if isinstance(x, (int, Integer, float, Float)):
return False
a = Coefficient(v, x, S(0))
b = Coefficient(v, x, S(1))
c = Coefficient(v, x, S(2))
d = Coefficient(v, x, S(3))
e = Coefficient(v, x, S(4))
if And(NonzeroQ(a), NonzeroQ(b), ZeroQ(S(8)*a**S(2)*d - S(4)*a*b*c + b**S(3))):
return True
return False
pattern1801 = Pattern(Integral(v_**p_, x_), cons5, cons1017, cons1018, cons246, CustomConstraint(With1801))
def replacement1801(v, x, p):
a = Coefficient(v, x, S(0))
b = Coefficient(v, x, S(1))
c = Coefficient(v, x, S(2))
d = Coefficient(v, x, S(3))
e = Coefficient(v, x, S(4))
rubi.append(1801)
return Dist(-S(16)*a**S(2), Subst(Int((a*(S(256)*a**S(4)*x**S(4) + S(256)*a**S(3)*e - S(64)*a**S(2)*b*d - S(32)*a**S(2)*x**S(2)*(-S(8)*a*c + S(3)*b**S(2)) + S(16)*a*b**S(2)*c - S(3)*b**S(4))/(-S(4)*a*x + b)**S(4))**p/(-S(4)*a*x + b)**S(2), x), x, S(1)/x + b/(S(4)*a)), x)
rule1801 = ReplacementRule(pattern1801, replacement1801)
def With1802(B, C, e, b, D, d, c, a, x, A):
q = sqrt(S(8)*a**S(2) - S(4)*a*c + b**S(2))
rubi.append(1802)
return -Dist(S(1)/q, Int((A*b - A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a - S(2)*C*a + D*b - D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b - q)), x), x) + Dist(S(1)/q, Int((A*b + A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a - S(2)*C*a + D*b + D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b + q)), x), x)
pattern1802 = Pattern(Integral((x_**S(3)*WC('D', S(1)) + x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons34, cons35, cons36, cons1023, cons1020, cons1021, cons1022)
rule1802 = ReplacementRule(pattern1802, With1802)
def With1803(B, e, b, D, d, c, a, x, A):
q = sqrt(S(8)*a**S(2) - S(4)*a*c + b**S(2))
rubi.append(1803)
return -Dist(S(1)/q, Int((A*b - A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a + D*b - D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b - q)), x), x) + Dist(S(1)/q, Int((A*b + A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a + D*b + D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b + q)), x), x)
pattern1803 = Pattern(Integral((x_**S(3)*WC('D', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons34, cons35, cons1023, cons1020, cons1021, cons1022)
rule1803 = ReplacementRule(pattern1803, With1803)
def With1804(B, C, e, m, b, D, d, c, a, x, A):
q = sqrt(S(8)*a**S(2) - S(4)*a*c + b**S(2))
rubi.append(1804)
return -Dist(S(1)/q, Int(x**m*(A*b - A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a - S(2)*C*a + D*b - D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b - q)), x), x) + Dist(S(1)/q, Int(x**m*(A*b + A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a - S(2)*C*a + D*b + D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b + q)), x), x)
pattern1804 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(3)*WC('D', S(1)) + x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons34, cons35, cons36, cons1023, cons21, cons1020, cons1021, cons1022)
rule1804 = ReplacementRule(pattern1804, With1804)
def With1805(B, e, m, b, D, d, c, a, x, A):
q = sqrt(S(8)*a**S(2) - S(4)*a*c + b**S(2))
rubi.append(1805)
return -Dist(S(1)/q, Int(x**m*(A*b - A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a + D*b - D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b - q)), x), x) + Dist(S(1)/q, Int(x**m*(A*b + A*q - S(2)*B*a + S(2)*D*a + x*(S(2)*A*a + D*b + D*q))/(S(2)*a*x**S(2) + S(2)*a + x*(b + q)), x), x)
pattern1805 = Pattern(Integral(x_**WC('m', S(1))*(x_**S(3)*WC('D', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons34, cons35, cons1023, cons21, cons1020, cons1021, cons1022)
rule1805 = ReplacementRule(pattern1805, With1805)
def With1806(B, C, e, b, d, c, a, x, A):
q = Rt(C*(C*(-S(4)*c*e + d**S(2)) + S(2)*e*(-S(4)*A*e + B*d)), S(2))
rubi.append(1806)
return Simp(-S(2)*C**S(2)*atanh((-B*e + C*d + S(2)*C*e*x)/q)/q, x) + Simp(S(2)*C**S(2)*atanh(C*(S(12)*A*B*e - S(4)*A*C*d - S(3)*B**S(2)*d + S(4)*B*C*c + S(8)*C**S(2)*e*x**S(3) + S(4)*C*x**S(2)*(-B*e + S(2)*C*d) + S(4)*C*x*(S(2)*A*e - B*d + S(2)*C*c))/(q*(-S(4)*A*C + B**S(2))))/q, x)
pattern1806 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1024, cons1025, cons1026)
rule1806 = ReplacementRule(pattern1806, With1806)
def With1807(C, e, b, d, c, a, x, A):
q = Rt(C*(-S(8)*A*e**S(2) + C*(-S(4)*c*e + d**S(2))), S(2))
rubi.append(1807)
return Simp(-S(2)*C**S(2)*atanh(C*(d + S(2)*e*x)/q)/q, x) + Simp(S(2)*C**S(2)*atanh(C*(A*d - S(2)*C*d*x**S(2) - S(2)*C*e*x**S(3) - S(2)*x*(A*e + C*c))/(A*q))/q, x)
pattern1807 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons1027, cons1028, cons1029)
rule1807 = ReplacementRule(pattern1807, With1807)
def With1808(B, C, e, b, d, c, a, x, A):
q = Rt(-C*(C*(-S(4)*c*e + d**S(2)) + S(2)*e*(-S(4)*A*e + B*d)), S(2))
rubi.append(1808)
return Simp(S(2)*C**S(2)*ArcTan((-B*e + C*d + S(2)*C*e*x)/q)/q, x) - Simp(S(2)*C**S(2)*ArcTan(C*(S(12)*A*B*e - S(4)*A*C*d - S(3)*B**S(2)*d + S(4)*B*C*c + S(8)*C**S(2)*e*x**S(3) + S(4)*C*x**S(2)*(-B*e + S(2)*C*d) + S(4)*C*x*(S(2)*A*e - B*d + S(2)*C*c))/(q*(-S(4)*A*C + B**S(2))))/q, x)
pattern1808 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1024, cons1025, cons1030)
rule1808 = ReplacementRule(pattern1808, With1808)
def With1809(C, e, b, d, c, a, x, A):
q = Rt(-C*(-S(8)*A*e**S(2) + C*(-S(4)*c*e + d**S(2))), S(2))
rubi.append(1809)
return Simp(S(2)*C**S(2)*ArcTan((C*d + S(2)*C*e*x)/q)/q, x) - Simp(S(2)*C**S(2)*ArcTan(-C*(-A*d + S(2)*C*d*x**S(2) + S(2)*C*e*x**S(3) + S(2)*x*(A*e + C*c))/(A*q))/q, x)
pattern1809 = Pattern(Integral((x_**S(2)*WC('C', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons36, cons1027, cons1028, cons1031)
rule1809 = ReplacementRule(pattern1809, With1809)
pattern1810 = Pattern(Integral((x_**S(3)*WC('D', S(1)) + x_**S(2)*WC('C', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons36, cons1023, cons1032, cons1033)
def replacement1810(B, C, e, b, D, d, c, a, x, A):
rubi.append(1810)
return -Dist(S(1)/(S(4)*e), Int((-S(4)*A*e + D*b + x**S(2)*(-S(4)*C*e + S(3)*D*d) + S(2)*x*(-S(2)*B*e + D*c))/(a + b*x + c*x**S(2) + d*x**S(3) + e*x**S(4)), x), x) + Simp(D*log(a + b*x + c*x**S(2) + d*x**S(3) + e*x**S(4))/(S(4)*e), x)
rule1810 = ReplacementRule(pattern1810, replacement1810)
pattern1811 = Pattern(Integral((x_**S(3)*WC('D', S(1)) + x_*WC('B', S(1)) + WC('A', S(0)))/(a_ + x_**S(4)*WC('e', S(1)) + x_**S(3)*WC('d', S(1)) + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons1023, cons1034, cons1035)
def replacement1811(B, e, b, D, d, c, a, x, A):
rubi.append(1811)
return -Dist(S(1)/(S(4)*e), Int((-S(4)*A*e + D*b + S(3)*D*d*x**S(2) + S(2)*x*(-S(2)*B*e + D*c))/(a + b*x + c*x**S(2) + d*x**S(3) + e*x**S(4)), x), x) + Simp(D*log(a + b*x + c*x**S(2) + d*x**S(3) + e*x**S(4))/(S(4)*e), x)
rule1811 = ReplacementRule(pattern1811, replacement1811)
pattern1812 = Pattern(Integral(u_/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1036)
def replacement1812(u, f, b, d, c, a, x, e):
rubi.append(1812)
return -Dist(a/(f*(-a*d + b*c)), Int(u*sqrt(c + d*x)/x, x), x) + Dist(c/(e*(-a*d + b*c)), Int(u*sqrt(a + b*x)/x, x), x)
rule1812 = ReplacementRule(pattern1812, replacement1812)
pattern1813 = Pattern(Integral(u_/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons71, cons1037)
def replacement1813(u, f, b, d, c, a, x, e):
rubi.append(1813)
return Dist(b/(f*(-a*d + b*c)), Int(u*sqrt(c + d*x), x), x) - Dist(d/(e*(-a*d + b*c)), Int(u*sqrt(a + b*x), x), x)
rule1813 = ReplacementRule(pattern1813, replacement1813)
pattern1814 = Pattern(Integral(u_/(sqrt(x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1)) + sqrt(x_*WC('d', S(1)) + WC('c', S(0)))*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1038, cons1039)
def replacement1814(u, f, b, d, c, a, x, e):
rubi.append(1814)
return Dist(e, Int(u*sqrt(a + b*x)/(a*e**S(2) - c*f**S(2) + x*(b*e**S(2) - d*f**S(2))), x), x) - Dist(f, Int(u*sqrt(c + d*x)/(a*e**S(2) - c*f**S(2) + x*(b*e**S(2) - d*f**S(2))), x), x)
rule1814 = ReplacementRule(pattern1814, replacement1814)
pattern1815 = Pattern(Integral(WC('u', S(1))/(x_**WC('n', S(1))*WC('d', S(1)) + sqrt(x_**WC('p', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons4, cons1040, cons1041)
def replacement1815(p, u, b, d, c, n, a, x):
rubi.append(1815)
return Dist(S(1)/(a*c), Int(u*sqrt(a + b*x**(S(2)*n)), x), x) - Dist(b/(a*d), Int(u*x**n, x), x)
rule1815 = ReplacementRule(pattern1815, replacement1815)
pattern1816 = Pattern(Integral(x_**WC('m', S(1))/(x_**WC('n', S(1))*WC('d', S(1)) + sqrt(x_**WC('p', S(1))*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1040, cons1042)
def replacement1816(p, m, b, d, c, n, a, x):
rubi.append(1816)
return Dist(c, Int(x**m*sqrt(a + b*x**(S(2)*n))/(a*c**S(2) + x**(S(2)*n)*(b*c**S(2) - d**S(2))), x), x) - Dist(d, Int(x**(m + n)/(a*c**S(2) + x**(S(2)*n)*(b*c**S(2) - d**S(2))), x), x)
rule1816 = ReplacementRule(pattern1816, replacement1816)
def With1817(f, b, d, a, x, e):
r = Numerator(Rt(a/b, S(3)))
s = Denominator(Rt(a/b, S(3)))
rubi.append(1817)
return Dist(r/(S(3)*a), Int(S(1)/((r + s*x)*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(r/(S(3)*a), Int((S(2)*r - s*x)/(sqrt(d + e*x + f*x**S(2))*(r**S(2) - r*s*x + s**S(2)*x**S(2))), x), x)
pattern1817 = Pattern(Integral(S(1)/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons468)
rule1817 = ReplacementRule(pattern1817, With1817)
def With1818(f, b, d, a, x):
r = Numerator(Rt(a/b, S(3)))
s = Denominator(Rt(a/b, S(3)))
rubi.append(1818)
return Dist(r/(S(3)*a), Int(S(1)/(sqrt(d + f*x**S(2))*(r + s*x)), x), x) + Dist(r/(S(3)*a), Int((S(2)*r - s*x)/(sqrt(d + f*x**S(2))*(r**S(2) - r*s*x + s**S(2)*x**S(2))), x), x)
pattern1818 = Pattern(Integral(S(1)/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons27, cons125, cons468)
rule1818 = ReplacementRule(pattern1818, With1818)
def With1819(f, b, d, a, x, e):
r = Numerator(Rt(-a/b, S(3)))
s = Denominator(Rt(-a/b, S(3)))
rubi.append(1819)
return Dist(r/(S(3)*a), Int(S(1)/((r - s*x)*sqrt(d + e*x + f*x**S(2))), x), x) + Dist(r/(S(3)*a), Int((S(2)*r + s*x)/(sqrt(d + e*x + f*x**S(2))*(r**S(2) + r*s*x + s**S(2)*x**S(2))), x), x)
pattern1819 = Pattern(Integral(S(1)/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons27, cons48, cons125, cons469)
rule1819 = ReplacementRule(pattern1819, With1819)
def With1820(f, b, d, a, x):
r = Numerator(Rt(-a/b, S(3)))
s = Denominator(Rt(-a/b, S(3)))
rubi.append(1820)
return Dist(r/(S(3)*a), Int(S(1)/(sqrt(d + f*x**S(2))*(r - s*x)), x), x) + Dist(r/(S(3)*a), Int((S(2)*r + s*x)/(sqrt(d + f*x**S(2))*(r**S(2) + r*s*x + s**S(2)*x**S(2))), x), x)
pattern1820 = Pattern(Integral(S(1)/((a_ + x_**S(3)*WC('b', S(1)))*sqrt(x_**S(2)*WC('f', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons27, cons125, cons469)
rule1820 = ReplacementRule(pattern1820, With1820)
pattern1821 = Pattern(Integral(S(1)/((d_ + x_*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1043)
def replacement1821(b, d, c, a, x, e):
rubi.append(1821)
return Dist(d, Int(S(1)/((d**S(2) - e**S(2)*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x) - Dist(e, Int(x/((d**S(2) - e**S(2)*x**S(2))*sqrt(a + b*x**S(2) + c*x**S(4))), x), x)
rule1821 = ReplacementRule(pattern1821, replacement1821)
pattern1822 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons297)
def replacement1822(d, c, a, x, e):
rubi.append(1822)
return Dist(d, Int(S(1)/(sqrt(a + c*x**S(4))*(d**S(2) - e**S(2)*x**S(2))), x), x) - Dist(e, Int(x/(sqrt(a + c*x**S(4))*(d**S(2) - e**S(2)*x**S(2))), x), x)
rule1822 = ReplacementRule(pattern1822, replacement1822)
pattern1823 = Pattern(Integral(S(1)/((d_ + x_*WC('e', S(1)))**S(2)*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1044, cons1045)
def replacement1823(b, d, c, a, x, e):
rubi.append(1823)
return -Dist(c/(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4)), Int((d**S(2) - e**S(2)*x**S(2))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) - Simp(e**S(3)*sqrt(a + b*x**S(2) + c*x**S(4))/((d + e*x)*(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4))), x)
rule1823 = ReplacementRule(pattern1823, replacement1823)
pattern1824 = Pattern(Integral(S(1)/((d_ + x_*WC('e', S(1)))**S(2)*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1044, cons1046)
def replacement1824(b, d, c, a, x, e):
rubi.append(1824)
return -Dist(c/(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4)), Int((d**S(2) - e**S(2)*x**S(2))/sqrt(a + b*x**S(2) + c*x**S(4)), x), x) + Dist((b*d*e**S(2) + S(2)*c*d**S(3))/(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4)), Int(S(1)/((d + e*x)*sqrt(a + b*x**S(2) + c*x**S(4))), x), x) - Simp(e**S(3)*sqrt(a + b*x**S(2) + c*x**S(4))/((d + e*x)*(a*e**S(4) + b*d**S(2)*e**S(2) + c*d**S(4))), x)
rule1824 = ReplacementRule(pattern1824, replacement1824)
pattern1825 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_*WC('e', S(1)))**S(2)), x_), cons2, cons7, cons27, cons48, cons1047)
def replacement1825(d, c, a, x, e):
rubi.append(1825)
return -Dist(c/(a*e**S(4) + c*d**S(4)), Int((d**S(2) - e**S(2)*x**S(2))/sqrt(a + c*x**S(4)), x), x) + Dist(S(2)*c*d**S(3)/(a*e**S(4) + c*d**S(4)), Int(S(1)/(sqrt(a + c*x**S(4))*(d + e*x)), x), x) - Simp(e**S(3)*sqrt(a + c*x**S(4))/((d + e*x)*(a*e**S(4) + c*d**S(4))), x)
rule1825 = ReplacementRule(pattern1825, replacement1825)
pattern1826 = Pattern(Integral((A_ + x_**S(2)*WC('B', S(1)))/((d_ + x_**S(2)*WC('e', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons34, cons35, cons1048, cons705)
def replacement1826(B, e, b, d, c, a, x, A):
rubi.append(1826)
return Dist(A, Subst(Int(S(1)/(d - x**S(2)*(-S(2)*a*e + b*d)), x), x, x/sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1826 = ReplacementRule(pattern1826, replacement1826)
pattern1827 = Pattern(Integral((A_ + x_**S(2)*WC('B', S(1)))/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(2)*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons34, cons35, cons1048, cons705)
def replacement1827(B, e, d, c, a, x, A):
rubi.append(1827)
return Dist(A, Subst(Int(S(1)/(S(2)*a*e*x**S(2) + d), x), x, x/sqrt(a + c*x**S(4))), x)
rule1827 = ReplacementRule(pattern1827, replacement1827)
pattern1828 = Pattern(Integral((A_ + x_**S(4)*WC('B', S(1)))/(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))*(d_ + x_**S(4)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons382, cons1049)
def replacement1828(B, f, b, d, c, a, A, x, e):
rubi.append(1828)
return Dist(A, Subst(Int(S(1)/(d - x**S(2)*(-a*e + b*d)), x), x, x/sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1828 = ReplacementRule(pattern1828, replacement1828)
pattern1829 = Pattern(Integral((A_ + x_**S(4)*WC('B', S(1)))/(sqrt(a_ + x_**S(4)*WC('c', S(1)))*(d_ + x_**S(4)*WC('f', S(1)) + x_**S(2)*WC('e', S(1)))), x_), cons2, cons7, cons27, cons48, cons125, cons34, cons35, cons382, cons1049)
def replacement1829(B, e, f, d, c, a, x, A):
rubi.append(1829)
return Dist(A, Subst(Int(S(1)/(a*e*x**S(2) + d), x), x, x/sqrt(a + c*x**S(4))), x)
rule1829 = ReplacementRule(pattern1829, replacement1829)
pattern1830 = Pattern(Integral((A_ + x_**S(4)*WC('B', S(1)))/((d_ + x_**S(4)*WC('f', S(1)))*sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))), x_), cons2, cons3, cons7, cons27, cons125, cons34, cons35, cons382, cons1049)
def replacement1830(B, f, b, d, c, a, x, A):
rubi.append(1830)
return Dist(A, Subst(Int(S(1)/(-b*d*x**S(2) + d), x), x, x/sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1830 = ReplacementRule(pattern1830, replacement1830)
pattern1831 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))/(d_ + x_**S(4)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1050, cons697)
def replacement1831(b, d, c, a, x, e):
rubi.append(1831)
return Dist(a/d, Subst(Int(S(1)/(-S(2)*b*x**S(2) + x**S(4)*(-S(4)*a*c + b**S(2)) + S(1)), x), x, x/sqrt(a + b*x**S(2) + c*x**S(4))), x)
rule1831 = ReplacementRule(pattern1831, replacement1831)
def With1832(b, d, c, a, x, e):
q = sqrt(-S(4)*a*c + b**S(2))
rubi.append(1832)
return Simp(sqrt(S(2))*a*sqrt(-b + q)*atanh(sqrt(S(2))*x*sqrt(-b + q)*(b + S(2)*c*x**S(2) + q)/(S(4)*sqrt(a + b*x**S(2) + c*x**S(4))*Rt(-a*c, S(2))))/(S(4)*d*Rt(-a*c, S(2))), x) - Simp(sqrt(S(2))*a*sqrt(b + q)*ArcTan(sqrt(S(2))*x*sqrt(b + q)*(b + S(2)*c*x**S(2) - q)/(S(4)*sqrt(a + b*x**S(2) + c*x**S(4))*Rt(-a*c, S(2))))/(S(4)*d*Rt(-a*c, S(2))), x)
pattern1832 = Pattern(Integral(sqrt(a_ + x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)))/(d_ + x_**S(4)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons1050, cons709)
rule1832 = ReplacementRule(pattern1832, With1832)
pattern1833 = Pattern(Integral(S(1)/((a_ + x_*WC('b', S(1)))*sqrt(c_ + x_**S(2)*WC('d', S(1)))*sqrt(e_ + x_**S(2)*WC('f', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons153)
def replacement1833(f, b, d, a, c, x, e):
rubi.append(1833)
return Dist(a, Int(S(1)/((a**S(2) - b**S(2)*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x) - Dist(b, Int(x/((a**S(2) - b**S(2)*x**S(2))*sqrt(c + d*x**S(2))*sqrt(e + f*x**S(2))), x), x)
rule1833 = ReplacementRule(pattern1833, replacement1833)
pattern1834 = Pattern(Integral((x_*WC('h', S(1)) + WC('g', S(0)))*sqrt(x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons1051, cons1052)
def replacement1834(g, f, b, d, a, c, x, h, e):
rubi.append(1834)
return Simp(S(2)*sqrt(d + e*x + f*sqrt(a + b*x + c*x**S(2)))*(S(9)*c**S(2)*f*g*h*x**S(2) + S(3)*c**S(2)*f*h**S(2)*x**S(3) + c*f*x*(a*h**S(2) - b*g*h + S(10)*c*g**S(2)) + f*(S(2)*a*b*h**S(2) - S(3)*a*c*g*h - S(2)*b**S(2)*g*h + S(5)*b*c*g**S(2)) - (-d*h + e*g)*sqrt(a + b*x + c*x**S(2))*(-S(2)*b*h + S(5)*c*g + c*h*x))/(S(15)*c**S(2)*f*(g + h*x)), x)
rule1834 = ReplacementRule(pattern1834, replacement1834)
pattern1835 = Pattern(Integral((u_ + (sqrt(v_)*WC('k', S(1)) + WC('j', S(0)))*WC('f', S(1)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons125, cons208, cons209, cons796, cons797, cons21, cons4, cons68, cons818, cons1053, cons1054)
def replacement1835(v, u, j, k, m, g, f, n, x, h):
rubi.append(1835)
return Int((g + h*x)**m*(f*k*sqrt(ExpandToSum(v, x)) + ExpandToSum(f*j + u, x))**n, x)
rule1835 = ReplacementRule(pattern1835, replacement1835)
pattern1836 = Pattern(Integral(((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0)))**n_*WC('h', S(1)) + WC('g', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons1055, cons38)
def replacement1836(p, g, f, b, d, a, c, n, x, h, e):
rubi.append(1836)
return Dist(S(2), Subst(Int((g + h*x**n)**p*(d**S(2)*e + e*x**S(2) - f**S(2)*(-a*e + b*d) - x*(-b*f**S(2) + S(2)*d*e))/(b*f**S(2) - S(2)*d*e + S(2)*e*x)**S(2), x), x, d + e*x + f*sqrt(a + b*x + c*x**S(2))), x)
rule1836 = ReplacementRule(pattern1836, replacement1836)
pattern1837 = Pattern(Integral(((x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)) + WC('d', S(0)))**n_*WC('h', S(1)) + WC('g', S(0)))**WC('p', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons209, cons4, cons1055, cons38)
def replacement1837(p, g, f, d, c, a, n, x, h, e):
rubi.append(1837)
return Dist(S(1)/(S(2)*e), Subst(Int((g + h*x**n)**p*(a*f**S(2) + d**S(2) - S(2)*d*x + x**S(2))/(d - x)**S(2), x), x, d + e*x + f*sqrt(a + c*x**S(2))), x)
rule1837 = ReplacementRule(pattern1837, replacement1837)
pattern1838 = Pattern(Integral(((u_ + sqrt(v_)*WC('f', S(1)))**n_*WC('h', S(1)) + WC('g', S(0)))**WC('p', S(1)), x_), cons125, cons208, cons209, cons4, cons68, cons818, cons819, cons1056, cons38)
def replacement1838(v, p, u, g, f, n, x, h):
rubi.append(1838)
return Int((g + h*(f*sqrt(ExpandToSum(v, x)) + ExpandToSum(u, x))**n)**p, x)
rule1838 = ReplacementRule(pattern1838, replacement1838)
pattern1839 = Pattern(Integral((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + WC('a', S(0)))*WC('f', S(1)))**WC('n', S(1))*(x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons2, cons7, cons48, cons125, cons208, cons209, cons4, cons1055, cons17)
def replacement1839(m, g, f, a, c, n, x, h, e):
rubi.append(1839)
return Dist(S(2)**(-m + S(-1))*e**(-m + S(-1)), Subst(Int(x**(-m + n + S(-2))*(a*f**S(2) + x**S(2))*(-a*f**S(2)*h + S(2)*e*g*x + h*x**S(2))**m, x), x, e*x + f*sqrt(a + c*x**S(2))), x)
rule1839 = ReplacementRule(pattern1839, replacement1839)
pattern1840 = Pattern(Integral(x_**WC('p', S(1))*(g_ + x_**S(2)*WC('i', S(1)))**WC('m', S(1))*(x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)))**WC('n', S(1)), x_), cons2, cons7, cons48, cons125, cons208, cons224, cons4, cons1055, cons1057, cons1058, cons1059)
def replacement1840(p, m, f, g, i, c, n, a, x, e):
rubi.append(1840)
return Dist(S(2)**(-S(2)*m - p + S(-1))*e**(-p + S(-1))*f**(-S(2)*m)*(i/c)**m, Subst(Int(x**(-S(2)*m + n - p + S(-2))*(-a*f**S(2) + x**S(2))**p*(a*f**S(2) + x**S(2))**(S(2)*m + S(1)), x), x, e*x + f*sqrt(a + c*x**S(2))), x)
rule1840 = ReplacementRule(pattern1840, replacement1840)
pattern1841 = Pattern(Integral((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1))*(x_**S(2)*WC('i', S(1)) + x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons4, cons1055, cons1057, cons1060, cons515, cons1059)
def replacement1841(m, g, f, b, i, d, a, c, n, x, h, e):
rubi.append(1841)
return Dist(S(2)*f**(-S(2)*m)*(i/c)**m, Subst(Int(x**n*(b*f**S(2) - S(2)*d*e + S(2)*e*x)**(-S(2)*m + S(-2))*(d**S(2)*e + e*x**S(2) - f**S(2)*(-a*e + b*d) - x*(-b*f**S(2) + S(2)*d*e))**(S(2)*m + S(1)), x), x, d + e*x + f*sqrt(a + b*x + c*x**S(2))), x)
rule1841 = ReplacementRule(pattern1841, replacement1841)
pattern1842 = Pattern(Integral((g_ + x_**S(2)*WC('i', S(1)))**WC('m', S(1))*(x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons224, cons4, cons1055, cons1057, cons515, cons1059)
def replacement1842(m, f, g, i, d, c, n, a, x, e):
rubi.append(1842)
return Dist(S(2)**(-S(2)*m + S(-1))*f**(-S(2)*m)*(i/c)**m/e, Subst(Int(x**n*(-d + x)**(-S(2)*m + S(-2))*(a*f**S(2) + d**S(2) - S(2)*d*x + x**S(2))**(S(2)*m + S(1)), x), x, d + e*x + f*sqrt(a + c*x**S(2))), x)
rule1842 = ReplacementRule(pattern1842, replacement1842)
pattern1843 = Pattern(Integral((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1))*(x_**S(2)*WC('i', S(1)) + x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons4, cons1055, cons1057, cons1060, cons73, cons1061)
def replacement1843(m, g, f, b, i, d, a, c, n, x, h, e):
rubi.append(1843)
return Dist((i/c)**(m + S(-1)/2)*sqrt(g + h*x + i*x**S(2))/sqrt(a + b*x + c*x**S(2)), Int((a + b*x + c*x**S(2))**m*(d + e*x + f*sqrt(a + b*x + c*x**S(2)))**n, x), x)
rule1843 = ReplacementRule(pattern1843, replacement1843)
pattern1844 = Pattern(Integral((g_ + x_**S(2)*WC('i', S(1)))**WC('m', S(1))*(x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons224, cons4, cons1055, cons1057, cons73, cons1061)
def replacement1844(m, f, g, i, d, c, n, a, x, e):
rubi.append(1844)
return Dist((i/c)**(m + S(-1)/2)*sqrt(g + i*x**S(2))/sqrt(a + c*x**S(2)), Int((a + c*x**S(2))**m*(d + e*x + f*sqrt(a + c*x**S(2)))**n, x), x)
rule1844 = ReplacementRule(pattern1844, replacement1844)
pattern1845 = Pattern(Integral((x_*WC('e', S(1)) + sqrt(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1))*(x_**S(2)*WC('i', S(1)) + x_*WC('h', S(1)) + WC('g', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons4, cons1055, cons1057, cons1060, cons951, cons1061)
def replacement1845(m, g, f, b, i, d, a, c, n, x, h, e):
rubi.append(1845)
return Dist((i/c)**(m + S(1)/2)*sqrt(a + b*x + c*x**S(2))/sqrt(g + h*x + i*x**S(2)), Int((a + b*x + c*x**S(2))**m*(d + e*x + f*sqrt(a + b*x + c*x**S(2)))**n, x), x)
rule1845 = ReplacementRule(pattern1845, replacement1845)
pattern1846 = Pattern(Integral((g_ + x_**S(2)*WC('i', S(1)))**WC('m', S(1))*(x_*WC('e', S(1)) + sqrt(a_ + x_**S(2)*WC('c', S(1)))*WC('f', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons224, cons4, cons1055, cons1057, cons951, cons1061)
def replacement1846(m, f, g, i, d, c, n, a, x, e):
rubi.append(1846)
return Dist((i/c)**(m + S(1)/2)*sqrt(a + c*x**S(2))/sqrt(g + i*x**S(2)), Int((a + c*x**S(2))**m*(d + e*x + f*sqrt(a + c*x**S(2)))**n, x), x)
rule1846 = ReplacementRule(pattern1846, replacement1846)
pattern1847 = Pattern(Integral(w_**WC('m', S(1))*(u_ + (sqrt(v_)*WC('k', S(1)) + WC('j', S(0)))*WC('f', S(1)))**WC('n', S(1)), x_), cons125, cons796, cons797, cons21, cons4, cons68, cons1062, cons1063, cons1064)
def replacement1847(v, w, u, j, k, m, f, n, x):
rubi.append(1847)
return Int((f*k*sqrt(ExpandToSum(v, x)) + ExpandToSum(f*j + u, x))**n*ExpandToSum(w, x)**m, x)
rule1847 = ReplacementRule(pattern1847, replacement1847)
pattern1848 = Pattern(Integral(S(1)/((a_ + x_**WC('n', S(1))*WC('b', S(1)))*sqrt(x_**S(2)*WC('c', S(1)) + (a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons4, cons1065)
def replacement1848(p, b, d, c, n, a, x):
rubi.append(1848)
return Dist(S(1)/a, Subst(Int(S(1)/(-c*x**S(2) + S(1)), x), x, x/sqrt(c*x**S(2) + d*(a + b*x**n)**(S(2)/n))), x)
rule1848 = ReplacementRule(pattern1848, replacement1848)
pattern1849 = Pattern(Integral(sqrt(a_ + sqrt(c_ + x_**S(2)*WC('d', S(1)))*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons1066)
def replacement1849(b, d, c, a, x):
rubi.append(1849)
return Simp(S(2)*a*x/sqrt(a + b*sqrt(c + d*x**S(2))), x) + Simp(S(2)*b**S(2)*d*x**S(3)/(S(3)*(a + b*sqrt(c + d*x**S(2)))**(S(3)/2)), x)
rule1849 = ReplacementRule(pattern1849, replacement1849)
pattern1850 = Pattern(Integral(sqrt(x_**S(2)*WC('a', S(1)) + x_*sqrt(c_ + x_**S(2)*WC('d', S(1)))*WC('b', S(1)))/(x_*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons1067, cons1068)
def replacement1850(b, d, c, a, x):
rubi.append(1850)
return Dist(sqrt(S(2))*b/a, Subst(Int(S(1)/sqrt(S(1) + x**S(2)/a), x), x, a*x + b*sqrt(c + d*x**S(2))), x)
rule1850 = ReplacementRule(pattern1850, replacement1850)
pattern1851 = Pattern(Integral(sqrt(x_*(x_*WC('a', S(1)) + sqrt(c_ + x_**S(2)*WC('d', S(1)))*WC('b', S(1)))*WC('e', S(1)))/(x_*sqrt(c_ + x_**S(2)*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons1067, cons1069)
def replacement1851(b, d, a, c, x, e):
rubi.append(1851)
return Int(sqrt(a*e*x**S(2) + b*e*x*sqrt(c + d*x**S(2)))/(x*sqrt(c + d*x**S(2))), x)
rule1851 = ReplacementRule(pattern1851, replacement1851)
pattern1852 = Pattern(Integral(sqrt(x_**S(2)*WC('c', S(1)) + sqrt(a_ + x_**S(4)*WC('b', S(1)))*WC('d', S(1)))/sqrt(a_ + x_**S(4)*WC('b', S(1))), x_), cons2, cons3, cons7, cons27, cons1070)
def replacement1852(b, d, c, a, x):
rubi.append(1852)
return Dist(d, Subst(Int(S(1)/(-S(2)*c*x**S(2) + S(1)), x), x, x/sqrt(c*x**S(2) + d*sqrt(a + b*x**S(4)))), x)
rule1852 = ReplacementRule(pattern1852, replacement1852)
pattern1853 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sqrt(x_**S(2)*WC('b', S(1)) + sqrt(a_ + x_**S(4)*WC('e', S(1))))/sqrt(a_ + x_**S(4)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons21, cons1071, cons43)
def replacement1853(m, b, d, c, a, x, e):
rubi.append(1853)
return Dist(S(1)/2 - I/S(2), Int((c + d*x)**m/sqrt(sqrt(a) - I*b*x**S(2)), x), x) + Dist(S(1)/2 + I/S(2), Int((c + d*x)**m/sqrt(sqrt(a) + I*b*x**S(2)), x), x)
rule1853 = ReplacementRule(pattern1853, replacement1853)
def With1854(b, d, c, a, x):
q = Rt(b/a, S(3))
rubi.append(1854)
return Dist(d/(-c*q + d*(S(1) + sqrt(S(3)))), Int((q*x + S(1) + sqrt(S(3)))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) - Dist(q/(-c*q + d*(S(1) + sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
pattern1854 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons481, cons479)
rule1854 = ReplacementRule(pattern1854, With1854)
def With1855(b, d, c, a, x):
q = Rt(-b/a, S(3))
rubi.append(1855)
return Dist(d/(c*q + d*(S(1) + sqrt(S(3)))), Int((-q*x + S(1) + sqrt(S(3)))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist(q/(c*q + d*(S(1) + sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
pattern1855 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons481, cons480)
rule1855 = ReplacementRule(pattern1855, With1855)
def With1856(b, d, c, a, x):
q = Rt(-b/a, S(3))
rubi.append(1856)
return Dist(d/(c*q + d*(-sqrt(S(3)) + S(1))), Int((-q*x - sqrt(S(3)) + S(1))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist(q/(c*q + d*(-sqrt(S(3)) + S(1))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
pattern1856 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons482, cons479)
rule1856 = ReplacementRule(pattern1856, With1856)
def With1857(b, d, c, a, x):
q = Rt(b/a, S(3))
rubi.append(1857)
return Dist(d/(-c*q + d*(-sqrt(S(3)) + S(1))), Int((q*x - sqrt(S(3)) + S(1))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) - Dist(q/(-c*q + d*(-sqrt(S(3)) + S(1))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
pattern1857 = Pattern(Integral(S(1)/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons482, cons480)
rule1857 = ReplacementRule(pattern1857, With1857)
def With1858(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(b/a, S(3))
if ZeroQ(-e*q + f*(S(1) + sqrt(S(3)))):
return True
return False
pattern1858 = Pattern(Integral((e_ + x_*WC('f', S(1)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons481, cons479, CustomConstraint(With1858))
def replacement1858(f, b, d, c, a, x, e):
q = Rt(b/a, S(3))
rubi.append(1858)
return Dist(S(4)*S(3)**(S(1)/4)*f*sqrt((q**S(2)*x**S(2) - q*x + S(1))/(q*x + S(1) + sqrt(S(3)))**S(2))*sqrt(-sqrt(S(3)) + S(2))*(q*x + S(1))/(q*sqrt((q*x + S(1))/(q*x + S(1) + sqrt(S(3)))**S(2))*sqrt(a + b*x**S(3))), Subst(Int(S(1)/(sqrt(-x**S(2) + S(1))*sqrt(x**S(2) - S(4)*sqrt(S(3)) + S(7))*(-c*q + d*(-sqrt(S(3)) + S(1)) + x*(-c*q + d*(S(1) + sqrt(S(3)))))), x), x, (-q*x + S(-1) + sqrt(S(3)))/(q*x + S(1) + sqrt(S(3)))), x)
rule1858 = ReplacementRule(pattern1858, replacement1858)
def With1859(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(b/a, S(3))
if NonzeroQ(-e*q + f*(S(1) + sqrt(S(3)))):
return True
return False
pattern1859 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons481, cons479, CustomConstraint(With1859))
def replacement1859(f, b, d, c, a, x, e):
q = Rt(b/a, S(3))
rubi.append(1859)
return Dist((-c*f + d*e)/(-c*q + d*(S(1) + sqrt(S(3)))), Int((q*x + S(1) + sqrt(S(3)))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist((-e*q + f*(S(1) + sqrt(S(3))))/(-c*q + d*(S(1) + sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
rule1859 = ReplacementRule(pattern1859, replacement1859)
def With1860(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-b/a, S(3))
if ZeroQ(e*q + f*(S(1) + sqrt(S(3)))):
return True
return False
pattern1860 = Pattern(Integral((e_ + x_*WC('f', S(1)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons481, cons480, CustomConstraint(With1860))
def replacement1860(f, b, d, c, a, x, e):
q = Rt(-b/a, S(3))
rubi.append(1860)
return Dist(-S(4)*S(3)**(S(1)/4)*f*sqrt((q**S(2)*x**S(2) + q*x + S(1))/(-q*x + S(1) + sqrt(S(3)))**S(2))*sqrt(-sqrt(S(3)) + S(2))*(-q*x + S(1))/(q*sqrt((-q*x + S(1))/(-q*x + S(1) + sqrt(S(3)))**S(2))*sqrt(a + b*x**S(3))), Subst(Int(S(1)/(sqrt(-x**S(2) + S(1))*sqrt(x**S(2) - S(4)*sqrt(S(3)) + S(7))*(c*q + d*(-sqrt(S(3)) + S(1)) + x*(c*q + d*(S(1) + sqrt(S(3)))))), x), x, (q*x + S(-1) + sqrt(S(3)))/(-q*x + S(1) + sqrt(S(3)))), x)
rule1860 = ReplacementRule(pattern1860, replacement1860)
def With1861(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-b/a, S(3))
if NonzeroQ(e*q + f*(S(1) + sqrt(S(3)))):
return True
return False
pattern1861 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons481, cons480, CustomConstraint(With1861))
def replacement1861(f, b, d, c, a, x, e):
q = Rt(-b/a, S(3))
rubi.append(1861)
return Dist((-c*f + d*e)/(c*q + d*(S(1) + sqrt(S(3)))), Int((-q*x + S(1) + sqrt(S(3)))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist((e*q + f*(S(1) + sqrt(S(3))))/(c*q + d*(S(1) + sqrt(S(3)))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
rule1861 = ReplacementRule(pattern1861, replacement1861)
def With1862(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-b/a, S(3))
if ZeroQ(e*q + f*(-sqrt(S(3)) + S(1))):
return True
return False
pattern1862 = Pattern(Integral((e_ + x_*WC('f', S(1)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons482, cons479, CustomConstraint(With1862))
def replacement1862(f, b, d, c, a, x, e):
q = Rt(-b/a, S(3))
rubi.append(1862)
return Dist(S(4)*S(3)**(S(1)/4)*f*sqrt((q**S(2)*x**S(2) + q*x + S(1))/(-q*x - sqrt(S(3)) + S(1))**S(2))*sqrt(sqrt(S(3)) + S(2))*(-q*x + S(1))/(q*sqrt(-(-q*x + S(1))/(-q*x - sqrt(S(3)) + S(1))**S(2))*sqrt(a + b*x**S(3))), Subst(Int(S(1)/(sqrt(-x**S(2) + S(1))*sqrt(x**S(2) + S(4)*sqrt(S(3)) + S(7))*(c*q + d*(S(1) + sqrt(S(3))) + x*(c*q + d*(-sqrt(S(3)) + S(1))))), x), x, (-q*x + S(1) + sqrt(S(3)))/(q*x + S(-1) + sqrt(S(3)))), x)
rule1862 = ReplacementRule(pattern1862, replacement1862)
def With1863(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(-b/a, S(3))
if NonzeroQ(e*q + f*(-sqrt(S(3)) + S(1))):
return True
return False
pattern1863 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons482, cons479, CustomConstraint(With1863))
def replacement1863(f, b, d, c, a, x, e):
q = Rt(-b/a, S(3))
rubi.append(1863)
return Dist((-c*f + d*e)/(c*q + d*(-sqrt(S(3)) + S(1))), Int((-q*x - sqrt(S(3)) + S(1))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist((e*q + f*(-sqrt(S(3)) + S(1)))/(c*q + d*(-sqrt(S(3)) + S(1))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
rule1863 = ReplacementRule(pattern1863, replacement1863)
def With1864(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(b/a, S(3))
if ZeroQ(-e*q + f*(-sqrt(S(3)) + S(1))):
return True
return False
pattern1864 = Pattern(Integral((e_ + x_*WC('f', S(1)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons482, cons480, CustomConstraint(With1864))
def replacement1864(f, b, d, c, a, x, e):
q = Rt(b/a, S(3))
rubi.append(1864)
return Dist(-S(4)*S(3)**(S(1)/4)*f*sqrt((q**S(2)*x**S(2) - q*x + S(1))/(q*x - sqrt(S(3)) + S(1))**S(2))*sqrt(sqrt(S(3)) + S(2))*(q*x + S(1))/(q*sqrt(-(q*x + S(1))/(q*x - sqrt(S(3)) + S(1))**S(2))*sqrt(a + b*x**S(3))), Subst(Int(S(1)/(sqrt(-x**S(2) + S(1))*sqrt(x**S(2) + S(4)*sqrt(S(3)) + S(7))*(-c*q + d*(S(1) + sqrt(S(3))) + x*(-c*q + d*(-sqrt(S(3)) + S(1))))), x), x, (q*x + S(1) + sqrt(S(3)))/(-q*x + S(-1) + sqrt(S(3)))), x)
rule1864 = ReplacementRule(pattern1864, replacement1864)
def With1865(f, b, d, c, a, x, e):
if isinstance(x, (int, Integer, float, Float)):
return False
q = Rt(b/a, S(3))
if NonzeroQ(-e*q + f*(-sqrt(S(3)) + S(1))):
return True
return False
pattern1865 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))/(sqrt(a_ + x_**S(3)*WC('b', S(1)))*(c_ + x_*WC('d', S(1)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons482, cons480, CustomConstraint(With1865))
def replacement1865(f, b, d, c, a, x, e):
q = Rt(b/a, S(3))
rubi.append(1865)
return Dist((-c*f + d*e)/(-c*q + d*(-sqrt(S(3)) + S(1))), Int((q*x - sqrt(S(3)) + S(1))/(sqrt(a + b*x**S(3))*(c + d*x)), x), x) + Dist((-e*q + f*(-sqrt(S(3)) + S(1)))/(-c*q + d*(-sqrt(S(3)) + S(1))), Int(S(1)/sqrt(a + b*x**S(3)), x), x)
rule1865 = ReplacementRule(pattern1865, replacement1865)
pattern1866 = Pattern(Integral(x_**WC('m', S(1))/(c_ + x_**n_*WC('d', S(1)) + sqrt(a_ + x_**n_*WC('b', S(1)))*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons4, cons1072, cons500)
def replacement1866(m, b, d, c, n, a, x, e):
rubi.append(1866)
return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)/(c + d*x + e*sqrt(a + b*x)), x), x, x**n), x)
rule1866 = ReplacementRule(pattern1866, replacement1866)
pattern1867 = Pattern(Integral(WC('u', S(1))/(c_ + x_**n_*WC('d', S(1)) + sqrt(a_ + x_**n_*WC('b', S(1)))*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons4, cons1072)
def replacement1867(u, b, d, c, n, a, x, e):
rubi.append(1867)
return Dist(c, Int(u/(-a*e**S(2) + c**S(2) + c*d*x**n), x), x) - Dist(a*e, Int(u/(sqrt(a + b*x**n)*(-a*e**S(2) + c**S(2) + c*d*x**n)), x), x)
rule1867 = ReplacementRule(pattern1867, replacement1867)
pattern1868 = Pattern(Integral((A_ + x_**n_*WC('B', S(1)))/(a_ + x_**S(2)*WC('b', S(1)) + x_**n2_*WC('d', S(1)) + x_**n_*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons4, cons46, cons963, cons1073, cons1074)
def replacement1868(B, b, n2, d, c, n, a, x, A):
rubi.append(1868)
return Dist(A**S(2)*(n + S(-1)), Subst(Int(S(1)/(A**S(2)*b*x**S(2)*(n + S(-1))**S(2) + a), x), x, x/(A*(n + S(-1)) - B*x**n)), x)
rule1868 = ReplacementRule(pattern1868, replacement1868)
pattern1869 = Pattern(Integral(x_**WC('m', S(1))*(A_ + x_**WC('n', S(1))*WC('B', S(1)))/(a_ + x_**n2_*WC('d', S(1)) + x_**WC('k', S(1))*WC('b', S(1)) + x_**WC('n', S(1))*WC('c', S(1))), x_), cons2, cons3, cons7, cons27, cons34, cons35, cons21, cons4, cons46, cons1075, cons1076, cons1077)
def replacement1869(B, k, m, b, n2, d, c, n, a, x, A):
rubi.append(1869)
return Dist(A**S(2)*(m - n + S(1))/(m + S(1)), Subst(Int(S(1)/(A**S(2)*b*x**S(2)*(m - n + S(1))**S(2) + a), x), x, x**(m + S(1))/(A*(m - n + S(1)) + B*x**n*(m + S(1)))), x)
rule1869 = ReplacementRule(pattern1869, replacement1869)
pattern1870 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n2_*WC('f', S(1)) + x_**n_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons46, cons931, cons226, cons702)
def replacement1870(p, f, b, n2, g, d, n3, c, a, n, x, e):
rubi.append(1870)
return -Dist(S(1)/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a*b*(a*g + c*e) - S(2)*a*c*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))) - b**S(2)*c*d*(n*p + n + S(1)) + x**n*(a*b**S(2)*g*(n*(p + S(2)) + S(1)) - S(2)*a*c*(a*g*(n + S(1)) - c*e*(n*(S(2)*p + S(3)) + S(1))) - b*c*(a*f + c*d)*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*(a*g + c*e) - S(2)*a*c*(-a*f + c*d) + b**S(2)*c*d + x**n*(-a*b**S(2)*g - S(2)*a*c*(-a*g + c*e) + b*c*(a*f + c*d)))/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1870 = ReplacementRule(pattern1870, replacement1870)
pattern1871 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_*(d_ + x_**n2_*WC('f', S(1)) + x_**n_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons46, cons226, cons702)
def replacement1871(p, f, b, n2, d, c, a, n, x, e):
rubi.append(1871)
return -Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a*b*e - S(2)*a*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))) - b**S(2)*d*(n*p + n + S(1)) - x**n*(-S(2)*a*c*e*(n*(S(2)*p + S(3)) + S(1)) + b*(a*f + c*d)*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*e - S(2)*a*(-a*f + c*d) + b**S(2)*d + x**n*(-S(2)*a*c*e + b*(a*f + c*d)))/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1871 = ReplacementRule(pattern1871, replacement1871)
pattern1872 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n_*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons208, cons4, cons46, cons931, cons226, cons702)
def replacement1872(p, g, b, n2, d, n3, c, a, n, x, e):
rubi.append(1872)
return -Dist(S(1)/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a*b*(a*g + c*e) + S(2)*a*c**S(2)*d*(S(2)*n*(p + S(1)) + S(1)) - b**S(2)*c*d*(n*p + n + S(1)) + x**n*(a*b**S(2)*g*(n*(p + S(2)) + S(1)) - S(2)*a*c*(a*g*(n + S(1)) - c*e*(n*(S(2)*p + S(3)) + S(1))) - b*c**S(2)*d*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a*b*(a*g + c*e) - S(2)*a*c**S(2)*d + b**S(2)*c*d + x**n*(-a*b**S(2)*g - S(2)*a*c*(-a*g + c*e) + b*c**S(2)*d))/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1872 = ReplacementRule(pattern1872, replacement1872)
pattern1873 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n2_*WC('f', S(1))), x_), cons2, cons3, cons7, cons27, cons125, cons208, cons4, cons46, cons931, cons226, cons702)
def replacement1873(p, f, b, n2, g, d, n3, c, a, n, x):
rubi.append(1873)
return -Dist(S(1)/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a**S(2)*b*g - S(2)*a*c*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))) - b**S(2)*c*d*(n*p + n + S(1)) + x**n*(-S(2)*a**S(2)*c*g*(n + S(1)) + a*b**S(2)*g*(n*(p + S(2)) + S(1)) - b*c*(a*f + c*d)*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a**S(2)*b*g - S(2)*a*c*(-a*f + c*d) + b**S(2)*c*d + x**n*(S(2)*a**S(2)*c*g - a*b**S(2)*g + b*c*(a*f + c*d)))/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1873 = ReplacementRule(pattern1873, replacement1873)
pattern1874 = Pattern(Integral((d_ + x_**n2_*WC('f', S(1)))*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons4, cons46, cons226, cons702)
def replacement1874(p, f, b, n2, d, c, a, n, x):
rubi.append(1874)
return Dist(S(1)/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(S(2)*a*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))) + b**S(2)*d*(n*p + n + S(1)) + b*x**n*(a*f + c*d)*(n*(S(2)*p + S(3)) + S(1)), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*(-a*f + c*d) + b**S(2)*d + b*x**n*(a*f + c*d))/(a*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1874 = ReplacementRule(pattern1874, replacement1874)
pattern1875 = Pattern(Integral((d_ + g_*x_**n3_)*(a_ + x_**n2_*WC('c', S(1)) + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons7, cons27, cons208, cons4, cons46, cons931, cons226, cons702)
def replacement1875(p, g, b, n2, d, n3, c, n, a, x):
rubi.append(1875)
return -Dist(S(1)/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), Int((a + b*x**n + c*x**(S(2)*n))**(p + S(1))*Simp(a**S(2)*b*g + S(2)*a*c**S(2)*d*(S(2)*n*(p + S(1)) + S(1)) - b**S(2)*c*d*(n*p + n + S(1)) + x**n*(-S(2)*a**S(2)*c*g*(n + S(1)) + a*b**S(2)*g*(n*(p + S(2)) + S(1)) - b*c**S(2)*d*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(x*(a + b*x**n + c*x**(S(2)*n))**(p + S(1))*(-a**S(2)*b*g - S(2)*a*c**S(2)*d + b**S(2)*c*d + x**n*(S(2)*a**S(2)*c*g - a*b**S(2)*g + b*c**S(2)*d))/(a*c*n*(p + S(1))*(-S(4)*a*c + b**S(2))), x)
rule1875 = ReplacementRule(pattern1875, replacement1875)
pattern1876 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n2_*WC('f', S(1)) + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons4, cons46, cons931, cons702)
def replacement1876(p, f, g, n2, d, n3, c, a, n, x, e):
rubi.append(1876)
return -Dist(-S(1)/(S(4)*a**S(2)*c**S(2)*n*(p + S(1))), Int((a + c*x**(S(2)*n))**(p + S(1))*Simp(-S(2)*a*c*x**n*(a*g*(n + S(1)) - c*e*(n*(S(2)*p + S(3)) + S(1))) - S(2)*a*c*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))), x), x), x) - Simp(-x*(a + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c*x**n*(-a*g + c*e) - S(2)*a*c*(-a*f + c*d))/(S(4)*a**S(2)*c**S(2)*n*(p + S(1))), x)
rule1876 = ReplacementRule(pattern1876, replacement1876)
pattern1877 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + x_**n2_*WC('f', S(1)) + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons4, cons46, cons702)
def replacement1877(p, f, n2, d, c, a, n, x, e):
rubi.append(1877)
return -Dist(-S(1)/(S(4)*a**S(2)*c*n*(p + S(1))), Int((a + c*x**(S(2)*n))**(p + S(1))*Simp(S(2)*a*c*e*x**n*(n*(S(2)*p + S(3)) + S(1)) - S(2)*a*(a*f - c*d*(S(2)*n*(p + S(1)) + S(1))), x), x), x) - Simp(-x*(a + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c*e*x**n - S(2)*a*(-a*f + c*d))/(S(4)*a**S(2)*c*n*(p + S(1))), x)
rule1877 = ReplacementRule(pattern1877, replacement1877)
pattern1878 = Pattern(Integral((a_ + x_**n2_*WC('c', S(1)))**p_*(d_ + g_*x_**n3_ + x_**n_*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons208, cons4, cons46, cons931, cons702)
def replacement1878(p, g, n2, d, n3, c, a, n, x, e):
rubi.append(1878)
return -Dist(-S(1)/(S(4)*a**S(2)*c**S(2)*n*(p + S(1))), Int((a + c*x**(S(2)*n))**(p + S(1))*Simp(S(2)*a*c**S(2)*d*(S(2)*n*(p + S(1)) + S(1)) - S(2)*a*c*x**n*(a*g*(n + S(1)) - c*e*(n*(S(2)*p + S(3)) + S(1))), x), x), x) - Simp(-x*(a + c*x**(S(2)*n))**(p + S(1))*(-S(2)*a*c**S(2)*d - S(2)*a*c*x**n*(-a*g + c*e))/(S(4)*a**S(2)*c**S(2)*n*(p + S(1))), x)
rule1878 = ReplacementRule(pattern1878, replacement1878)
def With1879(f, b, g, d, c, a, x, e):
q = Rt((S(12)*a**S(2)*g**S(2) - a*c*f**S(2) + f*(-S(2)*a*b*g + S(3)*c**S(2)*d))/(c*g*(-a*f + S(3)*c*d)), S(2))
r = Rt((a*c*f**S(2) - f*(S(2)*a*b*g + S(3)*c**S(2)*d) + S(4)*g*(a**S(2)*g + b*c*d))/(c*g*(-a*f + S(3)*c*d)), S(2))
rubi.append(1879)
return -Simp(c*ArcTan((r - S(2)*x)/q)/(g*q), x) + Simp(c*ArcTan((r + S(2)*x)/q)/(g*q), x) - Simp(c*ArcTan(x*(-a*f + S(3)*c*d)*(S(6)*a**S(2)*b*g**S(2) - S(2)*a**S(2)*c*f*g - a*b**S(2)*f*g + b*c**S(2)*d*f + c**S(2)*g*x**S(4)*(-a*f + S(3)*c*d) + c*x**S(2)*(S(2)*a**S(2)*g**S(2) - a*c*f**S(2) - b*c*d*g + S(3)*c**S(2)*d*f))/(g*q*(-S(2)*a**S(2)*g + b*c*d)*(S(4)*a**S(2)*g - a*b*f + b*c*d)))/(g*q), x)
pattern1879 = Pattern(Integral((x_**S(4)*WC('c', S(1)) + x_**S(2)*WC('b', S(1)) + WC('a', S(0)))/(d_ + x_**S(6)*WC('g', S(1)) + x_**S(4)*WC('f', S(1)) + x_**S(2)*WC('e', S(1))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1078, cons1079, cons1080, cons1081, cons1082, cons1083)
rule1879 = ReplacementRule(pattern1879, With1879)
def With1880(f, g, d, a, c, x, e):
q = Rt((S(12)*a**S(2)*g**S(2) - a*c*f**S(2) + S(3)*c**S(2)*d*f)/(c*g*(-a*f + S(3)*c*d)), S(2))
r = Rt((S(4)*a**S(2)*g**S(2) + a*c*f**S(2) - S(3)*c**S(2)*d*f)/(c*g*(-a*f + S(3)*c*d)), S(2))
rubi.append(1880)
return -Simp(c*ArcTan((r - S(2)*x)/q)/(g*q), x) + Simp(c*ArcTan((r + S(2)*x)/q)/(g*q), x) - Simp(c*ArcTan(c*x*(-a*f + S(3)*c*d)*(S(2)*a**S(2)*f*g - c*g*x**S(4)*(-a*f + S(3)*c*d) - x**S(2)*(S(2)*a**S(2)*g**S(2) - a*c*f**S(2) + S(3)*c**S(2)*d*f))/(S(8)*a**S(4)*g**S(3)*q))/(g*q), x)
pattern1880 = Pattern(Integral((x_**S(4)*WC('c', S(1)) + WC('a', S(0)))/(d_ + x_**S(6)*WC('g', S(1)) + x_**S(4)*WC('f', S(1)) + x_**S(2)*WC('e', S(1))), x_), cons2, cons7, cons27, cons48, cons125, cons208, cons1084, cons1085, cons1080, cons1086)
rule1880 = ReplacementRule(pattern1880, With1880)
def With1881(v, x, p, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
m = Exponent(u, x)
n = Exponent(v, x)
c = Coefficient(u, x, m)/((m + n*p + S(1))*Coefficient(v, x, n))
c = Coefficient(u, x, m)/((m + n*p + S(1))*Coefficient(v, x, n))
w = Apart(-c*x**(m - n)*(v*(m - n + S(1)) + x*(p + S(1))*D(v, x)) + u, x)
res = And(Inequality(S(1), Less, n, LessEqual, m + S(1)), Less(m + n*p, S(-1)), FalseQ(DerivativeDivides(v, u, x)))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern1881 = Pattern(Integral(u_*v_**p_, x_), cons13, cons137, cons804, cons1017, cons1087, cons1088, cons1089, CustomConstraint(With1881))
def replacement1881(v, x, p, u):
m = Exponent(u, x)
n = Exponent(v, x)
c = Coefficient(u, x, m)/((m + n*p + S(1))*Coefficient(v, x, n))
c = Coefficient(u, x, m)/((m + n*p + S(1))*Coefficient(v, x, n))
w = Apart(-c*x**(m - n)*(v*(m - n + S(1)) + x*(p + S(1))*D(v, x)) + u, x)
rubi.append(1881)
return Simp(If(ZeroQ(w), c*v**(p + 1)*x**(m - n + 1), c*v**(p + 1)*x**(m - n + 1) + Int(v**p*w, x)), x)
rule1881 = ReplacementRule(pattern1881, replacement1881)
return [rule1473, rule1474, rule1475, rule1476, rule1477, rule1478, rule1479, rule1480, rule1481, rule1482, rule1483, rule1484, rule1485, rule1486, rule1487, rule1488, rule1489, rule1490, rule1491, rule1492, rule1493, rule1494, rule1495, rule1496, rule1497, rule1498, rule1499, rule1500, rule1501, rule1502, rule1503, rule1504, rule1505, rule1506, rule1507, rule1508, rule1509, rule1510, rule1511, rule1512, rule1513, rule1514, rule1515, rule1516, rule1517, rule1518, rule1519, rule1520, rule1521, rule1522, rule1523, rule1524, rule1525, rule1526, rule1527, rule1528, rule1529, rule1530, rule1531, rule1532, rule1533, rule1534, rule1535, rule1536, rule1537, rule1538, rule1539, rule1540, rule1541, rule1542, rule1543, rule1544, rule1545, rule1546, rule1547, rule1548, rule1549, rule1550, rule1551, rule1552, rule1553, rule1554, rule1555, rule1556, rule1557, rule1558, rule1559, rule1560, rule1561, rule1562, rule1563, rule1564, rule1565, rule1566, rule1567, rule1568, rule1569, rule1570, rule1571, rule1572, rule1573, rule1574, rule1575, rule1576, rule1577, rule1578, rule1579, rule1580, rule1581, rule1582, rule1583, rule1584, rule1585, rule1586, rule1587, rule1588, rule1589, rule1590, rule1591, rule1592, rule1593, rule1594, rule1595, rule1596, rule1597, rule1598, rule1599, rule1600, rule1601, rule1602, rule1603, rule1604, rule1605, rule1606, rule1607, rule1608, rule1609, rule1610, rule1611, rule1612, rule1613, rule1614, rule1615, rule1616, rule1617, rule1618, rule1619, rule1620, rule1621, rule1622, rule1623, rule1624, rule1625, rule1626, rule1627, rule1628, rule1629, rule1630, rule1631, rule1632, rule1633, rule1634, rule1635, rule1636, rule1637, rule1638, rule1639, rule1640, rule1641, rule1642, rule1643, rule1644, rule1645, rule1646, rule1647, rule1648, rule1649, rule1650, rule1651, rule1652, rule1653, rule1654, rule1655, rule1656, rule1657, rule1658, rule1659, rule1660, rule1661, rule1662, rule1663, rule1664, rule1665, rule1666, rule1667, rule1668, rule1669, rule1670, rule1671, rule1672, rule1673, rule1674, rule1675, rule1676, rule1677, rule1678, rule1679, rule1680, rule1681, rule1682, rule1683, rule1684, rule1685, rule1686, rule1687, rule1688, rule1689, rule1690, rule1691, rule1692, rule1693, rule1694, rule1695, rule1696, rule1697, rule1698, rule1699, rule1700, rule1701, rule1702, rule1703, rule1704, rule1705, rule1706, rule1707, rule1708, rule1709, rule1710, rule1711, rule1712, rule1713, rule1714, rule1715, rule1716, rule1717, rule1718, rule1719, rule1720, rule1721, rule1722, rule1723, rule1724, rule1725, rule1726, rule1727, rule1728, rule1729, rule1730, rule1731, rule1732, rule1733, rule1734, rule1735, rule1736, rule1737, rule1738, rule1739, rule1740, rule1741, rule1742, rule1743, rule1744, rule1745, rule1746, rule1747, rule1748, rule1749, rule1750, rule1751, rule1752, rule1753, rule1754, rule1755, rule1756, rule1757, rule1758, rule1759, rule1760, rule1761, rule1762, rule1763, rule1764, rule1765, rule1766, rule1767, rule1768, rule1769, rule1770, rule1771, rule1772, rule1773, rule1774, rule1775, rule1776, rule1777, rule1778, rule1779, rule1780, rule1781, rule1782, rule1783, rule1784, rule1785, rule1786, rule1787, rule1788, rule1789, rule1790, rule1791, rule1792, rule1793, rule1794, rule1795, rule1796, rule1797, rule1798, rule1799, rule1800, rule1801, rule1802, rule1803, rule1804, rule1805, rule1806, rule1807, rule1808, rule1809, rule1810, rule1811, rule1812, rule1813, rule1814, rule1815, rule1816, rule1817, rule1818, rule1819, rule1820, rule1821, rule1822, rule1823, rule1824, rule1825, rule1826, rule1827, rule1828, rule1829, rule1830, rule1831, rule1832, rule1833, rule1834, rule1835, rule1836, rule1837, rule1838, rule1839, rule1840, rule1841, rule1842, rule1843, rule1844, rule1845, rule1846, rule1847, rule1848, rule1849, rule1850, rule1851, rule1852, rule1853, rule1854, rule1855, rule1856, rule1857, rule1858, rule1859, rule1860, rule1861, rule1862, rule1863, rule1864, rule1865, rule1866, rule1867, rule1868, rule1869, rule1870, rule1871, rule1872, rule1873, rule1874, rule1875, rule1876, rule1877, rule1878, rule1879, rule1880, rule1881, ]
|
71d4bd69a78328045c28c9ebc67389e45b58e192a3056ddce46274439c890da8 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def integrand_simplification(rubi):
from sympy.integrals.rubi.constraints import cons1, cons2, cons3, cons4, cons5, cons6, cons7, cons8, cons9, cons10, cons11, cons12, cons13, cons14, cons15, cons16, cons17, cons18, cons19, cons20, cons21, cons22, cons23, cons24, cons25, cons26, cons27, cons28, cons29, cons30, cons31, cons32, cons33, cons34, cons35, cons36, cons37, cons38, cons39, cons40, cons41, cons42, cons43, cons44, cons45, cons46, cons47, cons48, cons49, cons50, cons51, cons52, cons53, cons54, cons55, cons56, cons57, cons58, cons59, cons60, cons61, cons62, cons63, cons64, cons65
pattern1 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons4, cons5, cons1)
def replacement1(p, u, b, a, n, x):
rubi.append(1)
return Int(u*(b*x**n)**p, x)
rule1 = ReplacementRule(pattern1, replacement1)
pattern2 = Pattern(Integral((a_ + x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons6, cons1)
def replacement2(p, u, j, b, c, n, a, x):
rubi.append(2)
return Int(u*(b*x**n + c*x**(S(2)*n))**p, x)
rule2 = ReplacementRule(pattern2, replacement2)
pattern3 = Pattern(Integral((x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons6, cons8)
def replacement3(p, u, j, b, a, c, n, x):
rubi.append(3)
return Int(u*(a + c*x**(S(2)*n))**p, x)
rule3 = ReplacementRule(pattern3, replacement3)
pattern4 = Pattern(Integral((x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons6, cons9)
def replacement4(p, u, j, b, a, c, n, x):
rubi.append(4)
return Int(u*(a + b*x**n)**p, x)
rule4 = ReplacementRule(pattern4, replacement4)
pattern5 = Pattern(Integral((v_*WC('a', S(1)) + v_*WC('b', S(1)) + WC('w', S(0)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons10)
def replacement5(v, w, p, u, b, a, x):
rubi.append(5)
return Int(u*(v*(a + b) + w)**p, x)
rule5 = ReplacementRule(pattern5, replacement5)
pattern6 = Pattern(Integral(Pm_**p_*WC('u', S(1)), x_), cons11, cons12, cons13)
def replacement6(x, p, Pm, u):
rubi.append(6)
return Int(Pm**p*u, x)
rule6 = ReplacementRule(pattern6, replacement6)
pattern7 = Pattern(Integral(a_, x_), cons2, cons2)
def replacement7(x, a):
rubi.append(7)
return Simp(a*x, x)
rule7 = ReplacementRule(pattern7, replacement7)
pattern8 = Pattern(Integral(a_*(b_ + x_*WC('c', S(1))), x_), cons2, cons3, cons7, cons14)
def replacement8(x, c, b, a):
rubi.append(8)
return Simp(a*(b + c*x)**S(2)/(S(2)*c), x)
rule8 = ReplacementRule(pattern8, replacement8)
pattern9 = Pattern(Integral(-u_, x_))
def replacement9(x, u):
rubi.append(9)
return Dist(S(-1), Int(u, x), x)
rule9 = ReplacementRule(pattern9, replacement9)
pattern10 = Pattern(Integral(u_*Complex(S(0), a_), x_), cons2, cons15)
def replacement10(x, a, u):
rubi.append(10)
return Dist(Complex(S(0), a), Int(u, x), x)
rule10 = ReplacementRule(pattern10, replacement10)
pattern11 = Pattern(Integral(a_*u_, x_), cons2, cons2)
def replacement11(x, a, u):
rubi.append(11)
return Dist(a, Int(u, x), x)
rule11 = ReplacementRule(pattern11, replacement11)
pattern12 = Pattern(Integral(u_, x_), cons16)
def replacement12(x, u):
rubi.append(12)
return Simp(IntSum(u, x), x)
rule12 = ReplacementRule(pattern12, replacement12)
pattern13 = Pattern(Integral(v_**WC('m', S(1))*(b_*v_)**n_*WC('u', S(1)), x_), cons3, cons4, cons17)
def replacement13(v, u, m, b, n, x):
rubi.append(13)
return Dist(b**(-m), Int(u*(b*v)**(m + n), x), x)
rule13 = ReplacementRule(pattern13, replacement13)
pattern14 = Pattern(Integral((v_*WC('a', S(1)))**m_*(v_*WC('b', S(1)))**n_*WC('u', S(1)), x_), cons2, cons3, cons21, cons18, cons19, cons20)
def replacement14(v, u, m, b, a, n, x):
rubi.append(14)
return Dist(a**(m + S(1)/2)*b**(n + S(-1)/2)*sqrt(b*v)/sqrt(a*v), Int(u*v**(m + n), x), x)
rule14 = ReplacementRule(pattern14, replacement14)
pattern15 = Pattern(Integral((v_*WC('a', S(1)))**m_*(v_*WC('b', S(1)))**n_*WC('u', S(1)), x_), cons2, cons3, cons21, cons18, cons22, cons20)
def replacement15(v, u, m, b, a, n, x):
rubi.append(15)
return Dist(a**(m + S(-1)/2)*b**(n + S(1)/2)*sqrt(a*v)/sqrt(b*v), Int(u*v**(m + n), x), x)
rule15 = ReplacementRule(pattern15, replacement15)
pattern16 = Pattern(Integral((v_*WC('a', S(1)))**m_*(v_*WC('b', S(1)))**n_*WC('u', S(1)), x_), cons2, cons3, cons21, cons4, cons18, cons23, cons20)
def replacement16(v, u, m, b, a, n, x):
rubi.append(16)
return Dist(a**(m + n)*(a*v)**(-n)*(b*v)**n, Int(u*v**(m + n), x), x)
rule16 = ReplacementRule(pattern16, replacement16)
pattern17 = Pattern(Integral((v_*WC('a', S(1)))**m_*(v_*WC('b', S(1)))**n_*WC('u', S(1)), x_), cons2, cons3, cons21, cons4, cons18, cons23, cons24)
def replacement17(v, u, m, b, a, n, x):
rubi.append(17)
return Dist(a**(-IntPart(n))*b**IntPart(n)*(a*v)**(-FracPart(n))*(b*v)**FracPart(n), Int(u*(a*v)**(m + n), x), x)
rule17 = ReplacementRule(pattern17, replacement17)
pattern18 = Pattern(Integral((a_ + v_*WC('b', S(1)))**WC('m', S(1))*(c_ + v_*WC('d', S(1)))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons25, cons17, cons26)
def replacement18(v, u, m, b, d, a, n, c, x):
rubi.append(18)
return Dist((b/d)**m, Int(u*(c + d*v)**(m + n), x), x)
rule18 = ReplacementRule(pattern18, replacement18)
pattern19 = Pattern(Integral((a_ + v_*WC('b', S(1)))**m_*(c_ + v_*WC('d', S(1)))**n_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons25, cons28, cons29)
def replacement19(v, u, m, b, d, a, c, n, x):
rubi.append(19)
return Dist((b/d)**m, Int(u*(c + d*v)**(m + n), x), x)
rule19 = ReplacementRule(pattern19, replacement19)
pattern20 = Pattern(Integral((a_ + v_*WC('b', S(1)))**m_*(c_ + v_*WC('d', S(1)))**n_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons25, cons30)
def replacement20(v, u, m, b, d, a, c, n, x):
rubi.append(20)
return Dist((a + b*v)**m*(c + d*v)**(-m), Int(u*(c + d*v)**(m + n), x), x)
rule20 = ReplacementRule(pattern20, replacement20)
pattern21 = Pattern(Integral((v_*WC('a', S(1)))**m_*(v_**S(2)*WC('c', S(1)) + v_*WC('b', S(1)))*WC('u', S(1)), x_), cons2, cons3, cons7, cons31, cons32)
def replacement21(v, u, m, b, c, a, x):
rubi.append(21)
return Dist(S(1)/a, Int(u*(a*v)**(m + S(1))*(b + c*v), x), x)
rule21 = ReplacementRule(pattern21, replacement21)
pattern22 = Pattern(Integral((a_ + v_*WC('b', S(1)))**m_*(v_**S(2)*WC('C', S(1)) + v_*WC('B', S(1)) + WC('A', S(0)))*WC('u', S(1)), x_), cons2, cons3, cons34, cons35, cons36, cons33, cons31, cons32)
def replacement22(B, v, C, u, m, b, a, x, A):
rubi.append(22)
return Dist(b**(S(-2)), Int(u*(a + b*v)**(m + S(1))*Simp(B*b - C*a + C*b*v, x), x), x)
rule22 = ReplacementRule(pattern22, replacement22)
pattern23 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('m', S(1))*(c_ + x_**WC('q', S(1))*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons37, cons38, cons39, cons40)
def replacement23(p, u, m, b, d, a, n, c, x, q):
rubi.append(23)
return Dist((d/a)**p, Int(u*x**(-n*p)*(a + b*x**n)**(m + p), x), x)
rule23 = ReplacementRule(pattern23, replacement23)
pattern24 = Pattern(Integral((a_ + x_**WC('n', S(1))*WC('b', S(1)))**WC('m', S(1))*(c_ + x_**j_*WC('d', S(1)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons5, cons6, cons41, cons42, cons43, cons44)
def replacement24(p, u, j, m, b, d, a, n, c, x):
rubi.append(24)
return Dist((-b**S(2)/d)**m, Int(u*(a - b*x**n)**(-m), x), x)
rule24 = ReplacementRule(pattern24, replacement24)
pattern25 = Pattern(Integral((a_ + x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons45, cons38)
def replacement25(p, u, b, c, a, x):
rubi.append(25)
return Int(S(2)**(-S(2)*p)*c**(-p)*u*(b + S(2)*c*x)**(S(2)*p), x)
rule25 = ReplacementRule(pattern25, replacement25)
pattern26 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)) + x_**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons4, cons46, cons45, cons38)
def replacement26(p, u, b, n2, c, a, n, x):
rubi.append(26)
return Dist(c**(-p), Int(u*(b/S(2) + c*x**n)**(S(2)*p), x), x)
rule26 = ReplacementRule(pattern26, replacement26)
pattern27 = Pattern(Integral((d_ + x_*WC('e', S(1)))*(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons47)
def replacement27(p, b, d, c, a, x, e):
rubi.append(27)
return Dist(d/b, Subst(Int(x**p, x), x, a + b*x + c*x**S(2)), x)
rule27 = ReplacementRule(pattern27, replacement27)
pattern28 = Pattern(Integral((x_**WC('p', S(1))*WC('a', S(1)) + x_**WC('q', S(1))*WC('b', S(1)))**WC('m', S(1))*WC('u', S(1)), x_), cons2, cons3, cons5, cons50, cons17, cons49)
def replacement28(p, u, m, b, a, x, q):
rubi.append(28)
return Int(u*x**(m*p)*(a + b*x**(-p + q))**m, x)
rule28 = ReplacementRule(pattern28, replacement28)
pattern29 = Pattern(Integral((x_**WC('p', S(1))*WC('a', S(1)) + x_**WC('q', S(1))*WC('b', S(1)) + x_**WC('r', S(1))*WC('c', S(1)))**WC('m', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons5, cons50, cons52, cons17, cons49, cons51)
def replacement29(p, u, m, b, r, a, c, x, q):
rubi.append(29)
return Int(u*x**(m*p)*(a + b*x**(-p + q) + c*x**(-p + r))**m, x)
rule29 = ReplacementRule(pattern29, replacement29)
pattern30 = Pattern(Integral(x_**WC('m', S(1))/(a_ + x_**n_*WC('b', S(1))), x_), cons2, cons3, cons21, cons4, cons53)
def replacement30(m, b, a, n, x):
rubi.append(30)
return Simp(rubi_log(RemoveContent(a + b*x**n, x))/(b*n), x)
rule30 = ReplacementRule(pattern30, replacement30)
pattern31 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons21, cons4, cons5, cons53, cons54)
def replacement31(p, m, b, a, n, x):
rubi.append(31)
return Simp((a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x)
rule31 = ReplacementRule(pattern31, replacement31)
pattern32 = Pattern(Integral(x_**WC('m', S(1))*(a1_ + x_**WC('n', S(1))*WC('b1', S(1)))**p_*(a2_ + x_**WC('n', S(1))*WC('b2', S(1)))**p_, x_), cons57, cons58, cons59, cons60, cons21, cons4, cons5, cons55, cons56, cons54)
def replacement32(p, a2, m, b2, b1, n, x, a1):
rubi.append(32)
return Simp((a1 + b1*x**n)**(p + S(1))*(a2 + b2*x**n)**(p + S(1))/(S(2)*b1*b2*n*(p + S(1))), x)
rule32 = ReplacementRule(pattern32, replacement32)
def With33(p, Pm, b, a, n, Qm, x):
if isinstance(x, (int, Integer, float, Float)):
return False
m = Expon(Pm, x)
if And(Equal(Expon(Qm, x), m + S(-1)), ZeroQ(-Qm*m*Coeff(Pm, x, m) + Coeff(Qm, x, m + S(-1))*D(Pm, x))):
return True
return False
pattern33 = Pattern(Integral(Qm_*(Pm_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons4, cons5, cons11, cons61, CustomConstraint(With33))
def replacement33(p, Pm, b, a, n, Qm, x):
m = Expon(Pm, x)
rubi.append(33)
return Dist(Coeff(Qm, x, m + S(-1))/(m*Coeff(Pm, x, m)), Subst(Int((a + b*x**n)**p, x), x, Pm), x)
rule33 = ReplacementRule(pattern33, replacement33)
def With34(p, Pm, b, n2, c, a, n, Qm, x):
if isinstance(x, (int, Integer, float, Float)):
return False
m = Expon(Pm, x)
if And(Equal(Expon(Qm, x), m + S(-1)), ZeroQ(-Qm*m*Coeff(Pm, x, m) + Coeff(Qm, x, m + S(-1))*D(Pm, x))):
return True
return False
pattern34 = Pattern(Integral(Qm_*(Pm_**WC('n', S(1))*WC('b', S(1)) + Pm_**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons46, cons11, cons61, CustomConstraint(With34))
def replacement34(p, Pm, b, n2, c, a, n, Qm, x):
m = Expon(Pm, x)
rubi.append(34)
return Dist(Coeff(Qm, x, m + S(-1))/(m*Coeff(Pm, x, m)), Subst(Int((a + b*x**n + c*x**(S(2)*n))**p, x), x, Pm), x)
rule34 = ReplacementRule(pattern34, replacement34)
def With35(p, u, m, Pq, Qr, x):
if isinstance(x, (int, Integer, float, Float)):
return False
gcd = PolyGCD(Pq, Qr, x)
if NonzeroQ(gcd + S(-1)):
return True
return False
pattern35 = Pattern(Integral(Pq_**m_*Qr_**p_*WC('u', S(1)), x_), cons62, cons63, cons64, cons65, CustomConstraint(With35))
def replacement35(p, u, m, Pq, Qr, x):
gcd = PolyGCD(Pq, Qr, x)
rubi.append(35)
return Int(gcd**(m + p)*u*PolynomialQuotient(Pq, gcd, x)**m*PolynomialQuotient(Qr, gcd, x)**p, x)
rule35 = ReplacementRule(pattern35, replacement35)
def With36(p, u, Pq, Qr, x):
if isinstance(x, (int, Integer, float, Float)):
return False
gcd = PolyGCD(Pq, Qr, x)
if NonzeroQ(gcd + S(-1)):
return True
return False
pattern36 = Pattern(Integral(Pq_*Qr_**p_*WC('u', S(1)), x_), cons63, cons64, cons65, CustomConstraint(With36))
def replacement36(p, u, Pq, Qr, x):
gcd = PolyGCD(Pq, Qr, x)
rubi.append(36)
return Int(gcd**(p + S(1))*u*PolynomialQuotient(Pq, gcd, x)*PolynomialQuotient(Qr, gcd, x)**p, x)
rule36 = ReplacementRule(pattern36, replacement36)
return [rule1, rule2, rule3, rule4, rule5, rule6, rule7, rule8, rule9, rule10, rule11, rule12, rule13, rule14, rule15, rule16, rule17, rule18, rule19, rule20, rule21, rule22, rule23, rule24, rule25, rule26, rule27, rule28, rule29, rule30, rule31, rule32, rule33, rule34, rule35, rule36, ]
|
5b080cc982cc42d02752b10802d6684b6888a1bbd25ae9ca8921b6921ba83e57 | '''
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
if matchpy:
from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match
from sympy.integrals.rubi.utility_function import (
Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,
ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,
PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,
ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,
NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,
FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,
SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,
Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,
IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,
ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,
ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,
Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,
PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,
ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,
Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,
SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,
NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,
EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,
PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,
TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,
NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,
ExpandLinearProduct, GCD, ContentFactor, NumericFactor,
NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,
ExpandExpression, Apart, SmartApart, MatchQ,
PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,
RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,
ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,
AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,
RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,
MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,
GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,
PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,
RationalFunctionFactors, NonrationalFunctionFactors, Reverse,
RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,
SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,
CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,
GeneralizedBinomialParts, GeneralizedTrinomialDegree,
GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,
MinimumMonomialExponent, MonomialExponent, LinearMatchQ,
PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,
TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,
QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,
NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,
PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,
AlgebraicFunctionFactors, NonalgebraicFunctionFactors,
QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,
Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,
NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,
CombineExponents, FactorInteger, FactorAbsurdNumber,
SubstForInverseFunction, SubstForFractionalPower,
SubstForFractionalPowerOfQuotientOfLinears,
FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,
SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,
FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,
MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,
TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,
OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,
PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,
PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,
FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,
FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,
NormalizeIntegrandAux, NormalizeIntegrandFactor,
NormalizeIntegrandFactorBase, NormalizeTogether,
NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,
SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,
TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,
UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,
PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,
IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,
FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,
FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,
SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,
SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,
SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,
InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,
FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,
PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,
KnownTangentIntegrandQ, KnownCotangentIntegrandQ,
KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,
AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,
ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,
ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,
FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,
PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,
FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,
FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,
FunctionOfLog, PowerVariableExpn, PowerVariableDegree,
PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,
SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,
Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,
SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,
ConstantFactor, SameQ, ReplacePart, CommonFactors,
MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,
FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,
FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,
rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,
RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,
SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,
FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,
_SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,
_SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,
_SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,
TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,
PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,
CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,
Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant,
Negative, Quotient
)
from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul,
Add, Pow, sign, EulerGamma)
from sympy.integrals.rubi.symbol import WC
from sympy.core.symbol import symbols, Symbol
from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)
from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2)
from sympy import pi as Pi
A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz']
a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']]
i, ii , Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u')
_UseGamma = False
ShowSteps = False
StepCounter = None
def miscellaneous_trig(rubi):
from sympy.integrals.rubi.constraints import cons1646, cons18, cons2, cons3, cons7, cons27, cons21, cons4, cons34, cons35, cons36, cons1647, cons1648, cons1649, cons1650, cons1651, cons1652, cons1653, cons25, cons1654, cons1408, cons208, cons38, cons48, cons125, cons147, cons343, cons5, cons240, cons244, cons1333, cons137, cons1655, cons1288, cons166, cons319, cons1656, cons31, cons249, cons94, cons253, cons13, cons163, cons246, cons1278, cons1657, cons1658, cons1659, cons170, cons1660, cons1661, cons93, cons89, cons1662, cons162, cons88, cons1663, cons1664, cons85, cons128, cons1479, cons744, cons1482, cons1665, cons23, cons1666, cons1667, cons1668, cons1669, cons1247, cons1670, cons1671, cons1672, cons1673, cons555, cons1674, cons628, cons10, cons1675, cons1676, cons1677, cons66, cons1230, cons376, cons49, cons50, cons51, cons52, cons1678, cons1439, cons1679, cons1680, cons1681, cons1682, cons62, cons584, cons464, cons1683, cons168, cons1684, cons1685, cons1686, cons1687, cons1688, cons812, cons813, cons17, cons1689, cons1690, cons1691, cons1692, cons1099, cons1693, cons87, cons165, cons1694, cons1695, cons1395, cons1696, cons1442, cons1697, cons1502, cons963, cons1698, cons1644, cons1699, cons196, cons1700, cons1011, cons150, cons1551, cons1701, cons1702, cons209, cons224, cons1703, cons810, cons811, cons148, cons528, cons1704, cons1705, cons1706, cons1707, cons1708, cons54, cons1709, cons1710, cons146, cons1711, cons1505, cons1712, cons1713, cons1714, cons1715, cons1716, cons1717, cons1718, cons1719, cons1645, cons1720, cons1721, cons1722, cons1723, cons1724, cons338, cons53, cons627, cons71, cons1725, cons1726, cons1727, cons1728, cons1360, cons1478, cons463, cons1729, cons1730, cons1731, cons1732, cons1265, cons1267, cons1474, cons1481, cons1733
pattern4685 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1646, cons18)
def replacement4685(u, m, b, d, c, a, n, x):
rubi.append(4685)
return Dist((c*tan(a + b*x))**m*(d*sin(a + b*x))**(-m)*(d*cos(a + b*x))**m, Int((d*sin(a + b*x))**(m + n)*(d*cos(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4685 = ReplacementRule(pattern4685, replacement4685)
pattern4686 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1646, cons18)
def replacement4686(u, m, b, d, c, a, n, x):
rubi.append(4686)
return Dist((c*tan(a + b*x))**m*(d*sin(a + b*x))**(-m)*(d*cos(a + b*x))**m, Int((d*sin(a + b*x))**m*(d*cos(a + b*x))**(-m + n)*ActivateTrig(u), x), x)
rule4686 = ReplacementRule(pattern4686, replacement4686)
pattern4687 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1646, cons18)
def replacement4687(u, m, b, d, c, a, n, x):
rubi.append(4687)
return Dist((c/tan(a + b*x))**m*(d*sin(a + b*x))**m*(d*cos(a + b*x))**(-m), Int((d*sin(a + b*x))**(-m + n)*(d*cos(a + b*x))**m*ActivateTrig(u), x), x)
rule4687 = ReplacementRule(pattern4687, replacement4687)
pattern4688 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1646, cons18)
def replacement4688(u, m, b, d, c, a, n, x):
rubi.append(4688)
return Dist((c/tan(a + b*x))**m*(d*sin(a + b*x))**m*(d*cos(a + b*x))**(-m), Int((d*sin(a + b*x))**(-m)*(d*cos(a + b*x))**(m + n)*ActivateTrig(u), x), x)
rule4688 = ReplacementRule(pattern4688, replacement4688)
pattern4689 = Pattern(Integral(u_*(WC('c', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1646)
def replacement4689(u, m, b, d, c, a, n, x):
rubi.append(4689)
return Dist((c/sin(a + b*x))**m*(d*sin(a + b*x))**m, Int((d*sin(a + b*x))**(-m + n)*ActivateTrig(u), x), x)
rule4689 = ReplacementRule(pattern4689, replacement4689)
pattern4690 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1646)
def replacement4690(u, m, b, c, a, x):
rubi.append(4690)
return Dist((c*sin(a + b*x))**(-m)*(c*cos(a + b*x))**m*(c*tan(a + b*x))**m, Int((c*sin(a + b*x))**m*(c*cos(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4690 = ReplacementRule(pattern4690, replacement4690)
pattern4691 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1646)
def replacement4691(u, m, b, c, a, x):
rubi.append(4691)
return Dist((c*sin(a + b*x))**m*(c*cos(a + b*x))**(-m)*(c/tan(a + b*x))**m, Int((c*sin(a + b*x))**(-m)*(c*cos(a + b*x))**m*ActivateTrig(u), x), x)
rule4691 = ReplacementRule(pattern4691, replacement4691)
pattern4692 = Pattern(Integral(u_*(WC('c', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1646)
def replacement4692(u, m, b, c, a, x):
rubi.append(4692)
return Dist((c/cos(a + b*x))**m*(c*cos(a + b*x))**m, Int((c*cos(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4692 = ReplacementRule(pattern4692, replacement4692)
pattern4693 = Pattern(Integral(u_*(WC('c', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1646)
def replacement4693(u, m, b, c, a, x):
rubi.append(4693)
return Dist((c/sin(a + b*x))**m*(c*sin(a + b*x))**m, Int((c*sin(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4693 = ReplacementRule(pattern4693, replacement4693)
pattern4694 = Pattern(Integral(u_*(WC('c', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons1646)
def replacement4694(B, u, b, c, a, n, x, A):
rubi.append(4694)
return Dist(c, Int((c*sin(a + b*x))**(n + S(-1))*(A*sin(a + b*x) + B)*ActivateTrig(u), x), x)
rule4694 = ReplacementRule(pattern4694, replacement4694)
pattern4695 = Pattern(Integral(u_*(WC('c', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons1646)
def replacement4695(B, u, b, c, a, n, x, A):
rubi.append(4695)
return Dist(c, Int((c*cos(a + b*x))**(n + S(-1))*(A*cos(a + b*x) + B)*ActivateTrig(u), x), x)
rule4695 = ReplacementRule(pattern4695, replacement4695)
pattern4696 = Pattern(Integral(u_*(A_ + WC('B', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons1646)
def replacement4696(B, u, b, a, x, A):
rubi.append(4696)
return Int((A*sin(a + b*x) + B)*ActivateTrig(u)/sin(a + b*x), x)
rule4696 = ReplacementRule(pattern4696, replacement4696)
pattern4697 = Pattern(Integral(u_*(A_ + WC('B', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons1646)
def replacement4697(B, u, b, a, x, A):
rubi.append(4697)
return Int((A*cos(a + b*x) + B)*ActivateTrig(u)/cos(a + b*x), x)
rule4697 = ReplacementRule(pattern4697, replacement4697)
pattern4698 = Pattern(Integral((WC('c', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons36, cons4, cons1646)
def replacement4698(B, C, u, b, a, c, n, x, A):
rubi.append(4698)
return Dist(c**S(2), Int((c*sin(a + b*x))**(n + S(-2))*(A*sin(a + b*x)**S(2) + B*sin(a + b*x) + C)*ActivateTrig(u), x), x)
rule4698 = ReplacementRule(pattern4698, replacement4698)
pattern4699 = Pattern(Integral((WC('c', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons36, cons4, cons1646)
def replacement4699(B, C, u, b, c, a, n, x, A):
rubi.append(4699)
return Dist(c**S(2), Int((c*cos(a + b*x))**(n + S(-2))*(A*cos(a + b*x)**S(2) + B*cos(a + b*x) + C)*ActivateTrig(u), x), x)
rule4699 = ReplacementRule(pattern4699, replacement4699)
pattern4700 = Pattern(Integral((WC('c', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('C', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons36, cons4, cons1646)
def replacement4700(C, u, b, c, a, n, x, A):
rubi.append(4700)
return Dist(c**S(2), Int((c*sin(a + b*x))**(n + S(-2))*(A*sin(a + b*x)**S(2) + C)*ActivateTrig(u), x), x)
rule4700 = ReplacementRule(pattern4700, replacement4700)
pattern4701 = Pattern(Integral((WC('c', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('C', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons36, cons4, cons1646)
def replacement4701(C, u, b, c, a, n, x, A):
rubi.append(4701)
return Dist(c**S(2), Int((c*cos(a + b*x))**(n + S(-2))*(A*cos(a + b*x)**S(2) + C)*ActivateTrig(u), x), x)
rule4701 = ReplacementRule(pattern4701, replacement4701)
pattern4702 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons35, cons36, cons1646)
def replacement4702(B, C, u, b, a, x, A):
rubi.append(4702)
return Int((A*sin(a + b*x)**S(2) + B*sin(a + b*x) + C)*ActivateTrig(u)/sin(a + b*x)**S(2), x)
rule4702 = ReplacementRule(pattern4702, replacement4702)
pattern4703 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons35, cons36, cons1646)
def replacement4703(B, C, u, b, a, x, A):
rubi.append(4703)
return Int((A*cos(a + b*x)**S(2) + B*cos(a + b*x) + C)*ActivateTrig(u)/cos(a + b*x)**S(2), x)
rule4703 = ReplacementRule(pattern4703, replacement4703)
pattern4704 = Pattern(Integral(u_*(A_ + WC('C', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons36, cons1646)
def replacement4704(C, u, b, a, x, A):
rubi.append(4704)
return Int((A*sin(a + b*x)**S(2) + C)*ActivateTrig(u)/sin(a + b*x)**S(2), x)
rule4704 = ReplacementRule(pattern4704, replacement4704)
pattern4705 = Pattern(Integral(u_*(A_ + WC('C', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons36, cons1646)
def replacement4705(C, u, b, a, x, A):
rubi.append(4705)
return Int((A*cos(a + b*x)**S(2) + C)*ActivateTrig(u)/cos(a + b*x)**S(2), x)
rule4705 = ReplacementRule(pattern4705, replacement4705)
pattern4706 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons36, cons1647)
def replacement4706(B, C, u, b, a, x, A):
rubi.append(4706)
return Int((A*sin(a + b*x) + B*sin(a + b*x)**S(2) + C)*ActivateTrig(u)/sin(a + b*x), x)
rule4706 = ReplacementRule(pattern4706, replacement4706)
pattern4707 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons36, cons1647)
def replacement4707(B, C, u, b, a, x, A):
rubi.append(4707)
return Int((A*cos(a + b*x) + B*cos(a + b*x)**S(2) + C)*ActivateTrig(u)/cos(a + b*x), x)
rule4707 = ReplacementRule(pattern4707, replacement4707)
pattern4708 = Pattern(Integral(u_*(WC('A', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)) + WC('B', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**n1_ + WC('C', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**n2_), x_), cons2, cons3, cons34, cons35, cons36, cons4, cons1648, cons1649)
def replacement4708(B, C, u, n1, b, n2, a, n, x, A):
rubi.append(4708)
return Int((A + B*sin(a + b*x) + C*sin(a + b*x)**S(2))*ActivateTrig(u)*sin(a + b*x)**n, x)
rule4708 = ReplacementRule(pattern4708, replacement4708)
pattern4709 = Pattern(Integral(u_*(WC('A', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)) + WC('B', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**n1_ + WC('C', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**n2_), x_), cons2, cons3, cons34, cons35, cons36, cons4, cons1648, cons1649)
def replacement4709(B, C, u, n1, b, n2, a, n, x, A):
rubi.append(4709)
return Int((A + B*cos(a + b*x) + C*cos(a + b*x)**S(2))*ActivateTrig(u)*cos(a + b*x)**n, x)
rule4709 = ReplacementRule(pattern4709, replacement4709)
pattern4710 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1650)
def replacement4710(u, m, b, d, c, a, n, x):
rubi.append(4710)
return Dist((c/tan(a + b*x))**m*(d*tan(a + b*x))**m, Int((d*tan(a + b*x))**(-m + n)*ActivateTrig(u), x), x)
rule4710 = ReplacementRule(pattern4710, replacement4710)
pattern4711 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1651)
def replacement4711(u, m, b, d, c, a, n, x):
rubi.append(4711)
return Dist((c*tan(a + b*x))**m*(d*sin(a + b*x))**(-m)*(d*cos(a + b*x))**m, Int((d*sin(a + b*x))**m*(d*cos(a + b*x))**(-m + n)*ActivateTrig(u), x), x)
rule4711 = ReplacementRule(pattern4711, replacement4711)
pattern4712 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1650)
def replacement4712(u, m, b, c, a, x):
rubi.append(4712)
return Dist((c/tan(a + b*x))**m*(c*tan(a + b*x))**m, Int((c*tan(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4712 = ReplacementRule(pattern4712, replacement4712)
pattern4713 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1651)
def replacement4713(u, m, b, c, a, x):
rubi.append(4713)
return Dist((c/tan(a + b*x))**m*(c*tan(a + b*x))**m, Int((c/tan(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4713 = ReplacementRule(pattern4713, replacement4713)
pattern4714 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons1650)
def replacement4714(B, u, b, c, a, n, x, A):
rubi.append(4714)
return Dist(c, Int((c*tan(a + b*x))**(n + S(-1))*(A*tan(a + b*x) + B)*ActivateTrig(u), x), x)
rule4714 = ReplacementRule(pattern4714, replacement4714)
pattern4715 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons1651)
def replacement4715(B, u, b, c, a, n, x, A):
rubi.append(4715)
return Dist(c, Int((c/tan(a + b*x))**(n + S(-1))*(A/tan(a + b*x) + B)*ActivateTrig(u), x), x)
rule4715 = ReplacementRule(pattern4715, replacement4715)
pattern4716 = Pattern(Integral(u_*(A_ + WC('B', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons1650)
def replacement4716(B, u, b, a, x, A):
rubi.append(4716)
return Int((A*tan(a + b*x) + B)*ActivateTrig(u)/tan(a + b*x), x)
rule4716 = ReplacementRule(pattern4716, replacement4716)
pattern4717 = Pattern(Integral(u_*(A_ + WC('B', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons1651)
def replacement4717(B, u, b, a, x, A):
rubi.append(4717)
return Int((A/tan(a + b*x) + B)*ActivateTrig(u)*tan(a + b*x), x)
rule4717 = ReplacementRule(pattern4717, replacement4717)
pattern4718 = Pattern(Integral((WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons36, cons4, cons1650)
def replacement4718(B, C, u, b, a, c, n, x, A):
rubi.append(4718)
return Dist(c**S(2), Int((c*tan(a + b*x))**(n + S(-2))*(A*tan(a + b*x)**S(2) + B*tan(a + b*x) + C)*ActivateTrig(u), x), x)
rule4718 = ReplacementRule(pattern4718, replacement4718)
pattern4719 = Pattern(Integral((WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons36, cons4, cons1651)
def replacement4719(B, C, u, b, c, a, n, x, A):
rubi.append(4719)
return Dist(c**S(2), Int((c/tan(a + b*x))**(n + S(-2))*(A/tan(a + b*x)**S(2) + B/tan(a + b*x) + C)*ActivateTrig(u), x), x)
rule4719 = ReplacementRule(pattern4719, replacement4719)
pattern4720 = Pattern(Integral((WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('C', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons36, cons4, cons1650)
def replacement4720(C, u, b, c, a, n, x, A):
rubi.append(4720)
return Dist(c**S(2), Int((c*tan(a + b*x))**(n + S(-2))*(A*tan(a + b*x)**S(2) + C)*ActivateTrig(u), x), x)
rule4720 = ReplacementRule(pattern4720, replacement4720)
pattern4721 = Pattern(Integral((WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('C', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons36, cons4, cons1651)
def replacement4721(C, u, b, c, a, n, x, A):
rubi.append(4721)
return Dist(c**S(2), Int((c/tan(a + b*x))**(n + S(-2))*(A/tan(a + b*x)**S(2) + C)*ActivateTrig(u), x), x)
rule4721 = ReplacementRule(pattern4721, replacement4721)
pattern4722 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons35, cons36, cons1650)
def replacement4722(B, C, u, b, a, x, A):
rubi.append(4722)
return Int((A*tan(a + b*x)**S(2) + B*tan(a + b*x) + C)*ActivateTrig(u)/tan(a + b*x)**S(2), x)
rule4722 = ReplacementRule(pattern4722, replacement4722)
pattern4723 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons35, cons36, cons1651)
def replacement4723(B, C, u, b, a, x, A):
rubi.append(4723)
return Int((A/tan(a + b*x)**S(2) + B/tan(a + b*x) + C)*ActivateTrig(u)*tan(a + b*x)**S(2), x)
rule4723 = ReplacementRule(pattern4723, replacement4723)
pattern4724 = Pattern(Integral(u_*(A_ + WC('C', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons36, cons1650)
def replacement4724(C, u, b, a, x, A):
rubi.append(4724)
return Int((A*tan(a + b*x)**S(2) + C)*ActivateTrig(u)/tan(a + b*x)**S(2), x)
rule4724 = ReplacementRule(pattern4724, replacement4724)
pattern4725 = Pattern(Integral(u_*(A_ + WC('C', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons36, cons1651)
def replacement4725(C, u, b, a, x, A):
rubi.append(4725)
return Int((A/tan(a + b*x)**S(2) + C)*ActivateTrig(u)*tan(a + b*x)**S(2), x)
rule4725 = ReplacementRule(pattern4725, replacement4725)
pattern4726 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons36, cons1647)
def replacement4726(B, C, u, b, a, x, A):
rubi.append(4726)
return Int((A*tan(a + b*x) + B*tan(a + b*x)**S(2) + C)*ActivateTrig(u)/tan(a + b*x), x)
rule4726 = ReplacementRule(pattern4726, replacement4726)
pattern4727 = Pattern(Integral(u_*(WC('A', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)) + WC('B', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**n1_ + WC('C', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**n2_), x_), cons2, cons3, cons34, cons35, cons36, cons4, cons1648, cons1649)
def replacement4727(B, C, u, n1, b, n2, a, n, x, A):
rubi.append(4727)
return Int((A + B*tan(a + b*x) + C*tan(a + b*x)**S(2))*ActivateTrig(u)*tan(a + b*x)**n, x)
rule4727 = ReplacementRule(pattern4727, replacement4727)
pattern4728 = Pattern(Integral(u_*((S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**n1_*WC('B', S(1)) + (S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**n2_*WC('C', S(1)) + (S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*WC('A', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons4, cons1648, cons1649)
def replacement4728(B, C, u, n1, b, n2, a, n, x, A):
rubi.append(4728)
return Int((A + B/tan(a + b*x) + C/tan(a + b*x)**S(2))*(S(1)/tan(a + b*x))**n*ActivateTrig(u), x)
rule4728 = ReplacementRule(pattern4728, replacement4728)
pattern4729 = Pattern(Integral(u_*(WC('c', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1652)
def replacement4729(u, m, b, d, c, a, n, x):
rubi.append(4729)
return Dist((c*sin(a + b*x))**m*(d/sin(a + b*x))**m, Int((d/sin(a + b*x))**(-m + n)*ActivateTrig(u), x), x)
rule4729 = ReplacementRule(pattern4729, replacement4729)
pattern4730 = Pattern(Integral(u_*(WC('c', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1652)
def replacement4730(u, m, b, d, c, a, n, x):
rubi.append(4730)
return Dist((c*cos(a + b*x))**m*(d/cos(a + b*x))**m, Int((d/cos(a + b*x))**(-m + n)*ActivateTrig(u), x), x)
rule4730 = ReplacementRule(pattern4730, replacement4730)
pattern4731 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1652, cons18)
def replacement4731(u, m, b, d, c, a, n, x):
rubi.append(4731)
return Dist((c*tan(a + b*x))**m*(d/sin(a + b*x))**m*(d/cos(a + b*x))**(-m), Int((d/sin(a + b*x))**(-m)*(d/cos(a + b*x))**(m + n)*ActivateTrig(u), x), x)
rule4731 = ReplacementRule(pattern4731, replacement4731)
pattern4732 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1652, cons18)
def replacement4732(u, m, b, d, c, a, n, x):
rubi.append(4732)
return Dist((c*tan(a + b*x))**m*(d/sin(a + b*x))**m*(d/cos(a + b*x))**(-m), Int((d/sin(a + b*x))**(-m + n)*(d/cos(a + b*x))**m*ActivateTrig(u), x), x)
rule4732 = ReplacementRule(pattern4732, replacement4732)
pattern4733 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1652, cons18)
def replacement4733(u, m, b, d, c, a, n, x):
rubi.append(4733)
return Dist((c/tan(a + b*x))**m*(d/sin(a + b*x))**(-m)*(d/cos(a + b*x))**m, Int((d/sin(a + b*x))**m*(d/cos(a + b*x))**(-m + n)*ActivateTrig(u), x), x)
rule4733 = ReplacementRule(pattern4733, replacement4733)
pattern4734 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('d', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1652, cons18)
def replacement4734(u, m, b, d, c, a, n, x):
rubi.append(4734)
return Dist((c/tan(a + b*x))**m*(d/sin(a + b*x))**(-m)*(d/cos(a + b*x))**m, Int((d/sin(a + b*x))**(m + n)*(d/cos(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4734 = ReplacementRule(pattern4734, replacement4734)
pattern4735 = Pattern(Integral(u_*(WC('c', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1652)
def replacement4735(u, m, b, c, a, x):
rubi.append(4735)
return Dist((c/sin(a + b*x))**m*(c*sin(a + b*x))**m, Int((c/sin(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4735 = ReplacementRule(pattern4735, replacement4735)
pattern4736 = Pattern(Integral(u_*(WC('c', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1652)
def replacement4736(u, m, b, c, a, x):
rubi.append(4736)
return Dist((c/cos(a + b*x))**m*(c*cos(a + b*x))**m, Int((c/cos(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4736 = ReplacementRule(pattern4736, replacement4736)
pattern4737 = Pattern(Integral(u_*(WC('c', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1652)
def replacement4737(u, m, b, c, a, x):
rubi.append(4737)
return Dist((c/sin(a + b*x))**m*(c/cos(a + b*x))**(-m)*(c*tan(a + b*x))**m, Int((c/sin(a + b*x))**(-m)*(c/cos(a + b*x))**m*ActivateTrig(u), x), x)
rule4737 = ReplacementRule(pattern4737, replacement4737)
pattern4738 = Pattern(Integral(u_*(WC('c', S(1))/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons7, cons21, cons18, cons1652)
def replacement4738(u, m, b, c, a, x):
rubi.append(4738)
return Dist((c/sin(a + b*x))**(-m)*(c/cos(a + b*x))**m*(c/tan(a + b*x))**m, Int((c/sin(a + b*x))**m*(c/cos(a + b*x))**(-m)*ActivateTrig(u), x), x)
rule4738 = ReplacementRule(pattern4738, replacement4738)
pattern4739 = Pattern(Integral(u_*(WC('c', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons1652)
def replacement4739(B, u, b, c, a, n, x, A):
rubi.append(4739)
return Dist(c, Int((c/cos(a + b*x))**(n + S(-1))*(A/cos(a + b*x) + B)*ActivateTrig(u), x), x)
rule4739 = ReplacementRule(pattern4739, replacement4739)
pattern4740 = Pattern(Integral(u_*(WC('c', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('B', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons34, cons35, cons4, cons1652)
def replacement4740(B, u, b, c, a, n, x, A):
rubi.append(4740)
return Dist(c, Int((c/sin(a + b*x))**(n + S(-1))*(A/sin(a + b*x) + B)*ActivateTrig(u), x), x)
rule4740 = ReplacementRule(pattern4740, replacement4740)
pattern4741 = Pattern(Integral(u_*(A_ + WC('B', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons1652)
def replacement4741(B, u, b, a, x, A):
rubi.append(4741)
return Int((A/cos(a + b*x) + B)*ActivateTrig(u)*cos(a + b*x), x)
rule4741 = ReplacementRule(pattern4741, replacement4741)
pattern4742 = Pattern(Integral(u_*(A_ + WC('B', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons34, cons35, cons1652)
def replacement4742(B, u, b, a, x, A):
rubi.append(4742)
return Int((A/sin(a + b*x) + B)*ActivateTrig(u)*sin(a + b*x), x)
rule4742 = ReplacementRule(pattern4742, replacement4742)
pattern4743 = Pattern(Integral((WC('c', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons36, cons4, cons1652)
def replacement4743(B, C, u, b, a, c, n, x, A):
rubi.append(4743)
return Dist(c**S(2), Int((c/cos(a + b*x))**(n + S(-2))*(A/cos(a + b*x)**S(2) + B/cos(a + b*x) + C)*ActivateTrig(u), x), x)
rule4743 = ReplacementRule(pattern4743, replacement4743)
pattern4744 = Pattern(Integral((WC('c', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons35, cons36, cons4, cons1652)
def replacement4744(B, C, u, b, c, a, n, x, A):
rubi.append(4744)
return Dist(c**S(2), Int((c/sin(a + b*x))**(n + S(-2))*(A/sin(a + b*x)**S(2) + B/sin(a + b*x) + C)*ActivateTrig(u), x), x)
rule4744 = ReplacementRule(pattern4744, replacement4744)
pattern4745 = Pattern(Integral((WC('c', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('C', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons36, cons4, cons1652)
def replacement4745(C, u, b, c, a, n, x, A):
rubi.append(4745)
return Dist(c**S(2), Int((c/cos(a + b*x))**(n + S(-2))*(A/cos(a + b*x)**S(2) + C)*ActivateTrig(u), x), x)
rule4745 = ReplacementRule(pattern4745, replacement4745)
pattern4746 = Pattern(Integral((WC('c', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(A_ + WC('C', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2))*WC('u', S(1)), x_), cons2, cons3, cons7, cons34, cons36, cons4, cons1652)
def replacement4746(C, u, b, c, a, n, x, A):
rubi.append(4746)
return Dist(c**S(2), Int((c/sin(a + b*x))**(n + S(-2))*(A/sin(a + b*x)**S(2) + C)*ActivateTrig(u), x), x)
rule4746 = ReplacementRule(pattern4746, replacement4746)
pattern4747 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons35, cons36, cons1652)
def replacement4747(B, C, u, b, a, x, A):
rubi.append(4747)
return Int((A/cos(a + b*x)**S(2) + B/cos(a + b*x) + C)*ActivateTrig(u)*cos(a + b*x)**S(2), x)
rule4747 = ReplacementRule(pattern4747, replacement4747)
pattern4748 = Pattern(Integral(u_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))) + WC('C', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons35, cons36, cons1652)
def replacement4748(B, C, u, b, a, x, A):
rubi.append(4748)
return Int((A/sin(a + b*x)**S(2) + B/sin(a + b*x) + C)*ActivateTrig(u)*sin(a + b*x)**S(2), x)
rule4748 = ReplacementRule(pattern4748, replacement4748)
pattern4749 = Pattern(Integral(u_*(A_ + WC('C', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons36, cons1652)
def replacement4749(C, u, b, a, x, A):
rubi.append(4749)
return Int((A/cos(a + b*x)**S(2) + C)*ActivateTrig(u)*cos(a + b*x)**S(2), x)
rule4749 = ReplacementRule(pattern4749, replacement4749)
pattern4750 = Pattern(Integral(u_*(A_ + WC('C', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)), x_), cons2, cons3, cons34, cons36, cons1652)
def replacement4750(C, u, b, a, x, A):
rubi.append(4750)
return Int((A/sin(a + b*x)**S(2) + C)*ActivateTrig(u)*sin(a + b*x)**S(2), x)
rule4750 = ReplacementRule(pattern4750, replacement4750)
pattern4751 = Pattern(Integral(u_*((S(1)/cos(x_*WC('b', S(1)) + WC('a', S(0))))**n1_*WC('B', S(1)) + (S(1)/cos(x_*WC('b', S(1)) + WC('a', S(0))))**n2_*WC('C', S(1)) + (S(1)/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*WC('A', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons4, cons1648, cons1649)
def replacement4751(B, C, u, n1, b, n2, a, n, x, A):
rubi.append(4751)
return Int((A + B/cos(a + b*x) + C/cos(a + b*x)**S(2))*(S(1)/cos(a + b*x))**n*ActivateTrig(u), x)
rule4751 = ReplacementRule(pattern4751, replacement4751)
pattern4752 = Pattern(Integral(u_*((S(1)/sin(x_*WC('b', S(1)) + WC('a', S(0))))**n1_*WC('B', S(1)) + (S(1)/sin(x_*WC('b', S(1)) + WC('a', S(0))))**n2_*WC('C', S(1)) + (S(1)/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*WC('A', S(1))), x_), cons2, cons3, cons34, cons35, cons36, cons4, cons1648, cons1649)
def replacement4752(B, C, u, n1, b, n2, a, n, x, A):
rubi.append(4752)
return Int((A + B/sin(a + b*x) + C/sin(a + b*x)**S(2))*(S(1)/sin(a + b*x))**n*ActivateTrig(u), x)
rule4752 = ReplacementRule(pattern4752, replacement4752)
pattern4753 = Pattern(Integral(sin(x_*WC('b', S(1)) + WC('a', S(0)))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1653)
def replacement4753(b, d, c, a, x):
rubi.append(4753)
return Simp(sin(a - c + x*(b - d))/(S(2)*b - S(2)*d), x) - Simp(sin(a + c + x*(b + d))/(S(2)*b + S(2)*d), x)
rule4753 = ReplacementRule(pattern4753, replacement4753)
pattern4754 = Pattern(Integral(cos(x_*WC('b', S(1)) + WC('a', S(0)))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1653)
def replacement4754(b, d, c, a, x):
rubi.append(4754)
return Simp(sin(a - c + x*(b - d))/(S(2)*b - S(2)*d), x) + Simp(sin(a + c + x*(b + d))/(S(2)*b + S(2)*d), x)
rule4754 = ReplacementRule(pattern4754, replacement4754)
pattern4755 = Pattern(Integral(sin(x_*WC('b', S(1)) + WC('a', S(0)))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1653)
def replacement4755(b, d, c, a, x):
rubi.append(4755)
return -Simp(cos(a - c + x*(b - d))/(S(2)*b - S(2)*d), x) - Simp(cos(a + c + x*(b + d))/(S(2)*b + S(2)*d), x)
rule4755 = ReplacementRule(pattern4755, replacement4755)
pattern4756 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons208, cons25, cons1654, cons1408)
def replacement4756(p, g, b, d, c, a, x):
rubi.append(4756)
return Dist(S(1)/2, Int((g*sin(c + d*x))**p, x), x) + Dist(S(1)/2, Int((g*sin(c + d*x))**p*cos(c + d*x), x), x)
rule4756 = ReplacementRule(pattern4756, replacement4756)
pattern4757 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons208, cons25, cons1654, cons1408)
def replacement4757(p, g, b, d, c, a, x):
rubi.append(4757)
return Dist(S(1)/2, Int((g*sin(c + d*x))**p, x), x) - Dist(S(1)/2, Int((g*sin(c + d*x))**p*cos(c + d*x), x), x)
rule4757 = ReplacementRule(pattern4757, replacement4757)
pattern4758 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons25, cons1654, cons38)
def replacement4758(p, m, b, d, a, c, x, e):
rubi.append(4758)
return Dist(S(2)**p*e**(-p), Int((e*cos(a + b*x))**(m + p)*sin(a + b*x)**p, x), x)
rule4758 = ReplacementRule(pattern4758, replacement4758)
pattern4759 = Pattern(Integral((WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons125, cons4, cons25, cons1654, cons38)
def replacement4759(p, f, b, d, c, a, n, x):
rubi.append(4759)
return Dist(S(2)**p*f**(-p), Int((f*sin(a + b*x))**(n + p)*cos(a + b*x)**p, x), x)
rule4759 = ReplacementRule(pattern4759, replacement4759)
pattern4760 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons21, cons5, cons25, cons1654, cons147, cons343)
def replacement4760(p, m, g, b, d, c, a, x, e):
rubi.append(4760)
return Simp(e**S(2)*(e*cos(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(p + S(1))), x)
rule4760 = ReplacementRule(pattern4760, replacement4760)
pattern4761 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons21, cons5, cons25, cons1654, cons147, cons343)
def replacement4761(p, m, g, b, d, c, a, x, e):
rubi.append(4761)
return -Simp(e**S(2)*(e*sin(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(p + S(1))), x)
rule4761 = ReplacementRule(pattern4761, replacement4761)
pattern4762 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons21, cons5, cons25, cons1654, cons147, cons240)
def replacement4762(p, m, g, b, d, a, c, x, e):
rubi.append(4762)
return -Simp((e*cos(a + b*x))**m*(g*sin(c + d*x))**(p + S(1))/(b*g*m), x)
rule4762 = ReplacementRule(pattern4762, replacement4762)
pattern4763 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons21, cons5, cons25, cons1654, cons147, cons240)
def replacement4763(p, m, g, b, d, a, c, x, e):
rubi.append(4763)
return Simp((e*sin(a + b*x))**m*(g*sin(c + d*x))**(p + S(1))/(b*g*m), x)
rule4763 = ReplacementRule(pattern4763, replacement4763)
pattern4764 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons25, cons1654, cons147, cons244, cons1333, cons137, cons1655, cons1288)
def replacement4764(p, m, g, b, d, c, a, x, e):
rubi.append(4764)
return Dist(e**S(4)*(m + p + S(-1))/(S(4)*g**S(2)*(p + S(1))), Int((e*cos(a + b*x))**(m + S(-4))*(g*sin(c + d*x))**(p + S(2)), x), x) + Simp(e**S(2)*(e*cos(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(p + S(1))), x)
rule4764 = ReplacementRule(pattern4764, replacement4764)
pattern4765 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons25, cons1654, cons147, cons244, cons1333, cons137, cons1655, cons1288)
def replacement4765(p, m, g, b, d, c, a, x, e):
rubi.append(4765)
return Dist(e**S(4)*(m + p + S(-1))/(S(4)*g**S(2)*(p + S(1))), Int((e*sin(a + b*x))**(m + S(-4))*(g*sin(c + d*x))**(p + S(2)), x), x) - Simp(e**S(2)*(e*sin(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(p + S(1))), x)
rule4765 = ReplacementRule(pattern4765, replacement4765)
pattern4766 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons25, cons1654, cons147, cons244, cons166, cons137, cons319, cons1656, cons1288)
def replacement4766(p, m, g, b, d, c, a, x, e):
rubi.append(4766)
return Dist(e**S(2)*(m + S(2)*p + S(2))/(S(4)*g**S(2)*(p + S(1))), Int((e*cos(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**(p + S(2)), x), x) + Simp((e*cos(a + b*x))**m*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(p + S(1))), x)
rule4766 = ReplacementRule(pattern4766, replacement4766)
pattern4767 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons25, cons1654, cons147, cons244, cons166, cons137, cons319, cons1656, cons1288)
def replacement4767(p, m, g, b, d, c, a, x, e):
rubi.append(4767)
return Dist(e**S(2)*(m + S(2)*p + S(2))/(S(4)*g**S(2)*(p + S(1))), Int((e*sin(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**(p + S(2)), x), x) - Simp((e*sin(a + b*x))**m*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(p + S(1))), x)
rule4767 = ReplacementRule(pattern4767, replacement4767)
pattern4768 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons5, cons25, cons1654, cons147, cons31, cons166, cons249, cons1288)
def replacement4768(p, m, g, b, d, c, a, x, e):
rubi.append(4768)
return Dist(e**S(2)*(m + p + S(-1))/(m + S(2)*p), Int((e*cos(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**p, x), x) + Simp(e**S(2)*(e*cos(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(m + S(2)*p)), x)
rule4768 = ReplacementRule(pattern4768, replacement4768)
pattern4769 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons5, cons25, cons1654, cons147, cons31, cons166, cons249, cons1288)
def replacement4769(p, m, g, b, d, c, a, x, e):
rubi.append(4769)
return Dist(e**S(2)*(m + p + S(-1))/(m + S(2)*p), Int((e*sin(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**p, x), x) - Simp(e**S(2)*(e*sin(a + b*x))**(m + S(-2))*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(m + S(2)*p)), x)
rule4769 = ReplacementRule(pattern4769, replacement4769)
pattern4770 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons5, cons25, cons1654, cons147, cons31, cons94, cons319, cons253, cons1288)
def replacement4770(p, m, g, b, d, c, a, x, e):
rubi.append(4770)
return Dist((m + S(2)*p + S(2))/(e**S(2)*(m + p + S(1))), Int((e*cos(a + b*x))**(m + S(2))*(g*sin(c + d*x))**p, x), x) - Simp((e*cos(a + b*x))**m*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(m + p + S(1))), x)
rule4770 = ReplacementRule(pattern4770, replacement4770)
pattern4771 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons5, cons25, cons1654, cons147, cons31, cons94, cons319, cons253, cons1288)
def replacement4771(p, m, g, b, d, c, a, x, e):
rubi.append(4771)
return Dist((m + S(2)*p + S(2))/(e**S(2)*(m + p + S(1))), Int((e*sin(a + b*x))**(m + S(2))*(g*sin(c + d*x))**p, x), x) + Simp((e*sin(a + b*x))**m*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(m + p + S(1))), x)
rule4771 = ReplacementRule(pattern4771, replacement4771)
pattern4772 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons208, cons25, cons1654, cons147, cons13, cons163, cons246)
def replacement4772(p, g, b, d, c, a, x):
rubi.append(4772)
return Dist(S(2)*g*p/(S(2)*p + S(1)), Int((g*sin(c + d*x))**(p + S(-1))*sin(a + b*x), x), x) + Simp(S(2)*(g*sin(c + d*x))**p*sin(a + b*x)/(d*(S(2)*p + S(1))), x)
rule4772 = ReplacementRule(pattern4772, replacement4772)
pattern4773 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons208, cons25, cons1654, cons147, cons13, cons163, cons246)
def replacement4773(p, g, b, d, c, a, x):
rubi.append(4773)
return Dist(S(2)*g*p/(S(2)*p + S(1)), Int((g*sin(c + d*x))**(p + S(-1))*cos(a + b*x), x), x) + Simp(-S(2)*(g*sin(c + d*x))**p*cos(a + b*x)/(d*(S(2)*p + S(1))), x)
rule4773 = ReplacementRule(pattern4773, replacement4773)
pattern4774 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons208, cons25, cons1654, cons147, cons13, cons137, cons246)
def replacement4774(p, g, b, d, c, a, x):
rubi.append(4774)
return Dist((S(2)*p + S(3))/(S(2)*g*(p + S(1))), Int((g*sin(c + d*x))**(p + S(1))*sin(a + b*x), x), x) + Simp((g*sin(c + d*x))**(p + S(1))*cos(a + b*x)/(S(2)*b*g*(p + S(1))), x)
rule4774 = ReplacementRule(pattern4774, replacement4774)
pattern4775 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons208, cons25, cons1654, cons147, cons13, cons137, cons246)
def replacement4775(p, g, b, d, c, a, x):
rubi.append(4775)
return Dist((S(2)*p + S(3))/(S(2)*g*(p + S(1))), Int((g*sin(c + d*x))**(p + S(1))*cos(a + b*x), x), x) - Simp((g*sin(c + d*x))**(p + S(1))*sin(a + b*x)/(S(2)*b*g*(p + S(1))), x)
rule4775 = ReplacementRule(pattern4775, replacement4775)
pattern4776 = Pattern(Integral(cos(x_*WC('b', S(1)) + WC('a', S(0)))/sqrt(sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons25, cons1654)
def replacement4776(b, d, c, a, x):
rubi.append(4776)
return Simp(log(sin(a + b*x) + sqrt(sin(c + d*x)) + cos(a + b*x))/d, x) - Simp(-asin(sin(a + b*x) - cos(a + b*x))/d, x)
rule4776 = ReplacementRule(pattern4776, replacement4776)
pattern4777 = Pattern(Integral(sin(x_*WC('b', S(1)) + WC('a', S(0)))/sqrt(sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons25, cons1654)
def replacement4777(b, d, c, a, x):
rubi.append(4777)
return -Simp(log(sin(a + b*x) + sqrt(sin(c + d*x)) + cos(a + b*x))/d, x) - Simp(-asin(sin(a + b*x) - cos(a + b*x))/d, x)
rule4777 = ReplacementRule(pattern4777, replacement4777)
pattern4778 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_/cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons208, cons5, cons25, cons1654, cons147, cons246)
def replacement4778(p, g, b, d, c, a, x):
rubi.append(4778)
return Dist(S(2)*g, Int((g*sin(c + d*x))**(p + S(-1))*sin(a + b*x), x), x)
rule4778 = ReplacementRule(pattern4778, replacement4778)
pattern4779 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_/sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons208, cons5, cons25, cons1654, cons147, cons246)
def replacement4779(p, g, b, d, c, a, x):
rubi.append(4779)
return Dist(S(2)*g, Int((g*sin(c + d*x))**(p + S(-1))*cos(a + b*x), x), x)
rule4779 = ReplacementRule(pattern4779, replacement4779)
pattern4780 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons208, cons21, cons5, cons25, cons1654, cons147)
def replacement4780(p, m, g, b, d, a, c, x, e):
rubi.append(4780)
return Dist((e*cos(a + b*x))**(-p)*(g*sin(c + d*x))**p*sin(a + b*x)**(-p), Int((e*cos(a + b*x))**(m + p)*sin(a + b*x)**p, x), x)
rule4780 = ReplacementRule(pattern4780, replacement4780)
pattern4781 = Pattern(Integral((WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons125, cons208, cons4, cons5, cons25, cons1654, cons147)
def replacement4781(p, g, f, b, d, a, n, c, x):
rubi.append(4781)
return Dist((f*sin(a + b*x))**(-p)*(g*sin(c + d*x))**p*cos(a + b*x)**(-p), Int((f*sin(a + b*x))**(n + p)*cos(a + b*x)**p, x), x)
rule4781 = ReplacementRule(pattern4781, replacement4781)
pattern4782 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2)*cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons208, cons25, cons1654, cons1408)
def replacement4782(p, g, b, d, c, a, x):
rubi.append(4782)
return Dist(S(1)/4, Int((g*sin(c + d*x))**p, x), x) - Dist(S(1)/4, Int((g*sin(c + d*x))**p*cos(c + d*x)**S(2), x), x)
rule4782 = ReplacementRule(pattern4782, replacement4782)
pattern4783 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons25, cons1654, cons38)
def replacement4783(p, m, f, b, d, a, n, c, x, e):
rubi.append(4783)
return Dist(S(2)**p*e**(-p)*f**(-p), Int((e*cos(a + b*x))**(m + p)*(f*sin(a + b*x))**(n + p), x), x)
rule4783 = ReplacementRule(pattern4783, replacement4783)
pattern4784 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons25, cons1654, cons147, cons1278)
def replacement4784(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4784)
return Simp(e*(e*cos(a + b*x))**(m + S(-1))*(f*sin(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*f*(n + p + S(1))), x)
rule4784 = ReplacementRule(pattern4784, replacement4784)
pattern4785 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons25, cons1654, cons147, cons1278)
def replacement4785(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4785)
return -Simp(e*(e*sin(a + b*x))**(m + S(-1))*(f*cos(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*f*(n + p + S(1))), x)
rule4785 = ReplacementRule(pattern4785, replacement4785)
pattern4786 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons25, cons1654, cons147, cons1657, cons253)
def replacement4786(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4786)
return -Simp((e*cos(a + b*x))**(m + S(1))*(f*sin(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*e*f*(m + p + S(1))), x)
rule4786 = ReplacementRule(pattern4786, replacement4786)
pattern4787 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons25, cons1654, cons147, cons244, cons1658, cons137, cons1659, cons170)
def replacement4787(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4787)
return Dist(e**S(4)*(m + p + S(-1))/(S(4)*g**S(2)*(n + p + S(1))), Int((e*cos(a + b*x))**(m + S(-4))*(f*sin(a + b*x))**n*(g*sin(c + d*x))**(p + S(2)), x), x) + Simp(e**S(2)*(e*cos(a + b*x))**(m + S(-2))*(f*sin(a + b*x))**n*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(n + p + S(1))), x)
rule4787 = ReplacementRule(pattern4787, replacement4787)
pattern4788 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons25, cons1654, cons147, cons244, cons1658, cons137, cons1659, cons170)
def replacement4788(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4788)
return Dist(e**S(4)*(m + p + S(-1))/(S(4)*g**S(2)*(n + p + S(1))), Int((e*sin(a + b*x))**(m + S(-4))*(f*cos(a + b*x))**n*(g*sin(c + d*x))**(p + S(2)), x), x) - Simp(e**S(2)*(e*sin(a + b*x))**(m + S(-2))*(f*cos(a + b*x))**n*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(n + p + S(1))), x)
rule4788 = ReplacementRule(pattern4788, replacement4788)
pattern4789 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons25, cons1654, cons147, cons244, cons166, cons137, cons1660, cons1659, cons170, cons1661)
def replacement4789(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4789)
return Dist(e**S(2)*(m + n + S(2)*p + S(2))/(S(4)*g**S(2)*(n + p + S(1))), Int((e*cos(a + b*x))**(m + S(-2))*(f*sin(a + b*x))**n*(g*sin(c + d*x))**(p + S(2)), x), x) + Simp((e*cos(a + b*x))**m*(f*sin(a + b*x))**n*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(n + p + S(1))), x)
rule4789 = ReplacementRule(pattern4789, replacement4789)
pattern4790 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons25, cons1654, cons147, cons244, cons166, cons137, cons1660, cons1659, cons170, cons1661)
def replacement4790(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4790)
return Dist(e**S(2)*(m + n + S(2)*p + S(2))/(S(4)*g**S(2)*(n + p + S(1))), Int((e*sin(a + b*x))**(m + S(-2))*(f*cos(a + b*x))**n*(g*sin(c + d*x))**(p + S(2)), x), x) - Simp((e*sin(a + b*x))**m*(f*cos(a + b*x))**n*(g*sin(c + d*x))**(p + S(1))/(S(2)*b*g*(n + p + S(1))), x)
rule4790 = ReplacementRule(pattern4790, replacement4790)
pattern4791 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons25, cons1654, cons147, cons93, cons166, cons89, cons1659, cons170)
def replacement4791(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4791)
return Dist(e**S(2)*(m + p + S(-1))/(f**S(2)*(n + p + S(1))), Int((e*cos(a + b*x))**(m + S(-2))*(f*sin(a + b*x))**(n + S(2))*(g*sin(c + d*x))**p, x), x) + Simp(e*(e*cos(a + b*x))**(m + S(-1))*(f*sin(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*f*(n + p + S(1))), x)
rule4791 = ReplacementRule(pattern4791, replacement4791)
pattern4792 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons5, cons25, cons1654, cons147, cons93, cons166, cons89, cons1659, cons170)
def replacement4792(p, m, f, b, g, d, a, c, n, x, e):
rubi.append(4792)
return Dist(e**S(2)*(m + p + S(-1))/(f**S(2)*(n + p + S(1))), Int((e*sin(a + b*x))**(m + S(-2))*(f*cos(a + b*x))**(n + S(2))*(g*sin(c + d*x))**p, x), x) - Simp(e*(e*sin(a + b*x))**(m + S(-1))*(f*cos(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*f*(n + p + S(1))), x)
rule4792 = ReplacementRule(pattern4792, replacement4792)
pattern4793 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons25, cons1654, cons147, cons31, cons166, cons1662, cons170)
def replacement4793(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4793)
return Dist(e**S(2)*(m + p + S(-1))/(m + n + S(2)*p), Int((e*cos(a + b*x))**(m + S(-2))*(f*sin(a + b*x))**n*(g*sin(c + d*x))**p, x), x) + Simp(e*(e*cos(a + b*x))**(m + S(-1))*(f*sin(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*f*(m + n + S(2)*p)), x)
rule4793 = ReplacementRule(pattern4793, replacement4793)
pattern4794 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons25, cons1654, cons147, cons31, cons166, cons1662, cons170)
def replacement4794(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4794)
return Dist(e**S(2)*(m + p + S(-1))/(m + n + S(2)*p), Int((e*sin(a + b*x))**(m + S(-2))*(f*cos(a + b*x))**n*(g*sin(c + d*x))**p, x), x) - Simp(e*(e*sin(a + b*x))**(m + S(-1))*(f*cos(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*f*(m + n + S(2)*p)), x)
rule4794 = ReplacementRule(pattern4794, replacement4794)
pattern4795 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons25, cons1654, cons147, cons162, cons94, cons88, cons163, cons1662, cons170)
def replacement4795(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4795)
return Dist(S(2)*f*g*(n + p + S(-1))/(e*(m + n + S(2)*p)), Int((e*cos(a + b*x))**(m + S(1))*(f*sin(a + b*x))**(n + S(-1))*(g*sin(c + d*x))**(p + S(-1)), x), x) - Simp(f*(e*cos(a + b*x))**(m + S(1))*(f*sin(a + b*x))**(n + S(-1))*(g*sin(c + d*x))**p/(b*e*(m + n + S(2)*p)), x)
rule4795 = ReplacementRule(pattern4795, replacement4795)
pattern4796 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons25, cons1654, cons147, cons162, cons94, cons88, cons163, cons1662, cons170)
def replacement4796(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4796)
return Dist(S(2)*f*g*(n + p + S(-1))/(e*(m + n + S(2)*p)), Int((e*sin(a + b*x))**(m + S(1))*(f*cos(a + b*x))**(n + S(-1))*(g*sin(c + d*x))**(p + S(-1)), x), x) + Simp(f*(e*sin(a + b*x))**(m + S(1))*(f*cos(a + b*x))**(n + S(-1))*(g*sin(c + d*x))**p/(b*e*(m + n + S(2)*p)), x)
rule4796 = ReplacementRule(pattern4796, replacement4796)
pattern4797 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons25, cons1654, cons147, cons162, cons94, cons88, cons137, cons1660, cons253, cons170)
def replacement4797(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4797)
return Dist(f*(m + n + S(2)*p + S(2))/(S(2)*e*g*(m + p + S(1))), Int((e*cos(a + b*x))**(m + S(1))*(f*sin(a + b*x))**(n + S(-1))*(g*sin(c + d*x))**(p + S(1)), x), x) - Simp((e*cos(a + b*x))**(m + S(1))*(f*sin(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*e*f*(m + p + S(1))), x)
rule4797 = ReplacementRule(pattern4797, replacement4797)
pattern4798 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons25, cons1654, cons147, cons162, cons94, cons88, cons137, cons1660, cons253, cons170)
def replacement4798(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4798)
return Dist(f*(m + n + S(2)*p + S(2))/(S(2)*e*g*(m + p + S(1))), Int((e*sin(a + b*x))**(m + S(1))*(f*cos(a + b*x))**(n + S(-1))*(g*sin(c + d*x))**(p + S(1)), x), x) + Simp((e*sin(a + b*x))**(m + S(1))*(f*cos(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*e*f*(m + p + S(1))), x)
rule4798 = ReplacementRule(pattern4798, replacement4798)
pattern4799 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons25, cons1654, cons147, cons31, cons94, cons1660, cons253, cons170)
def replacement4799(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4799)
return Dist((m + n + S(2)*p + S(2))/(e**S(2)*(m + p + S(1))), Int((e*cos(a + b*x))**(m + S(2))*(f*sin(a + b*x))**n*(g*sin(c + d*x))**p, x), x) - Simp((e*cos(a + b*x))**(m + S(1))*(f*sin(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*e*f*(m + p + S(1))), x)
rule4799 = ReplacementRule(pattern4799, replacement4799)
pattern4800 = Pattern(Integral((WC('e', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**m_*(WC('f', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons4, cons5, cons25, cons1654, cons147, cons31, cons94, cons1660, cons253, cons170)
def replacement4800(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4800)
return Dist((m + n + S(2)*p + S(2))/(e**S(2)*(m + p + S(1))), Int((e*sin(a + b*x))**(m + S(2))*(f*cos(a + b*x))**n*(g*sin(c + d*x))**p, x), x) + Simp((e*sin(a + b*x))**(m + S(1))*(f*cos(a + b*x))**(n + S(1))*(g*sin(c + d*x))**p/(b*e*f*(m + p + S(1))), x)
rule4800 = ReplacementRule(pattern4800, replacement4800)
pattern4801 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*(WC('f', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons21, cons4, cons5, cons25, cons1654, cons147)
def replacement4801(p, m, f, b, g, d, a, n, c, x, e):
rubi.append(4801)
return Dist((e*cos(a + b*x))**(-p)*(f*sin(a + b*x))**(-p)*(g*sin(c + d*x))**p, Int((e*cos(a + b*x))**(m + p)*(f*sin(a + b*x))**(n + p), x), x)
rule4801 = ReplacementRule(pattern4801, replacement4801)
pattern4802 = Pattern(Integral((WC('e', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('m', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons21, cons25, cons1663)
def replacement4802(m, b, d, a, c, x, e):
rubi.append(4802)
return -Simp((e*cos(a + b*x))**(m + S(1))*(m + S(2))*cos((a + b*x)*(m + S(1)))/(d*e*(m + S(1))), x)
rule4802 = ReplacementRule(pattern4802, replacement4802)
pattern4803 = Pattern(Integral((F_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + a_)**p_, x_), cons2, cons3, cons7, cons27, cons1664, cons85, cons128)
def replacement4803(p, b, d, c, a, n, x, F):
rubi.append(4803)
return Int((a + b*F(c + d*x)**n)**p, x)
rule4803 = ReplacementRule(pattern4803, replacement4803)
pattern4804 = Pattern(Integral(S(1)/(F_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + a_), x_), cons2, cons3, cons7, cons27, cons1664, cons1479, cons744)
def replacement4804(b, d, c, a, n, x, F):
rubi.append(4804)
return Dist(S(2)/(a*n), Sum_doit(Int(S(1)/(S(1) - (S(-1))**(-S(4)*k/n)*F(c + d*x)**S(2)/Rt(-a/b, n/S(2))), x), List(k, S(1), n/S(2))), x)
rule4804 = ReplacementRule(pattern4804, replacement4804)
pattern4805 = Pattern(Integral(S(1)/(F_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + a_), x_), cons2, cons3, cons7, cons27, cons1664, cons1482, cons744)
def replacement4805(b, d, c, a, n, x, F):
rubi.append(4805)
return Int(ExpandTrig(S(1)/(a + b*F(c + d*x)**n), x), x)
rule4805 = ReplacementRule(pattern4805, replacement4805)
pattern4806 = Pattern(Integral(G_**(x_*WC('d', S(1)) + WC('c', S(0)))/(F_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + a_), x_), cons2, cons3, cons7, cons27, cons21, cons1665, cons85, cons744)
def replacement4806(m, b, G, d, c, a, n, x, F):
rubi.append(4806)
return Int(ExpandTrig(G(c + d*x)**m, S(1)/(a + b*F(c + d*x)**n), x), x)
rule4806 = ReplacementRule(pattern4806, replacement4806)
def With4807(p, d, a, c, n, x, F):
v = ActivateTrig(F(c + d*x))
rubi.append(4807)
return Dist(a**IntPart(n)*(a*v**p)**FracPart(n)*(v/NonfreeFactors(v, x))**(p*IntPart(n))*NonfreeFactors(v, x)**(-p*FracPart(n)), Int(NonfreeFactors(v, x)**(n*p), x), x)
pattern4807 = Pattern(Integral((F_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('a', S(1)))**n_, x_), cons2, cons7, cons27, cons4, cons5, cons1664, cons23, cons38)
rule4807 = ReplacementRule(pattern4807, With4807)
def With4808(p, b, d, c, a, n, x, F):
v = ActivateTrig(F(c + d*x))
rubi.append(4808)
return Dist(a**IntPart(n)*(a*(b*v)**p)**FracPart(n)*(b*v)**(-p*FracPart(n)), Int((b*v)**(n*p), x), x)
pattern4808 = Pattern(Integral(((F_*(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)))**p_*WC('a', S(1)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1664, cons23, cons147)
rule4808 = ReplacementRule(pattern4808, With4808)
def With4809(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(sin(c*(a + b*x)), x)
if FunctionOfQ(sin(c*(a + b*x))/d, u, x, True):
return True
return False
pattern4809 = Pattern(Integral(F_*u_*(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)), x_), cons2, cons3, cons7, cons1666, CustomConstraint(With4809))
def replacement4809(u, b, c, a, x, F):
d = FreeFactors(sin(c*(a + b*x)), x)
rubi.append(4809)
return Dist(d/(b*c), Subst(Int(SubstFor(S(1), sin(c*(a + b*x))/d, u, x), x), x, sin(c*(a + b*x))/d), x)
rule4809 = ReplacementRule(pattern4809, replacement4809)
def With4810(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(cos(c*(a + b*x)), x)
if FunctionOfQ(cos(c*(a + b*x))/d, u, x, True):
return True
return False
pattern4810 = Pattern(Integral(F_*u_*(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)), x_), cons2, cons3, cons7, cons1667, CustomConstraint(With4810))
def replacement4810(u, b, c, a, x, F):
d = FreeFactors(cos(c*(a + b*x)), x)
rubi.append(4810)
return -Dist(d/(b*c), Subst(Int(SubstFor(S(1), cos(c*(a + b*x))/d, u, x), x), x, cos(c*(a + b*x))/d), x)
rule4810 = ReplacementRule(pattern4810, replacement4810)
def With4811(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(sin(c*(a + b*x)), x)
if FunctionOfQ(sin(c*(a + b*x))/d, u, x, True):
return True
return False
pattern4811 = Pattern(Integral(F_*u_*(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)), x_), cons2, cons3, cons7, cons1668, CustomConstraint(With4811))
def replacement4811(u, b, c, a, x, F):
d = FreeFactors(sin(c*(a + b*x)), x)
rubi.append(4811)
return Dist(S(1)/(b*c), Subst(Int(SubstFor(S(1)/x, sin(c*(a + b*x))/d, u, x), x), x, sin(c*(a + b*x))/d), x)
rule4811 = ReplacementRule(pattern4811, replacement4811)
def With4812(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(cos(c*(a + b*x)), x)
if FunctionOfQ(cos(c*(a + b*x))/d, u, x, True):
return True
return False
pattern4812 = Pattern(Integral(F_*u_*(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)), x_), cons2, cons3, cons7, cons1669, CustomConstraint(With4812))
def replacement4812(u, b, c, a, x, F):
d = FreeFactors(cos(c*(a + b*x)), x)
rubi.append(4812)
return -Dist(S(1)/(b*c), Subst(Int(SubstFor(S(1)/x, cos(c*(a + b*x))/d, u, x), x), x, cos(c*(a + b*x))/d), x)
rule4812 = ReplacementRule(pattern4812, replacement4812)
def With4813(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(tan(c*(a + b*x)), x)
if FunctionOfQ(tan(c*(a + b*x))/d, u, x, True):
return True
return False
pattern4813 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons1247, cons1670, CustomConstraint(With4813))
def replacement4813(u, b, c, a, x, F):
d = FreeFactors(tan(c*(a + b*x)), x)
rubi.append(4813)
return Dist(d/(b*c), Subst(Int(SubstFor(S(1), tan(c*(a + b*x))/d, u, x), x), x, tan(c*(a + b*x))/d), x)
rule4813 = ReplacementRule(pattern4813, replacement4813)
def With4814(u, b, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(tan(c*(a + b*x)), x)
if FunctionOfQ(tan(c*(a + b*x))/d, u, x, True):
return True
return False
pattern4814 = Pattern(Integral(u_/cos((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**S(2), x_), cons2, cons3, cons7, cons1247, CustomConstraint(With4814))
def replacement4814(u, b, c, a, x):
d = FreeFactors(tan(c*(a + b*x)), x)
rubi.append(4814)
return Dist(d/(b*c), Subst(Int(SubstFor(S(1), tan(c*(a + b*x))/d, u, x), x), x, tan(c*(a + b*x))/d), x)
rule4814 = ReplacementRule(pattern4814, replacement4814)
def With4815(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(S(1)/tan(c*(a + b*x)), x)
if FunctionOfQ(S(1)/(d*tan(c*(a + b*x))), u, x, True):
return True
return False
pattern4815 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons1247, cons1671, CustomConstraint(With4815))
def replacement4815(u, b, c, a, x, F):
d = FreeFactors(S(1)/tan(c*(a + b*x)), x)
rubi.append(4815)
return -Dist(d/(b*c), Subst(Int(SubstFor(S(1), S(1)/(d*tan(c*(a + b*x))), u, x), x), x, S(1)/(d*tan(c*(a + b*x)))), x)
rule4815 = ReplacementRule(pattern4815, replacement4815)
def With4816(u, b, c, a, x):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(S(1)/tan(c*(a + b*x)), x)
if FunctionOfQ(S(1)/(d*tan(c*(a + b*x))), u, x, True):
return True
return False
pattern4816 = Pattern(Integral(u_/sin((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**S(2), x_), cons2, cons3, cons7, cons1247, CustomConstraint(With4816))
def replacement4816(u, b, c, a, x):
d = FreeFactors(S(1)/tan(c*(a + b*x)), x)
rubi.append(4816)
return -Dist(d/(b*c), Subst(Int(SubstFor(S(1), S(1)/(d*tan(c*(a + b*x))), u, x), x), x, S(1)/(d*tan(c*(a + b*x)))), x)
rule4816 = ReplacementRule(pattern4816, replacement4816)
def With4817(u, b, c, n, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(tan(c*(a + b*x)), x)
if And(FunctionOfQ(tan(c*(a + b*x))/d, u, x, True), TryPureTanSubst((S(1)/tan(c*(a + b*x)))**n*ActivateTrig(u), x)):
return True
return False
pattern4817 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons85, cons1668, CustomConstraint(With4817))
def replacement4817(u, b, c, n, a, x, F):
d = FreeFactors(tan(c*(a + b*x)), x)
rubi.append(4817)
return Dist(d**(-n + S(1))/(b*c), Subst(Int(SubstFor(x**(-n)/(d**S(2)*x**S(2) + S(1)), tan(c*(a + b*x))/d, u, x), x), x, tan(c*(a + b*x))/d), x)
rule4817 = ReplacementRule(pattern4817, replacement4817)
def With4818(u, b, c, n, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(S(1)/tan(c*(a + b*x)), x)
if And(FunctionOfQ(S(1)/(d*tan(c*(a + b*x))), u, x, True), TryPureTanSubst(ActivateTrig(u)*tan(c*(a + b*x))**n, x)):
return True
return False
pattern4818 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons85, cons1669, CustomConstraint(With4818))
def replacement4818(u, b, c, n, a, x, F):
d = FreeFactors(S(1)/tan(c*(a + b*x)), x)
rubi.append(4818)
return -Dist(d**(-n + S(1))/(b*c), Subst(Int(SubstFor(x**(-n)/(d**S(2)*x**S(2) + S(1)), S(1)/(d*tan(c*(a + b*x))), u, x), x), x, S(1)/(d*tan(c*(a + b*x)))), x)
rule4818 = ReplacementRule(pattern4818, replacement4818)
def With4819(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
v = FunctionOfTrig(u, x)
d = FreeFactors(S(1)/tan(v), x)
res = And(Not(FalseQ(v)), FunctionOfQ(NonfreeFactors(S(1)/tan(v), x), u, x, True), TryPureTanSubst(ActivateTrig(u), x))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4819 = Pattern(Integral(u_, x_), CustomConstraint(With4819))
def replacement4819(x, u):
v = FunctionOfTrig(u, x)
d = FreeFactors(S(1)/tan(v), x)
rubi.append(4819)
return Simp(With(List(Set(d, FreeFactors(S(1)/tan(v), x))), Dist(-d/Coefficient(v, x, S(1)), Subst(Int(SubstFor(S(1)/(d**S(2)*x**S(2) + S(1)), S(1)/(d*tan(v)), u, x), x), x, S(1)/(d*tan(v))), x)), x)
rule4819 = ReplacementRule(pattern4819, replacement4819)
def With4820(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
v = FunctionOfTrig(u, x)
d = FreeFactors(tan(v), x)
res = And(Not(FalseQ(v)), FunctionOfQ(NonfreeFactors(tan(v), x), u, x, True), TryPureTanSubst(ActivateTrig(u), x))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4820 = Pattern(Integral(u_, x_), CustomConstraint(With4820))
def replacement4820(x, u):
v = FunctionOfTrig(u, x)
d = FreeFactors(tan(v), x)
rubi.append(4820)
return Simp(With(List(Set(d, FreeFactors(tan(v), x))), Dist(d/Coefficient(v, x, S(1)), Subst(Int(SubstFor(S(1)/(d**S(2)*x**S(2) + S(1)), tan(v)/d, u, x), x), x, tan(v)/d), x)), x)
rule4820 = ReplacementRule(pattern4820, replacement4820)
pattern4821 = Pattern(Integral(F_**(x_*WC('b', S(1)) + WC('a', S(0)))*G_**(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons1672, cons1673, cons555)
def replacement4821(p, b, G, d, c, a, x, q, F):
rubi.append(4821)
return Int(ExpandTrigReduce(ActivateTrig(F(a + b*x)**p*G(c + d*x)**q), x), x)
rule4821 = ReplacementRule(pattern4821, replacement4821)
pattern4822 = Pattern(Integral(F_**(x_*WC('b', S(1)) + WC('a', S(0)))*G_**(x_*WC('d', S(1)) + WC('c', S(0)))*H_**(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons1672, cons1673, cons1674, cons628)
def replacement4822(p, f, b, r, G, d, c, a, H, x, q, e, F):
rubi.append(4822)
return Int(ExpandTrigReduce(ActivateTrig(F(a + b*x)**p*G(c + d*x)**q*H(e + f*x)**r), x), x)
rule4822 = ReplacementRule(pattern4822, replacement4822)
def With4823(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(sin(c*(a + b*x)), x)
if FunctionOfQ(sin(c*(a + b*x))/d, u, x):
return True
return False
pattern4823 = Pattern(Integral(F_*u_*(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)), x_), cons2, cons3, cons7, cons1666, CustomConstraint(With4823))
def replacement4823(u, b, c, a, x, F):
d = FreeFactors(sin(c*(a + b*x)), x)
rubi.append(4823)
return Dist(d/(b*c), Subst(Int(SubstFor(S(1), sin(c*(a + b*x))/d, u, x), x), x, sin(c*(a + b*x))/d), x)
rule4823 = ReplacementRule(pattern4823, replacement4823)
def With4824(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(cos(c*(a + b*x)), x)
if FunctionOfQ(cos(c*(a + b*x))/d, u, x):
return True
return False
pattern4824 = Pattern(Integral(F_*u_*(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)), x_), cons2, cons3, cons7, cons1667, CustomConstraint(With4824))
def replacement4824(u, b, c, a, x, F):
d = FreeFactors(cos(c*(a + b*x)), x)
rubi.append(4824)
return -Dist(d/(b*c), Subst(Int(SubstFor(S(1), cos(c*(a + b*x))/d, u, x), x), x, cos(c*(a + b*x))/d), x)
rule4824 = ReplacementRule(pattern4824, replacement4824)
def With4825(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(sin(c*(a + b*x)), x)
if FunctionOfQ(sin(c*(a + b*x))/d, u, x):
return True
return False
pattern4825 = Pattern(Integral(F_*u_*(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)), x_), cons2, cons3, cons7, cons1668, CustomConstraint(With4825))
def replacement4825(u, b, c, a, x, F):
d = FreeFactors(sin(c*(a + b*x)), x)
rubi.append(4825)
return Dist(S(1)/(b*c), Subst(Int(SubstFor(S(1)/x, sin(c*(a + b*x))/d, u, x), x), x, sin(c*(a + b*x))/d), x)
rule4825 = ReplacementRule(pattern4825, replacement4825)
def With4826(u, b, c, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(cos(c*(a + b*x)), x)
if FunctionOfQ(cos(c*(a + b*x))/d, u, x):
return True
return False
pattern4826 = Pattern(Integral(F_*u_*(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)), x_), cons2, cons3, cons7, cons1669, CustomConstraint(With4826))
def replacement4826(u, b, c, a, x, F):
d = FreeFactors(cos(c*(a + b*x)), x)
rubi.append(4826)
return -Dist(S(1)/(b*c), Subst(Int(SubstFor(S(1)/x, cos(c*(a + b*x))/d, u, x), x), x, cos(c*(a + b*x))/d), x)
rule4826 = ReplacementRule(pattern4826, replacement4826)
def With4827(u, b, c, a, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(sin(c*(a + b*x)), x)
if FunctionOfQ(sin(c*(a + b*x))/d, u, x):
return True
return False
pattern4827 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons1482, cons1247, cons1666, CustomConstraint(With4827))
def replacement4827(u, b, c, a, n, x, F):
d = FreeFactors(sin(c*(a + b*x)), x)
rubi.append(4827)
return Dist(d/(b*c), Subst(Int(SubstFor((-d**S(2)*x**S(2) + S(1))**(n/S(2) + S(-1)/2), sin(c*(a + b*x))/d, u, x), x), x, sin(c*(a + b*x))/d), x)
rule4827 = ReplacementRule(pattern4827, replacement4827)
def With4828(u, b, c, a, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(sin(c*(a + b*x)), x)
if FunctionOfQ(sin(c*(a + b*x))/d, u, x):
return True
return False
pattern4828 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons1482, cons1247, cons1670, CustomConstraint(With4828))
def replacement4828(u, b, c, a, n, x, F):
d = FreeFactors(sin(c*(a + b*x)), x)
rubi.append(4828)
return Dist(d/(b*c), Subst(Int(SubstFor((-d**S(2)*x**S(2) + S(1))**(-n/S(2) + S(-1)/2), sin(c*(a + b*x))/d, u, x), x), x, sin(c*(a + b*x))/d), x)
rule4828 = ReplacementRule(pattern4828, replacement4828)
def With4829(u, b, c, a, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(cos(c*(a + b*x)), x)
if FunctionOfQ(cos(c*(a + b*x))/d, u, x):
return True
return False
pattern4829 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons1482, cons1247, cons1667, CustomConstraint(With4829))
def replacement4829(u, b, c, a, n, x, F):
d = FreeFactors(cos(c*(a + b*x)), x)
rubi.append(4829)
return -Dist(d/(b*c), Subst(Int(SubstFor((-d**S(2)*x**S(2) + S(1))**(n/S(2) + S(-1)/2), cos(c*(a + b*x))/d, u, x), x), x, cos(c*(a + b*x))/d), x)
rule4829 = ReplacementRule(pattern4829, replacement4829)
def With4830(u, b, c, a, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(cos(c*(a + b*x)), x)
if FunctionOfQ(cos(c*(a + b*x))/d, u, x):
return True
return False
pattern4830 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons1482, cons1247, cons1671, CustomConstraint(With4830))
def replacement4830(u, b, c, a, n, x, F):
d = FreeFactors(cos(c*(a + b*x)), x)
rubi.append(4830)
return -Dist(d/(b*c), Subst(Int(SubstFor((-d**S(2)*x**S(2) + S(1))**(-n/S(2) + S(-1)/2), cos(c*(a + b*x))/d, u, x), x), x, cos(c*(a + b*x))/d), x)
rule4830 = ReplacementRule(pattern4830, replacement4830)
def With4831(u, b, c, a, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(sin(c*(a + b*x)), x)
if FunctionOfQ(sin(c*(a + b*x))/d, u, x):
return True
return False
pattern4831 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons1482, cons1247, cons1668, CustomConstraint(With4831))
def replacement4831(u, b, c, a, n, x, F):
d = FreeFactors(sin(c*(a + b*x)), x)
rubi.append(4831)
return Dist(d**(-n + S(1))/(b*c), Subst(Int(SubstFor(x**(-n)*(-d**S(2)*x**S(2) + S(1))**(n/S(2) + S(-1)/2), sin(c*(a + b*x))/d, u, x), x), x, sin(c*(a + b*x))/d), x)
rule4831 = ReplacementRule(pattern4831, replacement4831)
def With4832(u, b, c, a, n, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
d = FreeFactors(cos(c*(a + b*x)), x)
if FunctionOfQ(cos(c*(a + b*x))/d, u, x):
return True
return False
pattern4832 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*u_, x_), cons2, cons3, cons7, cons1482, cons1247, cons1669, CustomConstraint(With4832))
def replacement4832(u, b, c, a, n, x, F):
d = FreeFactors(cos(c*(a + b*x)), x)
rubi.append(4832)
return -Dist(d**(-n + S(1))/(b*c), Subst(Int(SubstFor(x**(-n)*(-d**S(2)*x**S(2) + S(1))**(n/S(2) + S(-1)/2), cos(c*(a + b*x))/d, u, x), x), x, cos(c*(a + b*x))/d), x)
rule4832 = ReplacementRule(pattern4832, replacement4832)
def With4833(v, u, b, d, c, n, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
e = FreeFactors(sin(c*(a + b*x)), x)
if FunctionOfQ(sin(c*(a + b*x))/e, u, x):
return True
return False
pattern4833 = Pattern(Integral(u_*(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*WC('d', S(1)) + v_), x_), cons2, cons3, cons7, cons27, cons10, cons1482, cons1247, cons1666, CustomConstraint(With4833))
def replacement4833(v, u, b, d, c, n, a, x, F):
e = FreeFactors(sin(c*(a + b*x)), x)
rubi.append(4833)
return Dist(d, Int(ActivateTrig(u)*cos(c*(a + b*x))**n, x), x) + Int(ActivateTrig(u*v), x)
rule4833 = ReplacementRule(pattern4833, replacement4833)
def With4834(v, u, b, d, c, n, a, x, F):
if isinstance(x, (int, Integer, float, Float)):
return False
e = FreeFactors(cos(c*(a + b*x)), x)
if FunctionOfQ(cos(c*(a + b*x))/e, u, x):
return True
return False
pattern4834 = Pattern(Integral(u_*(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*WC('d', S(1)) + v_), x_), cons2, cons3, cons7, cons27, cons10, cons1482, cons1247, cons1667, CustomConstraint(With4834))
def replacement4834(v, u, b, d, c, n, a, x, F):
e = FreeFactors(cos(c*(a + b*x)), x)
rubi.append(4834)
return Dist(d, Int(ActivateTrig(u)*sin(c*(a + b*x))**n, x), x) + Int(ActivateTrig(u*v), x)
rule4834 = ReplacementRule(pattern4834, replacement4834)
def With4835(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
v = FunctionOfTrig(u, x)
d = FreeFactors(sin(v), x)
res = And(Not(FalseQ(v)), FunctionOfQ(NonfreeFactors(sin(v), x), u/cos(v), x))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4835 = Pattern(Integral(u_, x_), CustomConstraint(With4835))
def replacement4835(x, u):
v = FunctionOfTrig(u, x)
d = FreeFactors(sin(v), x)
rubi.append(4835)
return Simp(With(List(Set(d, FreeFactors(sin(v), x))), Dist(d/Coefficient(v, x, S(1)), Subst(Int(SubstFor(S(1), sin(v)/d, u/cos(v), x), x), x, sin(v)/d), x)), x)
rule4835 = ReplacementRule(pattern4835, replacement4835)
def With4836(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
v = FunctionOfTrig(u, x)
d = FreeFactors(cos(v), x)
res = And(Not(FalseQ(v)), FunctionOfQ(NonfreeFactors(cos(v), x), u/sin(v), x))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4836 = Pattern(Integral(u_, x_), CustomConstraint(With4836))
def replacement4836(x, u):
v = FunctionOfTrig(u, x)
d = FreeFactors(cos(v), x)
rubi.append(4836)
return Simp(With(List(Set(d, FreeFactors(cos(v), x))), Dist(-d/Coefficient(v, x, S(1)), Subst(Int(SubstFor(S(1), cos(v)/d, u/sin(v), x), x), x, cos(v)/d), x)), x)
rule4836 = ReplacementRule(pattern4836, replacement4836)
pattern4837 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1675)
def replacement4837(p, u, b, d, a, c, x, e):
rubi.append(4837)
return Dist((a + c)**p, Int(ActivateTrig(u), x), x)
rule4837 = ReplacementRule(pattern4837, replacement4837)
pattern4838 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('c', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1676)
def replacement4838(p, u, b, d, a, c, x, e):
rubi.append(4838)
return Dist((a + c)**p, Int(ActivateTrig(u), x), x)
rule4838 = ReplacementRule(pattern4838, replacement4838)
pattern4839 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('c', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**WC('p', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons1676)
def replacement4839(p, u, b, d, c, a, x, e):
rubi.append(4839)
return Dist((a + c)**p, Int(ActivateTrig(u), x), x)
rule4839 = ReplacementRule(pattern4839, replacement4839)
def With4840(y, x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(ActivateTrig(y), ActivateTrig(u), x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4840 = Pattern(Integral(u_/y_, x_), cons1677, CustomConstraint(With4840))
def replacement4840(y, x, u):
q = DerivativeDivides(ActivateTrig(y), ActivateTrig(u), x)
rubi.append(4840)
return Simp(q*log(RemoveContent(ActivateTrig(y), x)), x)
rule4840 = ReplacementRule(pattern4840, replacement4840)
def With4841(y, w, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(ActivateTrig(w*y), ActivateTrig(u), x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4841 = Pattern(Integral(u_/(w_*y_), x_), cons1677, CustomConstraint(With4841))
def replacement4841(y, w, u, x):
q = DerivativeDivides(ActivateTrig(w*y), ActivateTrig(u), x)
rubi.append(4841)
return Simp(q*log(RemoveContent(ActivateTrig(w*y), x)), x)
rule4841 = ReplacementRule(pattern4841, replacement4841)
def With4842(y, m, u, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(ActivateTrig(y), ActivateTrig(u), x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4842 = Pattern(Integral(u_*y_**WC('m', S(1)), x_), cons21, cons66, cons1677, CustomConstraint(With4842))
def replacement4842(y, m, u, x):
q = DerivativeDivides(ActivateTrig(y), ActivateTrig(u), x)
rubi.append(4842)
return Simp(q*ActivateTrig(y**(m + S(1)))/(m + S(1)), x)
rule4842 = ReplacementRule(pattern4842, replacement4842)
def With4843(z, y, u, m, n, x):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
q = DerivativeDivides(ActivateTrig(y*z), ActivateTrig(u*z**(-m + n)), x)
res = Not(FalseQ(q))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4843 = Pattern(Integral(u_*y_**WC('m', S(1))*z_**WC('n', S(1)), x_), cons21, cons4, cons66, cons1677, CustomConstraint(With4843))
def replacement4843(z, y, u, m, n, x):
q = DerivativeDivides(ActivateTrig(y*z), ActivateTrig(u*z**(-m + n)), x)
rubi.append(4843)
return Simp(q*ActivateTrig(y**(m + S(1))*z**(m + S(1)))/(m + S(1)), x)
rule4843 = ReplacementRule(pattern4843, replacement4843)
def With4844(p, u, d, a, c, n, x, F):
v = ActivateTrig(F(c + d*x))
rubi.append(4844)
return Dist(a**IntPart(n)*(a*v**p)**FracPart(n)*(v/NonfreeFactors(v, x))**(p*IntPart(n))*NonfreeFactors(v, x)**(-p*FracPart(n)), Int(ActivateTrig(u)*NonfreeFactors(v, x)**(n*p), x), x)
pattern4844 = Pattern(Integral((F_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('a', S(1)))**n_*WC('u', S(1)), x_), cons2, cons7, cons27, cons4, cons5, cons1664, cons23, cons38)
rule4844 = ReplacementRule(pattern4844, With4844)
def With4845(p, u, b, d, c, a, n, x, F):
v = ActivateTrig(F(c + d*x))
rubi.append(4845)
return Dist(a**IntPart(n)*(a*(b*v)**p)**FracPart(n)*(b*v)**(-p*FracPart(n)), Int((b*v)**(n*p)*ActivateTrig(u), x), x)
pattern4845 = Pattern(Integral(((F_*(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)))**p_*WC('a', S(1)))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons1664, cons23, cons147)
rule4845 = ReplacementRule(pattern4845, With4845)
def With4846(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
v = FunctionOfTrig(u, x)
d = FreeFactors(tan(v), x)
res = And(Not(FalseQ(v)), FunctionOfQ(NonfreeFactors(tan(v), x), u, x))
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4846 = Pattern(Integral(u_, x_), cons1230, CustomConstraint(With4846))
def replacement4846(x, u):
v = FunctionOfTrig(u, x)
d = FreeFactors(tan(v), x)
rubi.append(4846)
return Dist(d/Coefficient(v, x, 1), Subst(Int(SubstFor(1/(d**2*x**2 + 1), tan(v)/d, u, x), x), x, tan(v)/d), x)
rule4846 = ReplacementRule(pattern4846, replacement4846)
pattern4847 = Pattern(Integral(((S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)))**p_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons376)
def replacement4847(p, u, b, d, c, n, a, x):
rubi.append(4847)
return Int((a*sin(c + d*x)**n + b)**p*(S(1)/cos(c + d*x))**(n*p)*ActivateTrig(u), x)
rule4847 = ReplacementRule(pattern4847, replacement4847)
pattern4848 = Pattern(Integral(((S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*WC('a', S(1)))**p_*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons376)
def replacement4848(p, u, b, d, c, n, a, x):
rubi.append(4848)
return Int((a*cos(c + d*x)**n + b)**p*(S(1)/sin(c + d*x))**(n*p)*ActivateTrig(u), x)
rule4848 = ReplacementRule(pattern4848, replacement4848)
pattern4849 = Pattern(Integral(u_*(F_**(x_*WC('d', S(1)) + WC('c', S(0)))*a_ + F_**(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons5, cons50, cons1664, cons85, cons49)
def replacement4849(p, u, b, d, c, n, a, x, q, F):
rubi.append(4849)
return Int(ActivateTrig(u*(a + b*F(c + d*x)**(-p + q))**n*F(c + d*x)**(n*p)), x)
rule4849 = ReplacementRule(pattern4849, replacement4849)
pattern4850 = Pattern(Integral(u_*(F_**(x_*WC('e', S(1)) + WC('d', S(0)))*a_ + F_**(x_*WC('e', S(1)) + WC('d', S(0)))*WC('b', S(1)) + F_**(x_*WC('e', S(1)) + WC('d', S(0)))*WC('c', S(1)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons52, cons1664, cons85, cons49, cons51)
def replacement4850(p, u, b, r, d, c, n, a, x, q, e, F):
rubi.append(4850)
return Int(ActivateTrig(u*(a + b*F(d + e*x)**(-p + q) + c*F(d + e*x)**(-p + r))**n*F(d + e*x)**(n*p)), x)
rule4850 = ReplacementRule(pattern4850, replacement4850)
pattern4851 = Pattern(Integral(u_*(F_**(x_*WC('e', S(1)) + WC('d', S(0)))*WC('b', S(1)) + F_**(x_*WC('e', S(1)) + WC('d', S(0)))*WC('c', S(1)) + a_)**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons5, cons50, cons1664, cons85, cons1678)
def replacement4851(p, u, b, d, c, n, a, x, q, e, F):
rubi.append(4851)
return Int(ActivateTrig(u*(a*F(d + e*x)**(-p) + b + c*F(d + e*x)**(-p + q))**n*F(d + e*x)**(n*p)), x)
rule4851 = ReplacementRule(pattern4851, replacement4851)
pattern4852 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1439)
def replacement4852(u, b, d, c, a, n, x):
rubi.append(4852)
return Int((a*exp(-a*(c + d*x)/b))**n*ActivateTrig(u), x)
rule4852 = ReplacementRule(pattern4852, replacement4852)
pattern4853 = Pattern(Integral(u_, x_), cons1679)
def replacement4853(x, u):
rubi.append(4853)
return Int(TrigSimplify(u), x)
rule4853 = ReplacementRule(pattern4853, replacement4853)
def With4854(v, p, u, a, x):
uu = ActivateTrig(u)
vv = ActivateTrig(v)
rubi.append(4854)
return Dist(a**IntPart(p)*vv**(-FracPart(p))*(a*vv)**FracPart(p), Int(uu*vv**p, x), x)
pattern4854 = Pattern(Integral((a_*v_)**p_*WC('u', S(1)), x_), cons2, cons5, cons147, cons1680)
rule4854 = ReplacementRule(pattern4854, With4854)
def With4855(v, p, u, m, x):
uu = ActivateTrig(u)
vv = ActivateTrig(v)
rubi.append(4855)
return Dist(vv**(-m*FracPart(p))*(vv**m)**FracPart(p), Int(uu*vv**(m*p), x), x)
pattern4855 = Pattern(Integral((v_**m_)**p_*WC('u', S(1)), x_), cons21, cons5, cons147, cons1680)
rule4855 = ReplacementRule(pattern4855, With4855)
def With4856(v, w, p, u, m, n, x):
uu = ActivateTrig(u)
vv = ActivateTrig(v)
ww = ActivateTrig(w)
rubi.append(4856)
return Dist(vv**(-m*FracPart(p))*ww**(-n*FracPart(p))*(vv**m*ww**n)**FracPart(p), Int(uu*vv**(m*p)*ww**(n*p), x), x)
pattern4856 = Pattern(Integral((v_**WC('m', S(1))*w_**WC('n', S(1)))**p_*WC('u', S(1)), x_), cons21, cons4, cons5, cons147, cons1681)
rule4856 = ReplacementRule(pattern4856, With4856)
def With4857(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
v = ExpandTrig(u, x)
if SumQ(v):
return True
return False
pattern4857 = Pattern(Integral(u_, x_), cons1677, CustomConstraint(With4857))
def replacement4857(x, u):
v = ExpandTrig(u, x)
rubi.append(4857)
return Int(v, x)
rule4857 = ReplacementRule(pattern4857, replacement4857)
def With4858(x, u):
if isinstance(x, (int, Integer, float, Float)):
return False
try:
w = With(List(Set(ShowSteps, False), Set(StepCounter, Null)), Int(SubstFor(S(1)/(x**S(2)*FreeFactors(tan(FunctionOfTrig(u, x)/S(2)), x)**S(2) + S(1)), tan(FunctionOfTrig(u, x)/S(2))/FreeFactors(tan(FunctionOfTrig(u, x)/S(2)), x), u, x), x))
v = FunctionOfTrig(u, x)
d = FreeFactors(tan(v/S(2)), x)
res = FreeQ(w, Int)
except (TypeError, AttributeError):
return False
if res:
return True
return False
pattern4858 = Pattern(Integral(u_, x_), cons1230, cons1682, CustomConstraint(With4858))
def replacement4858(x, u):
w = With(List(Set(ShowSteps, False), Set(StepCounter, Null)), Int(SubstFor(S(1)/(x**S(2)*FreeFactors(tan(FunctionOfTrig(u, x)/S(2)), x)**S(2) + S(1)), tan(FunctionOfTrig(u, x)/S(2))/FreeFactors(tan(FunctionOfTrig(u, x)/S(2)), x), u, x), x))
v = FunctionOfTrig(u, x)
d = FreeFactors(tan(v/S(2)), x)
rubi.append(4858)
return Simp(Dist(2*d/Coefficient(v, x, 1), Subst(Int(SubstFor(1/(d**2*x**2 + 1), tan(v/2)/d, u, x), x), x, tan(v/2)/d), x), x)
rule4858 = ReplacementRule(pattern4858, replacement4858)
def With4859(x, u):
v = ActivateTrig(u)
rubi.append(4859)
return Int(v, x)
pattern4859 = Pattern(Integral(u_, x_), cons1677)
rule4859 = ReplacementRule(pattern4859, With4859)
pattern4860 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons4, cons62, cons584)
def replacement4860(m, b, d, c, a, n, x):
rubi.append(4860)
return -Dist(d*m/(b*(n + S(1))), Int((c + d*x)**(m + S(-1))*sin(a + b*x)**(n + S(1)), x), x) + Simp((c + d*x)**m*sin(a + b*x)**(n + S(1))/(b*(n + S(1))), x)
rule4860 = ReplacementRule(pattern4860, replacement4860)
pattern4861 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons62, cons584)
def replacement4861(m, b, d, c, a, n, x):
rubi.append(4861)
return Dist(d*m/(b*(n + S(1))), Int((c + d*x)**(m + S(-1))*cos(a + b*x)**(n + S(1)), x), x) - Simp((c + d*x)**m*cos(a + b*x)**(n + S(1))/(b*(n + S(1))), x)
rule4861 = ReplacementRule(pattern4861, replacement4861)
pattern4862 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons464)
def replacement4862(p, m, b, d, c, a, n, x):
rubi.append(4862)
return Int(ExpandTrigReduce((c + d*x)**m, sin(a + b*x)**n*cos(a + b*x)**p, x), x)
rule4862 = ReplacementRule(pattern4862, replacement4862)
pattern4863 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons464)
def replacement4863(p, m, b, d, c, a, n, x):
rubi.append(4863)
return -Int((c + d*x)**m*sin(a + b*x)**n*tan(a + b*x)**(p + S(-2)), x) + Int((c + d*x)**m*sin(a + b*x)**(n + S(-2))*tan(a + b*x)**p, x)
rule4863 = ReplacementRule(pattern4863, replacement4863)
pattern4864 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons21, cons464)
def replacement4864(p, m, b, d, c, a, n, x):
rubi.append(4864)
return Int((c + d*x)**m*(S(1)/tan(a + b*x))**p*cos(a + b*x)**(n + S(-2)), x) - Int((c + d*x)**m*(S(1)/tan(a + b*x))**(p + S(-2))*cos(a + b*x)**n, x)
rule4864 = ReplacementRule(pattern4864, replacement4864)
pattern4865 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1683, cons31, cons168)
def replacement4865(p, m, b, d, c, a, n, x):
rubi.append(4865)
return -Dist(d*m/(b*n), Int((c + d*x)**(m + S(-1))*(S(1)/cos(a + b*x))**n, x), x) + Simp((c + d*x)**m*(S(1)/cos(a + b*x))**n/(b*n), x)
rule4865 = ReplacementRule(pattern4865, replacement4865)
pattern4866 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons1683, cons31, cons168)
def replacement4866(p, m, b, d, c, a, n, x):
rubi.append(4866)
return Dist(d*m/(b*n), Int((c + d*x)**(m + S(-1))*(S(1)/sin(a + b*x))**n, x), x) - Simp((c + d*x)**m*(S(1)/sin(a + b*x))**n/(b*n), x)
rule4866 = ReplacementRule(pattern4866, replacement4866)
pattern4867 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))/cos(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons4, cons62, cons584)
def replacement4867(m, b, d, c, a, n, x):
rubi.append(4867)
return -Dist(d*m/(b*(n + S(1))), Int((c + d*x)**(m + S(-1))*tan(a + b*x)**(n + S(1)), x), x) + Simp((c + d*x)**m*tan(a + b*x)**(n + S(1))/(b*(n + S(1))), x)
rule4867 = ReplacementRule(pattern4867, replacement4867)
pattern4868 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))/sin(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons4, cons62, cons584)
def replacement4868(m, b, d, c, a, n, x):
rubi.append(4868)
return Dist(d*m/(b*(n + S(1))), Int((c + d*x)**(m + S(-1))*(S(1)/tan(a + b*x))**(n + S(1)), x), x) - Simp((c + d*x)**m*(S(1)/tan(a + b*x))**(n + S(1))/(b*(n + S(1))), x)
rule4868 = ReplacementRule(pattern4868, replacement4868)
pattern4869 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**p_/cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons1408)
def replacement4869(p, m, b, d, c, a, x):
rubi.append(4869)
return Int((c + d*x)**m*tan(a + b*x)**(p + S(-2))/cos(a + b*x)**S(3), x) - Int((c + d*x)**m*tan(a + b*x)**(p + S(-2))/cos(a + b*x), x)
rule4869 = ReplacementRule(pattern4869, replacement4869)
pattern4870 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1408)
def replacement4870(p, m, b, d, c, a, n, x):
rubi.append(4870)
return -Int((c + d*x)**m*(S(1)/cos(a + b*x))**n*tan(a + b*x)**(p + S(-2)), x) + Int((c + d*x)**m*(S(1)/cos(a + b*x))**(n + S(2))*tan(a + b*x)**(p + S(-2)), x)
rule4870 = ReplacementRule(pattern4870, replacement4870)
pattern4871 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**p_/sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons21, cons1408)
def replacement4871(p, m, b, d, c, a, x):
rubi.append(4871)
return Int((c + d*x)**m*(S(1)/tan(a + b*x))**(p + S(-2))/sin(a + b*x)**S(3), x) - Int((c + d*x)**m*(S(1)/tan(a + b*x))**(p + S(-2))/sin(a + b*x), x)
rule4871 = ReplacementRule(pattern4871, replacement4871)
pattern4872 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**p_, x_), cons2, cons3, cons7, cons27, cons21, cons4, cons1408)
def replacement4872(p, m, b, d, c, a, n, x):
rubi.append(4872)
return -Int((c + d*x)**m*(S(1)/sin(a + b*x))**n*(S(1)/tan(a + b*x))**(p + S(-2)), x) + Int((c + d*x)**m*(S(1)/sin(a + b*x))**(n + S(2))*(S(1)/tan(a + b*x))**(p + S(-2)), x)
rule4872 = ReplacementRule(pattern4872, replacement4872)
def With4873(p, m, b, d, c, a, n, x):
u = IntHide((S(1)/cos(a + b*x))**n*tan(a + b*x)**p, x)
rubi.append(4873)
return -Dist(d*m, Int(u*(c + d*x)**(m + S(-1)), x), x) + Dist((c + d*x)**m, u, x)
pattern4873 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*tan(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons62, cons1684)
rule4873 = ReplacementRule(pattern4873, With4873)
def With4874(p, m, b, d, c, a, n, x):
u = IntHide((S(1)/sin(a + b*x))**n*(S(1)/tan(a + b*x))**p, x)
rubi.append(4874)
return -Dist(d*m, Int(u*(c + d*x)**(m + S(-1)), x), x) + Dist((c + d*x)**m, u, x)
pattern4874 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/tan(x_*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons4, cons5, cons62, cons1684)
rule4874 = ReplacementRule(pattern4874, With4874)
pattern4875 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons31, cons85)
def replacement4875(m, b, d, c, a, n, x):
rubi.append(4875)
return Dist(S(2)**n, Int((c + d*x)**m*(S(1)/sin(S(2)*a + S(2)*b*x))**n, x), x)
rule4875 = ReplacementRule(pattern4875, replacement4875)
def With4876(p, m, b, d, c, a, n, x):
u = IntHide((S(1)/sin(a + b*x))**n*(S(1)/cos(a + b*x))**p, x)
rubi.append(4876)
return -Dist(d*m, Int(u*(c + d*x)**(m + S(-1)), x), x) + Dist((c + d*x)**m, u, x)
pattern4876 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(S(1)/sin(x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1))*(S(1)/cos(x_*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons376, cons31, cons168, cons1685)
rule4876 = ReplacementRule(pattern4876, With4876)
pattern4877 = Pattern(Integral(F_**v_*G_**w_*u_**WC('m', S(1)), x_), cons21, cons4, cons5, cons1686, cons1687, cons1688, cons812, cons813)
def replacement4877(v, w, p, u, m, G, n, x, F):
rubi.append(4877)
return Int(ExpandToSum(u, x)**m*F(ExpandToSum(v, x))**n*G(ExpandToSum(v, x))**p, x)
rule4877 = ReplacementRule(pattern4877, replacement4877)
pattern4878 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement4878(m, f, b, d, c, n, a, x, e):
rubi.append(4878)
return -Dist(f*m/(b*d*(n + S(1))), Int((a + b*sin(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) + Simp((a + b*sin(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule4878 = ReplacementRule(pattern4878, replacement4878)
pattern4879 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement4879(m, f, b, d, c, n, a, x, e):
rubi.append(4879)
return Dist(f*m/(b*d*(n + S(1))), Int((a + b*cos(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) - Simp((a + b*cos(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule4879 = ReplacementRule(pattern4879, replacement4879)
pattern4880 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement4880(m, f, b, d, c, n, a, x, e):
rubi.append(4880)
return -Dist(f*m/(b*d*(n + S(1))), Int((a + b*tan(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) + Simp((a + b*tan(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule4880 = ReplacementRule(pattern4880, replacement4880)
pattern4881 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement4881(m, f, b, d, c, n, a, x, e):
rubi.append(4881)
return Dist(f*m/(b*d*(n + S(1))), Int((a + b/tan(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) - Simp((a + b/tan(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule4881 = ReplacementRule(pattern4881, replacement4881)
pattern4882 = Pattern(Integral((a_ + WC('b', S(1))/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))/cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement4882(m, f, b, d, c, n, a, x, e):
rubi.append(4882)
return -Dist(f*m/(b*d*(n + S(1))), Int((a + b/cos(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) + Simp((a + b/cos(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule4882 = ReplacementRule(pattern4882, replacement4882)
pattern4883 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))/(sin(x_*WC('d', S(1)) + WC('c', S(0)))*tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons584)
def replacement4883(m, f, b, d, c, n, a, x, e):
rubi.append(4883)
return Dist(f*m/(b*d*(n + S(1))), Int((a + b/sin(c + d*x))**(n + S(1))*(e + f*x)**(m + S(-1)), x), x) - Simp((a + b/sin(c + d*x))**(n + S(1))*(e + f*x)**m/(b*d*(n + S(1))), x)
rule4883 = ReplacementRule(pattern4883, replacement4883)
pattern4884 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons555, cons17)
def replacement4884(p, m, f, b, d, a, c, x, q, e):
rubi.append(4884)
return Int(ExpandTrigReduce((e + f*x)**m, sin(a + b*x)**p*sin(c + d*x)**q, x), x)
rule4884 = ReplacementRule(pattern4884, replacement4884)
pattern4885 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons555, cons17)
def replacement4885(p, m, f, b, d, c, a, x, q, e):
rubi.append(4885)
return Int(ExpandTrigReduce((e + f*x)**m, cos(a + b*x)**p*cos(c + d*x)**q, x), x)
rule4885 = ReplacementRule(pattern4885, replacement4885)
pattern4886 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons555)
def replacement4886(p, m, f, b, d, c, a, x, q, e):
rubi.append(4886)
return Int(ExpandTrigReduce((e + f*x)**m, sin(a + b*x)**p*cos(c + d*x)**q, x), x)
rule4886 = ReplacementRule(pattern4886, replacement4886)
pattern4887 = Pattern(Integral(F_**(x_*WC('b', S(1)) + WC('a', S(0)))*G_**(x_*WC('d', S(1)) + WC('c', S(0)))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons1689, cons1690, cons555, cons25, cons1691)
def replacement4887(p, m, f, b, G, d, a, c, x, q, e, F):
rubi.append(4887)
return Int(ExpandTrigExpand((e + f*x)**m*G(c + d*x)**q, F, c + d*x, p, b/d, x), x)
rule4887 = ReplacementRule(pattern4887, replacement4887)
pattern4888 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sin(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1692)
def replacement4888(b, d, c, a, x, F, e):
rubi.append(4888)
return -Simp(F**(c*(a + b*x))*e*cos(d + e*x)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)), x) + Simp(F**(c*(a + b*x))*b*c*log(F)*sin(d + e*x)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)), x)
rule4888 = ReplacementRule(pattern4888, replacement4888)
pattern4889 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cos(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1692)
def replacement4889(b, d, c, a, x, F, e):
rubi.append(4889)
return Simp(F**(c*(a + b*x))*e*sin(d + e*x)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)), x) + Simp(F**(c*(a + b*x))*b*c*log(F)*cos(d + e*x)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)), x)
rule4889 = ReplacementRule(pattern4889, replacement4889)
pattern4890 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1693, cons87, cons165)
def replacement4890(b, d, c, a, n, x, F, e):
rubi.append(4890)
return Dist(e**S(2)*n*(n + S(-1))/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), Int(F**(c*(a + b*x))*sin(d + e*x)**(n + S(-2)), x), x) + Simp(F**(c*(a + b*x))*b*c*log(F)*sin(d + e*x)**n/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x) - Simp(F**(c*(a + b*x))*e*n*sin(d + e*x)**(n + S(-1))*cos(d + e*x)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x)
rule4890 = ReplacementRule(pattern4890, replacement4890)
pattern4891 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1694, cons31, cons166)
def replacement4891(m, b, d, c, a, x, F, e):
rubi.append(4891)
return Dist(e**S(2)*m*(m + S(-1))/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*m**S(2)), Int(F**(c*(a + b*x))*cos(d + e*x)**(m + S(-2)), x), x) + Simp(F**(c*(a + b*x))*b*c*log(F)*cos(d + e*x)**m/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*m**S(2)), x) + Simp(F**(c*(a + b*x))*e*m*sin(d + e*x)*cos(d + e*x)**(m + S(-1))/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*m**S(2)), x)
rule4891 = ReplacementRule(pattern4891, replacement4891)
pattern4892 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons1695, cons584, cons1395)
def replacement4892(b, d, c, a, n, x, F, e):
rubi.append(4892)
return Simp(F**(c*(a + b*x))*sin(d + e*x)**(n + S(1))*cos(d + e*x)/(e*(n + S(1))), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*sin(d + e*x)**(n + S(2))/(e**S(2)*(n + S(1))*(n + S(2))), x)
rule4892 = ReplacementRule(pattern4892, replacement4892)
pattern4893 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons1695, cons584, cons1395)
def replacement4893(b, d, c, a, n, x, F, e):
rubi.append(4893)
return -Simp(F**(c*(a + b*x))*sin(d + e*x)*cos(d + e*x)**(n + S(1))/(e*(n + S(1))), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*cos(d + e*x)**(n + S(2))/(e**S(2)*(n + S(1))*(n + S(2))), x)
rule4893 = ReplacementRule(pattern4893, replacement4893)
pattern4894 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1696, cons87, cons89, cons1442)
def replacement4894(b, d, c, a, n, x, F, e):
rubi.append(4894)
return Dist((b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(2))**S(2))/(e**S(2)*(n + S(1))*(n + S(2))), Int(F**(c*(a + b*x))*sin(d + e*x)**(n + S(2)), x), x) + Simp(F**(c*(a + b*x))*sin(d + e*x)**(n + S(1))*cos(d + e*x)/(e*(n + S(1))), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*sin(d + e*x)**(n + S(2))/(e**S(2)*(n + S(1))*(n + S(2))), x)
rule4894 = ReplacementRule(pattern4894, replacement4894)
pattern4895 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1696, cons87, cons89, cons1442)
def replacement4895(b, d, c, a, n, x, F, e):
rubi.append(4895)
return Dist((b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(2))**S(2))/(e**S(2)*(n + S(1))*(n + S(2))), Int(F**(c*(a + b*x))*cos(d + e*x)**(n + S(2)), x), x) - Simp(F**(c*(a + b*x))*sin(d + e*x)*cos(d + e*x)**(n + S(1))/(e*(n + S(1))), x) - Simp(F**(c*(a + b*x))*b*c*log(F)*cos(d + e*x)**(n + S(2))/(e**S(2)*(n + S(1))*(n + S(2))), x)
rule4895 = ReplacementRule(pattern4895, replacement4895)
pattern4896 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons23)
def replacement4896(b, d, c, a, n, x, F, e):
rubi.append(4896)
return Dist((exp(S(2)*I*(d + e*x)) + S(-1))**(-n)*exp(I*n*(d + e*x))*sin(d + e*x)**n, Int(F**(c*(a + b*x))*(exp(S(2)*I*(d + e*x)) + S(-1))**n*exp(-I*n*(d + e*x)), x), x)
rule4896 = ReplacementRule(pattern4896, replacement4896)
pattern4897 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons23)
def replacement4897(b, d, c, a, n, x, F, e):
rubi.append(4897)
return Dist((exp(S(2)*I*(d + e*x)) + S(1))**(-n)*exp(I*n*(d + e*x))*cos(d + e*x)**n, Int(F**(c*(a + b*x))*(exp(S(2)*I*(d + e*x)) + S(1))**n*exp(-I*n*(d + e*x)), x), x)
rule4897 = ReplacementRule(pattern4897, replacement4897)
pattern4898 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons85)
def replacement4898(b, d, c, a, n, x, F, e):
rubi.append(4898)
return Dist(I**n, Int(ExpandIntegrand(F**(c*(a + b*x))*(-exp(S(2)*I*(d + e*x)) + S(1))**n*(exp(S(2)*I*(d + e*x)) + S(1))**(-n), x), x), x)
rule4898 = ReplacementRule(pattern4898, replacement4898)
pattern4899 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons85)
def replacement4899(b, d, c, n, a, x, F, e):
rubi.append(4899)
return Dist((-I)**n, Int(ExpandIntegrand(F**(c*(a + b*x))*(-exp(S(2)*I*(d + e*x)) + S(1))**(-n)*(exp(S(2)*I*(d + e*x)) + S(1))**n, x), x), x)
rule4899 = ReplacementRule(pattern4899, replacement4899)
pattern4900 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1693, cons87, cons89)
def replacement4900(b, d, c, a, n, x, F, e):
rubi.append(4900)
return Dist(e**S(2)*n*(n + S(1))/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), Int(F**(c*(a + b*x))*(S(1)/cos(d + e*x))**(n + S(2)), x), x) + Simp(F**(c*(a + b*x))*b*c*(S(1)/cos(d + e*x))**n*log(F)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x) - Simp(F**(c*(a + b*x))*e*n*(S(1)/cos(d + e*x))**(n + S(1))*sin(d + e*x)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x)
rule4900 = ReplacementRule(pattern4900, replacement4900)
pattern4901 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1693, cons87, cons89)
def replacement4901(b, d, c, a, n, x, F, e):
rubi.append(4901)
return Dist(e**S(2)*n*(n + S(1))/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), Int(F**(c*(a + b*x))*(S(1)/sin(d + e*x))**(n + S(2)), x), x) + Simp(F**(c*(a + b*x))*b*c*(S(1)/sin(d + e*x))**n*log(F)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x) + Simp(F**(c*(a + b*x))*e*n*(S(1)/sin(d + e*x))**(n + S(1))*cos(d + e*x)/(b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*n**S(2)), x)
rule4901 = ReplacementRule(pattern4901, replacement4901)
pattern4902 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons1697, cons1502, cons963)
def replacement4902(b, d, c, a, n, x, F, e):
rubi.append(4902)
return Simp(F**(c*(a + b*x))*(S(1)/cos(d + e*x))**(n + S(-1))*sin(d + e*x)/(e*(n + S(-1))), x) - Simp(F**(c*(a + b*x))*b*c*(S(1)/cos(d + e*x))**(n + S(-2))*log(F)/(e**S(2)*(n + S(-2))*(n + S(-1))), x)
rule4902 = ReplacementRule(pattern4902, replacement4902)
pattern4903 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons4, cons1697, cons1502, cons963)
def replacement4903(b, d, c, a, n, x, F, e):
rubi.append(4903)
return Simp(F**(c*(a + b*x))*(S(1)/sin(d + e*x))**(n + S(-1))*cos(d + e*x)/(e*(n + S(-1))), x) - Simp(F**(c*(a + b*x))*b*c*(S(1)/sin(d + e*x))**(n + S(-2))*log(F)/(e**S(2)*(n + S(-2))*(n + S(-1))), x)
rule4903 = ReplacementRule(pattern4903, replacement4903)
pattern4904 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1698, cons87, cons165, cons1644)
def replacement4904(b, d, c, a, n, x, F, e):
rubi.append(4904)
return Dist((b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(-2))**S(2))/(e**S(2)*(n + S(-2))*(n + S(-1))), Int(F**(c*(a + b*x))*(S(1)/cos(d + e*x))**(n + S(-2)), x), x) + Simp(F**(c*(a + b*x))*(S(1)/cos(d + e*x))**(n + S(-1))*sin(d + e*x)/(e*(n + S(-1))), x) - Simp(F**(c*(a + b*x))*b*c*(S(1)/cos(d + e*x))**(n + S(-2))*log(F)/(e**S(2)*(n + S(-2))*(n + S(-1))), x)
rule4904 = ReplacementRule(pattern4904, replacement4904)
pattern4905 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons1698, cons87, cons165, cons1644)
def replacement4905(b, d, c, a, n, x, F, e):
rubi.append(4905)
return Dist((b**S(2)*c**S(2)*log(F)**S(2) + e**S(2)*(n + S(-2))**S(2))/(e**S(2)*(n + S(-2))*(n + S(-1))), Int(F**(c*(a + b*x))*(S(1)/sin(d + e*x))**(n + S(-2)), x), x) - Simp(F**(c*(a + b*x))*(S(1)/sin(d + e*x))**(n + S(-1))*cos(d + e*x)/(e*(n + S(-1))), x) - Simp(F**(c*(a + b*x))*b*c*(S(1)/sin(d + e*x))**(n + S(-2))*log(F)/(e**S(2)*(n + S(-2))*(n + S(-1))), x)
rule4905 = ReplacementRule(pattern4905, replacement4905)
pattern4906 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons85)
def replacement4906(b, d, c, a, n, x, F, e):
rubi.append(4906)
return Simp(S(2)**n*F**(c*(a + b*x))*Hypergeometric2F1(n, -I*b*c*log(F)/(S(2)*e) + n/S(2), -I*b*c*log(F)/(S(2)*e) + n/S(2) + S(1), -exp(S(2)*I*(d + e*x)))*exp(I*n*(d + e*x))/(b*c*log(F) + I*e*n), x)
rule4906 = ReplacementRule(pattern4906, replacement4906)
pattern4907 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons85)
def replacement4907(b, d, c, n, a, x, F, e):
rubi.append(4907)
return Simp(F**(c*(a + b*x))*(-S(2)*I)**n*Hypergeometric2F1(n, -I*b*c*log(F)/(S(2)*e) + n/S(2), -I*b*c*log(F)/(S(2)*e) + n/S(2) + S(1), exp(S(2)*I*(d + e*x)))*exp(I*n*(d + e*x))/(b*c*log(F) + I*e*n), x)
rule4907 = ReplacementRule(pattern4907, replacement4907)
pattern4908 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons23)
def replacement4908(b, d, c, a, n, x, F, e):
rubi.append(4908)
return Dist((exp(S(2)*I*(d + e*x)) + S(1))**n*(S(1)/cos(d + e*x))**n*exp(-I*n*(d + e*x)), Int(SimplifyIntegrand(F**(c*(a + b*x))*(exp(S(2)*I*(d + e*x)) + S(1))**(-n)*exp(I*n*(d + e*x)), x), x), x)
rule4908 = ReplacementRule(pattern4908, replacement4908)
pattern4909 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons23)
def replacement4909(b, d, c, n, a, x, F, e):
rubi.append(4909)
return Dist((S(1) - exp(-S(2)*I*(d + e*x)))**n*(S(1)/sin(d + e*x))**n*exp(I*n*(d + e*x)), Int(SimplifyIntegrand(F**(c*(a + b*x))*(S(1) - exp(-S(2)*I*(d + e*x)))**(-n)*exp(-I*n*(d + e*x)), x), x), x)
rule4909 = ReplacementRule(pattern4909, replacement4909)
pattern4910 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1699, cons196)
def replacement4910(g, b, f, d, c, a, n, x, F, e):
rubi.append(4910)
return Dist(S(2)**n*f**n, Int(F**(c*(a + b*x))*cos(-Pi*f/(S(4)*g) + d/S(2) + e*x/S(2))**(S(2)*n), x), x)
rule4910 = ReplacementRule(pattern4910, replacement4910)
pattern4911 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1700, cons196)
def replacement4911(g, b, f, d, c, n, a, x, F, e):
rubi.append(4911)
return Dist(S(2)**n*f**n, Int(F**(c*(a + b*x))*cos(d/S(2) + e*x/S(2))**(S(2)*n), x), x)
rule4911 = ReplacementRule(pattern4911, replacement4911)
pattern4912 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1011, cons196)
def replacement4912(g, b, f, d, c, n, a, x, F, e):
rubi.append(4912)
return Dist(S(2)**n*f**n, Int(F**(c*(a + b*x))*sin(d/S(2) + e*x/S(2))**(S(2)*n), x), x)
rule4912 = ReplacementRule(pattern4912, replacement4912)
pattern4913 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1699, cons150, cons1551)
def replacement4913(m, g, b, f, d, c, a, n, x, F, e):
rubi.append(4913)
return Dist(g**n, Int(F**(c*(a + b*x))*(-tan(-Pi*f/(S(4)*g) + d/S(2) + e*x/S(2)))**m, x), x)
rule4913 = ReplacementRule(pattern4913, replacement4913)
pattern4914 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1700, cons150, cons1551)
def replacement4914(m, g, b, f, d, c, n, a, x, F, e):
rubi.append(4914)
return Dist(f**n, Int(F**(c*(a + b*x))*tan(d/S(2) + e*x/S(2))**m, x), x)
rule4914 = ReplacementRule(pattern4914, replacement4914)
pattern4915 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(f_ + WC('g', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1011, cons150, cons1551)
def replacement4915(m, g, b, f, d, c, n, a, x, F, e):
rubi.append(4915)
return Dist(f**n, Int(F**(c*(a + b*x))*(S(1)/tan(d/S(2) + e*x/S(2)))**m, x), x)
rule4915 = ReplacementRule(pattern4915, replacement4915)
pattern4916 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(h_ + WC('i', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(f_ + WC('g', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons1699, cons1701, cons1702)
def replacement4916(g, b, f, i, d, c, a, x, h, F, e):
rubi.append(4916)
return Dist(S(2)*i, Int(F**(c*(a + b*x))*cos(d + e*x)/(f + g*sin(d + e*x)), x), x) + Int(F**(c*(a + b*x))*(h - i*cos(d + e*x))/(f + g*sin(d + e*x)), x)
rule4916 = ReplacementRule(pattern4916, replacement4916)
pattern4917 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(h_ + WC('i', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(f_ + WC('g', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons209, cons224, cons1699, cons1701, cons1703)
def replacement4917(g, b, f, i, d, c, a, x, h, F, e):
rubi.append(4917)
return Dist(S(2)*i, Int(F**(c*(a + b*x))*sin(d + e*x)/(f + g*cos(d + e*x)), x), x) + Int(F**(c*(a + b*x))*(h - i*sin(d + e*x))/(f + g*cos(d + e*x)), x)
rule4917 = ReplacementRule(pattern4917, replacement4917)
pattern4918 = Pattern(Integral(F_**(u_*WC('c', S(1)))*G_**v_, x_), cons1099, cons7, cons4, cons1687, cons810, cons811)
def replacement4918(v, u, G, c, n, x, F):
rubi.append(4918)
return Int(F**(c*ExpandToSum(u, x))*G(ExpandToSum(v, x))**n, x)
rule4918 = ReplacementRule(pattern4918, replacement4918)
def With4919(m, b, d, c, a, n, x, F, e):
u = IntHide(F**(c*(a + b*x))*sin(d + e*x)**n, x)
rubi.append(4919)
return -Dist(m, Int(u*x**(m + S(-1)), x), x) + Dist(x**m, u, x)
pattern4919 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*x_**WC('m', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons31, cons168, cons148)
rule4919 = ReplacementRule(pattern4919, With4919)
def With4920(m, b, d, c, n, a, x, F, e):
u = IntHide(F**(c*(a + b*x))*cos(d + e*x)**n, x)
rubi.append(4920)
return -Dist(m, Int(u*x**(m + S(-1)), x), x) + Dist(x**m, u, x)
pattern4920 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*x_**WC('m', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons31, cons168, cons148)
rule4920 = ReplacementRule(pattern4920, With4920)
pattern4921 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cos(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons528)
def replacement4921(m, f, g, b, d, c, n, a, x, F, e):
rubi.append(4921)
return Int(ExpandTrigReduce(F**(c*(a + b*x)), sin(d + e*x)**m*cos(f + g*x)**n, x), x)
rule4921 = ReplacementRule(pattern4921, replacement4921)
pattern4922 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*x_**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cos(x_*WC('g', S(1)) + WC('f', S(0)))**WC('n', S(1)), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons1704)
def replacement4922(p, m, f, g, b, d, c, n, a, x, F, e):
rubi.append(4922)
return Int(ExpandTrigReduce(F**(c*(a + b*x))*x**p, sin(d + e*x)**m*cos(f + g*x)**n, x), x)
rule4922 = ReplacementRule(pattern4922, replacement4922)
pattern4923 = Pattern(Integral(F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*G_**(x_*WC('e', S(1)) + WC('d', S(0)))*H_**(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons1099, cons2, cons3, cons7, cons27, cons48, cons528, cons1687, cons1705)
def replacement4923(m, b, G, d, c, a, n, H, x, F, e):
rubi.append(4923)
return Int(ExpandTrigToExp(F**(c*(a + b*x)), G(d + e*x)**m*H(d + e*x)**n, x), x)
rule4923 = ReplacementRule(pattern4923, replacement4923)
pattern4924 = Pattern(Integral(F_**u_*sin(v_)**WC('n', S(1)), x_), cons1099, cons1706, cons1707, cons148)
def replacement4924(v, u, n, x, F):
rubi.append(4924)
return Int(ExpandTrigToExp(F**u, sin(v)**n, x), x)
rule4924 = ReplacementRule(pattern4924, replacement4924)
pattern4925 = Pattern(Integral(F_**u_*cos(v_)**WC('n', S(1)), x_), cons1099, cons1706, cons1707, cons148)
def replacement4925(v, u, n, x, F):
rubi.append(4925)
return Int(ExpandTrigToExp(F**u, cos(v)**n, x), x)
rule4925 = ReplacementRule(pattern4925, replacement4925)
pattern4926 = Pattern(Integral(F_**u_*sin(v_)**WC('m', S(1))*cos(v_)**WC('n', S(1)), x_), cons1099, cons1706, cons1707, cons528)
def replacement4926(v, u, m, n, x, F):
rubi.append(4926)
return Int(ExpandTrigToExp(F**u, sin(v)**m*cos(v)**n, x), x)
rule4926 = ReplacementRule(pattern4926, replacement4926)
pattern4927 = Pattern(Integral(sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1708, cons54)
def replacement4927(p, b, a, n, c, x):
rubi.append(4927)
return Simp(x*(p + S(2))*sin(a + b*log(c*x**n))**(p + S(2))/(p + S(1)), x) + Simp(x*sin(a + b*log(c*x**n))**(p + S(2))/(b*n*(p + S(1))*tan(a + b*log(c*x**n))), x)
rule4927 = ReplacementRule(pattern4927, replacement4927)
pattern4928 = Pattern(Integral(cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1708, cons54)
def replacement4928(p, b, a, n, c, x):
rubi.append(4928)
return Simp(x*(p + S(2))*cos(a + b*log(c*x**n))**(p + S(2))/(p + S(1)), x) - Simp(x*cos(a + b*log(c*x**n))**(p + S(2))*tan(a + b*log(c*x**n))/(b*n*(p + S(1))), x)
rule4928 = ReplacementRule(pattern4928, replacement4928)
pattern4929 = Pattern(Integral(sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons128, cons1709)
def replacement4929(p, b, a, n, c, x):
rubi.append(4929)
return Int(ExpandIntegrand((-(c*x**n)**(S(1)/(n*p))*exp(-a*b*n*p)/(S(2)*b*n*p) + (c*x**n)**(-S(1)/(n*p))*exp(a*b*n*p)/(S(2)*b*n*p))**p, x), x)
rule4929 = ReplacementRule(pattern4929, replacement4929)
pattern4930 = Pattern(Integral(cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons128, cons1709)
def replacement4930(p, b, a, n, c, x):
rubi.append(4930)
return Int(ExpandIntegrand((-(c*x**n)**(S(1)/(n*p))*exp(-a*b*n*p)/S(2) + (c*x**n)**(-S(1)/(n*p))*exp(a*b*n*p)/S(2))**p, x), x)
rule4930 = ReplacementRule(pattern4930, replacement4930)
pattern4931 = Pattern(Integral(sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1710)
def replacement4931(b, a, n, c, x):
rubi.append(4931)
return Simp(x*sin(a + b*log(c*x**n))/(b**S(2)*n**S(2) + S(1)), x) - Simp(b*n*x*cos(a + b*log(c*x**n))/(b**S(2)*n**S(2) + S(1)), x)
rule4931 = ReplacementRule(pattern4931, replacement4931)
pattern4932 = Pattern(Integral(cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1710)
def replacement4932(b, a, n, c, x):
rubi.append(4932)
return Simp(x*cos(a + b*log(c*x**n))/(b**S(2)*n**S(2) + S(1)), x) + Simp(b*n*x*sin(a + b*log(c*x**n))/(b**S(2)*n**S(2) + S(1)), x)
rule4932 = ReplacementRule(pattern4932, replacement4932)
pattern4933 = Pattern(Integral(sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons146, cons1711)
def replacement4933(p, b, a, n, c, x):
rubi.append(4933)
return Dist(b**S(2)*n**S(2)*p*(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + S(1)), Int(sin(a + b*log(c*x**n))**(p + S(-2)), x), x) + Simp(x*sin(a + b*log(c*x**n))**p/(b**S(2)*n**S(2)*p**S(2) + S(1)), x) - Simp(b*n*p*x*sin(a + b*log(c*x**n))**(p + S(-1))*cos(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + S(1)), x)
rule4933 = ReplacementRule(pattern4933, replacement4933)
pattern4934 = Pattern(Integral(cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons146, cons1711)
def replacement4934(p, b, a, n, c, x):
rubi.append(4934)
return Dist(b**S(2)*n**S(2)*p*(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + S(1)), Int(cos(a + b*log(c*x**n))**(p + S(-2)), x), x) + Simp(x*cos(a + b*log(c*x**n))**p/(b**S(2)*n**S(2)*p**S(2) + S(1)), x) + Simp(b*n*p*x*sin(a + b*log(c*x**n))*cos(a + b*log(c*x**n))**(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + S(1)), x)
rule4934 = ReplacementRule(pattern4934, replacement4934)
pattern4935 = Pattern(Integral(sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons137, cons1505, cons1712)
def replacement4935(p, b, a, n, c, x):
rubi.append(4935)
return Dist((b**S(2)*n**S(2)*(p + S(2))**S(2) + S(1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(sin(a + b*log(c*x**n))**(p + S(2)), x), x) - Simp(x*sin(a + b*log(c*x**n))**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x) + Simp(x*sin(a + b*log(c*x**n))**(p + S(2))/(b*n*(p + S(1))*tan(a + b*log(c*x**n))), x)
rule4935 = ReplacementRule(pattern4935, replacement4935)
pattern4936 = Pattern(Integral(cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons137, cons1505, cons1712)
def replacement4936(p, b, a, n, c, x):
rubi.append(4936)
return Dist((b**S(2)*n**S(2)*(p + S(2))**S(2) + S(1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(cos(a + b*log(c*x**n))**(p + S(2)), x), x) - Simp(x*cos(a + b*log(c*x**n))**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x) - Simp(x*cos(a + b*log(c*x**n))**(p + S(2))*tan(a + b*log(c*x**n))/(b*n*(p + S(1))), x)
rule4936 = ReplacementRule(pattern4936, replacement4936)
pattern4937 = Pattern(Integral(sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1711)
def replacement4937(p, b, a, n, c, x):
rubi.append(4937)
return Simp(x*(-S(2)*(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(2))**(-p)*(-I*(c*x**n)**(I*b)*exp(I*a) + I*(c*x**n)**(-I*b)*exp(-I*a))**p*Hypergeometric2F1(-p, -I*(-I*b*n*p + S(1))/(S(2)*b*n), S(1) - I*(-I*b*n*p + S(1))/(S(2)*b*n), (c*x**n)**(S(2)*I*b)*exp(S(2)*I*a))/(-I*b*n*p + S(1)), x)
rule4937 = ReplacementRule(pattern4937, replacement4937)
pattern4938 = Pattern(Integral(cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1711)
def replacement4938(p, b, a, n, c, x):
rubi.append(4938)
return Simp(x*((c*x**n)**(I*b)*exp(I*a) + (c*x**n)**(-I*b)*exp(-I*a))**p*(S(2)*(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(2))**(-p)*Hypergeometric2F1(-p, -I*(-I*b*n*p + S(1))/(S(2)*b*n), S(1) - I*(-I*b*n*p + S(1))/(S(2)*b*n), -(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a))/(-I*b*n*p + S(1)), x)
rule4938 = ReplacementRule(pattern4938, replacement4938)
pattern4939 = Pattern(Integral(x_**WC('m', S(1))*sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1713, cons54, cons66)
def replacement4939(p, m, b, c, n, a, x):
rubi.append(4939)
return Simp(x**(m + S(1))*(p + S(2))*sin(a + b*log(c*x**n))**(p + S(2))/((m + S(1))*(p + S(1))), x) + Simp(x**(m + S(1))*sin(a + b*log(c*x**n))**(p + S(2))/(b*n*(p + S(1))*tan(a + b*log(c*x**n))), x)
rule4939 = ReplacementRule(pattern4939, replacement4939)
pattern4940 = Pattern(Integral(x_**WC('m', S(1))*cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1713, cons54, cons66)
def replacement4940(p, m, b, a, n, c, x):
rubi.append(4940)
return Simp(x**(m + S(1))*(p + S(2))*cos(a + b*log(c*x**n))**(p + S(2))/((m + S(1))*(p + S(1))), x) - Simp(x**(m + S(1))*cos(a + b*log(c*x**n))**(p + S(2))*tan(a + b*log(c*x**n))/(b*n*(p + S(1))), x)
rule4940 = ReplacementRule(pattern4940, replacement4940)
pattern4941 = Pattern(Integral(x_**WC('m', S(1))*sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons128, cons1714)
def replacement4941(p, m, b, c, n, a, x):
rubi.append(4941)
return Dist(S(2)**(-p), Int(ExpandIntegrand(x**m*(-(c*x**n)**((m + S(1))/(n*p))*(m + S(1))*exp(-a*b*n*p/(m + S(1)))/(b*n*p) + (c*x**n)**(-(m + S(1))/(n*p))*(m + S(1))*exp(a*b*n*p/(m + S(1)))/(b*n*p))**p, x), x), x)
rule4941 = ReplacementRule(pattern4941, replacement4941)
pattern4942 = Pattern(Integral(x_**WC('m', S(1))*cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons128, cons1714)
def replacement4942(p, m, b, a, n, c, x):
rubi.append(4942)
return Dist(S(2)**(-p), Int(ExpandIntegrand(x**m*(-(c*x**n)**((m + S(1))/(n*p))*exp(-a*b*n*p/(m + S(1))) + (c*x**n)**(-(m + S(1))/(n*p))*exp(a*b*n*p/(m + S(1))))**p, x), x), x)
rule4942 = ReplacementRule(pattern4942, replacement4942)
pattern4943 = Pattern(Integral(x_**WC('m', S(1))*sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons4, cons1715)
def replacement4943(m, b, c, n, a, x):
rubi.append(4943)
return Simp(x**(m + S(1))*(m + S(1))*sin(a + b*log(c*x**n))/(b**S(2)*n**S(2) + (m + S(1))**S(2)), x) - Simp(b*n*x**(m + S(1))*cos(a + b*log(c*x**n))/(b**S(2)*n**S(2) + (m + S(1))**S(2)), x)
rule4943 = ReplacementRule(pattern4943, replacement4943)
pattern4944 = Pattern(Integral(x_**WC('m', S(1))*cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons4, cons1715)
def replacement4944(m, b, a, n, c, x):
rubi.append(4944)
return Simp(x**(m + S(1))*(m + S(1))*cos(a + b*log(c*x**n))/(b**S(2)*n**S(2) + (m + S(1))**S(2)), x) + Simp(b*n*x**(m + S(1))*sin(a + b*log(c*x**n))/(b**S(2)*n**S(2) + (m + S(1))**S(2)), x)
rule4944 = ReplacementRule(pattern4944, replacement4944)
pattern4945 = Pattern(Integral(x_**WC('m', S(1))*sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons146, cons1716)
def replacement4945(p, m, b, c, n, a, x):
rubi.append(4945)
return Dist(b**S(2)*n**S(2)*p*(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), Int(x**m*sin(a + b*log(c*x**n))**(p + S(-2)), x), x) + Simp(x**(m + S(1))*(m + S(1))*sin(a + b*log(c*x**n))**p/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), x) - Simp(b*n*p*x**(m + S(1))*sin(a + b*log(c*x**n))**(p + S(-1))*cos(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), x)
rule4945 = ReplacementRule(pattern4945, replacement4945)
pattern4946 = Pattern(Integral(x_**WC('m', S(1))*cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons146, cons1716)
def replacement4946(p, m, b, a, n, c, x):
rubi.append(4946)
return Dist(b**S(2)*n**S(2)*p*(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), Int(x**m*cos(a + b*log(c*x**n))**(p + S(-2)), x), x) + Simp(x**(m + S(1))*(m + S(1))*cos(a + b*log(c*x**n))**p/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), x) + Simp(b*n*p*x**(m + S(1))*sin(a + b*log(c*x**n))*cos(a + b*log(c*x**n))**(p + S(-1))/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), x)
rule4946 = ReplacementRule(pattern4946, replacement4946)
pattern4947 = Pattern(Integral(x_**WC('m', S(1))*sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons137, cons1505, cons1717)
def replacement4947(p, m, b, c, n, a, x):
rubi.append(4947)
return Dist((b**S(2)*n**S(2)*(p + S(2))**S(2) + (m + S(1))**S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**m*sin(a + b*log(c*x**n))**(p + S(2)), x), x) + Simp(x**(m + S(1))*sin(a + b*log(c*x**n))**(p + S(2))/(b*n*(p + S(1))*tan(a + b*log(c*x**n))), x) - Simp(x**(m + S(1))*(m + S(1))*sin(a + b*log(c*x**n))**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x)
rule4947 = ReplacementRule(pattern4947, replacement4947)
pattern4948 = Pattern(Integral(x_**WC('m', S(1))*cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons137, cons1505, cons1717)
def replacement4948(p, m, b, a, n, c, x):
rubi.append(4948)
return Dist((b**S(2)*n**S(2)*(p + S(2))**S(2) + (m + S(1))**S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**m*cos(a + b*log(c*x**n))**(p + S(2)), x), x) - Simp(x**(m + S(1))*cos(a + b*log(c*x**n))**(p + S(2))*tan(a + b*log(c*x**n))/(b*n*(p + S(1))), x) - Simp(x**(m + S(1))*(m + S(1))*cos(a + b*log(c*x**n))**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x)
rule4948 = ReplacementRule(pattern4948, replacement4948)
pattern4949 = Pattern(Integral(x_**WC('m', S(1))*sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1716)
def replacement4949(p, m, b, c, n, a, x):
rubi.append(4949)
return Simp(x**(m + S(1))*(-S(2)*(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(2))**(-p)*(-I*(c*x**n)**(I*b)*exp(I*a) + I*(c*x**n)**(-I*b)*exp(-I*a))**p*Hypergeometric2F1(-p, -I*(-I*b*n*p + m + S(1))/(S(2)*b*n), S(1) - I*(-I*b*n*p + m + S(1))/(S(2)*b*n), (c*x**n)**(S(2)*I*b)*exp(S(2)*I*a))/(-I*b*n*p + m + S(1)), x)
rule4949 = ReplacementRule(pattern4949, replacement4949)
pattern4950 = Pattern(Integral(x_**WC('m', S(1))*cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1716)
def replacement4950(p, m, b, a, n, c, x):
rubi.append(4950)
return Simp(x**(m + S(1))*((c*x**n)**(I*b)*exp(I*a) + (c*x**n)**(-I*b)*exp(-I*a))**p*(S(2)*(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(2))**(-p)*Hypergeometric2F1(-p, -I*(-I*b*n*p + m + S(1))/(S(2)*b*n), S(1) - I*(-I*b*n*p + m + S(1))/(S(2)*b*n), -(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a))/(-I*b*n*p + m + S(1)), x)
rule4950 = ReplacementRule(pattern4950, replacement4950)
pattern4951 = Pattern(Integral(S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1718)
def replacement4951(b, a, n, c, x):
rubi.append(4951)
return Dist(S(2)*exp(a*b*n), Int((c*x**n)**(S(1)/n)/((c*x**n)**(S(2)/n) + exp(S(2)*a*b*n)), x), x)
rule4951 = ReplacementRule(pattern4951, replacement4951)
pattern4952 = Pattern(Integral(S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons4, cons1718)
def replacement4952(b, a, n, c, x):
rubi.append(4952)
return Dist(S(2)*b*n*exp(a*b*n), Int((c*x**n)**(S(1)/n)/(-(c*x**n)**(S(2)/n) + exp(S(2)*a*b*n)), x), x)
rule4952 = ReplacementRule(pattern4952, replacement4952)
pattern4953 = Pattern(Integral((S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1719, cons1645)
def replacement4953(p, b, a, n, c, x):
rubi.append(4953)
return Simp(x*(p + S(-2))*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2))/(p + S(-1)), x) + Simp(x*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2))*tan(a + b*log(c*x**n))/(b*n*(p + S(-1))), x)
rule4953 = ReplacementRule(pattern4953, replacement4953)
pattern4954 = Pattern(Integral((S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons5, cons1719, cons1645)
def replacement4954(p, b, a, n, c, x):
rubi.append(4954)
return Simp(x*(p + S(-2))*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2))/(p + S(-1)), x) - Simp(x*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2))/(b*n*(p + S(-1))*tan(a + b*log(c*x**n))), x)
rule4954 = ReplacementRule(pattern4954, replacement4954)
pattern4955 = Pattern(Integral((S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons146, cons1720, cons1721)
def replacement4955(p, b, a, n, c, x):
rubi.append(4955)
return Dist((b**S(2)*n**S(2)*(p + S(-2))**S(2) + S(1))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), Int((S(1)/cos(a + b*log(c*x**n)))**(p + S(-2)), x), x) - Simp(x*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), x) + Simp(x*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2))*tan(a + b*log(c*x**n))/(b*n*(p + S(-1))), x)
rule4955 = ReplacementRule(pattern4955, replacement4955)
pattern4956 = Pattern(Integral((S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons146, cons1720, cons1721)
def replacement4956(p, b, a, n, c, x):
rubi.append(4956)
return Dist((b**S(2)*n**S(2)*(p + S(-2))**S(2) + S(1))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), Int((S(1)/sin(a + b*log(c*x**n)))**(p + S(-2)), x), x) - Simp(x*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), x) - Simp(x*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2))/(b*n*(p + S(-1))*tan(a + b*log(c*x**n))), x)
rule4956 = ReplacementRule(pattern4956, replacement4956)
pattern4957 = Pattern(Integral((S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons137, cons1711)
def replacement4957(p, b, a, n, c, x):
rubi.append(4957)
return Dist(b**S(2)*n**S(2)*p*(p + S(1))/(b**S(2)*n**S(2)*p**S(2) + S(1)), Int((S(1)/cos(a + b*log(c*x**n)))**(p + S(2)), x), x) + Simp(x*(S(1)/cos(a + b*log(c*x**n)))**p/(b**S(2)*n**S(2)*p**S(2) + S(1)), x) - Simp(b*n*p*x*(S(1)/cos(a + b*log(c*x**n)))**(p + S(1))*sin(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + S(1)), x)
rule4957 = ReplacementRule(pattern4957, replacement4957)
pattern4958 = Pattern(Integral((S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons4, cons13, cons137, cons1711)
def replacement4958(p, b, a, n, c, x):
rubi.append(4958)
return Dist(b**S(2)*n**S(2)*p*(p + S(1))/(b**S(2)*n**S(2)*p**S(2) + S(1)), Int((S(1)/sin(a + b*log(c*x**n)))**(p + S(2)), x), x) + Simp(x*(S(1)/sin(a + b*log(c*x**n)))**p/(b**S(2)*n**S(2)*p**S(2) + S(1)), x) + Simp(b*n*p*x*(S(1)/sin(a + b*log(c*x**n)))**(p + S(1))*cos(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + S(1)), x)
rule4958 = ReplacementRule(pattern4958, replacement4958)
pattern4959 = Pattern(Integral((S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons1711)
def replacement4959(p, b, a, n, c, x):
rubi.append(4959)
return Simp(x*((c*x**n)**(I*b)*exp(I*a)/((c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(1)))**p*(S(2)*(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(2))**p*Hypergeometric2F1(p, -I*(I*b*n*p + S(1))/(S(2)*b*n), S(1) - I*(I*b*n*p + S(1))/(S(2)*b*n), -(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a))/(I*b*n*p + S(1)), x)
rule4959 = ReplacementRule(pattern4959, replacement4959)
pattern4960 = Pattern(Integral((S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons4, cons5, cons1711)
def replacement4960(p, b, a, n, c, x):
rubi.append(4960)
return Simp(x*(-I*(c*x**n)**(I*b)*exp(I*a)/(-(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(1)))**p*(-S(2)*(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(2))**p*Hypergeometric2F1(p, -I*(I*b*n*p + S(1))/(S(2)*b*n), S(1) - I*(I*b*n*p + S(1))/(S(2)*b*n), (c*x**n)**(S(2)*I*b)*exp(S(2)*I*a))/(I*b*n*p + S(1)), x)
rule4960 = ReplacementRule(pattern4960, replacement4960)
pattern4961 = Pattern(Integral(x_**WC('m', S(1))/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons4, cons1722)
def replacement4961(m, b, c, n, a, x):
rubi.append(4961)
return Dist(S(2)*exp(a*b*n/(m + S(1))), Int(x**m*(c*x**n)**((m + S(1))/n)/((c*x**n)**(S(2)*(m + S(1))/n) + exp(S(2)*a*b*n/(m + S(1)))), x), x)
rule4961 = ReplacementRule(pattern4961, replacement4961)
pattern4962 = Pattern(Integral(x_**WC('m', S(1))/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))), x_), cons2, cons3, cons7, cons21, cons4, cons1722)
def replacement4962(m, b, a, n, c, x):
rubi.append(4962)
return Dist(S(2)*b*n*exp(a*b*n/(m + S(1)))/(m + S(1)), Int(x**m*(c*x**n)**((m + S(1))/n)/(-(c*x**n)**(S(2)*(m + S(1))/n) + exp(S(2)*a*b*n/(m + S(1)))), x), x)
rule4962 = ReplacementRule(pattern4962, replacement4962)
pattern4963 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1723, cons66, cons1645)
def replacement4963(p, m, b, c, n, a, x):
rubi.append(4963)
return Simp(x**(m + S(1))*(p + S(-2))*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2))/((m + S(1))*(p + S(-1))), x) + Simp(x**(m + S(1))*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2))*tan(a + b*log(c*x**n))/(b*n*(p + S(-1))), x)
rule4963 = ReplacementRule(pattern4963, replacement4963)
pattern4964 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1723, cons66, cons1645)
def replacement4964(p, m, b, a, n, c, x):
rubi.append(4964)
return Simp(x**(m + S(1))*(p + S(-2))*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2))/((m + S(1))*(p + S(-1))), x) - Simp(x**(m + S(1))*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2))/(b*n*(p + S(-1))*tan(a + b*log(c*x**n))), x)
rule4964 = ReplacementRule(pattern4964, replacement4964)
pattern4965 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons146, cons1720, cons1724)
def replacement4965(p, m, b, c, n, a, x):
rubi.append(4965)
return Dist((b**S(2)*n**S(2)*(p + S(-2))**S(2) + (m + S(1))**S(2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), Int(x**m*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2)), x), x) + Simp(x**(m + S(1))*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2))*tan(a + b*log(c*x**n))/(b*n*(p + S(-1))), x) - Simp(x**(m + S(1))*(m + S(1))*(S(1)/cos(a + b*log(c*x**n)))**(p + S(-2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), x)
rule4965 = ReplacementRule(pattern4965, replacement4965)
pattern4966 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons146, cons1720, cons1724)
def replacement4966(p, m, b, a, n, c, x):
rubi.append(4966)
return Dist((b**S(2)*n**S(2)*(p + S(-2))**S(2) + (m + S(1))**S(2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), Int(x**m*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2)), x), x) - Simp(x**(m + S(1))*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2))/(b*n*(p + S(-1))*tan(a + b*log(c*x**n))), x) - Simp(x**(m + S(1))*(m + S(1))*(S(1)/sin(a + b*log(c*x**n)))**(p + S(-2))/(b**S(2)*n**S(2)*(p + S(-2))*(p + S(-1))), x)
rule4966 = ReplacementRule(pattern4966, replacement4966)
pattern4967 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons137, cons1716)
def replacement4967(p, m, b, c, n, a, x):
rubi.append(4967)
return Dist(b**S(2)*n**S(2)*p*(p + S(1))/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), Int(x**m*(S(1)/cos(a + b*log(c*x**n)))**(p + S(2)), x), x) + Simp(x**(m + S(1))*(m + S(1))*(S(1)/cos(a + b*log(c*x**n)))**p/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), x) - Simp(b*n*p*x**(m + S(1))*(S(1)/cos(a + b*log(c*x**n)))**(p + S(1))*sin(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), x)
rule4967 = ReplacementRule(pattern4967, replacement4967)
pattern4968 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**p_, x_), cons2, cons3, cons7, cons21, cons4, cons13, cons137, cons1716)
def replacement4968(p, m, b, a, n, c, x):
rubi.append(4968)
return Dist(b**S(2)*n**S(2)*p*(p + S(1))/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), Int(x**m*(S(1)/sin(a + b*log(c*x**n)))**(p + S(2)), x), x) + Simp(x**(m + S(1))*(m + S(1))*(S(1)/sin(a + b*log(c*x**n)))**p/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), x) + Simp(b*n*p*x**(m + S(1))*(S(1)/sin(a + b*log(c*x**n)))**(p + S(1))*cos(a + b*log(c*x**n))/(b**S(2)*n**S(2)*p**S(2) + (m + S(1))**S(2)), x)
rule4968 = ReplacementRule(pattern4968, replacement4968)
pattern4969 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cos(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1716)
def replacement4969(p, m, b, c, n, a, x):
rubi.append(4969)
return Simp(x**(m + S(1))*((c*x**n)**(I*b)*exp(I*a)/((c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(1)))**p*(S(2)*(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(2))**p*Hypergeometric2F1(p, -I*(I*b*n*p + m + S(1))/(S(2)*b*n), S(1) - I*(I*b*n*p + m + S(1))/(S(2)*b*n), -(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a))/(I*b*n*p + m + S(1)), x)
rule4969 = ReplacementRule(pattern4969, replacement4969)
pattern4970 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sin(WC('a', S(0)) + WC('b', S(1))*log(x_**WC('n', S(1))*WC('c', S(1)))))**WC('p', S(1)), x_), cons2, cons3, cons7, cons21, cons4, cons5, cons1716)
def replacement4970(p, m, b, a, n, c, x):
rubi.append(4970)
return Simp(x**(m + S(1))*(-I*(c*x**n)**(I*b)*exp(I*a)/(-(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(1)))**p*(-S(2)*(c*x**n)**(S(2)*I*b)*exp(S(2)*I*a) + S(2))**p*Hypergeometric2F1(p, -I*(I*b*n*p + m + S(1))/(S(2)*b*n), S(1) - I*(I*b*n*p + m + S(1))/(S(2)*b*n), (c*x**n)**(S(2)*I*b)*exp(S(2)*I*a))/(I*b*n*p + m + S(1)), x)
rule4970 = ReplacementRule(pattern4970, replacement4970)
pattern4971 = Pattern(Integral(log(x_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons13, cons163)
def replacement4971(x, a, p, b):
rubi.append(4971)
return -Dist(p, Int(log(b*x)**(p + S(-1))*sin(a*x*log(b*x)**p), x), x) - Simp(cos(a*x*log(b*x)**p)/a, x)
rule4971 = ReplacementRule(pattern4971, replacement4971)
pattern4972 = Pattern(Integral(log(x_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons13, cons163)
def replacement4972(x, a, p, b):
rubi.append(4972)
return -Dist(p, Int(log(b*x)**(p + S(-1))*cos(a*x*log(b*x)**p), x), x) + Simp(sin(a*x*log(b*x)**p)/a, x)
rule4972 = ReplacementRule(pattern4972, replacement4972)
pattern4973 = Pattern(Integral(log(x_*WC('b', S(1)))**WC('p', S(1))*sin(x_**n_*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons338, cons163)
def replacement4973(p, b, a, n, x):
rubi.append(4973)
return -Dist(p/n, Int(log(b*x)**(p + S(-1))*sin(a*x**n*log(b*x)**p), x), x) - Dist((n + S(-1))/(a*n), Int(x**(-n)*cos(a*x**n*log(b*x)**p), x), x) - Simp(x**(-n + S(1))*cos(a*x**n*log(b*x)**p)/(a*n), x)
rule4973 = ReplacementRule(pattern4973, replacement4973)
pattern4974 = Pattern(Integral(log(x_*WC('b', S(1)))**WC('p', S(1))*cos(x_**n_*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons338, cons163)
def replacement4974(p, b, a, n, x):
rubi.append(4974)
return -Dist(p/n, Int(log(b*x)**(p + S(-1))*cos(a*x**n*log(b*x)**p), x), x) + Dist((n + S(-1))/(a*n), Int(x**(-n)*sin(a*x**n*log(b*x)**p), x), x) + Simp(x**(-n + S(1))*sin(a*x**n*log(b*x)**p)/(a*n), x)
rule4974 = ReplacementRule(pattern4974, replacement4974)
pattern4975 = Pattern(Integral(x_**WC('m', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))*sin(x_**WC('n', S(1))*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons21, cons4, cons53, cons13, cons163)
def replacement4975(p, m, b, a, n, x):
rubi.append(4975)
return -Dist(p/n, Int(x**m*log(b*x)**(p + S(-1))*sin(a*x**n*log(b*x)**p), x), x) - Simp(cos(a*x**n*log(b*x)**p)/(a*n), x)
rule4975 = ReplacementRule(pattern4975, replacement4975)
pattern4976 = Pattern(Integral(x_**WC('m', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))*cos(x_**WC('n', S(1))*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons21, cons4, cons53, cons13, cons163)
def replacement4976(p, m, b, a, n, x):
rubi.append(4976)
return -Dist(p/n, Int(x**m*log(b*x)**(p + S(-1))*cos(a*x**n*log(b*x)**p), x), x) + Simp(sin(a*x**n*log(b*x)**p)/(a*n), x)
rule4976 = ReplacementRule(pattern4976, replacement4976)
pattern4977 = Pattern(Integral(x_**WC('m', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))*sin(x_**WC('n', S(1))*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons21, cons4, cons13, cons163, cons627)
def replacement4977(p, m, b, a, n, x):
rubi.append(4977)
return -Dist(p/n, Int(x**m*log(b*x)**(p + S(-1))*sin(a*x**n*log(b*x)**p), x), x) + Dist((m - n + S(1))/(a*n), Int(x**(m - n)*cos(a*x**n*log(b*x)**p), x), x) - Simp(x**(m - n + S(1))*cos(a*x**n*log(b*x)**p)/(a*n), x)
rule4977 = ReplacementRule(pattern4977, replacement4977)
pattern4978 = Pattern(Integral(x_**m_*log(x_*WC('b', S(1)))**WC('p', S(1))*cos(x_**WC('n', S(1))*WC('a', S(1))*log(x_*WC('b', S(1)))**WC('p', S(1))), x_), cons2, cons3, cons21, cons4, cons13, cons163, cons627)
def replacement4978(p, m, b, a, n, x):
rubi.append(4978)
return -Dist(p/n, Int(x**m*log(b*x)**(p + S(-1))*cos(a*x**n*log(b*x)**p), x), x) - Dist((m - n + S(1))/(a*n), Int(x**(m - n)*sin(a*x**n*log(b*x)**p), x), x) + Simp(x**(m - n + S(1))*sin(a*x**n*log(b*x)**p)/(a*n), x)
rule4978 = ReplacementRule(pattern4978, replacement4978)
pattern4979 = Pattern(Integral(sin(WC('a', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons7, cons27, cons148)
def replacement4979(d, a, n, c, x):
rubi.append(4979)
return -Dist(S(1)/d, Subst(Int(sin(a*x)**n/x**S(2), x), x, S(1)/(c + d*x)), x)
rule4979 = ReplacementRule(pattern4979, replacement4979)
pattern4980 = Pattern(Integral(cos(WC('a', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons7, cons27, cons148)
def replacement4980(d, a, n, c, x):
rubi.append(4980)
return -Dist(S(1)/d, Subst(Int(cos(a*x)**n/x**S(2), x), x, S(1)/(c + d*x)), x)
rule4980 = ReplacementRule(pattern4980, replacement4980)
pattern4981 = Pattern(Integral(sin((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons148, cons71)
def replacement4981(b, d, c, a, n, x, e):
rubi.append(4981)
return -Dist(S(1)/d, Subst(Int(sin(b*e/d - e*x*(-a*d + b*c)/d)**n/x**S(2), x), x, S(1)/(c + d*x)), x)
rule4981 = ReplacementRule(pattern4981, replacement4981)
pattern4982 = Pattern(Integral(cos((x_*WC('b', S(1)) + WC('a', S(0)))*WC('e', S(1))/(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons148, cons71)
def replacement4982(b, d, c, a, n, x, e):
rubi.append(4982)
return -Dist(S(1)/d, Subst(Int(cos(b*e/d - e*x*(-a*d + b*c)/d)**n/x**S(2), x), x, S(1)/(c + d*x)), x)
rule4982 = ReplacementRule(pattern4982, replacement4982)
def With4983(x, n, u):
lst = QuotientOfLinearsParts(u, x)
rubi.append(4983)
return Int(sin((x*Part(lst, S(2)) + Part(lst, S(1)))/(x*Part(lst, S(4)) + Part(lst, S(3))))**n, x)
pattern4983 = Pattern(Integral(sin(u_)**WC('n', S(1)), x_), cons148, cons1725)
rule4983 = ReplacementRule(pattern4983, With4983)
def With4984(x, n, u):
lst = QuotientOfLinearsParts(u, x)
rubi.append(4984)
return Int(cos((x*Part(lst, S(2)) + Part(lst, S(1)))/(x*Part(lst, S(4)) + Part(lst, S(3))))**n, x)
pattern4984 = Pattern(Integral(cos(u_)**WC('n', S(1)), x_), cons148, cons1725)
rule4984 = ReplacementRule(pattern4984, With4984)
pattern4985 = Pattern(Integral(WC('u', S(1))*sin(v_)**WC('p', S(1))*sin(w_)**WC('q', S(1)), x_), cons1688)
def replacement4985(v, w, p, u, x, q):
rubi.append(4985)
return Int(u*sin(v)**(p + q), x)
rule4985 = ReplacementRule(pattern4985, replacement4985)
pattern4986 = Pattern(Integral(WC('u', S(1))*cos(v_)**WC('p', S(1))*cos(w_)**WC('q', S(1)), x_), cons1688)
def replacement4986(v, w, p, u, x, q):
rubi.append(4986)
return Int(u*cos(v)**(p + q), x)
rule4986 = ReplacementRule(pattern4986, replacement4986)
pattern4987 = Pattern(Integral(sin(v_)**WC('p', S(1))*sin(w_)**WC('q', S(1)), x_), cons1726, cons555)
def replacement4987(v, w, p, x, q):
rubi.append(4987)
return Int(ExpandTrigReduce(sin(v)**p*sin(w)**q, x), x)
rule4987 = ReplacementRule(pattern4987, replacement4987)
pattern4988 = Pattern(Integral(cos(v_)**WC('p', S(1))*cos(w_)**WC('q', S(1)), x_), cons1726, cons555)
def replacement4988(v, w, p, x, q):
rubi.append(4988)
return Int(ExpandTrigReduce(cos(v)**p*cos(w)**q, x), x)
rule4988 = ReplacementRule(pattern4988, replacement4988)
pattern4989 = Pattern(Integral(x_**WC('m', S(1))*sin(v_)**WC('p', S(1))*sin(w_)**WC('q', S(1)), x_), cons1727, cons1726)
def replacement4989(v, w, p, m, x, q):
rubi.append(4989)
return Int(ExpandTrigReduce(x**m, sin(v)**p*sin(w)**q, x), x)
rule4989 = ReplacementRule(pattern4989, replacement4989)
pattern4990 = Pattern(Integral(x_**WC('m', S(1))*cos(v_)**WC('p', S(1))*cos(w_)**WC('q', S(1)), x_), cons1727, cons1726)
def replacement4990(v, w, p, m, x, q):
rubi.append(4990)
return Int(ExpandTrigReduce(x**m, cos(v)**p*cos(w)**q, x), x)
rule4990 = ReplacementRule(pattern4990, replacement4990)
pattern4991 = Pattern(Integral(WC('u', S(1))*sin(v_)**WC('p', S(1))*cos(w_)**WC('p', S(1)), x_), cons1688, cons38)
def replacement4991(v, w, p, u, x):
rubi.append(4991)
return Dist(S(2)**(-p), Int(u*sin(S(2)*v)**p, x), x)
rule4991 = ReplacementRule(pattern4991, replacement4991)
pattern4992 = Pattern(Integral(sin(v_)**WC('p', S(1))*cos(w_)**WC('q', S(1)), x_), cons555, cons1726)
def replacement4992(v, w, p, x, q):
rubi.append(4992)
return Int(ExpandTrigReduce(sin(v)**p*cos(w)**q, x), x)
rule4992 = ReplacementRule(pattern4992, replacement4992)
pattern4993 = Pattern(Integral(x_**WC('m', S(1))*sin(v_)**WC('p', S(1))*cos(w_)**WC('q', S(1)), x_), cons1727, cons1726)
def replacement4993(v, w, p, m, x, q):
rubi.append(4993)
return Int(ExpandTrigReduce(x**m, sin(v)**p*cos(w)**q, x), x)
rule4993 = ReplacementRule(pattern4993, replacement4993)
pattern4994 = Pattern(Integral(sin(v_)*tan(w_)**WC('n', S(1)), x_), cons87, cons88, cons1728)
def replacement4994(v, w, n, x):
rubi.append(4994)
return Dist(cos(v - w), Int(tan(w)**(n + S(-1))/cos(w), x), x) - Int(cos(v)*tan(w)**(n + S(-1)), x)
rule4994 = ReplacementRule(pattern4994, replacement4994)
pattern4995 = Pattern(Integral((S(1)/tan(w_))**WC('n', S(1))*cos(v_), x_), cons87, cons88, cons1728)
def replacement4995(v, w, n, x):
rubi.append(4995)
return Dist(cos(v - w), Int((S(1)/tan(w))**(n + S(-1))/sin(w), x), x) - Int((S(1)/tan(w))**(n + S(-1))*sin(v), x)
rule4995 = ReplacementRule(pattern4995, replacement4995)
pattern4996 = Pattern(Integral((S(1)/tan(w_))**WC('n', S(1))*sin(v_), x_), cons87, cons88, cons1728)
def replacement4996(v, w, n, x):
rubi.append(4996)
return Dist(sin(v - w), Int((S(1)/tan(w))**(n + S(-1))/sin(w), x), x) + Int((S(1)/tan(w))**(n + S(-1))*cos(v), x)
rule4996 = ReplacementRule(pattern4996, replacement4996)
pattern4997 = Pattern(Integral(cos(v_)*tan(w_)**WC('n', S(1)), x_), cons87, cons88, cons1728)
def replacement4997(v, w, n, x):
rubi.append(4997)
return -Dist(sin(v - w), Int(tan(w)**(n + S(-1))/cos(w), x), x) + Int(sin(v)*tan(w)**(n + S(-1)), x)
rule4997 = ReplacementRule(pattern4997, replacement4997)
pattern4998 = Pattern(Integral((S(1)/cos(w_))**WC('n', S(1))*sin(v_), x_), cons87, cons88, cons1728)
def replacement4998(v, w, n, x):
rubi.append(4998)
return Dist(sin(v - w), Int((S(1)/cos(w))**(n + S(-1)), x), x) + Dist(cos(v - w), Int((S(1)/cos(w))**(n + S(-1))*tan(w), x), x)
rule4998 = ReplacementRule(pattern4998, replacement4998)
pattern4999 = Pattern(Integral((S(1)/sin(w_))**WC('n', S(1))*cos(v_), x_), cons87, cons88, cons1728)
def replacement4999(v, w, n, x):
rubi.append(4999)
return -Dist(sin(v - w), Int((S(1)/sin(w))**(n + S(-1)), x), x) + Dist(cos(v - w), Int((S(1)/sin(w))**(n + S(-1))/tan(w), x), x)
rule4999 = ReplacementRule(pattern4999, replacement4999)
pattern5000 = Pattern(Integral((S(1)/sin(w_))**WC('n', S(1))*sin(v_), x_), cons87, cons88, cons1728)
def replacement5000(v, w, n, x):
rubi.append(5000)
return Dist(sin(v - w), Int((S(1)/sin(w))**(n + S(-1))/tan(w), x), x) + Dist(cos(v - w), Int((S(1)/sin(w))**(n + S(-1)), x), x)
rule5000 = ReplacementRule(pattern5000, replacement5000)
pattern5001 = Pattern(Integral((S(1)/cos(w_))**WC('n', S(1))*cos(v_), x_), cons87, cons88, cons1728)
def replacement5001(v, w, n, x):
rubi.append(5001)
return -Dist(sin(v - w), Int((S(1)/cos(w))**(n + S(-1))*tan(w), x), x) + Dist(cos(v - w), Int((S(1)/cos(w))**(n + S(-1)), x), x)
rule5001 = ReplacementRule(pattern5001, replacement5001)
pattern5002 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons21, cons4, cons1360)
def replacement5002(m, f, b, d, c, n, a, x, e):
rubi.append(5002)
return Int((a + b*sin(S(2)*c + S(2)*d*x)/S(2))**n*(e + f*x)**m, x)
rule5002 = ReplacementRule(pattern5002, replacement5002)
pattern5003 = Pattern(Integral(x_**WC('m', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons1478, cons150, cons168, cons463, cons1729)
def replacement5003(m, b, d, c, a, n, x):
rubi.append(5003)
return Dist(S(2)**(-n), Int(x**m*(S(2)*a - b*cos(S(2)*c + S(2)*d*x) + b)**n, x), x)
rule5003 = ReplacementRule(pattern5003, replacement5003)
pattern5004 = Pattern(Integral(x_**WC('m', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**n_, x_), cons2, cons3, cons7, cons27, cons1478, cons150, cons168, cons463, cons1729)
def replacement5004(m, b, d, c, a, n, x):
rubi.append(5004)
return Dist(S(2)**(-n), Int(x**m*(S(2)*a + b*cos(S(2)*c + S(2)*d*x) + b)**n, x), x)
rule5004 = ReplacementRule(pattern5004, replacement5004)
pattern5005 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sin((c_ + x_*WC('d', S(1)))**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons13)
def replacement5005(p, m, f, b, d, a, c, n, x, e):
rubi.append(5005)
return Dist(d**(-m + S(-1)), Subst(Int((-c*f + d*e + f*x)**m*sin(a + b*x**n)**p, x), x, c + d*x), x)
rule5005 = ReplacementRule(pattern5005, replacement5005)
pattern5006 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cos((c_ + x_*WC('d', S(1)))**n_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons4, cons62, cons13)
def replacement5006(p, m, f, b, d, a, c, n, x, e):
rubi.append(5006)
return Dist(d**(-m + S(-1)), Subst(Int((-c*f + d*e + f*x)**m*cos(a + b*x**n)**p, x), x, c + d*x), x)
rule5006 = ReplacementRule(pattern5006, replacement5006)
pattern5007 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons62, cons1478, cons1730)
def replacement5007(m, f, g, b, d, a, c, x, e):
rubi.append(5007)
return Dist(S(2), Int((f + g*x)**m/(S(2)*a + b + c + (b - c)*cos(S(2)*d + S(2)*e*x)), x), x)
rule5007 = ReplacementRule(pattern5007, replacement5007)
pattern5008 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/((b_ + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons3, cons7, cons27, cons48, cons125, cons208, cons62)
def replacement5008(m, f, g, b, d, c, x, e):
rubi.append(5008)
return Dist(S(2), Int((f + g*x)**m/(b + c + (b - c)*cos(S(2)*d + S(2)*e*x)), x), x)
rule5008 = ReplacementRule(pattern5008, replacement5008)
pattern5009 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/((WC('a', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('b', S(0)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons62, cons1478, cons1730)
def replacement5009(m, f, g, b, d, a, c, x, e):
rubi.append(5009)
return Dist(S(2), Int((f + g*x)**m/(S(2)*a + b + c + (b - c)*cos(S(2)*d + S(2)*e*x)), x), x)
rule5009 = ReplacementRule(pattern5009, replacement5009)
pattern5010 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/((c_ + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons3, cons7, cons27, cons48, cons125, cons208, cons62)
def replacement5010(m, f, b, g, d, c, x, e):
rubi.append(5010)
return Dist(S(2), Int((f + g*x)**m/(b + c + (b - c)*cos(S(2)*d + S(2)*e*x)), x), x)
rule5010 = ReplacementRule(pattern5010, replacement5010)
pattern5011 = Pattern(Integral((x_*WC('g', S(1)) + WC('f', S(0)))**WC('m', S(1))/((WC('a', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2) + WC('c', S(0)))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons208, cons62, cons1478, cons1730)
def replacement5011(m, f, b, g, d, c, a, x, e):
rubi.append(5011)
return Dist(S(2), Int((f + g*x)**m/(S(2)*a + b + c + (b - c)*cos(S(2)*d + S(2)*e*x)), x), x)
rule5011 = ReplacementRule(pattern5011, replacement5011)
pattern5012 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1731)
def replacement5012(m, f, b, d, c, a, x, e):
rubi.append(5012)
return Int((e + f*x)**m*exp(I*(c + d*x))/(a - I*b*exp(I*(c + d*x)) - Rt(a**S(2) - b**S(2), S(2))), x) + Int((e + f*x)**m*exp(I*(c + d*x))/(a - I*b*exp(I*(c + d*x)) + Rt(a**S(2) - b**S(2), S(2))), x) - Simp(I*(e + f*x)**(m + S(1))/(b*f*(m + S(1))), x)
rule5012 = ReplacementRule(pattern5012, replacement5012)
pattern5013 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1731)
def replacement5013(m, f, b, d, c, a, x, e):
rubi.append(5013)
return -Dist(I, Int((e + f*x)**m*exp(I*(c + d*x))/(a + b*exp(I*(c + d*x)) - Rt(a**S(2) - b**S(2), S(2))), x), x) - Dist(I, Int((e + f*x)**m*exp(I*(c + d*x))/(a + b*exp(I*(c + d*x)) + Rt(a**S(2) - b**S(2), S(2))), x), x) + Simp(I*(e + f*x)**(m + S(1))/(b*f*(m + S(1))), x)
rule5013 = ReplacementRule(pattern5013, replacement5013)
pattern5014 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1732)
def replacement5014(m, f, b, d, c, a, x, e):
rubi.append(5014)
return Dist(I, Int((e + f*x)**m*exp(I*(c + d*x))/(I*a + b*exp(I*(c + d*x)) - Rt(-a**S(2) + b**S(2), S(2))), x), x) + Dist(I, Int((e + f*x)**m*exp(I*(c + d*x))/(I*a + b*exp(I*(c + d*x)) + Rt(-a**S(2) + b**S(2), S(2))), x), x) - Simp(I*(e + f*x)**(m + S(1))/(b*f*(m + S(1))), x)
rule5014 = ReplacementRule(pattern5014, replacement5014)
pattern5015 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons1732)
def replacement5015(m, f, b, d, c, a, x, e):
rubi.append(5015)
return Int((e + f*x)**m*exp(I*(c + d*x))/(I*a + I*b*exp(I*(c + d*x)) - Rt(-a**S(2) + b**S(2), S(2))), x) + Int((e + f*x)**m*exp(I*(c + d*x))/(I*a + I*b*exp(I*(c + d*x)) + Rt(-a**S(2) + b**S(2), S(2))), x) + Simp(I*(e + f*x)**(m + S(1))/(b*f*(m + S(1))), x)
rule5015 = ReplacementRule(pattern5015, replacement5015)
pattern5016 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons85, cons165, cons1265)
def replacement5016(m, f, b, d, c, n, a, x, e):
rubi.append(5016)
return Dist(S(1)/a, Int((e + f*x)**m*cos(c + d*x)**(n + S(-2)), x), x) - Dist(S(1)/b, Int((e + f*x)**m*sin(c + d*x)*cos(c + d*x)**(n + S(-2)), x), x)
rule5016 = ReplacementRule(pattern5016, replacement5016)
pattern5017 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons85, cons165, cons1265)
def replacement5017(m, f, b, d, c, a, n, x, e):
rubi.append(5017)
return Dist(S(1)/a, Int((e + f*x)**m*sin(c + d*x)**(n + S(-2)), x), x) - Dist(S(1)/b, Int((e + f*x)**m*sin(c + d*x)**(n + S(-2))*cos(c + d*x), x), x)
rule5017 = ReplacementRule(pattern5017, replacement5017)
pattern5018 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons85, cons165, cons1267)
def replacement5018(m, f, b, d, c, n, a, x, e):
rubi.append(5018)
return -Dist(S(1)/b, Int((e + f*x)**m*sin(c + d*x)*cos(c + d*x)**(n + S(-2)), x), x) + Dist(a/b**S(2), Int((e + f*x)**m*cos(c + d*x)**(n + S(-2)), x), x) - Dist((a**S(2) - b**S(2))/b**S(2), Int((e + f*x)**m*cos(c + d*x)**(n + S(-2))/(a + b*sin(c + d*x)), x), x)
rule5018 = ReplacementRule(pattern5018, replacement5018)
pattern5019 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons62, cons85, cons165, cons1267)
def replacement5019(m, f, b, d, c, a, n, x, e):
rubi.append(5019)
return -Dist(S(1)/b, Int((e + f*x)**m*sin(c + d*x)**(n + S(-2))*cos(c + d*x), x), x) + Dist(a/b**S(2), Int((e + f*x)**m*sin(c + d*x)**(n + S(-2)), x), x) - Dist((a**S(2) - b**S(2))/b**S(2), Int((e + f*x)**m*sin(c + d*x)**(n + S(-2))/(a + b*cos(c + d*x)), x), x)
rule5019 = ReplacementRule(pattern5019, replacement5019)
pattern5020 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))*(x_*WC('f', S(1)) + WC('e', S(0)))/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1474)
def replacement5020(B, f, b, d, c, a, A, x, e):
rubi.append(5020)
return Dist(B*f/(a*d), Int(cos(c + d*x)/(a + b*sin(c + d*x)), x), x) - Simp(B*(e + f*x)*cos(c + d*x)/(a*d*(a + b*sin(c + d*x))), x)
rule5020 = ReplacementRule(pattern5020, replacement5020)
pattern5021 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))*(x_*WC('f', S(1)) + WC('e', S(0)))/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons7, cons27, cons48, cons125, cons34, cons35, cons1474)
def replacement5021(B, f, b, d, c, a, A, x, e):
rubi.append(5021)
return -Dist(B*f/(a*d), Int(sin(c + d*x)/(a + b*cos(c + d*x)), x), x) + Simp(B*(e + f*x)*sin(c + d*x)/(a*d*(a + b*cos(c + d*x))), x)
rule5021 = ReplacementRule(pattern5021, replacement5021)
pattern5022 = Pattern(Integral((a_ + WC('b', S(1))*tan(v_))**WC('n', S(1))*(S(1)/cos(v_))**WC('m', S(1)), x_), cons2, cons3, cons150, cons1551, cons1481)
def replacement5022(v, m, b, a, n, x):
rubi.append(5022)
return Int((a*cos(v) + b*sin(v))**n, x)
rule5022 = ReplacementRule(pattern5022, replacement5022)
pattern5023 = Pattern(Integral((a_ + WC('b', S(1))/tan(v_))**WC('n', S(1))*(S(1)/sin(v_))**WC('m', S(1)), x_), cons2, cons3, cons150, cons1551, cons1481)
def replacement5023(v, m, b, a, n, x):
rubi.append(5023)
return Int((a*sin(v) + b*cos(v))**n, x)
rule5023 = ReplacementRule(pattern5023, replacement5023)
pattern5024 = Pattern(Integral(WC('u', S(1))*sin(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons528)
def replacement5024(u, m, b, d, a, c, n, x):
rubi.append(5024)
return Int(ExpandTrigReduce(u, sin(a + b*x)**m*sin(c + d*x)**n, x), x)
rule5024 = ReplacementRule(pattern5024, replacement5024)
pattern5025 = Pattern(Integral(WC('u', S(1))*cos(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons7, cons27, cons528)
def replacement5025(u, m, b, d, a, c, n, x):
rubi.append(5025)
return Int(ExpandTrigReduce(u, cos(a + b*x)**m*cos(c + d*x)**n, x), x)
rule5025 = ReplacementRule(pattern5025, replacement5025)
pattern5026 = Pattern(Integral(S(1)/(cos(c_ + x_*WC('d', S(1)))*cos(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1733, cons71)
def replacement5026(b, d, c, a, x):
rubi.append(5026)
return Dist(S(1)/sin((-a*d + b*c)/b), Int(tan(c + d*x), x), x) - Dist(S(1)/sin((-a*d + b*c)/d), Int(tan(a + b*x), x), x)
rule5026 = ReplacementRule(pattern5026, replacement5026)
pattern5027 = Pattern(Integral(S(1)/(sin(c_ + x_*WC('d', S(1)))*sin(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1733, cons71)
def replacement5027(b, d, c, a, x):
rubi.append(5027)
return Dist(S(1)/sin((-a*d + b*c)/b), Int(S(1)/tan(a + b*x), x), x) - Dist(S(1)/sin((-a*d + b*c)/d), Int(S(1)/tan(c + d*x), x), x)
rule5027 = ReplacementRule(pattern5027, replacement5027)
pattern5028 = Pattern(Integral(tan(c_ + x_*WC('d', S(1)))*tan(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons7, cons27, cons1733, cons71)
def replacement5028(b, d, c, a, x):
rubi.append(5028)
return Dist(b*cos((-a*d + b*c)/d)/d, Int(S(1)/(cos(a + b*x)*cos(c + d*x)), x), x) - Simp(b*x/d, x)
rule5028 = ReplacementRule(pattern5028, replacement5028)
pattern5029 = Pattern(Integral(S(1)/(tan(c_ + x_*WC('d', S(1)))*tan(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons7, cons27, cons1733, cons71)
def replacement5029(b, d, c, a, x):
rubi.append(5029)
return Dist(cos((-a*d + b*c)/d), Int(S(1)/(sin(a + b*x)*sin(c + d*x)), x), x) - Simp(b*x/d, x)
rule5029 = ReplacementRule(pattern5029, replacement5029)
pattern5030 = Pattern(Integral((WC('a', S(1))*cos(v_) + WC('b', S(1))*sin(v_))**WC('n', S(1))*WC('u', S(1)), x_), cons2, cons3, cons4, cons1439)
def replacement5030(v, u, b, a, n, x):
rubi.append(5030)
return Int(u*(a*exp(-a*v/b))**n, x)
rule5030 = ReplacementRule(pattern5030, replacement5030)
return [rule4685, rule4686, rule4687, rule4688, rule4689, rule4690, rule4691, rule4692, rule4693, rule4694, rule4695, rule4696, rule4697, rule4698, rule4699, rule4700, rule4701, rule4702, rule4703, rule4704, rule4705, rule4706, rule4707, rule4708, rule4709, rule4710, rule4711, rule4712, rule4713, rule4714, rule4715, rule4716, rule4717, rule4718, rule4719, rule4720, rule4721, rule4722, rule4723, rule4724, rule4725, rule4726, rule4727, rule4728, rule4729, rule4730, rule4731, rule4732, rule4733, rule4734, rule4735, rule4736, rule4737, rule4738, rule4739, rule4740, rule4741, rule4742, rule4743, rule4744, rule4745, rule4746, rule4747, rule4748, rule4749, rule4750, rule4751, rule4752, rule4753, rule4754, rule4755, rule4756, rule4757, rule4758, rule4759, rule4760, rule4761, rule4762, rule4763, rule4764, rule4765, rule4766, rule4767, rule4768, rule4769, rule4770, rule4771, rule4772, rule4773, rule4774, rule4775, rule4776, rule4777, rule4778, rule4779, rule4780, rule4781, rule4782, rule4783, rule4784, rule4785, rule4786, rule4787, rule4788, rule4789, rule4790, rule4791, rule4792, rule4793, rule4794, rule4795, rule4796, rule4797, rule4798, rule4799, rule4800, rule4801, rule4802, rule4803, rule4804, rule4805, rule4806, rule4807, rule4808, rule4809, rule4810, rule4811, rule4812, rule4813, rule4814, rule4815, rule4816, rule4817, rule4818, rule4819, rule4820, rule4821, rule4822, rule4823, rule4824, rule4825, rule4826, rule4827, rule4828, rule4829, rule4830, rule4831, rule4832, rule4833, rule4834, rule4835, rule4836, rule4837, rule4838, rule4839, rule4840, rule4841, rule4842, rule4843, rule4844, rule4845, rule4846, rule4847, rule4848, rule4849, rule4850, rule4851, rule4852, rule4853, rule4854, rule4855, rule4856, rule4857, rule4858, rule4859, rule4860, rule4861, rule4862, rule4863, rule4864, rule4865, rule4866, rule4867, rule4868, rule4869, rule4870, rule4871, rule4872, rule4873, rule4874, rule4875, rule4876, rule4877, rule4878, rule4879, rule4880, rule4881, rule4882, rule4883, rule4884, rule4885, rule4886, rule4887, rule4888, rule4889, rule4890, rule4891, rule4892, rule4893, rule4894, rule4895, rule4896, rule4897, rule4898, rule4899, rule4900, rule4901, rule4902, rule4903, rule4904, rule4905, rule4906, rule4907, rule4908, rule4909, rule4910, rule4911, rule4912, rule4913, rule4914, rule4915, rule4916, rule4917, rule4918, rule4919, rule4920, rule4921, rule4922, rule4923, rule4924, rule4925, rule4926, rule4927, rule4928, rule4929, rule4930, rule4931, rule4932, rule4933, rule4934, rule4935, rule4936, rule4937, rule4938, rule4939, rule4940, rule4941, rule4942, rule4943, rule4944, rule4945, rule4946, rule4947, rule4948, rule4949, rule4950, rule4951, rule4952, rule4953, rule4954, rule4955, rule4956, rule4957, rule4958, rule4959, rule4960, rule4961, rule4962, rule4963, rule4964, rule4965, rule4966, rule4967, rule4968, rule4969, rule4970, rule4971, rule4972, rule4973, rule4974, rule4975, rule4976, rule4977, rule4978, rule4979, rule4980, rule4981, rule4982, rule4983, rule4984, rule4985, rule4986, rule4987, rule4988, rule4989, rule4990, rule4991, rule4992, rule4993, rule4994, rule4995, rule4996, rule4997, rule4998, rule4999, rule5000, rule5001, rule5002, rule5003, rule5004, rule5005, rule5006, rule5007, rule5008, rule5009, rule5010, rule5011, rule5012, rule5013, rule5014, rule5015, rule5016, rule5017, rule5018, rule5019, rule5020, rule5021, rule5022, rule5023, rule5024, rule5025, rule5026, rule5027, rule5028, rule5029, rule5030, ]
|
cbf55f21a849209c3eaef41ecb66f7eacd9040ad30995ebed0599b7d2700b634 | '''
Parser for FullForm[Downvalues[]] of Mathematica rules.
This parser is customised to parse the output in MatchPy rules format. Multiple
`Constraints` are divided into individual `Constraints` because it helps the
MatchPy's `ManyToOneReplacer` to backtrack earlier and improve the speed.
Parsed output is formatted into readable format by using `sympify` and print the
expression using `sstr`. This replaces `And`, `Mul`, 'Pow' by their respective
symbols.
Mathematica
===========
To get the full form from Wolfram Mathematica, type:
```
ShowSteps = False
Import["RubiLoader.m"]
Export["output.txt", ToString@FullForm@DownValues@Int]
```
The file ``output.txt`` will then contain the rules in parseable format.
References
==========
[1] http://reference.wolfram.com/language/ref/FullForm.html
[2] http://reference.wolfram.com/language/ref/DownValues.html
[3] https://gist.github.com/Upabjojr/bc07c49262944f9c1eb0
'''
import re
import os
import inspect
from sympy import sympify, Function, Set, Symbol
from sympy.core.compatibility import string_types
from sympy.printing import StrPrinter
from sympy.utilities.misc import debug
class RubiStrPrinter(StrPrinter):
def _print_Not(self, expr):
return "Not(%s)" % self._print(expr.args[0])
def rubi_printer(expr, **settings):
return RubiStrPrinter(settings).doprint(expr)
replacements = dict( # Mathematica equivalent functions in SymPy
Times="Mul",
Plus="Add",
Power="Pow",
Log='log',
Exp='exp',
Sqrt='sqrt',
Cos='cos',
Sin='sin',
Tan='tan',
Cot='1/tan',
cot='1/tan',
Sec='1/cos',
sec='1/cos',
Csc='1/sin',
csc='1/sin',
ArcSin='asin',
ArcCos='acos',
# ArcTan='atan',
ArcCot='acot',
ArcSec='asec',
ArcCsc='acsc',
Sinh='sinh',
Cosh='cosh',
Tanh='tanh',
Coth='1/tanh',
coth='1/tanh',
Sech='1/cosh',
sech='1/cosh',
Csch='1/sinh',
csch='1/sinh',
ArcSinh='asinh',
ArcCosh='acosh',
ArcTanh='atanh',
ArcCoth='acoth',
ArcSech='asech',
ArcCsch='acsch',
Expand='expand',
Im='im',
Re='re',
Flatten='flatten',
Polylog='polylog',
Cancel='cancel',
#Gamma='gamma',
TrigExpand='expand_trig',
Sign='sign',
Simplify='simplify',
Defer='UnevaluatedExpr',
Identity = 'S',
Sum = 'Sum_doit',
Module = 'With',
Block = 'With',
Null = 'None'
)
temporary_variable_replacement = { # Temporarily rename because it can raise errors while sympifying
'gcd' : "_gcd",
'jn' : "_jn",
}
permanent_variable_replacement = { # Permamenely rename these variables
r"\[ImaginaryI]" : 'ImaginaryI',
"$UseGamma": '_UseGamma',
}
#these functions have different return type in different cases. So better to use a try and except in the constraints, when any of these appear
f_diff_return_type = ['BinomialParts', 'BinomialDegree', 'TrinomialParts', 'GeneralizedBinomialParts', 'GeneralizedTrinomialParts', 'PseudoBinomialParts', 'PerfectPowerTest',
'SquareFreeFactorTest', 'SubstForFractionalPowerOfQuotientOfLinears', 'FractionalPowerOfQuotientOfLinears', 'InverseFunctionOfQuotientOfLinears',
'FractionalPowerOfSquareQ', 'FunctionOfLinear', 'FunctionOfInverseLinear', 'FunctionOfTrig', 'FindTrigFactor', 'FunctionOfLog',
'PowerVariableExpn', 'FunctionOfSquareRootOfQuadratic', 'SubstForFractionalPowerOfLinear', 'FractionalPowerOfLinear', 'InverseFunctionOfLinear',
'Divides', 'DerivativeDivides', 'TrigSquare', 'SplitProduct', 'SubstForFractionalPowerOfQuotientOfLinears', 'InverseFunctionOfQuotientOfLinears',
'FunctionOfHyperbolic', 'SplitSum']
def contains_diff_return_type(a):
'''
This function returns whether an expression contains functions which have different return types in
diiferent cases.
'''
if isinstance(a, list):
for i in a:
if contains_diff_return_type(i):
return True
elif type(a) == Function('With') or type(a) == Function('Module'):
for i in f_diff_return_type:
if a.has(Function(i)):
return True
else:
if a in f_diff_return_type:
return True
return False
def parse_full_form(wmexpr):
'''
Parses FullForm[Downvalues[]] generated by Mathematica
'''
out = []
stack = [out]
generator = re.finditer(r'[\[\],]', wmexpr)
last_pos = 0
for match in generator:
if match is None:
break
position = match.start()
last_expr = wmexpr[last_pos:position].replace(',', '').replace(']', '').replace('[', '').strip()
if match.group() == ',':
if last_expr != '':
stack[-1].append(last_expr)
elif match.group() == ']':
if last_expr != '':
stack[-1].append(last_expr)
stack.pop()
current_pos = stack[-1]
elif match.group() == '[':
stack[-1].append([last_expr])
stack.append(stack[-1][-1])
last_pos = match.end()
return out[0]
def get_default_values(parsed, default_values={}):
'''
Returns Optional variables and their values in the pattern
'''
if not isinstance(parsed, list):
return default_values
if parsed[0] == "Times": # find Default arguments for "Times"
for i in parsed[1:]:
if i[0] == "Optional":
default_values[(i[1][1])] = 1
if parsed[0] == "Plus": # find Default arguments for "Plus"
for i in parsed[1:]:
if i[0] == "Optional":
default_values[(i[1][1])] = 0
if parsed[0] == "Power": # find Default arguments for "Power"
for i in parsed[1:]:
if i[0] == "Optional":
default_values[(i[1][1])] = 1
if len(parsed) == 1:
return default_values
for i in parsed:
default_values = get_default_values(i, default_values)
return default_values
def add_wildcards(string, optional={}):
'''
Replaces `Pattern(variable)` by `variable` in `string`.
Returns the free symbols present in the string.
'''
symbols = [] # stores symbols present in the expression
p = r'(Optional\(Pattern\((\w+), Blank\)\))'
matches = re.findall(p, string)
for i in matches:
string = string.replace(i[0], "WC('{}', S({}))".format(i[1], optional[i[1]]))
symbols.append(i[1])
p = r'(Pattern\((\w+), Blank\))'
matches = re.findall(p, string)
for i in matches:
string = string.replace(i[0], i[1] + '_')
symbols.append(i[1])
p = r'(Pattern\((\w+), Blank\(Symbol\)\))'
matches = re.findall(p, string)
for i in matches:
string = string.replace(i[0], i[1] + '_')
symbols.append(i[1])
return string, symbols
def seperate_freeq(s, variables=[], x=None):
'''
Returns list of symbols in FreeQ.
'''
if s[0] == 'FreeQ':
if len(s[1]) == 1:
variables = [s[1]]
else:
variables = s[1][1:]
x = s[2]
else:
for i in s[1:]:
variables, x = seperate_freeq(i, variables, x)
return variables, x
return variables, x
def parse_freeq(l, x, cons_index, cons_dict, cons_import, symbols=None):
'''
Converts FreeQ constraints into MatchPy constraint
'''
res = []
cons = ''
for i in l:
if isinstance(i, string_types):
r = ' return FreeQ({}, {})'.format(i, x)
# First it checks if a constraint is already present in `cons_dict`, If yes, use it else create a new one.
if r not in cons_dict.values():
cons_index += 1
c = '\n def cons_f{}({}, {}):\n'.format(cons_index, i, x)
c += r
c += '\n\n cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
cons_name = 'cons{}'.format(cons_index)
cons_dict[cons_name] = r
else:
c = ''
cons_name = next(key for key, value in cons_dict.items() if value == r)
elif isinstance(i, list):
s = list(set(get_free_symbols(i, symbols)))
s = ', '.join(s)
r = ' return FreeQ({}, {})'.format(generate_sympy_from_parsed(i), x)
if r not in cons_dict.values():
cons_index += 1
c = '\n def cons_f{}({}):\n'.format(cons_index, s)
c += r
c += '\n\n cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
cons_name = 'cons{}'.format(cons_index)
cons_dict[cons_name] = r
else:
c = ''
cons_name = next(key for key, value in cons_dict.items() if value == r)
if cons_name not in cons_import:
cons_import.append(cons_name)
res.append(cons_name)
cons += c
if res != []:
return ', ' + ', '.join(res), cons, cons_index
return '', cons, cons_index
def generate_sympy_from_parsed(parsed, wild=False, symbols=[], replace_Int=False):
'''
Parses list into Python syntax.
Parameters
==========
wild : When set to True, the symbols are replaced as wild symbols.
symbols : Symbols already present in the pattern.
replace_Int: when set to True, `Int` is replaced by `Integral`(used to parse pattern).
'''
out = ""
if not isinstance(parsed, list):
try: #return S(number) if parsed is Number
float(parsed)
return "S({})".format(parsed)
except:
pass
if parsed in symbols:
if wild:
return parsed + '_'
return parsed
if parsed[0] == 'Rational':
return 'S({})/S({})'.format(generate_sympy_from_parsed(parsed[1], wild=wild, symbols=symbols, replace_Int=replace_Int), generate_sympy_from_parsed(parsed[2], wild=wild, symbols=symbols, replace_Int=replace_Int))
if parsed[0] in replacements:
out += replacements[parsed[0]]
elif parsed[0] == 'Int' and replace_Int:
out += 'Integral'
else:
out += parsed[0]
if len(parsed) == 1:
return out
result = [generate_sympy_from_parsed(i, wild=wild, symbols=symbols, replace_Int=replace_Int) for i in parsed[1:]]
if '' in result:
result.remove('')
out += "("
out += ", ".join(result)
out += ")"
return out
def get_free_symbols(s, symbols, free_symbols=None):
'''
Returns free_symbols present in `s`.
'''
free_symbols = free_symbols or []
if not isinstance(s, list):
if s in symbols:
free_symbols.append(s)
return free_symbols
for i in s:
free_symbols = get_free_symbols(i, symbols, free_symbols)
return free_symbols
def set_matchq_in_constraint(a, cons_index):
'''
Takes care of the case, when a pattern matching has to be done inside a constraint.
'''
lst = []
res = ''
if isinstance(a, list):
if a[0] == 'MatchQ':
s = a
optional = get_default_values(s, {})
r = generate_sympy_from_parsed(s, replace_Int=True)
r, free_symbols = add_wildcards(r, optional=optional)
free_symbols = list(set(free_symbols)) #remove common symbols
r = sympify(r, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")})
pattern = r.args[1].args[0]
cons = r.args[1].args[1]
pattern = rubi_printer(pattern, sympy_integers=True)
pattern = setWC(pattern)
res = ' def _cons_f_{}({}):\n return {}\n'.format(cons_index, ', '.join(free_symbols), cons)
res += ' _cons_{} = CustomConstraint(_cons_f_{})\n'.format(cons_index, cons_index)
res += ' pat = Pattern(UtilityOperator({}, x), _cons_{})\n'.format(pattern, cons_index)
res += ' result_matchq = is_match(UtilityOperator({}, x), pat)'.format(r.args[0])
return "result_matchq", res
else:
for i in a:
if isinstance(i, list):
r = set_matchq_in_constraint(i, cons_index)
lst.append(r[0])
res = r[1]
else:
lst.append(i)
return (lst, res)
def _divide_constriant(s, symbols, cons_index, cons_dict, cons_import):
# Creates a CustomConstraint of the form `CustomConstraint(lambda a, x: FreeQ(a, x))`
lambda_symbols = list(set(get_free_symbols(s, symbols, [])))
r = generate_sympy_from_parsed(s)
r = sympify(r, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")})
if r.has(Function('MatchQ')):
match_res = set_matchq_in_constraint(s, cons_index)
res = match_res[1]
res += '\n return {}'.format(rubi_printer(sympify(generate_sympy_from_parsed(match_res[0]), locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")}), sympy_integers = True))
elif contains_diff_return_type(s):
res = ' try:\n return {}\n except (TypeError, AttributeError):\n return False'.format(rubi_printer(r, sympy_integers=True))
else:
res = ' return {}'.format(rubi_printer(r, sympy_integers=True))
# First it checks if a constraint is already present in `cons_dict`, If yes, use it else create a new one.
if not res in cons_dict.values():
cons_index += 1
cons = '\n def cons_f{}({}):\n'.format(cons_index, ', '.join(lambda_symbols))
if 'x' in lambda_symbols:
cons += ' if isinstance(x, (int, Integer, float, Float)):\n return False\n'
cons += res
cons += '\n\n cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
cons_name = 'cons{}'.format(cons_index)
cons_dict[cons_name] = res
else:
cons = ''
cons_name = next(key for key, value in cons_dict.items() if value == res)
if cons_name not in cons_import:
cons_import.append(cons_name)
return (cons_name, cons, cons_index)
def divide_constraint(s, symbols, cons_index, cons_dict, cons_import):
'''
Divides multiple constraints into smaller constraints.
Parameters
==========
s : constraint as list
symbols : all the symbols present in the expression
'''
result =[]
cons = ''
if s[0] == 'And':
for i in s[1:]:
if i[0]!= 'FreeQ':
a = _divide_constriant(i, symbols, cons_index, cons_dict, cons_import)
result.append(a[0])
cons += a[1]
cons_index = a[2]
else:
a = _divide_constriant(s, symbols, cons_index, cons_dict, cons_import)
result.append(a[0])
cons += a[1]
cons_index = a[2]
r = ['']
for i in result:
if i != '':
r.append(i)
return ', '.join(r),cons, cons_index
def setWC(string):
'''
Replaces `WC(a, b)` by `WC('a', S(b))`
'''
p = r'(WC\((\w+), S\(([-+]?\d)\)\))'
matches = re.findall(p, string)
for i in matches:
string = string.replace(i[0], "WC('{}', S({}))".format(i[1], i[2]))
return string
def process_return_type(a1, L):
'''
Functions like `Set`, `With` and `CompoundExpression` has to be taken special care.
'''
a = sympify(a1[1])
x =''
processed = False
return_value = ''
if type(a) == Function('With') or type(a) == Function('Module'):
for i in a.args:
for s in i.args:
if isinstance(s, Set) and not s in L:
x += '\n {} = {}'.format(s.args[0], rubi_printer(s.args[1], sympy_integers=True))
if not type(i) in (Function('List'), Function('CompoundExpression')) and not i.has(Function('CompoundExpression')):
return_value = i
processed = True
elif type(i) ==Function('CompoundExpression'):
return_value = i.args[-1]
processed = True
elif type(i.args[0]) == Function('CompoundExpression'):
C = i.args[0]
return_value = '{}({}, {})'.format(i.func, C.args[-1], i.args[1])
processed = True
return x, return_value, processed
def extract_set(s, L):
'''
this function extracts all `Set` functions
'''
lst = []
if isinstance(s, Set) and not s in L:
lst.append(s)
else:
try:
for i in s.args:
lst += extract_set(i, L)
except: #when s has no attribute args (like `bool`)
pass
return lst
def replaceWith(s, symbols, index):
'''
Replaces `With` and `Module by python functions`
'''
return_type = None
with_value = ''
if type(s) == Function('With') or type(s) == Function('Module'):
constraints = ' '
result = ' def With{}({}):'.format(index, ', '.join(symbols))
if type(s.args[0]) == Function('List'): # get all local variables of With and Module
L = list(s.args[0].args)
else:
L = [s.args[0]]
lst = []
for i in s.args[1:]:
lst+=extract_set(i, L)
L+=lst
for i in L: # define local variables
if isinstance(i, Set):
with_value += '\n {} = {}'.format(i.args[0], rubi_printer(i.args[1], sympy_integers=True))
elif isinstance(i, Symbol):
with_value += "\n {} = Symbol('{}')".format(i, i)
#result += with_value
if type(s.args[1]) == Function('CompoundExpression'): # Expand CompoundExpression
C = s.args[1]
result += with_value
if isinstance(C.args[0], Set):
result += '\n {} = {}'.format(C.args[0].args[0], C.args[0].args[1])
result += '\n rubi.append({})\n return {}'.format(index, rubi_printer(C.args[1], sympy_integers=True))
return result, constraints, return_type
elif type(s.args[1]) == Function('Condition'):
C = s.args[1]
if len(C.args) == 2:
if all(j in symbols for j in [str(i) for i in C.free_symbols]):
result += with_value
#constraints += 'CustomConstraint(lambda {}: {})'.format(', '.join([str(i) for i in C.free_symbols]), sstr(C.args[1], sympy_integers=True))
result += '\n rubi.append({})\n return {}'.format(index, rubi_printer(C.args[0], sympy_integers=True))
else:
if 'x' in symbols:
result += '\n if isinstance(x, (int, Integer, float, Float)):\n return False'
if contains_diff_return_type(s):
n_with_value = with_value.replace('\n', '\n ')
result += '\n try:{}\n res = {}'.format(n_with_value, rubi_printer(C.args[1], sympy_integers=True))
result += '\n except (TypeError, AttributeError):\n return False'
result += '\n if res:'
else:
result+=with_value
result += '\n if {}:'.format(rubi_printer(C.args[1], sympy_integers=True))
return_type = (with_value, rubi_printer(C.args[0], sympy_integers=True))
return_type1 = process_return_type(return_type, L)
if return_type1[2]:
return_type = ( with_value+return_type1[0], rubi_printer(return_type1[1]))
result += '\n return True'
result += '\n return False'
constraints = ', CustomConstraint(With{})'.format(index)
return result, constraints, return_type
elif type(s.args[1]) == Function('Module') or type(s.args[1]) == Function('With'):
C = s.args[1]
result += with_value
return_type = (with_value, rubi_printer(C, sympy_integers=True))
return_type1 = process_return_type(return_type, L)
if return_type1[2]:
return_type = ( with_value+return_type1[0], rubi_printer(return_type1[1]))
result+=return_type1[0]
result+='\n rubi.append({})\n return {}'.format(index, rubi_printer(return_type1[1]))
return result, constraints, None
elif s.args[1].has(Function("CompoundExpression")):
C = s.args[1].args[0]
result += with_value
if isinstance(C.args[0], Set):
result += '\n {} = {}'.format(C.args[0].args[0], C.args[0].args[1])
result += '\n return {}({}, {})'.format(s.args[1].func, C.args[-1], s.args[1].args[1])
return result, constraints, None
result += with_value
result += '\n rubi.append({})\n return {}'.format(index, rubi_printer(s.args[1], sympy_integers=True))
return result, constraints, return_type
else:
return rubi_printer(s, sympy_integers=True), '', return_type
def downvalues_rules(r, header, cons_dict, cons_index, index):
'''
Function which generates parsed rules by substituting all possible
combinations of default values.
'''
rules = '['
parsed = '\n\n'
cons = ''
cons_import = [] # it contains name of constraints that need to be imported for rules.
for i in r:
debug('parsing rule {}'.format(r.index(i) + 1))
# Parse Pattern
if i[1][1][0] == 'Condition':
p = i[1][1][1].copy()
else:
p = i[1][1].copy()
optional = get_default_values(p, {})
pattern = generate_sympy_from_parsed(p.copy(), replace_Int=True)
pattern, free_symbols = add_wildcards(pattern, optional=optional)
free_symbols = list(set(free_symbols)) #remove common symbols
# Parse Transformed Expression and Constraints
if i[2][0] == 'Condition': # parse rules without constraints separately
constriant, constraint_def, cons_index = divide_constraint(i[2][2], free_symbols, cons_index, cons_dict, cons_import) # separate And constraints into individual constraints
FreeQ_vars, FreeQ_x = seperate_freeq(i[2][2].copy()) # separate FreeQ into individual constraints
transformed = generate_sympy_from_parsed(i[2][1].copy(), symbols=free_symbols)
else:
constriant = ''
constraint_def = ''
FreeQ_vars, FreeQ_x = [], []
transformed = generate_sympy_from_parsed(i[2].copy(), symbols=free_symbols)
FreeQ_constraint, free_cons_def, cons_index = parse_freeq(FreeQ_vars, FreeQ_x, cons_index, cons_dict, cons_import, free_symbols)
pattern = sympify(pattern, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not") })
pattern = rubi_printer(pattern, sympy_integers=True)
pattern = setWC(pattern)
transformed = sympify(transformed, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not") })
constraint_def = constraint_def + free_cons_def
cons+=constraint_def
index += 1
# below are certain if - else condition depending on various situation that may be encountered
if type(transformed) == Function('With') or type(transformed) == Function('Module'): # define separate function when With appears
transformed, With_constraints, return_type = replaceWith(transformed, free_symbols, index)
if return_type is None:
parsed += '{}'.format(transformed)
parsed += '\n pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
parsed += '\n ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', With{}'.format(index) + ')\n'
else:
parsed += '{}'.format(transformed)
parsed += '\n pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + With_constraints + ')'
parsed += '\n def replacement{}({}):\n'.format(index, ', '.join(free_symbols)) + return_type[0] + '\n rubi.append({})\n return '.format(index) + return_type[1]
parsed += '\n ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', replacement{}'.format(index) + ')\n'
else:
transformed = rubi_printer(transformed, sympy_integers=True)
parsed += ' pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
parsed += '\n def replacement{}({}):\n rubi.append({})\n return '.format(index, ', '.join(free_symbols), index) + transformed
parsed += '\n ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', replacement{}'.format(index) + ')\n'
rules += 'rule{}, '.format(index)
rules += ']'
parsed += ' return ' + rules +'\n'
header += ' from sympy.integrals.rubi.constraints import ' + ', '.join(word for word in cons_import)
parsed = header + parsed
return parsed, cons_index, cons, index
def rubi_rule_parser(fullform, header=None, module_name='rubi_object'):
'''
Parses rules in MatchPy format.
Parameters
==========
fullform : FullForm of the rule as string.
header : Header imports for the file. Uses default imports if None.
module_name : name of RUBI module
References
==========
[1] http://reference.wolfram.com/language/ref/FullForm.html
[2] http://reference.wolfram.com/language/ref/DownValues.html
[3] https://gist.github.com/Upabjojr/bc07c49262944f9c1eb0
'''
if header is None: # use default header values
path_header = os.path.dirname(os.path.abspath(inspect.getfile(inspect.currentframe())))
header = open(os.path.join(path_header, "header.py.txt"), "r").read()
header = header.format(module_name)
cons_dict = {} # dict keeps track of constraints that has been encountered, thus avoids repetition of constraints.
cons_index =0 # for index of a constraint
index = 0 # indicates the number of a rule.
cons = ''
# Temporarily rename these variables because it
# can raise errors while sympifying
for i in temporary_variable_replacement:
fullform = fullform.replace(i, temporary_variable_replacement[i])
# Permanently rename these variables
for i in permanent_variable_replacement:
fullform = fullform.replace(i, permanent_variable_replacement[i])
rules = []
for i in parse_full_form(fullform): # separate all rules
if i[0] == 'RuleDelayed':
rules.append(i)
parsed = downvalues_rules(rules, header, cons_dict, cons_index, index)
result = parsed[0].strip() + '\n'
cons_index = parsed[1]
cons += parsed[2]
index = parsed[3]
# Replace temporary variables by actual values
for i in temporary_variable_replacement:
cons = cons.replace(temporary_variable_replacement[i], i)
result = result.replace(temporary_variable_replacement[i], i)
cons = "\n".join(header.split("\n")[:-2])+ '\n' + cons
return result, cons
|
012525c5c64cf99172d9ffd2844ab0ee30a6a74747e760693af2d3704203f235 | import sys
from sympy.external import import_module
matchpy = import_module("matchpy")
if not matchpy:
#bin/test will not execute any tests now
disabled = True
if sys.version_info[:2] < (3, 6):
disabled = True
from sympy.integrals.rubi.utility_function import (Int, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, PolynomialQuotient,
EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor,
DerivativeDivides, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, PureFunctionOfCothQ, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, IntegralFreeQ, Sum_doit, rubi_exp, rubi_log, rubi_log as log,
PolynomialRemainder, CoprimeQ, Distribute, ProductLog, Floor, PolyGamma, process_trig, replace_pow_exp)
from sympy.core.symbol import symbols, S
from sympy.functions.elementary.trigonometric import atan, acsc, asin, acot, acos, asec, atan2
from sympy.functions.elementary.hyperbolic import acosh, asinh, atanh, acsch, cosh, sinh, tanh, coth, sech, csch, acoth
from sympy.functions import (sin, cos, tan, cot, sec, csc, sqrt, log as sym_log)
from sympy import (I, E, pi, hyper, Add, Wild, simplify, Symbol, exp, UnevaluatedExpr, Pow, li, Ei, expint,
Si, Ci, Shi, Chi, loggamma, zeta, zoo, gamma, polylog, oo, polygamma)
from sympy import Integral, nsimplify, Min
A, B, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B a b c d e f g h y z m n p q u v w F', real=True, imaginary=False)
x = Symbol('x')
def test_ZeroQ():
e = b*(n*p + n + 1)
d = a
assert ZeroQ(a*e - b*d*(n*(p + S(1)) + S(1)))
assert ZeroQ(S(0))
assert not ZeroQ(S(10))
assert not ZeroQ(S(-2))
assert ZeroQ(0, 2-2)
assert ZeroQ([S(2), (4), S(0), S(8)]) == [False, False, True, False]
assert ZeroQ([S(2), S(4), S(8)]) == [False, False, False]
def test_NonzeroQ():
assert NonzeroQ(S(1)) == True
def test_FreeQ():
l = [a*b, x, a + b]
assert FreeQ(l, x) == False
l = [a*b, a + b]
assert FreeQ(l, x) == True
def test_List():
assert List(a, b, c) == [a, b, c]
def test_Log():
assert Log(a) == log(a)
def test_PositiveIntegerQ():
assert PositiveIntegerQ(S(1))
assert not PositiveIntegerQ(S(-3))
assert not PositiveIntegerQ(S(0))
def test_NegativeIntegerQ():
assert not NegativeIntegerQ(S(1))
assert NegativeIntegerQ(S(-3))
assert not NegativeIntegerQ(S(0))
def test_PositiveQ():
assert PositiveQ(S(1))
assert not PositiveQ(S(-3))
assert not PositiveQ(S(0))
assert not PositiveQ(zoo)
assert not PositiveQ(I)
assert PositiveQ(b/(b*(b*c/(-a*d + b*c)) - a*(b*d/(-a*d + b*c))))
def test_IntegerQ():
assert IntegerQ(S(1))
assert not IntegerQ(S(-1.9))
assert not IntegerQ(S(0.0))
assert IntegerQ(S(-1))
def test_FracPart():
assert FracPart(S(10)) == 0
assert FracPart(S(10)+0.5) == 10.5
def test_IntPart():
assert IntPart(m*n) == 0
assert IntPart(S(10)) == 10
assert IntPart(1 + m) == 1
def test_NegQ():
assert NegQ(-S(3))
assert not NegQ(S(0))
assert not NegQ(S(0))
def test_RationalQ():
assert RationalQ(S(5)/6)
assert RationalQ(S(5)/6, S(4)/5)
assert not RationalQ(Sqrt(1.6))
assert not RationalQ(Sqrt(1.6), S(5)/6)
assert not RationalQ(log(2))
def test_ArcCosh():
assert ArcCosh(x) == acosh(x)
def test_LinearQ():
assert not LinearQ(a, x)
assert LinearQ(3*x + y**2, x)
assert not LinearQ(3*x + y**2, y)
assert not LinearQ(S(3), x)
def test_Sqrt():
assert Sqrt(x) == sqrt(x)
assert Sqrt(25) == 5
def test_Util_Coefficient():
from sympy.integrals.rubi.utility_function import Util_Coefficient
assert Util_Coefficient(a + b*x + c*x**3, x, a) == Util_Coefficient(a + b*x + c*x**3, x, a)
assert Util_Coefficient(a + b*x + c*x**3, x, 4).doit() == 0
def test_Coefficient():
assert Coefficient(7 + 2*x + 4*x**3, x, 1) == 2
assert Coefficient(a + b*x + c*x**3, x, 0) == a
assert Coefficient(a + b*x + c*x**3, x, 4) == 0
assert Coefficient(b*x + c*x**3, x, 3) == c
assert Coefficient(x, x, -1) == 0
def test_Denominator():
assert Denominator((-S(1)/S(2) + I/3)) == 6
assert Denominator((-a/b)**3) == (b)**(3)
assert Denominator(S(3)/2) == 2
assert Denominator(x/y) == y
assert Denominator(S(4)/5) == 5
def test_Hypergeometric2F1():
assert Hypergeometric2F1(1, 2, 3, x) == hyper((1, 2), (3,), x)
def test_ArcTan():
assert ArcTan(x) == atan(x)
assert ArcTan(x, y) == atan2(x, y)
def test_Not():
a = 10
assert Not(a == 2)
def test_FractionalPart():
assert FractionalPart(S(3.0)) == 0.0
def test_IntegerPart():
assert IntegerPart(3.6) == 3
assert IntegerPart(-3.6) == -4
def test_AppellF1():
assert AppellF1(1,0,0.5,1,0.5,0.25).evalf() == 1.154700538379251529018298
assert AppellF1(a, b, c, d, e, f) == AppellF1(a, b, c, d, e, f)
def test_Simplify():
assert Simplify(sin(x)**2 + cos(x)**2) == 1
assert Simplify((x**3 + x**2 - x - 1)/(x**2 + 2*x + 1)) == x - 1
def test_ArcTanh():
assert ArcTanh(a) == atanh(a)
def test_ArcSin():
assert ArcSin(a) == asin(a)
def test_ArcSinh():
assert ArcSinh(a) == asinh(a)
def test_ArcCos():
assert ArcCos(a) == acos(a)
def test_ArcCsc():
assert ArcCsc(a) == acsc(a)
def test_ArcCsch():
assert ArcCsch(a) == acsch(a)
def test_Equal():
assert Equal(a, a)
assert not Equal(a, b)
def test_LessEqual():
assert LessEqual(1, 2, 3)
assert LessEqual(1, 1)
assert not LessEqual(3, 2, 1)
def test_With():
assert With(Set(x, 3), x + y) == 3 + y
assert With(List(Set(x, 3), Set(y, c)), x + y) == 3 + c
def test_Less():
assert Less(1, 2, 3)
assert not Less(1, 1, 3)
def test_Greater():
assert Greater(3, 2, 1)
assert not Greater(3, 2, 2)
def test_GreaterEqual():
assert GreaterEqual(3, 2, 1)
assert GreaterEqual(3, 2, 2)
assert not GreaterEqual(2, 3)
def test_Unequal():
assert Unequal(1, 2)
assert not Unequal(1, 1)
def test_FractionQ():
assert not FractionQ(S('3'))
assert FractionQ(S('3')/S('2'))
def test_Expand():
assert Expand((1 + x)**10) == x**10 + 10*x**9 + 45*x**8 + 120*x**7 + 210*x**6 + 252*x**5 + 210*x**4 + 120*x**3 + 45*x**2 + 10*x + 1
def test_Scan():
assert list(Scan(sin, [a, b])) == [sin(a), sin(b)]
def test_MapAnd():
assert MapAnd(PositiveQ, [S(1), S(2), S(3), S(0)]) == False
assert MapAnd(PositiveQ, [S(1), S(2), S(3)]) == True
def test_FalseQ():
assert FalseQ(True) == False
assert FalseQ(False) == True
def test_ComplexNumberQ():
assert ComplexNumberQ(1 + I*2, I) == True
assert ComplexNumberQ(a + b, I) == False
def test_Re():
assert Re(1 + I) == 1
def test_Im():
assert Im(1 + 2*I) == 2
assert Im(a*I) == a
def test_PositiveOrZeroQ():
assert PositiveOrZeroQ(S(0)) == True
assert PositiveOrZeroQ(S(1)) == True
assert PositiveOrZeroQ(-S(1)) == False
def test_RealNumericQ():
assert RealNumericQ(S(1)) == True
assert RealNumericQ(-S(1)) == True
def test_NegativeOrZeroQ():
assert NegativeOrZeroQ(S(0)) == True
assert NegativeOrZeroQ(-S(1)) == True
assert NegativeOrZeroQ(S(1)) == False
def test_FractionOrNegativeQ():
assert FractionOrNegativeQ(S(1)/2) == True
assert FractionOrNegativeQ(-S(1)) == True
def test_ProductQ():
assert ProductQ(a*b) == True
assert ProductQ(a + b) == False
def test_SumQ():
assert SumQ(a*b) == False
assert SumQ(a + b) == True
def test_NonsumQ():
assert NonsumQ(a*b) == True
assert NonsumQ(a + b) == False
def test_SqrtNumberQ():
assert SqrtNumberQ(sqrt(2)) == True
def test_IntLinearcQ():
assert IntLinearcQ(1, 2, 3, 4, 5, 6, x) == True
assert IntLinearcQ(S(1)/100, S(2)/100, S(3)/100, S(4)/100, S(5)/100, S(6)/100, x) == False
def test_IndependentQ():
assert IndependentQ(a + b*x, x) == False
assert IndependentQ(a + b, x) == True
def test_PowerQ():
assert PowerQ(a**b) == True
assert PowerQ(a + b) == False
def test_IntegerPowerQ():
assert IntegerPowerQ(a**2) == True
assert IntegerPowerQ(a**0.5) == False
def test_PositiveIntegerPowerQ():
assert PositiveIntegerPowerQ(a**3) == True
assert PositiveIntegerPowerQ(a**(-2)) == False
def test_FractionalPowerQ():
assert FractionalPowerQ(a**(S(2)/S(3)))
assert FractionalPowerQ(a**sqrt(2)) == False
def test_AtomQ():
assert AtomQ(x)
assert not AtomQ(x+1)
assert not AtomQ([a, b])
def test_ExpQ():
assert ExpQ(E**2)
assert not ExpQ(2**E)
def test_LogQ():
assert LogQ(log(x))
assert not LogQ(sin(x) + log(x))
def test_Head():
assert Head(sin(x)) == sin
assert Head(log(x**3 + 3)) in (sym_log, log)
def test_MemberQ():
assert MemberQ([a, b, c], b)
assert MemberQ([sin, cos, log, tan], Head(sin(x)))
assert MemberQ([[sin, cos], [tan, cot]], [sin, cos])
assert not MemberQ([[sin, cos], [tan, cot]], [sin, tan])
def test_TrigQ():
assert TrigQ(sin(x))
assert TrigQ(tan(x**2 + 2))
assert not TrigQ(sin(x) + tan(x))
def test_SinQ():
assert SinQ(sin(x))
assert not SinQ(tan(x))
def test_CosQ():
assert CosQ(cos(x))
assert not CosQ(csc(x))
def test_TanQ():
assert TanQ(tan(x))
assert not TanQ(cot(x))
def test_CotQ():
assert not CotQ(tan(x))
assert CotQ(cot(x))
def test_SecQ():
assert SecQ(sec(x))
assert not SecQ(csc(x))
def test_CscQ():
assert not CscQ(sec(x))
assert CscQ(csc(x))
def test_HyperbolicQ():
assert HyperbolicQ(sinh(x))
assert HyperbolicQ(cosh(x))
assert HyperbolicQ(tanh(x))
assert not HyperbolicQ(sinh(x) + cosh(x) + tanh(x))
def test_SinhQ():
assert SinhQ(sinh(x))
assert not SinhQ(cosh(x))
def test_CoshQ():
assert not CoshQ(sinh(x))
assert CoshQ(cosh(x))
def test_TanhQ():
assert TanhQ(tanh(x))
assert not TanhQ(coth(x))
def test_CothQ():
assert not CothQ(tanh(x))
assert CothQ(coth(x))
def test_SechQ():
assert SechQ(sech(x))
assert not SechQ(csch(x))
def test_CschQ():
assert not CschQ(sech(x))
assert CschQ(csch(x))
def test_InverseTrigQ():
assert InverseTrigQ(acot(x))
assert InverseTrigQ(asec(x))
assert not InverseTrigQ(acsc(x) + asec(x))
def test_SinCosQ():
assert SinCosQ(sin(x))
assert SinCosQ(cos(x))
assert SinCosQ(sec(x))
assert not SinCosQ(acsc(x))
def test_SinhCoshQ():
assert not SinhCoshQ(sin(x))
assert SinhCoshQ(cosh(x))
assert SinhCoshQ(sech(x))
assert SinhCoshQ(csch(x))
def test_LeafCount():
assert LeafCount(1 + a + x**2) == 6
def test_Numerator():
assert Numerator((-S(1)/S(2) + I/3)) == -3 + 2*I
assert Numerator((-a/b)**3) == (-a)**(3)
assert Numerator(S(3)/2) == 3
assert Numerator(x/y) == x
def test_Length():
assert Length(a + b) == 2
assert Length(sin(a)*cos(a)) == 2
def test_ListQ():
assert ListQ([1, 2])
assert not ListQ(a)
def test_InverseHyperbolicQ():
assert InverseHyperbolicQ(acosh(a))
def test_InverseFunctionQ():
assert InverseFunctionQ(log(a))
assert InverseFunctionQ(acos(a))
assert not InverseFunctionQ(a)
assert InverseFunctionQ(acosh(a))
assert InverseFunctionQ(polylog(a, b))
def test_EqQ():
assert EqQ(a, a)
assert not EqQ(a, b)
def test_FactorSquareFree():
assert FactorSquareFree(x**5 - x**3 - x**2 + 1) == (x**3 + 2*x**2 + 2*x + 1)*(x - 1)**2
def test_FactorSquareFreeList():
assert FactorSquareFreeList(x**5-x**3-x**2 + 1) == [[1, 1], [x**3 + 2*x**2 + 2*x + 1, 1], [x - 1, 2]]
assert FactorSquareFreeList(x**4 - 2*x**2 + 1) == [[1, 1], [x**2 - 1, 2]]
def test_PerfectPowerTest():
assert not PerfectPowerTest(sqrt(x), x)
assert not PerfectPowerTest(x**5-x**3-x**2 + 1, x)
assert PerfectPowerTest(x**4 - 2*x**2 + 1, x) == (x**2 - 1)**2
def test_SquareFreeFactorTest():
assert not SquareFreeFactorTest(sqrt(x), x)
assert SquareFreeFactorTest(x**5 - x**3 - x**2 + 1, x) == (x**3 + 2*x**2 + 2*x + 1)*(x - 1)**2
def test_Rest():
assert Rest([2, 3, 5, 7]) == [3, 5, 7]
assert Rest(a + b + c) == b + c
assert Rest(a*b*c) == b*c
assert Rest(1/b) == -1
def test_First():
assert First([2, 3, 5, 7]) == 2
assert First(y**S(2)) == y
assert First(a + b + c) == a
assert First(a*b*c) == a
def test_ComplexFreeQ():
assert ComplexFreeQ(a)
assert not ComplexFreeQ(a + 2*I)
def test_FractionalPowerFreeQ():
assert not FractionalPowerFreeQ(x**(S(2)/3))
assert FractionalPowerFreeQ(x)
def test_Exponent():
assert Exponent(x**2 + x + 1 + 5, x, Min) == 0
assert Exponent(x**2 + x + 1 + 5, x, List) == [0, 1, 2]
assert Exponent(x**2 + x + 1, x, List) == [0, 1, 2]
assert Exponent(x**2 + 2*x + 1, x, List) == [0, 1, 2]
assert Exponent(x**3 + x + 1, x) == 3
assert Exponent(x**2 + 2*x + 1, x) == 2
assert Exponent(x**3, x, List) == [3]
assert Exponent(S(1), x) == 0
assert Exponent(x**(-3), x) == 0
def test_Expon():
assert Expon(x**2+2*x+1, x) == 2
assert Expon(x**3, x, List) == [3]
def test_QuadraticQ():
assert not QuadraticQ([x**2+x+1, 5*x**2], x)
assert QuadraticQ([x**2+x+1, 5*x**2+3*x+6], x)
assert not QuadraticQ(x**2+1+x**3, x)
assert QuadraticQ(x**2+1+x, x)
assert not QuadraticQ(x**2, x)
def test_BinomialQ():
assert BinomialQ(x**9, x)
assert not BinomialQ((1 + x)**3, x)
def test_BinomialParts():
assert BinomialParts(2 + x*(9*x), x) == [2, 9, 2]
assert BinomialParts(x**9, x) == [0, 1, 9]
assert BinomialParts(2*x**3, x) == [0, 2, 3]
assert BinomialParts(2 + x, x) == [2, 1, 1]
def test_BinomialDegree():
assert BinomialDegree(b + 2*c*x**n, x) == n
assert BinomialDegree(2 + x*(9*x), x) == 2
assert BinomialDegree(x**9, x) == 9
def test_PolynomialQ():
assert not PolynomialQ(x*(-1 + x**2), (1 + x)**(S(1)/2))
assert not PolynomialQ((16*x + 1)/((x + 5)**2*(x**2 + x + 1)), 2*x)
C = Symbol('C')
assert not PolynomialQ(A + b*x + c*x**2, x**2)
assert PolynomialQ(A + B*x + C*x**2)
assert PolynomialQ(A + B*x**4 + C*x**2, x**2)
assert PolynomialQ(x**3, x)
assert not PolynomialQ(sqrt(x), x)
def test_PolyQ():
assert PolyQ(-2*a*d**3*e**2 + x**6*(a*e**5 - b*d*e**4 + c*d**2*e**3)\
+ x**4*(-2*a*d*e**4 + 2*b*d**2*e**3 - 2*c*d**3*e**2) + x**2*(2*a*d**2*e**3 - 2*b*d**3*e**2), x)
assert not PolyQ(1/sqrt(a + b*x**2 - c*x**4), x**2)
assert PolyQ(x, x, 1)
assert PolyQ(x**2, x, 2)
assert not PolyQ(x**3, x, 2)
def test_EvenQ():
assert EvenQ(S(2))
assert not EvenQ(S(1))
def test_OddQ():
assert OddQ(S(1))
assert not OddQ(S(2))
def test_PerfectSquareQ():
assert PerfectSquareQ(S(4))
assert PerfectSquareQ(a**S(2)*b**S(4))
assert not PerfectSquareQ(S(1)/3)
def test_NiceSqrtQ():
assert NiceSqrtQ(S(1)/3)
assert not NiceSqrtQ(-S(1))
assert NiceSqrtQ(pi**2)
assert NiceSqrtQ(pi**2*sin(4)**4)
assert not NiceSqrtQ(pi**2*sin(4)**3)
def test_Together():
assert Together(1/a + b/2) == (a*b + 2)/(2*a)
def test_PosQ():
#assert not PosQ((b*e - c*d)/(c*e))
assert not PosQ(S(0))
assert PosQ(S(1))
assert PosQ(pi)
assert PosQ(pi**3)
assert PosQ((-pi)**4)
assert PosQ(sin(1)**2*pi**4)
def test_NumericQ():
assert NumericQ(sin(cos(2)))
def test_NumberQ():
assert NumberQ(pi)
def test_CoefficientList():
assert CoefficientList(1 + a*x, x) == [1, a]
assert CoefficientList(1 + a*x**3, x) == [1, 0, 0, a]
assert CoefficientList(sqrt(x), x) == []
def test_ReplaceAll():
assert ReplaceAll(x, {x: a}) == a
assert ReplaceAll(a*x, {x: a + b}) == a*(a + b)
assert ReplaceAll(a*x, {a: b, x: a + b}) == b*(a + b)
def test_ExpandLinearProduct():
assert ExpandLinearProduct(log(x), x**2, a, b, x) == a**2*log(x)/b**2 - 2*a*(a + b*x)*log(x)/b**2 + (a + b*x)**2*log(x)/b**2
assert ExpandLinearProduct((a + b*x)**n, x**3, a, b, x) == -a**3*(a + b*x)**n/b**3 + 3*a**2*(a + b*x)**(n + 1)/b**3 - 3*a*(a + b*x)**(n + 2)/b**3 + (a + b*x)**(n + 3)/b**3
def test_PolynomialDivide():
assert PolynomialDivide((a*c - b*c*x)**2, (a + b*x)**2, x) == -4*a*b*c**2*x/(a + b*x)**2 + c**2
assert PolynomialDivide(x + x**2, x, x) == x + 1
assert PolynomialDivide((1 + x)**3, (1 + x)**2, x) == x + 1
assert PolynomialDivide((a + b*x)**3, x**3, x) == a*(a**2 + 3*a*b*x + 3*b**2*x**2)/x**3 + b**3
assert PolynomialDivide(x**3*(a + b*x), S(1), x) == b*x**4 + a*x**3
assert PolynomialDivide(x**6, (a + b*x)**2, x) == -a**5*(5*a + 6*b*x)/(b**6*(a + b*x)**2) + 5*a**4/b**6 - 4*a**3*x/b**5 + 3*a**2*x**2/b**4 - 2*a*x**3/b**3 + x**4/b**2
def test_MatchQ():
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x])
c_ = Wild('c', exclude=[x])
assert MatchQ(a*b + c, a_*b_ + c_, a_, b_, c_) == (a, b, c)
def test_PolynomialQuotientRemainder():
assert PolynomialQuotientRemainder(x**2, x+a, x) == [-a + x, a**2]
def test_FreeFactors():
assert FreeFactors(a, x) == a
assert FreeFactors(x + a, x) == 1
assert FreeFactors(a*b*x, x) == a*b
def test_NonfreeFactors():
assert NonfreeFactors(a, x) == 1
assert NonfreeFactors(x + a, x) == x + a
assert NonfreeFactors(a*b*x, x) == x
def test_FreeTerms():
assert FreeTerms(a, x) == a
assert FreeTerms(x*a, x) == 0
assert FreeTerms(a*x + b, x) == b
def test_NonfreeTerms():
assert NonfreeTerms(a, x) == 0
assert NonfreeTerms(a*x, x) == a*x
assert NonfreeTerms(a*x + b, x) == a*x
def test_RemoveContent():
assert RemoveContent(a + b*x, x) == a + b*x
def test_ExpandAlgebraicFunction():
assert ExpandAlgebraicFunction((a + b)*x, x) == a*x + b*x
assert ExpandAlgebraicFunction((a + b)**2*x, x)== a**2*x + 2*a*b*x + b**2*x
assert ExpandAlgebraicFunction((a + b)**2*x**2, x) == a**2*x**2 + 2*a*b*x**2 + b**2*x**2
def test_CollectReciprocals():
assert CollectReciprocals(-1/(1 + 1*x) - 1/(1 - 1*x), x) == -2/(-x**2 + 1)
assert CollectReciprocals(1/(1 + 1*x) - 1/(1 - 1*x), x) == -2*x/(-x**2 + 1)
def test_ExpandCleanup():
assert ExpandCleanup(a + b, x) == a*(1 + b/a)
assert ExpandCleanup(b**2/(a**2*(a + b*x)**2) + 1/(a**2*x**2) + 2*b**2/(a**3*(a + b*x)) - 2*b/(a**3*x), x) == b**2/(a**2*(a + b*x)**2) + 1/(a**2*x**2) + 2*b**2/(a**3*(a + b*x)) - 2*b/(a**3*x)
def test_AlgebraicFunctionQ():
assert not AlgebraicFunctionQ(1/(a + c*x**(2*n)), x)
assert AlgebraicFunctionQ(a, x) == True
assert AlgebraicFunctionQ(a*b, x) == True
assert AlgebraicFunctionQ(x**2, x) == True
assert AlgebraicFunctionQ(x**2*a, x) == True
assert AlgebraicFunctionQ(x**2 + a, x) == True
assert AlgebraicFunctionQ(sin(x), x) == False
assert AlgebraicFunctionQ([], x) == True
assert AlgebraicFunctionQ([a, a*b], x) == True
assert AlgebraicFunctionQ([sin(x)], x) == False
def test_MonomialQ():
assert not MonomialQ(2*x**7 + 6, x)
assert MonomialQ(2*x**7, x)
assert not MonomialQ(2*x**7 + 5*x**3, x)
assert not MonomialQ([2*x**7 + 6, 2*x**7], x)
assert MonomialQ([2*x**7, 5*x**3], x)
def test_MonomialSumQ():
assert MonomialSumQ(2*x**7 + 6, x) == True
assert MonomialSumQ(x**2 + x**3 + 5*x, x) == True
def test_MinimumMonomialExponent():
assert MinimumMonomialExponent(x**2 + 5*x**2 + 3*x**5, x) == 2
assert MinimumMonomialExponent(x**2 + 5*x**2 + 1, x) == 0
def test_MonomialExponent():
assert MonomialExponent(3*x**7, x) == 7
assert not MonomialExponent(3+x**3, x)
def test_LinearMatchQ():
assert LinearMatchQ(2 + 3*x, x)
assert LinearMatchQ(3*x, x)
assert not LinearMatchQ(3*x**2, x)
def test_SimplerQ():
a1, b1 = symbols('a1 b1')
assert SimplerQ(a1, b1)
assert SimplerQ(2*a, a + 2)
assert SimplerQ(2, x)
assert not SimplerQ(x**2, x)
assert SimplerQ(2*x, x + 2 + 6*x**3)
def test_GeneralizedTrinomialParts():
assert not GeneralizedTrinomialParts((7 + 2*x**6 + 3*x**12), x)
assert GeneralizedTrinomialParts(x**2 + x**3 + x**4, x) == [1, 1, 1, 3, 2]
assert not GeneralizedTrinomialParts(2*x + 3*x + 4*x, x)
def test_TrinomialQ():
assert TrinomialQ((7 + 2*x**6 + 3*x**12), x)
assert not TrinomialQ(x**2, x)
def test_GeneralizedTrinomialDegree():
assert not GeneralizedTrinomialDegree((7 + 2*x**6 + 3*x**12), x)
assert GeneralizedTrinomialDegree(x**2 + x**3 + x**4, x) == 1
def test_GeneralizedBinomialParts():
assert GeneralizedBinomialParts(3*x*(3 + x**6), x) == [9, 3, 7, 1]
assert GeneralizedBinomialParts((3*x + x**7), x) == [3, 1, 7, 1]
def test_GeneralizedBinomialDegree():
assert GeneralizedBinomialDegree(3*x*(3 + x**6), x) == 6
assert GeneralizedBinomialDegree((3*x + x**7), x) == 6
def test_PowerOfLinearQ():
assert PowerOfLinearQ((6*x), x)
assert not PowerOfLinearQ((3 + 6*x**3), x)
assert PowerOfLinearQ((3 + 6*x)**3, x)
def test_LinearPairQ():
assert not LinearPairQ(6*x**2 + 4, 3*x**2 + 2, x)
assert LinearPairQ(6*x + 4, 3*x + 2, x)
assert not LinearPairQ(6*x, 3*x + 2, x)
assert LinearPairQ(6*x, 3*x, x)
def test_LeadTerm():
assert LeadTerm(a*b*c) == a*b*c
assert LeadTerm(a + b + c) == a
def test_RemainingTerms():
assert RemainingTerms(a*b*c) == a*b*c
assert RemainingTerms(a + b + c) == b + c
def test_LeadFactor():
assert LeadFactor(a*b*c) == a
assert LeadFactor(a + b + c) == a + b + c
assert LeadFactor(b*I) == I
assert LeadFactor(c*a**b) == a**b
assert LeadFactor(S(2)) == S(2)
def test_RemainingFactors():
assert RemainingFactors(a*b*c) == b*c
assert RemainingFactors(a + b + c) == 1
assert RemainingFactors(a*I) == a
def test_LeadBase():
assert LeadBase(a**b) == a
assert LeadBase(a**b*c) == a
def test_LeadDegree():
assert LeadDegree(a**b) == b
assert LeadDegree(a**b*c) == b
def test_Numer():
assert Numer(a/b) == a
assert Numer(a**(-2)) == 1
assert Numer(a**(-2)*a/b) == 1
def test_Denom():
assert Denom(a/b) == b
assert Denom(a**(-2)) == a**2
assert Denom(a**(-2)*a/b) == a*b
def test_Coeff():
assert Coeff(7 + 2*x + 4*x**3, x, 1) == 2
assert Coeff(a + b*x + c*x**3, x, 0) == a
assert Coeff(a + b*x + c*x**3, x, 4) == 0
assert Coeff(b*x + c*x**3, x, 3) == c
def test_MergeMonomials():
assert MergeMonomials(x**2*(1 + 1*x)**3*(1 + 1*x)**n, x) == x**2*(x + 1)**(n + 3)
assert MergeMonomials(x**2*(1 + 1*x)**2*(1*(1 + 1*x)**1)**2, x) == x**2*(x + 1)**4
assert MergeMonomials(b**2/a**3, x) == b**2/a**3
def test_RationalFunctionQ():
assert RationalFunctionQ(a, x)
assert RationalFunctionQ(x**2, x)
assert RationalFunctionQ(x**3 + x**4, x)
assert RationalFunctionQ(x**3*S(2), x)
assert not RationalFunctionQ(x**3 + x**(0.5), x)
assert not RationalFunctionQ(x**(S(2)/3)*(a + b*x)**2, x)
def test_Apart():
assert Apart(1/(x**2*(a + b*x)**2), x) == b**2/(a**2*(a + b*x)**2) + 1/(a**2*x**2) + 2*b**2/(a**3*(a + b*x)) - 2*b/(a**3*x)
assert Apart(x**(S(2)/3)*(a + b*x)**2, x) == x**(S(2)/3)*(a + b*x)**2
def test_RationalFunctionFactors():
assert RationalFunctionFactors(a, x) == a
assert RationalFunctionFactors(sqrt(x), x) == 1
assert RationalFunctionFactors(x*x**3, x) == x*x**3
assert RationalFunctionFactors(x*sqrt(x), x) == 1
def test_NonrationalFunctionFactors():
assert NonrationalFunctionFactors(x, x) == 1
assert NonrationalFunctionFactors(sqrt(x), x) == sqrt(x)
assert NonrationalFunctionFactors(sqrt(x)*log(x), x) == sqrt(x)*log(x)
def test_Reverse():
assert Reverse([1, 2, 3]) == [3, 2, 1]
assert Reverse(a**b) == b**a
def test_RationalFunctionExponents():
assert RationalFunctionExponents(sqrt(x), x) == [0, 0]
assert RationalFunctionExponents(a, x) == [0, 0]
assert RationalFunctionExponents(x, x) == [1, 0]
assert RationalFunctionExponents(x**(-1), x)== [0, 1]
assert RationalFunctionExponents(x**(-1)*a, x) == [0, 1]
assert RationalFunctionExponents(x**(-1) + a, x) == [1, 1]
def test_PolynomialGCD():
assert PolynomialGCD(x**2 - 1, x**2 - 3*x + 2) == x - 1
def test_PolyGCD():
assert PolyGCD(x**2 - 1, x**2 - 3*x + 2, x) == x - 1
def test_AlgebraicFunctionFactors():
assert AlgebraicFunctionFactors(sin(x)*x, x) == x
assert AlgebraicFunctionFactors(sin(x), x) == 1
assert AlgebraicFunctionFactors(x, x) == x
def test_NonalgebraicFunctionFactors():
assert NonalgebraicFunctionFactors(sin(x)*x, x) == sin(x)
assert NonalgebraicFunctionFactors(sin(x), x) == sin(x)
assert NonalgebraicFunctionFactors(x, x) == 1
def test_QuotientOfLinearsP():
assert QuotientOfLinearsP((a + b*x)/(x), x)
assert QuotientOfLinearsP(x*a, x)
assert not QuotientOfLinearsP(x**2*a, x)
assert not QuotientOfLinearsP(x**2 + a, x)
assert QuotientOfLinearsP(x + a, x)
assert QuotientOfLinearsP(x, x)
assert QuotientOfLinearsP(1 + x, x)
def test_QuotientOfLinearsParts():
assert QuotientOfLinearsParts((b*x)/(c), x) == [0, b/c, 1, 0]
assert QuotientOfLinearsParts((b*x)/(c + x), x) == [0, b, c, 1]
assert QuotientOfLinearsParts((b*x)/(c + d*x), x) == [0, b, c, d]
assert QuotientOfLinearsParts((a + b*x)/(c + d*x), x) == [a, b, c, d]
assert QuotientOfLinearsParts(x**2 + a, x) == [a + x**2, 0, 1, 0]
assert QuotientOfLinearsParts(a/x, x) == [a, 0, 0, 1]
assert QuotientOfLinearsParts(1/x, x) == [1, 0, 0, 1]
assert QuotientOfLinearsParts(a*x + 1, x) == [1, a, 1, 0]
assert QuotientOfLinearsParts(x, x) == [0, 1, 1, 0]
assert QuotientOfLinearsParts(a, x) == [a, 0, 1, 0]
def test_QuotientOfLinearsQ():
assert not QuotientOfLinearsQ((a + x), x)
assert QuotientOfLinearsQ((a + x)/(x), x)
assert QuotientOfLinearsQ((a + b*x)/(x), x)
def test_Flatten():
assert Flatten([a, b, [c, [d, e]]]) == [a, b, c, d, e]
def test_Sort():
assert Sort([b, a, c]) == [a, b, c]
assert Sort([b, a, c], True) == [c, b, a]
def test_AbsurdNumberQ():
assert AbsurdNumberQ(S(1))
assert not AbsurdNumberQ(a*x)
assert not AbsurdNumberQ(a**(S(1)/2))
assert AbsurdNumberQ((S(1)/3)**(S(1)/3))
def test_AbsurdNumberFactors():
assert AbsurdNumberFactors(S(1)) == S(1)
assert AbsurdNumberFactors((S(1)/3)**(S(1)/3)) == S(3)**(S(2)/3)/S(3)
assert AbsurdNumberFactors(a) == S(1)
def test_NonabsurdNumberFactors():
assert NonabsurdNumberFactors(a) == a
assert NonabsurdNumberFactors(S(1)) == S(1)
assert NonabsurdNumberFactors(a*S(2)) == a
def test_NumericFactor():
assert NumericFactor(S(1)) == S(1)
assert NumericFactor(1*I) == S(1)
assert NumericFactor(S(1) + I) == S(1)
assert NumericFactor(a**(S(1)/3)) == S(1)
assert NumericFactor(a*S(3)) == S(3)
assert NumericFactor(a + b) == S(1)
def test_NonnumericFactors():
assert NonnumericFactors(S(3)) == S(1)
assert NonnumericFactors(I) == I
assert NonnumericFactors(S(3) + I) == S(3) + I
assert NonnumericFactors((S(1)/3)**(S(1)/3)) == S(1)
assert NonnumericFactors(log(a)) == log(a)
def test_Prepend():
assert Prepend([1, 2, 3], [4, 5]) == [4, 5, 1, 2, 3]
def test_SumSimplerQ():
assert not SumSimplerQ(S(4 + x),S(3 + x**3))
assert SumSimplerQ(S(4 + x), S(3 - x))
def test_SumSimplerAuxQ():
assert SumSimplerAuxQ(S(4 + x), S(3 - x))
assert not SumSimplerAuxQ(S(4), S(3))
def test_SimplerSqrtQ():
assert SimplerSqrtQ(S(2), S(16*x**3))
assert not SimplerSqrtQ(S(x*2), S(16))
assert not SimplerSqrtQ(S(-4), S(16))
assert SimplerSqrtQ(S(4), S(16))
assert not SimplerSqrtQ(S(4), S(0))
def test_TrinomialParts():
assert TrinomialParts((1 + 5*x**3)**2, x) == [1, 10, 25, 3]
assert TrinomialParts(1 + 5*x**3 + 2*x**6, x) == [1, 5, 2, 3]
assert TrinomialParts(((1 + 5*x**3)**2) + 6, x) == [7, 10, 25, 3]
assert not TrinomialParts(1 + 5*x**3 + 2*x**5, x)
def test_TrinomialDegree():
assert TrinomialDegree((7 + 2*x**6)**2, x) == 6
assert TrinomialDegree(1 + 5*x**3 + 2*x**6, x) == 3
assert not TrinomialDegree(1 + 5*x**3 + 2*x**5, x)
def test_CubicMatchQ():
assert not CubicMatchQ(S(3 + x**6), x)
assert CubicMatchQ(S(x**3), x)
assert not CubicMatchQ(S(3), x)
assert CubicMatchQ(S(3 + x**3), x)
assert CubicMatchQ(S(3 + x**3 + 2*x), x)
def test_BinomialMatchQ():
assert BinomialMatchQ(x, x)
assert BinomialMatchQ(2 + 3*x**5, x)
assert BinomialMatchQ(3*x**5, x)
assert BinomialMatchQ(3*x, x)
assert not BinomialMatchQ(x + x**2 + x**3, x)
def test_TrinomialMatchQ():
assert not TrinomialMatchQ((5 + 2*x**6)**2, x)
assert not TrinomialMatchQ((7 + 8*x**6), x)
assert TrinomialMatchQ((7 + 2*x**6 + 3*x**3), x)
assert TrinomialMatchQ(b*x**2 + c*x**4, x)
def test_GeneralizedBinomialMatchQ():
assert not GeneralizedBinomialMatchQ((1 + x**4), x)
assert GeneralizedBinomialMatchQ((3*x + x**7), x)
def test_QuadraticMatchQ():
assert not QuadraticMatchQ((a + b*x)*(c + d*x), x)
assert QuadraticMatchQ(x**2 + x, x)
assert QuadraticMatchQ(x**2+1+x, x)
assert QuadraticMatchQ(x**2, x)
def test_PowerOfLinearMatchQ():
assert PowerOfLinearMatchQ(x, x)
assert not PowerOfLinearMatchQ(S(6)**3, x)
assert not PowerOfLinearMatchQ(S(6 + 3*x**2)**3, x)
assert PowerOfLinearMatchQ(S(6 + 3*x)**3, x)
def test_GeneralizedTrinomialMatchQ():
assert not GeneralizedTrinomialMatchQ(7 + 2*x**6 + 3*x**12, x)
assert not GeneralizedTrinomialMatchQ(7 + 2*x**6 + 3*x**3, x)
assert not GeneralizedTrinomialMatchQ(7 + 2*x**6 + 3*x**5, x)
assert GeneralizedTrinomialMatchQ(x**2 + x**3 + x**4, x)
def test_QuotientOfLinearsMatchQ():
assert QuotientOfLinearsMatchQ((1 + x)*(3 + 4*x**2)/(2 + 4*x), x)
assert not QuotientOfLinearsMatchQ(x*(3 + 4*x**2)/(2 + 4*x**3), x)
assert QuotientOfLinearsMatchQ(x*(3 + 4*x)/(2 + 4*x), x)
assert QuotientOfLinearsMatchQ(2*(3 + 4*x)/(2 + 4*x), x)
def test_PolynomialTermQ():
assert not PolynomialTermQ(S(3), x)
assert PolynomialTermQ(3*x**6, x)
assert not PolynomialTermQ(3*x**6+5*x, x)
def test_PolynomialTerms():
assert PolynomialTerms(x + 6*x**3 + log(x), x) == 6*x**3 + x
assert PolynomialTerms(x + 6*x**3 + 6*x, x) == 6*x**3 + 7*x
assert PolynomialTerms(x + 6*x**3 + 6, x) == 6*x**3 + x
def test_NonpolynomialTerms():
assert NonpolynomialTerms(x + 6*x**3 + log(x), x) == log(x)
assert NonpolynomialTerms(x + 6*x**3 + 6*x, x) == 0
assert NonpolynomialTerms(x + 6*x**3 + 6, x) == 6
def test_PseudoBinomialQ():
assert PseudoBinomialQ(3 + 5*(x)**6, x)
assert PseudoBinomialQ(3 + 5*(2 + 5*x)**6, x)
def test_PseudoBinomialParts():
assert PseudoBinomialParts(3 + 7*(1 + x)**6, x) == [3, 1, 7**(S(1)/S(6)), 7**(S(1)/S(6)), 6]
assert PseudoBinomialParts(3 + 7*(1 + x)**3, x) == [3, 1, 7**(S(1)/S(3)), 7**(S(1)/S(3)), 3]
assert not PseudoBinomialParts(3 + 7*(1 + x)**2, x)
assert PseudoBinomialParts(3 + 7*(x)**5, x) == [3, 1, 0, 7**(S(1)/S(5)), 5]
def test_PseudoBinomialPairQ():
assert not PseudoBinomialPairQ(3 + 5*(x)**6,3 + (x)**6, x)
assert not PseudoBinomialPairQ(3 + 5*(1 + x)**6,3 + (1 + x)**6, x)
def test_NormalizePseudoBinomial():
assert NormalizePseudoBinomial(3 + 5*(1 + x)**6, x) == 3+(5**(S(1)/S(6))+5**(S(1)/S(6))*x)**S(6)
assert NormalizePseudoBinomial(3 + 5*(x)**6, x) == 3+5*x**6
def test_CancelCommonFactors():
assert CancelCommonFactors(S(x*y*S(6))**S(6), S(x*y*S(6))) == [46656*x**6*y**6, 6*x*y]
assert CancelCommonFactors(S(y*6)**S(6), S(x*y*S(6))) == [46656*y**6, 6*x*y]
assert CancelCommonFactors(S(6), S(3)) == [6, 3]
def test_SimplerIntegrandQ():
assert SimplerIntegrandQ(S(5), 4*x, x)
assert not SimplerIntegrandQ(S(x + 5*x**3), S(x**2 + 3*x), x)
assert SimplerIntegrandQ(S(x + 8), S(x**2 + 3*x), x)
def test_Drop():
assert Drop([1, 2, 3, 4, 5, 6], [2, 4]) == [1, 5, 6]
assert Drop([1, 2, 3, 4, 5, 6], -3) == [1, 2, 3]
assert Drop([1, 2, 3, 4, 5, 6], 2) == [3, 4, 5, 6]
assert Drop(a*b*c, 1) == b*c
def test_SubstForInverseFunction():
assert SubstForInverseFunction(x, a, b, x) == b
assert SubstForInverseFunction(a, a, b, x) == a
assert SubstForInverseFunction(x**a, x**a, b, x) == x
assert SubstForInverseFunction(a*x**a, a, b, x) == a*b**a
def test_SubstForFractionalPower():
assert SubstForFractionalPower(a, b, n, c, x) == a
assert SubstForFractionalPower(x, b, n, c, x) == c
assert SubstForFractionalPower(a**(S(1)/2), a, n, b, x) == x**(n/2)
def test_CombineExponents():
assert True
def test_FractionalPowerOfSquareQ():
assert not FractionalPowerOfSquareQ(x)
assert not FractionalPowerOfSquareQ((a + b)**(S(2)/S(3)))
assert not FractionalPowerOfSquareQ((a + b)**(S(2)/S(3))*c)
assert FractionalPowerOfSquareQ(((a + b*x)**(S(2)))**(S(1)/3)) == (a + b*x)**S(2)
def test_FractionalPowerSubexpressionQ():
assert not FractionalPowerSubexpressionQ(x, a, x)
assert FractionalPowerSubexpressionQ(x**(S(2)/S(3)), a, x)
assert not FractionalPowerSubexpressionQ(b*a, a, x)
def test_FactorNumericGcd():
assert FactorNumericGcd(5*a**2*e**4 + 2*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 - 6*b*c*d**3*e + 21*c**2*d**4) ==\
5*a**2*e**4 + 2*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 - 6*b*c*d**3*e + 21*c**2*d**4
assert FactorNumericGcd(x**(S(2))) == x**S(2)
assert FactorNumericGcd(log(x)) == log(x)
assert FactorNumericGcd(log(x)*x) == x*log(x)
assert FactorNumericGcd(log(x) + x**S(2)) == log(x) + x**S(2)
def test_Apply():
assert Apply(List, [a, b, c]) == [a, b, c]
def test_TrigSimplify():
assert TrigSimplify(a*sin(x)**2 + a*cos(x)**2 + v) == a + v
assert TrigSimplify(a*sec(x)**2 - a*tan(x)**2 + v) == a + v
assert TrigSimplify(a*csc(x)**2 - a*cot(x)**2 + v) == a + v
assert TrigSimplify(S(1) - sin(x)**2) == cos(x)**2
assert TrigSimplify(1 + tan(x)**2) == sec(x)**2
assert TrigSimplify(1 + cot(x)**2) == csc(x)**2
assert TrigSimplify(-S(1) + sec(x)**2) == tan(x)**2
assert TrigSimplify(-1 + csc(x)**2) == cot(x)**2
def test_MergeFactors():
assert simplify(MergeFactors(b/(a - c)**3 , 8*c**3*(b*x + c)**(3/2)/(3*b**4) - 24*c**2*(b*x + c)**(5/2)/(5*b**4) + \
24*c*(b*x + c)**(7/2)/(7*b**4) - 8*(b*x + c)**(9/2)/(9*b**4)) - (8*c**3*(b*x + c)**1.5/(3*b**3) - 24*c**2*(b*x + c)**2.5/(5*b**3) + \
24*c*(b*x + c)**3.5/(7*b**3) - 8*(b*x + c)**4.5/(9*b**3))/(a - c)**3) == 0
assert MergeFactors(x, x) == x**2
assert MergeFactors(x*y, x) == x**2*y
def test_FactorInteger():
assert FactorInteger(2434500) == [(2, 2), (3, 2), (5, 3), (541, 1)]
def test_ContentFactor():
assert ContentFactor(a*b + a*c) == a*(b + c)
def test_Order():
assert Order(a, b) == 1
assert Order(b, a) == -1
assert Order(a, a) == 0
def test_FactorOrder():
assert FactorOrder(1, 1) == 0
assert FactorOrder(1, 2) == -1
assert FactorOrder(2, 1) == 1
assert FactorOrder(a, b) == 1
def test_Smallest():
assert Smallest([2, 1, 3, 4]) == 1
assert Smallest(1, 2) == 1
assert Smallest(-1, -2) == -2
def test_MostMainFactorPosition():
assert MostMainFactorPosition([S(1), S(2), S(3)]) == 1
assert MostMainFactorPosition([S(1), S(7), S(3), S(4), S(5)]) == 2
def test_OrderedQ():
assert OrderedQ([a, b])
assert not OrderedQ([b, a])
def test_MinimumDegree():
assert MinimumDegree(S(1), S(2)) == 1
assert MinimumDegree(S(1), sqrt(2)) == 1
assert MinimumDegree(sqrt(2), S(1)) == 1
assert MinimumDegree(sqrt(3), sqrt(2)) == sqrt(2)
assert MinimumDegree(sqrt(2), sqrt(2)) == sqrt(2)
def test_PositiveFactors():
assert PositiveFactors(S(0)) == 1
assert PositiveFactors(-S(1)) == S(1)
assert PositiveFactors(sqrt(2)) == sqrt(2)
assert PositiveFactors(-log(2)) == log(2)
assert PositiveFactors(sqrt(2)*S(-1)) == sqrt(2)
def test_NonpositiveFactors():
assert NonpositiveFactors(S(0)) == 0
assert NonpositiveFactors(-S(1)) == -1
assert NonpositiveFactors(sqrt(2)) == 1
assert NonpositiveFactors(-log(2)) == -1
def test_Sign():
assert Sign(S(0)) == 0
assert Sign(S(1)) == 1
assert Sign(-S(1)) == -1
def test_PolynomialInQ():
v = log(x)
assert PolynomialInQ(S(1), v, x)
assert PolynomialInQ(v, v, x)
assert PolynomialInQ(1 + v**2, v, x)
assert PolynomialInQ(1 + a*v**2, v, x)
assert not PolynomialInQ(sqrt(v), v, x)
def test_ExponentIn():
v = log(x)
assert ExponentIn(S(1), log(x), x) == 0
assert ExponentIn(S(1) + v, log(x), x) == 1
assert ExponentIn(S(1) + v + v**3, log(x), x) == 3
assert ExponentIn(S(2)*sqrt(v)*v**3, log(x), x) == 3.5
def test_PolynomialInSubst():
v = log(x)
assert PolynomialInSubst(S(1) + log(x)**3, log(x), x) == 1 + x**3
assert PolynomialInSubst(S(1) + log(x), log(x), x) == x + 1
def test_Distrib():
assert Distrib(x, a) == x*a
assert Distrib(x, a + b) == a*x + b*x
def test_DistributeDegree():
assert DistributeDegree(x, m) == x**m
assert DistributeDegree(x**a, m) == x**(a*m)
assert DistributeDegree(a*b, m) == a**m * b**m
def test_FunctionOfPower():
assert FunctionOfPower(a, x) == None
assert FunctionOfPower(x, x) == 1
assert FunctionOfPower(x**3, x) == 3
assert FunctionOfPower(x**3*cos(x**6), x) == 3
def test_DivideDegreesOfFactors():
assert DivideDegreesOfFactors(a**b, S(3)) == a**(b/3)
assert DivideDegreesOfFactors(a**b*c, S(3)) == a**(b/3)*c**(c/3)
def test_MonomialFactor():
assert MonomialFactor(a, x) == [0, a]
assert MonomialFactor(x, x) == [1, 1]
assert MonomialFactor(x + y, x) == [0, x + y]
assert MonomialFactor(log(x), x) == [0, log(x)]
assert MonomialFactor(log(x)*x, x) == [1, log(x)]
def test_NormalizeIntegrand():
assert NormalizeIntegrand((x**2 + 8), x) == x**2 + 8
assert NormalizeIntegrand((x**2 + 3*x)**2, x) == x**2*(x + 3)**2
assert NormalizeIntegrand(a**2*(a + b*x)**2, x) == a**2*(a + b*x)**2
assert NormalizeIntegrand(b**2/(a**2*(a + b*x)**2), x) == b**2/(a**2*(a + b*x)**2)
def test_NormalizeIntegrandAux():
v = (6*A*a*c - 2*A*b**2 + B*a*b)/(a*x**2) - (6*A*a**2*c**2 - 10*A*a*b**2*c - 8*A*a*b*c**2*x + 2*A*b**4 + 2*A*b**3*c*x + 5*B*a**2*b*c + 4*B*a**2*c**2*x - B*a*b**3 - B*a*b**2*c*x)/(a**2*(a + b*x + c*x**2)) + (-2*A*b + B*a)*(4*a*c - b**2)/(a**2*x)
assert NormalizeIntegrandAux(v, x) == (6*A*a*c - 2*A*b**2 + B*a*b)/(a*x**2) - (6*A*a**2*c**2 - 10*A*a*b**2*c + 2*A*b**4 + 5*B*a**2*b*c - B*a*b**3 + x*(-8*A*a*b*c**2 + 2*A*b**3*c + 4*B*a**2*c**2 - B*a*b**2*c))/(a**2*(a + b*x + c*x**2)) + (-2*A*b + B*a)*(4*a*c - b**2)/(a**2*x)
assert NormalizeIntegrandAux((x**2 + 3*x)**2, x) == x**2*(x + 3)**2
assert NormalizeIntegrandAux((x**2 + 8), x) == x**2 + 8
def test_NormalizeIntegrandFactor():
assert NormalizeIntegrandFactor((3*x + x**3)**2, x) == x**2*(x**2 + 3)**2
assert NormalizeIntegrandFactor((x**2 + 8), x) == x**2 + 8
def test_NormalizeIntegrandFactorBase():
assert NormalizeIntegrandFactorBase((x**2 + 8)**3, x) == (x**2 + 8)**3
assert NormalizeIntegrandFactorBase((x**2 + 8), x) == x**2 + 8
assert NormalizeIntegrandFactorBase(a**2*(a + b*x)**2, x) == a**2*(a + b*x)**2
def test_AbsorbMinusSign():
assert AbsorbMinusSign((x + 2)**5*(x + 3)**5) == (-x - 3)**5*(x + 2)**5
assert AbsorbMinusSign((x + 2)**5*(x + 3)**2) == -(x + 2)**5*(x + 3)**2
def test_NormalizeLeadTermSigns():
assert NormalizeLeadTermSigns((-x + 3)*(x**2 + 3)) == (-x + 3)*(x**2 + 3)
assert NormalizeLeadTermSigns(x + 3) == x + 3
def test_SignOfFactor():
assert SignOfFactor(S(-x + 3)) == [1, -x + 3]
assert SignOfFactor(S(-x)) == [-1, x]
def test_NormalizePowerOfLinear():
assert NormalizePowerOfLinear((x + 3)**5, x) == (x + 3)**5
assert NormalizePowerOfLinear(((x + 3)**2) + 3, x) == x**2 + 6*x + 12
def test_SimplifyIntegrand():
assert SimplifyIntegrand((x**2 + 3)**2, x) == (x**2 + 3)**2
assert SimplifyIntegrand(x**2 + 3 + (x**6) + 6, x) == x**6 + x**2 + 9
def test_SimplifyTerm():
assert SimplifyTerm(a**2/b**2, x) == a**2/b**2
assert SimplifyTerm(-6*x/5 + (5*x + 3)**2/25 - 9/25, x) == x**2
def test_togetherSimplify():
assert TogetherSimplify(-6*x/5 + (5*x + 3)**2/25 - 9/25) == x**2
def test_ExpandToSum():
qq = 6
Pqq = e**3
Pq = (d+e*x**2)**3
aa = 2
nn = 2
cc = 1
pp = -1/2
bb = 3
assert nsimplify(ExpandToSum(Pq - Pqq*x**qq - Pqq*(aa*x**(-2*nn + qq)*(-2*nn + qq + 1) + bb*x**(-nn + qq)*(nn*(pp - 1) + qq + 1))/(cc*(2*nn*pp + qq + 1)), x) - \
(d**3 + x**4*(3*d*e**2 - 2.4*e**3) + x**2*(3*d**2*e - 1.2*e**3))) == 0
assert ExpandToSum(x**2 + 3*x + 3, x**3 + 3, x) == x**3*(x**2 + 3*x + 3) + 3*x**2 + 9*x + 9
assert ExpandToSum(x**3 + 6, x) == x**3 + 6
assert ExpandToSum(S(x**2 + 3*x + 3)*3, x) == 3*x**2 + 9*x + 9
assert ExpandToSum((a + b*x), x) == a + b*x
def test_UnifySum():
assert UnifySum((3 + x + 6*x**3 + sin(x)), x) == 6*x**3 + x + sin(x) + 3
assert UnifySum((3 + x + 6*x**3)*3, x) == 18*x**3 + 3*x + 9
def test_FunctionOfInverseLinear():
assert FunctionOfInverseLinear((x)/(a + b*x), x) == [a, b]
assert FunctionOfInverseLinear((c + d*x)/(a + b*x), x) == [a, b]
assert not FunctionOfInverseLinear(1/(a + b*x), x)
def test_PureFunctionOfSinhQ():
v = log(x)
f = sinh(v)
assert PureFunctionOfSinhQ(f, v, x)
assert not PureFunctionOfSinhQ(cosh(v), v, x)
assert PureFunctionOfSinhQ(f**2, v, x)
def test_PureFunctionOfTanhQ():
v = log(x)
f = tanh(v)
assert PureFunctionOfTanhQ(f, v, x)
assert not PureFunctionOfTanhQ(cosh(v), v, x)
assert PureFunctionOfTanhQ(f**2, v, x)
def test_PureFunctionOfCoshQ():
v = log(x)
f = cosh(v)
assert PureFunctionOfCoshQ(f, v, x)
assert not PureFunctionOfCoshQ(sinh(v), v, x)
assert PureFunctionOfCoshQ(f**2, v, x)
def test_IntegerQuotientQ():
u = S(2)*sin(x)
v = sin(x)
assert IntegerQuotientQ(u, v)
assert IntegerQuotientQ(u, u)
assert not IntegerQuotientQ(S(1), S(2))
def test_OddQuotientQ():
u = S(3)*sin(x)
v = sin(x)
assert OddQuotientQ(u, v)
assert OddQuotientQ(u, u)
assert not OddQuotientQ(S(1), S(2))
def test_EvenQuotientQ():
u = S(2)*sin(x)
v = sin(x)
assert EvenQuotientQ(u, v)
assert not EvenQuotientQ(u, u)
assert not EvenQuotientQ(S(1), S(2))
def test_FunctionOfSinhQ():
v = log(x)
assert FunctionOfSinhQ(cos(sinh(v)), v, x)
assert FunctionOfSinhQ(sinh(v), v, x)
assert FunctionOfSinhQ(sinh(v)*cos(sinh(v)), v, x)
def test_FunctionOfCoshQ():
v = log(x)
assert FunctionOfCoshQ(cos(cosh(v)), v, x)
assert FunctionOfCoshQ(cosh(v), v, x)
assert FunctionOfCoshQ(cosh(v)*cos(cosh(v)), v, x)
def test_FunctionOfTanhQ():
v = log(x)
t = Tanh(v)
c = Coth(v)
assert FunctionOfTanhQ(t, v, x)
assert FunctionOfTanhQ(c, v, x)
assert FunctionOfTanhQ(t + c, v, x)
assert FunctionOfTanhQ(t*c, v, x)
assert not FunctionOfTanhQ(sin(x), v, x)
def test_FunctionOfTanhWeight():
v = log(x)
t = Tanh(v)
c = Coth(v)
assert FunctionOfTanhWeight(x, v, x) == 0
assert FunctionOfTanhWeight(sinh(v), v, x) == 0
assert FunctionOfTanhWeight(tanh(v), v, x) == 1
assert FunctionOfTanhWeight(coth(v), v, x) == -1
assert FunctionOfTanhWeight(t**2, v, x) == 1
assert FunctionOfTanhWeight(sinh(v)**2, v, x) == -1
assert FunctionOfTanhWeight(coth(v)*sinh(v)**2, v, x) == -2
def test_FunctionOfHyperbolicQ():
v = log(x)
s = Sinh(v)
t = Tanh(v)
assert not FunctionOfHyperbolicQ(x, v, x)
assert FunctionOfHyperbolicQ(s + t, v, x)
assert FunctionOfHyperbolicQ(sinh(t), v, x)
def test_SmartNumerator():
assert SmartNumerator(x**(-2)) == 1
assert SmartNumerator(x**(2)*a) == x**2*a
def test_SmartDenominator():
assert SmartDenominator(x**(-2)) == x**2
assert SmartDenominator(x**(-2)*1/S(3)) == x**2*3
def test_SubstForAux():
v = log(x)
assert SubstForAux(v, v, x) == x
assert SubstForAux(v**2, v, x) == x**2
assert SubstForAux(x, v, x) == x
assert SubstForAux(v**2, v**4, x) == sqrt(x)
assert SubstForAux(v**2*v, v, x) == x**3
def test_SubstForTrig():
v = log(x)
s, c, t = sin(v), cos(v), tan(v)
assert SubstForTrig(cos(a/2 + b*x/2), x/sqrt(x**2 + 1), 1/sqrt(x**2 + 1), a/2 + b*x/2, x) == 1/sqrt(x**2 + 1)
assert SubstForTrig(s, sin, cos, v, x) == sin
assert SubstForTrig(t, sin(v), cos(v), v, x) == sin(log(x))/cos(log(x))
assert SubstForTrig(sin(2*v), sin(x), cos(x), v, x) == 2*sin(x)*cos(x)
assert SubstForTrig(s*t, sin(x), cos(x), v, x) == sin(x)**2/cos(x)
def test_SubstForHyperbolic():
v = log(x)
s, c, t = sinh(v), cosh(v), tanh(v)
assert SubstForHyperbolic(s, sinh(x), cosh(x), v, x) == sinh(x)
assert SubstForHyperbolic(t, sinh(x), cosh(x), v, x) == sinh(x)/cosh(x)
assert SubstForHyperbolic(sinh(2*v), sinh(x), cosh(x), v, x) == 2*sinh(x)*cosh(x)
assert SubstForHyperbolic(s*t, sinh(x), cosh(x), v, x) == sinh(x)**2/cosh(x)
def test_SubstForFractionalPowerOfLinear():
u = a + b*x
assert not SubstForFractionalPowerOfLinear(u, x)
assert not SubstForFractionalPowerOfLinear(u**(S(2)), x)
assert SubstForFractionalPowerOfLinear(u**(S(1)/2), x) == [x**2, 2, a + b*x, 1/b]
def test_InverseFunctionOfLinear():
u = a + b*x
assert InverseFunctionOfLinear(log(u)*sin(x), x) == log(u)
assert InverseFunctionOfLinear(log(u), x) == log(u)
def test_InertTrigQ():
s = sin(x)
c = cos(x)
assert not InertTrigQ(sin(x), csc(x), cos(h))
assert InertTrigQ(sin(x), csc(x))
assert not InertTrigQ(s, c)
assert InertTrigQ(c)
def test_PowerOfInertTrigSumQ():
func = sin
assert PowerOfInertTrigSumQ((1 + S(2)*(S(3)*func(x**2))**S(5))**3, func, x)
assert PowerOfInertTrigSumQ((1 + 2*(S(3)*func(x**2))**3 + 4*(S(5)*func(x**2))**S(3))**2, func, x)
def test_PiecewiseLinearQ():
assert PiecewiseLinearQ(a + b*x, x)
assert not PiecewiseLinearQ(Log(c*sin(a)**S(3)), x)
assert not PiecewiseLinearQ(x**3, x)
assert PiecewiseLinearQ(atanh(tanh(a + b*x)), x)
assert PiecewiseLinearQ(tanh(atanh(a + b*x)), x)
assert not PiecewiseLinearQ(coth(atanh(a + b*x)), x)
def test_KnownTrigIntegrandQ():
func = sin(a + b*x)
assert KnownTrigIntegrandQ([sin], S(1), x)
assert KnownTrigIntegrandQ([sin], (a + b*func)**m, x)
assert KnownTrigIntegrandQ([sin], (a + b*func)**m*(1 + 2*func), x)
assert KnownTrigIntegrandQ([sin], a + c*func**2, x)
assert KnownTrigIntegrandQ([sin], a + b*func + c*func**2, x)
assert KnownTrigIntegrandQ([sin], (a + b*func)**m*(c + d*func**2), x)
assert KnownTrigIntegrandQ([sin], (a + b*func)**m*(c + d*func + e*func**2), x)
assert not KnownTrigIntegrandQ([cos], (a + b*func)**m, x)
def test_KnownSineIntegrandQ():
assert KnownSineIntegrandQ((a + b*sin(a + b*x))**m, x)
def test_KnownTangentIntegrandQ():
assert KnownTangentIntegrandQ((a + b*tan(a + b*x))**m, x)
def test_KnownCotangentIntegrandQ():
assert KnownCotangentIntegrandQ((a + b*cot(a + b*x))**m, x)
def test_KnownSecantIntegrandQ():
assert KnownSecantIntegrandQ((a + b*sec(a + b*x))**m, x)
def test_TryPureTanSubst():
assert TryPureTanSubst(atan(c*(a + b*tan(a + b*x))), x)
assert TryPureTanSubst(atanh(c*(a + b*cot(a + b*x))), x)
assert not TryPureTanSubst(tan(c*(a + b*cot(a + b*x))), x)
def test_TryPureTanhSubst():
assert not TryPureTanhSubst(log(x), x)
assert TryPureTanhSubst(sin(x), x)
assert not TryPureTanhSubst(atanh(a*tanh(x)), x)
assert not TryPureTanhSubst((a + b*x)**S(2), x)
def test_TryTanhSubst():
assert not TryTanhSubst(log(x), x)
assert not TryTanhSubst(a*(b + c)**3, x)
assert not TryTanhSubst(1/(a + b*sinh(x)**S(3)), x)
assert not TryTanhSubst(sinh(S(3)*x)*cosh(S(4)*x), x)
assert not TryTanhSubst(a*(b*sech(x)**3)**c, x)
def test_GeneralizedBinomialQ():
assert GeneralizedBinomialQ(a*x**q + b*x**n, x)
assert not GeneralizedBinomialQ(a*x**q, x)
def test_GeneralizedTrinomialQ():
assert not GeneralizedTrinomialQ(7 + 2*x**6 + 3*x**12, x)
assert not GeneralizedTrinomialQ(a*x**q + c*x**(2*n-q), x)
def test_SubstForFractionalPowerOfQuotientOfLinears():
assert SubstForFractionalPowerOfQuotientOfLinears(((a + b*x)/(c + d*x))**(S(3)/2), x) == [x**4/(b - d*x**2)**2, 2, (a + b*x)/(c + d*x), -a*d + b*c]
def test_SubstForFractionalPowerQ():
assert SubstForFractionalPowerQ(x, sin(x), x)
assert SubstForFractionalPowerQ(x**2, sin(x), x)
assert not SubstForFractionalPowerQ(x**(S(3)/2), sin(x), x)
assert SubstForFractionalPowerQ(sin(x)**(S(3)/2), sin(x), x)
def test_AbsurdNumberGCD():
assert AbsurdNumberGCD(S(4)) == 4
assert AbsurdNumberGCD(S(4), S(8), S(12)) == 4
assert AbsurdNumberGCD(S(2), S(3), S(12)) == 1
def test_TrigReduce():
assert TrigReduce(cos(x)**2) == cos(2*x)/2 + 1/2
assert TrigReduce(cos(x)**2*sin(x)) == sin(x)/4 + sin(3*x)/4
assert TrigReduce(cos(x)**2+sin(x)) == sin(x) + cos(2*x)/2 + 1/2
assert TrigReduce(cos(x)**2*sin(x)**5) == 5*sin(x)/64 + sin(3*x)/64 - 3*sin(5*x)/64 + sin(7*x)/64
assert TrigReduce(2*sin(x)*cos(x) + 2*cos(x)**2) == sin(2*x) + cos(2*x) + 1
assert TrigReduce(sinh(a + b*x)**2) == cosh(2*a + 2*b*x)/2 - 1/2
assert TrigReduce(sinh(a + b*x)*cosh(a + b*x)) == sinh(2*a + 2*b*x)/2
def test_FunctionOfDensePolynomialsQ():
assert FunctionOfDensePolynomialsQ(x**2 + 3, x)
assert not FunctionOfDensePolynomialsQ(x**2, x)
assert not FunctionOfDensePolynomialsQ(x, x)
assert FunctionOfDensePolynomialsQ(S(2), x)
def test_PureFunctionOfSinQ():
v = log(x)
f = sin(v)
assert PureFunctionOfSinQ(f, v, x)
assert not PureFunctionOfSinQ(cos(v), v, x)
assert PureFunctionOfSinQ(f**2, v, x)
def test_PureFunctionOfTanQ():
v = log(x)
f = tan(v)
assert PureFunctionOfTanQ(f, v, x)
assert not PureFunctionOfTanQ(cos(v), v, x)
assert PureFunctionOfTanQ(f**2, v, x)
def test_PowerVariableSubst():
assert PowerVariableSubst((2*x)**3, 2, x) == 8*x**(3/2)
assert PowerVariableSubst((2*x)**3, 2, x) == 8*x**(3/2)
assert PowerVariableSubst((2*x), 2, x) == 2*x
assert PowerVariableSubst((2*x)**3, 2, x) == 8*x**(3/2)
assert PowerVariableSubst((2*x)**7, 2, x) == 128*x**(7/2)
assert PowerVariableSubst((6+2*x)**7, 2, x) == (2*x + 6)**7
assert PowerVariableSubst((2*x)**7+3, 2, x) == 128*x**(7/2) + 3
def test_PowerVariableDegree():
assert PowerVariableDegree(S(2), 0, 2*x, x) == [0, 2*x]
assert PowerVariableDegree((2*x)**2, 0, 2*x, x) == [2, 1]
assert PowerVariableDegree(x**2, 0, 2*x, x) == [2, 1]
assert PowerVariableDegree(S(4), 0, 2*x, x) == [0, 2*x]
def test_PowerVariableExpn():
assert not PowerVariableExpn((x)**3, 2, x)
assert not PowerVariableExpn((2*x)**3, 2, x)
assert PowerVariableExpn((2*x)**2, 4, x) == [4*x**3, 2, 1]
def test_FunctionOfQ():
assert FunctionOfQ(x**2, sqrt(-exp(2*x**2) + 1)*exp(x**2),x)
assert not FunctionOfQ(S(x**3), x*2, x)
assert FunctionOfQ(S(a), x*2, x)
assert FunctionOfQ(S(3*x), x*2, x)
def test_ExpandTrigExpand():
assert ExpandTrigExpand(1, cos(x), x**2, 2, 2, x) == 4*cos(x**2)**4 - 4*cos(x**2)**2 + 1
assert ExpandTrigExpand(1, cos(x) + sin(x), x**2, 2, 2, x) == 4*sin(x**2)**2*cos(x**2)**2 + 8*sin(x**2)*cos(x**2)**3 - 4*sin(x**2)*cos(x**2) + 4*cos(x**2)**4 - 4*cos(x**2)**2 + 1
def test_TrigToExp():
from sympy.integrals.rubi.utility_function import rubi_exp as exp
assert TrigToExp(sin(x)) == -I*(exp(I*x) - exp(-I*x))/2
assert TrigToExp(cos(x)) == exp(I*x)/2 + exp(-I*x)/2
assert TrigToExp(cos(x)*tan(x**2)) == I*(exp(I*x)/2 + exp(-I*x)/2)*(-exp(I*x**2) + exp(-I*x**2))/(exp(I*x**2) + exp(-I*x**2))
assert TrigToExp(cos(x) + sin(x)**2) == -(exp(I*x) - exp(-I*x))**2/4 + exp(I*x)/2 + exp(-I*x)/2
assert Simplify(TrigToExp(cos(x)*tan(x**S(2))*sin(x)**S(2))-(-I*(exp(I*x)/S(2) + exp(-I*x)/S(2))*(exp(I*x) - exp(-I*x))**S(2)*(-exp(I*x**S(2)) + exp(-I*x**S(2)))/(S(4)*(exp(I*x**S(2)) + exp(-I*x**S(2)))))) == 0
def test_ExpandTrigReduce():
assert ExpandTrigReduce(2*cos(3 + x)**3, x) == 3*cos(x + 3)/2 + cos(3*x + 9)/2
assert ExpandTrigReduce(2*sin(x)**3+cos(2 + x), x) == 3*sin(x)/2 - sin(3*x)/2 + cos(x + 2)
assert ExpandTrigReduce(cos(x + 3)**2, x) == cos(2*x + 6)/2 + 1/2
def test_NormalizeTrig():
assert NormalizeTrig(S(2*sin(2 + x)), x) == 2*sin(x + 2)
assert NormalizeTrig(S(2*sin(2 + x)**3), x) == 2*sin(x + 2)**3
assert NormalizeTrig(S(2*sin((2 + x)**2)**3), x) == 2*sin(x**2 + 4*x + 4)**3
def test_FunctionOfTrigQ():
v = log(x)
s = sin(v)
t = tan(v)
assert not FunctionOfTrigQ(x, v, x)
assert FunctionOfTrigQ(s + t, v, x)
assert FunctionOfTrigQ(sin(t), v, x)
def test_RationalFunctionExpand():
assert RationalFunctionExpand(x**S(5)*(e + f*x)**n/(a + b*x**S(3)), x) == -a*x**2*(e + f*x)**n/(b*(a + b*x**3)) +\
e**2*(e + f*x)**n/(b*f**2) - 2*e*(e + f*x)**(n + 1)/(b*f**2) + (e + f*x)**(n + 2)/(b*f**2)
assert RationalFunctionExpand(x**S(3)*(S(2)*x + 2)**S(2)/(2*x**2 + 1), x) == 2*x**3 + 4*x**2 + x + (- x + 2)/(2*x**2 + 1) - 2
assert RationalFunctionExpand((a + b*x + c*x**4)*log(x)**3, x) == a*log(x)**3 + b*x*log(x)**3 + c*x**4*log(x)**3
assert RationalFunctionExpand(a + b*x + c*x**4, x) == a + b*x + c*x**4
def test_SameQ():
assert SameQ(1, 1, 1)
assert not SameQ(1, 1, 2)
def test_Map2():
assert Map2(Add, [a, b, c], [x, y, z]) == [a + x, b + y, c + z]
def test_ConstantFactor():
assert ConstantFactor(a + a*x**3, x) == [a, x**3 + 1]
assert ConstantFactor(a, x) == [a, 1]
assert ConstantFactor(x, x) == [1, x]
assert ConstantFactor(x**S(3), x) == [1, x**3]
assert ConstantFactor(x**(S(3)/2), x) == [1, x**(3/2)]
assert ConstantFactor(a*x**3, x) == [a, x**3]
assert ConstantFactor(a + x**3, x) == [1, a + x**3]
def test_CommonFactors():
assert CommonFactors([a, a, a]) == [a, 1, 1, 1]
assert CommonFactors([x*S(2), x**S(3)*S(2), sin(x)*x*S(2)]) == [2, x, x**3, x*sin(x)]
assert CommonFactors([x, x**S(3), sin(x)*x]) == [1, x, x**3, x*sin(x)]
assert CommonFactors([S(2), S(4), S(6)]) == [2, 1, 2, 3]
def test_FunctionOfLinear():
f = sin(a + b*x)
assert FunctionOfLinear(f, x) == [sin(x), a, b]
assert FunctionOfLinear(a + b*x, x) == [x, a, b]
assert not FunctionOfLinear(a, x)
def test_FunctionOfExponentialQ():
assert FunctionOfExponentialQ(exp(x + exp(x) + exp(exp(x))), x)
assert FunctionOfExponentialQ(a**(a + b*x), x)
assert FunctionOfExponentialQ(a**(b*x), x)
assert not FunctionOfExponentialQ(a**sin(a + b*x), x)
def test_FunctionOfExponential():
assert FunctionOfExponential(a**(a + b*x), x)
def test_FunctionOfExponentialFunction():
assert FunctionOfExponentialFunction(a**(a + b*x), x) == x
assert FunctionOfExponentialFunction(S(2)*a**(a + b*x), x) == 2*x
def test_FunctionOfTrig():
assert FunctionOfTrig(sin(x + 1), x + 1, x) == x + 1
assert FunctionOfTrig(sin(x), x) == x
assert not FunctionOfTrig(cos(x**2 + 1), x)
assert FunctionOfTrig(sin(a+b*x)**3, x) == a+b*x
def test_AlgebraicTrigFunctionQ():
assert AlgebraicTrigFunctionQ(sin(x + 3), x)
assert AlgebraicTrigFunctionQ(x, x)
assert AlgebraicTrigFunctionQ(x + 1, x)
assert AlgebraicTrigFunctionQ(sinh(x + 1), x)
assert AlgebraicTrigFunctionQ(sinh(x + 1)**2, x)
assert not AlgebraicTrigFunctionQ(sinh(x**2 + 1)**2, x)
def test_FunctionOfHyperbolic():
assert FunctionOfTrig(sin(x + 1), x + 1, x) == x + 1
assert FunctionOfTrig(sin(x), x) == x
assert not FunctionOfTrig(cos(x**2 + 1), x)
def test_FunctionOfExpnQ():
assert FunctionOfExpnQ(x, x, x) == 1
assert FunctionOfExpnQ(x**2, x, x) == 2
assert FunctionOfExpnQ(x**2.1, x, x) == 1
assert not FunctionOfExpnQ(x, x**2, x)
assert not FunctionOfExpnQ(x + 1, (x + 5)**2, x)
assert not FunctionOfExpnQ(x + 1, (x + 1)**2, x)
def test_PureFunctionOfCosQ():
v = log(x)
f = cos(v)
assert PureFunctionOfCosQ(f, v, x)
assert not PureFunctionOfCosQ(sin(v), v, x)
assert PureFunctionOfCosQ(f**2, v, x)
def test_PureFunctionOfCotQ():
v = log(x)
f = cot(v)
assert PureFunctionOfCotQ(f, v, x)
assert not PureFunctionOfCotQ(sin(v), v, x)
assert PureFunctionOfCotQ(f**2, v, x)
def test_FunctionOfSinQ():
v = log(x)
assert FunctionOfSinQ(cos(sin(v)), v, x)
assert FunctionOfSinQ(sin(v), v, x)
assert FunctionOfSinQ(sin(v)*cos(sin(v)), v, x)
def test_FunctionOfCosQ():
v = log(x)
assert FunctionOfCosQ(cos(cos(v)), v, x)
assert FunctionOfCosQ(cos(v), v, x)
assert FunctionOfCosQ(cos(v)*cos(cos(v)), v, x)
def test_FunctionOfTanQ():
v = log(x)
t = tan(v)
c = cot(v)
assert FunctionOfTanQ(t, v, x)
assert FunctionOfTanQ(c, v, x)
assert FunctionOfTanQ(t + c, v, x)
assert FunctionOfTanQ(t*c, v, x)
assert not FunctionOfTanQ(sin(x), v, x)
def test_FunctionOfTanWeight():
v = log(x)
t = tan(v)
c = cot(v)
assert FunctionOfTanWeight(x, v, x) == 0
assert FunctionOfTanWeight(sin(v), v, x) == 0
assert FunctionOfTanWeight(tan(v), v, x) == 1
assert FunctionOfTanWeight(cot(v), v, x) == -1
assert FunctionOfTanWeight(t**2, v, x) == 1
assert FunctionOfTanWeight(sin(v)**2, v, x) == -1
assert FunctionOfTanWeight(cot(v)*sin(v)**2, v, x) == -2
def test_OddTrigPowerQ():
assert not OddTrigPowerQ(sin(x)**3, 1, x)
assert OddTrigPowerQ(sin(3),1,x)
assert OddTrigPowerQ(sin(3*x),x,x)
assert OddTrigPowerQ(sin(3*x)**3,x,x)
def test_FunctionOfLog():
assert not FunctionOfLog(x**2*(a + b*x)**3*exp(-a - b*x) ,False, False, x)
assert FunctionOfLog(log(2*x**8)*2 + log(2*x**8) + 1, x) == [3*x + 1, 2*x**8, 8]
assert FunctionOfLog(log(2*x)**2,x) == [x**2, 2*x, 1]
assert FunctionOfLog(log(3*x**3)**2 + 1,x) == [x**2 + 1, 3*x**3, 3]
assert FunctionOfLog(log(2*x**8)*2,x) == [2*x, 2*x**8, 8]
assert not FunctionOfLog(2*sin(x)*2,x)
def test_EulerIntegrandQ():
assert EulerIntegrandQ((2*x + 3*((x + 1)**3)**1.5)**(-3), x)
assert not EulerIntegrandQ((2*x + (2*x**2)**2)**3, x)
assert not EulerIntegrandQ(3*x**2 + 5*x + 1, x)
def test_Divides():
assert not Divides(x, a*x**2, x)
assert Divides(x, a*x, x) == a
def test_EasyDQ():
assert EasyDQ(3*x**2, x)
assert EasyDQ(3*x**3 - 6, x)
assert EasyDQ(x**3, x)
assert EasyDQ(sin(x**log(3)), x)
def test_ProductOfLinearPowersQ():
assert ProductOfLinearPowersQ(S(1), x)
assert ProductOfLinearPowersQ((x + 1)**3, x)
assert not ProductOfLinearPowersQ((x**2 + 1)**3, x)
assert ProductOfLinearPowersQ(x + 1, x)
def test_Rt():
b = symbols('b')
assert Rt(-b**2, 4) == (-b**2)**(S(1)/S(4))
assert Rt(x**2, 2) == x
assert Rt(S(2 + 3*I), S(8)) == (2 + 3*I)**(1/8)
assert Rt(x**2 + 4 + 4*x, 2) == x + 2
assert Rt(S(8), S(3)) == 2
assert Rt(S(16807), S(5)) == 7
def test_NthRoot():
assert NthRoot(S(14580), S(3)) == 9*2**(S(2)/S(3))*5**(S(1)/S(3))
assert NthRoot(9, 2) == 3.0
assert NthRoot(81, 2) == 9.0
assert NthRoot(81, 4) == 3.0
def test_AtomBaseQ():
assert not AtomBaseQ(x**2)
assert AtomBaseQ(x**3)
assert AtomBaseQ(x)
assert AtomBaseQ(S(2)**3)
assert not AtomBaseQ(sin(x))
def test_SumBaseQ():
assert not SumBaseQ((x + 1)**2)
assert SumBaseQ((x + 1)**3)
assert SumBaseQ((3*x+3))
assert not SumBaseQ(x)
def test_NegSumBaseQ():
assert not NegSumBaseQ(-x + 1)
assert NegSumBaseQ(x - 1)
assert not NegSumBaseQ((x - 1)**2)
assert NegSumBaseQ((x - 1)**3)
def test_AllNegTermQ():
x = Symbol('x', negative=True)
assert AllNegTermQ(x)
assert not AllNegTermQ(x + 2)
assert AllNegTermQ(x - 2)
assert AllNegTermQ((x - 2)**3)
assert not AllNegTermQ((x - 2)**2)
def test_TrigSquareQ():
assert TrigSquareQ(sin(x)**2)
assert TrigSquareQ(cos(x)**2)
assert not TrigSquareQ(tan(x)**2)
def test_Inequality():
assert not Inequality(S('0'), Less, m, LessEqual, S('1'))
assert Inequality(S('0'), Less, S('1'))
assert Inequality(S('0'), Less, S('1'), LessEqual, S('5'))
def test_SplitProduct():
assert SplitProduct(OddQ, S(3)*x) == [3, x]
assert not SplitProduct(OddQ, S(2)*x)
def test_SplitSum():
assert SplitSum(FracPart, sin(x)) == [sin(x), 0]
assert SplitSum(FracPart, sin(x) + S(2)) == [sin(x), S(2)]
def test_Complex():
assert Complex(a, b) == a + I*b
def test_SimpFixFactor():
assert SimpFixFactor((a*c + b*c)**S(4), x) == (a*c + b*c)**4
assert SimpFixFactor((a*Complex(0, c) + b*Complex(0, d))**S(3), x) == -I*(a*c + b*d)**3
assert SimpFixFactor((a*Complex(0, d) + b*Complex(0, e) + c*Complex(0, f))**S(2), x) == -(a*d + b*e + c*f)**2
assert SimpFixFactor((a + b*x**(-1/S(2))*x**S(3))**S(3), x) == (a + b*x**(5/2))**3
assert SimpFixFactor((a*c + b*c**S(2)*x**S(2))**S(3), x) == c**3*(a + b*c*x**2)**3
assert SimpFixFactor((a*c**S(2) + b*c**S(1)*x**S(2))**S(3), x) == c**3*(a*c + b*x**2)**3
assert SimpFixFactor(a*cos(x)**2 + a*sin(x)**2 + v, x) == a*cos(x)**2 + a*sin(x)**2 + v
def test_SimplifyAntiderivative():
assert SimplifyAntiderivative(acoth(coth(x)), x) == x
assert SimplifyAntiderivative(a*x, x) == a*x
assert SimplifyAntiderivative(atanh(cot(x)), x) == atanh(2*sin(x)*cos(x))/2
assert SimplifyAntiderivative(a*cos(x)**2 + a*sin(x)**2 + v, x) == a*cos(x)**2 + a*sin(x)**2
def test_FixSimplify():
assert FixSimplify(x*Complex(0, a)*(v*Complex(0, b) + w)**S(3)) == a*x*(b*v - I*w)**3
def test_TrigSimplifyAux():
assert TrigSimplifyAux(a*cos(x)**2 + a*sin(x)**2 + v) == a + v
assert TrigSimplifyAux(x**2) == x**2
def test_SubstFor():
assert SubstFor(x**2 + 1, tanh(x), x) == tanh(x)
assert SubstFor(x**2, sinh(x), x) == sinh(sqrt(x))
def test_FresnelS():
assert FresnelS(oo) == 1/2
assert FresnelS(0) == 0
def test_FresnelC():
assert FresnelC(0) == 0
assert FresnelC(oo) == 1/2
def test_Erfc():
assert Erfc(0) == 1
assert Erfc(oo) == 0
def test_Erfi():
assert Erfi(oo) == oo
assert Erfi(0) == 0
def test_Gamma():
assert Gamma(u) == gamma(u)
def test_ElementaryFunctionQ():
assert ElementaryFunctionQ(x + y)
assert ElementaryFunctionQ(sin(x + y))
assert ElementaryFunctionQ(E**(x*a))
def test_Util_Part():
from sympy.integrals.rubi.utility_function import Util_Part
assert Util_Part(1, a + b).doit() == a
assert Util_Part(c, a + b).doit() == Util_Part(c, a + b)
def test_Part():
assert Part([1, 2, 3], 1) == 1
assert Part(a*b, 1) == a
def test_PolyLog():
assert PolyLog(a, b) == polylog(a, b)
def test_PureFunctionOfCothQ():
v = log(x)
assert PureFunctionOfCothQ(coth(v), v, x)
assert PureFunctionOfCothQ(a + coth(v), v, x)
assert not PureFunctionOfCothQ(sin(v), v, x)
def test_ExpandIntegrand():
assert ExpandIntegrand(sqrt(a + b*x**S(2) + c*x**S(4)), (f*x)**(S(3)/2)*(d + e*x**S(2)), x) == \
d*(f*x)**(3/2)*sqrt(a + b*x**2 + c*x**4) + e*(f*x)**(7/2)*sqrt(a + b*x**2 + c*x**4)/f**2
assert ExpandIntegrand((6*A*a*c - 2*A*b**2 + B*a*b - 2*c*x*(A*b - 2*B*a))/(x**2*(a + b*x + c*x**2)), x) == \
(6*A*a*c - 2*A*b**2 + B*a*b)/(a*x**2) + (-6*A*a**2*c**2 + 10*A*a*b**2*c - 2*A*b**4 - 5*B*a**2*b*c + B*a*b**3 + x*(8*A*a*b*c**2 - 2*A*b**3*c - 4*B*a**2*c**2 + B*a*b**2*c))/(a**2*(a + b*x + c*x**2)) + (-2*A*b + B*a)*(4*a*c - b**2)/(a**2*x)
assert ExpandIntegrand(x**2*(e + f*x)**3*F**(a + b*(c + d*x)**1), x) == F**(a + b*(c + d*x))*e**2*(e + f*x)**3/f**2 - 2*F**(a + b*(c + d*x))*e*(e + f*x)**4/f**2 + F**(a + b*(c + d*x))*(e + f*x)**5/f**2
assert ExpandIntegrand((x)*(a + b*x)**2*f**(e*(c + d*x)**n), x) == a**2*f**(e*(c + d*x)**n)*x + 2*a*b*f**(e*(c + d*x)**n)*x**2 + b**2*f**(e*(c + d*x)**n)*x**3
assert ExpandIntegrand(sin(x)**3*(a + b*(1/sin(x)))**2, x) == a**2*sin(x)**3 + 2*a*b*sin(x)**2 + b**2*sin(x)
assert ExpandIntegrand(x*(a + b*ArcSin(c + d*x))**n, x) == -c*(a + b*asin(c + d*x))**n/d + (a + b*asin(c + d*x))**n*(c + d*x)/d
assert ExpandIntegrand((a + b*x)**S(3)*(A + B*x)/(c + d*x), x) == B*(a + b*x)**3/d + b*(a + b*x)**2*(A*d - B*c)/d**2 + b*(a + b*x)*(A*d - B*c)*(a*d - b*c)/d**3 + b*(A*d - B*c)*(a*d - b*c)**2/d**4 + (A*d - B*c)*(a*d - b*c)**3/(d**4*(c + d*x))
assert ExpandIntegrand((x**2)*(S(3)*x)**(S(1)/2), x) ==sqrt(3)*x**(5/2)
assert ExpandIntegrand((x)*(sin(x))**(S(1)/2), x) == x*sqrt(sin(x))
assert ExpandIntegrand(x*(e + f*x)**2*F**(b*(c + d*x)), x) == -F**(b*(c + d*x))*e*(e + f*x)**2/f + F**(b*(c + d*x))*(e + f*x)**3/f
assert ExpandIntegrand(x**m*(e + f*x)**2*F**(b*(c + d*x)**n), x) == F**(b*(c + d*x)**n)*e**2*x**m + 2*F**(b*(c + d*x)**n)*e*f*x*x**m + F**(b*(c + d*x)**n)*f**2*x**2*x**m
assert simplify(ExpandIntegrand((S(1) - S(1)*x**S(2))**(-S(3)), x) - (-S(3)/(8*(x**2 - 1)) + S(3)/(16*(x + 1)**2) + S(1)/(S(8)*(x + 1)**3) + S(3)/(S(16)*(x - 1)**2) - S(1)/(S(8)*(x - 1)**3))) == 0
assert ExpandIntegrand(-S(1), 1/((-q - x)**3*(q - x)**3), x) == 1/(8*q**3*(q + x)**3) - 1/(8*q**3*(-q + x)**3) - 3/(8*q**4*(-q**2 + x**2)) + 3/(16*q**4*(q + x)**2) + 3/(16*q**4*(-q + x)**2)
assert ExpandIntegrand((1 + 1*x)**(3)/(2 + 1*x), x) == x**2 + x + 1 - 1/(x + 2)
assert ExpandIntegrand((c + d*x**1 + e*x**2)/(1 - x**3), x) == (c - (-1)**(S(1)/3)*d + (-1)**(S(2)/3)*e)/(-3*(-1)**(S(2)/3)*x + 3) + (c + (-1)**(S(2)/3)*d - (-1)**(S(1)/3)*e)/(3*(-1)**(S(1)/3)*x + 3) + (c + d + e)/(-3*x + 3)
assert ExpandIntegrand((c + d*x**1 + e*x**2 + f*x**3)/(1 - x**4), x) == (c + I*d - e - I*f)/(4*I*x + 4) + (c - I*d - e + I*f)/(-4*I*x + 4) + (c - d + e - f)/(4*x + 4) + (c + d + e + f)/(-4*x + 4)
assert ExpandIntegrand((d + e*(f + g*x))/(2 + 3*x + 1*x**2), x) == (-2*d - 2*e*f + 4*e*g)/(2*x + 4) + (2*d + 2*e*f - 2*e*g)/(2*x + 2)
assert ExpandIntegrand(x/(a*x**3 + b*Sqrt(c + d*x**6)), x) == a*x**4/(-b**2*c + x**6*(a**2 - b**2*d)) + b*x*sqrt(c + d*x**6)/(b**2*c + x**6*(-a**2 + b**2*d))
assert simplify(ExpandIntegrand(x**1*(1 - x**4)**(-2), x) - (x/(S(4)*(x**2 + 1)) + x/(S(4)*(x**2 + 1)**2) - x/(S(4)*(x**2 - 1)) + x/(S(4)*(x**2 - 1)**2))) == 0
assert simplify(ExpandIntegrand((-1 + x**S(6))**(-3), x) - (S(3)/(S(8)*(x**6 - 1)) - S(3)/(S(16)*(x**S(3) + S(1))**S(2)) - S(1)/(S(8)*(x**S(3) + S(1))**S(3)) - S(3)/(S(16)*(x**S(3) - S(1))**S(2)) + S(1)/(S(8)*(x**S(3) - S(1))**S(3)))) == 0
assert simplify(ExpandIntegrand(u**1*(a + b*u**2 + c*u**4)**(-1), x)) == simplify(1/(2*b*(u + sqrt(-(a + c*u**4)/b))) - 1/(2*b*(-u + sqrt(-(a + c*u**4)/b))))
assert simplify(ExpandIntegrand((1 + 1*u + 1*u**2)**(-2), x) - (S(1)/(S(2)*(-u - 1)*(-u**2 - u - 1)) + S(1)/(S(4)*(-u - 1)*(u + sqrt(-u - 1))**2) + S(1)/(S(4)*(-u - 1)*(u - sqrt(-u - 1))**2))) == 0
assert ExpandIntegrand(x*(a + b*Log(c*(d*(e + f*x)**p)**q))**n, x) == -e*(a + b*log(c*(d*(e + f*x)**p)**q))**n/f + (a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)/f
assert ExpandIntegrand(x*f**(e*(c + d*x)*S(1)), x) == f**(e*(c + d*x))*x
assert simplify(ExpandIntegrand((x)*(a + b*x)**m*Log(c*(d + e*x**n)**p), x) - (-a*(a + b*x)**m*log(c*(d + e*x**n)**p)/b + (a + b*x)**(m + S(1))*log(c*(d + e*x**n)**p)/b)) == 0
assert simplify(ExpandIntegrand(u*(a + b*F**v)**S(2)*(c + d*F**v)**S(-3), x) - (b**2*u/(d**2*(F**v*d + c)) + 2*b*u*(a*d - b*c)/(d**2*(F**v*d + c)**2) + u*(a*d - b*c)**2/(d**2*(F**v*d + c)**3))) == 0
assert ExpandIntegrand((S(1) + 1*x)**S(2)*f**(e*(1 + S(1)*x)**n)/(g + h*x), x) == f**(e*(x + 1)**n)*(x + 1)/h + f**(e*(x + 1)**n)*(-g + h)/h**2 + f**(e*(x + 1)**n)*(g - h)**2/(h**2*(g + h*x))
assert ExpandIntegrand((a*c - b*c*x)**2/(a + b*x)**2, x) == 4*a**2*c**2/(a + b*x)**2 - 4*a*c**2/(a + b*x) + c**2
assert simplify(ExpandIntegrand(x**2*(1 - 1*x**2)**(-2), x) - (1/(S(2)*(x**2 - 1)) + 1/(S(4)*(x + 1)**2) + 1/(S(4)*(x - 1)**2))) == 0
assert ExpandIntegrand((a + x)**2, x) == a**2 + 2*a*x + x**2
assert ExpandIntegrand((a + b*x)**S(2)/x**3, x) == a**2/x**3 + 2*a*b/x**2 + b**2/x
assert ExpandIntegrand(1/(x**2*(a + b*x)**2), x) == b**2/(a**2*(a + b*x)**2) + 1/(a**2*x**2) + 2*b**2/(a**3*(a + b*x)) - 2*b/(a**3*x)
assert ExpandIntegrand((1 + x)**3/x, x) == x**2 + 3*x + 3 + 1/x
assert ExpandIntegrand((1 + 2*(3 + 4*x**2))/(2 + 3*x**2 + 1*x**4), x) == 18/(2*x**2 + 4) - 2/(2*x**2 + 2)
assert ExpandIntegrand((c + d*x**2 + e*x**3)/(1 - 1*x**4), x) == (c - d - I*e)/(4*I*x + 4) + (c - d + I*e)/(-4*I*x + 4) + (c + d - e)/(4*x + 4) + (c + d + e)/(-4*x + 4)
assert ExpandIntegrand((a + b*x)**2/(c + d*x), x) == b*(a + b*x)/d + b*(a*d - b*c)/d**2 + (a*d - b*c)**2/(d**2*(c + d*x))
assert ExpandIntegrand(x**2*(a + b*Log(c*(d*(e + f*x)**p)**q))**n, x) == e**2*(a + b*log(c*(d*(e + f*x)**p)**q))**n/f**2 - 2*e*(a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)/f**2 + (a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)**2/f**2
assert ExpandIntegrand(x*(1 + 2*x)**3*log(2*(1 + 1*x**2)**1), x) == 8*x**4*log(2*x**2 + 2) + 12*x**3*log(2*x**2 + 2) + 6*x**2*log(2*x**2 + 2) + x*log(2*x**2 + 2)
assert ExpandIntegrand((1 + 1*x)**S(3)*f**(e*(1 + 1*x)**n)/(g + h*x), x) == f**(e*(x + 1)**n)*(x + 1)**2/h + f**(e*(x + 1)**n)*(-g + h)*(x + 1)/h**2 + f**(e*(x + 1)**n)*(-g + h)**2/h**3 - f**(e*(x + 1)**n)*(g - h)**3/(h**3*(g + h*x))
def test_Dist():
assert Dist(x, a + b, x) == a*x + b*x
assert Dist(x, Integral(a + b , x), x) == x*Integral(a + b, x)
assert Dist(3*x,(a+b), x) - Dist(2*x, (a+b), x) == a*x + b*x
assert Dist(3*x,(a+b), x) + Dist(2*x, (a+b), x) == 5*a*x + 5*b*x
assert Dist(x, c*Integral((a + b), x), x) == c*x*Integral(a + b, x)
def test_IntegralFreeQ():
assert not IntegralFreeQ(Integral(a, x))
assert IntegralFreeQ(a + b)
def test_OneQ():
from sympy.integrals.rubi.utility_function import OneQ
assert OneQ(S(1))
assert not OneQ(S(2))
def test_DerivativeDivides():
assert not DerivativeDivides(x, x, x)
assert not DerivativeDivides(a, x + y, b)
assert DerivativeDivides(a + x, a, x) == a
assert DerivativeDivides(a + b, x + y, b) == x + y
def test_LogIntegral():
from sympy.integrals.rubi.utility_function import LogIntegral
assert LogIntegral(a) == li(a)
def test_SinIntegral():
from sympy.integrals.rubi.utility_function import SinIntegral
assert SinIntegral(a) == Si(a)
def test_CosIntegral():
from sympy.integrals.rubi.utility_function import CosIntegral
assert CosIntegral(a) == Ci(a)
def test_SinhIntegral():
from sympy.integrals.rubi.utility_function import SinhIntegral
assert SinhIntegral(a) == Shi(a)
def test_CoshIntegral():
from sympy.integrals.rubi.utility_function import CoshIntegral
assert CoshIntegral(a) == Chi(a)
def test_ExpIntegralEi():
from sympy.integrals.rubi.utility_function import ExpIntegralEi
assert ExpIntegralEi(a) == Ei(a)
def test_ExpIntegralE():
from sympy.integrals.rubi.utility_function import ExpIntegralE
assert ExpIntegralE(a, z) == expint(a, z)
def test_LogGamma():
from sympy.integrals.rubi.utility_function import LogGamma
assert LogGamma(a) == loggamma(a)
def test_Factorial():
from sympy.integrals.rubi.utility_function import Factorial
assert Factorial(S(5)) == 120
def test_Zeta():
from sympy.integrals.rubi.utility_function import Zeta
assert Zeta(a, z) == zeta(a, z)
def test_HypergeometricPFQ():
from sympy.integrals.rubi.utility_function import HypergeometricPFQ
assert HypergeometricPFQ([a, b], [c], z) == hyper([a, b], [c], z)
def test_PolyGamma():
assert PolyGamma(S(2), S(3)) == polygamma(2, 3)
def test_ProductLog():
from sympy import N
assert N(ProductLog(S(5.0)), 5) == N(1.32672466524220, 5)
assert N(ProductLog(S(2), S(3.5)), 5) == N(-1.14064876353898 + 10.8912237027092*I, 5)
def test_PolynomialQuotient():
assert PolynomialQuotient(log((-a*d + b*c)/(b*(c + d*x)))/(c + d*x), a + b*x, e) == log((-a*d + b*c)/(b*(c + d*x)))/((a + b*x)*(c + d*x))
assert PolynomialQuotient(x**2, x + a, x) == -a + x
def test_PolynomialRemainder():
assert PolynomialRemainder(log((-a*d + b*c)/(b*(c + d*x)))/(c + d*x), a + b*x, e) == 0
assert PolynomialRemainder(x**2, x + a, x) == a**2
def test_Floor():
assert Floor(S(7.5)) == 7
assert Floor(S(15.5), S(6)) == 12
def test_Factor():
from sympy.integrals.rubi.utility_function import Factor
assert Factor(a*b + a*c) == a*(b + c)
def test_Rule():
from sympy.integrals.rubi.utility_function import Rule
assert Rule(x, S(5)) == {x: 5}
def test_Distribute():
assert Distribute((a + b)*c + (a + b)*d, Add) == c*(a + b) + d*(a + b)
assert Distribute((a + b)*(c + e), Add) == a*c + a*e + b*c + b*e
def test_CoprimeQ():
assert CoprimeQ(S(7), S(5))
assert not CoprimeQ(S(6), S(3))
def test_Discriminant():
from sympy.integrals.rubi.utility_function import Discriminant
assert Discriminant(a*x**2 + b*x + c, x) == b**2 - 4*a*c
assert Discriminant(1/x, x) == Discriminant(1/x, x)
def test_Sum_doit():
assert Sum_doit(2*x + 2, [x, 0, 1.7]) == 6
def test_DeactivateTrig():
assert DeactivateTrig(sec(a + b*x), x) == sec(a + b*x)
def test_Negative():
from sympy.integrals.rubi.utility_function import Negative
assert Negative(S(-2))
assert not Negative(S(0))
def test_Quotient():
from sympy.integrals.rubi.utility_function import Quotient
assert Quotient(17, 5) == 3
def test_process_trig():
assert process_trig(x*cot(x)) == x/tan(x)
assert process_trig(coth(x)*csc(x)) == S(1)/(tanh(x)*sin(x))
def test_replace_pow_exp():
assert replace_pow_exp(rubi_exp(S(5))) == exp(S(5))
def test_rubi_unevaluated_expr():
from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr
assert rubi_unevaluated_expr(a)*rubi_unevaluated_expr(b) == rubi_unevaluated_expr(b)*rubi_unevaluated_expr(a)
def test_rubi_exp():
# class name in utility_function is `exp`. To avoid confusion `rubi_exp` has been used here
assert isinstance(rubi_exp(a), Pow)
def test_rubi_log():
# class name in utility_function is `log`. To avoid confusion `rubi_log` has been used here
assert rubi_log(rubi_exp(S(a))) == a
|
8a2310632849c8155302b0eecbefe8b6db6dffde8af35617309bc9518a014f94 | from sympy import (Add, Basic, Expr, S, Symbol, Wild, Float, Integer, Rational, I,
sin, cos, tan, exp, log, nan, oo, sqrt, symbols, Integral, sympify,
WildFunction, Poly, Function, Derivative, Number, pi, NumberSymbol, zoo,
Piecewise, Mul, Pow, nsimplify, ratsimp, trigsimp, radsimp, powsimp,
simplify, together, collect, factorial, apart, combsimp, factor, refine,
cancel, Tuple, default_sort_key, DiracDelta, gamma, Dummy, Sum, E,
exp_polar, expand, diff, O, Heaviside, Si, Max, UnevaluatedExpr,
integrate, gammasimp)
from sympy.core.expr import ExprBuilder
from sympy.core.function import AppliedUndef
from sympy.core.compatibility import range
from sympy.physics.secondquant import FockState
from sympy.physics.units import meter
from sympy.utilities.pytest import raises, XFAIL
from sympy.abc import a, b, c, n, t, u, x, y, z
class DummyNumber(object):
"""
Minimal implementation of a number that works with SymPy.
If one has a Number class (e.g. Sage Integer, or some other custom class)
that one wants to work well with SymPy, one has to implement at least the
methods of this class DummyNumber, resp. its subclasses I5 and F1_1.
Basically, one just needs to implement either __int__() or __float__() and
then one needs to make sure that the class works with Python integers and
with itself.
"""
def __radd__(self, a):
if isinstance(a, (int, float)):
return a + self.number
return NotImplemented
def __truediv__(a, b):
return a.__div__(b)
def __rtruediv__(a, b):
return a.__rdiv__(b)
def __add__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number + a
return NotImplemented
def __rsub__(self, a):
if isinstance(a, (int, float)):
return a - self.number
return NotImplemented
def __sub__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number - a
return NotImplemented
def __rmul__(self, a):
if isinstance(a, (int, float)):
return a * self.number
return NotImplemented
def __mul__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number * a
return NotImplemented
def __rdiv__(self, a):
if isinstance(a, (int, float)):
return a / self.number
return NotImplemented
def __div__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number / a
return NotImplemented
def __rpow__(self, a):
if isinstance(a, (int, float)):
return a ** self.number
return NotImplemented
def __pow__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number ** a
return NotImplemented
def __pos__(self):
return self.number
def __neg__(self):
return - self.number
class I5(DummyNumber):
number = 5
def __int__(self):
return self.number
class F1_1(DummyNumber):
number = 1.1
def __float__(self):
return self.number
i5 = I5()
f1_1 = F1_1()
# basic sympy objects
basic_objs = [
Rational(2),
Float("1.3"),
x,
y,
pow(x, y)*y,
]
# all supported objects
all_objs = basic_objs + [
5,
5.5,
i5,
f1_1
]
def dotest(s):
for x in all_objs:
for y in all_objs:
s(x, y)
return True
def test_basic():
def j(a, b):
x = a
x = +a
x = -a
x = a + b
x = a - b
x = a*b
x = a/b
x = a**b
assert dotest(j)
def test_ibasic():
def s(a, b):
x = a
x += b
x = a
x -= b
x = a
x *= b
x = a
x /= b
assert dotest(s)
def test_relational():
from sympy import Lt
assert (pi < 3) is S.false
assert (pi <= 3) is S.false
assert (pi > 3) is S.true
assert (pi >= 3) is S.true
assert (-pi < 3) is S.true
assert (-pi <= 3) is S.true
assert (-pi > 3) is S.false
assert (-pi >= 3) is S.false
r = Symbol('r', real=True)
assert (r - 2 < r - 3) is S.false
assert Lt(x + I, x + I + 2).func == Lt # issue 8288
def test_relational_assumptions():
from sympy import Lt, Gt, Le, Ge
m1 = Symbol("m1", nonnegative=False)
m2 = Symbol("m2", positive=False)
m3 = Symbol("m3", nonpositive=False)
m4 = Symbol("m4", negative=False)
assert (m1 < 0) == Lt(m1, 0)
assert (m2 <= 0) == Le(m2, 0)
assert (m3 > 0) == Gt(m3, 0)
assert (m4 >= 0) == Ge(m4, 0)
m1 = Symbol("m1", nonnegative=False, real=True)
m2 = Symbol("m2", positive=False, real=True)
m3 = Symbol("m3", nonpositive=False, real=True)
m4 = Symbol("m4", negative=False, real=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=True)
m2 = Symbol("m2", nonpositive=True)
m3 = Symbol("m3", positive=True)
m4 = Symbol("m4", nonnegative=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=False, real=True)
m2 = Symbol("m2", nonpositive=False, real=True)
m3 = Symbol("m3", positive=False, real=True)
m4 = Symbol("m4", nonnegative=False, real=True)
assert (m1 < 0) is S.false
assert (m2 <= 0) is S.false
assert (m3 > 0) is S.false
assert (m4 >= 0) is S.false
def test_relational_noncommutative():
from sympy import Lt, Gt, Le, Ge
A, B = symbols('A,B', commutative=False)
assert (A < B) == Lt(A, B)
assert (A <= B) == Le(A, B)
assert (A > B) == Gt(A, B)
assert (A >= B) == Ge(A, B)
def test_basic_nostr():
for obj in basic_objs:
raises(TypeError, lambda: obj + '1')
raises(TypeError, lambda: obj - '1')
if obj == 2:
assert obj * '1' == '11'
else:
raises(TypeError, lambda: obj * '1')
raises(TypeError, lambda: obj / '1')
raises(TypeError, lambda: obj ** '1')
def test_series_expansion_for_uniform_order():
assert (1/x + y + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + x).series(x, 0, 1) == 1/x + y + O(x)
assert (1/x + 1 + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + 1 + x).series(x, 0, 1) == 1/x + 1 + O(x)
assert (1/x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 1) == 1/x + y + O(x)
def test_leadterm():
assert (3 + 2*x**(log(3)/log(2) - 1)).leadterm(x) == (3, 0)
assert (1/x**2 + 1 + x + x**2).leadterm(x)[1] == -2
assert (1/x + 1 + x + x**2).leadterm(x)[1] == -1
assert (x**2 + 1/x).leadterm(x)[1] == -1
assert (1 + x**2).leadterm(x)[1] == 0
assert (x + 1).leadterm(x)[1] == 0
assert (x + x**2).leadterm(x)[1] == 1
assert (x**2).leadterm(x)[1] == 2
def test_as_leading_term():
assert (3 + 2*x**(log(3)/log(2) - 1)).as_leading_term(x) == 3
assert (1/x**2 + 1 + x + x**2).as_leading_term(x) == 1/x**2
assert (1/x + 1 + x + x**2).as_leading_term(x) == 1/x
assert (x**2 + 1/x).as_leading_term(x) == 1/x
assert (1 + x**2).as_leading_term(x) == 1
assert (x + 1).as_leading_term(x) == 1
assert (x + x**2).as_leading_term(x) == x
assert (x**2).as_leading_term(x) == x**2
assert (x + oo).as_leading_term(x) == oo
raises(ValueError, lambda: (x + 1).as_leading_term(1))
def test_leadterm2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).leadterm(x) == \
(sin(1 + sin(1)), 0)
def test_leadterm3():
assert (y + z + x).leadterm(x) == (y + z, 0)
def test_as_leading_term2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).as_leading_term(x) == \
sin(1 + sin(1))
def test_as_leading_term3():
assert (2 + pi + x).as_leading_term(x) == 2 + pi
assert (2*x + pi*x + x**2).as_leading_term(x) == (2 + pi)*x
def test_as_leading_term4():
# see issue 6843
n = Symbol('n', integer=True, positive=True)
r = -n**3/(2*n**2 + 4*n + 2) - n**2/(n**2 + 2*n + 1) + \
n**2/(n + 1) - n/(2*n**2 + 4*n + 2) + n/(n*x + x) + 2*n/(n + 1) - \
1 + 1/(n*x + x) + 1/(n + 1) - 1/x
assert r.as_leading_term(x).cancel() == n/2
def test_as_leading_term_stub():
class foo(Function):
pass
assert foo(1/x).as_leading_term(x) == foo(1/x)
assert foo(1).as_leading_term(x) == foo(1)
raises(NotImplementedError, lambda: foo(x).as_leading_term(x))
def test_as_leading_term_deriv_integral():
# related to issue 11313
assert Derivative(x ** 3, x).as_leading_term(x) == 3*x**2
assert Derivative(x ** 3, y).as_leading_term(x) == 0
assert Integral(x ** 3, x).as_leading_term(x) == x**4/4
assert Integral(x ** 3, y).as_leading_term(x) == y*x**3
assert Derivative(exp(x), x).as_leading_term(x) == 1
assert Derivative(log(x), x).as_leading_term(x) == (1/x).as_leading_term(x)
def test_atoms():
assert x.atoms() == {x}
assert (1 + x).atoms() == {x, S(1)}
assert (1 + 2*cos(x)).atoms(Symbol) == {x}
assert (1 + 2*cos(x)).atoms(Symbol, Number) == {S(1), S(2), x}
assert (2*(x**(y**x))).atoms() == {S(2), x, y}
assert Rational(1, 2).atoms() == {S.Half}
assert Rational(1, 2).atoms(Symbol) == set([])
assert sin(oo).atoms(oo) == set()
assert Poly(0, x).atoms() == {S.Zero}
assert Poly(1, x).atoms() == {S.One}
assert Poly(x, x).atoms() == {x}
assert Poly(x, x, y).atoms() == {x}
assert Poly(x + y, x, y).atoms() == {x, y}
assert Poly(x + y, x, y, z).atoms() == {x, y}
assert Poly(x + y*t, x, y, z).atoms() == {t, x, y}
assert (I*pi).atoms(NumberSymbol) == {pi}
assert (I*pi).atoms(NumberSymbol, I) == \
(I*pi).atoms(I, NumberSymbol) == {pi, I}
assert exp(exp(x)).atoms(exp) == {exp(exp(x)), exp(x)}
assert (1 + x*(2 + y) + exp(3 + z)).atoms(Add) == \
{1 + x*(2 + y) + exp(3 + z), 2 + y, 3 + z}
# issue 6132
f = Function('f')
e = (f(x) + sin(x) + 2)
assert e.atoms(AppliedUndef) == \
{f(x)}
assert e.atoms(AppliedUndef, Function) == \
{f(x), sin(x)}
assert e.atoms(Function) == \
{f(x), sin(x)}
assert e.atoms(AppliedUndef, Number) == \
{f(x), S(2)}
assert e.atoms(Function, Number) == \
{S(2), sin(x), f(x)}
def test_is_polynomial():
k = Symbol('k', nonnegative=True, integer=True)
assert Rational(2).is_polynomial(x, y, z) is True
assert (S.Pi).is_polynomial(x, y, z) is True
assert x.is_polynomial(x) is True
assert x.is_polynomial(y) is True
assert (x**2).is_polynomial(x) is True
assert (x**2).is_polynomial(y) is True
assert (x**(-2)).is_polynomial(x) is False
assert (x**(-2)).is_polynomial(y) is True
assert (2**x).is_polynomial(x) is False
assert (2**x).is_polynomial(y) is True
assert (x**k).is_polynomial(x) is False
assert (x**k).is_polynomial(k) is False
assert (x**x).is_polynomial(x) is False
assert (k**k).is_polynomial(k) is False
assert (k**x).is_polynomial(k) is False
assert (x**(-k)).is_polynomial(x) is False
assert ((2*x)**k).is_polynomial(x) is False
assert (x**2 + 3*x - 8).is_polynomial(x) is True
assert (x**2 + 3*x - 8).is_polynomial(y) is True
assert (x**2 + 3*x - 8).is_polynomial() is True
assert sqrt(x).is_polynomial(x) is False
assert (sqrt(x)**3).is_polynomial(x) is False
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(x) is True
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(y) is False
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial() is True
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial() is False
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial(x, y) is True
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial(x, y) is False
def test_is_rational_function():
assert Integer(1).is_rational_function() is True
assert Integer(1).is_rational_function(x) is True
assert Rational(17, 54).is_rational_function() is True
assert Rational(17, 54).is_rational_function(x) is True
assert (12/x).is_rational_function() is True
assert (12/x).is_rational_function(x) is True
assert (x/y).is_rational_function() is True
assert (x/y).is_rational_function(x) is True
assert (x/y).is_rational_function(x, y) is True
assert (x**2 + 1/x/y).is_rational_function() is True
assert (x**2 + 1/x/y).is_rational_function(x) is True
assert (x**2 + 1/x/y).is_rational_function(x, y) is True
assert (sin(y)/x).is_rational_function() is False
assert (sin(y)/x).is_rational_function(y) is False
assert (sin(y)/x).is_rational_function(x) is True
assert (sin(y)/x).is_rational_function(x, y) is False
assert (S.NaN).is_rational_function() is False
assert (S.Infinity).is_rational_function() is False
assert (-S.Infinity).is_rational_function() is False
assert (S.ComplexInfinity).is_rational_function() is False
def test_is_algebraic_expr():
assert sqrt(3).is_algebraic_expr(x) is True
assert sqrt(3).is_algebraic_expr() is True
eq = ((1 + x**2)/(1 - y**2))**(S(1)/3)
assert eq.is_algebraic_expr(x) is True
assert eq.is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(x) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr() is True
assert (cos(y)/sqrt(x)).is_algebraic_expr() is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x) is True
assert (cos(y)/sqrt(x)).is_algebraic_expr(y) is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x, y) is False
def test_SAGE1():
#see https://github.com/sympy/sympy/issues/3346
class MyInt:
def _sympy_(self):
return Integer(5)
m = MyInt()
e = Rational(2)*m
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE2():
class MyInt(object):
def __int__(self):
return 5
assert sympify(MyInt()) == 5
e = Rational(2)*MyInt()
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE3():
class MySymbol:
def __rmul__(self, other):
return ('mys', other, self)
o = MySymbol()
e = x*o
assert e == ('mys', x, o)
def test_len():
e = x*y
assert len(e.args) == 2
e = x + y + z
assert len(e.args) == 3
def test_doit():
a = Integral(x**2, x)
assert isinstance(a.doit(), Integral) is False
assert isinstance(a.doit(integrals=True), Integral) is False
assert isinstance(a.doit(integrals=False), Integral) is True
assert (2*Integral(x, x)).doit() == x**2
def test_attribute_error():
raises(AttributeError, lambda: x.cos())
raises(AttributeError, lambda: x.sin())
raises(AttributeError, lambda: x.exp())
def test_args():
assert (x*y).args in ((x, y), (y, x))
assert (x + y).args in ((x, y), (y, x))
assert (x*y + 1).args in ((x*y, 1), (1, x*y))
assert sin(x*y).args == (x*y,)
assert sin(x*y).args[0] == x*y
assert (x**y).args == (x, y)
assert (x**y).args[0] == x
assert (x**y).args[1] == y
def test_noncommutative_expand_issue_3757():
A, B, C = symbols('A,B,C', commutative=False)
assert A*B - B*A != 0
assert (A*(A + B)*B).expand() == A**2*B + A*B**2
assert (A*(A + B + C)*B).expand() == A**2*B + A*B**2 + A*C*B
def test_as_numer_denom():
a, b, c = symbols('a, b, c')
assert nan.as_numer_denom() == (nan, 1)
assert oo.as_numer_denom() == (oo, 1)
assert (-oo).as_numer_denom() == (-oo, 1)
assert zoo.as_numer_denom() == (zoo, 1)
assert (-zoo).as_numer_denom() == (zoo, 1)
assert x.as_numer_denom() == (x, 1)
assert (1/x).as_numer_denom() == (1, x)
assert (x/y).as_numer_denom() == (x, y)
assert (x/2).as_numer_denom() == (x, 2)
assert (x*y/z).as_numer_denom() == (x*y, z)
assert (x/(y*z)).as_numer_denom() == (x, y*z)
assert Rational(1, 2).as_numer_denom() == (1, 2)
assert (1/y**2).as_numer_denom() == (1, y**2)
assert (x/y**2).as_numer_denom() == (x, y**2)
assert ((x**2 + 1)/y).as_numer_denom() == (x**2 + 1, y)
assert (x*(y + 1)/y**7).as_numer_denom() == (x*(y + 1), y**7)
assert (x**-2).as_numer_denom() == (1, x**2)
assert (a/x + b/2/x + c/3/x).as_numer_denom() == \
(6*a + 3*b + 2*c, 6*x)
assert (a/x + b/2/x + c/3/y).as_numer_denom() == \
(2*c*x + y*(6*a + 3*b), 6*x*y)
assert (a/x + b/2/x + c/.5/x).as_numer_denom() == \
(2*a + b + 4.0*c, 2*x)
# this should take no more than a few seconds
assert int(log(Add(*[Dummy()/i/x for i in range(1, 705)]
).as_numer_denom()[1]/x).n(4)) == 705
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).as_numer_denom() == \
(x + i, 3)
assert (S.Infinity + x/3 + y/4).as_numer_denom() == \
(4*x + 3*y + S.Infinity, 12)
assert (oo*x + zoo*y).as_numer_denom() == \
(zoo*y + oo*x, 1)
A, B, C = symbols('A,B,C', commutative=False)
assert (A*B*C**-1).as_numer_denom() == (A*B*C**-1, 1)
assert (A*B*C**-1/x).as_numer_denom() == (A*B*C**-1, x)
assert (C**-1*A*B).as_numer_denom() == (C**-1*A*B, 1)
assert (C**-1*A*B/x).as_numer_denom() == (C**-1*A*B, x)
assert ((A*B*C)**-1).as_numer_denom() == ((A*B*C)**-1, 1)
assert ((A*B*C)**-1/x).as_numer_denom() == ((A*B*C)**-1, x)
def test_trunc():
import math
x, y = symbols('x y')
assert math.trunc(2) == 2
assert math.trunc(4.57) == 4
assert math.trunc(-5.79) == -5
assert math.trunc(pi) == 3
assert math.trunc(log(7)) == 1
assert math.trunc(exp(5)) == 148
assert math.trunc(cos(pi)) == -1
assert math.trunc(sin(5)) == 0
raises(TypeError, lambda: math.trunc(x))
raises(TypeError, lambda: math.trunc(x + y**2))
raises(TypeError, lambda: math.trunc(oo))
def test_as_independent():
assert S.Zero.as_independent(x, as_Add=True) == (0, 0)
assert S.Zero.as_independent(x, as_Add=False) == (0, 0)
assert (2*x*sin(x) + y + x).as_independent(x) == (y, x + 2*x*sin(x))
assert (2*x*sin(x) + y + x).as_independent(y) == (x + 2*x*sin(x), y)
assert (2*x*sin(x) + y + x).as_independent(x, y) == (0, y + x + 2*x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(x) == (cos(y), x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(y) == (x*sin(x), cos(y))
assert (x*sin(x)*cos(y)).as_independent(x, y) == (1, x*sin(x)*cos(y))
assert (sin(x)).as_independent(x) == (1, sin(x))
assert (sin(x)).as_independent(y) == (sin(x), 1)
assert (2*sin(x)).as_independent(x) == (2, sin(x))
assert (2*sin(x)).as_independent(y) == (2*sin(x), 1)
# issue 4903 = 1766b
n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
assert (n1 + n1*n2).as_independent(n2) == (n1, n1*n2)
assert (n2*n1 + n1*n2).as_independent(n2) == (0, n1*n2 + n2*n1)
assert (n1*n2*n1).as_independent(n2) == (n1, n2*n1)
assert (n1*n2*n1).as_independent(n1) == (1, n1*n2*n1)
assert (3*x).as_independent(x, as_Add=True) == (0, 3*x)
assert (3*x).as_independent(x, as_Add=False) == (3, x)
assert (3 + x).as_independent(x, as_Add=True) == (3, x)
assert (3 + x).as_independent(x, as_Add=False) == (1, 3 + x)
# issue 5479
assert (3*x).as_independent(Symbol) == (3, x)
# issue 5648
assert (n1*x*y).as_independent(x) == (n1*y, x)
assert ((x + n1)*(x - y)).as_independent(x) == (1, (x + n1)*(x - y))
assert ((x + n1)*(x - y)).as_independent(y) == (x + n1, x - y)
assert (DiracDelta(x - n1)*DiracDelta(x - y)).as_independent(x) \
== (1, DiracDelta(x - n1)*DiracDelta(x - y))
assert (x*y*n1*n2*n3).as_independent(n2) == (x*y*n1, n2*n3)
assert (x*y*n1*n2*n3).as_independent(n1) == (x*y, n1*n2*n3)
assert (x*y*n1*n2*n3).as_independent(n3) == (x*y*n1*n2, n3)
assert (DiracDelta(x - n1)*DiracDelta(y - n1)*DiracDelta(x - n2)).as_independent(y) == \
(DiracDelta(x - n1)*DiracDelta(x - n2), DiracDelta(y - n1))
# issue 5784
assert (x + Integral(x, (x, 1, 2))).as_independent(x, strict=True) == \
(Integral(x, (x, 1, 2)), x)
eq = Add(x, -x, 2, -3, evaluate=False)
assert eq.as_independent(x) == (-1, Add(x, -x, evaluate=False))
eq = Mul(x, 1/x, 2, -3, evaluate=False)
eq.as_independent(x) == (-6, Mul(x, 1/x, evaluate=False))
assert (x*y).as_independent(z, as_Add=True) == (x*y, 0)
@XFAIL
def test_call_2():
# TODO UndefinedFunction does not subclass Expr
f = Function('f')
assert (2*f)(x) == 2*f(x)
def test_replace():
f = log(sin(x)) + tan(sin(x**2))
assert f.replace(sin, cos) == log(cos(x)) + tan(cos(x**2))
assert f.replace(
sin, lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
a = Wild('a')
b = Wild('b')
assert f.replace(sin(a), cos(a)) == log(cos(x)) + tan(cos(x**2))
assert f.replace(
sin(a), lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
# test exact
assert (2*x).replace(a*x + b, b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, b - a) == 2*x
assert (2*x).replace(a*x + b, b - a, exact=False) == 2/x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=False) == 2/x
g = 2*sin(x**3)
assert g.replace(
lambda expr: expr.is_Number, lambda expr: expr**2) == 4*sin(x**9)
assert cos(x).replace(cos, sin, map=True) == (sin(x), {cos(x): sin(x)})
assert sin(x).replace(cos, sin) == sin(x)
cond, func = lambda x: x.is_Mul, lambda x: 2*x
assert (x*y).replace(cond, func, map=True) == (2*x*y, {x*y: 2*x*y})
assert (x*(1 + x*y)).replace(cond, func, map=True) == \
(2*x*(2*x*y + 1), {x*(2*x*y + 1): 2*x*(2*x*y + 1), x*y: 2*x*y})
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y, map=True) == \
(sin(x), {sin(x): sin(x)/y})
# if not simultaneous then y*sin(x) -> y*sin(x)/y = sin(x) -> sin(x)/y
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y,
simultaneous=False) == sin(x)/y
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e) == O(1, x)
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e,
simultaneous=False) == x**2/2 + O(x**3)
assert (x*(x*y + 3)).replace(lambda x: x.is_Mul, lambda x: 2 + x) == \
x*(x*y + 5) + 2
e = (x*y + 1)*(2*x*y + 1) + 1
assert e.replace(cond, func, map=True) == (
2*((2*x*y + 1)*(4*x*y + 1)) + 1,
{2*x*y: 4*x*y, x*y: 2*x*y, (2*x*y + 1)*(4*x*y + 1):
2*((2*x*y + 1)*(4*x*y + 1))})
assert x.replace(x, y) == y
assert (x + 1).replace(1, 2) == x + 2
# https://groups.google.com/forum/#!topic/sympy/8wCgeC95tz0
n1, n2, n3 = symbols('n1:4', commutative=False)
f = Function('f')
assert (n1*f(n2)).replace(f, lambda x: x) == n1*n2
assert (n3*f(n2)).replace(f, lambda x: x) == n3*n2
def test_find():
expr = (x + y + 2 + sin(3*x))
assert expr.find(lambda u: u.is_Integer) == {S(2), S(3)}
assert expr.find(lambda u: u.is_Symbol) == {x, y}
assert expr.find(lambda u: u.is_Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(lambda u: u.is_Symbol, group=True) == {x: 2, y: 1}
assert expr.find(Integer) == {S(2), S(3)}
assert expr.find(Symbol) == {x, y}
assert expr.find(Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(Symbol, group=True) == {x: 2, y: 1}
a = Wild('a')
expr = sin(sin(x)) + sin(x) + cos(x) + x
assert expr.find(lambda u: type(u) is sin) == {sin(x), sin(sin(x))}
assert expr.find(
lambda u: type(u) is sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin(a)) == {sin(x), sin(sin(x))}
assert expr.find(sin(a), group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin) == {sin(x), sin(sin(x))}
assert expr.find(sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
def test_count():
expr = (x + y + 2 + sin(3*x))
assert expr.count(lambda u: u.is_Integer) == 2
assert expr.count(lambda u: u.is_Symbol) == 3
assert expr.count(Integer) == 2
assert expr.count(Symbol) == 3
assert expr.count(2) == 1
a = Wild('a')
assert expr.count(sin) == 1
assert expr.count(sin(a)) == 1
assert expr.count(lambda u: type(u) is sin) == 1
f = Function('f')
assert f(x).count(f(x)) == 1
assert f(x).diff(x).count(f(x)) == 1
assert f(x).diff(x).count(x) == 2
def test_has_basics():
f = Function('f')
g = Function('g')
p = Wild('p')
assert sin(x).has(x)
assert sin(x).has(sin)
assert not sin(x).has(y)
assert not sin(x).has(cos)
assert f(x).has(x)
assert f(x).has(f)
assert not f(x).has(y)
assert not f(x).has(g)
assert f(x).diff(x).has(x)
assert f(x).diff(x).has(f)
assert f(x).diff(x).has(Derivative)
assert not f(x).diff(x).has(y)
assert not f(x).diff(x).has(g)
assert not f(x).diff(x).has(sin)
assert (x**2).has(Symbol)
assert not (x**2).has(Wild)
assert (2*p).has(Wild)
assert not x.has()
def test_has_multiple():
f = x**2*y + sin(2**t + log(z))
assert f.has(x)
assert f.has(y)
assert f.has(z)
assert f.has(t)
assert not f.has(u)
assert f.has(x, y, z, t)
assert f.has(x, y, z, t, u)
i = Integer(4400)
assert not i.has(x)
assert (i*x**i).has(x)
assert not (i*y**i).has(x)
assert (i*y**i).has(x, y)
assert not (i*y**i).has(x, z)
def test_has_piecewise():
f = (x*y + 3/y)**(3 + 2)
g = Function('g')
h = Function('h')
p = Piecewise((g(x), x < -1), (1, x <= 1), (f, True))
assert p.has(x)
assert p.has(y)
assert not p.has(z)
assert p.has(1)
assert p.has(3)
assert not p.has(4)
assert p.has(f)
assert p.has(g)
assert not p.has(h)
def test_has_iterative():
A, B, C = symbols('A,B,C', commutative=False)
f = x*gamma(x)*sin(x)*exp(x*y)*A*B*C*cos(x*A*B)
assert f.has(x)
assert f.has(x*y)
assert f.has(x*sin(x))
assert not f.has(x*sin(y))
assert f.has(x*A)
assert f.has(x*A*B)
assert not f.has(x*A*C)
assert f.has(x*A*B*C)
assert not f.has(x*A*C*B)
assert f.has(x*sin(x)*A*B*C)
assert not f.has(x*sin(x)*A*C*B)
assert not f.has(x*sin(y)*A*B*C)
assert f.has(x*gamma(x))
assert not f.has(x + sin(x))
assert (x & y & z).has(x & z)
def test_has_integrals():
f = Integral(x**2 + sin(x*y*z), (x, 0, x + y + z))
assert f.has(x + y)
assert f.has(x + z)
assert f.has(y + z)
assert f.has(x*y)
assert f.has(x*z)
assert f.has(y*z)
assert not f.has(2*x + y)
assert not f.has(2*x*y)
def test_has_tuple():
f = Function('f')
g = Function('g')
h = Function('h')
assert Tuple(x, y).has(x)
assert not Tuple(x, y).has(z)
assert Tuple(f(x), g(x)).has(x)
assert not Tuple(f(x), g(x)).has(y)
assert Tuple(f(x), g(x)).has(f)
assert Tuple(f(x), g(x)).has(f(x))
assert not Tuple(f, g).has(x)
assert Tuple(f, g).has(f)
assert not Tuple(f, g).has(h)
assert Tuple(True).has(True) is True # .has(1) will also be True
def test_has_units():
from sympy.physics.units import m, s
assert (x*m/s).has(x)
assert (x*m/s).has(y, z) is False
def test_has_polys():
poly = Poly(x**2 + x*y*sin(z), x, y, t)
assert poly.has(x)
assert poly.has(x, y, z)
assert poly.has(x, y, z, t)
def test_has_physics():
assert FockState((x, y)).has(x)
def test_as_poly_as_expr():
f = x**2 + 2*x*y
assert f.as_poly().as_expr() == f
assert f.as_poly(x, y).as_expr() == f
assert (f + sin(x)).as_poly(x, y) is None
p = Poly(f, x, y)
assert p.as_poly() == p
def test_nonzero():
assert bool(S.Zero) is False
assert bool(S.One) is True
assert bool(x) is True
assert bool(x + y) is True
assert bool(x - x) is False
assert bool(x*y) is True
assert bool(x*1) is True
assert bool(x*0) is False
def test_is_number():
assert Float(3.14).is_number is True
assert Integer(737).is_number is True
assert Rational(3, 2).is_number is True
assert Rational(8).is_number is True
assert x.is_number is False
assert (2*x).is_number is False
assert (x + y).is_number is False
assert log(2).is_number is True
assert log(x).is_number is False
assert (2 + log(2)).is_number is True
assert (8 + log(2)).is_number is True
assert (2 + log(x)).is_number is False
assert (8 + log(2) + x).is_number is False
assert (1 + x**2/x - x).is_number is True
assert Tuple(Integer(1)).is_number is False
assert Add(2, x).is_number is False
assert Mul(3, 4).is_number is True
assert Pow(log(2), 2).is_number is True
assert oo.is_number is True
g = WildFunction('g')
assert g.is_number is False
assert (2*g).is_number is False
assert (x**2).subs(x, 3).is_number is True
# test extensibility of .is_number
# on subinstances of Basic
class A(Basic):
pass
a = A()
assert a.is_number is False
def test_as_coeff_add():
assert S(2).as_coeff_add() == (2, ())
assert S(3.0).as_coeff_add() == (0, (S(3.0),))
assert S(-3.0).as_coeff_add() == (0, (S(-3.0),))
assert x.as_coeff_add() == (0, (x,))
assert (x - 1).as_coeff_add() == (-1, (x,))
assert (x + 1).as_coeff_add() == (1, (x,))
assert (x + 2).as_coeff_add() == (2, (x,))
assert (x + y).as_coeff_add(y) == (x, (y,))
assert (3*x).as_coeff_add(y) == (3*x, ())
# don't do expansion
e = (x + y)**2
assert e.as_coeff_add(y) == (0, (e,))
def test_as_coeff_mul():
assert S(2).as_coeff_mul() == (2, ())
assert S(3.0).as_coeff_mul() == (1, (S(3.0),))
assert S(-3.0).as_coeff_mul() == (-1, (S(3.0),))
assert S(-3.0).as_coeff_mul(rational=False) == (-S(3.0), ())
assert x.as_coeff_mul() == (1, (x,))
assert (-x).as_coeff_mul() == (-1, (x,))
assert (2*x).as_coeff_mul() == (2, (x,))
assert (x*y).as_coeff_mul(y) == (x, (y,))
assert (3 + x).as_coeff_mul() == (1, (3 + x,))
assert (3 + x).as_coeff_mul(y) == (3 + x, ())
# don't do expansion
e = exp(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
e = 2**(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
assert (1.1*x).as_coeff_mul(rational=False) == (1.1, (x,))
assert (1.1*x).as_coeff_mul() == (1, (1.1, x))
assert (-oo*x).as_coeff_mul(rational=True) == (-1, (oo, x))
def test_as_coeff_exponent():
assert (3*x**4).as_coeff_exponent(x) == (3, 4)
assert (2*x**3).as_coeff_exponent(x) == (2, 3)
assert (4*x**2).as_coeff_exponent(x) == (4, 2)
assert (6*x**1).as_coeff_exponent(x) == (6, 1)
assert (3*x**0).as_coeff_exponent(x) == (3, 0)
assert (2*x**0).as_coeff_exponent(x) == (2, 0)
assert (1*x**0).as_coeff_exponent(x) == (1, 0)
assert (0*x**0).as_coeff_exponent(x) == (0, 0)
assert (-1*x**0).as_coeff_exponent(x) == (-1, 0)
assert (-2*x**0).as_coeff_exponent(x) == (-2, 0)
assert (2*x**3 + pi*x**3).as_coeff_exponent(x) == (2 + pi, 3)
assert (x*log(2)/(2*x + pi*x)).as_coeff_exponent(x) == \
(log(2)/(2 + pi), 0)
# issue 4784
D = Derivative
f = Function('f')
fx = D(f(x), x)
assert fx.as_coeff_exponent(f(x)) == (fx, 0)
def test_extractions():
assert ((x*y)**3).extract_multiplicatively(x**2 * y) == x*y**2
assert ((x*y)**3).extract_multiplicatively(x**4 * y) is None
assert (2*x).extract_multiplicatively(2) == x
assert (2*x).extract_multiplicatively(3) is None
assert (2*x).extract_multiplicatively(-1) is None
assert (Rational(1, 2)*x).extract_multiplicatively(3) == x/6
assert (sqrt(x)).extract_multiplicatively(x) is None
assert (sqrt(x)).extract_multiplicatively(1/x) is None
assert x.extract_multiplicatively(-x) is None
assert (-2 - 4*I).extract_multiplicatively(-2) == 1 + 2*I
assert (-2 - 4*I).extract_multiplicatively(3) is None
assert (-2*x - 4*y - 8).extract_multiplicatively(-2) == x + 2*y + 4
assert (-2*x*y - 4*x**2*y).extract_multiplicatively(-2*y) == 2*x**2 + x
assert (2*x*y + 4*x**2*y).extract_multiplicatively(2*y) == 2*x**2 + x
assert (-4*y**2*x).extract_multiplicatively(-3*y) is None
assert (2*x).extract_multiplicatively(1) == 2*x
assert (-oo).extract_multiplicatively(5) == -oo
assert (oo).extract_multiplicatively(5) == oo
assert ((x*y)**3).extract_additively(1) is None
assert (x + 1).extract_additively(x) == 1
assert (x + 1).extract_additively(2*x) is None
assert (x + 1).extract_additively(-x) is None
assert (-x + 1).extract_additively(2*x) is None
assert (2*x + 3).extract_additively(x) == x + 3
assert (2*x + 3).extract_additively(2) == 2*x + 1
assert (2*x + 3).extract_additively(3) == 2*x
assert (2*x + 3).extract_additively(-2) is None
assert (2*x + 3).extract_additively(3*x) is None
assert (2*x + 3).extract_additively(2*x) == 3
assert x.extract_additively(0) == x
assert S(2).extract_additively(x) is None
assert S(2.).extract_additively(2) == S.Zero
assert S(2*x + 3).extract_additively(x + 1) == x + 2
assert S(2*x + 3).extract_additively(y + 1) is None
assert S(2*x - 3).extract_additively(x + 1) is None
assert S(2*x - 3).extract_additively(y + z) is None
assert ((a + 1)*x*4 + y).extract_additively(x).expand() == \
4*a*x + 3*x + y
assert ((a + 1)*x*4 + 3*y).extract_additively(x + 2*y).expand() == \
4*a*x + 3*x + y
assert (y*(x + 1)).extract_additively(x + 1) is None
assert ((y + 1)*(x + 1) + 3).extract_additively(x + 1) == \
y*(x + 1) + 3
assert ((x + y)*(x + 1) + x + y + 3).extract_additively(x + y) == \
x*(x + y) + 3
assert (x + y + 2*((x + y)*(x + 1)) + 3).extract_additively((x + y)*(x + 1)) == \
x + y + (x + 1)*(x + y) + 3
assert ((y + 1)*(x + 2*y + 1) + 3).extract_additively(y + 1) == \
(x + 2*y)*(y + 1) + 3
n = Symbol("n", integer=True)
assert (Integer(-3)).could_extract_minus_sign() is True
assert (-n*x + x).could_extract_minus_sign() != \
(n*x - x).could_extract_minus_sign()
assert (x - y).could_extract_minus_sign() != \
(-x + y).could_extract_minus_sign()
assert (1 - x - y).could_extract_minus_sign() is True
assert (1 - x + y).could_extract_minus_sign() is False
assert ((-x - x*y)/y).could_extract_minus_sign() is True
assert (-(x + x*y)/y).could_extract_minus_sign() is True
assert ((x + x*y)/(-y)).could_extract_minus_sign() is True
assert ((x + x*y)/y).could_extract_minus_sign() is False
assert (x*(-x - x**3)).could_extract_minus_sign() is True
assert ((-x - y)/(x + y)).could_extract_minus_sign() is True
class sign_invariant(Function, Expr):
nargs = 1
def __neg__(self):
return self
foo = sign_invariant(x)
assert foo == -foo
assert foo.could_extract_minus_sign() is False
# The results of each of these will vary on different machines, e.g.
# the first one might be False and the other (then) is true or vice versa,
# so both are included.
assert ((-x - y)/(x - y)).could_extract_minus_sign() is False or \
((-x - y)/(y - x)).could_extract_minus_sign() is False
assert (x - y).could_extract_minus_sign() is False
assert (-x + y).could_extract_minus_sign() is True
def test_nan_extractions():
for r in (1, 0, I, nan):
assert nan.extract_additively(r) is None
assert nan.extract_multiplicatively(r) is None
def test_coeff():
assert (x + 1).coeff(x + 1) == 1
assert (3*x).coeff(0) == 0
assert (z*(1 + x)*x**2).coeff(1 + x) == z*x**2
assert (1 + 2*x*x**(1 + x)).coeff(x*x**(1 + x)) == 2
assert (1 + 2*x**(y + z)).coeff(x**(y + z)) == 2
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (3 + 2*x + 4*x**2).coeff(-1) == 0
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (-x/8 + x*y).coeff(x) == -S(1)/8 + y
assert (-x/8 + x*y).coeff(-x) == S(1)/8
assert (4*x).coeff(2*x) == 0
assert (2*x).coeff(2*x) == 1
assert (-oo*x).coeff(x*oo) == -1
assert (10*x).coeff(x, 0) == 0
assert (10*x).coeff(10*x, 0) == 0
n1, n2 = symbols('n1 n2', commutative=False)
assert (n1*n2).coeff(n1) == 1
assert (n1*n2).coeff(n2) == n1
assert (n1*n2 + x*n1).coeff(n1) == 1 # 1*n1*(n2+x)
assert (n2*n1 + x*n1).coeff(n1) == n2 + x
assert (n2*n1 + x*n1**2).coeff(n1) == n2
assert (n1**x).coeff(n1) == 0
assert (n1*n2 + n2*n1).coeff(n1) == 0
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=1) == n2
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=0) == 2
f = Function('f')
assert (2*f(x) + 3*f(x).diff(x)).coeff(f(x)) == 2
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr.coeff(x + y) == 0
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
assert (x + y + 3*z).coeff(1) == x + y
assert (-x + 2*y).coeff(-1) == x
assert (x - 2*y).coeff(-1) == 2*y
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (-x - 2*y).coeff(2) == -y
assert (x + sqrt(2)*x).coeff(sqrt(2)) == x
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (z*(x + y)**2).coeff((x + y)**2) == z
assert (z*(x + y)**2).coeff(x + y) == 0
assert (2 + 2*x + (x + 1)*y).coeff(x + 1) == y
assert (x + 2*y + 3).coeff(1) == x
assert (x + 2*y + 3).coeff(x, 0) == 2*y + 3
assert (x**2 + 2*y + 3*x).coeff(x**2, 0) == 2*y + 3*x
assert x.coeff(0, 0) == 0
assert x.coeff(x, 0) == 0
n, m, o, l = symbols('n m o l', commutative=False)
assert n.coeff(n) == 1
assert y.coeff(n) == 0
assert (3*n).coeff(n) == 3
assert (2 + n).coeff(x*m) == 0
assert (2*x*n*m).coeff(x) == 2*n*m
assert (2 + n).coeff(x*m*n + y) == 0
assert (2*x*n*m).coeff(3*n) == 0
assert (n*m + m*n*m).coeff(n) == 1 + m
assert (n*m + m*n*m).coeff(n, right=True) == m # = (1 + m)*n*m
assert (n*m + m*n).coeff(n) == 0
assert (n*m + o*m*n).coeff(m*n) == o
assert (n*m + o*m*n).coeff(m*n, right=1) == 1
assert (n*m + n*m*n).coeff(n*m, right=1) == 1 + n # = n*m*(n + 1)
assert (x*y).coeff(z, 0) == x*y
def test_coeff2():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff((psi(r).diff(r))) == 2/r
def test_coeff2_0():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff(psi(r).diff(r, 2)) == 1
def test_coeff_expand():
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
def test_integrate():
assert x.integrate(x) == x**2/2
assert x.integrate((x, 0, 1)) == S(1)/2
def test_as_base_exp():
assert x.as_base_exp() == (x, S.One)
assert (x*y*z).as_base_exp() == (x*y*z, S.One)
assert (x + y + z).as_base_exp() == (x + y + z, S.One)
assert ((x + y)**z).as_base_exp() == (x + y, z)
def test_issue_4963():
assert hasattr(Mul(x, y), "is_commutative")
assert hasattr(Mul(x, y, evaluate=False), "is_commutative")
assert hasattr(Pow(x, y), "is_commutative")
assert hasattr(Pow(x, y, evaluate=False), "is_commutative")
expr = Mul(Pow(2, 2, evaluate=False), 3, evaluate=False) + 1
assert hasattr(expr, "is_commutative")
def test_action_verbs():
assert nsimplify((1/(exp(3*pi*x/5) + 1))) == \
(1/(exp(3*pi*x/5) + 1)).nsimplify()
assert ratsimp(1/x + 1/y) == (1/x + 1/y).ratsimp()
assert trigsimp(log(x), deep=True) == (log(x)).trigsimp(deep=True)
assert radsimp(1/(2 + sqrt(2))) == (1/(2 + sqrt(2))).radsimp()
assert radsimp(1/(a + b*sqrt(c)), symbolic=False) == \
(1/(a + b*sqrt(c))).radsimp(symbolic=False)
assert powsimp(x**y*x**z*y**z, combine='all') == \
(x**y*x**z*y**z).powsimp(combine='all')
assert (x**t*y**t).powsimp(force=True) == (x*y)**t
assert simplify(x**y*x**z*y**z) == (x**y*x**z*y**z).simplify()
assert together(1/x + 1/y) == (1/x + 1/y).together()
assert collect(a*x**2 + b*x**2 + a*x - b*x + c, x) == \
(a*x**2 + b*x**2 + a*x - b*x + c).collect(x)
assert apart(y/(y + 2)/(y + 1), y) == (y/(y + 2)/(y + 1)).apart(y)
assert combsimp(y/(x + 2)/(x + 1)) == (y/(x + 2)/(x + 1)).combsimp()
assert gammasimp(gamma(x)/gamma(x-5)) == (gamma(x)/gamma(x-5)).gammasimp()
assert factor(x**2 + 5*x + 6) == (x**2 + 5*x + 6).factor()
assert refine(sqrt(x**2)) == sqrt(x**2).refine()
assert cancel((x**2 + 5*x + 6)/(x + 2)) == ((x**2 + 5*x + 6)/(x + 2)).cancel()
def test_as_powers_dict():
assert x.as_powers_dict() == {x: 1}
assert (x**y*z).as_powers_dict() == {x: y, z: 1}
assert Mul(2, 2, evaluate=False).as_powers_dict() == {S(2): S(2)}
assert (x*y).as_powers_dict()[z] == 0
assert (x + y).as_powers_dict()[z] == 0
def test_as_coefficients_dict():
check = [S(1), x, y, x*y, 1]
assert [Add(3*x, 2*x, y, 3).as_coefficients_dict()[i] for i in check] == \
[3, 5, 1, 0, 3]
assert [Add(3*x, 2*x, y, 3, evaluate=False).as_coefficients_dict()[i]
for i in check] == [3, 5, 1, 0, 3]
assert [(3*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3, 0]
assert [(3.0*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3.0, 0]
assert (3.0*x*y).as_coefficients_dict()[3.0*x*y] == 0
def test_args_cnc():
A = symbols('A', commutative=False)
assert (x + A).args_cnc() == \
[[], [x + A]]
assert (x + a).args_cnc() == \
[[a + x], []]
assert (x*a).args_cnc() == \
[[a, x], []]
assert (x*y*A*(A + 1)).args_cnc(cset=True) == \
[{x, y}, [A, 1 + A]]
assert Mul(x, x, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x}, []]
assert Mul(x, x**2, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x, x**2}, []]
raises(ValueError, lambda: Mul(x, x, evaluate=False).args_cnc(cset=True))
assert Mul(x, y, x, evaluate=False).args_cnc() == \
[[x, y, x], []]
# always split -1 from leading number
assert (-1.*x).args_cnc() == [[-1, 1.0, x], []]
def test_new_rawargs():
n = Symbol('n', commutative=False)
a = x + n
assert a.is_commutative is False
assert a._new_rawargs(x).is_commutative
assert a._new_rawargs(x, y).is_commutative
assert a._new_rawargs(x, n).is_commutative is False
assert a._new_rawargs(x, y, n).is_commutative is False
m = x*n
assert m.is_commutative is False
assert m._new_rawargs(x).is_commutative
assert m._new_rawargs(n).is_commutative is False
assert m._new_rawargs(x, y).is_commutative
assert m._new_rawargs(x, n).is_commutative is False
assert m._new_rawargs(x, y, n).is_commutative is False
assert m._new_rawargs(x, n, reeval=False).is_commutative is False
assert m._new_rawargs(S.One) is S.One
def test_issue_5226():
assert Add(evaluate=False) == 0
assert Mul(evaluate=False) == 1
assert Mul(x + y, evaluate=False).is_Add
def test_free_symbols():
# free_symbols should return the free symbols of an object
assert S(1).free_symbols == set()
assert (x).free_symbols == {x}
assert Integral(x, (x, 1, y)).free_symbols == {y}
assert (-Integral(x, (x, 1, y))).free_symbols == {y}
assert meter.free_symbols == set()
assert (meter**x).free_symbols == {x}
def test_issue_5300():
x = Symbol('x', commutative=False)
assert x*sqrt(2)/sqrt(6) == x*sqrt(3)/3
def test_floordiv():
from sympy.functions.elementary.integers import floor
assert x // y == floor(x / y)
def test_as_coeff_Mul():
assert S(0).as_coeff_Mul() == (S.One, S.Zero)
assert Integer(3).as_coeff_Mul() == (Integer(3), Integer(1))
assert Rational(3, 4).as_coeff_Mul() == (Rational(3, 4), Integer(1))
assert Float(5.0).as_coeff_Mul() == (Float(5.0), Integer(1))
assert (Integer(3)*x).as_coeff_Mul() == (Integer(3), x)
assert (Rational(3, 4)*x).as_coeff_Mul() == (Rational(3, 4), x)
assert (Float(5.0)*x).as_coeff_Mul() == (Float(5.0), x)
assert (Integer(3)*x*y).as_coeff_Mul() == (Integer(3), x*y)
assert (Rational(3, 4)*x*y).as_coeff_Mul() == (Rational(3, 4), x*y)
assert (Float(5.0)*x*y).as_coeff_Mul() == (Float(5.0), x*y)
assert (x).as_coeff_Mul() == (S.One, x)
assert (x*y).as_coeff_Mul() == (S.One, x*y)
assert (-oo*x).as_coeff_Mul(rational=True) == (-1, oo*x)
def test_as_coeff_Add():
assert Integer(3).as_coeff_Add() == (Integer(3), Integer(0))
assert Rational(3, 4).as_coeff_Add() == (Rational(3, 4), Integer(0))
assert Float(5.0).as_coeff_Add() == (Float(5.0), Integer(0))
assert (Integer(3) + x).as_coeff_Add() == (Integer(3), x)
assert (Rational(3, 4) + x).as_coeff_Add() == (Rational(3, 4), x)
assert (Float(5.0) + x).as_coeff_Add() == (Float(5.0), x)
assert (Float(5.0) + x).as_coeff_Add(rational=True) == (0, Float(5.0) + x)
assert (Integer(3) + x + y).as_coeff_Add() == (Integer(3), x + y)
assert (Rational(3, 4) + x + y).as_coeff_Add() == (Rational(3, 4), x + y)
assert (Float(5.0) + x + y).as_coeff_Add() == (Float(5.0), x + y)
assert (x).as_coeff_Add() == (S.Zero, x)
assert (x*y).as_coeff_Add() == (S.Zero, x*y)
def test_expr_sorting():
f, g = symbols('f,g', cls=Function)
exprs = [1/x**2, 1/x, sqrt(sqrt(x)), sqrt(x), x, sqrt(x)**3, x**2]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x, 2*x, 2*x**2, 2*x**3, x**n, 2*x**n, sin(x), sin(x)**n,
sin(x**2), cos(x), cos(x**2), tan(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x + 1, x**2 + x + 1, x**3 + x**2 + x + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [S(4), x - 3*I/2, x + 3*I/2, x - 4*I + 1, x + 4*I + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(x), g(x), exp(x), sin(x), cos(x), factorial(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [Tuple(x, y), Tuple(x, z), Tuple(x, y, z)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[3], [1, 2]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [1, 2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{x: -y}, {x: y}]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{1}, {1, 2}]
assert sorted(exprs, key=default_sort_key) == exprs
a, b = exprs = [Dummy('x'), Dummy('x')]
assert sorted([b, a], key=default_sort_key) == exprs
def test_as_ordered_factors():
f, g = symbols('f,g', cls=Function)
assert x.as_ordered_factors() == [x]
assert (2*x*x**n*sin(x)*cos(x)).as_ordered_factors() \
== [Integer(2), x, x**n, sin(x), cos(x)]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Mul(*args)
assert expr.as_ordered_factors() == args
A, B = symbols('A,B', commutative=False)
assert (A*B).as_ordered_factors() == [A, B]
assert (B*A).as_ordered_factors() == [B, A]
def test_as_ordered_terms():
f, g = symbols('f,g', cls=Function)
assert x.as_ordered_terms() == [x]
assert (sin(x)**2*cos(x) + sin(x)*cos(x)**2 + 1).as_ordered_terms() \
== [sin(x)**2*cos(x), sin(x)*cos(x)**2, 1]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Add(*args)
assert expr.as_ordered_terms() == args
assert (1 + 4*sqrt(3)*pi*x).as_ordered_terms() == [4*pi*x*sqrt(3), 1]
assert ( 2 + 3*I).as_ordered_terms() == [2, 3*I]
assert (-2 + 3*I).as_ordered_terms() == [-2, 3*I]
assert ( 2 - 3*I).as_ordered_terms() == [2, -3*I]
assert (-2 - 3*I).as_ordered_terms() == [-2, -3*I]
assert ( 4 + 3*I).as_ordered_terms() == [4, 3*I]
assert (-4 + 3*I).as_ordered_terms() == [-4, 3*I]
assert ( 4 - 3*I).as_ordered_terms() == [4, -3*I]
assert (-4 - 3*I).as_ordered_terms() == [-4, -3*I]
f = x**2*y**2 + x*y**4 + y + 2
assert f.as_ordered_terms(order="lex") == [x**2*y**2, x*y**4, y, 2]
assert f.as_ordered_terms(order="grlex") == [x*y**4, x**2*y**2, y, 2]
assert f.as_ordered_terms(order="rev-lex") == [2, y, x*y**4, x**2*y**2]
assert f.as_ordered_terms(order="rev-grlex") == [2, y, x**2*y**2, x*y**4]
k = symbols('k')
assert k.as_ordered_terms(data=True) == ([(k, ((1.0, 0.0), (1,), ()))], [k])
def test_sort_key_atomic_expr():
from sympy.physics.units import m, s
assert sorted([-m, s], key=lambda arg: arg.sort_key()) == [-m, s]
def test_eval_interval():
assert exp(x)._eval_interval(*Tuple(x, 0, 1)) == exp(1) - exp(0)
# issue 4199
# first subs and limit gives NaN
a = x/y
assert a._eval_interval(x, S(0), oo)._eval_interval(y, oo, S(0)) is S.NaN
# second subs and limit gives NaN
assert a._eval_interval(x, S(0), oo)._eval_interval(y, S(0), oo) is S.NaN
# difference gives S.NaN
a = x - y
assert a._eval_interval(x, S(1), oo)._eval_interval(y, oo, S(1)) is S.NaN
raises(ValueError, lambda: x._eval_interval(x, None, None))
a = -y*Heaviside(x - y)
assert a._eval_interval(x, -oo, oo) == -y
assert a._eval_interval(x, oo, -oo) == y
def test_eval_interval_zoo():
# Test that limit is used when zoo is returned
assert Si(1/x)._eval_interval(x, S(0), S(1)) == -pi/2 + Si(1)
def test_primitive():
assert (3*(x + 1)**2).primitive() == (3, (x + 1)**2)
assert (6*x + 2).primitive() == (2, 3*x + 1)
assert (x/2 + 3).primitive() == (S(1)/2, x + 6)
eq = (6*x + 2)*(x/2 + 3)
assert eq.primitive()[0] == 1
eq = (2 + 2*x)**2
assert eq.primitive()[0] == 1
assert (4.0*x).primitive() == (1, 4.0*x)
assert (4.0*x + y/2).primitive() == (S.Half, 8.0*x + y)
assert (-2*x).primitive() == (2, -x)
assert Add(5*z/7, 0.5*x, 3*y/2, evaluate=False).primitive() == \
(S(1)/14, 7.0*x + 21*y + 10*z)
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).primitive() == \
(S(1)/3, i + x)
assert (S.Infinity + 2*x/3 + 4*y/7).primitive() == \
(S(1)/21, 14*x + 12*y + oo)
assert S.Zero.primitive() == (S.One, S.Zero)
def test_issue_5843():
a = 1 + x
assert (2*a).extract_multiplicatively(a) == 2
assert (4*a).extract_multiplicatively(2*a) == 2
assert ((3*a)*(2*a)).extract_multiplicatively(a) == 6*a
def test_is_constant():
from sympy.solvers.solvers import checksol
Sum(x, (x, 1, 10)).is_constant() is True
Sum(x, (x, 1, n)).is_constant() is False
Sum(x, (x, 1, n)).is_constant(y) is True
Sum(x, (x, 1, n)).is_constant(n) is False
Sum(x, (x, 1, n)).is_constant(x) is True
eq = a*cos(x)**2 + a*sin(x)**2 - a
eq.is_constant() is True
assert eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
assert x.is_constant() is False
assert x.is_constant(y) is True
assert checksol(x, x, Sum(x, (x, 1, n))) is False
assert checksol(x, x, Sum(x, (x, 1, n))) is False
f = Function('f')
assert f(1).is_constant
assert checksol(x, x, f(x)) is False
assert Pow(x, S(0), evaluate=False).is_constant() is True # == 1
assert Pow(S(0), x, evaluate=False).is_constant() is False # == 0 or 1
assert (2**x).is_constant() is False
assert Pow(S(2), S(3), evaluate=False).is_constant() is True
z1, z2 = symbols('z1 z2', zero=True)
assert (z1 + 2*z2).is_constant() is True
assert meter.is_constant() is True
assert (3*meter).is_constant() is True
assert (x*meter).is_constant() is False
assert Poly(3,x).is_constant() is True
def test_equals():
assert (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2).equals(0)
assert (x**2 - 1).equals((x + 1)*(x - 1))
assert (cos(x)**2 + sin(x)**2).equals(1)
assert (a*cos(x)**2 + a*sin(x)**2).equals(a)
r = sqrt(2)
assert (-1/(r + r*x) + 1/r/(1 + x)).equals(0)
assert factorial(x + 1).equals((x + 1)*factorial(x))
assert sqrt(3).equals(2*sqrt(3)) is False
assert (sqrt(5)*sqrt(3)).equals(sqrt(3)) is False
assert (sqrt(5) + sqrt(3)).equals(0) is False
assert (sqrt(5) + pi).equals(0) is False
assert meter.equals(0) is False
assert (3*meter**2).equals(0) is False
eq = -(-1)**(S(3)/4)*6**(S(1)/4) + (-6)**(S(1)/4)*I
if eq != 0: # if canonicalization makes this zero, skip the test
assert eq.equals(0)
assert sqrt(x).equals(0) is False
# from integrate(x*sqrt(1 + 2*x), x);
# diff is zero only when assumptions allow
i = 2*sqrt(2)*x**(S(5)/2)*(1 + 1/(2*x))**(S(5)/2)/5 + \
2*sqrt(2)*x**(S(3)/2)*(1 + 1/(2*x))**(S(5)/2)/(-6 - 3/x)
ans = sqrt(2*x + 1)*(6*x**2 + x - 1)/15
diff = i - ans
assert diff.equals(0) is False
assert diff.subs(x, -S.Half/2) == 7*sqrt(2)/120
# there are regions for x for which the expression is True, for
# example, when x < -1/2 or x > 0 the expression is zero
p = Symbol('p', positive=True)
assert diff.subs(x, p).equals(0) is True
assert diff.subs(x, -1).equals(0) is True
# prove via minimal_polynomial or self-consistency
eq = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert eq.equals(0)
q = 3**Rational(1, 3) + 3
p = expand(q**3)**Rational(1, 3)
assert (p - q).equals(0)
# issue 6829
# eq = q*x + q/4 + x**4 + x**3 + 2*x**2 - S(1)/3
# z = eq.subs(x, solve(eq, x)[0])
q = symbols('q')
z = (q*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/12)/2 - sqrt((2*q - S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S(1)/3) - S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S(1)/3) - S(13)/6)/2 - S(1)/4) + q/4 + (-sqrt(-2*(-(q
- S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) - S(13)/12)/2 - sqrt((2*q
- S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/6)/2 - S(1)/4)**4 + (-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S(1)/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/6)/2 - S(1)/4)**3 + 2*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S(1)/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/6)/2 - S(1)/4)**2 - S(1)/3)
assert z.equals(0)
def test_random():
from sympy import posify, lucas
assert posify(x)[0]._random() is not None
assert lucas(n)._random(2, -2, 0, -1, 1) is None
# issue 8662
assert Piecewise((Max(x, y), z))._random() is None
def test_round():
from sympy.abc import x
assert Float('0.1249999').round(2) == 0.12
d20 = 12345678901234567890
ans = S(d20).round(2)
assert ans.is_Float and ans == d20
ans = S(d20).round(-2)
assert ans.is_Float and ans == 12345678901234567900
assert S('1/7').round(4) == 0.1429
assert S('.[12345]').round(4) == 0.1235
assert S('.1349').round(2) == 0.13
n = S(12345)
ans = n.round()
assert ans.is_Float
assert ans == n
ans = n.round(1)
assert ans.is_Float
assert ans == n
ans = n.round(4)
assert ans.is_Float
assert ans == n
assert n.round(-1) == 12350
r = n.round(-4)
assert r == 10000
# in fact, it should equal many values since __eq__
# compares at equal precision
assert all(r == i for i in range(9984, 10049))
assert n.round(-5) == 0
assert (pi + sqrt(2)).round(2) == 4.56
assert (10*(pi + sqrt(2))).round(-1) == 50
raises(TypeError, lambda: round(x + 2, 2))
assert S(2.3).round(1) == 2.3
e = S(12.345).round(2)
assert e == round(12.345, 2)
assert type(e) is Float
assert (Float(.3, 3) + 2*pi).round() == 7
assert (Float(.3, 3) + 2*pi*100).round() == 629
assert (Float(.03, 3) + 2*pi/100).round(5) == 0.09283
assert (Float(.03, 3) + 2*pi/100).round(4) == 0.0928
assert (pi + 2*E*I).round() == 3 + 5*I
assert S.Zero.round() == 0
a = (Add(1, Float('1.' + '9'*27, ''), evaluate=0))
assert a.round(10) == Float('3.0000000000', '')
assert a.round(25) == Float('3.0000000000000000000000000', '')
assert a.round(26) == Float('3.00000000000000000000000000', '')
assert a.round(27) == Float('2.999999999999999999999999999', '')
assert a.round(30) == Float('2.999999999999999999999999999', '')
raises(TypeError, lambda: x.round())
f = Function('f')
raises(TypeError, lambda: f(1).round())
# exact magnitude of 10
assert str(S(1).round()) == '1.'
assert str(S(100).round()) == '100.'
# applied to real and imaginary portions
assert (2*pi + E*I).round() == 6 + 3*I
assert (2*pi + I/10).round() == 6
assert (pi/10 + 2*I).round() == 2*I
# the lhs re and im parts are Float with dps of 2
# and those on the right have dps of 15 so they won't compare
# equal unless we use string or compare components (which will
# then coerce the floats to the same precision) or re-create
# the floats
assert str((pi/10 + E*I).round(2)) == '0.31 + 2.72*I'
assert (pi/10 + E*I).round(2).as_real_imag() == (0.31, 2.72)
assert (pi/10 + E*I).round(2) == Float(0.31, 2) + I*Float(2.72, 3)
# issue 6914
assert (I**(I + 3)).round(3) == Float('-0.208', '')*I
# issue 8720
assert S(-123.6).round() == -124.
assert S(-1.5).round() == -2.
assert S(-100.5).round() == -101.
assert S(-1.5 - 10.5*I).round() == -2.0 - 11.0*I
# issue 7961
assert str(S(0.006).round(2)) == '0.01'
assert str(S(0.00106).round(4)) == '0.0011'
# issue 8147
assert S.NaN.round() == S.NaN
assert S.Infinity.round() == S.Infinity
assert S.NegativeInfinity.round() == S.NegativeInfinity
assert S.ComplexInfinity.round() == S.ComplexInfinity
def test_held_expression_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
e1 = x*he
assert isinstance(e1, Mul)
assert e1.args == (x, he)
assert e1.doit() == 1
assert UnevaluatedExpr(Derivative(x, x)).doit(deep=False
) == Derivative(x, x)
assert UnevaluatedExpr(Derivative(x, x)).doit() == 1
xx = Mul(x, x, evaluate=False)
assert xx != x**2
ue2 = UnevaluatedExpr(xx)
assert isinstance(ue2, UnevaluatedExpr)
assert ue2.args == (xx,)
assert ue2.doit() == x**2
assert ue2.doit(deep=False) == xx
x2 = UnevaluatedExpr(2)*2
assert type(x2) is Mul
assert x2.args == (2, UnevaluatedExpr(2))
def test_round_exception_nostr():
# Don't use the string form of the expression in the round exception, as
# it's too slow
s = Symbol('bad')
try:
s.round()
except TypeError as e:
assert 'bad' not in str(e)
else:
# Did not raise
raise AssertionError("Did not raise")
def test_extract_branch_factor():
assert exp_polar(2.0*I*pi).extract_branch_factor() == (1, 1)
def test_identity_removal():
assert Add.make_args(x + 0) == (x,)
assert Mul.make_args(x*1) == (x,)
def test_float_0():
assert Float(0.0) + 1 == Float(1.0)
@XFAIL
def test_float_0_fail():
assert Float(0.0)*x == Float(0.0)
assert (x + Float(0.0)).is_Add
def test_issue_6325():
ans = (b**2 + z**2 - (b*(a + b*t) + z*(c + t*z))**2/(
(a + b*t)**2 + (c + t*z)**2))/sqrt((a + b*t)**2 + (c + t*z)**2)
e = sqrt((a + b*t)**2 + (c + z*t)**2)
assert diff(e, t, 2) == ans
e.diff(t, 2) == ans
assert diff(e, t, 2, simplify=False) != ans
def test_issue_7426():
f1 = a % c
f2 = x % z
assert f1.equals(f2) is None
def test_issue_1112():
x = Symbol('x', positive=False)
assert (x > 0) is S.false
def test_issue_10161():
x = symbols('x', real=True)
assert x*abs(x)*abs(x) == x**3
def test_issue_10755():
x = symbols('x')
raises(TypeError, lambda: int(log(x)))
raises(TypeError, lambda: log(x).round(2))
def test_issue_11877():
x = symbols('x')
assert integrate(log(S(1)/2 - x), (x, 0, S(1)/2)) == -S(1)/2 -log(2)/2
def test_normal():
x = symbols('x')
e = Mul(S.Half, 1 + x, evaluate=False)
assert e.normal() == e
def test_ExprBuilder():
eb = ExprBuilder(Mul)
eb.args.extend([x, x])
assert eb.build() == x**2
|
a5bfe411021317080721997d3c4ed5f2719ee0a92f615ab9d1289c9f1acb971d | """Test whether all elements of cls.args are instances of Basic. """
# NOTE: keep tests sorted by (module, class name) key. If a class can't
# be instantiated, add it here anyway with @SKIP("abstract class) (see
# e.g. Function).
import os
import re
import io
from sympy import (Basic, S, symbols, sqrt, sin, oo, Interval, exp, Lambda, pi,
Eq, log, Function)
from sympy.core.compatibility import range
from sympy.utilities.pytest import XFAIL, SKIP
x, y, z = symbols('x,y,z')
def test_all_classes_are_tested():
this = os.path.split(__file__)[0]
path = os.path.join(this, os.pardir, os.pardir)
sympy_path = os.path.abspath(path)
prefix = os.path.split(sympy_path)[0] + os.sep
re_cls = re.compile(r"^class ([A-Za-z][A-Za-z0-9_]*)\s*\(", re.MULTILINE)
modules = {}
for root, dirs, files in os.walk(sympy_path):
module = root.replace(prefix, "").replace(os.sep, ".")
for file in files:
if file.startswith(("_", "test_", "bench_")):
continue
if not file.endswith(".py"):
continue
with io.open(os.path.join(root, file), "r", encoding='utf-8') as f:
text = f.read()
submodule = module + '.' + file[:-3]
names = re_cls.findall(text)
if not names:
continue
try:
mod = __import__(submodule, fromlist=names)
except ImportError:
continue
def is_Basic(name):
cls = getattr(mod, name)
if hasattr(cls, '_sympy_deprecated_func'):
cls = cls._sympy_deprecated_func
return issubclass(cls, Basic)
names = list(filter(is_Basic, names))
if names:
modules[submodule] = names
ns = globals()
failed = []
for module, names in modules.items():
mod = module.replace('.', '__')
for name in names:
test = 'test_' + mod + '__' + name
if test not in ns:
failed.append(module + '.' + name)
assert not failed, "Missing classes: %s. Please add tests for these to sympy/core/tests/test_args.py." % ", ".join(failed)
def _test_args(obj):
return all(isinstance(arg, Basic) for arg in obj.args)
def test_sympy__assumptions__assume__AppliedPredicate():
from sympy.assumptions.assume import AppliedPredicate, Predicate
from sympy import Q
assert _test_args(AppliedPredicate(Predicate("test"), 2))
assert _test_args(Q.is_true(True))
def test_sympy__assumptions__assume__Predicate():
from sympy.assumptions.assume import Predicate
assert _test_args(Predicate("test"))
def test_sympy__assumptions__sathandlers__UnevaluatedOnFree():
from sympy.assumptions.sathandlers import UnevaluatedOnFree
from sympy import Q
assert _test_args(UnevaluatedOnFree(Q.positive))
assert _test_args(UnevaluatedOnFree(Q.positive(x)))
assert _test_args(UnevaluatedOnFree(Q.positive(x * y)))
def test_sympy__assumptions__sathandlers__AllArgs():
from sympy.assumptions.sathandlers import AllArgs
from sympy import Q
assert _test_args(AllArgs(Q.positive))
assert _test_args(AllArgs(Q.positive(x)))
assert _test_args(AllArgs(Q.positive(x*y)))
def test_sympy__assumptions__sathandlers__AnyArgs():
from sympy.assumptions.sathandlers import AnyArgs
from sympy import Q
assert _test_args(AnyArgs(Q.positive))
assert _test_args(AnyArgs(Q.positive(x)))
assert _test_args(AnyArgs(Q.positive(x*y)))
def test_sympy__assumptions__sathandlers__ExactlyOneArg():
from sympy.assumptions.sathandlers import ExactlyOneArg
from sympy import Q
assert _test_args(ExactlyOneArg(Q.positive))
assert _test_args(ExactlyOneArg(Q.positive(x)))
assert _test_args(ExactlyOneArg(Q.positive(x*y)))
def test_sympy__assumptions__sathandlers__CheckOldAssump():
from sympy.assumptions.sathandlers import CheckOldAssump
from sympy import Q
assert _test_args(CheckOldAssump(Q.positive))
assert _test_args(CheckOldAssump(Q.positive(x)))
assert _test_args(CheckOldAssump(Q.positive(x*y)))
def test_sympy__assumptions__sathandlers__CheckIsPrime():
from sympy.assumptions.sathandlers import CheckIsPrime
from sympy import Q
# Input must be a number
assert _test_args(CheckIsPrime(Q.positive))
assert _test_args(CheckIsPrime(Q.positive(5)))
@SKIP("abstract Class")
def test_sympy__codegen__ast__AssignmentBase():
from sympy.codegen.ast import AssignmentBase
assert _test_args(AssignmentBase(x, 1))
@SKIP("abstract Class")
def test_sympy__codegen__ast__AugmentedAssignment():
from sympy.codegen.ast import AugmentedAssignment
assert _test_args(AugmentedAssignment(x, 1))
def test_sympy__codegen__ast__AddAugmentedAssignment():
from sympy.codegen.ast import AddAugmentedAssignment
assert _test_args(AddAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__SubAugmentedAssignment():
from sympy.codegen.ast import SubAugmentedAssignment
assert _test_args(SubAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__MulAugmentedAssignment():
from sympy.codegen.ast import MulAugmentedAssignment
assert _test_args(MulAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__DivAugmentedAssignment():
from sympy.codegen.ast import DivAugmentedAssignment
assert _test_args(DivAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__ModAugmentedAssignment():
from sympy.codegen.ast import ModAugmentedAssignment
assert _test_args(ModAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__CodeBlock():
from sympy.codegen.ast import CodeBlock, Assignment
assert _test_args(CodeBlock(Assignment(x, 1), Assignment(y, 2)))
def test_sympy__codegen__ast__For():
from sympy.codegen.ast import For, CodeBlock, AddAugmentedAssignment
from sympy import Range
assert _test_args(For(x, Range(10), CodeBlock(AddAugmentedAssignment(y, 1))))
def test_sympy__codegen__ast__Token():
from sympy.codegen.ast import Token
assert _test_args(Token())
def test_sympy__codegen__ast__ContinueToken():
from sympy.codegen.ast import ContinueToken
assert _test_args(ContinueToken())
def test_sympy__codegen__ast__BreakToken():
from sympy.codegen.ast import BreakToken
assert _test_args(BreakToken())
def test_sympy__codegen__ast__NoneToken():
from sympy.codegen.ast import NoneToken
assert _test_args(NoneToken())
def test_sympy__codegen__ast__String():
from sympy.codegen.ast import String
assert _test_args(String('foobar'))
def test_sympy__codegen__ast__QuotedString():
from sympy.codegen.ast import QuotedString
assert _test_args(QuotedString('foobar'))
def test_sympy__codegen__ast__Comment():
from sympy.codegen.ast import Comment
assert _test_args(Comment('this is a comment'))
def test_sympy__codegen__ast__Node():
from sympy.codegen.ast import Node
assert _test_args(Node())
assert _test_args(Node(attrs={1, 2, 3}))
def test_sympy__codegen__ast__Type():
from sympy.codegen.ast import Type
assert _test_args(Type('float128'))
def test_sympy__codegen__ast__IntBaseType():
from sympy.codegen.ast import IntBaseType
assert _test_args(IntBaseType('bigint'))
def test_sympy__codegen__ast___SizedIntType():
from sympy.codegen.ast import _SizedIntType
assert _test_args(_SizedIntType('int128', 128))
def test_sympy__codegen__ast__SignedIntType():
from sympy.codegen.ast import SignedIntType
assert _test_args(SignedIntType('int128_with_sign', 128))
def test_sympy__codegen__ast__UnsignedIntType():
from sympy.codegen.ast import UnsignedIntType
assert _test_args(UnsignedIntType('unt128', 128))
def test_sympy__codegen__ast__FloatBaseType():
from sympy.codegen.ast import FloatBaseType
assert _test_args(FloatBaseType('positive_real'))
def test_sympy__codegen__ast__FloatType():
from sympy.codegen.ast import FloatType
assert _test_args(FloatType('float242', 242, nmant=142, nexp=99))
def test_sympy__codegen__ast__ComplexBaseType():
from sympy.codegen.ast import ComplexBaseType
assert _test_args(ComplexBaseType('positive_cmplx'))
def test_sympy__codegen__ast__ComplexType():
from sympy.codegen.ast import ComplexType
assert _test_args(ComplexType('complex42', 42, nmant=15, nexp=5))
def test_sympy__codegen__ast__Attribute():
from sympy.codegen.ast import Attribute
assert _test_args(Attribute('noexcept'))
def test_sympy__codegen__ast__Variable():
from sympy.codegen.ast import Variable, Type, value_const
assert _test_args(Variable(x))
assert _test_args(Variable(y, Type('float32'), {value_const}))
assert _test_args(Variable(z, type=Type('float64')))
def test_sympy__codegen__ast__Pointer():
from sympy.codegen.ast import Pointer, Type, pointer_const
assert _test_args(Pointer(x))
assert _test_args(Pointer(y, type=Type('float32')))
assert _test_args(Pointer(z, Type('float64'), {pointer_const}))
def test_sympy__codegen__ast__Declaration():
from sympy.codegen.ast import Declaration, Variable, Type
vx = Variable(x, type=Type('float'))
assert _test_args(Declaration(vx))
def test_sympy__codegen__ast__While():
from sympy.codegen.ast import While, AddAugmentedAssignment
assert _test_args(While(abs(x) < 1, [AddAugmentedAssignment(x, -1)]))
def test_sympy__codegen__ast__Scope():
from sympy.codegen.ast import Scope, AddAugmentedAssignment
assert _test_args(Scope([AddAugmentedAssignment(x, -1)]))
def test_sympy__codegen__ast__Stream():
from sympy.codegen.ast import Stream
assert _test_args(Stream('stdin'))
def test_sympy__codegen__ast__Print():
from sympy.codegen.ast import Print
assert _test_args(Print([x, y]))
assert _test_args(Print([x, y], "%d %d"))
def test_sympy__codegen__ast__FunctionPrototype():
from sympy.codegen.ast import FunctionPrototype, real, Declaration, Variable
inp_x = Declaration(Variable(x, type=real))
assert _test_args(FunctionPrototype(real, 'pwer', [inp_x]))
def test_sympy__codegen__ast__FunctionDefinition():
from sympy.codegen.ast import FunctionDefinition, real, Declaration, Variable, Assignment
inp_x = Declaration(Variable(x, type=real))
assert _test_args(FunctionDefinition(real, 'pwer', [inp_x], [Assignment(x, x**2)]))
def test_sympy__codegen__ast__Return():
from sympy.codegen.ast import Return
assert _test_args(Return(x))
def test_sympy__codegen__ast__FunctionCall():
from sympy.codegen.ast import FunctionCall
assert _test_args(FunctionCall('pwer', [x]))
def test_sympy__codegen__ast__Element():
from sympy.codegen.ast import Element
assert _test_args(Element('x', range(3)))
def test_sympy__codegen__cnodes__CommaOperator():
from sympy.codegen.cnodes import CommaOperator
assert _test_args(CommaOperator(1, 2))
def test_sympy__codegen__cnodes__goto():
from sympy.codegen.cnodes import goto
assert _test_args(goto('early_exit'))
def test_sympy__codegen__cnodes__Label():
from sympy.codegen.cnodes import Label
assert _test_args(Label('early_exit'))
def test_sympy__codegen__cnodes__PreDecrement():
from sympy.codegen.cnodes import PreDecrement
assert _test_args(PreDecrement(x))
def test_sympy__codegen__cnodes__PostDecrement():
from sympy.codegen.cnodes import PostDecrement
assert _test_args(PostDecrement(x))
def test_sympy__codegen__cnodes__PreIncrement():
from sympy.codegen.cnodes import PreIncrement
assert _test_args(PreIncrement(x))
def test_sympy__codegen__cnodes__PostIncrement():
from sympy.codegen.cnodes import PostIncrement
assert _test_args(PostIncrement(x))
def test_sympy__codegen__cnodes__struct():
from sympy.codegen.ast import real, Variable
from sympy.codegen.cnodes import struct
assert _test_args(struct(declarations=[
Variable(x, type=real),
Variable(y, type=real)
]))
def test_sympy__codegen__cnodes__union():
from sympy.codegen.ast import float32, int32, Variable
from sympy.codegen.cnodes import union
assert _test_args(union(declarations=[
Variable(x, type=float32),
Variable(y, type=int32)
]))
def test_sympy__codegen__cxxnodes__using():
from sympy.codegen.cxxnodes import using
assert _test_args(using('std::vector'))
assert _test_args(using('std::vector', 'vec'))
def test_sympy__codegen__fnodes__Program():
from sympy.codegen.fnodes import Program
assert _test_args(Program('foobar', []))
def test_sympy__codegen__fnodes__Module():
from sympy.codegen.fnodes import Module
assert _test_args(Module('foobar', [], []))
def test_sympy__codegen__fnodes__Subroutine():
from sympy.codegen.fnodes import Subroutine
x = symbols('x', real=True)
assert _test_args(Subroutine('foo', [x], []))
def test_sympy__codegen__fnodes__GoTo():
from sympy.codegen.fnodes import GoTo
assert _test_args(GoTo([10]))
assert _test_args(GoTo([10, 20], x > 1))
def test_sympy__codegen__fnodes__FortranReturn():
from sympy.codegen.fnodes import FortranReturn
assert _test_args(FortranReturn(10))
def test_sympy__codegen__fnodes__Extent():
from sympy.codegen.fnodes import Extent
assert _test_args(Extent())
assert _test_args(Extent(None))
assert _test_args(Extent(':'))
assert _test_args(Extent(-3, 4))
assert _test_args(Extent(x, y))
def test_sympy__codegen__fnodes__use_rename():
from sympy.codegen.fnodes import use_rename
assert _test_args(use_rename('loc', 'glob'))
def test_sympy__codegen__fnodes__use():
from sympy.codegen.fnodes import use
assert _test_args(use('modfoo', only='bar'))
def test_sympy__codegen__fnodes__SubroutineCall():
from sympy.codegen.fnodes import SubroutineCall
assert _test_args(SubroutineCall('foo', ['bar', 'baz']))
def test_sympy__codegen__fnodes__Do():
from sympy.codegen.fnodes import Do
assert _test_args(Do([], 'i', 1, 42))
def test_sympy__codegen__fnodes__ImpliedDoLoop():
from sympy.codegen.fnodes import ImpliedDoLoop
assert _test_args(ImpliedDoLoop('i', 'i', 1, 42))
def test_sympy__codegen__fnodes__ArrayConstructor():
from sympy.codegen.fnodes import ArrayConstructor
assert _test_args(ArrayConstructor([1, 2, 3]))
from sympy.codegen.fnodes import ImpliedDoLoop
idl = ImpliedDoLoop('i', 'i', 1, 42)
assert _test_args(ArrayConstructor([1, idl, 3]))
def test_sympy__codegen__fnodes__sum_():
from sympy.codegen.fnodes import sum_
assert _test_args(sum_('arr'))
def test_sympy__codegen__fnodes__product_():
from sympy.codegen.fnodes import product_
assert _test_args(product_('arr'))
@XFAIL
def test_sympy__combinatorics__graycode__GrayCode():
from sympy.combinatorics.graycode import GrayCode
# an integer is given and returned from GrayCode as the arg
assert _test_args(GrayCode(3, start='100'))
assert _test_args(GrayCode(3, rank=1))
def test_sympy__combinatorics__subsets__Subset():
from sympy.combinatorics.subsets import Subset
assert _test_args(Subset([0, 1], [0, 1, 2, 3]))
assert _test_args(Subset(['c', 'd'], ['a', 'b', 'c', 'd']))
@XFAIL
def test_sympy__combinatorics__permutations__Permutation():
from sympy.combinatorics.permutations import Permutation
assert _test_args(Permutation([0, 1, 2, 3]))
def test_sympy__combinatorics__perm_groups__PermutationGroup():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.perm_groups import PermutationGroup
assert _test_args(PermutationGroup([Permutation([0, 1])]))
def test_sympy__combinatorics__polyhedron__Polyhedron():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.polyhedron import Polyhedron
from sympy.abc import w, x, y, z
pgroup = [Permutation([[0, 1, 2], [3]]),
Permutation([[0, 1, 3], [2]]),
Permutation([[0, 2, 3], [1]]),
Permutation([[1, 2, 3], [0]]),
Permutation([[0, 1], [2, 3]]),
Permutation([[0, 2], [1, 3]]),
Permutation([[0, 3], [1, 2]]),
Permutation([[0, 1, 2, 3]])]
corners = [w, x, y, z]
faces = [(w, x, y), (w, y, z), (w, z, x), (x, y, z)]
assert _test_args(Polyhedron(corners, faces, pgroup))
@XFAIL
def test_sympy__combinatorics__prufer__Prufer():
from sympy.combinatorics.prufer import Prufer
assert _test_args(Prufer([[0, 1], [0, 2], [0, 3]], 4))
def test_sympy__combinatorics__partitions__Partition():
from sympy.combinatorics.partitions import Partition
assert _test_args(Partition([1]))
@XFAIL
def test_sympy__combinatorics__partitions__IntegerPartition():
from sympy.combinatorics.partitions import IntegerPartition
assert _test_args(IntegerPartition([1]))
def test_sympy__concrete__products__Product():
from sympy.concrete.products import Product
assert _test_args(Product(x, (x, 0, 10)))
assert _test_args(Product(x, (x, 0, y), (y, 0, 10)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_limits__ExprWithLimits():
from sympy.concrete.expr_with_limits import ExprWithLimits
assert _test_args(ExprWithLimits(x, (x, 0, 10)))
assert _test_args(ExprWithLimits(x*y, (x, 0, 10.),(y,1.,3)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_limits__AddWithLimits():
from sympy.concrete.expr_with_limits import AddWithLimits
assert _test_args(AddWithLimits(x, (x, 0, 10)))
assert _test_args(AddWithLimits(x*y, (x, 0, 10),(y,1,3)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_intlimits__ExprWithIntLimits():
from sympy.concrete.expr_with_intlimits import ExprWithIntLimits
assert _test_args(ExprWithIntLimits(x, (x, 0, 10)))
assert _test_args(ExprWithIntLimits(x*y, (x, 0, 10),(y,1,3)))
def test_sympy__concrete__summations__Sum():
from sympy.concrete.summations import Sum
assert _test_args(Sum(x, (x, 0, 10)))
assert _test_args(Sum(x, (x, 0, y), (y, 0, 10)))
def test_sympy__core__add__Add():
from sympy.core.add import Add
assert _test_args(Add(x, y, z, 2))
def test_sympy__core__basic__Atom():
from sympy.core.basic import Atom
assert _test_args(Atom())
def test_sympy__core__basic__Basic():
from sympy.core.basic import Basic
assert _test_args(Basic())
def test_sympy__core__containers__Dict():
from sympy.core.containers import Dict
assert _test_args(Dict({x: y, y: z}))
def test_sympy__core__containers__Tuple():
from sympy.core.containers import Tuple
assert _test_args(Tuple(x, y, z, 2))
def test_sympy__core__expr__AtomicExpr():
from sympy.core.expr import AtomicExpr
assert _test_args(AtomicExpr())
def test_sympy__core__expr__Expr():
from sympy.core.expr import Expr
assert _test_args(Expr())
def test_sympy__core__expr__UnevaluatedExpr():
from sympy.core.expr import UnevaluatedExpr
from sympy.abc import x
assert _test_args(UnevaluatedExpr(x))
def test_sympy__core__function__Application():
from sympy.core.function import Application
assert _test_args(Application(1, 2, 3))
def test_sympy__core__function__AppliedUndef():
from sympy.core.function import AppliedUndef
assert _test_args(AppliedUndef(1, 2, 3))
def test_sympy__core__function__Derivative():
from sympy.core.function import Derivative
assert _test_args(Derivative(2, x, y, 3))
@SKIP("abstract class")
def test_sympy__core__function__Function():
pass
def test_sympy__core__function__Lambda():
assert _test_args(Lambda((x, y), x + y + z))
def test_sympy__core__function__Subs():
from sympy.core.function import Subs
assert _test_args(Subs(x + y, x, 2))
def test_sympy__core__function__WildFunction():
from sympy.core.function import WildFunction
assert _test_args(WildFunction('f'))
def test_sympy__core__mod__Mod():
from sympy.core.mod import Mod
assert _test_args(Mod(x, 2))
def test_sympy__core__mul__Mul():
from sympy.core.mul import Mul
assert _test_args(Mul(2, x, y, z))
def test_sympy__core__numbers__Catalan():
from sympy.core.numbers import Catalan
assert _test_args(Catalan())
def test_sympy__core__numbers__ComplexInfinity():
from sympy.core.numbers import ComplexInfinity
assert _test_args(ComplexInfinity())
def test_sympy__core__numbers__EulerGamma():
from sympy.core.numbers import EulerGamma
assert _test_args(EulerGamma())
def test_sympy__core__numbers__Exp1():
from sympy.core.numbers import Exp1
assert _test_args(Exp1())
def test_sympy__core__numbers__Float():
from sympy.core.numbers import Float
assert _test_args(Float(1.23))
def test_sympy__core__numbers__GoldenRatio():
from sympy.core.numbers import GoldenRatio
assert _test_args(GoldenRatio())
def test_sympy__core__numbers__TribonacciConstant():
from sympy.core.numbers import TribonacciConstant
assert _test_args(TribonacciConstant())
def test_sympy__core__numbers__Half():
from sympy.core.numbers import Half
assert _test_args(Half())
def test_sympy__core__numbers__ImaginaryUnit():
from sympy.core.numbers import ImaginaryUnit
assert _test_args(ImaginaryUnit())
def test_sympy__core__numbers__Infinity():
from sympy.core.numbers import Infinity
assert _test_args(Infinity())
def test_sympy__core__numbers__Integer():
from sympy.core.numbers import Integer
assert _test_args(Integer(7))
@SKIP("abstract class")
def test_sympy__core__numbers__IntegerConstant():
pass
def test_sympy__core__numbers__NaN():
from sympy.core.numbers import NaN
assert _test_args(NaN())
def test_sympy__core__numbers__NegativeInfinity():
from sympy.core.numbers import NegativeInfinity
assert _test_args(NegativeInfinity())
def test_sympy__core__numbers__NegativeOne():
from sympy.core.numbers import NegativeOne
assert _test_args(NegativeOne())
def test_sympy__core__numbers__Number():
from sympy.core.numbers import Number
assert _test_args(Number(1, 7))
def test_sympy__core__numbers__NumberSymbol():
from sympy.core.numbers import NumberSymbol
assert _test_args(NumberSymbol())
def test_sympy__core__numbers__One():
from sympy.core.numbers import One
assert _test_args(One())
def test_sympy__core__numbers__Pi():
from sympy.core.numbers import Pi
assert _test_args(Pi())
def test_sympy__core__numbers__Rational():
from sympy.core.numbers import Rational
assert _test_args(Rational(1, 7))
@SKIP("abstract class")
def test_sympy__core__numbers__RationalConstant():
pass
def test_sympy__core__numbers__Zero():
from sympy.core.numbers import Zero
assert _test_args(Zero())
@SKIP("abstract class")
def test_sympy__core__operations__AssocOp():
pass
@SKIP("abstract class")
def test_sympy__core__operations__LatticeOp():
pass
def test_sympy__core__power__Pow():
from sympy.core.power import Pow
assert _test_args(Pow(x, 2))
def test_sympy__algebras__quaternion__Quaternion():
from sympy.algebras.quaternion import Quaternion
assert _test_args(Quaternion(x, 1, 2, 3))
def test_sympy__core__relational__Equality():
from sympy.core.relational import Equality
assert _test_args(Equality(x, 2))
def test_sympy__core__relational__GreaterThan():
from sympy.core.relational import GreaterThan
assert _test_args(GreaterThan(x, 2))
def test_sympy__core__relational__LessThan():
from sympy.core.relational import LessThan
assert _test_args(LessThan(x, 2))
@SKIP("abstract class")
def test_sympy__core__relational__Relational():
pass
def test_sympy__core__relational__StrictGreaterThan():
from sympy.core.relational import StrictGreaterThan
assert _test_args(StrictGreaterThan(x, 2))
def test_sympy__core__relational__StrictLessThan():
from sympy.core.relational import StrictLessThan
assert _test_args(StrictLessThan(x, 2))
def test_sympy__core__relational__Unequality():
from sympy.core.relational import Unequality
assert _test_args(Unequality(x, 2))
def test_sympy__sandbox__indexed_integrals__IndexedIntegral():
from sympy.tensor import IndexedBase, Idx
from sympy.sandbox.indexed_integrals import IndexedIntegral
A = IndexedBase('A')
i, j = symbols('i j', integer=True)
a1, a2 = symbols('a1:3', cls=Idx)
assert _test_args(IndexedIntegral(A[a1], A[a2]))
assert _test_args(IndexedIntegral(A[i], A[j]))
def test_sympy__calculus__util__AccumulationBounds():
from sympy.calculus.util import AccumulationBounds
assert _test_args(AccumulationBounds(0, 1))
def test_sympy__sets__ordinals__OmegaPower():
from sympy.sets.ordinals import OmegaPower
assert _test_args(OmegaPower(1, 1))
def test_sympy__sets__ordinals__Ordinal():
from sympy.sets.ordinals import Ordinal, OmegaPower
assert _test_args(Ordinal(OmegaPower(2, 1)))
def test_sympy__sets__ordinals__OrdinalOmega():
from sympy.sets.ordinals import OrdinalOmega
assert _test_args(OrdinalOmega())
def test_sympy__sets__ordinals__OrdinalZero():
from sympy.sets.ordinals import OrdinalZero
assert _test_args(OrdinalZero())
def test_sympy__sets__sets__EmptySet():
from sympy.sets.sets import EmptySet
assert _test_args(EmptySet())
def test_sympy__sets__sets__UniversalSet():
from sympy.sets.sets import UniversalSet
assert _test_args(UniversalSet())
def test_sympy__sets__sets__FiniteSet():
from sympy.sets.sets import FiniteSet
assert _test_args(FiniteSet(x, y, z))
def test_sympy__sets__sets__Interval():
from sympy.sets.sets import Interval
assert _test_args(Interval(0, 1))
def test_sympy__sets__sets__ProductSet():
from sympy.sets.sets import ProductSet, Interval
assert _test_args(ProductSet(Interval(0, 1), Interval(0, 1)))
@SKIP("does it make sense to test this?")
def test_sympy__sets__sets__Set():
from sympy.sets.sets import Set
assert _test_args(Set())
def test_sympy__sets__sets__Intersection():
from sympy.sets.sets import Intersection, Interval
assert _test_args(Intersection(Interval(0, 3), Interval(2, 4),
evaluate=False))
def test_sympy__sets__sets__Union():
from sympy.sets.sets import Union, Interval
assert _test_args(Union(Interval(0, 1), Interval(2, 3)))
def test_sympy__sets__sets__Complement():
from sympy.sets.sets import Complement
assert _test_args(Complement(Interval(0, 2), Interval(0, 1)))
def test_sympy__sets__sets__SymmetricDifference():
from sympy.sets.sets import FiniteSet, SymmetricDifference
assert _test_args(SymmetricDifference(FiniteSet(1, 2, 3), \
FiniteSet(2, 3, 4)))
def test_sympy__core__trace__Tr():
from sympy.core.trace import Tr
a, b = symbols('a b')
assert _test_args(Tr(a + b))
def test_sympy__sets__setexpr__SetExpr():
from sympy.sets.setexpr import SetExpr
assert _test_args(SetExpr(Interval(0, 1)))
def test_sympy__sets__fancysets__Naturals():
from sympy.sets.fancysets import Naturals
assert _test_args(Naturals())
def test_sympy__sets__fancysets__Naturals0():
from sympy.sets.fancysets import Naturals0
assert _test_args(Naturals0())
def test_sympy__sets__fancysets__Integers():
from sympy.sets.fancysets import Integers
assert _test_args(Integers())
def test_sympy__sets__fancysets__Reals():
from sympy.sets.fancysets import Reals
assert _test_args(Reals())
def test_sympy__sets__fancysets__Complexes():
from sympy.sets.fancysets import Complexes
assert _test_args(Complexes())
def test_sympy__sets__fancysets__ComplexRegion():
from sympy.sets.fancysets import ComplexRegion
from sympy import S
from sympy.sets import Interval
a = Interval(0, 1)
b = Interval(2, 3)
theta = Interval(0, 2*S.Pi)
assert _test_args(ComplexRegion(a*b))
assert _test_args(ComplexRegion(a*theta, polar=True))
def test_sympy__sets__fancysets__ImageSet():
from sympy.sets.fancysets import ImageSet
from sympy import S, Symbol
x = Symbol('x')
assert _test_args(ImageSet(Lambda(x, x**2), S.Naturals))
def test_sympy__sets__fancysets__Range():
from sympy.sets.fancysets import Range
assert _test_args(Range(1, 5, 1))
def test_sympy__sets__conditionset__ConditionSet():
from sympy.sets.conditionset import ConditionSet
from sympy import S, Symbol
x = Symbol('x')
assert _test_args(ConditionSet(x, Eq(x**2, 1), S.Reals))
def test_sympy__sets__contains__Contains():
from sympy.sets.fancysets import Range
from sympy.sets.contains import Contains
assert _test_args(Contains(x, Range(0, 10, 2)))
# STATS
from sympy.stats.crv_types import NormalDistribution
nd = NormalDistribution(0, 1)
from sympy.stats.frv_types import DieDistribution
die = DieDistribution(6)
def test_sympy__stats__crv__ContinuousDomain():
from sympy.stats.crv import ContinuousDomain
assert _test_args(ContinuousDomain({x}, Interval(-oo, oo)))
def test_sympy__stats__crv__SingleContinuousDomain():
from sympy.stats.crv import SingleContinuousDomain
assert _test_args(SingleContinuousDomain(x, Interval(-oo, oo)))
def test_sympy__stats__crv__ProductContinuousDomain():
from sympy.stats.crv import SingleContinuousDomain, ProductContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
E = SingleContinuousDomain(y, Interval(0, oo))
assert _test_args(ProductContinuousDomain(D, E))
def test_sympy__stats__crv__ConditionalContinuousDomain():
from sympy.stats.crv import (SingleContinuousDomain,
ConditionalContinuousDomain)
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ConditionalContinuousDomain(D, x > 0))
def test_sympy__stats__crv__ContinuousPSpace():
from sympy.stats.crv import ContinuousPSpace, SingleContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ContinuousPSpace(D, nd))
def test_sympy__stats__crv__SingleContinuousPSpace():
from sympy.stats.crv import SingleContinuousPSpace
assert _test_args(SingleContinuousPSpace(x, nd))
@SKIP("abstract class")
def test_sympy__stats__crv__SingleContinuousDistribution():
pass
def test_sympy__stats__drv__SingleDiscreteDomain():
from sympy.stats.drv import SingleDiscreteDomain
assert _test_args(SingleDiscreteDomain(x, S.Naturals))
def test_sympy__stats__drv__ProductDiscreteDomain():
from sympy.stats.drv import SingleDiscreteDomain, ProductDiscreteDomain
X = SingleDiscreteDomain(x, S.Naturals)
Y = SingleDiscreteDomain(y, S.Integers)
assert _test_args(ProductDiscreteDomain(X, Y))
def test_sympy__stats__drv__SingleDiscretePSpace():
from sympy.stats.drv import SingleDiscretePSpace
from sympy.stats.drv_types import PoissonDistribution
assert _test_args(SingleDiscretePSpace(x, PoissonDistribution(1)))
def test_sympy__stats__drv__DiscretePSpace():
from sympy.stats.drv import DiscretePSpace, SingleDiscreteDomain
density = Lambda(x, 2**(-x))
domain = SingleDiscreteDomain(x, S.Naturals)
assert _test_args(DiscretePSpace(domain, density))
def test_sympy__stats__drv__ConditionalDiscreteDomain():
from sympy.stats.drv import ConditionalDiscreteDomain, SingleDiscreteDomain
X = SingleDiscreteDomain(x, S.Naturals0)
assert _test_args(ConditionalDiscreteDomain(X, x > 2))
def test_sympy__stats__joint_rv__JointPSpace():
from sympy.stats.joint_rv import JointPSpace, JointDistribution
assert _test_args(JointPSpace('X', JointDistribution(1)))
def test_sympy__stats__joint_rv__JointRandomSymbol():
from sympy.stats.joint_rv import JointRandomSymbol
assert _test_args(JointRandomSymbol(x))
def test_sympy__stats__joint_rv__JointDistributionHandmade():
from sympy import Indexed
from sympy.stats.joint_rv import JointDistributionHandmade
x1, x2 = (Indexed('x', i) for i in (1, 2))
assert _test_args(JointDistributionHandmade(x1 + x2, S.Reals**2))
def test_sympy__stats__joint_rv__MarginalDistribution():
from sympy.stats.rv import RandomSymbol
from sympy.stats.joint_rv import MarginalDistribution
r = RandomSymbol(S('r'))
assert _test_args(MarginalDistribution(r, (r,)))
def test_sympy__stats__joint_rv__CompoundDistribution():
from sympy.stats.joint_rv import CompoundDistribution
from sympy.stats.drv_types import PoissonDistribution
r = PoissonDistribution(x)
assert _test_args(CompoundDistribution(PoissonDistribution(r)))
@SKIP("abstract class")
def test_sympy__stats__drv__SingleDiscreteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__drv__DiscreteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__drv__DiscreteDomain():
pass
def test_sympy__stats__rv__RandomDomain():
from sympy.stats.rv import RandomDomain
from sympy.sets.sets import FiniteSet
assert _test_args(RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3)))
def test_sympy__stats__rv__SingleDomain():
from sympy.stats.rv import SingleDomain
from sympy.sets.sets import FiniteSet
assert _test_args(SingleDomain(x, FiniteSet(1, 2, 3)))
def test_sympy__stats__rv__ConditionalDomain():
from sympy.stats.rv import ConditionalDomain, RandomDomain
from sympy.sets.sets import FiniteSet
D = RandomDomain(FiniteSet(x), FiniteSet(1, 2))
assert _test_args(ConditionalDomain(D, x > 1))
def test_sympy__stats__rv__PSpace():
from sympy.stats.rv import PSpace, RandomDomain
from sympy import FiniteSet
D = RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3, 4, 5, 6))
assert _test_args(PSpace(D, die))
@SKIP("abstract Class")
def test_sympy__stats__rv__SinglePSpace():
pass
def test_sympy__stats__rv__RandomSymbol():
from sympy.stats.rv import RandomSymbol
from sympy.stats.crv import SingleContinuousPSpace
A = SingleContinuousPSpace(x, nd)
assert _test_args(RandomSymbol(x, A))
@SKIP("abstract Class")
def test_sympy__stats__rv__ProductPSpace():
pass
def test_sympy__stats__rv__IndependentProductPSpace():
from sympy.stats.rv import IndependentProductPSpace
from sympy.stats.crv import SingleContinuousPSpace
A = SingleContinuousPSpace(x, nd)
B = SingleContinuousPSpace(y, nd)
assert _test_args(IndependentProductPSpace(A, B))
def test_sympy__stats__rv__ProductDomain():
from sympy.stats.rv import ProductDomain, SingleDomain
D = SingleDomain(x, Interval(-oo, oo))
E = SingleDomain(y, Interval(0, oo))
assert _test_args(ProductDomain(D, E))
def test_sympy__stats__symbolic_probability__Probability():
from sympy.stats.symbolic_probability import Probability
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Probability(X > 0))
def test_sympy__stats__symbolic_probability__Expectation():
from sympy.stats.symbolic_probability import Expectation
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Expectation(X > 0))
def test_sympy__stats__symbolic_probability__Covariance():
from sympy.stats.symbolic_probability import Covariance
from sympy.stats import Normal
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 3)
assert _test_args(Covariance(X, Y))
def test_sympy__stats__symbolic_probability__Variance():
from sympy.stats.symbolic_probability import Variance
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Variance(X))
def test_sympy__stats__frv_types__DiscreteUniformDistribution():
from sympy.stats.frv_types import DiscreteUniformDistribution
from sympy.core.containers import Tuple
assert _test_args(DiscreteUniformDistribution(Tuple(*list(range(6)))))
def test_sympy__stats__frv_types__DieDistribution():
assert _test_args(die)
def test_sympy__stats__frv_types__BernoulliDistribution():
from sympy.stats.frv_types import BernoulliDistribution
assert _test_args(BernoulliDistribution(S.Half, 0, 1))
def test_sympy__stats__frv_types__BinomialDistribution():
from sympy.stats.frv_types import BinomialDistribution
assert _test_args(BinomialDistribution(5, S.Half, 1, 0))
def test_sympy__stats__frv_types__HypergeometricDistribution():
from sympy.stats.frv_types import HypergeometricDistribution
assert _test_args(HypergeometricDistribution(10, 5, 3))
def test_sympy__stats__frv_types__RademacherDistribution():
from sympy.stats.frv_types import RademacherDistribution
assert _test_args(RademacherDistribution())
def test_sympy__stats__frv__FiniteDomain():
from sympy.stats.frv import FiniteDomain
assert _test_args(FiniteDomain({(x, 1), (x, 2)})) # x can be 1 or 2
def test_sympy__stats__frv__SingleFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain
assert _test_args(SingleFiniteDomain(x, {1, 2})) # x can be 1 or 2
def test_sympy__stats__frv__ProductFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain, ProductFiniteDomain
xd = SingleFiniteDomain(x, {1, 2})
yd = SingleFiniteDomain(y, {1, 2})
assert _test_args(ProductFiniteDomain(xd, yd))
def test_sympy__stats__frv__ConditionalFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain, ConditionalFiniteDomain
xd = SingleFiniteDomain(x, {1, 2})
assert _test_args(ConditionalFiniteDomain(xd, x > 1))
def test_sympy__stats__frv__FinitePSpace():
from sympy.stats.frv import FinitePSpace, SingleFiniteDomain
xd = SingleFiniteDomain(x, {1, 2, 3, 4, 5, 6})
p = 1.0/6
xd = SingleFiniteDomain(x, {1, 2})
assert _test_args(FinitePSpace(xd, {(x, 1): S.Half, (x, 2): S.Half}))
def test_sympy__stats__frv__SingleFinitePSpace():
from sympy.stats.frv import SingleFinitePSpace
from sympy import Symbol
assert _test_args(SingleFinitePSpace(Symbol('x'), die))
def test_sympy__stats__frv__ProductFinitePSpace():
from sympy.stats.frv import SingleFinitePSpace, ProductFinitePSpace
from sympy import Symbol
xp = SingleFinitePSpace(Symbol('x'), die)
yp = SingleFinitePSpace(Symbol('y'), die)
assert _test_args(ProductFinitePSpace(xp, yp))
@SKIP("abstract class")
def test_sympy__stats__frv__SingleFiniteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__crv__ContinuousDistribution():
pass
def test_sympy__stats__frv_types__FiniteDistributionHandmade():
from sympy.stats.frv_types import FiniteDistributionHandmade
assert _test_args(FiniteDistributionHandmade({1: 1}))
def test_sympy__stats__crv__ContinuousDistributionHandmade():
from sympy.stats.crv import ContinuousDistributionHandmade
from sympy import Symbol, Interval
assert _test_args(ContinuousDistributionHandmade(Symbol('x'),
Interval(0, 2)))
def test_sympy__stats__drv__DiscreteDistributionHandmade():
from sympy.stats.drv import DiscreteDistributionHandmade
assert _test_args(DiscreteDistributionHandmade(x, S.Naturals))
def test_sympy__stats__rv__Density():
from sympy.stats.rv import Density
from sympy.stats.crv_types import Normal
assert _test_args(Density(Normal('x', 0, 1)))
def test_sympy__stats__crv_types__ArcsinDistribution():
from sympy.stats.crv_types import ArcsinDistribution
assert _test_args(ArcsinDistribution(0, 1))
def test_sympy__stats__crv_types__BeniniDistribution():
from sympy.stats.crv_types import BeniniDistribution
assert _test_args(BeniniDistribution(1, 1, 1))
def test_sympy__stats__crv_types__BetaDistribution():
from sympy.stats.crv_types import BetaDistribution
assert _test_args(BetaDistribution(1, 1))
def test_sympy__stats__crv_types__BetaPrimeDistribution():
from sympy.stats.crv_types import BetaPrimeDistribution
assert _test_args(BetaPrimeDistribution(1, 1))
def test_sympy__stats__crv_types__CauchyDistribution():
from sympy.stats.crv_types import CauchyDistribution
assert _test_args(CauchyDistribution(0, 1))
def test_sympy__stats__crv_types__ChiDistribution():
from sympy.stats.crv_types import ChiDistribution
assert _test_args(ChiDistribution(1))
def test_sympy__stats__crv_types__ChiNoncentralDistribution():
from sympy.stats.crv_types import ChiNoncentralDistribution
assert _test_args(ChiNoncentralDistribution(1,1))
def test_sympy__stats__crv_types__ChiSquaredDistribution():
from sympy.stats.crv_types import ChiSquaredDistribution
assert _test_args(ChiSquaredDistribution(1))
def test_sympy__stats__crv_types__DagumDistribution():
from sympy.stats.crv_types import DagumDistribution
assert _test_args(DagumDistribution(1, 1, 1))
def test_sympy__stats__crv_types__ExponentialDistribution():
from sympy.stats.crv_types import ExponentialDistribution
assert _test_args(ExponentialDistribution(1))
def test_sympy__stats__crv_types__FDistributionDistribution():
from sympy.stats.crv_types import FDistributionDistribution
assert _test_args(FDistributionDistribution(1, 1))
def test_sympy__stats__crv_types__FisherZDistribution():
from sympy.stats.crv_types import FisherZDistribution
assert _test_args(FisherZDistribution(1, 1))
def test_sympy__stats__crv_types__FrechetDistribution():
from sympy.stats.crv_types import FrechetDistribution
assert _test_args(FrechetDistribution(1, 1, 1))
def test_sympy__stats__crv_types__GammaInverseDistribution():
from sympy.stats.crv_types import GammaInverseDistribution
assert _test_args(GammaInverseDistribution(1, 1))
def test_sympy__stats__crv_types__GammaDistribution():
from sympy.stats.crv_types import GammaDistribution
assert _test_args(GammaDistribution(1, 1))
def test_sympy__stats__crv_types__GumbelDistribution():
from sympy.stats.crv_types import GumbelDistribution
assert _test_args(GumbelDistribution(1, 1))
def test_sympy__stats__crv_types__GompertzDistribution():
from sympy.stats.crv_types import GompertzDistribution
assert _test_args(GompertzDistribution(1, 1))
def test_sympy__stats__crv_types__KumaraswamyDistribution():
from sympy.stats.crv_types import KumaraswamyDistribution
assert _test_args(KumaraswamyDistribution(1, 1))
def test_sympy__stats__crv_types__LaplaceDistribution():
from sympy.stats.crv_types import LaplaceDistribution
assert _test_args(LaplaceDistribution(0, 1))
def test_sympy__stats__crv_types__LogisticDistribution():
from sympy.stats.crv_types import LogisticDistribution
assert _test_args(LogisticDistribution(0, 1))
def test_sympy__stats__crv_types__LogNormalDistribution():
from sympy.stats.crv_types import LogNormalDistribution
assert _test_args(LogNormalDistribution(0, 1))
def test_sympy__stats__crv_types__MaxwellDistribution():
from sympy.stats.crv_types import MaxwellDistribution
assert _test_args(MaxwellDistribution(1))
def test_sympy__stats__crv_types__NakagamiDistribution():
from sympy.stats.crv_types import NakagamiDistribution
assert _test_args(NakagamiDistribution(1, 1))
def test_sympy__stats__crv_types__NormalDistribution():
from sympy.stats.crv_types import NormalDistribution
assert _test_args(NormalDistribution(0, 1))
def test_sympy__stats__crv_types__ParetoDistribution():
from sympy.stats.crv_types import ParetoDistribution
assert _test_args(ParetoDistribution(1, 1))
def test_sympy__stats__crv_types__QuadraticUDistribution():
from sympy.stats.crv_types import QuadraticUDistribution
assert _test_args(QuadraticUDistribution(1, 2))
def test_sympy__stats__crv_types__RaisedCosineDistribution():
from sympy.stats.crv_types import RaisedCosineDistribution
assert _test_args(RaisedCosineDistribution(1, 1))
def test_sympy__stats__crv_types__RayleighDistribution():
from sympy.stats.crv_types import RayleighDistribution
assert _test_args(RayleighDistribution(1))
def test_sympy__stats__crv_types__ShiftedGompertzDistribution():
from sympy.stats.crv_types import ShiftedGompertzDistribution
assert _test_args(ShiftedGompertzDistribution(1, 1))
def test_sympy__stats__crv_types__StudentTDistribution():
from sympy.stats.crv_types import StudentTDistribution
assert _test_args(StudentTDistribution(1))
def test_sympy__stats__crv_types__TrapezoidalDistribution():
from sympy.stats.crv_types import TrapezoidalDistribution
assert _test_args(TrapezoidalDistribution(1, 2, 3, 4))
def test_sympy__stats__crv_types__TriangularDistribution():
from sympy.stats.crv_types import TriangularDistribution
assert _test_args(TriangularDistribution(-1, 0, 1))
def test_sympy__stats__crv_types__UniformDistribution():
from sympy.stats.crv_types import UniformDistribution
assert _test_args(UniformDistribution(0, 1))
def test_sympy__stats__crv_types__UniformSumDistribution():
from sympy.stats.crv_types import UniformSumDistribution
assert _test_args(UniformSumDistribution(1))
def test_sympy__stats__crv_types__VonMisesDistribution():
from sympy.stats.crv_types import VonMisesDistribution
assert _test_args(VonMisesDistribution(1, 1))
def test_sympy__stats__crv_types__WeibullDistribution():
from sympy.stats.crv_types import WeibullDistribution
assert _test_args(WeibullDistribution(1, 1))
def test_sympy__stats__crv_types__WignerSemicircleDistribution():
from sympy.stats.crv_types import WignerSemicircleDistribution
assert _test_args(WignerSemicircleDistribution(1))
def test_sympy__stats__drv_types__GeometricDistribution():
from sympy.stats.drv_types import GeometricDistribution
assert _test_args(GeometricDistribution(.5))
def test_sympy__stats__drv_types__LogarithmicDistribution():
from sympy.stats.drv_types import LogarithmicDistribution
assert _test_args(LogarithmicDistribution(.5))
def test_sympy__stats__drv_types__NegativeBinomialDistribution():
from sympy.stats.drv_types import NegativeBinomialDistribution
assert _test_args(NegativeBinomialDistribution(.5, .5))
def test_sympy__stats__drv_types__PoissonDistribution():
from sympy.stats.drv_types import PoissonDistribution
assert _test_args(PoissonDistribution(1))
def test_sympy__stats__drv_types__YuleSimonDistribution():
from sympy.stats.drv_types import YuleSimonDistribution
assert _test_args(YuleSimonDistribution(.5))
def test_sympy__stats__drv_types__ZetaDistribution():
from sympy.stats.drv_types import ZetaDistribution
assert _test_args(ZetaDistribution(1.5))
def test_sympy__stats__joint_rv__JointDistribution():
from sympy.stats.joint_rv import JointDistribution
assert _test_args(JointDistribution(1, 2, 3, 4))
def test_sympy__stats__joint_rv_types__MultivariateNormalDistribution():
from sympy.stats.joint_rv_types import MultivariateNormalDistribution
assert _test_args(
MultivariateNormalDistribution([0, 1], [[1, 0],[0, 1]]))
def test_sympy__stats__joint_rv_types__MultivariateLaplaceDistribution():
from sympy.stats.joint_rv_types import MultivariateLaplaceDistribution
assert _test_args(MultivariateLaplaceDistribution([0, 1], [[1, 0],[0, 1]]))
def test_sympy__stats__joint_rv_types__MultivariateTDistribution():
from sympy.stats.joint_rv_types import MultivariateTDistribution
assert _test_args(MultivariateTDistribution([0, 1], [[1, 0],[0, 1]], 1))
def test_sympy__stats__joint_rv_types__NormalGammaDistribution():
from sympy.stats.joint_rv_types import NormalGammaDistribution
assert _test_args(NormalGammaDistribution(1, 2, 3, 4))
def test_sympy__core__symbol__Dummy():
from sympy.core.symbol import Dummy
assert _test_args(Dummy('t'))
def test_sympy__core__symbol__Symbol():
from sympy.core.symbol import Symbol
assert _test_args(Symbol('t'))
def test_sympy__core__symbol__Wild():
from sympy.core.symbol import Wild
assert _test_args(Wild('x', exclude=[x]))
@SKIP("abstract class")
def test_sympy__functions__combinatorial__factorials__CombinatorialFunction():
pass
def test_sympy__functions__combinatorial__factorials__FallingFactorial():
from sympy.functions.combinatorial.factorials import FallingFactorial
assert _test_args(FallingFactorial(2, x))
def test_sympy__functions__combinatorial__factorials__MultiFactorial():
from sympy.functions.combinatorial.factorials import MultiFactorial
assert _test_args(MultiFactorial(x))
def test_sympy__functions__combinatorial__factorials__RisingFactorial():
from sympy.functions.combinatorial.factorials import RisingFactorial
assert _test_args(RisingFactorial(2, x))
def test_sympy__functions__combinatorial__factorials__binomial():
from sympy.functions.combinatorial.factorials import binomial
assert _test_args(binomial(2, x))
def test_sympy__functions__combinatorial__factorials__subfactorial():
from sympy.functions.combinatorial.factorials import subfactorial
assert _test_args(subfactorial(1))
def test_sympy__functions__combinatorial__factorials__factorial():
from sympy.functions.combinatorial.factorials import factorial
assert _test_args(factorial(x))
def test_sympy__functions__combinatorial__factorials__factorial2():
from sympy.functions.combinatorial.factorials import factorial2
assert _test_args(factorial2(x))
def test_sympy__functions__combinatorial__numbers__bell():
from sympy.functions.combinatorial.numbers import bell
assert _test_args(bell(x, y))
def test_sympy__functions__combinatorial__numbers__bernoulli():
from sympy.functions.combinatorial.numbers import bernoulli
assert _test_args(bernoulli(x))
def test_sympy__functions__combinatorial__numbers__catalan():
from sympy.functions.combinatorial.numbers import catalan
assert _test_args(catalan(x))
def test_sympy__functions__combinatorial__numbers__genocchi():
from sympy.functions.combinatorial.numbers import genocchi
assert _test_args(genocchi(x))
def test_sympy__functions__combinatorial__numbers__euler():
from sympy.functions.combinatorial.numbers import euler
assert _test_args(euler(x))
def test_sympy__functions__combinatorial__numbers__carmichael():
from sympy.functions.combinatorial.numbers import carmichael
assert _test_args(carmichael(x))
def test_sympy__functions__combinatorial__numbers__fibonacci():
from sympy.functions.combinatorial.numbers import fibonacci
assert _test_args(fibonacci(x))
def test_sympy__functions__combinatorial__numbers__tribonacci():
from sympy.functions.combinatorial.numbers import tribonacci
assert _test_args(tribonacci(x))
def test_sympy__functions__combinatorial__numbers__harmonic():
from sympy.functions.combinatorial.numbers import harmonic
assert _test_args(harmonic(x, 2))
def test_sympy__functions__combinatorial__numbers__lucas():
from sympy.functions.combinatorial.numbers import lucas
assert _test_args(lucas(x))
def test_sympy__functions__combinatorial__numbers__partition():
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.numbers import partition
assert _test_args(partition(Symbol('a', integer=True)))
def test_sympy__functions__elementary__complexes__Abs():
from sympy.functions.elementary.complexes import Abs
assert _test_args(Abs(x))
def test_sympy__functions__elementary__complexes__adjoint():
from sympy.functions.elementary.complexes import adjoint
assert _test_args(adjoint(x))
def test_sympy__functions__elementary__complexes__arg():
from sympy.functions.elementary.complexes import arg
assert _test_args(arg(x))
def test_sympy__functions__elementary__complexes__conjugate():
from sympy.functions.elementary.complexes import conjugate
assert _test_args(conjugate(x))
def test_sympy__functions__elementary__complexes__im():
from sympy.functions.elementary.complexes import im
assert _test_args(im(x))
def test_sympy__functions__elementary__complexes__re():
from sympy.functions.elementary.complexes import re
assert _test_args(re(x))
def test_sympy__functions__elementary__complexes__sign():
from sympy.functions.elementary.complexes import sign
assert _test_args(sign(x))
def test_sympy__functions__elementary__complexes__polar_lift():
from sympy.functions.elementary.complexes import polar_lift
assert _test_args(polar_lift(x))
def test_sympy__functions__elementary__complexes__periodic_argument():
from sympy.functions.elementary.complexes import periodic_argument
assert _test_args(periodic_argument(x, y))
def test_sympy__functions__elementary__complexes__principal_branch():
from sympy.functions.elementary.complexes import principal_branch
assert _test_args(principal_branch(x, y))
def test_sympy__functions__elementary__complexes__transpose():
from sympy.functions.elementary.complexes import transpose
assert _test_args(transpose(x))
def test_sympy__functions__elementary__exponential__LambertW():
from sympy.functions.elementary.exponential import LambertW
assert _test_args(LambertW(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__exponential__ExpBase():
pass
def test_sympy__functions__elementary__exponential__exp():
from sympy.functions.elementary.exponential import exp
assert _test_args(exp(2))
def test_sympy__functions__elementary__exponential__exp_polar():
from sympy.functions.elementary.exponential import exp_polar
assert _test_args(exp_polar(2))
def test_sympy__functions__elementary__exponential__log():
from sympy.functions.elementary.exponential import log
assert _test_args(log(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__HyperbolicFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__ReciprocalHyperbolicFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__InverseHyperbolicFunction():
pass
def test_sympy__functions__elementary__hyperbolic__acosh():
from sympy.functions.elementary.hyperbolic import acosh
assert _test_args(acosh(2))
def test_sympy__functions__elementary__hyperbolic__acoth():
from sympy.functions.elementary.hyperbolic import acoth
assert _test_args(acoth(2))
def test_sympy__functions__elementary__hyperbolic__asinh():
from sympy.functions.elementary.hyperbolic import asinh
assert _test_args(asinh(2))
def test_sympy__functions__elementary__hyperbolic__atanh():
from sympy.functions.elementary.hyperbolic import atanh
assert _test_args(atanh(2))
def test_sympy__functions__elementary__hyperbolic__asech():
from sympy.functions.elementary.hyperbolic import asech
assert _test_args(asech(2))
def test_sympy__functions__elementary__hyperbolic__acsch():
from sympy.functions.elementary.hyperbolic import acsch
assert _test_args(acsch(2))
def test_sympy__functions__elementary__hyperbolic__cosh():
from sympy.functions.elementary.hyperbolic import cosh
assert _test_args(cosh(2))
def test_sympy__functions__elementary__hyperbolic__coth():
from sympy.functions.elementary.hyperbolic import coth
assert _test_args(coth(2))
def test_sympy__functions__elementary__hyperbolic__csch():
from sympy.functions.elementary.hyperbolic import csch
assert _test_args(csch(2))
def test_sympy__functions__elementary__hyperbolic__sech():
from sympy.functions.elementary.hyperbolic import sech
assert _test_args(sech(2))
def test_sympy__functions__elementary__hyperbolic__sinh():
from sympy.functions.elementary.hyperbolic import sinh
assert _test_args(sinh(2))
def test_sympy__functions__elementary__hyperbolic__tanh():
from sympy.functions.elementary.hyperbolic import tanh
assert _test_args(tanh(2))
@SKIP("does this work at all?")
def test_sympy__functions__elementary__integers__RoundFunction():
from sympy.functions.elementary.integers import RoundFunction
assert _test_args(RoundFunction())
def test_sympy__functions__elementary__integers__ceiling():
from sympy.functions.elementary.integers import ceiling
assert _test_args(ceiling(x))
def test_sympy__functions__elementary__integers__floor():
from sympy.functions.elementary.integers import floor
assert _test_args(floor(x))
def test_sympy__functions__elementary__integers__frac():
from sympy.functions.elementary.integers import frac
assert _test_args(frac(x))
def test_sympy__functions__elementary__miscellaneous__IdentityFunction():
from sympy.functions.elementary.miscellaneous import IdentityFunction
assert _test_args(IdentityFunction())
def test_sympy__functions__elementary__miscellaneous__Max():
from sympy.functions.elementary.miscellaneous import Max
assert _test_args(Max(x, 2))
def test_sympy__functions__elementary__miscellaneous__Min():
from sympy.functions.elementary.miscellaneous import Min
assert _test_args(Min(x, 2))
@SKIP("abstract class")
def test_sympy__functions__elementary__miscellaneous__MinMaxBase():
pass
def test_sympy__functions__elementary__piecewise__ExprCondPair():
from sympy.functions.elementary.piecewise import ExprCondPair
assert _test_args(ExprCondPair(1, True))
def test_sympy__functions__elementary__piecewise__Piecewise():
from sympy.functions.elementary.piecewise import Piecewise
assert _test_args(Piecewise((1, x >= 0), (0, True)))
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__TrigonometricFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__ReciprocalTrigonometricFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__InverseTrigonometricFunction():
pass
def test_sympy__functions__elementary__trigonometric__acos():
from sympy.functions.elementary.trigonometric import acos
assert _test_args(acos(2))
def test_sympy__functions__elementary__trigonometric__acot():
from sympy.functions.elementary.trigonometric import acot
assert _test_args(acot(2))
def test_sympy__functions__elementary__trigonometric__asin():
from sympy.functions.elementary.trigonometric import asin
assert _test_args(asin(2))
def test_sympy__functions__elementary__trigonometric__asec():
from sympy.functions.elementary.trigonometric import asec
assert _test_args(asec(2))
def test_sympy__functions__elementary__trigonometric__acsc():
from sympy.functions.elementary.trigonometric import acsc
assert _test_args(acsc(2))
def test_sympy__functions__elementary__trigonometric__atan():
from sympy.functions.elementary.trigonometric import atan
assert _test_args(atan(2))
def test_sympy__functions__elementary__trigonometric__atan2():
from sympy.functions.elementary.trigonometric import atan2
assert _test_args(atan2(2, 3))
def test_sympy__functions__elementary__trigonometric__cos():
from sympy.functions.elementary.trigonometric import cos
assert _test_args(cos(2))
def test_sympy__functions__elementary__trigonometric__csc():
from sympy.functions.elementary.trigonometric import csc
assert _test_args(csc(2))
def test_sympy__functions__elementary__trigonometric__cot():
from sympy.functions.elementary.trigonometric import cot
assert _test_args(cot(2))
def test_sympy__functions__elementary__trigonometric__sin():
assert _test_args(sin(2))
def test_sympy__functions__elementary__trigonometric__sinc():
from sympy.functions.elementary.trigonometric import sinc
assert _test_args(sinc(2))
def test_sympy__functions__elementary__trigonometric__sec():
from sympy.functions.elementary.trigonometric import sec
assert _test_args(sec(2))
def test_sympy__functions__elementary__trigonometric__tan():
from sympy.functions.elementary.trigonometric import tan
assert _test_args(tan(2))
@SKIP("abstract class")
def test_sympy__functions__special__bessel__BesselBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__bessel__SphericalBesselBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__bessel__SphericalHankelBase():
pass
def test_sympy__functions__special__bessel__besseli():
from sympy.functions.special.bessel import besseli
assert _test_args(besseli(x, 1))
def test_sympy__functions__special__bessel__besselj():
from sympy.functions.special.bessel import besselj
assert _test_args(besselj(x, 1))
def test_sympy__functions__special__bessel__besselk():
from sympy.functions.special.bessel import besselk
assert _test_args(besselk(x, 1))
def test_sympy__functions__special__bessel__bessely():
from sympy.functions.special.bessel import bessely
assert _test_args(bessely(x, 1))
def test_sympy__functions__special__bessel__hankel1():
from sympy.functions.special.bessel import hankel1
assert _test_args(hankel1(x, 1))
def test_sympy__functions__special__bessel__hankel2():
from sympy.functions.special.bessel import hankel2
assert _test_args(hankel2(x, 1))
def test_sympy__functions__special__bessel__jn():
from sympy.functions.special.bessel import jn
assert _test_args(jn(0, x))
def test_sympy__functions__special__bessel__yn():
from sympy.functions.special.bessel import yn
assert _test_args(yn(0, x))
def test_sympy__functions__special__bessel__hn1():
from sympy.functions.special.bessel import hn1
assert _test_args(hn1(0, x))
def test_sympy__functions__special__bessel__hn2():
from sympy.functions.special.bessel import hn2
assert _test_args(hn2(0, x))
def test_sympy__functions__special__bessel__AiryBase():
pass
def test_sympy__functions__special__bessel__airyai():
from sympy.functions.special.bessel import airyai
assert _test_args(airyai(2))
def test_sympy__functions__special__bessel__airybi():
from sympy.functions.special.bessel import airybi
assert _test_args(airybi(2))
def test_sympy__functions__special__bessel__airyaiprime():
from sympy.functions.special.bessel import airyaiprime
assert _test_args(airyaiprime(2))
def test_sympy__functions__special__bessel__airybiprime():
from sympy.functions.special.bessel import airybiprime
assert _test_args(airybiprime(2))
def test_sympy__functions__special__elliptic_integrals__elliptic_k():
from sympy.functions.special.elliptic_integrals import elliptic_k as K
assert _test_args(K(x))
def test_sympy__functions__special__elliptic_integrals__elliptic_f():
from sympy.functions.special.elliptic_integrals import elliptic_f as F
assert _test_args(F(x, y))
def test_sympy__functions__special__elliptic_integrals__elliptic_e():
from sympy.functions.special.elliptic_integrals import elliptic_e as E
assert _test_args(E(x))
assert _test_args(E(x, y))
def test_sympy__functions__special__elliptic_integrals__elliptic_pi():
from sympy.functions.special.elliptic_integrals import elliptic_pi as P
assert _test_args(P(x, y))
assert _test_args(P(x, y, z))
def test_sympy__functions__special__delta_functions__DiracDelta():
from sympy.functions.special.delta_functions import DiracDelta
assert _test_args(DiracDelta(x, 1))
def test_sympy__functions__special__singularity_functions__SingularityFunction():
from sympy.functions.special.singularity_functions import SingularityFunction
assert _test_args(SingularityFunction(x, y, z))
def test_sympy__functions__special__delta_functions__Heaviside():
from sympy.functions.special.delta_functions import Heaviside
assert _test_args(Heaviside(x))
def test_sympy__functions__special__error_functions__erf():
from sympy.functions.special.error_functions import erf
assert _test_args(erf(2))
def test_sympy__functions__special__error_functions__erfc():
from sympy.functions.special.error_functions import erfc
assert _test_args(erfc(2))
def test_sympy__functions__special__error_functions__erfi():
from sympy.functions.special.error_functions import erfi
assert _test_args(erfi(2))
def test_sympy__functions__special__error_functions__erf2():
from sympy.functions.special.error_functions import erf2
assert _test_args(erf2(2, 3))
def test_sympy__functions__special__error_functions__erfinv():
from sympy.functions.special.error_functions import erfinv
assert _test_args(erfinv(2))
def test_sympy__functions__special__error_functions__erfcinv():
from sympy.functions.special.error_functions import erfcinv
assert _test_args(erfcinv(2))
def test_sympy__functions__special__error_functions__erf2inv():
from sympy.functions.special.error_functions import erf2inv
assert _test_args(erf2inv(2, 3))
@SKIP("abstract class")
def test_sympy__functions__special__error_functions__FresnelIntegral():
pass
def test_sympy__functions__special__error_functions__fresnels():
from sympy.functions.special.error_functions import fresnels
assert _test_args(fresnels(2))
def test_sympy__functions__special__error_functions__fresnelc():
from sympy.functions.special.error_functions import fresnelc
assert _test_args(fresnelc(2))
def test_sympy__functions__special__error_functions__erfs():
from sympy.functions.special.error_functions import _erfs
assert _test_args(_erfs(2))
def test_sympy__functions__special__error_functions__Ei():
from sympy.functions.special.error_functions import Ei
assert _test_args(Ei(2))
def test_sympy__functions__special__error_functions__li():
from sympy.functions.special.error_functions import li
assert _test_args(li(2))
def test_sympy__functions__special__error_functions__Li():
from sympy.functions.special.error_functions import Li
assert _test_args(Li(2))
@SKIP("abstract class")
def test_sympy__functions__special__error_functions__TrigonometricIntegral():
pass
def test_sympy__functions__special__error_functions__Si():
from sympy.functions.special.error_functions import Si
assert _test_args(Si(2))
def test_sympy__functions__special__error_functions__Ci():
from sympy.functions.special.error_functions import Ci
assert _test_args(Ci(2))
def test_sympy__functions__special__error_functions__Shi():
from sympy.functions.special.error_functions import Shi
assert _test_args(Shi(2))
def test_sympy__functions__special__error_functions__Chi():
from sympy.functions.special.error_functions import Chi
assert _test_args(Chi(2))
def test_sympy__functions__special__error_functions__expint():
from sympy.functions.special.error_functions import expint
assert _test_args(expint(y, x))
def test_sympy__functions__special__gamma_functions__gamma():
from sympy.functions.special.gamma_functions import gamma
assert _test_args(gamma(x))
def test_sympy__functions__special__gamma_functions__loggamma():
from sympy.functions.special.gamma_functions import loggamma
assert _test_args(loggamma(2))
def test_sympy__functions__special__gamma_functions__lowergamma():
from sympy.functions.special.gamma_functions import lowergamma
assert _test_args(lowergamma(x, 2))
def test_sympy__functions__special__gamma_functions__polygamma():
from sympy.functions.special.gamma_functions import polygamma
assert _test_args(polygamma(x, 2))
def test_sympy__functions__special__gamma_functions__uppergamma():
from sympy.functions.special.gamma_functions import uppergamma
assert _test_args(uppergamma(x, 2))
def test_sympy__functions__special__beta_functions__beta():
from sympy.functions.special.beta_functions import beta
assert _test_args(beta(x, x))
def test_sympy__functions__special__mathieu_functions__MathieuBase():
pass
def test_sympy__functions__special__mathieu_functions__mathieus():
from sympy.functions.special.mathieu_functions import mathieus
assert _test_args(mathieus(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieuc():
from sympy.functions.special.mathieu_functions import mathieuc
assert _test_args(mathieuc(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieusprime():
from sympy.functions.special.mathieu_functions import mathieusprime
assert _test_args(mathieusprime(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieucprime():
from sympy.functions.special.mathieu_functions import mathieucprime
assert _test_args(mathieucprime(1, 1, 1))
@SKIP("abstract class")
def test_sympy__functions__special__hyper__TupleParametersBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__hyper__TupleArg():
pass
def test_sympy__functions__special__hyper__hyper():
from sympy.functions.special.hyper import hyper
assert _test_args(hyper([1, 2, 3], [4, 5], x))
def test_sympy__functions__special__hyper__meijerg():
from sympy.functions.special.hyper import meijerg
assert _test_args(meijerg([1, 2, 3], [4, 5], [6], [], x))
@SKIP("abstract class")
def test_sympy__functions__special__hyper__HyperRep():
pass
def test_sympy__functions__special__hyper__HyperRep_power1():
from sympy.functions.special.hyper import HyperRep_power1
assert _test_args(HyperRep_power1(x, y))
def test_sympy__functions__special__hyper__HyperRep_power2():
from sympy.functions.special.hyper import HyperRep_power2
assert _test_args(HyperRep_power2(x, y))
def test_sympy__functions__special__hyper__HyperRep_log1():
from sympy.functions.special.hyper import HyperRep_log1
assert _test_args(HyperRep_log1(x))
def test_sympy__functions__special__hyper__HyperRep_atanh():
from sympy.functions.special.hyper import HyperRep_atanh
assert _test_args(HyperRep_atanh(x))
def test_sympy__functions__special__hyper__HyperRep_asin1():
from sympy.functions.special.hyper import HyperRep_asin1
assert _test_args(HyperRep_asin1(x))
def test_sympy__functions__special__hyper__HyperRep_asin2():
from sympy.functions.special.hyper import HyperRep_asin2
assert _test_args(HyperRep_asin2(x))
def test_sympy__functions__special__hyper__HyperRep_sqrts1():
from sympy.functions.special.hyper import HyperRep_sqrts1
assert _test_args(HyperRep_sqrts1(x, y))
def test_sympy__functions__special__hyper__HyperRep_sqrts2():
from sympy.functions.special.hyper import HyperRep_sqrts2
assert _test_args(HyperRep_sqrts2(x, y))
def test_sympy__functions__special__hyper__HyperRep_log2():
from sympy.functions.special.hyper import HyperRep_log2
assert _test_args(HyperRep_log2(x))
def test_sympy__functions__special__hyper__HyperRep_cosasin():
from sympy.functions.special.hyper import HyperRep_cosasin
assert _test_args(HyperRep_cosasin(x, y))
def test_sympy__functions__special__hyper__HyperRep_sinasin():
from sympy.functions.special.hyper import HyperRep_sinasin
assert _test_args(HyperRep_sinasin(x, y))
def test_sympy__functions__special__hyper__appellf1():
from sympy.functions.special.hyper import appellf1
a, b1, b2, c, x, y = symbols('a b1 b2 c x y')
assert _test_args(appellf1(a, b1, b2, c, x, y))
@SKIP("abstract class")
def test_sympy__functions__special__polynomials__OrthogonalPolynomial():
pass
def test_sympy__functions__special__polynomials__jacobi():
from sympy.functions.special.polynomials import jacobi
assert _test_args(jacobi(x, 2, 2, 2))
def test_sympy__functions__special__polynomials__gegenbauer():
from sympy.functions.special.polynomials import gegenbauer
assert _test_args(gegenbauer(x, 2, 2))
def test_sympy__functions__special__polynomials__chebyshevt():
from sympy.functions.special.polynomials import chebyshevt
assert _test_args(chebyshevt(x, 2))
def test_sympy__functions__special__polynomials__chebyshevt_root():
from sympy.functions.special.polynomials import chebyshevt_root
assert _test_args(chebyshevt_root(3, 2))
def test_sympy__functions__special__polynomials__chebyshevu():
from sympy.functions.special.polynomials import chebyshevu
assert _test_args(chebyshevu(x, 2))
def test_sympy__functions__special__polynomials__chebyshevu_root():
from sympy.functions.special.polynomials import chebyshevu_root
assert _test_args(chebyshevu_root(3, 2))
def test_sympy__functions__special__polynomials__hermite():
from sympy.functions.special.polynomials import hermite
assert _test_args(hermite(x, 2))
def test_sympy__functions__special__polynomials__legendre():
from sympy.functions.special.polynomials import legendre
assert _test_args(legendre(x, 2))
def test_sympy__functions__special__polynomials__assoc_legendre():
from sympy.functions.special.polynomials import assoc_legendre
assert _test_args(assoc_legendre(x, 0, y))
def test_sympy__functions__special__polynomials__laguerre():
from sympy.functions.special.polynomials import laguerre
assert _test_args(laguerre(x, 2))
def test_sympy__functions__special__polynomials__assoc_laguerre():
from sympy.functions.special.polynomials import assoc_laguerre
assert _test_args(assoc_laguerre(x, 0, y))
def test_sympy__functions__special__spherical_harmonics__Ynm():
from sympy.functions.special.spherical_harmonics import Ynm
assert _test_args(Ynm(1, 1, x, y))
def test_sympy__functions__special__spherical_harmonics__Znm():
from sympy.functions.special.spherical_harmonics import Znm
assert _test_args(Znm(1, 1, x, y))
def test_sympy__functions__special__tensor_functions__LeviCivita():
from sympy.functions.special.tensor_functions import LeviCivita
assert _test_args(LeviCivita(x, y, 2))
def test_sympy__functions__special__tensor_functions__KroneckerDelta():
from sympy.functions.special.tensor_functions import KroneckerDelta
assert _test_args(KroneckerDelta(x, y))
def test_sympy__functions__special__zeta_functions__dirichlet_eta():
from sympy.functions.special.zeta_functions import dirichlet_eta
assert _test_args(dirichlet_eta(x))
def test_sympy__functions__special__zeta_functions__zeta():
from sympy.functions.special.zeta_functions import zeta
assert _test_args(zeta(101))
def test_sympy__functions__special__zeta_functions__lerchphi():
from sympy.functions.special.zeta_functions import lerchphi
assert _test_args(lerchphi(x, y, z))
def test_sympy__functions__special__zeta_functions__polylog():
from sympy.functions.special.zeta_functions import polylog
assert _test_args(polylog(x, y))
def test_sympy__functions__special__zeta_functions__stieltjes():
from sympy.functions.special.zeta_functions import stieltjes
assert _test_args(stieltjes(x, y))
def test_sympy__integrals__integrals__Integral():
from sympy.integrals.integrals import Integral
assert _test_args(Integral(2, (x, 0, 1)))
def test_sympy__integrals__risch__NonElementaryIntegral():
from sympy.integrals.risch import NonElementaryIntegral
assert _test_args(NonElementaryIntegral(exp(-x**2), x))
@SKIP("abstract class")
def test_sympy__integrals__transforms__IntegralTransform():
pass
def test_sympy__integrals__transforms__MellinTransform():
from sympy.integrals.transforms import MellinTransform
assert _test_args(MellinTransform(2, x, y))
def test_sympy__integrals__transforms__InverseMellinTransform():
from sympy.integrals.transforms import InverseMellinTransform
assert _test_args(InverseMellinTransform(2, x, y, 0, 1))
def test_sympy__integrals__transforms__LaplaceTransform():
from sympy.integrals.transforms import LaplaceTransform
assert _test_args(LaplaceTransform(2, x, y))
def test_sympy__integrals__transforms__InverseLaplaceTransform():
from sympy.integrals.transforms import InverseLaplaceTransform
assert _test_args(InverseLaplaceTransform(2, x, y, 0))
@SKIP("abstract class")
def test_sympy__integrals__transforms__FourierTypeTransform():
pass
def test_sympy__integrals__transforms__InverseFourierTransform():
from sympy.integrals.transforms import InverseFourierTransform
assert _test_args(InverseFourierTransform(2, x, y))
def test_sympy__integrals__transforms__FourierTransform():
from sympy.integrals.transforms import FourierTransform
assert _test_args(FourierTransform(2, x, y))
@SKIP("abstract class")
def test_sympy__integrals__transforms__SineCosineTypeTransform():
pass
def test_sympy__integrals__transforms__InverseSineTransform():
from sympy.integrals.transforms import InverseSineTransform
assert _test_args(InverseSineTransform(2, x, y))
def test_sympy__integrals__transforms__SineTransform():
from sympy.integrals.transforms import SineTransform
assert _test_args(SineTransform(2, x, y))
def test_sympy__integrals__transforms__InverseCosineTransform():
from sympy.integrals.transforms import InverseCosineTransform
assert _test_args(InverseCosineTransform(2, x, y))
def test_sympy__integrals__transforms__CosineTransform():
from sympy.integrals.transforms import CosineTransform
assert _test_args(CosineTransform(2, x, y))
@SKIP("abstract class")
def test_sympy__integrals__transforms__HankelTypeTransform():
pass
def test_sympy__integrals__transforms__InverseHankelTransform():
from sympy.integrals.transforms import InverseHankelTransform
assert _test_args(InverseHankelTransform(2, x, y, 0))
def test_sympy__integrals__transforms__HankelTransform():
from sympy.integrals.transforms import HankelTransform
assert _test_args(HankelTransform(2, x, y, 0))
@XFAIL
def test_sympy__liealgebras__cartan_type__CartanType_generator():
from sympy.liealgebras.cartan_type import CartanType_generator
assert _test_args(CartanType_generator("A2"))
@XFAIL
def test_sympy__liealgebras__cartan_type__Standard_Cartan():
from sympy.liealgebras.cartan_type import Standard_Cartan
assert _test_args(Standard_Cartan("A", 2))
@XFAIL
def test_sympy__liealgebras__weyl_group__WeylGroup():
from sympy.liealgebras.weyl_group import WeylGroup
assert _test_args(WeylGroup("B4"))
@XFAIL
def test_sympy__liealgebras__root_system__RootSystem():
from sympy.liealgebras.root_system import RootSystem
assert _test_args(RootSystem("A2"))
@XFAIL
def test_sympy__liealgebras__type_a__TypeA():
from sympy.liealgebras.type_a import TypeA
assert _test_args(TypeA(2))
@XFAIL
def test_sympy__liealgebras__type_b__TypeB():
from sympy.liealgebras.type_b import TypeB
assert _test_args(TypeB(4))
@XFAIL
def test_sympy__liealgebras__type_c__TypeC():
from sympy.liealgebras.type_c import TypeC
assert _test_args(TypeC(4))
@XFAIL
def test_sympy__liealgebras__type_d__TypeD():
from sympy.liealgebras.type_d import TypeD
assert _test_args(TypeD(4))
@XFAIL
def test_sympy__liealgebras__type_e__TypeE():
from sympy.liealgebras.type_e import TypeE
assert _test_args(TypeE(6))
@XFAIL
def test_sympy__liealgebras__type_f__TypeF():
from sympy.liealgebras.type_f import TypeF
assert _test_args(TypeF(4))
@XFAIL
def test_sympy__liealgebras__type_g__TypeG():
from sympy.liealgebras.type_g import TypeG
assert _test_args(TypeG(2))
def test_sympy__logic__boolalg__And():
from sympy.logic.boolalg import And
assert _test_args(And(x, y, 1))
@SKIP("abstract class")
def test_sympy__logic__boolalg__Boolean():
pass
def test_sympy__logic__boolalg__BooleanFunction():
from sympy.logic.boolalg import BooleanFunction
assert _test_args(BooleanFunction(1, 2, 3))
@SKIP("abstract class")
def test_sympy__logic__boolalg__BooleanAtom():
pass
def test_sympy__logic__boolalg__BooleanTrue():
from sympy.logic.boolalg import true
assert _test_args(true)
def test_sympy__logic__boolalg__BooleanFalse():
from sympy.logic.boolalg import false
assert _test_args(false)
def test_sympy__logic__boolalg__Equivalent():
from sympy.logic.boolalg import Equivalent
assert _test_args(Equivalent(x, 2))
def test_sympy__logic__boolalg__ITE():
from sympy.logic.boolalg import ITE
assert _test_args(ITE(x, y, 1))
def test_sympy__logic__boolalg__Implies():
from sympy.logic.boolalg import Implies
assert _test_args(Implies(x, y))
def test_sympy__logic__boolalg__Nand():
from sympy.logic.boolalg import Nand
assert _test_args(Nand(x, y, 1))
def test_sympy__logic__boolalg__Nor():
from sympy.logic.boolalg import Nor
assert _test_args(Nor(x, y))
def test_sympy__logic__boolalg__Not():
from sympy.logic.boolalg import Not
assert _test_args(Not(x))
def test_sympy__logic__boolalg__Or():
from sympy.logic.boolalg import Or
assert _test_args(Or(x, y))
def test_sympy__logic__boolalg__Xor():
from sympy.logic.boolalg import Xor
assert _test_args(Xor(x, y, 2))
def test_sympy__logic__boolalg__Xnor():
from sympy.logic.boolalg import Xnor
assert _test_args(Xnor(x, y, 2))
def test_sympy__matrices__matrices__DeferredVector():
from sympy.matrices.matrices import DeferredVector
assert _test_args(DeferredVector("X"))
@SKIP("abstract class")
def test_sympy__matrices__expressions__matexpr__MatrixBase():
pass
def test_sympy__matrices__immutable__ImmutableDenseMatrix():
from sympy.matrices.immutable import ImmutableDenseMatrix
m = ImmutableDenseMatrix([[1, 2], [3, 4]])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableDenseMatrix(1, 1, [1])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableDenseMatrix(2, 2, lambda i, j: 1)
assert m[0, 0] is S.One
m = ImmutableDenseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j))
assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified
assert _test_args(m)
assert _test_args(Basic(*list(m)))
def test_sympy__matrices__immutable__ImmutableSparseMatrix():
from sympy.matrices.immutable import ImmutableSparseMatrix
m = ImmutableSparseMatrix([[1, 2], [3, 4]])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(1, 1, {(0, 0): 1})
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(1, 1, [1])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(2, 2, lambda i, j: 1)
assert m[0, 0] is S.One
m = ImmutableSparseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j))
assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified
assert _test_args(m)
assert _test_args(Basic(*list(m)))
def test_sympy__matrices__expressions__slice__MatrixSlice():
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', 4, 4)
assert _test_args(MatrixSlice(X, (0, 2), (0, 2)))
def test_sympy__matrices__expressions__applyfunc__ElementwiseApplyFunction():
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol("X", x, x)
func = Lambda(x, x**2)
assert _test_args(ElementwiseApplyFunction(func, X))
def test_sympy__matrices__expressions__blockmatrix__BlockDiagMatrix():
from sympy.matrices.expressions.blockmatrix import BlockDiagMatrix
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
assert _test_args(BlockDiagMatrix(X, Y))
def test_sympy__matrices__expressions__blockmatrix__BlockMatrix():
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
Z = MatrixSymbol('Z', x, y)
O = ZeroMatrix(y, x)
assert _test_args(BlockMatrix([[X, Z], [O, Y]]))
def test_sympy__matrices__expressions__inverse__Inverse():
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Inverse(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__matadd__MatAdd():
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(MatAdd(X, Y))
def test_sympy__matrices__expressions__matexpr__Identity():
from sympy.matrices.expressions.matexpr import Identity
assert _test_args(Identity(3))
def test_sympy__matrices__expressions__matexpr__GenericIdentity():
from sympy.matrices.expressions.matexpr import GenericIdentity
assert _test_args(GenericIdentity())
@SKIP("abstract class")
def test_sympy__matrices__expressions__matexpr__MatrixExpr():
pass
def test_sympy__matrices__expressions__matexpr__MatrixElement():
from sympy.matrices.expressions.matexpr import MatrixSymbol, MatrixElement
from sympy import S
assert _test_args(MatrixElement(MatrixSymbol('A', 3, 5), S(2), S(3)))
def test_sympy__matrices__expressions__matexpr__MatrixSymbol():
from sympy.matrices.expressions.matexpr import MatrixSymbol
assert _test_args(MatrixSymbol('A', 3, 5))
def test_sympy__matrices__expressions__matexpr__ZeroMatrix():
from sympy.matrices.expressions.matexpr import ZeroMatrix
assert _test_args(ZeroMatrix(3, 5))
def test_sympy__matrices__expressions__matexpr__GenericZeroMatrix():
from sympy.matrices.expressions.matexpr import GenericZeroMatrix
assert _test_args(GenericZeroMatrix())
def test_sympy__matrices__expressions__matmul__MatMul():
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', y, x)
assert _test_args(MatMul(X, Y))
def test_sympy__matrices__expressions__dotproduct__DotProduct():
from sympy.matrices.expressions.dotproduct import DotProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, 1)
Y = MatrixSymbol('Y', x, 1)
assert _test_args(DotProduct(X, Y))
def test_sympy__matrices__expressions__diagonal__DiagonalMatrix():
from sympy.matrices.expressions.diagonal import DiagonalMatrix
from sympy.matrices.expressions import MatrixSymbol
x = MatrixSymbol('x', 10, 1)
assert _test_args(DiagonalMatrix(x))
def test_sympy__matrices__expressions__diagonal__DiagonalOf():
from sympy.matrices.expressions.diagonal import DiagonalOf
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('x', 10, 10)
assert _test_args(DiagonalOf(X))
def test_sympy__matrices__expressions__diagonal__DiagonalizeVector():
from sympy.matrices.expressions.diagonal import DiagonalizeVector
from sympy.matrices.expressions import MatrixSymbol
x = MatrixSymbol('x', 10, 1)
assert _test_args(DiagonalizeVector(x))
def test_sympy__matrices__expressions__hadamard__HadamardProduct():
from sympy.matrices.expressions.hadamard import HadamardProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(HadamardProduct(X, Y))
def test_sympy__matrices__expressions__hadamard__HadamardPower():
from sympy.matrices.expressions.hadamard import HadamardPower
from sympy.matrices.expressions import MatrixSymbol
from sympy import Symbol
X = MatrixSymbol('X', x, y)
n = Symbol("n")
assert _test_args(HadamardPower(X, n))
def test_sympy__matrices__expressions__kronecker__KroneckerProduct():
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(KroneckerProduct(X, Y))
def test_sympy__matrices__expressions__matpow__MatPow():
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, x)
assert _test_args(MatPow(X, 2))
def test_sympy__matrices__expressions__transpose__Transpose():
from sympy.matrices.expressions.transpose import Transpose
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Transpose(MatrixSymbol('A', 3, 5)))
def test_sympy__matrices__expressions__adjoint__Adjoint():
from sympy.matrices.expressions.adjoint import Adjoint
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Adjoint(MatrixSymbol('A', 3, 5)))
def test_sympy__matrices__expressions__trace__Trace():
from sympy.matrices.expressions.trace import Trace
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Trace(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__determinant__Determinant():
from sympy.matrices.expressions.determinant import Determinant
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Determinant(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__funcmatrix__FunctionMatrix():
from sympy.matrices.expressions.funcmatrix import FunctionMatrix
from sympy import symbols
i, j = symbols('i,j')
assert _test_args(FunctionMatrix(3, 3, Lambda((i, j), i - j) ))
def test_sympy__matrices__expressions__fourier__DFT():
from sympy.matrices.expressions.fourier import DFT
from sympy import S
assert _test_args(DFT(S(2)))
def test_sympy__matrices__expressions__fourier__IDFT():
from sympy.matrices.expressions.fourier import IDFT
from sympy import S
assert _test_args(IDFT(S(2)))
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', 10, 10)
def test_sympy__matrices__expressions__factorizations__LofLU():
from sympy.matrices.expressions.factorizations import LofLU
assert _test_args(LofLU(X))
def test_sympy__matrices__expressions__factorizations__UofLU():
from sympy.matrices.expressions.factorizations import UofLU
assert _test_args(UofLU(X))
def test_sympy__matrices__expressions__factorizations__QofQR():
from sympy.matrices.expressions.factorizations import QofQR
assert _test_args(QofQR(X))
def test_sympy__matrices__expressions__factorizations__RofQR():
from sympy.matrices.expressions.factorizations import RofQR
assert _test_args(RofQR(X))
def test_sympy__matrices__expressions__factorizations__LofCholesky():
from sympy.matrices.expressions.factorizations import LofCholesky
assert _test_args(LofCholesky(X))
def test_sympy__matrices__expressions__factorizations__UofCholesky():
from sympy.matrices.expressions.factorizations import UofCholesky
assert _test_args(UofCholesky(X))
def test_sympy__matrices__expressions__factorizations__EigenVectors():
from sympy.matrices.expressions.factorizations import EigenVectors
assert _test_args(EigenVectors(X))
def test_sympy__matrices__expressions__factorizations__EigenValues():
from sympy.matrices.expressions.factorizations import EigenValues
assert _test_args(EigenValues(X))
def test_sympy__matrices__expressions__factorizations__UofSVD():
from sympy.matrices.expressions.factorizations import UofSVD
assert _test_args(UofSVD(X))
def test_sympy__matrices__expressions__factorizations__VofSVD():
from sympy.matrices.expressions.factorizations import VofSVD
assert _test_args(VofSVD(X))
def test_sympy__matrices__expressions__factorizations__SofSVD():
from sympy.matrices.expressions.factorizations import SofSVD
assert _test_args(SofSVD(X))
@SKIP("abstract class")
def test_sympy__matrices__expressions__factorizations__Factorization():
pass
def test_sympy__physics__vector__frame__CoordinateSym():
from sympy.physics.vector import CoordinateSym
from sympy.physics.vector import ReferenceFrame
assert _test_args(CoordinateSym('R_x', ReferenceFrame('R'), 0))
def test_sympy__physics__paulialgebra__Pauli():
from sympy.physics.paulialgebra import Pauli
assert _test_args(Pauli(1))
def test_sympy__physics__quantum__anticommutator__AntiCommutator():
from sympy.physics.quantum.anticommutator import AntiCommutator
assert _test_args(AntiCommutator(x, y))
def test_sympy__physics__quantum__cartesian__PositionBra3D():
from sympy.physics.quantum.cartesian import PositionBra3D
assert _test_args(PositionBra3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PositionKet3D():
from sympy.physics.quantum.cartesian import PositionKet3D
assert _test_args(PositionKet3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PositionState3D():
from sympy.physics.quantum.cartesian import PositionState3D
assert _test_args(PositionState3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PxBra():
from sympy.physics.quantum.cartesian import PxBra
assert _test_args(PxBra(x, y, z))
def test_sympy__physics__quantum__cartesian__PxKet():
from sympy.physics.quantum.cartesian import PxKet
assert _test_args(PxKet(x, y, z))
def test_sympy__physics__quantum__cartesian__PxOp():
from sympy.physics.quantum.cartesian import PxOp
assert _test_args(PxOp(x, y, z))
def test_sympy__physics__quantum__cartesian__XBra():
from sympy.physics.quantum.cartesian import XBra
assert _test_args(XBra(x))
def test_sympy__physics__quantum__cartesian__XKet():
from sympy.physics.quantum.cartesian import XKet
assert _test_args(XKet(x))
def test_sympy__physics__quantum__cartesian__XOp():
from sympy.physics.quantum.cartesian import XOp
assert _test_args(XOp(x))
def test_sympy__physics__quantum__cartesian__YOp():
from sympy.physics.quantum.cartesian import YOp
assert _test_args(YOp(x))
def test_sympy__physics__quantum__cartesian__ZOp():
from sympy.physics.quantum.cartesian import ZOp
assert _test_args(ZOp(x))
def test_sympy__physics__quantum__cg__CG():
from sympy.physics.quantum.cg import CG
from sympy import S
assert _test_args(CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1))
def test_sympy__physics__quantum__cg__Wigner3j():
from sympy.physics.quantum.cg import Wigner3j
assert _test_args(Wigner3j(6, 0, 4, 0, 2, 0))
def test_sympy__physics__quantum__cg__Wigner6j():
from sympy.physics.quantum.cg import Wigner6j
assert _test_args(Wigner6j(1, 2, 3, 2, 1, 2))
def test_sympy__physics__quantum__cg__Wigner9j():
from sympy.physics.quantum.cg import Wigner9j
assert _test_args(Wigner9j(2, 1, 1, S(3)/2, S(1)/2, 1, S(1)/2, S(1)/2, 0))
def test_sympy__physics__quantum__circuitplot__Mz():
from sympy.physics.quantum.circuitplot import Mz
assert _test_args(Mz(0))
def test_sympy__physics__quantum__circuitplot__Mx():
from sympy.physics.quantum.circuitplot import Mx
assert _test_args(Mx(0))
def test_sympy__physics__quantum__commutator__Commutator():
from sympy.physics.quantum.commutator import Commutator
A, B = symbols('A,B', commutative=False)
assert _test_args(Commutator(A, B))
def test_sympy__physics__quantum__constants__HBar():
from sympy.physics.quantum.constants import HBar
assert _test_args(HBar())
def test_sympy__physics__quantum__dagger__Dagger():
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.state import Ket
assert _test_args(Dagger(Dagger(Ket('psi'))))
def test_sympy__physics__quantum__gate__CGate():
from sympy.physics.quantum.gate import CGate, Gate
assert _test_args(CGate((0, 1), Gate(2)))
def test_sympy__physics__quantum__gate__CGateS():
from sympy.physics.quantum.gate import CGateS, Gate
assert _test_args(CGateS((0, 1), Gate(2)))
def test_sympy__physics__quantum__gate__CNotGate():
from sympy.physics.quantum.gate import CNotGate
assert _test_args(CNotGate(0, 1))
def test_sympy__physics__quantum__gate__Gate():
from sympy.physics.quantum.gate import Gate
assert _test_args(Gate(0))
def test_sympy__physics__quantum__gate__HadamardGate():
from sympy.physics.quantum.gate import HadamardGate
assert _test_args(HadamardGate(0))
def test_sympy__physics__quantum__gate__IdentityGate():
from sympy.physics.quantum.gate import IdentityGate
assert _test_args(IdentityGate(0))
def test_sympy__physics__quantum__gate__OneQubitGate():
from sympy.physics.quantum.gate import OneQubitGate
assert _test_args(OneQubitGate(0))
def test_sympy__physics__quantum__gate__PhaseGate():
from sympy.physics.quantum.gate import PhaseGate
assert _test_args(PhaseGate(0))
def test_sympy__physics__quantum__gate__SwapGate():
from sympy.physics.quantum.gate import SwapGate
assert _test_args(SwapGate(0, 1))
def test_sympy__physics__quantum__gate__TGate():
from sympy.physics.quantum.gate import TGate
assert _test_args(TGate(0))
def test_sympy__physics__quantum__gate__TwoQubitGate():
from sympy.physics.quantum.gate import TwoQubitGate
assert _test_args(TwoQubitGate(0))
def test_sympy__physics__quantum__gate__UGate():
from sympy.physics.quantum.gate import UGate
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy import Integer, Tuple
assert _test_args(
UGate(Tuple(Integer(1)), ImmutableDenseMatrix([[1, 0], [0, 2]])))
def test_sympy__physics__quantum__gate__XGate():
from sympy.physics.quantum.gate import XGate
assert _test_args(XGate(0))
def test_sympy__physics__quantum__gate__YGate():
from sympy.physics.quantum.gate import YGate
assert _test_args(YGate(0))
def test_sympy__physics__quantum__gate__ZGate():
from sympy.physics.quantum.gate import ZGate
assert _test_args(ZGate(0))
@SKIP("TODO: sympy.physics")
def test_sympy__physics__quantum__grover__OracleGate():
from sympy.physics.quantum.grover import OracleGate
assert _test_args(OracleGate())
def test_sympy__physics__quantum__grover__WGate():
from sympy.physics.quantum.grover import WGate
assert _test_args(WGate(1))
def test_sympy__physics__quantum__hilbert__ComplexSpace():
from sympy.physics.quantum.hilbert import ComplexSpace
assert _test_args(ComplexSpace(x))
def test_sympy__physics__quantum__hilbert__DirectSumHilbertSpace():
from sympy.physics.quantum.hilbert import DirectSumHilbertSpace, ComplexSpace, FockSpace
c = ComplexSpace(2)
f = FockSpace()
assert _test_args(DirectSumHilbertSpace(c, f))
def test_sympy__physics__quantum__hilbert__FockSpace():
from sympy.physics.quantum.hilbert import FockSpace
assert _test_args(FockSpace())
def test_sympy__physics__quantum__hilbert__HilbertSpace():
from sympy.physics.quantum.hilbert import HilbertSpace
assert _test_args(HilbertSpace())
def test_sympy__physics__quantum__hilbert__L2():
from sympy.physics.quantum.hilbert import L2
from sympy import oo, Interval
assert _test_args(L2(Interval(0, oo)))
def test_sympy__physics__quantum__hilbert__TensorPowerHilbertSpace():
from sympy.physics.quantum.hilbert import TensorPowerHilbertSpace, FockSpace
f = FockSpace()
assert _test_args(TensorPowerHilbertSpace(f, 2))
def test_sympy__physics__quantum__hilbert__TensorProductHilbertSpace():
from sympy.physics.quantum.hilbert import TensorProductHilbertSpace, FockSpace, ComplexSpace
c = ComplexSpace(2)
f = FockSpace()
assert _test_args(TensorProductHilbertSpace(f, c))
def test_sympy__physics__quantum__innerproduct__InnerProduct():
from sympy.physics.quantum import Bra, Ket, InnerProduct
b = Bra('b')
k = Ket('k')
assert _test_args(InnerProduct(b, k))
def test_sympy__physics__quantum__operator__DifferentialOperator():
from sympy.physics.quantum.operator import DifferentialOperator
from sympy import Derivative, Function
f = Function('f')
assert _test_args(DifferentialOperator(1/x*Derivative(f(x), x), f(x)))
def test_sympy__physics__quantum__operator__HermitianOperator():
from sympy.physics.quantum.operator import HermitianOperator
assert _test_args(HermitianOperator('H'))
def test_sympy__physics__quantum__operator__IdentityOperator():
from sympy.physics.quantum.operator import IdentityOperator
assert _test_args(IdentityOperator(5))
def test_sympy__physics__quantum__operator__Operator():
from sympy.physics.quantum.operator import Operator
assert _test_args(Operator('A'))
def test_sympy__physics__quantum__operator__OuterProduct():
from sympy.physics.quantum.operator import OuterProduct
from sympy.physics.quantum import Ket, Bra
b = Bra('b')
k = Ket('k')
assert _test_args(OuterProduct(k, b))
def test_sympy__physics__quantum__operator__UnitaryOperator():
from sympy.physics.quantum.operator import UnitaryOperator
assert _test_args(UnitaryOperator('U'))
def test_sympy__physics__quantum__piab__PIABBra():
from sympy.physics.quantum.piab import PIABBra
assert _test_args(PIABBra('B'))
def test_sympy__physics__quantum__boson__BosonOp():
from sympy.physics.quantum.boson import BosonOp
assert _test_args(BosonOp('a'))
assert _test_args(BosonOp('a', False))
def test_sympy__physics__quantum__boson__BosonFockKet():
from sympy.physics.quantum.boson import BosonFockKet
assert _test_args(BosonFockKet(1))
def test_sympy__physics__quantum__boson__BosonFockBra():
from sympy.physics.quantum.boson import BosonFockBra
assert _test_args(BosonFockBra(1))
def test_sympy__physics__quantum__boson__BosonCoherentKet():
from sympy.physics.quantum.boson import BosonCoherentKet
assert _test_args(BosonCoherentKet(1))
def test_sympy__physics__quantum__boson__BosonCoherentBra():
from sympy.physics.quantum.boson import BosonCoherentBra
assert _test_args(BosonCoherentBra(1))
def test_sympy__physics__quantum__fermion__FermionOp():
from sympy.physics.quantum.fermion import FermionOp
assert _test_args(FermionOp('c'))
assert _test_args(FermionOp('c', False))
def test_sympy__physics__quantum__fermion__FermionFockKet():
from sympy.physics.quantum.fermion import FermionFockKet
assert _test_args(FermionFockKet(1))
def test_sympy__physics__quantum__fermion__FermionFockBra():
from sympy.physics.quantum.fermion import FermionFockBra
assert _test_args(FermionFockBra(1))
def test_sympy__physics__quantum__pauli__SigmaOpBase():
from sympy.physics.quantum.pauli import SigmaOpBase
assert _test_args(SigmaOpBase())
def test_sympy__physics__quantum__pauli__SigmaX():
from sympy.physics.quantum.pauli import SigmaX
assert _test_args(SigmaX())
def test_sympy__physics__quantum__pauli__SigmaY():
from sympy.physics.quantum.pauli import SigmaY
assert _test_args(SigmaY())
def test_sympy__physics__quantum__pauli__SigmaZ():
from sympy.physics.quantum.pauli import SigmaZ
assert _test_args(SigmaZ())
def test_sympy__physics__quantum__pauli__SigmaMinus():
from sympy.physics.quantum.pauli import SigmaMinus
assert _test_args(SigmaMinus())
def test_sympy__physics__quantum__pauli__SigmaPlus():
from sympy.physics.quantum.pauli import SigmaPlus
assert _test_args(SigmaPlus())
def test_sympy__physics__quantum__pauli__SigmaZKet():
from sympy.physics.quantum.pauli import SigmaZKet
assert _test_args(SigmaZKet(0))
def test_sympy__physics__quantum__pauli__SigmaZBra():
from sympy.physics.quantum.pauli import SigmaZBra
assert _test_args(SigmaZBra(0))
def test_sympy__physics__quantum__piab__PIABHamiltonian():
from sympy.physics.quantum.piab import PIABHamiltonian
assert _test_args(PIABHamiltonian('P'))
def test_sympy__physics__quantum__piab__PIABKet():
from sympy.physics.quantum.piab import PIABKet
assert _test_args(PIABKet('K'))
def test_sympy__physics__quantum__qexpr__QExpr():
from sympy.physics.quantum.qexpr import QExpr
assert _test_args(QExpr(0))
def test_sympy__physics__quantum__qft__Fourier():
from sympy.physics.quantum.qft import Fourier
assert _test_args(Fourier(0, 1))
def test_sympy__physics__quantum__qft__IQFT():
from sympy.physics.quantum.qft import IQFT
assert _test_args(IQFT(0, 1))
def test_sympy__physics__quantum__qft__QFT():
from sympy.physics.quantum.qft import QFT
assert _test_args(QFT(0, 1))
def test_sympy__physics__quantum__qft__RkGate():
from sympy.physics.quantum.qft import RkGate
assert _test_args(RkGate(0, 1))
def test_sympy__physics__quantum__qubit__IntQubit():
from sympy.physics.quantum.qubit import IntQubit
assert _test_args(IntQubit(0))
def test_sympy__physics__quantum__qubit__IntQubitBra():
from sympy.physics.quantum.qubit import IntQubitBra
assert _test_args(IntQubitBra(0))
def test_sympy__physics__quantum__qubit__IntQubitState():
from sympy.physics.quantum.qubit import IntQubitState, QubitState
assert _test_args(IntQubitState(QubitState(0, 1)))
def test_sympy__physics__quantum__qubit__Qubit():
from sympy.physics.quantum.qubit import Qubit
assert _test_args(Qubit(0, 0, 0))
def test_sympy__physics__quantum__qubit__QubitBra():
from sympy.physics.quantum.qubit import QubitBra
assert _test_args(QubitBra('1', 0))
def test_sympy__physics__quantum__qubit__QubitState():
from sympy.physics.quantum.qubit import QubitState
assert _test_args(QubitState(0, 1))
def test_sympy__physics__quantum__density__Density():
from sympy.physics.quantum.density import Density
from sympy.physics.quantum.state import Ket
assert _test_args(Density([Ket(0), 0.5], [Ket(1), 0.5]))
@SKIP("TODO: sympy.physics.quantum.shor: Cmod Not Implemented")
def test_sympy__physics__quantum__shor__CMod():
from sympy.physics.quantum.shor import CMod
assert _test_args(CMod())
def test_sympy__physics__quantum__spin__CoupledSpinState():
from sympy.physics.quantum.spin import CoupledSpinState
assert _test_args(CoupledSpinState(1, 0, (1, 1)))
assert _test_args(CoupledSpinState(1, 0, (1, S(1)/2, S(1)/2)))
assert _test_args(CoupledSpinState(
1, 0, (1, S(1)/2, S(1)/2), ((2, 3, S(1)/2), (1, 2, 1)) ))
j, m, j1, j2, j3, j12, x = symbols('j m j1:4 j12 x')
assert CoupledSpinState(
j, m, (j1, j2, j3)).subs(j2, x) == CoupledSpinState(j, m, (j1, x, j3))
assert CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, j12), (1, 2, j)) ).subs(j12, x) == \
CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, x), (1, 2, j)) )
def test_sympy__physics__quantum__spin__J2Op():
from sympy.physics.quantum.spin import J2Op
assert _test_args(J2Op('J'))
def test_sympy__physics__quantum__spin__JminusOp():
from sympy.physics.quantum.spin import JminusOp
assert _test_args(JminusOp('J'))
def test_sympy__physics__quantum__spin__JplusOp():
from sympy.physics.quantum.spin import JplusOp
assert _test_args(JplusOp('J'))
def test_sympy__physics__quantum__spin__JxBra():
from sympy.physics.quantum.spin import JxBra
assert _test_args(JxBra(1, 0))
def test_sympy__physics__quantum__spin__JxBraCoupled():
from sympy.physics.quantum.spin import JxBraCoupled
assert _test_args(JxBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JxKet():
from sympy.physics.quantum.spin import JxKet
assert _test_args(JxKet(1, 0))
def test_sympy__physics__quantum__spin__JxKetCoupled():
from sympy.physics.quantum.spin import JxKetCoupled
assert _test_args(JxKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JxOp():
from sympy.physics.quantum.spin import JxOp
assert _test_args(JxOp('J'))
def test_sympy__physics__quantum__spin__JyBra():
from sympy.physics.quantum.spin import JyBra
assert _test_args(JyBra(1, 0))
def test_sympy__physics__quantum__spin__JyBraCoupled():
from sympy.physics.quantum.spin import JyBraCoupled
assert _test_args(JyBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JyKet():
from sympy.physics.quantum.spin import JyKet
assert _test_args(JyKet(1, 0))
def test_sympy__physics__quantum__spin__JyKetCoupled():
from sympy.physics.quantum.spin import JyKetCoupled
assert _test_args(JyKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JyOp():
from sympy.physics.quantum.spin import JyOp
assert _test_args(JyOp('J'))
def test_sympy__physics__quantum__spin__JzBra():
from sympy.physics.quantum.spin import JzBra
assert _test_args(JzBra(1, 0))
def test_sympy__physics__quantum__spin__JzBraCoupled():
from sympy.physics.quantum.spin import JzBraCoupled
assert _test_args(JzBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JzKet():
from sympy.physics.quantum.spin import JzKet
assert _test_args(JzKet(1, 0))
def test_sympy__physics__quantum__spin__JzKetCoupled():
from sympy.physics.quantum.spin import JzKetCoupled
assert _test_args(JzKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JzOp():
from sympy.physics.quantum.spin import JzOp
assert _test_args(JzOp('J'))
def test_sympy__physics__quantum__spin__Rotation():
from sympy.physics.quantum.spin import Rotation
assert _test_args(Rotation(pi, 0, pi/2))
def test_sympy__physics__quantum__spin__SpinState():
from sympy.physics.quantum.spin import SpinState
assert _test_args(SpinState(1, 0))
def test_sympy__physics__quantum__spin__WignerD():
from sympy.physics.quantum.spin import WignerD
assert _test_args(WignerD(0, 1, 2, 3, 4, 5))
def test_sympy__physics__quantum__state__Bra():
from sympy.physics.quantum.state import Bra
assert _test_args(Bra(0))
def test_sympy__physics__quantum__state__BraBase():
from sympy.physics.quantum.state import BraBase
assert _test_args(BraBase(0))
def test_sympy__physics__quantum__state__Ket():
from sympy.physics.quantum.state import Ket
assert _test_args(Ket(0))
def test_sympy__physics__quantum__state__KetBase():
from sympy.physics.quantum.state import KetBase
assert _test_args(KetBase(0))
def test_sympy__physics__quantum__state__State():
from sympy.physics.quantum.state import State
assert _test_args(State(0))
def test_sympy__physics__quantum__state__StateBase():
from sympy.physics.quantum.state import StateBase
assert _test_args(StateBase(0))
def test_sympy__physics__quantum__state__TimeDepBra():
from sympy.physics.quantum.state import TimeDepBra
assert _test_args(TimeDepBra('psi', 't'))
def test_sympy__physics__quantum__state__TimeDepKet():
from sympy.physics.quantum.state import TimeDepKet
assert _test_args(TimeDepKet('psi', 't'))
def test_sympy__physics__quantum__state__TimeDepState():
from sympy.physics.quantum.state import TimeDepState
assert _test_args(TimeDepState('psi', 't'))
def test_sympy__physics__quantum__state__Wavefunction():
from sympy.physics.quantum.state import Wavefunction
from sympy.functions import sin
from sympy import Piecewise
n = 1
L = 1
g = Piecewise((0, x < 0), (0, x > L), (sqrt(2//L)*sin(n*pi*x/L), True))
assert _test_args(Wavefunction(g, x))
def test_sympy__physics__quantum__tensorproduct__TensorProduct():
from sympy.physics.quantum.tensorproduct import TensorProduct
assert _test_args(TensorProduct(x, y))
def test_sympy__physics__quantum__identitysearch__GateIdentity():
from sympy.physics.quantum.gate import X
from sympy.physics.quantum.identitysearch import GateIdentity
assert _test_args(GateIdentity(X(0), X(0)))
def test_sympy__physics__quantum__sho1d__SHOOp():
from sympy.physics.quantum.sho1d import SHOOp
assert _test_args(SHOOp('a'))
def test_sympy__physics__quantum__sho1d__RaisingOp():
from sympy.physics.quantum.sho1d import RaisingOp
assert _test_args(RaisingOp('a'))
def test_sympy__physics__quantum__sho1d__LoweringOp():
from sympy.physics.quantum.sho1d import LoweringOp
assert _test_args(LoweringOp('a'))
def test_sympy__physics__quantum__sho1d__NumberOp():
from sympy.physics.quantum.sho1d import NumberOp
assert _test_args(NumberOp('N'))
def test_sympy__physics__quantum__sho1d__Hamiltonian():
from sympy.physics.quantum.sho1d import Hamiltonian
assert _test_args(Hamiltonian('H'))
def test_sympy__physics__quantum__sho1d__SHOState():
from sympy.physics.quantum.sho1d import SHOState
assert _test_args(SHOState(0))
def test_sympy__physics__quantum__sho1d__SHOKet():
from sympy.physics.quantum.sho1d import SHOKet
assert _test_args(SHOKet(0))
def test_sympy__physics__quantum__sho1d__SHOBra():
from sympy.physics.quantum.sho1d import SHOBra
assert _test_args(SHOBra(0))
def test_sympy__physics__secondquant__AnnihilateBoson():
from sympy.physics.secondquant import AnnihilateBoson
assert _test_args(AnnihilateBoson(0))
def test_sympy__physics__secondquant__AnnihilateFermion():
from sympy.physics.secondquant import AnnihilateFermion
assert _test_args(AnnihilateFermion(0))
@SKIP("abstract class")
def test_sympy__physics__secondquant__Annihilator():
pass
def test_sympy__physics__secondquant__AntiSymmetricTensor():
from sympy.physics.secondquant import AntiSymmetricTensor
i, j = symbols('i j', below_fermi=True)
a, b = symbols('a b', above_fermi=True)
assert _test_args(AntiSymmetricTensor('v', (a, i), (b, j)))
def test_sympy__physics__secondquant__BosonState():
from sympy.physics.secondquant import BosonState
assert _test_args(BosonState((0, 1)))
@SKIP("abstract class")
def test_sympy__physics__secondquant__BosonicOperator():
pass
def test_sympy__physics__secondquant__Commutator():
from sympy.physics.secondquant import Commutator
assert _test_args(Commutator(x, y))
def test_sympy__physics__secondquant__CreateBoson():
from sympy.physics.secondquant import CreateBoson
assert _test_args(CreateBoson(0))
def test_sympy__physics__secondquant__CreateFermion():
from sympy.physics.secondquant import CreateFermion
assert _test_args(CreateFermion(0))
@SKIP("abstract class")
def test_sympy__physics__secondquant__Creator():
pass
def test_sympy__physics__secondquant__Dagger():
from sympy.physics.secondquant import Dagger
from sympy import I
assert _test_args(Dagger(2*I))
def test_sympy__physics__secondquant__FermionState():
from sympy.physics.secondquant import FermionState
assert _test_args(FermionState((0, 1)))
def test_sympy__physics__secondquant__FermionicOperator():
from sympy.physics.secondquant import FermionicOperator
assert _test_args(FermionicOperator(0))
def test_sympy__physics__secondquant__FockState():
from sympy.physics.secondquant import FockState
assert _test_args(FockState((0, 1)))
def test_sympy__physics__secondquant__FockStateBosonBra():
from sympy.physics.secondquant import FockStateBosonBra
assert _test_args(FockStateBosonBra((0, 1)))
def test_sympy__physics__secondquant__FockStateBosonKet():
from sympy.physics.secondquant import FockStateBosonKet
assert _test_args(FockStateBosonKet((0, 1)))
def test_sympy__physics__secondquant__FockStateBra():
from sympy.physics.secondquant import FockStateBra
assert _test_args(FockStateBra((0, 1)))
def test_sympy__physics__secondquant__FockStateFermionBra():
from sympy.physics.secondquant import FockStateFermionBra
assert _test_args(FockStateFermionBra((0, 1)))
def test_sympy__physics__secondquant__FockStateFermionKet():
from sympy.physics.secondquant import FockStateFermionKet
assert _test_args(FockStateFermionKet((0, 1)))
def test_sympy__physics__secondquant__FockStateKet():
from sympy.physics.secondquant import FockStateKet
assert _test_args(FockStateKet((0, 1)))
def test_sympy__physics__secondquant__InnerProduct():
from sympy.physics.secondquant import InnerProduct
from sympy.physics.secondquant import FockStateKet, FockStateBra
assert _test_args(InnerProduct(FockStateBra((0, 1)), FockStateKet((0, 1))))
def test_sympy__physics__secondquant__NO():
from sympy.physics.secondquant import NO, F, Fd
assert _test_args(NO(Fd(x)*F(y)))
def test_sympy__physics__secondquant__PermutationOperator():
from sympy.physics.secondquant import PermutationOperator
assert _test_args(PermutationOperator(0, 1))
def test_sympy__physics__secondquant__SqOperator():
from sympy.physics.secondquant import SqOperator
assert _test_args(SqOperator(0))
def test_sympy__physics__secondquant__TensorSymbol():
from sympy.physics.secondquant import TensorSymbol
assert _test_args(TensorSymbol(x))
def test_sympy__physics__units__dimensions__Dimension():
from sympy.physics.units.dimensions import Dimension
assert _test_args(Dimension("length", "L"))
def test_sympy__physics__units__dimensions__DimensionSystem():
from sympy.physics.units.dimensions import DimensionSystem
from sympy.physics.units.dimensions import length, time, velocity
assert _test_args(DimensionSystem((length, time), (velocity,)))
def test_sympy__physics__units__quantities__Quantity():
from sympy.physics.units.quantities import Quantity
from sympy.physics.units import length
assert _test_args(Quantity("dam"))
def test_sympy__physics__units__prefixes__Prefix():
from sympy.physics.units.prefixes import Prefix
assert _test_args(Prefix('kilo', 'k', 3))
def test_sympy__core__numbers__AlgebraicNumber():
from sympy.core.numbers import AlgebraicNumber
assert _test_args(AlgebraicNumber(sqrt(2), [1, 2, 3]))
def test_sympy__polys__polytools__GroebnerBasis():
from sympy.polys.polytools import GroebnerBasis
assert _test_args(GroebnerBasis([x, y, z], x, y, z))
def test_sympy__polys__polytools__Poly():
from sympy.polys.polytools import Poly
assert _test_args(Poly(2, x, y))
def test_sympy__polys__polytools__PurePoly():
from sympy.polys.polytools import PurePoly
assert _test_args(PurePoly(2, x, y))
@SKIP('abstract class')
def test_sympy__polys__rootoftools__RootOf():
pass
def test_sympy__polys__rootoftools__ComplexRootOf():
from sympy.polys.rootoftools import ComplexRootOf
assert _test_args(ComplexRootOf(x**3 + x + 1, 0))
def test_sympy__polys__rootoftools__RootSum():
from sympy.polys.rootoftools import RootSum
assert _test_args(RootSum(x**3 + x + 1, sin))
def test_sympy__series__limits__Limit():
from sympy.series.limits import Limit
assert _test_args(Limit(x, x, 0, dir='-'))
def test_sympy__series__order__Order():
from sympy.series.order import Order
assert _test_args(Order(1, x, y))
@SKIP('Abstract Class')
def test_sympy__series__sequences__SeqBase():
pass
def test_sympy__series__sequences__EmptySequence():
from sympy.series.sequences import EmptySequence
assert _test_args(EmptySequence())
@SKIP('Abstract Class')
def test_sympy__series__sequences__SeqExpr():
pass
def test_sympy__series__sequences__SeqPer():
from sympy.series.sequences import SeqPer
assert _test_args(SeqPer((1, 2, 3), (0, 10)))
def test_sympy__series__sequences__SeqFormula():
from sympy.series.sequences import SeqFormula
assert _test_args(SeqFormula(x**2, (0, 10)))
def test_sympy__series__sequences__RecursiveSeq():
from sympy.series.sequences import RecursiveSeq
y = Function("y")
n = symbols("n")
assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y, n, (0, 1)))
assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y, n))
def test_sympy__series__sequences__SeqExprOp():
from sympy.series.sequences import SeqExprOp, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqExprOp(s1, s2))
def test_sympy__series__sequences__SeqAdd():
from sympy.series.sequences import SeqAdd, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqAdd(s1, s2))
def test_sympy__series__sequences__SeqMul():
from sympy.series.sequences import SeqMul, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqMul(s1, s2))
@SKIP('Abstract Class')
def test_sympy__series__series_class__SeriesBase():
pass
def test_sympy__series__fourier__FourierSeries():
from sympy.series.fourier import fourier_series
assert _test_args(fourier_series(x, (x, -pi, pi)))
def test_sympy__series__fourier__FiniteFourierSeries():
from sympy.series.fourier import fourier_series
assert _test_args(fourier_series(sin(pi*x), (x, -1, 1)))
def test_sympy__series__formal__FormalPowerSeries():
from sympy.series.formal import fps
assert _test_args(fps(log(1 + x), x))
def test_sympy__simplify__hyperexpand__Hyper_Function():
from sympy.simplify.hyperexpand import Hyper_Function
assert _test_args(Hyper_Function([2], [1]))
def test_sympy__simplify__hyperexpand__G_Function():
from sympy.simplify.hyperexpand import G_Function
assert _test_args(G_Function([2], [1], [], []))
@SKIP("abstract class")
def test_sympy__tensor__array__ndim_array__ImmutableNDimArray():
pass
def test_sympy__tensor__array__dense_ndim_array__ImmutableDenseNDimArray():
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
densarr = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4))
assert _test_args(densarr)
def test_sympy__tensor__array__sparse_ndim_array__ImmutableSparseNDimArray():
from sympy.tensor.array.sparse_ndim_array import ImmutableSparseNDimArray
sparr = ImmutableSparseNDimArray(range(10, 34), (2, 3, 4))
assert _test_args(sparr)
def test_sympy__tensor__functions__TensorProduct():
from sympy.tensor.functions import TensorProduct
tp = TensorProduct(3, 4, evaluate=False)
assert _test_args(tp)
def test_sympy__tensor__indexed__Idx():
from sympy.tensor.indexed import Idx
assert _test_args(Idx('test'))
assert _test_args(Idx(1, (0, 10)))
def test_sympy__tensor__indexed__Indexed():
from sympy.tensor.indexed import Indexed, Idx
assert _test_args(Indexed('A', Idx('i'), Idx('j')))
def test_sympy__tensor__indexed__IndexedBase():
from sympy.tensor.indexed import IndexedBase
assert _test_args(IndexedBase('A', shape=(x, y)))
assert _test_args(IndexedBase('A', 1))
assert _test_args(IndexedBase('A')[0, 1])
def test_sympy__tensor__tensor__TensorIndexType():
from sympy.tensor.tensor import TensorIndexType
assert _test_args(TensorIndexType('Lorentz', metric=False))
def test_sympy__tensor__tensor__TensorSymmetry():
from sympy.tensor.tensor import TensorSymmetry, get_symmetric_group_sgs
assert _test_args(TensorSymmetry(get_symmetric_group_sgs(2)))
def test_sympy__tensor__tensor__TensorType():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, TensorType
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
sym = TensorSymmetry(get_symmetric_group_sgs(1))
assert _test_args(TensorType([Lorentz], sym))
def test_sympy__tensor__tensor__TensorHead():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs, TensorHead
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
sym = TensorSymmetry(get_symmetric_group_sgs(1))
S1 = TensorType([Lorentz], sym)
assert _test_args(TensorHead('p', S1, 0))
def test_sympy__tensor__tensor__TensorIndex():
from sympy.tensor.tensor import TensorIndexType, TensorIndex
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
assert _test_args(TensorIndex('i', Lorentz))
@SKIP("abstract class")
def test_sympy__tensor__tensor__TensExpr():
pass
def test_sympy__tensor__tensor__TensAdd():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs, tensor_indices, TensAdd
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
S1 = TensorType([Lorentz], sym)
p, q = S1('p,q')
t1 = p(a)
t2 = q(a)
assert _test_args(TensAdd(t1, t2))
def test_sympy__tensor__tensor__Tensor():
from sympy.core import S
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs, tensor_indices, TensMul
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
S1 = TensorType([Lorentz], sym)
p = S1('p')
assert _test_args(p(a))
def test_sympy__tensor__tensor__TensMul():
from sympy.core import S
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs, tensor_indices, TensMul
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
S1 = TensorType([Lorentz], sym)
p = S1('p')
q = S1('q')
assert _test_args(3*p(a)*q(b))
def test_sympy__tensor__tensor__TensorElement():
from sympy.tensor.tensor import TensorIndexType, tensorhead, TensorElement
L = TensorIndexType("L")
A = tensorhead("A", [L, L], [[1], [1]])
telem = TensorElement(A(x, y), {x: 1})
assert _test_args(telem)
def test_sympy__tensor__toperators__PartialDerivative():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
from sympy.tensor.toperators import PartialDerivative
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
A = tensorhead("A", [Lorentz], [[1]])
assert _test_args(PartialDerivative(A(a), A(b)))
def test_as_coeff_add():
assert (7, (3*x, 4*x**2)) == (7 + 3*x + 4*x**2).as_coeff_add()
def test_sympy__geometry__curve__Curve():
from sympy.geometry.curve import Curve
assert _test_args(Curve((x, 1), (x, 0, 1)))
def test_sympy__geometry__point__Point():
from sympy.geometry.point import Point
assert _test_args(Point(0, 1))
def test_sympy__geometry__point__Point2D():
from sympy.geometry.point import Point2D
assert _test_args(Point2D(0, 1))
def test_sympy__geometry__point__Point3D():
from sympy.geometry.point import Point3D
assert _test_args(Point3D(0, 1, 2))
def test_sympy__geometry__ellipse__Ellipse():
from sympy.geometry.ellipse import Ellipse
assert _test_args(Ellipse((0, 1), 2, 3))
def test_sympy__geometry__ellipse__Circle():
from sympy.geometry.ellipse import Circle
assert _test_args(Circle((0, 1), 2))
def test_sympy__geometry__parabola__Parabola():
from sympy.geometry.parabola import Parabola
from sympy.geometry.line import Line
assert _test_args(Parabola((0, 0), Line((2, 3), (4, 3))))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity():
pass
def test_sympy__geometry__line__Line():
from sympy.geometry.line import Line
assert _test_args(Line((0, 1), (2, 3)))
def test_sympy__geometry__line__Ray():
from sympy.geometry.line import Ray
assert _test_args(Ray((0, 1), (2, 3)))
def test_sympy__geometry__line__Segment():
from sympy.geometry.line import Segment
assert _test_args(Segment((0, 1), (2, 3)))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity2D():
pass
def test_sympy__geometry__line__Line2D():
from sympy.geometry.line import Line2D
assert _test_args(Line2D((0, 1), (2, 3)))
def test_sympy__geometry__line__Ray2D():
from sympy.geometry.line import Ray2D
assert _test_args(Ray2D((0, 1), (2, 3)))
def test_sympy__geometry__line__Segment2D():
from sympy.geometry.line import Segment2D
assert _test_args(Segment2D((0, 1), (2, 3)))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity3D():
pass
def test_sympy__geometry__line__Line3D():
from sympy.geometry.line import Line3D
assert _test_args(Line3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__line__Segment3D():
from sympy.geometry.line import Segment3D
assert _test_args(Segment3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__line__Ray3D():
from sympy.geometry.line import Ray3D
assert _test_args(Ray3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__plane__Plane():
from sympy.geometry.plane import Plane
assert _test_args(Plane((1, 1, 1), (-3, 4, -2), (1, 2, 3)))
def test_sympy__geometry__polygon__Polygon():
from sympy.geometry.polygon import Polygon
assert _test_args(Polygon((0, 1), (2, 3), (4, 5), (6, 7)))
def test_sympy__geometry__polygon__RegularPolygon():
from sympy.geometry.polygon import RegularPolygon
assert _test_args(RegularPolygon((0, 1), 2, 3, 4))
def test_sympy__geometry__polygon__Triangle():
from sympy.geometry.polygon import Triangle
assert _test_args(Triangle((0, 1), (2, 3), (4, 5)))
def test_sympy__geometry__entity__GeometryEntity():
from sympy.geometry.entity import GeometryEntity
from sympy.geometry.point import Point
assert _test_args(GeometryEntity(Point(1, 0), 1, [1, 2]))
@SKIP("abstract class")
def test_sympy__geometry__entity__GeometrySet():
pass
def test_sympy__diffgeom__diffgeom__Manifold():
from sympy.diffgeom import Manifold
assert _test_args(Manifold('name', 3))
def test_sympy__diffgeom__diffgeom__Patch():
from sympy.diffgeom import Manifold, Patch
assert _test_args(Patch('name', Manifold('name', 3)))
def test_sympy__diffgeom__diffgeom__CoordSystem():
from sympy.diffgeom import Manifold, Patch, CoordSystem
assert _test_args(CoordSystem('name', Patch('name', Manifold('name', 3))))
@XFAIL
def test_sympy__diffgeom__diffgeom__Point():
from sympy.diffgeom import Manifold, Patch, CoordSystem, Point
assert _test_args(Point(
CoordSystem('name', Patch('name', Manifold('name', 3))), [x, y]))
def test_sympy__diffgeom__diffgeom__BaseScalarField():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseScalarField(cs, 0))
def test_sympy__diffgeom__diffgeom__BaseVectorField():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseVectorField(cs, 0))
def test_sympy__diffgeom__diffgeom__Differential():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(Differential(BaseScalarField(cs, 0)))
def test_sympy__diffgeom__diffgeom__Commutator():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, Commutator
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
cs1 = CoordSystem('name1', Patch('name', Manifold('name', 3)))
v = BaseVectorField(cs, 0)
v1 = BaseVectorField(cs1, 0)
assert _test_args(Commutator(v, v1))
def test_sympy__diffgeom__diffgeom__TensorProduct():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, TensorProduct
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
assert _test_args(TensorProduct(d, d))
def test_sympy__diffgeom__diffgeom__WedgeProduct():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, WedgeProduct
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
d1 = Differential(BaseScalarField(cs, 1))
assert _test_args(WedgeProduct(d, d1))
def test_sympy__diffgeom__diffgeom__LieDerivative():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, BaseVectorField, LieDerivative
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
v = BaseVectorField(cs, 0)
assert _test_args(LieDerivative(v, d))
@XFAIL
def test_sympy__diffgeom__diffgeom__BaseCovarDerivativeOp():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseCovarDerivativeOp
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseCovarDerivativeOp(cs, 0, [[[0, ]*3, ]*3, ]*3))
def test_sympy__diffgeom__diffgeom__CovarDerivativeOp():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, CovarDerivativeOp
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
v = BaseVectorField(cs, 0)
_test_args(CovarDerivativeOp(v, [[[0, ]*3, ]*3, ]*3))
def test_sympy__categories__baseclasses__Class():
from sympy.categories.baseclasses import Class
assert _test_args(Class())
def test_sympy__categories__baseclasses__Object():
from sympy.categories import Object
assert _test_args(Object("A"))
@XFAIL
def test_sympy__categories__baseclasses__Morphism():
from sympy.categories import Object, Morphism
assert _test_args(Morphism(Object("A"), Object("B")))
def test_sympy__categories__baseclasses__IdentityMorphism():
from sympy.categories import Object, IdentityMorphism
assert _test_args(IdentityMorphism(Object("A")))
def test_sympy__categories__baseclasses__NamedMorphism():
from sympy.categories import Object, NamedMorphism
assert _test_args(NamedMorphism(Object("A"), Object("B"), "f"))
def test_sympy__categories__baseclasses__CompositeMorphism():
from sympy.categories import Object, NamedMorphism, CompositeMorphism
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
assert _test_args(CompositeMorphism(f, g))
def test_sympy__categories__baseclasses__Diagram():
from sympy.categories import Object, NamedMorphism, Diagram
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
d = Diagram([f])
assert _test_args(d)
def test_sympy__categories__baseclasses__Category():
from sympy.categories import Object, NamedMorphism, Diagram, Category
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d1 = Diagram([f, g])
d2 = Diagram([f])
K = Category("K", commutative_diagrams=[d1, d2])
assert _test_args(K)
def test_sympy__ntheory__factor___totient():
from sympy.ntheory.factor_ import totient
k = symbols('k', integer=True)
t = totient(k)
assert _test_args(t)
def test_sympy__ntheory__factor___reduced_totient():
from sympy.ntheory.factor_ import reduced_totient
k = symbols('k', integer=True)
t = reduced_totient(k)
assert _test_args(t)
def test_sympy__ntheory__factor___divisor_sigma():
from sympy.ntheory.factor_ import divisor_sigma
k = symbols('k', integer=True)
n = symbols('n', integer=True)
t = divisor_sigma(n, k)
assert _test_args(t)
def test_sympy__ntheory__factor___udivisor_sigma():
from sympy.ntheory.factor_ import udivisor_sigma
k = symbols('k', integer=True)
n = symbols('n', integer=True)
t = udivisor_sigma(n, k)
assert _test_args(t)
def test_sympy__ntheory__factor___primenu():
from sympy.ntheory.factor_ import primenu
n = symbols('n', integer=True)
t = primenu(n)
assert _test_args(t)
def test_sympy__ntheory__factor___primeomega():
from sympy.ntheory.factor_ import primeomega
n = symbols('n', integer=True)
t = primeomega(n)
assert _test_args(t)
def test_sympy__ntheory__residue_ntheory__mobius():
from sympy.ntheory import mobius
assert _test_args(mobius(2))
def test_sympy__ntheory__generate__primepi():
from sympy.ntheory import primepi
n = symbols('n')
t = primepi(n)
assert _test_args(t)
def test_sympy__physics__optics__waves__TWave():
from sympy.physics.optics import TWave
A, f, phi = symbols('A, f, phi')
assert _test_args(TWave(A, f, phi))
def test_sympy__physics__optics__gaussopt__BeamParameter():
from sympy.physics.optics import BeamParameter
assert _test_args(BeamParameter(530e-9, 1, w=1e-3))
def test_sympy__physics__optics__medium__Medium():
from sympy.physics.optics import Medium
assert _test_args(Medium('m'))
def test_sympy__codegen__array_utils__CodegenArrayContraction():
from sympy.codegen.array_utils import CodegenArrayContraction
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayContraction(A, (0, 1)))
def test_sympy__codegen__array_utils__CodegenArrayDiagonal():
from sympy.codegen.array_utils import CodegenArrayDiagonal
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayDiagonal(A, (0, 1)))
def test_sympy__codegen__array_utils__CodegenArrayTensorProduct():
from sympy.codegen.array_utils import CodegenArrayTensorProduct
from sympy import IndexedBase
A, B = symbols("A B", cls=IndexedBase)
assert _test_args(CodegenArrayTensorProduct(A, B))
def test_sympy__codegen__array_utils__CodegenArrayElementwiseAdd():
from sympy.codegen.array_utils import CodegenArrayElementwiseAdd
from sympy import IndexedBase
A, B = symbols("A B", cls=IndexedBase)
assert _test_args(CodegenArrayElementwiseAdd(A, B))
def test_sympy__codegen__array_utils__CodegenArrayPermuteDims():
from sympy.codegen.array_utils import CodegenArrayPermuteDims
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayPermuteDims(A, (1, 0)))
def test_sympy__codegen__ast__Assignment():
from sympy.codegen.ast import Assignment
assert _test_args(Assignment(x, y))
def test_sympy__codegen__cfunctions__expm1():
from sympy.codegen.cfunctions import expm1
assert _test_args(expm1(x))
def test_sympy__codegen__cfunctions__log1p():
from sympy.codegen.cfunctions import log1p
assert _test_args(log1p(x))
def test_sympy__codegen__cfunctions__exp2():
from sympy.codegen.cfunctions import exp2
assert _test_args(exp2(x))
def test_sympy__codegen__cfunctions__log2():
from sympy.codegen.cfunctions import log2
assert _test_args(log2(x))
def test_sympy__codegen__cfunctions__fma():
from sympy.codegen.cfunctions import fma
assert _test_args(fma(x, y, z))
def test_sympy__codegen__cfunctions__log10():
from sympy.codegen.cfunctions import log10
assert _test_args(log10(x))
def test_sympy__codegen__cfunctions__Sqrt():
from sympy.codegen.cfunctions import Sqrt
assert _test_args(Sqrt(x))
def test_sympy__codegen__cfunctions__Cbrt():
from sympy.codegen.cfunctions import Cbrt
assert _test_args(Cbrt(x))
def test_sympy__codegen__cfunctions__hypot():
from sympy.codegen.cfunctions import hypot
assert _test_args(hypot(x, y))
def test_sympy__codegen__fnodes__FFunction():
from sympy.codegen.fnodes import FFunction
assert _test_args(FFunction('f'))
def test_sympy__codegen__fnodes__F95Function():
from sympy.codegen.fnodes import F95Function
assert _test_args(F95Function('f'))
def test_sympy__codegen__fnodes__isign():
from sympy.codegen.fnodes import isign
assert _test_args(isign(1, x))
def test_sympy__codegen__fnodes__dsign():
from sympy.codegen.fnodes import dsign
assert _test_args(dsign(1, x))
def test_sympy__codegen__fnodes__cmplx():
from sympy.codegen.fnodes import cmplx
assert _test_args(cmplx(x, y))
def test_sympy__codegen__fnodes__kind():
from sympy.codegen.fnodes import kind
assert _test_args(kind(x))
def test_sympy__codegen__fnodes__merge():
from sympy.codegen.fnodes import merge
assert _test_args(merge(1, 2, Eq(x, 0)))
def test_sympy__codegen__fnodes___literal():
from sympy.codegen.fnodes import _literal
assert _test_args(_literal(1))
def test_sympy__codegen__fnodes__literal_sp():
from sympy.codegen.fnodes import literal_sp
assert _test_args(literal_sp(1))
def test_sympy__codegen__fnodes__literal_dp():
from sympy.codegen.fnodes import literal_dp
assert _test_args(literal_dp(1))
def test_sympy__vector__coordsysrect__CoordSys3D():
from sympy.vector.coordsysrect import CoordSys3D
assert _test_args(CoordSys3D('C'))
def test_sympy__vector__point__Point():
from sympy.vector.point import Point
assert _test_args(Point('P'))
def test_sympy__vector__basisdependent__BasisDependent():
from sympy.vector.basisdependent import BasisDependent
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
def test_sympy__vector__basisdependent__BasisDependentMul():
from sympy.vector.basisdependent import BasisDependentMul
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
def test_sympy__vector__basisdependent__BasisDependentAdd():
from sympy.vector.basisdependent import BasisDependentAdd
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
def test_sympy__vector__basisdependent__BasisDependentZero():
from sympy.vector.basisdependent import BasisDependentZero
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
def test_sympy__vector__vector__BaseVector():
from sympy.vector.vector import BaseVector
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseVector(0, C, ' ', ' '))
def test_sympy__vector__vector__VectorAdd():
from sympy.vector.vector import VectorAdd, VectorMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
from sympy.abc import a, b, c, x, y, z
v1 = a*C.i + b*C.j + c*C.k
v2 = x*C.i + y*C.j + z*C.k
assert _test_args(VectorAdd(v1, v2))
assert _test_args(VectorMul(x, v1))
def test_sympy__vector__vector__VectorMul():
from sympy.vector.vector import VectorMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
from sympy.abc import a
assert _test_args(VectorMul(a, C.i))
def test_sympy__vector__vector__VectorZero():
from sympy.vector.vector import VectorZero
assert _test_args(VectorZero())
def test_sympy__vector__vector__Vector():
from sympy.vector.vector import Vector
#Vector is never to be initialized using args
pass
def test_sympy__vector__vector__Cross():
from sympy.vector.vector import Cross
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
_test_args(Cross(C.i, C.j))
def test_sympy__vector__vector__Dot():
from sympy.vector.vector import Dot
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
_test_args(Dot(C.i, C.j))
def test_sympy__vector__dyadic__Dyadic():
from sympy.vector.dyadic import Dyadic
#Dyadic is never to be initialized using args
pass
def test_sympy__vector__dyadic__BaseDyadic():
from sympy.vector.dyadic import BaseDyadic
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseDyadic(C.i, C.j))
def test_sympy__vector__dyadic__DyadicMul():
from sympy.vector.dyadic import BaseDyadic, DyadicMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(DyadicMul(3, BaseDyadic(C.i, C.j)))
def test_sympy__vector__dyadic__DyadicAdd():
from sympy.vector.dyadic import BaseDyadic, DyadicAdd
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(2 * DyadicAdd(BaseDyadic(C.i, C.i),
BaseDyadic(C.i, C.j)))
def test_sympy__vector__dyadic__DyadicZero():
from sympy.vector.dyadic import DyadicZero
assert _test_args(DyadicZero())
def test_sympy__vector__deloperator__Del():
from sympy.vector.deloperator import Del
assert _test_args(Del())
def test_sympy__vector__operators__Curl():
from sympy.vector.operators import Curl
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Curl(C.i))
def test_sympy__vector__operators__Laplacian():
from sympy.vector.operators import Laplacian
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Laplacian(C.i))
def test_sympy__vector__operators__Divergence():
from sympy.vector.operators import Divergence
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Divergence(C.i))
def test_sympy__vector__operators__Gradient():
from sympy.vector.operators import Gradient
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Gradient(C.x))
def test_sympy__vector__orienters__Orienter():
from sympy.vector.orienters import Orienter
#Not to be initialized
def test_sympy__vector__orienters__ThreeAngleOrienter():
from sympy.vector.orienters import ThreeAngleOrienter
#Not to be initialized
def test_sympy__vector__orienters__AxisOrienter():
from sympy.vector.orienters import AxisOrienter
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(AxisOrienter(x, C.i))
def test_sympy__vector__orienters__BodyOrienter():
from sympy.vector.orienters import BodyOrienter
assert _test_args(BodyOrienter(x, y, z, '123'))
def test_sympy__vector__orienters__SpaceOrienter():
from sympy.vector.orienters import SpaceOrienter
assert _test_args(SpaceOrienter(x, y, z, '123'))
def test_sympy__vector__orienters__QuaternionOrienter():
from sympy.vector.orienters import QuaternionOrienter
a, b, c, d = symbols('a b c d')
assert _test_args(QuaternionOrienter(a, b, c, d))
def test_sympy__vector__scalar__BaseScalar():
from sympy.vector.scalar import BaseScalar
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseScalar(0, C, ' ', ' '))
def test_sympy__physics__wigner__Wigner3j():
from sympy.physics.wigner import Wigner3j
assert _test_args(Wigner3j(0, 0, 0, 0, 0, 0))
def test_sympy__integrals__rubi__symbol__matchpyWC():
from sympy.integrals.rubi.symbol import matchpyWC
assert _test_args(matchpyWC(1, True, 'a'))
def test_sympy__integrals__rubi__utility_function__rubi_unevaluated_expr():
from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr
a = symbols('a')
assert _test_args(rubi_unevaluated_expr(a))
def test_sympy__integrals__rubi__utility_function__rubi_exp():
from sympy.integrals.rubi.utility_function import rubi_exp
assert _test_args(rubi_exp(5))
def test_sympy__integrals__rubi__utility_function__rubi_log():
from sympy.integrals.rubi.utility_function import rubi_log
assert _test_args(rubi_log(5))
def test_sympy__integrals__rubi__utility_function__Int():
from sympy.integrals.rubi.utility_function import Int
assert _test_args(Int(5, x))
def test_sympy__integrals__rubi__utility_function__Util_Coefficient():
from sympy.integrals.rubi.utility_function import Util_Coefficient
a, x = symbols('a x')
assert _test_args(Util_Coefficient(a, x))
def test_sympy__integrals__rubi__utility_function__Gamma():
from sympy.integrals.rubi.utility_function import Gamma
assert _test_args(Gamma(5))
def test_sympy__integrals__rubi__utility_function__Util_Part():
from sympy.integrals.rubi.utility_function import Util_Part
a, b = symbols('a b')
assert _test_args(Util_Part(a + b, 0))
def test_sympy__integrals__rubi__utility_function__PolyGamma():
from sympy.integrals.rubi.utility_function import PolyGamma
assert _test_args(PolyGamma(1, 1))
def test_sympy__integrals__rubi__utility_function__ProductLog():
from sympy.integrals.rubi.utility_function import ProductLog
assert _test_args(ProductLog(1))
|
60a2c0ad8fcedca8bb8d4eeb442b26ee4f858d5010068aa3bfad52a9eda6d952 | import decimal
from sympy import (Rational, Symbol, Float, I, sqrt, cbrt, oo, nan, pi, E,
Integer, S, factorial, Catalan, EulerGamma, GoldenRatio,
TribonacciConstant, cos, exp,
Number, zoo, log, Mul, Pow, Tuple, latex, Gt, Lt, Ge, Le,
AlgebraicNumber, simplify, sin, fibonacci, RealField,
sympify, srepr)
from sympy.core.compatibility import long
from sympy.core.power import integer_nthroot, isqrt, integer_log
from sympy.core.logic import fuzzy_not
from sympy.core.numbers import (igcd, ilcm, igcdex, seterr,
igcd2, igcd_lehmer, mpf_norm, comp, mod_inverse)
from sympy.core.mod import Mod
from sympy.polys.domains.groundtypes import PythonRational
from sympy.utilities.decorator import conserve_mpmath_dps
from sympy.utilities.iterables import permutations
from sympy.utilities.pytest import XFAIL, raises
from mpmath import mpf
from mpmath.rational import mpq
import mpmath
t = Symbol('t', real=False)
def same_and_same_prec(a, b):
# stricter matching for Floats
return a == b and a._prec == b._prec
def test_seterr():
seterr(divide=True)
raises(ValueError, lambda: S.Zero/S.Zero)
seterr(divide=False)
assert S.Zero / S.Zero == S.NaN
def test_mod():
x = Rational(1, 2)
y = Rational(3, 4)
z = Rational(5, 18043)
assert x % x == 0
assert x % y == 1/S(2)
assert x % z == 3/S(36086)
assert y % x == 1/S(4)
assert y % y == 0
assert y % z == 9/S(72172)
assert z % x == 5/S(18043)
assert z % y == 5/S(18043)
assert z % z == 0
a = Float(2.6)
assert (a % .2) == 0
assert (a % 2).round(15) == 0.6
assert (a % 0.5).round(15) == 0.1
p = Symbol('p', infinite=True)
assert oo % oo == nan
assert zoo % oo == nan
assert 5 % oo == nan
assert p % 5 == nan
# In these two tests, if the precision of m does
# not match the precision of the ans, then it is
# likely that the change made now gives an answer
# with degraded accuracy.
r = Rational(500, 41)
f = Float('.36', 3)
m = r % f
ans = Float(r % Rational(f), 3)
assert m == ans and m._prec == ans._prec
f = Float('8.36', 3)
m = f % r
ans = Float(Rational(f) % r, 3)
assert m == ans and m._prec == ans._prec
s = S.Zero
assert s % float(1) == S.Zero
# No rounding required since these numbers can be represented
# exactly.
assert Rational(3, 4) % Float(1.1) == 0.75
assert Float(1.5) % Rational(5, 4) == 0.25
assert Rational(5, 4).__rmod__(Float('1.5')) == 0.25
assert Float('1.5').__rmod__(Float('2.75')) == Float('1.25')
assert 2.75 % Float('1.5') == Float('1.25')
a = Integer(7)
b = Integer(4)
assert type(a % b) == Integer
assert a % b == Integer(3)
assert Integer(1) % Rational(2, 3) == Rational(1, 3)
assert Rational(7, 5) % Integer(1) == Rational(2, 5)
assert Integer(2) % 1.5 == 0.5
assert Integer(3).__rmod__(Integer(10)) == Integer(1)
assert Integer(10) % 4 == Integer(2)
assert 15 % Integer(4) == Integer(3)
def test_divmod():
assert divmod(S(12), S(8)) == Tuple(1, 4)
assert divmod(-S(12), S(8)) == Tuple(-2, 4)
assert divmod(S(0), S(1)) == Tuple(0, 0)
raises(ZeroDivisionError, lambda: divmod(S(0), S(0)))
raises(ZeroDivisionError, lambda: divmod(S(1), S(0)))
assert divmod(S(12), 8) == Tuple(1, 4)
assert divmod(12, S(8)) == Tuple(1, 4)
assert divmod(S("2"), S("3/2")) == Tuple(S("1"), S("1/2"))
assert divmod(S("3/2"), S("2")) == Tuple(S("0"), S("3/2"))
assert divmod(S("2"), S("3.5")) == Tuple(S("0"), S("2"))
assert divmod(S("3.5"), S("2")) == Tuple(S("1"), S("1.5"))
assert divmod(S("2"), S("1/3")) == Tuple(S("6"), S("0"))
assert divmod(S("1/3"), S("2")) == Tuple(S("0"), S("1/3"))
assert divmod(S("2"), S("0.1")) == Tuple(S("20"), S("0"))
assert divmod(S("0.1"), S("2")) == Tuple(S("0"), S("0.1"))
assert divmod(S("2"), 2) == Tuple(S("1"), S("0"))
assert divmod(2, S("2")) == Tuple(S("1"), S("0"))
assert divmod(S("2"), 1.5) == Tuple(S("1"), S("0.5"))
assert divmod(1.5, S("2")) == Tuple(S("0"), S("1.5"))
assert divmod(0.3, S("2")) == Tuple(S("0"), S("0.3"))
assert divmod(S("3/2"), S("3.5")) == Tuple(S("0"), S("3/2"))
assert divmod(S("3.5"), S("3/2")) == Tuple(S("2"), S("0.5"))
assert divmod(S("3/2"), S("1/3")) == Tuple(S("4"), Float("1/6"))
assert divmod(S("1/3"), S("3/2")) == Tuple(S("0"), S("1/3"))
assert divmod(S("3/2"), S("0.1")) == Tuple(S("15"), S("0"))
assert divmod(S("0.1"), S("3/2")) == Tuple(S("0"), S("0.1"))
assert divmod(S("3/2"), 2) == Tuple(S("0"), S("3/2"))
assert divmod(2, S("3/2")) == Tuple(S("1"), S("0.5"))
assert divmod(S("3/2"), 1.5) == Tuple(S("1"), S("0"))
assert divmod(1.5, S("3/2")) == Tuple(S("1"), S("0"))
assert divmod(S("3/2"), 0.3) == Tuple(S("5"), S("0"))
assert divmod(0.3, S("3/2")) == Tuple(S("0"), S("0.3"))
assert divmod(S("1/3"), S("3.5")) == Tuple(S("0"), S("1/3"))
assert divmod(S("3.5"), S("0.1")) == Tuple(S("35"), S("0"))
assert divmod(S("0.1"), S("3.5")) == Tuple(S("0"), S("0.1"))
assert divmod(S("3.5"), 2) == Tuple(S("1"), S("1.5"))
assert divmod(2, S("3.5")) == Tuple(S("0"), S("2"))
assert divmod(S("3.5"), 1.5) == Tuple(S("2"), S("0.5"))
assert divmod(1.5, S("3.5")) == Tuple(S("0"), S("1.5"))
assert divmod(0.3, S("3.5")) == Tuple(S("0"), S("0.3"))
assert divmod(S("0.1"), S("1/3")) == Tuple(S("0"), S("0.1"))
assert divmod(S("1/3"), 2) == Tuple(S("0"), S("1/3"))
assert divmod(2, S("1/3")) == Tuple(S("6"), S("0"))
assert divmod(S("1/3"), 1.5) == Tuple(S("0"), S("1/3"))
assert divmod(0.3, S("1/3")) == Tuple(S("0"), S("0.3"))
assert divmod(S("0.1"), 2) == Tuple(S("0"), S("0.1"))
assert divmod(2, S("0.1")) == Tuple(S("20"), S("0"))
assert divmod(S("0.1"), 1.5) == Tuple(S("0"), S("0.1"))
assert divmod(1.5, S("0.1")) == Tuple(S("15"), S("0"))
assert divmod(S("0.1"), 0.3) == Tuple(S("0"), S("0.1"))
assert str(divmod(S("2"), 0.3)) == '(6, 0.2)'
assert str(divmod(S("3.5"), S("1/3"))) == '(10, 0.166666666666667)'
assert str(divmod(S("3.5"), 0.3)) == '(11, 0.2)'
assert str(divmod(S("1/3"), S("0.1"))) == '(3, 0.0333333333333333)'
assert str(divmod(1.5, S("1/3"))) == '(4, 0.166666666666667)'
assert str(divmod(S("1/3"), 0.3)) == '(1, 0.0333333333333333)'
assert str(divmod(0.3, S("0.1"))) == '(2, 0.1)'
assert divmod(-3, S(2)) == (-2, 1)
assert divmod(S(-3), S(2)) == (-2, 1)
assert divmod(S(-3), 2) == (-2, 1)
assert divmod(S(4), S(-3.1)) == Tuple(-2, -2.2)
assert divmod(S(4), S(-2.1)) == divmod(4, -2.1)
assert divmod(S(-8), S(-2.5) ) == Tuple(3 , -0.5)
def test_igcd():
assert igcd(0, 0) == 0
assert igcd(0, 1) == 1
assert igcd(1, 0) == 1
assert igcd(0, 7) == 7
assert igcd(7, 0) == 7
assert igcd(7, 1) == 1
assert igcd(1, 7) == 1
assert igcd(-1, 0) == 1
assert igcd(0, -1) == 1
assert igcd(-1, -1) == 1
assert igcd(-1, 7) == 1
assert igcd(7, -1) == 1
assert igcd(8, 2) == 2
assert igcd(4, 8) == 4
assert igcd(8, 16) == 8
assert igcd(7, -3) == 1
assert igcd(-7, 3) == 1
assert igcd(-7, -3) == 1
assert igcd(*[10, 20, 30]) == 10
raises(TypeError, lambda: igcd())
raises(TypeError, lambda: igcd(2))
raises(ValueError, lambda: igcd(0, None))
raises(ValueError, lambda: igcd(1, 2.2))
for args in permutations((45.1, 1, 30)):
raises(ValueError, lambda: igcd(*args))
for args in permutations((1, 2, None)):
raises(ValueError, lambda: igcd(*args))
def test_igcd_lehmer():
a, b = fibonacci(10001), fibonacci(10000)
# len(str(a)) == 2090
# small divisors, long Euclidean sequence
assert igcd_lehmer(a, b) == 1
c = fibonacci(100)
assert igcd_lehmer(a*c, b*c) == c
# big divisor
assert igcd_lehmer(a, 10**1000) == 1
# swapping argmument
assert igcd_lehmer(1, 2) == igcd_lehmer(2, 1)
def test_igcd2():
# short loop
assert igcd2(2**100 - 1, 2**99 - 1) == 1
# Lehmer's algorithm
a, b = int(fibonacci(10001)), int(fibonacci(10000))
assert igcd2(a, b) == 1
def test_ilcm():
assert ilcm(0, 0) == 0
assert ilcm(1, 0) == 0
assert ilcm(0, 1) == 0
assert ilcm(1, 1) == 1
assert ilcm(2, 1) == 2
assert ilcm(8, 2) == 8
assert ilcm(8, 6) == 24
assert ilcm(8, 7) == 56
assert ilcm(*[10, 20, 30]) == 60
raises(ValueError, lambda: ilcm(8.1, 7))
raises(ValueError, lambda: ilcm(8, 7.1))
raises(TypeError, lambda: ilcm(8))
def test_igcdex():
assert igcdex(2, 3) == (-1, 1, 1)
assert igcdex(10, 12) == (-1, 1, 2)
assert igcdex(100, 2004) == (-20, 1, 4)
assert igcdex(0, 0) == (0, 1, 0)
assert igcdex(1, 0) == (1, 0, 1)
def _strictly_equal(a, b):
return (a.p, a.q, type(a.p), type(a.q)) == \
(b.p, b.q, type(b.p), type(b.q))
def _test_rational_new(cls):
"""
Tests that are common between Integer and Rational.
"""
assert cls(0) is S.Zero
assert cls(1) is S.One
assert cls(-1) is S.NegativeOne
# These look odd, but are similar to int():
assert cls('1') is S.One
assert cls(u'-1') is S.NegativeOne
i = Integer(10)
assert _strictly_equal(i, cls('10'))
assert _strictly_equal(i, cls(u'10'))
assert _strictly_equal(i, cls(long(10)))
assert _strictly_equal(i, cls(i))
raises(TypeError, lambda: cls(Symbol('x')))
def test_Integer_new():
"""
Test for Integer constructor
"""
_test_rational_new(Integer)
assert _strictly_equal(Integer(0.9), S.Zero)
assert _strictly_equal(Integer(10.5), Integer(10))
raises(ValueError, lambda: Integer("10.5"))
assert Integer(Rational('1.' + '9'*20)) == 1
def test_Rational_new():
""""
Test for Rational constructor
"""
_test_rational_new(Rational)
n1 = Rational(1, 2)
assert n1 == Rational(Integer(1), 2)
assert n1 == Rational(Integer(1), Integer(2))
assert n1 == Rational(1, Integer(2))
assert n1 == Rational(Rational(1, 2))
assert 1 == Rational(n1, n1)
assert Rational(3, 2) == Rational(Rational(1, 2), Rational(1, 3))
assert Rational(3, 1) == Rational(1, Rational(1, 3))
n3_4 = Rational(3, 4)
assert Rational('3/4') == n3_4
assert -Rational('-3/4') == n3_4
assert Rational('.76').limit_denominator(4) == n3_4
assert Rational(19, 25).limit_denominator(4) == n3_4
assert Rational('19/25').limit_denominator(4) == n3_4
assert Rational(1.0, 3) == Rational(1, 3)
assert Rational(1, 3.0) == Rational(1, 3)
assert Rational(Float(0.5)) == Rational(1, 2)
assert Rational('1e2/1e-2') == Rational(10000)
assert Rational('1 234') == Rational(1234)
assert Rational('1/1 234') == Rational(1, 1234)
assert Rational(-1, 0) == S.ComplexInfinity
assert Rational(1, 0) == S.ComplexInfinity
# Make sure Rational doesn't lose precision on Floats
assert Rational(pi.evalf(100)).evalf(100) == pi.evalf(100)
raises(TypeError, lambda: Rational('3**3'))
raises(TypeError, lambda: Rational('1/2 + 2/3'))
# handle fractions.Fraction instances
try:
import fractions
assert Rational(fractions.Fraction(1, 2)) == Rational(1, 2)
except ImportError:
pass
assert Rational(mpq(2, 6)) == Rational(1, 3)
assert Rational(PythonRational(2, 6)) == Rational(1, 3)
def test_Number_new():
""""
Test for Number constructor
"""
# Expected behavior on numbers and strings
assert Number(1) is S.One
assert Number(2).__class__ is Integer
assert Number(-622).__class__ is Integer
assert Number(5, 3).__class__ is Rational
assert Number(5.3).__class__ is Float
assert Number('1') is S.One
assert Number('2').__class__ is Integer
assert Number('-622').__class__ is Integer
assert Number('5/3').__class__ is Rational
assert Number('5.3').__class__ is Float
raises(ValueError, lambda: Number('cos'))
raises(TypeError, lambda: Number(cos))
a = Rational(3, 5)
assert Number(a) is a # Check idempotence on Numbers
def test_Number_cmp():
n1 = Number(1)
n2 = Number(2)
n3 = Number(-3)
assert n1 < n2
assert n1 <= n2
assert n3 < n1
assert n2 > n3
assert n2 >= n3
raises(TypeError, lambda: n1 < S.NaN)
raises(TypeError, lambda: n1 <= S.NaN)
raises(TypeError, lambda: n1 > S.NaN)
raises(TypeError, lambda: n1 >= S.NaN)
def test_Rational_cmp():
n1 = Rational(1, 4)
n2 = Rational(1, 3)
n3 = Rational(2, 4)
n4 = Rational(2, -4)
n5 = Rational(0)
n6 = Rational(1)
n7 = Rational(3)
n8 = Rational(-3)
assert n8 < n5
assert n5 < n6
assert n6 < n7
assert n8 < n7
assert n7 > n8
assert (n1 + 1)**n2 < 2
assert ((n1 + n6)/n7) < 1
assert n4 < n3
assert n2 < n3
assert n1 < n2
assert n3 > n1
assert not n3 < n1
assert not (Rational(-1) > 0)
assert Rational(-1) < 0
raises(TypeError, lambda: n1 < S.NaN)
raises(TypeError, lambda: n1 <= S.NaN)
raises(TypeError, lambda: n1 > S.NaN)
raises(TypeError, lambda: n1 >= S.NaN)
def test_Float():
def eq(a, b):
t = Float("1.0E-15")
return (-t < a - b < t)
a = Float(2) ** Float(3)
assert eq(a.evalf(), Float(8))
assert eq((pi ** -1).evalf(), Float("0.31830988618379067"))
a = Float(2) ** Float(4)
assert eq(a.evalf(), Float(16))
assert (S(.3) == S(.5)) is False
x_str = Float((0, '13333333333333', -52, 53))
x2_str = Float((0, '26666666666666', -53, 53))
x_hex = Float((0, long(0x13333333333333), -52, 53))
x_dec = Float((0, 5404319552844595, -52, 53))
assert x_str == x_hex == x_dec == Float(1.2)
# This looses a binary digit of precision, so it isn't equal to the above,
# but check that it normalizes correctly
x2_hex = Float((0, long(0x13333333333333)*2, -53, 53))
assert x2_hex._mpf_ == (0, 5404319552844595, -52, 52)
# XXX: Should this test also hold?
# assert x2_hex._prec == 52
# x2_str and 1.2 are superficially the same
assert str(x2_str) == str(Float(1.2))
# but are different at the mpf level
assert Float(1.2)._mpf_ == (0, long(5404319552844595), -52, 53)
assert x2_str._mpf_ == (0, long(10808639105689190), -53, 53)
assert Float((0, long(0), -123, -1)) == Float('nan')
assert Float((0, long(0), -456, -2)) == Float('inf') == Float('+inf')
assert Float((1, long(0), -789, -3)) == Float('-inf')
raises(ValueError, lambda: Float((0, 7, 1, 3), ''))
assert Float('+inf').is_finite is False
assert Float('+inf').is_negative is False
assert Float('+inf').is_positive is True
assert Float('+inf').is_infinite is True
assert Float('+inf').is_zero is False
assert Float('-inf').is_finite is False
assert Float('-inf').is_negative is True
assert Float('-inf').is_positive is False
assert Float('-inf').is_infinite is True
assert Float('-inf').is_zero is False
assert Float('0.0').is_finite is True
assert Float('0.0').is_negative is False
assert Float('0.0').is_positive is False
assert Float('0.0').is_infinite is False
assert Float('0.0').is_zero is True
# rationality properties
assert Float(1).is_rational is None
assert Float(1).is_irrational is None
assert sqrt(2).n(15).is_rational is None
assert sqrt(2).n(15).is_irrational is None
# do not automatically evalf
def teq(a):
assert (a.evalf() == a) is False
assert (a.evalf() != a) is True
assert (a == a.evalf()) is False
assert (a != a.evalf()) is True
teq(pi)
teq(2*pi)
teq(cos(0.1, evaluate=False))
# long integer
i = 12345678901234567890
assert same_and_same_prec(Float(12, ''), Float('12', ''))
assert same_and_same_prec(Float(Integer(i), ''), Float(i, ''))
assert same_and_same_prec(Float(i, ''), Float(str(i), 20))
assert same_and_same_prec(Float(str(i)), Float(i, ''))
assert same_and_same_prec(Float(i), Float(i, ''))
# inexact floats (repeating binary = denom not multiple of 2)
# cannot have precision greater than 15
assert Float(.125, 22) == .125
assert Float(2.0, 22) == 2
assert float(Float('.12500000000000001', '')) == .125
raises(ValueError, lambda: Float(.12500000000000001, ''))
# allow spaces
Float('123 456.123 456') == Float('123456.123456')
Integer('123 456') == Integer('123456')
Rational('123 456.123 456') == Rational('123456.123456')
assert Float(' .3e2') == Float('0.3e2')
# allow underscore
assert Float('1_23.4_56') == Float('123.456')
assert Float('1_23.4_5_6', 12) == Float('123.456', 12)
# ...but not in all cases (per Py 3.6)
raises(ValueError, lambda: Float('_1'))
raises(ValueError, lambda: Float('1_'))
raises(ValueError, lambda: Float('1_.'))
raises(ValueError, lambda: Float('1._'))
raises(ValueError, lambda: Float('1__2'))
# allow auto precision detection
assert Float('.1', '') == Float(.1, 1)
assert Float('.125', '') == Float(.125, 3)
assert Float('.100', '') == Float(.1, 3)
assert Float('2.0', '') == Float('2', 2)
raises(ValueError, lambda: Float("12.3d-4", ""))
raises(ValueError, lambda: Float(12.3, ""))
raises(ValueError, lambda: Float('.'))
raises(ValueError, lambda: Float('-.'))
zero = Float('0.0')
assert Float('-0') == zero
assert Float('.0') == zero
assert Float('-.0') == zero
assert Float('-0.0') == zero
assert Float(0.0) == zero
assert Float(0) == zero
assert Float(0, '') == Float('0', '')
assert Float(1) == Float(1.0)
assert Float(S.Zero) == zero
assert Float(S.One) == Float(1.0)
assert Float(decimal.Decimal('0.1'), 3) == Float('.1', 3)
assert Float(decimal.Decimal('nan')) == S.NaN
assert Float(decimal.Decimal('Infinity')) == S.Infinity
assert Float(decimal.Decimal('-Infinity')) == S.NegativeInfinity
assert '{0:.3f}'.format(Float(4.236622)) == '4.237'
assert '{0:.35f}'.format(Float(pi.n(40), 40)) == \
'3.14159265358979323846264338327950288'
assert Float(oo) == Float('+inf')
assert Float(-oo) == Float('-inf')
# unicode
assert Float(u'0.73908513321516064100000000') == \
Float('0.73908513321516064100000000')
assert Float(u'0.73908513321516064100000000', 28) == \
Float('0.73908513321516064100000000', 28)
# binary precision
# Decimal value 0.1 cannot be expressed precisely as a base 2 fraction
a = Float(S(1)/10, dps=15)
b = Float(S(1)/10, dps=16)
p = Float(S(1)/10, precision=53)
q = Float(S(1)/10, precision=54)
assert a._mpf_ == p._mpf_
assert not a._mpf_ == q._mpf_
assert not b._mpf_ == q._mpf_
# Precision specifying errors
raises(ValueError, lambda: Float("1.23", dps=3, precision=10))
raises(ValueError, lambda: Float("1.23", dps="", precision=10))
raises(ValueError, lambda: Float("1.23", dps=3, precision=""))
raises(ValueError, lambda: Float("1.23", dps="", precision=""))
# from NumberSymbol
assert same_and_same_prec(Float(pi, 32), pi.evalf(32))
assert same_and_same_prec(Float(Catalan), Catalan.evalf())
@conserve_mpmath_dps
def test_float_mpf():
import mpmath
mpmath.mp.dps = 100
mp_pi = mpmath.pi()
assert Float(mp_pi, 100) == Float(mp_pi._mpf_, 100) == pi.evalf(100)
mpmath.mp.dps = 15
assert Float(mp_pi, 100) == Float(mp_pi._mpf_, 100) == pi.evalf(100)
def test_Float_RealElement():
repi = RealField(dps=100)(pi.evalf(100))
# We still have to pass the precision because Float doesn't know what
# RealElement is, but make sure it keeps full precision from the result.
assert Float(repi, 100) == pi.evalf(100)
def test_Float_default_to_highprec_from_str():
s = str(pi.evalf(128))
assert same_and_same_prec(Float(s), Float(s, ''))
def test_Float_eval():
a = Float(3.2)
assert (a**2).is_Float
def test_Float_issue_2107():
a = Float(0.1, 10)
b = Float("0.1", 10)
assert a - a == 0
assert a + (-a) == 0
assert S.Zero + a - a == 0
assert S.Zero + a + (-a) == 0
assert b - b == 0
assert b + (-b) == 0
assert S.Zero + b - b == 0
assert S.Zero + b + (-b) == 0
def test_issue_14289():
from sympy.polys.numberfields import to_number_field
a = 1 - sqrt(2)
b = to_number_field(a)
assert b.as_expr() == a
assert b.minpoly(a).expand() == 0
def test_Float_from_tuple():
a = Float((0, '1L', 0, 1))
b = Float((0, '1', 0, 1))
assert a == b
def test_Infinity():
assert oo != 1
assert 1*oo == oo
assert 1 != oo
assert oo != -oo
assert oo != Symbol("x")**3
assert oo + 1 == oo
assert 2 + oo == oo
assert 3*oo + 2 == oo
assert S.Half**oo == 0
assert S.Half**(-oo) == oo
assert -oo*3 == -oo
assert oo + oo == oo
assert -oo + oo*(-5) == -oo
assert 1/oo == 0
assert 1/(-oo) == 0
assert 8/oo == 0
assert oo % 2 == nan
assert 2 % oo == nan
assert oo/oo == nan
assert oo/-oo == nan
assert -oo/oo == nan
assert -oo/-oo == nan
assert oo - oo == nan
assert oo - -oo == oo
assert -oo - oo == -oo
assert -oo - -oo == nan
assert oo + -oo == nan
assert -oo + oo == nan
assert oo + oo == oo
assert -oo + oo == nan
assert oo + -oo == nan
assert -oo + -oo == -oo
assert oo*oo == oo
assert -oo*oo == -oo
assert oo*-oo == -oo
assert -oo*-oo == oo
assert oo/0 == oo
assert -oo/0 == -oo
assert 0/oo == 0
assert 0/-oo == 0
assert oo*0 == nan
assert -oo*0 == nan
assert 0*oo == nan
assert 0*-oo == nan
assert oo + 0 == oo
assert -oo + 0 == -oo
assert 0 + oo == oo
assert 0 + -oo == -oo
assert oo - 0 == oo
assert -oo - 0 == -oo
assert 0 - oo == -oo
assert 0 - -oo == oo
assert oo/2 == oo
assert -oo/2 == -oo
assert oo/-2 == -oo
assert -oo/-2 == oo
assert oo*2 == oo
assert -oo*2 == -oo
assert oo*-2 == -oo
assert 2/oo == 0
assert 2/-oo == 0
assert -2/oo == 0
assert -2/-oo == 0
assert 2*oo == oo
assert 2*-oo == -oo
assert -2*oo == -oo
assert -2*-oo == oo
assert 2 + oo == oo
assert 2 - oo == -oo
assert -2 + oo == oo
assert -2 - oo == -oo
assert 2 + -oo == -oo
assert 2 - -oo == oo
assert -2 + -oo == -oo
assert -2 - -oo == oo
assert S(2) + oo == oo
assert S(2) - oo == -oo
assert oo/I == -oo*I
assert -oo/I == oo*I
assert oo*float(1) == Float('inf') and (oo*float(1)).is_Float
assert -oo*float(1) == Float('-inf') and (-oo*float(1)).is_Float
assert oo/float(1) == Float('inf') and (oo/float(1)).is_Float
assert -oo/float(1) == Float('-inf') and (-oo/float(1)).is_Float
assert oo*float(-1) == Float('-inf') and (oo*float(-1)).is_Float
assert -oo*float(-1) == Float('inf') and (-oo*float(-1)).is_Float
assert oo/float(-1) == Float('-inf') and (oo/float(-1)).is_Float
assert -oo/float(-1) == Float('inf') and (-oo/float(-1)).is_Float
assert oo + float(1) == Float('inf') and (oo + float(1)).is_Float
assert -oo + float(1) == Float('-inf') and (-oo + float(1)).is_Float
assert oo - float(1) == Float('inf') and (oo - float(1)).is_Float
assert -oo - float(1) == Float('-inf') and (-oo - float(1)).is_Float
assert float(1)*oo == Float('inf') and (float(1)*oo).is_Float
assert float(1)*-oo == Float('-inf') and (float(1)*-oo).is_Float
assert float(1)/oo == 0
assert float(1)/-oo == 0
assert float(-1)*oo == Float('-inf') and (float(-1)*oo).is_Float
assert float(-1)*-oo == Float('inf') and (float(-1)*-oo).is_Float
assert float(-1)/oo == 0
assert float(-1)/-oo == 0
assert float(1) + oo == Float('inf')
assert float(1) + -oo == Float('-inf')
assert float(1) - oo == Float('-inf')
assert float(1) - -oo == Float('inf')
assert Float('nan') == nan
assert nan*1.0 == nan
assert -1.0*nan == nan
assert nan*oo == nan
assert nan*-oo == nan
assert nan/oo == nan
assert nan/-oo == nan
assert nan + oo == nan
assert nan + -oo == nan
assert nan - oo == nan
assert nan - -oo == nan
assert -oo * S.Zero == nan
assert oo*nan == nan
assert -oo*nan == nan
assert oo/nan == nan
assert -oo/nan == nan
assert oo + nan == nan
assert -oo + nan == nan
assert oo - nan == nan
assert -oo - nan == nan
assert S.Zero * oo == nan
assert oo.is_Rational is False
assert isinstance(oo, Rational) is False
assert S.One/oo == 0
assert -S.One/oo == 0
assert S.One/-oo == 0
assert -S.One/-oo == 0
assert S.One*oo == oo
assert -S.One*oo == -oo
assert S.One*-oo == -oo
assert -S.One*-oo == oo
assert S.One/nan == nan
assert S.One - -oo == oo
assert S.One + nan == nan
assert S.One - nan == nan
assert nan - S.One == nan
assert nan/S.One == nan
assert -oo - S.One == -oo
def test_Infinity_2():
x = Symbol('x')
assert oo*x != oo
assert oo*(pi - 1) == oo
assert oo*(1 - pi) == -oo
assert (-oo)*x != -oo
assert (-oo)*(pi - 1) == -oo
assert (-oo)*(1 - pi) == oo
assert (-1)**S.NaN is S.NaN
assert oo - Float('inf') is S.NaN
assert oo + Float('-inf') is S.NaN
assert oo*0 is S.NaN
assert oo/Float('inf') is S.NaN
assert oo/Float('-inf') is S.NaN
assert oo**S.NaN is S.NaN
assert -oo + Float('inf') is S.NaN
assert -oo - Float('-inf') is S.NaN
assert -oo*S.NaN is S.NaN
assert -oo*0 is S.NaN
assert -oo/Float('inf') is S.NaN
assert -oo/Float('-inf') is S.NaN
assert -oo/S.NaN is S.NaN
assert abs(-oo) == oo
assert all((-oo)**i is S.NaN for i in (oo, -oo, S.NaN))
assert (-oo)**3 == -oo
assert (-oo)**2 == oo
assert abs(S.ComplexInfinity) == oo
def test_Mul_Infinity_Zero():
assert 0*Float('inf') == nan
assert 0*Float('-inf') == nan
assert 0*Float('inf') == nan
assert 0*Float('-inf') == nan
assert Float('inf')*0 == nan
assert Float('-inf')*0 == nan
assert Float('inf')*0 == nan
assert Float('-inf')*0 == nan
assert Float(0)*Float('inf') == nan
assert Float(0)*Float('-inf') == nan
assert Float(0)*Float('inf') == nan
assert Float(0)*Float('-inf') == nan
assert Float('inf')*Float(0) == nan
assert Float('-inf')*Float(0) == nan
assert Float('inf')*Float(0) == nan
assert Float('-inf')*Float(0) == nan
def test_Div_By_Zero():
assert 1/S(0) == zoo
assert 1/Float(0) == Float('inf')
assert 0/S(0) == nan
assert 0/Float(0) == nan
assert S(0)/0 == nan
assert Float(0)/0 == nan
assert -1/S(0) == zoo
assert -1/Float(0) == Float('-inf')
def test_Infinity_inequations():
assert oo > pi
assert not (oo < pi)
assert exp(-3) < oo
assert Float('+inf') > pi
assert not (Float('+inf') < pi)
assert exp(-3) < Float('+inf')
raises(TypeError, lambda: oo < I)
raises(TypeError, lambda: oo <= I)
raises(TypeError, lambda: oo > I)
raises(TypeError, lambda: oo >= I)
raises(TypeError, lambda: -oo < I)
raises(TypeError, lambda: -oo <= I)
raises(TypeError, lambda: -oo > I)
raises(TypeError, lambda: -oo >= I)
raises(TypeError, lambda: I < oo)
raises(TypeError, lambda: I <= oo)
raises(TypeError, lambda: I > oo)
raises(TypeError, lambda: I >= oo)
raises(TypeError, lambda: I < -oo)
raises(TypeError, lambda: I <= -oo)
raises(TypeError, lambda: I > -oo)
raises(TypeError, lambda: I >= -oo)
assert oo > -oo and oo >= -oo
assert (oo < -oo) == False and (oo <= -oo) == False
assert -oo < oo and -oo <= oo
assert (-oo > oo) == False and (-oo >= oo) == False
assert (oo < oo) == False # issue 7775
assert (oo > oo) == False
assert (-oo > -oo) == False and (-oo < -oo) == False
assert oo >= oo and oo <= oo and -oo >= -oo and -oo <= -oo
assert (-oo < -Float('inf')) == False
assert (oo > Float('inf')) == False
assert -oo >= -Float('inf')
assert oo <= Float('inf')
x = Symbol('x')
b = Symbol('b', finite=True, real=True)
assert (x < oo) == Lt(x, oo) # issue 7775
assert b < oo and b > -oo and b <= oo and b >= -oo
assert oo > b and oo >= b and (oo < b) == False and (oo <= b) == False
assert (-oo > b) == False and (-oo >= b) == False and -oo < b and -oo <= b
assert (oo < x) == Lt(oo, x) and (oo > x) == Gt(oo, x)
assert (oo <= x) == Le(oo, x) and (oo >= x) == Ge(oo, x)
assert (-oo < x) == Lt(-oo, x) and (-oo > x) == Gt(-oo, x)
assert (-oo <= x) == Le(-oo, x) and (-oo >= x) == Ge(-oo, x)
def test_NaN():
assert nan == nan
assert nan != 1
assert 1*nan == nan
assert 1 != nan
assert nan == -nan
assert oo != Symbol("x")**3
assert nan + 1 == nan
assert 2 + nan == nan
assert 3*nan + 2 == nan
assert -nan*3 == nan
assert nan + nan == nan
assert -nan + nan*(-5) == nan
assert 1/nan == nan
assert 1/(-nan) == nan
assert 8/nan == nan
raises(TypeError, lambda: nan > 0)
raises(TypeError, lambda: nan < 0)
raises(TypeError, lambda: nan >= 0)
raises(TypeError, lambda: nan <= 0)
raises(TypeError, lambda: 0 < nan)
raises(TypeError, lambda: 0 > nan)
raises(TypeError, lambda: 0 <= nan)
raises(TypeError, lambda: 0 >= nan)
assert S.One + nan == nan
assert S.One - nan == nan
assert S.One*nan == nan
assert S.One/nan == nan
assert nan - S.One == nan
assert nan*S.One == nan
assert nan + S.One == nan
assert nan/S.One == nan
assert nan**0 == 1 # as per IEEE 754
assert 1**nan == nan # IEEE 754 is not the best choice for symbolic work
# test Pow._eval_power's handling of NaN
assert Pow(nan, 0, evaluate=False)**2 == 1
def test_special_numbers():
assert isinstance(S.NaN, Number) is True
assert isinstance(S.Infinity, Number) is True
assert isinstance(S.NegativeInfinity, Number) is True
assert S.NaN.is_number is True
assert S.Infinity.is_number is True
assert S.NegativeInfinity.is_number is True
assert S.ComplexInfinity.is_number is True
assert isinstance(S.NaN, Rational) is False
assert isinstance(S.Infinity, Rational) is False
assert isinstance(S.NegativeInfinity, Rational) is False
assert S.NaN.is_rational is not True
assert S.Infinity.is_rational is not True
assert S.NegativeInfinity.is_rational is not True
def test_powers():
assert integer_nthroot(1, 2) == (1, True)
assert integer_nthroot(1, 5) == (1, True)
assert integer_nthroot(2, 1) == (2, True)
assert integer_nthroot(2, 2) == (1, False)
assert integer_nthroot(2, 5) == (1, False)
assert integer_nthroot(4, 2) == (2, True)
assert integer_nthroot(123**25, 25) == (123, True)
assert integer_nthroot(123**25 + 1, 25) == (123, False)
assert integer_nthroot(123**25 - 1, 25) == (122, False)
assert integer_nthroot(1, 1) == (1, True)
assert integer_nthroot(0, 1) == (0, True)
assert integer_nthroot(0, 3) == (0, True)
assert integer_nthroot(10000, 1) == (10000, True)
assert integer_nthroot(4, 2) == (2, True)
assert integer_nthroot(16, 2) == (4, True)
assert integer_nthroot(26, 2) == (5, False)
assert integer_nthroot(1234567**7, 7) == (1234567, True)
assert integer_nthroot(1234567**7 + 1, 7) == (1234567, False)
assert integer_nthroot(1234567**7 - 1, 7) == (1234566, False)
b = 25**1000
assert integer_nthroot(b, 1000) == (25, True)
assert integer_nthroot(b + 1, 1000) == (25, False)
assert integer_nthroot(b - 1, 1000) == (24, False)
c = 10**400
c2 = c**2
assert integer_nthroot(c2, 2) == (c, True)
assert integer_nthroot(c2 + 1, 2) == (c, False)
assert integer_nthroot(c2 - 1, 2) == (c - 1, False)
assert integer_nthroot(2, 10**10) == (1, False)
p, r = integer_nthroot(int(factorial(10000)), 100)
assert p % (10**10) == 5322420655
assert not r
# Test that this is fast
assert integer_nthroot(2, 10**10) == (1, False)
# output should be int if possible
assert type(integer_nthroot(2**61, 2)[0]) is int
def test_integer_nthroot_overflow():
assert integer_nthroot(10**(50*50), 50) == (10**50, True)
assert integer_nthroot(10**100000, 10000) == (10**10, True)
def test_integer_log():
raises(ValueError, lambda: integer_log(2, 1))
raises(ValueError, lambda: integer_log(0, 2))
raises(ValueError, lambda: integer_log(1.1, 2))
raises(ValueError, lambda: integer_log(1, 2.2))
assert integer_log(1, 2) == (0, True)
assert integer_log(1, 3) == (0, True)
assert integer_log(2, 3) == (0, False)
assert integer_log(3, 3) == (1, True)
assert integer_log(3*2, 3) == (1, False)
assert integer_log(3**2, 3) == (2, True)
assert integer_log(3*4, 3) == (2, False)
assert integer_log(3**3, 3) == (3, True)
assert integer_log(27, 5) == (2, False)
assert integer_log(2, 3) == (0, False)
assert integer_log(-4, -2) == (2, False)
assert integer_log(27, -3) == (3, False)
assert integer_log(-49, 7) == (0, False)
assert integer_log(-49, -7) == (2, False)
def test_isqrt():
from math import sqrt as _sqrt
limit = 17984395633462800708566937239551
assert int(_sqrt(limit)) == integer_nthroot(limit, 2)[0]
assert int(_sqrt(limit + 1)) != integer_nthroot(limit + 1, 2)[0]
assert isqrt(limit + 1) == integer_nthroot(limit + 1, 2)[0]
assert isqrt(limit + 1 + S.Half) == integer_nthroot(limit + 1, 2)[0]
def test_powers_Integer():
"""Test Integer._eval_power"""
# check infinity
assert S(1) ** S.Infinity == S.NaN
assert S(-1)** S.Infinity == S.NaN
assert S(2) ** S.Infinity == S.Infinity
assert S(-2)** S.Infinity == S.Infinity + S.Infinity * S.ImaginaryUnit
assert S(0) ** S.Infinity == 0
# check Nan
assert S(1) ** S.NaN == S.NaN
assert S(-1) ** S.NaN == S.NaN
# check for exact roots
assert S(-1) ** Rational(6, 5) == - (-1)**(S(1)/5)
assert sqrt(S(4)) == 2
assert sqrt(S(-4)) == I * 2
assert S(16) ** Rational(1, 4) == 2
assert S(-16) ** Rational(1, 4) == 2 * (-1)**Rational(1, 4)
assert S(9) ** Rational(3, 2) == 27
assert S(-9) ** Rational(3, 2) == -27*I
assert S(27) ** Rational(2, 3) == 9
assert S(-27) ** Rational(2, 3) == 9 * (S(-1) ** Rational(2, 3))
assert (-2) ** Rational(-2, 1) == Rational(1, 4)
# not exact roots
assert sqrt(-3) == I*sqrt(3)
assert (3) ** (S(3)/2) == 3 * sqrt(3)
assert (-3) ** (S(3)/2) == - 3 * sqrt(-3)
assert (-3) ** (S(5)/2) == 9 * I * sqrt(3)
assert (-3) ** (S(7)/2) == - I * 27 * sqrt(3)
assert (2) ** (S(3)/2) == 2 * sqrt(2)
assert (2) ** (S(-3)/2) == sqrt(2) / 4
assert (81) ** (S(2)/3) == 9 * (S(3) ** (S(2)/3))
assert (-81) ** (S(2)/3) == 9 * (S(-3) ** (S(2)/3))
assert (-3) ** Rational(-7, 3) == \
-(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == \
-(-1)**Rational(1, 3)*3**Rational(1, 3)/3
# join roots
assert sqrt(6) + sqrt(24) == 3*sqrt(6)
assert sqrt(2) * sqrt(3) == sqrt(6)
# separate symbols & constansts
x = Symbol("x")
assert sqrt(49 * x) == 7 * sqrt(x)
assert sqrt((3 - sqrt(pi)) ** 2) == 3 - sqrt(pi)
# check that it is fast for big numbers
assert (2**64 + 1) ** Rational(4, 3)
assert (2**64 + 1) ** Rational(17, 25)
# negative rational power and negative base
assert (-3) ** Rational(-7, 3) == \
-(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == \
-(-1)**Rational(1, 3)*3**Rational(1, 3)/3
assert (-2) ** Rational(-10, 3) == \
(-1)**Rational(2, 3)*2**Rational(2, 3)/16
assert abs(Pow(-2, Rational(-10, 3)).n() -
Pow(-2, Rational(-10, 3), evaluate=False).n()) < 1e-16
# negative base and rational power with some simplification
assert (-8) ** Rational(2, 5) == \
2*(-1)**Rational(2, 5)*2**Rational(1, 5)
assert (-4) ** Rational(9, 5) == \
-8*(-1)**Rational(4, 5)*2**Rational(3, 5)
assert S(1234).factors() == {617: 1, 2: 1}
assert Rational(2*3, 3*5*7).factors() == {2: 1, 5: -1, 7: -1}
# test that eval_power factors numbers bigger than
# the current limit in factor_trial_division (2**15)
from sympy import nextprime
n = nextprime(2**15)
assert sqrt(n**2) == n
assert sqrt(n**3) == n*sqrt(n)
assert sqrt(4*n) == 2*sqrt(n)
# check that factors of base with powers sharing gcd with power are removed
assert (2**4*3)**Rational(1, 6) == 2**Rational(2, 3)*3**Rational(1, 6)
assert (2**4*3)**Rational(5, 6) == 8*2**Rational(1, 3)*3**Rational(5, 6)
# check that bases sharing a gcd are exptracted
assert 2**Rational(1, 3)*3**Rational(1, 4)*6**Rational(1, 5) == \
2**Rational(8, 15)*3**Rational(9, 20)
assert sqrt(8)*24**Rational(1, 3)*6**Rational(1, 5) == \
4*2**Rational(7, 10)*3**Rational(8, 15)
assert sqrt(8)*(-24)**Rational(1, 3)*(-6)**Rational(1, 5) == \
4*(-3)**Rational(8, 15)*2**Rational(7, 10)
assert 2**Rational(1, 3)*2**Rational(8, 9) == 2*2**Rational(2, 9)
assert 2**Rational(2, 3)*6**Rational(1, 3) == 2*3**Rational(1, 3)
assert 2**Rational(2, 3)*6**Rational(8, 9) == \
2*2**Rational(5, 9)*3**Rational(8, 9)
assert (-2)**Rational(2, S(3))*(-4)**Rational(1, S(3)) == -2*2**Rational(1, 3)
assert 3*Pow(3, 2, evaluate=False) == 3**3
assert 3*Pow(3, -1/S(3), evaluate=False) == 3**(2/S(3))
assert (-2)**(1/S(3))*(-3)**(1/S(4))*(-5)**(5/S(6)) == \
-(-1)**Rational(5, 12)*2**Rational(1, 3)*3**Rational(1, 4) * \
5**Rational(5, 6)
assert Integer(-2)**Symbol('', even=True) == \
Integer(2)**Symbol('', even=True)
assert (-1)**Float(.5) == 1.0*I
def test_powers_Rational():
"""Test Rational._eval_power"""
# check infinity
assert Rational(1, 2) ** S.Infinity == 0
assert Rational(3, 2) ** S.Infinity == S.Infinity
assert Rational(-1, 2) ** S.Infinity == 0
assert Rational(-3, 2) ** S.Infinity == \
S.Infinity + S.Infinity * S.ImaginaryUnit
# check Nan
assert Rational(3, 4) ** S.NaN == S.NaN
assert Rational(-2, 3) ** S.NaN == S.NaN
# exact roots on numerator
assert sqrt(Rational(4, 3)) == 2 * sqrt(3) / 3
assert Rational(4, 3) ** Rational(3, 2) == 8 * sqrt(3) / 9
assert sqrt(Rational(-4, 3)) == I * 2 * sqrt(3) / 3
assert Rational(-4, 3) ** Rational(3, 2) == - I * 8 * sqrt(3) / 9
assert Rational(27, 2) ** Rational(1, 3) == 3 * (2 ** Rational(2, 3)) / 2
assert Rational(5**3, 8**3) ** Rational(4, 3) == Rational(5**4, 8**4)
# exact root on denominator
assert sqrt(Rational(1, 4)) == Rational(1, 2)
assert sqrt(Rational(1, -4)) == I * Rational(1, 2)
assert sqrt(Rational(3, 4)) == sqrt(3) / 2
assert sqrt(Rational(3, -4)) == I * sqrt(3) / 2
assert Rational(5, 27) ** Rational(1, 3) == (5 ** Rational(1, 3)) / 3
# not exact roots
assert sqrt(Rational(1, 2)) == sqrt(2) / 2
assert sqrt(Rational(-4, 7)) == I * sqrt(Rational(4, 7))
assert Rational(-3, 2)**Rational(-7, 3) == \
-4*(-1)**Rational(2, 3)*2**Rational(1, 3)*3**Rational(2, 3)/27
assert Rational(-3, 2)**Rational(-2, 3) == \
-(-1)**Rational(1, 3)*2**Rational(2, 3)*3**Rational(1, 3)/3
assert Rational(-3, 2)**Rational(-10, 3) == \
8*(-1)**Rational(2, 3)*2**Rational(1, 3)*3**Rational(2, 3)/81
assert abs(Pow(Rational(-2, 3), Rational(-7, 4)).n() -
Pow(Rational(-2, 3), Rational(-7, 4), evaluate=False).n()) < 1e-16
# negative integer power and negative rational base
assert Rational(-2, 3) ** Rational(-2, 1) == Rational(9, 4)
a = Rational(1, 10)
assert a**Float(a, 2) == Float(a, 2)**Float(a, 2)
assert Rational(-2, 3)**Symbol('', even=True) == \
Rational(2, 3)**Symbol('', even=True)
def test_powers_Float():
assert str((S('-1/10')**S('3/10')).n()) == str(Float(-.1)**(.3))
def test_abs1():
assert Rational(1, 6) != Rational(-1, 6)
assert abs(Rational(1, 6)) == abs(Rational(-1, 6))
def test_accept_int():
assert Float(4) == 4
def test_dont_accept_str():
assert Float("0.2") != "0.2"
assert not (Float("0.2") == "0.2")
def test_int():
a = Rational(5)
assert int(a) == 5
a = Rational(9, 10)
assert int(a) == int(-a) == 0
assert 1/(-1)**Rational(2, 3) == -(-1)**Rational(1, 3)
assert int(pi) == 3
assert int(E) == 2
assert int(GoldenRatio) == 1
assert int(TribonacciConstant) == 2
# issue 10368
a = S(32442016954)/78058255275
assert type(int(a)) is type(int(-a)) is int
def test_long():
a = Rational(5)
assert long(a) == 5
a = Rational(9, 10)
assert long(a) == long(-a) == 0
a = Integer(2**100)
assert long(a) == a
assert long(pi) == 3
assert long(E) == 2
assert long(GoldenRatio) == 1
assert long(TribonacciConstant) == 2
def test_real_bug():
x = Symbol("x")
assert str(2.0*x*x) in ["(2.0*x)*x", "2.0*x**2", "2.00000000000000*x**2"]
assert str(2.1*x*x) != "(2.0*x)*x"
def test_bug_sqrt():
assert ((sqrt(Rational(2)) + 1)*(sqrt(Rational(2)) - 1)).expand() == 1
def test_pi_Pi():
"Test that pi (instance) is imported, but Pi (class) is not"
from sympy import pi
with raises(ImportError):
from sympy import Pi
def test_no_len():
# there should be no len for numbers
raises(TypeError, lambda: len(Rational(2)))
raises(TypeError, lambda: len(Rational(2, 3)))
raises(TypeError, lambda: len(Integer(2)))
def test_issue_3321():
assert sqrt(Rational(1, 5)) == sqrt(Rational(1, 5))
assert 5 * sqrt(Rational(1, 5)) == sqrt(5)
def test_issue_3692():
assert ((-1)**Rational(1, 6)).expand(complex=True) == I/2 + sqrt(3)/2
assert ((-5)**Rational(1, 6)).expand(complex=True) == \
5**Rational(1, 6)*I/2 + 5**Rational(1, 6)*sqrt(3)/2
assert ((-64)**Rational(1, 6)).expand(complex=True) == I + sqrt(3)
def test_issue_3423():
x = Symbol("x")
assert sqrt(x - 1).as_base_exp() == (x - 1, S.Half)
assert sqrt(x - 1) != I*sqrt(1 - x)
def test_issue_3449():
x = Symbol("x")
assert sqrt(x - 1).subs(x, 5) == 2
def test_issue_13890():
x = Symbol("x")
e = (-x/4 - S(1)/12)**x - 1
f = simplify(e)
a = S(9)/5
assert abs(e.subs(x,a).evalf() - f.subs(x,a).evalf()) < 1e-15
def test_Integer_factors():
def F(i):
return Integer(i).factors()
assert F(1) == {}
assert F(2) == {2: 1}
assert F(3) == {3: 1}
assert F(4) == {2: 2}
assert F(5) == {5: 1}
assert F(6) == {2: 1, 3: 1}
assert F(7) == {7: 1}
assert F(8) == {2: 3}
assert F(9) == {3: 2}
assert F(10) == {2: 1, 5: 1}
assert F(11) == {11: 1}
assert F(12) == {2: 2, 3: 1}
assert F(13) == {13: 1}
assert F(14) == {2: 1, 7: 1}
assert F(15) == {3: 1, 5: 1}
assert F(16) == {2: 4}
assert F(17) == {17: 1}
assert F(18) == {2: 1, 3: 2}
assert F(19) == {19: 1}
assert F(20) == {2: 2, 5: 1}
assert F(21) == {3: 1, 7: 1}
assert F(22) == {2: 1, 11: 1}
assert F(23) == {23: 1}
assert F(24) == {2: 3, 3: 1}
assert F(25) == {5: 2}
assert F(26) == {2: 1, 13: 1}
assert F(27) == {3: 3}
assert F(28) == {2: 2, 7: 1}
assert F(29) == {29: 1}
assert F(30) == {2: 1, 3: 1, 5: 1}
assert F(31) == {31: 1}
assert F(32) == {2: 5}
assert F(33) == {3: 1, 11: 1}
assert F(34) == {2: 1, 17: 1}
assert F(35) == {5: 1, 7: 1}
assert F(36) == {2: 2, 3: 2}
assert F(37) == {37: 1}
assert F(38) == {2: 1, 19: 1}
assert F(39) == {3: 1, 13: 1}
assert F(40) == {2: 3, 5: 1}
assert F(41) == {41: 1}
assert F(42) == {2: 1, 3: 1, 7: 1}
assert F(43) == {43: 1}
assert F(44) == {2: 2, 11: 1}
assert F(45) == {3: 2, 5: 1}
assert F(46) == {2: 1, 23: 1}
assert F(47) == {47: 1}
assert F(48) == {2: 4, 3: 1}
assert F(49) == {7: 2}
assert F(50) == {2: 1, 5: 2}
assert F(51) == {3: 1, 17: 1}
def test_Rational_factors():
def F(p, q, visual=None):
return Rational(p, q).factors(visual=visual)
assert F(2, 3) == {2: 1, 3: -1}
assert F(2, 9) == {2: 1, 3: -2}
assert F(2, 15) == {2: 1, 3: -1, 5: -1}
assert F(6, 10) == {3: 1, 5: -1}
def test_issue_4107():
assert pi*(E + 10) + pi*(-E - 10) != 0
assert pi*(E + 10**10) + pi*(-E - 10**10) != 0
assert pi*(E + 10**20) + pi*(-E - 10**20) != 0
assert pi*(E + 10**80) + pi*(-E - 10**80) != 0
assert (pi*(E + 10) + pi*(-E - 10)).expand() == 0
assert (pi*(E + 10**10) + pi*(-E - 10**10)).expand() == 0
assert (pi*(E + 10**20) + pi*(-E - 10**20)).expand() == 0
assert (pi*(E + 10**80) + pi*(-E - 10**80)).expand() == 0
def test_IntegerInteger():
a = Integer(4)
b = Integer(a)
assert a == b
def test_Rational_gcd_lcm_cofactors():
assert Integer(4).gcd(2) == Integer(2)
assert Integer(4).lcm(2) == Integer(4)
assert Integer(4).gcd(Integer(2)) == Integer(2)
assert Integer(4).lcm(Integer(2)) == Integer(4)
a, b = 720**99911, 480**12342
assert Integer(a).lcm(b) == a*b/Integer(a).gcd(b)
assert Integer(4).gcd(3) == Integer(1)
assert Integer(4).lcm(3) == Integer(12)
assert Integer(4).gcd(Integer(3)) == Integer(1)
assert Integer(4).lcm(Integer(3)) == Integer(12)
assert Rational(4, 3).gcd(2) == Rational(2, 3)
assert Rational(4, 3).lcm(2) == Integer(4)
assert Rational(4, 3).gcd(Integer(2)) == Rational(2, 3)
assert Rational(4, 3).lcm(Integer(2)) == Integer(4)
assert Integer(4).gcd(Rational(2, 9)) == Rational(2, 9)
assert Integer(4).lcm(Rational(2, 9)) == Integer(4)
assert Rational(4, 3).gcd(Rational(2, 9)) == Rational(2, 9)
assert Rational(4, 3).lcm(Rational(2, 9)) == Rational(4, 3)
assert Rational(4, 5).gcd(Rational(2, 9)) == Rational(2, 45)
assert Rational(4, 5).lcm(Rational(2, 9)) == Integer(4)
assert Rational(5, 9).lcm(Rational(3, 7)) == Rational(Integer(5).lcm(3),Integer(9).gcd(7))
assert Integer(4).cofactors(2) == (Integer(2), Integer(2), Integer(1))
assert Integer(4).cofactors(Integer(2)) == \
(Integer(2), Integer(2), Integer(1))
assert Integer(4).gcd(Float(2.0)) == S.One
assert Integer(4).lcm(Float(2.0)) == Float(8.0)
assert Integer(4).cofactors(Float(2.0)) == (S.One, Integer(4), Float(2.0))
assert Rational(1, 2).gcd(Float(2.0)) == S.One
assert Rational(1, 2).lcm(Float(2.0)) == Float(1.0)
assert Rational(1, 2).cofactors(Float(2.0)) == \
(S.One, Rational(1, 2), Float(2.0))
def test_Float_gcd_lcm_cofactors():
assert Float(2.0).gcd(Integer(4)) == S.One
assert Float(2.0).lcm(Integer(4)) == Float(8.0)
assert Float(2.0).cofactors(Integer(4)) == (S.One, Float(2.0), Integer(4))
assert Float(2.0).gcd(Rational(1, 2)) == S.One
assert Float(2.0).lcm(Rational(1, 2)) == Float(1.0)
assert Float(2.0).cofactors(Rational(1, 2)) == \
(S.One, Float(2.0), Rational(1, 2))
def test_issue_4611():
assert abs(pi._evalf(50) - 3.14159265358979) < 1e-10
assert abs(E._evalf(50) - 2.71828182845905) < 1e-10
assert abs(Catalan._evalf(50) - 0.915965594177219) < 1e-10
assert abs(EulerGamma._evalf(50) - 0.577215664901533) < 1e-10
assert abs(GoldenRatio._evalf(50) - 1.61803398874989) < 1e-10
assert abs(TribonacciConstant._evalf(50) - 1.83928675521416) < 1e-10
x = Symbol("x")
assert (pi + x).evalf() == pi.evalf() + x
assert (E + x).evalf() == E.evalf() + x
assert (Catalan + x).evalf() == Catalan.evalf() + x
assert (EulerGamma + x).evalf() == EulerGamma.evalf() + x
assert (GoldenRatio + x).evalf() == GoldenRatio.evalf() + x
assert (TribonacciConstant + x).evalf() == TribonacciConstant.evalf() + x
@conserve_mpmath_dps
def test_conversion_to_mpmath():
assert mpmath.mpmathify(Integer(1)) == mpmath.mpf(1)
assert mpmath.mpmathify(Rational(1, 2)) == mpmath.mpf(0.5)
assert mpmath.mpmathify(Float('1.23', 15)) == mpmath.mpf('1.23')
assert mpmath.mpmathify(I) == mpmath.mpc(1j)
assert mpmath.mpmathify(1 + 2*I) == mpmath.mpc(1 + 2j)
assert mpmath.mpmathify(1.0 + 2*I) == mpmath.mpc(1 + 2j)
assert mpmath.mpmathify(1 + 2.0*I) == mpmath.mpc(1 + 2j)
assert mpmath.mpmathify(1.0 + 2.0*I) == mpmath.mpc(1 + 2j)
assert mpmath.mpmathify(Rational(1, 2) + Rational(1, 2)*I) == mpmath.mpc(0.5 + 0.5j)
assert mpmath.mpmathify(2*I) == mpmath.mpc(2j)
assert mpmath.mpmathify(2.0*I) == mpmath.mpc(2j)
assert mpmath.mpmathify(Rational(1, 2)*I) == mpmath.mpc(0.5j)
mpmath.mp.dps = 100
assert mpmath.mpmathify(pi.evalf(100) + pi.evalf(100)*I) == mpmath.pi + mpmath.pi*mpmath.j
assert mpmath.mpmathify(pi.evalf(100)*I) == mpmath.pi*mpmath.j
def test_relational():
# real
x = S(.1)
assert (x != cos) is True
assert (x == cos) is False
# rational
x = Rational(1, 3)
assert (x != cos) is True
assert (x == cos) is False
# integer defers to rational so these tests are omitted
# number symbol
x = pi
assert (x != cos) is True
assert (x == cos) is False
def test_Integer_as_index():
assert 'hello'[Integer(2):] == 'llo'
def test_Rational_int():
assert int( Rational(7, 5)) == 1
assert int( Rational(1, 2)) == 0
assert int(-Rational(1, 2)) == 0
assert int(-Rational(7, 5)) == -1
def test_zoo():
b = Symbol('b', finite=True)
nz = Symbol('nz', nonzero=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
im = Symbol('i', imaginary=True)
c = Symbol('c', complex=True)
pb = Symbol('pb', positive=True, finite=True)
nb = Symbol('nb', negative=True, finite=True)
imb = Symbol('ib', imaginary=True, finite=True)
for i in [I, S.Infinity, S.NegativeInfinity, S.Zero, S.One, S.Pi, S.Half, S(3), log(3),
b, nz, p, n, im, pb, nb, imb, c]:
if i.is_finite and (i.is_real or i.is_imaginary):
assert i + zoo is zoo
assert i - zoo is zoo
assert zoo + i is zoo
assert zoo - i is zoo
elif i.is_finite is not False:
assert (i + zoo).is_Add
assert (i - zoo).is_Add
assert (zoo + i).is_Add
assert (zoo - i).is_Add
else:
assert (i + zoo) is S.NaN
assert (i - zoo) is S.NaN
assert (zoo + i) is S.NaN
assert (zoo - i) is S.NaN
if fuzzy_not(i.is_zero) and (i.is_real or i.is_imaginary):
assert i*zoo is zoo
assert zoo*i is zoo
elif i.is_zero:
assert i*zoo is S.NaN
assert zoo*i is S.NaN
else:
assert (i*zoo).is_Mul
assert (zoo*i).is_Mul
if fuzzy_not((1/i).is_zero) and (i.is_real or i.is_imaginary):
assert zoo/i is zoo
elif (1/i).is_zero:
assert zoo/i is S.NaN
elif i.is_zero:
assert zoo/i is zoo
else:
assert (zoo/i).is_Mul
assert (I*oo).is_Mul # allow directed infinity
assert zoo + zoo is S.NaN
assert zoo * zoo is zoo
assert zoo - zoo is S.NaN
assert zoo/zoo is S.NaN
assert zoo**zoo is S.NaN
assert zoo**0 is S.One
assert zoo**2 is zoo
assert 1/zoo is S.Zero
assert Mul.flatten([S(-1), oo, S(0)]) == ([S.NaN], [], None)
def test_issue_4122():
x = Symbol('x', nonpositive=True)
assert (oo + x).is_Add
x = Symbol('x', finite=True)
assert (oo + x).is_Add # x could be imaginary
x = Symbol('x', nonnegative=True)
assert oo + x == oo
x = Symbol('x', finite=True, real=True)
assert oo + x == oo
# similarly for negative infinity
x = Symbol('x', nonnegative=True)
assert (-oo + x).is_Add
x = Symbol('x', finite=True)
assert (-oo + x).is_Add
x = Symbol('x', nonpositive=True)
assert -oo + x == -oo
x = Symbol('x', finite=True, real=True)
assert -oo + x == -oo
def test_GoldenRatio_expand():
assert GoldenRatio.expand(func=True) == S.Half + sqrt(5)/2
def test_TribonacciConstant_expand():
assert TribonacciConstant.expand(func=True) == \
(1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3
def test_as_content_primitive():
assert S.Zero.as_content_primitive() == (1, 0)
assert S.Half.as_content_primitive() == (S.Half, 1)
assert (-S.Half).as_content_primitive() == (S.Half, -1)
assert S(3).as_content_primitive() == (3, 1)
assert S(3.1).as_content_primitive() == (1, 3.1)
def test_hashing_sympy_integers():
# Test for issue 5072
assert set([Integer(3)]) == set([int(3)])
assert hash(Integer(4)) == hash(int(4))
def test_issue_4172():
assert int((E**100).round()) == \
26881171418161354484126255515800135873611119
assert int((pi**100).round()) == \
51878483143196131920862615246303013562686760680406
assert int((Rational(1)/EulerGamma**100).round()) == \
734833795660954410469466
@XFAIL
def test_mpmath_issues():
from mpmath.libmp.libmpf import _normalize
import mpmath.libmp as mlib
rnd = mlib.round_nearest
mpf = (0, long(0), -123, -1, 53, rnd) # nan
assert _normalize(mpf, 53) != (0, long(0), 0, 0)
mpf = (0, long(0), -456, -2, 53, rnd) # +inf
assert _normalize(mpf, 53) != (0, long(0), 0, 0)
mpf = (1, long(0), -789, -3, 53, rnd) # -inf
assert _normalize(mpf, 53) != (0, long(0), 0, 0)
from mpmath.libmp.libmpf import fnan
assert mlib.mpf_eq(fnan, fnan)
def test_Catalan_EulerGamma_prec():
n = GoldenRatio
f = Float(n.n(), 5)
assert f._mpf_ == (0, long(212079), -17, 18)
assert f._prec == 20
assert n._as_mpf_val(20) == f._mpf_
n = EulerGamma
f = Float(n.n(), 5)
assert f._mpf_ == (0, long(302627), -19, 19)
assert f._prec == 20
assert n._as_mpf_val(20) == f._mpf_
def test_Float_eq():
assert Float(.12, 3) != Float(.12, 4)
assert Float(.12, 3) == .12
assert 0.12 == Float(.12, 3)
assert Float('.12', 22) != .12
def test_int_NumberSymbols():
assert [int(i) for i in [pi, EulerGamma, E, GoldenRatio, Catalan]] == \
[3, 0, 2, 1, 0]
def test_issue_6640():
from mpmath.libmp.libmpf import finf, fninf
# fnan is not included because Float no longer returns fnan,
# but otherwise, the same sort of test could apply
assert Float(finf).is_zero is False
assert Float(fninf).is_zero is False
assert bool(Float(0)) is False
def test_issue_6349():
assert Float('23.e3', '')._prec == 10
assert Float('23e3', '')._prec == 20
assert Float('23000', '')._prec == 20
assert Float('-23000', '')._prec == 20
def test_mpf_norm():
assert mpf_norm((1, 0, 1, 0), 10) == mpf('0')._mpf_
assert Float._new((1, 0, 1, 0), 10)._mpf_ == mpf('0')._mpf_
def test_latex():
assert latex(pi) == r"\pi"
assert latex(E) == r"e"
assert latex(GoldenRatio) == r"\phi"
assert latex(TribonacciConstant) == r"\text{TribonacciConstant}"
assert latex(EulerGamma) == r"\gamma"
assert latex(oo) == r"\infty"
assert latex(-oo) == r"-\infty"
assert latex(zoo) == r"\tilde{\infty}"
assert latex(nan) == r"\text{NaN}"
assert latex(I) == r"i"
def test_issue_7742():
assert -oo % 1 == nan
def test_simplify_AlgebraicNumber():
A = AlgebraicNumber
e = 3**(S(1)/6)*(3 + (135 + 78*sqrt(3))**(S(2)/3))/(45 + 26*sqrt(3))**(S(1)/3)
assert simplify(A(e)) == A(12) # wester test_C20
e = (41 + 29*sqrt(2))**(S(1)/5)
assert simplify(A(e)) == A(1 + sqrt(2)) # wester test_C21
e = (3 + 4*I)**(Rational(3, 2))
assert simplify(A(e)) == A(2 + 11*I) # issue 4401
def test_Float_idempotence():
x = Float('1.23', '')
y = Float(x)
z = Float(x, 15)
assert same_and_same_prec(y, x)
assert not same_and_same_prec(z, x)
x = Float(10**20)
y = Float(x)
z = Float(x, 15)
assert same_and_same_prec(y, x)
assert not same_and_same_prec(z, x)
def test_comp():
# sqrt(2) = 1.414213 5623730950...
a = sqrt(2).n(7)
assert comp(a, 1.41421346) is False
assert comp(a, 1.41421347)
assert comp(a, 1.41421366)
assert comp(a, 1.41421367) is False
assert comp(sqrt(2).n(2), '1.4')
assert comp(sqrt(2).n(2), Float(1.4, 2), '')
raises(ValueError, lambda: comp(sqrt(2).n(2), 1.4, ''))
assert comp(sqrt(2).n(2), Float(1.4, 3), '') is False
def test_issue_9491():
assert oo**zoo == nan
def test_issue_10063():
assert 2**Float(3) == Float(8)
def test_issue_10020():
assert oo**I is S.NaN
assert oo**(1 + I) is S.ComplexInfinity
assert oo**(-1 + I) is S.Zero
assert (-oo)**I is S.NaN
assert (-oo)**(-1 + I) is S.Zero
assert oo**t == Pow(oo, t, evaluate=False)
assert (-oo)**t == Pow(-oo, t, evaluate=False)
def test_invert_numbers():
assert S(2).invert(5) == 3
assert S(2).invert(S(5)/2) == S.Half
assert S(2).invert(5.) == 0.5
assert S(2).invert(S(5)) == 3
assert S(2.).invert(5) == 0.5
assert S(sqrt(2)).invert(5) == 1/sqrt(2)
assert S(sqrt(2)).invert(sqrt(3)) == 1/sqrt(2)
def test_mod_inverse():
assert mod_inverse(3, 11) == 4
assert mod_inverse(5, 11) == 9
assert mod_inverse(21124921, 521512) == 7713
assert mod_inverse(124215421, 5125) == 2981
assert mod_inverse(214, 12515) == 1579
assert mod_inverse(5823991, 3299) == 1442
assert mod_inverse(123, 44) == 39
assert mod_inverse(2, 5) == 3
assert mod_inverse(-2, 5) == 2
assert mod_inverse(2, -5) == -2
assert mod_inverse(-2, -5) == -3
assert mod_inverse(-3, -7) == -5
x = Symbol('x')
assert S(2).invert(x) == S.Half
raises(TypeError, lambda: mod_inverse(2, x))
raises(ValueError, lambda: mod_inverse(2, S.Half))
raises(ValueError, lambda: mod_inverse(2, cos(1)**2 + sin(1)**2))
def test_golden_ratio_rewrite_as_sqrt():
assert GoldenRatio.rewrite(sqrt) == S.Half + sqrt(5)*S.Half
def test_tribonacci_constant_rewrite_as_sqrt():
assert TribonacciConstant.rewrite(sqrt) == \
(1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3
def test_comparisons_with_unknown_type():
class Foo(object):
"""
Class that is unaware of Basic, and relies on both classes returning
the NotImplemented singleton for equivalence to evaluate to False.
"""
ni, nf, nr = Integer(3), Float(1.0), Rational(1, 3)
foo = Foo()
for n in ni, nf, nr, oo, -oo, zoo, nan:
assert n != foo
assert foo != n
assert not n == foo
assert not foo == n
raises(TypeError, lambda: n < foo)
raises(TypeError, lambda: foo > n)
raises(TypeError, lambda: n > foo)
raises(TypeError, lambda: foo < n)
raises(TypeError, lambda: n <= foo)
raises(TypeError, lambda: foo >= n)
raises(TypeError, lambda: n >= foo)
raises(TypeError, lambda: foo <= n)
class Bar(object):
"""
Class that considers itself equal to any instance of Number except
infinities and nans, and relies on sympy types returning the
NotImplemented singleton for symmetric equality relations.
"""
def __eq__(self, other):
if other in (oo, -oo, zoo, nan):
return False
if isinstance(other, Number):
return True
return NotImplemented
def __ne__(self, other):
return not self == other
bar = Bar()
for n in ni, nf, nr:
assert n == bar
assert bar == n
assert not n != bar
assert not bar != n
for n in oo, -oo, zoo, nan:
assert n != bar
assert bar != n
assert not n == bar
assert not bar == n
for n in ni, nf, nr, oo, -oo, zoo, nan:
raises(TypeError, lambda: n < bar)
raises(TypeError, lambda: bar > n)
raises(TypeError, lambda: n > bar)
raises(TypeError, lambda: bar < n)
raises(TypeError, lambda: n <= bar)
raises(TypeError, lambda: bar >= n)
raises(TypeError, lambda: n >= bar)
raises(TypeError, lambda: bar <= n)
def test_NumberSymbol_comparison():
rpi = Rational('905502432259640373/288230376151711744')
fpi = Float(float(pi))
assert (rpi == pi) == (pi == rpi)
assert (rpi != pi) == (pi != rpi)
assert (rpi < pi) == (pi > rpi)
assert (rpi <= pi) == (pi >= rpi)
assert (rpi > pi) == (pi < rpi)
assert (rpi >= pi) == (pi <= rpi)
assert (fpi == pi) == (pi == fpi)
assert (fpi != pi) == (pi != fpi)
assert (fpi < pi) == (pi > fpi)
assert (fpi <= pi) == (pi >= fpi)
assert (fpi > pi) == (pi < fpi)
assert (fpi >= pi) == (pi <= fpi)
def test_Integer_precision():
# Make sure Integer inputs for keyword args work
assert Float('1.0', dps=Integer(15))._prec == 53
assert Float('1.0', precision=Integer(15))._prec == 15
assert type(Float('1.0', precision=Integer(15))._prec) == int
assert sympify(srepr(Float('1.0', precision=15))) == Float('1.0', precision=15)
def test_numpy_to_float():
from sympy.utilities.pytest import skip
from sympy.external import import_module
np = import_module('numpy')
if not np:
skip('numpy not installed. Abort numpy tests.')
def check_prec_and_relerr(npval, ratval):
prec = np.finfo(npval).nmant + 1
x = Float(npval)
assert x._prec == prec
y = Float(ratval, precision=prec)
assert abs((x - y)/y) < 2**(-(prec + 1))
check_prec_and_relerr(np.float16(2.0/3), S(2)/3)
check_prec_and_relerr(np.float32(2.0/3), S(2)/3)
check_prec_and_relerr(np.float64(2.0/3), S(2)/3)
# extended precision, on some arch/compilers:
x = np.longdouble(2)/3
check_prec_and_relerr(x, S(2)/3)
y = Float(x, precision=10)
assert same_and_same_prec(y, Float(S(2)/3, precision=10))
raises(TypeError, lambda: Float(np.complex64(1+2j)))
raises(TypeError, lambda: Float(np.complex128(1+2j)))
def test_Integer_ceiling_floor():
a = Integer(4)
assert(a.floor() == a)
assert(a.ceiling() == a)
def test_ComplexInfinity():
assert((zoo).floor() == zoo)
assert((zoo).ceiling() == zoo)
assert(zoo**zoo == S.NaN)
def test_Infinity_floor_ceiling_power():
assert((oo).floor() == oo)
assert((oo).ceiling() == oo)
assert((oo)**S.NaN == S.NaN)
assert((oo)**zoo == S.NaN)
def test_One_power():
assert((S.One)**12 == S.One)
assert((S.NegativeOne)**S.NaN == S.NaN)
def test_NegativeInfinity():
assert((-oo).floor() == -oo)
assert((-oo).ceiling() == -oo)
assert((-oo)**11 == -oo)
assert((-oo)**12 == oo)
def test_issue_6133():
raises(TypeError, lambda: (-oo < None))
raises(TypeError, lambda: (S(-2) < None))
raises(TypeError, lambda: (oo < None))
raises(TypeError, lambda: (oo > None))
raises(TypeError, lambda: (S(2) < None))
|
5ba5e0553501189ccfd7bfd4e535246fcc58993080d65e929050b6d5ab66abc4 | from sympy.utilities.pytest import XFAIL, raises, warns_deprecated_sympy
from sympy import (S, Symbol, symbols, nan, oo, I, pi, Float, And, Or,
Not, Implies, Xor, zoo, sqrt, Rational, simplify, Function,
log, cos, sin, Add, floor, ceiling)
from sympy.core.compatibility import range
from sympy.core.relational import (Relational, Equality, Unequality,
GreaterThan, LessThan, StrictGreaterThan,
StrictLessThan, Rel, Eq, Lt, Le,
Gt, Ge, Ne)
from sympy.sets.sets import Interval, FiniteSet
from itertools import combinations
x, y, z, t = symbols('x,y,z,t')
def test_rel_ne():
assert Relational(x, y, '!=') == Ne(x, y)
# issue 6116
p = Symbol('p', positive=True)
assert Ne(p, 0) is S.true
def test_rel_subs():
e = Relational(x, y, '==')
e = e.subs(x, z)
assert isinstance(e, Equality)
assert e.lhs == z
assert e.rhs == y
e = Relational(x, y, '>=')
e = e.subs(x, z)
assert isinstance(e, GreaterThan)
assert e.lhs == z
assert e.rhs == y
e = Relational(x, y, '<=')
e = e.subs(x, z)
assert isinstance(e, LessThan)
assert e.lhs == z
assert e.rhs == y
e = Relational(x, y, '>')
e = e.subs(x, z)
assert isinstance(e, StrictGreaterThan)
assert e.lhs == z
assert e.rhs == y
e = Relational(x, y, '<')
e = e.subs(x, z)
assert isinstance(e, StrictLessThan)
assert e.lhs == z
assert e.rhs == y
e = Eq(x, 0)
assert e.subs(x, 0) is S.true
assert e.subs(x, 1) is S.false
def test_wrappers():
e = x + x**2
res = Relational(y, e, '==')
assert Rel(y, x + x**2, '==') == res
assert Eq(y, x + x**2) == res
res = Relational(y, e, '<')
assert Lt(y, x + x**2) == res
res = Relational(y, e, '<=')
assert Le(y, x + x**2) == res
res = Relational(y, e, '>')
assert Gt(y, x + x**2) == res
res = Relational(y, e, '>=')
assert Ge(y, x + x**2) == res
res = Relational(y, e, '!=')
assert Ne(y, x + x**2) == res
def test_Eq():
assert Eq(x, x) # issue 5719
with warns_deprecated_sympy():
assert Eq(x) == Eq(x, 0)
# issue 6116
p = Symbol('p', positive=True)
assert Eq(p, 0) is S.false
# issue 13348
assert Eq(True, 1) is S.false
def test_rel_Infinity():
# NOTE: All of these are actually handled by sympy.core.Number, and do
# not create Relational objects.
assert (oo > oo) is S.false
assert (oo > -oo) is S.true
assert (oo > 1) is S.true
assert (oo < oo) is S.false
assert (oo < -oo) is S.false
assert (oo < 1) is S.false
assert (oo >= oo) is S.true
assert (oo >= -oo) is S.true
assert (oo >= 1) is S.true
assert (oo <= oo) is S.true
assert (oo <= -oo) is S.false
assert (oo <= 1) is S.false
assert (-oo > oo) is S.false
assert (-oo > -oo) is S.false
assert (-oo > 1) is S.false
assert (-oo < oo) is S.true
assert (-oo < -oo) is S.false
assert (-oo < 1) is S.true
assert (-oo >= oo) is S.false
assert (-oo >= -oo) is S.true
assert (-oo >= 1) is S.false
assert (-oo <= oo) is S.true
assert (-oo <= -oo) is S.true
assert (-oo <= 1) is S.true
def test_bool():
assert Eq(0, 0) is S.true
assert Eq(1, 0) is S.false
assert Ne(0, 0) is S.false
assert Ne(1, 0) is S.true
assert Lt(0, 1) is S.true
assert Lt(1, 0) is S.false
assert Le(0, 1) is S.true
assert Le(1, 0) is S.false
assert Le(0, 0) is S.true
assert Gt(1, 0) is S.true
assert Gt(0, 1) is S.false
assert Ge(1, 0) is S.true
assert Ge(0, 1) is S.false
assert Ge(1, 1) is S.true
assert Eq(I, 2) is S.false
assert Ne(I, 2) is S.true
raises(TypeError, lambda: Gt(I, 2))
raises(TypeError, lambda: Ge(I, 2))
raises(TypeError, lambda: Lt(I, 2))
raises(TypeError, lambda: Le(I, 2))
a = Float('.000000000000000000001', '')
b = Float('.0000000000000000000001', '')
assert Eq(pi + a, pi + b) is S.false
def test_rich_cmp():
assert (x < y) == Lt(x, y)
assert (x <= y) == Le(x, y)
assert (x > y) == Gt(x, y)
assert (x >= y) == Ge(x, y)
def test_doit():
from sympy import Symbol
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
np = Symbol('np', nonpositive=True)
nn = Symbol('nn', nonnegative=True)
assert Gt(p, 0).doit() is S.true
assert Gt(p, 1).doit() == Gt(p, 1)
assert Ge(p, 0).doit() is S.true
assert Le(p, 0).doit() is S.false
assert Lt(n, 0).doit() is S.true
assert Le(np, 0).doit() is S.true
assert Gt(nn, 0).doit() == Gt(nn, 0)
assert Lt(nn, 0).doit() is S.false
assert Eq(x, 0).doit() == Eq(x, 0)
def test_new_relational():
x = Symbol('x')
assert Eq(x, 0) == Relational(x, 0) # None ==> Equality
assert Eq(x, 0) == Relational(x, 0, '==')
assert Eq(x, 0) == Relational(x, 0, 'eq')
assert Eq(x, 0) == Equality(x, 0)
assert Eq(x, 0) != Relational(x, 1) # None ==> Equality
assert Eq(x, 0) != Relational(x, 1, '==')
assert Eq(x, 0) != Relational(x, 1, 'eq')
assert Eq(x, 0) != Equality(x, 1)
assert Eq(x, -1) == Relational(x, -1) # None ==> Equality
assert Eq(x, -1) == Relational(x, -1, '==')
assert Eq(x, -1) == Relational(x, -1, 'eq')
assert Eq(x, -1) == Equality(x, -1)
assert Eq(x, -1) != Relational(x, 1) # None ==> Equality
assert Eq(x, -1) != Relational(x, 1, '==')
assert Eq(x, -1) != Relational(x, 1, 'eq')
assert Eq(x, -1) != Equality(x, 1)
assert Ne(x, 0) == Relational(x, 0, '!=')
assert Ne(x, 0) == Relational(x, 0, '<>')
assert Ne(x, 0) == Relational(x, 0, 'ne')
assert Ne(x, 0) == Unequality(x, 0)
assert Ne(x, 0) != Relational(x, 1, '!=')
assert Ne(x, 0) != Relational(x, 1, '<>')
assert Ne(x, 0) != Relational(x, 1, 'ne')
assert Ne(x, 0) != Unequality(x, 1)
assert Ge(x, 0) == Relational(x, 0, '>=')
assert Ge(x, 0) == Relational(x, 0, 'ge')
assert Ge(x, 0) == GreaterThan(x, 0)
assert Ge(x, 1) != Relational(x, 0, '>=')
assert Ge(x, 1) != Relational(x, 0, 'ge')
assert Ge(x, 1) != GreaterThan(x, 0)
assert (x >= 1) == Relational(x, 1, '>=')
assert (x >= 1) == Relational(x, 1, 'ge')
assert (x >= 1) == GreaterThan(x, 1)
assert (x >= 0) != Relational(x, 1, '>=')
assert (x >= 0) != Relational(x, 1, 'ge')
assert (x >= 0) != GreaterThan(x, 1)
assert Le(x, 0) == Relational(x, 0, '<=')
assert Le(x, 0) == Relational(x, 0, 'le')
assert Le(x, 0) == LessThan(x, 0)
assert Le(x, 1) != Relational(x, 0, '<=')
assert Le(x, 1) != Relational(x, 0, 'le')
assert Le(x, 1) != LessThan(x, 0)
assert (x <= 1) == Relational(x, 1, '<=')
assert (x <= 1) == Relational(x, 1, 'le')
assert (x <= 1) == LessThan(x, 1)
assert (x <= 0) != Relational(x, 1, '<=')
assert (x <= 0) != Relational(x, 1, 'le')
assert (x <= 0) != LessThan(x, 1)
assert Gt(x, 0) == Relational(x, 0, '>')
assert Gt(x, 0) == Relational(x, 0, 'gt')
assert Gt(x, 0) == StrictGreaterThan(x, 0)
assert Gt(x, 1) != Relational(x, 0, '>')
assert Gt(x, 1) != Relational(x, 0, 'gt')
assert Gt(x, 1) != StrictGreaterThan(x, 0)
assert (x > 1) == Relational(x, 1, '>')
assert (x > 1) == Relational(x, 1, 'gt')
assert (x > 1) == StrictGreaterThan(x, 1)
assert (x > 0) != Relational(x, 1, '>')
assert (x > 0) != Relational(x, 1, 'gt')
assert (x > 0) != StrictGreaterThan(x, 1)
assert Lt(x, 0) == Relational(x, 0, '<')
assert Lt(x, 0) == Relational(x, 0, 'lt')
assert Lt(x, 0) == StrictLessThan(x, 0)
assert Lt(x, 1) != Relational(x, 0, '<')
assert Lt(x, 1) != Relational(x, 0, 'lt')
assert Lt(x, 1) != StrictLessThan(x, 0)
assert (x < 1) == Relational(x, 1, '<')
assert (x < 1) == Relational(x, 1, 'lt')
assert (x < 1) == StrictLessThan(x, 1)
assert (x < 0) != Relational(x, 1, '<')
assert (x < 0) != Relational(x, 1, 'lt')
assert (x < 0) != StrictLessThan(x, 1)
# finally, some fuzz testing
from random import randint
from sympy.core.compatibility import unichr
for i in range(100):
while 1:
strtype, length = (unichr, 65535) if randint(0, 1) else (chr, 255)
relation_type = strtype(randint(0, length))
if randint(0, 1):
relation_type += strtype(randint(0, length))
if relation_type not in ('==', 'eq', '!=', '<>', 'ne', '>=', 'ge',
'<=', 'le', '>', 'gt', '<', 'lt', ':=',
'+=', '-=', '*=', '/=', '%='):
break
raises(ValueError, lambda: Relational(x, 1, relation_type))
assert all(Relational(x, 0, op).rel_op == '==' for op in ('eq', '=='))
assert all(Relational(x, 0, op).rel_op == '!='
for op in ('ne', '<>', '!='))
assert all(Relational(x, 0, op).rel_op == '>' for op in ('gt', '>'))
assert all(Relational(x, 0, op).rel_op == '<' for op in ('lt', '<'))
assert all(Relational(x, 0, op).rel_op == '>=' for op in ('ge', '>='))
assert all(Relational(x, 0, op).rel_op == '<=' for op in ('le', '<='))
def test_relational_bool_output():
# https://github.com/sympy/sympy/issues/5931
raises(TypeError, lambda: bool(x > 3))
raises(TypeError, lambda: bool(x >= 3))
raises(TypeError, lambda: bool(x < 3))
raises(TypeError, lambda: bool(x <= 3))
raises(TypeError, lambda: bool(Eq(x, 3)))
raises(TypeError, lambda: bool(Ne(x, 3)))
def test_relational_logic_symbols():
# See issue 6204
assert (x < y) & (z < t) == And(x < y, z < t)
assert (x < y) | (z < t) == Or(x < y, z < t)
assert ~(x < y) == Not(x < y)
assert (x < y) >> (z < t) == Implies(x < y, z < t)
assert (x < y) << (z < t) == Implies(z < t, x < y)
assert (x < y) ^ (z < t) == Xor(x < y, z < t)
assert isinstance((x < y) & (z < t), And)
assert isinstance((x < y) | (z < t), Or)
assert isinstance(~(x < y), GreaterThan)
assert isinstance((x < y) >> (z < t), Implies)
assert isinstance((x < y) << (z < t), Implies)
assert isinstance((x < y) ^ (z < t), (Or, Xor))
def test_univariate_relational_as_set():
assert (x > 0).as_set() == Interval(0, oo, True, True)
assert (x >= 0).as_set() == Interval(0, oo)
assert (x < 0).as_set() == Interval(-oo, 0, True, True)
assert (x <= 0).as_set() == Interval(-oo, 0)
assert Eq(x, 0).as_set() == FiniteSet(0)
assert Ne(x, 0).as_set() == Interval(-oo, 0, True, True) + \
Interval(0, oo, True, True)
assert (x**2 >= 4).as_set() == Interval(-oo, -2) + Interval(2, oo)
@XFAIL
def test_multivariate_relational_as_set():
assert (x*y >= 0).as_set() == Interval(0, oo)*Interval(0, oo) + \
Interval(-oo, 0)*Interval(-oo, 0)
def test_Not():
assert Not(Equality(x, y)) == Unequality(x, y)
assert Not(Unequality(x, y)) == Equality(x, y)
assert Not(StrictGreaterThan(x, y)) == LessThan(x, y)
assert Not(StrictLessThan(x, y)) == GreaterThan(x, y)
assert Not(GreaterThan(x, y)) == StrictLessThan(x, y)
assert Not(LessThan(x, y)) == StrictGreaterThan(x, y)
def test_evaluate():
assert str(Eq(x, x, evaluate=False)) == 'Eq(x, x)'
assert Eq(x, x, evaluate=False).doit() == S.true
assert str(Ne(x, x, evaluate=False)) == 'Ne(x, x)'
assert Ne(x, x, evaluate=False).doit() == S.false
assert str(Ge(x, x, evaluate=False)) == 'x >= x'
assert str(Le(x, x, evaluate=False)) == 'x <= x'
assert str(Gt(x, x, evaluate=False)) == 'x > x'
assert str(Lt(x, x, evaluate=False)) == 'x < x'
def assert_all_ineq_raise_TypeError(a, b):
raises(TypeError, lambda: a > b)
raises(TypeError, lambda: a >= b)
raises(TypeError, lambda: a < b)
raises(TypeError, lambda: a <= b)
raises(TypeError, lambda: b > a)
raises(TypeError, lambda: b >= a)
raises(TypeError, lambda: b < a)
raises(TypeError, lambda: b <= a)
def assert_all_ineq_give_class_Inequality(a, b):
"""All inequality operations on `a` and `b` result in class Inequality."""
from sympy.core.relational import _Inequality as Inequality
assert isinstance(a > b, Inequality)
assert isinstance(a >= b, Inequality)
assert isinstance(a < b, Inequality)
assert isinstance(a <= b, Inequality)
assert isinstance(b > a, Inequality)
assert isinstance(b >= a, Inequality)
assert isinstance(b < a, Inequality)
assert isinstance(b <= a, Inequality)
def test_imaginary_compare_raises_TypeError():
# See issue #5724
assert_all_ineq_raise_TypeError(I, x)
def test_complex_compare_not_real():
# two cases which are not real
y = Symbol('y', imaginary=True)
z = Symbol('z', complex=True, real=False)
for w in (y, z):
assert_all_ineq_raise_TypeError(2, w)
# some cases which should remain un-evaluated
t = Symbol('t')
x = Symbol('x', real=True)
z = Symbol('z', complex=True)
for w in (x, z, t):
assert_all_ineq_give_class_Inequality(2, w)
def test_imaginary_and_inf_compare_raises_TypeError():
# See pull request #7835
y = Symbol('y', imaginary=True)
assert_all_ineq_raise_TypeError(oo, y)
assert_all_ineq_raise_TypeError(-oo, y)
def test_complex_pure_imag_not_ordered():
raises(TypeError, lambda: 2*I < 3*I)
# more generally
x = Symbol('x', real=True, nonzero=True)
y = Symbol('y', imaginary=True)
z = Symbol('z', complex=True)
assert_all_ineq_raise_TypeError(I, y)
t = I*x # an imaginary number, should raise errors
assert_all_ineq_raise_TypeError(2, t)
t = -I*y # a real number, so no errors
assert_all_ineq_give_class_Inequality(2, t)
t = I*z # unknown, should be unevaluated
assert_all_ineq_give_class_Inequality(2, t)
def test_x_minus_y_not_same_as_x_lt_y():
"""
A consequence of pull request #7792 is that `x - y < 0` and `x < y`
are not synonymous.
"""
x = I + 2
y = I + 3
raises(TypeError, lambda: x < y)
assert x - y < 0
ineq = Lt(x, y, evaluate=False)
raises(TypeError, lambda: ineq.doit())
assert ineq.lhs - ineq.rhs < 0
t = Symbol('t', imaginary=True)
x = 2 + t
y = 3 + t
ineq = Lt(x, y, evaluate=False)
raises(TypeError, lambda: ineq.doit())
assert ineq.lhs - ineq.rhs < 0
# this one should give error either way
x = I + 2
y = 2*I + 3
raises(TypeError, lambda: x < y)
raises(TypeError, lambda: x - y < 0)
def test_nan_equality_exceptions():
# See issue #7774
import random
assert Equality(nan, nan) is S.false
assert Unequality(nan, nan) is S.true
# See issue #7773
A = (x, S(0), S(1)/3, pi, oo, -oo)
assert Equality(nan, random.choice(A)) is S.false
assert Equality(random.choice(A), nan) is S.false
assert Unequality(nan, random.choice(A)) is S.true
assert Unequality(random.choice(A), nan) is S.true
def test_nan_inequality_raise_errors():
# See discussion in pull request #7776. We test inequalities with
# a set including examples of various classes.
for q in (x, S(0), S(10), S(1)/3, pi, S(1.3), oo, -oo, nan):
assert_all_ineq_raise_TypeError(q, nan)
def test_nan_complex_inequalities():
# Comparisons of NaN with non-real raise errors, we're not too
# fussy whether its the NaN error or complex error.
for r in (I, zoo, Symbol('z', imaginary=True)):
assert_all_ineq_raise_TypeError(r, nan)
def test_complex_infinity_inequalities():
raises(TypeError, lambda: zoo > 0)
raises(TypeError, lambda: zoo >= 0)
raises(TypeError, lambda: zoo < 0)
raises(TypeError, lambda: zoo <= 0)
def test_inequalities_symbol_name_same():
"""Using the operator and functional forms should give same results."""
# We test all combinations from a set
# FIXME: could replace with random selection after test passes
A = (x, y, S(0), S(1)/3, pi, oo, -oo)
for a in A:
for b in A:
assert Gt(a, b) == (a > b)
assert Lt(a, b) == (a < b)
assert Ge(a, b) == (a >= b)
assert Le(a, b) == (a <= b)
for b in (y, S(0), S(1)/3, pi, oo, -oo):
assert Gt(x, b, evaluate=False) == (x > b)
assert Lt(x, b, evaluate=False) == (x < b)
assert Ge(x, b, evaluate=False) == (x >= b)
assert Le(x, b, evaluate=False) == (x <= b)
for b in (y, S(0), S(1)/3, pi, oo, -oo):
assert Gt(b, x, evaluate=False) == (b > x)
assert Lt(b, x, evaluate=False) == (b < x)
assert Ge(b, x, evaluate=False) == (b >= x)
assert Le(b, x, evaluate=False) == (b <= x)
def test_inequalities_symbol_name_same_complex():
"""Using the operator and functional forms should give same results.
With complex non-real numbers, both should raise errors.
"""
# FIXME: could replace with random selection after test passes
for a in (x, S(0), S(1)/3, pi, oo):
raises(TypeError, lambda: Gt(a, I))
raises(TypeError, lambda: a > I)
raises(TypeError, lambda: Lt(a, I))
raises(TypeError, lambda: a < I)
raises(TypeError, lambda: Ge(a, I))
raises(TypeError, lambda: a >= I)
raises(TypeError, lambda: Le(a, I))
raises(TypeError, lambda: a <= I)
def test_inequalities_cant_sympify_other():
# see issue 7833
from operator import gt, lt, ge, le
bar = "foo"
for a in (x, S(0), S(1)/3, pi, I, zoo, oo, -oo, nan):
for op in (lt, gt, le, ge):
raises(TypeError, lambda: op(a, bar))
def test_ineq_avoid_wild_symbol_flip():
# see issue #7951, we try to avoid this internally, e.g., by using
# __lt__ instead of "<".
from sympy.core.symbol import Wild
p = symbols('p', cls=Wild)
# x > p might flip, but Gt should not:
assert Gt(x, p) == Gt(x, p, evaluate=False)
# Previously failed as 'p > x':
e = Lt(x, y).subs({y: p})
assert e == Lt(x, p, evaluate=False)
# Previously failed as 'p <= x':
e = Ge(x, p).doit()
assert e == Ge(x, p, evaluate=False)
def test_issue_8245():
a = S("6506833320952669167898688709329/5070602400912917605986812821504")
q = a.n(10)
assert (a == q) is True
assert (a != q) is False
assert (a > q) == False
assert (a < q) == False
assert (a >= q) == True
assert (a <= q) == True
a = sqrt(2)
r = Rational(str(a.n(30)))
assert (r == a) is False
assert (r != a) is True
assert (r > a) == True
assert (r < a) == False
assert (r >= a) == True
assert (r <= a) == False
a = sqrt(2)
r = Rational(str(a.n(29)))
assert (r == a) is False
assert (r != a) is True
assert (r > a) == False
assert (r < a) == True
assert (r >= a) == False
assert (r <= a) == True
assert Eq(log(cos(2)**2 + sin(2)**2), 0) == True
def test_issue_8449():
p = Symbol('p', nonnegative=True)
assert Lt(-oo, p)
assert Ge(-oo, p) is S.false
assert Gt(oo, -p)
assert Le(oo, -p) is S.false
def test_simplify_relational():
assert simplify(x*(y + 1) - x*y - x + 1 < x) == (x > 1)
r = S(1) < x
# canonical operations are not the same as simplification,
# so if there is no simplification, canonicalization will
# be done unless the measure forbids it
assert simplify(r) == r.canonical
assert simplify(r, ratio=0) != r.canonical
# this is not a random test; in _eval_simplify
# this will simplify to S.false and that is the
# reason for the 'if r.is_Relational' in Relational's
# _eval_simplify routine
assert simplify(-(2**(3*pi/2) + 6**pi)**(1/pi) +
2*(2**(pi/2) + 3**pi)**(1/pi) < 0) is S.false
# canonical at least
for f in (Eq, Ne):
f(y, x).simplify() == f(x, y)
f(x - 1, 0).simplify() == f(x, 1)
f(x - 1, x).simplify() == S.false
f(2*x - 1, x).simplify() == f(x, 1)
f(2*x, 4).simplify() == f(x, 2)
z = cos(1)**2 + sin(1)**2 - 1 # z.is_zero is None
f(z*x, 0).simplify() == f(z*x, 0)
def test_equals():
w, x, y, z = symbols('w:z')
f = Function('f')
assert Eq(x, 1).equals(Eq(x*(y + 1) - x*y - x + 1, x))
assert Eq(x, y).equals(x < y, True) == False
assert Eq(x, f(1)).equals(Eq(x, f(2)), True) == f(1) - f(2)
assert Eq(f(1), y).equals(Eq(f(2), y), True) == f(1) - f(2)
assert Eq(x, f(1)).equals(Eq(f(2), x), True) == f(1) - f(2)
assert Eq(f(1), x).equals(Eq(x, f(2)), True) == f(1) - f(2)
assert Eq(w, x).equals(Eq(y, z), True) == False
assert Eq(f(1), f(2)).equals(Eq(f(3), f(4)), True) == f(1) - f(3)
assert (x < y).equals(y > x, True) == True
assert (x < y).equals(y >= x, True) == False
assert (x < y).equals(z < y, True) == False
assert (x < y).equals(x < z, True) == False
assert (x < f(1)).equals(x < f(2), True) == f(1) - f(2)
assert (f(1) < x).equals(f(2) < x, True) == f(1) - f(2)
def test_reversed():
assert (x < y).reversed == (y > x)
assert (x <= y).reversed == (y >= x)
assert Eq(x, y, evaluate=False).reversed == Eq(y, x, evaluate=False)
assert Ne(x, y, evaluate=False).reversed == Ne(y, x, evaluate=False)
assert (x >= y).reversed == (y <= x)
assert (x > y).reversed == (y < x)
def test_canonical():
c = [i.canonical for i in (
x + y < z,
x + 2 > 3,
x < 2,
S(2) > x,
x**2 > -x/y,
Gt(3, 2, evaluate=False)
)]
assert [i.canonical for i in c] == c
assert [i.reversed.canonical for i in c] == c
assert not any(i.lhs.is_Number and not i.rhs.is_Number for i in c)
c = [i.reversed.func(i.rhs, i.lhs, evaluate=False).canonical for i in c]
assert [i.canonical for i in c] == c
assert [i.reversed.canonical for i in c] == c
assert not any(i.lhs.is_Number and not i.rhs.is_Number for i in c)
@XFAIL
def test_issue_8444_nonworkingtests():
x = symbols('x', real=True)
assert (x <= oo) == (x >= -oo) == True
x = symbols('x')
assert x >= floor(x)
assert (x < floor(x)) == False
assert x <= ceiling(x)
assert (x > ceiling(x)) == False
def test_issue_8444_workingtests():
x = symbols('x')
assert Gt(x, floor(x)) == Gt(x, floor(x), evaluate=False)
assert Ge(x, floor(x)) == Ge(x, floor(x), evaluate=False)
assert Lt(x, ceiling(x)) == Lt(x, ceiling(x), evaluate=False)
assert Le(x, ceiling(x)) == Le(x, ceiling(x), evaluate=False)
i = symbols('i', integer=True)
assert (i > floor(i)) == False
assert (i < ceiling(i)) == False
def test_issue_10304():
d = cos(1)**2 + sin(1)**2 - 1
assert d.is_comparable is False # if this fails, find a new d
e = 1 + d*I
assert simplify(Eq(e, 0)) is S.false
def test_issue_10401():
x = symbols('x')
fin = symbols('inf', finite=True)
inf = symbols('inf', infinite=True)
inf2 = symbols('inf2', infinite=True)
zero = symbols('z', zero=True)
nonzero = symbols('nz', zero=False, finite=True)
assert Eq(1/(1/x + 1), 1).func is Eq
assert Eq(1/(1/x + 1), 1).subs(x, S.ComplexInfinity) is S.true
assert Eq(1/(1/fin + 1), 1) is S.false
T, F = S.true, S.false
assert Eq(fin, inf) is F
assert Eq(inf, inf2) is T and inf != inf2
assert Eq(inf/inf2, 0) is F
assert Eq(inf/fin, 0) is F
assert Eq(fin/inf, 0) is T
assert Eq(zero/nonzero, 0) is T and ((zero/nonzero) != 0)
assert Eq(inf, -inf) is F
assert Eq(fin/(fin + 1), 1) is S.false
o = symbols('o', odd=True)
assert Eq(o, 2*o) is S.false
p = symbols('p', positive=True)
assert Eq(p/(p - 1), 1) is F
def test_issue_10633():
assert Eq(True, False) == False
assert Eq(False, True) == False
assert Eq(True, True) == True
assert Eq(False, False) == True
def test_issue_10927():
x = symbols('x')
assert str(Eq(x, oo)) == 'Eq(x, oo)'
assert str(Eq(x, -oo)) == 'Eq(x, -oo)'
def test_issues_13081_12583_12534():
# 13081
r = Rational('905502432259640373/288230376151711744')
assert (r < pi) is S.false
assert (r > pi) is S.true
# 12583
v = sqrt(2)
u = sqrt(v) + 2/sqrt(10 - 8/sqrt(2 - v) + 4*v*(1/sqrt(2 - v) - 1))
assert (u >= 0) is S.true
# 12534; Rational vs NumberSymbol
# here are some precisions for which Rational forms
# at a lower and higher precision bracket the value of pi
# e.g. for p = 20:
# Rational(pi.n(p + 1)).n(25) = 3.14159265358979323846 2834
# pi.n(25) = 3.14159265358979323846 2643
# Rational(pi.n(p )).n(25) = 3.14159265358979323846 1987
assert [p for p in range(20, 50) if
(Rational(pi.n(p)) < pi) and
(pi < Rational(pi.n(p + 1)))] == [20, 24, 27, 33, 37, 43, 48]
# pick one such precision and affirm that the reversed operation
# gives the opposite result, i.e. if x < y is true then x > y
# must be false
p = 20
# Rational vs NumberSymbol
G = [Rational(pi.n(i)) > pi for i in (p, p + 1)]
L = [Rational(pi.n(i)) < pi for i in (p, p + 1)]
assert G == [False, True]
assert all(i is not j for i, j in zip(L, G))
# Float vs NumberSymbol
G = [pi.n(i) > pi for i in (p, p + 1)]
L = [pi.n(i) < pi for i in (p, p + 1)]
assert G == [False, True]
assert all(i is not j for i, j in zip(L, G))
# Float vs Float
G = [pi.n(p) > pi.n(p + 1)]
L = [pi.n(p) < pi.n(p + 1)]
assert G == [True]
assert all(i is not j for i, j in zip(L, G))
# Float vs Rational
# the rational form is less than the floating representation
# at the same precision
assert [i for i in range(15, 50) if Rational(pi.n(i)) > pi.n(i)] == []
# this should be the same if we reverse the relational
assert [i for i in range(15, 50) if pi.n(i) < Rational(pi.n(i))] == []
def test_binary_symbols():
ans = set([x])
for f in Eq, Ne:
for t in S.true, S.false:
eq = f(x, S.true)
assert eq.binary_symbols == ans
assert eq.reversed.binary_symbols == ans
assert f(x, 1).binary_symbols == set()
def test_rel_args():
# can't have Boolean args; this is automatic with Python 3
# so this test and the __lt__, etc..., definitions in
# relational.py and boolalg.py which are marked with ///
# can be removed.
for op in ['<', '<=', '>', '>=']:
for b in (S.true, x < 1, And(x, y)):
for v in (0.1, 1, 2**32, t, S(1)):
raises(TypeError, lambda: Relational(b, v, op))
def test_Equality_rewrite_as_Add():
eq = Eq(x + y, y - x)
assert eq.rewrite(Add) == 2*x
assert eq.rewrite(Add, evaluate=None).args == (x, x, y, -y)
assert eq.rewrite(Add, evaluate=False).args == (x, y, x, -y)
def test_issue_15847():
a = Ne(x*(x+y), x**2 + x*y)
assert simplify(a) == False
def test_negated_property():
eq = Eq(x, y)
assert eq.negated == Ne(x, y)
eq = Ne(x, y)
assert eq.negated == Eq(x, y)
eq = Ge(x + y, y - x)
assert eq.negated == Lt(x + y, y - x)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, y).negated.negated == f(x, y)
def test_reversedsign_property():
eq = Eq(x, y)
assert eq.reversedsign == Eq(-x, -y)
eq = Ne(x, y)
assert eq.reversedsign == Ne(-x, -y)
eq = Ge(x + y, y - x)
assert eq.reversedsign == Le(-x - y, x - y)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, y).reversedsign.reversedsign == f(x, y)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(-x, y).reversedsign.reversedsign == f(-x, y)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, -y).reversedsign.reversedsign == f(x, -y)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(-x, -y).reversedsign.reversedsign == f(-x, -y)
def test_reversed_reversedsign_property():
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, y).reversed.reversedsign == f(x, y).reversedsign.reversed
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(-x, y).reversed.reversedsign == f(-x, y).reversedsign.reversed
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, -y).reversed.reversedsign == f(x, -y).reversedsign.reversed
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(-x, -y).reversed.reversedsign == \
f(-x, -y).reversedsign.reversed
def test_improved_canonical():
def test_different_forms(listofforms):
for form1, form2 in combinations(listofforms, 2):
assert form1.canonical == form2.canonical
def generate_forms(expr):
return [expr, expr.reversed, expr.reversedsign,
expr.reversed.reversedsign]
test_different_forms(generate_forms(x > -y))
test_different_forms(generate_forms(x >= -y))
test_different_forms(generate_forms(Eq(x, -y)))
test_different_forms(generate_forms(Ne(x, -y)))
test_different_forms(generate_forms(pi < x))
test_different_forms(generate_forms(pi - 5*y < -x + 2*y**2 - 7))
assert (pi >= x).canonical == (x <= pi)
|
e3fa52f52e96676032c4b089ed1a0f5f8d033ae0d3b5948c67a615b9a00a2645 | """Tests for tools and arithmetics for monomials of distributed polynomials. """
from sympy.polys.monomials import (
itermonomials, monomial_count,
monomial_mul, monomial_div,
monomial_gcd, monomial_lcm,
monomial_max, monomial_min,
monomial_divides, monomial_pow,
Monomial,
)
from sympy.polys.polyerrors import ExactQuotientFailed
from sympy.abc import a, b, c, x, y, z
from sympy.core import S, symbols
from sympy.utilities.pytest import raises
def test_monomials():
assert itermonomials([], -1) == set()
assert itermonomials([], 0) == {S(1)}
assert itermonomials([], 1) == {S(1)}
assert itermonomials([], 2) == {S(1)}
assert itermonomials([], 3) == {S(1)}
assert itermonomials([x], -1) == set()
assert itermonomials([x], 0) == {S(1)}
assert itermonomials([x], 1) == {S(1), x}
assert itermonomials([x], 2) == {S(1), x, x**2}
assert itermonomials([x], 3) == {S(1), x, x**2, x**3}
assert itermonomials([x, y], 0) == {S(1)}
assert itermonomials([x, y], 1) == {S(1), x, y}
assert itermonomials([x, y], 2) == {S(1), x, y, x**2, y**2, x*y}
assert itermonomials([x, y], 3) == \
{S(1), x, y, x**2, x**3, y**2, y**3, x*y, x*y**2, y*x**2}
i, j, k = symbols('i j k', commutative=False)
assert itermonomials([i, j, k], 0) == {S(1)}
assert itermonomials([i, j, k], 1) == {S(1), i, j, k}
assert itermonomials([i, j, k], 2) == \
{S(1), i, j, k, i**2, j**2, k**2, i*j, i*k, j*i, j*k, k*i, k*j}
assert itermonomials([i, j, k], 3) == \
{S(1), i, j, k, i**2, j**2, k**2, i*j, i*k, j*i, j*k, k*i, k*j,
i**3, j**3, k**3,
i**2 * j, i**2 * k, j * i**2, k * i**2,
j**2 * i, j**2 * k, i * j**2, k * j**2,
k**2 * i, k**2 * j, i * k**2, j * k**2,
i*j*i, i*k*i, j*i*j, j*k*j, k*i*k, k*j*k,
i*j*k, i*k*j, j*i*k, j*k*i, k*i*j, k*j*i,
}
assert itermonomials([x, i, j], 0) == {S(1)}
assert itermonomials([x, i, j], 1) == {S(1), x, i, j}
assert itermonomials([x, i, j], 2) == {S(1), x, i, j, x*i, x*j, i*j, j*i, x**2, i**2, j**2}
assert itermonomials([x, i, j], 3) == \
{S(1), x, i, j, x*i, x*j, i*j, j*i, x**2, i**2, j**2,
x**3, i**3, j**3,
x**2 * i, x**2 * j,
x * i**2, j * i**2, i**2 * j, i*j*i,
x * j**2, i * j**2, j**2 * i, j*i*j,
x * i * j, x * j * i,
}
def test_monomial_count():
assert monomial_count(2, 2) == 6
assert monomial_count(2, 3) == 10
def test_monomial_mul():
assert monomial_mul((3, 4, 1), (1, 2, 0)) == (4, 6, 1)
def test_monomial_div():
assert monomial_div((3, 4, 1), (1, 2, 0)) == (2, 2, 1)
def test_monomial_gcd():
assert monomial_gcd((3, 4, 1), (1, 2, 0)) == (1, 2, 0)
def test_monomial_lcm():
assert monomial_lcm((3, 4, 1), (1, 2, 0)) == (3, 4, 1)
def test_monomial_max():
assert monomial_max((3, 4, 5), (0, 5, 1), (6, 3, 9)) == (6, 5, 9)
def test_monomial_pow():
assert monomial_pow((1, 2, 3), 3) == (3, 6, 9)
def test_monomial_min():
assert monomial_min((3, 4, 5), (0, 5, 1), (6, 3, 9)) == (0, 3, 1)
def test_monomial_divides():
assert monomial_divides((1, 2, 3), (4, 5, 6)) is True
assert monomial_divides((1, 2, 3), (0, 5, 6)) is False
def test_Monomial():
m = Monomial((3, 4, 1), (x, y, z))
n = Monomial((1, 2, 0), (x, y, z))
assert m.as_expr() == x**3*y**4*z
assert n.as_expr() == x**1*y**2
assert m.as_expr(a, b, c) == a**3*b**4*c
assert n.as_expr(a, b, c) == a**1*b**2
assert m.exponents == (3, 4, 1)
assert m.gens == (x, y, z)
assert n.exponents == (1, 2, 0)
assert n.gens == (x, y, z)
assert m == (3, 4, 1)
assert n != (3, 4, 1)
assert m != (1, 2, 0)
assert n == (1, 2, 0)
assert (m == 1) is False
assert m[0] == m[-3] == 3
assert m[1] == m[-2] == 4
assert m[2] == m[-1] == 1
assert n[0] == n[-3] == 1
assert n[1] == n[-2] == 2
assert n[2] == n[-1] == 0
assert m[:2] == (3, 4)
assert n[:2] == (1, 2)
assert m*n == Monomial((4, 6, 1))
assert m/n == Monomial((2, 2, 1))
assert m*(1, 2, 0) == Monomial((4, 6, 1))
assert m/(1, 2, 0) == Monomial((2, 2, 1))
assert m.gcd(n) == Monomial((1, 2, 0))
assert m.lcm(n) == Monomial((3, 4, 1))
assert m.gcd((1, 2, 0)) == Monomial((1, 2, 0))
assert m.lcm((1, 2, 0)) == Monomial((3, 4, 1))
assert m**0 == Monomial((0, 0, 0))
assert m**1 == m
assert m**2 == Monomial((6, 8, 2))
assert m**3 == Monomial((9, 12, 3))
raises(ExactQuotientFailed, lambda: m/Monomial((5, 2, 0)))
mm = Monomial((1, 2, 3))
raises(ValueError, lambda: mm.as_expr())
assert str(mm) == 'Monomial((1, 2, 3))'
assert str(m) == 'x**3*y**4*z**1'
raises(NotImplementedError, lambda: m*1)
raises(NotImplementedError, lambda: m/1)
raises(ValueError, lambda: m**-1)
raises(TypeError, lambda: m.gcd(3))
raises(TypeError, lambda: m.lcm(3))
|
3893394d89af30c906df0f7723fb5163479cc38cf4e93824fe41fa755e990583 | from sympy import Add, Basic, symbols, Symbol
from sympy.unify.core import Compound, Variable
from sympy.unify.usympy import (deconstruct, construct, unify, is_associative,
is_commutative)
from sympy.abc import x, y, z, n
from sympy.utilities.pytest import XFAIL
def test_deconstruct():
expr = Basic(1, 2, 3)
expected = Compound(Basic, (1, 2, 3))
assert deconstruct(expr) == expected
assert deconstruct(1) == 1
assert deconstruct(x) == x
assert deconstruct(x, variables=(x,)) == Variable(x)
assert deconstruct(Add(1, x, evaluate=False)) == Compound(Add, (1, x))
assert deconstruct(Add(1, x, evaluate=False), variables=(x,)) == \
Compound(Add, (1, Variable(x)))
def test_construct():
expr = Compound(Basic, (1, 2, 3))
expected = Basic(1, 2, 3)
assert construct(expr) == expected
def test_nested():
expr = Basic(1, Basic(2), 3)
cmpd = Compound(Basic, (1, Compound(Basic, (2,)), 3))
assert deconstruct(expr) == cmpd
assert construct(cmpd) == expr
def test_unify():
expr = Basic(1, 2, 3)
a, b, c = map(Symbol, 'abc')
pattern = Basic(a, b, c)
assert list(unify(expr, pattern, {}, (a, b, c))) == [{a: 1, b: 2, c: 3}]
assert list(unify(expr, pattern, variables=(a, b, c))) == \
[{a: 1, b: 2, c: 3}]
def test_unify_variables():
assert list(unify(Basic(1, 2), Basic(1, x), {}, variables=(x,))) == [{x: 2}]
def test_s_input():
expr = Basic(1, 2)
a, b = map(Symbol, 'ab')
pattern = Basic(a, b)
assert list(unify(expr, pattern, {}, (a, b))) == [{a: 1, b: 2}]
assert list(unify(expr, pattern, {a: 5}, (a, b))) == []
def iterdicteq(a, b):
a = tuple(a)
b = tuple(b)
return len(a) == len(b) and all(x in b for x in a)
def test_unify_commutative():
expr = Add(1, 2, 3, evaluate=False)
a, b, c = map(Symbol, 'abc')
pattern = Add(a, b, c, evaluate=False)
result = tuple(unify(expr, pattern, {}, (a, b, c)))
expected = ({a: 1, b: 2, c: 3},
{a: 1, b: 3, c: 2},
{a: 2, b: 1, c: 3},
{a: 2, b: 3, c: 1},
{a: 3, b: 1, c: 2},
{a: 3, b: 2, c: 1})
assert iterdicteq(result, expected)
def test_unify_iter():
expr = Add(1, 2, 3, evaluate=False)
a, b, c = map(Symbol, 'abc')
pattern = Add(a, c, evaluate=False)
assert is_associative(deconstruct(pattern))
assert is_commutative(deconstruct(pattern))
result = list(unify(expr, pattern, {}, (a, c)))
expected = [{a: 1, c: Add(2, 3, evaluate=False)},
{a: 1, c: Add(3, 2, evaluate=False)},
{a: 2, c: Add(1, 3, evaluate=False)},
{a: 2, c: Add(3, 1, evaluate=False)},
{a: 3, c: Add(1, 2, evaluate=False)},
{a: 3, c: Add(2, 1, evaluate=False)},
{a: Add(1, 2, evaluate=False), c: 3},
{a: Add(2, 1, evaluate=False), c: 3},
{a: Add(1, 3, evaluate=False), c: 2},
{a: Add(3, 1, evaluate=False), c: 2},
{a: Add(2, 3, evaluate=False), c: 1},
{a: Add(3, 2, evaluate=False), c: 1}]
assert iterdicteq(result, expected)
def test_hard_match():
from sympy import sin, cos
expr = sin(x) + cos(x)**2
p, q = map(Symbol, 'pq')
pattern = sin(p) + cos(p)**2
assert list(unify(expr, pattern, {}, (p, q))) == [{p: x}]
def test_matrix():
from sympy import MatrixSymbol
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 2, 2)
Z = MatrixSymbol('Z', 2, 3)
assert list(unify(X, Y, {}, variables=[n, Symbol('X')])) == [{Symbol('X'): Symbol('Y'), n: 2}]
assert list(unify(X, Z, {}, variables=[n, Symbol('X')])) == []
def test_non_frankenAdds():
# the is_commutative property used to fail because of Basic.__new__
# This caused is_commutative and str calls to fail
expr = x+y*2
rebuilt = construct(deconstruct(expr))
# Ensure that we can run these commands without causing an error
str(rebuilt)
rebuilt.is_commutative
def test_FiniteSet_commutivity():
from sympy import FiniteSet
a, b, c, x, y = symbols('a,b,c,x,y')
s = FiniteSet(a, b, c)
t = FiniteSet(x, y)
variables = (x, y)
assert {x: FiniteSet(a, c), y: b} in tuple(unify(s, t, variables=variables))
def test_FiniteSet_complex():
from sympy import FiniteSet
a, b, c, x, y, z = symbols('a,b,c,x,y,z')
expr = FiniteSet(Basic(1, x), y, Basic(x, z))
pattern = FiniteSet(a, Basic(x, b))
variables = a, b
expected = tuple([{b: 1, a: FiniteSet(y, Basic(x, z))},
{b: z, a: FiniteSet(y, Basic(1, x))}])
assert iterdicteq(unify(expr, pattern, variables=variables), expected)
@XFAIL
def test_and():
variables = x, y
str(list(unify((x>0) & (z<3), pattern, variables=variables)))
def test_Union():
from sympy import Interval
assert list(unify(Interval(0, 1) + Interval(10, 11),
Interval(0, 1) + Interval(12, 13),
variables=(Interval(12, 13),)))
def test_is_commutative():
assert is_commutative(deconstruct(x+y))
assert is_commutative(deconstruct(x*y))
assert not is_commutative(deconstruct(x**y))
def test_commutative_in_commutative():
from sympy.abc import a,b,c,d
from sympy import sin, cos
eq = sin(3)*sin(4)*sin(5) + 4*cos(3)*cos(4)
pat = a*cos(b)*cos(c) + d*sin(b)*sin(c)
assert next(unify(eq, pat, variables=(a,b,c,d)))
|
77bdce8d1bf1039176512d21f552f703a9e9ca11d1cba39d1a728c2b09e2e1d7 | from sympy import Abs, Rational, Float, S, Symbol, symbols, cos, pi, sqrt, oo
from sympy.functions.elementary.trigonometric import tan
from sympy.geometry import (Circle, Ellipse, GeometryError, Point, Point2D, \
Polygon, Ray, RegularPolygon, Segment, Triangle, \
are_similar,convex_hull, intersection, Line)
from sympy.utilities.pytest import raises, slow, warns
from sympy.utilities.randtest import verify_numerically
from sympy.geometry.polygon import rad, deg
from sympy import integrate
def feq(a, b):
"""Test if two floating point values are 'equal'."""
t_float = Float("1.0E-10")
return -t_float < a - b < t_float
@slow
def test_polygon():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
q = Symbol('q', real=True)
u = Symbol('u', real=True)
v = Symbol('v', real=True)
w = Symbol('w', real=True)
x1 = Symbol('x1', real=True)
half = Rational(1, 2)
a, b, c = Point(0, 0), Point(2, 0), Point(3, 3)
t = Triangle(a, b, c)
assert Polygon(a, Point(1, 0), b, c) == t
assert Polygon(Point(1, 0), b, c, a) == t
assert Polygon(b, c, a, Point(1, 0)) == t
# 2 "remove folded" tests
assert Polygon(a, Point(3, 0), b, c) == t
assert Polygon(a, b, Point(3, -1), b, c) == t
# remove multiple collinear points
assert Polygon(Point(-4, 15), Point(-11, 15), Point(-15, 15),
Point(-15, 33/5), Point(-15, -87/10), Point(-15, -15),
Point(-42/5, -15), Point(-2, -15), Point(7, -15), Point(15, -15),
Point(15, -3), Point(15, 10), Point(15, 15)) == \
Polygon(Point(-15,-15), Point(15,-15), Point(15,15), Point(-15,15))
p1 = Polygon(
Point(0, 0), Point(3, -1),
Point(6, 0), Point(4, 5),
Point(2, 3), Point(0, 3))
p2 = Polygon(
Point(6, 0), Point(3, -1),
Point(0, 0), Point(0, 3),
Point(2, 3), Point(4, 5))
p3 = Polygon(
Point(0, 0), Point(3, 0),
Point(5, 2), Point(4, 4))
p4 = Polygon(
Point(0, 0), Point(4, 4),
Point(5, 2), Point(3, 0))
p5 = Polygon(
Point(0, 0), Point(4, 4),
Point(0, 4))
p6 = Polygon(
Point(-11, 1), Point(-9, 6.6),
Point(-4, -3), Point(-8.4, -8.7))
p7 = Polygon(
Point(x, y), Point(q, u),
Point(v, w))
p8 = Polygon(
Point(x, y), Point(v, w),
Point(q, u))
p9 = Polygon(
Point(0, 0), Point(4, 4),
Point(3, 0), Point(5, 2))
p10 = Polygon(
Point(0, 2), Point(2, 2),
Point(0, 0), Point(2, 0))
p11 = Polygon(Point(0, 0), 1, n=3)
r = Ray(Point(-9,6.6), Point(-9,5.5))
#
# General polygon
#
assert p1 == p2
assert len(p1.args) == 6
assert len(p1.sides) == 6
assert p1.perimeter == 5 + 2*sqrt(10) + sqrt(29) + sqrt(8)
assert p1.area == 22
assert not p1.is_convex()
assert Polygon((-1, 1), (2, -1), (2, 1), (-1, -1), (3, 0)
).is_convex() is False
# ensure convex for both CW and CCW point specification
assert p3.is_convex()
assert p4.is_convex()
dict5 = p5.angles
assert dict5[Point(0, 0)] == pi / 4
assert dict5[Point(0, 4)] == pi / 2
assert p5.encloses_point(Point(x, y)) is None
assert p5.encloses_point(Point(1, 3))
assert p5.encloses_point(Point(0, 0)) is False
assert p5.encloses_point(Point(4, 0)) is False
assert p1.encloses(Circle(Point(2.5,2.5),5)) is False
assert p1.encloses(Ellipse(Point(2.5,2),5,6)) is False
p5.plot_interval('x') == [x, 0, 1]
assert p5.distance(
Polygon(Point(10, 10), Point(14, 14), Point(10, 14))) == 6 * sqrt(2)
assert p5.distance(
Polygon(Point(1, 8), Point(5, 8), Point(8, 12), Point(1, 12))) == 4
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
Polygon(Point(0, 0), Point(1, 0), Point(1, 1)).distance(
Polygon(Point(0, 0), Point(0, 1), Point(1, 1)))
assert hash(p5) == hash(Polygon(Point(0, 0), Point(4, 4), Point(0, 4)))
assert hash(p1) == hash(p2)
assert hash(p7) == hash(p8)
assert hash(p3) != hash(p9)
assert p5 == Polygon(Point(4, 4), Point(0, 4), Point(0, 0))
assert Polygon(Point(4, 4), Point(0, 4), Point(0, 0)) in p5
assert p5 != Point(0, 4)
assert Point(0, 1) in p5
assert p5.arbitrary_point('t').subs(Symbol('t', real=True), 0) == \
Point(0, 0)
raises(ValueError, lambda: Polygon(
Point(x, 0), Point(0, y), Point(x, y)).arbitrary_point('x'))
assert p6.intersection(r) == [Point(-9, -S(84)/13), Point(-9, S(33)/5)]
assert p10.area == 0
assert p11 == RegularPolygon(Point(0, 0), 1, 3, 0)
assert p11.vertices[0] == Point(1, 0)
assert p11.args[0] == Point(0, 0)
p11.spin(pi/2)
assert p11.vertices[0] == Point(0, 1)
#
# Regular polygon
#
p1 = RegularPolygon(Point(0, 0), 10, 5)
p2 = RegularPolygon(Point(0, 0), 5, 5)
raises(GeometryError, lambda: RegularPolygon(Point(0, 0), Point(0,
1), Point(1, 1)))
raises(GeometryError, lambda: RegularPolygon(Point(0, 0), 1, 2))
raises(ValueError, lambda: RegularPolygon(Point(0, 0), 1, 2.5))
assert p1 != p2
assert p1.interior_angle == 3*pi/5
assert p1.exterior_angle == 2*pi/5
assert p2.apothem == 5*cos(pi/5)
assert p2.circumcenter == p1.circumcenter == Point(0, 0)
assert p1.circumradius == p1.radius == 10
assert p2.circumcircle == Circle(Point(0, 0), 5)
assert p2.incircle == Circle(Point(0, 0), p2.apothem)
assert p2.inradius == p2.apothem == (5 * (1 + sqrt(5)) / 4)
p2.spin(pi / 10)
dict1 = p2.angles
assert dict1[Point(0, 5)] == 3 * pi / 5
assert p1.is_convex()
assert p1.rotation == 0
assert p1.encloses_point(Point(0, 0))
assert p1.encloses_point(Point(11, 0)) is False
assert p2.encloses_point(Point(0, 4.9))
p1.spin(pi/3)
assert p1.rotation == pi/3
assert p1.vertices[0] == Point(5, 5*sqrt(3))
for var in p1.args:
if isinstance(var, Point):
assert var == Point(0, 0)
else:
assert var == 5 or var == 10 or var == pi / 3
assert p1 != Point(0, 0)
assert p1 != p5
# while spin works in place (notice that rotation is 2pi/3 below)
# rotate returns a new object
p1_old = p1
assert p1.rotate(pi/3) == RegularPolygon(Point(0, 0), 10, 5, 2*pi/3)
assert p1 == p1_old
assert p1.area == (-250*sqrt(5) + 1250)/(4*tan(pi/5))
assert p1.length == 20*sqrt(-sqrt(5)/8 + S(5)/8)
assert p1.scale(2, 2) == \
RegularPolygon(p1.center, p1.radius*2, p1._n, p1.rotation)
assert RegularPolygon((0, 0), 1, 4).scale(2, 3) == \
Polygon(Point(2, 0), Point(0, 3), Point(-2, 0), Point(0, -3))
assert repr(p1) == str(p1)
#
# Angles
#
angles = p4.angles
assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))
angles = p3.angles
assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))
#
# Triangle
#
p1 = Point(0, 0)
p2 = Point(5, 0)
p3 = Point(0, 5)
t1 = Triangle(p1, p2, p3)
t2 = Triangle(p1, p2, Point(Rational(5, 2), sqrt(Rational(75, 4))))
t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
s1 = t1.sides
assert Triangle(p1, p2, p1) == Polygon(p1, p2, p1) == Segment(p1, p2)
raises(GeometryError, lambda: Triangle(Point(0, 0)))
# Basic stuff
assert Triangle(p1, p1, p1) == p1
assert Triangle(p2, p2*2, p2*3) == Segment(p2, p2*3)
assert t1.area == Rational(25, 2)
assert t1.is_right()
assert t2.is_right() is False
assert t3.is_right()
assert p1 in t1
assert t1.sides[0] in t1
assert Segment((0, 0), (1, 0)) in t1
assert Point(5, 5) not in t2
assert t1.is_convex()
assert feq(t1.angles[p1].evalf(), pi.evalf()/2)
assert t1.is_equilateral() is False
assert t2.is_equilateral()
assert t3.is_equilateral() is False
assert are_similar(t1, t2) is False
assert are_similar(t1, t3)
assert are_similar(t2, t3) is False
assert t1.is_similar(Point(0, 0)) is False
assert t1.is_similar(t2) is False
# Bisectors
bisectors = t1.bisectors()
assert bisectors[p1] == Segment(
p1, Point(Rational(5, 2), Rational(5, 2)))
assert t2.bisectors()[p2] == Segment(
Point(5, 0), Point(Rational(5, 4), 5*sqrt(3)/4))
p4 = Point(0, x1)
assert t3.bisectors()[p4] == Segment(p4, Point(x1*(sqrt(2) - 1), 0))
ic = (250 - 125*sqrt(2))/50
assert t1.incenter == Point(ic, ic)
# Inradius
assert t1.inradius == t1.incircle.radius == 5 - 5*sqrt(2)/2
assert t2.inradius == t2.incircle.radius == 5*sqrt(3)/6
assert t3.inradius == t3.incircle.radius == x1**2/((2 + sqrt(2))*Abs(x1))
# Exradius
assert t1.exradii[t1.sides[2]] == 5*sqrt(2)/2
# Circumcircle
assert t1.circumcircle.center == Point(2.5, 2.5)
# Medians + Centroid
m = t1.medians
assert t1.centroid == Point(Rational(5, 3), Rational(5, 3))
assert m[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
assert t3.medians[p1] == Segment(p1, Point(x1/2, x1/2))
assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]
assert t1.medial == Triangle(Point(2.5, 0), Point(0, 2.5), Point(2.5, 2.5))
# Nine-point circle
assert t1.nine_point_circle == Circle(Point(2.5, 0),
Point(0, 2.5), Point(2.5, 2.5))
assert t1.nine_point_circle == Circle(Point(0, 0),
Point(0, 2.5), Point(2.5, 2.5))
# Perpendicular
altitudes = t1.altitudes
assert altitudes[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
assert altitudes[p2].equals(s1[0])
assert altitudes[p3] == s1[2]
assert t1.orthocenter == p1
t = S('''Triangle(
Point(100080156402737/5000000000000, 79782624633431/500000000000),
Point(39223884078253/2000000000000, 156345163124289/1000000000000),
Point(31241359188437/1250000000000, 338338270939941/1000000000000000))''')
assert t.orthocenter == S('''Point(-780660869050599840216997'''
'''79471538701955848721853/80368430960602242240789074233100000000000000,'''
'''20151573611150265741278060334545897615974257/16073686192120448448157'''
'''8148466200000000000)''')
# Ensure
assert len(intersection(*bisectors.values())) == 1
assert len(intersection(*altitudes.values())) == 1
assert len(intersection(*m.values())) == 1
# Distance
p1 = Polygon(
Point(0, 0), Point(1, 0),
Point(1, 1), Point(0, 1))
p2 = Polygon(
Point(0, Rational(5)/4), Point(1, Rational(5)/4),
Point(1, Rational(9)/4), Point(0, Rational(9)/4))
p3 = Polygon(
Point(1, 2), Point(2, 2),
Point(2, 1))
p4 = Polygon(
Point(1, 1), Point(Rational(6)/5, 1),
Point(1, Rational(6)/5))
pt1 = Point(half, half)
pt2 = Point(1, 1)
'''Polygon to Point'''
assert p1.distance(pt1) == half
assert p1.distance(pt2) == 0
assert p2.distance(pt1) == Rational(3)/4
assert p3.distance(pt2) == sqrt(2)/2
'''Polygon to Polygon'''
# p1.distance(p2) emits a warning
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
assert p1.distance(p2) == half/2
assert p1.distance(p3) == sqrt(2)/2
# p3.distance(p4) emits a warning
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
assert p3.distance(p4) == (sqrt(2)/2 - sqrt(Rational(2)/25)/2)
def test_convex_hull():
p = [Point(-5, -1), Point(-2, 1), Point(-2, -1), Point(-1, -3), \
Point(0, 0), Point(1, 1), Point(2, 2), Point(2, -1), Point(3, 1), \
Point(4, -1), Point(6, 2)]
ch = Polygon(p[0], p[3], p[9], p[10], p[6], p[1])
#test handling of duplicate points
p.append(p[3])
#more than 3 collinear points
another_p = [Point(-45, -85), Point(-45, 85), Point(-45, 26), \
Point(-45, -24)]
ch2 = Segment(another_p[0], another_p[1])
assert convex_hull(*another_p) == ch2
assert convex_hull(*p) == ch
assert convex_hull(p[0]) == p[0]
assert convex_hull(p[0], p[1]) == Segment(p[0], p[1])
# no unique points
assert convex_hull(*[p[-1]]*3) == p[-1]
# collection of items
assert convex_hull(*[Point(0, 0), \
Segment(Point(1, 0), Point(1, 1)), \
RegularPolygon(Point(2, 0), 2, 4)]) == \
Polygon(Point(0, 0), Point(2, -2), Point(4, 0), Point(2, 2))
def test_encloses():
# square with a dimpled left side
s = Polygon(Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1), \
Point(S.Half, S.Half))
# the following is True if the polygon isn't treated as closing on itself
assert s.encloses(Point(0, S.Half)) is False
assert s.encloses(Point(S.Half, S.Half)) is False # it's a vertex
assert s.encloses(Point(Rational(3, 4), S.Half)) is True
def test_triangle_kwargs():
assert Triangle(sss=(3, 4, 5)) == \
Triangle(Point(0, 0), Point(3, 0), Point(3, 4))
assert Triangle(asa=(30, 2, 30)) == \
Triangle(Point(0, 0), Point(2, 0), Point(1, sqrt(3)/3))
assert Triangle(sas=(1, 45, 2)) == \
Triangle(Point(0, 0), Point(2, 0), Point(sqrt(2)/2, sqrt(2)/2))
assert Triangle(sss=(1, 2, 5)) is None
assert deg(rad(180)) == 180
def test_transform():
pts = [Point(0, 0), Point(S(1)/2, S(1)/4), Point(1, 1)]
pts_out = [Point(-4, -10), Point(-3, -S(37)/4), Point(-2, -7)]
assert Triangle(*pts).scale(2, 3, (4, 5)) == Triangle(*pts_out)
assert RegularPolygon((0, 0), 1, 4).scale(2, 3, (4, 5)) == \
Polygon(Point(-2, -10), Point(-4, -7), Point(-6, -10), Point(-4, -13))
def test_reflect():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
b = Symbol('b')
m = Symbol('m')
l = Line((0, b), slope=m)
p = Point(x, y)
r = p.reflect(l)
dp = l.perpendicular_segment(p).length
dr = l.perpendicular_segment(r).length
assert verify_numerically(dp, dr)
t = Triangle((0, 0), (1, 0), (2, 3))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=oo)) \
== Triangle(Point(5, 0), Point(4, 0), Point(4, 2))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=oo)) \
== Triangle(Point(-1, 0), Point(-2, 0), Point(-2, 2))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=0)) \
== Triangle(Point(1, 6), Point(2, 6), Point(2, 4))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=0)) \
== Triangle(Point(1, 0), Point(2, 0), Point(2, -2))
def test_bisectors():
p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
t = Triangle(p1, p2, p3)
assert t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1))
def test_incenter():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).incenter \
== Point(1 - sqrt(2)/2, 1 - sqrt(2)/2)
def test_inradius():
assert Triangle(Point(0, 0), Point(4, 0), Point(0, 3)).inradius == 1
def test_incircle():
assert Triangle(Point(0, 0), Point(2, 0), Point(0, 2)).incircle \
== Circle(Point(2 - sqrt(2), 2 - sqrt(2)), 2 - sqrt(2))
def test_exradii():
t = Triangle(Point(0, 0), Point(6, 0), Point(0, 2))
assert t.exradii[t.sides[2]] == (-2 + sqrt(10))
def test_medians():
t = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
assert t.medians[Point(0, 0)] == Segment(Point(0, 0), Point(S(1)/2, S(1)/2))
def test_medial():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).medial \
== Triangle(Point(S(1)/2, 0), Point(S(1)/2, S(1)/2), Point(0, S(1)/2))
def test_nine_point_circle():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).nine_point_circle \
== Circle(Point2D(S(1)/4, S(1)/4), sqrt(2)/4)
def test_eulerline():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).eulerline \
== Line(Point2D(0, 0), Point2D(S(1)/2, S(1)/2))
assert Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3))).eulerline \
== Point2D(5, 5*sqrt(3)/3)
assert Triangle(Point(4, -6), Point(4, -1), Point(-3, 3)).eulerline \
== Line(Point2D(S(64)/7, 3), Point2D(-S(29)/14, -S(7)/2))
def test_intersection():
poly1 = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
poly2 = Polygon(Point(0, 1), Point(-5, 0),
Point(0, -4), Point(0, S(1)/5),
Point(S(1)/2, -0.1), Point(1,0), Point(0, 1))
assert poly1.intersection(poly2) == [Point2D(S(1)/3, 0),
Segment(Point(0, S(1)/5), Point(0, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly1) == [Point(S(1)/3, 0),
Segment(Point(0, 0), Point(0, S(1)/5)),
Segment(Point(1, 0), Point(0, 1))]
assert poly1.intersection(Point(0, 0)) == [Point(0, 0)]
assert poly1.intersection(Point(-12, -43)) == []
assert poly2.intersection(Line((-12, 0), (12, 0))) == [Point(-5, 0),
Point(0, 0),Point(S(1)/3, 0), Point(1, 0)]
assert poly2.intersection(Line((-12, 12), (12, 12))) == []
assert poly2.intersection(Ray((-3,4), (1,0))) == [Segment(Point(1, 0),
Point(0, 1))]
assert poly2.intersection(Circle((0, -1), 1)) == [Point(0, -2),
Point(0, 0)]
assert poly1.intersection(poly1) == [Segment(Point(0, 0), Point(1, 0)),
Segment(Point(0, 1), Point(0, 0)), Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly2) == [Segment(Point(-5, 0), Point(0, -4)),
Segment(Point(0, -4), Point(0, S(1)/5)),
Segment(Point(0, S(1)/5), Point(S(1)/2, -S(1)/10)),
Segment(Point(0, 1), Point(-5, 0)),
Segment(Point(S(1)/2, -S(1)/10), Point(1, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(Triangle(Point(0, 1), Point(1, 0), Point(-1, 1))) \
== [Point(-S(5)/7, S(6)/7), Segment(Point2D(0, 1), Point(1, 0))]
assert poly1.intersection(RegularPolygon((-12, -15), 3, 3)) == []
def test_parameter_value():
t = Symbol('t')
sq = Polygon((0, 0), (0, 1), (1, 1), (1, 0))
assert sq.parameter_value((0.5, 1), t) == {t: S(3)/8}
q = Polygon((0, 0), (2, 1), (2, 4), (4, 0))
assert q.parameter_value((4, 0), t) == {t: -6 + 3*sqrt(5)} # ~= 0.708
raises(ValueError, lambda: sq.parameter_value((5, 6), t))
def test_issue_12966():
poly = Polygon(Point(0, 0), Point(0, 10), Point(5, 10), Point(5, 5),
Point(10, 5), Point(10, 0))
t = Symbol('t')
pt = poly.arbitrary_point(t)
DELTA = 5/poly.perimeter
assert [pt.subs(t, DELTA*i) for i in range(int(1/DELTA))] == [
Point(0, 0), Point(0, 5), Point(0, 10), Point(5, 10),
Point(5, 5), Point(10, 5), Point(10, 0), Point(5, 0)]
def test_second_moment_of_area():
x, y = symbols('x, y')
# triangle
p1, p2, p3 = [(0, 0), (4, 0), (0, 2)]
p = (0, 0)
# equation of hypotenuse
eq_y = (1-x/4)*2
I_yy = integrate((x**2) * (integrate(1, (y, 0, eq_y))), (x, 0, 4))
I_xx = integrate(1 * (integrate(y**2, (y, 0, eq_y))), (x, 0, 4))
I_xy = integrate(x * (integrate(y, (y, 0, eq_y))), (x, 0, 4))
triangle = Polygon(p1, p2, p3)
assert (I_xx - triangle.second_moment_of_area(p)[0]) == 0
assert (I_yy - triangle.second_moment_of_area(p)[1]) == 0
assert (I_xy - triangle.second_moment_of_area(p)[2]) == 0
# rectangle
p1, p2, p3, p4=[(0, 0), (4, 0), (4, 2), (0, 2)]
I_yy = integrate((x**2) * integrate(1, (y, 0, 2)), (x, 0, 4))
I_xx = integrate(1 * integrate(y**2, (y, 0, 2)), (x, 0, 4))
I_xy = integrate(x * integrate(y, (y, 0, 2)), (x, 0, 4))
rectangle = Polygon(p1, p2, p3, p4)
assert (I_xx - rectangle.second_moment_of_area(p)[0]) == 0
assert (I_yy - rectangle.second_moment_of_area(p)[1]) == 0
assert (I_xy - rectangle.second_moment_of_area(p)[2]) == 0
|
e0167d4739c5b12f09c2ec0502a4e1764cae32e26b40493f5cfc867356c7455f | from sympy import Rational, oo, sqrt, S
from sympy import Line, Point, Point2D, Parabola, Segment2D, Ray2D
from sympy import Circle, Ellipse, symbols, sign
from sympy.utilities.pytest import raises
def test_parabola_geom():
a, b = symbols('a b')
p1 = Point(0, 0)
p2 = Point(3, 7)
p3 = Point(0, 4)
p4 = Point(6, 0)
p5 = Point(a, a)
d1 = Line(Point(4, 0), Point(4, 9))
d2 = Line(Point(7, 6), Point(3, 6))
d3 = Line(Point(4, 0), slope=oo)
d4 = Line(Point(7, 6), slope=0)
d5 = Line(Point(b, a), slope=oo)
d6 = Line(Point(a, b), slope=0)
half = Rational(1, 2)
pa1 = Parabola(None, d2)
pa2 = Parabola(directrix=d1)
pa3 = Parabola(p1, d1)
pa4 = Parabola(p2, d2)
pa5 = Parabola(p2, d4)
pa6 = Parabola(p3, d2)
pa7 = Parabola(p2, d1)
pa8 = Parabola(p4, d1)
pa9 = Parabola(p4, d3)
pa10 = Parabola(p5, d5)
pa11 = Parabola(p5, d6)
raises(ValueError, lambda:
Parabola(Point(7, 8, 9), Line(Point(6, 7), Point(7, 7))))
raises(NotImplementedError, lambda:
Parabola(Point(7, 8), Line(Point(3, 7), Point(2, 9))))
raises(ValueError, lambda:
Parabola(Point(0, 2), Line(Point(7, 2), Point(6, 2))))
raises(ValueError, lambda: Parabola(Point(7, 8), Point(3, 8)))
# Basic Stuff
assert pa1.focus == Point(0, 0)
assert pa2 == pa3
assert pa4 != pa7
assert pa6 != pa7
assert pa6.focus == Point2D(0, 4)
assert pa6.focal_length == 1
assert pa6.p_parameter == -1
assert pa6.vertex == Point2D(0, 5)
assert pa6.eccentricity == 1
assert pa7.focus == Point2D(3, 7)
assert pa7.focal_length == half
assert pa7.p_parameter == -half
assert pa7.vertex == Point2D(7*half, 7)
assert pa4.focal_length == half
assert pa4.p_parameter == half
assert pa4.vertex == Point2D(3, 13*half)
assert pa8.focal_length == 1
assert pa8.p_parameter == 1
assert pa8.vertex == Point2D(5, 0)
assert pa4.focal_length == pa5.focal_length
assert pa4.p_parameter == pa5.p_parameter
assert pa4.vertex == pa5.vertex
assert pa4.equation() == pa5.equation()
assert pa8.focal_length == pa9.focal_length
assert pa8.p_parameter == pa9.p_parameter
assert pa8.vertex == pa9.vertex
assert pa8.equation() == pa9.equation()
assert pa10.focal_length == pa11.focal_length == sqrt((a - b) ** 2) / 2 # if a, b real == abs(a - b)/2
assert pa11.vertex == Point(*pa10.vertex[::-1]) == Point(a,
a - sqrt((a - b)**2)*sign(a - b)/2) # change axis x->y, y->x on pa10
def test_parabola_intersection():
l1 = Line(Point(1, -2), Point(-1,-2))
l2 = Line(Point(1, 2), Point(-1,2))
l3 = Line(Point(1, 0), Point(-1,0))
p1 = Point(0,0)
p2 = Point(0, -2)
p3 = Point(120, -12)
parabola1 = Parabola(p1, l1)
# parabola with parabola
assert parabola1.intersection(parabola1) == [parabola1]
assert parabola1.intersection(Parabola(p1, l2)) == [Point2D(-2, 0), Point2D(2, 0)]
assert parabola1.intersection(Parabola(p2, l3)) == [Point2D(0, -1)]
assert parabola1.intersection(Parabola(Point(16, 0), l1)) == [Point2D(8, 15)]
assert parabola1.intersection(Parabola(Point(0, 16), l1)) == [Point2D(-6, 8), Point2D(6, 8)]
assert parabola1.intersection(Parabola(p3, l3)) == []
# parabola with point
assert parabola1.intersection(p1) == []
assert parabola1.intersection(Point2D(0, -1)) == [Point2D(0, -1)]
assert parabola1.intersection(Point2D(4, 3)) == [Point2D(4, 3)]
# parabola with line
assert parabola1.intersection(Line(Point2D(-7, 3), Point(12, 3))) == [Point2D(-4, 3), Point2D(4, 3)]
assert parabola1.intersection(Line(Point(-4, -1), Point(4, -1))) == [Point(0, -1)]
assert parabola1.intersection(Line(Point(2, 0), Point(0, -2))) == [Point2D(2, 0)]
# parabola with segment
assert parabola1.intersection(Segment2D((-4, -5), (4, 3))) == [Point2D(0, -1), Point2D(4, 3)]
assert parabola1.intersection(Segment2D((0, -5), (0, 6))) == [Point2D(0, -1)]
assert parabola1.intersection(Segment2D((-12, -65), (14, -68))) == []
# parabola with ray
assert parabola1.intersection(Ray2D((-4, -5), (4, 3))) == [Point2D(0, -1), Point2D(4, 3)]
assert parabola1.intersection(Ray2D((0, 7), (1, 14))) == [Point2D(14 + 2*sqrt(57), 105 + 14*sqrt(57))]
assert parabola1.intersection(Ray2D((0, 7), (0, 14))) == []
# parabola with ellipse/circle
assert parabola1.intersection(Circle(p1, 2)) == [Point2D(-2, 0), Point2D(2, 0)]
assert parabola1.intersection(Circle(p2, 1)) == [Point2D(0, -1), Point2D(0, -1)]
assert parabola1.intersection(Ellipse(p2, 2, 1)) == [Point2D(0, -1), Point2D(0, -1)]
assert parabola1.intersection(Ellipse(Point(0, 19), 5, 7)) == []
assert parabola1.intersection(Ellipse((0, 3), 12, 4)) == \
[Point2D(0, -1), Point2D(0, -1), Point2D(-4*sqrt(17)/3, S(59)/9), Point2D(4*sqrt(17)/3, S(59)/9)]
|
ef91496f98495beba62d33ca9cc1f674f6f799adeca8ada11c9d004225b0c432 | """Utilities to deal with sympy.Matrix, numpy and scipy.sparse."""
from __future__ import print_function, division
from sympy import MatrixBase, I, Expr, Integer
from sympy.matrices import eye, zeros
from sympy.external import import_module
__all__ = [
'numpy_ndarray',
'scipy_sparse_matrix',
'sympy_to_numpy',
'sympy_to_scipy_sparse',
'numpy_to_sympy',
'scipy_sparse_to_sympy',
'flatten_scalar',
'matrix_dagger',
'to_sympy',
'to_numpy',
'to_scipy_sparse',
'matrix_tensor_product',
'matrix_zeros'
]
# Conditionally define the base classes for numpy and scipy.sparse arrays
# for use in isinstance tests.
np = import_module('numpy')
if not np:
class numpy_ndarray(object):
pass
else:
numpy_ndarray = np.ndarray
scipy = import_module('scipy', __import__kwargs={'fromlist': ['sparse']})
if not scipy:
class scipy_sparse_matrix(object):
pass
sparse = None
else:
sparse = scipy.sparse
# Try to find spmatrix.
if hasattr(sparse, 'base'):
# Newer versions have it under scipy.sparse.base.
scipy_sparse_matrix = sparse.base.spmatrix
elif hasattr(sparse, 'sparse'):
# Older versions have it under scipy.sparse.sparse.
scipy_sparse_matrix = sparse.sparse.spmatrix
def sympy_to_numpy(m, **options):
"""Convert a sympy Matrix/complex number to a numpy matrix or scalar."""
if not np:
raise ImportError
dtype = options.get('dtype', 'complex')
if isinstance(m, MatrixBase):
return np.matrix(m.tolist(), dtype=dtype)
elif isinstance(m, Expr):
if m.is_Number or m.is_NumberSymbol or m == I:
return complex(m)
raise TypeError('Expected MatrixBase or complex scalar, got: %r' % m)
def sympy_to_scipy_sparse(m, **options):
"""Convert a sympy Matrix/complex number to a numpy matrix or scalar."""
if not np or not sparse:
raise ImportError
dtype = options.get('dtype', 'complex')
if isinstance(m, MatrixBase):
return sparse.csr_matrix(np.matrix(m.tolist(), dtype=dtype))
elif isinstance(m, Expr):
if m.is_Number or m.is_NumberSymbol or m == I:
return complex(m)
raise TypeError('Expected MatrixBase or complex scalar, got: %r' % m)
def scipy_sparse_to_sympy(m, **options):
"""Convert a scipy.sparse matrix to a sympy matrix."""
return MatrixBase(m.todense())
def numpy_to_sympy(m, **options):
"""Convert a numpy matrix to a sympy matrix."""
return MatrixBase(m)
def to_sympy(m, **options):
"""Convert a numpy/scipy.sparse matrix to a sympy matrix."""
if isinstance(m, MatrixBase):
return m
elif isinstance(m, numpy_ndarray):
return numpy_to_sympy(m)
elif isinstance(m, scipy_sparse_matrix):
return scipy_sparse_to_sympy(m)
elif isinstance(m, Expr):
return m
raise TypeError('Expected sympy/numpy/scipy.sparse matrix, got: %r' % m)
def to_numpy(m, **options):
"""Convert a sympy/scipy.sparse matrix to a numpy matrix."""
dtype = options.get('dtype', 'complex')
if isinstance(m, (MatrixBase, Expr)):
return sympy_to_numpy(m, dtype=dtype)
elif isinstance(m, numpy_ndarray):
return m
elif isinstance(m, scipy_sparse_matrix):
return m.todense()
raise TypeError('Expected sympy/numpy/scipy.sparse matrix, got: %r' % m)
def to_scipy_sparse(m, **options):
"""Convert a sympy/numpy matrix to a scipy.sparse matrix."""
dtype = options.get('dtype', 'complex')
if isinstance(m, (MatrixBase, Expr)):
return sympy_to_scipy_sparse(m, dtype=dtype)
elif isinstance(m, numpy_ndarray):
if not sparse:
raise ImportError
return sparse.csr_matrix(m)
elif isinstance(m, scipy_sparse_matrix):
return m
raise TypeError('Expected sympy/numpy/scipy.sparse matrix, got: %r' % m)
def flatten_scalar(e):
"""Flatten a 1x1 matrix to a scalar, return larger matrices unchanged."""
if isinstance(e, MatrixBase):
if e.shape == (1, 1):
e = e[0]
if isinstance(e, (numpy_ndarray, scipy_sparse_matrix)):
if e.shape == (1, 1):
e = complex(e[0, 0])
return e
def matrix_dagger(e):
"""Return the dagger of a sympy/numpy/scipy.sparse matrix."""
if isinstance(e, MatrixBase):
return e.H
elif isinstance(e, (numpy_ndarray, scipy_sparse_matrix)):
return e.conjugate().transpose()
raise TypeError('Expected sympy/numpy/scipy.sparse matrix, got: %r' % e)
# TODO: Move this into sympy.matricies.
def _sympy_tensor_product(*matrices):
"""Compute the kronecker product of a sequence of sympy Matrices.
"""
from sympy.matrices.expressions.kronecker import matrix_kronecker_product
return matrix_kronecker_product(*matrices)
def _numpy_tensor_product(*product):
"""numpy version of tensor product of multiple arguments."""
if not np:
raise ImportError
answer = product[0]
for item in product[1:]:
answer = np.kron(answer, item)
return answer
def _scipy_sparse_tensor_product(*product):
"""scipy.sparse version of tensor product of multiple arguments."""
if not sparse:
raise ImportError
answer = product[0]
for item in product[1:]:
answer = sparse.kron(answer, item)
# The final matrices will just be multiplied, so csr is a good final
# sparse format.
return sparse.csr_matrix(answer)
def matrix_tensor_product(*product):
"""Compute the matrix tensor product of sympy/numpy/scipy.sparse matrices."""
if isinstance(product[0], MatrixBase):
return _sympy_tensor_product(*product)
elif isinstance(product[0], numpy_ndarray):
return _numpy_tensor_product(*product)
elif isinstance(product[0], scipy_sparse_matrix):
return _scipy_sparse_tensor_product(*product)
def _numpy_eye(n):
"""numpy version of complex eye."""
if not np:
raise ImportError
return np.matrix(np.eye(n, dtype='complex'))
def _scipy_sparse_eye(n):
"""scipy.sparse version of complex eye."""
if not sparse:
raise ImportError
return sparse.eye(n, n, dtype='complex')
def matrix_eye(n, **options):
"""Get the version of eye and tensor_product for a given format."""
format = options.get('format', 'sympy')
if format == 'sympy':
return eye(n)
elif format == 'numpy':
return _numpy_eye(n)
elif format == 'scipy.sparse':
return _scipy_sparse_eye(n)
raise NotImplementedError('Invalid format: %r' % format)
def _numpy_zeros(m, n, **options):
"""numpy version of zeros."""
dtype = options.get('dtype', 'float64')
if not np:
raise ImportError
return np.zeros((m, n), dtype=dtype)
def _scipy_sparse_zeros(m, n, **options):
"""scipy.sparse version of zeros."""
spmatrix = options.get('spmatrix', 'csr')
dtype = options.get('dtype', 'float64')
if not sparse:
raise ImportError
if spmatrix == 'lil':
return sparse.lil_matrix((m, n), dtype=dtype)
elif spmatrix == 'csr':
return sparse.csr_matrix((m, n), dtype=dtype)
def matrix_zeros(m, n, **options):
""""Get a zeros matrix for a given format."""
format = options.get('format', 'sympy')
dtype = options.get('dtype', 'float64')
spmatrix = options.get('spmatrix', 'csr')
if format == 'sympy':
return zeros(m, n)
elif format == 'numpy':
return _numpy_zeros(m, n, **options)
elif format == 'scipy.sparse':
return _scipy_sparse_zeros(m, n, **options)
raise NotImplementedError('Invaild format: %r' % format)
def _numpy_matrix_to_zero(e):
"""Convert a numpy zero matrix to the zero scalar."""
if not np:
raise ImportError
test = np.zeros_like(e)
if np.allclose(e, test):
return 0.0
else:
return e
def _scipy_sparse_matrix_to_zero(e):
"""Convert a scipy.sparse zero matrix to the zero scalar."""
if not np:
raise ImportError
edense = e.todense()
test = np.zeros_like(edense)
if np.allclose(edense, test):
return 0.0
else:
return e
def matrix_to_zero(e):
"""Convert a zero matrix to the scalar zero."""
if isinstance(e, MatrixBase):
if zeros(*e.shape) == e:
e = Integer(0)
elif isinstance(e, numpy_ndarray):
e = _numpy_matrix_to_zero(e)
elif isinstance(e, scipy_sparse_matrix):
e = _scipy_sparse_matrix_to_zero(e)
return e
|
60121ed3d120a0fcf6668835e154a36c32089a5c390073e167b012094315746b | from sympy import Rational, pi, sqrt, S
from sympy.physics.units.quantities import Quantity
from sympy.physics.units.dimensions import (dimsys_default, Dimension,
acceleration, action, amount_of_substance, capacitance, charge,
conductance, current, energy, force, frequency, information, impedance,
inductance, length, luminous_intensity, magnetic_density, magnetic_flux,
mass, power, pressure, temperature, time, velocity, voltage)
from sympy.physics.units.prefixes import (
centi, deci, kilo, micro, milli, nano, pico,
kibi, mebi, gibi, tebi, pebi, exbi)
One = S.One
#### UNITS ####
# Dimensionless:
percent = percents = Quantity("percent", latex_repr=r"\%")
percent.set_dimension(One)
percent.set_scale_factor(Rational(1, 100))
permille = Quantity("permille")
permille.set_dimension(One)
permille.set_scale_factor(Rational(1, 1000))
# Angular units (dimensionless)
rad = radian = radians = Quantity("radian", abbrev="rad")
radian.set_dimension(One)
radian.set_scale_factor(One)
deg = degree = degrees = Quantity("degree", abbrev="deg", latex_repr=r"^\circ")
degree.set_dimension(One)
degree.set_scale_factor(pi/180)
sr = steradian = steradians = Quantity("steradian", abbrev="sr")
steradian.set_dimension(One)
steradian.set_scale_factor(One)
mil = angular_mil = angular_mils = Quantity("angular_mil", abbrev="mil")
angular_mil.set_dimension(One)
angular_mil.set_scale_factor(2*pi/6400)
# Base units:
m = meter = meters = Quantity("meter", abbrev="m")
meter.set_dimension(length)
meter.set_scale_factor(One)
# NOTE: the `kilogram` has scale factor of 1 in SI.
# The current state of the code assumes SI unit dimensions, in
# the future this module will be modified in order to be unit system-neutral
# (that is, support all kinds of unit systems).
kg = kilogram = kilograms = Quantity("kilogram", abbrev="kg")
kilogram.set_dimension(mass)
kilogram.set_scale_factor(One)
s = second = seconds = Quantity("second", abbrev="s")
second.set_dimension(time)
second.set_scale_factor(One)
A = ampere = amperes = Quantity("ampere", abbrev='A')
ampere.set_dimension(current)
ampere.set_scale_factor(One)
K = kelvin = kelvins = Quantity("kelvin", abbrev='K')
kelvin.set_dimension(temperature)
kelvin.set_scale_factor(One)
mol = mole = moles = Quantity("mole", abbrev="mol")
mole.set_dimension(amount_of_substance)
mole.set_scale_factor(One)
cd = candela = candelas = Quantity("candela", abbrev="cd")
candela.set_dimension(luminous_intensity)
candela.set_scale_factor(One)
g = gram = grams = Quantity("gram", abbrev="g")
gram.set_dimension(mass)
gram.set_scale_factor(kilogram/kilo)
mg = milligram = milligrams = Quantity("milligram", abbrev="mg")
milligram.set_dimension(mass)
milligram.set_scale_factor(milli*gram)
ug = microgram = micrograms = Quantity("microgram", abbrev="ug", latex_repr=r"\mu\text{g}")
microgram.set_dimension(mass)
microgram.set_scale_factor(micro*gram)
# derived units
newton = newtons = N = Quantity("newton", abbrev="N")
newton.set_dimension(force)
newton.set_scale_factor(kilogram*meter/second**2)
joule = joules = J = Quantity("joule", abbrev="J")
joule.set_dimension(energy)
joule.set_scale_factor(newton*meter)
watt = watts = W = Quantity("watt", abbrev="W")
watt.set_dimension(power)
watt.set_scale_factor(joule/second)
pascal = pascals = Pa = pa = Quantity("pascal", abbrev="Pa")
pascal.set_dimension(pressure)
pascal.set_scale_factor(newton/meter**2)
hertz = hz = Hz = Quantity("hertz", abbrev="Hz")
hertz.set_dimension(frequency)
hertz.set_scale_factor(One)
# MKSA extension to MKS: derived units
coulomb = coulombs = C = Quantity("coulomb", abbrev='C')
coulomb.set_dimension(charge)
coulomb.set_scale_factor(One)
volt = volts = v = V = Quantity("volt", abbrev='V')
volt.set_dimension(voltage)
volt.set_scale_factor(joule/coulomb)
ohm = ohms = Quantity("ohm", abbrev='ohm', latex_repr=r"\Omega")
ohm.set_dimension(impedance)
ohm.set_scale_factor(volt/ampere)
siemens = S = mho = mhos = Quantity("siemens", abbrev='S')
siemens.set_dimension(conductance)
siemens.set_scale_factor(ampere/volt)
farad = farads = F = Quantity("farad", abbrev='F')
farad.set_dimension(capacitance)
farad.set_scale_factor(coulomb/volt)
henry = henrys = H = Quantity("henry", abbrev='H')
henry.set_dimension(inductance)
henry.set_scale_factor(volt*second/ampere)
tesla = teslas = T = Quantity("tesla", abbrev='T')
tesla.set_dimension(magnetic_density)
tesla.set_scale_factor(volt*second/meter**2)
weber = webers = Wb = wb = Quantity("weber", abbrev='Wb')
weber.set_dimension(magnetic_flux)
weber.set_scale_factor(joule/ampere)
# Other derived units:
optical_power = dioptre = diopter = D = Quantity("dioptre")
dioptre.set_dimension(1/length)
dioptre.set_scale_factor(1/meter)
lux = lx = Quantity("lux", abbrev="lx")
lux.set_dimension(luminous_intensity/length**2)
lux.set_scale_factor(steradian*candela/meter**2)
# katal is the SI unit of catalytic activity
katal = kat = Quantity("katal", abbrev="kat")
katal.set_dimension(amount_of_substance/time)
katal.set_scale_factor(mol/second)
# gray is the SI unit of absorbed dose
gray = Gy = Quantity("gray")
gray.set_dimension(energy/mass)
gray.set_scale_factor(meter**2/second**2)
# becquerel is the SI unit of radioactivity
becquerel = Bq = Quantity("becquerel", abbrev="Bq")
becquerel.set_dimension(1/time)
becquerel.set_scale_factor(1/second)
# Common length units
km = kilometer = kilometers = Quantity("kilometer", abbrev="km")
kilometer.set_dimension(length)
kilometer.set_scale_factor(kilo*meter)
dm = decimeter = decimeters = Quantity("decimeter", abbrev="dm")
decimeter.set_dimension(length)
decimeter.set_scale_factor(deci*meter)
cm = centimeter = centimeters = Quantity("centimeter", abbrev="cm")
centimeter.set_dimension(length)
centimeter.set_scale_factor(centi*meter)
mm = millimeter = millimeters = Quantity("millimeter", abbrev="mm")
millimeter.set_dimension(length)
millimeter.set_scale_factor(milli*meter)
um = micrometer = micrometers = micron = microns = \
Quantity("micrometer", abbrev="um", latex_repr=r'\mu\text{m}')
micrometer.set_dimension(length)
micrometer.set_scale_factor(micro*meter)
nm = nanometer = nanometers = Quantity("nanometer", abbrev="nm")
nanometer.set_dimension(length)
nanometer.set_scale_factor(nano*meter)
pm = picometer = picometers = Quantity("picometer", abbrev="pm")
picometer.set_dimension(length)
picometer.set_scale_factor(pico*meter)
ft = foot = feet = Quantity("foot", abbrev="ft")
foot.set_dimension(length)
foot.set_scale_factor(Rational(3048, 10000)*meter)
inch = inches = Quantity("inch")
inch.set_dimension(length)
inch.set_scale_factor(foot/12)
yd = yard = yards = Quantity("yard", abbrev="yd")
yard.set_dimension(length)
yard.set_scale_factor(3*feet)
mi = mile = miles = Quantity("mile")
mile.set_dimension(length)
mile.set_scale_factor(5280*feet)
nmi = nautical_mile = nautical_miles = Quantity("nautical_mile")
nautical_mile.set_dimension(length)
nautical_mile.set_scale_factor(6076*feet)
# Common volume and area units
l = liter = liters = Quantity("liter")
liter.set_dimension(length**3)
liter.set_scale_factor(meter**3 / 1000)
dl = deciliter = deciliters = Quantity("deciliter")
deciliter.set_dimension(length**3)
deciliter.set_scale_factor(liter / 10)
cl = centiliter = centiliters = Quantity("centiliter")
centiliter.set_dimension(length**3)
centiliter.set_scale_factor(liter / 100)
ml = milliliter = milliliters = Quantity("milliliter")
milliliter.set_dimension(length**3)
milliliter.set_scale_factor(liter / 1000)
# Common time units
ms = millisecond = milliseconds = Quantity("millisecond", abbrev="ms")
millisecond.set_dimension(time)
millisecond.set_scale_factor(milli*second)
us = microsecond = microseconds = Quantity("microsecond", abbrev="us", latex_repr=r'\mu\text{s}')
microsecond.set_dimension(time)
microsecond.set_scale_factor(micro*second)
ns = nanosecond = nanoseconds = Quantity("nanosecond", abbrev="ns")
nanosecond.set_dimension(time)
nanosecond.set_scale_factor(nano*second)
ps = picosecond = picoseconds = Quantity("picosecond", abbrev="ps")
picosecond.set_dimension(time)
picosecond.set_scale_factor(pico*second)
minute = minutes = Quantity("minute")
minute.set_dimension(time)
minute.set_scale_factor(60*second)
h = hour = hours = Quantity("hour")
hour.set_dimension(time)
hour.set_scale_factor(60*minute)
day = days = Quantity("day")
day.set_dimension(time)
day.set_scale_factor(24*hour)
anomalistic_year = anomalistic_years = Quantity("anomalistic_year")
anomalistic_year.set_dimension(time)
anomalistic_year.set_scale_factor(365.259636*day)
sidereal_year = sidereal_years = Quantity("sidereal_year")
sidereal_year.set_dimension(time)
sidereal_year.set_scale_factor(31558149.540)
tropical_year = tropical_years = Quantity("tropical_year")
tropical_year.set_dimension(time)
tropical_year.set_scale_factor(365.24219*day)
common_year = common_years = Quantity("common_year")
common_year.set_dimension(time)
common_year.set_scale_factor(365*day)
julian_year = julian_years = Quantity("julian_year")
julian_year.set_dimension(time)
julian_year.set_scale_factor((365 + One/4)*day)
draconic_year = draconic_years = Quantity("draconic_year")
draconic_year.set_dimension(time)
draconic_year.set_scale_factor(346.62*day)
gaussian_year = gaussian_years = Quantity("gaussian_year")
gaussian_year.set_dimension(time)
gaussian_year.set_scale_factor(365.2568983*day)
full_moon_cycle = full_moon_cycles = Quantity("full_moon_cycle")
full_moon_cycle.set_dimension(time)
full_moon_cycle.set_scale_factor(411.78443029*day)
year = years = tropical_year
#### CONSTANTS ####
# Newton constant
G = gravitational_constant = Quantity("gravitational_constant", abbrev="G")
gravitational_constant.set_dimension(length**3*mass**-1*time**-2)
gravitational_constant.set_scale_factor(6.67408e-11*m**3/(kg*s**2))
# speed of light
c = speed_of_light = Quantity("speed_of_light", abbrev="c")
speed_of_light.set_dimension(velocity)
speed_of_light.set_scale_factor(299792458*meter/second)
# Reduced Planck constant
hbar = Quantity("hbar", abbrev="hbar")
hbar.set_dimension(action)
hbar.set_scale_factor(1.05457266e-34*joule*second)
# Planck constant
planck = Quantity("planck", abbrev="h")
planck.set_dimension(action)
planck.set_scale_factor(2*pi*hbar)
# Electronvolt
eV = electronvolt = electronvolts = Quantity("electronvolt", abbrev="eV")
electronvolt.set_dimension(energy)
electronvolt.set_scale_factor(1.60219e-19*joule)
# Avogadro number
avogadro_number = Quantity("avogadro_number")
avogadro_number.set_dimension(One)
avogadro_number.set_scale_factor(6.022140857e23)
# Avogadro constant
avogadro = avogadro_constant = Quantity("avogadro_constant")
avogadro_constant.set_dimension(amount_of_substance**-1)
avogadro_constant.set_scale_factor(avogadro_number / mol)
# Boltzmann constant
boltzmann = boltzmann_constant = Quantity("boltzmann_constant")
boltzmann_constant.set_dimension(energy/temperature)
boltzmann_constant.set_scale_factor(1.38064852e-23*joule/kelvin)
# Stefan-Boltzmann constant
stefan = stefan_boltzmann_constant = Quantity("stefan_boltzmann_constant")
stefan_boltzmann_constant.set_dimension(energy*time**-1*length**-2*temperature**-4)
stefan_boltzmann_constant.set_scale_factor(5.670367e-8*joule/(s*m**2*kelvin**4))
# Atomic mass
amu = amus = atomic_mass_unit = atomic_mass_constant = Quantity("atomic_mass_constant")
atomic_mass_constant.set_dimension(mass)
atomic_mass_constant.set_scale_factor(1.660539040e-24*gram)
# Molar gas constant
R = molar_gas_constant = Quantity("molar_gas_constant", abbrev="R")
molar_gas_constant.set_dimension(energy/(temperature * amount_of_substance))
molar_gas_constant.set_scale_factor(8.3144598*joule/kelvin/mol)
# Faraday constant
faraday_constant = Quantity("faraday_constant")
faraday_constant.set_dimension(charge/amount_of_substance)
faraday_constant.set_scale_factor(96485.33289*C/mol)
# Josephson constant
josephson_constant = Quantity("josephson_constant", abbrev="K_j")
josephson_constant.set_dimension(frequency/voltage)
josephson_constant.set_scale_factor(483597.8525e9*hertz/V)
# Von Klitzing constant
von_klitzing_constant = Quantity("von_klitzing_constant", abbrev="R_k")
von_klitzing_constant.set_dimension(voltage/current)
von_klitzing_constant.set_scale_factor(25812.8074555*ohm)
# Acceleration due to gravity (on the Earth surface)
gee = gees = acceleration_due_to_gravity = Quantity("acceleration_due_to_gravity", abbrev="g")
acceleration_due_to_gravity.set_dimension(acceleration)
acceleration_due_to_gravity.set_scale_factor(9.80665*meter/second**2)
# magnetic constant:
u0 = magnetic_constant = vacuum_permeability = Quantity("magnetic_constant")
magnetic_constant.set_dimension(force/current**2)
magnetic_constant.set_scale_factor(4*pi/10**7 * newton/ampere**2)
# electric constat:
e0 = electric_constant = vacuum_permittivity = Quantity("vacuum_permittivity")
vacuum_permittivity.set_dimension(capacitance/length)
vacuum_permittivity.set_scale_factor(1/(u0 * c**2))
# vacuum impedance:
Z0 = vacuum_impedance = Quantity("vacuum_impedance", abbrev='Z_0', latex_repr=r'Z_{0}')
vacuum_impedance.set_dimension(impedance)
vacuum_impedance.set_scale_factor(u0 * c)
# Coulomb's constant:
coulomb_constant = coulombs_constant = electric_force_constant = \
Quantity("coulomb_constant", abbrev="k_e")
coulomb_constant.set_dimension(force*length**2/charge**2)
coulomb_constant.set_scale_factor(1/(4*pi*vacuum_permittivity))
atmosphere = atmospheres = atm = Quantity("atmosphere", abbrev="atm")
atmosphere.set_dimension(pressure)
atmosphere.set_scale_factor(101325 * pascal)
kPa = kilopascal = Quantity("kilopascal", abbrev="kPa")
kilopascal.set_dimension(pressure)
kilopascal.set_scale_factor(kilo*Pa)
bar = bars = Quantity("bar", abbrev="bar")
bar.set_dimension(pressure)
bar.set_scale_factor(100*kPa)
pound = pounds = Quantity("pound") # exact
pound.set_dimension(mass)
pound.set_scale_factor(Rational(45359237, 100000000) * kg)
psi = Quantity("psi")
psi.set_dimension(pressure)
psi.set_scale_factor(pound * gee / inch ** 2)
dHg0 = 13.5951 # approx value at 0 C
mmHg = torr = Quantity("mmHg")
mmHg.set_dimension(pressure)
mmHg.set_scale_factor(dHg0 * acceleration_due_to_gravity * kilogram / meter**2)
mmu = mmus = milli_mass_unit = Quantity("milli_mass_unit")
milli_mass_unit.set_dimension(mass)
milli_mass_unit.set_scale_factor(atomic_mass_unit/1000)
quart = quarts = Quantity("quart")
quart.set_dimension(length**3)
quart.set_scale_factor(Rational(231, 4) * inch**3)
# Other convenient units and magnitudes
ly = lightyear = lightyears = Quantity("lightyear", abbrev="ly")
lightyear.set_dimension(length)
lightyear.set_scale_factor(speed_of_light*julian_year)
au = astronomical_unit = astronomical_units = Quantity("astronomical_unit", abbrev="AU")
astronomical_unit.set_dimension(length)
astronomical_unit.set_scale_factor(149597870691*meter)
# Fundamental Planck units:
planck_mass = Quantity("planck_mass", abbrev="m_P", latex_repr=r'm_\text{P}')
planck_mass.set_dimension(mass)
planck_mass.set_scale_factor(sqrt(hbar*speed_of_light/G))
planck_time = Quantity("planck_time", abbrev="t_P", latex_repr=r't_\text{P}')
planck_time.set_dimension(time)
planck_time.set_scale_factor(sqrt(hbar*G/speed_of_light**5))
planck_temperature = Quantity("planck_temperature", abbrev="T_P",
latex_repr=r'T_\text{P}')
planck_temperature.set_dimension(temperature)
planck_temperature.set_scale_factor(sqrt(hbar*speed_of_light**5/G/boltzmann**2))
planck_length = Quantity("planck_length", abbrev="l_P", latex_repr=r'l_\text{P}')
planck_length.set_dimension(length)
planck_length.set_scale_factor(sqrt(hbar*G/speed_of_light**3))
planck_charge = Quantity("planck_charge", abbrev="q_P", latex_repr=r'q_\text{P}')
planck_charge.set_dimension(charge)
planck_charge.set_scale_factor(sqrt(4*pi*electric_constant*hbar*speed_of_light))
# Derived Planck units:
planck_area = Quantity("planck_area")
planck_area.set_dimension(length**2)
planck_area.set_scale_factor(planck_length**2)
planck_volume = Quantity("planck_volume")
planck_volume.set_dimension(length**3)
planck_volume.set_scale_factor(planck_length**3)
planck_momentum = Quantity("planck_momentum")
planck_momentum.set_dimension(mass*velocity)
planck_momentum.set_scale_factor(planck_mass * speed_of_light)
planck_energy = Quantity("planck_energy", abbrev="E_P", latex_repr=r'E_\text{P}')
planck_energy.set_dimension(energy)
planck_energy.set_scale_factor(planck_mass * speed_of_light**2)
planck_force = Quantity("planck_force", abbrev="F_P", latex_repr=r'F_\text{P}')
planck_force.set_dimension(force)
planck_force.set_scale_factor(planck_energy / planck_length)
planck_power = Quantity("planck_power", abbrev="P_P", latex_repr=r'P_\text{P}')
planck_power.set_dimension(power)
planck_power.set_scale_factor(planck_energy / planck_time)
planck_density = Quantity("planck_density", abbrev="rho_P", latex_repr=r'\rho_\text{P}')
planck_density.set_dimension(mass/length**3)
planck_density.set_scale_factor(planck_mass / planck_length**3)
planck_energy_density = Quantity("planck_energy_density", abbrev="rho^E_P")
planck_energy_density.set_dimension(energy / length**3)
planck_energy_density.set_scale_factor(planck_energy / planck_length**3)
planck_intensity = Quantity("planck_intensity", abbrev="I_P", latex_repr=r'I_\text{P}')
planck_intensity.set_dimension(mass * time**(-3))
planck_intensity.set_scale_factor(planck_energy_density * speed_of_light)
planck_angular_frequency = Quantity("planck_angular_frequency", abbrev="omega_P",
latex_repr=r'\omega_\text{P}')
planck_angular_frequency.set_dimension(1 / time)
planck_angular_frequency.set_scale_factor(1 / planck_time)
planck_pressure = Quantity("planck_pressure", abbrev="p_P", latex_repr=r'p_\text{P}')
planck_pressure.set_dimension(pressure)
planck_pressure.set_scale_factor(planck_force / planck_length**2)
planck_current = Quantity("planck_current", abbrev="I_P", latex_repr=r'I_\text{P}')
planck_current.set_dimension(current)
planck_current.set_scale_factor(planck_charge / planck_time)
planck_voltage = Quantity("planck_voltage", abbrev="V_P", latex_repr=r'V_\text{P}')
planck_voltage.set_dimension(voltage)
planck_voltage.set_scale_factor(planck_energy / planck_charge)
planck_impedance = Quantity("planck_impedance", abbrev="Z_P", latex_repr=r'Z_\text{P}')
planck_impedance.set_dimension(impedance)
planck_impedance.set_scale_factor(planck_voltage / planck_current)
planck_acceleration = Quantity("planck_acceleration", abbrev="a_P",
latex_repr=r'a_\text{P}')
planck_acceleration.set_dimension(acceleration)
planck_acceleration.set_scale_factor(speed_of_light / planck_time)
# Information theory units:
bit = bits = Quantity("bit")
bit.set_dimension(information)
bit.set_scale_factor(One)
byte = bytes = Quantity("byte")
byte.set_dimension(information)
byte.set_scale_factor(8*bit)
kibibyte = kibibytes = Quantity("kibibyte")
kibibyte.set_dimension(information)
kibibyte.set_scale_factor(kibi*byte)
mebibyte = mebibytes = Quantity("mebibyte")
mebibyte.set_dimension(information)
mebibyte.set_scale_factor(mebi*byte)
gibibyte = gibibytes = Quantity("gibibyte")
gibibyte.set_dimension(information)
gibibyte.set_scale_factor(gibi*byte)
tebibyte = tebibytes = Quantity("tebibyte")
tebibyte.set_dimension(information)
tebibyte.set_scale_factor(tebi*byte)
pebibyte = pebibytes = Quantity("pebibyte")
pebibyte.set_dimension(information)
pebibyte.set_scale_factor(pebi*byte)
exbibyte = exbibytes = Quantity("exbibyte")
exbibyte.set_dimension(information)
exbibyte.set_scale_factor(exbi*byte)
# Older units for radioactivity
curie = Ci = Quantity("curie", abbrev="Ci")
curie.set_dimension(1/time)
curie.set_scale_factor(37000000000*becquerel)
rutherford = Rd = Quantity("rutherford", abbrev="Rd")
rutherford.set_dimension(1/time)
rutherford.set_scale_factor(1000000*becquerel)
# check that scale factors are the right SI dimensions:
for _scale_factor, _dimension in zip(
Quantity.SI_quantity_scale_factors.values(),
Quantity.SI_quantity_dimension_map.values()):
dimex = Quantity.get_dimensional_expr(_scale_factor)
if dimex != 1:
if not dimsys_default.equivalent_dims(_dimension, Dimension(dimex)):
raise ValueError("quantity value and dimension mismatch")
del _scale_factor, _dimension
|
09712b76f8010af32a09016e52da4a4caf0603b7d513ee98e71eb746e66d30b5 | """
Unit system for physical quantities; include definition of constants.
"""
from __future__ import division
from sympy.utilities.exceptions import SymPyDeprecationWarning
from .dimensions import DimensionSystem
class UnitSystem(object):
"""
UnitSystem represents a coherent set of units.
A unit system is basically a dimension system with notions of scales. Many
of the methods are defined in the same way.
It is much better if all base units have a symbol.
"""
def __init__(self, base, units=(), name="", descr=""):
self.name = name
self.descr = descr
# construct the associated dimension system
base_dims = [u.dimension for u in base]
derived_dims = [u.dimension for u in units if u.dimension not in base_dims]
self._system = DimensionSystem(base_dims, derived_dims)
if not self.is_consistent:
raise ValueError("UnitSystem is not consistent")
self._units = tuple(set(base) | set(units))
# create a dict linkin
# this is possible since we have already verified that the base units
# form a coherent system
base_dict = dict((u.dimension, u) for u in base)
# order the base units in the same order than the dimensions in the
# associated system, in order to ensure that we get always the same
self._base_units = tuple(base_dict[d] for d in self._system.base_dims)
def __str__(self):
"""
Return the name of the system.
If it does not exist, then it makes a list of symbols (or names) of
the base dimensions.
"""
if self.name != "":
return self.name
else:
return "UnitSystem((%s))" % ", ".join(
str(d) for d in self._base_units)
def __repr__(self):
return '<UnitSystem: %s>' % repr(self._base_units)
def extend(self, base, units=(), name="", description=""):
"""Extend the current system into a new one.
Take the base and normal units of the current system to merge
them to the base and normal units given in argument.
If not provided, name and description are overridden by empty strings.
"""
base = self._base_units + tuple(base)
units = self._units + tuple(units)
return UnitSystem(base, units, name, description)
def print_unit_base(self, unit):
"""
Useless method.
DO NOT USE, use instead ``convert_to``.
Give the string expression of a unit in term of the basis.
Units are displayed by decreasing power.
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="print_unit_base",
useinstead="convert_to",
).warn()
from sympy.physics.units import convert_to
return convert_to(unit, self._base_units)
@property
def dim(self):
"""
Give the dimension of the system.
That is return the number of units forming the basis.
"""
return self._system.dim
@property
def is_consistent(self):
"""
Check if the underlying dimension system is consistent.
"""
# test is performed in DimensionSystem
return self._system.is_consistent
|
fde34f1112e31d54c8c933cb5a6c57f1c3706d0188015f42a80ade5e6f484066 | """
Definition of physical dimensions.
Unit systems will be constructed on top of these dimensions.
Most of the examples in the doc use MKS system and are presented from the
computer point of view: from a human point, adding length to time is not legal
in MKS but it is in natural system; for a computer in natural system there is
no time dimension (but a velocity dimension instead) - in the basis - so the
question of adding time to length has no meaning.
"""
from __future__ import division
import collections
from sympy import Integer, Matrix, S, Symbol, sympify, Basic, Tuple, Dict, default_sort_key
from sympy.core.compatibility import reduce, string_types
from sympy.core.expr import Expr
from sympy.core.power import Pow
from sympy.utilities.exceptions import SymPyDeprecationWarning
class Dimension(Expr):
"""
This class represent the dimension of a physical quantities.
The ``Dimension`` constructor takes as parameters a name and an optional
symbol.
For example, in classical mechanics we know that time is different from
temperature and dimensions make this difference (but they do not provide
any measure of these quantites.
>>> from sympy.physics.units import Dimension
>>> length = Dimension('length')
>>> length
Dimension(length)
>>> time = Dimension('time')
>>> time
Dimension(time)
Dimensions can be composed using multiplication, division and
exponentiation (by a number) to give new dimensions. Addition and
subtraction is defined only when the two objects are the same dimension.
>>> velocity = length / time
>>> velocity
Dimension(length/time)
It is possible to use a dimension system object to get the dimensionsal
dependencies of a dimension, for example the dimension system used by the
SI units convention can be used:
>>> from sympy.physics.units.dimensions import dimsys_SI
>>> dimsys_SI.get_dimensional_dependencies(velocity)
{'length': 1, 'time': -1}
>>> length + length
Dimension(length)
>>> l2 = length**2
>>> l2
Dimension(length**2)
>>> dimsys_SI.get_dimensional_dependencies(l2)
{'length': 2}
"""
_op_priority = 13.0
_dimensional_dependencies = dict()
is_commutative = True
is_number = False
# make sqrt(M**2) --> M
is_positive = True
is_real = True
def __new__(cls, name, symbol=None):
if isinstance(name, string_types):
name = Symbol(name)
else:
name = sympify(name)
if not isinstance(name, Expr):
raise TypeError("Dimension name needs to be a valid math expression")
if isinstance(symbol, string_types):
symbol = Symbol(symbol)
elif symbol is not None:
assert isinstance(symbol, Symbol)
if symbol is not None:
obj = Expr.__new__(cls, name, symbol)
else:
obj = Expr.__new__(cls, name)
obj._name = name
obj._symbol = symbol
return obj
@property
def name(self):
return self._name
@property
def symbol(self):
return self._symbol
def __hash__(self):
return Expr.__hash__(self)
def __eq__(self, other):
if isinstance(other, Dimension):
return self.name == other.name
return False
def __str__(self):
"""
Display the string representation of the dimension.
"""
if self.symbol is None:
return "Dimension(%s)" % (self.name)
else:
return "Dimension(%s, %s)" % (self.name, self.symbol)
def __repr__(self):
return self.__str__()
def __neg__(self):
return self
def _register_as_base_dim(self):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="do not call ._register_as_base_dim()",
useinstead="DimensionSystem"
).warn()
if not self.name.is_Symbol:
raise TypeError("Base dimensions need to have symbolic name")
name = self.name
if name in dimsys_default.dimensional_dependencies:
raise IndexError("already in dependencies dict")
# Horrible code:
d = dict(dimsys_default.dimensional_dependencies)
d[name] = Dict({name: 1})
dimsys_default._args = (dimsys_default.args[:2] + (Dict(d),))
def __add__(self, other):
from sympy.physics.units.quantities import Quantity
other = sympify(other)
if isinstance(other, Basic):
if other.has(Quantity):
other = Dimension(Quantity.get_dimensional_expr(other))
if isinstance(other, Dimension) and self == other:
return self
return super(Dimension, self).__add__(other)
return self
def __radd__(self, other):
return self + other
def __sub__(self, other):
# there is no notion of ordering (or magnitude) among dimension,
# subtraction is equivalent to addition when the operation is legal
return self + other
def __rsub__(self, other):
# there is no notion of ordering (or magnitude) among dimension,
# subtraction is equivalent to addition when the operation is legal
return self + other
def __pow__(self, other):
return self._eval_power(other)
def _eval_power(self, other):
other = sympify(other)
return Dimension(self.name**other)
def __mul__(self, other):
from sympy.physics.units.quantities import Quantity
if isinstance(other, Basic):
if other.has(Quantity):
other = Dimension(Quantity.get_dimensional_expr(other))
if isinstance(other, Dimension):
return Dimension(self.name*other.name)
if not other.free_symbols: # other.is_number cannot be used
return self
return super(Dimension, self).__mul__(other)
return self
def __rmul__(self, other):
return self*other
def __div__(self, other):
return self*Pow(other, -1)
def __rdiv__(self, other):
return other * pow(self, -1)
__truediv__ = __div__
__rtruediv__ = __rdiv__
def get_dimensional_dependencies(self, mark_dimensionless=False):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="do not call",
useinstead="DimensionSystem"
).warn()
name = self.name
dimdep = dimsys_default.get_dimensional_dependencies(name)
if mark_dimensionless and dimdep == {}:
return {'dimensionless': 1}
return {str(i): j for i, j in dimdep.items()}
@classmethod
def _from_dimensional_dependencies(cls, dependencies):
return reduce(lambda x, y: x * y, (
Dimension(d)**e for d, e in dependencies.items()
))
@classmethod
def _get_dimensional_dependencies_for_name(cls, name):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="do not call from `Dimension` objects.",
useinstead="DimensionSystem"
).warn()
return dimsys_default.get_dimensional_dependencies(name)
@property
def is_dimensionless(self, dimensional_dependencies=None):
"""
Check if the dimension object really has a dimension.
A dimension should have at least one component with non-zero power.
"""
if self.name == 1:
return True
if dimensional_dependencies is None:
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="wrong class",
).warn()
dimensional_dependencies=dimsys_default
return dimensional_dependencies.get_dimensional_dependencies(self) == {}
def has_integer_powers(self, dim_sys):
"""
Check if the dimension object has only integer powers.
All the dimension powers should be integers, but rational powers may
appear in intermediate steps. This method may be used to check that the
final result is well-defined.
"""
for dpow in dim_sys.get_dimensional_dependencies(self).values():
if not isinstance(dpow, (int, Integer)):
return False
return True
# base dimensions (MKS)
length = Dimension(name="length", symbol="L")
mass = Dimension(name="mass", symbol="M")
time = Dimension(name="time", symbol="T")
# base dimensions (MKSA not in MKS)
current = Dimension(name='current', symbol='I')
# other base dimensions:
temperature = Dimension("temperature", "T")
amount_of_substance = Dimension("amount_of_substance")
luminous_intensity = Dimension("luminous_intensity")
# derived dimensions (MKS)
velocity = Dimension(name="velocity")
acceleration = Dimension(name="acceleration")
momentum = Dimension(name="momentum")
force = Dimension(name="force", symbol="F")
energy = Dimension(name="energy", symbol="E")
power = Dimension(name="power")
pressure = Dimension(name="pressure")
frequency = Dimension(name="frequency", symbol="f")
action = Dimension(name="action", symbol="A")
volume = Dimension("volume")
# derived dimensions (MKSA not in MKS)
voltage = Dimension(name='voltage', symbol='U')
impedance = Dimension(name='impedance', symbol='Z')
conductance = Dimension(name='conductance', symbol='G')
capacitance = Dimension(name='capacitance')
inductance = Dimension(name='inductance')
charge = Dimension(name='charge', symbol='Q')
magnetic_density = Dimension(name='magnetic_density', symbol='B')
magnetic_flux = Dimension(name='magnetic_flux')
# Dimensions in information theory:
information = Dimension(name='information')
# Create dimensions according the the base units in MKSA.
# For other unit systems, they can be derived by transforming the base
# dimensional dependency dictionary.
class DimensionSystem(Basic):
r"""
DimensionSystem represents a coherent set of dimensions.
The constructor takes three parameters:
- base dimensions;
- derived dimensions: these are defined in terms of the base dimensions
(for example velocity is defined from the division of length by time);
- dependency of dimensions: how the derived dimensions depend
on the base dimensions.
Optionally either the ``derived_dims`` or the ``dimensional_dependencies``
may be omitted.
"""
def __new__(cls, base_dims, derived_dims=[], dimensional_dependencies={}, name=None, descr=None):
dimensional_dependencies = dict(dimensional_dependencies)
if (name is not None) or (descr is not None):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
useinstead="do not define a `name` or `descr`",
).warn()
def parse_dim(dim):
if isinstance(dim, string_types):
dim = Dimension(Symbol(dim))
elif isinstance(dim, Dimension):
pass
elif isinstance(dim, Symbol):
dim = Dimension(dim)
else:
raise TypeError("%s wrong type" % dim)
return dim
base_dims = [parse_dim(i) for i in base_dims]
derived_dims = [parse_dim(i) for i in derived_dims]
for dim in base_dims:
dim = dim.name
if (dim in dimensional_dependencies
and (len(dimensional_dependencies[dim]) != 1 or
dimensional_dependencies[dim].get(dim, None) != 1)):
raise IndexError("Repeated value in base dimensions")
dimensional_dependencies[dim] = Dict({dim: 1})
def parse_dim_name(dim):
if isinstance(dim, Dimension):
return dim.name
elif isinstance(dim, string_types):
return Symbol(dim)
elif isinstance(dim, Symbol):
return dim
else:
raise TypeError("unrecognized type %s for %s" % (type(dim), dim))
for dim in dimensional_dependencies.keys():
dim = parse_dim(dim)
if (dim not in derived_dims) and (dim not in base_dims):
derived_dims.append(dim)
def parse_dict(d):
return Dict({parse_dim_name(i): j for i, j in d.items()})
# Make sure everything is a SymPy type:
dimensional_dependencies = {parse_dim_name(i): parse_dict(j) for i, j in
dimensional_dependencies.items()}
for dim in derived_dims:
if dim in base_dims:
raise ValueError("Dimension %s both in base and derived" % dim)
if dim.name not in dimensional_dependencies:
# TODO: should this raise a warning?
dimensional_dependencies[dim] = Dict({dim.name: 1})
base_dims.sort(key=default_sort_key)
derived_dims.sort(key=default_sort_key)
base_dims = Tuple(*base_dims)
derived_dims = Tuple(*derived_dims)
dimensional_dependencies = Dict({i: Dict(j) for i, j in dimensional_dependencies.items()})
obj = Basic.__new__(cls, base_dims, derived_dims, dimensional_dependencies)
return obj
@property
def base_dims(self):
return self.args[0]
@property
def derived_dims(self):
return self.args[1]
@property
def dimensional_dependencies(self):
return self.args[2]
def _get_dimensional_dependencies_for_name(self, name):
if name.is_Symbol:
return dict(self.dimensional_dependencies.get(name, {}))
if name.is_Number:
return {}
get_for_name = dimsys_default._get_dimensional_dependencies_for_name
if name.is_Mul:
ret = collections.defaultdict(int)
dicts = [get_for_name(i) for i in name.args]
for d in dicts:
for k, v in d.items():
ret[k] += v
return {k: v for (k, v) in ret.items() if v != 0}
if name.is_Pow:
dim = get_for_name(name.base)
return {k: v*name.exp for (k, v) in dim.items()}
if name.is_Function:
args = (Dimension._from_dimensional_dependencies(
get_for_name(arg)) for arg in name.args)
result = name.func(*args)
if isinstance(result, Dimension):
return dimsys_default.get_dimensional_dependencies(result)
elif result.func == name.func:
return {}
else:
return get_for_name(result)
def get_dimensional_dependencies(self, name, mark_dimensionless=False):
if isinstance(name, Dimension):
name = name.name
if isinstance(name, string_types):
name = Symbol(name)
dimdep = self._get_dimensional_dependencies_for_name(name)
if mark_dimensionless and dimdep == {}:
return {'dimensionless': 1}
return {str(i): j for i, j in dimdep.items()}
def equivalent_dims(self, dim1, dim2):
deps1 = self.get_dimensional_dependencies(dim1)
deps2 = self.get_dimensional_dependencies(dim2)
return deps1 == deps2
def extend(self, new_base_dims, new_derived_dims=[], new_dim_deps={}, name=None, description=None):
if (name is not None) or (description is not None):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="name and descriptions of DimensionSystem",
useinstead="do not specify `name` or `description`",
).warn()
deps = dict(self.dimensional_dependencies)
deps.update(new_dim_deps)
return DimensionSystem(
tuple(self.base_dims) + tuple(new_base_dims),
tuple(self.derived_dims) + tuple(new_derived_dims),
deps
)
@staticmethod
def sort_dims(dims):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Sort dimensions given in argument using their str function.
This function will ensure that we get always the same tuple for a given
set of dimensions.
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="sort_dims",
useinstead="sorted(..., key=default_sort_key)",
).warn()
return tuple(sorted(dims, key=str))
def __getitem__(self, key):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Shortcut to the get_dim method, using key access.
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="the get [ ] operator",
useinstead="the dimension definition",
).warn()
d = self.get_dim(key)
#TODO: really want to raise an error?
if d is None:
raise KeyError(key)
return d
def __call__(self, unit):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Wrapper to the method print_dim_base
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="call DimensionSystem",
useinstead="the dimension definition",
).warn()
return self.print_dim_base(unit)
def is_dimensionless(self, dimension):
"""
Check if the dimension object really has a dimension.
A dimension should have at least one component with non-zero power.
"""
if dimension.name == 1:
return True
return self.get_dimensional_dependencies(dimension) == {}
@property
def list_can_dims(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
List all canonical dimension names.
"""
dimset = set([])
for i in self.base_dims:
dimset.update(set(dimsys_default.get_dimensional_dependencies(i).keys()))
return tuple(sorted(dimset, key=str))
@property
def inv_can_transf_matrix(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Compute the inverse transformation matrix from the base to the
canonical dimension basis.
It corresponds to the matrix where columns are the vector of base
dimensions in canonical basis.
This matrix will almost never be used because dimensions are always
defined with respect to the canonical basis, so no work has to be done
to get them in this basis. Nonetheless if this matrix is not square
(or not invertible) it means that we have chosen a bad basis.
"""
matrix = reduce(lambda x, y: x.row_join(y),
[self.dim_can_vector(d) for d in self.base_dims])
return matrix
@property
def can_transf_matrix(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Return the canonical transformation matrix from the canonical to the
base dimension basis.
It is the inverse of the matrix computed with inv_can_transf_matrix().
"""
#TODO: the inversion will fail if the system is inconsistent, for
# example if the matrix is not a square
return reduce(lambda x, y: x.row_join(y),
[self.dim_can_vector(d) for d in sorted(self.base_dims, key=str)]
).inv()
def dim_can_vector(self, dim):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Dimensional representation in terms of the canonical base dimensions.
"""
vec = []
for d in self.list_can_dims:
vec.append(dimsys_default.get_dimensional_dependencies(dim).get(d, 0))
return Matrix(vec)
def dim_vector(self, dim):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Vector representation in terms of the base dimensions.
"""
return self.can_transf_matrix * Matrix(self.dim_can_vector(dim))
def print_dim_base(self, dim):
"""
Give the string expression of a dimension in term of the basis symbols.
"""
dims = self.dim_vector(dim)
symbols = [i.symbol if i.symbol is not None else i.name for i in self.base_dims]
res = S.One
for (s, p) in zip(symbols, dims):
res *= s**p
return res
@property
def dim(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Give the dimension of the system.
That is return the number of dimensions forming the basis.
"""
return len(self.base_dims)
@property
def is_consistent(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Check if the system is well defined.
"""
# not enough or too many base dimensions compared to independent
# dimensions
# in vector language: the set of vectors do not form a basis
return self.inv_can_transf_matrix.is_square
dimsys_MKS = DimensionSystem([
# Dimensional dependencies for MKS base dimensions
length,
mass,
time,
], dimensional_dependencies=dict(
# Dimensional dependencies for derived dimensions
velocity=dict(length=1, time=-1),
acceleration=dict(length=1, time=-2),
momentum=dict(mass=1, length=1, time=-1),
force=dict(mass=1, length=1, time=-2),
energy=dict(mass=1, length=2, time=-2),
power=dict(length=2, mass=1, time=-3),
pressure=dict(mass=1, length=-1, time=-2),
frequency=dict(time=-1),
action=dict(length=2, mass=1, time=-1),
volume=dict(length=3),
))
dimsys_MKSA = dimsys_MKS.extend([
# Dimensional dependencies for base dimensions (MKSA not in MKS)
current,
], new_dim_deps=dict(
# Dimensional dependencies for derived dimensions
voltage=dict(mass=1, length=2, current=-1, time=-3),
impedance=dict(mass=1, length=2, current=-2, time=-3),
conductance=dict(mass=-1, length=-2, current=2, time=3),
capacitance=dict(mass=-1, length=-2, current=2, time=4),
inductance=dict(mass=1, length=2, current=-2, time=-2),
charge=dict(current=1, time=1),
magnetic_density=dict(mass=1, current=-1, time=-2),
magnetic_flux=dict(length=2, mass=1, current=-1, time=-2),
))
dimsys_SI = dimsys_MKSA.extend(
[
# Dimensional dependencies for other base dimensions:
temperature,
amount_of_substance,
luminous_intensity,
])
dimsys_default = dimsys_SI.extend(
[information],
)
|
c91c1c7198fcc3ab63b20bc2204b0046096664edcc7313af023bfac114816d61 | """
Several methods to simplify expressions involving unit objects.
"""
from __future__ import division
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy import Add, Mul, Pow, Tuple, sympify
from sympy.core.compatibility import reduce, Iterable, ordered
from sympy.physics.units.dimensions import Dimension, dimsys_default
from sympy.physics.units.prefixes import Prefix
from sympy.physics.units.quantities import Quantity
from sympy.utilities.iterables import sift
def dim_simplify(expr):
"""
NOTE: this function could be deprecated in the future.
Simplify expression by recursively evaluating the dimension arguments.
This function proceeds to a very rough dimensional analysis. It tries to
simplify expression with dimensions, and it deletes all what multiplies a
dimension without being a dimension. This is necessary to avoid strange
behavior when Add(L, L) be transformed into Mul(2, L).
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
feature="dimensional simplification function",
issue=13336,
useinstead="don't use",
).warn()
_, expr = Quantity._collect_factor_and_dimension(expr)
return expr
def _get_conversion_matrix_for_expr(expr, target_units):
from sympy import Matrix
expr_dim = Dimension(Quantity.get_dimensional_expr(expr))
dim_dependencies = dimsys_default.get_dimensional_dependencies(expr_dim, mark_dimensionless=True)
target_dims = [Dimension(Quantity.get_dimensional_expr(x)) for x in target_units]
canon_dim_units = {i for x in target_dims for i in dimsys_default.get_dimensional_dependencies(x, mark_dimensionless=True)}
canon_expr_units = {i for i in dim_dependencies}
if not canon_expr_units.issubset(canon_dim_units):
return None
canon_dim_units = sorted(canon_dim_units)
camat = Matrix([[dimsys_default.get_dimensional_dependencies(i, mark_dimensionless=True).get(j, 0) for i in target_dims] for j in canon_dim_units])
exprmat = Matrix([dim_dependencies.get(k, 0) for k in canon_dim_units])
res_exponents = camat.solve_least_squares(exprmat, method=None)
return res_exponents
def convert_to(expr, target_units):
"""
Convert ``expr`` to the same expression with all of its units and quantities
represented as factors of ``target_units``, whenever the dimension is compatible.
``target_units`` may be a single unit/quantity, or a collection of
units/quantities.
Examples
========
>>> from sympy.physics.units import speed_of_light, meter, gram, second, day
>>> from sympy.physics.units import mile, newton, kilogram, atomic_mass_constant
>>> from sympy.physics.units import kilometer, centimeter
>>> from sympy.physics.units import convert_to
>>> convert_to(mile, kilometer)
25146*kilometer/15625
>>> convert_to(mile, kilometer).n()
1.609344*kilometer
>>> convert_to(speed_of_light, meter/second)
299792458*meter/second
>>> convert_to(day, second)
86400*second
>>> 3*newton
3*newton
>>> convert_to(3*newton, kilogram*meter/second**2)
3*kilogram*meter/second**2
>>> convert_to(atomic_mass_constant, gram)
1.66053904e-24*gram
Conversion to multiple units:
>>> convert_to(speed_of_light, [meter, second])
299792458*meter/second
>>> convert_to(3*newton, [centimeter, gram, second])
300000*centimeter*gram/second**2
Conversion to Planck units:
>>> from sympy.physics.units import gravitational_constant, hbar
>>> convert_to(atomic_mass_constant, [gravitational_constant, speed_of_light, hbar]).n()
7.62950196312651e-20*gravitational_constant**(-0.5)*hbar**0.5*speed_of_light**0.5
"""
if not isinstance(target_units, (Iterable, Tuple)):
target_units = [target_units]
if isinstance(expr, Add):
return Add.fromiter(convert_to(i, target_units) for i in expr.args)
expr = sympify(expr)
if not isinstance(expr, Quantity) and expr.has(Quantity):
expr = expr.replace(lambda x: isinstance(x, Quantity), lambda x: x.convert_to(target_units))
def get_total_scale_factor(expr):
if isinstance(expr, Mul):
return reduce(lambda x, y: x * y, [get_total_scale_factor(i) for i in expr.args])
elif isinstance(expr, Pow):
return get_total_scale_factor(expr.base) ** expr.exp
elif isinstance(expr, Quantity):
return expr.scale_factor
return expr
depmat = _get_conversion_matrix_for_expr(expr, target_units)
if depmat is None:
return expr
expr_scale_factor = get_total_scale_factor(expr)
return expr_scale_factor * Mul.fromiter((1/get_total_scale_factor(u) * u) ** p for u, p in zip(target_units, depmat))
def quantity_simplify(expr):
"""Return an equivalent expression in which prefixes are replaced
with numerical values and all units of a given dimension are the
unified in a canonical manner.
Examples
========
>>> from sympy.physics.units.util import quantity_simplify
>>> from sympy.physics.units.prefixes import kilo
>>> from sympy.physics.units import foot, inch
>>> quantity_simplify(kilo*foot*inch)
250*foot**2/3
>>> quantity_simplify(foot - 6*inch)
foot/2
"""
if expr.is_Atom or not expr.has(Prefix, Quantity):
return expr
# replace all prefixes with numerical values
p = expr.atoms(Prefix)
expr = expr.xreplace({p: p.scale_factor for p in p})
# replace all quantities of given dimension with a canonical
# quantity, chosen from those in the expression
d = sift(expr.atoms(Quantity), lambda i: i.dimension)
for k in d:
if len(d[k]) == 1:
continue
v = list(ordered(d[k]))
ref = v[0]/v[0].scale_factor
expr = expr.xreplace({vi: ref*vi.scale_factor for vi in v[1:]})
return expr
def check_dimensions(expr):
"""Return expr if there are not unitless values added to
dimensional quantities, else raise a ValueError."""
from sympy.solvers.solveset import _term_factors
# the case of adding a number to a dimensional quantity
# is ignored for the sake of SymPy core routines, so this
# function will raise an error now if such an addend is
# found.
# Also, when doing substitutions, multiplicative constants
# might be introduced, so remove those now
adds = expr.atoms(Add)
DIM_OF = dimsys_default.get_dimensional_dependencies
for a in adds:
deset = set()
for ai in a.args:
if ai.is_number:
deset.add(())
continue
dims = []
skip = False
for i in Mul.make_args(ai):
if i.has(Quantity):
i = Dimension(Quantity.get_dimensional_expr(i))
if i.has(Dimension):
dims.extend(DIM_OF(i).items())
elif i.free_symbols:
skip = True
break
if not skip:
deset.add(tuple(sorted(dims)))
if len(deset) > 1:
raise ValueError(
"addends have incompatible dimensions")
# clear multiplicative constants on Dimensions which may be
# left after substitution
reps = {}
for m in expr.atoms(Mul):
if any(isinstance(i, Dimension) for i in m.args):
reps[m] = m.func(*[
i for i in m.args if not i.is_number])
return expr.xreplace(reps)
|
1c6d58caa137cededca6f99c902842105b7cdfa9221ab7b28042fad3f658f4c6 | """
Physical quantities.
"""
from __future__ import division
from sympy import (Abs, Add, AtomicExpr, Derivative, Function, Mul,
Pow, S, Symbol, sympify)
from sympy.core.compatibility import string_types
from sympy.physics.units import Dimension, dimensions
from sympy.physics.units.prefixes import Prefix
from sympy.utilities.exceptions import SymPyDeprecationWarning
class Quantity(AtomicExpr):
"""
Physical quantity: can be a unit of measure, a constant or a generic quantity.
"""
is_commutative = True
is_real = True
is_number = False
is_nonzero = True
_diff_wrt = True
def __new__(cls, name, abbrev=None, dimension=None, scale_factor=None,
latex_repr=None, pretty_unicode_repr=None,
pretty_ascii_repr=None, mathml_presentation_repr=None,
**assumptions):
if not isinstance(name, Symbol):
name = Symbol(name)
# For Quantity(name, dim, scale, abbrev) to work like in the
# old version of Sympy:
if not isinstance(abbrev, string_types) and not \
isinstance(abbrev, Symbol):
dimension, scale_factor, abbrev = abbrev, dimension, scale_factor
if dimension is not None:
SymPyDeprecationWarning(
deprecated_since_version="1.3",
issue=14319,
feature="Quantity arguments",
useinstead="SI_quantity_dimension_map",
).warn()
if scale_factor is not None:
SymPyDeprecationWarning(
deprecated_since_version="1.3",
issue=14319,
feature="Quantity arguments",
useinstead="SI_quantity_scale_factors",
).warn()
if abbrev is None:
abbrev = name
elif isinstance(abbrev, string_types):
abbrev = Symbol(abbrev)
obj = AtomicExpr.__new__(cls, name, abbrev)
obj._name = name
obj._abbrev = abbrev
obj._latex_repr = latex_repr
obj._unicode_repr = pretty_unicode_repr
obj._ascii_repr = pretty_ascii_repr
obj._mathml_repr = mathml_presentation_repr
if dimension is not None:
# TODO: remove after deprecation:
obj.set_dimension(dimension)
if scale_factor is not None:
# TODO: remove after deprecation:
obj.set_scale_factor(scale_factor)
return obj
### Currently only SI is supported: ###
# Dimensional representations for the SI units:
SI_quantity_dimension_map = {}
# Scale factors in SI units:
SI_quantity_scale_factors = {}
def set_dimension(self, dimension, unit_system="SI"):
from sympy.physics.units.dimensions import dimsys_default, DimensionSystem
if unit_system != "SI":
# TODO: add support for more units and dimension systems:
raise NotImplementedError("Currently only SI is supported")
dim_sys = dimsys_default
if not isinstance(dimension, dimensions.Dimension):
if dimension == 1:
dimension = Dimension(1)
else:
raise ValueError("expected dimension or 1")
else:
for dim_sym in dimension.name.atoms(Dimension):
if dim_sym not in [i.name for i in dim_sys._dimensional_dependencies]:
raise ValueError("Dimension %s is not registered in the "
"dimensional dependency tree." % dim_sym)
Quantity.SI_quantity_dimension_map[self] = dimension
def set_scale_factor(self, scale_factor, unit_system="SI"):
if unit_system != "SI":
# TODO: add support for more units and dimension systems:
raise NotImplementedError("Currently only SI is supported")
scale_factor = sympify(scale_factor)
# replace all prefixes by their ratio to canonical units:
scale_factor = scale_factor.replace(lambda x: isinstance(x, Prefix), lambda x: x.scale_factor)
# replace all quantities by their ratio to canonical units:
scale_factor = scale_factor.replace(lambda x: isinstance(x, Quantity), lambda x: x.scale_factor)
Quantity.SI_quantity_scale_factors[self] = scale_factor
@property
def name(self):
return self._name
@property
def dimension(self):
# TODO: add support for units other than SI:
return Quantity.SI_quantity_dimension_map[self]
@property
def abbrev(self):
"""
Symbol representing the unit name.
Prepend the abbreviation with the prefix symbol if it is defines.
"""
return self._abbrev
@property
def scale_factor(self):
"""
Overall magnitude of the quantity as compared to the canonical units.
"""
return Quantity.SI_quantity_scale_factors.get(self, S.One)
def _eval_is_positive(self):
return self.scale_factor.is_positive
def _eval_is_constant(self):
return self.scale_factor.is_constant()
def _eval_Abs(self):
scale_factor = Abs(self.scale_factor)
if scale_factor == self.scale_factor:
return self
return None
q = self.func(self.name, self.abbrev)
def _eval_subs(self, old, new):
if isinstance(new, Quantity) and self != old:
return self
@staticmethod
def get_dimensional_expr(expr):
if isinstance(expr, Mul):
return Mul(*[Quantity.get_dimensional_expr(i) for i in expr.args])
elif isinstance(expr, Pow):
return Quantity.get_dimensional_expr(expr.base) ** expr.exp
elif isinstance(expr, Add):
return Quantity.get_dimensional_expr(expr.args[0])
elif isinstance(expr, Derivative):
dim = Quantity.get_dimensional_expr(expr.expr)
for independent, count in expr.variable_count:
dim /= Quantity.get_dimensional_expr(independent)**count
return dim
elif isinstance(expr, Function):
args = [Quantity.get_dimensional_expr(arg) for arg in expr.args]
if all(i == 1 for i in args):
return S.One
return expr.func(*args)
elif isinstance(expr, Quantity):
return expr.dimension.name
return S.One
@staticmethod
def _collect_factor_and_dimension(expr):
"""Return tuple with factor expression and dimension expression."""
if isinstance(expr, Quantity):
return expr.scale_factor, expr.dimension
elif isinstance(expr, Mul):
factor = 1
dimension = Dimension(1)
for arg in expr.args:
arg_factor, arg_dim = Quantity._collect_factor_and_dimension(arg)
factor *= arg_factor
dimension *= arg_dim
return factor, dimension
elif isinstance(expr, Pow):
factor, dim = Quantity._collect_factor_and_dimension(expr.base)
exp_factor, exp_dim = Quantity._collect_factor_and_dimension(expr.exp)
if exp_dim.is_dimensionless:
exp_dim = 1
return factor ** exp_factor, dim ** (exp_factor * exp_dim)
elif isinstance(expr, Add):
factor, dim = Quantity._collect_factor_and_dimension(expr.args[0])
for addend in expr.args[1:]:
addend_factor, addend_dim = \
Quantity._collect_factor_and_dimension(addend)
if dim != addend_dim:
raise ValueError(
'Dimension of "{0}" is {1}, '
'but it should be {2}'.format(
addend, addend_dim.name, dim.name))
factor += addend_factor
return factor, dim
elif isinstance(expr, Derivative):
factor, dim = Quantity._collect_factor_and_dimension(expr.args[0])
for independent, count in expr.variable_count:
ifactor, idim = Quantity._collect_factor_and_dimension(independent)
factor /= ifactor**count
dim /= idim**count
return factor, dim
elif isinstance(expr, Function):
fds = [Quantity._collect_factor_and_dimension(
arg) for arg in expr.args]
return (expr.func(*(f[0] for f in fds)),
expr.func(*(d[1] for d in fds)))
elif isinstance(expr, Dimension):
return 1, expr
else:
return expr, Dimension(1)
def _latex(self, printer):
if self._latex_repr:
return self._latex_repr
else:
return r'\text{{{}}}'.format(self.args[1] \
if len(self.args) >= 2 else self.args[0])
def convert_to(self, other):
"""
Convert the quantity to another quantity of same dimensions.
Examples
========
>>> from sympy.physics.units import speed_of_light, meter, second
>>> speed_of_light
speed_of_light
>>> speed_of_light.convert_to(meter/second)
299792458*meter/second
>>> from sympy.physics.units import liter
>>> liter.convert_to(meter**3)
meter**3/1000
"""
from .util import convert_to
return convert_to(self, other)
@property
def free_symbols(self):
"""Return free symbols from quantity."""
return self.scale_factor.free_symbols
|
5aef15b61af882cc75cdaa6c0eeaf6593affb0cb3cac0be1793b60f7825482e2 | from sympy.physics.secondquant import (
Dagger, Bd, VarBosonicBasis, BBra, B, BKet, FixedBosonicBasis,
matrix_rep, apply_operators, InnerProduct, Commutator, KroneckerDelta,
AnnihilateBoson, CreateBoson, BosonicOperator,
F, Fd, FKet, BosonState, CreateFermion, AnnihilateFermion,
evaluate_deltas, AntiSymmetricTensor, contraction, NO, wicks,
PermutationOperator, simplify_index_permutations,
_sort_anticommuting_fermions, _get_ordered_dummies,
substitute_dummies, FockState, FockStateBosonKet,
ContractionAppliesOnlyToFermions
)
from sympy import (Dummy, expand, Function, I, Rational, simplify, sqrt, Sum,
Symbol, symbols, srepr)
from sympy.core.compatibility import range
from sympy.utilities.pytest import XFAIL, slow, raises
from sympy.printing.latex import latex
def test_PermutationOperator():
p, q, r, s = symbols('p,q,r,s')
f, g, h, i = map(Function, 'fghi')
P = PermutationOperator
assert P(p, q).get_permuted(f(p)*g(q)) == -f(q)*g(p)
assert P(p, q).get_permuted(f(p, q)) == -f(q, p)
assert P(p, q).get_permuted(f(p)) == f(p)
expr = (f(p)*g(q)*h(r)*i(s)
- f(q)*g(p)*h(r)*i(s)
- f(p)*g(q)*h(s)*i(r)
+ f(q)*g(p)*h(s)*i(r))
perms = [P(p, q), P(r, s)]
assert (simplify_index_permutations(expr, perms) ==
P(p, q)*P(r, s)*f(p)*g(q)*h(r)*i(s))
assert latex(P(p, q)) == 'P(pq)'
def test_index_permutations_with_dummies():
a, b, c, d = symbols('a b c d')
p, q, r, s = symbols('p q r s', cls=Dummy)
f, g = map(Function, 'fg')
P = PermutationOperator
# No dummy substitution necessary
expr = f(a, b, p, q) - f(b, a, p, q)
assert simplify_index_permutations(
expr, [P(a, b)]) == P(a, b)*f(a, b, p, q)
# Cases where dummy substitution is needed
expected = P(a, b)*substitute_dummies(f(a, b, p, q))
expr = f(a, b, p, q) - f(b, a, q, p)
result = simplify_index_permutations(expr, [P(a, b)])
assert expected == substitute_dummies(result)
expr = f(a, b, q, p) - f(b, a, p, q)
result = simplify_index_permutations(expr, [P(a, b)])
assert expected == substitute_dummies(result)
# A case where nothing can be done
expr = f(a, b, q, p) - g(b, a, p, q)
result = simplify_index_permutations(expr, [P(a, b)])
assert expr == result
def test_dagger():
i, j, n, m = symbols('i,j,n,m')
assert Dagger(1) == 1
assert Dagger(1.0) == 1.0
assert Dagger(2*I) == -2*I
assert Dagger(Rational(1, 2)*I/3.0) == -Rational(1, 2)*I/3.0
assert Dagger(BKet([n])) == BBra([n])
assert Dagger(B(0)) == Bd(0)
assert Dagger(Bd(0)) == B(0)
assert Dagger(B(n)) == Bd(n)
assert Dagger(Bd(n)) == B(n)
assert Dagger(B(0) + B(1)) == Bd(0) + Bd(1)
assert Dagger(n*m) == Dagger(n)*Dagger(m) # n, m commute
assert Dagger(B(n)*B(m)) == Bd(m)*Bd(n)
assert Dagger(B(n)**10) == Dagger(B(n))**10
assert Dagger('a') == Dagger(Symbol('a'))
assert Dagger(Dagger('a')) == Symbol('a')
def test_operator():
i, j = symbols('i,j')
o = BosonicOperator(i)
assert o.state == i
assert o.is_symbolic
o = BosonicOperator(1)
assert o.state == 1
assert not o.is_symbolic
def test_create():
i, j, n, m = symbols('i,j,n,m')
o = Bd(i)
assert latex(o) == "b^\\dagger_{i}"
assert isinstance(o, CreateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Bd(0)
assert o.apply_operator(BKet([n])) == sqrt(n + 1)*BKet([n + 1])
o = Bd(n)
assert o.apply_operator(BKet([n])) == o*BKet([n])
def test_annihilate():
i, j, n, m = symbols('i,j,n,m')
o = B(i)
assert latex(o) == "b_{i}"
assert isinstance(o, AnnihilateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = B(0)
assert o.apply_operator(BKet([n])) == sqrt(n)*BKet([n - 1])
o = B(n)
assert o.apply_operator(BKet([n])) == o*BKet([n])
def test_basic_state():
i, j, n, m = symbols('i,j,n,m')
s = BosonState([0, 1, 2, 3, 4])
assert len(s) == 5
assert s.args[0] == tuple(range(5))
assert s.up(0) == BosonState([1, 1, 2, 3, 4])
assert s.down(4) == BosonState([0, 1, 2, 3, 3])
for i in range(5):
assert s.up(i).down(i) == s
assert s.down(0) == 0
for i in range(5):
assert s[i] == i
s = BosonState([n, m])
assert s.down(0) == BosonState([n - 1, m])
assert s.up(0) == BosonState([n + 1, m])
@XFAIL
def test_move1():
i, j = symbols('i,j')
A, C = symbols('A,C', cls=Function)
o = A(i)*C(j)
# This almost works, but has a minus sign wrong
assert move(o, 0, 1) == KroneckerDelta(i, j) + C(j)*A(i)
@XFAIL
def test_move2():
i, j = symbols('i,j')
A, C = symbols('A,C', cls=Function)
o = C(j)*A(i)
# This almost works, but has a minus sign wrong
assert move(o, 0, 1) == -KroneckerDelta(i, j) + A(i)*C(j)
def test_basic_apply():
n = symbols("n")
e = B(0)*BKet([n])
assert apply_operators(e) == sqrt(n)*BKet([n - 1])
e = Bd(0)*BKet([n])
assert apply_operators(e) == sqrt(n + 1)*BKet([n + 1])
def test_complex_apply():
n, m = symbols("n,m")
o = Bd(0)*B(0)*Bd(1)*B(0)
e = apply_operators(o*BKet([n, m]))
answer = sqrt(n)*sqrt(m + 1)*(-1 + n)*BKet([-1 + n, 1 + m])
assert expand(e) == expand(answer)
def test_number_operator():
n = symbols("n")
o = Bd(0)*B(0)
e = apply_operators(o*BKet([n]))
assert e == n*BKet([n])
def test_inner_product():
i, j, k, l = symbols('i,j,k,l')
s1 = BBra([0])
s2 = BKet([1])
assert InnerProduct(s1, Dagger(s1)) == 1
assert InnerProduct(s1, s2) == 0
s1 = BBra([i, j])
s2 = BKet([k, l])
r = InnerProduct(s1, s2)
assert r == KroneckerDelta(i, k)*KroneckerDelta(j, l)
def test_symbolic_matrix_elements():
n, m = symbols('n,m')
s1 = BBra([n])
s2 = BKet([m])
o = B(0)
e = apply_operators(s1*o*s2)
assert e == sqrt(m)*KroneckerDelta(n, m - 1)
def test_matrix_elements():
b = VarBosonicBasis(5)
o = B(0)
m = matrix_rep(o, b)
for i in range(4):
assert m[i, i + 1] == sqrt(i + 1)
o = Bd(0)
m = matrix_rep(o, b)
for i in range(4):
assert m[i + 1, i] == sqrt(i + 1)
def test_fixed_bosonic_basis():
b = FixedBosonicBasis(2, 2)
# assert b == [FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]
state = b.state(1)
assert state == FockStateBosonKet((1, 1))
assert b.index(state) == 1
assert b.state(1) == b[1]
assert len(b) == 3
assert str(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]'
assert repr(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]'
assert srepr(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]'
@slow
def test_sho():
n, m = symbols('n,m')
h_n = Bd(n)*B(n)*(n + Rational(1, 2))
H = Sum(h_n, (n, 0, 5))
o = H.doit(deep=False)
b = FixedBosonicBasis(2, 6)
m = matrix_rep(o, b)
# We need to double check these energy values to make sure that they
# are correct and have the proper degeneracies!
diag = [1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11]
for i in range(len(diag)):
assert diag[i] == m[i, i]
def test_commutation():
n, m = symbols("n,m", above_fermi=True)
c = Commutator(B(0), Bd(0))
assert c == 1
c = Commutator(Bd(0), B(0))
assert c == -1
c = Commutator(B(n), Bd(0))
assert c == KroneckerDelta(n, 0)
c = Commutator(B(0), B(0))
assert c == 0
c = Commutator(B(0), Bd(0))
e = simplify(apply_operators(c*BKet([n])))
assert e == BKet([n])
c = Commutator(B(0), B(1))
e = simplify(apply_operators(c*BKet([n, m])))
assert e == 0
c = Commutator(F(m), Fd(m))
assert c == +1 - 2*NO(Fd(m)*F(m))
c = Commutator(Fd(m), F(m))
assert c.expand() == -1 + 2*NO(Fd(m)*F(m))
C = Commutator
X, Y, Z = symbols('X,Y,Z', commutative=False)
assert C(C(X, Y), Z) != 0
assert C(C(X, Z), Y) != 0
assert C(Y, C(X, Z)) != 0
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
D = KroneckerDelta
assert C(Fd(a), F(i)) == -2*NO(F(i)*Fd(a))
assert C(Fd(j), NO(Fd(a)*F(i))).doit(wicks=True) == -D(j, i)*Fd(a)
assert C(Fd(a)*F(i), Fd(b)*F(j)).doit(wicks=True) == 0
c1 = Commutator(F(a), Fd(a))
assert Commutator.eval(c1, c1) == 0
c = Commutator(Fd(a)*F(i),Fd(b)*F(j))
assert latex(c) == r'\left[a^\dagger_{a} a_{i},a^\dagger_{b} a_{j}\right]'
assert repr(c) == 'Commutator(CreateFermion(a)*AnnihilateFermion(i),CreateFermion(b)*AnnihilateFermion(j))'
assert str(c) == '[CreateFermion(a)*AnnihilateFermion(i),CreateFermion(b)*AnnihilateFermion(j)]'
def test_create_f():
i, j, n, m = symbols('i,j,n,m')
o = Fd(i)
assert isinstance(o, CreateFermion)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Fd(1)
assert o.apply_operator(FKet([n])) == FKet([1, n])
assert o.apply_operator(FKet([n])) == -FKet([n, 1])
o = Fd(n)
assert o.apply_operator(FKet([])) == FKet([n])
vacuum = FKet([], fermi_level=4)
assert vacuum == FKet([], fermi_level=4)
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
assert Fd(i).apply_operator(FKet([i, j, k], 4)) == FKet([j, k], 4)
assert Fd(a).apply_operator(FKet([i, b, k], 4)) == FKet([a, i, b, k], 4)
assert Dagger(B(p)).apply_operator(q) == q*CreateBoson(p)
assert repr(Fd(p)) == 'CreateFermion(p)'
assert srepr(Fd(p)) == "CreateFermion(Symbol('p'))"
assert latex(Fd(p)) == r'a^\dagger_{p}'
def test_annihilate_f():
i, j, n, m = symbols('i,j,n,m')
o = F(i)
assert isinstance(o, AnnihilateFermion)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = F(1)
assert o.apply_operator(FKet([1, n])) == FKet([n])
assert o.apply_operator(FKet([n, 1])) == -FKet([n])
o = F(n)
assert o.apply_operator(FKet([n])) == FKet([])
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
assert F(i).apply_operator(FKet([i, j, k], 4)) == 0
assert F(a).apply_operator(FKet([i, b, k], 4)) == 0
assert F(l).apply_operator(FKet([i, j, k], 3)) == 0
assert F(l).apply_operator(FKet([i, j, k], 4)) == FKet([l, i, j, k], 4)
assert str(F(p)) == 'f(p)'
assert repr(F(p)) == 'AnnihilateFermion(p)'
assert srepr(F(p)) == "AnnihilateFermion(Symbol('p'))"
assert latex(F(p)) == 'a_{p}'
def test_create_b():
i, j, n, m = symbols('i,j,n,m')
o = Bd(i)
assert isinstance(o, CreateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = Bd(0)
assert o.apply_operator(BKet([n])) == sqrt(n + 1)*BKet([n + 1])
o = Bd(n)
assert o.apply_operator(BKet([n])) == o*BKet([n])
def test_annihilate_b():
i, j, n, m = symbols('i,j,n,m')
o = B(i)
assert isinstance(o, AnnihilateBoson)
o = o.subs(i, j)
assert o.atoms(Symbol) == {j}
o = B(0)
def test_wicks():
p, q, r, s = symbols('p,q,r,s', above_fermi=True)
# Testing for particles only
str = F(p)*Fd(q)
assert wicks(str) == NO(F(p)*Fd(q)) + KroneckerDelta(p, q)
str = Fd(p)*F(q)
assert wicks(str) == NO(Fd(p)*F(q))
str = F(p)*Fd(q)*F(r)*Fd(s)
nstr = wicks(str)
fasit = NO(
KroneckerDelta(p, q)*KroneckerDelta(r, s)
+ KroneckerDelta(p, q)*AnnihilateFermion(r)*CreateFermion(s)
+ KroneckerDelta(r, s)*AnnihilateFermion(p)*CreateFermion(q)
- KroneckerDelta(p, s)*AnnihilateFermion(r)*CreateFermion(q)
- AnnihilateFermion(p)*AnnihilateFermion(r)*CreateFermion(q)*CreateFermion(s))
assert nstr == fasit
assert (p*q*nstr).expand() == wicks(p*q*str)
assert (nstr*p*q*2).expand() == wicks(str*p*q*2)
# Testing CC equations particles and holes
i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy)
p, q, r, s = symbols('p q r s', cls=Dummy)
assert (wicks(F(a)*NO(F(i)*F(j))*Fd(b)) ==
NO(F(a)*F(i)*F(j)*Fd(b)) +
KroneckerDelta(a, b)*NO(F(i)*F(j)))
assert (wicks(F(a)*NO(F(i)*F(j)*F(k))*Fd(b)) ==
NO(F(a)*F(i)*F(j)*F(k)*Fd(b)) -
KroneckerDelta(a, b)*NO(F(i)*F(j)*F(k)))
expr = wicks(Fd(i)*NO(Fd(j)*F(k))*F(l))
assert (expr ==
-KroneckerDelta(i, k)*NO(Fd(j)*F(l)) -
KroneckerDelta(j, l)*NO(Fd(i)*F(k)) -
KroneckerDelta(i, k)*KroneckerDelta(j, l) +
KroneckerDelta(i, l)*NO(Fd(j)*F(k)) +
NO(Fd(i)*Fd(j)*F(k)*F(l)))
expr = wicks(F(a)*NO(F(b)*Fd(c))*Fd(d))
assert (expr ==
-KroneckerDelta(a, c)*NO(F(b)*Fd(d)) -
KroneckerDelta(b, d)*NO(F(a)*Fd(c)) -
KroneckerDelta(a, c)*KroneckerDelta(b, d) +
KroneckerDelta(a, d)*NO(F(b)*Fd(c)) +
NO(F(a)*F(b)*Fd(c)*Fd(d)))
def test_NO():
i, j, k, l = symbols('i j k l', below_fermi=True)
a, b, c, d = symbols('a b c d', above_fermi=True)
p, q, r, s = symbols('p q r s', cls=Dummy)
assert (NO(Fd(p)*F(q) + Fd(a)*F(b)) ==
NO(Fd(p)*F(q)) + NO(Fd(a)*F(b)))
assert (NO(Fd(i)*NO(F(j)*Fd(a))) ==
NO(Fd(i)*F(j)*Fd(a)))
assert NO(1) == 1
assert NO(i) == i
assert (NO(Fd(a)*Fd(b)*(F(c) + F(d))) ==
NO(Fd(a)*Fd(b)*F(c)) +
NO(Fd(a)*Fd(b)*F(d)))
assert NO(Fd(a)*F(b))._remove_brackets() == Fd(a)*F(b)
assert NO(F(j)*Fd(i))._remove_brackets() == F(j)*Fd(i)
assert (NO(Fd(p)*F(q)).subs(Fd(p), Fd(a) + Fd(i)) ==
NO(Fd(a)*F(q)) + NO(Fd(i)*F(q)))
assert (NO(Fd(p)*F(q)).subs(F(q), F(a) + F(i)) ==
NO(Fd(p)*F(a)) + NO(Fd(p)*F(i)))
expr = NO(Fd(p)*F(q))._remove_brackets()
assert wicks(expr) == NO(expr)
assert NO(Fd(a)*F(b)) == - NO(F(b)*Fd(a))
no = NO(Fd(a)*F(i)*F(b)*Fd(j))
l1 = [ ind for ind in no.iter_q_creators() ]
assert l1 == [0, 1]
l2 = [ ind for ind in no.iter_q_annihilators() ]
assert l2 == [3, 2]
no = NO(Fd(a)*Fd(i))
assert no.has_q_creators == 1
assert no.has_q_annihilators == -1
assert str(no) == ':CreateFermion(a)*CreateFermion(i):'
assert repr(no) == 'NO(CreateFermion(a)*CreateFermion(i))'
assert latex(no) == r'\left\{a^\dagger_{a} a^\dagger_{i}\right\}'
raises(NotImplementedError, lambda: NO(Bd(p)*F(q)))
def test_sorting():
i, j = symbols('i,j', below_fermi=True)
a, b = symbols('a,b', above_fermi=True)
p, q = symbols('p,q')
# p, q
assert _sort_anticommuting_fermions([Fd(p), F(q)]) == ([Fd(p), F(q)], 0)
assert _sort_anticommuting_fermions([F(p), Fd(q)]) == ([Fd(q), F(p)], 1)
# i, p
assert _sort_anticommuting_fermions([F(p), Fd(i)]) == ([F(p), Fd(i)], 0)
assert _sort_anticommuting_fermions([Fd(i), F(p)]) == ([F(p), Fd(i)], 1)
assert _sort_anticommuting_fermions([Fd(p), Fd(i)]) == ([Fd(p), Fd(i)], 0)
assert _sort_anticommuting_fermions([Fd(i), Fd(p)]) == ([Fd(p), Fd(i)], 1)
assert _sort_anticommuting_fermions([F(p), F(i)]) == ([F(i), F(p)], 1)
assert _sort_anticommuting_fermions([F(i), F(p)]) == ([F(i), F(p)], 0)
assert _sort_anticommuting_fermions([Fd(p), F(i)]) == ([F(i), Fd(p)], 1)
assert _sort_anticommuting_fermions([F(i), Fd(p)]) == ([F(i), Fd(p)], 0)
# a, p
assert _sort_anticommuting_fermions([F(p), Fd(a)]) == ([Fd(a), F(p)], 1)
assert _sort_anticommuting_fermions([Fd(a), F(p)]) == ([Fd(a), F(p)], 0)
assert _sort_anticommuting_fermions([Fd(p), Fd(a)]) == ([Fd(a), Fd(p)], 1)
assert _sort_anticommuting_fermions([Fd(a), Fd(p)]) == ([Fd(a), Fd(p)], 0)
assert _sort_anticommuting_fermions([F(p), F(a)]) == ([F(p), F(a)], 0)
assert _sort_anticommuting_fermions([F(a), F(p)]) == ([F(p), F(a)], 1)
assert _sort_anticommuting_fermions([Fd(p), F(a)]) == ([Fd(p), F(a)], 0)
assert _sort_anticommuting_fermions([F(a), Fd(p)]) == ([Fd(p), F(a)], 1)
# i, a
assert _sort_anticommuting_fermions([F(i), Fd(j)]) == ([F(i), Fd(j)], 0)
assert _sort_anticommuting_fermions([Fd(j), F(i)]) == ([F(i), Fd(j)], 1)
assert _sort_anticommuting_fermions([Fd(a), Fd(i)]) == ([Fd(a), Fd(i)], 0)
assert _sort_anticommuting_fermions([Fd(i), Fd(a)]) == ([Fd(a), Fd(i)], 1)
assert _sort_anticommuting_fermions([F(a), F(i)]) == ([F(i), F(a)], 1)
assert _sort_anticommuting_fermions([F(i), F(a)]) == ([F(i), F(a)], 0)
def test_contraction():
i, j, k, l = symbols('i,j,k,l', below_fermi=True)
a, b, c, d = symbols('a,b,c,d', above_fermi=True)
p, q, r, s = symbols('p,q,r,s')
assert contraction(Fd(i), F(j)) == KroneckerDelta(i, j)
assert contraction(F(a), Fd(b)) == KroneckerDelta(a, b)
assert contraction(F(a), Fd(i)) == 0
assert contraction(Fd(a), F(i)) == 0
assert contraction(F(i), Fd(a)) == 0
assert contraction(Fd(i), F(a)) == 0
assert contraction(Fd(i), F(p)) == KroneckerDelta(i, p)
restr = evaluate_deltas(contraction(Fd(p), F(q)))
assert restr.is_only_below_fermi
restr = evaluate_deltas(contraction(F(p), Fd(q)))
assert restr.is_only_above_fermi
raises(ContractionAppliesOnlyToFermions, lambda: contraction(B(a), Fd(b)))
def test_evaluate_deltas():
i, j, k = symbols('i,j,k')
r = KroneckerDelta(i, j) * KroneckerDelta(j, k)
assert evaluate_deltas(r) == KroneckerDelta(i, k)
r = KroneckerDelta(i, 0) * KroneckerDelta(j, k)
assert evaluate_deltas(r) == KroneckerDelta(i, 0) * KroneckerDelta(j, k)
r = KroneckerDelta(1, j) * KroneckerDelta(j, k)
assert evaluate_deltas(r) == KroneckerDelta(1, k)
r = KroneckerDelta(j, 2) * KroneckerDelta(k, j)
assert evaluate_deltas(r) == KroneckerDelta(2, k)
r = KroneckerDelta(i, 0) * KroneckerDelta(i, j) * KroneckerDelta(j, 1)
assert evaluate_deltas(r) == 0
r = (KroneckerDelta(0, i) * KroneckerDelta(0, j)
* KroneckerDelta(1, j) * KroneckerDelta(1, j))
assert evaluate_deltas(r) == 0
def test_Tensors():
i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy)
p, q, r, s = symbols('p q r s')
AT = AntiSymmetricTensor
assert AT('t', (a, b), (i, j)) == -AT('t', (b, a), (i, j))
assert AT('t', (a, b), (i, j)) == AT('t', (b, a), (j, i))
assert AT('t', (a, b), (i, j)) == -AT('t', (a, b), (j, i))
assert AT('t', (a, a), (i, j)) == 0
assert AT('t', (a, b), (i, i)) == 0
assert AT('t', (a, b, c), (i, j)) == -AT('t', (b, a, c), (i, j))
assert AT('t', (a, b, c), (i, j, k)) == AT('t', (b, a, c), (i, k, j))
tabij = AT('t', (a, b), (i, j))
assert tabij.has(a)
assert tabij.has(b)
assert tabij.has(i)
assert tabij.has(j)
assert tabij.subs(b, c) == AT('t', (a, c), (i, j))
assert (2*tabij).subs(i, c) == 2*AT('t', (a, b), (c, j))
assert tabij.symbol == Symbol('t')
assert latex(tabij) == 't^{ab}_{ij}'
assert str(tabij) == 't((_a, _b),(_i, _j))'
assert AT('t', (a, a), (i, j)).subs(a, b) == AT('t', (b, b), (i, j))
assert AT('t', (a, i), (a, j)).subs(a, b) == AT('t', (b, i), (b, j))
def test_fully_contracted():
i, j, k, l = symbols('i j k l', below_fermi=True)
a, b, c, d = symbols('a b c d', above_fermi=True)
p, q, r, s = symbols('p q r s', cls=Dummy)
Fock = (AntiSymmetricTensor('f', (p,), (q,))*
NO(Fd(p)*F(q)))
V = (AntiSymmetricTensor('v', (p, q), (r, s))*
NO(Fd(p)*Fd(q)*F(s)*F(r)))/4
Fai = wicks(NO(Fd(i)*F(a))*Fock,
keep_only_fully_contracted=True,
simplify_kronecker_deltas=True)
assert Fai == AntiSymmetricTensor('f', (a,), (i,))
Vabij = wicks(NO(Fd(i)*Fd(j)*F(b)*F(a))*V,
keep_only_fully_contracted=True,
simplify_kronecker_deltas=True)
assert Vabij == AntiSymmetricTensor('v', (a, b), (i, j))
def test_substitute_dummies_without_dummies():
i, j = symbols('i,j')
assert substitute_dummies(att(i, j) + 2) == att(i, j) + 2
assert substitute_dummies(att(i, j) + 1) == att(i, j) + 1
def test_substitute_dummies_NO_operator():
i, j = symbols('i j', cls=Dummy)
assert substitute_dummies(att(i, j)*NO(Fd(i)*F(j))
- att(j, i)*NO(Fd(j)*F(i))) == 0
def test_substitute_dummies_SQ_operator():
i, j = symbols('i j', cls=Dummy)
assert substitute_dummies(att(i, j)*Fd(i)*F(j)
- att(j, i)*Fd(j)*F(i)) == 0
def test_substitute_dummies_new_indices():
i, j = symbols('i j', below_fermi=True, cls=Dummy)
a, b = symbols('a b', above_fermi=True, cls=Dummy)
p, q = symbols('p q', cls=Dummy)
f = Function('f')
assert substitute_dummies(f(i, a, p) - f(j, b, q), new_indices=True) == 0
def test_substitute_dummies_substitution_order():
i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
f = Function('f')
from sympy.utilities.iterables import variations
for permut in variations([i, j, k, l], 4):
assert substitute_dummies(f(*permut) - f(i, j, k, l)) == 0
def test_dummy_order_inner_outer_lines_VT1T1T1():
ii = symbols('i', below_fermi=True)
aa = symbols('a', above_fermi=True)
k, l = symbols('k l', below_fermi=True, cls=Dummy)
c, d = symbols('c d', above_fermi=True, cls=Dummy)
v = Function('v')
t = Function('t')
dums = _get_ordered_dummies
# Coupled-Cluster T1 terms with V*T1*T1*T1
# t^{a}_{k} t^{c}_{i} t^{d}_{l} v^{lk}_{dc}
exprs = [
# permut v and t <=> swapping internal lines, equivalent
# irrespective of symmetries in v
v(k, l, c, d)*t(c, ii)*t(d, l)*t(aa, k),
v(l, k, c, d)*t(c, ii)*t(d, k)*t(aa, l),
v(k, l, d, c)*t(d, ii)*t(c, l)*t(aa, k),
v(l, k, d, c)*t(d, ii)*t(c, k)*t(aa, l),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_dummy_order_inner_outer_lines_VT1T1T1T1():
ii, jj = symbols('i j', below_fermi=True)
aa, bb = symbols('a b', above_fermi=True)
k, l = symbols('k l', below_fermi=True, cls=Dummy)
c, d = symbols('c d', above_fermi=True, cls=Dummy)
v = Function('v')
t = Function('t')
dums = _get_ordered_dummies
# Coupled-Cluster T2 terms with V*T1*T1*T1*T1
exprs = [
# permut t <=> swapping external lines, not equivalent
# except if v has certain symmetries.
v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
v(k, l, c, d)*t(c, jj)*t(d, ii)*t(aa, k)*t(bb, l),
v(k, l, c, d)*t(c, ii)*t(d, jj)*t(bb, k)*t(aa, l),
v(k, l, c, d)*t(c, jj)*t(d, ii)*t(bb, k)*t(aa, l),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [
# permut v <=> swapping external lines, not equivalent
# except if v has certain symmetries.
#
# Note that in contrast to above, these permutations have identical
# dummy order. That is because the proximity to external indices
# has higher influence on the canonical dummy ordering than the
# position of a dummy on the factors. In fact, the terms here are
# similar in structure as the result of the dummy substitions above.
v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
v(l, k, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
v(k, l, d, c)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
v(l, k, d, c)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
]
for permut in exprs[1:]:
assert dums(exprs[0]) == dums(permut)
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [
# permut t and v <=> swapping internal lines, equivalent.
# Canonical dummy order is different, and a consistent
# substitution reveals the equivalence.
v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l),
v(k, l, d, c)*t(c, jj)*t(d, ii)*t(aa, k)*t(bb, l),
v(l, k, c, d)*t(c, ii)*t(d, jj)*t(bb, k)*t(aa, l),
v(l, k, d, c)*t(c, jj)*t(d, ii)*t(bb, k)*t(aa, l),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_equivalent_internal_lines_VT1T1():
i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy)
v = Function('v')
t = Function('t')
dums = _get_ordered_dummies
exprs = [ # permute v. Different dummy order. Not equivalent.
v(i, j, a, b)*t(a, i)*t(b, j),
v(j, i, a, b)*t(a, i)*t(b, j),
v(i, j, b, a)*t(a, i)*t(b, j),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [ # permute v. Different dummy order. Equivalent
v(i, j, a, b)*t(a, i)*t(b, j),
v(j, i, b, a)*t(a, i)*t(b, j),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
exprs = [ # permute t. Same dummy order, not equivalent.
v(i, j, a, b)*t(a, i)*t(b, j),
v(i, j, a, b)*t(b, i)*t(a, j),
]
for permut in exprs[1:]:
assert dums(exprs[0]) == dums(permut)
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [ # permute v and t. Different dummy order, equivalent
v(i, j, a, b)*t(a, i)*t(b, j),
v(j, i, a, b)*t(a, j)*t(b, i),
v(i, j, b, a)*t(b, i)*t(a, j),
v(j, i, b, a)*t(b, j)*t(a, i),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_equivalent_internal_lines_VT2conjT2():
# this diagram requires special handling in TCE
i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy)
a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy)
p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)
from sympy.utilities.iterables import variations
v = Function('v')
t = Function('t')
dums = _get_ordered_dummies
# v(abcd)t(abij)t(ijcd)
template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(i, j, p3, p4)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert dums(base) != dums(expr)
assert substitute_dummies(expr) == substitute_dummies(base)
template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(j, i, p3, p4)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert dums(base) != dums(expr)
assert substitute_dummies(expr) == substitute_dummies(base)
# v(abcd)t(abij)t(jicd)
template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(j, i, p3, p4)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert dums(base) != dums(expr)
assert substitute_dummies(expr) == substitute_dummies(base)
template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(i, j, p3, p4)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert dums(base) != dums(expr)
assert substitute_dummies(expr) == substitute_dummies(base)
def test_equivalent_internal_lines_VT2conjT2_ambiguous_order():
# These diagrams invokes _determine_ambiguous() because the
# dummies can not be ordered unambiguously by the key alone
i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy)
a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy)
p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)
from sympy.utilities.iterables import variations
v = Function('v')
t = Function('t')
dums = _get_ordered_dummies
# v(abcd)t(abij)t(cdij)
template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(p3, p4, i, j)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert dums(base) != dums(expr)
assert substitute_dummies(expr) == substitute_dummies(base)
template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(p3, p4, i, j)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert dums(base) != dums(expr)
assert substitute_dummies(expr) == substitute_dummies(base)
def test_equivalent_internal_lines_VT2():
i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy)
v = Function('v')
t = Function('t')
dums = _get_ordered_dummies
exprs = [
# permute v. Same dummy order, not equivalent.
#
# This test show that the dummy order may not be sensitive to all
# index permutations. The following expressions have identical
# structure as the resulting terms from of the dummy subsitutions
# in the test above. Here, all expressions have the same dummy
# order, so they cannot be simplified by means of dummy
# substitution. In order to simplify further, it is necessary to
# exploit symmetries in the objects, for instance if t or v is
# antisymmetric.
v(i, j, a, b)*t(a, b, i, j),
v(j, i, a, b)*t(a, b, i, j),
v(i, j, b, a)*t(a, b, i, j),
v(j, i, b, a)*t(a, b, i, j),
]
for permut in exprs[1:]:
assert dums(exprs[0]) == dums(permut)
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [
# permute t.
v(i, j, a, b)*t(a, b, i, j),
v(i, j, a, b)*t(b, a, i, j),
v(i, j, a, b)*t(a, b, j, i),
v(i, j, a, b)*t(b, a, j, i),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [ # permute v and t. Relabelling of dummies should be equivalent.
v(i, j, a, b)*t(a, b, i, j),
v(j, i, a, b)*t(a, b, j, i),
v(i, j, b, a)*t(b, a, i, j),
v(j, i, b, a)*t(b, a, j, i),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_internal_external_VT2T2():
ii, jj = symbols('i j', below_fermi=True)
aa, bb = symbols('a b', above_fermi=True)
k, l = symbols('k l', below_fermi=True, cls=Dummy)
c, d = symbols('c d', above_fermi=True, cls=Dummy)
v = Function('v')
t = Function('t')
dums = _get_ordered_dummies
exprs = [
v(k, l, c, d)*t(aa, c, ii, k)*t(bb, d, jj, l),
v(l, k, c, d)*t(aa, c, ii, l)*t(bb, d, jj, k),
v(k, l, d, c)*t(aa, d, ii, k)*t(bb, c, jj, l),
v(l, k, d, c)*t(aa, d, ii, l)*t(bb, c, jj, k),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
exprs = [
v(k, l, c, d)*t(aa, c, ii, k)*t(d, bb, jj, l),
v(l, k, c, d)*t(aa, c, ii, l)*t(d, bb, jj, k),
v(k, l, d, c)*t(aa, d, ii, k)*t(c, bb, jj, l),
v(l, k, d, c)*t(aa, d, ii, l)*t(c, bb, jj, k),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
exprs = [
v(k, l, c, d)*t(c, aa, ii, k)*t(bb, d, jj, l),
v(l, k, c, d)*t(c, aa, ii, l)*t(bb, d, jj, k),
v(k, l, d, c)*t(d, aa, ii, k)*t(bb, c, jj, l),
v(l, k, d, c)*t(d, aa, ii, l)*t(bb, c, jj, k),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_internal_external_pqrs():
ii, jj = symbols('i j')
aa, bb = symbols('a b')
k, l = symbols('k l', cls=Dummy)
c, d = symbols('c d', cls=Dummy)
v = Function('v')
t = Function('t')
dums = _get_ordered_dummies
exprs = [
v(k, l, c, d)*t(aa, c, ii, k)*t(bb, d, jj, l),
v(l, k, c, d)*t(aa, c, ii, l)*t(bb, d, jj, k),
v(k, l, d, c)*t(aa, d, ii, k)*t(bb, c, jj, l),
v(l, k, d, c)*t(aa, d, ii, l)*t(bb, c, jj, k),
]
for permut in exprs[1:]:
assert dums(exprs[0]) != dums(permut)
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_dummy_order_well_defined():
aa, bb = symbols('a b', above_fermi=True)
k, l, m = symbols('k l m', below_fermi=True, cls=Dummy)
c, d = symbols('c d', above_fermi=True, cls=Dummy)
p, q = symbols('p q', cls=Dummy)
A = Function('A')
B = Function('B')
C = Function('C')
dums = _get_ordered_dummies
# We go through all key components in the order of increasing priority,
# and consider only fully orderable expressions. Non-orderable expressions
# are tested elsewhere.
# pos in first factor determines sort order
assert dums(A(k, l)*B(l, k)) == [k, l]
assert dums(A(l, k)*B(l, k)) == [l, k]
assert dums(A(k, l)*B(k, l)) == [k, l]
assert dums(A(l, k)*B(k, l)) == [l, k]
# factors involving the index
assert dums(A(k, l)*B(l, m)*C(k, m)) == [l, k, m]
assert dums(A(k, l)*B(l, m)*C(m, k)) == [l, k, m]
assert dums(A(l, k)*B(l, m)*C(k, m)) == [l, k, m]
assert dums(A(l, k)*B(l, m)*C(m, k)) == [l, k, m]
assert dums(A(k, l)*B(m, l)*C(k, m)) == [l, k, m]
assert dums(A(k, l)*B(m, l)*C(m, k)) == [l, k, m]
assert dums(A(l, k)*B(m, l)*C(k, m)) == [l, k, m]
assert dums(A(l, k)*B(m, l)*C(m, k)) == [l, k, m]
# same, but with factor order determined by non-dummies
assert dums(A(k, aa, l)*A(l, bb, m)*A(bb, k, m)) == [l, k, m]
assert dums(A(k, aa, l)*A(l, bb, m)*A(bb, m, k)) == [l, k, m]
assert dums(A(k, aa, l)*A(m, bb, l)*A(bb, k, m)) == [l, k, m]
assert dums(A(k, aa, l)*A(m, bb, l)*A(bb, m, k)) == [l, k, m]
assert dums(A(l, aa, k)*A(l, bb, m)*A(bb, k, m)) == [l, k, m]
assert dums(A(l, aa, k)*A(l, bb, m)*A(bb, m, k)) == [l, k, m]
assert dums(A(l, aa, k)*A(m, bb, l)*A(bb, k, m)) == [l, k, m]
assert dums(A(l, aa, k)*A(m, bb, l)*A(bb, m, k)) == [l, k, m]
# index range
assert dums(A(p, c, k)*B(p, c, k)) == [k, c, p]
assert dums(A(p, k, c)*B(p, c, k)) == [k, c, p]
assert dums(A(c, k, p)*B(p, c, k)) == [k, c, p]
assert dums(A(c, p, k)*B(p, c, k)) == [k, c, p]
assert dums(A(k, c, p)*B(p, c, k)) == [k, c, p]
assert dums(A(k, p, c)*B(p, c, k)) == [k, c, p]
assert dums(B(p, c, k)*A(p, c, k)) == [k, c, p]
assert dums(B(p, k, c)*A(p, c, k)) == [k, c, p]
assert dums(B(c, k, p)*A(p, c, k)) == [k, c, p]
assert dums(B(c, p, k)*A(p, c, k)) == [k, c, p]
assert dums(B(k, c, p)*A(p, c, k)) == [k, c, p]
assert dums(B(k, p, c)*A(p, c, k)) == [k, c, p]
def test_dummy_order_ambiguous():
aa, bb = symbols('a b', above_fermi=True)
i, j, k, l, m = symbols('i j k l m', below_fermi=True, cls=Dummy)
a, b, c, d, e = symbols('a b c d e', above_fermi=True, cls=Dummy)
p, q = symbols('p q', cls=Dummy)
p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
p5, p6, p7, p8 = symbols('p5 p6 p7 p8', above_fermi=True, cls=Dummy)
h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)
h5, h6, h7, h8 = symbols('h5 h6 h7 h8', below_fermi=True, cls=Dummy)
A = Function('A')
B = Function('B')
from sympy.utilities.iterables import variations
# A*A*A*A*B -- ordering of p5 and p4 is used to figure out the rest
template = A(p1, p2)*A(p4, p1)*A(p2, p3)*A(p3, p5)*B(p5, p4)
permutator = variations([a, b, c, d, e], 5)
base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4, p5], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
# A*A*A*A*A -- an arbitrary index is assigned and the rest are figured out
template = A(p1, p2)*A(p4, p1)*A(p2, p3)*A(p3, p5)*A(p5, p4)
permutator = variations([a, b, c, d, e], 5)
base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4, p5], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
# A*A*A -- ordering of p5 and p4 is used to figure out the rest
template = A(p1, p2, p4, p1)*A(p2, p3, p3, p5)*A(p5, p4)
permutator = variations([a, b, c, d, e], 5)
base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4, p5], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
def atv(*args):
return AntiSymmetricTensor('v', args[:2], args[2:] )
def att(*args):
if len(args) == 4:
return AntiSymmetricTensor('t', args[:2], args[2:] )
elif len(args) == 2:
return AntiSymmetricTensor('t', (args[0],), (args[1],))
def test_dummy_order_inner_outer_lines_VT1T1T1_AT():
ii = symbols('i', below_fermi=True)
aa = symbols('a', above_fermi=True)
k, l = symbols('k l', below_fermi=True, cls=Dummy)
c, d = symbols('c d', above_fermi=True, cls=Dummy)
# Coupled-Cluster T1 terms with V*T1*T1*T1
# t^{a}_{k} t^{c}_{i} t^{d}_{l} v^{lk}_{dc}
exprs = [
# permut v and t <=> swapping internal lines, equivalent
# irrespective of symmetries in v
atv(k, l, c, d)*att(c, ii)*att(d, l)*att(aa, k),
atv(l, k, c, d)*att(c, ii)*att(d, k)*att(aa, l),
atv(k, l, d, c)*att(d, ii)*att(c, l)*att(aa, k),
atv(l, k, d, c)*att(d, ii)*att(c, k)*att(aa, l),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_dummy_order_inner_outer_lines_VT1T1T1T1_AT():
ii, jj = symbols('i j', below_fermi=True)
aa, bb = symbols('a b', above_fermi=True)
k, l = symbols('k l', below_fermi=True, cls=Dummy)
c, d = symbols('c d', above_fermi=True, cls=Dummy)
# Coupled-Cluster T2 terms with V*T1*T1*T1*T1
# non-equivalent substitutions (change of sign)
exprs = [
# permut t <=> swapping external lines
atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(aa, k)*att(bb, l),
atv(k, l, c, d)*att(c, jj)*att(d, ii)*att(aa, k)*att(bb, l),
atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(bb, k)*att(aa, l),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == -substitute_dummies(permut)
# equivalent substitutions
exprs = [
atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(aa, k)*att(bb, l),
# permut t <=> swapping external lines
atv(k, l, c, d)*att(c, jj)*att(d, ii)*att(bb, k)*att(aa, l),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_equivalent_internal_lines_VT1T1_AT():
i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy)
exprs = [ # permute v. Different dummy order. Not equivalent.
atv(i, j, a, b)*att(a, i)*att(b, j),
atv(j, i, a, b)*att(a, i)*att(b, j),
atv(i, j, b, a)*att(a, i)*att(b, j),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [ # permute v. Different dummy order. Equivalent
atv(i, j, a, b)*att(a, i)*att(b, j),
atv(j, i, b, a)*att(a, i)*att(b, j),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
exprs = [ # permute t. Same dummy order, not equivalent.
atv(i, j, a, b)*att(a, i)*att(b, j),
atv(i, j, a, b)*att(b, i)*att(a, j),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [ # permute v and t. Different dummy order, equivalent
atv(i, j, a, b)*att(a, i)*att(b, j),
atv(j, i, a, b)*att(a, j)*att(b, i),
atv(i, j, b, a)*att(b, i)*att(a, j),
atv(j, i, b, a)*att(b, j)*att(a, i),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_equivalent_internal_lines_VT2conjT2_AT():
# this diagram requires special handling in TCE
i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy)
a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy)
p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)
from sympy.utilities.iterables import variations
# atv(abcd)att(abij)att(ijcd)
template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(i, j, p3, p4)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(j, i, p3, p4)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
# atv(abcd)att(abij)att(jicd)
template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(j, i, p3, p4)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(i, j, p3, p4)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
def test_equivalent_internal_lines_VT2conjT2_ambiguous_order_AT():
# These diagrams invokes _determine_ambiguous() because the
# dummies can not be ordered unambiguously by the key alone
i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy)
a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy)
p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy)
h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy)
from sympy.utilities.iterables import variations
# atv(abcd)att(abij)att(cdij)
template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(p3, p4, i, j)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(p3, p4, i, j)
permutator = variations([a, b, c, d], 4)
base = template.subs(zip([p1, p2, p3, p4], next(permutator)))
for permut in permutator:
subslist = zip([p1, p2, p3, p4], permut)
expr = template.subs(subslist)
assert substitute_dummies(expr) == substitute_dummies(base)
def test_equivalent_internal_lines_VT2_AT():
i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy)
a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy)
exprs = [
# permute v. Same dummy order, not equivalent.
atv(i, j, a, b)*att(a, b, i, j),
atv(j, i, a, b)*att(a, b, i, j),
atv(i, j, b, a)*att(a, b, i, j),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [
# permute t.
atv(i, j, a, b)*att(a, b, i, j),
atv(i, j, a, b)*att(b, a, i, j),
atv(i, j, a, b)*att(a, b, j, i),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) != substitute_dummies(permut)
exprs = [ # permute v and t. Relabelling of dummies should be equivalent.
atv(i, j, a, b)*att(a, b, i, j),
atv(j, i, a, b)*att(a, b, j, i),
atv(i, j, b, a)*att(b, a, i, j),
atv(j, i, b, a)*att(b, a, j, i),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_internal_external_VT2T2_AT():
ii, jj = symbols('i j', below_fermi=True)
aa, bb = symbols('a b', above_fermi=True)
k, l = symbols('k l', below_fermi=True, cls=Dummy)
c, d = symbols('c d', above_fermi=True, cls=Dummy)
exprs = [
atv(k, l, c, d)*att(aa, c, ii, k)*att(bb, d, jj, l),
atv(l, k, c, d)*att(aa, c, ii, l)*att(bb, d, jj, k),
atv(k, l, d, c)*att(aa, d, ii, k)*att(bb, c, jj, l),
atv(l, k, d, c)*att(aa, d, ii, l)*att(bb, c, jj, k),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
exprs = [
atv(k, l, c, d)*att(aa, c, ii, k)*att(d, bb, jj, l),
atv(l, k, c, d)*att(aa, c, ii, l)*att(d, bb, jj, k),
atv(k, l, d, c)*att(aa, d, ii, k)*att(c, bb, jj, l),
atv(l, k, d, c)*att(aa, d, ii, l)*att(c, bb, jj, k),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
exprs = [
atv(k, l, c, d)*att(c, aa, ii, k)*att(bb, d, jj, l),
atv(l, k, c, d)*att(c, aa, ii, l)*att(bb, d, jj, k),
atv(k, l, d, c)*att(d, aa, ii, k)*att(bb, c, jj, l),
atv(l, k, d, c)*att(d, aa, ii, l)*att(bb, c, jj, k),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_internal_external_pqrs_AT():
ii, jj = symbols('i j')
aa, bb = symbols('a b')
k, l = symbols('k l', cls=Dummy)
c, d = symbols('c d', cls=Dummy)
exprs = [
atv(k, l, c, d)*att(aa, c, ii, k)*att(bb, d, jj, l),
atv(l, k, c, d)*att(aa, c, ii, l)*att(bb, d, jj, k),
atv(k, l, d, c)*att(aa, d, ii, k)*att(bb, c, jj, l),
atv(l, k, d, c)*att(aa, d, ii, l)*att(bb, c, jj, k),
]
for permut in exprs[1:]:
assert substitute_dummies(exprs[0]) == substitute_dummies(permut)
def test_canonical_ordering_AntiSymmetricTensor():
v = symbols("v")
virtual_indices = ('c', 'd')
occupied_indices = ('k', 'l')
c, d = symbols(('c','d'), above_fermi=True,
cls=Dummy)
k, l = symbols(('k','l'), below_fermi=True,
cls=Dummy)
# formerly, the left gave either the left or the right
assert AntiSymmetricTensor(v, (k, l), (d, c)
) == -AntiSymmetricTensor(v, (l, k), (d, c))
|
14517edc24114c91f177c93f364e36a530035feceed13d9098f038be9844f48c | """
This module can be used to solve 2D beam bending problems with
singularity functions in mechanics.
"""
from __future__ import print_function, division
from sympy.core import S, Symbol, diff, symbols
from sympy.solvers import linsolve
from sympy.printing import sstr
from sympy.functions import SingularityFunction, Piecewise, factorial
from sympy.core import sympify
from sympy.integrals import integrate
from sympy.series import limit
from sympy.plotting import plot
from sympy.external import import_module
from sympy.utilities.decorator import doctest_depends_on
from sympy import lambdify
matplotlib = import_module('matplotlib', __import__kwargs={'fromlist':['pyplot']})
numpy = import_module('numpy', __import__kwargs={'fromlist':['linspace']})
__doctest_requires__ = {('Beam.plot_loading_results',): ['matplotlib']}
class Beam(object):
"""
A Beam is a structural element that is capable of withstanding load
primarily by resisting against bending. Beams are characterized by
their cross sectional profile(Second moment of area), their length
and their material.
.. note::
While solving a beam bending problem, a user should choose its
own sign convention and should stick to it. The results will
automatically follow the chosen sign convention.
Examples
========
There is a beam of length 4 meters. A constant distributed load of 6 N/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. The deflection of the beam at the end is restricted.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols, Piecewise
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(4, E, I)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(6, 2, 0)
>>> b.apply_load(R2, 4, -1)
>>> b.bc_deflection = [(0, 0), (4, 0)]
>>> b.boundary_conditions
{'deflection': [(0, 0), (4, 0)], 'slope': []}
>>> b.load
R1*SingularityFunction(x, 0, -1) + R2*SingularityFunction(x, 4, -1) + 6*SingularityFunction(x, 2, 0)
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.load
-3*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 2, 0) - 9*SingularityFunction(x, 4, -1)
>>> b.shear_force()
-3*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 2, 1) - 9*SingularityFunction(x, 4, 0)
>>> b.bending_moment()
-3*SingularityFunction(x, 0, 1) + 3*SingularityFunction(x, 2, 2) - 9*SingularityFunction(x, 4, 1)
>>> b.slope()
(-3*SingularityFunction(x, 0, 2)/2 + SingularityFunction(x, 2, 3) - 9*SingularityFunction(x, 4, 2)/2 + 7)/(E*I)
>>> b.deflection()
(7*x - SingularityFunction(x, 0, 3)/2 + SingularityFunction(x, 2, 4)/4 - 3*SingularityFunction(x, 4, 3)/2)/(E*I)
>>> b.deflection().rewrite(Piecewise)
(7*x - Piecewise((x**3, x > 0), (0, True))/2
- 3*Piecewise(((x - 4)**3, x - 4 > 0), (0, True))/2
+ Piecewise(((x - 2)**4, x - 2 > 0), (0, True))/4)/(E*I)
"""
def __init__(self, length, elastic_modulus, second_moment, variable=Symbol('x'), base_char='C'):
"""Initializes the class.
Parameters
==========
length : Sympifyable
A Symbol or value representing the Beam's length.
elastic_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of Elasticity.
It is a measure of the stiffness of the Beam material. It can
also be a continuous function of position along the beam.
second_moment : Sympifyable
A SymPy expression representing the Beam's Second moment of area.
It is a geometrical property of an area which reflects how its
points are distributed with respect to its neutral axis. It can
also be a continuous function of position along the beam.
variable : Symbol, optional
A Symbol object that will be used as the variable along the beam
while representing the load, shear, moment, slope and deflection
curve. By default, it is set to ``Symbol('x')``.
base_char : String, optional
A String that will be used as base character to generate sequential
symbols for integration constants in cases where boundary conditions
are not sufficient to solve them.
"""
self.length = length
self.elastic_modulus = elastic_modulus
self.second_moment = second_moment
self.variable = variable
self._base_char = base_char
self._boundary_conditions = {'deflection': [], 'slope': []}
self._load = 0
self._applied_loads = []
self._reaction_loads = {}
self._composite_type = None
self._hinge_position = None
def __str__(self):
str_sol = 'Beam({}, {}, {})'.format(sstr(self._length), sstr(self._elastic_modulus), sstr(self._second_moment))
return str_sol
@property
def reaction_loads(self):
""" Returns the reaction forces in a dictionary."""
return self._reaction_loads
@property
def length(self):
"""Length of the Beam."""
return self._length
@length.setter
def length(self, l):
self._length = sympify(l)
@property
def variable(self):
"""
A symbol that can be used as a variable along the length of the beam
while representing load distribution, shear force curve, bending
moment, slope curve and the deflection curve. By default, it is set
to ``Symbol('x')``, but this property is mutable.
Examples
========
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> x, y, z = symbols('x, y, z')
>>> b = Beam(4, E, I)
>>> b.variable
x
>>> b.variable = y
>>> b.variable
y
>>> b = Beam(4, E, I, z)
>>> b.variable
z
"""
return self._variable
@variable.setter
def variable(self, v):
if isinstance(v, Symbol):
self._variable = v
else:
raise TypeError("""The variable should be a Symbol object.""")
@property
def elastic_modulus(self):
"""Young's Modulus of the Beam. """
return self._elastic_modulus
@elastic_modulus.setter
def elastic_modulus(self, e):
self._elastic_modulus = sympify(e)
@property
def second_moment(self):
"""Second moment of area of the Beam. """
return self._second_moment
@second_moment.setter
def second_moment(self, i):
self._second_moment = sympify(i)
@property
def boundary_conditions(self):
"""
Returns a dictionary of boundary conditions applied on the beam.
The dictionary has three kewwords namely moment, slope and deflection.
The value of each keyword is a list of tuple, where each tuple
contains loaction and value of a boundary condition in the format
(location, value).
Examples
========
There is a beam of length 4 meters. The bending moment at 0 should be 4
and at 4 it should be 0. The slope of the beam should be 1 at 0. The
deflection should be 2 at 0.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.bc_deflection = [(0, 2)]
>>> b.bc_slope = [(0, 1)]
>>> b.boundary_conditions
{'deflection': [(0, 2)], 'slope': [(0, 1)]}
Here the deflection of the beam should be ``2`` at ``0``.
Similarly, the slope of the beam should be ``1`` at ``0``.
"""
return self._boundary_conditions
@property
def bc_slope(self):
return self._boundary_conditions['slope']
@bc_slope.setter
def bc_slope(self, s_bcs):
self._boundary_conditions['slope'] = s_bcs
@property
def bc_deflection(self):
return self._boundary_conditions['deflection']
@bc_deflection.setter
def bc_deflection(self, d_bcs):
self._boundary_conditions['deflection'] = d_bcs
def join(self, beam, via="fixed"):
"""
This method joins two beams to make a new composite beam system.
Passed Beam class instance is attached to the right end of calling
object. This method can be used to form beams having Discontinuous
values of Elastic modulus or Second moment.
Parameters
==========
beam : Beam class object
The Beam object which would be connected to the right of calling
object.
via : String
States the way two Beam object would get connected
- For axially fixed Beams, via="fixed"
- For Beams connected via hinge, via="hinge"
Examples
========
There is a cantilever beam of length 4 meters. For first 2 meters
its moment of inertia is `1.5*I` and `I` for the other end.
A pointload of magnitude 4 N is applied from the top at its free end.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b1 = Beam(2, E, 1.5*I)
>>> b2 = Beam(2, E, I)
>>> b = b1.join(b2, "fixed")
>>> b.apply_load(20, 4, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 0, -2)
>>> b.bc_slope = [(0, 0)]
>>> b.bc_deflection = [(0, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.load
80*SingularityFunction(x, 0, -2) - 20*SingularityFunction(x, 0, -1) + 20*SingularityFunction(x, 4, -1)
>>> b.slope()
(((80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))/I - 120/I)/E + 80.0/(E*I))*SingularityFunction(x, 2, 0)
+ 0.666666666666667*(80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 0, 0)/(E*I)
- 0.666666666666667*(80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 2, 0)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
new_length = self.length + beam.length
if self.second_moment != beam.second_moment:
new_second_moment = Piecewise((self.second_moment, x<=self.length),
(beam.second_moment, x<=new_length))
else:
new_second_moment = self.second_moment
if via == "fixed":
new_beam = Beam(new_length, E, new_second_moment, x)
new_beam._composite_type = "fixed"
return new_beam
if via == "hinge":
new_beam = Beam(new_length, E, new_second_moment, x)
new_beam._composite_type = "hinge"
new_beam._hinge_position = self.length
return new_beam
def apply_support(self, loc, type="fixed"):
"""
This method applies support to a particular beam object.
Parameters
==========
loc : Sympifyable
Location of point at which support is applied.
type : String
Determines type of Beam support applied. To apply support structure
with
- zero degree of freedom, type = "fixed"
- one degree of freedom, type = "pin"
- two degrees of freedom, type = "roller"
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(30, E, I)
>>> b.apply_support(10, 'roller')
>>> b.apply_support(30, 'roller')
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(120, 30, -2)
>>> R_10, R_30 = symbols('R_10, R_30')
>>> b.solve_for_reaction_loads(R_10, R_30)
>>> b.load
-8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1)
+ 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
>>> b.slope()
(-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)
"""
if type == "pin" or type == "roller":
reaction_load = Symbol('R_'+str(loc))
self.apply_load(reaction_load, loc, -1)
self.bc_deflection.append((loc, 0))
else:
reaction_load = Symbol('R_'+str(loc))
reaction_moment = Symbol('M_'+str(loc))
self.apply_load(reaction_load, loc, -1)
self.apply_load(reaction_moment, loc, -2)
self.bc_deflection.append((loc, 0))
self.bc_slope.append((loc, 0))
def apply_load(self, value, start, order, end=None):
"""
This method adds up the loads given to a particular beam object.
Parameters
==========
value : Sympifyable
The magnitude of an applied load.
start : Sympifyable
The starting point of the applied load. For point moments and
point forces this is the location of application.
order : Integer
The order of the applied load.
- For moments, order = -2
- For point loads, order =-1
- For constant distributed load, order = 0
- For ramp loads, order = 1
- For parabolic ramp loads, order = 2
- ... so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point
within the length of the beam.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A point load of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 2 meters to 3 meters
away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 2, 2, end=3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
self._applied_loads.append((value, start, order, end))
self._load += value*SingularityFunction(x, start, order)
if end:
if order.is_negative:
msg = ("If 'end' is provided the 'order' of the load cannot "
"be negative, i.e. 'end' is only valid for distributed "
"loads.")
raise ValueError(msg)
# NOTE : A Taylor series can be used to define the summation of
# singularity functions that subtract from the load past the end
# point such that it evaluates to zero past 'end'.
f = value * x**order
for i in range(0, order + 1):
self._load -= (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i) / factorial(i))
def remove_load(self, value, start, order, end=None):
"""
This method removes a particular load present on the beam object.
Returns a ValueError if the load passed as an argument is not
present on the beam.
Parameters
==========
value : Sympifyable
The magnitude of an applied load.
start : Sympifyable
The starting point of the applied load. For point moments and
point forces this is the location of application.
order : Integer
The order of the applied load.
- For moments, order= -2
- For point loads, order=-1
- For constant distributed load, order=0
- For ramp loads, order=1
- For parabolic ramp loads, order=2
- ... so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point
within the length of the beam.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A pointload of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 2 meters to 3 meters
away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 2, 2, end=3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
>>> b.remove_load(-2, 2, 2, end = 3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if (value, start, order, end) in self._applied_loads:
self._load -= value*SingularityFunction(x, start, order)
self._applied_loads.remove((value, start, order, end))
else:
msg = "No such load distribution exists on the beam object."
raise ValueError(msg)
if end:
# TODO : This is essentially duplicate code wrt to apply_load,
# would be better to move it to one location and both methods use
# it.
if order.is_negative:
msg = ("If 'end' is provided the 'order' of the load cannot "
"be negative, i.e. 'end' is only valid for distributed "
"loads.")
raise ValueError(msg)
# NOTE : A Taylor series can be used to define the summation of
# singularity functions that subtract from the load past the end
# point such that it evaluates to zero past 'end'.
f = value * x**order
for i in range(0, order + 1):
self._load += (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i) / factorial(i))
@property
def load(self):
"""
Returns a Singularity Function expression which represents
the load distribution curve of the Beam object.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A point load of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 3 meters away from the
starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 3, 2)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 3, 2)
"""
return self._load
@property
def applied_loads(self):
"""
Returns a list of all loads applied on the beam object.
Each load in the list is a tuple of form (value, start, order, end).
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A pointload of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point. Another pointload of magnitude 5 N
is applied at same position.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(5, 2, -1)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 9*SingularityFunction(x, 2, -1)
>>> b.applied_loads
[(-3, 0, -2, None), (4, 2, -1, None), (5, 2, -1, None)]
"""
return self._applied_loads
def _solve_hinge_beams(self, *reactions):
"""Method to find integration constants and reactional variables in a
composite beam connected via hinge.
This method resolves the composite Beam into its sub-beams and then
equations of shear force, bending moment, slope and deflection are
evaluated for both of them separately. These equations are then solved
for unknown reactions and integration constants using the boundary
conditions applied on the Beam. Equal deflection of both sub-beams
at the hinge joint gives us another equation to solve the system.
Examples
========
A combined beam, with constant fkexural rigidity E*I, is formed by joining
a Beam of length 2*l to the right of another Beam of length l. The whole beam
is fixed at both of its both end. A point load of magnitude P is also applied
from the top at a distance of 2*l from starting point.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> l=symbols('l', positive=True)
>>> b1=Beam(l ,E,I)
>>> b2=Beam(2*l ,E,I)
>>> b=b1.join(b2,"hinge")
>>> M1, A1, M2, A2, P = symbols('M1 A1 M2 A2 P')
>>> b.apply_load(A1,0,-1)
>>> b.apply_load(M1,0,-2)
>>> b.apply_load(P,2*l,-1)
>>> b.apply_load(A2,3*l,-1)
>>> b.apply_load(M2,3*l,-2)
>>> b.bc_slope=[(0,0), (3*l, 0)]
>>> b.bc_deflection=[(0,0), (3*l, 0)]
>>> b.solve_for_reaction_loads(M1, A1, M2, A2)
>>> b.reaction_loads
{A1: -5*P/18, A2: -13*P/18, M1: 5*P*l/18, M2: -4*P*l/9}
>>> b.slope()
(5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, 0, 0)/(E*I)
- (5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, l, 0)/(E*I)
+ (P*l**2/18 - 4*P*l*SingularityFunction(-l + x, 2*l, 1)/9 - 5*P*SingularityFunction(-l + x, 0, 2)/36 + P*SingularityFunction(-l + x, l, 2)/2
- 13*P*SingularityFunction(-l + x, 2*l, 2)/36)*SingularityFunction(x, l, 0)/(E*I)
>>> b.deflection()
(5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, 0, 0)/(E*I)
- (5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, l, 0)/(E*I)
+ (5*P*l**3/54 + P*l**2*(-l + x)/18 - 2*P*l*SingularityFunction(-l + x, 2*l, 2)/9 - 5*P*SingularityFunction(-l + x, 0, 3)/108 + P*SingularityFunction(-l + x, l, 3)/6
- 13*P*SingularityFunction(-l + x, 2*l, 3)/108)*SingularityFunction(x, l, 0)/(E*I)
"""
x = self.variable
l = self._hinge_position
E = self._elastic_modulus
I = self._second_moment
if isinstance(I, Piecewise):
I1 = I.args[0][0]
I2 = I.args[1][0]
else:
I1 = I2 = I
load_1 = 0 # Load equation on first segment of composite beam
load_2 = 0 # Load equation on second segment of composite beam
# Distributing load on both segments
for load in self.applied_loads:
if load[1] < l:
load_1 += load[0]*SingularityFunction(x, load[1], load[2])
if load[2] == 0:
load_1 -= load[0]*SingularityFunction(x, load[3], load[2])
elif load[2] > 0:
load_1 -= load[0]*SingularityFunction(x, load[3], load[2]) + load[0]*SingularityFunction(x, load[3], 0)
elif load[1] == l:
load_1 += load[0]*SingularityFunction(x, load[1], load[2])
load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
elif load[1] > l:
load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
if load[2] == 0:
load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2])
elif load[2] > 0:
load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2]) + load[0]*SingularityFunction(x, load[3] - l, 0)
h = Symbol('h') # Force due to hinge
load_1 += h*SingularityFunction(x, l, -1)
load_2 -= h*SingularityFunction(x, 0, -1)
eq = []
shear_1 = integrate(load_1, x)
shear_curve_1 = limit(shear_1, x, l)
eq.append(shear_curve_1)
bending_1 = integrate(shear_1, x)
moment_curve_1 = limit(bending_1, x, l)
eq.append(moment_curve_1)
shear_2 = integrate(load_2, x)
shear_curve_2 = limit(shear_2, x, self.length - l)
eq.append(shear_curve_2)
bending_2 = integrate(shear_2, x)
moment_curve_2 = limit(bending_2, x, self.length - l)
eq.append(moment_curve_2)
C1 = Symbol('C1')
C2 = Symbol('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
slope_1 = S(1)/(E*I1)*(integrate(bending_1, x) + C1)
def_1 = S(1)/(E*I1)*(integrate((E*I)*slope_1, x) + C1*x + C2)
slope_2 = S(1)/(E*I2)*(integrate(integrate(integrate(load_2, x), x), x) + C3)
def_2 = S(1)/(E*I2)*(integrate((E*I)*slope_2, x) + C4)
for position, value in self.bc_slope:
if position<l:
eq.append(slope_1.subs(x, position) - value)
else:
eq.append(slope_2.subs(x, position - l) - value)
for position, value in self.bc_deflection:
if position<l:
eq.append(def_1.subs(x, position) - value)
else:
eq.append(def_2.subs(x, position - l) - value)
eq.append(def_1.subs(x, l) - def_2.subs(x, 0)) # Deflection of both the segments at hinge would be equal
constants = list(linsolve(eq, C1, C2, C3, C4, h, *reactions))
reaction_values = list(constants[0])[5:]
self._reaction_loads = dict(zip(reactions, reaction_values))
self._load = self._load.subs(self._reaction_loads)
# Substituting constants and reactional load and moments with their corresponding values
slope_1 = slope_1.subs({C1: constants[0][0], h:constants[0][4]}).subs(self._reaction_loads)
def_1 = def_1.subs({C1: constants[0][0], C2: constants[0][1], h:constants[0][4]}).subs(self._reaction_loads)
slope_2 = slope_2.subs({x: x-l, C3: constants[0][2], h:constants[0][4]}).subs(self._reaction_loads)
def_2 = def_2.subs({x: x-l,C3: constants[0][2], C4: constants[0][3], h:constants[0][4]}).subs(self._reaction_loads)
self._hinge_beam_slope = slope_1*SingularityFunction(x, 0, 0) - slope_1*SingularityFunction(x, l, 0) + slope_2*SingularityFunction(x, l, 0)
self._hinge_beam_deflection = def_1*SingularityFunction(x, 0, 0) - def_1*SingularityFunction(x, l, 0) + def_2*SingularityFunction(x, l, 0)
def solve_for_reaction_loads(self, *reactions):
"""
Solves for the reaction forces.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols, linsolve, limit
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1) # Reaction force at x = 10
>>> b.apply_load(R2, 30, -1) # Reaction force at x = 30
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.load
R1*SingularityFunction(x, 10, -1) + R2*SingularityFunction(x, 30, -1)
- 8*SingularityFunction(x, 0, -1) + 120*SingularityFunction(x, 30, -2)
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.reaction_loads
{R1: 6, R2: 2}
>>> b.load
-8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1)
+ 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
"""
if self._composite_type == "hinge":
return self._solve_hinge_beams(*reactions)
x = self.variable
l = self.length
C3 = Symbol('C3')
C4 = Symbol('C4')
shear_curve = limit(self.shear_force(), x, l)
moment_curve = limit(self.bending_moment(), x, l)
slope_eqs = []
deflection_eqs = []
slope_curve = integrate(self.bending_moment(), x) + C3
for position, value in self._boundary_conditions['slope']:
eqs = slope_curve.subs(x, position) - value
slope_eqs.append(eqs)
deflection_curve = integrate(slope_curve, x) + C4
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
deflection_eqs.append(eqs)
solution = list((linsolve([shear_curve, moment_curve] + slope_eqs
+ deflection_eqs, (C3, C4) + reactions).args)[0])
solution = solution[2:]
self._reaction_loads = dict(zip(reactions, solution))
self._load = self._load.subs(self._reaction_loads)
def shear_force(self):
"""
Returns a Singularity Function expression which represents
the shear force curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.shear_force()
-8*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 10, 0) + 120*SingularityFunction(x, 30, -1) + 2*SingularityFunction(x, 30, 0)
"""
x = self.variable
return integrate(self.load, x)
def max_shear_force(self):
"""Returns maximum Shear force and its coordinate
in the Beam object."""
from sympy import solve, Mul, Interval
shear_curve = self.shear_force()
x = self.variable
terms = shear_curve.args
singularity = [] # Points at which shear function changes
for term in terms:
if isinstance(term, Mul):
term = term.args[-1] # SingularityFunction in the term
singularity.append(term.args[1])
singularity.sort()
singularity = list(set(singularity))
intervals = [] # List of Intervals with discrete value of shear force
shear_values = [] # List of values of shear force in each interval
for i, s in enumerate(singularity):
if s == 0:
continue
try:
shear_slope = Piecewise((float("nan"), x<=singularity[i-1]),(self._load.rewrite(Piecewise), x<s), (float("nan"), True))
points = solve(shear_slope, x)
val = []
for point in points:
val.append(shear_curve.subs(x, point))
points.extend([singularity[i-1], s])
val.extend([limit(shear_curve, x, singularity[i-1], '+'), limit(shear_curve, x, s, '-')])
val = list(map(abs, val))
max_shear = max(val)
shear_values.append(max_shear)
intervals.append(points[val.index(max_shear)])
# If shear force in a particular Interval has zero or constant
# slope, then above block gives NotImplementedError as
# solve can't represent Interval solutions.
except NotImplementedError:
initial_shear = limit(shear_curve, x, singularity[i-1], '+')
final_shear = limit(shear_curve, x, s, '-')
# If shear_curve has a constant slope(it is a line).
if shear_curve.subs(x, (singularity[i-1] + s)/2) == (initial_shear + final_shear)/2 and initial_shear != final_shear:
shear_values.extend([initial_shear, final_shear])
intervals.extend([singularity[i-1], s])
else: # shear_curve has same value in whole Interval
shear_values.append(final_shear)
intervals.append(Interval(singularity[i-1], s))
shear_values = list(map(abs, shear_values))
maximum_shear = max(shear_values)
point = intervals[shear_values.index(maximum_shear)]
return (point, maximum_shear)
def bending_moment(self):
"""
Returns a Singularity Function expression which represents
the bending moment curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.bending_moment()
-8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 0) + 2*SingularityFunction(x, 30, 1)
"""
x = self.variable
return integrate(self.shear_force(), x)
def max_bmoment(self):
"""Returns maximum Shear force and its coordinate
in the Beam object."""
from sympy import solve, Mul, Interval
bending_curve = self.bending_moment()
x = self.variable
terms = bending_curve.args
singularity = [] # Points at which bending moment changes
for term in terms:
if isinstance(term, Mul):
term = term.args[-1] # SingularityFunction in the term
singularity.append(term.args[1])
singularity.sort()
singularity = list(set(singularity))
intervals = [] # List of Intervals with discrete value of bending moment
moment_values = [] # List of values of bending moment in each interval
for i, s in enumerate(singularity):
if s == 0:
continue
try:
moment_slope = Piecewise((float("nan"), x<=singularity[i-1]),(self.shear_force().rewrite(Piecewise), x<s), (float("nan"), True))
points = solve(moment_slope, x)
val = []
for point in points:
val.append(bending_curve.subs(x, point))
points.extend([singularity[i-1], s])
val.extend([limit(bending_curve, x, singularity[i-1], '+'), limit(bending_curve, x, s, '-')])
val = list(map(abs, val))
max_moment = max(val)
moment_values.append(max_moment)
intervals.append(points[val.index(max_moment)])
# If bending moment in a particular Interval has zero or constant
# slope, then above block gives NotImplementedError as solve
# can't represent Interval solutions.
except NotImplementedError:
initial_moment = limit(bending_curve, x, singularity[i-1], '+')
final_moment = limit(bending_curve, x, s, '-')
# If bending_curve has a constant slope(it is a line).
if bending_curve.subs(x, (singularity[i-1] + s)/2) == (initial_moment + final_moment)/2 and initial_moment != final_moment:
moment_values.extend([initial_moment, final_moment])
intervals.extend([singularity[i-1], s])
else: # bending_curve has same value in whole Interval
moment_values.append(final_moment)
intervals.append(Interval(singularity[i-1], s))
moment_values = list(map(abs, moment_values))
maximum_moment = max(moment_values)
point = intervals[moment_values.index(maximum_moment)]
return (point, maximum_moment)
def point_cflexure(self):
"""
Returns a Set of point(s) with zero bending moment and
where bending moment curve of the beam object changes
its sign from negative to positive or vice versa.
Examples
========
There is is 10 meter long overhanging beam. There are
two simple supports below the beam. One at the start
and another one at a distance of 6 meters from the start.
Point loads of magnitude 10KN and 20KN are applied at
2 meters and 4 meters from start respectively. A Uniformly
distribute load of magnitude of magnitude 3KN/m is also
applied on top starting from 6 meters away from starting
point till end.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(10, E, I)
>>> b.apply_load(-4, 0, -1)
>>> b.apply_load(-46, 6, -1)
>>> b.apply_load(10, 2, -1)
>>> b.apply_load(20, 4, -1)
>>> b.apply_load(3, 6, 0)
>>> b.point_cflexure()
[10/3]
"""
from sympy import solve, Piecewise
# To restrict the range within length of the Beam
moment_curve = Piecewise((float("nan"), self.variable<=0),
(self.bending_moment(), self.variable<self.length),
(float("nan"), True))
points = solve(moment_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
return points
def slope(self):
"""
Returns a Singularity Function expression which represents
the slope the elastic curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.slope()
(-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
I = self.second_moment
if self._composite_type == "hinge":
return self._hinge_beam_slope
if not self._boundary_conditions['slope']:
return diff(self.deflection(), x)
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
slope = 0
prev_slope = 0
prev_end = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S(1)/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
if i != len(args) - 1:
slope += (prev_slope + slope_value)*SingularityFunction(x, prev_end, 0) - \
(prev_slope + slope_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
slope += (prev_slope + slope_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
return slope
C3 = Symbol('C3')
slope_curve = integrate(S(1)/(E*I)*self.bending_moment(), x) + C3
bc_eqs = []
for position, value in self._boundary_conditions['slope']:
eqs = slope_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, C3))
slope_curve = slope_curve.subs({C3: constants[0][0]})
return slope_curve
def deflection(self):
"""
Returns a Singularity Function expression which represents
the elastic curve or deflection of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.deflection()
(4000*x/3 - 4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3)
+ 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3 - 12000)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
I = self.second_moment
if self._composite_type == "hinge":
return self._hinge_beam_deflection
if not self._boundary_conditions['deflection'] and not self._boundary_conditions['slope']:
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S(1)/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
base_char = self._base_char
constants = symbols(base_char + '3:5')
return S(1)/(E*I)*integrate(integrate(self.bending_moment(), x), x) + constants[0]*x + constants[1]
elif not self._boundary_conditions['deflection']:
base_char = self._base_char
constant = symbols(base_char + '4')
return integrate(self.slope(), x) + constant
elif not self._boundary_conditions['slope'] and self._boundary_conditions['deflection']:
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S(1)/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
base_char = self._base_char
C3, C4 = symbols(base_char + '3:5') # Integration constants
slope_curve = integrate(self.bending_moment(), x) + C3
deflection_curve = integrate(slope_curve, x) + C4
bc_eqs = []
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, (C3, C4)))
deflection_curve = deflection_curve.subs({C3: constants[0][0], C4: constants[0][1]})
return S(1)/(E*I)*deflection_curve
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S(1)/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
C4 = Symbol('C4')
deflection_curve = integrate(self.slope(), x) + C4
bc_eqs = []
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, C4))
deflection_curve = deflection_curve.subs({C4: constants[0][0]})
return deflection_curve
def max_deflection(self):
"""
Returns point of max deflection and its coresponding deflection value
in a Beam object.
"""
from sympy import solve, Piecewise
# To restrict the range within length of the Beam
slope_curve = Piecewise((float("nan"), self.variable<=0),
(self.slope(), self.variable<self.length),
(float("nan"), True))
points = solve(slope_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
deflection_curve = self.deflection()
deflections = [deflection_curve.subs(self.variable, x) for x in points]
deflections = list(map(abs, deflections))
if len(deflections) != 0:
max_def = max(deflections)
return (points[deflections.index(max_def)], max_def)
else:
return None
def plot_shear_force(self, subs=None):
"""
Returns a plot for Shear force present in the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_shear_force()
Plot object containing:
[0]: cartesian line: -13750*SingularityFunction(x, 0, 0) + 5000*SingularityFunction(x, 2, 0)
+ 10000*SingularityFunction(x, 4, 1) - 31250*SingularityFunction(x, 8, 0)
- 10000*SingularityFunction(x, 8, 1) for x over (0.0, 8.0)
"""
shear_force = self.shear_force()
if subs is None:
subs = {}
for sym in shear_force.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(shear_force.subs(subs), (self.variable, 0, length), title='Shear Force',
xlabel='position', ylabel='Value', line_color='g')
def plot_bending_moment(self, subs=None):
"""
Returns a plot for Bending moment present in the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_bending_moment()
Plot object containing:
[0]: cartesian line: -13750*SingularityFunction(x, 0, 1) + 5000*SingularityFunction(x, 2, 1)
+ 5000*SingularityFunction(x, 4, 2) - 31250*SingularityFunction(x, 8, 1)
- 5000*SingularityFunction(x, 8, 2) for x over (0.0, 8.0)
"""
bending_moment = self.bending_moment()
if subs is None:
subs = {}
for sym in bending_moment.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(bending_moment.subs(subs), (self.variable, 0, length), title='Bending Moment',
xlabel='position', ylabel='Value', line_color='b')
def plot_slope(self, subs=None):
"""
Returns a plot for slope of deflection curve of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_slope()
Plot object containing:
[0]: cartesian line: -8.59375e-5*SingularityFunction(x, 0, 2) + 3.125e-5*SingularityFunction(x, 2, 2)
+ 2.08333333333333e-5*SingularityFunction(x, 4, 3) - 0.0001953125*SingularityFunction(x, 8, 2)
- 2.08333333333333e-5*SingularityFunction(x, 8, 3) + 0.00138541666666667 for x over (0.0, 8.0)
"""
slope = self.slope()
if subs is None:
subs = {}
for sym in slope.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(slope.subs(subs), (self.variable, 0, length), title='Slope',
xlabel='position', ylabel='Value', line_color='m')
def plot_deflection(self, subs=None):
"""
Returns a plot for deflection curve of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_deflection()
Plot object containing:
[0]: cartesian line: 0.00138541666666667*x - 2.86458333333333e-5*SingularityFunction(x, 0, 3)
+ 1.04166666666667e-5*SingularityFunction(x, 2, 3) + 5.20833333333333e-6*SingularityFunction(x, 4, 4)
- 6.51041666666667e-5*SingularityFunction(x, 8, 3) - 5.20833333333333e-6*SingularityFunction(x, 8, 4)
for x over (0.0, 8.0)
"""
deflection = self.deflection()
if subs is None:
subs = {}
for sym in deflection.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(deflection.subs(subs), (self.variable, 0, length),
title='Deflection', xlabel='position', ylabel='Value',
line_color='r')
@doctest_depends_on(modules=('numpy', 'matplotlib',))
def plot_loading_results(self, subs=None):
"""
Returns Axes object containing subplots of Shear Force, Bending Moment,
Slope and Deflection of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
.. note::
This method only works if numpy and matplotlib libraries
are installed on the system.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10
KN/m is applied from half of the beam till the end. There are two
simple supports below the beam, one at the starting point and another
at the ending point of the beam. A pointload of magnitude 5 KN is also
applied from top of the beam, at a distance of 4 meters from the
starting point. Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> axes = b.plot_loading_results()
"""
if matplotlib is None:
raise ImportError('Install matplotlib to use this method.')
else:
plt = matplotlib.pyplot
if numpy is None:
raise ImportError('Install numpy to use this method.')
else:
linspace = numpy.linspace
variable = self.variable
if subs is None:
subs = {}
for sym in self.deflection().atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' % sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
# As we are using matplotlib directly in this method, we need to change
# SymPy methods to numpy functions.
shear = lambdify(variable,
self.shear_force().subs(subs).rewrite(Piecewise),
'numpy')
moment = lambdify(variable,
self.bending_moment().subs(subs).rewrite(Piecewise),
'numpy')
slope = lambdify(variable, self.slope().subs(subs).rewrite(Piecewise),
'numpy')
deflection = lambdify(variable,
self.deflection().subs(subs).rewrite(Piecewise),
'numpy')
points = linspace(0, float(length), num=100*length)
# Creating a grid for subplots with 2 rows and 2 columns
fig, axs = plt.subplots(4, 1)
# axs is a 2D-numpy array containing axes
axs[0].plot(points, shear(points))
axs[0].set_title("Shear Force")
axs[1].plot(points, moment(points))
axs[1].set_title("Bending Moment")
axs[2].plot(points, slope(points))
axs[2].set_title("Slope")
axs[3].plot(points, deflection(points))
axs[3].set_title("Deflection")
fig.tight_layout() # For better spacing between subplots
return axs
class Beam3D(Beam):
"""
This class handles loads applied in any direction of a 3D space along
with unequal values of Second moment along different axes.
.. note::
While solving a beam bending problem, a user should choose its
own sign convention and should stick to it. The results will
automatically follow the chosen sign convention.
This class assumes that any kind of distributed load/moment is
applied through out the span of a beam.
Examples
========
There is a beam of l meters long. A constant distributed load of magnitude q
is applied along y-axis from start till the end of beam. A constant distributed
moment of magnitude m is also applied along z-axis from start till the end of beam.
Beam is fixed at both of its end. So, deflection of the beam at the both ends
is restricted.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols, simplify
>>> l, E, G, I, A = symbols('l, E, G, I, A')
>>> b = Beam3D(l, E, G, I, A)
>>> x, q, m = symbols('x, q, m')
>>> b.apply_load(q, 0, 0, dir="y")
>>> b.apply_moment_load(m, 0, -1, dir="z")
>>> b.shear_force()
[0, -q*x, 0]
>>> b.bending_moment()
[0, 0, -m*x + q*x**2/2]
>>> b.bc_slope = [(0, [0, 0, 0]), (l, [0, 0, 0])]
>>> b.bc_deflection = [(0, [0, 0, 0]), (l, [0, 0, 0])]
>>> b.solve_slope_deflection()
>>> b.slope()
[0, 0, l*x*(-l*q + 3*l*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I)) + 3*m)/(6*E*I)
+ q*x**3/(6*E*I) + x**2*(-l*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I))
- m)/(2*E*I)]
>>> dx, dy, dz = b.deflection()
>>> dx
0
>>> dz
0
>>> expectedy = (
... -l**2*q*x**2/(12*E*I) + l**2*x**2*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(8*E*I*(A*G*l**2 + 12*E*I))
... + l*m*x**2/(4*E*I) - l*x**3*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(12*E*I*(A*G*l**2 + 12*E*I)) - m*x**3/(6*E*I)
... + q*x**4/(24*E*I) + l*x*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*A*G*(A*G*l**2 + 12*E*I)) - q*x**2/(2*A*G)
... )
>>> simplify(dy - expectedy)
0
References
==========
.. [1] http://homes.civil.aau.dk/jc/FemteSemester/Beams3D.pdf
"""
def __init__(self, length, elastic_modulus, shear_modulus , second_moment, area, variable=Symbol('x')):
"""Initializes the class.
Parameters
==========
length : Sympifyable
A Symbol or value representing the Beam's length.
elastic_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of Elasticity.
It is a measure of the stiffness of the Beam material.
shear_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of rigidity.
It is a measure of rigidity of the Beam material.
second_moment : Sympifyable or list
A list of two elements having SymPy expression representing the
Beam's Second moment of area. First value represent Second moment
across y-axis and second across z-axis.
Single SymPy expression can be passed if both values are same
area : Sympifyable
A SymPy expression representing the Beam's cross-sectional area
in a plane prependicular to length of the Beam.
variable : Symbol, optional
A Symbol object that will be used as the variable along the beam
while representing the load, shear, moment, slope and deflection
curve. By default, it is set to ``Symbol('x')``.
"""
super(Beam3D, self).__init__(length, elastic_modulus, second_moment, variable)
self.shear_modulus = shear_modulus
self.area = area
self._load_vector = [0, 0, 0]
self._moment_load_vector = [0, 0, 0]
self._load_Singularity = [0, 0, 0]
self._slope = [0, 0, 0]
self._deflection = [0, 0, 0]
@property
def shear_modulus(self):
"""Young's Modulus of the Beam. """
return self._shear_modulus
@shear_modulus.setter
def shear_modulus(self, e):
self._shear_modulus = sympify(e)
@property
def second_moment(self):
"""Second moment of area of the Beam. """
return self._second_moment
@second_moment.setter
def second_moment(self, i):
if isinstance(i, list):
i = [sympify(x) for x in i]
self._second_moment = i
else:
self._second_moment = sympify(i)
@property
def area(self):
"""Cross-sectional area of the Beam. """
return self._area
@area.setter
def area(self, a):
self._area = sympify(a)
@property
def load_vector(self):
"""
Returns a three element list representing the load vector.
"""
return self._load_vector
@property
def moment_load_vector(self):
"""
Returns a three element list representing moment loads on Beam.
"""
return self._moment_load_vector
@property
def boundary_conditions(self):
"""
Returns a dictionary of boundary conditions applied on the beam.
The dictionary has two keywords namely slope and deflection.
The value of each keyword is a list of tuple, where each tuple
contains loaction and value of a boundary condition in the format
(location, value). Further each value is a list corresponding to
slope or deflection(s) values along three axes at that location.
Examples
========
There is a beam of length 4 meters. The slope at 0 should be 4 along
the x-axis and 0 along others. At the other end of beam, deflection
along all the three axes should be zero.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(30, E, G, I, A, x)
>>> b.bc_slope = [(0, (4, 0, 0))]
>>> b.bc_deflection = [(4, [0, 0, 0])]
>>> b.boundary_conditions
{'deflection': [(4, [0, 0, 0])], 'slope': [(0, (4, 0, 0))]}
Here the deflection of the beam should be ``0`` along all the three axes at ``4``.
Similarly, the slope of the beam should be ``4`` along x-axis and ``0``
along y and z axis at ``0``.
"""
return self._boundary_conditions
def apply_load(self, value, start, order, dir="y"):
"""
This method adds up the force load to a particular beam object.
Parameters
==========
value : Sympifyable
The magnitude of an applied load.
dir : String
Axis along which load is applied.
order : Integer
The order of the applied load.
- For point loads, order=-1
- For constant distributed load, order=0
- For ramp loads, order=1
- For parabolic ramp loads, order=2
- ... so on.
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if dir == "x":
if not order == -1:
self._load_vector[0] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
elif dir == "y":
if not order == -1:
self._load_vector[1] += value
self._load_Singularity[1] += value*SingularityFunction(x, start, order)
else:
if not order == -1:
self._load_vector[2] += value
self._load_Singularity[2] += value*SingularityFunction(x, start, order)
def apply_moment_load(self, value, start, order, dir="y"):
"""
This method adds up the moment loads to a particular beam object.
Parameters
==========
value : Sympifyable
The magnitude of an applied moment.
dir : String
Axis along which moment is applied.
order : Integer
The order of the applied load.
- For point moments, order=-2
- For constant distributed moment, order=-1
- For ramp moments, order=0
- For parabolic ramp moments, order=1
- ... so on.
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if dir == "x":
if not order == -2:
self._moment_load_vector[0] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
elif dir == "y":
if not order == -2:
self._moment_load_vector[1] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
else:
if not order == -2:
self._moment_load_vector[2] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
def apply_support(self, loc, type="fixed"):
if type == "pin" or type == "roller":
reaction_load = Symbol('R_'+str(loc))
self._reaction_loads[reaction_load] = reaction_load
self.bc_deflection.append((loc, [0, 0, 0]))
else:
reaction_load = Symbol('R_'+str(loc))
reaction_moment = Symbol('M_'+str(loc))
self._reaction_loads[reaction_load] = [reaction_load, reaction_moment]
self.bc_deflection.append((loc, [0, 0, 0]))
self.bc_slope.append((loc, [0, 0, 0]))
def solve_for_reaction_loads(self, *reaction):
"""
Solves for the reaction forces.
Examples
========
There is a beam of length 30 meters. It it supported by rollers at
of its end. A constant distributed load of magnitude 8 N is applied
from start till its end along y-axis. Another linear load having
slope equal to 9 is applied along z-axis.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(30, E, G, I, A, x)
>>> b.apply_load(8, start=0, order=0, dir="y")
>>> b.apply_load(9*x, start=0, order=0, dir="z")
>>> b.bc_deflection = [(0, [0, 0, 0]), (30, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="y")
>>> b.apply_load(R2, start=30, order=-1, dir="y")
>>> b.apply_load(R3, start=0, order=-1, dir="z")
>>> b.apply_load(R4, start=30, order=-1, dir="z")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.reaction_loads
{R1: -120, R2: -120, R3: -1350, R4: -2700}
"""
x = self.variable
l = self.length
q = self._load_Singularity
shear_curves = [integrate(load, x) for load in q]
moment_curves = [integrate(shear, x) for shear in shear_curves]
for i in range(3):
react = [r for r in reaction if (shear_curves[i].has(r) or moment_curves[i].has(r))]
if len(react) == 0:
continue
shear_curve = limit(shear_curves[i], x, l)
moment_curve = limit(moment_curves[i], x, l)
sol = list((linsolve([shear_curve, moment_curve], react).args)[0])
sol_dict = dict(zip(react, sol))
reaction_loads = self._reaction_loads
# Check if any of the evaluated rection exists in another direction
# and if it exists then it should have same value.
for key in sol_dict:
if key in reaction_loads and sol_dict[key] != reaction_loads[key]:
raise ValueError("Ambiguous solution for %s in different directions." % key)
self._reaction_loads.update(sol_dict)
def shear_force(self):
"""
Returns a list of three expressions which represents the shear force
curve of the Beam object along all three axes.
"""
x = self.variable
q = self._load_vector
return [integrate(-q[0], x), integrate(-q[1], x), integrate(-q[2], x)]
def axial_force(self):
"""
Returns expression of Axial shear force present inside the Beam object.
"""
return self.shear_force()[0]
def bending_moment(self):
"""
Returns a list of three expressions which represents the bending moment
curve of the Beam object along all three axes.
"""
x = self.variable
m = self._moment_load_vector
shear = self.shear_force()
return [integrate(-m[0], x), integrate(-m[1] + shear[2], x),
integrate(-m[2] - shear[1], x) ]
def torsional_moment(self):
"""
Returns expression of Torsional moment present inside the Beam object.
"""
return self.bending_moment()[0]
def solve_slope_deflection(self):
from sympy import dsolve, Function, Derivative, Eq
x = self.variable
l = self.length
E = self.elastic_modulus
G = self.shear_modulus
I = self.second_moment
if isinstance(I, list):
I_y, I_z = I[0], I[1]
else:
I_y = I_z = I
A = self.area
load = self._load_vector
moment = self._moment_load_vector
defl = Function('defl')
theta = Function('theta')
# Finding deflection along x-axis(and corresponding slope value by differentiating it)
# Equation used: Derivative(E*A*Derivative(def_x(x), x), x) + load_x = 0
eq = Derivative(E*A*Derivative(defl(x), x), x) + load[0]
def_x = dsolve(Eq(eq, 0), defl(x)).args[1]
# Solving constants originated from dsolve
C1 = Symbol('C1')
C2 = Symbol('C2')
constants = list((linsolve([def_x.subs(x, 0), def_x.subs(x, l)], C1, C2).args)[0])
def_x = def_x.subs({C1:constants[0], C2:constants[1]})
slope_x = def_x.diff(x)
self._deflection[0] = def_x
self._slope[0] = slope_x
# Finding deflection along y-axis and slope across z-axis. System of equation involved:
# 1: Derivative(E*I_z*Derivative(theta_z(x), x), x) + G*A*(Derivative(defl_y(x), x) - theta_z(x)) + moment_z = 0
# 2: Derivative(G*A*(Derivative(defl_y(x), x) - theta_z(x)), x) + load_y = 0
C_i = Symbol('C_i')
# Substitute value of `G*A*(Derivative(defl_y(x), x) - theta_z(x))` from (2) in (1)
eq1 = Derivative(E*I_z*Derivative(theta(x), x), x) + (integrate(-load[1], x) + C_i) + moment[2]
slope_z = dsolve(Eq(eq1, 0)).args[1]
# Solve for constants originated from using dsolve on eq1
constants = list((linsolve([slope_z.subs(x, 0), slope_z.subs(x, l)], C1, C2).args)[0])
slope_z = slope_z.subs({C1:constants[0], C2:constants[1]})
# Put value of slope obtained back in (2) to solve for `C_i` and find deflection across y-axis
eq2 = G*A*(Derivative(defl(x), x)) + load[1]*x - C_i - G*A*slope_z
def_y = dsolve(Eq(eq2, 0), defl(x)).args[1]
# Solve for constants originated from using dsolve on eq2
constants = list((linsolve([def_y.subs(x, 0), def_y.subs(x, l)], C1, C_i).args)[0])
self._deflection[1] = def_y.subs({C1:constants[0], C_i:constants[1]})
self._slope[2] = slope_z.subs(C_i, constants[1])
# Finding deflection along z-axis and slope across y-axis. System of equation involved:
# 1: Derivative(E*I_y*Derivative(theta_y(x), x), x) - G*A*(Derivative(defl_z(x), x) + theta_y(x)) + moment_y = 0
# 2: Derivative(G*A*(Derivative(defl_z(x), x) + theta_y(x)), x) + load_z = 0
# Substitute value of `G*A*(Derivative(defl_y(x), x) + theta_z(x))` from (2) in (1)
eq1 = Derivative(E*I_y*Derivative(theta(x), x), x) + (integrate(load[2], x) - C_i) + moment[1]
slope_y = dsolve(Eq(eq1, 0)).args[1]
# Solve for constants originated from using dsolve on eq1
constants = list((linsolve([slope_y.subs(x, 0), slope_y.subs(x, l)], C1, C2).args)[0])
slope_y = slope_y.subs({C1:constants[0], C2:constants[1]})
# Put value of slope obtained back in (2) to solve for `C_i` and find deflection across z-axis
eq2 = G*A*(Derivative(defl(x), x)) + load[2]*x - C_i + G*A*slope_y
def_z = dsolve(Eq(eq2,0)).args[1]
# Solve for constants originated from using dsolve on eq2
constants = list((linsolve([def_z.subs(x, 0), def_z.subs(x, l)], C1, C_i).args)[0])
self._deflection[2] = def_z.subs({C1:constants[0], C_i:constants[1]})
self._slope[1] = slope_y.subs(C_i, constants[1])
def slope(self):
"""
Returns a three element list representing slope of deflection curve
along all the three axes.
"""
return self._slope
def deflection(self):
"""
Returns a three element list representing deflection curve along all
the three axes.
"""
return self._deflection
|
bc3e0e8c34b6e1e8cf85f1e950d04f2a25b0bbd997e97befbee9f4075d9ee576 | """
**Contains**
* refraction_angle
* fresnel_coefficients
* deviation
* brewster_angle
* critical_angle
* lens_makers_formula
* mirror_formula
* lens_formula
* hyperfocal_distance
* transverse_magnification
"""
from __future__ import division
__all__ = ['refraction_angle',
'deviation',
'fresnel_coefficients',
'brewster_angle',
'critical_angle',
'lens_makers_formula',
'mirror_formula',
'lens_formula',
'hyperfocal_distance',
'transverse_magnification'
]
from sympy import Symbol, sympify, sqrt, Matrix, acos, oo, Limit, atan2, asin,\
cos, sin, tan, I, cancel
from sympy.core.compatibility import is_sequence
from sympy.geometry.line import Ray3D, Point3D
from sympy.geometry.util import intersection
from sympy.geometry.plane import Plane
from .medium import Medium
def refraction_angle(incident, medium1, medium2, normal=None, plane=None):
"""
This function calculates transmitted vector after refraction at planar
surface. `medium1` and `medium2` can be `Medium` or any sympifiable object.
If `incident` is an object of `Ray3D`, `normal` also has to be an instance
of `Ray3D` in order to get the output as a `Ray3D`. Please note that if
plane of separation is not provided and normal is an instance of `Ray3D`,
normal will be assumed to be intersecting incident ray at the plane of
separation. This will not be the case when `normal` is a `Matrix` or
any other sequence.
If `incident` is an instance of `Ray3D` and `plane` has not been provided
and `normal` is not `Ray3D`, output will be a `Matrix`.
Parameters
==========
incident : Matrix, Ray3D, or sequence
Incident vector
medium1 : sympy.physics.optics.medium.Medium or sympifiable
Medium 1 or its refractive index
medium2 : sympy.physics.optics.medium.Medium or sympifiable
Medium 2 or its refractive index
normal : Matrix, Ray3D, or sequence
Normal vector
plane : Plane
Plane of separation of the two media.
Examples
========
>>> from sympy.physics.optics import refraction_angle
>>> from sympy.geometry import Point3D, Ray3D, Plane
>>> from sympy.matrices import Matrix
>>> from sympy import symbols
>>> n = Matrix([0, 0, 1])
>>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
>>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
>>> refraction_angle(r1, 1, 1, n)
Matrix([
[ 1],
[ 1],
[-1]])
>>> refraction_angle(r1, 1, 1, plane=P)
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
With different index of refraction of the two media
>>> n1, n2 = symbols('n1, n2')
>>> refraction_angle(r1, n1, n2, n)
Matrix([
[ n1/n2],
[ n1/n2],
[-sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)]])
>>> refraction_angle(r1, n1, n2, plane=P)
Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)))
"""
# A flag to check whether to return Ray3D or not
return_ray = False
if plane is not None and normal is not None:
raise ValueError("Either plane or normal is acceptable.")
if not isinstance(incident, Matrix):
if is_sequence(incident):
_incident = Matrix(incident)
elif isinstance(incident, Ray3D):
_incident = Matrix(incident.direction_ratio)
else:
raise TypeError(
"incident should be a Matrix, Ray3D, or sequence")
else:
_incident = incident
# If plane is provided, get direction ratios of the normal
# to the plane from the plane else go with `normal` param.
if plane is not None:
if not isinstance(plane, Plane):
raise TypeError("plane should be an instance of geometry.plane.Plane")
# If we have the plane, we can get the intersection
# point of incident ray and the plane and thus return
# an instance of Ray3D.
if isinstance(incident, Ray3D):
return_ray = True
intersection_pt = plane.intersection(incident)[0]
_normal = Matrix(plane.normal_vector)
else:
if not isinstance(normal, Matrix):
if is_sequence(normal):
_normal = Matrix(normal)
elif isinstance(normal, Ray3D):
_normal = Matrix(normal.direction_ratio)
if isinstance(incident, Ray3D):
intersection_pt = intersection(incident, normal)
if len(intersection_pt) == 0:
raise ValueError(
"Normal isn't concurrent with the incident ray.")
else:
return_ray = True
intersection_pt = intersection_pt[0]
else:
raise TypeError(
"Normal should be a Matrix, Ray3D, or sequence")
else:
_normal = normal
n1, n2 = None, None
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
eta = n1/n2 # Relative index of refraction
# Calculating magnitude of the vectors
mag_incident = sqrt(sum([i**2 for i in _incident]))
mag_normal = sqrt(sum([i**2 for i in _normal]))
# Converting vectors to unit vectors by dividing
# them with their magnitudes
_incident /= mag_incident
_normal /= mag_normal
c1 = -_incident.dot(_normal) # cos(angle_of_incidence)
cs2 = 1 - eta**2*(1 - c1**2) # cos(angle_of_refraction)**2
if cs2.is_negative: # This is the case of total internal reflection(TIR).
return 0
drs = eta*_incident + (eta*c1 - sqrt(cs2))*_normal
# Multiplying unit vector by its magnitude
drs = drs*mag_incident
if not return_ray:
return drs
else:
return Ray3D(intersection_pt, direction_ratio=drs)
def fresnel_coefficients(angle_of_incidence, medium1, medium2):
"""
This function uses Fresnel equations to calculate reflection and
transmission coefficients. Those are obtained for both polarisations
when the electric field vector is in the plane of incidence (labelled 'p')
and when the electric field vector is perpendicular to the plane of
incidence (labelled 's'). There are four real coefficients unless the
incident ray reflects in total internal in which case there are two complex
ones. Angle of incidence is the angle between the incident ray and the
surface normal. ``medium1`` and ``medium2`` can be ``Medium`` or any
sympifiable object.
Parameters
==========
angle_of_incidence : sympifiable
medium1 : Medium or sympifiable
Medium 1 or its refractive index
medium2 : Medium or sympifiable
Medium 2 or its refractive index
Returns a list with four real Fresnel coefficients:
[reflection p (TM), reflection s (TE),
transmission p (TM), transmission s (TE)]
If the ray is undergoes total internal reflection then returns a
list of two complex Fresnel coefficients:
[reflection p (TM), reflection s (TE)]
Examples
========
>>> from sympy.physics.optics import fresnel_coefficients
>>> fresnel_coefficients(0.3, 1, 2)
[0.317843553417859, -0.348645229818821,
0.658921776708929, 0.651354770181179]
>>> fresnel_coefficients(0.6, 2, 1)
[-0.235625382192159 - 0.971843958291041*I,
0.816477005968898 - 0.577377951366403*I]
References
==========
https://en.wikipedia.org/wiki/Fresnel_equations
"""
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
angle_of_refraction = asin(n1*sin(angle_of_incidence)/n2)
try:
angle_of_total_internal_reflection_onset = critical_angle(n1, n2)
except ValueError:
angle_of_total_internal_reflection_onset = None
if angle_of_total_internal_reflection_onset == None or\
angle_of_total_internal_reflection_onset > angle_of_incidence:
R_s = -sin(angle_of_incidence - angle_of_refraction)\
/sin(angle_of_incidence + angle_of_refraction)
R_p = tan(angle_of_incidence - angle_of_refraction)\
/tan(angle_of_incidence + angle_of_refraction)
T_s = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\
/sin(angle_of_incidence + angle_of_refraction)
T_p = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\
/(sin(angle_of_incidence + angle_of_refraction)\
*cos(angle_of_incidence - angle_of_refraction))
return [R_p, R_s, T_p, T_s]
else:
n = n2/n1
R_s = cancel((cos(angle_of_incidence)-\
I*sqrt(sin(angle_of_incidence)**2 - n**2))\
/(cos(angle_of_incidence)+\
I*sqrt(sin(angle_of_incidence)**2 - n**2)))
R_p = cancel((n**2*cos(angle_of_incidence)-\
I*sqrt(sin(angle_of_incidence)**2 - n**2))\
/(n**2*cos(angle_of_incidence)+\
I*sqrt(sin(angle_of_incidence)**2 - n**2)))
return [R_p, R_s]
def deviation(incident, medium1, medium2, normal=None, plane=None):
"""
This function calculates the angle of deviation of a ray
due to refraction at planar surface.
Parameters
==========
incident : Matrix, Ray3D, or sequence
Incident vector
medium1 : sympy.physics.optics.medium.Medium or sympifiable
Medium 1 or its refractive index
medium2 : sympy.physics.optics.medium.Medium or sympifiable
Medium 2 or its refractive index
normal : Matrix, Ray3D, or sequence
Normal vector
plane : Plane
Plane of separation of the two media.
Examples
========
>>> from sympy.physics.optics import deviation
>>> from sympy.geometry import Point3D, Ray3D, Plane
>>> from sympy.matrices import Matrix
>>> from sympy import symbols
>>> n1, n2 = symbols('n1, n2')
>>> n = Matrix([0, 0, 1])
>>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
>>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
>>> deviation(r1, 1, 1, n)
0
>>> deviation(r1, n1, n2, plane=P)
-acos(-sqrt(-2*n1**2/(3*n2**2) + 1)) + acos(-sqrt(3)/3)
"""
refracted = refraction_angle(incident,
medium1,
medium2,
normal=normal,
plane=plane)
if refracted != 0:
if isinstance(refracted, Ray3D):
refracted = Matrix(refracted.direction_ratio)
if not isinstance(incident, Matrix):
if is_sequence(incident):
_incident = Matrix(incident)
elif isinstance(incident, Ray3D):
_incident = Matrix(incident.direction_ratio)
else:
raise TypeError(
"incident should be a Matrix, Ray3D, or sequence")
else:
_incident = incident
if plane is None:
if not isinstance(normal, Matrix):
if is_sequence(normal):
_normal = Matrix(normal)
elif isinstance(normal, Ray3D):
_normal = Matrix(normal.direction_ratio)
else:
raise TypeError(
"normal should be a Matrix, Ray3D, or sequence")
else:
_normal = normal
else:
_normal = Matrix(plane.normal_vector)
mag_incident = sqrt(sum([i**2 for i in _incident]))
mag_normal = sqrt(sum([i**2 for i in _normal]))
mag_refracted = sqrt(sum([i**2 for i in refracted]))
_incident /= mag_incident
_normal /= mag_normal
refracted /= mag_refracted
i = acos(_incident.dot(_normal))
r = acos(refracted.dot(_normal))
return i - r
def brewster_angle(medium1, medium2):
"""
This function calculates the Brewster's angle of incidence to Medium 2 from
Medium 1 in radians.
Parameters
==========
medium 1 : Medium or sympifiable
Refractive index of Medium 1
medium 2 : Medium or sympifiable
Refractive index of Medium 1
Examples
========
>>> from sympy.physics.optics import brewster_angle
>>> brewster_angle(1, 1.33)
0.926093295503462
"""
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
return atan2(n2, n1)
def critical_angle(medium1, medium2):
"""
This function calculates the critical angle of incidence (marking the onset
of total internal) to Medium 2 from Medium 1 in radians.
Parameters
==========
medium 1 : Medium or sympifiable
Refractive index of Medium 1
medium 2 : Medium or sympifiable
Refractive index of Medium 1
Examples
========
>>> from sympy.physics.optics import critical_angle
>>> critical_angle(1.33, 1)
0.850908514477849
"""
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
if n2 > n1:
raise ValueError('Total internal reflection impossible for n1 < n2')
else:
return asin(n2/n1)
def lens_makers_formula(n_lens, n_surr, r1, r2):
"""
This function calculates focal length of a thin lens.
It follows cartesian sign convention.
Parameters
==========
n_lens : Medium or sympifiable
Index of refraction of lens.
n_surr : Medium or sympifiable
Index of reflection of surrounding.
r1 : sympifiable
Radius of curvature of first surface.
r2 : sympifiable
Radius of curvature of second surface.
Examples
========
>>> from sympy.physics.optics import lens_makers_formula
>>> lens_makers_formula(1.33, 1, 10, -10)
15.1515151515151
"""
if isinstance(n_lens, Medium):
n_lens = n_lens.refractive_index
else:
n_lens = sympify(n_lens)
if isinstance(n_surr, Medium):
n_surr = n_surr.refractive_index
else:
n_surr = sympify(n_surr)
r1 = sympify(r1)
r2 = sympify(r2)
return 1/((n_lens - n_surr)/n_surr*(1/r1 - 1/r2))
def mirror_formula(focal_length=None, u=None, v=None):
"""
This function provides one of the three parameters
when two of them are supplied.
This is valid only for paraxial rays.
Parameters
==========
focal_length : sympifiable
Focal length of the mirror.
u : sympifiable
Distance of object from the pole on
the principal axis.
v : sympifiable
Distance of the image from the pole
on the principal axis.
Examples
========
>>> from sympy.physics.optics import mirror_formula
>>> from sympy.abc import f, u, v
>>> mirror_formula(focal_length=f, u=u)
f*u/(-f + u)
>>> mirror_formula(focal_length=f, v=v)
f*v/(-f + v)
>>> mirror_formula(u=u, v=v)
u*v/(u + v)
"""
if focal_length and u and v:
raise ValueError("Please provide only two parameters")
focal_length = sympify(focal_length)
u = sympify(u)
v = sympify(v)
if u == oo:
_u = Symbol('u')
if v == oo:
_v = Symbol('v')
if focal_length == oo:
_f = Symbol('f')
if focal_length is None:
if u == oo and v == oo:
return Limit(Limit(_v*_u/(_v + _u), _u, oo), _v, oo).doit()
if u == oo:
return Limit(v*_u/(v + _u), _u, oo).doit()
if v == oo:
return Limit(_v*u/(_v + u), _v, oo).doit()
return v*u/(v + u)
if u is None:
if v == oo and focal_length == oo:
return Limit(Limit(_v*_f/(_v - _f), _v, oo), _f, oo).doit()
if v == oo:
return Limit(_v*focal_length/(_v - focal_length), _v, oo).doit()
if focal_length == oo:
return Limit(v*_f/(v - _f), _f, oo).doit()
return v*focal_length/(v - focal_length)
if v is None:
if u == oo and focal_length == oo:
return Limit(Limit(_u*_f/(_u - _f), _u, oo), _f, oo).doit()
if u == oo:
return Limit(_u*focal_length/(_u - focal_length), _u, oo).doit()
if focal_length == oo:
return Limit(u*_f/(u - _f), _f, oo).doit()
return u*focal_length/(u - focal_length)
def lens_formula(focal_length=None, u=None, v=None):
"""
This function provides one of the three parameters
when two of them are supplied.
This is valid only for paraxial rays.
Parameters
==========
focal_length : sympifiable
Focal length of the mirror.
u : sympifiable
Distance of object from the optical center on
the principal axis.
v : sympifiable
Distance of the image from the optical center
on the principal axis.
Examples
========
>>> from sympy.physics.optics import lens_formula
>>> from sympy.abc import f, u, v
>>> lens_formula(focal_length=f, u=u)
f*u/(f + u)
>>> lens_formula(focal_length=f, v=v)
f*v/(f - v)
>>> lens_formula(u=u, v=v)
u*v/(u - v)
"""
if focal_length and u and v:
raise ValueError("Please provide only two parameters")
focal_length = sympify(focal_length)
u = sympify(u)
v = sympify(v)
if u == oo:
_u = Symbol('u')
if v == oo:
_v = Symbol('v')
if focal_length == oo:
_f = Symbol('f')
if focal_length is None:
if u == oo and v == oo:
return Limit(Limit(_v*_u/(_u - _v), _u, oo), _v, oo).doit()
if u == oo:
return Limit(v*_u/(_u - v), _u, oo).doit()
if v == oo:
return Limit(_v*u/(u - _v), _v, oo).doit()
return v*u/(u - v)
if u is None:
if v == oo and focal_length == oo:
return Limit(Limit(_v*_f/(_f - _v), _v, oo), _f, oo).doit()
if v == oo:
return Limit(_v*focal_length/(focal_length - _v), _v, oo).doit()
if focal_length == oo:
return Limit(v*_f/(_f - v), _f, oo).doit()
return v*focal_length/(focal_length - v)
if v is None:
if u == oo and focal_length == oo:
return Limit(Limit(_u*_f/(_u + _f), _u, oo), _f, oo).doit()
if u == oo:
return Limit(_u*focal_length/(_u + focal_length), _u, oo).doit()
if focal_length == oo:
return Limit(u*_f/(u + _f), _f, oo).doit()
return u*focal_length/(u + focal_length)
def hyperfocal_distance(f, N, c):
"""
Parameters
==========
f: sympifiable
Focal length of a given lens
N: sympifiable
F-number of a given lens
c: sympifiable
Circle of Confusion (CoC) of a given image format
Example
=======
>>> from sympy.physics.optics import hyperfocal_distance
>>> from sympy.abc import f, N, c
>>> round(hyperfocal_distance(f = 0.5, N = 8, c = 0.0033), 2)
9.47
"""
f = sympify(f)
N = sympify(N)
c = sympify(c)
return (1/(N * c))*(f**2)
def transverse_magnification(si, so):
"""
Calculates the transverse magnification, which is the ratio of the
image size to the object size.
Parameters
==========
so: sympifiable
Lens-object distance
si: sympifiable
Lens-image distance
Example
=======
>>> from sympy.physics.optics import transverse_magnification
>>> transverse_magnification(30, 15)
-2
"""
si = sympify(si)
so = sympify(so)
return (-(si/so))
|
ab826a98f089bb07cc32364224778d518b7e5061d4bc60fe255c682ee5180ee1 | from sympy import I, Mul, latex, Matrix
from sympy.physics.quantum import (Dagger, Commutator, AntiCommutator, qapply,
Operator, represent)
from sympy.physics.quantum.pauli import (SigmaOpBase, SigmaX, SigmaY, SigmaZ,
SigmaMinus, SigmaPlus,
qsimplify_pauli)
from sympy.physics.quantum.pauli import SigmaZKet, SigmaZBra
from sympy.utilities.pytest import raises
sx, sy, sz = SigmaX(), SigmaY(), SigmaZ()
sx1, sy1, sz1 = SigmaX(1), SigmaY(1), SigmaZ(1)
sx2, sy2, sz2 = SigmaX(2), SigmaY(2), SigmaZ(2)
sm, sp = SigmaMinus(), SigmaPlus()
sm1, sp1 = SigmaMinus(1), SigmaPlus(1)
A, B = Operator("A"), Operator("B")
def test_pauli_operators_types():
assert isinstance(sx, SigmaOpBase) and isinstance(sx, SigmaX)
assert isinstance(sy, SigmaOpBase) and isinstance(sy, SigmaY)
assert isinstance(sz, SigmaOpBase) and isinstance(sz, SigmaZ)
assert isinstance(sm, SigmaOpBase) and isinstance(sm, SigmaMinus)
assert isinstance(sp, SigmaOpBase) and isinstance(sp, SigmaPlus)
def test_pauli_operators_commutator():
assert Commutator(sx, sy).doit() == 2 * I * sz
assert Commutator(sy, sz).doit() == 2 * I * sx
assert Commutator(sz, sx).doit() == 2 * I * sy
def test_pauli_operators_commutator_with_labels():
assert Commutator(sx1, sy1).doit() == 2 * I * sz1
assert Commutator(sy1, sz1).doit() == 2 * I * sx1
assert Commutator(sz1, sx1).doit() == 2 * I * sy1
assert Commutator(sx2, sy2).doit() == 2 * I * sz2
assert Commutator(sy2, sz2).doit() == 2 * I * sx2
assert Commutator(sz2, sx2).doit() == 2 * I * sy2
assert Commutator(sx1, sy2).doit() == 0
assert Commutator(sy1, sz2).doit() == 0
assert Commutator(sz1, sx2).doit() == 0
def test_pauli_operators_anticommutator():
assert AntiCommutator(sy, sz).doit() == 0
assert AntiCommutator(sz, sx).doit() == 0
assert AntiCommutator(sx, sm).doit() == 1
assert AntiCommutator(sx, sp).doit() == 1
def test_pauli_operators_adjoint():
assert Dagger(sx) == sx
assert Dagger(sy) == sy
assert Dagger(sz) == sz
def test_pauli_operators_adjoint_with_labels():
assert Dagger(sx1) == sx1
assert Dagger(sy1) == sy1
assert Dagger(sz1) == sz1
assert Dagger(sx1) != sx2
assert Dagger(sy1) != sy2
assert Dagger(sz1) != sz2
def test_pauli_operators_multiplication():
assert qsimplify_pauli(sx * sx) == 1
assert qsimplify_pauli(sy * sy) == 1
assert qsimplify_pauli(sz * sz) == 1
assert qsimplify_pauli(sx * sy) == I * sz
assert qsimplify_pauli(sy * sz) == I * sx
assert qsimplify_pauli(sz * sx) == I * sy
assert qsimplify_pauli(sy * sx) == - I * sz
assert qsimplify_pauli(sz * sy) == - I * sx
assert qsimplify_pauli(sx * sz) == - I * sy
def test_pauli_operators_multiplication_with_labels():
assert qsimplify_pauli(sx1 * sx1) == 1
assert qsimplify_pauli(sy1 * sy1) == 1
assert qsimplify_pauli(sz1 * sz1) == 1
assert isinstance(sx1 * sx2, Mul)
assert isinstance(sy1 * sy2, Mul)
assert isinstance(sz1 * sz2, Mul)
assert qsimplify_pauli(sx1 * sy1 * sx2 * sy2) == - sz1 * sz2
assert qsimplify_pauli(sy1 * sz1 * sz2 * sx2) == - sx1 * sy2
def test_pauli_states():
sx, sz = SigmaX(), SigmaZ()
up = SigmaZKet(0)
down = SigmaZKet(1)
assert qapply(sx * up) == down
assert qapply(sx * down) == up
assert qapply(sz * up) == up
assert qapply(sz * down) == - down
up = SigmaZBra(0)
down = SigmaZBra(1)
assert qapply(up * sx, dagger=True) == down
assert qapply(down * sx, dagger=True) == up
assert qapply(up * sz, dagger=True) == up
assert qapply(down * sz, dagger=True) == - down
assert Dagger(SigmaZKet(0)) == SigmaZBra(0)
assert Dagger(SigmaZBra(1)) == SigmaZKet(1)
raises(ValueError, lambda: SigmaZBra(2))
raises(ValueError, lambda: SigmaZKet(2))
def test_use_name():
assert sm.use_name is False
assert sm1.use_name is True
assert sx.use_name is False
assert sx1.use_name is True
def test_printing():
assert latex(sx) == r'{\sigma_x}'
assert latex(sx1) == r'{\sigma_x^{(1)}}'
assert latex(sy) == r'{\sigma_y}'
assert latex(sy1) == r'{\sigma_y^{(1)}}'
assert latex(sz) == r'{\sigma_z}'
assert latex(sz1) == r'{\sigma_z^{(1)}}'
assert latex(sm) == r'{\sigma_-}'
assert latex(sm1) == r'{\sigma_-^{(1)}}'
assert latex(sp) == r'{\sigma_+}'
assert latex(sp1) == r'{\sigma_+^{(1)}}'
def test_represent():
represent(sx) == Matrix([[0, 1], [1, 0]])
represent(sy) == Matrix([[0, -I], [I, 0]])
represent(sz) == Matrix([[1, 0], [0, -1]])
represent(sm) == Matrix([[0, 0], [1, 0]])
represent(sp) == Matrix([[0, 1], [0, 0]])
|
324c6611beef855f6de8d0af041a0050c6b7c0b6f1ec6fa72ab5f02a371a0fad | """
MKS unit system.
MKS stands for "meter, kilogram, second".
"""
from __future__ import division
from sympy.physics.units import UnitSystem
from sympy.physics.units.definitions import G, Hz, J, N, Pa, W, c, g, kg, m, s
from sympy.physics.units.dimensions import (
acceleration, action, energy, force, frequency, momentum,
power, pressure, velocity, dimsys_MKS)
from sympy.physics.units.prefixes import PREFIXES, prefix_unit
dims = (velocity, acceleration, momentum, force, energy, power, pressure,
frequency, action)
# dimension system
_mks_dim = dimsys_MKS
units = [m, g, s, J, N, W, Pa, Hz]
all_units = []
# Prefixes of units like g, J, N etc get added using `prefix_unit`
# in the for loop, but the actual units have to be added manually.
all_units.extend([g, J, N, W, Pa, Hz])
for u in units:
all_units.extend(prefix_unit(u, PREFIXES))
all_units.extend([G, c])
# unit system
MKS = UnitSystem(base=(m, kg, s), units=all_units, name="MKS")
|
2facdd0057c4aec9b6171c8e3477a43331353f852bfb7b05236013aa50fd4a5f | """
MKS unit system.
MKS stands for "meter, kilogram, second, ampere".
"""
from __future__ import division
from sympy.physics.units.definitions import Z0, A, C, F, H, S, T, V, Wb, ohm
from sympy.physics.units.dimensions import (
capacitance, charge, conductance, current, impedance, inductance,
magnetic_density, magnetic_flux, voltage, dimsys_MKSA)
from sympy.physics.units.prefixes import PREFIXES, prefix_unit
from sympy.physics.units.systems.mks import MKS
dims = (voltage, impedance, conductance, current, capacitance, inductance, charge,
magnetic_density, magnetic_flux)
# dimension system
_mksa_dim = dimsys_MKSA
units = [A, V, ohm, S, F, H, C, T, Wb]
all_units = []
for u in units:
all_units.extend(prefix_unit(u, PREFIXES))
all_units.extend([Z0])
MKSA = MKS.extend(base=(A,), units=all_units, name='MKSA')
|
d4d441e2ba2dd0185032608815c8e4c30b760ef0cb02d14e6eafd1a86ac7a944 | """
SI unit system.
Based on MKSA, which stands for "meter, kilogram, second, ampere".
Added kelvin, candela and mole.
"""
from __future__ import division
from sympy.physics.units.definitions import (
K, cd, lux, mol,hertz, newton, pascal, joule, watt, coulomb, volt, farad,
ohm, siemens, weber, tesla, henry, candela, becquerel, gray, katal)
from sympy.physics.units.dimensions import (
amount_of_substance, temperature, dimsys_SI,
frequency, force, pressure, energy, power, charge, voltage, capacitance,
conductance, magnetic_flux, magnetic_density, inductance,
luminous_intensity)
from sympy.physics.units.prefixes import PREFIXES, prefix_unit
from sympy.physics.units.systems.mksa import MKSA
derived_dims = (frequency, force, pressure, energy, power, charge, voltage,
capacitance, conductance, magnetic_flux,
magnetic_density, inductance, luminous_intensity)
base_dims = (amount_of_substance, luminous_intensity, temperature)
# dimension system
_si_dim = dimsys_SI
units = [mol, cd, K, lux, hertz, newton, pascal, joule, watt, coulomb, volt,
farad, ohm, siemens, weber, tesla, henry, candela, lux, becquerel,
gray, katal]
all_units = []
for u in units:
all_units.extend(prefix_unit(u, PREFIXES))
all_units.extend([mol, cd, K, lux])
SI = MKSA.extend(base=(mol, cd, K), units=all_units, name='SI')
|
0300bd5a353eb14d00201eb6b15e8b78ae559d99919bf35582d80ec5497bd36b | from __future__ import print_function, division
import functools
from sympy import Basic, Tuple, S
from sympy.core.sympify import _sympify
from sympy.tensor.array.mutable_ndim_array import MutableNDimArray
from sympy.tensor.array.ndim_array import NDimArray, ImmutableNDimArray
class DenseNDimArray(NDimArray):
def __new__(self, *args, **kwargs):
return ImmutableDenseNDimArray(*args, **kwargs)
def __getitem__(self, index):
"""
Allows to get items from N-dim array.
Examples
========
>>> from sympy import MutableDenseNDimArray
>>> a = MutableDenseNDimArray([0, 1, 2, 3], (2, 2))
>>> a
[[0, 1], [2, 3]]
>>> a[0, 0]
0
>>> a[1, 1]
3
Symbolic index:
>>> from sympy.abc import i, j
>>> a[i, j]
[[0, 1], [2, 3]][i, j]
Replace `i` and `j` to get element `(1, 1)`:
>>> a[i, j].subs({i: 1, j: 1})
3
"""
syindex = self._check_symbolic_index(index)
if syindex is not None:
return syindex
if isinstance(index, tuple) and any([isinstance(i, slice) for i in index]):
sl_factors, eindices = self._get_slice_data_for_array_access(index)
array = [self._array[self._parse_index(i)] for i in eindices]
nshape = [len(el) for i, el in enumerate(sl_factors) if isinstance(index[i], slice)]
return type(self)(array, nshape)
else:
if isinstance(index, slice):
return self._array[index]
else:
index = self._parse_index(index)
return self._array[index]
@classmethod
def zeros(cls, *shape):
list_length = functools.reduce(lambda x, y: x*y, shape, S.One)
return cls._new(([0]*list_length,), shape)
def tomatrix(self):
"""
Converts MutableDenseNDimArray to Matrix. Can convert only 2-dim array, else will raise error.
Examples
========
>>> from sympy import MutableDenseNDimArray
>>> a = MutableDenseNDimArray([1 for i in range(9)], (3, 3))
>>> b = a.tomatrix()
>>> b
Matrix([
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])
"""
from sympy.matrices import Matrix
if self.rank() != 2:
raise ValueError('Dimensions must be of size of 2')
return Matrix(self.shape[0], self.shape[1], self._array)
def __iter__(self):
return self._array.__iter__()
def reshape(self, *newshape):
"""
Returns MutableDenseNDimArray instance with new shape. Elements number
must be suitable to new shape. The only argument of method sets
new shape.
Examples
========
>>> from sympy import MutableDenseNDimArray
>>> a = MutableDenseNDimArray([1, 2, 3, 4, 5, 6], (2, 3))
>>> a.shape
(2, 3)
>>> a
[[1, 2, 3], [4, 5, 6]]
>>> b = a.reshape(3, 2)
>>> b.shape
(3, 2)
>>> b
[[1, 2], [3, 4], [5, 6]]
"""
new_total_size = functools.reduce(lambda x,y: x*y, newshape)
if new_total_size != self._loop_size:
raise ValueError("Invalid reshape parameters " + newshape)
# there is no `.func` as this class does not subtype `Basic`:
return type(self)(self._array, newshape)
class ImmutableDenseNDimArray(DenseNDimArray, ImmutableNDimArray):
"""
"""
def __new__(cls, iterable, shape=None, **kwargs):
return cls._new(iterable, shape, **kwargs)
@classmethod
def _new(cls, iterable, shape, **kwargs):
from sympy.utilities.iterables import flatten
shape, flat_list = cls._handle_ndarray_creation_inputs(iterable, shape, **kwargs)
shape = Tuple(*map(_sympify, shape))
cls._check_special_bounds(flat_list, shape)
flat_list = flatten(flat_list)
flat_list = Tuple(*flat_list)
self = Basic.__new__(cls, flat_list, shape, **kwargs)
self._shape = shape
self._array = list(flat_list)
self._rank = len(shape)
self._loop_size = functools.reduce(lambda x,y: x*y, shape, 1)
return self
def __setitem__(self, index, value):
raise TypeError('immutable N-dim array')
def as_mutable(self):
return MutableDenseNDimArray(self)
class MutableDenseNDimArray(DenseNDimArray, MutableNDimArray):
def __new__(cls, iterable=None, shape=None, **kwargs):
return cls._new(iterable, shape, **kwargs)
@classmethod
def _new(cls, iterable, shape, **kwargs):
from sympy.utilities.iterables import flatten
shape, flat_list = cls._handle_ndarray_creation_inputs(iterable, shape, **kwargs)
flat_list = flatten(flat_list)
self = object.__new__(cls)
self._shape = shape
self._array = list(flat_list)
self._rank = len(shape)
self._loop_size = functools.reduce(lambda x,y: x*y, shape) if shape else 0
return self
def __setitem__(self, index, value):
"""Allows to set items to MutableDenseNDimArray.
Examples
========
>>> from sympy import MutableDenseNDimArray
>>> a = MutableDenseNDimArray.zeros(2, 2)
>>> a[0,0] = 1
>>> a[1,1] = 1
>>> a
[[1, 0], [0, 1]]
"""
if isinstance(index, tuple) and any([isinstance(i, slice) for i in index]):
value, eindices, slice_offsets = self._get_slice_data_for_array_assignment(index, value)
for i in eindices:
other_i = [ind - j for ind, j in zip(i, slice_offsets) if j is not None]
self._array[self._parse_index(i)] = value[other_i]
else:
index = self._parse_index(index)
self._setter_iterable_check(value)
value = _sympify(value)
self._array[index] = value
def as_immutable(self):
return ImmutableDenseNDimArray(self)
@property
def free_symbols(self):
return {i for j in self._array for i in j.free_symbols}
|
6aa85e9f2ec90e5ea13b8f1c52176fd378696c7219f8974d791170a78ae0d06f | from functools import wraps
from sympy import Matrix, eye, Integer, expand, Indexed, Sum
from sympy.combinatorics import Permutation
from sympy.core import S, Rational, Symbol, Basic, Add
from sympy.core.containers import Tuple
from sympy.core.symbol import symbols
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.printing.pretty.pretty import pretty
from sympy.tensor.array import Array
from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorSymmetry, \
get_symmetric_group_sgs, TensorType, TensorIndex, tensor_mul, TensAdd, \
riemann_cyclic_replace, riemann_cyclic, TensMul, tensorsymmetry, tensorhead, \
TensorManager, TensExpr, TensorHead, canon_bp
from sympy.utilities.pytest import raises, XFAIL, ignore_warnings
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.core.compatibility import range
from sympy.matrices import diag
def filter_warnings_decorator(f):
@wraps(f)
def wrapper():
with ignore_warnings(SymPyDeprecationWarning):
f()
return wrapper
def _is_equal(arg1, arg2):
if isinstance(arg1, TensExpr):
return arg1.equals(arg2)
elif isinstance(arg2, TensExpr):
return arg2.equals(arg1)
return arg1 == arg2
#################### Tests from tensor_can.py #######################
def test_canonicalize_no_slot_sym():
# A_d0 * B^d0; T_c = A^d0*B_d0
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, d0, d1 = tensor_indices('a,b,d0,d1', Lorentz)
sym1 = tensorsymmetry([1])
S1 = TensorType([Lorentz], sym1)
A, B = S1('A,B')
t = A(-d0)*B(d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0)*B(-L_0)'
# A^a * B^b; T_c = T
t = A(a)*B(b)
tc = t.canon_bp()
assert tc == t
# B^b * A^a
t1 = B(b)*A(a)
tc = t1.canon_bp()
assert str(tc) == 'A(a)*B(b)'
# A symmetric
# A^{b}_{d0}*A^{d0, a}; T_c = A^{a d0}*A{b}_{d0}
sym2 = tensorsymmetry([1]*2)
S2 = TensorType([Lorentz]*2, sym2)
A = S2('A')
t = A(b, -d0)*A(d0, a)
tc = t.canon_bp()
assert str(tc) == 'A(a, L_0)*A(b, -L_0)'
# A^{d1}_{d0}*B^d0*C_d1
# T_c = A^{d0 d1}*B_d0*C_d1
B, C = S1('B,C')
t = A(d1, -d0)*B(d0)*C(-d1)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, L_1)*B(-L_0)*C(-L_1)'
# A without symmetry
# A^{d1}_{d0}*B^d0*C_d1 ord=[d0,-d0,d1,-d1]; g = [2,1,0,3,4,5]
# T_c = A^{d0 d1}*B_d1*C_d0; can = [0,2,3,1,4,5]
nsym2 = tensorsymmetry([1],[1])
NS2 = TensorType([Lorentz]*2, nsym2)
A = NS2('A')
B, C = S1('B, C')
t = A(d1, -d0)*B(d0)*C(-d1)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, L_1)*B(-L_1)*C(-L_0)'
# A, B without symmetry
# A^{d1}_{d0}*B_{d1}^{d0}
# T_c = A^{d0 d1}*B_{d0 d1}
B = NS2('B')
t = A(d1, -d0)*B(-d1, d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, L_1)*B(-L_0, -L_1)'
# A_{d0}^{d1}*B_{d1}^{d0}
# T_c = A^{d0 d1}*B_{d1 d0}
t = A(-d0, d1)*B(-d1, d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, L_1)*B(-L_1, -L_0)'
# A, B, C without symmetry
# A^{d1 d0}*B_{a d0}*C_{d1 b}
# T_c=A^{d0 d1}*B_{a d1}*C_{d0 b}
C = NS2('C')
t = A(d1, d0)*B(-a, -d0)*C(-d1, -b)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, L_1)*B(-a, -L_1)*C(-L_0, -b)'
# A symmetric, B and C without symmetry
# A^{d1 d0}*B_{a d0}*C_{d1 b}
# T_c = A^{d0 d1}*B_{a d0}*C_{d1 b}
A = S2('A')
t = A(d1, d0)*B(-a, -d0)*C(-d1, -b)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, L_1)*B(-a, -L_0)*C(-L_1, -b)'
# A and C symmetric, B without symmetry
# A^{d1 d0}*B_{a d0}*C_{d1 b} ord=[a,b,d0,-d0,d1,-d1]
# T_c = A^{d0 d1}*B_{a d0}*C_{b d1}
C = S2('C')
t = A(d1, d0)*B(-a, -d0)*C(-d1, -b)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, L_1)*B(-a, -L_0)*C(-b, -L_1)'
def test_canonicalize_no_dummies():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a, b, c, d', Lorentz)
sym1 = tensorsymmetry([1])
sym2 = tensorsymmetry([1]*2)
sym2a = tensorsymmetry([2])
# A commuting
# A^c A^b A^a
# T_c = A^a A^b A^c
S1 = TensorType([Lorentz], sym1)
A = S1('A')
t = A(c)*A(b)*A(a)
tc = t.canon_bp()
assert str(tc) == 'A(a)*A(b)*A(c)'
# A anticommuting
# A^c A^b A^a
# T_c = -A^a A^b A^c
A = S1('A', 1)
t = A(c)*A(b)*A(a)
tc = t.canon_bp()
assert str(tc) == '-A(a)*A(b)*A(c)'
# A commuting and symmetric
# A^{b,d}*A^{c,a}
# T_c = A^{a c}*A^{b d}
S2 = TensorType([Lorentz]*2, sym2)
A = S2('A')
t = A(b, d)*A(c, a)
tc = t.canon_bp()
assert str(tc) == 'A(a, c)*A(b, d)'
# A anticommuting and symmetric
# A^{b,d}*A^{c,a}
# T_c = -A^{a c}*A^{b d}
A = S2('A', 1)
t = A(b, d)*A(c, a)
tc = t.canon_bp()
assert str(tc) == '-A(a, c)*A(b, d)'
# A^{c,a}*A^{b,d}
# T_c = A^{a c}*A^{b d}
t = A(c, a)*A(b, d)
tc = t.canon_bp()
assert str(tc) == 'A(a, c)*A(b, d)'
def test_tensorhead_construction_without_symmetry():
L = Lorentz = TensorIndexType('Lorentz')
A1 = tensorhead('A', [L, L])
A2 = tensorhead('A', [L, L], [[1], [1]])
assert A1 == A2
A3 = tensorhead('A', [L, L], [[1, 1]]) # Symmetric
assert A1 != A3
def test_no_metric_symmetry():
# no metric symmetry; A no symmetry
# A^d1_d0 * A^d0_d1
# T_c = A^d0_d1 * A^d1_d0
Lorentz = TensorIndexType('Lorentz', metric=None, dummy_fmt='L')
d0, d1, d2, d3 = tensor_indices('d:4', Lorentz)
A = tensorhead('A', [Lorentz]*2, [[1], [1]])
t = A(d1, -d0)*A(d0, -d1)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_0)'
# A^d1_d2 * A^d0_d3 * A^d2_d1 * A^d3_d0
# T_c = A^d0_d1 * A^d1_d0 * A^d2_d3 * A^d3_d2
t = A(d1, -d2)*A(d0, -d3)*A(d2,-d1)*A(d3,-d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_0)*A(L_2, -L_3)*A(L_3, -L_2)'
# A^d0_d2 * A^d1_d3 * A^d3_d0 * A^d2_d1
# T_c = A^d0_d1 * A^d1_d2 * A^d2_d3 * A^d3_d0
t = A(d0, -d1)*A(d1, -d2)*A(d2, -d3)*A(d3,-d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, -L_1)*A(L_1, -L_2)*A(L_2, -L_3)*A(L_3, -L_0)'
def test_canonicalize1():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \
tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Lorentz)
sym1 = tensorsymmetry([1])
base3, gens3 = get_symmetric_group_sgs(3)
sym2 = tensorsymmetry([1]*2)
sym2a = tensorsymmetry([2])
sym3 = tensorsymmetry([1]*3)
sym3a = tensorsymmetry([3])
# A_d0*A^d0; ord = [d0,-d0]
# T_c = A^d0*A_d0
S1 = TensorType([Lorentz], sym1)
A = S1('A')
t = A(-d0)*A(d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0)*A(-L_0)'
# A commuting
# A_d0*A_d1*A_d2*A^d2*A^d1*A^d0
# T_c = A^d0*A_d0*A^d1*A_d1*A^d2*A_d2
t = A(-d0)*A(-d1)*A(-d2)*A(d2)*A(d1)*A(d0)
tc = t.canon_bp()
assert str(tc) == 'A(L_0)*A(-L_0)*A(L_1)*A(-L_1)*A(L_2)*A(-L_2)'
# A anticommuting
# A_d0*A_d1*A_d2*A^d2*A^d1*A^d0
# T_c 0
A = S1('A', 1)
t = A(-d0)*A(-d1)*A(-d2)*A(d2)*A(d1)*A(d0)
tc = t.canon_bp()
assert tc == 0
# A commuting symmetric
# A^{d0 b}*A^a_d1*A^d1_d0
# T_c = A^{a d0}*A^{b d1}*A_{d0 d1}
S2 = TensorType([Lorentz]*2, sym2)
A = S2('A')
t = A(d0, b)*A(a, -d1)*A(d1, -d0)
tc = t.canon_bp()
assert str(tc) == 'A(a, L_0)*A(b, L_1)*A(-L_0, -L_1)'
# A, B commuting symmetric
# A^{d0 b}*A^d1_d0*B^a_d1
# T_c = A^{b d0}*A_d0^d1*B^a_d1
B = S2('B')
t = A(d0, b)*A(d1, -d0)*B(a, -d1)
tc = t.canon_bp()
assert str(tc) == 'A(b, L_0)*A(-L_0, L_1)*B(a, -L_1)'
# A commuting symmetric
# A^{d1 d0 b}*A^{a}_{d1 d0}; ord=[a,b, d0,-d0,d1,-d1]
# T_c = A^{a d0 d1}*A^{b}_{d0 d1}
S3 = TensorType([Lorentz]*3, sym3)
A = S3('A')
t = A(d1, d0, b)*A(a, -d1, -d0)
tc = t.canon_bp()
assert str(tc) == 'A(a, L_0, L_1)*A(b, -L_0, -L_1)'
# A^{d3 d0 d2}*A^a0_{d1 d2}*A^d1_d3^a1*A^{a2 a3}_d0
# T_c = A^{a0 d0 d1}*A^a1_d0^d2*A^{a2 a3 d3}*A_{d1 d2 d3}
t = A(d3, d0, d2)*A(a0, -d1, -d2)*A(d1, -d3, a1)*A(a2, a3, -d0)
tc = t.canon_bp()
assert str(tc) == 'A(a0, L_0, L_1)*A(a1, -L_0, L_2)*A(a2, a3, L_3)*A(-L_1, -L_2, -L_3)'
# A commuting symmetric, B antisymmetric
# A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3
# in this esxample and in the next three,
# renaming dummy indices and using symmetry of A,
# T = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3
# can = 0
S2a = TensorType([Lorentz]*2, sym2a)
A = S3('A')
B = S2a('B')
t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3)
tc = t.canon_bp()
assert tc == 0
# A anticommuting symmetric, B anticommuting
# A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3
# T_c = A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3}
A = S3('A', 1)
B = S2a('B')
t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3)
tc = t.canon_bp()
assert str(tc) == 'A(L_0, L_1, L_2)*A(-L_0, -L_1, L_3)*B(-L_2, -L_3)'
# A anticommuting symmetric, B antisymmetric commuting, antisymmetric metric
# A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3
# T_c = -A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3}
Spinor = TensorIndexType('Spinor', metric=1, dummy_fmt='S')
a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \
tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Spinor)
S3 = TensorType([Spinor]*3, sym3)
S2a = TensorType([Spinor]*2, sym2a)
A = S3('A', 1)
B = S2a('B')
t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3)
tc = t.canon_bp()
assert str(tc) == '-A(S_0, S_1, S_2)*A(-S_0, -S_1, S_3)*B(-S_2, -S_3)'
# A anticommuting symmetric, B antisymmetric anticommuting,
# no metric symmetry
# A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3
# T_c = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3
Mat = TensorIndexType('Mat', metric=None, dummy_fmt='M')
a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \
tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Mat)
S3 = TensorType([Mat]*3, sym3)
S2a = TensorType([Mat]*2, sym2a)
A = S3('A', 1)
B = S2a('B')
t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3)
tc = t.canon_bp()
assert str(tc) == 'A(M_0, M_1, M_2)*A(-M_0, -M_1, -M_3)*B(-M_2, M_3)'
# Gamma anticommuting
# Gamma_{mu nu} * gamma^rho * Gamma^{nu mu alpha}
# T_c = -Gamma^{mu nu} * gamma^rho * Gamma_{alpha mu nu}
S1 = TensorType([Lorentz], sym1)
S2a = TensorType([Lorentz]*2, sym2a)
S3a = TensorType([Lorentz]*3, sym3a)
alpha, beta, gamma, mu, nu, rho = \
tensor_indices('alpha,beta,gamma,mu,nu,rho', Lorentz)
Gamma = S1('Gamma', 2)
Gamma2 = S2a('Gamma', 2)
Gamma3 = S3a('Gamma', 2)
t = Gamma2(-mu,-nu)*Gamma(rho)*Gamma3(nu, mu, alpha)
tc = t.canon_bp()
assert str(tc) == '-Gamma(L_0, L_1)*Gamma(rho)*Gamma(alpha, -L_0, -L_1)'
# Gamma_{mu nu} * Gamma^{gamma beta} * gamma_rho * Gamma^{nu mu alpha}
# T_c = Gamma^{mu nu} * Gamma^{beta gamma} * gamma_rho * Gamma^alpha_{mu nu}
t = Gamma2(mu, nu)*Gamma2(beta, gamma)*Gamma(-rho)*Gamma3(alpha, -mu, -nu)
tc = t.canon_bp()
assert str(tc) == 'Gamma(L_0, L_1)*Gamma(beta, gamma)*Gamma(-rho)*Gamma(alpha, -L_0, -L_1)'
# f^a_{b,c} antisymmetric in b,c; A_mu^a no symmetry
# f^c_{d a} * f_{c e b} * A_mu^d * A_nu^a * A^{nu e} * A^{mu b}
# g = [8,11,5, 9,13,7, 1,10, 3,4, 2,12, 0,6, 14,15]
# T_c = -f^{a b c} * f_a^{d e} * A^mu_b * A_{mu d} * A^nu_c * A_{nu e}
Flavor = TensorIndexType('Flavor', dummy_fmt='F')
a, b, c, d, e, ff = tensor_indices('a,b,c,d,e,f', Flavor)
mu, nu = tensor_indices('mu,nu', Lorentz)
sym_f = tensorsymmetry([1], [2])
S_f = TensorType([Flavor]*3, sym_f)
sym_A = tensorsymmetry([1], [1])
S_A = TensorType([Lorentz, Flavor], sym_A)
f = S_f('f')
A = S_A('A')
t = f(c, -d, -a)*f(-c, -e, -b)*A(-mu, d)*A(-nu, a)*A(nu, e)*A(mu, b)
tc = t.canon_bp()
assert str(tc) == '-f(F_0, F_1, F_2)*f(-F_0, F_3, F_4)*A(L_0, -F_1)*A(-L_0, -F_3)*A(L_1, -F_2)*A(-L_1, -F_4)'
def test_bug_correction_tensor_indices():
# to make sure that tensor_indices does not return a list if creating
# only one index:
from sympy.tensor.tensor import tensor_indices, TensorIndexType, TensorIndex
A = TensorIndexType("A")
i = tensor_indices('i', A)
assert not isinstance(i, (tuple, list))
assert isinstance(i, TensorIndex)
def test_riemann_invariants():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
d0, d1, d2, d3, d4, d5, d6, d7, d8, d9, d10, d11 = \
tensor_indices('d0:12', Lorentz)
# R^{d0 d1}_{d1 d0}; ord = [d0,-d0,d1,-d1]
# T_c = -R^{d0 d1}_{d0 d1}
R = tensorhead('R', [Lorentz]*4, [[2, 2]])
t = R(d0, d1, -d1, -d0)
tc = t.canon_bp()
assert str(tc) == '-R(L_0, L_1, -L_0, -L_1)'
# R_d11^d1_d0^d5 * R^{d6 d4 d0}_d5 * R_{d7 d2 d8 d9} *
# R_{d10 d3 d6 d4} * R^{d2 d7 d11}_d1 * R^{d8 d9 d3 d10}
# can = [0,2,4,6, 1,3,8,10, 5,7,12,14, 9,11,16,18, 13,15,20,22,
# 17,19,21<F10,23, 24,25]
# T_c = R^{d0 d1 d2 d3} * R_{d0 d1}^{d4 d5} * R_{d2 d3}^{d6 d7} *
# R_{d4 d5}^{d8 d9} * R_{d6 d7}^{d10 d11} * R_{d8 d9 d10 d11}
t = R(-d11,d1,-d0,d5)*R(d6,d4,d0,-d5)*R(-d7,-d2,-d8,-d9)* \
R(-d10,-d3,-d6,-d4)*R(d2,d7,d11,-d1)*R(d8,d9,d3,d10)
tc = t.canon_bp()
assert str(tc) == 'R(L_0, L_1, L_2, L_3)*R(-L_0, -L_1, L_4, L_5)*R(-L_2, -L_3, L_6, L_7)*R(-L_4, -L_5, L_8, L_9)*R(-L_6, -L_7, L_10, L_11)*R(-L_8, -L_9, -L_10, -L_11)'
def test_riemann_products():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
d0, d1, d2, d3, d4, d5, d6 = tensor_indices('d0:7', Lorentz)
a0, a1, a2, a3, a4, a5 = tensor_indices('a0:6', Lorentz)
a, b = tensor_indices('a,b', Lorentz)
R = tensorhead('R', [Lorentz]*4, [[2, 2]])
# R^{a b d0}_d0 = 0
t = R(a, b, d0, -d0)
tc = t.canon_bp()
assert tc == 0
# R^{d0 b a}_d0
# T_c = -R^{a d0 b}_d0
t = R(d0, b, a, -d0)
tc = t.canon_bp()
assert str(tc) == '-R(a, L_0, b, -L_0)'
# R^d1_d2^b_d0 * R^{d0 a}_d1^d2; ord=[a,b,d0,-d0,d1,-d1,d2,-d2]
# T_c = -R^{a d0 d1 d2}* R^b_{d0 d1 d2}
t = R(d1, -d2, b, -d0)*R(d0, a, -d1, d2)
tc = t.canon_bp()
assert str(tc) == '-R(a, L_0, L_1, L_2)*R(b, -L_0, -L_1, -L_2)'
# A symmetric commuting
# R^{d6 d5}_d2^d1 * R^{d4 d0 d2 d3} * A_{d6 d0} A_{d3 d1} * A_{d4 d5}
# g = [12,10,5,2, 8,0,4,6, 13,1, 7,3, 9,11,14,15]
# T_c = -R^{d0 d1 d2 d3} * R_d0^{d4 d5 d6} * A_{d1 d4}*A_{d2 d5}*A_{d3 d6}
V = tensorhead('V', [Lorentz]*2, [[1]*2])
t = R(d6, d5, -d2, d1)*R(d4, d0, d2, d3)*V(-d6, -d0)*V(-d3, -d1)*V(-d4, -d5)
tc = t.canon_bp()
assert str(tc) == '-R(L_0, L_1, L_2, L_3)*R(-L_0, L_4, L_5, L_6)*V(-L_1, -L_4)*V(-L_2, -L_5)*V(-L_3, -L_6)'
# R^{d2 a0 a2 d0} * R^d1_d2^{a1 a3} * R^{a4 a5}_{d0 d1}
# T_c = R^{a0 d0 a2 d1}*R^{a1 a3}_d0^d2*R^{a4 a5}_{d1 d2}
t = R(d2, a0, a2, d0)*R(d1, -d2, a1, a3)*R(a4, a5, -d0, -d1)
tc = t.canon_bp()
assert str(tc) == 'R(a0, L_0, a2, L_1)*R(a1, a3, -L_0, L_2)*R(a4, a5, -L_1, -L_2)'
######################################################################
def test_canonicalize2():
D = Symbol('D')
Eucl = TensorIndexType('Eucl', metric=0, dim=D, dummy_fmt='E')
i0,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14 = \
tensor_indices('i0:15', Eucl)
A = tensorhead('A', [Eucl]*3, [[3]])
# two examples from Cvitanovic, Group Theory page 59
# of identities for antisymmetric tensors of rank 3
# contracted according to the Kuratowski graph eq.(6.59)
t = A(i0,i1,i2)*A(-i1,i3,i4)*A(-i3,i7,i5)*A(-i2,-i5,i6)*A(-i4,-i6,i8)
t1 = t.canon_bp()
assert t1 == 0
# eq.(6.60)
#t = A(i0,i1,i2)*A(-i1,i3,i4)*A(-i2,i5,i6)*A(-i3,i7,i8)*A(-i6,-i7,i9)*
# A(-i8,i10,i13)*A(-i5,-i10,i11)*A(-i4,-i11,i12)*A(-i3,-i12,i14)
t = A(i0,i1,i2)*A(-i1,i3,i4)*A(-i2,i5,i6)*A(-i3,i7,i8)*A(-i6,-i7,i9)*\
A(-i8,i10,i13)*A(-i5,-i10,i11)*A(-i4,-i11,i12)*A(-i9,-i12,i14)
t1 = t.canon_bp()
assert t1 == 0
def test_canonicalize3():
D = Symbol('D')
Spinor = TensorIndexType('Spinor', dim=D, metric=True, dummy_fmt='S')
a0,a1,a2,a3,a4 = tensor_indices('a0:5', Spinor)
C = Spinor.metric
chi, psi = tensorhead('chi,psi', [Spinor], [[1]], 1)
t = chi(a1)*psi(a0)
t1 = t.canon_bp()
assert t1 == t
t = psi(a1)*chi(a0)
t1 = t.canon_bp()
assert t1 == -chi(a0)*psi(a1)
class Metric(Basic):
def __new__(cls, name, antisym, **kwargs):
obj = Basic.__new__(cls, name, antisym, **kwargs)
obj.name = name
obj.antisym = antisym
return obj
def test_TensorIndexType():
D = Symbol('D')
G = Metric('g', False)
Lorentz = TensorIndexType('Lorentz', metric=G, dim=D, dummy_fmt='L')
m0, m1, m2, m3, m4 = tensor_indices('m0:5', Lorentz)
sym2 = tensorsymmetry([1]*2)
sym2n = tensorsymmetry(*get_symmetric_group_sgs(2))
assert sym2 == sym2n
g = Lorentz.metric
assert str(g) == 'g(Lorentz,Lorentz)'
assert Lorentz.eps_dim == Lorentz.dim
TSpace = TensorIndexType('TSpace')
i0, i1 = tensor_indices('i0 i1', TSpace)
g = TSpace.metric
A = tensorhead('A', [TSpace]*2, [[1]*2])
assert str(A(i0,-i0).canon_bp()) == 'A(TSpace_0, -TSpace_0)'
def test_indices():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
assert a.tensor_index_type == Lorentz
assert a != -a
A, B = tensorhead('A B', [Lorentz]*2, [[1]*2])
t = A(a,b)*B(-b,c)
indices = t.get_indices()
L_0 = TensorIndex('L_0', Lorentz)
assert indices == [a, L_0, -L_0, c]
raises(ValueError, lambda: tensor_indices(3, Lorentz))
raises(ValueError, lambda: A(a,b,c))
def test_tensorsymmetry():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
sym = tensorsymmetry([1]*2)
sym1 = TensorSymmetry(get_symmetric_group_sgs(2))
assert sym == sym1
sym = tensorsymmetry([2])
sym1 = TensorSymmetry(get_symmetric_group_sgs(2, 1))
assert sym == sym1
sym2 = tensorsymmetry()
assert sym2.base == Tuple() and sym2.generators == Tuple(Permutation(1))
raises(NotImplementedError, lambda: tensorsymmetry([2, 1]))
def test_TensorType():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
sym = tensorsymmetry([1]*2)
A = tensorhead('A', [Lorentz]*2, [[1]*2])
assert A.typ == TensorType([Lorentz]*2, sym)
assert A.types == [Lorentz]
assert A.index_types == Tuple(*[Lorentz, Lorentz])
typ = TensorType([Lorentz]*2, sym)
assert str(typ) == "TensorType(['Lorentz', 'Lorentz'])"
raises(ValueError, lambda: typ(2))
def test_TensExpr():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
g = Lorentz.metric
A, B = tensorhead('A B', [Lorentz]*2, [[1]*2])
raises(ValueError, lambda: g(c, d)/g(a, b))
raises(ValueError, lambda: S.One/g(a, b))
raises(ValueError, lambda: (A(c, d) + g(c, d))/g(a, b))
raises(ValueError, lambda: S.One/(A(c, d) + g(c, d)))
raises(ValueError, lambda: A(a, b) + A(a, c))
t = A(a, b) + B(a, b)
#raises(NotImplementedError, lambda: TensExpr.__mul__(t, 'a'))
#raises(NotImplementedError, lambda: TensExpr.__add__(t, 'a'))
#raises(NotImplementedError, lambda: TensExpr.__radd__(t, 'a'))
#raises(NotImplementedError, lambda: TensExpr.__sub__(t, 'a'))
#raises(NotImplementedError, lambda: TensExpr.__rsub__(t, 'a'))
#raises(NotImplementedError, lambda: TensExpr.__div__(t, 'a'))
#raises(NotImplementedError, lambda: TensExpr.__rdiv__(t, 'a'))
with ignore_warnings(SymPyDeprecationWarning):
# DO NOT REMOVE THIS AFTER DEPRECATION REMOVED:
raises(ValueError, lambda: A(a, b)**2)
raises(NotImplementedError, lambda: 2**A(a, b))
raises(NotImplementedError, lambda: abs(A(a, b)))
def test_TensorHead():
# simple example of algebraic expression
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a,b = tensor_indices('a,b', Lorentz)
# A, B symmetric
A = tensorhead('A', [Lorentz]*2, [[1]*2])
assert A.rank == 2
assert A.symmetry == tensorsymmetry([1]*2)
def test_add1():
assert TensAdd().args == ()
assert TensAdd().doit() == 0
# simple example of algebraic expression
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a,b,d0,d1,i,j,k = tensor_indices('a,b,d0,d1,i,j,k', Lorentz)
# A, B symmetric
A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
t1 = A(b,-d0)*B(d0,a)
assert TensAdd(t1).equals(t1)
t2a = B(d0,a) + A(d0, a)
t2 = A(b,-d0)*t2a
assert str(t2) == 'A(b, -L_0)*(A(L_0, a) + B(L_0, a))'
t2 = t2.expand()
assert str(t2) == 'A(b, -L_0)*A(L_0, a) + A(b, -L_0)*B(L_0, a)'
t2 = t2.canon_bp()
assert str(t2) == 'A(a, L_0)*A(b, -L_0) + A(b, L_0)*B(a, -L_0)'
t2b = t2 + t1
assert str(t2b) == 'A(a, L_0)*A(b, -L_0) + A(b, -L_0)*B(L_0, a) + A(b, L_0)*B(a, -L_0)'
t2b = t2b.canon_bp()
assert str(t2b) == '2*A(b, L_0)*B(a, -L_0) + A(a, L_0)*A(b, -L_0)'
p, q, r = tensorhead('p,q,r', [Lorentz], [[1]])
t = q(d0)*2
assert str(t) == '2*q(d0)'
t = 2*q(d0)
assert str(t) == '2*q(d0)'
t1 = p(d0) + 2*q(d0)
assert str(t1) == '2*q(d0) + p(d0)'
t2 = p(-d0) + 2*q(-d0)
assert str(t2) == '2*q(-d0) + p(-d0)'
t1 = p(d0)
t3 = t1*t2
assert str(t3) == 'p(L_0)*(2*q(-L_0) + p(-L_0))'
t3 = t3.expand()
assert str(t3) == '2*p(L_0)*q(-L_0) + p(L_0)*p(-L_0)'
t3 = t2*t1
t3 = t3.expand()
assert str(t3) == '2*q(-L_0)*p(L_0) + p(-L_0)*p(L_0)'
t3 = t3.canon_bp()
assert str(t3) == '2*p(L_0)*q(-L_0) + p(L_0)*p(-L_0)'
t1 = p(d0) + 2*q(d0)
t3 = t1*t2
t3 = t3.canon_bp()
assert str(t3) == '4*p(L_0)*q(-L_0) + 4*q(L_0)*q(-L_0) + p(L_0)*p(-L_0)'
t1 = p(d0) - 2*q(d0)
assert str(t1) == '-2*q(d0) + p(d0)'
t2 = p(-d0) + 2*q(-d0)
t3 = t1*t2
t3 = t3.canon_bp()
assert t3 == p(d0)*p(-d0) - 4*q(d0)*q(-d0)
t = p(i)*p(j)*(p(k) + q(k)) + p(i)*(p(j) + q(j))*(p(k) - 3*q(k))
t = t.canon_bp()
assert t == 2*p(i)*p(j)*p(k) - 2*p(i)*p(j)*q(k) + p(i)*p(k)*q(j) - 3*p(i)*q(j)*q(k)
t1 = (p(i) + q(i) + 2*r(i))*(p(j) - q(j))
t2 = (p(j) + q(j) + 2*r(j))*(p(i) - q(i))
t = t1 + t2
t = t.canon_bp()
assert t == 2*p(i)*p(j) + 2*p(i)*r(j) + 2*p(j)*r(i) - 2*q(i)*q(j) - 2*q(i)*r(j) - 2*q(j)*r(i)
t = p(i)*q(j)/2
assert 2*t == p(i)*q(j)
t = (p(i) + q(i))/2
assert 2*t == p(i) + q(i)
t = S.One - p(i)*p(-i)
t = t.canon_bp()
tz1 = t + p(-j)*p(j)
assert tz1 != 1
tz1 = tz1.canon_bp()
assert tz1.equals(1)
t = S.One + p(i)*p(-i)
assert (t - p(-j)*p(j)).canon_bp().equals(1)
t = A(a, b) + B(a, b)
assert t.rank == 2
t1 = t - A(a, b) - B(a, b)
assert t1 == 0
t = 1 - (A(a, -a) + B(a, -a))
t1 = 1 + (A(a, -a) + B(a, -a))
assert (t + t1).expand().equals(2)
t2 = 1 + A(a, -a)
assert t1 != t2
assert t2 != TensMul.from_data(0, [], [], [])
t = p(i) + q(i)
raises(ValueError, lambda: t(i, j))
def test_special_eq_ne():
# test special equality cases:
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a,b,d0,d1,i,j,k = tensor_indices('a,b,d0,d1,i,j,k', Lorentz)
# A, B symmetric
A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
p, q, r = tensorhead('p,q,r', [Lorentz], [[1]])
t = 0*A(a, b)
assert _is_equal(t, 0)
assert _is_equal(t, S.Zero)
assert p(i) != A(a, b)
assert A(a, -a) != A(a, b)
assert 0*(A(a, b) + B(a, b)) == 0
assert 0*(A(a, b) + B(a, b)) == S.Zero
assert 3*(A(a, b) - A(a, b)) == S.Zero
assert p(i) + q(i) != A(a, b)
assert p(i) + q(i) != A(a, b) + B(a, b)
assert p(i) - p(i) == 0
assert p(i) - p(i) == S.Zero
assert _is_equal(A(a, b), A(b, a))
def test_add2():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
m, n, p, q = tensor_indices('m,n,p,q', Lorentz)
R = tensorhead('R', [Lorentz]*4, [[2, 2]])
A = tensorhead('A', [Lorentz]*3, [[3]])
t1 = 2*R(m, n, p, q) - R(m, q, n, p) + R(m, p, n, q)
t2 = t1*A(-n, -p, -q)
t2 = t2.canon_bp()
assert t2 == 0
t1 = S(2)/3*R(m,n,p,q) - S(1)/3*R(m,q,n,p) + S(1)/3*R(m,p,n,q)
t2 = t1*A(-n, -p, -q)
t2 = t2.canon_bp()
assert t2 == 0
t = A(m, -m, n) + A(n, p, -p)
t = t.canon_bp()
assert t == 0
def test_add3():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
i0, i1 = tensor_indices('i0:2', Lorentz)
E, px, py, pz = symbols('E px py pz')
A = tensorhead('A', [Lorentz], [[1]])
B = tensorhead('B', [Lorentz], [[1]])
expr1 = A(i0)*A(-i0) - (E**2 - px**2 - py**2 - pz**2)
assert expr1.args == (px**2, py**2, pz**2, -E**2, A(i0)*A(-i0))
expr2 = E**2 - px**2 - py**2 - pz**2 - A(i0)*A(-i0)
assert expr2.args == (E**2, -px**2, -py**2, -pz**2, -A(i0)*A(-i0))
expr3 = A(i0)*A(-i0) - E**2 + px**2 + py**2 + pz**2
assert expr3.args == (px**2, py**2, pz**2, -E**2, A(i0)*A(-i0))
expr4 = B(i1)*B(-i1) + 2*E**2 - 2*px**2 - 2*py**2 - 2*pz**2 - A(i0)*A(-i0)
assert expr4.args == (-2*px**2, -2*py**2, -2*pz**2, 2*E**2, -A(i0)*A(-i0), B(i1)*B(-i1))
def test_mul():
from sympy.abc import x
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
sym = tensorsymmetry([1]*2)
t = TensMul.from_data(S.One, [], [], [])
assert str(t) == '1'
A, B = tensorhead('A B', [Lorentz]*2, [[1]*2])
t = (1 + x)*A(a, b)
assert str(t) == '(x + 1)*A(a, b)'
assert t.index_types == [Lorentz, Lorentz]
assert t.rank == 2
assert t.dum == []
assert t.coeff == 1 + x
assert sorted(t.free) == [(a, 0), (b, 1)]
assert t.components == [A]
ts = A(a, b)
assert str(ts) == 'A(a, b)'
assert ts.index_types == [Lorentz, Lorentz]
assert ts.rank == 2
assert ts.dum == []
assert ts.coeff == 1
assert sorted(ts.free) == [(a, 0), (b, 1)]
assert ts.components == [A]
t = A(-b, a)*B(-a, c)*A(-c, d)
t1 = tensor_mul(*t.split())
assert t == t(-b, d)
assert t == t1
assert tensor_mul(*[]) == TensMul.from_data(S.One, [], [], [])
t = TensMul.from_data(1, [], [], [])
zsym = tensorsymmetry()
typ = TensorType([], zsym)
C = typ('C')
assert str(C()) == 'C'
assert str(t) == '1'
assert t == 1
raises(ValueError, lambda: A(a, b)*A(a, c))
t = A(a, b)*A(-a, c)
raises(ValueError, lambda: t(a, b, c))
def test_substitute_indices():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
i, j, k, l, m, n, p, q = tensor_indices('i,j,k,l,m,n,p,q', Lorentz)
A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
t = A(i, k)*B(-k, -j)
t1 = t.substitute_indices((i, j), (j, k))
t1a = A(j, l)*B(-l, -k)
assert t1 == t1a
p = tensorhead('p', [Lorentz], [[1]])
t = p(i)
t1 = t.substitute_indices((j, k))
assert t1 == t
t1 = t.substitute_indices((i, j))
assert t1 == p(j)
t1 = t.substitute_indices((i, -j))
assert t1 == p(-j)
t1 = t.substitute_indices((-i, j))
assert t1 == p(-j)
t1 = t.substitute_indices((-i, -j))
assert t1 == p(j)
A_tmul = A(m, n)
A_c = A_tmul(m, -m)
assert _is_equal(A_c, A(n, -n))
ABm = A(i, j)*B(m, n)
ABc1 = ABm(i, j, -i, -j)
assert _is_equal(ABc1, A(i, -j)*B(-i, j))
ABc2 = ABm(i, -i, j, -j)
assert _is_equal(ABc2, A(m, -m)*B(-n, n))
asum = A(i, j) + B(i, j)
asc1 = asum(i, -i)
assert _is_equal(asc1, A(i, -i) + B(i, -i))
assert A(i, -i) == A(i, -i)()
assert canon_bp(A(i, -i) + B(-j, j) - (A(i, -i) + B(i, -i))()) == 0
assert _is_equal(A(i, j)*B(-j, k), (A(m, -j)*B(j, n))(i, k))
raises(ValueError, lambda: A(i, -i)(j, k))
def test_riemann_cyclic_replace():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
m0, m1, m2, m3 = tensor_indices('m:4', Lorentz)
symr = tensorsymmetry([2, 2])
R = tensorhead('R', [Lorentz]*4, [[2, 2]])
t = R(m0, m2, m1, m3)
t1 = riemann_cyclic_replace(t)
t1a = -S.One/3*R(m0, m3, m2, m1) + S.One/3*R(m0, m1, m2, m3) + Rational(2, 3)*R(m0, m2, m1, m3)
assert t1 == t1a
def test_riemann_cyclic():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
i, j, k, l, m, n, p, q = tensor_indices('i,j,k,l,m,n,p,q', Lorentz)
R = tensorhead('R', [Lorentz]*4, [[2, 2]])
t = R(i,j,k,l) + R(i,l,j,k) + R(i,k,l,j) - \
R(i,j,l,k) - R(i,l,k,j) - R(i,k,j,l)
t2 = t*R(-i,-j,-k,-l)
t3 = riemann_cyclic(t2)
assert t3 == 0
t = R(i,j,k,l)*(R(-i,-j,-k,-l) - 2*R(-i,-k,-j,-l))
t1 = riemann_cyclic(t)
assert t1 == 0
t = R(i,j,k,l)
t1 = riemann_cyclic(t)
assert t1 == -S(1)/3*R(i, l, j, k) + S(1)/3*R(i, k, j, l) + S(2)/3*R(i, j, k, l)
t = R(i,j,k,l)*R(-k,-l,m,n)*(R(-m,-n,-i,-j) + 2*R(-m,-j,-n,-i))
t1 = riemann_cyclic(t)
assert t1 == 0
@XFAIL
def test_div():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
m0,m1,m2,m3 = tensor_indices('m0:4', Lorentz)
R = tensorhead('R', [Lorentz]*4, [[2, 2]])
t = R(m0,m1,-m1,m3)
t1 = t/S(4)
assert str(t1) == '(1/4)*R(m0, L_0, -L_0, m3)'
t = t.canon_bp()
assert not t1._is_canon_bp
t1 = t*4
assert t1._is_canon_bp
t1 = t1/4
assert t1._is_canon_bp
def test_contract_metric1():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a, b, c, d, e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
p = tensorhead('p', [Lorentz], [[1]])
t = g(a, b)*p(-b)
t1 = t.contract_metric(g)
assert t1 == p(a)
A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
# case with g with all free indices
t1 = A(a,b)*B(-b,c)*g(d, e)
t2 = t1.contract_metric(g)
assert t1 == t2
# case of g(d, -d)
t1 = A(a,b)*B(-b,c)*g(-d, d)
t2 = t1.contract_metric(g)
assert t2 == D*A(a, d)*B(-d, c)
# g with one free index
t1 = A(a,b)*B(-b,-c)*g(c, d)
t2 = t1.contract_metric(g)
assert t2 == A(a, c)*B(-c, d)
# g with both indices contracted with another tensor
t1 = A(a,b)*B(-b,-c)*g(c, -a)
t2 = t1.contract_metric(g)
assert _is_equal(t2, A(a, b)*B(-b, -a))
t1 = A(a,b)*B(-b,-c)*g(c, d)*g(-a, -d)
t2 = t1.contract_metric(g)
assert _is_equal(t2, A(a,b)*B(-b,-a))
t1 = A(a,b)*g(-a,-b)
t2 = t1.contract_metric(g)
assert _is_equal(t2, A(a, -a))
assert not t2.free
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
g = Lorentz.metric
raises(ValueError, lambda: g(a, -a).contract_metric(g)) # no dim
def test_contract_metric2():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a, b, c, d, e, L_0 = tensor_indices('a,b,c,d,e,L_0', Lorentz)
g = Lorentz.metric
p, q = tensorhead('p,q', [Lorentz], [[1]])
t1 = g(a,b)*p(c)*p(-c)
t2 = 3*g(-a,-b)*q(c)*q(-c)
t = t1*t2
t = t.contract_metric(g)
assert t == 3*D*p(a)*p(-a)*q(b)*q(-b)
t1 = g(a,b)*p(c)*p(-c)
t2 = 3*q(-a)*q(-b)
t = t1*t2
t = t.contract_metric(g)
t = t.canon_bp()
assert t == 3*p(a)*p(-a)*q(b)*q(-b)
t1 = 2*g(a,b)*p(c)*p(-c)
t2 = - 3*g(-a,-b)*q(c)*q(-c)
t = t1*t2
t = t.contract_metric(g)
t = 6*g(a,b)*g(-a,-b)*p(c)*p(-c)*q(d)*q(-d)
t = t.contract_metric(g)
t1 = 2*g(a,b)*p(c)*p(-c)
t2 = q(-a)*q(-b) + 3*g(-a,-b)*q(c)*q(-c)
t = t1*t2
t = t.contract_metric(g)
assert t == (2 + 6*D)*p(a)*p(-a)*q(b)*q(-b)
t1 = p(a)*p(b) + p(a)*q(b) + 2*g(a,b)*p(c)*p(-c)
t2 = q(-a)*q(-b) - g(-a,-b)*q(c)*q(-c)
t = t1*t2
t = t.contract_metric(g)
t1 = (1 - 2*D)*p(a)*p(-a)*q(b)*q(-b) + p(a)*q(-a)*p(b)*q(-b)
assert canon_bp(t - t1) == 0
t = g(a,b)*g(c,d)*g(-b,-c)
t1 = t.contract_metric(g)
assert t1 == g(a, d)
t1 = g(a,b)*g(c,d) + g(a,c)*g(b,d) + g(a,d)*g(b,c)
t2 = t1.substitute_indices((a,-a),(b,-b),(c,-c),(d,-d))
t = t1*t2
t = t.contract_metric(g)
assert t.equals(3*D**2 + 6*D)
t = 2*p(a)*g(b,-b)
t1 = t.contract_metric(g)
assert t1.equals(2*D*p(a))
t = 2*p(a)*g(b,-a)
t1 = t.contract_metric(g)
assert t1 == 2*p(b)
M = Symbol('M')
t = (p(a)*p(b) + g(a, b)*M**2)*g(-a, -b) - D*M**2
t1 = t.contract_metric(g)
assert t1 == p(a)*p(-a)
A = tensorhead('A', [Lorentz]*2, [[1]*2])
t = A(a, b)*p(L_0)*g(-a, -b)
t1 = t.contract_metric(g)
assert str(t1) == 'A(L_1, -L_1)*p(L_0)' or str(t1) == 'A(-L_1, L_1)*p(L_0)'
def test_metric_contract3():
D = Symbol('D')
Spinor = TensorIndexType('Spinor', dim=D, metric=True, dummy_fmt='S')
a0,a1,a2,a3,a4 = tensor_indices('a0:5', Spinor)
C = Spinor.metric
chi, psi = tensorhead('chi,psi', [Spinor], [[1]], 1)
B = tensorhead('B', [Spinor]*2, [[1],[1]])
t = C(a0, -a0)
t1 = t.contract_metric(C)
assert t1.equals(-D)
t = C(-a0, a0)
t1 = t.contract_metric(C)
assert t1.equals(D)
t = C(a0,a1)*C(-a0,-a1)
t1 = t.contract_metric(C)
assert t1.equals(D)
t = C(a1,a0)*C(-a0,-a1)
t1 = t.contract_metric(C)
assert t1.equals(-D)
t = C(-a0,a1)*C(a0,-a1)
t1 = t.contract_metric(C)
assert t1.equals(-D)
t = C(a1,-a0)*C(a0,-a1)
t1 = t.contract_metric(C)
assert t1.equals(D)
t = C(a0,a1)*B(-a1,-a0)
t1 = t.contract_metric(C)
t1 = t1.canon_bp()
assert _is_equal(t1, B(a0, -a0))
t = C(a1,a0)*B(-a1,-a0)
t1 = t.contract_metric(C)
assert _is_equal(t1, -B(a0, -a0))
t = C(a0,-a1)*B(a1,-a0)
t1 = t.contract_metric(C)
assert _is_equal(t1, -B(a0, -a0))
t = C(-a0,a1)*B(-a1,a0)
t1 = t.contract_metric(C)
assert _is_equal(t1, -B(a0, -a0))
t = C(-a0,-a1)*B(a1,a0)
t1 = t.contract_metric(C)
assert _is_equal(t1, B(a0, -a0))
t = C(-a1, a0)*B(a1,-a0)
t1 = t.contract_metric(C)
assert _is_equal(t1, B(a0, -a0))
t = C(a0,a1)*psi(-a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, psi(a0))
t = C(a1,a0)*psi(-a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, -psi(a0))
t = C(a0,a1)*chi(-a0)*psi(-a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, -chi(a1)*psi(-a1))
t = C(a1,a0)*chi(-a0)*psi(-a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, chi(a1)*psi(-a1))
t = C(-a1,a0)*chi(-a0)*psi(a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, chi(-a1)*psi(a1))
t = C(a0, -a1)*chi(-a0)*psi(a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, -chi(-a1)*psi(a1))
t = C(-a0,-a1)*chi(a0)*psi(a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, chi(-a1)*psi(a1))
t = C(-a1,-a0)*chi(a0)*psi(a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, -chi(-a1)*psi(a1))
t = C(-a1,-a0)*B(a0,a2)*psi(a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, -B(-a1,a2)*psi(a1))
t = C(a1,a0)*B(-a2,-a0)*psi(-a1)
t1 = t.contract_metric(C)
assert _is_equal(t1, B(-a2,a1)*psi(-a1))
def test_epsilon():
Lorentz = TensorIndexType('Lorentz', dim=4, dummy_fmt='L')
a, b, c, d, e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
epsilon = Lorentz.epsilon
p, q, r, s = tensorhead('p,q,r,s', [Lorentz], [[1]])
t = epsilon(b,a,c,d)
t1 = t.canon_bp()
assert t1 == -epsilon(a,b,c,d)
t = epsilon(c,b,d,a)
t1 = t.canon_bp()
assert t1 == epsilon(a,b,c,d)
t = epsilon(c,a,d,b)
t1 = t.canon_bp()
assert t1 == -epsilon(a,b,c,d)
t = epsilon(a,b,c,d)*p(-a)*q(-b)
t1 = t.canon_bp()
assert t1 == epsilon(c, d, a, b)*p(-a)*q(-b)
t = epsilon(c,b,d,a)*p(-a)*q(-b)
t1 = t.canon_bp()
assert t1 == epsilon(c, d, a, b)*p(-a)*q(-b)
t = epsilon(c,a,d,b)*p(-a)*q(-b)
t1 = t.canon_bp()
assert t1 == -epsilon(c, d, a, b)*p(-a)*q(-b)
t = epsilon(c,a,d,b)*p(-a)*p(-b)
t1 = t.canon_bp()
assert t1 == 0
t = epsilon(c,a,d,b)*p(-a)*q(-b) + epsilon(a,b,c,d)*p(-b)*q(-a)
t1 = t.canon_bp()
assert t1 == -2*epsilon(c, d, a, b)*p(-a)*q(-b)
# Test that epsilon can be create with a SymPy integer:
Lorentz = TensorIndexType('Lorentz', dim=Integer(4), dummy_fmt='L')
epsilon = Lorentz.epsilon
assert isinstance(epsilon, TensorHead)
def test_contract_delta1():
# see Group Theory by Cvitanovic page 9
n = Symbol('n')
Color = TensorIndexType('Color', metric=None, dim=n, dummy_fmt='C')
a, b, c, d, e, f = tensor_indices('a,b,c,d,e,f', Color)
delta = Color.delta
def idn(a, b, d, c):
assert a.is_up and d.is_up
assert not (b.is_up or c.is_up)
return delta(a, c)*delta(d, b)
def T(a, b, d, c):
assert a.is_up and d.is_up
assert not (b.is_up or c.is_up)
return delta(a, b)*delta(d, c)
def P1(a, b, c, d):
return idn(a,b,c,d) - 1/n*T(a,b,c,d)
def P2(a, b, c, d):
return 1/n*T(a,b,c,d)
t = P1(a, -b, e, -f)*P1(f, -e, d, -c)
t1 = t.contract_delta(delta)
assert canon_bp(t1 - P1(a, -b, d, -c)) == 0
t = P2(a, -b, e, -f)*P2(f, -e, d, -c)
t1 = t.contract_delta(delta)
assert t1 == P2(a, -b, d, -c)
t = P1(a, -b, e, -f)*P2(f, -e, d, -c)
t1 = t.contract_delta(delta)
assert t1 == 0
t = P1(a, -b, b, -a)
t1 = t.contract_delta(delta)
assert t1.equals(n**2 - 1)
def test_fun():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a,b,c,d,e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
p, q = tensorhead('p q', [Lorentz], [[1]])
t = q(c)*p(a)*q(b) + g(a,b)*g(c,d)*q(-d)
assert t(a,b,c) == t
assert canon_bp(t - t(b,a,c) - q(c)*p(a)*q(b) + q(c)*p(b)*q(a)) == 0
assert t(b,c,d) == q(d)*p(b)*q(c) + g(b,c)*g(d,e)*q(-e)
t1 = t.fun_eval((a,b),(b,a))
assert canon_bp(t1 - q(c)*p(b)*q(a) - g(a,b)*g(c,d)*q(-d)) == 0
# check that g_{a b; c} = 0
# example taken from L. Brewin
# "A brief introduction to Cadabra" arxiv:0903.2085
# dg_{a b c} = \partial_{a} g_{b c} is symmetric in b, c
dg = tensorhead('dg', [Lorentz]*3, [[1], [1]*2])
# gamma^a_{b c} is the Christoffel symbol
gamma = S.Half*g(a,d)*(dg(-b,-d,-c) + dg(-c,-b,-d) - dg(-d,-b,-c))
# t = g_{a b; c}
t = dg(-c,-a,-b) - g(-a,-d)*gamma(d,-b,-c) - g(-b,-d)*gamma(d,-a,-c)
t = t.contract_metric(g)
assert t == 0
t = q(c)*p(a)*q(b)
assert t(b,c,d) == q(d)*p(b)*q(c)
def test_TensorManager():
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
LorentzH = TensorIndexType('LorentzH', dummy_fmt='LH')
i, j = tensor_indices('i,j', Lorentz)
ih, jh = tensor_indices('ih,jh', LorentzH)
p, q = tensorhead('p q', [Lorentz], [[1]])
ph, qh = tensorhead('ph qh', [LorentzH], [[1]])
Gsymbol = Symbol('Gsymbol')
GHsymbol = Symbol('GHsymbol')
TensorManager.set_comm(Gsymbol, GHsymbol, 0)
G = tensorhead('G', [Lorentz], [[1]], Gsymbol)
assert TensorManager._comm_i2symbol[G.comm] == Gsymbol
GH = tensorhead('GH', [LorentzH], [[1]], GHsymbol)
ps = G(i)*p(-i)
psh = GH(ih)*ph(-ih)
t = ps + psh
t1 = t*t
assert canon_bp(t1 - ps*ps - 2*ps*psh - psh*psh) == 0
qs = G(i)*q(-i)
qsh = GH(ih)*qh(-ih)
assert _is_equal(ps*qsh, qsh*ps)
assert not _is_equal(ps*qs, qs*ps)
n = TensorManager.comm_symbols2i(Gsymbol)
assert TensorManager.comm_i2symbol(n) == Gsymbol
assert GHsymbol in TensorManager._comm_symbols2i
raises(ValueError, lambda: TensorManager.set_comm(GHsymbol, 1, 2))
TensorManager.set_comms((Gsymbol,GHsymbol,0),(Gsymbol,1,1))
assert TensorManager.get_comm(n, 1) == TensorManager.get_comm(1, n) == 1
TensorManager.clear()
assert TensorManager.comm == [{0:0, 1:0, 2:0}, {0:0, 1:1, 2:None}, {0:0, 1:None}]
assert GHsymbol not in TensorManager._comm_symbols2i
nh = TensorManager.comm_symbols2i(GHsymbol)
assert GHsymbol in TensorManager._comm_symbols2i
def test_hash():
D = Symbol('D')
Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
a,b,c,d,e = tensor_indices('a,b,c,d,e', Lorentz)
g = Lorentz.metric
p, q = tensorhead('p q', [Lorentz], [[1]])
p_type = p.args[1]
t1 = p(a)*q(b)
t2 = p(a)*p(b)
assert hash(t1) != hash(t2)
t3 = p(a)*p(b) + g(a,b)
t4 = p(a)*p(b) - g(a,b)
assert hash(t3) != hash(t4)
assert a.func(*a.args) == a
assert Lorentz.func(*Lorentz.args) == Lorentz
assert g.func(*g.args) == g
assert p.func(*p.args) == p
assert p_type.func(*p_type.args) == p_type
assert p(a).func(*(p(a)).args) == p(a)
assert t1.func(*t1.args) == t1
assert t2.func(*t2.args) == t2
assert t3.func(*t3.args) == t3
assert t4.func(*t4.args) == t4
assert hash(a.func(*a.args)) == hash(a)
assert hash(Lorentz.func(*Lorentz.args)) == hash(Lorentz)
assert hash(g.func(*g.args)) == hash(g)
assert hash(p.func(*p.args)) == hash(p)
assert hash(p_type.func(*p_type.args)) == hash(p_type)
assert hash(p(a).func(*(p(a)).args)) == hash(p(a))
assert hash(t1.func(*t1.args)) == hash(t1)
assert hash(t2.func(*t2.args)) == hash(t2)
assert hash(t3.func(*t3.args)) == hash(t3)
assert hash(t4.func(*t4.args)) == hash(t4)
def check_all(obj):
return all([isinstance(_, Basic) for _ in obj.args])
assert check_all(a)
assert check_all(Lorentz)
assert check_all(g)
assert check_all(p)
assert check_all(p_type)
assert check_all(p(a))
assert check_all(t1)
assert check_all(t2)
assert check_all(t3)
assert check_all(t4)
tsymmetry = tensorsymmetry([2], [1], [1, 1, 1])
assert tsymmetry.func(*tsymmetry.args) == tsymmetry
assert hash(tsymmetry.func(*tsymmetry.args)) == hash(tsymmetry)
assert check_all(tsymmetry)
### TEST VALUED TENSORS ###
def _get_valued_base_test_variables():
minkowski = Matrix((
(1, 0, 0, 0),
(0, -1, 0, 0),
(0, 0, -1, 0),
(0, 0, 0, -1),
))
Lorentz = TensorIndexType('Lorentz', dim=4)
Lorentz.data = minkowski
i0, i1, i2, i3, i4 = tensor_indices('i0:5', Lorentz)
E, px, py, pz = symbols('E px py pz')
A = tensorhead('A', [Lorentz], [[1]])
A.data = [E, px, py, pz]
B = tensorhead('B', [Lorentz], [[1]], 'Gcomm')
B.data = range(4)
AB = tensorhead("AB", [Lorentz] * 2, [[1]]*2)
AB.data = minkowski
ba_matrix = Matrix((
(1, 2, 3, 4),
(5, 6, 7, 8),
(9, 0, -1, -2),
(-3, -4, -5, -6),
))
BA = tensorhead("BA", [Lorentz] * 2, [[1]]*2)
BA.data = ba_matrix
# Let's test the diagonal metric, with inverted Minkowski metric:
LorentzD = TensorIndexType('LorentzD')
LorentzD.data = [-1, 1, 1, 1]
mu0, mu1, mu2 = tensor_indices('mu0:3', LorentzD)
C = tensorhead('C', [LorentzD], [[1]])
C.data = [E, px, py, pz]
### non-diagonal metric ###
ndm_matrix = (
(1, 1, 0,),
(1, 0, 1),
(0, 1, 0,),
)
ndm = TensorIndexType("ndm")
ndm.data = ndm_matrix
n0, n1, n2 = tensor_indices('n0:3', ndm)
NA = tensorhead('NA', [ndm], [[1]])
NA.data = range(10, 13)
NB = tensorhead('NB', [ndm]*2, [[1]]*2)
NB.data = [[i+j for j in range(10, 13)] for i in range(10, 13)]
NC = tensorhead('NC', [ndm]*3, [[1]]*3)
NC.data = [[[i+j+k for k in range(4, 7)] for j in range(1, 4)] for i in range(2, 5)]
return (A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4)
@filter_warnings_decorator
def test_valued_tensor_iter():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
# iteration on VTensorHead
assert list(A) == [E, px, py, pz]
assert list(ba_matrix) == list(BA)
# iteration on VTensMul
assert list(A(i1)) == [E, px, py, pz]
assert list(BA(i1, i2)) == list(ba_matrix)
assert list(3 * BA(i1, i2)) == [3 * i for i in list(ba_matrix)]
assert list(-5 * BA(i1, i2)) == [-5 * i for i in list(ba_matrix)]
# iteration on VTensAdd
# A(i1) + A(i1)
assert list(A(i1) + A(i1)) == [2*E, 2*px, 2*py, 2*pz]
assert BA(i1, i2) - BA(i1, i2) == 0
assert list(BA(i1, i2) - 2 * BA(i1, i2)) == [-i for i in list(ba_matrix)]
@filter_warnings_decorator
def test_valued_tensor_covariant_contravariant_elements():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
assert A(-i0)[0] == A(i0)[0]
assert A(-i0)[1] == -A(i0)[1]
assert AB(i0, i1)[1, 1] == -1
assert AB(i0, -i1)[1, 1] == 1
assert AB(-i0, -i1)[1, 1] == -1
assert AB(-i0, i1)[1, 1] == 1
@filter_warnings_decorator
def test_valued_tensor_get_matrix():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
matab = AB(i0, i1).get_matrix()
assert matab == Matrix([
[1, 0, 0, 0],
[0, -1, 0, 0],
[0, 0, -1, 0],
[0, 0, 0, -1],
])
# when alternating contravariant/covariant with [1, -1, -1, -1] metric
# it becomes the identity matrix:
assert AB(i0, -i1).get_matrix() == eye(4)
# covariant and contravariant forms:
assert A(i0).get_matrix() == Matrix([E, px, py, pz])
assert A(-i0).get_matrix() == Matrix([E, -px, -py, -pz])
@filter_warnings_decorator
def test_valued_tensor_contraction():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
assert (A(i0) * A(-i0)).data == E ** 2 - px ** 2 - py ** 2 - pz ** 2
assert (A(i0) * A(-i0)).data == A ** 2
assert (A(i0) * A(-i0)).data == A(i0) ** 2
assert (A(i0) * B(-i0)).data == -px - 2 * py - 3 * pz
for i in range(4):
for j in range(4):
assert (A(i0) * B(-i1))[i, j] == [E, px, py, pz][i] * [0, -1, -2, -3][j]
# test contraction on the alternative Minkowski metric: [-1, 1, 1, 1]
assert (C(mu0) * C(-mu0)).data == -E ** 2 + px ** 2 + py ** 2 + pz ** 2
contrexp = A(i0) * AB(i1, -i0)
assert A(i0).rank == 1
assert AB(i1, -i0).rank == 2
assert contrexp.rank == 1
for i in range(4):
assert contrexp[i] == [E, px, py, pz][i]
@filter_warnings_decorator
def test_valued_tensor_self_contraction():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
assert AB(i0, -i0).data == 4
assert BA(i0, -i0).data == 2
@filter_warnings_decorator
def test_valued_tensor_pow():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
assert C**2 == -E**2 + px**2 + py**2 + pz**2
assert C**1 == sqrt(-E**2 + px**2 + py**2 + pz**2)
assert C(mu0)**2 == C**2
assert C(mu0)**1 == C**1
@filter_warnings_decorator
def test_valued_tensor_expressions():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
x1, x2, x3 = symbols('x1:4')
# test coefficient in contraction:
rank2coeff = x1 * A(i3) * B(i2)
assert rank2coeff[1, 1] == x1 * px
assert rank2coeff[3, 3] == 3 * pz * x1
coeff_expr = ((x1 * A(i4)) * (B(-i4) / x2)).data
assert coeff_expr.expand() == -px*x1/x2 - 2*py*x1/x2 - 3*pz*x1/x2
add_expr = A(i0) + B(i0)
assert add_expr[0] == E
assert add_expr[1] == px + 1
assert add_expr[2] == py + 2
assert add_expr[3] == pz + 3
sub_expr = A(i0) - B(i0)
assert sub_expr[0] == E
assert sub_expr[1] == px - 1
assert sub_expr[2] == py - 2
assert sub_expr[3] == pz - 3
assert (add_expr * B(-i0)).data == -px - 2*py - 3*pz - 14
expr1 = x1*A(i0) + x2*B(i0)
expr2 = expr1 * B(i1) * (-4)
expr3 = expr2 + 3*x3*AB(i0, i1)
expr4 = expr3 / 2
assert expr4 * 2 == expr3
expr5 = (expr4 * BA(-i1, -i0))
assert expr5.data.expand() == 28*E*x1 + 12*px*x1 + 20*py*x1 + 28*pz*x1 + 136*x2 + 3*x3
@filter_warnings_decorator
def test_valued_tensor_add_scalar():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
# one scalar summand after the contracted tensor
expr1 = A(i0)*A(-i0) - (E**2 - px**2 - py**2 - pz**2)
assert expr1.data == 0
# multiple scalar summands in front of the contracted tensor
expr2 = E**2 - px**2 - py**2 - pz**2 - A(i0)*A(-i0)
assert expr2.data == 0
# multiple scalar summands after the contracted tensor
expr3 = A(i0)*A(-i0) - E**2 + px**2 + py**2 + pz**2
assert expr3.data == 0
# multiple scalar summands and multiple tensors
expr4 = C(mu0)*C(-mu0) + 2*E**2 - 2*px**2 - 2*py**2 - 2*pz**2 - A(i0)*A(-i0)
assert expr4.data == 0
@filter_warnings_decorator
def test_noncommuting_components():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
euclid = TensorIndexType('Euclidean')
euclid.data = [1, 1]
i1, i2, i3 = tensor_indices('i1:4', euclid)
a, b, c, d = symbols('a b c d', commutative=False)
V1 = tensorhead('V1', [euclid] * 2, [[1]]*2)
V1.data = [[a, b], (c, d)]
V2 = tensorhead('V2', [euclid] * 2, [[1]]*2)
V2.data = [[a, c], [b, d]]
vtp = V1(i1, i2) * V2(-i2, -i1)
assert vtp.data == a**2 + b**2 + c**2 + d**2
assert vtp.data != a**2 + 2*b*c + d**2
vtp2 = V1(i1, i2)*V1(-i2, -i1)
assert vtp2.data == a**2 + b*c + c*b + d**2
assert vtp2.data != a**2 + 2*b*c + d**2
Vc = (b * V1(i1, -i1)).data
assert Vc.expand() == b * a + b * d
@filter_warnings_decorator
def test_valued_non_diagonal_metric():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
mmatrix = Matrix(ndm_matrix)
assert (NA(n0)*NA(-n0)).data == (NA(n0).get_matrix().T * mmatrix * NA(n0).get_matrix())[0, 0]
@filter_warnings_decorator
def test_valued_assign_numpy_ndarray():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
# this is needed to make sure that a numpy.ndarray can be assigned to a
# tensor.
arr = [E+1, px-1, py, pz]
A.data = Array(arr)
for i in range(4):
assert A(i0).data[i] == arr[i]
qx, qy, qz = symbols('qx qy qz')
A(-i0).data = Array([E, qx, qy, qz])
for i in range(4):
assert A(i0).data[i] == [E, -qx, -qy, -qz][i]
assert A.data[i] == [E, -qx, -qy, -qz][i]
# test on multi-indexed tensors.
random_4x4_data = [[(i**3-3*i**2)%(j+7) for i in range(4)] for j in range(4)]
AB(-i0, -i1).data = random_4x4_data
for i in range(4):
for j in range(4):
assert AB(i0, i1).data[i, j] == random_4x4_data[i][j]*(-1 if i else 1)*(-1 if j else 1)
assert AB(-i0, i1).data[i, j] == random_4x4_data[i][j]*(-1 if j else 1)
assert AB(i0, -i1).data[i, j] == random_4x4_data[i][j]*(-1 if i else 1)
assert AB(-i0, -i1).data[i, j] == random_4x4_data[i][j]
AB(-i0, i1).data = random_4x4_data
for i in range(4):
for j in range(4):
assert AB(i0, i1).data[i, j] == random_4x4_data[i][j]*(-1 if i else 1)
assert AB(-i0, i1).data[i, j] == random_4x4_data[i][j]
assert AB(i0, -i1).data[i, j] == random_4x4_data[i][j]*(-1 if i else 1)*(-1 if j else 1)
assert AB(-i0, -i1).data[i, j] == random_4x4_data[i][j]*(-1 if j else 1)
@filter_warnings_decorator
def test_valued_metric_inverse():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
# let's assign some fancy matrix, just to verify it:
# (this has no physical sense, it's just testing sympy);
# it is symmetrical:
md = [[2, 2, 2, 1], [2, 3, 1, 0], [2, 1, 2, 3], [1, 0, 3, 2]]
Lorentz.data = md
m = Matrix(md)
metric = Lorentz.metric
minv = m.inv()
meye = eye(4)
# the Kronecker Delta:
KD = Lorentz.get_kronecker_delta()
for i in range(4):
for j in range(4):
assert metric(i0, i1).data[i, j] == m[i, j]
assert metric(-i0, -i1).data[i, j] == minv[i, j]
assert metric(i0, -i1).data[i, j] == meye[i, j]
assert metric(-i0, i1).data[i, j] == meye[i, j]
assert metric(i0, i1)[i, j] == m[i, j]
assert metric(-i0, -i1)[i, j] == minv[i, j]
assert metric(i0, -i1)[i, j] == meye[i, j]
assert metric(-i0, i1)[i, j] == meye[i, j]
assert KD(i0, -i1)[i, j] == meye[i, j]
@filter_warnings_decorator
def test_valued_canon_bp_swapaxes():
(A, B, AB, BA, C, Lorentz, E, px, py, pz, LorentzD, mu0, mu1, mu2, ndm, n0, n1,
n2, NA, NB, NC, minkowski, ba_matrix, ndm_matrix, i0, i1, i2, i3, i4) = _get_valued_base_test_variables()
e1 = A(i1)*A(i0)
e2 = e1.canon_bp()
assert e2 == A(i0)*A(i1)
for i in range(4):
for j in range(4):
assert e1[i, j] == e2[j, i]
o1 = B(i2)*A(i1)*B(i0)
o2 = o1.canon_bp()
for i in range(4):
for j in range(4):
for k in range(4):
assert o1[i, j, k] == o2[j, i, k]
def test_pprint():
Lorentz = TensorIndexType('Lorentz')
A = tensorhead('A', [Lorentz], [[1]])
assert pretty(A) == "A(Lorentz)"
@filter_warnings_decorator
def test_valued_components_with_wrong_symmetry():
IT = TensorIndexType('IT', dim=3)
i0, i1, i2, i3 = tensor_indices('i0:4', IT)
IT.data = [1, 1, 1]
A_nosym = tensorhead('A', [IT]*2, [[1]]*2)
A_sym = tensorhead('A', [IT]*2, [[1]*2])
A_antisym = tensorhead('A', [IT]*2, [[2]])
mat_nosym = Matrix([[1,2,3],[4,5,6],[7,8,9]])
mat_sym = mat_nosym + mat_nosym.T
mat_antisym = mat_nosym - mat_nosym.T
A_nosym.data = mat_nosym
A_nosym.data = mat_sym
A_nosym.data = mat_antisym
def assign(A, dat):
A.data = dat
A_sym.data = mat_sym
raises(ValueError, lambda: assign(A_sym, mat_nosym))
raises(ValueError, lambda: assign(A_sym, mat_antisym))
A_antisym.data = mat_antisym
raises(ValueError, lambda: assign(A_antisym, mat_sym))
raises(ValueError, lambda: assign(A_antisym, mat_nosym))
A_sym.data = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]
A_antisym.data = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]
@filter_warnings_decorator
def test_issue_10972_TensMul_data():
Lorentz = TensorIndexType('Lorentz', metric=False, dummy_fmt='i', dim=2)
Lorentz.data = [-1, 1]
mu, nu, alpha, beta = tensor_indices('\\mu, \\nu, \\alpha, \\beta',
Lorentz)
Vec = TensorType([Lorentz], tensorsymmetry([1]))
A2 = TensorType([Lorentz] * 2, tensorsymmetry([2]))
u = Vec('u')
u.data = [1, 0]
F = A2('F')
F.data = [[0, 1],
[-1, 0]]
mul_1 = F(mu, alpha) * u(-alpha) * F(nu, beta) * u(-beta)
assert (mul_1.data == Array([[0, 0], [0, 1]]))
mul_2 = F(mu, alpha) * F(nu, beta) * u(-alpha) * u(-beta)
assert (mul_2.data == mul_1.data)
assert ((mul_1 + mul_1).data == 2 * mul_1.data)
@filter_warnings_decorator
def test_TensMul_data():
Lorentz = TensorIndexType('Lorentz', metric=False, dummy_fmt='L', dim=4)
Lorentz.data = [-1, 1, 1, 1]
mu, nu, alpha, beta = tensor_indices('\\mu, \\nu, \\alpha, \\beta',
Lorentz)
Vec = TensorType([Lorentz], tensorsymmetry([1]))
A2 = TensorType([Lorentz] * 2, tensorsymmetry([2]))
u = Vec('u')
u.data = [1, 0, 0, 0]
F = A2('F')
Ex, Ey, Ez, Bx, By, Bz = symbols('E_x E_y E_z B_x B_y B_z')
F.data = [
[0, Ex, Ey, Ez],
[-Ex, 0, Bz, -By],
[-Ey, -Bz, 0, Bx],
[-Ez, By, -Bx, 0]]
E = F(mu, nu) * u(-nu)
assert ((E(mu) * E(nu)).data ==
Array([[0, 0, 0, 0],
[0, Ex ** 2, Ex * Ey, Ex * Ez],
[0, Ex * Ey, Ey ** 2, Ey * Ez],
[0, Ex * Ez, Ey * Ez, Ez ** 2]])
)
assert ((E(mu) * E(nu)).canon_bp().data == (E(mu) * E(nu)).data)
assert ((F(mu, alpha) * F(beta, nu) * u(-alpha) * u(-beta)).data ==
- (E(mu) * E(nu)).data
)
assert ((F(alpha, mu) * F(beta, nu) * u(-alpha) * u(-beta)).data ==
(E(mu) * E(nu)).data
)
S2 = TensorType([Lorentz] * 2, tensorsymmetry([1] * 2))
g = S2('g')
g.data = Lorentz.data
# tensor 'perp' is orthogonal to vector 'u'
perp = u(mu) * u(nu) + g(mu, nu)
mul_1 = u(-mu) * perp(mu, nu)
assert (mul_1.data == Array([0, 0, 0, 0]))
mul_2 = u(-mu) * perp(mu, alpha) * perp(nu, beta)
assert (mul_2.data == Array.zeros(4, 4, 4))
Fperp = perp(mu, alpha) * perp(nu, beta) * F(-alpha, -beta)
assert (Fperp.data[0, :] == Array([0, 0, 0, 0]))
assert (Fperp.data[:, 0] == Array([0, 0, 0, 0]))
mul_3 = u(-mu) * Fperp(mu, nu)
assert (mul_3.data == Array([0, 0, 0, 0]))
@filter_warnings_decorator
def test_issue_11020_TensAdd_data():
Lorentz = TensorIndexType('Lorentz', metric=False, dummy_fmt='i', dim=2)
Lorentz.data = [-1, 1]
a, b, c, d = tensor_indices('a, b, c, d', Lorentz)
i0, i1 = tensor_indices('i_0:2', Lorentz)
Vec = TensorType([Lorentz], tensorsymmetry([1]))
S2 = TensorType([Lorentz] * 2, tensorsymmetry([1] * 2))
# metric tensor
g = S2('g')
g.data = Lorentz.data
u = Vec('u')
u.data = [1, 0]
add_1 = g(b, c) * g(d, i0) * u(-i0) - g(b, c) * u(d)
assert (add_1.data == Array.zeros(2, 2, 2))
# Now let us replace index `d` with `a`:
add_2 = g(b, c) * g(a, i0) * u(-i0) - g(b, c) * u(a)
assert (add_2.data == Array.zeros(2, 2, 2))
# some more tests
# perp is tensor orthogonal to u^\mu
perp = u(a) * u(b) + g(a, b)
mul_1 = u(-a) * perp(a, b)
assert (mul_1.data == Array([0, 0]))
mul_2 = u(-c) * perp(c, a) * perp(d, b)
assert (mul_2.data == Array.zeros(2, 2, 2))
def test_index_iteration():
L = TensorIndexType("Lorentz", dummy_fmt="L")
i0,i1,i2,i3,i4 = tensor_indices('i0:5', L)
L0 = tensor_indices('L_0', L)
L1 = tensor_indices('L_1', L)
A = tensorhead("A", [L, L], [[1], [1]])
B = tensorhead("B", [L, L], [[1, 1]])
C = tensorhead("C", [L], [[1]])
e1 = A(i0, i2)
e2 = A(i0, -i0)
e3 = A(i0, i1)*B(i2, i3)
e4 = A(i0, i1)*B(i2, -i1)
e5 = A(i0, i1)*B(-i0, -i1)
e6 = e1 + e4
assert list(e1._iterate_free_indices) == [(i0, (1, 0)), (i2, (1, 1))]
assert list(e1._iterate_dummy_indices) == []
assert list(e1._iterate_indices) == [(i0, (1, 0)), (i2, (1, 1))]
assert list(e2._iterate_free_indices) == []
assert list(e2._iterate_dummy_indices) == [(L0, (1, 0)), (-L0, (1, 1))]
assert list(e2._iterate_indices) == [(L0, (1, 0)), (-L0, (1, 1))]
assert list(e3._iterate_free_indices) == [(i0, (0, 1, 0)), (i1, (0, 1, 1)), (i2, (1, 1, 0)), (i3, (1, 1, 1))]
assert list(e3._iterate_dummy_indices) == []
assert list(e3._iterate_indices) == [(i0, (0, 1, 0)), (i1, (0, 1, 1)), (i2, (1, 1, 0)), (i3, (1, 1, 1))]
assert list(e4._iterate_free_indices) == [(i0, (0, 1, 0)), (i2, (1, 1, 0))]
assert list(e4._iterate_dummy_indices) == [(L0, (0, 1, 1)), (-L0, (1, 1, 1))]
assert list(e4._iterate_indices) == [(i0, (0, 1, 0)), (L0, (0, 1, 1)), (i2, (1, 1, 0)), (-L0, (1, 1, 1))]
assert list(e5._iterate_free_indices) == []
assert list(e5._iterate_dummy_indices) == [(L0, (0, 1, 0)), (L1, (0, 1, 1)), (-L0, (1, 1, 0)), (-L1, (1, 1, 1))]
assert list(e5._iterate_indices) == [(L0, (0, 1, 0)), (L1, (0, 1, 1)), (-L0, (1, 1, 0)), (-L1, (1, 1, 1))]
assert list(e6._iterate_free_indices) == [(i0, (0, 1, 0)), (i2, (0, 1, 1)), (i0, (1, 0, 1, 0)), (i2, (1, 1, 1, 0))]
assert list(e6._iterate_dummy_indices) == [(L0, (1, 0, 1, 1)), (-L0, (1, 1, 1, 1))]
assert list(e6._iterate_indices) == [(i0, (0, 1, 0)), (i2, (0, 1, 1)), (i0, (1, 0, 1, 0)), (L0, (1, 0, 1, 1)), (i2, (1, 1, 1, 0)), (-L0, (1, 1, 1, 1))]
assert e1.get_indices() == [i0, i2]
assert e1.get_free_indices() == [i0, i2]
assert e2.get_indices() == [L0, -L0]
assert e2.get_free_indices() == []
assert e3.get_indices() == [i0, i1, i2, i3]
assert e3.get_free_indices() == [i0, i1, i2, i3]
assert e4.get_indices() == [i0, L0, i2, -L0]
assert e4.get_free_indices() == [i0, i2]
assert e5.get_indices() == [L0, L1, -L0, -L1]
assert e5.get_free_indices() == []
def test_tensor_expand():
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
i0 = tensor_indices("i0", L)
L_0 = TensorIndex("L_0", L)
L_1 = TensorIndex("L_1", L)
A, B, C, D = tensorhead("A B C D", [L], [[1]])
H = tensorhead("H", [L, L], [[1], [1]])
assert isinstance(Add(A(i), B(i)), TensAdd)
assert isinstance(expand(A(i)+B(i)), TensAdd)
expr = A(i)*(A(-i)+B(-i))
assert expr.args == (A(L_0), A(-L_0) + B(-L_0))
assert expr != A(i)*A(-i) + A(i)*B(-i)
assert expr.expand() == A(i)*A(-i) + A(i)*B(-i)
assert str(expr) == "A(L_0)*(A(-L_0) + B(-L_0))"
expr = A(i)*A(j) + A(i)*B(j)
assert str(expr) == "A(i)*A(j) + A(i)*B(j)"
expr = A(-i)*(A(i)*A(j) + A(i)*B(j)*C(k)*C(-k))
assert expr != A(-i)*A(i)*A(j) + A(-i)*A(i)*B(j)*C(k)*C(-k)
assert expr.expand() == A(-i)*A(i)*A(j) + A(-i)*A(i)*B(j)*C(k)*C(-k)
assert str(expr) == "A(-L_0)*(A(L_0)*A(j) + A(L_0)*B(j)*C(L_1)*C(-L_1))"
assert str(expr.canon_bp()) == 'A(L_0)*A(-L_0)*B(j)*C(L_1)*C(-L_1) + A(j)*A(L_0)*A(-L_0)'
expr = A(-i)*(2*A(i)*A(j) + A(i)*B(j))
assert expr.expand() == 2*A(-i)*A(i)*A(j) + A(-i)*A(i)*B(j)
expr = 2*A(i)*A(-i)
assert expr.coeff == 2
expr = A(i)*(B(j)*C(k) + C(j)*(A(k) + D(k)))
assert str(expr) == "A(i)*(B(j)*C(k) + C(j)*(A(k) + D(k)))"
assert str(expr.expand()) == "A(i)*B(j)*C(k) + A(i)*C(j)*A(k) + A(i)*C(j)*D(k)"
assert isinstance(TensMul(3), TensMul)
tm = TensMul(3).doit()
assert tm == 3
assert isinstance(tm, Integer)
p1 = B(j)*B(-j) + B(j)*C(-j)
p2 = C(-i)*p1
p3 = A(i)*p2
expr = A(i)*(B(-i) + C(-i)*(B(j)*B(-j) + B(j)*C(-j)))
assert expr.expand() == A(i)*B(-i) + A(i)*C(-i)*B(j)*B(-j) + A(i)*C(-i)*B(j)*C(-j)
expr = C(-i)*(B(j)*B(-j) + B(j)*C(-j))
assert expr.expand() == C(-i)*B(j)*B(-j) + C(-i)*B(j)*C(-j)
def test_tensor_alternative_construction():
L = TensorIndexType("L")
i0, i1, i2, i3 = tensor_indices('i0:4', L)
A = tensorhead("A", [L], [[1]])
x, y = symbols("x y")
assert A(i0) == A(Symbol("i0"))
assert A(-i0) == A(-Symbol("i0"))
raises(TypeError, lambda: A(x+y))
raises(ValueError, lambda: A(2*x))
def test_tensor_replacement():
L = TensorIndexType("L")
L2 = TensorIndexType("L2", dim=2)
i, j, k, l = tensor_indices("i j k l", L)
i0 = tensor_indices("i0", L)
A, B, C, D = tensorhead("A B C D", [L], [[1]])
H = tensorhead("H", [L, L], [[1], [1]])
K = tensorhead("K", [L, L, L, L], [[1], [1], [1], [1]])
expr = H(i, j)
repl = {H(i,-j): [[1,2],[3,4]], L: diag(1, -1)}
assert expr._extract_data(repl) == ([i, j], Array([[1, -2], [3, -4]]))
assert expr.replace_with_arrays(repl) == Array([[1, -2], [3, -4]])
assert expr.replace_with_arrays(repl, [i, j]) == Array([[1, -2], [3, -4]])
assert expr.replace_with_arrays(repl, [i, -j]) == Array([[1, 2], [3, 4]])
assert expr.replace_with_arrays(repl, [-i, j]) == Array([[1, -2], [-3, 4]])
assert expr.replace_with_arrays(repl, [-i, -j]) == Array([[1, 2], [-3, -4]])
assert expr.replace_with_arrays(repl, [j, i]) == Array([[1, 3], [-2, -4]])
assert expr.replace_with_arrays(repl, [j, -i]) == Array([[1, -3], [-2, 4]])
assert expr.replace_with_arrays(repl, [-j, i]) == Array([[1, 3], [2, 4]])
assert expr.replace_with_arrays(repl, [-j, -i]) == Array([[1, -3], [2, -4]])
# Test stability of optional parameter 'indices'
assert expr.replace_with_arrays(repl) == Array([[1, -2], [3, -4]])
expr = H(i,j)
repl = {H(i,j): [[1,2],[3,4]], L: diag(1, -1)}
assert expr._extract_data(repl) == ([i, j], Array([[1, 2], [3, 4]]))
assert expr.replace_with_arrays(repl) == Array([[1, 2], [3, 4]])
assert expr.replace_with_arrays(repl, [i, j]) == Array([[1, 2], [3, 4]])
assert expr.replace_with_arrays(repl, [i, -j]) == Array([[1, -2], [3, -4]])
assert expr.replace_with_arrays(repl, [-i, j]) == Array([[1, 2], [-3, -4]])
assert expr.replace_with_arrays(repl, [-i, -j]) == Array([[1, -2], [-3, 4]])
assert expr.replace_with_arrays(repl, [j, i]) == Array([[1, 3], [2, 4]])
assert expr.replace_with_arrays(repl, [j, -i]) == Array([[1, -3], [2, -4]])
assert expr.replace_with_arrays(repl, [-j, i]) == Array([[1, 3], [-2, -4]])
assert expr.replace_with_arrays(repl, [-j, -i]) == Array([[1, -3], [-2, 4]])
# Not the same indices:
expr = H(i,k)
repl = {H(i,j): [[1,2],[3,4]], L: diag(1, -1)}
assert expr._extract_data(repl) == ([i, k], Array([[1, 2], [3, 4]]))
expr = A(i)*A(-i)
repl = {A(i): [1,2], L: diag(1, -1)}
assert expr._extract_data(repl) == ([], -3)
assert expr.replace_with_arrays(repl, []) == -3
expr = K(i, j, -j, k)*A(-i)*A(-k)
repl = {A(i): [1, 2], K(i,j,k,l): Array([1]*2**4).reshape(2,2,2,2), L: diag(1, -1)}
assert expr._extract_data(repl)
expr = H(j, k)
repl = {H(i,j): [[1,2],[3,4]], L: diag(1, -1)}
raises(ValueError, lambda: expr._extract_data(repl))
expr = A(i)
repl = {B(i): [1, 2]}
raises(ValueError, lambda: expr._extract_data(repl))
expr = A(i)
repl = {A(i): [[1, 2], [3, 4]]}
raises(ValueError, lambda: expr._extract_data(repl))
# TensAdd:
expr = A(k)*H(i, j) + B(k)*H(i, j)
repl = {A(k): [1], B(k): [1], H(i, j): [[1, 2],[3,4]], L:diag(1,1)}
assert expr._extract_data(repl) == ([k, i, j], Array([[[2, 4], [6, 8]]]))
assert expr.replace_with_arrays(repl, [k, i, j]) == Array([[[2, 4], [6, 8]]])
assert expr.replace_with_arrays(repl, [k, j, i]) == Array([[[2, 6], [4, 8]]])
expr = A(k)*A(-k) + 100
repl = {A(k): [2, 3], L: diag(1, 1)}
assert expr.replace_with_arrays(repl, []) == 113
## Symmetrization:
expr = H(i, j) + H(j, i)
repl = {H(i, j): [[1, 2], [3, 4]]}
assert expr._extract_data(repl) == ([i, j], Array([[2, 5], [5, 8]]))
assert expr.replace_with_arrays(repl, [i, j]) == Array([[2, 5], [5, 8]])
assert expr.replace_with_arrays(repl, [j, i]) == Array([[2, 5], [5, 8]])
## Anti-symmetrization:
expr = H(i, j) - H(j, i)
repl = {H(i, j): [[1, 2], [3, 4]]}
assert expr.replace_with_arrays(repl, [i, j]) == Array([[0, -1], [1, 0]])
assert expr.replace_with_arrays(repl, [j, i]) == Array([[0, 1], [-1, 0]])
# Tensors with contractions in replacements:
expr = K(i, j, k, -k)
repl = {K(i, j, k, -k): [[1, 2], [3, 4]]}
assert expr._extract_data(repl) == ([i, j], Array([[1, 2], [3, 4]]))
expr = H(i, -i)
repl = {H(i, -i): 42}
assert expr._extract_data(repl) == ([], 42)
# Replace with array, raise exception if indices are not compatible:
expr = A(i)*A(j)
repl = {A(i): [1, 2]}
raises(ValueError, lambda: expr.replace_with_arrays(repl, [j]))
# Raise exception if array dimension is not compatible:
expr = A(i)
repl = {A(i): [[1, 2]]}
raises(ValueError, lambda: expr.replace_with_arrays(repl, [i]))
# TensorIndexType with dimension, wrong dimension in replacement array:
u1, u2, u3 = tensor_indices("u1:4", L2)
U = tensorhead("U", [L2], [[1]])
expr = U(u1)*U(-u2)
repl = {U(u1): [[1]]}
raises(ValueError, lambda: expr.replace_with_arrays(repl, [u1, -u2]))
def test_rewrite_tensor_to_Indexed():
L = TensorIndexType("L", dim=4)
A = tensorhead("A", [L, L, L, L], [[1], [1], [1], [1]])
B = tensorhead("B", [L], [[1]])
i0, i1, i2, i3 = symbols("i0:4")
L_0, L_1 = symbols("L_0:2")
a1 = A(i0, i1, i2, i3)
assert a1.rewrite(Indexed) == Indexed(Symbol("A"), i0, i1, i2, i3)
a2 = A(i0, -i0, i2, i3)
assert a2.rewrite(Indexed) == Sum(Indexed(Symbol("A"), L_0, L_0, i2, i3), (L_0, 0, 3))
a3 = a2 + A(i2, i3, i0, -i0)
assert a3.rewrite(Indexed) == \
Sum(Indexed(Symbol("A"), L_0, L_0, i2, i3), (L_0, 0, 3)) +\
Sum(Indexed(Symbol("A"), i2, i3, L_0, L_0), (L_0, 0, 3))
b1 = B(-i0)*a1
assert b1.rewrite(Indexed) == Sum(Indexed(Symbol("B"), L_0)*Indexed(Symbol("A"), L_0, i1, i2, i3), (L_0, 0, 3))
b2 = B(-i3)*a2
assert b2.rewrite(Indexed) == Sum(Indexed(Symbol("B"), L_1)*Indexed(Symbol("A"), L_0, L_0, i2, L_1), (L_0, 0, 3), (L_1, 0, 3))
|
422e0695ff6bc4c77b346f1cdfe657e78095a9ddb194633cd61e007de6feacf7 | import collections
import random
from sympy.assumptions import Q
from sympy.core.add import Add
from sympy.core.compatibility import range
from sympy.core.function import (Function, diff)
from sympy.core.numbers import (E, Float, I, Integer, oo, pi)
from sympy.core.relational import (Eq, Lt)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, sin, tan)
from sympy.logic.boolalg import (And, Or)
from sympy.matrices.common import (ShapeError, MatrixError, NonSquareMatrixError,
_MinimalMatrix, MatrixShaping, MatrixProperties, MatrixOperations, MatrixArithmetic,
MatrixSpecial)
from sympy.matrices.matrices import (MatrixDeterminant,
MatrixReductions, MatrixSubspaces, MatrixEigen, MatrixCalculus)
from sympy.matrices import (Matrix, diag, eye,
matrix_multiply_elementwise, ones, zeros, SparseMatrix)
from sympy.polys.polytools import Poly
from sympy.simplify.simplify import simplify
from sympy.simplify.trigsimp import trigsimp
from sympy.utilities.iterables import flatten
from sympy.utilities.pytest import (raises, XFAIL, slow, skip,
warns_deprecated_sympy)
from sympy.abc import a, b, c, d, x, y, z
# classes to test the basic matrix classes
class ShapingOnlyMatrix(_MinimalMatrix, MatrixShaping):
pass
def eye_Shaping(n):
return ShapingOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Shaping(n):
return ShapingOnlyMatrix(n, n, lambda i, j: 0)
class PropertiesOnlyMatrix(_MinimalMatrix, MatrixProperties):
pass
def eye_Properties(n):
return PropertiesOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Properties(n):
return PropertiesOnlyMatrix(n, n, lambda i, j: 0)
class OperationsOnlyMatrix(_MinimalMatrix, MatrixOperations):
pass
def eye_Operations(n):
return OperationsOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Operations(n):
return OperationsOnlyMatrix(n, n, lambda i, j: 0)
class ArithmeticOnlyMatrix(_MinimalMatrix, MatrixArithmetic):
pass
def eye_Arithmetic(n):
return ArithmeticOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Arithmetic(n):
return ArithmeticOnlyMatrix(n, n, lambda i, j: 0)
class DeterminantOnlyMatrix(_MinimalMatrix, MatrixDeterminant):
pass
def eye_Determinant(n):
return DeterminantOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Determinant(n):
return DeterminantOnlyMatrix(n, n, lambda i, j: 0)
class ReductionsOnlyMatrix(_MinimalMatrix, MatrixReductions):
pass
def eye_Reductions(n):
return ReductionsOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Reductions(n):
return ReductionsOnlyMatrix(n, n, lambda i, j: 0)
class SpecialOnlyMatrix(_MinimalMatrix, MatrixSpecial):
pass
class SubspaceOnlyMatrix(_MinimalMatrix, MatrixSubspaces):
pass
class EigenOnlyMatrix(_MinimalMatrix, MatrixEigen):
pass
class CalculusOnlyMatrix(_MinimalMatrix, MatrixCalculus):
pass
def test__MinimalMatrix():
x = _MinimalMatrix(2, 3, [1, 2, 3, 4, 5, 6])
assert x.rows == 2
assert x.cols == 3
assert x[2] == 3
assert x[1, 1] == 5
assert list(x) == [1, 2, 3, 4, 5, 6]
assert list(x[1, :]) == [4, 5, 6]
assert list(x[:, 1]) == [2, 5]
assert list(x[:, :]) == list(x)
assert x[:, :] == x
assert _MinimalMatrix(x) == x
assert _MinimalMatrix([[1, 2, 3], [4, 5, 6]]) == x
assert _MinimalMatrix(([1, 2, 3], [4, 5, 6])) == x
assert _MinimalMatrix([(1, 2, 3), (4, 5, 6)]) == x
assert _MinimalMatrix(((1, 2, 3), (4, 5, 6))) == x
assert not (_MinimalMatrix([[1, 2], [3, 4], [5, 6]]) == x)
# ShapingOnlyMatrix tests
def test_vec():
m = ShapingOnlyMatrix(2, 2, [1, 3, 2, 4])
m_vec = m.vec()
assert m_vec.cols == 1
for i in range(4):
assert m_vec[i] == i + 1
def test_tolist():
lst = [[S.One, S.Half, x*y, S.Zero], [x, y, z, x**2], [y, -S.One, z*x, 3]]
flat_lst = [S.One, S.Half, x*y, S.Zero, x, y, z, x**2, y, -S.One, z*x, 3]
m = ShapingOnlyMatrix(3, 4, flat_lst)
assert m.tolist() == lst
def test_row_col_del():
e = ShapingOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
raises(ValueError, lambda: e.row_del(5))
raises(ValueError, lambda: e.row_del(-5))
raises(ValueError, lambda: e.col_del(5))
raises(ValueError, lambda: e.col_del(-5))
assert e.row_del(2) == e.row_del(-1) == Matrix([[1, 2, 3], [4, 5, 6]])
assert e.col_del(2) == e.col_del(-1) == Matrix([[1, 2], [4, 5], [7, 8]])
assert e.row_del(1) == e.row_del(-2) == Matrix([[1, 2, 3], [7, 8, 9]])
assert e.col_del(1) == e.col_del(-2) == Matrix([[1, 3], [4, 6], [7, 9]])
def test_get_diag_blocks1():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert a.get_diag_blocks() == [a]
assert b.get_diag_blocks() == [b]
assert c.get_diag_blocks() == [c]
def test_get_diag_blocks2():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
A, B, C, D = diag(a, b, b), diag(a, b, c), diag(a, c, b), diag(c, c, b)
A = ShapingOnlyMatrix(A.rows, A.cols, A)
B = ShapingOnlyMatrix(B.rows, B.cols, B)
C = ShapingOnlyMatrix(C.rows, C.cols, C)
D = ShapingOnlyMatrix(D.rows, D.cols, D)
assert A.get_diag_blocks() == [a, b, b]
assert B.get_diag_blocks() == [a, b, c]
assert C.get_diag_blocks() == [a, c, b]
assert D.get_diag_blocks() == [c, c, b]
def test_shape():
m = ShapingOnlyMatrix(1, 2, [0, 0])
m.shape == (1, 2)
def test_reshape():
m0 = eye_Shaping(3)
assert m0.reshape(1, 9) == Matrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = ShapingOnlyMatrix(3, 4, lambda i, j: i + j)
assert m1.reshape(
4, 3) == Matrix(((0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)))
assert m1.reshape(2, 6) == Matrix(((0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)))
def test_row_col():
m = ShapingOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert m.row(0) == Matrix(1, 3, [1, 2, 3])
assert m.col(0) == Matrix(3, 1, [1, 4, 7])
def test_row_join():
assert eye_Shaping(3).row_join(Matrix([7, 7, 7])) == \
Matrix([[1, 0, 0, 7],
[0, 1, 0, 7],
[0, 0, 1, 7]])
def test_col_join():
assert eye_Shaping(3).col_join(Matrix([[7, 7, 7]])) == \
Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[7, 7, 7]])
def test_row_insert():
r4 = Matrix([[4, 4, 4]])
for i in range(-4, 5):
l = [1, 0, 0]
l.insert(i, 4)
assert flatten(eye_Shaping(3).row_insert(i, r4).col(0).tolist()) == l
def test_col_insert():
c4 = Matrix([4, 4, 4])
for i in range(-4, 5):
l = [0, 0, 0]
l.insert(i, 4)
assert flatten(zeros_Shaping(3).col_insert(i, c4).row(0).tolist()) == l
# issue 13643
assert eye_Shaping(6).col_insert(3, Matrix([[2, 2], [2, 2], [2, 2], [2, 2], [2, 2], [2, 2]])) == \
Matrix([[1, 0, 0, 2, 2, 0, 0, 0],
[0, 1, 0, 2, 2, 0, 0, 0],
[0, 0, 1, 2, 2, 0, 0, 0],
[0, 0, 0, 2, 2, 1, 0, 0],
[0, 0, 0, 2, 2, 0, 1, 0],
[0, 0, 0, 2, 2, 0, 0, 1]])
def test_extract():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
assert m.extract([0, 1, 3], [0, 1]) == Matrix(3, 2, [0, 1, 3, 4, 9, 10])
assert m.extract([0, 3], [0, 0, 2]) == Matrix(2, 3, [0, 0, 2, 9, 9, 11])
assert m.extract(range(4), range(3)) == m
raises(IndexError, lambda: m.extract([4], [0]))
raises(IndexError, lambda: m.extract([0], [3]))
def test_hstack():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
m2 = ShapingOnlyMatrix(3, 4, lambda i, j: i*3 + j)
assert m == m.hstack(m)
assert m.hstack(m, m, m) == ShapingOnlyMatrix.hstack(m, m, m) == Matrix([
[0, 1, 2, 0, 1, 2, 0, 1, 2],
[3, 4, 5, 3, 4, 5, 3, 4, 5],
[6, 7, 8, 6, 7, 8, 6, 7, 8],
[9, 10, 11, 9, 10, 11, 9, 10, 11]])
raises(ShapeError, lambda: m.hstack(m, m2))
assert Matrix.hstack() == Matrix()
# test regression #12938
M1 = Matrix.zeros(0, 0)
M2 = Matrix.zeros(0, 1)
M3 = Matrix.zeros(0, 2)
M4 = Matrix.zeros(0, 3)
m = ShapingOnlyMatrix.hstack(M1, M2, M3, M4)
assert m.rows == 0 and m.cols == 6
def test_vstack():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
m2 = ShapingOnlyMatrix(3, 4, lambda i, j: i*3 + j)
assert m == m.vstack(m)
assert m.vstack(m, m, m) == ShapingOnlyMatrix.vstack(m, m, m) == Matrix([
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11],
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11],
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11]])
raises(ShapeError, lambda: m.vstack(m, m2))
assert Matrix.vstack() == Matrix()
# PropertiesOnlyMatrix tests
def test_atoms():
m = PropertiesOnlyMatrix(2, 2, [1, 2, x, 1 - 1/x])
assert m.atoms() == {S(1), S(2), S(-1), x}
assert m.atoms(Symbol) == {x}
def test_free_symbols():
assert PropertiesOnlyMatrix([[x], [0]]).free_symbols == {x}
def test_has():
A = PropertiesOnlyMatrix(((x, y), (2, 3)))
assert A.has(x)
assert not A.has(z)
assert A.has(Symbol)
A = PropertiesOnlyMatrix(((2, y), (2, 3)))
assert not A.has(x)
def test_is_anti_symmetric():
x = symbols('x')
assert PropertiesOnlyMatrix(2, 1, [1, 2]).is_anti_symmetric() is False
m = PropertiesOnlyMatrix(3, 3, [0, x**2 + 2*x + 1, y, -(x + 1)**2, 0, x*y, -y, -x*y, 0])
assert m.is_anti_symmetric() is True
assert m.is_anti_symmetric(simplify=False) is False
assert m.is_anti_symmetric(simplify=lambda x: x) is False
m = PropertiesOnlyMatrix(3, 3, [x.expand() for x in m])
assert m.is_anti_symmetric(simplify=False) is True
m = PropertiesOnlyMatrix(3, 3, [x.expand() for x in [S.One] + list(m)[1:]])
assert m.is_anti_symmetric() is False
def test_diagonal_symmetrical():
m = PropertiesOnlyMatrix(2, 2, [0, 1, 1, 0])
assert not m.is_diagonal()
assert m.is_symmetric()
assert m.is_symmetric(simplify=False)
m = PropertiesOnlyMatrix(2, 2, [1, 0, 0, 1])
assert m.is_diagonal()
m = PropertiesOnlyMatrix(3, 3, diag(1, 2, 3))
assert m.is_diagonal()
assert m.is_symmetric()
m = PropertiesOnlyMatrix(3, 3, [1, 0, 0, 0, 2, 0, 0, 0, 3])
assert m == diag(1, 2, 3)
m = PropertiesOnlyMatrix(2, 3, zeros(2, 3))
assert not m.is_symmetric()
assert m.is_diagonal()
m = PropertiesOnlyMatrix(((5, 0), (0, 6), (0, 0)))
assert m.is_diagonal()
m = PropertiesOnlyMatrix(((5, 0, 0), (0, 6, 0)))
assert m.is_diagonal()
m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2, 2, 0, y, 0, 3])
assert m.is_symmetric()
assert not m.is_symmetric(simplify=False)
assert m.expand().is_symmetric(simplify=False)
def test_is_hermitian():
a = PropertiesOnlyMatrix([[1, I], [-I, 1]])
assert a.is_hermitian
a = PropertiesOnlyMatrix([[2*I, I], [-I, 1]])
assert a.is_hermitian is False
a = PropertiesOnlyMatrix([[x, I], [-I, 1]])
assert a.is_hermitian is None
a = PropertiesOnlyMatrix([[x, 1], [-I, 1]])
assert a.is_hermitian is False
def test_is_Identity():
assert eye_Properties(3).is_Identity
assert not PropertiesOnlyMatrix(zeros(3)).is_Identity
assert not PropertiesOnlyMatrix(ones(3)).is_Identity
# issue 6242
assert not PropertiesOnlyMatrix([[1, 0, 0]]).is_Identity
def test_is_symbolic():
a = PropertiesOnlyMatrix([[x, x], [x, x]])
assert a.is_symbolic() is True
a = PropertiesOnlyMatrix([[1, 2, 3, 4], [5, 6, 7, 8]])
assert a.is_symbolic() is False
a = PropertiesOnlyMatrix([[1, 2, 3, 4], [5, 6, x, 8]])
assert a.is_symbolic() is True
a = PropertiesOnlyMatrix([[1, x, 3]])
assert a.is_symbolic() is True
a = PropertiesOnlyMatrix([[1, 2, 3]])
assert a.is_symbolic() is False
a = PropertiesOnlyMatrix([[1], [x], [3]])
assert a.is_symbolic() is True
a = PropertiesOnlyMatrix([[1], [2], [3]])
assert a.is_symbolic() is False
def test_is_upper():
a = PropertiesOnlyMatrix([[1, 2, 3]])
assert a.is_upper is True
a = PropertiesOnlyMatrix([[1], [2], [3]])
assert a.is_upper is False
def test_is_lower():
a = PropertiesOnlyMatrix([[1, 2, 3]])
assert a.is_lower is False
a = PropertiesOnlyMatrix([[1], [2], [3]])
assert a.is_lower is True
def test_is_square():
m = PropertiesOnlyMatrix([[1], [1]])
m2 = PropertiesOnlyMatrix([[2, 2], [2, 2]])
assert not m.is_square
assert m2.is_square
def test_is_symmetric():
m = PropertiesOnlyMatrix(2, 2, [0, 1, 1, 0])
assert m.is_symmetric()
m = PropertiesOnlyMatrix(2, 2, [0, 1, 0, 1])
assert not m.is_symmetric()
def test_is_hessenberg():
A = PropertiesOnlyMatrix([[3, 4, 1], [2, 4, 5], [0, 1, 2]])
assert A.is_upper_hessenberg
A = PropertiesOnlyMatrix(3, 3, [3, 2, 0, 4, 4, 1, 1, 5, 2])
assert A.is_lower_hessenberg
A = PropertiesOnlyMatrix(3, 3, [3, 2, -1, 4, 4, 1, 1, 5, 2])
assert A.is_lower_hessenberg is False
assert A.is_upper_hessenberg is False
A = PropertiesOnlyMatrix([[3, 4, 1], [2, 4, 5], [3, 1, 2]])
assert not A.is_upper_hessenberg
def test_is_zero():
assert PropertiesOnlyMatrix(0, 0, []).is_zero
assert PropertiesOnlyMatrix([[0, 0], [0, 0]]).is_zero
assert PropertiesOnlyMatrix(zeros(3, 4)).is_zero
assert not PropertiesOnlyMatrix(eye(3)).is_zero
assert PropertiesOnlyMatrix([[x, 0], [0, 0]]).is_zero == None
assert PropertiesOnlyMatrix([[x, 1], [0, 0]]).is_zero == False
a = Symbol('a', nonzero=True)
assert PropertiesOnlyMatrix([[a, 0], [0, 0]]).is_zero == False
def test_values():
assert set(PropertiesOnlyMatrix(2, 2, [0, 1, 2, 3]
).values()) == set([1, 2, 3])
x = Symbol('x', real=True)
assert set(PropertiesOnlyMatrix(2, 2, [x, 0, 0, 1]
).values()) == set([x, 1])
# OperationsOnlyMatrix tests
def test_applyfunc():
m0 = OperationsOnlyMatrix(eye(3))
assert m0.applyfunc(lambda x: 2*x) == eye(3)*2
assert m0.applyfunc(lambda x: 0) == zeros(3)
assert m0.applyfunc(lambda x: 1) == ones(3)
def test_adjoint():
dat = [[0, I], [1, 0]]
ans = OperationsOnlyMatrix([[0, 1], [-I, 0]])
assert ans.adjoint() == Matrix(dat)
def test_as_real_imag():
m1 = OperationsOnlyMatrix(2, 2, [1, 2, 3, 4])
m3 = OperationsOnlyMatrix(2, 2,
[1 + S.ImaginaryUnit, 2 + 2*S.ImaginaryUnit,
3 + 3*S.ImaginaryUnit, 4 + 4*S.ImaginaryUnit])
a, b = m3.as_real_imag()
assert a == m1
assert b == m1
def test_conjugate():
M = OperationsOnlyMatrix([[0, I, 5],
[1, 2, 0]])
assert M.T == Matrix([[0, 1],
[I, 2],
[5, 0]])
assert M.C == Matrix([[0, -I, 5],
[1, 2, 0]])
assert M.C == M.conjugate()
assert M.H == M.T.C
assert M.H == Matrix([[ 0, 1],
[-I, 2],
[ 5, 0]])
def test_doit():
a = OperationsOnlyMatrix([[Add(x, x, evaluate=False)]])
assert a[0] != 2*x
assert a.doit() == Matrix([[2*x]])
def test_evalf():
a = OperationsOnlyMatrix(2, 1, [sqrt(5), 6])
assert all(a.evalf()[i] == a[i].evalf() for i in range(2))
assert all(a.evalf(2)[i] == a[i].evalf(2) for i in range(2))
assert all(a.n(2)[i] == a[i].n(2) for i in range(2))
def test_expand():
m0 = OperationsOnlyMatrix([[x*(x + y), 2], [((x + y)*y)*x, x*(y + x*(x + y))]])
# Test if expand() returns a matrix
m1 = m0.expand()
assert m1 == Matrix(
[[x*y + x**2, 2], [x*y**2 + y*x**2, x*y + y*x**2 + x**3]])
a = Symbol('a', real=True)
assert OperationsOnlyMatrix(1, 1, [exp(I*a)]).expand(complex=True) == \
Matrix([cos(a) + I*sin(a)])
def test_refine():
m0 = OperationsOnlyMatrix([[Abs(x)**2, sqrt(x**2)],
[sqrt(x**2)*Abs(y)**2, sqrt(y**2)*Abs(x)**2]])
m1 = m0.refine(Q.real(x) & Q.real(y))
assert m1 == Matrix([[x**2, Abs(x)], [y**2*Abs(x), x**2*Abs(y)]])
m1 = m0.refine(Q.positive(x) & Q.positive(y))
assert m1 == Matrix([[x**2, x], [x*y**2, x**2*y]])
m1 = m0.refine(Q.negative(x) & Q.negative(y))
assert m1 == Matrix([[x**2, -x], [-x*y**2, -x**2*y]])
def test_replace():
F, G = symbols('F, G', cls=Function)
K = OperationsOnlyMatrix(2, 2, lambda i, j: G(i+j))
M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G)
assert N == K
def test_replace_map():
F, G = symbols('F, G', cls=Function)
K = OperationsOnlyMatrix(2, 2, [(G(0), {F(0): G(0)}), (G(1), {F(1): G(1)}), (G(1), {F(1) \
: G(1)}), (G(2), {F(2): G(2)})])
M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G, True)
assert N == K
def test_simplify():
n = Symbol('n')
f = Function('f')
M = OperationsOnlyMatrix([[ 1/x + 1/y, (x + x*y) / x ],
[ (f(x) + y*f(x))/f(x), 2 * (1/n - cos(n * pi)/n) / pi ]])
assert M.simplify() == Matrix([[ (x + y)/(x * y), 1 + y ],
[ 1 + y, 2*((1 - 1*cos(pi*n))/(pi*n)) ]])
eq = (1 + x)**2
M = OperationsOnlyMatrix([[eq]])
assert M.simplify() == Matrix([[eq]])
assert M.simplify(ratio=oo) == Matrix([[eq.simplify(ratio=oo)]])
def test_subs():
assert OperationsOnlyMatrix([[1, x], [x, 4]]).subs(x, 5) == Matrix([[1, 5], [5, 4]])
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs([[x, -1], [y, -2]]) == \
Matrix([[-1, 2], [-3, 4]])
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs([(x, -1), (y, -2)]) == \
Matrix([[-1, 2], [-3, 4]])
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs({x: -1, y: -2}) == \
Matrix([[-1, 2], [-3, 4]])
assert OperationsOnlyMatrix([[x*y]]).subs({x: y - 1, y: x - 1}, simultaneous=True) == \
Matrix([[(x - 1)*(y - 1)]])
def test_trace():
M = OperationsOnlyMatrix([[1, 0, 0],
[0, 5, 0],
[0, 0, 8]])
assert M.trace() == 14
def test_xreplace():
assert OperationsOnlyMatrix([[1, x], [x, 4]]).xreplace({x: 5}) == \
Matrix([[1, 5], [5, 4]])
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).xreplace({x: -1, y: -2}) == \
Matrix([[-1, 2], [-3, 4]])
def test_permute():
a = OperationsOnlyMatrix(3, 4, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])
raises(IndexError, lambda: a.permute([[0, 5]]))
b = a.permute_rows([[0, 2], [0, 1]])
assert a.permute([[0, 2], [0, 1]]) == b == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
b = a.permute_cols([[0, 2], [0, 1]])
assert a.permute([[0, 2], [0, 1]], orientation='cols') == b ==\
Matrix([
[ 2, 3, 1, 4],
[ 6, 7, 5, 8],
[10, 11, 9, 12]])
b = a.permute_cols([[0, 2], [0, 1]], direction='backward')
assert a.permute([[0, 2], [0, 1]], orientation='cols', direction='backward') == b ==\
Matrix([
[ 3, 1, 2, 4],
[ 7, 5, 6, 8],
[11, 9, 10, 12]])
assert a.permute([1, 2, 0, 3]) == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
from sympy.combinatorics import Permutation
assert a.permute(Permutation([1, 2, 0, 3])) == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
# ArithmeticOnlyMatrix tests
def test_abs():
m = ArithmeticOnlyMatrix([[1, -2], [x, y]])
assert abs(m) == ArithmeticOnlyMatrix([[1, 2], [Abs(x), Abs(y)]])
def test_add():
m = ArithmeticOnlyMatrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
assert m + m == ArithmeticOnlyMatrix([[2, 4, 6], [2*x, 2*y, 2*x], [4*y, -100, 2*z*x]])
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
raises(ShapeError, lambda: m + n)
def test_multiplication():
a = ArithmeticOnlyMatrix((
(1, 2),
(3, 1),
(0, 6),
))
b = ArithmeticOnlyMatrix((
(1, 2),
(3, 0),
))
raises(ShapeError, lambda: b*a)
raises(TypeError, lambda: a*{})
c = a*b
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
try:
eval('c = a @ b')
except SyntaxError:
pass
else:
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
h = a.multiply_elementwise(c)
assert h == matrix_multiply_elementwise(a, c)
assert h[0, 0] == 7
assert h[0, 1] == 4
assert h[1, 0] == 18
assert h[1, 1] == 6
assert h[2, 0] == 0
assert h[2, 1] == 0
raises(ShapeError, lambda: a.multiply_elementwise(b))
c = b * Symbol("x")
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == x
assert c[0, 1] == 2*x
assert c[1, 0] == 3*x
assert c[1, 1] == 0
c2 = x * b
assert c == c2
c = 5 * b
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
try:
eval('c = 5 @ b')
except SyntaxError:
pass
else:
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
def test_matmul():
a = Matrix([[1, 2], [3, 4]])
assert a.__matmul__(2) == NotImplemented
assert a.__rmatmul__(2) == NotImplemented
#This is done this way because @ is only supported in Python 3.5+
#To check 2@a case
try:
eval('2 @ a')
except SyntaxError:
pass
except TypeError: #TypeError is raised in case of NotImplemented is returned
pass
#Check a@2 case
try:
eval('a @ 2')
except SyntaxError:
pass
except TypeError: #TypeError is raised in case of NotImplemented is returned
pass
def test_power():
raises(NonSquareMatrixError, lambda: Matrix((1, 2))**2)
A = ArithmeticOnlyMatrix([[2, 3], [4, 5]])
assert (A**5)[:] == (6140, 8097, 10796, 14237)
A = ArithmeticOnlyMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
assert (A**3)[:] == (290, 262, 251, 448, 440, 368, 702, 954, 433)
assert A**0 == eye(3)
assert A**1 == A
assert (ArithmeticOnlyMatrix([[2]]) ** 100)[0, 0] == 2**100
assert ArithmeticOnlyMatrix([[1, 2], [3, 4]])**Integer(2) == ArithmeticOnlyMatrix([[7, 10], [15, 22]])
def test_neg():
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
assert -n == ArithmeticOnlyMatrix(1, 2, [-1, -2])
def test_sub():
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
assert n - n == ArithmeticOnlyMatrix(1, 2, [0, 0])
def test_div():
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
assert n/2 == ArithmeticOnlyMatrix(1, 2, [S(1)/2, S(2)/2])
# DeterminantOnlyMatrix tests
def test_det():
a = DeterminantOnlyMatrix(2, 3, [1, 2, 3, 4, 5, 6])
raises(NonSquareMatrixError, lambda: a.det())
z = zeros_Determinant(2)
ey = eye_Determinant(2)
assert z.det() == 0
assert ey.det() == 1
x = Symbol('x')
a = DeterminantOnlyMatrix(0, 0, [])
b = DeterminantOnlyMatrix(1, 1, [5])
c = DeterminantOnlyMatrix(2, 2, [1, 2, 3, 4])
d = DeterminantOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
e = DeterminantOnlyMatrix(4, 4,
[x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
# the method keyword for `det` doesn't kick in until 4x4 matrices,
# so there is no need to test all methods on smaller ones
assert a.det() == 1
assert b.det() == 5
assert c.det() == -2
assert d.det() == 3
assert e.det() == 4*x - 24
assert e.det(method='bareiss') == 4*x - 24
assert e.det(method='berkowitz') == 4*x - 24
raises(ValueError, lambda: e.det(iszerofunc="test"))
def test_adjugate():
x = Symbol('x')
e = DeterminantOnlyMatrix(4, 4,
[x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
adj = Matrix([
[ 4, -8, 4, 0],
[ 76, -14*x - 68, 14*x - 8, -4*x + 24],
[-122, 17*x + 142, -21*x + 4, 8*x - 48],
[ 48, -4*x - 72, 8*x, -4*x + 24]])
assert e.adjugate() == adj
assert e.adjugate(method='bareiss') == adj
assert e.adjugate(method='berkowitz') == adj
a = DeterminantOnlyMatrix(2, 3, [1, 2, 3, 4, 5, 6])
raises(NonSquareMatrixError, lambda: a.adjugate())
def test_cofactor_and_minors():
x = Symbol('x')
e = DeterminantOnlyMatrix(4, 4,
[x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
m = Matrix([
[ x, 1, 3],
[ 2, 9, 11],
[12, 13, 14]])
cm = Matrix([
[ 4, 76, -122, 48],
[-8, -14*x - 68, 17*x + 142, -4*x - 72],
[ 4, 14*x - 8, -21*x + 4, 8*x],
[ 0, -4*x + 24, 8*x - 48, -4*x + 24]])
sub = Matrix([
[x, 1, 2],
[4, 5, 6],
[2, 9, 10]])
assert e.minor_submatrix(1, 2) == m
assert e.minor_submatrix(-1, -1) == sub
assert e.minor(1, 2) == -17*x - 142
assert e.cofactor(1, 2) == 17*x + 142
assert e.cofactor_matrix() == cm
assert e.cofactor_matrix(method="bareiss") == cm
assert e.cofactor_matrix(method="berkowitz") == cm
raises(ValueError, lambda: e.cofactor(4, 5))
raises(ValueError, lambda: e.minor(4, 5))
raises(ValueError, lambda: e.minor_submatrix(4, 5))
a = DeterminantOnlyMatrix(2, 3, [1, 2, 3, 4, 5, 6])
assert a.minor_submatrix(0, 0) == Matrix([[5, 6]])
raises(ValueError, lambda:
DeterminantOnlyMatrix(0, 0, []).minor_submatrix(0, 0))
raises(NonSquareMatrixError, lambda: a.cofactor(0, 0))
raises(NonSquareMatrixError, lambda: a.minor(0, 0))
raises(NonSquareMatrixError, lambda: a.cofactor_matrix())
def test_charpoly():
x, y = Symbol('x'), Symbol('y')
m = DeterminantOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert eye_Determinant(3).charpoly(x) == Poly((x - 1)**3, x)
assert eye_Determinant(3).charpoly(y) == Poly((y - 1)**3, y)
assert m.charpoly() == Poly(x**3 - 15*x**2 - 18*x, x)
raises(NonSquareMatrixError, lambda: Matrix([[1], [2]]).charpoly())
# ReductionsOnlyMatrix tests
def test_row_op():
e = eye_Reductions(3)
raises(ValueError, lambda: e.elementary_row_op("abc"))
raises(ValueError, lambda: e.elementary_row_op())
raises(ValueError, lambda: e.elementary_row_op('n->kn', row=5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->kn', row=-5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=1, row2=5))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=5, row2=1))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=-5, row2=1))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=1, row2=-5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=5, row2=1, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=-5, row2=1, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=-5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=1, k=5))
# test various ways to set arguments
assert e.elementary_row_op("n->kn", 0, 5) == Matrix([[5, 0, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", 1, 5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", row=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", row1=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", 0, 1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", row1=0, row2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", row=0, row2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", 0, 5, 1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", row=0, k=5, row2=1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", row1=0, k=5, row2=1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
# make sure the matrix doesn't change size
a = ReductionsOnlyMatrix(2, 3, [0]*6)
assert a.elementary_row_op("n->kn", 1, 5) == Matrix(2, 3, [0]*6)
assert a.elementary_row_op("n<->m", 0, 1) == Matrix(2, 3, [0]*6)
assert a.elementary_row_op("n->n+km", 0, 5, 1) == Matrix(2, 3, [0]*6)
def test_col_op():
e = eye_Reductions(3)
raises(ValueError, lambda: e.elementary_col_op("abc"))
raises(ValueError, lambda: e.elementary_col_op())
raises(ValueError, lambda: e.elementary_col_op('n->kn', col=5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->kn', col=-5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=1, col2=5))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=5, col2=1))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=-5, col2=1))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=1, col2=-5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=5, col2=1, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=-5, col2=1, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=-5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=1, k=5))
# test various ways to set arguments
assert e.elementary_col_op("n->kn", 0, 5) == Matrix([[5, 0, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", 1, 5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", col=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", col1=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", 0, 1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", col1=0, col2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", col=0, col2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", 0, 5, 1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", col=0, k=5, col2=1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", col1=0, k=5, col2=1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
# make sure the matrix doesn't change size
a = ReductionsOnlyMatrix(2, 3, [0]*6)
assert a.elementary_col_op("n->kn", 1, 5) == Matrix(2, 3, [0]*6)
assert a.elementary_col_op("n<->m", 0, 1) == Matrix(2, 3, [0]*6)
assert a.elementary_col_op("n->n+km", 0, 5, 1) == Matrix(2, 3, [0]*6)
def test_is_echelon():
zro = zeros_Reductions(3)
ident = eye_Reductions(3)
assert zro.is_echelon
assert ident.is_echelon
a = ReductionsOnlyMatrix(0, 0, [])
assert a.is_echelon
a = ReductionsOnlyMatrix(2, 3, [3, 2, 1, 0, 0, 6])
assert a.is_echelon
a = ReductionsOnlyMatrix(2, 3, [0, 0, 6, 3, 2, 1])
assert not a.is_echelon
x = Symbol('x')
a = ReductionsOnlyMatrix(3, 1, [x, 0, 0])
assert a.is_echelon
a = ReductionsOnlyMatrix(3, 1, [x, x, 0])
assert not a.is_echelon
a = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 1, 2, 3, 0, 0, 0])
assert not a.is_echelon
def test_echelon_form():
# echelon form is not unique, but the result
# must be row-equivalent to the original matrix
# and it must be in echelon form.
a = zeros_Reductions(3)
e = eye_Reductions(3)
# we can assume the zero matrix and the identity matrix shouldn't change
assert a.echelon_form() == a
assert e.echelon_form() == e
a = ReductionsOnlyMatrix(0, 0, [])
assert a.echelon_form() == a
a = ReductionsOnlyMatrix(1, 1, [5])
assert a.echelon_form() == a
# now we get to the real tests
def verify_row_null_space(mat, rows, nulls):
for v in nulls:
assert all(t.is_zero for t in a_echelon*v)
for v in rows:
if not all(t.is_zero for t in v):
assert not all(t.is_zero for t in a_echelon*v.transpose())
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
nulls = [Matrix([
[ 1],
[-2],
[ 1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
nulls = []
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [2, 1, 3, 0, 0, 0, 2, 1, 3])
nulls = [Matrix([
[-S(1)/2],
[ 1],
[ 0]]),
Matrix([
[-S(3)/2],
[ 0],
[ 1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
# this one requires a row swap
a = ReductionsOnlyMatrix(3, 3, [2, 1, 3, 0, 0, 0, 1, 1, 3])
nulls = [Matrix([
[ 0],
[ -3],
[ 1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [0, 3, 3, 0, 2, 2, 0, 1, 1])
nulls = [Matrix([
[1],
[0],
[0]]),
Matrix([
[ 0],
[-1],
[ 1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(2, 3, [2, 2, 3, 3, 3, 0])
nulls = [Matrix([
[-1],
[1],
[0]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
def test_rref():
e = ReductionsOnlyMatrix(0, 0, [])
assert e.rref(pivots=False) == e
e = ReductionsOnlyMatrix(1, 1, [1])
a = ReductionsOnlyMatrix(1, 1, [5])
assert e.rref(pivots=False) == a.rref(pivots=False) == e
a = ReductionsOnlyMatrix(3, 1, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1], [0], [0]])
a = ReductionsOnlyMatrix(1, 3, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1, 2, 3]])
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert a.rref(pivots=False) == Matrix([
[1, 0, -1],
[0, 1, 2],
[0, 0, 0]])
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
b = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 0, 0, 0, 0, 0, 0])
c = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 1, 2, 3, 0, 0, 0])
d = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 0, 0, 0, 1, 2, 3])
assert a.rref(pivots=False) == \
b.rref(pivots=False) == \
c.rref(pivots=False) == \
d.rref(pivots=False) == b
e = eye_Reductions(3)
z = zeros_Reductions(3)
assert e.rref(pivots=False) == e
assert z.rref(pivots=False) == z
a = ReductionsOnlyMatrix([
[ 0, 0, 1, 2, 2, -5, 3],
[-1, 5, 2, 2, 1, -7, 5],
[ 0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
mat, pivot_offsets = a.rref()
assert mat == Matrix([
[1, -5, 0, 0, 1, 1, -1],
[0, 0, 1, 0, 0, -1, 1],
[0, 0, 0, 1, 1, -2, 1],
[0, 0, 0, 0, 0, 0, 0]])
assert pivot_offsets == (0, 2, 3)
a = ReductionsOnlyMatrix([[S(1)/19, S(1)/5, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[ 12, 13, 14, 15]])
assert a.rref(pivots=False) == Matrix([
[1, 0, 0, -S(76)/157],
[0, 1, 0, -S(5)/157],
[0, 0, 1, S(238)/157],
[0, 0, 0, 0]])
x = Symbol('x')
a = ReductionsOnlyMatrix(2, 3, [x, 1, 1, sqrt(x), x, 1])
for i, j in zip(a.rref(pivots=False),
[1, 0, sqrt(x)*(-x + 1)/(-x**(S(5)/2) + x),
0, 1, 1/(sqrt(x) + x + 1)]):
assert simplify(i - j).is_zero
# SpecialOnlyMatrix tests
def test_eye():
assert list(SpecialOnlyMatrix.eye(2, 2)) == [1, 0, 0, 1]
assert list(SpecialOnlyMatrix.eye(2)) == [1, 0, 0, 1]
assert type(SpecialOnlyMatrix.eye(2)) == SpecialOnlyMatrix
assert type(SpecialOnlyMatrix.eye(2, cls=Matrix)) == Matrix
def test_ones():
assert list(SpecialOnlyMatrix.ones(2, 2)) == [1, 1, 1, 1]
assert list(SpecialOnlyMatrix.ones(2)) == [1, 1, 1, 1]
assert SpecialOnlyMatrix.ones(2, 3) == Matrix([[1, 1, 1], [1, 1, 1]])
assert type(SpecialOnlyMatrix.ones(2)) == SpecialOnlyMatrix
assert type(SpecialOnlyMatrix.ones(2, cls=Matrix)) == Matrix
def test_zeros():
assert list(SpecialOnlyMatrix.zeros(2, 2)) == [0, 0, 0, 0]
assert list(SpecialOnlyMatrix.zeros(2)) == [0, 0, 0, 0]
assert SpecialOnlyMatrix.zeros(2, 3) == Matrix([[0, 0, 0], [0, 0, 0]])
assert type(SpecialOnlyMatrix.zeros(2)) == SpecialOnlyMatrix
assert type(SpecialOnlyMatrix.zeros(2, cls=Matrix)) == Matrix
def test_diag_make():
diag = SpecialOnlyMatrix.diag
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert diag(a, b, b) == Matrix([
[1, 2, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0],
[0, 0, 3, x, 0, 0],
[0, 0, y, 3, 0, 0],
[0, 0, 0, 0, 3, x],
[0, 0, 0, 0, y, 3],
])
assert diag(a, b, c) == Matrix([
[1, 2, 0, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0, 0],
[0, 0, 3, x, 0, 0, 0],
[0, 0, y, 3, 0, 0, 0],
[0, 0, 0, 0, 3, x, 3],
[0, 0, 0, 0, y, 3, z],
[0, 0, 0, 0, x, y, z],
])
assert diag(a, c, b) == Matrix([
[1, 2, 0, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0, 0],
[0, 0, 3, x, 3, 0, 0],
[0, 0, y, 3, z, 0, 0],
[0, 0, x, y, z, 0, 0],
[0, 0, 0, 0, 0, 3, x],
[0, 0, 0, 0, 0, y, 3],
])
a = Matrix([x, y, z])
b = Matrix([[1, 2], [3, 4]])
c = Matrix([[5, 6]])
# this "wandering diagonal" is what makes this
# a block diagonal where each block is independent
# of the others
assert diag(a, 7, b, c) == Matrix([
[x, 0, 0, 0, 0, 0],
[y, 0, 0, 0, 0, 0],
[z, 0, 0, 0, 0, 0],
[0, 7, 0, 0, 0, 0],
[0, 0, 1, 2, 0, 0],
[0, 0, 3, 4, 0, 0],
[0, 0, 0, 0, 5, 6]])
raises(ValueError, lambda: diag(a, 7, b, c, rows=5))
assert diag(1) == Matrix([[1]])
assert diag(1, rows=2) == Matrix([[1, 0], [0, 0]])
assert diag(1, cols=2) == Matrix([[1, 0], [0, 0]])
assert diag(1, rows=3, cols=2) == Matrix([[1, 0], [0, 0], [0, 0]])
assert diag(*[2, 3]) == Matrix([
[2, 0],
[0, 3]])
assert diag(Matrix([2, 3])) == Matrix([
[2],
[3]])
assert diag([1, [2, 3], 4], unpack=False) == \
diag([[1], [2, 3], [4]], unpack=False) == Matrix([
[1, 0],
[2, 3],
[4, 0]])
assert type(diag(1)) == SpecialOnlyMatrix
assert type(diag(1, cls=Matrix)) == Matrix
assert Matrix.diag([1, 2, 3]) == Matrix.diag(1, 2, 3)
assert Matrix.diag([1, 2, 3], unpack=False).shape == (3, 1)
assert Matrix.diag([[1, 2, 3]]).shape == (3, 1)
assert Matrix.diag([[1, 2, 3]], unpack=False).shape == (1, 3)
assert Matrix.diag([[[1, 2, 3]]]).shape == (1, 3)
# kerning can be used to move the starting point
assert Matrix.diag(ones(0, 2), 1, 2) == Matrix([
[0, 0, 1, 0],
[0, 0, 0, 2]])
assert Matrix.diag(ones(2, 0), 1, 2) == Matrix([
[0, 0],
[0, 0],
[1, 0],
[0, 2]])
def test_diagonal():
m = Matrix(3, 3, range(9))
d = m.diagonal()
assert d == m.diagonal(0)
assert tuple(d) == (0, 4, 8)
assert tuple(m.diagonal(1)) == (1, 5)
assert tuple(m.diagonal(-1)) == (3, 7)
assert tuple(m.diagonal(2)) == (2,)
assert type(m.diagonal()) == type(m)
s = SparseMatrix(3, 3, {(1, 1): 1})
assert type(s.diagonal()) == type(s)
assert type(m) != type(s)
raises(ValueError, lambda: m.diagonal(3))
raises(ValueError, lambda: m.diagonal(-3))
raises(ValueError, lambda: m.diagonal(pi))
def test_jordan_block():
assert SpecialOnlyMatrix.jordan_block(3, 2) == SpecialOnlyMatrix.jordan_block(3, eigenvalue=2) \
== SpecialOnlyMatrix.jordan_block(size=3, eigenvalue=2) \
== SpecialOnlyMatrix.jordan_block(3, 2, band='upper') \
== SpecialOnlyMatrix.jordan_block(
size=3, eigenval=2, eigenvalue=2) \
== Matrix([
[2, 1, 0],
[0, 2, 1],
[0, 0, 2]])
assert SpecialOnlyMatrix.jordan_block(3, 2, band='lower') == Matrix([
[2, 0, 0],
[1, 2, 0],
[0, 1, 2]])
# missing eigenvalue
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(2))
# non-integral size
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(3.5, 2))
# size not specified
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(eigenvalue=2))
# inconsistent eigenvalue
raises(ValueError,
lambda: SpecialOnlyMatrix.jordan_block(
eigenvalue=2, eigenval=4))
# Deprecated feature
with warns_deprecated_sympy():
assert (SpecialOnlyMatrix.jordan_block(cols=3, eigenvalue=2) ==
SpecialOnlyMatrix(3, 3, (2, 1, 0, 0, 2, 1, 0, 0, 2)))
with warns_deprecated_sympy():
assert (SpecialOnlyMatrix.jordan_block(rows=3, eigenvalue=2) ==
SpecialOnlyMatrix(3, 3, (2, 1, 0, 0, 2, 1, 0, 0, 2)))
with warns_deprecated_sympy():
assert SpecialOnlyMatrix.jordan_block(3, 2) == \
SpecialOnlyMatrix.jordan_block(cols=3, eigenvalue=2) == \
SpecialOnlyMatrix.jordan_block(rows=3, eigenvalue=2)
with warns_deprecated_sympy():
assert SpecialOnlyMatrix.jordan_block(
rows=4, cols=3, eigenvalue=2) == \
Matrix([
[2, 1, 0],
[0, 2, 1],
[0, 0, 2],
[0, 0, 0]])
# Using alias keyword
assert SpecialOnlyMatrix.jordan_block(size=3, eigenvalue=2) == \
SpecialOnlyMatrix.jordan_block(size=3, eigenval=2)
# SubspaceOnlyMatrix tests
def test_columnspace():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.columnspace()
assert basis[0] == Matrix([1, -2, 0, 3])
assert basis[1] == Matrix([2, -5, -3, 6])
assert basis[2] == Matrix([2, -1, 4, -7])
assert len(basis) == 3
assert Matrix.hstack(m, *basis).columnspace() == basis
def test_rowspace():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.rowspace()
assert basis[0] == Matrix([[1, 2, 0, 2, 5]])
assert basis[1] == Matrix([[0, -1, 1, 3, 2]])
assert basis[2] == Matrix([[0, 0, 0, 5, 5]])
assert len(basis) == 3
def test_nullspace():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.nullspace()
assert basis[0] == Matrix([-2, 1, 1, 0, 0])
assert basis[1] == Matrix([-1, -1, 0, -1, 1])
# make sure the null space is really gets zeroed
assert all(e.is_zero for e in m*basis[0])
assert all(e.is_zero for e in m*basis[1])
def test_orthogonalize():
m = Matrix([[1, 2], [3, 4]])
assert m.orthogonalize(Matrix([[2], [1]])) == [Matrix([[2], [1]])]
assert m.orthogonalize(Matrix([[2], [1]]), normalize=True) == \
[Matrix([[2*sqrt(5)/5], [sqrt(5)/5]])]
assert m.orthogonalize(Matrix([[1], [2]]), Matrix([[-1], [4]])) == \
[Matrix([[1], [2]]), Matrix([[-S(12)/5], [S(6)/5]])]
assert m.orthogonalize(Matrix([[0], [0]]), Matrix([[-1], [4]])) == \
[Matrix([[-1], [4]])]
assert m.orthogonalize(Matrix([[0], [0]])) == []
n = Matrix([[9, 1, 9], [3, 6, 10], [8, 5, 2]])
vecs = [Matrix([[-5], [1]]), Matrix([[-5], [2]]), Matrix([[-5], [-2]])]
assert n.orthogonalize(*vecs) == \
[Matrix([[-5], [1]]), Matrix([[S(5)/26], [S(25)/26]])]
vecs = [Matrix([0, 0, 0]), Matrix([1, 2, 3]), Matrix([1, 4, 5])]
raises(ValueError, lambda: Matrix.orthogonalize(*vecs, rankcheck=True))
vecs = [Matrix([1, 2, 3]), Matrix([4, 5, 6]), Matrix([7, 8, 9])]
raises(ValueError, lambda: Matrix.orthogonalize(*vecs, rankcheck=True))
# EigenOnlyMatrix tests
def test_eigenvals():
M = EigenOnlyMatrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}
# if we cannot factor the char poly, we raise an error
m = Matrix([
[3, 0, 0, 0, -3],
[0, -3, -3, 0, 3],
[0, 3, 0, 3, 0],
[0, 0, 3, 0, 3],
[3, 0, 0, 3, 0]])
raises(MatrixError, lambda: m.eigenvals())
def test_eigenvects():
M = EigenOnlyMatrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
vecs = M.eigenvects()
for val, mult, vec_list in vecs:
assert len(vec_list) == 1
assert M*vec_list[0] == val*vec_list[0]
def test_left_eigenvects():
M = EigenOnlyMatrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
vecs = M.left_eigenvects()
for val, mult, vec_list in vecs:
assert len(vec_list) == 1
assert vec_list[0]*M == val*vec_list[0]
def test_diagonalize():
m = EigenOnlyMatrix(2, 2, [0, -1, 1, 0])
raises(MatrixError, lambda: m.diagonalize(reals_only=True))
P, D = m.diagonalize()
assert D.is_diagonal()
assert D == Matrix([
[-I, 0],
[ 0, I]])
# make sure we use floats out if floats are passed in
m = EigenOnlyMatrix(2, 2, [0, .5, .5, 0])
P, D = m.diagonalize()
assert all(isinstance(e, Float) for e in D.values())
assert all(isinstance(e, Float) for e in P.values())
_, D2 = m.diagonalize(reals_only=True)
assert D == D2
def test_is_diagonalizable():
a, b, c = symbols('a b c')
m = EigenOnlyMatrix(2, 2, [a, c, c, b])
assert m.is_symmetric()
assert m.is_diagonalizable()
assert not EigenOnlyMatrix(2, 2, [1, 1, 0, 1]).is_diagonalizable()
m = EigenOnlyMatrix(2, 2, [0, -1, 1, 0])
assert m.is_diagonalizable()
assert not m.is_diagonalizable(reals_only=True)
def test_jordan_form():
m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
raises(NonSquareMatrixError, lambda: m.jordan_form())
# the next two tests test the cases where the old
# algorithm failed due to the fact that the block structure can
# *NOT* be determined from algebraic and geometric multiplicity alone
# This can be seen most easily when one lets compute the J.c.f. of a matrix that
# is in J.c.f already.
m = EigenOnlyMatrix(4, 4, [2, 1, 0, 0,
0, 2, 1, 0,
0, 0, 2, 0,
0, 0, 0, 2
])
P, J = m.jordan_form()
assert m == J
m = EigenOnlyMatrix(4, 4, [2, 1, 0, 0,
0, 2, 0, 0,
0, 0, 2, 1,
0, 0, 0, 2
])
P, J = m.jordan_form()
assert m == J
A = Matrix([[ 2, 4, 1, 0],
[-4, 2, 0, 1],
[ 0, 0, 2, 4],
[ 0, 0, -4, 2]])
P, J = A.jordan_form()
assert simplify(P*J*P.inv()) == A
assert EigenOnlyMatrix(1, 1, [1]).jordan_form() == (
Matrix([1]), Matrix([1]))
assert EigenOnlyMatrix(1, 1, [1]).jordan_form(
calc_transform=False) == Matrix([1])
# make sure if we cannot factor the characteristic polynomial, we raise an error
m = Matrix([[3, 0, 0, 0, -3], [0, -3, -3, 0, 3], [0, 3, 0, 3, 0], [0, 0, 3, 0, 3], [3, 0, 0, 3, 0]])
raises(MatrixError, lambda: m.jordan_form())
# make sure that if the input has floats, the output does too
m = Matrix([
[ 0.6875, 0.125 + 0.1875*sqrt(3)],
[0.125 + 0.1875*sqrt(3), 0.3125]])
P, J = m.jordan_form()
assert all(isinstance(x, Float) or x == 0 for x in P)
assert all(isinstance(x, Float) or x == 0 for x in J)
def test_singular_values():
x = Symbol('x', real=True)
A = EigenOnlyMatrix([[0, 1*I], [2, 0]])
# if singular values can be sorted, they should be in decreasing order
assert A.singular_values() == [2, 1]
A = eye(3)
A[1, 1] = x
A[2, 2] = 5
vals = A.singular_values()
# since Abs(x) cannot be sorted, test set equality
assert set(vals) == set([5, 1, Abs(x)])
A = EigenOnlyMatrix([[sin(x), cos(x)], [-cos(x), sin(x)]])
vals = [sv.trigsimp() for sv in A.singular_values()]
assert vals == [S(1), S(1)]
A = EigenOnlyMatrix([
[2, 4],
[1, 3],
[0, 0],
[0, 0]
])
assert A.singular_values() == \
[sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221))]
assert A.T.singular_values() == \
[sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221)), 0, 0]
# CalculusOnlyMatrix tests
@XFAIL
def test_diff():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [x, y])
# TODO: currently not working as ``_MinimalMatrix`` cannot be sympified:
assert m.diff(x) == Matrix(2, 1, [1, 0])
def test_integrate():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [x, y])
assert m.integrate(x) == Matrix(2, 1, [x**2/2, y*x])
def test_jacobian2():
rho, phi = symbols("rho,phi")
X = CalculusOnlyMatrix(3, 1, [rho*cos(phi), rho*sin(phi), rho**2])
Y = CalculusOnlyMatrix(2, 1, [rho, phi])
J = Matrix([
[cos(phi), -rho*sin(phi)],
[sin(phi), rho*cos(phi)],
[ 2*rho, 0],
])
assert X.jacobian(Y) == J
m = CalculusOnlyMatrix(2, 2, [1, 2, 3, 4])
m2 = CalculusOnlyMatrix(4, 1, [1, 2, 3, 4])
raises(TypeError, lambda: m.jacobian(Matrix([1, 2])))
raises(TypeError, lambda: m2.jacobian(m))
def test_limit():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [1/x, y])
assert m.limit(x, 5) == Matrix(2, 1, [S(1)/5, y])
def test_issue_13774():
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
v = [1, 1, 1]
raises(TypeError, lambda: M*v)
raises(TypeError, lambda: v*M)
def test___eq__():
assert (EigenOnlyMatrix(
[[0, 1, 1],
[1, 0, 0],
[1, 1, 1]]) == {}) is False
|
83d968d70a10bcb31e0a373992a04b58b122e90b7fd97ed7ff87fc9fa4bf2da2 | from sympy.matrices.sparsetools import _doktocsr, _csrtodok, banded
from sympy import eye, ones, zeros, Matrix, SparseMatrix
from sympy.utilities.pytest import raises
def test_doktocsr():
a = SparseMatrix([[1, 2, 0, 0], [0, 3, 9, 0], [0, 1, 4, 0]])
b = SparseMatrix(4, 6, [10, 20, 0, 0, 0, 0, 0, 30, 0, 40, 0, 0, 0, 0, 50,
60, 70, 0, 0, 0, 0, 0, 0, 80])
c = SparseMatrix(4, 4, [0, 0, 0, 0, 0, 12, 0, 2, 15, 0, 12, 0, 0, 0, 0, 4])
d = SparseMatrix(10, 10, {(1, 1): 12, (3, 5): 7, (7, 8): 12})
e = SparseMatrix([[0, 0, 0], [1, 0, 2], [3, 0, 0]])
f = SparseMatrix(7, 8, {(2, 3): 5, (4, 5):12})
assert _doktocsr(a) == [[1, 2, 3, 9, 1, 4], [0, 1, 1, 2, 1, 2],
[0, 2, 4, 6], [3, 4]]
assert _doktocsr(b) == [[10, 20, 30, 40, 50, 60, 70, 80],
[0, 1, 1, 3, 2, 3, 4, 5], [0, 2, 4, 7, 8], [4, 6]]
assert _doktocsr(c) == [[12, 2, 15, 12, 4], [1, 3, 0, 2, 3],
[0, 0, 2, 4, 5], [4, 4]]
assert _doktocsr(d) == [[12, 7, 12], [1, 5, 8],
[0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3], [10, 10]]
assert _doktocsr(e) == [[1, 2, 3], [0, 2, 0], [0, 0, 2, 3], [3, 3]]
assert _doktocsr(f) == [[5, 12], [3, 5], [0, 0, 0, 1, 1, 2, 2, 2], [7, 8]]
def test_csrtodok():
h = [[5, 7, 5], [2, 1, 3], [0, 1, 1, 3], [3, 4]]
g = [[12, 5, 4], [2, 4, 2], [0, 1, 2, 3], [3, 7]]
i = [[1, 3, 12], [0, 2, 4], [0, 2, 3], [2, 5]]
j = [[11, 15, 12, 15], [2, 4, 1, 2], [0, 1, 1, 2, 3, 4], [5, 8]]
k = [[1, 3], [2, 1], [0, 1, 1, 2], [3, 3]]
assert _csrtodok(h) == SparseMatrix(3, 4,
{(0, 2): 5, (2, 1): 7, (2, 3): 5})
assert _csrtodok(g) == SparseMatrix(3, 7,
{(0, 2): 12, (1, 4): 5, (2, 2): 4})
assert _csrtodok(i) == SparseMatrix([[1, 0, 3, 0, 0], [0, 0, 0, 0, 12]])
assert _csrtodok(j) == SparseMatrix(5, 8,
{(0, 2): 11, (2, 4): 15, (3, 1): 12, (4, 2): 15})
assert _csrtodok(k) == SparseMatrix(3, 3, {(0, 2): 1, (2, 1): 3})
def test_banded():
raises(TypeError, lambda: banded())
raises(TypeError, lambda: banded(1))
raises(TypeError, lambda: banded(1, 2))
raises(TypeError, lambda: banded(1, 2, 3))
raises(TypeError, lambda: banded(1, 2, 3, 4))
raises(ValueError, lambda: banded({0: (1, 2)}, rows=1))
raises(ValueError, lambda: banded({0: (1, 2)}, cols=1))
raises(ValueError, lambda: banded(1, {0: (1, 2)}))
raises(ValueError, lambda: banded(2, 1, {0: (1, 2)}))
raises(ValueError, lambda: banded(1, 2, {0: (1, 2)}))
assert isinstance(banded(2, 4, {}), SparseMatrix)
assert banded(2, 4, {}) == zeros(2, 4)
assert banded({0: 0, 1: 0}) == zeros(0)
assert banded({0: Matrix([1, 2])}) == Matrix([1, 2])
assert banded({1: [1, 2, 3, 0], -1: [4, 5, 6]}) == \
banded({1: (1, 2, 3), -1: (4, 5, 6)}) == \
Matrix([
[0, 1, 0, 0],
[4, 0, 2, 0],
[0, 5, 0, 3],
[0, 0, 6, 0]])
assert banded(3, 4, {-1: 1, 0: 2, 1: 3}) == \
Matrix([
[2, 3, 0, 0],
[1, 2, 3, 0],
[0, 1, 2, 3]])
s = lambda d: (1 + d)**2
assert banded(5, {0: s, 2: s}) == \
Matrix([
[1, 0, 1, 0, 0],
[0, 4, 0, 4, 0],
[0, 0, 9, 0, 9],
[0, 0, 0, 16, 0],
[0, 0, 0, 0, 25]])
assert banded(2, {0: 1}) == \
Matrix([
[1, 0],
[0, 1]])
assert banded(2, 3, {0: 1}) == \
Matrix([
[1, 0, 0],
[0, 1, 0]])
vert = Matrix([1, 2, 3])
assert banded({0: vert}, cols=3) == \
Matrix([
[1, 0, 0],
[2, 1, 0],
[3, 2, 1],
[0, 3, 2],
[0, 0, 3]])
assert banded(4, {0: ones(2)}) == \
Matrix([
[1, 1, 0, 0],
[1, 1, 0, 0],
[0, 0, 1, 1],
[0, 0, 1, 1]])
raises(ValueError, lambda: banded({0: 2, 1: ones(2)}, rows=5))
assert banded({0: 2, 2: (ones(2),)*3}) == \
Matrix([
[2, 0, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 1, 0, 0, 0, 0],
[0, 0, 2, 0, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 0, 2, 0, 1, 1],
[0, 0, 0, 0, 0, 2, 1, 1]])
raises(ValueError, lambda: banded({0: (2,)*5, 1: (ones(2),)*3}))
u2 = Matrix([[1, 1], [0, 1]])
assert banded({0: (2,)*5, 1: (u2,)*3}) == \
Matrix([
[2, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 0, 0, 0, 0],
[0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 0, 0],
[0, 0, 0, 0, 2, 1, 1],
[0, 0, 0, 0, 0, 0, 1]])
assert banded({0:(0, ones(2)), 2: 2}) == \
Matrix([
[0, 0, 2],
[0, 1, 1],
[0, 1, 1]])
raises(ValueError, lambda: banded({0: (0, ones(2)), 1: 2}))
assert banded({0: 1}, cols=3) == banded({0: 1}, rows=3) == eye(3)
assert banded({1: 1}, rows=3) == Matrix([
[0, 1, 0],
[0, 0, 1],
[0, 0, 0]])
|
840240cc507d655d329f58bfeb5372a2cb1b7f97891eb3792bc00ac31d59b499 | import random
from sympy import (
Abs, Add, E, Float, I, Integer, Max, Min, N, Poly, Pow, PurePoly, Rational,
S, Symbol, cos, exp, expand_mul, oo, pi, signsimp, simplify, sin, sqrt, symbols,
sympify, trigsimp, tan, sstr, diff, Function)
from sympy.matrices.matrices import (ShapeError, MatrixError,
NonSquareMatrixError, DeferredVector, _find_reasonable_pivot_naive,
_simplify)
from sympy.matrices import (
GramSchmidt, ImmutableMatrix, ImmutableSparseMatrix, Matrix,
SparseMatrix, casoratian, diag, eye, hessian,
matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2,
rot_axis3, wronskian, zeros, MutableDenseMatrix, ImmutableDenseMatrix, MatrixSymbol)
from sympy.core.compatibility import long, iterable, range, Hashable
from sympy.core import Tuple, Wild
from sympy.utilities.iterables import flatten, capture
from sympy.utilities.pytest import raises, XFAIL, slow, skip, warns_deprecated_sympy
from sympy.solvers import solve
from sympy.assumptions import Q
from sympy.tensor.array import Array
from sympy.matrices.expressions import MatPow
from sympy.abc import a, b, c, d, x, y, z, t
# don't re-order this list
classes = (Matrix, SparseMatrix, ImmutableMatrix, ImmutableSparseMatrix)
def test_args():
for c, cls in enumerate(classes):
m = cls.zeros(3, 2)
# all should give back the same type of arguments, e.g. ints for shape
assert m.shape == (3, 2) and all(type(i) is int for i in m.shape)
assert m.rows == 3 and type(m.rows) is int
assert m.cols == 2 and type(m.cols) is int
if not c % 2:
assert type(m._mat) in (list, tuple, Tuple)
else:
assert type(m._smat) is dict
def test_division():
v = Matrix(1, 2, [x, y])
assert v.__div__(z) == Matrix(1, 2, [x/z, y/z])
assert v.__truediv__(z) == Matrix(1, 2, [x/z, y/z])
assert v/z == Matrix(1, 2, [x/z, y/z])
def test_sum():
m = Matrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
assert m + m == Matrix([[2, 4, 6], [2*x, 2*y, 2*x], [4*y, -100, 2*z*x]])
n = Matrix(1, 2, [1, 2])
raises(ShapeError, lambda: m + n)
def test_abs():
m = Matrix(1, 2, [-3, x])
n = Matrix(1, 2, [3, Abs(x)])
assert abs(m) == n
def test_addition():
a = Matrix((
(1, 2),
(3, 1),
))
b = Matrix((
(1, 2),
(3, 0),
))
assert a + b == a.add(b) == Matrix([[2, 4], [6, 1]])
def test_fancy_index_matrix():
for M in (Matrix, SparseMatrix):
a = M(3, 3, range(9))
assert a == a[:, :]
assert a[1, :] == Matrix(1, 3, [3, 4, 5])
assert a[:, 1] == Matrix([1, 4, 7])
assert a[[0, 1], :] == Matrix([[0, 1, 2], [3, 4, 5]])
assert a[[0, 1], 2] == a[[0, 1], [2]]
assert a[2, [0, 1]] == a[[2], [0, 1]]
assert a[:, [0, 1]] == Matrix([[0, 1], [3, 4], [6, 7]])
assert a[0, 0] == 0
assert a[0:2, :] == Matrix([[0, 1, 2], [3, 4, 5]])
assert a[:, 0:2] == Matrix([[0, 1], [3, 4], [6, 7]])
assert a[::2, 1] == a[[0, 2], 1]
assert a[1, ::2] == a[1, [0, 2]]
a = M(3, 3, range(9))
assert a[[0, 2, 1, 2, 1], :] == Matrix([
[0, 1, 2],
[6, 7, 8],
[3, 4, 5],
[6, 7, 8],
[3, 4, 5]])
assert a[:, [0,2,1,2,1]] == Matrix([
[0, 2, 1, 2, 1],
[3, 5, 4, 5, 4],
[6, 8, 7, 8, 7]])
a = SparseMatrix.zeros(3)
a[1, 2] = 2
a[0, 1] = 3
a[2, 0] = 4
assert a.extract([1, 1], [2]) == Matrix([
[2],
[2]])
assert a.extract([1, 0], [2, 2, 2]) == Matrix([
[2, 2, 2],
[0, 0, 0]])
assert a.extract([1, 0, 1, 2], [2, 0, 1, 0]) == Matrix([
[2, 0, 0, 0],
[0, 0, 3, 0],
[2, 0, 0, 0],
[0, 4, 0, 4]])
def test_multiplication():
a = Matrix((
(1, 2),
(3, 1),
(0, 6),
))
b = Matrix((
(1, 2),
(3, 0),
))
c = a*b
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
try:
eval('c = a @ b')
except SyntaxError:
pass
else:
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
h = matrix_multiply_elementwise(a, c)
assert h == a.multiply_elementwise(c)
assert h[0, 0] == 7
assert h[0, 1] == 4
assert h[1, 0] == 18
assert h[1, 1] == 6
assert h[2, 0] == 0
assert h[2, 1] == 0
raises(ShapeError, lambda: matrix_multiply_elementwise(a, b))
c = b * Symbol("x")
assert isinstance(c, Matrix)
assert c[0, 0] == x
assert c[0, 1] == 2*x
assert c[1, 0] == 3*x
assert c[1, 1] == 0
c2 = x * b
assert c == c2
c = 5 * b
assert isinstance(c, Matrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
try:
eval('c = 5 @ b')
except SyntaxError:
pass
else:
assert isinstance(c, Matrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
def test_power():
raises(NonSquareMatrixError, lambda: Matrix((1, 2))**2)
R = Rational
A = Matrix([[2, 3], [4, 5]])
assert (A**-3)[:] == [R(-269)/8, R(153)/8, R(51)/2, R(-29)/2]
assert (A**5)[:] == [6140, 8097, 10796, 14237]
A = Matrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433]
assert A**0 == eye(3)
assert A**1 == A
assert (Matrix([[2]]) ** 100)[0, 0] == 2**100
assert eye(2)**10000000 == eye(2)
assert Matrix([[1, 2], [3, 4]])**Integer(2) == Matrix([[7, 10], [15, 22]])
A = Matrix([[33, 24], [48, 57]])
assert (A**(S(1)/2))[:] == [5, 2, 4, 7]
A = Matrix([[0, 4], [-1, 5]])
assert (A**(S(1)/2))**2 == A
assert Matrix([[1, 0], [1, 1]])**(S(1)/2) == Matrix([[1, 0], [S.Half, 1]])
assert Matrix([[1, 0], [1, 1]])**0.5 == Matrix([[1.0, 0], [0.5, 1.0]])
from sympy.abc import a, b, n
assert Matrix([[1, a], [0, 1]])**n == Matrix([[1, a*n], [0, 1]])
assert Matrix([[b, a], [0, b]])**n == Matrix([[b**n, a*b**(n-1)*n], [0, b**n]])
assert Matrix([[a, 1, 0], [0, a, 1], [0, 0, a]])**n == Matrix([
[a**n, a**(n-1)*n, a**(n-2)*(n-1)*n/2],
[0, a**n, a**(n-1)*n],
[0, 0, a**n]])
assert Matrix([[a, 1, 0], [0, a, 0], [0, 0, b]])**n == Matrix([
[a**n, a**(n-1)*n, 0],
[0, a**n, 0],
[0, 0, b**n]])
A = Matrix([[1, 0], [1, 7]])
assert A._matrix_pow_by_jordan_blocks(3) == A._eval_pow_by_recursion(3)
A = Matrix([[2]])
assert A**10 == Matrix([[2**10]]) == A._matrix_pow_by_jordan_blocks(10) == \
A._eval_pow_by_recursion(10)
# testing a matrix that cannot be jordan blocked issue 11766
m = Matrix([[3, 0, 0, 0, -3], [0, -3, -3, 0, 3], [0, 3, 0, 3, 0], [0, 0, 3, 0, 3], [3, 0, 0, 3, 0]])
raises(MatrixError, lambda: m._matrix_pow_by_jordan_blocks(10))
# test issue 11964
raises(ValueError, lambda: Matrix([[1, 1], [3, 3]])._matrix_pow_by_jordan_blocks(-10))
A = Matrix([[0, 1, 0], [0, 0, 1], [0, 0, 0]]) # Nilpotent jordan block size 3
assert A**10.0 == Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
raises(ValueError, lambda: A**2.1)
raises(ValueError, lambda: A**(S(3)/2))
A = Matrix([[8, 1], [3, 2]])
assert A**10.0 == Matrix([[1760744107, 272388050], [817164150, 126415807]])
A = Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]]) # Nilpotent jordan block size 1
assert A**10.2 == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
A = Matrix([[0, 1, 0], [0, 0, 1], [0, 0, 1]]) # Nilpotent jordan block size 2
assert A**10.0 == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
n = Symbol('n', integer=True)
assert isinstance(A**n, MatPow)
n = Symbol('n', integer=True, nonnegative=True)
raises(ValueError, lambda: A**n)
assert A**(n + 2) == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
raises(ValueError, lambda: A**(S(3)/2))
A = Matrix([[0, 0, 1], [3, 0, 1], [4, 3, 1]])
assert A**5.0 == Matrix([[168, 72, 89], [291, 144, 161], [572, 267, 329]])
assert A**5.0 == A**5
A = Matrix([[0, 1, 0],[-1, 0, 0],[0, 0, 0]])
n = Symbol("n")
An = A**n
assert An.subs(n, 2).doit() == A**2
raises(ValueError, lambda: An.subs(n, -2).doit())
assert An * An == A**(2*n)
def test_creation():
raises(ValueError, lambda: Matrix(5, 5, range(20)))
raises(ValueError, lambda: Matrix(5, -1, []))
raises(IndexError, lambda: Matrix((1, 2))[2])
with raises(IndexError):
Matrix((1, 2))[1:2] = 5
with raises(IndexError):
Matrix((1, 2))[3] = 5
assert Matrix() == Matrix([]) == Matrix([[]]) == Matrix(0, 0, [])
# anything can go into a matrix (laplace_transform uses tuples)
assert Matrix([[[], ()]]).tolist() == [[[], ()]]
assert Matrix([[[], ()]]).T.tolist() == [[[]], [()]]
a = Matrix([[x, 0], [0, 0]])
m = a
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
b = Matrix(2, 2, [x, 0, 0, 0])
m = b
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
assert a == b
assert Matrix(b) == b
c23 = Matrix(2, 3, range(1, 7))
c13 = Matrix(1, 3, range(7, 10))
c = Matrix([c23, c13])
assert c.cols == 3
assert c.rows == 3
assert c[:] == [1, 2, 3, 4, 5, 6, 7, 8, 9]
assert Matrix(eye(2)) == eye(2)
assert ImmutableMatrix(ImmutableMatrix(eye(2))) == ImmutableMatrix(eye(2))
assert ImmutableMatrix(c) == c.as_immutable()
assert Matrix(ImmutableMatrix(c)) == ImmutableMatrix(c).as_mutable()
assert c is not Matrix(c)
dat = [[ones(3,2), ones(3,3)*2], [ones(2,3)*3, ones(2,2)*4]]
M = Matrix(dat)
assert M == Matrix([
[1, 1, 2, 2, 2],
[1, 1, 2, 2, 2],
[1, 1, 2, 2, 2],
[3, 3, 3, 4, 4],
[3, 3, 3, 4, 4]])
assert M.tolist() != dat
# keep block form if evaluate=False
assert Matrix(dat, evaluate=False).tolist() == dat
A = MatrixSymbol("A", 2, 2)
dat = [ones(2), A]
assert Matrix(dat) == Matrix([
[ 1, 1],
[ 1, 1],
[A[0, 0], A[0, 1]],
[A[1, 0], A[1, 1]]])
assert Matrix(dat, evaluate=False).tolist() == [[i] for i in dat]
# 0-dim tolerance
assert Matrix([ones(2), ones(0)]) == Matrix([ones(2)])
raises(ValueError, lambda: Matrix([ones(2), ones(0, 3)]))
raises(ValueError, lambda: Matrix([ones(2), ones(3, 0)]))
def test_irregular_block():
assert Matrix.irregular(3, ones(2,1), ones(3,3)*2, ones(2,2)*3,
ones(1,1)*4, ones(2,2)*5, ones(1,2)*6, ones(1,2)*7) == Matrix([
[1, 2, 2, 2, 3, 3],
[1, 2, 2, 2, 3, 3],
[4, 2, 2, 2, 5, 5],
[6, 6, 7, 7, 5, 5]])
def test_tolist():
lst = [[S.One, S.Half, x*y, S.Zero], [x, y, z, x**2], [y, -S.One, z*x, 3]]
m = Matrix(lst)
assert m.tolist() == lst
def test_as_mutable():
assert zeros(0, 3).as_mutable() == zeros(0, 3)
assert zeros(0, 3).as_immutable() == ImmutableMatrix(zeros(0, 3))
assert zeros(3, 0).as_immutable() == ImmutableMatrix(zeros(3, 0))
def test_determinant():
for M in [Matrix(), Matrix([[1]])]:
assert (
M.det() ==
M._eval_det_bareiss() ==
M._eval_det_berkowitz() ==
M._eval_det_lu() ==
1)
M = Matrix(( (-3, 2),
( 8, -5) ))
assert M.det(method="bareiss") == -1
assert M.det(method="berkowitz") == -1
assert M.det(method="lu") == -1
M = Matrix(( (x, 1),
(y, 2*y) ))
assert M.det(method="bareiss") == 2*x*y - y
assert M.det(method="berkowitz") == 2*x*y - y
assert M.det(method="lu") == 2*x*y - y
M = Matrix(( (1, 1, 1),
(1, 2, 3),
(1, 3, 6) ))
assert M.det(method="bareiss") == 1
assert M.det(method="berkowitz") == 1
assert M.det(method="lu") == 1
M = Matrix(( ( 3, -2, 0, 5),
(-2, 1, -2, 2),
( 0, -2, 5, 0),
( 5, 0, 3, 4) ))
assert M.det(method="bareiss") == -289
assert M.det(method="berkowitz") == -289
assert M.det(method="lu") == -289
M = Matrix(( ( 1, 2, 3, 4),
( 5, 6, 7, 8),
( 9, 10, 11, 12),
(13, 14, 15, 16) ))
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(( (3, 2, 0, 0, 0),
(0, 3, 2, 0, 0),
(0, 0, 3, 2, 0),
(0, 0, 0, 3, 2),
(2, 0, 0, 0, 3) ))
assert M.det(method="bareiss") == 275
assert M.det(method="berkowitz") == 275
assert M.det(method="lu") == 275
M = Matrix(( (1, 0, 1, 2, 12),
(2, 0, 1, 1, 4),
(2, 1, 1, -1, 3),
(3, 2, -1, 1, 8),
(1, 1, 1, 0, 6) ))
assert M.det(method="bareiss") == -55
assert M.det(method="berkowitz") == -55
assert M.det(method="lu") == -55
M = Matrix(( (-5, 2, 3, 4, 5),
( 1, -4, 3, 4, 5),
( 1, 2, -3, 4, 5),
( 1, 2, 3, -2, 5),
( 1, 2, 3, 4, -1) ))
assert M.det(method="bareiss") == 11664
assert M.det(method="berkowitz") == 11664
assert M.det(method="lu") == 11664
M = Matrix(( ( 2, 7, -1, 3, 2),
( 0, 0, 1, 0, 1),
(-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3),
( 1, 0, 0, 0, 1) ))
assert M.det(method="bareiss") == 123
assert M.det(method="berkowitz") == 123
assert M.det(method="lu") == 123
M = Matrix(( (x, y, z),
(1, 0, 0),
(y, z, x) ))
assert M.det(method="bareiss") == z**2 - x*y
assert M.det(method="berkowitz") == z**2 - x*y
assert M.det(method="lu") == z**2 - x*y
# issue 13835
a = symbols('a')
M = lambda n: Matrix([[i + a*j for i in range(n)]
for j in range(n)])
assert M(5).det() == 0
assert M(6).det() == 0
assert M(7).det() == 0
def test_slicing():
m0 = eye(4)
assert m0[:3, :3] == eye(3)
assert m0[2:4, 0:2] == zeros(2)
m1 = Matrix(3, 3, lambda i, j: i + j)
assert m1[0, :] == Matrix(1, 3, (0, 1, 2))
assert m1[1:3, 1] == Matrix(2, 1, (2, 3))
m2 = Matrix([[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15]])
assert m2[:, -1] == Matrix(4, 1, [3, 7, 11, 15])
assert m2[-2:, :] == Matrix([[8, 9, 10, 11], [12, 13, 14, 15]])
def test_submatrix_assignment():
m = zeros(4)
m[2:4, 2:4] = eye(2)
assert m == Matrix(((0, 0, 0, 0),
(0, 0, 0, 0),
(0, 0, 1, 0),
(0, 0, 0, 1)))
m[:2, :2] = eye(2)
assert m == eye(4)
m[:, 0] = Matrix(4, 1, (1, 2, 3, 4))
assert m == Matrix(((1, 0, 0, 0),
(2, 1, 0, 0),
(3, 0, 1, 0),
(4, 0, 0, 1)))
m[:, :] = zeros(4)
assert m == zeros(4)
m[:, :] = [(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)]
assert m == Matrix(((1, 2, 3, 4),
(5, 6, 7, 8),
(9, 10, 11, 12),
(13, 14, 15, 16)))
m[:2, 0] = [0, 0]
assert m == Matrix(((0, 2, 3, 4),
(0, 6, 7, 8),
(9, 10, 11, 12),
(13, 14, 15, 16)))
def test_extract():
m = Matrix(4, 3, lambda i, j: i*3 + j)
assert m.extract([0, 1, 3], [0, 1]) == Matrix(3, 2, [0, 1, 3, 4, 9, 10])
assert m.extract([0, 3], [0, 0, 2]) == Matrix(2, 3, [0, 0, 2, 9, 9, 11])
assert m.extract(range(4), range(3)) == m
raises(IndexError, lambda: m.extract([4], [0]))
raises(IndexError, lambda: m.extract([0], [3]))
def test_reshape():
m0 = eye(3)
assert m0.reshape(1, 9) == Matrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = Matrix(3, 4, lambda i, j: i + j)
assert m1.reshape(
4, 3) == Matrix(((0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)))
assert m1.reshape(2, 6) == Matrix(((0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)))
def test_applyfunc():
m0 = eye(3)
assert m0.applyfunc(lambda x: 2*x) == eye(3)*2
assert m0.applyfunc(lambda x: 0) == zeros(3)
def test_expand():
m0 = Matrix([[x*(x + y), 2], [((x + y)*y)*x, x*(y + x*(x + y))]])
# Test if expand() returns a matrix
m1 = m0.expand()
assert m1 == Matrix(
[[x*y + x**2, 2], [x*y**2 + y*x**2, x*y + y*x**2 + x**3]])
a = Symbol('a', real=True)
assert Matrix([exp(I*a)]).expand(complex=True) == \
Matrix([cos(a) + I*sin(a)])
assert Matrix([[0, 1, 2], [0, 0, -1], [0, 0, 0]]).exp() == Matrix([
[1, 1, Rational(3, 2)],
[0, 1, -1],
[0, 0, 1]]
)
def test_refine():
m0 = Matrix([[Abs(x)**2, sqrt(x**2)],
[sqrt(x**2)*Abs(y)**2, sqrt(y**2)*Abs(x)**2]])
m1 = m0.refine(Q.real(x) & Q.real(y))
assert m1 == Matrix([[x**2, Abs(x)], [y**2*Abs(x), x**2*Abs(y)]])
m1 = m0.refine(Q.positive(x) & Q.positive(y))
assert m1 == Matrix([[x**2, x], [x*y**2, x**2*y]])
m1 = m0.refine(Q.negative(x) & Q.negative(y))
assert m1 == Matrix([[x**2, -x], [-x*y**2, -x**2*y]])
def test_random():
M = randMatrix(3, 3)
M = randMatrix(3, 3, seed=3)
assert M == randMatrix(3, 3, seed=3)
M = randMatrix(3, 4, 0, 150)
M = randMatrix(3, seed=4, symmetric=True)
assert M == randMatrix(3, seed=4, symmetric=True)
S = M.copy()
S.simplify()
assert S == M # doesn't fail when elements are Numbers, not int
rng = random.Random(4)
assert M == randMatrix(3, symmetric=True, prng=rng)
# Ensure symmetry
for size in (10, 11): # Test odd and even
for percent in (100, 70, 30):
M = randMatrix(size, symmetric=True, percent=percent, prng=rng)
assert M == M.T
M = randMatrix(10, min=1, percent=70)
zero_count = 0
for i in range(M.shape[0]):
for j in range(M.shape[1]):
if M[i, j] == 0:
zero_count += 1
assert zero_count == 30
def test_LUdecomp():
testmat = Matrix([[0, 2, 5, 3],
[3, 3, 7, 4],
[8, 4, 0, 2],
[-2, 6, 3, 4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
testmat = Matrix([[6, -2, 7, 4],
[0, 3, 6, 7],
[1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
# non-square
testmat = Matrix([[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
[10, 11, 12]])
L, U, p = testmat.LUdecomposition(rankcheck=False)
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4, 3)
# square and singular
testmat = Matrix([[1, 2, 3],
[2, 4, 6],
[4, 5, 6]])
L, U, p = testmat.LUdecomposition(rankcheck=False)
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(3)
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - M == zeros(3)
mL = Matrix((
(1, 0, 0),
(2, 3, 0),
))
assert mL.is_lower is True
assert mL.is_upper is False
mU = Matrix((
(1, 2, 3),
(0, 4, 5),
))
assert mU.is_lower is False
assert mU.is_upper is True
# test FF LUdecomp
M = Matrix([[1, 3, 3],
[3, 2, 6],
[3, 2, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
M = Matrix([[1, 2, 3, 4],
[3, -1, 2, 3],
[3, 1, 3, -2],
[6, -1, 0, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
M = Matrix([[0, 0, 1],
[2, 3, 0],
[3, 1, 4]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
# issue 15794
M = Matrix(
[[1, 2, 3],
[4, 5, 6],
[7, 8, 9]]
)
raises(ValueError, lambda : M.LUdecomposition_Simple(rankcheck=True))
def test_LUsolve():
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A*x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[2, 1], [1, 0], [1, 0]]) # issue 14548
b = Matrix([3, 1, 1])
assert A.LUsolve(b) == Matrix([1, 1])
b = Matrix([3, 1, 2]) # inconsistent
raises(ValueError, lambda: A.LUsolve(b))
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4],
[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix([2, 1, -4])
b = A*x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[0, -1, 2], [5, 10, 7]]) # underdetermined
x = Matrix([-1, 2, 0])
b = A*x
raises(NotImplementedError, lambda: A.LUsolve(b))
A = Matrix(4, 4, lambda i, j: 1/(i+j+1) if i != 3 else 0)
b = Matrix.zeros(4, 1)
raises(NotImplementedError, lambda: A.LUsolve(b))
def test_QRsolve():
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A*x
soln = A.QRsolve(b)
assert soln == x
x = Matrix([[1, 2], [3, 4], [5, 6]])
b = A*x
soln = A.QRsolve(b)
assert soln == x
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.QRsolve(b)
assert soln == x
x = Matrix([[7, 8], [9, 10], [11, 12]])
b = A*x
soln = A.QRsolve(b)
assert soln == x
def test_inverse():
A = eye(4)
assert A.inv() == eye(4)
assert A.inv(method="LU") == eye(4)
assert A.inv(method="ADJ") == eye(4)
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
Ainv = A.inv()
assert A*Ainv == eye(3)
assert A.inv(method="LU") == Ainv
assert A.inv(method="ADJ") == Ainv
# test that immutability is not a problem
cls = ImmutableMatrix
m = cls([[48, 49, 31],
[ 9, 71, 94],
[59, 28, 65]])
assert all(type(m.inv(s)) is cls for s in 'GE ADJ LU'.split())
cls = ImmutableSparseMatrix
m = cls([[48, 49, 31],
[ 9, 71, 94],
[59, 28, 65]])
assert all(type(m.inv(s)) is cls for s in 'CH LDL'.split())
def test_matrix_inverse_mod():
A = Matrix(2, 1, [1, 0])
raises(NonSquareMatrixError, lambda: A.inv_mod(2))
A = Matrix(2, 2, [1, 0, 0, 0])
raises(ValueError, lambda: A.inv_mod(2))
A = Matrix(2, 2, [1, 2, 3, 4])
Ai = Matrix(2, 2, [1, 1, 0, 1])
assert A.inv_mod(3) == Ai
A = Matrix(2, 2, [1, 0, 0, 1])
assert A.inv_mod(2) == A
A = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
raises(ValueError, lambda: A.inv_mod(5))
A = Matrix(3, 3, [5, 1, 3, 2, 6, 0, 2, 1, 1])
Ai = Matrix(3, 3, [6, 8, 0, 1, 5, 6, 5, 6, 4])
assert A.inv_mod(9) == Ai
A = Matrix(3, 3, [1, 6, -3, 4, 1, -5, 3, -5, 5])
Ai = Matrix(3, 3, [4, 3, 3, 1, 2, 5, 1, 5, 1])
assert A.inv_mod(6) == Ai
A = Matrix(3, 3, [1, 6, 1, 4, 1, 5, 3, 2, 5])
Ai = Matrix(3, 3, [6, 0, 3, 6, 6, 4, 1, 6, 1])
assert A.inv_mod(7) == Ai
def test_util():
R = Rational
v1 = Matrix(1, 3, [1, 2, 3])
v2 = Matrix(1, 3, [3, 4, 5])
assert v1.norm() == sqrt(14)
assert v1.project(v2) == Matrix(1, 3, [R(39)/25, R(52)/25, R(13)/5])
assert Matrix.zeros(1, 2) == Matrix(1, 2, [0, 0])
assert ones(1, 2) == Matrix(1, 2, [1, 1])
assert v1.copy() == v1
# cofactor
assert eye(3) == eye(3).cofactor_matrix()
test = Matrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
assert test.cofactor_matrix() == \
Matrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
test = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert test.cofactor_matrix() == \
Matrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])
def test_jacobian_hessian():
L = Matrix(1, 2, [x**2*y, 2*y**2 + x*y])
syms = [x, y]
assert L.jacobian(syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]])
L = Matrix(1, 2, [x, x**2*y**3])
assert L.jacobian(syms) == Matrix([[1, 0], [2*x*y**3, x**2*3*y**2]])
f = x**2*y
syms = [x, y]
assert hessian(f, syms) == Matrix([[2*y, 2*x], [2*x, 0]])
f = x**2*y**3
assert hessian(f, syms) == \
Matrix([[2*y**3, 6*x*y**2], [6*x*y**2, 6*x**2*y]])
f = z + x*y**2
g = x**2 + 2*y**3
ans = Matrix([[0, 2*y],
[2*y, 2*x]])
assert ans == hessian(f, Matrix([x, y]))
assert ans == hessian(f, Matrix([x, y]).T)
assert hessian(f, (y, x), [g]) == Matrix([
[ 0, 6*y**2, 2*x],
[6*y**2, 2*x, 2*y],
[ 2*x, 2*y, 0]])
def test_QR():
A = Matrix([[1, 2], [2, 3]])
Q, S = A.QRdecomposition()
R = Rational
assert Q == Matrix([
[ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)],
[2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
assert S == Matrix([[5**R(1, 2), 8*5**R(-1, 2)], [0, (R(1)/5)**R(1, 2)]])
assert Q*S == A
assert Q.T * Q == eye(2)
A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_non_square():
# Narrow (cols < rows) matrices
A = Matrix([[9, 0, 26], [12, 0, -7], [0, 4, 4], [0, -3, -3]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, -1, 4], [1, 4, -2], [1, 4, 2], [1, -1, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix(2, 1, [1, 2])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Wide (cols > rows) matrices
A = Matrix([[1, 2, 3], [4, 5, 6]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 2, 3, 4], [1, 4, 9, 16], [1, 8, 27, 64]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix(1, 2, [1, 2])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_trivial():
# Rank deficient matrices
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Zero rank matrices
A = Matrix([[0, 0, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Rank deficient matrices with zero norm from beginning columns
A = Matrix([[0, 0, 0], [1, 2, 3]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0], [2, 4, 6, 8]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 2, 3]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_nullspace():
# first test reduced row-ech form
R = Rational
M = Matrix([[5, 7, 2, 1],
[1, 6, 2, -1]])
out, tmp = M.rref()
assert out == Matrix([[1, 0, -R(2)/23, R(13)/23],
[0, 1, R(8)/23, R(-6)/23]])
M = Matrix([[-5, -1, 4, -3, -1],
[ 1, -1, -1, 1, 0],
[-1, 0, 0, 0, 0],
[ 4, 1, -4, 3, 1],
[-2, 0, 2, -2, -1]])
assert M*M.nullspace()[0] == Matrix(5, 1, [0]*5)
M = Matrix([[ 1, 3, 0, 2, 6, 3, 1],
[-2, -6, 0, -2, -8, 3, 1],
[ 3, 9, 0, 0, 6, 6, 2],
[-1, -3, 0, 1, 0, 9, 3]])
out, tmp = M.rref()
assert out == Matrix([[1, 3, 0, 0, 2, 0, 0],
[0, 0, 0, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 1, R(1)/3],
[0, 0, 0, 0, 0, 0, 0]])
# now check the vectors
basis = M.nullspace()
assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0])
assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0])
assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0])
assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1])
# issue 4797; just see that we can do it when rows > cols
M = Matrix([[1, 2], [2, 4], [3, 6]])
assert M.nullspace()
def test_columnspace():
M = Matrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
# now check the vectors
basis = M.columnspace()
assert basis[0] == Matrix([1, -2, 0, 3])
assert basis[1] == Matrix([2, -5, -3, 6])
assert basis[2] == Matrix([2, -1, 4, -7])
#check by columnspace definition
a, b, c, d, e = symbols('a b c d e')
X = Matrix([a, b, c, d, e])
for i in range(len(basis)):
eq=M*X-basis[i]
assert len(solve(eq, X)) != 0
#check if rank-nullity theorem holds
assert M.rank() == len(basis)
assert len(M.nullspace()) + len(M.columnspace()) == M.cols
def test_wronskian():
assert wronskian([cos(x), sin(x)], x) == cos(x)**2 + sin(x)**2
assert wronskian([exp(x), exp(2*x)], x) == exp(3*x)
assert wronskian([exp(x), x], x) == exp(x) - x*exp(x)
assert wronskian([1, x, x**2], x) == 2
w1 = -6*exp(x)*sin(x)*x + 6*cos(x)*exp(x)*x**2 - 6*exp(x)*cos(x)*x - \
exp(x)*cos(x)*x**3 + exp(x)*sin(x)*x**3
assert wronskian([exp(x), cos(x), x**3], x).expand() == w1
assert wronskian([exp(x), cos(x), x**3], x, method='berkowitz').expand() \
== w1
w2 = -x**3*cos(x)**2 - x**3*sin(x)**2 - 6*x*cos(x)**2 - 6*x*sin(x)**2
assert wronskian([sin(x), cos(x), x**3], x).expand() == w2
assert wronskian([sin(x), cos(x), x**3], x, method='berkowitz').expand() \
== w2
assert wronskian([], x) == 1
def test_eigen():
R = Rational
assert eye(3).charpoly(x) == Poly((x - 1)**3, x)
assert eye(3).charpoly(y) == Poly((y - 1)**3, y)
M = Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
assert M.eigenvals(multiple=False) == {S.One: 3}
assert M.eigenvals(multiple=True) == [1, 1, 1]
assert M.eigenvects() == (
[(1, 3, [Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])])
assert M.left_eigenvects() == (
[(1, 3, [Matrix([[1, 0, 0]]),
Matrix([[0, 1, 0]]),
Matrix([[0, 0, 1]])])])
M = Matrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}
assert M.eigenvects() == (
[
(-1, 1, [Matrix([-1, 1, 0])]),
( 0, 1, [Matrix([0, -1, 1])]),
( 2, 1, [Matrix([R(2, 3), R(1, 3), 1])])
])
assert M.left_eigenvects() == (
[
(-1, 1, [Matrix([[-2, 1, 1]])]),
(0, 1, [Matrix([[-1, -1, 1]])]),
(2, 1, [Matrix([[1, 1, 1]])])
])
a = Symbol('a')
M = Matrix([[a, 0],
[0, 1]])
assert M.eigenvals() == {a: 1, S.One: 1}
M = Matrix([[1, -1],
[1, 3]])
assert M.eigenvects() == ([(2, 2, [Matrix(2, 1, [-1, 1])])])
assert M.left_eigenvects() == ([(2, 2, [Matrix([[1, 1]])])])
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
a = R(15, 2)
b = 3*33**R(1, 2)
c = R(13, 2)
d = (R(33, 8) + 3*b/8)
e = (R(33, 8) - 3*b/8)
def NS(e, n):
return str(N(e, n))
r = [
(a - b/2, 1, [Matrix([(12 + 24/(c - b/2))/((c - b/2)*e) + 3/(c - b/2),
(6 + 12/(c - b/2))/e, 1])]),
( 0, 1, [Matrix([1, -2, 1])]),
(a + b/2, 1, [Matrix([(12 + 24/(c + b/2))/((c + b/2)*d) + 3/(c + b/2),
(6 + 12/(c + b/2))/d, 1])]),
]
r1 = [(NS(r[i][0], 2), NS(r[i][1], 2),
[NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
r = M.eigenvects()
r2 = [(NS(r[i][0], 2), NS(r[i][1], 2),
[NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
assert sorted(r1) == sorted(r2)
eps = Symbol('eps', real=True)
M = Matrix([[abs(eps), I*eps ],
[-I*eps, abs(eps) ]])
assert M.eigenvects() == (
[
( 0, 1, [Matrix([[-I*eps/abs(eps)], [1]])]),
( 2*abs(eps), 1, [ Matrix([[I*eps/abs(eps)], [1]]) ] ),
])
assert M.left_eigenvects() == (
[
(0, 1, [Matrix([[I*eps/Abs(eps), 1]])]),
(2*Abs(eps), 1, [Matrix([[-I*eps/Abs(eps), 1]])])
])
M = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
M._eigenvects = M.eigenvects(simplify=False)
assert max(i.q for i in M._eigenvects[0][2][0]) > 1
M._eigenvects = M.eigenvects(simplify=True)
assert max(i.q for i in M._eigenvects[0][2][0]) == 1
M = Matrix([[S(1)/4, 1], [1, 1]])
assert M.eigenvects(simplify=True) == [
(S(5)/8 - sqrt(73)/8, 1, [Matrix([[-sqrt(73)/8 - S(3)/8], [1]])]),
(S(5)/8 + sqrt(73)/8, 1, [Matrix([[-S(3)/8 + sqrt(73)/8], [1]])])]
assert M.eigenvects(simplify=False) ==[
(S(5)/8 - sqrt(73)/8, 1, [Matrix([[-1/(-S(3)/8 + sqrt(73)/8)],
[ 1]])]),
(S(5)/8 + sqrt(73)/8, 1, [Matrix([[-1/(-sqrt(73)/8 - S(3)/8)],
[ 1]])])]
m = Matrix([[1, .6, .6], [.6, .9, .9], [.9, .6, .6]])
evals = { S(5)/4 - sqrt(385)/20: 1, sqrt(385)/20 + S(5)/4: 1, S.Zero: 1}
assert m.eigenvals() == evals
nevals = list(sorted(m.eigenvals(rational=False).keys()))
sevals = list(sorted(evals.keys()))
assert all(abs(nevals[i] - sevals[i]) < 1e-9 for i in range(len(nevals)))
# issue 10719
assert Matrix([]).eigenvals() == {}
assert Matrix([]).eigenvects() == []
# issue 15119
raises(NonSquareMatrixError, lambda : Matrix([[1, 2], [0, 4], [0, 0]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 0], [3, 4], [5, 6]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals(error_when_incomplete = False))
raises(NonSquareMatrixError, lambda : Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals(error_when_incomplete = False))
# issue 15125
from sympy.core.function import count_ops
q = Symbol("q", positive = True)
m = Matrix([[-2, exp(-q), 1], [exp(q), -2, 1], [1, 1, -2]])
assert count_ops(m.eigenvals(simplify=False)) > count_ops(m.eigenvals(simplify=True))
assert count_ops(m.eigenvals(simplify=lambda x: x)) > count_ops(m.eigenvals(simplify=True))
assert isinstance(m.eigenvals(simplify=True, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=True, multiple=True), list)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=True), list)
def test_subs():
assert Matrix([[1, x], [x, 4]]).subs(x, 5) == Matrix([[1, 5], [5, 4]])
assert Matrix([[x, 2], [x + y, 4]]).subs([[x, -1], [y, -2]]) == \
Matrix([[-1, 2], [-3, 4]])
assert Matrix([[x, 2], [x + y, 4]]).subs([(x, -1), (y, -2)]) == \
Matrix([[-1, 2], [-3, 4]])
assert Matrix([[x, 2], [x + y, 4]]).subs({x: -1, y: -2}) == \
Matrix([[-1, 2], [-3, 4]])
assert Matrix([x*y]).subs({x: y - 1, y: x - 1}, simultaneous=True) == \
Matrix([(x - 1)*(y - 1)])
for cls in classes:
assert Matrix([[2, 0], [0, 2]]) == cls.eye(2).subs(1, 2)
def test_xreplace():
assert Matrix([[1, x], [x, 4]]).xreplace({x: 5}) == \
Matrix([[1, 5], [5, 4]])
assert Matrix([[x, 2], [x + y, 4]]).xreplace({x: -1, y: -2}) == \
Matrix([[-1, 2], [-3, 4]])
for cls in classes:
assert Matrix([[2, 0], [0, 2]]) == cls.eye(2).xreplace({1: 2})
def test_simplify():
n = Symbol('n')
f = Function('f')
M = Matrix([[ 1/x + 1/y, (x + x*y) / x ],
[ (f(x) + y*f(x))/f(x), 2 * (1/n - cos(n * pi)/n) / pi ]])
M.simplify()
assert M == Matrix([[ (x + y)/(x * y), 1 + y ],
[ 1 + y, 2*((1 - 1*cos(pi*n))/(pi*n)) ]])
eq = (1 + x)**2
M = Matrix([[eq]])
M.simplify()
assert M == Matrix([[eq]])
M.simplify(ratio=oo) == M
assert M == Matrix([[eq.simplify(ratio=oo)]])
def test_transpose():
M = Matrix([[1, 2, 3, 4, 5, 6, 7, 8, 9, 0],
[1, 2, 3, 4, 5, 6, 7, 8, 9, 0]])
assert M.T == Matrix( [ [1, 1],
[2, 2],
[3, 3],
[4, 4],
[5, 5],
[6, 6],
[7, 7],
[8, 8],
[9, 9],
[0, 0] ])
assert M.T.T == M
assert M.T == M.transpose()
def test_conjugate():
M = Matrix([[0, I, 5],
[1, 2, 0]])
assert M.T == Matrix([[0, 1],
[I, 2],
[5, 0]])
assert M.C == Matrix([[0, -I, 5],
[1, 2, 0]])
assert M.C == M.conjugate()
assert M.H == M.T.C
assert M.H == Matrix([[ 0, 1],
[-I, 2],
[ 5, 0]])
def test_conj_dirac():
raises(AttributeError, lambda: eye(3).D)
M = Matrix([[1, I, I, I],
[0, 1, I, I],
[0, 0, 1, I],
[0, 0, 0, 1]])
assert M.D == Matrix([[ 1, 0, 0, 0],
[-I, 1, 0, 0],
[-I, -I, -1, 0],
[-I, -I, I, -1]])
def test_trace():
M = Matrix([[1, 0, 0],
[0, 5, 0],
[0, 0, 8]])
assert M.trace() == 14
def test_shape():
M = Matrix([[x, 0, 0],
[0, y, 0]])
assert M.shape == (2, 3)
def test_col_row_op():
M = Matrix([[x, 0, 0],
[0, y, 0]])
M.row_op(1, lambda r, j: r + j + 1)
assert M == Matrix([[x, 0, 0],
[1, y + 2, 3]])
M.col_op(0, lambda c, j: c + y**j)
assert M == Matrix([[x + 1, 0, 0],
[1 + y, y + 2, 3]])
# neither row nor slice give copies that allow the original matrix to
# be changed
assert M.row(0) == Matrix([[x + 1, 0, 0]])
r1 = M.row(0)
r1[0] = 42
assert M[0, 0] == x + 1
r1 = M[0, :-1] # also testing negative slice
r1[0] = 42
assert M[0, 0] == x + 1
c1 = M.col(0)
assert c1 == Matrix([x + 1, 1 + y])
c1[0] = 0
assert M[0, 0] == x + 1
c1 = M[:, 0]
c1[0] = 42
assert M[0, 0] == x + 1
def test_zip_row_op():
for cls in classes[:2]: # XXX: immutable matrices don't support row ops
M = cls.eye(3)
M.zip_row_op(1, 0, lambda v, u: v + 2*u)
assert M == cls([[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
M = cls.eye(3)*2
M[0, 1] = -1
M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
assert M == cls([[2, -1, 0],
[4, 0, 0],
[0, 0, 2]])
def test_issue_3950():
m = Matrix([1, 2, 3])
a = Matrix([1, 2, 3])
b = Matrix([2, 2, 3])
assert not (m in [])
assert not (m in [1])
assert m != 1
assert m == a
assert m != b
def test_issue_3981():
class Index1(object):
def __index__(self):
return 1
class Index2(object):
def __index__(self):
return 2
index1 = Index1()
index2 = Index2()
m = Matrix([1, 2, 3])
assert m[index2] == 3
m[index2] = 5
assert m[2] == 5
m = Matrix([[1, 2, 3], [4, 5, 6]])
assert m[index1, index2] == 6
assert m[1, index2] == 6
assert m[index1, 2] == 6
m[index1, index2] = 4
assert m[1, 2] == 4
m[1, index2] = 6
assert m[1, 2] == 6
m[index1, 2] = 8
assert m[1, 2] == 8
def test_evalf():
a = Matrix([sqrt(5), 6])
assert all(a.evalf()[i] == a[i].evalf() for i in range(2))
assert all(a.evalf(2)[i] == a[i].evalf(2) for i in range(2))
assert all(a.n(2)[i] == a[i].n(2) for i in range(2))
def test_is_symbolic():
a = Matrix([[x, x], [x, x]])
assert a.is_symbolic() is True
a = Matrix([[1, 2, 3, 4], [5, 6, 7, 8]])
assert a.is_symbolic() is False
a = Matrix([[1, 2, 3, 4], [5, 6, x, 8]])
assert a.is_symbolic() is True
a = Matrix([[1, x, 3]])
assert a.is_symbolic() is True
a = Matrix([[1, 2, 3]])
assert a.is_symbolic() is False
a = Matrix([[1], [x], [3]])
assert a.is_symbolic() is True
a = Matrix([[1], [2], [3]])
assert a.is_symbolic() is False
def test_is_upper():
a = Matrix([[1, 2, 3]])
assert a.is_upper is True
a = Matrix([[1], [2], [3]])
assert a.is_upper is False
a = zeros(4, 2)
assert a.is_upper is True
def test_is_lower():
a = Matrix([[1, 2, 3]])
assert a.is_lower is False
a = Matrix([[1], [2], [3]])
assert a.is_lower is True
def test_is_nilpotent():
a = Matrix(4, 4, [0, 2, 1, 6, 0, 0, 1, 2, 0, 0, 0, 3, 0, 0, 0, 0])
assert a.is_nilpotent()
a = Matrix([[1, 0], [0, 1]])
assert not a.is_nilpotent()
a = Matrix([])
assert a.is_nilpotent()
def test_zeros_ones_fill():
n, m = 3, 5
a = zeros(n, m)
a.fill( 5 )
b = 5 * ones(n, m)
assert a == b
assert a.rows == b.rows == 3
assert a.cols == b.cols == 5
assert a.shape == b.shape == (3, 5)
assert zeros(2) == zeros(2, 2)
assert ones(2) == ones(2, 2)
assert zeros(2, 3) == Matrix(2, 3, [0]*6)
assert ones(2, 3) == Matrix(2, 3, [1]*6)
def test_empty_zeros():
a = zeros(0)
assert a == Matrix()
a = zeros(0, 2)
assert a.rows == 0
assert a.cols == 2
a = zeros(2, 0)
assert a.rows == 2
assert a.cols == 0
def test_issue_3749():
a = Matrix([[x**2, x*y], [x*sin(y), x*cos(y)]])
assert a.diff(x) == Matrix([[2*x, y], [sin(y), cos(y)]])
assert Matrix([
[x, -x, x**2],
[exp(x), 1/x - exp(-x), x + 1/x]]).limit(x, oo) == \
Matrix([[oo, -oo, oo], [oo, 0, oo]])
assert Matrix([
[(exp(x) - 1)/x, 2*x + y*x, x**x ],
[1/x, abs(x), abs(sin(x + 1))]]).limit(x, 0) == \
Matrix([[1, 0, 1], [oo, 0, sin(1)]])
assert a.integrate(x) == Matrix([
[Rational(1, 3)*x**3, y*x**2/2],
[x**2*sin(y)/2, x**2*cos(y)/2]])
def test_inv_iszerofunc():
A = eye(4)
A.col_swap(0, 1)
for method in "GE", "LU":
assert A.inv(method=method, iszerofunc=lambda x: x == 0) == \
A.inv(method="ADJ")
def test_jacobian_metrics():
rho, phi = symbols("rho,phi")
X = Matrix([rho*cos(phi), rho*sin(phi)])
Y = Matrix([rho, phi])
J = X.jacobian(Y)
assert J == X.jacobian(Y.T)
assert J == (X.T).jacobian(Y)
assert J == (X.T).jacobian(Y.T)
g = J.T*eye(J.shape[0])*J
g = g.applyfunc(trigsimp)
assert g == Matrix([[1, 0], [0, rho**2]])
def test_jacobian2():
rho, phi = symbols("rho,phi")
X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])
Y = Matrix([rho, phi])
J = Matrix([
[cos(phi), -rho*sin(phi)],
[sin(phi), rho*cos(phi)],
[ 2*rho, 0],
])
assert X.jacobian(Y) == J
def test_issue_4564():
X = Matrix([exp(x + y + z), exp(x + y + z), exp(x + y + z)])
Y = Matrix([x, y, z])
for i in range(1, 3):
for j in range(1, 3):
X_slice = X[:i, :]
Y_slice = Y[:j, :]
J = X_slice.jacobian(Y_slice)
assert J.rows == i
assert J.cols == j
for k in range(j):
assert J[:, k] == X_slice
def test_nonvectorJacobian():
X = Matrix([[exp(x + y + z), exp(x + y + z)],
[exp(x + y + z), exp(x + y + z)]])
raises(TypeError, lambda: X.jacobian(Matrix([x, y, z])))
X = X[0, :]
Y = Matrix([[x, y], [x, z]])
raises(TypeError, lambda: X.jacobian(Y))
raises(TypeError, lambda: X.jacobian(Matrix([ [x, y], [x, z] ])))
def test_vec():
m = Matrix([[1, 3], [2, 4]])
m_vec = m.vec()
assert m_vec.cols == 1
for i in range(4):
assert m_vec[i] == i + 1
def test_vech():
m = Matrix([[1, 2], [2, 3]])
m_vech = m.vech()
assert m_vech.cols == 1
for i in range(3):
assert m_vech[i] == i + 1
m_vech = m.vech(diagonal=False)
assert m_vech[0] == 2
m = Matrix([[1, x*(x + y)], [y*x + x**2, 1]])
m_vech = m.vech(diagonal=False)
assert m_vech[0] == x*(x + y)
m = Matrix([[1, x*(x + y)], [y*x, 1]])
m_vech = m.vech(diagonal=False, check_symmetry=False)
assert m_vech[0] == y*x
def test_vech_errors():
m = Matrix([[1, 3]])
raises(ShapeError, lambda: m.vech())
m = Matrix([[1, 3], [2, 4]])
raises(ValueError, lambda: m.vech())
raises(ShapeError, lambda: Matrix([ [1, 3] ]).vech())
raises(ValueError, lambda: Matrix([ [1, 3], [2, 4] ]).vech())
def test_diag():
# mostly tested in testcommonmatrix.py
assert diag([1, 2, 3]) == Matrix([1, 2, 3])
m = [1, 2, [3]]
raises(ValueError, lambda: diag(m))
assert diag(m, strict=False) == Matrix([1, 2, 3])
def test_get_diag_blocks1():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert a.get_diag_blocks() == [a]
assert b.get_diag_blocks() == [b]
assert c.get_diag_blocks() == [c]
def test_get_diag_blocks2():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert diag(a, b, b).get_diag_blocks() == [a, b, b]
assert diag(a, b, c).get_diag_blocks() == [a, b, c]
assert diag(a, c, b).get_diag_blocks() == [a, c, b]
assert diag(c, c, b).get_diag_blocks() == [c, c, b]
def test_inv_block():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
A = diag(a, b, b)
assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), b.inv())
A = diag(a, b, c)
assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), c.inv())
A = diag(a, c, b)
assert A.inv(try_block_diag=True) == diag(a.inv(), c.inv(), b.inv())
A = diag(a, a, b, a, c, a)
assert A.inv(try_block_diag=True) == diag(
a.inv(), a.inv(), b.inv(), a.inv(), c.inv(), a.inv())
assert A.inv(try_block_diag=True, method="ADJ") == diag(
a.inv(method="ADJ"), a.inv(method="ADJ"), b.inv(method="ADJ"),
a.inv(method="ADJ"), c.inv(method="ADJ"), a.inv(method="ADJ"))
def test_creation_args():
"""
Check that matrix dimensions can be specified using any reasonable type
(see issue 4614).
"""
raises(ValueError, lambda: zeros(3, -1))
raises(TypeError, lambda: zeros(1, 2, 3, 4))
assert zeros(long(3)) == zeros(3)
assert zeros(Integer(3)) == zeros(3)
raises(ValueError, lambda: zeros(3.))
assert eye(long(3)) == eye(3)
assert eye(Integer(3)) == eye(3)
raises(ValueError, lambda: eye(3.))
assert ones(long(3), Integer(4)) == ones(3, 4)
raises(TypeError, lambda: Matrix(5))
raises(TypeError, lambda: Matrix(1, 2))
raises(ValueError, lambda: Matrix([1, [2]]))
def test_diagonal_symmetrical():
m = Matrix(2, 2, [0, 1, 1, 0])
assert not m.is_diagonal()
assert m.is_symmetric()
assert m.is_symmetric(simplify=False)
m = Matrix(2, 2, [1, 0, 0, 1])
assert m.is_diagonal()
m = diag(1, 2, 3)
assert m.is_diagonal()
assert m.is_symmetric()
m = Matrix(3, 3, [1, 0, 0, 0, 2, 0, 0, 0, 3])
assert m == diag(1, 2, 3)
m = Matrix(2, 3, zeros(2, 3))
assert not m.is_symmetric()
assert m.is_diagonal()
m = Matrix(((5, 0), (0, 6), (0, 0)))
assert m.is_diagonal()
m = Matrix(((5, 0, 0), (0, 6, 0)))
assert m.is_diagonal()
m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2, 2, 0, y, 0, 3])
assert m.is_symmetric()
assert not m.is_symmetric(simplify=False)
assert m.expand().is_symmetric(simplify=False)
def test_diagonalization():
m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
assert not m.is_diagonalizable()
assert not m.is_symmetric()
raises(NonSquareMatrixError, lambda: m.diagonalize())
# diagonalizable
m = diag(1, 2, 3)
(P, D) = m.diagonalize()
assert P == eye(3)
assert D == m
m = Matrix(2, 2, [0, 1, 1, 0])
assert m.is_symmetric()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(2, 2, [1, 0, 0, 3])
assert m.is_symmetric()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
assert P == eye(2)
assert D == m
m = Matrix(2, 2, [1, 1, 0, 0])
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
for i in P:
assert i.as_numer_denom()[1] == 1
m = Matrix(2, 2, [1, 0, 0, 0])
assert m.is_diagonal()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
assert P == Matrix([[0, 1], [1, 0]])
# diagonalizable, complex only
m = Matrix(2, 2, [0, 1, -1, 0])
assert not m.is_diagonalizable(True)
raises(MatrixError, lambda: m.diagonalize(True))
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
# not diagonalizable
m = Matrix(2, 2, [0, 1, 0, 0])
assert not m.is_diagonalizable()
raises(MatrixError, lambda: m.diagonalize())
m = Matrix(3, 3, [-3, 1, -3, 20, 3, 10, 2, -2, 4])
assert not m.is_diagonalizable()
raises(MatrixError, lambda: m.diagonalize())
# symbolic
a, b, c, d = symbols('a b c d')
m = Matrix(2, 2, [a, c, c, b])
assert m.is_symmetric()
assert m.is_diagonalizable()
def test_issue_15887():
# Mutable matrix should not use cache
a = MutableDenseMatrix([[0, 1], [1, 0]])
assert a.is_diagonalizable() is True
a[1, 0] = 0
assert a.is_diagonalizable() is False
a = MutableDenseMatrix([[0, 1], [1, 0]])
a.diagonalize()
a[1, 0] = 0
raises(MatrixError, lambda: a.diagonalize())
# Test deprecated cache and kwargs
with warns_deprecated_sympy():
a._cache_eigenvects
with warns_deprecated_sympy():
a._cache_is_diagonalizable
with warns_deprecated_sympy():
a.is_diagonalizable(clear_cache=True)
with warns_deprecated_sympy():
a.is_diagonalizable(clear_subproducts=True)
@XFAIL
def test_eigen_vects():
m = Matrix(2, 2, [1, 0, 0, I])
raises(NotImplementedError, lambda: m.is_diagonalizable(True))
# !!! bug because of eigenvects() or roots(x**2 + (-1 - I)*x + I, x)
# see issue 5292
assert not m.is_diagonalizable(True)
raises(MatrixError, lambda: m.diagonalize(True))
(P, D) = m.diagonalize(True)
def test_jordan_form():
m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
raises(NonSquareMatrixError, lambda: m.jordan_form())
# diagonalizable
m = Matrix(3, 3, [7, -12, 6, 10, -19, 10, 12, -24, 13])
Jmust = Matrix(3, 3, [-1, 0, 0, 0, 1, 0, 0, 0, 1])
P, J = m.jordan_form()
assert Jmust == J
assert Jmust == m.diagonalize()[1]
# m = Matrix(3, 3, [0, 6, 3, 1, 3, 1, -2, 2, 1])
# m.jordan_form() # very long
# m.jordan_form() #
# diagonalizable, complex only
# Jordan cells
# complexity: one of eigenvalues is zero
m = Matrix(3, 3, [0, 1, 0, -4, 4, 0, -2, 1, 2])
# The blocks are ordered according to the value of their eigenvalues,
# in order to make the matrix compatible with .diagonalize()
Jmust = Matrix(3, 3, [2, 1, 0, 0, 2, 0, 0, 0, 2])
P, J = m.jordan_form()
assert Jmust == J
# complexity: all of eigenvalues are equal
m = Matrix(3, 3, [2, 6, -15, 1, 1, -5, 1, 2, -6])
# Jmust = Matrix(3, 3, [-1, 0, 0, 0, -1, 1, 0, 0, -1])
# same here see 1456ff
Jmust = Matrix(3, 3, [-1, 1, 0, 0, -1, 0, 0, 0, -1])
P, J = m.jordan_form()
assert Jmust == J
# complexity: two of eigenvalues are zero
m = Matrix(3, 3, [4, -5, 2, 5, -7, 3, 6, -9, 4])
Jmust = Matrix(3, 3, [0, 1, 0, 0, 0, 0, 0, 0, 1])
P, J = m.jordan_form()
assert Jmust == J
m = Matrix(4, 4, [6, 5, -2, -3, -3, -1, 3, 3, 2, 1, -2, -3, -1, 1, 5, 5])
Jmust = Matrix(4, 4, [2, 1, 0, 0,
0, 2, 0, 0,
0, 0, 2, 1,
0, 0, 0, 2]
)
P, J = m.jordan_form()
assert Jmust == J
m = Matrix(4, 4, [6, 2, -8, -6, -3, 2, 9, 6, 2, -2, -8, -6, -1, 0, 3, 4])
# Jmust = Matrix(4, 4, [2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, -2])
# same here see 1456ff
Jmust = Matrix(4, 4, [-2, 0, 0, 0,
0, 2, 1, 0,
0, 0, 2, 0,
0, 0, 0, 2])
P, J = m.jordan_form()
assert Jmust == J
m = Matrix(4, 4, [5, 4, 2, 1, 0, 1, -1, -1, -1, -1, 3, 0, 1, 1, -1, 2])
assert not m.is_diagonalizable()
Jmust = Matrix(4, 4, [1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 1, 0, 0, 0, 4])
P, J = m.jordan_form()
assert Jmust == J
# checking for maximum precision to remain unchanged
m = Matrix([[Float('1.0', precision=110), Float('2.0', precision=110)],
[Float('3.14159265358979323846264338327', precision=110), Float('4.0', precision=110)]])
P, J = m.jordan_form()
for term in J._mat:
if isinstance(term, Float):
assert term._prec == 110
def test_jordan_form_complex_issue_9274():
A = Matrix([[ 2, 4, 1, 0],
[-4, 2, 0, 1],
[ 0, 0, 2, 4],
[ 0, 0, -4, 2]])
p = 2 - 4*I;
q = 2 + 4*I;
Jmust1 = Matrix([[p, 1, 0, 0],
[0, p, 0, 0],
[0, 0, q, 1],
[0, 0, 0, q]])
Jmust2 = Matrix([[q, 1, 0, 0],
[0, q, 0, 0],
[0, 0, p, 1],
[0, 0, 0, p]])
P, J = A.jordan_form()
assert J == Jmust1 or J == Jmust2
assert simplify(P*J*P.inv()) == A
def test_issue_10220():
# two non-orthogonal Jordan blocks with eigenvalue 1
M = Matrix([[1, 0, 0, 1],
[0, 1, 1, 0],
[0, 0, 1, 1],
[0, 0, 0, 1]])
P, J = M.jordan_form()
assert P == Matrix([[0, 1, 0, 1],
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0]])
assert J == Matrix([
[1, 1, 0, 0],
[0, 1, 1, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]])
def test_jordan_form_issue_15858():
A = Matrix([
[1, 1, 1, 0],
[-2, -1, 0, -1],
[0, 0, -1, -1],
[0, 0, 2, 1]])
(P, J) = A.jordan_form()
assert simplify(P) == Matrix([
[-I, -I/2, I, I/2],
[-1 + I, 0, -1 - I, 0],
[0, I*(-1 + I)/2, 0, I*(1 + I)/2],
[0, 1, 0, 1]])
assert J == Matrix([
[-I, 1, 0, 0],
[0, -I, 0, 0],
[0, 0, I, 1],
[0, 0, 0, I]])
def test_Matrix_berkowitz_charpoly():
UA, K_i, K_w = symbols('UA K_i K_w')
A = Matrix([[-K_i - UA + K_i**2/(K_i + K_w), K_i*K_w/(K_i + K_w)],
[ K_i*K_w/(K_i + K_w), -K_w + K_w**2/(K_i + K_w)]])
charpoly = A.charpoly(x)
assert charpoly == \
Poly(x**2 + (K_i*UA + K_w*UA + 2*K_i*K_w)/(K_i + K_w)*x +
K_i*K_w*UA/(K_i + K_w), x, domain='ZZ(K_i,K_w,UA)')
assert type(charpoly) is PurePoly
A = Matrix([[1, 3], [2, 0]])
assert A.charpoly() == A.charpoly(x) == PurePoly(x**2 - x - 6)
A = Matrix([[1, 2], [x, 0]])
p = A.charpoly(x)
assert p.gen != x
assert p.as_expr().subs(p.gen, x) == x**2 - 3*x
def test_exp():
m = Matrix([[3, 4], [0, -2]])
m_exp = Matrix([[exp(3), -4*exp(-2)/5 + 4*exp(3)/5], [0, exp(-2)]])
assert m.exp() == m_exp
assert exp(m) == m_exp
m = Matrix([[1, 0], [0, 1]])
assert m.exp() == Matrix([[E, 0], [0, E]])
assert exp(m) == Matrix([[E, 0], [0, E]])
m = Matrix([[1, -1], [1, 1]])
assert m.exp() == Matrix([[E*cos(1), -E*sin(1)], [E*sin(1), E*cos(1)]])
def test_has():
A = Matrix(((x, y), (2, 3)))
assert A.has(x)
assert not A.has(z)
assert A.has(Symbol)
A = A.subs(x, 2)
assert not A.has(x)
def test_LUdecomposition_Simple_iszerofunc():
# Test if callable passed to matrices.LUdecomposition_Simple() as iszerofunc keyword argument is used inside
# matrices.LUdecomposition_Simple()
magic_string = "I got passed in!"
def goofyiszero(value):
raise ValueError(magic_string)
try:
lu, p = Matrix([[1, 0], [0, 1]]).LUdecomposition_Simple(iszerofunc=goofyiszero)
except ValueError as err:
assert magic_string == err.args[0]
return
assert False
def test_LUdecomposition_iszerofunc():
# Test if callable passed to matrices.LUdecomposition() as iszerofunc keyword argument is used inside
# matrices.LUdecomposition_Simple()
magic_string = "I got passed in!"
def goofyiszero(value):
raise ValueError(magic_string)
try:
l, u, p = Matrix([[1, 0], [0, 1]]).LUdecomposition(iszerofunc=goofyiszero)
except ValueError as err:
assert magic_string == err.args[0]
return
assert False
def test_find_reasonable_pivot_naive_finds_guaranteed_nonzero1():
# Test if matrices._find_reasonable_pivot_naive()
# finds a guaranteed non-zero pivot when the
# some of the candidate pivots are symbolic expressions.
# Keyword argument: simpfunc=None indicates that no simplifications
# should be performed during the search.
x = Symbol('x')
column = Matrix(3, 1, [x, cos(x)**2 + sin(x)**2, Rational(1, 2)])
pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
_find_reasonable_pivot_naive(column)
assert pivot_val == Rational(1, 2)
def test_find_reasonable_pivot_naive_finds_guaranteed_nonzero2():
# Test if matrices._find_reasonable_pivot_naive()
# finds a guaranteed non-zero pivot when the
# some of the candidate pivots are symbolic expressions.
# Keyword argument: simpfunc=_simplify indicates that the search
# should attempt to simplify candidate pivots.
x = Symbol('x')
column = Matrix(3, 1,
[x,
cos(x)**2+sin(x)**2+x**2,
cos(x)**2+sin(x)**2])
pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
_find_reasonable_pivot_naive(column, simpfunc=_simplify)
assert pivot_val == 1
def test_find_reasonable_pivot_naive_simplifies():
# Test if matrices._find_reasonable_pivot_naive()
# simplifies candidate pivots, and reports
# their offsets correctly.
x = Symbol('x')
column = Matrix(3, 1,
[x,
cos(x)**2+sin(x)**2+x,
cos(x)**2+sin(x)**2])
pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
_find_reasonable_pivot_naive(column, simpfunc=_simplify)
assert len(simplified) == 2
assert simplified[0][0] == 1
assert simplified[0][1] == 1+x
assert simplified[1][0] == 2
assert simplified[1][1] == 1
def test_errors():
raises(ValueError, lambda: Matrix([[1, 2], [1]]))
raises(IndexError, lambda: Matrix([[1, 2]])[1.2, 5])
raises(IndexError, lambda: Matrix([[1, 2]])[1, 5.2])
raises(ValueError, lambda: randMatrix(3, c=4, symmetric=True))
raises(ValueError, lambda: Matrix([1, 2]).reshape(4, 6))
raises(ShapeError,
lambda: Matrix([[1, 2], [3, 4]]).copyin_matrix([1, 0], Matrix([1, 2])))
raises(TypeError, lambda: Matrix([[1, 2], [3, 4]]).copyin_list([0,
1], set([])))
raises(NonSquareMatrixError, lambda: Matrix([[1, 2, 3], [2, 3, 0]]).inv())
raises(ShapeError,
lambda: Matrix(1, 2, [1, 2]).row_join(Matrix([[1, 2], [3, 4]])))
raises(
ShapeError, lambda: Matrix([1, 2]).col_join(Matrix([[1, 2], [3, 4]])))
raises(ShapeError, lambda: Matrix([1]).row_insert(1, Matrix([[1,
2], [3, 4]])))
raises(ShapeError, lambda: Matrix([1]).col_insert(1, Matrix([[1,
2], [3, 4]])))
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).trace())
raises(TypeError, lambda: Matrix([1]).applyfunc(1))
raises(ShapeError, lambda: Matrix([1]).LUsolve(Matrix([[1, 2], [3, 4]])))
raises(ValueError, lambda: Matrix([[1, 2], [3, 4]]).minor(4, 5))
raises(ValueError, lambda: Matrix([[1, 2], [3, 4]]).minor_submatrix(4, 5))
raises(TypeError, lambda: Matrix([1, 2, 3]).cross(1))
raises(TypeError, lambda: Matrix([1, 2, 3]).dot(1))
raises(ShapeError, lambda: Matrix([1, 2, 3]).dot(Matrix([1, 2])))
raises(ShapeError, lambda: Matrix([1, 2]).dot([]))
raises(TypeError, lambda: Matrix([1, 2]).dot('a'))
with warns_deprecated_sympy():
Matrix([[1, 2], [3, 4]]).dot(Matrix([[4, 3], [1, 2]]))
raises(ShapeError, lambda: Matrix([1, 2]).dot([1, 2, 3]))
raises(NonSquareMatrixError, lambda: Matrix([1, 2, 3]).exp())
raises(ShapeError, lambda: Matrix([[1, 2], [3, 4]]).normalized())
raises(ValueError, lambda: Matrix([1, 2]).inv(method='not a method'))
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_GE())
raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inverse_GE())
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_ADJ())
raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inverse_ADJ())
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_LU())
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).is_nilpotent())
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).det())
raises(ValueError,
lambda: Matrix([[1, 2], [3, 4]]).det(method='Not a real method'))
raises(ValueError,
lambda: Matrix([[1, 2, 3, 4], [5, 6, 7, 8],
[9, 10, 11, 12], [13, 14, 15, 16]]).det(iszerofunc="Not function"))
raises(ValueError,
lambda: Matrix([[1, 2, 3, 4], [5, 6, 7, 8],
[9, 10, 11, 12], [13, 14, 15, 16]]).det(iszerofunc=False))
raises(ValueError,
lambda: hessian(Matrix([[1, 2], [3, 4]]), Matrix([[1, 2], [2, 1]])))
raises(ValueError, lambda: hessian(Matrix([[1, 2], [3, 4]]), []))
raises(ValueError, lambda: hessian(Symbol('x')**2, 'a'))
raises(IndexError, lambda: eye(3)[5, 2])
raises(IndexError, lambda: eye(3)[2, 5])
M = Matrix(((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
raises(ValueError, lambda: M.det('method=LU_decomposition()'))
V = Matrix([[10, 10, 10]])
M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(ValueError, lambda: M.row_insert(4.7, V))
M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(ValueError, lambda: M.col_insert(-4.2, V))
def test_len():
assert len(Matrix()) == 0
assert len(Matrix([[1, 2]])) == len(Matrix([[1], [2]])) == 2
assert len(Matrix(0, 2, lambda i, j: 0)) == \
len(Matrix(2, 0, lambda i, j: 0)) == 0
assert len(Matrix([[0, 1, 2], [3, 4, 5]])) == 6
assert Matrix([1]) == Matrix([[1]])
assert not Matrix()
assert Matrix() == Matrix([])
def test_integrate():
A = Matrix(((1, 4, x), (y, 2, 4), (10, 5, x**2)))
assert A.integrate(x) == \
Matrix(((x, 4*x, x**2/2), (x*y, 2*x, 4*x), (10*x, 5*x, x**3/3)))
assert A.integrate(y) == \
Matrix(((y, 4*y, x*y), (y**2/2, 2*y, 4*y), (10*y, 5*y, y*x**2)))
def test_limit():
A = Matrix(((1, 4, sin(x)/x), (y, 2, 4), (10, 5, x**2 + 1)))
assert A.limit(x, 0) == Matrix(((1, 4, 1), (y, 2, 4), (10, 5, 1)))
def test_diff():
A = MutableDenseMatrix(((1, 4, x), (y, 2, 4), (10, 5, x**2 + 1)))
assert isinstance(A.diff(x), type(A))
assert A.diff(x) == MutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
assert A.diff(y) == MutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))
assert diff(A, x) == MutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
assert diff(A, y) == MutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))
A_imm = A.as_immutable()
assert isinstance(A_imm.diff(x), type(A_imm))
assert A_imm.diff(x) == ImmutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
assert A_imm.diff(y) == ImmutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))
assert diff(A_imm, x) == ImmutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
assert diff(A_imm, y) == ImmutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))
def test_diff_by_matrix():
# Derive matrix by matrix:
A = MutableDenseMatrix([[x, y], [z, t]])
assert A.diff(A) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
assert diff(A, A) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
A_imm = A.as_immutable()
assert A_imm.diff(A_imm) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
assert diff(A_imm, A_imm) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
# Derive a constant matrix:
assert A.diff(a) == MutableDenseMatrix([[0, 0], [0, 0]])
B = ImmutableDenseMatrix([a, b])
assert A.diff(B) == Array.zeros(2, 1, 2, 2)
assert A.diff(A) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
# Test diff with tuples:
dB = B.diff([[a, b]])
assert dB.shape == (2, 2, 1)
assert dB == Array([[[1], [0]], [[0], [1]]])
f = Function("f")
fxyz = f(x, y, z)
assert fxyz.diff([[x, y, z]]) == Array([fxyz.diff(x), fxyz.diff(y), fxyz.diff(z)])
assert fxyz.diff(([x, y, z], 2)) == Array([
[fxyz.diff(x, 2), fxyz.diff(x, y), fxyz.diff(x, z)],
[fxyz.diff(x, y), fxyz.diff(y, 2), fxyz.diff(y, z)],
[fxyz.diff(x, z), fxyz.diff(z, y), fxyz.diff(z, 2)],
])
expr = sin(x)*exp(y)
assert expr.diff([[x, y]]) == Array([cos(x)*exp(y), sin(x)*exp(y)])
assert expr.diff(y, ((x, y),)) == Array([cos(x)*exp(y), sin(x)*exp(y)])
assert expr.diff(x, ((x, y),)) == Array([-sin(x)*exp(y), cos(x)*exp(y)])
assert expr.diff(((y, x),), [[x, y]]) == Array([[cos(x)*exp(y), -sin(x)*exp(y)], [sin(x)*exp(y), cos(x)*exp(y)]])
# Test different notations:
fxyz.diff(x).diff(y).diff(x) == fxyz.diff(((x, y, z),), 3)[0, 1, 0]
fxyz.diff(z).diff(y).diff(x) == fxyz.diff(((x, y, z),), 3)[2, 1, 0]
fxyz.diff([[x, y, z]], ((z, y, x),)) == Array([[fxyz.diff(i).diff(j) for i in (x, y, z)] for j in (z, y, x)])
# Test scalar derived by matrix remains matrix:
res = x.diff(Matrix([[x, y]]))
assert isinstance(res, ImmutableDenseMatrix)
assert res == Matrix([[1, 0]])
res = (x**3).diff(Matrix([[x, y]]))
assert isinstance(res, ImmutableDenseMatrix)
assert res == Matrix([[3*x**2, 0]])
def test_getattr():
A = Matrix(((1, 4, x), (y, 2, 4), (10, 5, x**2 + 1)))
raises(AttributeError, lambda: A.nonexistantattribute)
assert getattr(A, 'diff')(x) == Matrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
def test_hessenberg():
A = Matrix([[3, 4, 1], [2, 4, 5], [0, 1, 2]])
assert A.is_upper_hessenberg
A = A.T
assert A.is_lower_hessenberg
A[0, -1] = 1
assert A.is_lower_hessenberg is False
A = Matrix([[3, 4, 1], [2, 4, 5], [3, 1, 2]])
assert not A.is_upper_hessenberg
A = zeros(5, 2)
assert A.is_upper_hessenberg
def test_cholesky():
raises(NonSquareMatrixError, lambda: Matrix((1, 2)).cholesky())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).cholesky())
raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).cholesky())
raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).cholesky())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).cholesky(hermitian=False))
assert Matrix(((5 + I, 0), (0, 1))).cholesky(hermitian=False) == Matrix([
[sqrt(5 + I), 0], [0, 1]])
A = Matrix(((1, 5), (5, 1)))
L = A.cholesky(hermitian=False)
assert L == Matrix([[1, 0], [5, 2*sqrt(6)*I]])
assert L*L.T == A
A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L = A.cholesky()
assert L * L.T == A
assert L.is_lower
assert L == Matrix([[5, 0, 0], [3, 3, 0], [-1, 1, 3]])
A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
assert A.cholesky() == Matrix(((2, 0, 0), (I, 1, 0), (1 - I, 0, 3)))
def test_LDLdecomposition():
raises(NonSquareMatrixError, lambda: Matrix((1, 2)).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition(hermitian=False))
A = Matrix(((1, 5), (5, 1)))
L, D = A.LDLdecomposition(hermitian=False)
assert L * D * L.T == A
A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L, D = A.LDLdecomposition()
assert L * D * L.T == A
assert L.is_lower
assert L == Matrix([[1, 0, 0], [ S(3)/5, 1, 0], [S(-1)/5, S(1)/3, 1]])
assert D.is_diagonal()
assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
L, D = A.LDLdecomposition()
assert expand_mul(L * D * L.H) == A
assert L == Matrix(((1, 0, 0), (I/2, 1, 0), (S(1)/2 - I/2, 0, 1)))
assert D == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))
def test_cholesky_solve():
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A*x
soln = A.cholesky_solve(b)
assert soln == x
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.cholesky_solve(b)
assert soln == x
A = Matrix(((1, 5), (5, 1)))
x = Matrix((4, -3))
b = A*x
soln = A.cholesky_solve(b)
assert soln == x
A = Matrix(((9, 3*I), (-3*I, 5)))
x = Matrix((-2, 1))
b = A*x
soln = A.cholesky_solve(b)
assert expand_mul(soln) == x
A = Matrix(((9*I, 3), (-3 + I, 5)))
x = Matrix((2 + 3*I, -1))
b = A*x
soln = A.cholesky_solve(b)
assert expand_mul(soln) == x
a00, a01, a11, b0, b1 = symbols('a00, a01, a11, b0, b1')
A = Matrix(((a00, a01), (a01, a11)))
b = Matrix((b0, b1))
x = A.cholesky_solve(b)
assert simplify(A*x) == b
def test_LDLsolve():
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A*x
soln = A.LDLsolve(b)
assert soln == x
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.LDLsolve(b)
assert soln == x
A = Matrix(((9, 3*I), (-3*I, 5)))
x = Matrix((-2, 1))
b = A*x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix(((9*I, 3), (-3 + I, 5)))
x = Matrix((2 + 3*I, -1))
b = A*x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix(((9, 3), (3, 9)))
x = Matrix((1, 1))
b = A * x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix([[-5, -3, -4], [-3, -7, 7]])
x = Matrix([[8], [7], [-2]])
b = A * x
raises(NotImplementedError, lambda: A.LDLsolve(b))
def test_lower_triangular_solve():
raises(NonSquareMatrixError,
lambda: Matrix([1, 0]).lower_triangular_solve(Matrix([0, 1])))
raises(ShapeError,
lambda: Matrix([[1, 0], [0, 1]]).lower_triangular_solve(Matrix([1])))
raises(ValueError,
lambda: Matrix([[2, 1], [1, 2]]).lower_triangular_solve(
Matrix([[1, 0], [0, 1]])))
A = Matrix([[1, 0], [0, 1]])
B = Matrix([[x, y], [y, x]])
C = Matrix([[4, 8], [2, 9]])
assert A.lower_triangular_solve(B) == B
assert A.lower_triangular_solve(C) == C
def test_upper_triangular_solve():
raises(NonSquareMatrixError,
lambda: Matrix([1, 0]).upper_triangular_solve(Matrix([0, 1])))
raises(TypeError,
lambda: Matrix([[1, 0], [0, 1]]).upper_triangular_solve(Matrix([1])))
raises(TypeError,
lambda: Matrix([[2, 1], [1, 2]]).upper_triangular_solve(
Matrix([[1, 0], [0, 1]])))
A = Matrix([[1, 0], [0, 1]])
B = Matrix([[x, y], [y, x]])
C = Matrix([[2, 4], [3, 8]])
assert A.upper_triangular_solve(B) == B
assert A.upper_triangular_solve(C) == C
def test_diagonal_solve():
raises(TypeError, lambda: Matrix([1, 1]).diagonal_solve(Matrix([1])))
A = Matrix([[1, 0], [0, 1]])*2
B = Matrix([[x, y], [y, x]])
assert A.diagonal_solve(B) == B/2
A = Matrix([[1, 0], [1, 2]])
raises(TypeError, lambda: A.diagonal_solve(B))
def test_matrix_norm():
# Vector Tests
# Test columns and symbols
x = Symbol('x', real=True)
v = Matrix([cos(x), sin(x)])
assert trigsimp(v.norm(2)) == 1
assert v.norm(10) == Pow(cos(x)**10 + sin(x)**10, S(1)/10)
# Test Rows
A = Matrix([[5, Rational(3, 2)]])
assert A.norm() == Pow(25 + Rational(9, 4), S(1)/2)
assert A.norm(oo) == max(A._mat)
assert A.norm(-oo) == min(A._mat)
# Matrix Tests
# Intuitive test
A = Matrix([[1, 1], [1, 1]])
assert A.norm(2) == 2
assert A.norm(-2) == 0
assert A.norm('frobenius') == 2
assert eye(10).norm(2) == eye(10).norm(-2) == 1
assert A.norm(oo) == 2
# Test with Symbols and more complex entries
A = Matrix([[3, y, y], [x, S(1)/2, -pi]])
assert (A.norm('fro')
== sqrt(S(37)/4 + 2*abs(y)**2 + pi**2 + x**2))
# Check non-square
A = Matrix([[1, 2, -3], [4, 5, Rational(13, 2)]])
assert A.norm(2) == sqrt(S(389)/8 + sqrt(78665)/8)
assert A.norm(-2) == S(0)
assert A.norm('frobenius') == sqrt(389)/2
# Test properties of matrix norms
# https://en.wikipedia.org/wiki/Matrix_norm#Definition
# Two matrices
A = Matrix([[1, 2], [3, 4]])
B = Matrix([[5, 5], [-2, 2]])
C = Matrix([[0, -I], [I, 0]])
D = Matrix([[1, 0], [0, -1]])
L = [A, B, C, D]
alpha = Symbol('alpha', real=True)
for order in ['fro', 2, -2]:
# Zero Check
assert zeros(3).norm(order) == S(0)
# Check Triangle Inequality for all Pairs of Matrices
for X in L:
for Y in L:
dif = (X.norm(order) + Y.norm(order) -
(X + Y).norm(order))
assert (dif >= 0)
# Scalar multiplication linearity
for M in [A, B, C, D]:
dif = simplify((alpha*M).norm(order) -
abs(alpha) * M.norm(order))
assert dif == 0
# Test Properties of Vector Norms
# https://en.wikipedia.org/wiki/Vector_norm
# Two column vectors
a = Matrix([1, 1 - 1*I, -3])
b = Matrix([S(1)/2, 1*I, 1])
c = Matrix([-1, -1, -1])
d = Matrix([3, 2, I])
e = Matrix([Integer(1e2), Rational(1, 1e2), 1])
L = [a, b, c, d, e]
alpha = Symbol('alpha', real=True)
for order in [1, 2, -1, -2, S.Infinity, S.NegativeInfinity, pi]:
# Zero Check
if order > 0:
assert Matrix([0, 0, 0]).norm(order) == S(0)
# Triangle inequality on all pairs
if order >= 1: # Triangle InEq holds only for these norms
for X in L:
for Y in L:
dif = (X.norm(order) + Y.norm(order) -
(X + Y).norm(order))
assert simplify(dif >= 0) is S.true
# Linear to scalar multiplication
if order in [1, 2, -1, -2, S.Infinity, S.NegativeInfinity]:
for X in L:
dif = simplify((alpha*X).norm(order) -
(abs(alpha) * X.norm(order)))
assert dif == 0
# ord=1
M = Matrix(3, 3, [1, 3, 0, -2, -1, 0, 3, 9, 6])
assert M.norm(1) == 13
def test_condition_number():
x = Symbol('x', real=True)
A = eye(3)
A[0, 0] = 10
A[2, 2] = S(1)/10
assert A.condition_number() == 100
A[1, 1] = x
assert A.condition_number() == Max(10, Abs(x)) / Min(S(1)/10, Abs(x))
M = Matrix([[cos(x), sin(x)], [-sin(x), cos(x)]])
Mc = M.condition_number()
assert all(Float(1.).epsilon_eq(Mc.subs(x, val).evalf()) for val in
[Rational(1, 5), Rational(1, 2), Rational(1, 10), pi/2, pi, 7*pi/4 ])
#issue 10782
assert Matrix([]).condition_number() == 0
def test_equality():
A = Matrix(((1, 2, 3), (4, 5, 6), (7, 8, 9)))
B = Matrix(((9, 8, 7), (6, 5, 4), (3, 2, 1)))
assert A == A[:, :]
assert not A != A[:, :]
assert not A == B
assert A != B
assert A != 10
assert not A == 10
# A SparseMatrix can be equal to a Matrix
C = SparseMatrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
D = Matrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
assert C == D
assert not C != D
def test_col_join():
assert eye(3).col_join(Matrix([[7, 7, 7]])) == \
Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[7, 7, 7]])
def test_row_insert():
r4 = Matrix([[4, 4, 4]])
for i in range(-4, 5):
l = [1, 0, 0]
l.insert(i, 4)
assert flatten(eye(3).row_insert(i, r4).col(0).tolist()) == l
def test_col_insert():
c4 = Matrix([4, 4, 4])
for i in range(-4, 5):
l = [0, 0, 0]
l.insert(i, 4)
assert flatten(zeros(3).col_insert(i, c4).row(0).tolist()) == l
def test_normalized():
assert Matrix([3, 4]).normalized() == \
Matrix([Rational(3, 5), Rational(4, 5)])
# Zero vector trivial cases
assert Matrix([0, 0, 0]).normalized() == Matrix([0, 0, 0])
# Machine precision error truncation trivial cases
m = Matrix([0,0,1.e-100])
assert m.normalized(
iszerofunc=lambda x: x.evalf(n=10, chop=True).is_zero
) == Matrix([0, 0, 0])
def test_print_nonzero():
assert capture(lambda: eye(3).print_nonzero()) == \
'[X ]\n[ X ]\n[ X]\n'
assert capture(lambda: eye(3).print_nonzero('.')) == \
'[. ]\n[ . ]\n[ .]\n'
def test_zeros_eye():
assert Matrix.eye(3) == eye(3)
assert Matrix.zeros(3) == zeros(3)
assert ones(3, 4) == Matrix(3, 4, [1]*12)
i = Matrix([[1, 0], [0, 1]])
z = Matrix([[0, 0], [0, 0]])
for cls in classes:
m = cls.eye(2)
assert i == m # but m == i will fail if m is immutable
assert i == eye(2, cls=cls)
assert type(m) == cls
m = cls.zeros(2)
assert z == m
assert z == zeros(2, cls=cls)
assert type(m) == cls
def test_is_zero():
assert Matrix().is_zero
assert Matrix([[0, 0], [0, 0]]).is_zero
assert zeros(3, 4).is_zero
assert not eye(3).is_zero
assert Matrix([[x, 0], [0, 0]]).is_zero == None
assert SparseMatrix([[x, 0], [0, 0]]).is_zero == None
assert ImmutableMatrix([[x, 0], [0, 0]]).is_zero == None
assert ImmutableSparseMatrix([[x, 0], [0, 0]]).is_zero == None
assert Matrix([[x, 1], [0, 0]]).is_zero == False
a = Symbol('a', nonzero=True)
assert Matrix([[a, 0], [0, 0]]).is_zero == False
def test_rotation_matrices():
# This tests the rotation matrices by rotating about an axis and back.
theta = pi/3
r3_plus = rot_axis3(theta)
r3_minus = rot_axis3(-theta)
r2_plus = rot_axis2(theta)
r2_minus = rot_axis2(-theta)
r1_plus = rot_axis1(theta)
r1_minus = rot_axis1(-theta)
assert r3_minus*r3_plus*eye(3) == eye(3)
assert r2_minus*r2_plus*eye(3) == eye(3)
assert r1_minus*r1_plus*eye(3) == eye(3)
# Check the correctness of the trace of the rotation matrix
assert r1_plus.trace() == 1 + 2*cos(theta)
assert r2_plus.trace() == 1 + 2*cos(theta)
assert r3_plus.trace() == 1 + 2*cos(theta)
# Check that a rotation with zero angle doesn't change anything.
assert rot_axis1(0) == eye(3)
assert rot_axis2(0) == eye(3)
assert rot_axis3(0) == eye(3)
def test_DeferredVector():
assert str(DeferredVector("vector")[4]) == "vector[4]"
assert sympify(DeferredVector("d")) == DeferredVector("d")
raises(IndexError, lambda: DeferredVector("d")[-1])
assert str(DeferredVector("d")) == "d"
assert repr(DeferredVector("test")) == "DeferredVector('test')"
def test_DeferredVector_not_iterable():
assert not iterable(DeferredVector('X'))
def test_DeferredVector_Matrix():
raises(TypeError, lambda: Matrix(DeferredVector("V")))
def test_GramSchmidt():
R = Rational
m1 = Matrix(1, 2, [1, 2])
m2 = Matrix(1, 2, [2, 3])
assert GramSchmidt([m1, m2]) == \
[Matrix(1, 2, [1, 2]), Matrix(1, 2, [R(2)/5, R(-1)/5])]
assert GramSchmidt([m1.T, m2.T]) == \
[Matrix(2, 1, [1, 2]), Matrix(2, 1, [R(2)/5, R(-1)/5])]
# from wikipedia
assert GramSchmidt([Matrix([3, 1]), Matrix([2, 2])], True) == [
Matrix([3*sqrt(10)/10, sqrt(10)/10]),
Matrix([-sqrt(10)/10, 3*sqrt(10)/10])]
def test_casoratian():
assert casoratian([1, 2, 3, 4], 1) == 0
assert casoratian([1, 2, 3, 4], 1, zero=False) == 0
def test_zero_dimension_multiply():
assert (Matrix()*zeros(0, 3)).shape == (0, 3)
assert zeros(3, 0)*zeros(0, 3) == zeros(3, 3)
assert zeros(0, 3)*zeros(3, 0) == Matrix()
def test_slice_issue_2884():
m = Matrix(2, 2, range(4))
assert m[1, :] == Matrix([[2, 3]])
assert m[-1, :] == Matrix([[2, 3]])
assert m[:, 1] == Matrix([[1, 3]]).T
assert m[:, -1] == Matrix([[1, 3]]).T
raises(IndexError, lambda: m[2, :])
raises(IndexError, lambda: m[2, 2])
def test_slice_issue_3401():
assert zeros(0, 3)[:, -1].shape == (0, 1)
assert zeros(3, 0)[0, :] == Matrix(1, 0, [])
def test_copyin():
s = zeros(3, 3)
s[3] = 1
assert s[:, 0] == Matrix([0, 1, 0])
assert s[3] == 1
assert s[3: 4] == [1]
s[1, 1] = 42
assert s[1, 1] == 42
assert s[1, 1:] == Matrix([[42, 0]])
s[1, 1:] = Matrix([[5, 6]])
assert s[1, :] == Matrix([[1, 5, 6]])
s[1, 1:] = [[42, 43]]
assert s[1, :] == Matrix([[1, 42, 43]])
s[0, 0] = 17
assert s[:, :1] == Matrix([17, 1, 0])
s[0, 0] = [1, 1, 1]
assert s[:, 0] == Matrix([1, 1, 1])
s[0, 0] = Matrix([1, 1, 1])
assert s[:, 0] == Matrix([1, 1, 1])
s[0, 0] = SparseMatrix([1, 1, 1])
assert s[:, 0] == Matrix([1, 1, 1])
def test_invertible_check():
# sometimes a singular matrix will have a pivot vector shorter than
# the number of rows in a matrix...
assert Matrix([[1, 2], [1, 2]]).rref() == (Matrix([[1, 2], [0, 0]]), (0,))
raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inv())
m = Matrix([
[-1, -1, 0],
[ x, 1, 1],
[ 1, x, -1],
])
assert len(m.rref()[1]) != m.rows
# in addition, unless simplify=True in the call to rref, the identity
# matrix will be returned even though m is not invertible
assert m.rref()[0] != eye(3)
assert m.rref(simplify=signsimp)[0] != eye(3)
raises(ValueError, lambda: m.inv(method="ADJ"))
raises(ValueError, lambda: m.inv(method="GE"))
raises(ValueError, lambda: m.inv(method="LU"))
def test_issue_3959():
x, y = symbols('x, y')
e = x*y
assert e.subs(x, Matrix([3, 5, 3])) == Matrix([3, 5, 3])*y
def test_issue_5964():
assert str(Matrix([[1, 2], [3, 4]])) == 'Matrix([[1, 2], [3, 4]])'
def test_issue_7604():
x, y = symbols(u"x y")
assert sstr(Matrix([[x, 2*y], [y**2, x + 3]])) == \
'Matrix([\n[ x, 2*y],\n[y**2, x + 3]])'
def test_is_Identity():
assert eye(3).is_Identity
assert eye(3).as_immutable().is_Identity
assert not zeros(3).is_Identity
assert not ones(3).is_Identity
# issue 6242
assert not Matrix([[1, 0, 0]]).is_Identity
# issue 8854
assert SparseMatrix(3,3, {(0,0):1, (1,1):1, (2,2):1}).is_Identity
assert not SparseMatrix(2,3, range(6)).is_Identity
assert not SparseMatrix(3,3, {(0,0):1, (1,1):1}).is_Identity
assert not SparseMatrix(3,3, {(0,0):1, (1,1):1, (2,2):1, (0,1):2, (0,2):3}).is_Identity
def test_dot():
assert ones(1, 3).dot(ones(3, 1)) == 3
assert ones(1, 3).dot([1, 1, 1]) == 3
assert Matrix([1, 2, 3]).dot(Matrix([1, 2, 3])) == 14
assert Matrix([1, 2, 3*I]).dot(Matrix([I, 2, 3*I])) == -5 + I
assert Matrix([1, 2, 3*I]).dot(Matrix([I, 2, 3*I]), hermitian=False) == -5 + I
assert Matrix([1, 2, 3*I]).dot(Matrix([I, 2, 3*I]), hermitian=True) == 13 + I
assert Matrix([1, 2, 3*I]).dot(Matrix([I, 2, 3*I]), hermitian=True, conjugate_convention="physics") == 13 - I
assert Matrix([1, 2, 3*I]).dot(Matrix([4, 5*I, 6]), hermitian=True, conjugate_convention="right") == 4 + 8*I
assert Matrix([1, 2, 3*I]).dot(Matrix([4, 5*I, 6]), hermitian=True, conjugate_convention="left") == 4 - 8*I
assert Matrix([I, 2*I]).dot(Matrix([I, 2*I]), hermitian=False, conjugate_convention="left") == -5
assert Matrix([I, 2*I]).dot(Matrix([I, 2*I]), conjugate_convention="left") == 5
raises(ValueError, lambda: Matrix([1, 2]).dot(Matrix([3, 4]), hermitian=True, conjugate_convention="test"))
def test_dual():
B_x, B_y, B_z, E_x, E_y, E_z = symbols(
'B_x B_y B_z E_x E_y E_z', real=True)
F = Matrix((
( 0, E_x, E_y, E_z),
(-E_x, 0, B_z, -B_y),
(-E_y, -B_z, 0, B_x),
(-E_z, B_y, -B_x, 0)
))
Fd = Matrix((
( 0, -B_x, -B_y, -B_z),
(B_x, 0, E_z, -E_y),
(B_y, -E_z, 0, E_x),
(B_z, E_y, -E_x, 0)
))
assert F.dual().equals(Fd)
assert eye(3).dual().equals(zeros(3))
assert F.dual().dual().equals(-F)
def test_anti_symmetric():
assert Matrix([1, 2]).is_anti_symmetric() is False
m = Matrix(3, 3, [0, x**2 + 2*x + 1, y, -(x + 1)**2, 0, x*y, -y, -x*y, 0])
assert m.is_anti_symmetric() is True
assert m.is_anti_symmetric(simplify=False) is False
assert m.is_anti_symmetric(simplify=lambda x: x) is False
# tweak to fail
m[2, 1] = -m[2, 1]
assert m.is_anti_symmetric() is False
# untweak
m[2, 1] = -m[2, 1]
m = m.expand()
assert m.is_anti_symmetric(simplify=False) is True
m[0, 0] = 1
assert m.is_anti_symmetric() is False
def test_normalize_sort_diogonalization():
A = Matrix(((1, 2), (2, 1)))
P, Q = A.diagonalize(normalize=True)
assert P*P.T == P.T*P == eye(P.cols)
P, Q = A.diagonalize(normalize=True, sort=True)
assert P*P.T == P.T*P == eye(P.cols)
assert P*Q*P.inv() == A
def test_issue_5321():
raises(ValueError, lambda: Matrix([[1, 2, 3], Matrix(0, 1, [])]))
def test_issue_5320():
assert Matrix.hstack(eye(2), 2*eye(2)) == Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2]
])
assert Matrix.vstack(eye(2), 2*eye(2)) == Matrix([
[1, 0],
[0, 1],
[2, 0],
[0, 2]
])
cls = SparseMatrix
assert cls.hstack(cls(eye(2)), cls(2*eye(2))) == Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2]
])
def test_issue_11944():
A = Matrix([[1]])
AIm = sympify(A)
assert Matrix.hstack(AIm, A) == Matrix([[1, 1]])
assert Matrix.vstack(AIm, A) == Matrix([[1], [1]])
def test_cross():
a = [1, 2, 3]
b = [3, 4, 5]
col = Matrix([-2, 4, -2])
row = col.T
def test(M, ans):
assert ans == M
assert type(M) == cls
for cls in classes:
A = cls(a)
B = cls(b)
test(A.cross(B), col)
test(A.cross(B.T), col)
test(A.T.cross(B.T), row)
test(A.T.cross(B), row)
raises(ShapeError, lambda:
Matrix(1, 2, [1, 1]).cross(Matrix(1, 2, [1, 1])))
def test_hash():
for cls in classes[-2:]:
s = {cls.eye(1), cls.eye(1)}
assert len(s) == 1 and s.pop() == cls.eye(1)
# issue 3979
for cls in classes[:2]:
assert not isinstance(cls.eye(1), Hashable)
@XFAIL
def test_issue_3979():
# when this passes, delete this and change the [1:2]
# to [:2] in the test_hash above for issue 3979
cls = classes[0]
raises(AttributeError, lambda: hash(cls.eye(1)))
def test_adjoint():
dat = [[0, I], [1, 0]]
ans = Matrix([[0, 1], [-I, 0]])
for cls in classes:
assert ans == cls(dat).adjoint()
def test_simplify_immutable():
from sympy import simplify, sin, cos
assert simplify(ImmutableMatrix([[sin(x)**2 + cos(x)**2]])) == \
ImmutableMatrix([[1]])
def test_rank():
from sympy.abc import x
m = Matrix([[1, 2], [x, 1 - 1/x]])
assert m.rank() == 2
n = Matrix(3, 3, range(1, 10))
assert n.rank() == 2
p = zeros(3)
assert p.rank() == 0
def test_issue_11434():
ax, ay, bx, by, cx, cy, dx, dy, ex, ey, t0, t1 = \
symbols('a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y t_0 t_1')
M = Matrix([[ax, ay, ax*t0, ay*t0, 0],
[bx, by, bx*t0, by*t0, 0],
[cx, cy, cx*t0, cy*t0, 1],
[dx, dy, dx*t0, dy*t0, 1],
[ex, ey, 2*ex*t1 - ex*t0, 2*ey*t1 - ey*t0, 0]])
assert M.rank() == 4
def test_rank_regression_from_so():
# see:
# https://stackoverflow.com/questions/19072700/why-does-sympy-give-me-the-wrong-answer-when-i-row-reduce-a-symbolic-matrix
nu, lamb = symbols('nu, lambda')
A = Matrix([[-3*nu, 1, 0, 0],
[ 3*nu, -2*nu - 1, 2, 0],
[ 0, 2*nu, (-1*nu) - lamb - 2, 3],
[ 0, 0, nu + lamb, -3]])
expected_reduced = Matrix([[1, 0, 0, 1/(nu**2*(-lamb - nu))],
[0, 1, 0, 3/(nu*(-lamb - nu))],
[0, 0, 1, 3/(-lamb - nu)],
[0, 0, 0, 0]])
expected_pivots = (0, 1, 2)
reduced, pivots = A.rref()
assert simplify(expected_reduced - reduced) == zeros(*A.shape)
assert pivots == expected_pivots
def test_replace():
from sympy import symbols, Function, Matrix
F, G = symbols('F, G', cls=Function)
K = Matrix(2, 2, lambda i, j: G(i+j))
M = Matrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G)
assert N == K
def test_replace_map():
from sympy import symbols, Function, Matrix
F, G = symbols('F, G', cls=Function)
K = Matrix(2, 2, [(G(0), {F(0): G(0)}), (G(1), {F(1): G(1)}), (G(1), {F(1)\
: G(1)}), (G(2), {F(2): G(2)})])
M = Matrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G, True)
assert N == K
def test_atoms():
m = Matrix([[1, 2], [x, 1 - 1/x]])
assert m.atoms() == {S(1),S(2),S(-1), x}
assert m.atoms(Symbol) == {x}
def test_pinv():
# Pseudoinverse of an invertible matrix is the inverse.
A1 = Matrix([[a, b], [c, d]])
assert simplify(A1.pinv()) == simplify(A1.inv())
# Test the four properties of the pseudoinverse for various matrices.
As = [Matrix([[13, 104], [2212, 3], [-3, 5]]),
Matrix([[1, 7, 9], [11, 17, 19]]),
Matrix([a, b])]
for A in As:
A_pinv = A.pinv()
AAp = A * A_pinv
ApA = A_pinv * A
assert simplify(AAp * A) == A
assert simplify(ApA * A_pinv) == A_pinv
assert AAp.H == AAp
assert ApA.H == ApA
def test_pinv_solve():
# Fully determined system (unique result, identical to other solvers).
A = Matrix([[1, 5], [7, 9]])
B = Matrix([12, 13])
assert A.pinv_solve(B) == A.cholesky_solve(B)
assert A.pinv_solve(B) == A.LDLsolve(B)
assert A.pinv_solve(B) == Matrix([sympify('-43/26'), sympify('71/26')])
assert A * A.pinv() * B == B
# Fully determined, with two-dimensional B matrix.
B = Matrix([[12, 13, 14], [15, 16, 17]])
assert A.pinv_solve(B) == A.cholesky_solve(B)
assert A.pinv_solve(B) == A.LDLsolve(B)
assert A.pinv_solve(B) == Matrix([[-33, -37, -41], [69, 75, 81]]) / 26
assert A * A.pinv() * B == B
# Underdetermined system (infinite results).
A = Matrix([[1, 0, 1], [0, 1, 1]])
B = Matrix([5, 7])
solution = A.pinv_solve(B)
w = {}
for s in solution.atoms(Symbol):
# Extract dummy symbols used in the solution.
w[s.name] = s
assert solution == Matrix([[w['w0_0']/3 + w['w1_0']/3 - w['w2_0']/3 + 1],
[w['w0_0']/3 + w['w1_0']/3 - w['w2_0']/3 + 3],
[-w['w0_0']/3 - w['w1_0']/3 + w['w2_0']/3 + 4]])
assert A * A.pinv() * B == B
# Overdetermined system (least squares results).
A = Matrix([[1, 0], [0, 0], [0, 1]])
B = Matrix([3, 2, 1])
assert A.pinv_solve(B) == Matrix([3, 1])
# Proof the solution is not exact.
assert A * A.pinv() * B != B
def test_pinv_rank_deficient():
# Test the four properties of the pseudoinverse for various matrices.
As = [Matrix([[1, 1, 1], [2, 2, 2]]),
Matrix([[1, 0], [0, 0]]),
Matrix([[1, 2], [2, 4], [3, 6]])]
for A in As:
A_pinv = A.pinv()
AAp = A * A_pinv
ApA = A_pinv * A
assert simplify(AAp * A) == A
assert simplify(ApA * A_pinv) == A_pinv
assert AAp.H == AAp
assert ApA.H == ApA
# Test solving with rank-deficient matrices.
A = Matrix([[1, 0], [0, 0]])
# Exact, non-unique solution.
B = Matrix([3, 0])
solution = A.pinv_solve(B)
w1 = solution.atoms(Symbol).pop()
assert w1.name == 'w1_0'
assert solution == Matrix([3, w1])
assert A * A.pinv() * B == B
# Least squares, non-unique solution.
B = Matrix([3, 1])
solution = A.pinv_solve(B)
w1 = solution.atoms(Symbol).pop()
assert w1.name == 'w1_0'
assert solution == Matrix([3, w1])
assert A * A.pinv() * B != B
@XFAIL
def test_pinv_rank_deficient_when_diagonalization_fails():
# Test the four properties of the pseudoinverse for matrices when
# diagonalization of A.H*A fails.
As = [Matrix([
[61, 89, 55, 20, 71, 0],
[62, 96, 85, 85, 16, 0],
[69, 56, 17, 4, 54, 0],
[10, 54, 91, 41, 71, 0],
[ 7, 30, 10, 48, 90, 0],
[0,0,0,0,0,0]])]
for A in As:
A_pinv = A.pinv()
AAp = A * A_pinv
ApA = A_pinv * A
assert simplify(AAp * A) == A
assert simplify(ApA * A_pinv) == A_pinv
assert AAp.H == AAp
assert ApA.H == ApA
def test_gauss_jordan_solve():
# Square, full rank, unique solution
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
b = Matrix([3, 6, 9])
sol, params = A.gauss_jordan_solve(b)
assert sol == Matrix([[-1], [2], [0]])
assert params == Matrix(0, 1, [])
# Square, full rank, unique solution, B has more columns than rows
A = eye(3)
B = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])
sol, params = A.gauss_jordan_solve(B)
assert sol == B
assert params == Matrix(0, 4, [])
# Square, reduced rank, parametrized solution
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
b = Matrix([3, 6, 9])
sol, params, freevar = A.gauss_jordan_solve(b, freevar=True)
w = {}
for s in sol.atoms(Symbol):
# Extract dummy symbols used in the solution.
w[s.name] = s
assert sol == Matrix([[w['tau0'] - 1], [-2*w['tau0'] + 2], [w['tau0']]])
assert params == Matrix([[w['tau0']]])
assert freevar == [2]
# Square, reduced rank, parametrized solution, B has two columns
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
B = Matrix([[3, 4], [6, 8], [9, 12]])
sol, params, freevar = A.gauss_jordan_solve(B, freevar=True)
w = {}
for s in sol.atoms(Symbol):
# Extract dummy symbols used in the solution.
w[s.name] = s
assert sol == Matrix([[w['tau0'] - 1, w['tau1'] - S(4)/3],
[-2*w['tau0'] + 2, -2*w['tau1'] + S(8)/3],
[w['tau0'], w['tau1']],])
assert params == Matrix([[w['tau0'], w['tau1']]])
assert freevar == [2]
# Square, reduced rank, parametrized solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 0])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[-2*w['tau0'] - 3*w['tau1']],
[w['tau0']], [w['tau1']]])
assert params == Matrix([[w['tau0']], [w['tau1']]])
# Square, reduced rank, parametrized solution
A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
b = Matrix([0, 0, 0])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[w['tau0']], [w['tau1']], [w['tau2']]])
assert params == Matrix([[w['tau0']], [w['tau1']], [w['tau2']]])
# Square, reduced rank, no solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 1])
raises(ValueError, lambda: A.gauss_jordan_solve(b))
# Rectangular, tall, full rank, unique solution
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
b = Matrix([0, 0, 1, 0])
sol, params = A.gauss_jordan_solve(b)
assert sol == Matrix([[-S(1)/2], [0], [S(1)/6]])
assert params == Matrix(0, 1, [])
# Rectangular, tall, full rank, unique solution, B has less columns than rows
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
B = Matrix([[0,0], [0, 0], [1, 2], [0, 0]])
sol, params = A.gauss_jordan_solve(B)
assert sol == Matrix([[-S(1)/2, -S(2)/2], [0, 0], [S(1)/6, S(2)/6]])
assert params == Matrix(0, 2, [])
# Rectangular, tall, full rank, no solution
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
b = Matrix([0, 0, 0, 1])
raises(ValueError, lambda: A.gauss_jordan_solve(b))
# Rectangular, tall, full rank, no solution, B has two columns (2nd has no solution)
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
B = Matrix([[0,0], [0, 0], [1, 0], [0, 1]])
raises(ValueError, lambda: A.gauss_jordan_solve(B))
# Rectangular, tall, full rank, no solution, B has two columns (1st has no solution)
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
B = Matrix([[0,0], [0, 0], [0, 1], [1, 0]])
raises(ValueError, lambda: A.gauss_jordan_solve(B))
# Rectangular, tall, reduced rank, parametrized solution
A = Matrix([[1, 5, 3], [2, 10, 6], [3, 15, 9], [1, 4, 3]])
b = Matrix([0, 0, 0, 1])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[-3*w['tau0'] + 5], [-1], [w['tau0']]])
assert params == Matrix([[w['tau0']]])
# Rectangular, tall, reduced rank, no solution
A = Matrix([[1, 5, 3], [2, 10, 6], [3, 15, 9], [1, 4, 3]])
b = Matrix([0, 0, 1, 1])
raises(ValueError, lambda: A.gauss_jordan_solve(b))
# Rectangular, wide, full rank, parametrized solution
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 1, 12]])
b = Matrix([1, 1, 1])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[2*w['tau0'] - 1], [-3*w['tau0'] + 1], [0],
[w['tau0']]])
assert params == Matrix([[w['tau0']]])
# Rectangular, wide, reduced rank, parametrized solution
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [2, 4, 6, 8]])
b = Matrix([0, 1, 0])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[w['tau0'] + 2*w['tau1'] + 1/S(2)],
[-2*w['tau0'] - 3*w['tau1'] - 1/S(4)],
[w['tau0']], [w['tau1']]])
assert params == Matrix([[w['tau0']], [w['tau1']]])
# watch out for clashing symbols
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau1')
M = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
A = M[:, :-1]
b = M[:, -1:]
sol, params = A.gauss_jordan_solve(b)
assert params == Matrix(3, 1, [x0, x1, x2])
assert sol == Matrix(5, 1, [x1, 0, x0, _x0, x2])
# Rectangular, wide, reduced rank, no solution
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [2, 4, 6, 8]])
b = Matrix([1, 1, 1])
raises(ValueError, lambda: A.gauss_jordan_solve(b))
def test_solve():
A = Matrix([[1,2], [2,4]])
b = Matrix([[3], [4]])
raises(ValueError, lambda: A.solve(b)) #no solution
b = Matrix([[ 4], [8]])
raises(ValueError, lambda: A.solve(b)) #infinite solution
def test_issue_7201():
assert ones(0, 1) + ones(0, 1) == Matrix(0, 1, [])
assert ones(1, 0) + ones(1, 0) == Matrix(1, 0, [])
def test_free_symbols():
for M in ImmutableMatrix, ImmutableSparseMatrix, Matrix, SparseMatrix:
assert M([[x], [0]]).free_symbols == {x}
def test_from_ndarray():
"""See issue 7465."""
try:
from numpy import array
except ImportError:
skip('NumPy must be available to test creating matrices from ndarrays')
assert Matrix(array([1, 2, 3])) == Matrix([1, 2, 3])
assert Matrix(array([[1, 2, 3]])) == Matrix([[1, 2, 3]])
assert Matrix(array([[1, 2, 3], [4, 5, 6]])) == \
Matrix([[1, 2, 3], [4, 5, 6]])
assert Matrix(array([x, y, z])) == Matrix([x, y, z])
raises(NotImplementedError, lambda: Matrix(array([[
[1, 2], [3, 4]], [[5, 6], [7, 8]]])))
def test_hermitian():
a = Matrix([[1, I], [-I, 1]])
assert a.is_hermitian
a[0, 0] = 2*I
assert a.is_hermitian is False
a[0, 0] = x
assert a.is_hermitian is None
a[0, 1] = a[1, 0]*I
assert a.is_hermitian is False
def test_doit():
a = Matrix([[Add(x,x, evaluate=False)]])
assert a[0] != 2*x
assert a.doit() == Matrix([[2*x]])
def test_issue_9457_9467_9876():
# for row_del(index)
M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
M.row_del(1)
assert M == Matrix([[1, 2, 3], [3, 4, 5]])
N = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
N.row_del(-2)
assert N == Matrix([[1, 2, 3], [3, 4, 5]])
O = Matrix([[1, 2, 3], [5, 6, 7], [9, 10, 11]])
O.row_del(-1)
assert O == Matrix([[1, 2, 3], [5, 6, 7]])
P = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(IndexError, lambda: P.row_del(10))
Q = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(IndexError, lambda: Q.row_del(-10))
# for col_del(index)
M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
M.col_del(1)
assert M == Matrix([[1, 3], [2, 4], [3, 5]])
N = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
N.col_del(-2)
assert N == Matrix([[1, 3], [2, 4], [3, 5]])
P = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(IndexError, lambda: P.col_del(10))
Q = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(IndexError, lambda: Q.col_del(-10))
def test_issue_9422():
x, y = symbols('x y', commutative=False)
a, b = symbols('a b')
M = eye(2)
M1 = Matrix(2, 2, [x, y, y, z])
assert y*x*M != x*y*M
assert b*a*M == a*b*M
assert x*M1 != M1*x
assert a*M1 == M1*a
assert y*x*M == Matrix([[y*x, 0], [0, y*x]])
def test_issue_10770():
M = Matrix([])
a = ['col_insert', 'row_join'], Matrix([9, 6, 3])
b = ['row_insert', 'col_join'], a[1].T
c = ['row_insert', 'col_insert'], Matrix([[1, 2], [3, 4]])
for ops, m in (a, b, c):
for op in ops:
f = getattr(M, op)
new = f(m) if 'join' in op else f(42, m)
assert new == m and id(new) != id(m)
def test_issue_10658():
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert A.extract([0, 1, 2], [True, True, False]) == \
Matrix([[1, 2], [4, 5], [7, 8]])
assert A.extract([0, 1, 2], [True, False, False]) == Matrix([[1], [4], [7]])
assert A.extract([True, False, False], [0, 1, 2]) == Matrix([[1, 2, 3]])
assert A.extract([True, False, True], [0, 1, 2]) == \
Matrix([[1, 2, 3], [7, 8, 9]])
assert A.extract([0, 1, 2], [False, False, False]) == Matrix(3, 0, [])
assert A.extract([False, False, False], [0, 1, 2]) == Matrix(0, 3, [])
assert A.extract([True, False, True], [False, True, False]) == \
Matrix([[2], [8]])
def test_opportunistic_simplification():
# this test relates to issue #10718, #9480, #11434
# issue #9480
m = Matrix([[-5 + 5*sqrt(2), -5], [-5*sqrt(2)/2 + 5, -5*sqrt(2)/2]])
assert m.rank() == 1
# issue #10781
m = Matrix([[3+3*sqrt(3)*I, -9],[4,-3+3*sqrt(3)*I]])
assert simplify(m.rref()[0] - Matrix([[1, -9/(3 + 3*sqrt(3)*I)], [0, 0]])) == zeros(2, 2)
# issue #11434
ax,ay,bx,by,cx,cy,dx,dy,ex,ey,t0,t1 = symbols('a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y t_0 t_1')
m = Matrix([[ax,ay,ax*t0,ay*t0,0],[bx,by,bx*t0,by*t0,0],[cx,cy,cx*t0,cy*t0,1],[dx,dy,dx*t0,dy*t0,1],[ex,ey,2*ex*t1-ex*t0,2*ey*t1-ey*t0,0]])
assert m.rank() == 4
def test_partial_pivoting():
# example from https://en.wikipedia.org/wiki/Pivot_element
# partial pivoting with back subsitution gives a perfect result
# naive pivoting give an error ~1e-13, so anything better than
# 1e-15 is good
mm=Matrix([[0.003 ,59.14, 59.17],[ 5.291, -6.13,46.78]])
assert (mm.rref()[0] - Matrix([[1.0, 0, 10.0], [ 0, 1.0, 1.0]])).norm() < 1e-15
# issue #11549
m_mixed = Matrix([[6e-17, 1.0, 4],[ -1.0, 0, 8],[ 0, 0, 1]])
m_float = Matrix([[6e-17, 1.0, 4.],[ -1.0, 0., 8.],[ 0., 0., 1.]])
m_inv = Matrix([[ 0, -1.0, 8.0],[1.0, 6.0e-17, -4.0],[ 0, 0, 1]])
# this example is numerically unstable and involves a matrix with a norm >= 8,
# this comparing the difference of the results with 1e-15 is numerically sound.
assert (m_mixed.inv() - m_inv).norm() < 1e-15
assert (m_float.inv() - m_inv).norm() < 1e-15
def test_iszero_substitution():
""" When doing numerical computations, all elements that pass
the iszerofunc test should be set to numerically zero if they
aren't already. """
# Matrix from issue #9060
m = Matrix([[0.9, -0.1, -0.2, 0],[-0.8, 0.9, -0.4, 0],[-0.1, -0.8, 0.6, 0]])
m_rref = m.rref(iszerofunc=lambda x: abs(x)<6e-15)[0]
m_correct = Matrix([[1.0, 0, -0.301369863013699, 0],[ 0, 1.0, -0.712328767123288, 0],[ 0, 0, 0, 0]])
m_diff = m_rref - m_correct
assert m_diff.norm() < 1e-15
# if a zero-substitution wasn't made, this entry will be -1.11022302462516e-16
assert m_rref[2,2] == 0
def test_rank_decomposition():
a = Matrix(0, 0, [])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix(1, 1, [5])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix([
[0, 0, 1, 2, 2, -5, 3],
[-1, 5, 2, 2, 1, -7, 5],
[0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
@slow
def test_issue_11238():
from sympy import Point
xx = 8*tan(13*pi/45)/(tan(13*pi/45) + sqrt(3))
yy = (-8*sqrt(3)*tan(13*pi/45)**2 + 24*tan(13*pi/45))/(-3 + tan(13*pi/45)**2)
p1 = Point(0, 0)
p2 = Point(1, -sqrt(3))
p0 = Point(xx,yy)
m1 = Matrix([p1 - simplify(p0), p2 - simplify(p0)])
m2 = Matrix([p1 - p0, p2 - p0])
m3 = Matrix([simplify(p1 - p0), simplify(p2 - p0)])
assert m1.rank(simplify=True) == 1
assert m2.rank(simplify=True) == 1
assert m3.rank(simplify=True) == 1
def test_as_real_imag():
m1 = Matrix(2,2,[1,2,3,4])
m2 = m1*S.ImaginaryUnit
m3 = m1 + m2
for kls in classes:
a,b = kls(m3).as_real_imag()
assert list(a) == list(m1)
assert list(b) == list(m1)
def test_deprecated():
# Maintain tests for deprecated functions. We must capture
# the deprecation warnings. When the deprecated functionality is
# removed, the corresponding tests should be removed.
m = Matrix(3, 3, [0, 1, 0, -4, 4, 0, -2, 1, 2])
P, Jcells = m.jordan_cells()
assert Jcells[1] == Matrix(1, 1, [2])
assert Jcells[0] == Matrix(2, 2, [2, 1, 0, 2])
with warns_deprecated_sympy():
assert Matrix([[1,2],[3,4]]).dot(Matrix([[1,3],[4,5]])) == [10, 19, 14, 28]
def test_issue_14489():
from sympy import Mod
A = Matrix([-1, 1, 2])
B = Matrix([10, 20, -15])
assert Mod(A, 3) == Matrix([2, 1, 2])
assert Mod(B, 4) == Matrix([2, 0, 1])
def test_issue_14517():
M = Matrix([
[ 0, 10*I, 10*I, 0],
[10*I, 0, 0, 10*I],
[10*I, 0, 5 + 2*I, 10*I],
[ 0, 10*I, 10*I, 5 + 2*I]])
ev = M.eigenvals()
# test one random eigenvalue, the computation is a little slow
test_ev = random.choice(list(ev.keys()))
assert (M - test_ev*eye(4)).det() == 0
def test_issue_14943():
# Test that __array__ accepts the optional dtype argument
try:
from numpy import array
except ImportError:
skip('NumPy must be available to test creating matrices from ndarrays')
M = Matrix([[1,2], [3,4]])
assert array(M, dtype=float).dtype.name == 'float64'
def test_issue_8240():
# Eigenvalues of large triangular matrices
n = 200
diagonal_variables = [Symbol('x%s' % i) for i in range(n)]
M = [[0 for i in range(n)] for j in range(n)]
for i in range(n):
M[i][i] = diagonal_variables[i]
M = Matrix(M)
eigenvals = M.eigenvals()
assert len(eigenvals) == n
for i in range(n):
assert eigenvals[diagonal_variables[i]] == 1
eigenvals = M.eigenvals(multiple=True)
assert set(eigenvals) == set(diagonal_variables)
# with multiplicity
M = Matrix([[x, 0, 0], [1, y, 0], [2, 3, x]])
eigenvals = M.eigenvals()
assert eigenvals == {x: 2, y: 1}
eigenvals = M.eigenvals(multiple=True)
assert len(eigenvals) == 3
assert eigenvals.count(x) == 2
assert eigenvals.count(y) == 1
def test_legacy_det():
# Minimal support for legacy keys for 'method' in det()
# Partially copied from test_determinant()
M = Matrix(( ( 3, -2, 0, 5),
(-2, 1, -2, 2),
( 0, -2, 5, 0),
( 5, 0, 3, 4) ))
assert M.det(method="bareis") == -289
assert M.det(method="det_lu") == -289
assert M.det(method="det_LU") == -289
M = Matrix(( (3, 2, 0, 0, 0),
(0, 3, 2, 0, 0),
(0, 0, 3, 2, 0),
(0, 0, 0, 3, 2),
(2, 0, 0, 0, 3) ))
assert M.det(method="bareis") == 275
assert M.det(method="det_lu") == 275
assert M.det(method="Bareis") == 275
M = Matrix(( (1, 0, 1, 2, 12),
(2, 0, 1, 1, 4),
(2, 1, 1, -1, 3),
(3, 2, -1, 1, 8),
(1, 1, 1, 0, 6) ))
assert M.det(method="bareis") == -55
assert M.det(method="det_lu") == -55
assert M.det(method="BAREISS") == -55
M = Matrix(( (-5, 2, 3, 4, 5),
( 1, -4, 3, 4, 5),
( 1, 2, -3, 4, 5),
( 1, 2, 3, -2, 5),
( 1, 2, 3, 4, -1) ))
assert M.det(method="bareis") == 11664
assert M.det(method="det_lu") == 11664
assert M.det(method="BERKOWITZ") == 11664
M = Matrix(( ( 2, 7, -1, 3, 2),
( 0, 0, 1, 0, 1),
(-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3),
( 1, 0, 0, 0, 1) ))
assert M.det(method="bareis") == 123
assert M.det(method="det_lu") == 123
assert M.det(method="LU") == 123
def test_case_6913():
m = MatrixSymbol('m', 1, 1)
a = Symbol("a")
a = m[0, 0]>0
assert str(a) == 'm[0, 0] > 0'
def test_issue_15872():
A = Matrix([[1, 1, 1, 0], [-2, -1, 0, -1], [0, 0, -1, -1], [0, 0, 2, 1]])
B = A - Matrix.eye(4) * I
assert B.rank() == 3
assert (B**2).rank() == 2
assert (B**3).rank() == 2
def test_issue_11948():
A = MatrixSymbol('A', 3, 3)
a = Wild('a')
assert A.match(a) == {a: A}
|
cf44c27b9f89ff9be2e3ffe8790aeaf65be29a8b8eed4727819a1c4f579accc7 | from sympy.matrices.expressions import MatrixExpr
from sympy import MatrixBase, Dummy, Lambda, Function, FunctionClass
from sympy.matrices.expressions.diagonal import diagonalize_vector
class ElementwiseApplyFunction(MatrixExpr):
r"""
Apply function to a matrix elementwise without evaluating.
Examples
========
It can be created by calling ``.applyfunc(<function>)`` on a matrix
expression:
>>> from sympy.matrices.expressions import MatrixSymbol
>>> from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
>>> from sympy import exp
>>> X = MatrixSymbol("X", 3, 3)
>>> X.applyfunc(exp)
exp(X...)
Otherwise using the class constructor:
>>> from sympy import eye
>>> expr = ElementwiseApplyFunction(exp, eye(3))
>>> expr
exp(Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])...)
>>> expr.doit()
Matrix([
[E, 1, 1],
[1, E, 1],
[1, 1, E]])
Notice the difference with the real mathematical functions:
>>> exp(eye(3))
Matrix([
[E, 0, 0],
[0, E, 0],
[0, 0, E]])
"""
def __new__(cls, function, expr):
obj = MatrixExpr.__new__(cls, expr)
if not isinstance(function, FunctionClass):
d = Dummy("d")
function = Lambda(d, function(d))
obj._function = function
obj._expr = expr
return obj
def _hashable_content(self):
return (self.function, self.expr)
@property
def function(self):
return self._function
@property
def expr(self):
return self._expr
@property
def shape(self):
return self.expr.shape
@property
def func(self):
# This strange construction is required by the assumptions:
# (.func needs to be a class)
class ElementwiseApplyFunction2(ElementwiseApplyFunction):
def __new__(obj, expr):
return ElementwiseApplyFunction(self.function, expr)
return ElementwiseApplyFunction2
def doit(self, **kwargs):
deep = kwargs.get("deep", True)
expr = self.expr
if deep:
expr = expr.doit(**kwargs)
if isinstance(expr, MatrixBase):
return expr.applyfunc(self.function)
else:
return self
def _entry(self, i, j, **kwargs):
return self.function(self.expr._entry(i, j, **kwargs))
def _eval_derivative_matrix_lines(self, x):
from sympy import HadamardProduct, hadamard_product, Mul, MatMul, Identity, Transpose
from sympy.matrices.expressions.diagonal import diagonalize_vector
from sympy.matrices.expressions.matmul import validate as matmul_validate
from sympy.core.expr import ExprBuilder
d = Dummy("d")
function = self.function(d)
fdiff = function.fdiff()
if isinstance(fdiff, Function):
fdiff = type(fdiff)
else:
fdiff = Lambda(d, fdiff)
lr = self.expr._eval_derivative_matrix_lines(x)
ewdiff = ElementwiseApplyFunction(fdiff, self.expr)
if 1 in x.shape:
# Vector:
iscolumn = self.shape[1] == 1
ewdiff = diagonalize_vector(ewdiff)
# TODO: check which axis is not 1
for i in lr:
if iscolumn:
ptr1 = i.first_pointer
ptr2 = Identity(ewdiff.shape[0])
else:
ptr1 = Identity(ewdiff.shape[1])
ptr2 = i.second_pointer
# TODO: check if pointers point to two different lines:
def mul(*args):
return Mul.fromiter(args)
def hadamard_or_mul(arg1, arg2):
if arg1.shape == arg2.shape:
return hadamard_product(arg1, arg2)
elif arg1.shape[1] == arg2.shape[0]:
return MatMul(arg1, arg2).doit()
elif arg1.shape[0] == arg2.shape[0]:
return MatMul(arg2.T, arg1).doit()
raise NotImplementedError
i._lines = [[hadamard_or_mul, [[mul, [ewdiff, ptr1]], ptr2]]]
i._first_pointer_parent = i._lines[0][1][0][1]
i._first_pointer_index = 1
i._second_pointer_parent = i._lines[0][1]
i._second_pointer_index = 1
else:
# Matrix case:
for i in lr:
ptr1 = i.first_pointer
ptr2 = i.second_pointer
newptr1 = Identity(ptr1.shape[1])
newptr2 = Identity(ptr2.shape[1])
subexpr1 = ExprBuilder(
MatMul,
[ptr1, ExprBuilder(diagonalize_vector, [newptr1])],
validator=matmul_validate,
)
subexpr2 = ExprBuilder(
Transpose,
[ExprBuilder(
MatMul,
[
ptr2,
ExprBuilder(diagonalize_vector, [newptr2])
,
],
)],
validator=matmul_validate,
)
# TODO: replace the expressions above with the following:
# subexpr = ExprBuilder(
# get_recognize([(1, 2, 8), (5, 6, 9)]),
# [ptr1, newptr1, ptr2, newptr2, ewdiff]
# )
# ... = recognize_matrix_expression(subexpr.build()).doit()
i.first_pointer = subexpr1
i.second_pointer = subexpr2
i._first_pointer_parent = subexpr1.args[1].args
i._first_pointer_index = 0
i._second_pointer_parent = subexpr2.args[0].args[1].args
i._second_pointer_index = 0
# TODO: check if pointers point to two different lines:
# Unify lines:
l = i._lines
# TODO: check nested fucntions, e.g. log(sin(...)), the second function should be a scalar one.
i._lines = [ExprBuilder(MatMul, [l[0], ewdiff, l[1]], validator=matmul_validate)]
return lr
|
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